statement stringlengths 1 4.15k | proof stringlengths 0 26.3k | type stringclasses 26
values | symbolic_name stringlengths 1 75 | library stringclasses 44
values | filename stringclasses 463
values | imports listlengths 0 23 | deps listlengths 0 64 | docstring stringlengths 0 2.4k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
square_spec n : square n = n * n. | Proof. reflexivity. Qed. | Lemma | square_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"square"
] | ** Square | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
Even n | := exists m, n = 2*m. | Definition | Even | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [] | ** Parity | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
Odd n | := exists m, n = 2*m+1. | Definition | Odd | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Even_0 : Even 0. | Proof. exists 0; reflexivity. Qed. | Lemma | Even_0 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Even_1 : ~ Even 1. | Proof.
intros ([|], H); try discriminate.
simpl in H.
now rewrite <- plus_n_Sm in H.
Qed. | Lemma | Even_1 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Even_2 n : Even n <-> Even (S (S n)). | Proof.
split; intros (m,H).
- exists (S m).
rewrite H; simpl.
now rewrite plus_n_Sm.
- destruct m as [|m]; try discriminate.
exists m.
simpl in H; rewrite <- plus_n_Sm in H.
now inversion H.
Qed. | Lemma | Even_2 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even",
"split"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Odd_0 : ~ Odd 0. | Proof. now intros ([|], H). Qed. | Lemma | Odd_0 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Odd"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Odd_1 : Odd 1. | Proof. exists 0; reflexivity. Qed. | Lemma | Odd_1 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Odd"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Odd_2 n : Odd n <-> Odd (S (S n)). | Proof.
split; intros (m,H).
- exists (S m).
rewrite H. simpl.
now rewrite <- (plus_n_Sm m).
- destruct m as [|m]; try discriminate.
exists m.
simpl in H; rewrite <- plus_n_Sm in H.
inversion H; simpl.
now rewrite <- !plus_n_Sm, <- !plus_n_O.
Qed. | Lemma | Odd_2 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Odd",
"split"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
even_spec : forall n, even n = true <-> Even n. | Proof.
fix even_spec 1.
intro n; destruct n as [|[|n]]; simpl.
- split; [ intros; apply Even_0 | trivial ].
- split; [ discriminate | intro H; elim (Even_1 H) ].
- rewrite even_spec.
apply Even_2.
Qed. | Lemma | even_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even",
"Even_0",
"Even_1",
"Even_2",
"even",
"split"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
odd_spec : forall n, odd n = true <-> Odd n. | Proof.
unfold odd.
fix odd_spec 1.
intro n; destruct n as [|[|n]]; simpl.
- split; [ discriminate | intro H; elim (Odd_0 H) ].
- split; [ intros; apply Odd_1 | trivial ].
- rewrite odd_spec.
apply Odd_2.
Qed. | Lemma | odd_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Odd",
"Odd_0",
"Odd_1",
"Odd_2",
"odd",
"split"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
divmod_spec : forall x y q u, u <= y ->
let (q',u') := divmod x y q u in
x + (S y)*q + (y-u) = (S y)*q' + (y-u') /\ u' <= y. | Proof.
intro x; induction x as [|x IHx].
- simpl; intuition.
- intros y q u H.
destruct u as [|u]; simpl divmod.
+ generalize (IHx y (S q) y (le_n y)).
destruct divmod as (q',u').
intros (EQ,LE); split; trivial.
rewrite <- EQ, sub_0_r, sub_diag, add_0_r.
now rewrite !add_succ_l, <-... | Lemma | divmod_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_0_r",
"add_assoc",
"add_succ_l",
"add_succ_r",
"induction",
"le_trans",
"mul_succ_r",
"split",
"sub_0_r",
"sub_diag",
"sub_succ_l"
] | ** Division | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
div_mod_eq x y : x = y*(x/y) + x mod y. | Proof.
destruct y as [|y]; [reflexivity | ].
unfold div, modulo.
generalize (divmod_spec x y 0 y (le_n y)).
destruct divmod as (q,u).
intros (U,V).
simpl in *.
now rewrite mul_0_r, sub_diag, !add_0_r in U.
