hcju/holps huggingface dataset:
UPPER_HEMICONTINUOUS |- (!f. (!t. (!s. ((!x. ((x IN s) ==> ((f x) SUBSET t))) ==> ((!u. (((open_in ((subtopology euclidean) t)) u) ==> ((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%880. (?x. (((SETSPEC GEN%PVAR%880) ((x IN s) /\ ((f x) SUBSET u))) x))))))) = (!u. (((closed_in ((subtopology euclidean) t)) u) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%881. (?x. (((SETSPEC GEN%PVAR%881) ((x IN s) /\ (~ (((f x) INTER u) = EMPTY)))) x))))))))))))
CONNECTED_OPEN_DIFF_CARD_LT |- (!s. (!t. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ ((open s) /\ ((connected s) /\ (t <_c UNIV)))) ==> (connected (s DIFF t)))))
OPEN_PATH_CONNECTED_COMPONENT |- (!s. (!x. ((open s) ==> (((path_component s) x) = ((connected_component s) x)))))
HOMEOMORPHIC_SIMPLE_PATH_ARC_EQ |- (!g. (!h. (((simple_path g) /\ ((simple_path h) /\ ((homeomorphic (path_image g)) (path_image h)))) ==> ((arc g) = (arc h)))))
COUNTABLE_ENR_PATH_COMPONENTS |- (!s. ((ENR s) ==> (COUNTABLE (GSPEC (\GEN%PVAR%2725. (?x. (((SETSPEC GEN%PVAR%2725) (x IN s)) ((path_component s) x))))))))
Lp_gen_theory.card_face_dart_iso |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!d. ((d IN (dart G)) ==> ((CARD ((IMAGE g) ((face G) d))) = (CARD ((face G) d)))))))))
SETDIST_UNIV |- ((!s. ((setdist (s , UNIV)) = (real_of_num (NUMERAL _0)))) /\ (!t. ((setdist (UNIV , t)) = (real_of_num (NUMERAL _0)))))
RELATIVE_FRONTIER_OF_TRIANGLE |- (!a. (!b. (!c. ((~ (collinear (a INSERT (b INSERT (c INSERT EMPTY))))) ==> ((relative_frontier ((hull convex) (a INSERT (b INSERT (c INSERT EMPTY))))) = ((closed_segment ((CONS (a , b)) NIL)) UNION ((closed_segment ((CONS (b , c)) NIL)) UNION (closed_segment ((CONS (c , a)) NIL)))))))))
Geomdetail.IN_SET4 |- ((x IN (a INSERT (b INSERT (c INSERT (d INSERT EMPTY))))) = ((x = a) \/ ((x = b) \/ ((x = c) \/ (x = d)))))
CARD_EQ_PERFECT_SET |- (!s. (((closed s) /\ ((!x. ((x IN s) ==> ((limit_point_of x) s))) /\ (~ (s = EMPTY)))) ==> (s =_c UNIV)))
HAUSDIST_STILL_NONEMPTY_INTERIOR |- (!s. (((bounded s) /\ ((convex s) /\ (~ ((interior s) = EMPTY)))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!s'. (((bounded s') /\ ((convex s') /\ ((~ (s' = EMPTY)) /\ ((real_lt (hausdist (s , s'))) e)))) ==> (~ ((interior s') = EMPTY))))))))
STARLIKE_IMP_CONNECTED |- (!s. ((starlike s) ==> (connected s)))
EXP_MONO_EQ |- (!x. (!y. (!n. (((x EXP n) = (y EXP n)) = ((x = y) \/ (n = (NUMERAL _0)))))))
CONVEX_HAS_BOUNDED_VARIATION |- (!f. (!a. (!b. (((convex_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((has_bounded_variation_on (lift o f)) (closed_interval ((CONS (a , b)) NIL)))))))
UNION_WITH_OUTSIDE |- (!s. ((s UNION (outside s)) = (UNIV DIFF (inside s))))
REAL_MEASURABLE_ON_MAX |- (!f. (!g. (!s. ((((real_measurable_on f) s) /\ ((real_measurable_on g) s)) ==> ((real_measurable_on (\x. ((real_max (f x)) (g x)))) s)))))
