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 |
|---|---|---|---|---|---|---|
doublen :=
match n with
| N0 => N0
| Npos p => Npos p~0
end. | Definition | Corelib | [
"From Corelib Require Export BinNums PosDef"
] | Corelib/BinNums/NatDef.v | double | |
succ_pos(n : N) : positive :=
match n with
| N0 => xH
| Npos p => Pos.succ p
end. | Definition | Corelib | [
"From Corelib Require Export BinNums PosDef"
] | Corelib/BinNums/NatDef.v | succ_pos | |
subn m :=
match n, m with
| N0, _ => N0
| n, N0 => n
| Npos n', Npos m' =>
match Pos.sub_mask n' m' with
| Pos.IsPos p => Npos p
| _ => N0
end
end. | Definition | Corelib | [
"From Corelib Require Export BinNums PosDef"
] | Corelib/BinNums/NatDef.v | sub | |
comparen m :=
match n, m with
| N0, N0 => Eq
| N0, Npos m' => Lt
| Npos n', N0 => Gt
| Npos n', Npos m' => Pos.compare n' m'
end. | Definition | Corelib | [
"From Corelib Require Export BinNums PosDef"
] | Corelib/BinNums/NatDef.v | compare | |
lebx y :=
match compare x y with Gt => false | _ => true end. | Definition | Corelib | [
"From Corelib Require Export BinNums PosDef"
] | Corelib/BinNums/NatDef.v | leb | |
pos_div_eucl(a:positive)(b:N) : N * N :=
match a with
| xH =>
match b with Npos 1 => (Npos 1, N0) | _ => (N0, Npos 1) end
| xO a' =>
let (q, r) := pos_div_eucl a' b in
let r' := double r in
if leb b r' then (succ_double q, sub r' b)
else (double q, r')
| xI a' =>
... | Fixpoint | Corelib | [
"From Corelib Require Export BinNums PosDef"
] | Corelib/BinNums/NatDef.v | pos_div_eucl | |
lorn m :=
match n, m with
| N0, _ => m
| _, N0 => n
| Npos p, Npos q => Npos (Pos.lor p q)
end. | Definition | Corelib | [
"From Corelib Require Export BinNums PosDef"
] | Corelib/BinNums/NatDef.v | lor | |
landn m :=
match n, m with
| N0, _ => N0
| _, N0 => N0
| Npos p, Npos q => Pos.land p q
end. | Definition | Corelib | [
"From Corelib Require Export BinNums PosDef"
] | Corelib/BinNums/NatDef.v | land | |
ldiffn m :=
match n, m with
| N0, _ => N0
| _, N0 => n
| Npos p, Npos q => Pos.ldiff p q
end. | Definition | Corelib | [
"From Corelib Require Export BinNums PosDef"
] | Corelib/BinNums/NatDef.v | ldiff | |
lxorn m :=
match n, m with
| N0, _ => m
| _, N0 => n
| Npos p, Npos q => Pos.lxor p q
end. | Definition | Corelib | [
"From Corelib Require Export BinNums PosDef"
] | Corelib/BinNums/NatDef.v | lxor | |
succx :=
match x with
| p~1 => (succ p)~0
| p~0 => p~1
| 1 => 1~0
end. | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | succ | |
addx y :=
match x, y with
| p~1, q~1 => (add_carry p q)~0
| p~1, q~0 => (add p q)~1
| p~1, 1 => (succ p)~0
| p~0, q~1 => (add p q)~1
| p~0, q~0 => (add p q)~0
| p~0, 1 => p~1
| 1, q~1 => (succ q)~0
| 1, q~0 => q~1
| 1, 1 => 1~0
end
with add_carry x y :=
match x, y with
| p... | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | add | |
pred_doublex :=
match x with
| p~1 => p~0~1
| p~0 => (pred_double p)~1
| 1 => 1
end. | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | pred_double | |
pred_Nx :=
match x with
| p~1 => Npos (p~0)
| p~0 => Npos (pred_double p)
| 1 => N0
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | pred_N | |
mask: Set :=
| IsNul : mask
| IsPos : positive -> mask
| IsNeg : mask. | Inductive | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | mask | |
succ_double_mask(x:mask) : mask :=
match x with
| IsNul => IsPos 1
| IsNeg => IsNeg
| IsPos p => IsPos p~1
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | succ_double_mask | |
double_mask(x:mask) : mask :=
match x with
| IsNul => IsNul
| IsNeg => IsNeg
| IsPos p => IsPos p~0
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | double_mask | |
double_pred_maskx : mask :=
match x with
| p~1 => IsPos p~0~0
| p~0 => IsPos (pred_double p)~0
