hcju/holps huggingface dataset:
Packing3.VORONOI_SET_2_BIS_LE |- (!V. (!u. (!v. (((u IN V) /\ (v IN V)) ==> (((voronoi_set V) (u INSERT (v INSERT EMPTY))) = (((voronoi_closed V) v) INTER ((bis_le u) v)))))))
INSIDE_INSIDE_COMPACT_CONNECTED |- (!s. (!t. (((closed s) /\ ((compact t) /\ ((s SUBSET (inside t)) /\ (connected t)))) ==> ((inside s) SUBSET (inside t)))))
Lp_ineqs_proofs.rhazim_fan_eq_rhazim |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((rhazim_fan (V , E)) d) = ((((((rhazim (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)))))))))))
Ssrnat.neq_ltn |- (!m. (!n. ((~ (m = n)) = ((m < n) \/ (n < m)))))
INT_MUL_ASSOC |- (!x. (!y. (!z. (((int_mul x) ((int_mul y) z)) = ((int_mul ((int_mul x) y)) z)))))
Ssrbool.orKb |- (!a. (!b. ((a /\ (b \/ a)) = a)))
ATN_LE_X |- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (atn x)) x)))
FORALL_ALL |- (!P. (!l. ((!x. ((ALL (P x)) l)) = ((ALL (\s. (!x. ((P x) s)))) l))))
REAL_SUMMABLE_ZETA |- (!n. (!x. (((real_lt (real_of_num (NUMERAL (BIT1 _0)))) x) ==> ((real_summable (from n)) (\k. (real_inv ((rpow (real_of_num k)) x)))))))
Seq2.list_of_empty_set |- ((list_of_set EMPTY) = NIL)
CONVEX_STARCENTRES |- (!s. (convex (GSPEC (\GEN%PVAR%1721. (?a. (((SETSPEC GEN%PVAR%1721) ((a IN s) /\ (!x. ((x IN s) ==> ((closed_segment ((CONS (a , x)) NIL)) SUBSET s))))) a))))))
RE_POS_SEGMENT |- (!a. (!b. (!x. (((x IN (closed_segment ((CONS (a , b)) NIL))) /\ (((real_lt (real_of_num (NUMERAL _0))) (Re a)) /\ ((real_lt (real_of_num (NUMERAL _0))) (Re b)))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re x))))))
Merge_ineq.nonf_flat_term2_234_x |- (!d. (!a. (!b. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((((flat_term2_234_x d) a) b) x1) x2) x3) x4) x5) x6) = (flat_term_x (((((((((edge2_234_x d) a) b) x1) x2) x3) x4) x5) x6))))))))))))
SUM_MULTICOUNT |- (!R. (!s. (!t. (!k. (((FINITE s) /\ ((FINITE t) /\ (!j. ((j IN t) ==> ((CARD (GSPEC (\GEN%PVAR%205. (?i. (((SETSPEC GEN%PVAR%205) ((i IN s) /\ ((R i) j))) i))))) = k))))) ==> (((sum s) (\i. (real_of_num (CARD (GSPEC (\GEN%PVAR%206. (?j. (((SETSPEC GEN%PVAR%206) ((j IN t) /\ ((R i) j))) j)))))))) = (real_of_num (k * (CARD t)))))))))
ANALYTIC_ON_COMPOSE_GEN |- (!f. (!g. (!s. (!t. ((((analytic_on f) s) /\ (((analytic_on g) t) /\ (!z. ((z IN s) ==> ((f z) IN t))))) ==> ((analytic_on (g o f)) s))))))
Merge_ineq.ineq_mp_simple |- (!a. (!b. (!c. (((ineq a) (b ==> c)) = (b ==> ((ineq a) c))))))
MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT_DENSE |- (!f. (!r. (((closure ((IMAGE lift) r)) = UNIV) ==> (((measurable_on f) UNIV) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ (a IN r))) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3601. (?x. (((SETSPEC GEN%PVAR%3601) ((real_gt ((f x) $ k)) a)) x))))))))))))
CROSS_INTER |- ((!s. (!t. (!u. ((s CROSS (t INTER u)) = ((s CROSS t) INTER (s CROSS u)))))) /\ (!s. (!t. (!u. (((s INTER t) CROSS u) = ((s CROSS u) INTER (t CROSS u)))))))