UNBOUNDED_HALFSPACE_COMPONENT_GT |- (!a. (!k. (~ (bounded (GSPEC (\GEN%PVAR%839. (?x. (((SETSPEC GEN%PVAR%839) ((real_gt (x $ k)) a)) x))))))))
REAL_SUMMABLE_ALTERNATING_SERIES |- (!a. (!m. (((!n. ((real_le (real_abs (a (SUC n)))) (real_abs (a n)))) /\ ((!n. ((EVEN n) ==> ((real_le (real_of_num (NUMERAL _0))) (a n)))) /\ ((!n. ((ODD n) ==> ((real_le (a n)) (real_of_num (NUMERAL _0))))) /\ (((---> a) (real_of_num (NUMERAL _0))) sequentially)))) ==> ((real_summable (from m)) a))))
Seq.eqseq_rcons |- (!s1. (!s2. (!x1. (!x2. ((((rcons s1) x1) = ((rcons s2) x2)) = ((s1 = s2) /\ (x1 = x2)))))))
REAL_CONTINUOUS_ON_DIV |- (!f. (!g. (!s. ((((real_continuous_on f) s) /\ (((real_continuous_on g) s) /\ (!x. ((x IN s) ==> (~ ((g x) = (real_of_num (NUMERAL _0)))))))) ==> ((real_continuous_on (\x. ((real_div (f x)) (g x)))) s)))))
Asfutbf.affine_hull_neg |- (!v. (((IMAGE vector_neg) ((hull affine) ((vec (NUMERAL _0)) INSERT (v INSERT EMPTY)))) = ((hull affine) ((vec (NUMERAL _0)) INSERT ((vector_neg v) INSERT EMPTY)))))
INTERIOR_STD_SIMPLEX |- ((interior ((hull convex) ((vec (NUMERAL _0)) INSERT (GSPEC (\GEN%PVAR%1807. (?i. (((SETSPEC GEN%PVAR%1807) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (basis i)))))))) = (GSPEC (\GEN%PVAR%1808. (?x. (((SETSPEC GEN%PVAR%1808) ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_lt (real_of_num (NUMERAL _0))) (x $ i)))) /\ ((real_lt ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (x $ i)))) (real_of_num (NUMERAL (BIT1 _0)))))) x)))))
HIGHER_COMPLEX_DERIVATIVE_1 |- (!f. (!z. ((((higher_complex_derivative (NUMERAL (BIT1 _0))) f) z) = ((complex_derivative f) z))))
CROSS_ROTATION_MATRIX |- (!A. (!x. (!y. ((rotation_matrix A) ==> (((cross ((matrix_vector_mul A) x)) ((matrix_vector_mul A) y)) = ((matrix_vector_mul A) ((cross x) y)))))))
Lp_ineqs_proofs.sol_sum |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((sum f) (azim_dart (V , E))) = ((real_add ((sol_fan (V , E)) f)) ((real_mul (real_of_num ((CARD f) - (NUMERAL (BIT0 (BIT1 _0)))))) pi)))))))))))
EVEN_EXP |- (!m. (!n. ((EVEN (m EXP n)) = ((EVEN m) /\ (~ (n = (NUMERAL _0)))))))
LINSEG_WOSET |- (!l. (!a. ((woset l) ==> (woset ((linseg l) a)))))
REAL_SGN_SQRT |- (!x. ((real_sgn (sqrt x)) = (real_sgn x)))
Conforming.lemma_card_node_eq_set_of_orbits |- (!x. (!V. (!E. (!f. (!y. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((f IN (node_set (hypermap1_of_fanx (x , (V , E))))) /\ (y IN f)))) ==> ((CARD (GSPEC (\GEN%PVAR%5670. (?z. (((SETSPEC GEN%PVAR%5670) (?i. ((i >= (NUMERAL _0)) /\ (z = (x , ((pr2 y) , ((((((((power_map_points sigma_fan) x) V) E) (pr2 y)) (pr3 y)) i) , (((((((power_map_points sigma_fan) x) V) E) (pr2 y)) (pr3 y)) (SUC i))))))))) z))))) = (CARD (((((set_of_orbits_points_fan x) V) E) (pr2 y)) (pr3 y))))))))))
