hcju/holps huggingface dataset:
COLLINEAR_STANDARD_HYPERPLANE_2 |- (!k. (!b. (((dimindex UNIV) <= (NUMERAL (BIT0 (BIT1 _0)))) ==> (collinear (GSPEC (\GEN%PVAR%1599. (?x. (((SETSPEC GEN%PVAR%1599) ((x $ k) = b)) x))))))))
List_hypermap_iso.set_of_edge_subset |- (!V. (!E. (!E'. (!x. ((E SUBSET E') ==> ((((set_of_edge x) V) E) SUBSET (((set_of_edge x) V) E')))))))
AFF_DIM_HALFSPACE_LT |- (!a. (!b. ((aff_dim (GSPEC (\GEN%PVAR%1548. (?x. (((SETSPEC GEN%PVAR%1548) ((real_lt ((dot a) x)) b)) x))))) = (((COND ((a = (vec (NUMERAL _0))) /\ ((real_le b) (real_of_num (NUMERAL _0))))) (int_neg (int_of_num (NUMERAL (BIT1 _0))))) (int_of_num (dimindex UNIV))))))
NSUM_INJECTION |- (!f. (!p. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> ((p x) IN s))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((p x) = (p y)))) ==> (x = y)))))) ==> (((nsum s) (f o p)) = ((nsum s) f))))))
Nonlinear_lemma.sqrt8_nn |- ((real_le (real_of_num (NUMERAL _0))) sqrt8)
NSUM_LE_GEN |- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) <= (g x)))) /\ (FINITE (GSPEC (\GEN%PVAR%188. (?x. (((SETSPEC GEN%PVAR%188) ((x IN s) /\ (~ ((g x) = (NUMERAL _0))))) x)))))) ==> (((nsum s) f) <= ((nsum s) g))))))
INESSENTIAL_ON_COMPONENTS |- (!f. (!s. (!t. ((((locally connected) s) /\ ((path_connected t) /\ (!c. ((c IN (components s)) ==> (?a. ((((homotopic_with (\x. T)) (c , t)) f) (\x. a))))))) ==> (?a. ((((homotopic_with (\x. T)) (s , t)) f) (\x. a)))))))
HAS_REAL_INTEGRAL_ALT |- (!f. (!s. (!i. ((((has_real_integral f) i) s) = ((!a. (!b. ((real_integrable_on (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) (closed_real_interval ((CONS (a , b)) NIL))))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!a. (!b. (((open_real_interval ((real_neg B) , B)) SUBSET (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_lt (real_abs ((real_sub ((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0)))))) i))) e)))))))))))))
MEASURABLE_INSERT |- (!x. (!s. ((measurable (x INSERT s)) = (measurable s))))
DIV_EQ_EXCLUSION |- ((((b * c) < ((a + (NUMERAL (BIT1 _0))) * d)) /\ ((a * d) < ((c + (NUMERAL (BIT1 _0))) * b))) ==> ((a DIV b) = (c DIV d)))
Hypermap_and_fan.FAN_PAIR_FIXED_POINT |- (!V. (!x. ((x IN (GSPEC (\GEN%PVAR%5632. (?v. (((SETSPEC GEN%PVAR%5632) ((v IN V) /\ ((((set_of_edge v) V) E) = EMPTY))) (v , v)))))) ==> ((((n_fan_pair_ext (V , E)) x) = x) /\ (((f_fan_pair (V , E)) x) = x)))))
INFSUM_CMUL |- (!s. (!x. (!c. (((summable s) x) ==> (((infsum s) (\n. ((% c) (x n)))) = ((% c) ((infsum s) x)))))))
HOMEOMORPHIC_FIXPOINT_PROPERTY |- (!s. (!t. (((homeomorphic s) t) ==> ((!f. ((((continuous_on f) s) /\ (((IMAGE f) s) SUBSET s)) ==> (?x. ((x IN s) /\ ((f x) = x))))) = (!g. ((((continuous_on g) t) /\ (((IMAGE g) t) SUBSET t)) ==> (?y. ((y IN t) /\ ((g y) = y)))))))))
LENGTH_MAP2 |- (!f. (!l. (!m. (((LENGTH l) = (LENGTH m)) ==> ((LENGTH (((MAP2 f) l) m)) = (LENGTH m))))))
EVENTUALLY_INV1_LT |- (!e. (((eventually (\n. ((real_lt (real_inv ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) e))) sequentially) = ((real_lt (real_of_num (NUMERAL _0))) e)))