Qed. | Lemma | div_mod_eq | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_0_r",
"div",
"divmod_spec",
"mod",
"modulo",
"mul_0_r",
"sub_diag"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
div_mod x y : y <> 0 -> x = y*(x/y) + x mod y. | Proof.
intros _; apply div_mod_eq.
Qed. | Lemma | div_mod | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div_mod_eq",
"mod"
] | The [y <> 0] hypothesis is needed to fit in [NAxiomsSig]. | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
mod_bound_pos x y : 0<=x -> 0<y -> 0 <= x mod y < y. | Proof.
intros Hx Hy.
split.
- apply le_0_l.
- destruct y; [ now elim Hy | clear Hy ].
unfold modulo.
apply lt_succ_r, le_sub_l.
Qed. | Lemma | mod_bound_pos | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"le_0_l",
"le_sub_l",
"lt_succ_r",
"mod",
"modulo",
"split"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
sqrt_iter_spec : forall k p q r,
q = p+p -> r<=q ->
let s := sqrt_iter k p q r in
s*s <= k + p*p + (q - r) < (S s)*(S s). | Proof.
intro k; induction k as [|k IHk].
- (* k = 0 *)
simpl; intros p q r Hq Hr.
split.
+ apply le_add_r.
+ apply lt_succ_r.
rewrite mul_succ_r, add_assoc, (add_comm p), <- add_assoc.
apply add_le_mono_l.
rewrite <- Hq.
apply le_sub_l.
- (* k = S k' *)
intros p q r; de... | Lemma | sqrt_iter_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_0_r",
"add_assoc",
"add_comm",
"add_le_mono_l",
"add_succ_r",
"induction",
"le_add_r",
"le_sub_l",
"le_trans",
"lt_succ_r",
"mul_succ_r",
"replace",
"split",
"sub_0_r",
"sub_diag",
"sub_succ_l"
] | ** Square root | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
sqrt_specif n : (sqrt n)*(sqrt n) <= n < S (sqrt n) * S (sqrt n). | Proof.
set (s:=sqrt n).
replace n with (n + 0*0 + (0-0)).
- apply sqrt_iter_spec; auto.
- simpl.
now rewrite !add_0_r.
Qed. | Lemma | sqrt_specif | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_0_r",
"replace",
"set",
"sqrt",
"sqrt_iter_spec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
sqrt_spec a (Ha:0<=a) | := sqrt_specif a. | Definition | sqrt_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"sqrt_specif"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
sqrt_neg a : a<0 -> sqrt a = 0. | Proof. inversion 1. Qed. | Lemma | sqrt_neg | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"sqrt"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
log2_iter_spec : forall k p q r,
2^(S p) = q + S r -> r < 2^p ->
let s := log2_iter k p q r in
2^s <= k + q < 2^(S s). | Proof.
intro k; induction k as [|k IHk].
- (* k = 0 *)
intros p q r EQ LT.
simpl log2_iter; cbv zeta.
split.
+ rewrite add_0_l, (add_le_mono_l _ _ (2^p)).
simpl pow in EQ.
rewrite add_0_r in EQ; rewrite EQ, add_comm.
apply add_le_mono_r, LT.
+ rewrite EQ, add_comm.
apply ... | Lemma | log2_iter_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_0_l",
"add_0_r",
"add_comm",
"add_le_mono_l",
"add_le_mono_r",
"add_lt_mono_l",
"add_succ_l",
"add_succ_r",
"induction",
"le_0_l",
"le_lt_trans",
"lt_succ_r",
"pow",
"split"
] | ** Logarithm | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
log2_spec n : 0<n ->
2^(log2 n) <= n < 2^(S (log2 n)). | Proof.
intros.
set (s:=log2 n).
replace n with (pred n + 1).
- apply log2_iter_spec; auto.
- rewrite add_1_r.
apply succ_pred.
now apply neq_sym, lt_neq.
