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 |
|---|---|---|---|---|---|---|
of_num_uint(d:Number.uint) :=
match d with
| Number.UIntDecimal d => of_uint d
| Number.UIntHexadecimal d => of_hex_uint d
end. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | of_num_uint | |
to_little_uintn acc :=
match n with
| O => acc
| S n => to_little_uint n (Decimal.Little.succ acc)
end. | Fixpoint | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | to_little_uint | |
to_uintn :=
Decimal.rev (to_little_uint n Decimal.zero). | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | to_uint | |
to_little_hex_uintn acc :=
match n with
| O => acc
| S n => to_little_hex_uint n (Hexadecimal.Little.succ acc)
end. | Fixpoint | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | to_little_hex_uint | |
to_hex_uintn :=
Hexadecimal.rev (to_little_hex_uint n Hexadecimal.zero). | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | to_hex_uint | |
to_num_uintn := Number.UIntDecimal (to_uint n). | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | to_num_uint | |
to_num_hex_uintn := Number.UIntHexadecimal (to_hex_uint n). | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | to_num_hex_uint | |
of_int(d:Decimal.int) : option nat :=
match Decimal.norm d with
| Decimal.Pos u => Some (of_uint u)
| _ => None
end. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | of_int | |
of_hex_int(d:Hexadecimal.int) : option nat :=
match Hexadecimal.norm d with
| Hexadecimal.Pos u => Some (of_hex_uint u)
| _ => None
end. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | of_hex_int | |
of_num_int(d:Number.int) : option nat :=
match d with
| Number.IntDecimal d => of_int d
| Number.IntHexadecimal d => of_hex_int d
end. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | of_num_int | |
to_intn := Decimal.Pos (to_uint n). | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | to_int | |
to_hex_intn := Hexadecimal.Pos (to_hex_uint n). | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | to_hex_int | |
to_num_intn := Number.IntDecimal (to_int n). | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | to_num_int | |
divmodx y q u :=
match x with
| 0 => (q,u)
| S x' => match u with
| 0 => divmod x' y (S q) y
| S u' => divmod x' y q u'
end
end. | Fixpoint | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | divmod | |
divx y :=
match y with
| 0 => y
| S y' => fst (divmod x y' 0 y')
end. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | div | |
modulox y :=
match y with
| 0 => x
| S y' => y' - snd (divmod x y' 0 y')
end.
Infix "/" := div : nat_scope.
Infix "mod" := modulo (at level 40, no associativity) : nat_scope. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | modulo | |
gcda b :=
match a with
| O => b
| S a' => gcd (b mod (S a')) (S a')
end. | Fixpoint | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | gcd | |
squaren := n * n. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | square | |
sqrt_iterk p q r :=
match k with
| O => p
| S k' => match r with
| O => sqrt_iter k' (S p) (S (S q)) (S (S q))
| S r' => sqrt_iter k' p q r'
end
end. | Fixpoint | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | sqrt_iter | |
sqrtn := sqrt_iter n 0 0 0. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | sqrt | |
log2_iterk p q r :=
match k with
| O => p
| S k' => match r with
| O => log2_iter k' (S p) (S q) q
| S r' => log2_iter k' p (S q) r'
end
end. | Fixpoint | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | log2_iter | |
log2n := log2_iter (pred n) 0 1 0. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | log2 | |
iter(n:nat) {A} (f:A->A) (x:A) : A :=
nat_rect (fun _ => A) x (fun _ => f) n. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | iter | |
div2n :=
match n with
| 0 => 0
| S 0 => 0
| S (S n') => S (div2 n')
end. | Fixpoint | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | div2 | |
testbita n : bool :=
match n with
| 0 => odd a
| S n => testbit (div2 a) n
end. | Fixpoint | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | testbit | |
shiftla := nat_rect _ a (fun _ => double). | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | shiftl | |
shiftra := nat_rect _ a (fun _ => div2). | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | shiftr | |
bitwise(op:bool->bool->bool) n a b :=
match n with
| 0 => 0
| S n' =>
(if op (odd a) (odd b) then 1 else 0) +
2*(bitwise op n' (div2 a) (div2 b))
end. | Fixpoint | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | bitwise | |
landa b := bitwise andb a a b. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | land | |
lora b := bitwise orb (max a b) a b. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | lor | |
ldiffa b := bitwise (fun b b' => andb b (negb b')) a a b. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | ldiff | |
lxora b := bitwise xorb (max a b) a b. | Definition | Corelib | [
"Require Import Notations Logic Datatypes",
"Require Decimal Hexadecimal Number"
] | Corelib/Init/Nat.v | lxor | |
uint:= UIntDecimal (u:Decimal.uint) | UIntHexadecimal (u:Hexadecimal.uint). | Variant | Corelib | [
"Require Import Decimal Hexadecimal"
] | Corelib/Init/Number.v | uint | |
signed_int:= IntDecimal (i:Decimal.int) | IntHexadecimal (i:Hexadecimal.int).