FRONTIER_SEGMENT |- ((!a. (!b. ((frontier (closed_segment ((CONS (a , b)) NIL))) = (((COND ((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV))) (closed_segment ((CONS (a , b)) NIL))) (a INSERT (b INSERT EMPTY)))))) /\ (!a. (!b. ((frontier (open_segment (a , b))) = (((COND (a = b)) EMPTY) (((COND ((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV))) (closed_segment ((CONS (a , b)) NIL))) (a INSERT (b INSERT EMPTY))))))))
ABSOLUTE_RETRACTION_CONVEX_CLOSED_RELATIVE |- (!s. (!t. (((convex s) /\ ((closed s) /\ ((~ (s = EMPTY)) /\ (s SUBSET t)))) ==> (?r. (((retraction (t , s)) r) /\ (!x. ((x IN (((hull affine) s) DIFF (relative_interior s))) ==> ((r x) IN (relative_frontier s)))))))))
Merge_ineq.nonf_ldih6_x_div_sqrtdelta_posbranch |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((ldih6_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((real_mul (lfun ((real_div (sqrt x6)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_div (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x6))) ((((((delta_x4 x6) x1) x5) x3) x4) x2))) (matan ((real_div ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x6) ((((((delta_x x1) x2) x3) x4) x5) x6)))) ((real_pow ((((((delta_x4 x6) x1) x5) x3) x4) x2)) (NUMERAL (BIT0 (BIT1 _0)))))))))))))))
MATRIX_CMUL_SUB_LDISTRIB |- (!A. (!B. (!c. (((%% c) ((matrix_sub A) B)) = ((matrix_sub ((%% c) A)) ((%% c) B))))))
REAL_COMPACT_UNION |- (!s. (!t. (((real_compact s) /\ (real_compact t)) ==> (real_compact (s UNION t)))))
CLOG_MUL_II |- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((clog ((complex_mul ii) z)) = (((COND (((real_le (real_of_num (NUMERAL _0))) (Re z)) \/ ((real_lt (Im z)) (real_of_num (NUMERAL _0))))) ((vector_add (clog z)) ((complex_mul ii) (Cx ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))) ((vector_sub (clog z)) ((complex_mul ii) (Cx ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))
Lp_ineqs_quads.ineq120_std |- (!V. (!E. ((lp_fan (V , E)) ==> (quad_nonlinear_v10 ==> ((!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((ineq ((CONS (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0) , (y4 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y5 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL))))))) ((real_lt ((((((delta_y y1) y2) y3) y4) y5) y6)) (real_of_num (NUMERAL _0)))))))))) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\ ((packing V) /\ ((V SUBSET ball_annulus) /\ (((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) <= (CARD V)) /\ (((CARD V) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) /\ ((!v. (!w. (((~ (v = w)) /\ ((v IN V) /\ ((w IN V) /\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (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 (d_tame (CARD V))) (((tau_fun V) E) FF)))))) ==> ((lp_tau (V , E)) ==> ((E = (ESTD V)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((real_ge ((real_add ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d))) ((real_sub ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) ((azim_dart (V , E)) d))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))) (real_of_num (NUMERAL _0)))))))))))))