Qed. | Lemma | log2_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_1_r",
"log2",
"log2_iter_spec",
"lt_neq",
"neq_sym",
"pred",
"replace",
"set",
"succ_pred"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
log2_nonpos n : n<=0 -> log2 n = 0. | Proof. inversion 1; now subst. Qed. | Lemma | log2_nonpos | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"log2"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
iter_swap_gen A B (f:A -> B) (g:A -> A) (h:B -> B) :
(forall a, f (g a) = h (f a)) -> forall n a,
f (iter n g a) = iter n h (f a). | Proof.
intros H n a.
induction n as [|n Hn].
- reflexivity.
- simpl. rewrite H, Hn. reflexivity.
Qed. | Lemma | iter_swap_gen | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"induction",
"iter"
] | ** Properties of [iter] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
iter_swap :
forall n (A:Type) (f:A -> A) (x:A),
iter n f (f x) = f (iter n f x). | Proof.
intros. symmetry. now apply iter_swap_gen.
Qed. | Lemma | iter_swap | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"iter",
"iter_swap_gen"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
iter_succ :
forall n (A:Type) (f:A -> A) (x:A),
iter (S n) f x = f (iter n f x). | Proof.
reflexivity.
Qed. | Lemma | iter_succ | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"iter"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
iter_succ_r :
forall n (A:Type) (f:A -> A) (x:A),
iter (S n) f x = iter n f (f x). | Proof.
intros; now rewrite iter_succ, iter_swap.
Qed. | Lemma | iter_succ_r | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"iter",
"iter_succ",
"iter_swap"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
iter_add :
forall p q (A:Type) (f:A -> A) (x:A),
iter (p+q) f x = iter p f (iter q f x). | Proof.
intro p. induction p as [|p IHp].
- reflexivity.
- intros q A f x. simpl. now rewrite IHp.
Qed. | Lemma | iter_add | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"induction",
"iter"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
iter_ind (A:Type) (f:A -> A) (a:A) (P:nat -> A -> Prop) :
P 0 a ->
(forall n a', P n a' -> P (S n) (f a')) ->
forall n, P n (iter n f a). | Proof.
intros H0 HS n. induction n as [|n Hn].
- exact H0.
- apply HS. exact Hn.
Qed. | Lemma | iter_ind | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"induction",
"iter"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
iter_rect (A:Type) (f:A -> A) (a:A) (P:nat -> A -> Type) :
P 0 a ->
(forall n a', P n a' -> P (S n) (f a')) ->
forall n, P n (iter n f a). | Proof.
intros H0 HS n. induction n as [|n Hn].
- exact H0.
- apply HS. exact Hn.
Defined. | Lemma | iter_rect | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"induction",
"iter"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
iter_invariant :
forall (n:nat) (A:Type) (f:A -> A) (Inv:A -> Prop),
(forall x:A, Inv x -> Inv (f x)) ->
forall x:A, Inv x -> Inv (iter n f x). | Proof.
intros; apply iter_ind; trivial.
Qed. | Lemma | iter_invariant | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"iter",
"iter_ind"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
divide x y | := exists z, y=z*x. | Definition | divide | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [] | ** Gcd | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
"( x | y )" | := (divide x y) (at level 0) : nat_scope. | Notation | ( x | y ) | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"divide"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
gcd_divide : forall a b, (gcd a b | a) /\ (gcd a b | b). | Proof.
fix gcd_divide 1.
intros [|a] b; simpl.
- split.
+ now exists 0.
+ exists 1; simpl.
now rewrite <- plus_n_O.
- fold (b mod (S a)).
destruct (gcd_divide (b mod (S a)) (S a)) as (H,H').
set (a':=S a) in *.
split; auto.
rewrite (div_mod_eq b a') at 2.