Abbreviation int := signed_int. | Variant | Corelib | [
"Require Import Decimal Hexadecimal"
] | Corelib/Init/Number.v | signed_int | |
number:= Decimal (d:Decimal.decimal) | Hexadecimal (h:Hexadecimal.hexadecimal). | Variant | Corelib | [
"Require Import Decimal Hexadecimal"
] | Corelib/Init/Number.v | number | |
Equalityfor uint. | Scheme | Corelib | [
"Require Import Decimal Hexadecimal"
] | Corelib/Init/Number.v | Equality | |
Equalityfor int. | Scheme | Corelib | [
"Require Import Decimal Hexadecimal"
] | Corelib/Init/Number.v | Equality | |
Equalityfor number.
Abbreviation int_eq_dec := signed_int_eq_dec.
Abbreviation int_beq := signed_int_beq.
Abbreviation internal_int_dec_lb := internal_signed_int_dec_lb.
Abbreviation internal_int_dec_bl := internal_signed_int_dec_bl.
Register uint as num.num_uint.type.
Register int as num.num_int.type.
Register number... | Scheme | Corelib | [
"Require Import Decimal Hexadecimal"
] | Corelib/Init/Number.v | Equality | |
uint_of_uint(i:uint) := i. | Definition | Corelib | [
"Require Import Decimal Hexadecimal"
] | Corelib/Init/Number.v | uint_of_uint | |
int_of_int(i:int) := i. | Definition | Corelib | [
"Require Import Decimal Hexadecimal"
] | Corelib/Init/Number.v | int_of_int | |
eq_S:= f_equal S. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | eq_S | |
f_equal_nat:= f_equal (A:=nat).
#[global]
Hint Resolve f_equal_nat: core. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | f_equal_nat | |
f_equal_pred:= f_equal pred. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | f_equal_pred | |
pred_Sn: forall n:nat, n = pred (S n).
Proof.
simpl; reflexivity.
Qed. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | pred_Sn | |
eq_add_Sn m (H: S n = S m): n = m := f_equal pred H.
#[global]
Hint Immediate eq_add_S: core. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | eq_add_S | |
not_eq_S: forall n m:nat, n <> m -> S n <> S m.
Proof.
red; auto.
Qed.
#[global]
Hint Resolve not_eq_S: core. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | not_eq_S | |
IsSucc(n:nat) : Prop :=
match n with
| O => False
| S p => True
end. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | IsSucc | |
O_S: forall n:nat, 0 <> S n.
Proof.
discriminate.
Qed.
#[global]
Hint Resolve O_S: core. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | O_S | |
n_Sn: forall n:nat, n <> S n.
Proof.
intro n; induction n; auto.
Qed.
#[global]
Hint Resolve n_Sn: core. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | n_Sn | |
f_equal2_plus:= f_equal2 plus. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | f_equal2_plus | |
f_equal2_nat:= f_equal2 (A1:=nat) (A2:=nat).
