hcju/holps huggingface dataset:
COMPLEX_TAYLOR_MVT |- (!f. (!w. (!z. (!n. ((!i. (!x. (((x IN (closed_segment ((CONS (w , z)) NIL))) /\ (i <= n)) ==> (((has_complex_derivative (f i)) ((f (i + (NUMERAL (BIT1 _0)))) x)) (at x))))) ==> (?u. ((u IN (closed_segment ((CONS (w , z)) NIL))) /\ ((Re ((f (NUMERAL _0)) z)) = (Re ((vector_add ((vsum ((NUMERAL _0) .. n)) (\i. ((complex_mul ((f i) w)) ((complex_div ((complex_pow ((vector_sub z) w)) i)) (Cx (real_of_num (FACT i)))))))) ((complex_mul ((complex_mul ((f (n + (NUMERAL (BIT1 _0)))) u)) ((complex_div ((complex_pow ((vector_sub z) u)) n)) (Cx (real_of_num (FACT n)))))) ((vector_sub z) w))))))))))))
Lp_main_estimate.v_prime_eq |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. (!k. ((d IN ((darts_k k) (hypermap_of_fan (V , E)))) ==> (((v_prime V) ((face (hypermap_of_fan (V , E))) d)) = (GSPEC (\GEN%PVAR%8731. (?i. (((SETSPEC GEN%PVAR%8731) (i < k)) (FST (((POWER (f_fan_pair_ext (V , E))) i) d)))))))))))))
COMPLEX_NOT_ROOT_UNITY |- (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (?u. (((vector_norm u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (~ (((complex_pow u) n) = (Cx (real_of_num (NUMERAL (BIT1 _0))))))))))
POWER_REAL_SERIES_CONV_IMP_ABSCONV_WEAK |- (!a. (!k. (!w. (!z. ((((real_summable k) (\n. ((real_mul (a n)) ((real_pow z) n)))) /\ ((real_lt (real_abs w)) (real_abs z))) ==> ((real_summable k) (\n. ((real_mul (real_abs (a n))) ((real_pow w) n)))))))))
TRACE_MATRIX_INV_RMUL |- (!A. ((trace ((matrix_mul A) (matrix_inv A))) = (real_of_num (rank A))))
Lp_ineqs_proofs.ln_fan_ly |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((ln_fan v) = (ly (vector_norm v)))))))))))
REALLIM_AT_ZERO |- (!f. (!l. (!a. ((((---> f) l) (at a)) = (((---> (\x. (f ((vector_add a) x)))) l) (at (vec (NUMERAL _0))))))))
VECTOR_EXPAND_4 |- (!x. (x = (vector ((CONS (x $ (NUMERAL (BIT1 _0)))) ((CONS (x $ (NUMERAL (BIT0 (BIT1 _0))))) ((CONS (x $ (NUMERAL (BIT1 (BIT1 _0))))) ((CONS (x $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) NIL)))))))
Pwssrat.rotate_to_hd |- (((rotate_to ((CONS h) t)) h) = ((CONS h) t))
Seq.eq_all_r |- (!s1. (!s2. ((!x. (((MEM x) s1) = ((MEM x) s2))) ==> (!a. (((all a) s1) = ((all a) s2))))))
SUBMETRIC_UNIV |- (((submetric m) UNIV) = m)
OPEN_NON_PATH_COMPONENT |- (!s. (!x. ((open s) ==> (open (s DIFF ((path_component s) x))))))
HAS_VECTOR_DERIVATIVE_VECTOR_POLYNOMIAL_FUNCTION |- (!p. ((vector_polynomial_function p) ==> (?p'. ((vector_polynomial_function p') /\ (!x. (((has_vector_derivative p) (p' x)) (at x)))))))
Pack2.measure_unions_sum_voronoi |- (!S. (!p. (!r. ((((real_le (real_of_num (NUMERAL _0))) r) /\ ((packing S) /\ (saturated S))) ==> ((measure (UNIONS (GSPEC (\GEN%PVAR%5985. (?w. (?v. (((SETSPEC GEN%PVAR%5985) ((v IN (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) /\ (w = S))) ((voronoi_closed w) v)))))))) = ((sum (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) (\v. (measure ((voronoi_closed S) v)))))))))
DIFFERENTIABLE_ON_COMPOSE |- (!f. (!g. (!s. ((((differentiable_on f) s) /\ ((differentiable_on g) ((IMAGE f) s))) ==> ((differentiable_on (g o f)) s)))))
INT_SUB_LDISTRIB |- (!x. (!y. (!z. (((int_mul x) ((int_sub y) z)) = ((int_sub ((int_mul x) y)) ((int_mul x) z))))))