Functional_equation.functional_proj_x1 |- (proj_x1 = (promote1_to_6 I))
Lp_ineqs_proofs.y8_list_def |- (!V. (!E. (!g. (!L. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((good_list L) ==> (((hyp_iso g) ((hypermap_of_list L) , (hypermap_of_fan (V , E)))) ==> ((ALL (\d. (((y8_list (g , (V , E))) d) = ((ye_list g) ((inverse (f_list_ext L)) ((inverse (f_list_ext L)) d)))))) (list_of_darts L)))))))))
CRAMER_MATRIX_LEFT_INVERSE |- (!A. (!A'. ((((matrix_mul A') A) = (mat (NUMERAL (BIT1 _0)))) = ((~ ((det A) = (real_of_num (NUMERAL _0)))) /\ (A' = (lambdak. (lambdal. ((real_div (det (lambdai. (lambdaj. (((COND (j = l)) (((COND (i = k)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))) ((A $ j) $ i)))))) (det A)))))))))
SUBSET_HYPERPLANES |- (!a. (!b. (!a'. (!b'. (((GSPEC (\GEN%PVAR%442. (?x. (((SETSPEC GEN%PVAR%442) (((dot a) x) = b)) x)))) SUBSET (GSPEC (\GEN%PVAR%443. (?x. (((SETSPEC GEN%PVAR%443) (((dot a') x) = b')) x))))) = (((GSPEC (\GEN%PVAR%444. (?x. (((SETSPEC GEN%PVAR%444) (((dot a) x) = b)) x)))) = EMPTY) \/ (((GSPEC (\GEN%PVAR%445. (?x. (((SETSPEC GEN%PVAR%445) (((dot a') x) = b')) x)))) = UNIV) \/ ((GSPEC (\GEN%PVAR%446. (?x. (((SETSPEC GEN%PVAR%446) (((dot a) x) = b)) x)))) = (GSPEC (\GEN%PVAR%447. (?x. (((SETSPEC GEN%PVAR%447) (((dot a') x) = b')) x))))))))))))
REAL_SERIES_LE |- (!f. (!g. (!s. (!y. (!z. (((((real_sums f) y) s) /\ ((((real_sums g) z) s) /\ (!i. ((i IN s) ==> ((real_le (f i)) (g i)))))) ==> ((real_le y) z)))))))
HAS_BOUNDED_REAL_VARIATION_TRANSLATION_EQ_INTERVAL |- (!a. (!f. (!u. (!v. (((has_bounded_real_variation_on (\x. (f ((real_add a) x)))) (closed_real_interval ((CONS (u , v)) NIL))) = ((has_bounded_real_variation_on f) (closed_real_interval ((CONS (((real_add a) u) , ((real_add a) v))) NIL))))))))
REAL_LOG_CONVEX_ON_SUBSET |- (!f. (!s. (!t. ((((real_log_convex_on f) t) /\ (s SUBSET t)) ==> ((real_log_convex_on f) s)))))
Merge_ineq.nonf_edge2_126_x |- (!d. (!x4'. (!x5'. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((((edge2_126_x d) x4') x5') x1) x2) x3) x4) x5) x6) = ((((((edge2_flatD_x1 d) x1) x2) x6) x4') x5')))))))))))
INT_ABS_NZ |- (!x. ((~ (x = (int_of_num (NUMERAL _0)))) = ((int_lt (int_of_num (NUMERAL _0))) (int_abs x))))
CONTENT_POS_LT_1 |- (!a. (!b. (((real_lt (drop a)) (drop b)) ==> ((real_lt (real_of_num (NUMERAL _0))) (content (closed_interval ((CONS (a , b)) NIL)))))))
HAS_BOUNDED_VARIATION_ON_COMPLEX_MUL |- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ (((has_bounded_variation_on g) s) /\ (is_interval s))) ==> ((has_bounded_variation_on (\x. ((complex_mul (f x)) (g x)))) s)))))
NPRODUCT_EQ_NUMSEG |- (!f. (!g. (!m. (!n. ((!i. (((m <= i) /\ (i <= n)) ==> ((f i) = (g i)))) ==> (((nproduct (m .. n)) f) = ((nproduct (m .. n)) g)))))))
Tame_list.INSERT_alt |- ((a INSERT B) = (GSPEC (\GEN%PVAR%8608. (?x. (((SETSPEC GEN%PVAR%8608) ((x = a) \/ (x IN B))) x)))))
Hxhytij.HXHYTIJ |- (!s. (!vv. (!ww. (((is_scs_v39 s) /\ (((BBprime2_v39 s) vv) /\ ((BBs_v39 s) ww))) ==> (((real_lt ((taustar_v39 s) vv)) ((taustar_v39 s) ww)) \/ (((BBindex_v39 s) vv) <= ((BBindex_v39 s) ww)))))))