NSUM_UNION_LZERO |- (!f. (!u. (!v. (((FINITE v) /\ (!x. (((x IN u) /\ (~ (x IN v))) ==> ((f x) = (NUMERAL _0))))) ==> (((nsum (u UNION v)) f) = ((nsum v) f))))))
Lp_gen_theory.iso_dart_pairs_trans |- (!H. (!G. (!g. (((hyp_iso g) (G , H)) ==> (!P. ((!p. ((p IN ((dart H) CROSS (dart H))) ==> (P p))) ==> (!p. ((p IN ((dart G) CROSS (dart G))) ==> (P ((g (FST p)) , (g (SND p))))))))))))
EVEN_AND_ODD |- (!n. (~ ((EVEN n) /\ (ODD n))))
EXISTS_TRIPLED_THM |- (!P. ((? (GABS (\f. (!x. (!y. (!z. ((GEQ (f (x , (y , z)))) (((P x) y) z)))))))) = (?x. (?y. (?z. (((P x) y) z))))))
COMPLEX_DERIVATIVE_TRANSFORM_WITHIN_OPEN |- (!f. (!g. (!s. (!z. (((open s) /\ (((holomorphic_on f) s) /\ (((holomorphic_on g) s) /\ ((z IN s) /\ (!w. ((w IN s) ==> ((f w) = (g w)))))))) ==> (((complex_derivative f) z) = ((complex_derivative g) z)))))))
REAL_DIFFERENTIABLE_WITHIN_LOG |- (!s. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_differentiable log) ((within (atreal x)) s)))))
FRECHET_DERIVATIVE_WITHIN_CLOSED_INTERVAL |- (!f. (!f'. (!x. (!a. (!b. (((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_lt (a $ i)) (b $ i)))) /\ ((x IN (closed_interval ((CONS (a , b)) NIL))) /\ (((has_derivative f) f') ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))))) ==> (((frechet_derivative f) ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))) = f')))))))
RESTRICTION_IDEMP |- (!s. (!f. (((RESTRICTION s) ((RESTRICTION s) f)) = ((RESTRICTION s) f))))
STC_INC_SC |- (!R. (!x. (!y. ((((SC R) x) y) ==> (((STC R) x) y)))))
REAL_STEINHAUS |- (!s. (((real_measurable s) /\ ((real_lt (real_of_num (NUMERAL _0))) (real_measure s))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ ((open_real_interval ((real_neg d) , d)) SUBSET (GSPEC (\GEN%PVAR%4987. (?x. (?y. (((SETSPEC GEN%PVAR%4987) ((x IN s) /\ (y IN s))) ((real_sub x) y)))))))))))
REAL_MEASURE_REAL_NEGLIGIBLE_UNIONS_IMAGE |- (!f. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> (real_measurable (f x)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> (real_negligible ((f x) INTER (f y)))))))) ==> ((real_measure (UNIONS ((IMAGE f) s))) = ((sum s) (\x. (real_measure (f x))))))))
Seq2.all_subseq |- (!s1. (!s2. (!P. ((((subseq s1) s2) /\ ((all P) s2)) ==> ((all P) s1)))))
Trigonometry.UIVNNRR1 |- (!V. (((aff_ge V) EMPTY) = ((aff_gt V) EMPTY)))
SYMMETRIC_LINEAR_IMAGE |- (!f. (!s. (((!x. ((x IN s) ==> ((vector_neg x) IN s))) /\ (linear f)) ==> (!x. ((x IN ((IMAGE f) s)) ==> ((vector_neg x) IN ((IMAGE f) s)))))))
Oxl_def.BIJ_NUMSEG |- (!s. (!a. (!b. (((BIJ (\j. (j + s))) (a .. b)) ((a + s) .. (b + s))))))
NORM_CSIN_POW_2 |- (!z. (((real_pow (vector_norm (csin z))) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_div ((real_add (exp ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (Im z)))) ((real_sub (real_inv (exp ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (Im z))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (cos ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (Re z))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))
