hcju/holps huggingface dataset:
Nonlinear_lemma.halfbump_x_expand |- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((halfbump_x x) = ((real_add (real_neg ((real_div (real_of_num (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))))))))))))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))))))))))))) ((real_sub ((real_mul ((real_div (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))))))))) (real_of_num (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))))))))) (sqrt x))) ((real_mul ((real_div (real_of_num (NUMERAL (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))))))))))) (real_of_num (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))))) x))))))
Functional_equation.functional_proj_y1 |- (proj_y1 = (promote1_to_6 sqrt))
MEASURABLE_CLOSED_IN |- (!s. (!t. ((((closed_in ((subtopology euclidean) s)) t) /\ (measurable s)) ==> (measurable t))))
REAL_MEASURABLE_ON_MIN |- (!f. (!g. (!s. ((((real_measurable_on f) s) /\ ((real_measurable_on g) s)) ==> ((real_measurable_on (\x. ((real_min (f x)) (g x)))) s)))))
DIFFERENTIABLE_ON_EQ |- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((differentiable_on f) s)) ==> ((differentiable_on g) s)))))
NORMAL_MATRIX_KERNEL_TRANSP |- (!A. ((((matrix_mul (transp A)) A) = ((matrix_mul A) (transp A))) ==> ((GSPEC (\GEN%PVAR%234. (?x. (((SETSPEC GEN%PVAR%234) (((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0)))) x)))) = (GSPEC (\GEN%PVAR%235. (?x. (((SETSPEC GEN%PVAR%235) (((matrix_vector_mul A) x) = (vec (NUMERAL _0)))) x)))))))
MATRIX_CMUL_ADD_RDISTRIB |- (!A. (!b. (!c. (((%% ((real_add b) c)) A) = ((matrix_add ((%% b) A)) ((%% c) A))))))
Lp_ineqs_proofs.tau_sum5 |- (!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 (BIT0 (BIT1 _0))))) ==> (((sum f) (rhazim_fan (V , E))) = ((real_add ((tau_fan (V , E)) f)) ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((real_add pi) sol0))))))))))))))
REALLIM_NULL_LMUL |- (!net. (!f. (!c. ((((---> f) (real_of_num (NUMERAL _0))) net) ==> (((---> (\x. ((real_mul c) (f x)))) (real_of_num (NUMERAL _0))) net)))))
Lp_gen_theory.ALL_list_dart_pairs_split |- (!L. (!P. (((ALL P) (list_dart_pairs L)) = ((ALL (\d1. ((ALL (\d2. (P (d1 , d2)))) (list_of_darts L)))) (list_of_darts L)))))
REAL_INTEGRABLE_RMUL_EQ |- (!f. (!s. (!c. (((real_integrable_on (\x. ((real_mul (f x)) c))) s) = ((c = (real_of_num (NUMERAL _0))) \/ ((real_integrable_on f) s))))))
AR_BALL |- (!a. (!r. ((AR (ball (a , r))) = ((real_lt (real_of_num (NUMERAL _0))) r))))
Functional_equation.functional_proj_y2 |- (proj_y2 = (rotate2 proj_y1))
Lp_ineqs_proofs.y5_hi_std |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((y5_fan (V , E)) d)) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))))))))))
ABSOLUTELY_INTEGRABLE_CHANGE_OF_VARIABLES |- (!f. (!g. (!g'. (!s. (((lebesgue_measurable s) /\ ((!x. ((x IN s) ==> (((has_derivative g) (g' x)) ((within (at x)) s)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((g x) = (g y)))) ==> (x = y)))))) ==> (((absolutely_integrable_on f) ((IMAGE g) s)) = ((absolutely_integrable_on (\x. ((% (real_abs (det (matrix (g' x))))) (f (g x))))) s)))))))