Seq2.list_sum_undup |- (!s. (!f. (((list_sum s) f) = ((list_sum (undup s)) (\x. ((real_mul (real_of_num ((count (pred1 x)) s))) (f x)))))))
PRODUCT_CLAUSES_RIGHT |- (!f. (!m. (!n. ((((NUMERAL _0) < n) /\ (m <= n)) ==> (((product (m .. n)) f) = ((real_mul ((product (m .. (n - (NUMERAL (BIT1 _0))))) f)) (f n)))))))
REAL_CONVEX_ON_EXP |- (!s. ((real_convex_on exp) s))
HOMEOMORPHIC_GRAPH |- (!f. (!s. (((continuous_on f) s) ==> ((homeomorphic (GSPEC (\GEN%PVAR%1035. (?x. (((SETSPEC GEN%PVAR%1035) (x IN s)) ((pastecart x) (f x))))))) s))))
MEASURABLE_OPEN_IN |- (!s. (!t. ((((open_in ((subtopology euclidean) s)) t) /\ (measurable s)) ==> (measurable t))))
ORDER_EXISTENCE_FINITE |- (!R. (!f. (!z. (((FINITE (GSPEC (\GEN%PVAR%77. (?n. (((SETSPEC GEN%PVAR%77) (n IN UNIV)) (R (((ITER n) f) z))))))) /\ (((R z) z) /\ ((!x. (!y. (((R x) y) ==> ((R y) x)))) /\ ((!x. (!y. (!z. ((((R x) y) /\ ((R y) z)) ==> ((R x) z))))) /\ (!x. (!y. (((R (f x)) (f y)) = ((R x) y)))))))) ==> (?d. (((NUMERAL _0) < d) /\ (!n. (((R (((ITER n) f) z)) z) = ((num_divides d) n)))))))))
RSTC_CASES_L |- (!R. (!x. (!z. ((((RSTC R) x) z) = ((x = z) \/ (((R x) z) \/ ((((RSTC R) z) x) \/ (?y. ((((RSTC R) x) y) /\ ((R y) z))))))))))
RSTC_INC_STC |- (!R. (!x. (!y. ((((STC R) x) y) ==> (((RSTC R) x) y)))))
NSUM_CONST_NUMSEG |- (!c. (!m. (!n. (((nsum (m .. n)) (\n. c)) = (((n + (NUMERAL (BIT1 _0))) - m) * c)))))
Deformation.COMPACT_SPHERE_0 |- (!a. (compact (GSPEC (\GEN%PVAR%7989. (?x. (((SETSPEC GEN%PVAR%7989) ((vector_norm x) = a)) x))))))
CONTINUOUS_ON_CPOW_RIGHT |- (!w. (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\z. ((cpow w) (f z)))) s)))))
CONTINUOUS_WITHIN_CEXP |- (!s. (!z. ((continuous cexp) ((within (at z)) s))))
ORTHOGONAL_LVSUM |- (!f. (!s. (!y. (((FINITE s) /\ (!x. ((x IN s) ==> ((orthogonal (f x)) y)))) ==> ((orthogonal ((vsum s) f)) y)))))
INJECTIVE_PREIMAGE |- (!f. ((!t. (!t'. (((GSPEC (\GEN%PVAR%145. (?x. (((SETSPEC GEN%PVAR%145) ((f x) IN t)) x)))) = (GSPEC (\GEN%PVAR%146. (?x. (((SETSPEC GEN%PVAR%146) ((f x) IN t')) x))))) ==> (t = t')))) = (((IMAGE f) UNIV) = UNIV)))
ISO_REFL |- ((ISO (\x. x)) (\x. x))
HOMOTOPIC_LINEAR_POSITIVE_SEMIDEFINITE_MAPS |- (!f. (!g. (((((homotopic_with (\f. ((linear f) /\ (positive_semidefinite (matrix f))))) (UNIV , UNIV)) f) g) = ((linear f) /\ ((linear g) /\ ((positive_semidefinite (matrix f)) /\ (positive_semidefinite (matrix g))))))))
CANTOR_BENDIXSON |- (!s. ((closed s) ==> (?t. (?u. ((closed t) /\ ((!x. ((x IN t) ==> ((limit_point_of x) t))) /\ ((COUNTABLE u) /\ (s = (t UNION u)))))))))
INT_EQ_ADD_RCANCEL |- (!x. (!y. (!z. ((((int_add x) z) = ((int_add y) z)) = (x = y)))))
COVERING_SPACE_LIFT |- (!p. (!c. (!s. (!f. (!u. ((((covering_space (c , p)) s) /\ ((simply_connected u) /\ (((locally path_connected) u) /\ (((continuous_on f) u) /\ (((IMAGE f) u) SUBSET s))))) ==> (?g. (((continuous_on g) u) /\ ((((IMAGE g) u) SUBSET c) /\ (!y. ((y IN u) ==> ((p (g y)) = (f y)))))))))))))
TENDSTO_LIM |- (!net. (!f. (!l. (((~ (trivial_limit net)) /\ (((--> f) l) net)) ==> (((lim net) f) = l)))))
CEXP_VSUM |- (!f. (!s. ((FINITE s) ==> ((cexp ((vsum s) f)) = ((cproduct s) (\x. (cexp (f x))))))))
