hcju/holps huggingface dataset:
Lp_ineqs_proofs.tau_sum3 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((CARD f) = (NUMERAL (BIT1 (BIT1 _0)))) ==> (((sum f) (rhazim_fan (V , E))) = ((real_add ((tau_fan (V , E)) f)) ((real_add pi) sol0)))))))))))))
GDELTA_UNIONS |- (!g. (((FINITE g) /\ (!u. ((u IN g) ==> (gdelta u)))) ==> (gdelta (UNIONS g))))
HOMEOMORPHISM_IMP_QUOTIENT_MAP |- (!f. (!g. (!s. (!t. (((homeomorphism (s , t)) (f , g)) ==> (!u. ((u SUBSET t) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%1030. (?x. (((SETSPEC GEN%PVAR%1030) ((x IN s) /\ ((f x) IN u))) x))))) = ((open_in ((subtopology euclidean) t)) u)))))))))
Dih2k_hypermap.VECMAT_VEC |- (!i. (!n. ((i < (dimindex UNIV)) ==> (((vecmat i) (vec n)) = (vec n)))))
Vasyyau.A_EQ2h0_IMP_B |- (((!i. ((((((scs_a_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((scs_b_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) \/ (((((scs_a_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ ((((scs_b_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = cstab)))) /\ ((((scs_a_v39 s) p) (p + (NUMERAL (BIT1 _0)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ==> ((((scs_b_v39 s) p) (p + (NUMERAL (BIT1 _0)))) = cstab))
SIN_VECTOR_ANGLE_RMUL |- (!a. (!x. (!y. ((sin ((vector_angle x) ((% a) y))) = (((COND (a = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL (BIT1 _0)))) (sin ((vector_angle x) y)))))))
CONIC_HULLS_EQ_IMP_SPANS_EQ |- (!s. (!t. ((((hull conic) s) = ((hull conic) t)) ==> ((span s) = (span t)))))
COUNTABLE_ISOLATED_SET |- (!s. ((!a. ((a IN s) ==> (~ ((limit_point_of a) s)))) ==> (COUNTABLE s)))
UNIFORM_LIM_COMPLEX_DIV |- (!net. (!P. (!f. (!g. (!l. (!m. (!b1. (!b2. ((((eventually (\x. (!n. ((P n) ==> ((real_le (vector_norm (l n))) b1))))) net) /\ (((real_lt (real_of_num (NUMERAL _0))) b2) /\ (((eventually (\x. (!n. ((P n) ==> ((real_le b2) (vector_norm (m n))))))) net) /\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((f n) x)) (l n)))) e))))) net))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((g n) x)) (m n)))) e))))) net))))))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((complex_div ((f n) x)) ((g n) x))) ((complex_div (l n)) (m n))))) e))))) net))))))))))))
Functional_equation.functional_dih2_x |- (dih2_x = (rotate2 dih_x))
MATRIX_CMUL_RZERO |- (!c. (((%% c) (mat (NUMERAL _0))) = (mat (NUMERAL _0))))
Tactics_jordan.switch_lemma_lt |- (!A. (!B. (!C. ((((real_lt (real_of_num (NUMERAL _0))) A) ==> (((real_lt (real_of_num (NUMERAL _0))) B) ==> C)) = (((real_lt (real_of_num (NUMERAL _0))) B) ==> (((real_lt (real_of_num (NUMERAL _0))) A) ==> C))))))
INT_LT_ADDNEG |- (!x. (!y. (!z. (((int_lt y) ((int_add x) (int_neg z))) = ((int_lt ((int_add y) z)) x)))))
INT_EQ_MUL_RCANCEL |- (!x. (!y. (!z. ((((int_mul x) z) = ((int_mul y) z)) = ((x = y) \/ (z = (int_of_num (NUMERAL _0))))))))