| 1 => IsNul
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | double_pred_mask | |
sub_mask(x y:positive) {struct y} : mask :=
match x, y with
| p~1, q~1 => double_mask (sub_mask p q)
| p~1, q~0 => succ_double_mask (sub_mask p q)
| p~1, 1 => IsPos p~0
| p~0, q~1 => succ_double_mask (sub_mask_carry p q)
| p~0, q~0 => double_mask (sub_mask p q)
| p~0, 1 => IsPos (pred_double p... | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | sub_mask | |
subx y :=
match sub_mask x y with
| IsPos z => z
| _ => 1
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | sub | |
mulx y :=
match x with
| p~1 => add y (mul p y)~0
| p~0 => (mul p y)~0
| 1 => y
end. | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | mul | |
iter{A} (f:A -> A) : A -> positive -> A :=
fix iter_fix x n := match n with
| xH => f x
| xO n' => iter_fix (iter_fix x n') n'
| xI n' => f (iter_fix (iter_fix x n') n')
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | iter | |
div2p :=
match p with
| 1 => 1
| p~0 => p
| p~1 => p
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | div2 | |
div2_upp :=
match p with
| 1 => 1
| p~0 => p
| p~1 => succ p
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | div2_up | |
compare_cont(r:comparison) (x y:positive) {struct y} : comparison :=
match x, y with
| p~1, q~1 => compare_cont r p q
| p~1, q~0 => compare_cont Gt p q
| p~1, 1 => Gt
| p~0, q~1 => compare_cont Lt p q
| p~0, q~0 => compare_cont r p q
| p~0, 1 => Gt
| 1, q~1 => Lt
| 1, q~0 => Lt
| 1... | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | compare_cont | |
compare:= compare_cont Eq. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | compare | |
eqbp q {struct q} :=
match p, q with
| p~1, q~1 => eqb p q
| p~0, q~0 => eqb p q
| 1, 1 => true
| _, _ => false
end. | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | eqb | |
lebx y :=
match compare x y with Gt => false | _ => true end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | leb | |
sqrtrem_step(f g:positive->positive) p :=
match p with
| (s, IsPos r) =>
let s' := s~0~1 in
let r' := g (f r) in
if leb s' r' then (s~1, sub_mask r' s')
else (s~0, IsPos r')
| (s,_) => (s~0, sub_mask (g (f 1)) 1~0~0)
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | sqrtrem_step | |
sqrtremp : positive * mask :=
match p with
| 1 => (1,IsNul)
| 1~0 => (1,IsPos 1)
| 1~1 => (1,IsPos 1~0)
| p~0~0 => sqrtrem_step xO xO (sqrtrem p)
| p~0~1 => sqrtrem_step xO xI (sqrtrem p)
| p~1~0 => sqrtrem_step xI xO (sqrtrem p)
| p~1~1 => sqrtrem_step xI xI (sqrtrem p)
end. | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | sqrtrem | |
sqrtp := fst (sqrtrem p). | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | sqrt | |
Nsucc_doublex :=
match x with
| N0 => Npos 1
| Npos p => Npos p~1
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | Nsucc_double | |
Ndoublen :=
match n with
| N0 => N0
| Npos p => Npos p~0
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | Ndouble | |
lor(p q : positive) : positive :=
match p, q with
| 1, q~0 => q~1
| 1, _ => q
| p~0, 1 => p~1
| _, 1 => p
| p~0, q~0 => (lor p q)~0
| p~0, q~1 => (lor p q)~1
| p~1, q~0 => (lor p q)~1
| p~1, q~1 => (lor p q)~1
end. | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | lor | |
land(p q : positive) : N :=
match p, q with
| 1, q~0 => N0
| 1, _ => Npos 1
| p~0, 1 => N0
| _, 1 => Npos 1
| p~0, q~0 => Ndouble (land p q)
| p~0, q~1 => Ndouble (land p q)
| p~1, q~0 => Ndouble (land p q)
| p~1, q~1 => Nsucc_double (land p q)
end. | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | land | |
ldiff(p q:positive) : N :=
match p, q with
| 1, q~0 => Npos 1
| 1, _ => N0
| _~0, 1 => Npos p
| p~1, 1 => Npos (p~0)
| p~0, q~0 => Ndouble (ldiff p q)
| p~0, q~1 => Ndouble (ldiff p q)
