hcju/holps huggingface dataset:
COHOMOTOPICALLY_TRIVIAL_ON_COMPONENTS_NULL |- (!s. (!t. (((((locally connected) s) \/ ((compact s) /\ (ANR t))) /\ (path_connected t)) ==> ((!f. ((((continuous_on f) s) /\ (((IMAGE f) s) SUBSET t)) ==> (?a. ((((homotopic_with (\x. T)) (s , t)) f) (\x. a))))) = (!c. ((c IN (components s)) ==> (!f. ((((continuous_on f) c) /\ (((IMAGE f) c) SUBSET t)) ==> (?a. ((((homotopic_with (\x. T)) (c , t)) f) (\x. a)))))))))))
Lp_ineqs_proofs.azim2_hi |- (!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))) ==> ((real_le ((azim2_fan (V , E)) d)) pi)))))))))
REFLECT_ALONG_EQ |- (!v. (!x. (!y. ((((reflect_along v) x) = ((reflect_along v) y)) = (x = y)))))
Rogers.KHEJKCI |- (!V. (!k. (!ul. (((saturated V) /\ ((packing V) /\ (((barV V) k) ul))) ==> ((face_of ((voronoi_list V) ul)) ((voronoi_closed V) (HD ul)))))))
HOLOMORPHIC_ON_CATN |- (!s. ((!z. (((z IN s) /\ ((Re z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_abs (Im z))) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((holomorphic_on catn) s)))
SEPARATING_HYPERPLANE_RELATIVE_INTERIORS |- (!s. (!t. (((convex s) /\ ((convex t) /\ ((~ (((s = EMPTY) /\ (t = UNIV)) \/ ((s = UNIV) /\ (t = EMPTY)))) /\ ((DISJOINT (relative_interior s)) (relative_interior t))))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ ((!x. ((x IN s) ==> ((real_le ((dot a) x)) b))) /\ (!x. ((x IN t) ==> ((real_ge ((dot a) x)) b))))))))))
Localization.localization |- (!V. (!E. (!FF. (((localization (V , E)) FF) = (((v_prime V) FF) , ((e_prime E) FF))))))
LOG_MONO_LE_IMP |- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_le x) y)) ==> ((real_le (log x)) (log y)))))
REAL_DIFFERENTIABLE_WITHIN_TAN |- (!s. (!x. ((~ ((cos x) = (real_of_num (NUMERAL _0)))) ==> ((real_differentiable tan) ((within (atreal x)) s)))))
CCOS_CONVERGES |- (!z. (((sums (\n. ((complex_mul ((complex_pow (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) n)) ((complex_div ((complex_pow z) ((NUMERAL (BIT0 (BIT1 _0))) * n))) (Cx (real_of_num (FACT ((NUMERAL (BIT0 (BIT1 _0))) * n)))))))) (ccos z)) (from (NUMERAL _0))))
MEASURABLE_TRIANGLE |- (!a. (!b. (!c. (measurable ((hull convex) (a INSERT (b INSERT (c INSERT EMPTY))))))))
DIFFERENTIABLE_ON_NEG |- (!f. (!s. (((differentiable_on f) s) ==> ((differentiable_on (\z. (vector_neg (f z)))) s))))
FACT_LE |- (!n. ((NUMERAL (BIT1 _0)) <= (FACT n)))
Ssrnat.minn0 |- (!n. (((minn n) (NUMERAL _0)) = (NUMERAL _0)))
REAL_MEASURABLE_COUNTABLE_UNIONS |- (!s. (!B. (((!n. (real_measurable (s n))) /\ (!n. ((real_le ((sum ((NUMERAL _0) .. n)) (\k. (real_measure (s k))))) B))) ==> (real_measurable (UNIONS (GSPEC (\GEN%PVAR%4952. (?n. (((SETSPEC GEN%PVAR%4952) (n IN UNIV)) (s n))))))))))
NEGLIGIBLE_CONVEX_HULL_1 |- (!a. (negligible ((hull convex) (a INSERT EMPTY))))
COMPLEX_INV_II |- ((complex_inv ii) = (vector_neg ii))
ONORM_TRIANGLE_LE |- (!f. (!g. (((linear f) /\ ((linear g) /\ ((real_le ((real_add (onorm f)) (onorm g))) e))) ==> ((real_le (onorm (\x. ((vector_add (f x)) (g x))))) e))))
Seq2.prev_el_rot |- (!x. (!s. (((MEM x) s) ==> (((prev_el s) x) = ((headl x) ((rotr (NUMERAL (BIT1 _0))) ((rot ((indexl x) s)) s)))))))