CARD_EQ_OPEN_IN_AFFINE |- (!u. (!s. (((affine u) /\ ((~ ((aff_dim u) = (int_of_num (NUMERAL _0)))) /\ (((open_in ((subtopology euclidean) u)) s) /\ (~ (s = EMPTY))))) ==> (s =_c UNIV))))
More_list_hypermap.contravening_dist_not_dart |- (!V. (!h. (!L. (!g. ((contravening V) ==> ((good_list L) ==> (((hyp_iso g) ((hypermap_of_list L) , (hypermap_of_fan (V , (ESTD V))))) ==> ((((BIJ h) (elements_of_list L)) V) ==> ((g = (\d. ((h (FST d)) , (h (SND d))))) ==> (!s. ((((ALL (\d. (((MEM (FST d)) (list_of_elements L)) /\ (((MEM (SND d)) (list_of_elements L)) /\ (((FST d) = (SND d)) = F))))) s) /\ ((ALL (\d. (((MEM d) (list_of_darts L)) = F))) s)) ==> ((ALL (\d. ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (distance (g d))))) s))))))))))))
RETRACTION_ARC |- (!p. ((arc p) ==> (?f. (((continuous_on f) UNIV) /\ ((((IMAGE f) UNIV) SUBSET (path_image p)) /\ (!x. ((x IN (path_image p)) ==> ((f x) = x))))))))
INVERTIBLE_MAT |- (!a. ((invertible (mat a)) = (~ (a = (NUMERAL _0)))))
OPEN_MAP_CLOSED_SUPERSET_PREIMAGE_EQ |- (!f. (!s. (!t. ((((IMAGE f) s) SUBSET t) ==> ((!k. (((open_in ((subtopology euclidean) s)) k) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) k)))) = (!u. (!w. ((((closed_in ((subtopology euclidean) s)) u) /\ ((w SUBSET t) /\ ((GSPEC (\GEN%PVAR%699. (?x. (((SETSPEC GEN%PVAR%699) ((x IN s) /\ ((f x) IN w))) x)))) SUBSET u))) ==> (?v. (((closed_in ((subtopology euclidean) t)) v) /\ ((w SUBSET v) /\ ((GSPEC (\GEN%PVAR%700. (?x. (((SETSPEC GEN%PVAR%700) ((x IN s) /\ ((f x) IN v))) x)))) SUBSET u))))))))))))
TC_SC |- (!R. (!x. (!y. ((((SC (TC R)) x) y) ==> (((TC (SC R)) x) y)))))
Optimize.EXPAND_1var_lemma |- (!f. (!a. (!b. ((!y. ((ineq ((CONS (a , (y , b))) NIL)) (f y))) = (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((ineq ((CONS (a , (x1 , b))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x2 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x3 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x4 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x5 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x6 , (real_of_num (NUMERAL (BIT1 _0)))))) NIL))))))) (f x1))))))))))))
INVERTIBLE_EIGENVALUES |- (!A. ((invertible A) = (!c. (!v. (((((matrix_vector_mul A) v) = ((% c) v)) /\ (~ (v = (vec (NUMERAL _0))))) ==> (~ (c = (real_of_num (NUMERAL _0)))))))))
AR_IMP_ABSOLUTE_RETRACT_UNIV |- (!s. (!s'. (((AR s) /\ (((homeomorphic s) s') /\ (closed s'))) ==> ((retract_of s') UNIV))))
INT_POW_NZ |- (!x. (!n. ((~ (x = (int_of_num (NUMERAL _0)))) ==> (~ (((int_pow x) n) = (int_of_num (NUMERAL _0)))))))
COLUMN_0 |- (!i. (((column i) (mat (NUMERAL _0))) = (vec (NUMERAL _0))))
SKOLEM_THM_GEN |- (!P. (!s. ((!x. ((P x) ==> (?y. ((R x) y)))) = (?f. (!x. ((P x) ==> ((R x) (f x))))))))
CONTINUOUS_AT_CX_RE |- (!z. ((continuous (Cx o Re)) (at z)))
