hcju/holps huggingface dataset:
Sphere.affsign
|- (!v. (!sgn. (!s. (!t. (((((affsign sgn) s) t) v) = (?f. ((v = ((lin_combo (s UNION t)) f)) /\ ((!w. ((t w) ==> (sgn (f w)))) /\ (((sum (s UNION t)) f) = (real_of_num (NUMERAL (BIT1 _0))))))))))))
REAL_DIFFERENTIABLE_AT_SIN
|- (!x. ((real_differentiable sin) (atreal x)))
REALLIM_1_OVER_N
|- (((---> (\n. (real_inv (real_of_num n)))) (real_of_num (NUMERAL _0))) sequentially)
DIAMETER_POS_LE
|- (!s. ((bounded s) ==> ((real_le (real_of_num (NUMERAL _0))) (diameter s))))
HAS_INTEGRAL_SPIKE_INTERIOR
|- (!f. (!g. (!a. (!b. (!y. (((!x. ((x IN (open_interval (a , b))) ==> ((g x) = (f x)))) /\ (((has_integral f) y) (closed_interval ((CONS (a , b)) NIL)))) ==> (((has_integral g) y) (closed_interval ((CONS (a , b)) NIL)))))))))
BERNOULLI_ALT
|- (!n. (!x. (((sum ((NUMERAL _0) .. n)) (\k. ((real_mul (real_of_num (binom ((n + (NUMERAL (BIT1 _0))) , k)))) ((bernoulli k) x)))) = ((real_mul ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))) ((real_pow x) n)))))
DIST_NZ
|- (!x. (!y. ((~ (x = y)) = ((real_lt (real_of_num (NUMERAL _0))) (distance (x , y))))))
VECTOR_DERIVATIVE_WORKS
|- (!net. (!f. (((differentiable f) net) = (((has_vector_derivative f) ((vector_derivative f) net)) net))))
IN_BALL
|- (!x. (!y. (!e. ((y IN (ball (x , e))) = ((real_lt (distance (x , y))) e)))))
VSUM_SUPERSET
|- (!f. (!u. (!v. (((u SUBSET v) /\ (!x. (((x IN v) /\ (~ (x IN u))) ==> ((f x) = (vec (NUMERAL _0)))))) ==> (((vsum v) f) = ((vsum u) f))))))
REAL_POW2_ABS
|- (!x. (((real_pow (real_abs x)) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))))
SUM_LE_NUMSEG
|- (!f. (!g. (!m. (!n. ((!i. (((m <= i) /\ (i <= n)) ==> ((real_le (f i)) (g i)))) ==> ((real_le ((sum (m .. n)) f)) ((sum (m .. n)) g)))))))
LOCALLY_COMPACT_INTER
|- (!s. (!t. ((((locally compact) s) /\ ((locally compact) t)) ==> ((locally compact) (s INTER t)))))
MAP
|- ((!f. (((MAP f) NIL) = NIL)) /\ (!f. (!h. (!t. (((MAP f) ((CONS h) t)) = ((CONS (f h)) ((MAP f) t)))))))
ORTHOGONAL_TRANSFORMATION_INVERSE
|- (!f. ((orthogonal_transformation f) ==> (?g. ((orthogonal_transformation g) /\ ((!x. ((g (f x)) = x)) /\ (!y. ((f (g y)) = y)))))))
DROP_EQ_0
|- (!x. (((drop x) = (real_of_num (NUMERAL _0))) = (x = (vec (NUMERAL _0)))))
HOMOTOPY_EQUIVALENT_TRANSLATION_LEFT_EQ
|- (!a. (!s. (!t. (((homotopy_equivalent ((IMAGE (\x. ((vector_add a) x))) s)) t) = ((homotopy_equivalent s) t)))))
Tame_defs2.sub
|- (!a. (!n. (((sub a) n) = (sub1 (a , n)))))
RELATIVE_FRONTIER_INJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((relative_frontier ((IMAGE f) s)) = ((IMAGE f) (relative_frontier s))))))
DIFFERENTIABLE_AT_IMP_DIFFERENTIABLE_ON
|- (!f. (!s. ((!x. ((x IN s) ==> ((differentiable f) (at x)))) ==> ((differentiable_on f) s))))
DIM_SUMS_INTER
|- (!s. (!t. (((subspace s) /\ (subspace t)) ==> (((dim (GSPEC (\GEN%PVAR%205. (?x. (?y. (((SETSPEC GEN%PVAR%205) ((x IN s) /\ (y IN t))) ((vector_add x) y))))))) + (dim (s INTER t))) = ((dim s) + (dim t))))))