REAL_POW |- (!z. (!n. ((real z) ==> (real ((complex_pow z) n)))))
BOUNDED_SPHERE |- (!a. (!r. (bounded (sphere (a , r)))))
NSUM_IMAGE_NONZERO |- (!d. (!i. (!s. (((FINITE s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((~ (x = y)) /\ ((i x) = (i y))))) ==> ((d (i x)) = (NUMERAL _0)))))) ==> (((nsum ((IMAGE i) s)) d) = ((nsum s) (d o i)))))))
STC_REFL |- (!R. ((!x. ((R x) x)) ==> (!x. (((STC R) x) x))))
HAS_DERIVATIVE_VSUM_NUMSEG |- (!f. (!net. (!m. (!n. ((!i. (((m <= i) /\ (i <= n)) ==> (((has_derivative (f i)) (f' i)) net))) ==> (((has_derivative (\x. ((vsum (m .. n)) (\i. ((f i) x))))) (\h. ((vsum (m .. n)) (\i. ((f' i) h))))) net))))))
EXTENSION_INTO_AR |- (!f. (!s. (!t. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((retract_of t) UNIV) /\ (closed s)))) ==> (?g. (((continuous_on g) UNIV) /\ ((((IMAGE g) UNIV) SUBSET t) /\ (!x. ((x IN s) ==> ((g x) = (f x)))))))))))
HULL_UNIONS_SUBSET |- (!P. (!f. ((UNIONS (GSPEC (\GEN%PVAR%96. (?s. (((SETSPEC GEN%PVAR%96) (s IN f)) ((hull P) s)))))) SUBSET ((hull P) (UNIONS f)))))
CONTINUOUS_WITHIN_CTAN |- (!s. (!z. ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) ==> ((continuous ctan) ((within (at z)) s)))))
Upfzbzm_support_lemmas.FINITE_PERMUTE_3 |- (FINITE (GSPEC (\GEN%PVAR%6211. (?p. (((SETSPEC GEN%PVAR%6211) ((permutes p) ((NUMERAL _0) INSERT ((NUMERAL (BIT1 _0)) INSERT ((NUMERAL (BIT0 (BIT1 _0))) INSERT EMPTY))))) p)))))
INTEGRAL_COMBINE_TAGGED_DIVISION_BOTTOMUP |- (!f. (!p. (!a. (!b. ((((tagged_division_of p) (closed_interval ((CONS (a , b)) NIL))) /\ (!x. (!k. (((x , k) IN p) ==> ((integrable_on f) k))))) ==> (((integral (closed_interval ((CONS (a , b)) NIL))) f) = ((vsum p) (GABS (\f. (!x. (!k. ((GEQ (f (x , k))) ((integral k) f)))))))))))))
INT_LE_LT |- (!x. (!y. (((int_le x) y) = (((int_lt x) y) \/ (x = y)))))
Nonlinear_lemma.sqrt_x6 |- (((((((sqrt_x6 x1) x2) x3) x4) x5) x6) = (sqrt x6))
AFFINE_DEPENDENT_BIGGERSET_GENERAL |- (!s. (((FINITE s) ==> ((CARD s) >= ((dim s) + (NUMERAL (BIT0 (BIT1 _0)))))) ==> (affine_dependent s)))
Lp_gen_theory.components_iso_image_face_map |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!d. ((d IN (dart G)) ==> ((((node H) (g ((face_map G) d))) = ((IMAGE g) ((node G) ((face_map G) d)))) /\ ((((face H) (g ((face_map G) d))) = ((IMAGE g) ((face G) ((face_map G) d)))) /\ ((((node H) (g ((inverse (face_map G)) d))) = ((IMAGE g) ((node G) ((inverse (face_map G)) d)))) /\ (((face H) (g ((inverse (face_map G)) d))) = ((IMAGE g) ((face G) ((inverse (face_map G)) d)))))))))))))
IM_COMPLEX_INV_GT_0 |- (!z. (((real_lt (real_of_num (NUMERAL _0))) (Im (complex_inv z))) = ((real_lt (Im z)) (real_of_num (NUMERAL _0)))))
List_hypermap.flatten_filter_empty |- (!L. ((flatten ((filter (\l. (~ (l = NIL)))) L)) = (flatten L)))
Collect_geom.IN_SET4 |- ((x IN (a INSERT (b INSERT (c INSERT (d INSERT EMPTY))))) = ((x = a) \/ ((x = b) \/ ((x = c) \/ (x = d)))))