Local_lemmas.AFF_GE_EQ |- ((((real_lt (real_of_num (NUMERAL _0))) t) /\ (((vector_sub a) x) = ((% t) ((vector_sub b) x)))) ==> (((aff_ge (x INSERT EMPTY)) (a INSERT EMPTY)) = ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))))
COMPLEX_POW_EQ_1 |- (!z. (!n. ((((complex_pow z) n) = (Cx (real_of_num (NUMERAL (BIT1 _0))))) ==> (((vector_norm z) = (real_of_num (NUMERAL (BIT1 _0)))) \/ (n = (NUMERAL _0))))))
INT_LTE_ADD2 |- (!w. (!x. (!y. (!z. ((((int_lt w) x) /\ ((int_le y) z)) ==> ((int_lt ((int_add w) y)) ((int_add x) z)))))))
INT_ABS_BETWEEN1 |- (!x. (!y. (!z. ((((int_lt x) z) /\ ((int_lt (int_abs ((int_sub y) x))) ((int_sub z) x))) ==> ((int_lt y) z)))))
INTEGRABLE_COMPLEX_RMUL_EQ |- (!f. (!s. (!c. (((integrable_on (\x. ((complex_mul (f x)) c))) s) = ((c = (Cx (real_of_num (NUMERAL _0)))) \/ ((integrable_on f) s))))))
CNJ_CPRODUCT |- (!f. (!s. ((FINITE s) ==> ((cnj ((cproduct s) f)) = ((cproduct s) (\i. (cnj (f i))))))))
REAL_CONTINUOUS_WITHIN_ACS |- (!s. (!x. (((real_lt (real_abs x)) (real_of_num (NUMERAL (BIT1 _0)))) ==> ((real_continuous acs) ((within (atreal x)) s)))))
REALLIM_1_OVER_N_OFFSET |- (!a. (((---> (\n. (real_inv ((real_add (real_of_num n)) a)))) (real_of_num (NUMERAL _0))) sequentially))
BETWEEN_NORM_LE |- (!r. (!a. (!b. (!x. ((((real_le (vector_norm a)) r) /\ (((real_le (vector_norm b)) r) /\ ((between x) (a , b)))) ==> ((real_le (vector_norm x)) r))))))
CONIC_CAP_DEGENERATE |- (!v0. (!r. (!a. (((((conic_cap v0) v0) r) a) = EMPTY))))
HOLOMORPHIC_ON_EMPTY |- (!f. ((holomorphic_on f) EMPTY))
REAL_COMPLEX_INTEGRAL |- (!f. (!s. ((((integrable_on f) s) /\ (!x. ((x IN s) ==> (real (f x))))) ==> (real ((integral s) f)))))
CONVEX_HULL_SUBSET_SPAN |- (!s. (((hull convex) s) SUBSET (span s)))
PSUBSET_SUBSET_TRANS |- (!s. (!t. (!u. (((s PSUBSET t) /\ (t SUBSET u)) ==> (s PSUBSET u)))))
Geomdetail.CONV_EM |- ((conv EMPTY) = EMPTY)
INT_ABS_BETWEEN2 |- (!x0. (!x. (!y0. (!y. ((((int_lt x0) y0) /\ (((int_lt ((int_mul (int_of_num (NUMERAL (BIT0 (BIT1 _0))))) (int_abs ((int_sub x) x0)))) ((int_sub y0) x0)) /\ ((int_lt ((int_mul (int_of_num (NUMERAL (BIT0 (BIT1 _0))))) (int_abs ((int_sub y) y0)))) ((int_sub y0) x0)))) ==> ((int_lt x) y))))))
Pqcsxwg.PQCSXWG2 |- (!v0. (!v1. (!v2. (!v3. (!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) /\ ((~ (collinear (v0 INSERT (v1 INSERT (v2 INSERT EMPTY))))) /\ ((x1 = ((real_pow (distance (v1 , v0))) (NUMERAL (BIT0 (BIT1 _0))))) /\ ((x2 = ((real_pow (distance (v2 , v0))) (NUMERAL (BIT0 (BIT1 _0))))) /\ ((x6 = ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\ (v3 = (((((((((mk_simplex1 v0) v1) v2) x1) x2) x3) x4) x5) x6))))))))))))) ==> ((continuous (\q. (((((((((mk_simplex1 v0) v1) v2) x1) x2) x3) x4) q) x6))) (atreal x5)))))))))))))
COMPLEX_DIFFERENTIABLE_WITHIN_CASN |- (!s. (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> ((complex_differentiable casn) ((within (at z)) s)))))