HAS_REAL_MEASURE_IMP_REAL_MEASURABLE |- (!s. (!m. (((has_real_measure s) m) ==> (real_measurable s))))
RIGHT_POLAR_DECOMPOSITION_INVERTIBLE_UNIQUE |- (!A. (!U. (!V. (!P. (!S. (((invertible A) /\ ((orthogonal_matrix U) /\ ((positive_semidefinite P) /\ ((orthogonal_matrix V) /\ ((positive_semidefinite S) /\ ((((matrix_mul U) P) = A) /\ (((matrix_mul V) S) = A))))))) ==> ((P = S) /\ (U = V))))))))
HAS_BOUNDED_VARIATION_ON_INDEFINITE_INTEGRAL_LEFT |- (!f. (!a. (!b. (((absolutely_integrable_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((has_bounded_variation_on (\c. ((integral (closed_interval ((CONS (c , b)) NIL))) f))) (closed_interval ((CONS (a , b)) NIL)))))))
REALLIM_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 (real_abs ((real_sub (f x)) l))) e))))))))))))
COLLINEAR_DESCALE |- (!a. (!b. (!c. (!x. (!y. (!z. (((~ (a = (real_of_num (NUMERAL _0)))) /\ ((~ (c = (real_of_num (NUMERAL _0)))) /\ ((collinear (((% a) x) INSERT (((% b) y) INSERT (((% c) z) INSERT EMPTY)))) /\ (~ ((vec (NUMERAL _0)) IN ((hull affine) (x INSERT (y INSERT (z INSERT EMPTY))))))))) ==> (collinear (x INSERT (y INSERT (z INSERT EMPTY)))))))))))
COVERING_SPACE_FINITE_SHEETS_EQ_CLOSED_MAP |- (!p. (!c. (!s. ((((covering_space (c , p)) s) /\ ((connected s) /\ (~ (?a. (s = (a INSERT EMPTY)))))) ==> ((!b. ((b IN s) ==> (FINITE (GSPEC (\GEN%PVAR%2081. (?x. (((SETSPEC GEN%PVAR%2081) ((x IN c) /\ ((p x) = b))) x))))))) = (!t. (((closed_in ((subtopology euclidean) c)) t) ==> ((closed_in ((subtopology euclidean) s)) ((IMAGE p) t)))))))))
REAL_DIFFERENTIABLE_CARATHEODORY_WITHINREAL |- (!f. (!z. (!s. (((real_differentiable f) ((within (atreal z)) s)) = (?g. ((!w. (((real_sub (f w)) (f z)) = ((real_mul (g w)) ((real_sub w) z)))) /\ ((real_continuous g) ((within (atreal z)) s))))))))
FACET_OF_HALFSPACE_GE |- (!f. (!a. (!b. (((facet_of f) (GSPEC (\GEN%PVAR%2218. (?x. (((SETSPEC GEN%PVAR%2218) ((real_ge ((dot a) x)) b)) x))))) = ((~ (a = (vec (NUMERAL _0)))) /\ (f = (GSPEC (\GEN%PVAR%2219. (?x. (((SETSPEC GEN%PVAR%2219) (((dot a) x) = b)) x))))))))))
HAS_BOUNDED_REAL_VARIATION_ON_EMPTY |- (!f. ((has_bounded_real_variation_on f) EMPTY))
Functional_equation.rotate5_dihatn_x |- ((rotate5 dihatn_x) = dih5atn_x)
INT_LT_NEG |- (!x. (!y. (((int_lt (int_neg x)) (int_neg y)) = ((int_lt y) x))))
NPRODUCT_POS_LT_NUMSEG |- (!f. (!m. (!n. ((!x. (((m <= x) /\ (x <= n)) ==> ((NUMERAL _0) < (f x)))) ==> ((NUMERAL _0) < ((nproduct (m .. n)) f))))))
Lp_main_estimate.ineq_tau5_diags_std |- (!V. (!E. ((lp_fan (V , E)) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\ ((packing V) /\ ((V SUBSET ball_annulus) /\ (((CARD V) = (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) /\ ((!v. (!w. (((~ (v = w)) /\ ((v IN V) /\ ((w IN V) /\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le sqrt8) (distance (v , w)))))) /\ (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v , w))) /\ ((real_le (distance (v , w))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))))))))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((tau_fun