INF_INSERT |- (!x. (!s. ((bounded ((IMAGE lift) s)) ==> ((inf (x INSERT s)) = (((COND (s = EMPTY)) x) ((real_min x) (inf s)))))))
Functional_equation.delta_all_atn |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\ (((real_lt (real_of_num (NUMERAL _0))) x1) /\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\ ((real_lt (real_of_num (NUMERAL _0))) x6))))))) ==> ((((((((rhazim_x x1) x2) x3) x4) x5) x6) = ((((((rhazimatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((rhazim2_x x1) x2) x3) x4) x5) x6) = ((((((rhazim2atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((rhazim3_x x1) x2) x3) x4) x5) x6) = ((((((rhazim3atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih_x x1) x2) x3) x4) x5) x6) = ((((((ldihatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih2_x x1) x2) x3) x4) x5) x6) = ((((((ldih2atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih3_x x1) x2) x3) x4) x5) x6) = ((((((ldih3atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih6_x x1) x2) x3) x4) x5) x6) = ((((((ldih6atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((gchi1_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x1))) ((((((dihatn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi2_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x2))) ((((((dih2atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi3_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x3))) ((((((dih3atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi4_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x4))) ((((((dih4atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi5_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x5))) ((((((dih5atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi6_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x6))) ((((((dih6atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((dih_x x1) x2) x3) x4) x5) x6) = ((((((dihatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih2_x x1) x2) x3) x4) x5) x6) = ((((((dih2atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih3_x x1) x2) x3) x4) x5) x6) = ((((((dih3atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih4_x x1) x2) x3) x4) x5) x6) = ((((((dih4atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih5_x x1) x2) x3) x4) x5) x6) = ((((((dih5atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih6_x x1) x2) x3) x4) x5) x6) = ((((((dih6atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((eta2_126 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x1) x2) x6)) (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((((((eta2_135 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x1) x3) x5)) (NUMERAL (BIT0 (BIT1 _0))))) /\ (((((((eta2_456 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x4) x5) x6)) (NUMERAL (BIT0 (BIT1 _0)))))))))))))))))))))))))))))))))
REAL_CONTINUOUS_AT_AZIM |- (!v. (!w. (!w1. (!w2. ((~ (coplanar (v INSERT (w INSERT (w1 INSERT (w2 INSERT EMPTY)))))) ==> ((real_continuous (((azim v) w) w1)) (at w2)))))))
BARYCENTRE_NOT_IN_SET |- (!s. (((~ (affine_dependent s)) /\ (~ (?a. (s = (a INSERT EMPTY))))) ==> (~ ((barycentre s) IN s))))