SEPARATING_HYPERPLANE_POLYHEDRA |- (!s. (!t. (((polyhedron s) /\ ((polyhedron t) /\ ((~ (s = EMPTY)) /\ ((~ (t = EMPTY)) /\ ((DISJOINT s) t))))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ ((!x. ((x IN s) ==> ((real_lt ((dot a) x)) b))) /\ (!x. ((x IN t) ==> ((real_gt ((dot a) x)) b))))))))))
ABSOLUTELY_SETCONTINUOUS_ON_LIFT_ABS |- (!f. (!s. (((absolutely_setcontinuous_on (\x. (lift (real_abs (f x))))) s) = ((absolutely_setcontinuous_on (\x. (lift (f x)))) s))))
INT_SGN_POW_2 |- (!x. ((int_sgn ((int_pow x) (NUMERAL (BIT0 (BIT1 _0))))) = (int_sgn (int_abs x))))
BOUNDED_EQ_TOTALLY_BOUNDED |- (!s. ((bounded s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?k. ((FINITE k) /\ ((k SUBSET s) /\ (s SUBSET (UNIONS ((IMAGE (\x. (ball (x , e)))) k))))))))))
BOUNDED_SUMS_IMAGES |- (!f. (!t. (!s. (((FINITE s) /\ (!a. ((a IN s) ==> (bounded (GSPEC (\GEN%PVAR%591. (?x. (((SETSPEC GEN%PVAR%591) (x IN t)) ((f x) a))))))))) ==> (bounded (GSPEC (\GEN%PVAR%592. (?x. (((SETSPEC GEN%PVAR%592) (x IN t)) ((vsum s) (f x)))))))))))
ANR_SIMPLICIAL_COMPLEX |- (!c. ((simplicial_complex c) ==> (ANR (UNIONS c))))
DIM_OPEN_IN |- (!s. (!t. (((~ (s = EMPTY)) /\ (((open_in ((subtopology euclidean) t)) s) /\ (subspace t))) ==> ((dim s) = (dim t)))))
ABSOLUTELY_INTEGRABLE_CONVOLUTION_LINF_L1 |- (!bop. (!f. (!g. (!x. (((bilinear bop) /\ (((measurable_on f) UNIV) /\ ((bounded ((IMAGE f) UNIV)) /\ ((absolutely_integrable_on g) UNIV)))) ==> ((absolutely_integrable_on (\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))
REAL_MEASURABLE_ON_MEASURABLE_SUBSET |- (!f. (!s. (!t. (((s SUBSET t) /\ (((real_measurable_on f) t) /\ (real_measurable s))) ==> ((real_measurable_on f) s)))))
OPEN_IN_CLOSED_IN |- (!s. ((s SUBSET (topspace top)) ==> (((open_in top) s) = ((closed_in top) ((topspace top) DIFF s)))))
REAL_BEPPO_LEVI_INCREASING |- (!f. (!s. (((!k. ((real_integrable_on (f k)) s)) /\ ((!k. (!x. ((x IN s) ==> ((real_le ((f k) x)) ((f (SUC k)) x))))) /\ (real_bounded (GSPEC (\GEN%PVAR%4926. (?k. (((SETSPEC GEN%PVAR%4926) (k IN UNIV)) ((real_integral s) (f k))))))))) ==> (?g. (?k. ((real_negligible k) /\ (!x. ((x IN (s DIFF k)) ==> (((---> (\k. ((f k) x))) (g x)) sequentially)))))))))
INTEGRAL_INTERVALS_DIFF_INCLUSION_EXCLUSION |- (!f. (!a. (!b. (!c. (!d. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((c IN (closed_interval ((CONS (a , b)) NIL))) /\ (d IN (closed_interval ((CONS (a , b)) NIL))))) ==> (((vector_sub ((integral (closed_interval ((CONS (a , d)) NIL))) f)) ((integral (closed_interval ((CONS (a , c)) NIL))) f)) = ((vsum (GSPEC (\GEN%PVAR%2930. (?s. (((SETSPEC GEN%PVAR%2930) ((~ (s = EMPTY)) /\ (s SUBSET ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))))) s))))) (\s. ((% ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) (CARD (GSPEC (\GEN%PVAR%2931. (?i. (((SETSPEC GEN%PVAR%2931) ((i IN s) /\ ((real_lt (d $ i)) (c $ i)))) i))))))) ((integral (closed_interval ((CONS ((lambdai. (((COND (i IN s)) ((real_min (c $ i)) (d $ i))) (a $ i))) , (lambdai. (((COND (i IN s)) ((real_max (c $ i)) (d $ i))) (c $ i))))) NIL))) f)))))))))))