HAS_REAL_DERIVATIVE_ZERO_UNIQUE_STRONG_INTERVAL |- (!f. (!a. (!b. (!k. (!y. (((COUNTABLE k) /\ (((real_continuous_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ (((f a) = y) /\ (!x. ((x IN ((closed_real_interval ((CONS (a , b)) NIL)) DIFF k)) ==> (((has_real_derivative f) (real_of_num (NUMERAL _0))) ((within (atreal x)) (closed_real_interval ((CONS (a , b)) NIL))))))))) ==> (!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> ((f x) = y)))))))))
REAL_INTEGRABLE_DECREASING_PRODUCT_UNIV |- (!f. (!g. (!B. ((((real_integrable_on f) UNIV) /\ ((!x. (!y. (((real_le x) y) ==> ((real_le (g y)) (g x))))) /\ (!x. ((real_le (real_abs (g x))) B)))) ==> ((real_integrable_on (\x. ((real_mul (g x)) (f x)))) UNIV)))))
OPEN_IN_TRANS_EQ |- (!s. (!t. ((!u. (((open_in ((subtopology euclidean) t)) u) ==> ((open_in ((subtopology euclidean) s)) t))) = ((open_in ((subtopology euclidean) s)) t))))
CONIC_IMP_CONNECTED |- (!s. ((conic s) ==> (connected s)))
OUTSIDE_CONNECTED_COMPONENT_LE |- (!s. ((outside s) = (GSPEC (\GEN%PVAR%2012. (?x. (((SETSPEC GEN%PVAR%2012) (!B. (?y. (((real_le B) (vector_norm y)) /\ (((connected_component (UNIV DIFF s)) x) y))))) x))))))
HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_WITHIN |- (!f. (!s. (!x. ((((holomorphic_on f) s) /\ (x IN s)) ==> ((complex_differentiable f) ((within (at x)) s))))))
INT_LE_ADDR |- (!x. (!y. (((int_le x) ((int_add x) y)) = ((int_le (int_of_num (NUMERAL _0))) y))))
Seq2.take_belast |- (!s. (!x. (!n. (((n < (sizel s)) /\ ((NUMERAL _0) < n)) ==> (((take n) ((belast x) s)) = ((CONS x) ((take (n - (NUMERAL (BIT1 _0)))) s)))))))
CLOSED_UNION_COMPLEMENT_COMPONENTS |- (!s. (!c. (((closed s) /\ (c SUBSET (components (UNIV DIFF s)))) ==> (closed (s UNION (UNIONS c))))))
POSITIVE_DEFINITE_MUL |- (!A. (!B. (((positive_definite A) /\ ((positive_definite B) /\ (((matrix_mul A) B) = ((matrix_mul B) A)))) ==> (positive_definite ((matrix_mul A) B)))))
AR_PCROSS_EQ |- (!s. (!t. ((AR (s PCROSS t)) = ((AR s) /\ ((AR t) /\ ((~ (s = EMPTY)) /\ (~ (t = EMPTY))))))))
Odxlstc.DEFORMATION_DIST_LE_ALL_COM_LE3 |- (((is_scs_v39 s) /\ (((MMs_v39 s) w) /\ (((scs_k_v39 s) = k) /\ (((NUMERAL (BIT1 (BIT1 _0))) <= k) /\ (!i. ((~ ((i MOD k) = (l MOD k))) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance ((w l) , (w i)))))))))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!t. (!i. ((((real_lt (real_of_num (NUMERAL _0))) t) /\ (((real_lt t) e) /\ (~ ((i MOD k) = (l MOD k))))) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance (((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) t)) (w l)) , (w i))))))))))