V) E) FF)))))) ==> ((E = (ESTD V)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le sqrt8) ((y4'_fan (V , E)) d)) /\ (((real_le sqrt8) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) /\ (((real_le sqrt8) ((y4'_fan (V , E)) (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) d))) /\ (((real_le sqrt8) ((y4'_fan (V , E)) (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) d))) /\ ((real_le sqrt8) ((y4'_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d))))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)))))))))))
CONVEX_HALFSPACE_RE_GT |- (!b. (convex (GSPEC (\GEN%PVAR%4769. (?z. (((SETSPEC GEN%PVAR%4769) ((real_gt (Re z)) b)) z))))))
Functional_equation.functional_x1_delta_x |- (x1_delta_x = ((mul6 proj_x1) delta_x))
INT_SGN_INT_SGN |- (!x. ((int_sgn (int_sgn x)) = (int_sgn x)))
REAL_VARIATION_TRANSLATION2 |- (!a. (!f. (!s. (((real_variation ((IMAGE (\x. ((real_add (real_neg a)) x))) s)) (\x. (f ((real_add a) x)))) = ((real_variation s) f)))))
INT_ADD_RDISTRIB |- (!x. (!y. (!z. (((int_mul ((int_add x) y)) z) = ((int_add ((int_mul x) z)) ((int_mul y) z))))))
RANK_TRIANGLE |- (!A. (!B. ((rank ((matrix_add A) B)) <= ((rank A) + (rank B)))))
HAS_COMPLEX_DERIVATIVE_SEQUENCE |- (!s. (!f. (!f'. (!g'. (((convex s) /\ ((!n. (!x. ((x IN s) ==> (((has_complex_derivative (f n)) ((f' n) x)) ((within (at x)) s))))) /\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\ (x IN s)) ==> ((real_le (vector_norm ((vector_sub ((f' n) x)) (g' x)))) e))))))) /\ (?x. (?l. ((x IN s) /\ (((--> (\n. ((f n) x))) l) sequentially))))))) ==> (?g. (!x. ((x IN s) ==> ((((--> (\n. ((f n) x))) (g x)) sequentially) /\ (((has_complex_derivative g) (g' x)) ((within (at x)) s)))))))))))
Merge_ineq.nonf_arclength_x_123' |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((arclength_x_123' x1) x2) x3) x4) x5) x6) = (acs ((real_div ((real_add x1) ((real_sub x2) x3))) (sqrt ((real_mul ((real_mul x1) x2)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))))))
REAL_MUL_SIN_SIN |- (!x. (!y. (((real_mul (sin x)) (sin y)) = ((real_div ((real_sub (cos ((real_sub x) y))) (cos ((real_add x) y)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))
INT_POW_ZERO |- (!n. (((int_pow (int_of_num (NUMERAL _0))) n) = (((COND (n = (NUMERAL _0))) (int_of_num (NUMERAL (BIT1 _0)))) (int_of_num (NUMERAL _0)))))
HAS_BOUNDED_VARIATION_INTEGRABLE_NORM_DERIVATIVE |- (!f. (!s. (!a. (!b. (((COUNTABLE s) /\ (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> ((differentiable f) (at x)))))) ==> (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) = ((integrable_on (\x. (lift (vector_norm ((vector_derivative f) (at x)))))) (closed_interval ((CONS (a , b)) NIL)))))))))
CONVEX_HULL_REDUNDANT_POINT |- (!s. (!a. ((((hull convex) (s DELETE a)) = ((hull convex) s)) = (~ ((extreme_point_of a) ((hull convex) s))))))
CONTINUOUS_DECREASING_IMAGE_INTERVAL_1 |- (!f. (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (!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 y))) (drop (f x)))))))) ==> (((IMAGE f) (closed_interval ((CONS (a , b)) NIL))) = (closed_interval ((CONS ((f b) , (f a))) NIL)))))))