CFUNSPACE_MDIST_LT |- (!top. (!m. (!f. (!g. (!a. (!x. ((((compact_in (topspace top)) top) /\ ((f IN (mspace ((cfunspace top) m))) /\ ((g IN (mspace ((cfunspace top) m))) /\ (((real_lt ((mdist ((cfunspace top) m)) (f , g))) a) /\ (x IN (topspace top)))))) ==> ((real_lt ((mdist m) ((f x) , (g x)))) a))))))))
COMPLEX_SUB_TRIANGLE |- (!a. (!b. (!c. (((vector_add ((vector_sub a) b)) ((vector_sub b) c)) = ((vector_sub a) c)))))
REAL_SECOND_MEAN_VALUE_THEOREM_GEN |- (!f. (!g. (!a. (!b. (!u. (!v. (((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (open_real_interval (a , b))) ==> (((real_le u) (g x)) /\ ((real_le (g x)) v)))) /\ (!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_le x) y))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_real_interval ((CONS (a , b)) NIL))) /\ (((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\x. ((real_mul (g x)) (f x)))) = ((real_add ((real_mul u) ((real_integral (closed_real_interval ((CONS (a , c)) NIL))) f))) ((real_mul v) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f)))))))))))))
REALLIM_AT_NEGINFINITY |- (!f. (!l. ((((---> f) l) at_neginfinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_le x) b) ==> ((real_lt (real_abs ((real_sub (f x)) l))) e)))))))))
REAL_NEGLIGIBLE_ON_INTERVALS |- (!s. ((real_negligible s) = (!a. (!b. (real_negligible (s INTER (closed_real_interval ((CONS (a , b)) NIL))))))))
REALLIM_WITHIN_OPEN |- (!f. (!l. (!a. (!s. (((a IN s) /\ (open s)) ==> ((((---> f) l) ((within (at a)) s)) = (((---> f) l) (at a))))))))
NO_RETRACTION_FRONTIER_BOUNDED |- (!s. (((bounded s) /\ (~ ((interior s) = EMPTY))) ==> (~ ((retract_of (frontier s)) s))))
Upfzbzm_support_lemmas.FINITE_PERMUTE_4 |- (FINITE (GSPEC (\GEN%PVAR%6212. (?p. (((SETSPEC GEN%PVAR%6212) ((permutes p) ((NUMERAL _0) INSERT ((NUMERAL (BIT1 _0)) INSERT ((NUMERAL (BIT0 (BIT1 _0))) INSERT ((NUMERAL (BIT1 (BIT1 _0))) INSERT EMPTY)))))) p)))))
SUMMABLE_SUBSET_ABSCONV |- (!x. (!s. (!t. ((((summable s) (\n. (lift (vector_norm (x n))))) /\ (t SUBSET s)) ==> ((summable t) (\n. (lift (vector_norm (x n)))))))))
BOHL_ALT |- (!f. (!s. (!a. ((((continuous_on f) s) /\ ((convex s) /\ ((compact s) /\ ((a IN (interior s)) /\ (((IMAGE f) s) SUBSET (UNIV DELETE a)))))) ==> (?x. ((x IN (frontier s)) /\ (a IN (open_segment (x , (f x))))))))))
Ssrbool.negbFE |- (!b. (((~ b) = F) ==> b))
Nonlinear_lemma.rho_sqrtx |- (!x. ((rho (sqrt x)) = ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul const1) ((real_div ((real_sub (sqrt x)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 _0)))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))
OPEN_OPEN_LEFT_PROJECTION |- (!s. (!t. (((open s) /\ (open t)) ==> (open (GSPEC (\GEN%PVAR%1073. (?x. (((SETSPEC GEN%PVAR%1073) ((x IN s) /\ (?y. (((pastecart x) y) IN t)))) x))))))))
Lp_ineqs_proofs.y8_def |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((y8_fan (V , E)) d) = ((y5_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))