ISOMETRY_SUBSET_SUBSPACE |- (!s. (!t. (((subspace s) /\ ((subspace t) /\ ((dim s) <= (dim t)))) ==> (?f. ((linear f) /\ ((((IMAGE f) s) SUBSET t) /\ (!x. ((x IN s) ==> ((vector_norm (f x)) = (vector_norm x))))))))))
EXISTS_SUBPATH_OF_PATH |- (!g. (!a. (!b. (((path g) /\ ((a IN (path_image g)) /\ (b IN (path_image g)))) ==> (?h. ((path h) /\ (((pathstart h) = a) /\ (((pathfinish h) = b) /\ ((path_image h) SUBSET (path_image g))))))))))
Constants_approx.INTERVAL_INV_NEG |- ((((interval_arith x) (a , b)) /\ ((real_lt b) (real_of_num (NUMERAL _0)))) ==> ((interval_arith (real_inv x)) ((real_inv b) , (real_inv a))))
HOMEOMORPHISM_CLOSED_IN_EQ |- (!f. (!g. (!s. (!t. (!u. (!v. ((((homeomorphism (s , t)) (f , g)) /\ ((u SUBSET s) /\ (v SUBSET s))) ==> (((closed_in ((subtopology euclidean) ((IMAGE f) u))) ((IMAGE f) v)) = ((closed_in ((subtopology euclidean) u)) v)))))))))
JOINABLE_COMPONENTS_EQ |- (!s. (!t. (!c1. (!c2. (((connected t) /\ ((t SUBSET s) /\ ((c1 IN (components s)) /\ ((c2 IN (components s)) /\ ((~ ((c1 INTER t) = EMPTY)) /\ (~ ((c2 INTER t) = EMPTY))))))) ==> (c1 = c2))))))
REALLIM_ZERO_NEGINFINITY |- (!f. (!l. ((((---> (\x. (f ((real_div (real_of_num (NUMERAL (BIT1 _0)))) x)))) l) (atreal (real_of_num (NUMERAL _0)))) ==> (((---> f) l) at_neginfinity))))
Miqmcsn.J_SCS_3M1_prime |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_3M1_prime) i)) i1) j) = F)
MEASURABLE_INTER_HALFSPACE_GE |- (!s. (!a. (!i. ((measurable s) ==> (measurable (s INTER (GSPEC (\GEN%PVAR%3589. (?x. (((SETSPEC GEN%PVAR%3589) ((real_ge (x $ i)) a)) x))))))))))
CONIC_HALFSPACE_GE |- (!a. (conic (GSPEC (\GEN%PVAR%1465. (?x. (((SETSPEC GEN%PVAR%1465) ((real_ge ((dot a) x)) (real_of_num (NUMERAL _0)))) x))))))
REAL_MEASURE_DIFF_SUBSET |- (!s. (!t. (((real_measurable s) /\ ((real_measurable t) /\ (t SUBSET s))) ==> ((real_measure (s DIFF t)) = ((real_sub (real_measure s)) (real_measure t))))))
MEASURABLE_ON_MEASURABLE_PREIMAGE_OPEN_INTERVAL |- (!f. (!s. ((measurable s) ==> (((measurable_on f) s) = (!a. (!b. (measurable (GSPEC (\GEN%PVAR%3653. (?x. (((SETSPEC GEN%PVAR%3653) ((x IN s) /\ ((f x) IN (open_interval (a , b))))) x)))))))))))
MBOUNDED_MBALL |- (!m. (!c. (!b. ((mbounded m) ((mball m) (c , b))))))
PATH_CONNECTED_PUNCTURED_BALL |- (!a. (!r. (((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) ==> (path_connected ((ball (a , r)) DELETE a)))))
BOUNDED_RECTIFIABLE_PATH_IMAGE |- (!g. ((rectifiable_path g) ==> (bounded (path_image g))))
Functional_equation.functional_acs_sqrt_x1_d4 |- (acs_sqrt_x1_d4 = (uni (acs , ((scalar6 proj_y1) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))
REAL_DERIVATIVE_POS_RIGHT_MAXIMUM |- (!f. (!f'. (!a. (!b. (!e. ((((real_lt a) b) /\ (((real_lt (real_of_num (NUMERAL _0))) e) /\ ((((has_real_derivative f) f') ((within (atreal b)) (closed_real_interval ((CONS (a , b)) NIL)))) /\ (!x. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_lt (real_abs ((real_sub x) b))) e)) ==> ((real_le (f x)) (f b))))))) ==> ((real_le (real_of_num (NUMERAL _0))) f')))))))
TAGGED_DIVISION_OF_ANOTHER |- (!p. (!s. (!s'. ((((tagged_partial_division_of p) s') /\ ((!t. (!k. (((t , k) IN p) ==> (k SUBSET s)))) /\ (!x. ((x IN s) ==> (?t. (?k. (((t , k) IN p) /\ (x IN k)))))))) ==> ((tagged_division_of p) s)))))