MATRIX_NEG_0 |- ((matrix_neg (mat (NUMERAL _0))) = (mat (NUMERAL _0)))
SIN_EQ_1 |- (!x. (((sin x) = (real_of_num (NUMERAL (BIT1 _0)))) = (?n. ((integer n) /\ (x = ((real_mul ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) pi))))))
COMPACT_ARC_IMAGE |- (!g. ((arc g) ==> (compact (path_image g))))
ENR_DELETE |- (!s. (!a. ((ENR s) ==> (ENR (s DELETE a)))))
Reduction3.size_mapAt |- (!ns. (!xs. (!f. ((sizel (((mapAt ns) f) xs)) = (sizel xs)))))
FRONTIER_FRONTIER_FRONTIER |- (!s. ((frontier (frontier (frontier s))) = (frontier (frontier s))))
INT_EQ_ADD_RCANCEL_0 |- (!x. (!y. ((((int_add x) y) = y) = (x = (int_of_num (NUMERAL _0))))))
REAL_LEBESGUE_MEASURABLE_DIFF |- (!s. (!t. (((real_lebesgue_measurable s) /\ (real_lebesgue_measurable t)) ==> (real_lebesgue_measurable (s DIFF t)))))
EULER_ROTATION_THEOREM_GEN |- (!A. (((ODD (dimindex UNIV)) /\ (rotation_matrix A)) ==> (?v. (((vector_norm v) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((matrix_vector_mul A) v) = v)))))
INDEPENDENT_2 |- (!a. (!b. (!x. (!y. (((independent (a INSERT (b INSERT EMPTY))) /\ (~ (a = b))) ==> ((((vector_add ((% x) a)) ((% y) b)) = (vec (NUMERAL _0))) = ((x = (real_of_num (NUMERAL _0))) /\ (y = (real_of_num (NUMERAL _0))))))))))
CONTINUOUS_ON_COMPLEX_DIV |- (!f. (!g. (!s. ((((continuous_on f) s) /\ (((continuous_on g) s) /\ (!x. ((x IN s) ==> (~ ((g x) = (Cx (real_of_num (NUMERAL _0))))))))) ==> ((continuous_on (\x. ((complex_div (f x)) (g x)))) s)))))
CONVEX_HULL_REDUNDANT_SUBSET |- (!s. (!t. (((compact s) /\ ((t SUBSET s) /\ ((s DIFF t) SUBSET (interior ((hull convex) s))))) ==> (((hull convex) s) = ((hull convex) t)))))
NORM_SEGMENT_LOWERBOUND |- (!a. (!b. (!x. (!r. (!d. ((((real_lt (real_of_num (NUMERAL _0))) r) /\ (((vector_norm a) = r) /\ (((vector_norm b) = r) /\ ((x IN (closed_segment ((CONS (a , b)) NIL))) /\ (((dot a) b) = ((real_mul d) ((real_pow r) (NUMERAL (BIT0 (BIT1 _0)))))))))) ==> ((real_le ((real_mul (sqrt ((real_div ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (real_abs d))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) r)) (vector_norm x))))))))
INT_LTE_ANTISYM |- (!x. (!y. (~ (((int_lt x) y) /\ ((int_le y) x)))))
Functional_equation.functional_delta_x1 |- (delta_x1 = (rotate4 delta_x4))
Functional_equation.nonf_gamma2_x1_div_a |- (!m. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((gamma2_x1_div_a m) x1) x2) x3) x4) x5) x6) = ((gamma2_x_div_azim m) x1)))))))))
CONTINUOUS_ON_COMPOSE_ARG |- (!s. (!p. ((((continuous_on (p o drop)) (closed_interval ((CONS ((vec (NUMERAL _0)) , (lift ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))) NIL))) /\ (((p ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) = (p (real_of_num (NUMERAL _0)))) /\ (~ ((Cx (real_of_num (NUMERAL _0))) IN s)))) ==> ((continuous_on (\z. (p (Arg z)))) s))))
Add_triangle.f_split_fan_face_eq |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (!d. ((((v , w) IN (dart_of_fan (V , E))) /\ ((d IN (dart_of_fan (V , E))) /\ (~ (d IN ((face (hypermap_of_fan (V , E))) (v , w)))))) ==> (((f_fan_pair ((split_fan_face (V , E)) (v , w))) d) = ((f_fan_pair (V , E)) d))))))))))