... | Lemma | gcd_divide | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div_mod_eq",
"fold",
"gcd",
"mod",
"mul_add_distr_r",
"mul_assoc",
"mul_comm",
"set",
"split"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
gcd_divide_l : forall a b, (gcd a b | a). | Proof. apply gcd_divide. Qed. | Lemma | gcd_divide_l | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"gcd",
"gcd_divide"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
gcd_divide_r : forall a b, (gcd a b | b). | Proof. apply gcd_divide. Qed. | Lemma | gcd_divide_r | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"gcd",
"gcd_divide"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
gcd_greatest : forall a b c, (c|a) -> (c|b) -> (c|gcd a b). | Proof.
fix gcd_greatest 1.
intros [|a] b; simpl; auto.
fold (b mod (S a)).
intros c H H'.
apply gcd_greatest; auto.
set (a':=S a) in *.
rewrite (div_mod_eq b a') in H'.
destruct H as (u,Hu), H' as (v,Hv).
exists (v - (b/a')*u).
rewrite mul_comm in Hv.
rewrite mul_sub_distr_r, <... | Lemma | gcd_greatest | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_comm",
"add_sub",
"div_mod_eq",
"fold",
"gcd",
"mod",
"mul_assoc",
"mul_comm",
"mul_sub_distr_r",
"set"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
gcd_nonneg a b : 0<=gcd a b. | Proof. apply le_0_l. Qed. | Lemma | gcd_nonneg | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"gcd",
"le_0_l"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
double_S : forall n, double (S n) = S (S (double n)) | := fun n => add_succ_r (S n) n. | Definition | double_S | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_succ_r",
"double"
] | ** Bitwise operations | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
double_add : forall n m, double (n + m) = double n + double m | := fun n m => add_shuffle1 n m n m. | Definition | double_add | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_shuffle1",
"double"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
double_twice : forall n, double n = 2*n. | Proof. simpl; intros; now rewrite add_0_r. Qed. | Lemma | double_twice | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_0_r",
"double"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
testbit_odd_0 : forall a : nat, testbit (add (mul 2 a) 1) 0 = true. | Parameter | testbit_odd_0 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add",
"mul",
"testbit"
] | needed to implement Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
testbit_even_0 : forall a : nat, testbit (mul 2 a) 0 = false. | Parameter | testbit_even_0 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"mul",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
testbit_odd_succ : forall a n : nat, le 0 n ->
testbit (add (mul 2 a) 1) (succ n) = testbit a n. | Parameter | testbit_odd_succ | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add",
"le",
"mul",
"succ",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
testbit_even_succ : forall a n : nat, le 0 n ->
testbit (mul 2 a) (succ n) = testbit a n. | Parameter | testbit_even_succ | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"le",
"mul",
"succ",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
testbit_neg_r : forall a n : nat, lt n 0 -> testbit a n = false. | Parameter | testbit_neg_r | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"lt",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
shiftr_spec : forall a n m : nat, le 0 m ->
testbit (shiftr a n) m = testbit a (add m n). | Parameter | shiftr_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add",
"le",
"shiftr",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
shiftl_spec_high :
forall a n m : nat, le 0 m ->
le n m -> testbit (shiftl a n) m = testbit a (sub m n). | Parameter | shiftl_spec_high | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"le",
"shiftl",
"sub",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
shiftl_spec_low :
forall a n m : nat, lt m n -> testbit (shiftl a n) m = false. | Parameter | shiftl_spec_low | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"lt",
"shiftl",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
land_spec :
forall a b n : nat, testbit (land a b) n = testbit a n && testbit b n. | Parameter | land_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"land",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
lor_spec :
forall a b n : nat, testbit (lor a b) n = testbit a n || testbit b n. | Parameter | lor_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"lor",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
ldiff_spec :
forall a b n : nat,
testbit (ldiff a b) n = testbit a n && negb (testbit b n). | Parameter | ldiff_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"ldiff",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
lxor_spec :
forall a b n : nat, testbit (lxor a b) n = xorb (testbit a n) (testbit b n). | Parameter | lxor_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"lxor",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
div2_spec :
forall a : nat, eq (div2 a) (shiftr a 1). | Parameter | div2_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div2",
"eq",
"shiftr"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
div2_double : forall n, div2 (2*n) = n. | Parameter | div2_double | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div2"
] | not yet generalized to Numbers.Natural.Abstract | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
div2_succ_double : forall n, div2 (S (2*n)) = n. | Parameter | div2_succ_double | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div2"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
div2_bitwise : forall op n a b,
div2 (bitwise op (S n) a b) = bitwise op n (div2 a) (div2 b). | Parameter | div2_bitwise | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"bitwise",
"div2"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
odd_bitwise : forall op n a b,
odd (bitwise op (S n) a b) = op (odd a) (odd b). | Parameter | odd_bitwise | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"bitwise",
"odd"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
testbit_bitwise_1 : forall op, (forall b, op false b = false) ->
forall n m a b, a<=n ->
testbit (bitwise op n a b) m = op (testbit a m) (testbit b m). | Parameter | testbit_bitwise_1 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"bitwise",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
testbit_bitwise_2 : forall op, op false false = false ->
forall n m a b, a<=n -> b<=n ->
testbit (bitwise op n a b) m = op (testbit a m) (testbit b m). | Parameter | testbit_bitwise_2 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"bitwise",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | ||
div2_double n : div2 (2*n) = n. | Proof.