HOLOMORPHIC_ON_CTAN |- (!s. ((!z. ((z IN s) ==> (~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((holomorphic_on ctan) s)))
COMPLEX_DIFFERENTIABLE_DIV_AT |- (!f. (!g. (!z. ((((complex_differentiable f) (at z)) /\ (((complex_differentiable g) (at z)) /\ (~ ((g z) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((complex_differentiable (\z. ((complex_div (f z)) (g z)))) (at z))))))
NSUM_PERMUTE_NUMSEG |- (!f. (!p. (!m. (!n. (((permutes p) (m .. n)) ==> (((nsum (m .. n)) f) = ((nsum (m .. n)) (f o p))))))))
REAL_ROOT_LE |- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ ((real_le (real_of_num (NUMERAL _0))) y)) ==> (((real_le ((root n) x)) y) = ((real_le x) ((real_pow y) n)))))))
ANALYTIC_ON_DIV |- (!f. (!g. (!s. ((((analytic_on f) s) /\ (((analytic_on g) s) /\ (!z. ((z IN s) ==> (~ ((g z) = (Cx (real_of_num (NUMERAL _0))))))))) ==> ((analytic_on (\z. ((complex_div (f z)) (g z)))) s)))))
Ssrnat.ltn_trans |- (!n. (!m. (!p. ((m < n) ==> ((n < p) ==> (m < p))))))
Ckqowsa_3_points.pos_vector_angle_bounds |- (!v. (!u. (((real_gt ((dot v) u)) (real_of_num (NUMERAL _0))) ==> (((real_le (real_of_num (NUMERAL _0))) ((vector_angle v) u)) /\ ((real_le ((vector_angle v) u)) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))
LIPSCHITZ_EXTENSION_EXISTS |- (!f. (!s. ((?B. (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y)))))))) ==> (?g. ((?B. (!x. (!y. ((real_le (vector_norm ((vector_sub (g x)) (g y)))) ((real_mul B) (vector_norm ((vector_sub x) y))))))) /\ (!x. ((x IN s) ==> ((g x) = (f x)))))))))
Pack2.sum_measure_voronoi_closed_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_closed S) v))))) (measure (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))))))
Lp_ineqs_proofs.dart4_azim2_eq_sum |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((azim2_fan (V , E)) d) = ((real_add ((((((dih_y (y2_fan d)) ((y3_fan (V , E)) d)) (y1_fan d)) ((y5_fan (V , E)) d)) (y6_fan d)) ((y4'_fan (V , E)) d))) ((((((dih_y (y2_fan d)) ((y7_fan (V , E)) d)) ((y3_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y9_fan (V , E)) d))))))))))))
Float.INT_POW_ODD_NEG1 |- (!x. (!n. (((int_pow (int_neg (int_of_num x))) (NUMERAL (BIT1 n))) = (int_neg ((int_pow (int_of_num x)) (NUMERAL (BIT1 n)))))))
PATH_CONNECTED_SUMS |- (!s. (!t. (((path_connected s) /\ (path_connected t)) ==> (path_connected (GSPEC (\GEN%PVAR%1816. (?x. (?y. (((SETSPEC GEN%PVAR%1816) ((x IN s) /\ (y IN t))) ((vector_add x) y))))))))))
FUNDAMENTAL_GROUP_SIMPLY_CONNECTED |- (!s. (!a. (((simply_connected s) /\ (a IN s)) ==> ((fundamental_group (s , a)) = (((homotopic_paths s) (linepath (a , a))) INSERT EMPTY)))))
HAS_REAL_MEASURE_SCALING_EQ |- (!s. (!m. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> (((has_real_measure ((IMAGE (\x. ((real_mul c) x))) s)) ((real_mul (real_abs c)) m)) = ((has_real_measure s) m))))))