UNION_INTERIOR_SUBSET |- (!s. (!t. (((interior s) UNION (interior t)) SUBSET (interior (s UNION t)))))
NEGATIONS_SPHERE |- (!r. (((IMAGE vector_neg) (sphere ((vec (NUMERAL _0)) , r))) = (sphere ((vec (NUMERAL _0)) , r))))
CARD_UNION_OVERLAP |- (!s. (!t. (((FINITE s) /\ ((FINITE t) /\ ((CARD (s UNION t)) < ((CARD s) + (CARD t))))) ==> (~ ((s INTER t) = EMPTY)))))
NEGLIGIBLE_IFF_MEASURABLE_SUBSETS |- (!s. ((negligible s) = (!t. ((t SUBSET s) ==> (measurable t)))))
ARC_JOIN_EQ_ALT |- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((arc ((++ g1) g2)) = ((arc g1) /\ ((arc g2) /\ (((path_image g1) INTER (path_image g2)) = ((pathstart g2) INSERT EMPTY))))))))
Appendix.LENGTH_s_init_list |- ((LENGTH s_init_list_v39) = (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))
Lp_ineqs_proofs.sol_sum3 |- (!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)))) ==> (((CARD f) = (NUMERAL (BIT1 (BIT1 _0)))) ==> (((sum f) (azim_dart (V , E))) = ((real_add ((sol_fan (V , E)) f)) pi)))))))))))
VSUM_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)))))) ==> (((vsum s) (f o p)) = ((vsum s) f))))))
NEIGHBOURHOOD_EXTENSION_INTO_ANR |- (!f. (!s. (!t. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ ((ANR t) /\ (closed s)))) ==> (?v. (?g. ((s SUBSET v) /\ ((open v) /\ (((continuous_on g) v) /\ ((((IMAGE g) v) SUBSET t) /\ (!x. ((x IN s) ==> ((g x) = (f x))))))))))))))
BERNOULLI_NUMBER_ALT |- (!n. (((sum ((NUMERAL _0) .. n)) (\k. ((real_mul (real_of_num (binom ((n + (NUMERAL (BIT1 _0))) , k)))) ((bernoulli k) (real_of_num (NUMERAL _0)))))) = (((COND (n = (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))
LIM_WITHIN_EMPTY |- (!f. (!l. (!x. (((--> f) l) ((within (at x)) EMPTY)))))
Miqmcsn.J_SCS_4M8_13 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_4M8_13) i)) i1) j) = F)
STC_IDEMP |- (!R. ((STC (STC R)) = (STC R)))
ROTATE2D_SUB_ARG |- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> ((rotate2d ((real_sub (Arg w)) (Arg z))) = (rotate2d (Arg ((complex_div w) z)))))))
HOMOTOPIC_COMPOSE |- (!f. (!f'. (!g. (!g'. (!s. (!t. (!u. ((((((homotopic_with (\x. T)) (s , t)) f) f') /\ ((((homotopic_with (\x. T)) (t , u)) g) g')) ==> ((((homotopic_with (\x. T)) (s , u)) (g o f)) (g' o f'))))))))))
CLOSEST_POINT_TRANSLATION |- (!s. (!a. (((convex s) /\ ((closed s) /\ (~ (s = EMPTY)))) ==> (((closest_point ((IMAGE (\x. ((vector_add a) x))) s)) ((vector_add a) x)) = ((vector_add a) ((closest_point s) x))))))
LT_MULT2 |- (!m. (!n. (!p. (!q. (((m < n) /\ (p < q)) ==> ((m * p) < (n * q)))))))
INT_MIN_LE |- (!x. (!y. (!z. (((int_le ((int_min x) y)) z) = (((int_le x) z) \/ ((int_le y) z))))))
COLLINEAR_TRANSLATION |- (!s. (!a. ((collinear s) ==> (collinear ((IMAGE (\x. ((vector_add a) x))) s)))))
Functional_equation.functional_proj_x2 |- (proj_x2 = (rotate2 proj_x1))
TIETZE_CLOSED_INTERVAL_1 |- (!f. (!u. (!s. (!a. (!b. ((((real_le (drop a)) (drop b)) /\ (((closed_in ((subtopology euclidean) u)) s) /\ (((continuous_on f) s) /\ (!x. ((x IN s) ==> ((f x) IN (closed_interval ((CONS (a , b)) NIL)))))))) ==> (?g. (((continuous_on g) u) /\ ((!x. ((x IN s) ==> ((g x) = (f x)))) /\ (!x. ((x IN u) ==> ((g x) IN (closed_interval ((CONS (a , b)) NIL))))))))))))))