Cuxvzoz.vv_split_azim_alt |- (!vv. (!k. (!i. (!j. ((LET (\V. (LET_END ((LET (\E. (LET_END ((LET (\f. (LET_END ((((periodic vv) k) /\ (((NUMERAL (BIT1 (BIT1 _0))) <= k) /\ ((~ (collinear ((vec (NUMERAL _0)) INSERT ((vv i) INSERT ((vv j) INSERT EMPTY))))) /\ ((~ (collinear ((vec (NUMERAL _0)) INSERT ((vv i) INSERT ((vv (i + (k - (NUMERAL (BIT1 _0))))) INSERT EMPTY))))) /\ ((~ (collinear ((vec (NUMERAL _0)) INSERT ((vv i) INSERT ((vv (SUC i)) INSERT EMPTY))))) /\ ((convex_local_fan (V , (E , f))) /\ (!i. (!j. (((i < k) /\ ((j < k) /\ ((vv i) = (vv j)))) ==> (i = j)))))))))) ==> (((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv (i + (k - (NUMERAL (BIT1 _0)))))) = ((real_add ((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv j))) ((((azim (vec (NUMERAL _0))) (vv i)) (vv j)) (vv (i + (k - (NUMERAL (BIT1 _0)))))))))))) ((IMAGE (\i. ((vv i) , (vv (SUC i))))) UNIV))))) ((IMAGE (\i. ((vv i) INSERT ((vv (SUC i)) INSERT EMPTY)))) UNIV))))) ((IMAGE vv) UNIV))))))
COMPLEX_IN_SPHERE_0 |- (!v. (!r. ((v IN (sphere ((Cx (real_of_num (NUMERAL _0))) , r))) = ((vector_norm v) = r))))
Ssrnat.minnAC |- (!m1. (!m2. (!m3. (((minn ((minn m1) m2)) m3) = ((minn ((minn m1) m3)) m2)))))
Lp_ineqs_proofs.dart3_azim3_fan_eq_dih3_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)))) ==> (((azim3_fan (V , E)) d) = ((((((dih3_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)))))))))))
LOCALLY_CONNECTED_SPHERE |- (!a. (!r. ((locally connected) (sphere (a , r)))))
Merge_ineq.nonf_mud_234_x_v1 |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a. (!b. (!c. (((real_le (real_of_num (NUMERAL _0))) a) ==> ((((((((((mud_234_x_v1 a) b) c) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((mu6_x ((real_mul a) a)) x2) x3) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0)))) (sqrt (((((((((delta_234_x ((real_mul a) a)) ((real_mul b) b)) ((real_mul c) c)) x1) x2) x3) x4) x5) x6))))))))))))))
ANTIDERIVATIVE_INTEGRAL_CONTINUOUS |- (!f. (!a. (!b. (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) ==> (?g. (!u. (!v. (((u IN (closed_interval ((CONS (a , b)) NIL))) /\ ((v IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le (drop u)) (drop v)))) ==> (((has_integral f) ((vector_sub (g v)) (g u))) (closed_interval ((CONS (u , v)) NIL)))))))))))
INT_LT_NEGTOTAL |- (!x. ((x = (int_of_num (NUMERAL _0))) \/ (((int_lt (int_of_num (NUMERAL _0))) x) \/ ((int_lt (int_of_num (NUMERAL _0))) (int_neg x)))))
TAN_PERIODIC_NPI |- (!x. (!n. ((tan ((real_add x) ((real_mul (real_of_num n)) pi))) = (tan x))))