Functional_equation.taum_x' |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL _0))) x1) /\ (((real_le (real_of_num (NUMERAL _0))) x2) /\ (((real_le (real_of_num (NUMERAL _0))) x3) /\ (((real_le (real_of_num (NUMERAL _0))) x4) /\ (((real_le (real_of_num (NUMERAL _0))) x5) /\ ((real_le (real_of_num (NUMERAL _0))) x6)))))) ==> (((((((taum_x x1) x2) x3) x4) x5) x6) = ((((((taum_x' x1) x2) x3) x4) x5) x6)))))))))
SECOND_MEAN_VALUE_THEOREM_BONNET |- (!f. (!g. (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_of_num (NUMERAL _0))) (g x)))) /\ (!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 (g x)) (g y)))))))) ==> (?c. ((c IN (closed_interval ((CONS (a , b)) NIL))) /\ (((integral (closed_interval ((CONS (a , b)) NIL))) (\x. ((% (g x)) (f x)))) = ((% (g b)) ((integral (closed_interval ((CONS (c , b)) NIL))) f))))))))))
ABSOLUTELY_REAL_INTEGRABLE_DECREASING |- (!f. (!a. (!b. ((!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 (f y)) (f x))))) ==> ((absolutely_real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL)))))))
BOUNDED_ARC_IMAGE |- (!g. ((arc g) ==> (bounded (path_image g))))
CNJ_CSQRT |- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_le (real_of_num (NUMERAL _0))) (Re z))) ==> ((cnj (csqrt z)) = (csqrt (cnj z)))))
Ssrnat.minn_maxr |- (!m. (!n1. (!n2. (((minn m) ((maxn n1) n2)) = ((maxn ((minn m) n1)) ((minn m) n2))))))
HAS_BOUNDED_REAL_VARIATION_ON_NEG |- (!f. (!s. (((has_bounded_real_variation_on f) s) ==> ((has_bounded_real_variation_on (\x. (real_neg (f x)))) s))))
AFFINE_HULL_EQ_SING |- (!s. (!a. ((((hull affine) s) = (a INSERT EMPTY)) = (s = (a INSERT EMPTY)))))
FUNSPACE_IMP_BOUNDED |- (!s. (!m. (!f. ((f IN (mspace ((funspace s) m))) ==> ((s = EMPTY) \/ (?c. (?b. (!x. ((x IN s) ==> ((real_le ((mdist m) (c , (f x)))) b))))))))))
DIV_ADD_MOD |- (!a. (!b. (!n. ((~ (n = (NUMERAL _0))) ==> ((((a + b) MOD n) = ((a MOD n) + (b MOD n))) = (((a + b) DIV n) = ((a DIV n) + (b DIV n))))))))
Lp_ineqs_proofs.sol_hi |- (!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)))) ==> ((real_le ((sol_fan (V , E)) f)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi))))))))))
ORTHOGONAL_TRANSFORMATION_ONTO_SUBSPACE |- (!s. (!t. (((subspace s) /\ ((subspace t) /\ ((dim s) = (dim t)))) ==> (?f. ((orthogonal_transformation f) /\ (((IMAGE f) s) = t))))))
PROPER_LOCAL_HOMEOMORPHISM_GLOBAL |- (!f. (!s. (!t. (((path_connected s) /\ ((simply_connected t) /\ (((continuous_on f) s) /\ ((((IMAGE f) s) = t) /\ ((!k. (((k SUBSET t) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%2104. (?x. (((SETSPEC GEN%PVAR%2104) ((x IN s) /\ ((f x) IN k))) x))))))) /\ (!x. ((x IN s) ==> (?u. (?v. (?q. ((x IN u) /\ (((open_in ((subtopology euclidean) s)) u) /\ (((open_in ((subtopology euclidean) t)) v) /\ ((homeomorphism (u , v)) (f , q))))))))))))))) ==> (?g. ((homeomorphism (s , t)) (f , g)))))))
FCONS_UNDO |- (!f. (f = ((FCONS (f (NUMERAL _0))) (f o SUC))))
ISOMETRY_IMP_AFFINITY |- (!f. ((!x. (!y. ((distance ((f x) , (f y))) = (distance (x , y))))) ==> (?h. ((linear h) /\ (!x. ((f x) = ((vector_add (f (vec (NUMERAL _0)))) (h x))))))))
