key stringlengths 12 67 | value stringlengths 29 186 |
|---|---|
add distributes over max | Definition Qmax_plus_distr_l : forall x y z : Q, Qmax y z + x == Qmax (y+x) (z+x) := fun a => @monotone_join_distr Qto _ (Qplus_monotone_l a). |
mean greater than min | Lemma Smallest_less_Average : forall a b : R, a [<] b -> a [<] (a[+]b) [/]TwoNZ. |
addition injective | Lemma Rplus_eq_reg_r : forall r r1 r2, r1 + r = r2 + r -> r1 = r2. |
decidable equality | Definition equiv_decb `{EqDec A} (x y : A) : bool := if x == y then true else false. |
less than transitivity | Lemma Zcompare_Lt_trans : forall n m p:Z, (n ?= m) = Lt -> (m ?= p) = Lt -> (n ?= p) = Lt. |
exponentiated fraction | Lemma nexp_distr_div : forall (x y : R) n y_ yn_, (x[/] y[//]y_) [^]n [=] (x[^]n[/] y[^]n[//]yn_). |
product of positives is positive | Lemma R_mult_resp_lt : forall x y, Rlt [0] x -> Rlt [0] y -> Rlt [0] (x[*]y). |
iff decomposition | Lemma iff_to_and : forall A B : Prop, (A <-> B) <-> (A -> B) /\ (B -> A). |
root x less than x | Lemma sqrt_less : forall x y ypos, x[^]2 [<] y -> x [<] sqrt y ypos. |
forall exists de morgan | Lemma neg_Forall_Exists_neg (P:A->Prop) (l:list A) : (forall x:A, {P x} + { ~ P x }) -> ~ Forall P l -> Exists (fun x => ~ P x) l. |
forall exists duality | Lemma neg_Forall_Exists_neg (P:A->Prop) (l:list A) : (forall x:A, {P x} + { ~ P x }) -> ~ Forall P l -> Exists (fun x => ~ P x) l. |
list sum order independent | Lemma sum_list_with_reverse (f : A → nat) l : sum_list_with f (reverse l) = sum_list_with f l. |
reverse of list same sum | Lemma sum_list_with_reverse (f : A → nat) l : sum_list_with f (reverse l) = sum_list_with f l. |
multiplication distributes over min | Lemma CReal_min_mult : forall (p q r:CReal), 0 <= r -> CReal_min (r * p) (r * q) == r * CReal_min p q. |
propagate error through addition | Lemma approximate_CRplus (x y: CR) (e: Qpos): approximate (x + y)%CR e = (approximate x ((1#2) * e)%Qpos + approximate y ((1#2) * e)%Qpos)%Q. |
multiply both sides of an inequality | Lemma Zmult_ge_compat_l n m p : n >= m -> p >= 0 -> p * n >= p * m. |
else-if chain identity | Lemma if_or b1 b2 T (x y : T) : (if b1 || b2 then x else y) = (if b1 then x else if b2 then x else y). |
product of odd numbers is odd | Theorem OddMult : forall n m : Z, Odd n -> Odd m -> Odd (n * m). |
similar matrix exponentiation | Lemma similar_exp m n (A : 'M[R]_m.+1) (B : 'M_n.+1) k: similar A B -> similar (A ^+ k) (B ^+ k). |
limit of a negated function | Lemma limit_Ropp : forall (f:R -> R) (D:R -> Prop) (l x0:R), limit1_in f D l x0 -> limit1_in (fun x:R => - f x) D (- l) x0. |
sum of empty range | Lemma Sum_empty : forall n f, 0 < n -> Sum n (pred n) f [=] [0]. |
vector list conversion | Lemma vec_to_list_inj2 {A n} (v : vec A n) (w : vec A n) : vec_to_list v = vec_to_list w → v = w. |
upper and lower bound equal | Lemma Rge_antisym : forall r1 r2, r1 >= r2 -> r2 >= r1 -> r1 = r2. |
positive rational induction | Lemma Qpos_positive_numerator_rect (P: Qpos -> Type): (forall (a b: positive), P (a # b)%Qpos) -> forall q, P q. |
plus minus cancellation | Lemma cg_cancel_mixed : forall x y : G, x [=] x [-] y[+]y. |
boolean negation function | Definition negb (b:bool) := if b then false else true. |
4 is positive | Lemma pos_four : ([0]:R) [<] Four. |
length of sublist | Lemma calc_Zlength_sublist : forall A (l : list A) len lo hi, Zlength l = len -> 0 <= lo <= hi -> hi <= len -> Zlength (sublist lo hi l) = hi - lo. |
superset of non-empty set is non-empty | Lemma subset_neq0 A B : A \subset B -> A != set0 -> B != set0. |
meet returns lesser | Lemma le_meet_r : forall x y : X, y <= x <-> meet x y == y. |
minimum of upper bounds is an upper bound | Lemma Rmin_glb_lt : forall x y z:R, z < x -> z < y -> z < Rmin x y. |
boolean inverse returns different value than input | Lemma no_fixpoint_negb : forall b:bool, negb b <> b. |