CONVEX_ON_IS_INDEFINITE_INTEGRAL |- (!f. (!s. ((((convex_on (drop o f)) s) /\ ((is_interval s) /\ (open s))) ==> (?f'. ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f' x))) (drop (f' y)))))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_le (drop x)) (drop y)))) ==> (((has_integral f') ((vector_sub (f y)) (f x))) (closed_interval ((CONS (x , y)) NIL)))))))))))
SERIES_DIRICHLET_COMPLEX_EXPLICIT |- (!f. (!g. (!p. (!q. ((((NUMERAL (BIT1 _0)) <= p) /\ ((bounded (GSPEC (\GEN%PVAR%4783. (?n. (((SETSPEC GEN%PVAR%4783) (n IN UNIV)) ((vsum (q .. n)) f)))))) /\ ((!n. ((p <= n) ==> ((real (g n)) /\ ((real_le (real_of_num (NUMERAL _0))) (Re (g n)))))) /\ (!n. ((p <= n) ==> ((real_le (Re (g (n + (NUMERAL (BIT1 _0)))))) (Re (g n)))))))) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!m. (!n. ((p <= m) ==> ((real_le (vector_norm ((vsum (m .. n)) (\k. ((complex_mul (f k)) (g k)))))) ((real_mul B) (vector_norm (g m))))))))))))))
REAL_UNIFORMLY_CONTINUOUS_ON_ID |- (!s. ((real_uniformly_continuous_on (\x. x)) s))
AFF_LE_INJECTIVE_LINEAR_IMAGE |- (!f. (!s. (!t. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((aff_le ((IMAGE f) s)) ((IMAGE f) t)) = ((IMAGE f) ((aff_le s) t)))))))
Planarity.remark012_fan |- (!x. (!V. (!E. (!u. (!w. (!a. (((FAN (x , (V , E))) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ (((real_lt (real_of_num (NUMERAL _0))) a) /\ ((real_lt a) (real_of_num (NUMERAL (BIT1 _0))))))) ==> (~ (((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) a)) u)) ((% a) w)) IN V)))))))))
Tame_list.indexf_mem1 |- (!x. (!y. (!s. ((~ ((MEM x) s)) ==> ((((indexf x) y) s) = (sizel s))))))
FIXED_POINT_INESSENTIAL_SPHERE_MAP |- (!f. (!a. (!r. (!c. ((((real_lt (real_of_num (NUMERAL _0))) r) /\ ((((homotopic_with (\x. T)) ((sphere (a , r)) , (sphere (a , r)))) f) (\x. c))) ==> (?x. ((x IN (sphere (a , r))) /\ ((f x) = x))))))))
Lp_ineqs_proofs.y1_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))) ==> ((y1_fan d) = (yn_fan (FST d)))))))))))
POLYTOPE_LINEAR_IMAGE_EQ |- (!f. (!s. (((linear f) /\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\ (!y. (?x. ((f x) = y))))) ==> ((polytope ((IMAGE f) s)) = (polytope s)))))
REAL_OPEN_DIFF |- (!s. (!t. (((real_open s) /\ (real_closed t)) ==> (real_open (s DIFF t)))))
Ssrnat.geqE |- (!m. (!n. ((m >= n) = (n <= m))))
TC_RTC_CASES_L |- ((((TC R) x) z) = (?y. ((((RTC R) x) y) /\ ((R y) z))))
HAS_BOUNDED_REAL_VARIATION_ON_COMBINE |- (!f. (!a. (!b. (!c. ((((real_le a) c) /\ ((real_le c) b)) ==> (((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , b)) NIL))) = (((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , c)) NIL))) /\ ((has_bounded_real_variation_on f) (closed_real_interval ((CONS (c , b)) NIL))))))))))