REALLIM_WITHINREAL
|- (!f. (!l. (!a. (!s. ((((---> f) l) ((within (atreal a)) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (((x IN s) /\ (((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\ ((real_lt (real_abs ((real_sub x) a))) d))) ==> ((real_lt (real_abs ((real_sub (f x)) l))) e))))))))))))
SPHERE_TRANSLATION
|- (!a. (!x. (!r. ((sphere (((vector_add a) x) , r)) = ((IMAGE (\y. ((vector_add a) y))) (sphere (x , r)))))))
LIM_Z_TIMES_CLOG
|- (((--> (\z. ((complex_mul z) (clog z)))) (Cx (real_of_num (NUMERAL _0)))) (at (Cx (real_of_num (NUMERAL _0)))))
orthogonal_transformation
|- (!f. ((orthogonal_transformation f) = ((linear f) /\ (!v. (!w. (((dot (f v)) (f w)) = ((dot v) w)))))))
COMPLEX_ADD_RID
|- (!x. (((vector_add x) (Cx (real_of_num (NUMERAL _0)))) = x))
RPOW_MINUS1_QUOTIENT_LT
|- (!a. (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) a) /\ ((~ (a = (real_of_num (NUMERAL (BIT1 _0))))) /\ (((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt x) y)))) ==> ((real_lt ((real_div ((real_sub ((rpow a) x)) (real_of_num (NUMERAL (BIT1 _0))))) x)) ((real_div ((real_sub ((rpow a) y)) (real_of_num (NUMERAL (BIT1 _0))))) y))))))
AFF_DIM_SING
|- (!a. ((aff_dim (a INSERT EMPTY)) = (int_of_num (NUMERAL _0))))
INSERT_AC
|- (((x INSERT (y INSERT s)) = (y INSERT (x INSERT s))) /\ ((x INSERT (x INSERT s)) = (x INSERT s)))
CONNECTED_TRANSLATION_EQ
|- (!a. (!s. ((connected ((IMAGE (\x. ((vector_add a) x))) s)) = (connected s))))
INTEGRABLE_SUB
|- (!f. (!g. (!s. ((((integrable_on f) s) /\ ((integrable_on g) s)) ==> ((integrable_on (\x. ((vector_sub (f x)) (g x)))) s)))))
Tame_inequalities.INEQ_ALT
|- (!A. (!bounds. (((ineq bounds) A) = (((ALL (GABS (\f. (!a. (!x. (!b. ((GEQ (f (a , (x , b)))) (((real_le a) x) /\ ((real_le x) b))))))))) bounds) ==> A))))
ITERATE_IMAGE_NONZERO
|- (!op. ((monoidal op) ==> (!g. (!f. (!s. (((FINITE s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((~ (x = y)) /\ ((f x) = (f y))))) ==> ((g (f x)) = (neutral op)))))) ==> ((((iterate op) ((IMAGE f) s)) g) = (((iterate op) s) (g o f)))))))))
LIM_NULL_COMPLEX
|- (!net. (!f. ((((--> f) l) net) = (((--> (\x. ((vector_sub (f x)) l))) (Cx (real_of_num (NUMERAL _0)))) net))))
CARD_LE_TRANS
|- (!s. (!t. (!u. (((s <=_c t) /\ (t <=_c u)) ==> (s <=_c u)))))
SIMPLEX_IMP_POLYTOPE
|- (!n. (!s. (((simplex n) s) ==> (polytope s))))
AFF_DIM_LE_CARD
|- (!s. ((FINITE s) ==> ((int_le (aff_dim s)) ((int_sub (int_of_num (CARD s))) (int_of_num (NUMERAL (BIT1 _0)))))))
PASTECART_IN_INTERIOR_SUBTOPOLOGY
|- (!s. (!t. (!u. (!x. (!y. (((((pastecart x) y) IN u) /\ ((open_in ((subtopology euclidean) (s PCROSS t))) u)) ==> (?v. (?w. (((open_in ((subtopology euclidean) s)) v) /\ ((x IN v) /\ (((open_in ((subtopology euclidean) t)) w) /\ ((y IN w) /\ ((v PCROSS w) SUBSET u)))))))))))))
INSIDE_INSIDE
|- (!s. (!t. ((s SUBSET (inside t)) ==> (((inside s) DIFF t) SUBSET (inside t)))))