IS_AFFINE_HULL |- (!s. ((affine s) = (?t. (s = ((hull affine) t)))))
Fnjlbxs.closedS |- (!k. (!S. ((((NUMERAL (BIT1 (BIT1 _0))) * k) <= (dimindex UNIV)) ==> ((!l. ((l IN S) = (((set_of_list l) SUBSET ball_annulus) /\ ((packing (set_of_list l)) /\ (((LENGTH l) = k) /\ (uniq l)))))) ==> (closed ((IMAGE (vector_of_list (NUMERAL (BIT1 (BIT1 _0))))) S))))))
REAL_MEASURABLE_TRANSLATION |- (!s. ((real_measurable ((IMAGE (\x. ((real_add a) x))) s)) = (real_measurable s)))
ZERO_AE_DERIVATIVE_IMP_CONSTANT |- (!f. (!a. (!b. (!s. ((((absolutely_continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((negligible s) /\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> (((has_vector_derivative f) (vec (NUMERAL _0))) (at x)))))) ==> (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((f x) = (f a)))))))))
CLOSURE_OF_EMPTY |- (!top. (((closure_of top) EMPTY) = EMPTY))
BETWEEN_COLLINEAR_DIST_EQ |- (!a. (!b. (!x. (((between x) (a , b)) = ((collinear (a INSERT (x INSERT (b INSERT EMPTY)))) /\ (((real_le (distance (x , a))) (distance (a , b))) /\ ((real_le (distance (x , b))) (distance (a , b)))))))))
EXPOSED_POINT_OF_INTER_SUPPORTING_HYPERPLANE_GE |- (!s. (!a. (!b. (!c. (((!x. ((x IN s) ==> ((real_ge ((dot a) x)) b))) /\ ((s INTER (GSPEC (\GEN%PVAR%2203. (?x. (((SETSPEC GEN%PVAR%2203) (((dot a) x) = b)) x))))) = (c INSERT EMPTY))) ==> ((exposed_face_of (c INSERT EMPTY)) s))))))
Ssrbool.addbT |- (!b. (((XOR b) T) = (~ b)))
Miqmcsn.SCS_4M8_13_ARROW_3T7_OPP |- ((scs_arrow_v39 (scs_4M8_02 INSERT EMPTY)) ((scs_opp_v39 scs_3T7) INSERT EMPTY))
Seq.constantP |- (!s. ((?x. (s = ((nseq (sizel s)) x))) = (constant s)))
OPEN_IN_FSIGMA |- (!s. (!t. ((((open_in ((subtopology euclidean) t)) s) /\ (fsigma t)) ==> (fsigma s))))
LIPSCHITZ_VECTOR_POLYNOMIAL_FUNCTION |- (!f. (!s. (((vector_polynomial_function f) /\ (bounded s)) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) 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))))))))))))
CONVEX_MAX |- (!f. (!g. (!s. ((((convex_on f) s) /\ ((convex_on g) s)) ==> ((convex_on (\x. ((real_max (f x)) (g x)))) s)))))
Dih2k_hypermap.EE_OF_HYP_HAS_ORDERS |- (((FAN ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l))))) /\ (((NUMERAL (BIT0 (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)))))) ==> ((has_orders (ee_of_hyp ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l)))))) (NUMERAL (BIT0 (BIT1 _0)))))
EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL_MINIMAL |- (!s. (((FINITE s) /\ (((hull affine) s) = UNIV)) ==> ((GSPEC (\GEN%PVAR%1795. (?y. (((SETSPEC GEN%PVAR%1795) (?u. ((!x. ((x IN s) ==> ((real_lt (real_of_num (NUMERAL _0))) (u x)))) /\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum s) (\x. ((% (u x)) x))) = y))))) y)))) SUBSET (interior ((hull convex) s)))))
INT_LE_ADD |- (!x. (!y. ((((int_le (int_of_num (NUMERAL _0))) x) /\ ((int_le (int_of_num (NUMERAL _0))) y)) ==> ((int_le (int_of_num (NUMERAL _0))) ((int_add x) y)))))