range of exponentiation is positive | Lemma Exp_pos' : forall x : IR, [0] [<] x -> [0] [<] Exp x. |
flip real inequality | Lemma inv_resp_leEq : forall x y : R, x [<=] y -> [--]y [<=] [--]x. |
forall elements in vector | Lemma Forall_vlookup_1 {A} (P : A → Prop) {n} (v : vec A n) i : Forall P (vec_to_list v) → P (v !!! i). |
positive integer conversion | Lemma positive_to_int_spec : forall p : positive, Zpos p = Z_of_N (fst (positive_to_int p)) * wB + to_Z (snd (positive_to_int p)). |
max of empty list | Lemma max_list_elem_of ns : ns ≠ [] → max_list ns ∈ ns. |
decrement both sides of an inequality | Lemma Zsucc_lt_reg n m : Z.succ n < Z.succ m -> n < m. |
bijection spec | Lemma bij_eq : bijective f -> forall x y, (f x == f y) = (x == y). |
integral of weighted sum of functions | Lemma antiderivative_P1 : forall (f g F G:R -> R) (l a b:R), antiderivative f F a b -> antiderivative g G a b -> antiderivative (fun x:R => l * f x + g x) (fun x:R => l * F x + G x) a b. |
n-differentiable function's derivative is (n-1)-differentiable | Lemma Diffble_I_imp_le : forall n, 0 < n -> forall F F', Diffble_I_n Hab' n F -> Derivative_I Hab' F F' -> Diffble_I_n Hab' (pred n) F'. |
map across zipped vectors spec | Lemma vlookup_zip_with `(f : A → B → C) {n} (v1 : vec A n) (v2 : vec B n) i : vzip_with f v1 v2 !!! i = f (v1 !!! i) (v2 !!! i). |
integer inequality injects to rational inequality | Lemma Zlt_Qlt (x y: Z): (x < y)%Z = (inject_Z x < inject_Z y). |
integers modulo their factors are zero | Lemma Zmod_divides : forall a b, b<>0 -> (a mod b = 0 <-> exists c, a = b*c). |
modular multiplication simplification | Lemma Zmod_mult_compat_lft : forall m a b : Z, (m > 0)%Z -> ((a * b) mod m)%Z = ((a mod m * b) mod m)%Z. |
integer less than decidable | Lemma Zlt_is_lt_bool n m : (n < m) <-> (n <? m) = true. |
positive bitwise xor | Lemma Pxor_semantics p p' n : N.testbit_nat (Pos.lxor p p') n = xorb (Pos.testbit_nat p n) (Pos.testbit_nat p' n). |
equivalence is an upper and lower bound | Lemma equiv_le_def : forall x y:X, x == y <-> (x <= y /\ y <= x). |
absolute value negates negative numbers | Lemma Rabs_left : forall r, r < 0 -> Rabs r = - r. |
strengthen inequality to strict inequality | Lemma leEq_not_eq : forall x y : R, x [<=] y -> x [#] y -> x [<] y. |
distribute negative through min | Definition Qmin_max_de_morgan : forall x y : Q, -(Qmin x y) == Qmax (-x) (-y) := @antitone_meet_join_distr Qto _ Qopp_le_compat. |
numbers divisible by powers of primes are divisible by those primes | Theorem Zdivide_power_2 x p n : 0 <= n -> 0 <= x -> prime p -> (x | p^n) -> exists m, x = p^m. |
cancel multiplication | Lemma Nmult_reg_r n m p : p <> 0 -> n * p = m * p -> n = m. |
divisibility preserved by multiplication | Lemma Zdivides_mult_cancel_lft : forall a b c : Z, Zdivides a b -> Zdivides (c * a) (c * b). |
equal functions have the same image | Lemma fun_image_wd : Feq I F G -> forall x, fun_image F I x -> fun_image G I x. |
iff composition | Lemma and_morph : forall (s1 s2 t1 t2:Prop), s1 <-> t1 -> s2 <-> t2 -> ((s1 /\ s2) <-> (t1 /\ t2)). |
remainder is modulo | Theorem Zquotrem_Zdiv_eucl_pos : forall a b:Z, 0 <= a -> 0 < b -> a÷b = a/b /\ Z.rem a b = a mod b. |
proposition de morgan | Lemma and_not_or : forall P Q:Prop, ~ P /\ ~ Q -> ~ (P \/ Q). |
division by product is division by the factors | Lemma divnMA m n p : m %/ (n * p) = m %/ n %/ p. |
sum of non-negatives is non-negative | Theorem Rplus_nonneg_nonneg : forall n m : R, 0 <= n -> 0 <= m -> 0 <= n + m. |
max of lower bounds is a lower bound | Definition Qmax_lub : forall x y z : Q, x <= z -> y <= z -> (Qmax x y) <= z := @join_lub Qto. |
strict inequality implies disequality | Lemma Z_lt_neq {x y: Z} : x < y -> y <> x. |
conjunctive hypothesis to hypotheses | Lemma uncurry_conj (A B C : Prop) : (A /\ B -> C) -> (A -> B -> C). |
tangent shift | Lemma tan_pi_minus : forall x, cos x <> 0 -> tan (PI - x) = - tan x. |