| p~1, q~1 => Ndouble (ldiff p q)
| p~1, q~0 => Nsucc_double (ldiff p q)
end. | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | ldiff | |
lxor(p q:positive) : N :=
match p, q with
| 1, 1 => N0
| 1, q~0 => Npos (q~1)
| 1, q~1 => Npos (q~0)
| p~0, 1 => Npos (p~1)
| p~0, q~0 => Ndouble (lxor p q)
| p~0, q~1 => Nsucc_double (lxor p q)
| p~1, 1 => Npos (p~0)
| p~1, q~0 => Nsucc_double (lxor p q)
| p~1, q~1 => Ndouble (lxo... | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | lxor | |
iter_op{A}(op:A->A->A) :=
fix iter (p:positive)(a:A) : A :=
match p with
| 1 => a
| p~0 => iter p (op a a)
| p~1 => op a (iter p (op a a))
end. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | iter_op | |
to_nat(x:positive) : nat := iter_op plus x (S O).
Arguments to_nat x: simpl never. | Definition | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | to_nat | |
of_succ_nat(n:nat) : positive :=
match n with
| O => 1
| S x => succ (of_succ_nat x)
end. | Fixpoint | Corelib | [
"From Corelib Require Export BinNums"
] | Corelib/BinNums/PosDef.v | of_succ_nat | |
Proper(R : crelation A) (m : A) :=
proper_prf : R m m. | Class | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | Proper | |
ProperProxy(R : crelation A) (m : A) :=
proper_proxy : R m m. | Class | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | ProperProxy | |
eq_proper_proxy(x : A) : ProperProxy (@eq A) x.
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | eq_proper_proxy | |
reflexive_proper_proxy`(Reflexive A R) (x : A) : ProperProxy R x.
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | reflexive_proper_proxy | |
proper_proper_proxyx `(Proper R x) : ProperProxy R x.
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_proper_proxy | |
respectful_hetero(A B : Type)
(C : A -> Type) (D : B -> Type)
(R : A -> B -> Type)
(R' : forall (x : A) (y : B), C x -> D y -> Type) :
(forall x : A, C x) -> (forall x : B, D x) -> Type :=
fun f g => forall x y, R x y -> R' x y (f x) (g y). | Definition | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | respectful_hetero | |
respectful{B} (R : crelation A) (R' : crelation B) : crelation (A -> B) :=
Eval compute in @respectful_hetero A A (fun _ => B) (fun _ => B) R (fun _ _ => R'). | Definition | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | respectful | |
solve_respectfult :=
match goal with
| |- respectful _ _ _ _ =>
let H := fresh "H" in
intros ? ? H; solve_respectful ltac:(setoid_rewrite H; t)
| _ => t; reflexivity
end. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | solve_respectful | |
solve_proper:= unfold Proper; solve_respectful ltac:(idtac). | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | solve_proper | |
f_equiv:=
match goal with
| |- ?R (?f ?x) (?f' _) =>
let T := type of x in
let Rx := fresh "R" in
evar (Rx : crelation T);
let H := fresh in
assert (H : (Rx==>R)%signatureT f f');
unfold Rx in *; clear Rx; [ f_equiv | apply H; clear H; try reflexivity ]
| |- ?R ?f ?f' =>
solve [change (... | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | f_equiv | |
forall_def(P : A -> Type) : Type := forall x : A, P x. | Definition | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | forall_def | |
forall_relation(P : A -> Type)
(sig : forall a, crelation (P a)) : crelation (forall x, P x) :=
fun f g => forall a, sig a (f a) (g a). | Definition | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | forall_relation | |
pointwise_relation{B} (R : crelation B) : crelation (A -> B) :=
fun f g => forall a, R (f a) (g a). | Definition | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | pointwise_relation | |
pointwise_pointwise{B} (R : crelation B) :
relation_equivalence (pointwise_relation R) (@eq A ==> R).