POSITIVE_SEMIDEFINITE_ZERO_COLUMN |- (!A. (!i. (((positive_semidefinite A) /\ (((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((A $ i) $ i) = (real_of_num (NUMERAL _0)))))) ==> (((column i) A) = (vec (NUMERAL _0))))))
UNCOUNTABLE_HAS_CONDENSATION_POINT |- (!s. ((~ (COUNTABLE s)) ==> (?x. ((condensation_point_of x) s))))
RANK_NEG |- (!A. ((rank (matrix_neg A)) = (rank A)))
LOCALLY_COMPACT_CLOSED_INTER_OPEN |- (!s. (((locally compact) s) = (?t. (?u. ((closed t) /\ ((open u) /\ (s = (t INTER u))))))))
CLOSED_MAP_FROM_COMPOSITION_SURJECTIVE |- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\ ((((IMAGE f) s) = t) /\ ((((IMAGE g) t) SUBSET u) /\ (!k. (((closed_in ((subtopology euclidean) s)) k) ==> ((closed_in ((subtopology euclidean) u)) ((IMAGE (g o f)) k))))))) ==> (!k. (((closed_in ((subtopology euclidean) t)) k) ==> ((closed_in ((subtopology euclidean) u)) ((IMAGE g) k))))))))))
HAS_REAL_DERIVATIVE_SERIES |- (!s. (!f. (!f'. (!g'. (!k. (((is_realinterval s) /\ ((!n. (!x. ((x IN s) ==> (((has_real_derivative (f n)) ((f' n) x)) ((within (atreal x)) s))))) /\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\ (x IN s)) ==> ((real_le (real_abs ((real_sub ((sum (k INTER ((NUMERAL _0) .. n))) (\i. ((f' i) x)))) (g' x)))) e))))))) /\ (?x. (?l. ((x IN s) /\ (((real_sums (\n. ((f n) x))) l) k))))))) ==> (?g. (!x. ((x IN s) ==> ((((real_sums (\n. ((f n) x))) (g x)) k) /\ (((has_real_derivative g) (g' x)) ((within (atreal x)) s))))))))))))
CONIC_HULL_POINTLESS_AS_IMAGE |- (!s. ((((hull conic) s) DELETE (vec (NUMERAL _0))) = ((IMAGE (\z. ((% (drop (fstcart z))) (sndcart z)))) ((GSPEC (\GEN%PVAR%1459. (?t. (((SETSPEC GEN%PVAR%1459) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t)))) PCROSS (s DELETE (vec (NUMERAL _0)))))))
HAS_INTEGRAL_COMPLEX_RMUL |- (!f. (!y. (!i. (!c. ((((has_integral f) y) i) ==> (((has_integral (\x. ((complex_mul (f x)) c))) ((complex_mul y) c)) i))))))
SUMMABLE_NEG |- (!x. (!s. (((summable s) x) ==> ((summable s) (\n. (vector_neg (x n)))))))
CENTRE_IN_MCBALL |- (!m. (!x. (!r. ((((real_le (real_of_num (NUMERAL _0))) r) /\ (x IN (mspace m))) ==> (x IN ((mcball m) (x , r)))))))
Nonlinear_lemma.ineq_expand10 |- (!a1. (!a2. (!a3. (!a4. (!a5. (!a6. (!a7. (!a8. (!a9. (!a10. (!b1. (!b2. (!b3. (!b4. (!b5. (!b6. (!b7. (!b8. (!b9. (!b10. (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!y7. (!y8. (!y9. (!y10. (!P. (((ineq ((CONS (a1 , (y1 , b1))) ((CONS (a2 , (y2 , b2))) ((CONS (a3 , (y3 , b3))) ((CONS (a4 , (y4 , b4))) ((CONS (a5 , (y5 , b5))) ((CONS (a6 , (y6 , b6))) ((CONS (a7 , (y7 , b7))) ((CONS (a8 , (y8 , b8))) ((CONS (a9 , (y9 , b9))) ((CONS (a10 , (y10 , b10))) NIL))))))))))) P) = ((((real_le a1) y1) /\ ((real_le y1) b1)) ==> ((((real_le a2) y2) /\ ((real_le y2) b2)) ==> ((((real_le a3) y3) /\ ((real_le y3) b3)) ==> ((((real_le a4) y4) /\ ((real_le y4) b4)) ==> ((((real_le a5) y5) /\ ((real_le y5) b5)) ==> ((((real_le a6) y6) /\ ((real_le y6) b6)) ==> ((((real_le a7) y7) /\ ((real_le y7) b7)) ==> ((((real_le a8) y8) /\ ((real_le y8) b8)) ==> ((((real_le a9) y9) /\ ((real_le y9) b9)) ==> ((((real_le a10) y10) /\ ((real_le y10) b10)) ==> P))))))))))))))))))))))))))))))))))))))))))