NPRODUCT_SUPPORT |- (!f. (!s. (((nproduct (((support *) f) s)) f) = ((nproduct s) f))))
SEPARATING_HYPERPLANE_AFFINE_HULLS |- (!s. (!t. (((~ (s = EMPTY)) /\ ((~ (t = EMPTY)) /\ ((DISJOINT ((hull affine) s)) ((hull affine) t)))) ==> (?a. (?b. (?c. ((~ (a = (vec (NUMERAL _0)))) /\ (((real_lt b) c) /\ ((!x. ((x IN s) ==> (((dot a) x) = b))) /\ (!x. ((x IN t) ==> (((dot a) x) = c))))))))))))
EVENTUALLY_FALSE |- (!net. (((eventually (\x. F)) net) = (trivial_limit net)))
PATH_COMPONENT_EMPTY |- (!x. (((path_component EMPTY) x) = EMPTY))
INFSUM_EQ |- (!f. (!g. (!k. ((((summable k) f) /\ (((summable k) g) /\ (!x. ((x IN k) ==> ((f x) = (g x)))))) ==> (((infsum k) f) = ((infsum k) g))))))
NPRODUCT_CONST_NUMSEG |- (!c. (!m. (!n. (((nproduct (m .. n)) (\x. c)) = (c EXP ((n + (NUMERAL (BIT1 _0))) - m))))))
MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL_EQ |- (!f. (!s. ((((measurable_on f) s) /\ (lebesgue_measurable s)) = (!t. ((borel t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3662. (?x. (((SETSPEC GEN%PVAR%3662) ((x IN s) /\ ((f x) IN t))) x))))))))))
REAL_INTEGRABLE_REFLECT_GEN |- (!f. (!s. (((real_integrable_on (\x. (f (real_neg x)))) s) = ((real_integrable_on f) ((IMAGE real_neg) s)))))
REALLIM_1_OVER_POW |- (!k. (((NUMERAL (BIT1 _0)) <= k) ==> (((---> (\n. (real_inv ((real_pow (real_of_num n)) k)))) (real_of_num (NUMERAL _0))) sequentially)))
COMPLEX_STONE_WEIERSTRASS |- (!P. (!s. (((compact s) /\ ((!f. ((P f) ==> ((continuous_on f) s))) /\ ((!c. (P (\x. c))) /\ ((!f. ((P f) ==> (P (\x. (cnj (f x)))))) /\ ((!f. (!g. (((P f) /\ (P g)) ==> (P (\x. ((vector_add (f x)) (g x))))))) /\ ((!f. (!g. (((P f) /\ (P g)) ==> (P (\x. ((complex_mul (f x)) (g x))))))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> (?f. ((P f) /\ (~ ((f x) = (f y)))))))))))))) ==> (!f. (!e. ((((continuous_on f) s) /\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?g. ((P g) /\ (!x. ((x IN s) ==> ((real_lt (vector_norm ((vector_sub (f x)) (g x)))) e)))))))))))
Nonlinear_lemma.lmdih5_0' |- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y5) ==> (((((((lmdih5_x_div_sqrtdelta_posbranch ((real_mul y1) y1)) ((real_mul y2) y2)) ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5)) ((real_mul y6) y6)) = (real_of_num (NUMERAL _0))))))))))
WF_EREC |- ((WF <<) ==> (!H. ((!f. (!g. (!x. ((!z. (((<< z) x) ==> ((f z) = (g z)))) ==> (((H f) x) = ((H g) x)))))) ==> (?!f. (!x. ((f x) = ((H f) x)))))))
Functional_equation.functional_asn797k |- (asn797k = ((mul6 proj_x1) (uni (asn , ((mul6 (constant6 cos797)) (uni (sin , ((div6 (constant6 pi)) proj_x1))))))))
ITSET_EQ |- (!s. (!f. (!g. (!b. (((FINITE s) /\ ((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((!x. (!y. (!s. ((~ (x = y)) ==> (((f x) ((f y) s)) = ((f y) ((f x) s))))))) /\ (!x. (!y. (!s. ((~ (x = y)) ==> (((g x) ((g y) s)) = ((g y) ((g x) s)))))))))) ==> ((((ITSET f) s) b) = (((ITSET g) s) b)))))))