Lp_ineqs_proofs.y5_hi_std2
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((y5_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))
Yxionxl2.LINAER_SYM_0
|- (linear (\x. (vector_neg x)))
dot
|- (!x. (!y. (((dot x) y) = ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. ((real_mul (x $ i)) (y $ i)))))))
DIST_TRIANGLE
|- (!x. (!y. (!z. ((real_le (distance (x , z))) ((real_add (distance (x , y))) (distance (y , z)))))))
REAL_MEASURABLE_ON_RESTRICT
|- (!f. (!s. ((((real_measurable_on f) UNIV) /\ (real_lebesgue_measurable s)) ==> ((real_measurable_on (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) UNIV))))
Tame_lemmas.fully_surrounded_sol
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> (((sol (vec (NUMERAL _0))) (((((dart_leads_into (vec (NUMERAL _0))) V) E) v) w)) = ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) ((sum ((face (hypermap_of_fan (V , E))) (v , w))) (\x. ((real_sub ((azim_dart (V , E)) x)) pi))))))))))))
CNJ_POW
|- (!z. (!n. ((cnj ((complex_pow z) n)) = ((complex_pow (cnj z)) n))))
LIM_COMPOSE_AT
|- (!net. (!f. (!g. (!y. (!z. (((((--> f) y) net) /\ (((eventually (\w. (((f w) = y) ==> ((g y) = z)))) net) /\ (((--> g) z) (at y)))) ==> (((--> (g o f)) z) net)))))))
INTEGRABLE_ADD
|- (!f. (!g. (!s. ((((integrable_on f) s) /\ ((integrable_on g) s)) ==> ((integrable_on (\x. ((vector_add (f x)) (g x)))) s)))))
REAL_LOG_CONVEX_CONST
|- (!s. (!a. (((real_le (real_of_num (NUMERAL _0))) a) ==> ((real_log_convex_on (\x. a)) s))))
CONTINUOUS_COMPLEX_DIV_WITHIN
|- (!f. (!g. (!s. (!a. ((((continuous f) ((within (at a)) s)) /\ (((continuous g) ((within (at a)) s)) /\ (~ ((g a) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((continuous (\x. ((complex_div (f x)) (g x)))) ((within (at a)) s)))))))
o_DEF
|- (!f. (!g. ((f o g) = (\x. (f (g x))))))
REAL_HALF
|- ((!e. (((real_lt (real_of_num (NUMERAL _0))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = ((real_lt (real_of_num (NUMERAL _0))) e))) /\ ((!e. (((real_add ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = e)) /\ (!e. (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = e))))
Seq.incr_nth
|- ((((incr_nth ((CONS n) v')) (SUC i)) = ((CONS n) ((incr_nth v') i))) /\ ((((incr_nth ((CONS n) v')) (NUMERAL _0)) = ((CONS (SUC n)) v')) /\ (((incr_nth NIL) i) = (((ncons i) (NUMERAL _0)) ((CONS (NUMERAL (BIT1 _0))) NIL)))))
OUTSIDE_SUBSET_CONVEX
|- (!s. (!t. (((convex t) /\ (s SUBSET t)) ==> ((UNIV DIFF t) SUBSET (outside s)))))
FINITE_POLYHEDRON_EXTREME_POINTS
|- (!s. ((polyhedron s) ==> (FINITE (GSPEC (\GEN%PVAR%2354. (?v. (((SETSPEC GEN%PVAR%2354) ((extreme_point_of v) s)) v)))))))
BIJ
|- (!f. (!s. (!t. ((((BIJ f) s) t) = ((((INJ f) s) t) /\ (((SURJ f) s) t))))))
FORALL_3
|- (!P. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (NUMERAL (BIT1 (BIT1 _0))))) ==> (P i))) = ((P (NUMERAL (BIT1 _0))) /\ ((P (NUMERAL (BIT0 (BIT1 _0)))) /\ (P (NUMERAL (BIT1 (BIT1 _0))))))))