induction n; trivial.
simpl mul.
rewrite add_succ_r; simpl.
now f_equal.
Qed. | Lemma | div2_double | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_succ_r",
"div2",
"induction",
"mul"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
div2_succ_double n : div2 (S (2*n)) = n. | Proof.
induction n; trivial.
simpl; f_equal.
now rewrite add_succ_r.
Qed. | Lemma | div2_succ_double | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_succ_r",
"div2",
"induction"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
le_div2 n : div2 (S n) <= n. | Proof.
revert n.
fix le_div2 1.
intro n; destruct n as [|n]; simpl; trivial.
apply lt_succ_r.
destruct n; [simpl|]; trivial.
now constructor.
Qed. | Lemma | le_div2 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div2",
"lt_succ_r"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
lt_div2 n : 0 < n -> div2 n < n. | Proof.
destruct n.
- inversion 1.
- intros _; apply lt_succ_r, le_div2.
Qed. | Lemma | lt_div2 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div2",
"le_div2",
"lt_succ_r"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
div2_decr a n : a <= S n -> div2 a <= n. | Proof.
destruct a as [|a]; intros H.
- simpl; apply le_0_l.
- apply succ_le_mono in H.
apply le_trans with a; [ apply le_div2 | trivial ].
Qed. | Lemma | div2_decr | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div2",
"le_0_l",
"le_div2",
"le_trans",
"succ_le_mono"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
testbit_0_l : forall n, testbit 0 n = false. | Proof. now intro n; induction n. Qed. | Lemma | testbit_0_l | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"induction",
"testbit"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
testbit_odd_0 a : testbit (2*a+1) 0 = true. | Proof. unfold testbit; rewrite odd_spec; now exists a. Qed. | Lemma | testbit_odd_0 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"odd_spec",
"testbit"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
testbit_even_0 a : testbit (2*a) 0 = false. | Proof.
unfold testbit, odd.
rewrite (proj2 (even_spec _)); trivial.
now exists a.
Qed. | Lemma | testbit_even_0 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"even_spec",
"odd",
"testbit"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
testbit_odd_succ' a n : testbit (2*a+1) (S n) = testbit a n. | Proof.
unfold testbit; fold testbit.
rewrite add_1_r; f_equal.
apply div2_succ_double.
Qed. | Lemma | testbit_odd_succ' | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_1_r",
"div2_succ_double",
"fold",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
testbit_even_succ' a n : testbit (2*a) (S n) = testbit a n. | Proof. unfold testbit; fold testbit; f_equal; apply div2_double. Qed. | Lemma | testbit_even_succ' | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div2_double",
"fold",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
shiftr_specif : forall a n m,
testbit (shiftr a n) m = testbit a (m+n). | Proof.
intros a n; induction n as [|n IHn]; intros m.
- now rewrite add_0_r.
- now rewrite add_succ_r, <- add_succ_l, <- IHn.