CLOSED_SIMPLE_PATH_IMAGE |- (!g. ((simple_path g) ==> (closed (path_image g))))
RSTC_RULES |- (!R. ((!x. (!y. (((R x) y) ==> (((RSTC R) x) y)))) /\ ((!x. (((RSTC R) x) x)) /\ ((!x. (!y. ((((RSTC R) x) y) ==> (((RSTC R) y) x)))) /\ (!x. (!y. (!z. (((((RSTC R) x) y) /\ (((RSTC R) y) z)) ==> (((RSTC R) x) z)))))))))
Lp_ineqs_proofs.rho_fan_eq_rho |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((rho_fan v) = (rho (vector_norm v)))))))))))
MEASURE_DISJOINT_UNIONS |- (!m. (!f. (((FINITE f) /\ ((!s. ((s IN f) ==> ((has_measure s) (m s)))) /\ (!s. (!t. (((s IN f) /\ ((t IN f) /\ (~ (s = t)))) ==> ((DISJOINT s) t)))))) ==> ((measure (UNIONS f)) = ((sum f) m)))))
CONVEX_HULL_INTERIOR_SUBSET |- (!s. (((hull convex) (interior s)) SUBSET (interior ((hull convex) s))))
Hypermap_iso.fan_hypermaps_iso2 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((iso (hypermap_of_fan (V , E))) (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))))))
LOG_DIV |- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((log ((real_div x) y)) = ((real_sub (log x)) (log y))))))
ROOT_MONO_LT_EQ |- (!n. (!x. (!y. ((~ (n = (NUMERAL _0))) ==> (((real_lt ((root n) x)) ((root n) y)) = ((real_lt x) y))))))
CONTINUOUS_FROM_CLOSED_GRAPH |- (!f. (!s. (!t. (((compact t) /\ ((((IMAGE f) s) SUBSET t) /\ (closed (GSPEC (\GEN%PVAR%967. (?x. (((SETSPEC GEN%PVAR%967) (x IN s)) ((pastecart x) (f x))))))))) ==> ((continuous_on f) s)))))
FINITE_CBALL |- (!a. (!r. ((FINITE (cball (a , r))) = ((real_le r) (real_of_num (NUMERAL _0))))))
Polar_fan.AZIM_CYCLE_SING |- (!x. (!u. (!v. (((((azim_cycle (v INSERT EMPTY)) x) u) v) = v))))
CONTINUOUS_WITHIN_CNJ |- (!s. (!z. ((continuous cnj) ((within (at z)) s))))
ODD_MOD |- (!n. ((ODD n) = ((n MOD (NUMERAL (BIT0 (BIT1 _0)))) = (NUMERAL (BIT1 _0)))))
LIPSCHITZ_IMP_RECTIFIABLE_PATH |- (!g. (!b. ((!x. (!y. (((x IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ (y IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))) ==> ((real_le (vector_norm ((vector_sub (g x)) (g y)))) ((real_mul b) (vector_norm ((vector_sub x) y))))))) ==> (rectifiable_path g))))
INT_MAX_LE |- (!x. (!y. (!z. (((int_le ((int_max x) y)) z) = (((int_le x) z) /\ ((int_le y) z))))))
MATRIX_ADD_LNEG |- (!A. (((matrix_add (matrix_neg A)) A) = (mat (NUMERAL _0))))
MEASURE_LINEAR_IMAGE_SAME |- (!f. (!s. (((linear f) /\ ((measurable s) /\ ((real_abs (det (matrix f))) = (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((measure ((IMAGE f) s)) = (measure s)))))
UNIFORMLY_CONTINUOUS_ON_MUL |- (!f. (!g. (!s. ((((uniformly_continuous_on (lift o f)) s) /\ (((uniformly_continuous_on g) s) /\ ((bounded ((IMAGE (lift o f)) s)) /\ (bounded ((IMAGE g) s))))) ==> ((uniformly_continuous_on (\x. ((% (f x)) (g x)))) s)))))