#[global]
Hint Resolve f_equal2_nat: core. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | f_equal2_nat | |
plus_n_O: forall n:nat, n = n + 0.
Proof.
intro n; induction n; simpl; auto.
Qed.
#[global]
Remove Hints eq_refl : core.
#[global]
Hint Resolve plus_n_O eq_refl: core. (* We want eq_refl to have higher priority than plus_n_O *) | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | plus_n_O | |
plus_O_n: forall n:nat, 0 + n = n.
Proof.
reflexivity.
Qed. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | plus_O_n | |
plus_n_Sm: forall n m:nat, S (n + m) = n + S m.
Proof.
intros n m; induction n; simpl; auto.
Qed.
#[global]
Hint Resolve plus_n_Sm: core. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | plus_n_Sm | |
plus_Sn_m: forall n m:nat, S n + m = S (n + m).
Proof.
reflexivity.
Qed. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | plus_Sn_m | |
f_equal2_mult:= f_equal2 mult.
#[global]
Hint Resolve f_equal2_mult: core. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | f_equal2_mult | |
mult_n_O: forall n:nat, 0 = n * 0.
Proof.
intro n; induction n; simpl; auto.
Qed.
#[global]
Hint Resolve mult_n_O: core. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | mult_n_O | |
mult_n_Sm: forall n m:nat, n * m + n = n * S m.
Proof.
intros n m; induction n as [| p H]; simpl; auto.
destruct H; rewrite <- plus_n_Sm; apply eq_S.
pattern m at 1 3; elim m; simpl; auto.
Qed.
#[global]
Hint Resolve mult_n_Sm: core. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | mult_n_Sm | |
le(n:nat) : nat -> Prop :=
| le_n : n <= n
| le_S : forall m:nat, n <= m -> n <= S m
where "n <= m" := (le n m) : nat_scope.
Register le_n as num.nat.le_n.
#[global]
Hint Constructors le: core. | Inductive | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | le | |
lt(n m:nat) := S n <= m.
#[global]
Hint Unfold lt: core.
Infix "<" := lt : nat_scope. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | lt | |
ge(n m:nat) := m <= n.
#[global]
Hint Unfold ge: core.
Infix ">=" := ge : nat_scope. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | ge | |
gt(n m:nat) := m < n.
#[global]
Hint Unfold gt: core.
Infix ">" := gt : nat_scope. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | gt | |
le_pred: forall n m, n <= m -> pred n <= pred m.
Proof.
induction 1 as [|m _]; auto. destruct m; simpl; auto.
Qed. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | le_pred | |
le_S_n: forall n m, S n <= S m -> n <= m.
Proof.
intros n m. exact (le_pred (S n) (S m)).
Qed. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | le_S_n | |
le_0_n: forall n, 0 <= n.
Proof.
intro n; induction n; constructor; trivial.
Qed. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | le_0_n | |
le_n_S: forall n m, n <= m -> S n <= S m.
Proof.
induction 1; constructor; trivial.
Qed. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | le_n_S | |
nat_case:
forall (n:nat) (P:nat -> Prop), P 0 -> (forall m:nat, P (S m)) -> P n.
Proof.
intros n P IH0 IHS; case n; auto.
Qed. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | nat_case | |
nat_double_ind:
forall R:nat -> nat -> Prop,
(forall n:nat, R 0 n) ->
(forall n:nat, R (S n) 0) ->
(forall n m:nat, R n m -> R (S n) (S m)) -> forall n m:nat, R n m.
Proof.
intros R ? ? ? n.
induction n; auto.
intro m; destruct m; auto.
Qed. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | nat_double_ind | |
max_ln m : m <= n -> Nat.max n m = n.
Proof.
revert m; induction n as [|n IHn]; intro m; destruct m; simpl; trivial.
- inversion 1.
- intros. apply f_equal, IHn, le_S_n; trivial.