Qed. | Lemma | shiftr_specif | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_0_r",
"add_succ_l",
"add_succ_r",
"induction",
"shiftr",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
shiftl_specif_high : forall a n m, n<=m ->
testbit (shiftl a n) m = testbit a (m-n). | Proof.
intros a n; induction n as [|n IHn]; intros m H; [ trivial | ].
- now rewrite sub_0_r.
- destruct m; [ inversion H | ].
simpl; apply succ_le_mono in H.
change (shiftl a (S n)) with (double (shiftl a n)).
rewrite double_twice, div2_double.
now apply IHn.
Qed. | Lemma | shiftl_specif_high | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div2_double",
"double",
"double_twice",
"induction",
"shiftl",
"sub_0_r",
"succ_le_mono",
"testbit"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
shiftl_spec_low : forall a n m, m<n ->
testbit (shiftl a n) m = false. | Proof.
intros a n; induction n as [|n IHn]; intros m H; [ inversion H | ].
change (shiftl a (S n)) with (double (shiftl a n)).
destruct m; simpl.
- unfold odd; apply negb_false_iff.
apply even_spec.
exists (shiftl a n).
apply double_twice.
- rewrite double_twice, div2_double.
apply IHn.
no... | Lemma | shiftl_spec_low | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div2_double",
"double",
"double_twice",
"even_spec",
"induction",
"negb_false_iff",
"odd",
"shiftl",
"succ_le_mono",
"testbit"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
div2_bitwise : forall op n a b,
div2 (bitwise op (S n) a b) = bitwise op n (div2 a) (div2 b). | Proof.
intros op n a b; unfold bitwise; fold bitwise.
destruct (op (odd a) (odd b)).
- now rewrite div2_succ_double.
- now rewrite add_0_l, div2_double.
Qed. | Lemma | div2_bitwise | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_0_l",
"bitwise",
"div2",
"div2_double",
"div2_succ_double",
"fold",
"odd"
] | not yet generalized, part of the interface at this point | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
odd_bitwise : forall op n a b,
odd (bitwise op (S n) a b) = op (odd a) (odd b). | Proof.
intros op n a b; unfold bitwise; fold bitwise.
destruct (op (odd a) (odd b)).
- apply odd_spec.
rewrite add_comm; eexists; eauto.
- unfold odd; apply negb_false_iff.
apply even_spec.
rewrite add_0_l; eexists; eauto.
Qed. | Lemma | odd_bitwise | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_0_l",
"add_comm",
"bitwise",
"even_spec",
"fold",
"negb_false_iff",
"odd",
"odd_spec"
] | not yet generalized, part of the interface at this point | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
testbit_bitwise_1 : forall op, (forall b, op false b = false) ->
forall n m a b, a<=n ->
testbit (bitwise op n a b) m = op (testbit a m) (testbit b m). | Proof.
intros op Hop.
intro n; induction n as [|n IHn]; intros m a b Ha.
- simpl; inversion Ha; subst.
now rewrite testbit_0_l.
- destruct m.
+ apply odd_bitwise.
+ unfold testbit; fold testbit; rewrite div2_bitwise.
apply IHn; now apply div2_decr.
Qed. | Lemma | testbit_bitwise_1 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"bitwise",
"div2_bitwise",
"div2_decr",
"fold",
"induction",
"odd_bitwise",
"testbit",
"testbit_0_l"
] | not yet generalized, part of the interface at this point | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
testbit_bitwise_2 : forall op, op false false = false ->
forall n m a b, a<=n -> b<=n ->
testbit (bitwise op n a b) m = op (testbit a m) (testbit b m). | Proof.
intros op Hop.
intro n; induction n as [|n IHn]; intros m a b Ha Hb.
- simpl; inversion Ha; inversion Hb; subst.
now rewrite testbit_0_l.
- destruct m.
+ apply odd_bitwise.
+ unfold testbit; fold testbit; rewrite div2_bitwise.
apply IHn; now apply div2_decr.