SUM_POS_BOUND
|- (!f. (!b. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))) /\ ((real_le ((sum s) f)) b))) ==> (!x. ((x IN s) ==> ((real_le (f x)) b)))))))
Planarity.aff_ge1_subset_aff_ge
|- (!x. (!v. (!u. (!v1. ((((DISJOINT (x INSERT EMPTY)) (v INSERT (u INSERT EMPTY))) /\ ((~ (collinear (x INSERT (v1 INSERT (u INSERT EMPTY))))) /\ (v1 IN ((aff_ge (x INSERT EMPTY)) (v INSERT (u INSERT EMPTY)))))) ==> (((aff_ge (x INSERT EMPTY)) (v1 INSERT (u INSERT EMPTY))) SUBSET ((aff_ge (x INSERT EMPTY)) (v INSERT (u INSERT EMPTY)))))))))
Tame_list.lower_case
|- ((MAP = map) /\ ((LENGTH = sizel) /\ ((FILTER = filter) /\ ((REVERSE = rev) /\ ((!s. (!x0. ((~ (NIL = s)) ==> ((LAST s) = ((last x0) s))))) /\ ((APPEND = cat) /\ (ALL = all)))))))
Pack1.sum_measure_voronoi_le_ball
|- (!S. (!p. (!r. ((((real_le (real_of_num (NUMERAL _0))) r) /\ ((packing S) /\ (saturated S))) ==> ((real_le ((sum (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) (\v. (measure ((voronoi_open S) v))))) (measure (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))))))
Dih2k_hypermap.DART_OF_HYP_SY_EQ
|- (((FAN ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l))))) /\ (((NUMERAL (BIT1 _0)) < (dimindex UNIV)) /\ (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ ((j <= (dimindex UNIV)) /\ (((row i) (vecmats l)) = ((row j) (vecmats l))))))) ==> (i = j)))))) ==> (((darts_of_hyp (E_SY (vecmats l))) (V_SY (vecmats l))) = ((F_SY (vecmats l)) UNION ((IMAGE (nn_of_hyp ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l)))))) (F_SY (vecmats l))))))
REAL_EQ_NEG2
|- (!x. (!y. (((real_neg x) = (real_neg y)) = (x = y))))
HAS_COMPLEX_DERIVATIVE_CACS
|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> (((has_complex_derivative cacs) (vector_neg (complex_inv (csin (cacs z))))) (at z))))
int_pow_th
|- (!x. (!n. ((real_of_int ((int_pow x) n)) = ((real_pow (real_of_int x)) n))))
LE_ADD2
|- (!m. (!n. (!p. (!q. (((m <= p) /\ (n <= q)) ==> ((m + n) <= (p + q)))))))
AFFINE_HULL_HALFSPACE_LT
|- (!a. (!b. (((hull affine) (GSPEC (\GEN%PVAR%1543. (?x. (((SETSPEC GEN%PVAR%1543) ((real_lt ((dot a) x)) b)) x))))) = (((COND ((a = (vec (NUMERAL _0))) /\ ((real_le b) (real_of_num (NUMERAL _0))))) EMPTY) UNIV))))
AFF_DIM_DIM_SUBSPACE
|- (!s. ((subspace s) ==> ((aff_dim s) = (int_of_num (dim s)))))
PERMUTATION_COMPOSE
|- (!p. (!q. (((permutation p) /\ (permutation q)) ==> (permutation (p o q)))))
Nonlinear_lemma.dih_x_sym
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dih_x x1) x2) x3) x4) x5) x6) = ((((((dih_x x1) x3) x2) x4) x6) x5))))))))
REAL_LE_LMUL_EQ
|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_le ((real_mul z) x)) ((real_mul z) y)) = ((real_le x) y))))))
NOT_CONS_NIL
|- (!h. (!t. (~ (((CONS h) t) = NIL))))
Seq.eqseq_cons
|- (!x1. (!x2. (!s1. (!s2. ((((CONS x1) s1) = ((CONS x2) s2)) = ((x1 = x2) /\ (s1 = s2)))))))
REAL_INTEGRABLE_INTEGRAL