AFF_LE_EQ_AFFINE_HULL |- (!s. (!t. (((aff_le s) EMPTY) = ((hull affine) s))))
INT_LNEG_UNIQ |- (!x. (!y. ((((int_add x) y) = (int_of_num (NUMERAL _0))) = (x = (int_neg y)))))
Topology.FINITE_SERIES |- (!n. (!f. (FINITE (GSPEC (\GEN%PVAR%5487. (?i. (((SETSPEC GEN%PVAR%5487) (i < n)) (f i))))))))
FORALL_INTEGER |- (!P. (((!n. (P (real_of_num n))) /\ (!x. ((P x) ==> (P (real_neg x))))) ==> (!x. ((integer x) ==> (P x)))))
MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED_EQ |- (!f. (!s. ((((measurable_on f) s) /\ (measurable s)) = (!t. ((closed t) ==> (measurable (GSPEC (\GEN%PVAR%3659. (?x. (((SETSPEC GEN%PVAR%3659) ((x IN s) /\ ((f x) IN t))) x))))))))))
SIMPLEX_FURTHEST_LE_EXISTS |- (!a. (!s. ((FINITE s) ==> (!x. ((x IN ((hull convex) s)) ==> (?y. ((y IN s) /\ ((real_le (vector_norm ((vector_sub x) a))) (vector_norm ((vector_sub y) a))))))))))
HOMEOMORPHIC_STANDARD_HYPERPLANE_HYPERPLANE |- (!a. (!b. (!k. (!c. (((~ (a = (vec (NUMERAL _0)))) /\ (((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV)))) ==> ((homeomorphic (GSPEC (\GEN%PVAR%1060. (?x. (((SETSPEC GEN%PVAR%1060) ((x $ k) = c)) x))))) (GSPEC (\GEN%PVAR%1061. (?x. (((SETSPEC GEN%PVAR%1061) (((dot a) x) = b)) x))))))))))
CARD_FACES_OF_SIMPLEX |- (!n. (!s. (((simplex n) s) ==> ((CARD (GSPEC (\GEN%PVAR%2481. (?f. (((SETSPEC GEN%PVAR%2481) ((face_of f) s)) f))))) = ((NUMERAL (BIT0 (BIT1 _0))) EXP (num_of_int ((int_add n) (int_of_num (NUMERAL (BIT1 _0))))))))))
REAL_SERIES_GOESTOZERO |- (!s. (!x. (((real_summable s) x) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\n. ((n IN s) ==> ((real_lt (real_abs (x n))) e)))) sequentially))))))
Localization.slicef |- (!E. (!FF. (!w. (!v. (((((slicef E) FF) v) w) = ((GSPEC (\GEN%PVAR%7541. (?f. (((SETSPEC GEN%PVAR%7541) (?u. ((u IN (((((slicev E) FF) v) w) DELETE w)) /\ (f = (u , ((rho_node1 FF) u)))))) f)))) UNION ((w , v) INSERT EMPTY)))))))
AFF_DIM_EQ_INTER_HYPERPLANE |- (!s. (!t. (((affine s) /\ ((affine t) /\ ((t SUBSET s) /\ (((int_add (aff_dim t)) (int_of_num (NUMERAL (BIT1 _0)))) = (aff_dim s))))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ (((GSPEC (\GEN%PVAR%1574. (?x. (((SETSPEC GEN%PVAR%1574) (((dot a) x) = b)) x)))) INTER s) = t)))))))
INFSUM_LINEAR |- (!f. (!h. (!s. ((((summable s) f) /\ (linear h)) ==> (((infsum s) (\n. (h (f n)))) = (h ((infsum s) f)))))))
Hypermap_iso.disjoint_diff |- (!s. (!t. ((DISJOINT s) (t DIFF s))))
ORTHOGONAL_MATRIX_ORTHONORMAL_COLUMNS_PAIRWISE |- (!A. ((orthogonal_matrix A) = (((CARD (columns A)) = (dimindex UNIV)) /\ ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((vector_norm ((column i) A)) = (real_of_num (NUMERAL (BIT1 _0)))))) /\ ((pairwise orthogonal) (columns A))))))
LIMPT_PCROSS |- (!s. (!t. (!x. (!y. ((((limit_point_of x) s) /\ ((limit_point_of y) t)) ==> ((limit_point_of ((pastecart x) y)) (s PCROSS t)))))))
Pack_defs.fcc_compatible |- (!S. (!f. (((fcc_compatible f) S) = (!v. ((v IN S) ==> ((real_le (sqrt (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_add (measure ((voronoi_open S) v))) (f v))))))))