REALLIM_AT_INFINITY_COMPLEX_0 |- (!f. (!l. ((((---> f) l) at_infinity) = (((---> (f o complex_inv)) l) (at (Cx (real_of_num (NUMERAL _0))))))))
LIM_INFINITY_POSINFINITY_LIFT |- (!f. (!l. ((((--> f) l) at_infinity) ==> (((--> (f o lift)) l) at_posinfinity))))
STONE_WEIERSTRASS_VECTOR_POLYNOMIAL_FUNCTION_AFFINE |- (!f. (!s. (!e. (!t. (((compact s) /\ (((continuous_on f) s) /\ (((real_lt (real_of_num (NUMERAL _0))) e) /\ ((affine t) /\ (((IMAGE f) s) SUBSET t))))) ==> (?g. ((vector_polynomial_function g) /\ ((((IMAGE g) s) SUBSET t) /\ (!x. ((x IN s) ==> ((real_lt (vector_norm ((vector_sub (f x)) (g x)))) e)))))))))))
LEFT_INVERSE_LINEAR |- (!f. (!g. (((linear f) /\ ((g o f) = I)) ==> (linear g))))
Add_triangle.contracted_in_dart |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (d_fan ((vec (NUMERAL _0)) , (V , E)))) ==> ((contracted_dart d) IN (dart_of_fan (V , E))))))))
Lp_ineqs_proofs.tau_hi |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> ((contravening V) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((real_le ((tau_fan (V , E)) f)) tgt)))))))))))
Lp_ineqs_proofs.face_last_dart |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) ==> ((((POWER (f_fan_pair_ext (V , E))) ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) - (NUMERAL (BIT1 _0)))) (v , w)) = ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v))))))))
BINOMIAL_THEOREM |- (!n. (!x. (!y. (((x + y) EXP n) = ((nsum ((NUMERAL _0) .. n)) (\k. ((binom (n , k)) * ((x EXP k) * (y EXP (n - k))))))))))
TRANSLATION_EQ_IMP |- (!P. ((!a. (!s. ((P ((IMAGE (\x. ((vector_add a) x))) s)) = (P s)))) = (!a. (!s. ((P s) ==> (P ((IMAGE (\x. ((vector_add a) x))) s)))))))
Fan.azim_is_zero_fan |- (!x. (!v. (!u. (((~ (v = x)) /\ ((~ (u = x)) /\ (~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))))) ==> (((((azim x) v) u) u) = (real_of_num (NUMERAL _0)))))))
RPOW_MINUS1_QUOTIENT_LE |- (!a. (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) a) /\ (((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_le x) y))) ==> ((real_le ((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))))))
RC_IDEMP |- (!R. ((RC (RC R)) = (RC R)))
Miqmcsn.J_SCS_3T4_prime2 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_3T4_prime2) i)) i1) j) = F)
LIM_WITHINREAL_LE |- (!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_le (real_abs ((real_sub x) a))) d))) ==> ((real_lt (distance ((f x) , l))) e))))))))))))
HAS_VECTOR_DERIVATIVE_NEG_EQ |- (!f. (!f'. (!net. ((((has_vector_derivative (\x. (vector_neg (f x)))) (vector_neg f')) net) = (((has_vector_derivative f) f') net)))))