Qed. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | max_l | |
max_rn m : n <= m -> Nat.max n m = m.
Proof.
revert m; induction n as [|n IHn]; intro m; destruct m; simpl; trivial.
- inversion 1.
- intros. apply f_equal, IHn, le_S_n; trivial.
Qed. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | max_r | |
min_ln m : n <= m -> Nat.min n m = n.
Proof.
revert m; induction n as [|n IHn]; intro m; destruct m; simpl; trivial.
- inversion 1.
- intros. apply f_equal, IHn, le_S_n; trivial.
Qed. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | min_l | |
min_rn m : m <= n -> Nat.min n m = m.
Proof.
revert m; induction n as [|n IHn]; intro m; destruct m; simpl; trivial.
- inversion 1.
- intros. apply f_equal, IHn, le_S_n; trivial.
Qed. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | min_r | |
nat_rect_succ_r{A} (f: A -> A) (x:A) n :
nat_rect (fun _ => A) x (fun _ => f) (S n) = nat_rect (fun _ => A) (f x) (fun _ => f) n.
Proof.
induction n as [|n IHn]; intros; simpl; rewrite <- ?IHn; trivial.
Qed. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | nat_rect_succ_r | |
nat_rect_plus:
forall (n m:nat) {A} (f:A -> A) (x:A),
nat_rect (fun _ => A) x (fun _ => f) (n + m) =
nat_rect (fun _ => A) (nat_rect (fun _ => A) x (fun _ => f) m) (fun _ => f) n.
Proof.
intro n; induction n as [|n IHn]; intros; simpl; rewrite ?IHn; trivial.
Qed. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic",
"Require Corelib.Init.Nat"
] | Corelib/Init/Peano.v | nat_rect_plus | |
reverse_coercion{T' T} (x' : T') (x : ReverseCoercionSource T)
: ReverseCoercionTarget T' := x'.
Register reverse_coercion as core.coercion.reverse_coercion. | Coercion | Corelib | [
"Require Export Notations",
"Require Export Equality",
"Require Export Logic",
"Require Export Datatypes",
"Require Export Specif",
"Require Corelib.Init.Byte",
"Require Corelib.Init.Decimal",
"Require Corelib.Init.Hexadecimal",
"Require Corelib.Init.Number",
"Require Corelib.Init.Nat",
"Require... | Corelib/Init/Prelude.v | reverse_coercion | |
sig(A:Type) (P:A -> Prop) : Type :=
exist : forall x:A, P x -> sig P.
Register sig as core.sig.type.
Register exist as core.sig.intro.
Register sig_rect as core.sig.rect.
#[universes(template)] | Inductive | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sig | |
sig2(A:Type) (P Q:A -> Prop) : Type :=
exist2 : forall x:A, P x -> Q x -> sig2 P Q. | Inductive | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sig2 | |
sigT(A:Type) (P:A -> Type) : Type :=
existT : forall x:A, P x -> sigT P.
Register sigT as core.sigT.type.
Register existT as core.sigT.intro.
Register sigT_rect as core.sigT.rect.
#[universes(template)] | Inductive | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sigT | |
sigT2(A:Type) (P Q:A -> Type) : Type :=
existT2 : forall x:A, P x -> Q x -> sigT2 P Q. | Inductive | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sigT2 | |
proj1_sig(e:sig P) := match e with
| exist _ a b => a
end. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | proj1_sig | |
proj2_sig(e:sig P) :=
match e return P (proj1_sig e) with
| exist _ a b => b
end.
Register proj1_sig as core.sig.proj1.
Register proj2_sig as core.sig.proj2. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | proj2_sig | |
sig_of_sig2(A : Type) (P Q : A -> Prop) (X : sig2 P Q) : sig P
:= exist P
(let (a, _, _) := X in a)
(let (x, p, _) as s return (P (let (a, _, _) := s in a)) := X in p). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sig_of_sig2 | |
proj3_sig(e : sig2 P Q) :=
let (a, b, c) return Q (proj1_sig (sig_of_sig2 e)) := e in c. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | proj3_sig | |
projT1(x:sigT P) : A := match x with
| existT _ a _ => a
end. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | projT1 | |
projT2(x:sigT P) : P (projT1 x) :=
match x return P (projT1 x) with
| existT _ _ h => h
end.