Trigonometry.QAFHJNM |- (!v. (!w. (!x. ((LET (\e3. (LET_END ((LET (\r. (LET_END ((LET (\phi. (LET_END (((~ (v = x)) /\ (~ (v = w))) ==> (?u'. ((((dot u') e3) = (real_of_num (NUMERAL _0))) /\ (x = ((vector_add v) ((vector_add u') ((% ((real_mul r) (cos phi))) e3)))))))))) (((arcV v) w) x))))) (vector_norm ((vector_sub x) v)))))) ((% ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (vector_norm ((vector_sub w) v)))) ((vector_sub w) v))))))
HAS_BOUNDED_REAL_VARIATION_AFFINITY_EQ |- (!m. (!c. (!f. (!s. (((has_bounded_real_variation_on (\x. (f ((real_add ((real_mul m) x)) c)))) s) = ((m = (real_of_num (NUMERAL _0))) \/ ((has_bounded_real_variation_on f) ((IMAGE (\x. ((real_add ((real_mul m) x)) c))) s))))))))
Lp_ineqs_proofs.SURJ_pair |- (!h1. (!h2. (!g. ((!x. ((g x) = ((h1 (FST x)) , (h2 (SND x))))) ==> (!s1. (!t1. (!s2. (!t2. (((((SURJ h1) s1) t1) /\ (((SURJ h2) s2) t2)) ==> (((SURJ g) (s1 CROSS s2)) (t1 CROSS t2)))))))))))
CARD_LE_COMPONENTS |- (!f. (!s. (((continuous_on f) s) ==> ((components ((IMAGE f) s)) <=_c (components s)))))
Tame_inequalities.DELTA_X4_MONO_LE_4 |- (!x1. (!x2. (!x3. (!a. (!x5. (!x6. (!b. ((((real_lt a) b) /\ ((real_le (real_of_num (NUMERAL _0))) x1)) ==> ((real_le ((((((delta_x4 x1) x2) x3) b) x5) x6)) ((((((delta_x4 x1) x2) x3) a) x5) x6))))))))))
REAL_DIFFERENTIABLE_WITHIN_EXP |- (!s. (!x. ((real_differentiable exp) ((within (atreal x)) s))))
Seq.eq_filter |- (!a1. (!a2. ((!x. ((a1 x) = (a2 x))) ==> (!s. (((filter a1) s) = ((filter a2) s))))))
SUMMABLE_SUB |- (!x. (!y. (!s. ((((summable s) x) /\ ((summable s) y)) ==> ((summable s) (\n. ((vector_sub (x n)) (y n))))))))
HOMOTOPIC_INVERTIBLE_LINEAR_MAPS |- (!f. (!g. (((((homotopic_with (\h. ((linear h) /\ (invertible (matrix h))))) (UNIV , UNIV)) f) g) = ((linear f) /\ ((linear g) /\ ((real_lt (real_of_num (NUMERAL _0))) ((real_mul (det (matrix f))) (det (matrix g)))))))))
INT_LT_POW2 |- (!n. ((int_lt (int_of_num (NUMERAL _0))) ((int_pow (int_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)))
INT_SGN_POW |- (!x. (!n. ((int_sgn ((int_pow x) n)) = ((int_pow (int_sgn x)) n))))
CONTINUOUS_TRANSFORM_WITHINREAL_SET_IMP |- (!f. (!a. (!s. (!t. ((((eventually (\x. ((x IN t) ==> (x IN s)))) (atreal a)) /\ ((continuous f) ((within (atreal a)) s))) ==> ((continuous f) ((within (atreal a)) t)))))))
CONIC_IMP_SIMPLY_CONNECTED |- (!s. ((conic s) ==> (simply_connected s)))
More_list_hypermap.contravening_dist_dart_le |- (!V. (!g. (!L. ((contravening V) ==> ((good_list L) ==> (((hyp_iso g) ((hypermap_of_list L) , (hypermap_of_fan (V , (ESTD V))))) ==> ((ALL (\d. ((real_le (distance (g d))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (list_of_darts L))))))))
AFFINE_EXPLICIT |- (!s. ((affine s) = (!t. (!u. (((FINITE t) /\ ((t SUBSET s) /\ (((sum t) u) = (real_of_num (NUMERAL (BIT1 _0)))))) ==> (((vsum t) (\x. ((% (u x)) x))) IN s))))))