Proof.
intros. split.
- simpl_crelation.
- firstorder.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | pointwise_pointwise | |
subrelation_respectful`(subl : subrelation A RA' RA, subr : subrelation B RB RB') :
subrelation (RA ==> RB) (RA' ==> RB').
Proof. simpl_crelation. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | subrelation_respectful | |
subrelation_reflR : @subrelation A R R.
Proof. simpl_crelation. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | subrelation_refl | |
subrelation_proper`(mor : Proper A R' m)
`(unc : Unconvertible (crelation A) R R')
`(sub : subrelation A R' R) : Proper R m.
Proof.
intros. apply sub. apply mor.
Qed.
Global Instance proper_subrelation_proper_arrow :
Proper (subrelation ++> eq ==> arrow) (@Proper A).
Proof. reduce. subs... | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | subrelation_proper | |
forall_subrelation(P : A -> Type) (R S : forall x : A, crelation (P x)) :
(forall a, subrelation (R a) (S a)) ->
subrelation (forall_relation P R) (forall_relation P S).
Proof. reduce. firstorder. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | forall_subrelation | |
subrelation_tacT U :=
(is_ground T ; is_ground U ; class_apply @subrelation_refl) ||
class_apply @subrelation_respectful || class_apply @subrelation_refl.
#[global]
Hint Extern 3 (@subrelation _ ?T ?U) => subrelation_tac T U : typeclass_instances. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | subrelation_tac | |
apply_subrelation: Prop := do_subrelation. | CoInductive | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | apply_subrelation | |
proper_subrelation:=
match goal with
[ H : apply_subrelation |- _ ] => clear H ; class_apply @subrelation_proper
end.
#[global]
Hint Extern 5 (@Proper _ ?H _) => proper_subrelation : typeclass_instances. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_subrelation | |
iff_impl_subrelation: subrelation iff impl | 2.
Proof. firstorder. Qed.
#[global] | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | iff_impl_subrelation | |
iff_flip_impl_subrelation: subrelation iff (flip impl) | 2.
Proof. firstorder. Qed. | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | iff_flip_impl_subrelation | |
iffT_arrow_subrelation: subrelation iffT arrow | 2.
Proof. firstorder. Qed.
#[global] | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | iffT_arrow_subrelation | |
iffT_flip_arrow_subrelation: subrelation iffT (flip arrow) | 2.
Proof. firstorder. Qed. | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | iffT_flip_arrow_subrelation | |
trans_contra_co_type_morphism`(Transitive A R) : Proper (R --> R ++> arrow) R.
Next Obligation.
Proof.
intros A R H x y X x0 y0 X0 X1.
apply transitivity with x; auto.
apply transitivity with x0; auto.
Qed. | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | trans_contra_co_type_morphism | |
trans_contra_inv_impl_type_morphism`(Transitive A R) {x} : Proper (R --> flip arrow) (R x) | 3.
Next Obligation.
Proof.
intros A R H x x0 y X X0.
apply transitivity with y; auto.
Qed.
Global Program | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | trans_contra_inv_impl_type_morphism | |
trans_co_impl_type_morphism`(Transitive A R) {x} : Proper (R ++> arrow) (R x) | 3.
Next Obligation.
Proof.
intros A R H x x0 y X X0.
apply transitivity with x0; auto.
Qed.
Global Program | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | trans_co_impl_type_morphism | |
trans_sym_co_inv_impl_type_morphism`(PER A R) {x} : Proper (R ++> flip arrow) (R x) | 3.
Next Obligation.
Proof.
intros A R H x x0 y X X0.
apply transitivity with y; auto. apply symmetry; auto.