PATH_CONNECTED_COMPLEMENT_CARD_LT |- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ (s <_c UNIV)) ==> (path_connected (UNIV DIFF s))))
LIM_NULL_COMPLEX_LMUL |- (!net. (!f. (!c. ((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) ==> (((--> (\x. ((complex_mul c) (f x)))) (Cx (real_of_num (NUMERAL _0)))) net)))))
REAL_NEGLIGIBLE_DIFF |- (!s. (!t. ((real_negligible s) ==> (real_negligible (s DIFF t)))))
CLOSED_CONNECTED_PREIMAGES_IMP_CONTINUOUS_ON |- (!f. (!s. (!t. (((((IMAGE f) s) SUBSET t) /\ ((compact t) /\ ((!y. (connected (GSPEC (\GEN%PVAR%1942. (?x. (((SETSPEC GEN%PVAR%1942) ((x IN s) /\ ((f x) = y))) x)))))) /\ ((!c. (((closed_in ((subtopology euclidean) s)) c) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) c)))) /\ (!c. (((c SUBSET s) /\ (connected c)) ==> (connected ((IMAGE f) c)))))))) ==> ((continuous_on f) s)))))
NOT_ON_PATH_CBALL |- (!g. (!z. (((path g) /\ (~ (z IN (path_image g)))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (((cball (z , e)) INTER (path_image g)) = EMPTY))))))
CONTINUOUS_COUNTABLE_RANGE_CONSTANT |- (!f. (!s. (((connected s) /\ (((continuous_on f) s) /\ (COUNTABLE ((IMAGE f) s)))) ==> (?a. (!x. ((x IN s) ==> ((f x) = a)))))))
ARG_ROTATE2D_UNIQUE_2PI |- (!s. (!t. (!z. (((~ (z = (Cx (real_of_num (NUMERAL _0))))) /\ (((real_le (real_of_num (NUMERAL _0))) s) /\ (((real_lt s) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ (((real_le (real_of_num (NUMERAL _0))) t) /\ (((real_lt t) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ ((Arg ((rotate2d s) z)) = (Arg ((rotate2d t) z)))))))) ==> (s = t)))))
INSIDE_CONNECTED_COMPONENT_LE |- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ (bounded s)) ==> ((inside s) = (GSPEC (\GEN%PVAR%2016. (?x. (((SETSPEC GEN%PVAR%2016) ((~ (x IN s)) /\ (!B. (?y. (((real_le B) (vector_norm y)) /\ (~ (((connected_component (UNIV DIFF s)) x) y))))))) x)))))))
CLOG_EQ |- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> (((clog w) = (clog z)) = (w = z)))))
Ssrnat.distnn |- (!m. (((distn m) m) = (NUMERAL _0)))
REAL_SGN_SIGN |- (!p. ((real_sgn (sign p)) = (sign p)))
REAL_SERIES_RATIO |- (!c. (!a. (!s. (!N. ((((real_lt c) (real_of_num (NUMERAL (BIT1 _0)))) /\ (!n. ((n >= N) ==> ((real_le (real_abs (a (SUC n)))) ((real_mul c) (real_abs (a n))))))) ==> (?l. (((real_sums a) l) s)))))))
NORM_EQ_0_DOT |- (!x. (((vector_norm x) = (real_of_num (NUMERAL _0))) = (((dot x) x) = (real_of_num (NUMERAL _0)))))
COMPLEX_SUB_NEG2 |- (!x. (!y. (((vector_sub (vector_neg x)) (vector_neg y)) = ((vector_sub y) x))))
RATIONAL_ALT |- (!x. ((rational x) = (?p. (?q. ((~ (q = (NUMERAL _0))) /\ ((real_abs x) = ((real_div (real_of_num p)) (real_of_num q))))))))