CONVEX |- (!V. ((convex V) = (!s. (!u. (((FINITE s) /\ ((~ (s = EMPTY)) /\ ((s SUBSET V) /\ ((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (u x)))) /\ (((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> (((vsum s) (\x. ((% (u x)) x))) IN V))))))
COMPLEX_NEG_RMUL |- (!x. (!y. ((vector_neg ((complex_mul x) y)) = ((complex_mul x) (vector_neg y)))))
INT_EQ_MUL_LCANCEL |- (!x. (!y. (!z. ((((int_mul x) y) = ((int_mul x) z)) = ((x = (int_of_num (NUMERAL _0))) \/ (y = z))))))
Seq2.butlast_cons |- (!h. (!t. ((butlast ((CONS h) t)) = (((COND (t = NIL)) NIL) ((CONS h) (butlast t))))))
FRONTIER_SPHERE |- (!a. (!r. ((frontier (sphere (a , r))) = (sphere (a , r)))))
SUMMABLE_COMPLEX_RMUL |- (!f. (!c. (!s. (((summable s) f) ==> ((summable s) (\x. ((complex_mul (f x)) c)))))))
HOMOTOPIC_PATHS_RCANCEL_EQ |- (!p. (!q. (!r. (!s. ((((pathfinish p) = (pathstart r)) /\ ((pathfinish q) = (pathstart r))) ==> ((((homotopic_paths s) ((++ p) r)) ((++ q) r)) = ((((homotopic_paths s) p) q) /\ ((path r) /\ ((path_image r) SUBSET s)))))))))
DEFORMATION_RETRACT_OF_CONTRACTIBLE_SING |- (!s. (!a. (((contractible s) /\ (a IN s)) ==> (?r. (((((homotopic_with (\x. T)) (s , s)) (\x. x)) r) /\ ((retraction (s , (a INSERT EMPTY))) r))))))
IS_INTERVAL_SIMPLY_CONNECTED_1 |- (!s. ((is_interval s) = (simply_connected s)))
REGULAR_OPEN_INTER |- (!s. (!t. ((((interior (closure s)) = s) /\ ((interior (closure t)) = t)) ==> ((interior (closure (s INTER t))) = (s INTER t)))))
HAS_INTEGRAL_LIM_SEQUENTIALLY |- (!f. (!l. (((((--> (f o lift)) (vec (NUMERAL _0))) at_posinfinity) /\ ((!n. ((integrable_on f) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec n))) NIL)))) /\ (((--> (\n. ((integral (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec n))) NIL))) f))) l) sequentially))) ==> (((has_integral f) l) (GSPEC (\GEN%PVAR%2954. (?t. (((SETSPEC GEN%PVAR%2954) ((real_le (real_of_num (NUMERAL _0))) (drop t))) t))))))))
INT_LE_NEGR |- (!x. (((int_le x) (int_neg x)) = ((int_le x) (int_of_num (NUMERAL _0)))))
Lp_ineqs_proofs.rho_lo |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (real_of_num (NUMERAL (BIT1 _0)))) (rho_fan v))))))))))
VITALI_COVERING_LEMMA_CBALLS |- (!s. (!a. (!r. (!k. (!B. (((s SUBSET (UNIONS ((IMAGE (\i. (cball ((a i) , (r i))))) k))) /\ (!i. ((i IN k) ==> (((real_lt (real_of_num (NUMERAL _0))) (r i)) /\ ((real_le (r i)) B))))) ==> (?c. ((COUNTABLE c) /\ ((c SUBSET k) /\ (((pairwise (\i. (\j. ((DISJOINT (cball ((a i) , (r i)))) (cball ((a j) , (r j))))))) c) /\ (s SUBSET (UNIONS ((IMAGE (\i. (cball ((a i) , ((real_mul (real_of_num (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) (r i)))))) c)))))))))))))
VECTOR_EXPAND_1 |- (!x. (x = (vector ((CONS (x $ (NUMERAL (BIT1 _0)))) NIL))))
STEINHAUS_DIFFS |- (!s. (!t. (((lebesgue_measurable s) /\ ((~ (negligible s)) /\ ((lebesgue_measurable t) /\ (~ (negligible t))))) ==> (~ ((interior (GSPEC (\GEN%PVAR%4608. (?x. (?y. (((SETSPEC GEN%PVAR%4608) ((x IN s) /\ (y IN t))) ((vector_sub x) y))))))) = EMPTY)))))
INTEGRABLE_UNION_EQ |- (!f. (!s. (!t. ((((integrable_on f) s) /\ ((integrable_on f) t)) ==> (((integrable_on f) (s UNION t)) = ((integrable_on f) (s INTER t)))))))