NSUM_ADD_GEN |- (!f. (!g. (!s. (((FINITE (GSPEC (\GEN%PVAR%172. (?x. (((SETSPEC GEN%PVAR%172) ((x IN s) /\ (~ ((f x) = (NUMERAL _0))))) x))))) /\ (FINITE (GSPEC (\GEN%PVAR%173. (?x. (((SETSPEC GEN%PVAR%173) ((x IN s) /\ (~ ((g x) = (NUMERAL _0))))) x)))))) ==> (((nsum s) (\x. ((f x) + (g x)))) = (((nsum s) f) + ((nsum s) g)))))))
Imjxphr.DEFORMATION_DIST_LE_V3_DEFOR_A |- (((is_scs_v39 s) /\ (((MMs_v39 s) w) /\ (((scs_k_v39 s) = k) /\ (((NUMERAL (BIT1 (BIT1 _0))) < k) /\ (((w l) = v2) /\ (((w (SUC l)) = v1) /\ ((((real_pow (vector_norm v1)) (NUMERAL (BIT0 (BIT1 _0)))) = x1) /\ ((((real_pow (vector_norm v2)) (NUMERAL (BIT0 (BIT1 _0)))) = x2) /\ ((((real_pow (vector_norm ((vector_sub v1) v2))) (NUMERAL (BIT0 (BIT1 _0)))) = x6) /\ ((a = (real_neg (real_of_num (NUMERAL (BIT1 _0))))) /\ (!i. (((~ ((l MOD k) = (i MOD k))) /\ (~ ((l MOD k) = ((SUC i) MOD k)))) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance (v2 , (w i)))))))))))))))) ==> (!i. (((~ ((l MOD k) = (i MOD k))) /\ (~ ((l MOD k) = ((SUC i) MOD k)))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!t. ((((real_lt (real_of_num (NUMERAL _0))) t) /\ ((real_lt t) e)) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance (((((((((v3_defor_v1 a) v1) v2) x1) x2) x6) x6) ((real_sub x2) t)) , (w i)))))))))))
CBALL_INTERVAL_0 |- (!e. ((cball ((vec (NUMERAL _0)) , e)) = (closed_interval ((CONS ((vector_neg (lift e)) , (lift e))) NIL))))
ABSOLUTELY_CONTINUOUS_ON_DIVISION |- (!f. (!a. (!b. (!d. (((division_of d) (closed_interval ((CONS (a , b)) NIL))) ==> ((!k. ((k IN d) ==> ((absolutely_continuous_on f) k))) = ((absolutely_continuous_on f) (closed_interval ((CONS (a , b)) NIL)))))))))
Merge_ineq.eta135_x_bd |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!c. ((~ ((real_lt ((real_add ((real_mul ((((((unit6 x1) x2) x3) x4) x5) x6)) c)) ((real_mul ((real_pow (((eta_x x1) x3) x5)) (NUMERAL (BIT0 (BIT1 _0))))) (real_neg (real_of_num (NUMERAL (BIT1 _0))))))) (real_of_num (NUMERAL _0)))) ==> (((real_lt c) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ==> ((real_lt ((real_pow (((eta_x x1) x3) x5)) (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))))))
ABSOLUTELY_CONTINUOUS_ON_TRANSLATION |- (!f. (!s. (!a. (((absolutely_continuous_on (\x. ((vector_add a) (f x)))) s) = ((absolutely_continuous_on f) s)))))
RE_CSQRT |- (!z. ((real_le (real_of_num (NUMERAL _0))) (Re (csqrt z))))
RSC_INC_RC |- (!R. (!x. (!y. ((((RC R) x) y) ==> (((RSC R) x) y)))))
RE_COMPLEX_DIV_LE_0 |- (!a. (!b. (((real_le (Re ((complex_div a) b))) (real_of_num (NUMERAL _0))) = ((real_le (Re ((complex_mul a) (cnj b)))) (real_of_num (NUMERAL _0))))))
AFFINE_AFFINITY |- (!s. (!a. (!c. ((affine s) ==> (affine ((IMAGE (\x. ((vector_add a) ((% c) x)))) s))))))
ABSOLUTELY_CONTINUOUS_DIFFERENTIABLE_BV |- (!f. (!a. (!b. ((((differentiable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL)))) ==> ((absolutely_continuous_on f) (closed_interval ((CONS (a , b)) NIL)))))))
SUBSPACE_ORTHOGONAL_TO_VECTORS |- (!s. (subspace (GSPEC (\GEN%PVAR%163. (?y. (((SETSPEC GEN%PVAR%163) (!x. ((x IN s) ==> ((orthogonal x) y)))) y))))))