Merge_ineq.nonf_mudLs_126_x |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a. (!b. (!c. (!d. (!e. ((((((((((((mudLs_126_x a) b) c) d) e) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((mu6_x ((real_mul c) c)) x1) x2) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0)))) ((real_add ((real_mul ((real_div (real_of_num (NUMERAL (BIT1 _0)))) ((real_add a) b))) ((real_sub (((((((((delta_126_x ((real_mul c) c)) ((real_mul d) d)) ((real_mul e) e)) x1) x2) x3) x4) x5) x6)) ((real_mul a) a)))) a))))))))))))))
REALLIM_TRANSFORM_WITHIN_SET_IMP |- (!f. (!l. (!a. (!s. (!t. ((((eventually (\x. ((x IN t) ==> (x IN s)))) (at a)) /\ (((---> f) l) ((within (at a)) s))) ==> (((---> f) l) ((within (at a)) t))))))))
REAL_CONVEX_ON_DERIVATIVE_INCREASING_IMP |- (!f. (!f'x. (!f'y. (!s. (!x. (!y. ((((real_convex_on f) s) /\ (((closed_real_interval ((CONS (x , y)) NIL)) SUBSET s) /\ ((((has_real_derivative f) f'x) ((within (atreal x)) s)) /\ ((((has_real_derivative f) f'y) ((within (atreal y)) s)) /\ ((real_lt x) y))))) ==> ((real_le f'x) f'y))))))))
CONTINUOUS_ON_DIST_CLOSEST_POINT |- (!s. (!t. (((closed s) /\ (~ (s = EMPTY))) ==> ((continuous_on (\x. (lift (distance (x , ((closest_point s) x)))))) t))))
Lp_ineqs_proofs.dart3_azim_fan_eq_dih_y |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT1 _0)))) (hypermap_of_fan (V , E)))) ==> (((azim_dart (V , E)) d) = ((((((dih_y (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)))))))))))
POSITIVE_SEMIDEFINITE_MUL |- (!A. (!B. (((positive_semidefinite A) /\ ((positive_semidefinite B) /\ (((matrix_mul A) B) = ((matrix_mul B) A)))) ==> (positive_semidefinite ((matrix_mul A) B)))))
DIFFERENTIABLE_ON_ADD |- (!f. (!g. (!s. ((((differentiable_on f) s) /\ ((differentiable_on g) s)) ==> ((differentiable_on (\z. ((vector_add (f z)) (g z)))) s)))))
CONTINUOUS_ON_CACS |- (!s. ((!z. (((z IN s) /\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((continuous_on cacs) s)))
SETDIST_RELATIVE_INTERIOR |- ((!s. (!t. ((convex s) ==> ((setdist ((relative_interior s) , t)) = (setdist (s , t)))))) /\ (!s. (!t. ((convex t) ==> ((setdist (s , (relative_interior t))) = (setdist (s , t)))))))
ABSOLUTELY_INTEGRABLE_CONVOLUTION_L2 |- (!bop. (!f. (!g. (!x. (((bilinear bop) /\ (((measurable_on f) UNIV) /\ (((measurable_on g) UNIV) /\ (((absolutely_integrable_on (\x. (lift ((real_pow (vector_norm (f x))) (NUMERAL (BIT0 (BIT1 _0))))))) UNIV) /\ ((absolutely_integrable_on (\x. (lift ((real_pow (vector_norm (g x))) (NUMERAL (BIT0 (BIT1 _0))))))) UNIV))))) ==> ((absolutely_integrable_on (\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))
CCOS_CSIN_CSQRT |- (!z. ((((real_lt (real_of_num (NUMERAL _0))) (cos (Re z))) \/ (((cos (Re z)) = (real_of_num (NUMERAL _0))) /\ ((real_le ((real_mul (Im z)) (sin (Re z)))) (real_of_num (NUMERAL _0))))) ==> ((ccos z) = (csqrt ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow (csin z)) (NUMERAL (BIT0 (BIT1 _0)))))))))
one_Axiom |- (!e. (?!fn. ((fn one) = e)))
Ssrbool.contraFN |- (!c. (!b. ((c ==> b) ==> ((b = F) ==> (~ c)))))
Hijqaha.J_SCS_4M5 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_4M5') i)) i1) j) = F)