gaussian sum | Lemma textbook_triangular_sum n : \sum_(0 <= i < n) i = 'C(n, 2). |
sum of first n naturals | Lemma textbook_triangular_sum n : \sum_(0 <= i < n) i = 'C(n, 2). |
naturals surject to the positive integers | Lemma IZN : forall n:Z, (0 <= n)%Z -> exists m : nat, n = Z.of_nat m. |
lower bound of a minimum is a lower bound of the min args | Lemma Nmin_le_3 a b c : Nleb a (N.min b c) = true -> Nleb a b = true. |
every real has a positive square root | Lemma sqrt_nonneg : forall x xpos, [0] [<=] sqrt x xpos. |
sum of powers | Lemma Sum_c_exp : forall H m n, Sum m n (fun i => c[^]i) [=] (c[^]S n[-]c[^]m[/] c[-][1][//]H). |
multiplying by zero is zero | Lemma zmult_zero : forall x:G, zmult x 0 [=] [0]. |
nonzero product has nonzero factors | Lemma R_mult_ap_zero' : forall x y, R_mult x y [#] [0] -> x [#] [0]. |
list concatenation preserves inclusion | Lemma incl_app : forall l m n:list A, incl l n -> incl m n -> incl (l ++ m) n. |
if a product is zero then one of the factors is zero | Theorem Qcmult_integral_l : forall x y, ~ x = 0 -> x*y = 0 -> y = 0. |
no integer is less than itself | Definition Zlt_irref : forall a : Z, ~ (a < a)%Z := Z.lt_irrefl. |
list range of numbers | Lemma seq_S : forall len start, seq start (S len) = seq start len ++ [start + len]. |
distribute negation through multiplication | Lemma CRopp_mult_distr_l : forall (r1 r2 : CR), - (r1 * r2) == (-r1) * r2. |
quotient of powers cancel | Theorem pow_RN_plus : forall (e : R) (n m : nat), e <> 0%R -> (e ^ n)%R = (e ^ (n + m) * / e ^ m)%R. |
injection to reals preserves subtraction | Lemma minus_INR : forall n m:nat, (m <= n)%nat -> INR (n - m) = INR n - INR m. |
integers are divisible by the negation of their factors | Lemma Zdivide_opp_r a b : (a | b) -> (a | - b). |
weaken strong inequality | Lemma Zgt_le_succ n m : m > n -> Z.succ n <= m. |
exp monotone | Lemma Exp_cancel_leEq : forall x y : IR, Exp x [<=] Exp y -> x [<=] y. |
subtract from both sides of an inequality | Lemma minus_resp_less : forall x y z : R, x [<] y -> x[-]z [<] y[-]z. |
can't surject nats onto reals | Theorem reals_not_countable : forall (f : nat -> IR),{x :IR | forall n : nat, x [#] (f n)}. |
shift cosine | Lemma Cos_plus_HalfPi : forall x : IR, Cos (x[+]Pi [/]TwoNZ) [=] [--] (Sin x). |
successor cancels predecessor | Theorem S_predn : forall m : nat, m <> 0 -> S (pred m) = m. |
integers are greater than their factors | Lemma Zdivisor_pos_le : forall a b : Z, (a > 0)%Z -> Zdivides b a -> (a >= b)%Z. |
powers of bases greater than one grow | Lemma Qpower_1_lt: forall (q : Q) (n : Z), (1<q)%Q -> (0<n)%Z -> (1 < q ^ n)%Q. |
boolean de morgan | Lemma negb_andb : forall b1 b2:bool, negb (b1 && b2) = negb b1 || negb b2. |
quotient of negatives is positive | Lemma Zdiv_opp_opp : forall a b:Z, (-a)/(-b) = a/b. |
sine and cosine taylor expansion coefficients | Lemma sin_cos_seq : forall n : nat, sin_seq n [=] [--] (cos_seq (S n)). |
equal concatenations have equal parts | Lemma app_inv_tail: forall l l1 l2 : list A, l1 ++ l = l2 ++ l -> l1 = l2. |
division decreases numerator | Theorem Zdivide_Zdiv_lt_pos a b : 1 < a -> 0 < b -> (a | b) -> 0 < b / a < b. |
range of modulo | Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. |
maximum rank of matrix | Lemma rank_leq_col m n (A : 'M_(m, n)) : \rank A <= n. |
binomial theorem | Theorem Pascal a b n : (a + b) ^ n = \sum_(i < n.+1) 'C(n, i) * (a ^ (n - i) * b ^ i). |
cos positive domain | Lemma Cos_nonneg : forall x, [--] (Pi [/]TwoNZ) [<=] x -> x [<=] Pi [/]TwoNZ -> [0] [<=] Cos x. |
cos non-negative domain | Lemma Cos_nonneg : forall x, [--] (Pi [/]TwoNZ) [<=] x -> x [<=] Pi [/]TwoNZ -> [0] [<=] Cos x. |
head of empty list | Theorem hd_error_nil : hd_error (@nil A) = None. |
filter elements in a list of lists | Lemma concat_filter_map : forall (l : list (list A)), concat (map filter l) = filter (concat l). |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 4