SING_STRONG_DEFORMATION_RETRACT_OF_AR |- (!s. (!a. (((AR s) /\ (a IN s)) ==> (?r. (((((homotopic_with (\h. ((h a) = a))) (s , s)) (\x. x)) r) /\ ((retraction (s , (a INSERT EMPTY))) r))))))
INT_SUB_ADD2 |- (!x. (!y. (((int_add y) ((int_sub x) y)) = x)))
DIFFERENTIABLE_ON_EMPTY |- (!f. ((differentiable_on f) EMPTY))
REAL_MEASURE_DISJOINT_UNIONS |- (!m. (!f. (((FINITE f) /\ ((!s. ((s IN f) ==> ((has_real_measure s) (m s)))) /\ (!s. (!t. (((s IN f) /\ ((t IN f) /\ (~ (s = t)))) ==> ((DISJOINT s) t)))))) ==> ((real_measure (UNIONS f)) = ((sum f) m)))))
IN_BALL_RE |- (!x. (!z. (!e. ((x IN (ball (z , e))) ==> ((real_lt (real_abs ((real_sub (Re x)) (Re z)))) e)))))
Tame_list.rev2 |- (!x. (!y. ((rev ((CONS x) ((CONS y) NIL))) = ((CONS y) ((CONS x) NIL)))))
INT_SGN_INEQS |- ((!x. (((int_le (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_le (int_of_num (NUMERAL _0))) x))) /\ ((!x. (((int_lt (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_lt (int_of_num (NUMERAL _0))) x))) /\ ((!x. (((int_ge (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_ge (int_of_num (NUMERAL _0))) x))) /\ ((!x. (((int_gt (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_gt (int_of_num (NUMERAL _0))) x))) /\ ((!x. (((int_of_num (NUMERAL _0)) = (int_sgn x)) = ((int_of_num (NUMERAL _0)) = x))) /\ ((!x. (((int_le (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_le x) (int_of_num (NUMERAL _0))))) /\ ((!x. (((int_lt (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_lt x) (int_of_num (NUMERAL _0))))) /\ ((!x. (((int_ge (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_ge x) (int_of_num (NUMERAL _0))))) /\ ((!x. (((int_gt (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_gt x) (int_of_num (NUMERAL _0))))) /\ (!x. (((int_sgn x) = (int_of_num (NUMERAL _0))) = (x = (int_of_num (NUMERAL _0))))))))))))))
CONVEX_CONE_HULL_NONEMPTY |- (!s. (~ (((hull convex_cone) s) = EMPTY)))
INT_LT_ADD1 |- (!x. (!y. (((int_le x) y) ==> ((int_lt x) ((int_add y) (int_of_num (NUMERAL (BIT1 _0))))))))
Seq.subseq_uniqP |- (!s1. (!s2. ((uniq s2) ==> (((subseq s1) s2) = (s1 = ((filter (\x. ((MEM x) s1))) s2))))))
Lp_ineqs_proofs.azim3c |- (!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))) ==> (((azim3_fan (V , E)) d) = ((azim_dart (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))
Tactics_jordan.switch_lemma_le |- (!A. (!B. (!C. ((((real_le (real_of_num (NUMERAL _0))) A) ==> (((real_le (real_of_num (NUMERAL _0))) B) ==> C)) = (((real_le (real_of_num (NUMERAL _0))) B) ==> (((real_le (real_of_num (NUMERAL _0))) A) ==> C))))))
HOLOMORPHIC_ON_DIV |- (!f. (!g. (!s. ((((holomorphic_on f) s) /\ (((holomorphic_on g) s) /\ (!z. ((z IN s) ==> (~ ((g z) = (Cx (real_of_num (NUMERAL _0))))))))) ==> ((holomorphic_on (\z. ((complex_div (f z)) (g z)))) s)))))
ABSOLUTE_RETRACT_FROM_UNION_AND_INTER |- (!s. (!t. ((((retract_of (s UNION t)) UNIV) /\ (((retract_of (s INTER t)) UNIV) /\ ((closed s) /\ (closed t)))) ==> ((retract_of s) UNIV))))
INTEGRABLE_SPIKE_EQ |- (!f. (!g. (!s. (!t. (((negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((integrable_on f) t) = ((integrable_on g) t)))))))