ANALYTIC_ON_SUB |- (!f. (!g. (!s. ((((analytic_on f) s) /\ ((analytic_on g) s)) ==> ((analytic_on (\z. ((vector_sub (f z)) (g z)))) s)))))
LOG_ROOT |- (!n. (!x. (((~ (n = (NUMERAL _0))) /\ ((real_lt (real_of_num (NUMERAL _0))) x)) ==> ((log ((root n) x)) = ((real_div (log x)) (real_of_num n))))))
CONTINUOUS_ON_ABS |- (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\x. (lambdai. (real_abs ((f x) $ i))))) s))))
COMPLEX_DIFFERENTIABLE_WITHIN_CSIN |- (!s. (!z. ((complex_differentiable csin) ((within (at z)) s))))
INSEG_ORDINAL |- (!l. (!m. ((((inseg m) l) /\ (ordinal l)) ==> (ordinal m))))
DIV_MULT2 |- (!m. (!n. (!p. ((~ ((m * p) = (NUMERAL _0))) ==> (((m * n) DIV (m * p)) = (n DIV p))))))
Seq.mask_cons |- (!b. (!m. (!x. (!s. (((mask ((CONS b) m)) ((CONS x) s)) = ((cat ((nseq (((COND b) (NUMERAL (BIT1 _0))) (NUMERAL _0))) x)) ((mask m) s)))))))
REAL_INTEGRAL_RESTRICT |- (!f. (!s. (!t. ((s SUBSET t) ==> (((real_integral t) (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) = ((real_integral s) f))))))
AFF_LE_SPECIAL_SCALE |- (!s. (!t. (!a. (!v. (((FINITE s) /\ ((FINITE t) /\ ((~ ((vec (NUMERAL _0)) IN t)) /\ ((~ (v IN t)) /\ ((~ (((% a) v) IN t)) /\ ((real_lt (real_of_num (NUMERAL _0))) a)))))) ==> (((aff_le ((vec (NUMERAL _0)) INSERT (((% a) v) INSERT s))) t) = ((aff_le ((vec (NUMERAL _0)) INSERT (v INSERT s))) t)))))))
Ssrnat.subn_sqr |- (!m. (!n. (((m EXP (NUMERAL (BIT0 (BIT1 _0)))) - (n EXP (NUMERAL (BIT0 (BIT1 _0))))) = ((m - n) * (m + n)))))
AFF_DIM_PSUBSET |- (!s. (!t. ((((hull affine) s) PSUBSET ((hull affine) t)) ==> ((int_lt (aff_dim s)) (aff_dim t)))))
INSIDE_EQ_OUTSIDE |- (!s. (((inside s) = (outside s)) = (s = UNIV)))
VSUM_SUPPORT |- (!f. (!s. (((vsum (GSPEC (\GEN%PVAR%110. (?x. (((SETSPEC GEN%PVAR%110) ((x IN s) /\ (~ ((f x) = (vec (NUMERAL _0)))))) x))))) f) = ((vsum s) f))))
DIFFERENTIABLE_ON_NORM |- (!s. ((~ ((vec (NUMERAL _0)) IN s)) ==> ((differentiable_on (\x. (lift (vector_norm x)))) s)))
DIM_OPEN |- (!s. (((open s) /\ (~ (s = EMPTY))) ==> ((dim s) = (dimindex UNIV))))
CONVEX_FINITE |- (!s. ((FINITE s) ==> ((convex s) = (!u. (((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (u x)))) /\ (((sum s) u) = (real_of_num (NUMERAL (BIT1 _0))))) ==> (((vsum s) (\x. ((% (u x)) x))) IN s))))))
Misc_defs_and_lemmas.SURJ_SECTION |- (!a. (!b. (!f. ((((SURJ f) a) b) ==> (?g. (((INJ g) b) a))))))
Nonlinear_lemma.sqrt_x4 |- (((((((sqrt_x4 x1) x2) x3) x4) x5) x6) = (sqrt x4))
INT_MIN_MIN |- (!x. (!y. (((int_le ((int_min x) y)) x) /\ ((int_le ((int_min x) y)) y))))
ANALYTIC_AT_MUL |- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\ ((analytic_on g) (z INSERT EMPTY))) ==> ((analytic_on (\w. ((complex_mul (f w)) (g w)))) (z INSERT EMPTY))))))
CASN_UNIQUE |- (!w. (!z. ((((csin z) = w) /\ (((real_lt (real_abs (Re z))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) \/ (((real_abs (Re z)) = ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) /\ ((Im z) = (real_of_num (NUMERAL _0)))))) ==> ((casn w) = z))))