Qed. | Lemma | testbit_bitwise_2 | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"bitwise",
"div2_bitwise",
"div2_decr",
"fold",
"induction",
"odd_bitwise",
"testbit",
"testbit_0_l"
] | not yet generalized, part of the interface at this point | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
land_spec a b n :
testbit (land a b) n = testbit a n && testbit b n. | Proof. unfold land; apply testbit_bitwise_1; trivial. Qed. | Lemma | land_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"land",
"testbit",
"testbit_bitwise_1"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
ldiff_spec a b n :
testbit (ldiff a b) n = testbit a n && negb (testbit b n). | Proof. unfold ldiff; apply testbit_bitwise_1; trivial. Qed. | Lemma | ldiff_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"ldiff",
"testbit",
"testbit_bitwise_1"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
lor_spec a b n :
testbit (lor a b) n = testbit a n || testbit b n. | Proof.
unfold lor; apply testbit_bitwise_2.
- trivial.
- destruct (compare_spec a b) as [H|H|H].
+ rewrite max_l; subst; trivial.
+ now apply lt_le_incl in H; rewrite max_r.
+ now apply lt_le_incl in H; rewrite max_l.
- destruct (compare_spec a b) as [H|H|H].
+ rewrite max_r; subst; trivial.
... | Lemma | lor_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"compare_spec",
"lor",
"lt_le_incl",
"max_l",
"max_r",
"testbit",
"testbit_bitwise_2"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
lxor_spec a b n :
testbit (lxor a b) n = xorb (testbit a n) (testbit b n). | Proof.
unfold lxor; apply testbit_bitwise_2.
- trivial.
- destruct (compare_spec a b) as [H|H|H].
+ rewrite max_l; subst; trivial.
+ now apply lt_le_incl in H; rewrite max_r.
+ now apply lt_le_incl in H; rewrite max_l.
- destruct (compare_spec a b) as [H|H|H].
+ rewrite max_r; subst; trivial.
... | Lemma | lxor_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"compare_spec",
"lt_le_incl",
"lxor",
"max_l",
"max_r",
"testbit",
"testbit_bitwise_2"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
div2_spec a : div2 a = shiftr a 1. | Proof. reflexivity. Qed. | Lemma | div2_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"div2",
"shiftr"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
testbit_odd_succ a n (_:0<=n) | := testbit_odd_succ' a n. | Definition | testbit_odd_succ | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"testbit_odd_succ'"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
testbit_even_succ a n (_:0<=n) | := testbit_even_succ' a n. | Definition | testbit_even_succ | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"testbit_even_succ'"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
testbit_neg_r a n (H:n<0) : testbit a n = false. | Proof. inversion H. Qed. | Lemma | testbit_neg_r | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"testbit"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
shiftl_spec_high a n m (_:0<=m) | := shiftl_specif_high a n m. | Definition | shiftl_spec_high | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"shiftl_specif_high"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
shiftr_spec a n m (_:0<=m) | := shiftr_specif a n m. | Definition | shiftr_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"shiftr_specif"
] | needed to implement Stdlib.Numbers.NatInt.NZBitsSpec | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
div_0_r a : a / 0 = 0. | Proof. reflexivity. Qed. | Lemma | div_0_r | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
mod_0_r a : a mod 0 = a. | Proof. reflexivity. Qed. | Lemma | mod_0_r | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"mod"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
binary_induction (A : nat -> Prop) :
A 0 -> (forall n, A n -> A (2 * n)) -> (forall n, A n -> A (2 * n + 1))
-> forall n, A n. | Proof. apply Private_binary_induction; intros x y ->; reflexivity. Qed. | Lemma | binary_induction | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Private_binary_induction"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
tail_add_spec n m : tail_add n m = n + m. | Proof.
revert m; induction n as [|n IH]; simpl; trivial; intros.
now rewrite IH, add_succ_r.
Qed. | Lemma | tail_add_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_succ_r",
"induction"
] | Properties of tail-recursive addition and multiplication | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
tail_addmul_spec r n m : tail_addmul r n m = r + n * m. | Proof.
revert m r; induction n as [| n IH]; simpl; trivial; intros.
rewrite IH, tail_add_spec.
rewrite add_assoc.
f_equal; apply add_comm.
