tomreichel/proofdb-human-eval · Datasets at Hugging Face

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).

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).