COMPLEX_NEG_ADD |- (!x. (!y. ((vector_neg ((vector_add x) y)) = ((vector_add (vector_neg x)) (vector_neg y)))))
REAL_DIV_SQRT |- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((real_div x) (sqrt x)) = (sqrt x))))
REAL_CARD_INTSEG_INT |- (!a. (!b. (((integer a) /\ (integer b)) ==> ((real_of_num (CARD (GSPEC (\GEN%PVAR%86. (?x. (((SETSPEC GEN%PVAR%86) ((integer x) /\ (((real_le a) x) /\ ((real_le x) b)))) x)))))) = (((COND ((real_lt b) a)) (real_of_num (NUMERAL _0))) ((real_add ((real_sub b) a)) (real_of_num (NUMERAL (BIT1 _0)))))))))
AFF_DIM_CONVEX_INTER_NONEMPTY_INTERIOR |- (!s. (!t. (((convex s) /\ (~ ((s INTER (interior t)) = EMPTY))) ==> ((aff_dim (s INTER t)) = (aff_dim s)))))
SYMMETRIC_MATRIX_ORTHOGONAL_EIGENVECTORS |- (!A. (!v. (!w. (!a. (!b. ((((transp A) = A) /\ ((((matrix_vector_mul A) v) = ((% a) v)) /\ ((((matrix_vector_mul A) w) = ((% b) w)) /\ (~ (a = b))))) ==> ((orthogonal v) w)))))))
REAL_INTEGRABLE_ON_EMPTY |- (!f. ((real_integrable_on f) EMPTY))
REAL_UNIFORMLY_CONTINUOUS_ON_SUM |- (!t. (!f. (!s. (((FINITE s) /\ (!a. ((a IN s) ==> ((real_uniformly_continuous_on (f a)) t)))) ==> ((real_uniformly_continuous_on (\x. ((sum s) (\a. ((f a) x))))) t)))))
AR_IMP_PATH_CONNECTED |- (!s. ((AR s) ==> (path_connected s)))
REAL_SUM_INTEGRAL_BOUNDS_INCREASING |- (!f. (!g. (!m. (!n. (((m <= n) /\ ((!x. ((x IN (closed_real_interval ((CONS (((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0)))) , ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) NIL))) ==> (((has_real_derivative g) (f x)) ((within (atreal x)) (closed_real_interval ((CONS (((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0)))) , ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) NIL)))))) /\ (!x. (!y. ((((real_le ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))) x) /\ (((real_le x) y) /\ ((real_le y) ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) ==> ((real_le (f x)) (f y))))))) ==> (((real_le ((real_sub (g (real_of_num n))) (g ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))))) ((sum (m .. n)) (\k. (f (real_of_num k))))) /\ ((real_le ((sum (m .. n)) (\k. (f (real_of_num k))))) ((real_sub (g ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) (g (real_of_num m))))))))))
SUMMABLE_IMP_SUMS_BOUNDED |- (!f. (!k. (((summable (from k)) f) ==> (bounded (GSPEC (\GEN%PVAR%1120. (?n. (((SETSPEC GEN%PVAR%1120) (n IN UNIV)) ((vsum (k .. n)) f)))))))))
RETRACT_OF_HOMOTOPICALLY_TRIVIAL_NULL |- (!s. (!t. (!u. ((((retract_of t) s) /\ (!f. ((((continuous_on f) u) /\ (((IMAGE f) u) SUBSET s)) ==> (?c. ((((homotopic_with (\x. T)) (u , s)) f) (\x. c)))))) ==> (!f. ((((continuous_on f) u) /\ (((IMAGE f) u) SUBSET t)) ==> (?c. ((((homotopic_with (\x. T)) (u , t)) f) (\x. c)))))))))
RELATIVE_FRONTIER_RELATIVE_INTERIOR |- (!s. ((convex s) ==> ((relative_frontier (relative_interior s)) = (relative_frontier s))))
DIST_TRIANGLE_ADD_HALF |- (!x. (!x'. (!y. (!y'. ((((real_lt (distance (x , x'))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) /\ ((real_lt (distance (y , y'))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> ((real_lt (distance (((vector_add x) y) , ((vector_add x') y')))) e))))))
ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_LEFT |- (!f. (!a. (!b. (((absolutely_integrable_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((absolutely_continuous_on (\x. ((integral (closed_interval ((CONS (x , b)) NIL))) f))) (closed_interval ((CONS (a , b)) NIL)))))))
ORTHOGONAL_MATRIX_ORTHOGONAL_EIGENVECTORS |- (!A. (!v. (!w. (!a. (!b. (((orthogonal_matrix A) /\ ((((matrix_vector_mul A) v) = ((% a) v)) /\ ((((matrix_vector_mul A) w) = ((% b) w)) /\ (~ (a = b))))) ==> ((orthogonal v) w)))))))
REAL_CONTINUOUS_ON_ACS |- (!s. ((!x. ((x IN s) ==> ((real_le (real_abs x)) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((real_continuous_on acs) s)))
COMPLEX_DERIVATIVE_SUB_AT |- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\ ((analytic_on g) (z INSERT EMPTY))) ==> (((complex_derivative (\w. ((vector_sub (f w)) (g w)))) z) = ((vector_sub ((complex_derivative f) z)) ((complex_derivative g) z)))))))
MEASURE_INTERVAL_4_ALT |- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND (((real_lt (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ (((real_lt (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_lt (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\ ((real_lt (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 (BIT1 _0))))) (a $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (NUMERAL _0)))))) /\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND (((real_lt (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ (((real_lt (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_lt (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\ ((real_lt (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 (BIT1 _0))))) (a $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (NUMERAL _0)))))))
HAS_REAL_MEASURE_UNION_REAL_NEGLIGIBLE_EQ |- (!s. (!t. (!m. ((real_negligible t) ==> (((has_real_measure (s UNION t)) m) = ((has_real_measure s) m))))))
INDEPENDENT_EQ_DIM_EQ_CARD |- (!s. ((independent s) = ((FINITE s) /\ ((dim s) = (CARD s)))))
NONNEGATIVE_INTEGER |- (!x. (((integer x) /\ ((real_le (real_of_num (NUMERAL _0))) x)) = (?n. (x = (real_of_num n)))))
CLOSED_INTERVAL_RIGHT |- (!a. (closed (GSPEC (\GEN%PVAR%762. (?x. (((SETSPEC GEN%PVAR%762) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (a $ i)) (x $ i))))) x))))))
COMMUTING_WITH_SQUARE_ROOT_MATRIX |- (!A. (!S. (!B. (((positive_semidefinite S) /\ ((((matrix_mul S) S) = A) /\ (((matrix_mul A) B) = ((matrix_mul B) A)))) ==> (((matrix_mul S) B) = ((matrix_mul B) S))))))
RESTRICTION_EQ |- (!s. (!f. (!x. (!y. (((x IN s) /\ ((f x) = y)) ==> ((((RESTRICTION s) f) x) = y))))))
Functional_equation.functional_halfbump_x4 |- (halfbump_x4 = (rotate4 halfbump_x1))
POSITIVE_SEMIDEFINITE_2 |- (!A. ((positive_semidefinite A) = ((((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0))) = ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_le (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) /\ (((real_le (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\ ((real_le ((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)))))))))))