Register projT1 as core.sigT.proj1.
Register projT2 as core.sigT.proj2. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | projT2 | |
sigT_of_sigT2(A : Type) (P Q : A -> Type) (X : sigT2 P Q) : sigT P
:= existT P
(let (a, _, _) := X in a)
(let (x, p, _) as s return (P (let (a, _, _) := s in a)) := X in p). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sigT_of_sigT2 | |
projT3(e : sigT2 P Q) :=
let (a, b, c) return Q (projT1 (sigT_of_sigT2 e)) := e in c. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | projT3 | |
sig_of_sigT(A : Type) (P : A -> Prop) (X : sigT P) : sig P
:= exist P (projT1 X) (projT2 X). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sig_of_sigT | |
sigT_of_sig(A : Type) (P : A -> Prop) (X : sig P) : sigT P
:= existT P (proj1_sig X) (proj2_sig X). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sigT_of_sig | |
sig2_of_sigT2(A : Type) (P Q : A -> Prop) (X : sigT2 P Q) : sig2 P Q
:= exist2 P Q (projT1 (sigT_of_sigT2 X)) (projT2 (sigT_of_sigT2 X)) (projT3 X). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sig2_of_sigT2 | |
sigT2_of_sig2(A : Type) (P Q : A -> Prop) (X : sig2 P Q) : sigT2 P Q
:= existT2 P Q (proj1_sig (sig_of_sig2 X)) (proj2_sig (sig_of_sig2 X)) (proj3_sig X). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sigT2_of_sig2 | |
ex_of_sig(A : Type) (P : A -> Prop) (X : sig P) : ex P
:= ex_intro P (proj1_sig X) (proj2_sig X). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | ex_of_sig | |
ex_of_sigT(A : Type) (P : A -> Prop) (X : sigT P) : ex P
:= ex_of_sig (sig_of_sigT X). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | ex_of_sigT | |
ex2_of_sig2(A : Type) (P Q : A -> Prop) (X : sig2 P Q) : ex2 P Q
:= ex_intro2 P Q (proj1_sig (sig_of_sig2 X)) (proj2_sig (sig_of_sig2 X)) (proj3_sig X). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | ex2_of_sig2 | |
ex2_of_sigT2(A : Type) (P Q : A -> Prop) (X : sigT2 P Q) : ex2 P Q
:= ex2_of_sig2 (sig2_of_sigT2 X). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | ex2_of_sigT2 | |
sigT_eta{A P} (p : { a : A & P a })
: p = existT _ (projT1 p) (projT2 p).
Proof. destruct p; reflexivity. Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sigT_eta | |
sig_eta{A P} (p : { a : A | P a })
: p = exist _ (proj1_sig p) (proj2_sig p).
Proof. destruct p; reflexivity. Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sig_eta | |
sigT2_eta{A P Q} (p : { a : A & P a & Q a })
: p = existT2 _ _ (projT1 (sigT_of_sigT2 p)) (projT2 (sigT_of_sigT2 p)) (projT3 p).
Proof. destruct p; reflexivity. Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sigT2_eta | |
sig2_eta{A P Q} (p : { a : A | P a & Q a })
: p = exist2 _ _ (proj1_sig (sig_of_sig2 p)) (proj2_sig (sig_of_sig2 p)) (proj3_sig p).
Proof. destruct p; reflexivity. Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sig2_eta | |
exists_to_inhabited_sig{A P} : (exists x : A, P x) -> inhabited {x : A | P x}.
Proof.
intros [x y]. exact (inhabits (exist _ x y)).
Qed. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | exists_to_inhabited_sig |
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