Qed.
Global Program Instance trans_sym_contra_arrow_morphism
`(PER A R) {x} : Proper (R --> arrow) (R x)... | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | trans_sym_co_inv_impl_type_morphism | |
trans_co_eq_inv_arrow_morphism`(Transitive A R) : Proper (R ==> (@eq A) ==> flip arrow) R | 2.
Next Obligation.
Proof.
intros A R H x y X y0 y1 e X0; destruct e.
apply transitivity with y; auto.
Qed. | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | trans_co_eq_inv_arrow_morphism | |
PER_type_morphism`(PER A R) : Proper (R ==> R ==> iffT) R | 1.
Next Obligation.
Proof.
intros A R H x y X x0 y0 X0.
split ; intros.
- apply transitivity with x0; auto.
apply transitivity with x; auto. apply symmetry; auto.
- apply transitivity with y; auto. apply transitivity with y0; auto.
... | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | PER_type_morphism | |
symmetric_equiv_flip`(Symmetric A R) : relation_equivalence R (flip R).
Proof. firstorder. Qed.
Global Program Instance compose_proper A B C RA RB RC :
Proper ((RB ==> RC) ==> (RA ==> RB) ==> (RA ==> RC)) (@compose A B C).
Next Obligation.
Proof.
simpl_crelation.
unfold compose. firstorder.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | symmetric_equiv_flip | |
Reflexive_partial_app_morphism`(Proper (A -> B) (R ==> R') m, ProperProxy A R x) :
Proper R' (m x).
Proof. simpl_crelation. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | Reflexive_partial_app_morphism | |
Params{A} (of : A) (arity : nat). | Class | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | Params | |
flip_respectful{A B} (R : crelation A) (R' : crelation B) :
relation_equivalence (flip (R ==> R')) (flip R ==> flip R').
Proof.
intros.
unfold flip, respectful.
split ; intros ; intuition.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | flip_respectful | |
flip1`(subrelation A R' R) : subrelation (flip (flip R')) R.
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | flip1 | |
flip2`(subrelation A R R') : subrelation R (flip (flip R')).
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | flip2 | |
eq_subrelation`(Reflexive A R) : subrelation (@eq A) R.
Proof. simpl_crelation. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | eq_subrelation | |
proper_flip_proper`(mor : Proper A R m) : Proper (flip R) m := mor. | Definition | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_flip_proper | |
reflexive_proper`{Reflexive A R} (x : A) : Proper R x.
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | reflexive_proper | |
proper_eq{A} (x : A) : Proper (@eq A) x.
Proof. intros. apply reflexive_proper. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_eq | |
PartialApplication. | Class | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | PartialApplication | |
normalization_done: Prop := did_normalization. | CoInductive | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | normalization_done | |
partial_application_tactic:=
let rec do_partial_apps H m cont :=
match m with
| ?m' ?x => class_apply @Reflexive_partial_app_morphism ;
[(do_partial_apps H m' ltac:(idtac))|clear H]
| _ => cont
end
in
let rec do_partial H ar m :=
lazymatch ar with
| 0%nat => do_partial_apps H... | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | partial_application_tactic | |
proper_proper{A} : Proper (relation_equivalence ==> eq ==> iffT) (@Proper A).
Proof.
intros R R' HRR' x y <-. red in HRR'.
split ; red ; intros.
- now apply (fst (HRR' _ _)).
- now apply (snd (HRR' _ _)).
Qed. | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_proper | |
proper_reflexive:=
match goal with
| [ _ : normalization_done |- _ ] => fail 1
| _ => class_apply proper_eq || class_apply @reflexive_proper
end.
#[global]
Hint Extern 1 (subrelation (flip _) _) => class_apply @flip1 : typeclass_instances.
#[global]
Hint Extern 1 (subrelation _ (flip _)) => class_apply @f... | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_reflexive | |
Normalizes(m : crelation A) (m' : crelation A) :=
normalizes : relation_equivalence m m'. | Class | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | Normalizes | |
proper_normalizes_proper`(Normalizes R0 R1, Proper A R1 m) : Proper R0 m.