COUNTABLE_CASES |- (!s. ((COUNTABLE s) = ((FINITE s) \/ (s =_c UNIV))))
AFF_LOWDIM_SUBSET_HYPERPLANE |- (!s. (((int_lt (aff_dim s)) (int_of_num (dimindex UNIV))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ (s SUBSET (GSPEC (\GEN%PVAR%1597. (?x. (((SETSPEC GEN%PVAR%1597) (((dot a) x) = b)) x))))))))))
HAS_INTEGRAL_SPIKE_FINITE_EQ |- (!f. (!g. (!s. (!y. (((FINITE s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> ((((has_integral f) y) t) = (((has_integral g) y) t)))))))
AFFINE_SCALING_EQ |- (!s. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> ((affine ((IMAGE (\x. ((% c) x))) s)) = (affine s)))))
ABSOLUTELY_CONTINUOUS_ON_VMUL |- (!f. (!s. (!v. (((absolutely_continuous_on (\x. (lift (f x)))) s) ==> ((absolutely_continuous_on (\x. ((% (f x)) v))) s)))))
MANHATTAN |- (((mspace manhattan) = UNIV) /\ (!x. (!y. (((mdist manhattan) (x , y)) = ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (real_abs ((real_sub (x $ i)) (y $ i)))))))))
VECTOR_POLYNOMIAL_FUNCTION_o |- (!f. (!g. (((vector_polynomial_function f) /\ (vector_polynomial_function g)) ==> (vector_polynomial_function (g o f)))))
Tactics_jordan.switch_lemma_let |- (!A. (!B. (!C. ((((real_lt (real_of_num (NUMERAL _0))) A) ==> (((real_le (real_of_num (NUMERAL _0))) B) ==> C)) = (((real_le (real_of_num (NUMERAL _0))) B) ==> (((real_lt (real_of_num (NUMERAL _0))) A) ==> C))))))
CONTINUOUS_ON_LIFT_ABS_COMPONENT |- (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\x. (lift (real_abs ((f x) $ k))))) s))))
REAL_SERIES_BOUND |- (!f. (!g. (!s. (!a. (!b. (((((real_sums f) a) s) /\ ((((real_sums g) b) s) /\ (!i. ((i IN s) ==> ((real_le (real_abs (f i))) (g i)))))) ==> ((real_le (real_abs a)) b)))))))
MEASURE_CBALL_SCALING |- (!a. (!c. (!r. (((real_le (real_of_num (NUMERAL _0))) c) ==> ((measure (cball (a , ((real_mul c) r)))) = ((real_mul ((real_pow c) (dimindex UNIV))) (measure (cball (a , r)))))))))
REAL_MEASURABLE_SCALING_EQ |- (!s. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> ((real_measurable ((IMAGE (\x. ((real_mul c) x))) s)) = (real_measurable s)))))
LIM_INFINITY_SEQUENTIALLY_COMPLEX |- (!f. (!l. ((((--> f) l) at_infinity) ==> (((--> (\n. (f (Cx (real_of_num n))))) l) sequentially))))
VECTOR_AFFINITY_EQ |- (!m. (!c. (!x. (!y. ((~ (m = (real_of_num (NUMERAL _0)))) ==> ((((vector_add ((% m) x)) c) = y) = (x = ((vector_add ((% (real_inv m)) y)) (vector_neg ((% (real_inv m)) c))))))))))
IS_INTERVAL_IMP_FSIGMA |- (!s. ((is_interval s) ==> (fsigma s)))
FORALL_TRIPLED_THM |- (!P. ((! (GABS (\f. (!x. (!y. (!z. ((GEQ (f (x , (y , z)))) (((P x) y) z)))))))) = (!x. (!y. (!z. (((P x) y) z))))))
INT_SUB_SUB2 |- (!x. (!y. (((int_sub x) ((int_sub x) y)) = y)))
CONTINUOUS_ON_MDIST |- (!m. (!a. ((a IN (mspace m)) ==> (((topcontinuous (mtopology m)) euclideanreal) (\x. ((mdist m) (a , x)))))))