Tame_list.mapAt_ALT |- (((((mapAt NIL) f) xs) = xs) /\ ((((mapAt ((CONS n) ns)) f) xs) = (((COND (n < (LENGTH xs))) (((mapAt ns) f) (((list_update xs) n) (f ((#!# xs) n))))) (((mapAt ns) f) xs))))
COUNTABLE_OPEN_CONNECTED_COMPONENTS |- (!s. (!t. ((open s) ==> (COUNTABLE (GSPEC (\GEN%PVAR%1917. (?x. (((SETSPEC GEN%PVAR%1917) (x IN t)) ((connected_component s) x)))))))))
WOSET_INSEG_ORDINAL |- (woset (GABS (\f. (!x. (!y. ((GEQ (f (x , y))) ((ordinal x) /\ ((ordinal y) /\ ((inseg x) y)))))))))
IMAGE_STRETCH_REAL_INTERVAL |- (!a. (!b. (!m. (((IMAGE (\x. ((real_mul m) x))) (closed_real_interval ((CONS (a , b)) NIL))) = (((COND ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) EMPTY) (((COND ((real_le (real_of_num (NUMERAL _0))) m)) (closed_real_interval ((CONS (((real_mul m) a) , ((real_mul m) b))) NIL))) (closed_real_interval ((CONS (((real_mul m) b) , ((real_mul m) a))) NIL))))))))
Merge_ineq.delta_pos_4_784 |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x1) /\ (((real_le x1) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x2) /\ (((real_le x2) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x3) /\ (((real_le x3) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x4) /\ (((real_le x4) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x5) /\ (((real_le x5) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x6) /\ ((real_le x6) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))))) ==> (((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))))))))))))))
INT_POW_EQ_0 |- (!x. (!n. ((((int_pow x) n) = (int_of_num (NUMERAL _0))) = ((x = (int_of_num (NUMERAL _0))) /\ (~ (n = (NUMERAL _0)))))))
FACE_OF_HALFSPACE_GE |- (!f. (!a. (!b. (((face_of f) (GSPEC (\GEN%PVAR%2156. (?x. (((SETSPEC GEN%PVAR%2156) ((real_ge ((dot a) x)) b)) x))))) = ((f = EMPTY) \/ ((f = (GSPEC (\GEN%PVAR%2157. (?x. (((SETSPEC GEN%PVAR%2157) (((dot a) x) = b)) x))))) \/ (f = (GSPEC (\GEN%PVAR%2158. (?x. (((SETSPEC GEN%PVAR%2158) ((real_ge ((dot a) x)) b)) x)))))))))))
MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_1D |- (!f. (!s. ((((measurable_on (lift o f)) s) /\ ((convex s) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (f (midpoint (x , y)))) ((real_div ((real_add (f x)) (f y))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))) ==> ((convex_on f) s))))
CONTINUOUS_ON_LIFT_PRODUCT |- (!f. (!s. (!t. (((FINITE t) /\ (!i. ((i IN t) ==> ((continuous_on (\x. (lift ((f x) i)))) s)))) ==> ((continuous_on (\x. (lift ((product t) (f x))))) s)))))
CONNECTED_NEST_GEN |- (!s. (((!n. ((closed (s n)) /\ (connected (s n)))) /\ ((?n. (compact (s n))) /\ (!m. (!n. ((m <= n) ==> ((s n) SUBSET (s m))))))) ==> (connected (INTERS (GSPEC (\GEN%PVAR%1155. (?n. (((SETSPEC GEN%PVAR%1155) (n IN UNIV)) (s n)))))))))
REALLIM_ATREAL_ID |- (((---> (\x. x)) a) (atreal a))