|- (!f. (!i. (((real_integrable_on f) i) ==> (((has_real_integral f) ((real_integral i) f)) i))))
PRE_ELIM_THM
|- ((P (PRE n)) = (!m. (((n = (SUC m)) \/ ((m = (NUMERAL _0)) /\ (n = (NUMERAL _0)))) ==> (P m))))
HAS_REAL_FRECHET_DERIVATIVE_WITHIN
|- ((((has_real_derivative f) f') ((within (atreal x)) s)) = (((has_derivative (lift o (f o drop))) (\x. ((% f') x))) ((within (at (lift x))) ((IMAGE lift) s))))
INTERVAL_NE_EMPTY_1
|- ((!a. (!b. ((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) = ((real_le (drop a)) (drop b))))) /\ (!a. (!b. ((~ ((open_interval (a , b)) = EMPTY)) = ((real_lt (drop a)) (drop b))))))
VSUM_CLAUSES_LEFT
|- (!f. (!m. (!n. ((m <= n) ==> (((vsum (m .. n)) f) = ((vector_add (f m)) ((vsum ((m + (NUMERAL (BIT1 _0))) .. n)) f)))))))
REAL_INTERVAL_NE_EMPTY
|- ((!a. (!b. ((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) = ((real_le a) b)))) /\ (!a. (!b. ((~ ((open_real_interval (a , b)) = EMPTY)) = ((real_lt a) b)))))
Seq2.mem_flatten
|- (!x. (!L. (((MEM x) (flatten L)) = (?l. (((MEM l) L) /\ ((MEM x) l))))))
VSUM_COMPONENT
|- (!s. (!f. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((((vsum s) f) $ i) = ((sum s) (\x. ((f x) $ i))))))))
REAL_NOT_LT
|- (!x. (!y. ((~ ((real_lt x) y)) = ((real_le y) x))))
CONVEX_HULL_EMPTY
|- (((hull convex) EMPTY) = EMPTY)
MATRIX_VECTOR_MUL_ADD_RDISTRIB
|- (!A. (!B. (!x. (((matrix_vector_mul ((matrix_add A) B)) x) = ((vector_add ((matrix_vector_mul A) x)) ((matrix_vector_mul B) x))))))
OPEN_INTERVAL
|- (!a. (!b. (open (open_interval (a , b)))))
Collect_geom2.NOT_COPLANAR_IMP_RADV_PROPERTIES
|- ((~ (coplanar_alt (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))) ==> (!w. (((v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))) w) ==> ((radV (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) = (distance ((circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) , w))))))
BASIS_INJ_EQ
|- (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ (j <= (dimindex UNIV))))) ==> (((basis i) = (basis j)) = (i = j)))))
SERIES_SUB
|- (!x. (!x0. (!y. (!y0. (!s. (((((sums x) x0) s) /\ (((sums y) y0) s)) ==> (((sums (\n. ((vector_sub (x n)) (y n)))) ((vector_sub x0) y0)) s)))))))
MULT_SYM
|- (!m. (!n. ((m * n) = (n * m))))
REAL_CONTINUOUS_ON_UNION
|- (!f. (!s. (!t. (((real_closed s) /\ ((real_closed t) /\ (((real_continuous_on f) s) /\ ((real_continuous_on f) t)))) ==> ((real_continuous_on f) (s UNION t))))))
Fan.sum4_azim_fan
|- (!x. (!v. (!u. (!w1. (!w2. ((((real_le ((((azim x) v) u) w1)) ((((azim x) v) u) w2)) /\ ((~ (collinear (x INSERT (v INSERT (w1 INSERT EMPTY))))) /\ ((~ (collinear (x INSERT (v INSERT (w2 INSERT EMPTY))))) /\ (~ (collinear (x INSERT (v INSERT (u INSERT EMPTY)))))))) ==> (((((azim x) v) u) w2) = ((real_add ((((azim x) v) u) w1)) ((((azim x) v) w1) w2)))))))))
HAS_REAL_MEASURE_MEASURE
|- (!s. ((real_measurable s) = ((has_real_measure s) (real_measure s))))
real_polynomial_function_RULES
|- ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (real_polynomial_function (\x. (x $ i))))) /\ ((!c. (real_polynomial_function (\x. c))) /\ ((!f. (!g. (((real_polynomial_function f) /\ (real_polynomial_function g)) ==> (real_polynomial_function (\x. ((real_add (f x)) (g x))))))) /\ (!f. (!g. (((real_polynomial_function f) /\ (real_polynomial_function g)) ==> (real_polynomial_function (\x. ((real_mul (f x)) (g x))))))))))
LE_EXISTS
|- (!m. (!n. ((m <= n) = (?d. (n = (m + d))))))
SETDIST_LE_SING
|- (!s. (!t. (!x. ((x IN s) ==> ((real_le (setdist (s , t))) (setdist ((x INSERT EMPTY) , t)))))))
HOMOTOPIC_INVERTIBLE_LINEAR_MAPS_ALT
|- (!f. (!g. (((((homotopic_with (\h. ((linear h) /\ (invertible (matrix h))))) (UNIV , UNIV)) f) g) = ((linear f) /\ ((linear g) /\ ((real_lt (real_of_num (NUMERAL _0))) ((real_mul (real_sgn (det (matrix f)))) (real_sgn (det (matrix g))))))))))
LEBESGUE_MEASURABLE_COMPL
|- (!s. ((lebesgue_measurable (UNIV DIFF s)) = (lebesgue_measurable s)))
ENR_IMP_FSGIMA
|- (!s. ((ENR s) ==> (fsigma s)))
Seq.nth0
|- (!x0. (!s. ((((nth x0) s) (NUMERAL _0)) = ((headl x0) s))))
Ssrfun.injective
|- (!f. ((injective f) = (!x1. (!x2. (((f x1) = (f x2)) ==> (x1 = x2))))))
Ssrbool.negbK
|- (!b. ((~ (~ b)) = b))
HOMOTOPIC_WITH_TRANS
|- (!P. (!X. (!Y. (!f. (!g. (!h. ((((((homotopic_with P) (X , Y)) f) g) /\ ((((homotopic_with P) (X , Y)) g) h)) ==> ((((homotopic_with P) (X , Y)) f) h))))))))
Ssrbool.addbb
|- (!b. (((XOR b) b) = F))
CONNECTED_OPEN_PATH_CONNECTED
|- (!s. (((open s) /\ (connected s)) ==> (path_connected s)))
ANR_COVERING_SPACE_EQ
|- (!p. (!s. (!c. (((covering_space (c , p)) s) ==> ((ANR s) = (ANR c))))))
NOT_EXISTS_THM
|- (!P. ((~ (?x. (P x))) = (!x. (~ (P x)))))
REAL_LE_SUP
|- (!s. (!a. (!b. (!y. (((y IN s) /\ (((real_le a) y) /\ (!x. ((x IN s) ==> ((real_le x) b))))) ==> ((real_le a) (sup s)))))))
REALLIM_TRANSFORM_EVENTUALLY
|- (!net. (!f. (!g. (!l. ((((eventually (\x. ((f x) = (g x)))) net) /\ (((---> f) l) net)) ==> (((---> g) l) net))))))
EXISTS_PATH_SUBPATH_TO_FRONTIER
|- (!g. (!s. (((path g) /\ (((pathstart g) IN s) /\ (~ ((pathfinish g) IN s)))) ==> (?h. ((path h) /\ (((pathstart h) = (pathstart g)) /\ (((path_image h) SUBSET (path_image g)) /\ ((((path_image h) DELETE (pathfinish h)) SUBSET (interior s)) /\ ((pathfinish h) IN (frontier s))))))))))
AZIM_EXISTS
|- (!v. (!w. (!w1. (!w2. (?theta. (((real_le (real_of_num (NUMERAL _0))) theta) /\ (((real_lt theta) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ (?h1. (?h2. (!e1. (!e2. (!e3. (((((orthonormal e1) e2) e3) /\ ((((% (distance (w , v))) e3) = ((vector_sub w) v)) /\ (~ (w = v)))) ==> (?psi. (?r1. (?r2. ((((vector_sub w1) v) = ((vector_add ((% ((real_mul r1) (cos psi))) e1)) ((vector_add ((% ((real_mul r1) (sin psi))) e2)) ((% h1) ((vector_sub w) v))))) /\ ((((vector_sub w2) v) = ((vector_add ((% ((real_mul r2) (cos ((real_add psi) theta)))) e1)) ((vector_add ((% ((real_mul r2) (sin ((real_add psi) theta)))) e2)) ((% h2) ((vector_sub w) v))))) /\ (((~ (collinear (v INSERT (w INSERT (w1 INSERT EMPTY))))) ==> ((real_lt (real_of_num (NUMERAL _0))) r1)) /\ ((~ (collinear (v INSERT (w INSERT (w2 INSERT EMPTY))))) ==> ((real_lt (real_of_num (NUMERAL _0))) r2)))))))))))))))))))))