CSQRT_EQ_0 |- (!z. (((csqrt z) = (Cx (real_of_num (NUMERAL _0)))) = (z = (Cx (real_of_num (NUMERAL _0))))))
LIM_COMPLEX_REAL_0 |- (!f. (!g. ((((eventually (\n. ((Re (g n)) = (f n)))) sequentially) /\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) sequentially)) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (real_abs (f n))) e)))))))))
CONTINUOUS_ON_CX_NORM |- (!s. ((continuous_on (\z. (Cx (vector_norm z)))) s))
Add_triangle.ext_in_dart1 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> (((ext_dart (V , E)) d) IN (d1_fan ((vec (NUMERAL _0)) , (V , E)))))))))
INFSUM_0 |- (((infsum s) (\i. (vec (NUMERAL _0)))) = (vec (NUMERAL _0)))
Local_lemmas1.CON_ATREAL_REAL_CON2 |- (!f. ((((continuous f) (atreal t)) /\ ((continuous g) (atreal t))) ==> ((real_continuous (\t. (distance ((f t) , (g t))))) (atreal t))))
Seq2.foldr |- (!x. (!s. (!f. (!z0. (((((foldr f) z0) ((CONS x) s)) = ((f x) (((foldr f) z0) s))) /\ ((((foldr f) z0) NIL) = z0))))))
REAL_INTEGRABLE_SPIKE_FINITE |- (!f. (!g. (!s. (((FINITE s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((real_integrable_on f) t) ==> ((real_integrable_on g) t))))))
NSUM_UNIONS_NONZERO |- (!f. (!s. (((FINITE s) /\ ((!t. ((t IN s) ==> (FINITE t))) /\ (!t1. (!t2. (!x. (((t1 IN s) /\ ((t2 IN s) /\ ((~ (t1 = t2)) /\ ((x IN t1) /\ (x IN t2))))) ==> ((f x) = (NUMERAL _0)))))))) ==> (((nsum (UNIONS s)) f) = ((nsum s) (\t. ((nsum t) f)))))))
Localization.order |- (!f. (!x. (!y. ((((order f) x) y) = (@n. (((((ITER n) f) x) = y) /\ (!i. ((((NUMERAL _0) < i) /\ (i < n)) ==> (~ ((((ITER i) f) x) = y))))))))))
COMPLEX_DIFFERENTIABLE_WITHIN_CLOG |- (!s. (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> ((complex_differentiable clog) ((within (at z)) s)))))
INTEGRAL_PASTECART_CONST |- (!a. (!b. (!c. (!d. (!k. (((integral (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) (\x. k)) = ((integral (closed_interval ((CONS (a , b)) NIL))) (\x. ((integral (closed_interval ((CONS (c , d)) NIL))) (\y. k))))))))))
FSIGMA_PCROSS_EQ |- (!s. (!t. ((fsigma (s PCROSS t)) = ((s = EMPTY) \/ ((t = EMPTY) \/ ((fsigma s) /\ (fsigma t)))))))
FSIGMA_LINEAR_IMAGE |- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((fsigma ((IMAGE f) s)) = (fsigma s)))))
INTEGER_SUB_EQ |- ((!x. (!y. ((integer x) ==> ((integer ((real_sub x) y)) = (integer y))))) /\ (!x. (!y. ((integer y) ==> ((integer ((real_sub x) y)) = (integer x))))))
Topology.union_aff |- (!x. (!v. (UNIV = ((aff (x INSERT (v INSERT EMPTY))) UNION (complement_set (x INSERT (v INSERT EMPTY)))))))
ANR_IMP_CLOSED_NEIGHBOURHOOD_RETRACT |- (!s. (!u. (((ANR s) /\ ((closed_in ((subtopology euclidean) u)) s)) ==> (?v. (?w. (((open_in ((subtopology euclidean) u)) v) /\ (((closed_in ((subtopology euclidean) u)) w) /\ ((s SUBSET v) /\ ((v SUBSET w) /\ ((retract_of s) w))))))))))
DIVMOD_ELIM_THM' |- (((P (m DIV n)) (m MOD n)) = (?q. (?r. ((((n = (NUMERAL _0)) /\ ((q = (NUMERAL _0)) /\ (r = m))) \/ ((m = ((q * n) + r)) /\ (r < n))) /\ ((P q) r)))))