Add_triangle.in_face_lemma |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) ==> (((v , w) IN ((face (hypermap_of_fan (V , E))) (v , w))) /\ ((((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v) IN ((face (hypermap_of_fan (V , E))) (v , w))) /\ (((f_fan_pair (V , E)) (v , w)) IN ((face (hypermap_of_fan (V , E))) (v , w)))))))))))
Hypermap.lemma_map_loop_map |- (!H. (!NF. (!L. (!x. ((((is_normal H) NF) /\ ((L IN NF) /\ (((in_dart_of_loop x) L) /\ (((loop_map L) x) IN (((atom H) L) x))))) ==> (((loop_map L) x) = ((inverse (node_map H)) x)))))))
BETWEEN_NORM_LT |- (!r. (!a. (!b. (!x. ((((real_lt (vector_norm a)) r) /\ (((real_lt (vector_norm b)) r) /\ ((between x) (a , b)))) ==> ((real_lt (vector_norm x)) r))))))
Collect_geom2.c_coef |- (!x12. (!x13. (!x14. (!x23. (!x24. (((((((b_coef x12) x13) x14) x23) x24) = ((real_add ((real_mul (real_neg x13)) x14)) ((real_add ((real_mul (real_neg x23)) x24)) ((real_add ((real_mul x13) x24)) ((real_add ((real_mul x14) x23)) ((real_add ((real_mul (real_neg x12)) x12)) ((real_add ((real_mul x12) x14)) ((real_add ((real_mul x12) x24)) ((real_add ((real_mul x12) x13)) ((real_mul x12) x23)))))))))) /\ ((((((c_coef x12) x13) x14) x23) x24) = ((real_add ((real_mul ((real_mul x14) x23)) x12)) ((real_add ((real_mul ((real_mul x14) x23)) x13)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x14)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x23)) ((real_add ((real_mul ((real_mul x14) x23)) x24)) ((real_add ((real_mul ((real_mul (real_neg x12)) x13)) x23)) ((real_add ((real_mul ((real_mul (real_neg x12)) x14)) x24)) ((real_add ((real_mul ((real_mul x13) x24)) x12)) ((real_add ((real_mul ((real_mul (real_neg x13)) x24)) x13)) ((real_add ((real_mul ((real_mul x13) x24)) x14)) ((real_add ((real_mul ((real_mul x13) x24)) x23)) ((real_mul ((real_mul (real_neg x13)) x24)) x24)))))))))))))))))))
CROSS_JACOBI |- (!x. (!y. (!z. (((vector_add ((cross x) ((cross y) z))) ((vector_add ((cross y) ((cross z) x))) ((cross z) ((cross x) y)))) = (vec (NUMERAL _0))))))
COVERING_SPACE_COMPACT |- (!p. (!c. (!s. (((covering_space (c , p)) s) ==> ((compact c) = ((compact s) /\ (!b. ((b IN s) ==> (FINITE (GSPEC (\GEN%PVAR%2085. (?x. (((SETSPEC GEN%PVAR%2085) ((x IN c) /\ ((p x) = b))) x)))))))))))))
BOREL_UNION |- (!s. (!t. (((borel s) /\ (borel t)) ==> (borel (s UNION t)))))
REAL_DIFFERENTIABLE_EQ |- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((real_differentiable_on f) s) = ((real_differentiable_on g) s))))))
Calc_derivative.imp_eq |- (!p. (!p'. (!x. (!a. (!a'. (!b. (!b'. (((x = (real_of_num (NUMERAL _0))) /\ (((((ratform p) x) a) b) /\ ((p = p') /\ ((a = a') /\ (b = b'))))) ==> ((p' /\ (~ (b' = (real_of_num (NUMERAL _0))))) ==> (a' = (real_of_num (NUMERAL _0))))))))))))
Nonlinear_lemma.gcy_low_hminus |- (!y. (((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) hminus)) ==> ((gcy y) = (gchi y))))
REAL_SECOND_MEAN_VALUE_THEOREM |- (!f. (!g. (!a. (!b. (((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ (!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 (g a)) ((real_integral (closed_real_interval ((CONS (a , c)) NIL))) f))) ((real_mul (g b)) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f)))))))))))