ITERATE_EQ_GENERAL_INVERSES
|- (!op. ((monoidal op) ==> (!s. (!t. (!f. (!g. (!h. (!k. (((!y. ((y IN t) ==> (((k y) IN s) /\ ((h (k y)) = y)))) /\ (!x. ((x IN s) ==> (((h x) IN t) /\ (((k (h x)) = x) /\ ((g (h x)) = (f x))))))) ==> ((((iterate op) s) f) = (((iterate op) t) g)))))))))))
List_hypermap.fst_n_list_ext_power
|- (!x. (!y. (!L. (!n. (((good_list L) /\ ((x , y) IN (darts_of_list L))) ==> ((FST (((POWER (n_list_ext L)) n) (x , y))) = x))))))
CROSS_RMUL
|- (!c. (!x. (!y. (((cross x) ((% c) y)) = ((% c) ((cross x) y))))))
Ssrnat.iter
|- (((((iter (SUC n)) f) x) = (f (((iter n) f) x))) /\ ((((iter (NUMERAL _0)) f) x) = x))
LIFT_IN_IMAGE_LIFT
|- (!x. (!s. (((lift x) IN ((IMAGE lift) s)) = (x IN s))))
Ssrnat.half0
|- ((half (NUMERAL _0)) = (NUMERAL _0))
Trigonometry.CFXEKKP1
|- ((cos ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (real_of_num (NUMERAL _0)))
Hypermap_iso.ext_dart_eq_inv_contracted_dart
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((ext_dart (V , E)) d) = (((res_inv contracted_dart) (d_fan ((vec (NUMERAL _0)) , (V , E)))) d)))))))
CONTINUOUS_ON_UPPERHALF_ARG
|- ((continuous_on (Cx o Arg)) ((GSPEC (\GEN%PVAR%4840. (?z. (((SETSPEC GEN%PVAR%4840) ((real_le (real_of_num (NUMERAL _0))) (Im z))) z)))) DIFF ((Cx (real_of_num (NUMERAL _0))) INSERT EMPTY)))
Packing3.BARV_INITIAL_SUBLIST
|- (!V. (!k. (!ul. (!vl. (((((barV V) k) ul) /\ (((initial_sublist vl) ul) /\ ((NUMERAL _0) < (LENGTH vl)))) ==> (((barV V) ((LENGTH vl) - (NUMERAL (BIT1 _0)))) vl))))))
INT_OF_NUM_LE
|- (!m. (!n. (((int_le (int_of_num m)) (int_of_num n)) = (m <= n))))
REAL_LEBESGUE_MEASURABLE_COUNTABLE_INTERS
|- (!f. (((COUNTABLE f) /\ (!s. ((s IN f) ==> (real_lebesgue_measurable s)))) ==> (real_lebesgue_measurable (INTERS f))))
Lp_ineqs_proofs.ys_eq
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> (((y1_fan (v , w)) = (vector_norm v)) /\ (((y2_fan (v , w)) = (vector_norm w)) /\ ((((y3_fan (V , E)) (v , w)) = (vector_norm (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))) /\ ((((y4'_fan (V , E)) (v , w)) = (distance (w , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))) /\ ((((y5_fan (V , E)) (v , w)) = (distance (v , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))) /\ ((y6_fan (v , w)) = (distance (v , w)))))))))))))))))
Trigonometry2.POW2_1
|- (((real_pow (real_of_num (NUMERAL (BIT1 _0)))) (NUMERAL (BIT0 (BIT1 _0)))) = (real_of_num (NUMERAL (BIT1 _0))))
eventually
|- (!net. (!p. (((eventually p) net) = ((trivial_limit net) \/ (?y. ((?x. (((netord net) x) y)) /\ (!x. ((((netord net) x) y) ==> (p x)))))))))
VECTOR_MUL_LZERO