Qed. | Lemma | tail_addmul_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"add_assoc",
"add_comm",
"induction",
"tail_add_spec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
tail_mul_spec n m : tail_mul n m = n * m. | Proof. unfold tail_mul; now rewrite tail_addmul_spec. Qed. | Lemma | tail_mul_spec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"tail_addmul_spec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Even_Odd_dec n : {Even n} + {Odd n}. | Proof.
induction n as [|n IHn].
- left; apply Even_0.
- elim IHn; intros.
+ right; apply Even_succ, Even_succ_succ; assumption.
+ left; apply Odd_succ, Odd_succ_succ; assumption.
Defined. | Definition | Even_Odd_dec | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even",
"Even_0",
"Even_succ",
"Even_succ_succ",
"Odd",
"Odd_succ",
"Odd_succ_succ",
"induction",
"left",
"right"
] | Additional results about [Even] and [Odd] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
Even_add_split n m :
Even (n + m) -> Even n /\ Even m \/ Odd n /\ Odd m. | Proof.
rewrite <- ? even_spec, <- ? odd_spec, even_add; unfold odd; do 2 destruct even; auto.
Qed. | Lemma | Even_add_split | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even",
"Odd",
"even",
"even_add",
"even_spec",
"odd",
"odd_spec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Odd_add_split n m :
Odd (n + m) -> Odd n /\ Even m \/ Even n /\ Odd m. | Proof.
rewrite <- ? even_spec, <- ? odd_spec, odd_add; unfold odd; do 2 destruct even; auto.
Qed. | Lemma | Odd_add_split | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even",
"Odd",
"even",
"even_spec",
"odd",
"odd_add",
"odd_spec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Even_Even_add n m: Even n -> Even m -> Even (n + m). | Proof. rewrite <- ? even_spec, even_add; do 2 destruct even; auto. Qed. | Lemma | Even_Even_add | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even",
"even",
"even_add",
"even_spec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Odd_add_l n m : Odd n -> Even m -> Odd (n + m). | Proof.
rewrite <- ? even_spec, <- ? odd_spec, odd_add; unfold odd; do 2 destruct even; auto.
Qed. | Lemma | Odd_add_l | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even",
"Odd",
"even",
"even_spec",
"odd",
"odd_add",
"odd_spec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Odd_add_r n m : Even n -> Odd m -> Odd (n + m). | Proof.
rewrite <- ? even_spec, <- ? odd_spec, odd_add; unfold odd; do 2 destruct even; auto.
Qed. | Lemma | Odd_add_r | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even",
"Odd",
"even",
"even_spec",
"odd",
"odd_add",
"odd_spec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Odd_Odd_add n m : Odd n -> Odd m -> Even (n + m). | Proof.
rewrite <- ? even_spec, <- ? odd_spec, even_add; unfold odd; do 2 destruct even; auto.
Qed. | Lemma | Odd_Odd_add | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even",
"Odd",
"even",
"even_add",
"even_spec",
"odd",
"odd_spec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Even_add_aux n m :
(Odd (n + m) <-> Odd n /\ Even m \/ Even n /\ Odd m) /\
(Even (n + m) <-> Even n /\ Even m \/ Odd n /\ Odd m). | Proof.
split; split.
- apply Odd_add_split.
- intros [[HO HE]|[HE HO]]; [ apply Odd_add_l | apply Odd_add_r ]; assumption.
- apply Even_add_split.
- intros [[HO HE]|[HE HO]]; [ apply Even_Even_add | apply Odd_Odd_add ]; assumption.
Qed. | Lemma | Even_add_aux | Arith | theories/Arith/PeanoNat.v | [
"Stdlib",
"NAxioms",
"NProperties",
"OrdersFacts",
"DecidableClass",
"Private_Parity"
] | [
"Even",
"Even_Even_add",
"Even_add_split",
"Odd",
"Odd_Odd_add",
"Odd_add_l",
"Odd_add_r",
"Odd_add_split",
"split"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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