Proof.
apply (_ : Normalizes R0 R1). assumption.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_normalizes_proper | |
flip_atomR : Normalizes R (flip (flip R)).
Proof.
firstorder.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | flip_atom | |
flip_arrow`(NA : Normalizes A R (flip R'''), NB : Normalizes B R' (flip R'')) :
Normalizes (A -> B) (R ==> R') (flip (R''' ==> R'')%signatureT).
Proof.
unfold Normalizes in *. intros.
eapply transitivity; [|eapply symmetry, flip_respectful].
now apply respectful_morphism.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | flip_arrow | |
normalizes:=
match goal with
| [ |- Normalizes _ (respectful _ _) _ ] => class_apply @flip_arrow
| _ => class_apply @flip_atom
end. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | normalizes | |
proper_normalization:=
match goal with
| [ _ : normalization_done |- _ ] => fail 1
| [ _ : apply_subrelation |- @Proper _ ?R _ ] =>
let H := fresh "H" in
set(H:=did_normalization) ; class_apply @proper_normalizes_proper
end.
#[global]
Hint Extern 1 (Normalizes _ _ _) => normalizes : typeclass_i... | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_normalization | |
proper_sym_flip:
forall `(Symmetric A R1)`(Proper (A->B) (R1==>R2) f),
Proper (R1==>flip R2) f.
Proof.
intros A R1 Sym B R2 f Hf.
intros x x' Hxx'. apply Hf, Sym, Hxx'.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_sym_flip | |
proper_sym_flip_2:
forall `(Symmetric A R1)`(Symmetric B R2)`(Proper (A->B->C) (R1==>R2==>R3) f),
Proper (R1==>R2==>flip R3) f.
Proof.
intros A R1 Sym1 B R2 Sym2 C R3 f Hf.
intros x x' Hxx' y y' Hyy'. apply Hf; auto.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_sym_flip_2 | |
proper_sym_impl_iff: forall `(Symmetric A R)`(Proper _ (R==>impl) f),
Proper (R==>iff) f.
Proof.
intros A R Sym f Hf x x' Hxx'. repeat red in Hf. split; eauto.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_sym_impl_iff | |
proper_sym_arrow_iffT: forall `(Symmetric A R)`(Proper _ (R==>arrow) f),
Proper (R==>iffT) f.
Proof.
intros A R Sym f Hf x x' Hxx'. repeat red in Hf. split; eauto.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_sym_arrow_iffT | |
proper_sym_impl_iff_2:
forall `(Symmetric A R)`(Symmetric B R')`(Proper _ (R==>R'==>impl) f),
Proper (R==>R'==>iff) f.
Proof.
intros A R Sym B R' Sym' f Hf x x' Hxx' y y' Hyy'.
repeat red in Hf. split; eauto.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_sym_impl_iff_2 | |
proper_sym_arrow_iffT_2:
forall `(Symmetric A R)`(Symmetric B R')`(Proper _ (R==>R'==>arrow) f),
Proper (R==>R'==>iffT) f.
Proof.
intros A R Sym B R' Sym' f Hf x x' Hxx' y y' Hyy'.
repeat red in Hf. split; eauto.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | proper_sym_arrow_iffT_2 | |
PartialOrder_proper_type`(PartialOrder A eqA R) :
Proper (eqA==>eqA==>iffT) R.
Proof.
intros.
apply proper_sym_arrow_iffT_2. 1-2: typeclasses eauto.
intros x x' Hx y y' Hy Hr.
apply transitivity with x.
- generalize (partial_order_equivalence x x'); compute; intuition.
- apply transitivity with y; auto.
generalize ... | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | PartialOrder_proper_type | |
PartialOrder_StrictOrder`(PartialOrder A eqA R) :
StrictOrder (relation_conjunction R (complement eqA)).
Proof.
split; compute.
- intros x (_,Hx). apply Hx, Equivalence_Reflexive.
- intros x y z (Hxy,Hxy') (Hyz,Hyz'). split.
+ apply PreOrder_Transitive with y; assumption.
+ intro Hxz.
apply Hxy'.
apply pa... | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | PartialOrder_StrictOrder |
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