|- (!x. (((% (real_of_num (NUMERAL _0))) x) = (vec (NUMERAL _0))))
Seq.nth
|- (!x0. (!h. (!t. (!n. (((((nth x0) NIL) n) = x0) /\ (((((nth x0) ((CONS h) t)) (NUMERAL _0)) = h) /\ ((((nth x0) ((CONS h) t)) (SUC n)) = (((nth x0) t) n))))))))
Sphere.rad2_y
|- (rad2_y = (y_of_x rad2_x))
Hypermap_and_fan.COMPONENTS_HYPERMAP_OF_FAN
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (((dart (hypermap_of_fan (V , E))) = (dart_of_fan (V , E))) /\ (((edge_map (hypermap_of_fan (V , E))) = (e_fan_pair_ext (V , E))) /\ (((node_map (hypermap_of_fan (V , E))) = (n_fan_pair_ext (V , E))) /\ ((face_map (hypermap_of_fan (V , E))) = (f_fan_pair_ext (V , E)))))))))
EVEN
|- (((EVEN (NUMERAL _0)) = T) /\ (!n. ((EVEN (SUC n)) = (~ (EVEN n)))))
POSITIVE_SEMIDEFINITE_DIAGONAL_INEQUALITY
|- (!A. (!i. (!j. (((positive_semidefinite A) /\ (((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ (j <= (dimindex UNIV)))))) ==> ((real_le ((real_pow ((A $ i) $ j)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ i) $ i)) ((A $ j) $ j)))))))
int_of_num_th
|- (!n. ((real_of_int (int_of_num n)) = (real_of_num n)))
MATRIX_INV_MUL_INNER
|- (!A. (((matrix_mul A) ((matrix_mul (matrix_inv A)) A)) = A))
CSQRT_UNIQUE
|- (!s. (!z. (((((complex_pow s) (NUMERAL (BIT0 (BIT1 _0)))) = z) /\ (((real_lt (real_of_num (NUMERAL _0))) (Re s)) \/ (((Re s) = (real_of_num (NUMERAL _0))) /\ ((real_le (real_of_num (NUMERAL _0))) (Im s))))) ==> ((csqrt z) = s))))
REAL_ADD
|- (!w. (!z. (((real w) /\ (real z)) ==> (real ((vector_add w) z)))))
CARD_INTSEG_INT
|- (!a. (!b. (((integer a) /\ (integer b)) ==> ((CARD (GSPEC (\GEN%PVAR%85. (?x. (((SETSPEC GEN%PVAR%85) ((integer x) /\ (((real_le a) x) /\ ((real_le x) b)))) x))))) = (((COND ((real_lt b) a)) (NUMERAL _0)) (num_of_int (int_of_real ((real_add ((real_sub b) a)) (real_of_num (NUMERAL (BIT1 _0)))))))))))
INCREASING_BOUNDED_VARIATION
|- (!f. (!a. (!b. ((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f x))) (drop (f y)))))) ==> ((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL)))))))
real_ge
|- (!y. (!x. (((real_ge x) y) = ((real_le y) x))))
Hypermap.orbit_one_point
|- (!f. (!x. (((f x) = x) = (((orbit_map f) x) = (x INSERT EMPTY)))))
num_INDUCTION
|- (!P. (((P (NUMERAL _0)) /\ (!n. ((P n) ==> (P (SUC n))))) ==> (!n. (P n))))
RPOW_SQRT
|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((rpow x) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (sqrt x))))
COUNTABLE_INSERT
|- (!x. (!s. ((COUNTABLE (x INSERT s)) = (COUNTABLE s))))
REAL_LE_LT
|- (!x. (!y. (((real_le x) y) = (((real_lt x) y) \/ (x = y)))))
REAL_LT_BETWEEN
|- (!a. (!b. (((real_lt a) b) = (?x. (((real_lt a) x) /\ ((real_lt x) b))))))
REAL_ABS_SUB
|- (!x. (!y. ((real_abs ((real_sub x) y)) = (real_abs ((real_sub y) x)))))
Tame_defs2.sub1
|- (!n. (!xs. ((sub1 (xs , n)) = ((EL n) xs))))
CONTINUOUS_COMPLEX_POW
|- (!net. (!f. (!n. (((continuous f) net) ==> ((continuous (\x. ((complex_pow (f x)) n))) net)))))