SUMS_COMPLEX_0 |- (!f. (!s. ((!n. ((n IN s) ==> ((f n) = (Cx (real_of_num (NUMERAL _0)))))) ==> (((sums f) (Cx (real_of_num (NUMERAL _0)))) s))))
MBOUNDED_SUBSET_MSPACE |- (!m. (!s. (((mbounded m) s) ==> (s SUBSET (mspace m)))))
INT_ADD_LDISTRIB |- (!x. (!y. (!z. (((int_mul x) ((int_add y) z)) = ((int_add ((int_mul x) y)) ((int_mul x) z))))))
TC_TC_RTC_CASES |- (!R. (!x. (!z. ((((TC R) x) z) = (?y. ((((TC R) x) y) /\ (((RTC R) y) z)))))))
LOCALLY_COMPACT_IMP_BOREL |- (!s. (((locally compact) s) ==> (borel s)))
LIM_CONG_WITHINREAL |- ((!x. ((~ (x = a)) ==> ((f x) = (g x)))) ==> ((((--> (\x. (f x))) l) ((within (atreal a)) s)) = (((--> g) l) ((within (atreal a)) s))))
NOT_PSUBSET_EMPTY |- (!s. (~ (s PSUBSET EMPTY)))
LOG_LE |- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (log ((real_add (real_of_num (NUMERAL (BIT1 _0)))) x))) x)))
CONVERGENT_REAL_BOUNDED_MONOTONE |- (!s. (((real_bounded ((IMAGE s) UNIV)) /\ ((!n. ((real_le (s n)) (s (SUC n)))) \/ (!n. ((real_le (s (SUC n))) (s n))))) ==> (?l. (((---> s) l) sequentially))))
CENTRE_IN_MCBALL_EQ |- (!m. (!x. (!r. ((x IN ((mcball m) (x , r))) = ((x IN (mspace m)) /\ ((real_le (real_of_num (NUMERAL _0))) r))))))
SETDIST_FRONTIERS |- (!s. (!t. ((setdist (s , t)) = (((COND ((DISJOINT s) t)) (setdist ((frontier s) , (frontier t)))) (real_of_num (NUMERAL _0))))))
Nonlinear_lemma.sqrt_x1 |- (((((((sqrt_x1 x1) x2) x3) x4) x5) x6) = (sqrt x1))
Ssrnat.odd_double |- (!n. ((odd (double n)) = F))
Seq.nth_incr_nth |- (!v. (!i. (!j. ((((nth (NUMERAL _0)) ((incr_nth v) i)) j) = ((((COND (i = j)) (NUMERAL (BIT1 _0))) (NUMERAL _0)) + (((nth (NUMERAL _0)) v) j))))))
Lp_main_estimate.ineq_tau6 |- (!V. (!E. ((lp_fan (V , E)) ==> ((!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)))))) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le ((y5_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (y6_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le ((y8_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le ((y9_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (ye_fan (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) d))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ ((real_le (ye_fan (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) d))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (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))))))))))
INT_POW_LT |- (!x. (!n. (((int_lt (int_of_num (NUMERAL _0))) x) ==> ((int_lt (int_of_num (NUMERAL _0))) ((int_pow x) n)))))
NPRODUCT_CONST_NUMSEG_1 |- (!c. (!n. (((nproduct ((NUMERAL (BIT1 _0)) .. n)) (\x. c)) = (c EXP n))))
COMPLEX_NEG_LMUL |- (!x. (!y. ((vector_neg ((complex_mul x) y)) = ((complex_mul (vector_neg x)) y))))
COMPLEX_DIFFERENTIABLE_INV_AT |- (!f. (!z. ((((complex_differentiable f) (at z)) /\ (~ ((f z) = (Cx (real_of_num (NUMERAL _0)))))) ==> ((complex_differentiable (\z. (complex_inv (f z)))) (at z)))))
REAL_VARIATION_TRIANGLE |- (!f. (!g. (!s. ((((has_bounded_real_variation_on f) s) /\ ((has_bounded_real_variation_on g) s)) ==> ((real_le ((real_variation s) (\x. ((real_add (f x)) (g x))))) ((real_add ((real_variation s) f)) ((real_variation s) g)))))))