ABSOLUTELY_CONTINUOUS_ISOMETRIC_COMPOSE |- (!f. (!g. (!s. ((!x. (!y. ((distance ((f x) , (f y))) = (distance (x , y))))) ==> (((absolutely_continuous_on (f o g)) s) = ((absolutely_continuous_on g) s))))))
MEASURABLE_SCALING_EQ |- (!s. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> ((measurable ((IMAGE (\x. ((% c) x))) s)) = (measurable s)))))
NEGATIONS_CBALL |- (!r. (((IMAGE vector_neg) (cball ((vec (NUMERAL _0)) , r))) = (cball ((vec (NUMERAL _0)) , r))))
TAN_SUB |- (!x. (!y. (((~ ((cos x) = (real_of_num (NUMERAL _0)))) /\ ((~ ((cos y) = (real_of_num (NUMERAL _0)))) /\ (~ ((cos ((real_sub x) y)) = (real_of_num (NUMERAL _0)))))) ==> ((tan ((real_sub x) y)) = ((real_div ((real_sub (tan x)) (tan y))) ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul (tan x)) (tan y))))))))
MEASURABLE_TETRAHEDRON |- (!a. (!b. (!c. (!d. (measurable ((hull convex) (a INSERT (b INSERT (c INSERT (d INSERT EMPTY))))))))))
SUBSET_CONVEX_HULL_FRONTIER |- (!s. ((bounded s) ==> (s SUBSET ((hull convex) (frontier s)))))
RTC_NE_IMP_TC |- (!R. (!x. (!y. (((((RTC R) x) y) /\ (~ (x = y))) ==> (((TC R) x) y)))))
NPRODUCT_CLOSED |- (!P. (!f. (!s. (((P (NUMERAL (BIT1 _0))) /\ ((!x. (!y. (((P x) /\ (P y)) ==> (P (x * y))))) /\ (!a. ((a IN s) ==> (P (f a)))))) ==> (P ((nproduct s) f))))))
CX_COSH |- ((Cx ((real_div ((real_add (exp x)) (real_inv (exp x)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (ccos ((complex_mul ii) (Cx x))))
ITERATE_DIFF_GEN |- (!op. ((monoidal op) ==> (!f. (!s. (!t. (((FINITE (((support op) f) s)) /\ ((((support op) f) t) SUBSET (((support op) f) s))) ==> (((op (((iterate op) (s DIFF t)) f)) (((iterate op) t) f)) = (((iterate op) s) f))))))))
POSITIVE_DEFINITE_2 |- (!A. ((positive_definite A) = ((((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0))) = ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_lt (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) /\ (((real_lt (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\ ((real_lt ((real_pow ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0)))))))))))
Misc_defs_and_lemmas.UNIONS_SUBSET |- (!U. (!X. ((!A. ((A IN U) ==> (A SUBSET X))) ==> ((UNIONS U) SUBSET X))))
CARD_LT_REFL |- (!s. (~ (s <_c s)))
REAL_INTEGRAL_SUBSTITUTION_SIMPLE |- (!f. (!g. (!g'. (!a. (!b. (!c. (!d. ((((real_continuous_on f) (closed_real_interval ((CONS (c , d)) NIL))) /\ ((!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> (((has_real_derivative g) (g' x)) ((within (atreal x)) (closed_real_interval ((CONS (a , b)) NIL)))))) /\ ((((IMAGE g) (closed_real_interval ((CONS (a , b)) NIL))) SUBSET (closed_real_interval ((CONS (c , d)) NIL))) /\ (((real_le a) b) /\ (((real_le c) d) /\ ((real_le (g a)) (g b))))))) ==> (((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\x. ((real_mul (f (g x))) (g' x)))) = ((real_integral (closed_real_interval ((CONS ((g a) , (g b))) NIL))) f))))))))))