hcju/holps huggingface dataset:
MEASURABLE_CONTINUOUS_COMPOSE |- (!f. (!g. ((((measurable_on f) UNIV) /\ (((continuous_on g) UNIV) /\ (!k. ((negligible k) ==> (negligible (GSPEC (\GEN%PVAR%3702. (?x. (((SETSPEC GEN%PVAR%3702) ((g x) IN k)) x))))))))) ==> ((measurable_on (f o g)) UNIV))))
CONTINUOUS_ON_COMPLEX_RMUL |- (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\x. ((complex_mul (f x)) c))) s))))
VECTOR_VARIATION_LIFT_ABS |- (!f. (!s. (((has_bounded_variation_on (\x. (lift (f x)))) s) ==> ((real_le ((vector_variation s) (\x. (lift (real_abs (f x)))))) ((vector_variation s) (\x. (lift (f x))))))))
INCREASING_BOUNDED_REAL_VARIATION |- (!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 x)) (f y))))) ==> ((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , b)) NIL)))))))
TRIANGULATION_INTER_SIMPLEX |- (!tr. (!t. (!t'. (((triangulation tr) /\ ((t IN tr) /\ (t' IN tr))) ==> ((t INTER t') = ((hull convex) ((GSPEC (\GEN%PVAR%2491. (?x. (((SETSPEC GEN%PVAR%2491) ((extreme_point_of x) t)) x)))) INTER (GSPEC (\GEN%PVAR%2492. (?x. (((SETSPEC GEN%PVAR%2492) ((extreme_point_of x) t')) x)))))))))))
Functional_equation.functional_halfbump_x1 |- (halfbump_x1 = (promote1_to_6 halfbump_x))
Nonlinear_lemma.vol3f_lm0 |- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y1) /\ (((real_le y2) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ ((real_le y6) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ==> ((((((vol3f y1) y2) y6) sqrt2) lmfun) = ((((((vol3f_sqrt2_lmplus y1) y2) y3) y4) y5) y6)))))))))
INT_SUB_LNEG |- (!x. (!y. (((int_sub (int_neg x)) y) = (int_neg ((int_add x) y)))))
Lp_ineqs_proofs.lp_cond_trans |- (!L. (!V. (!E. (!g. (!h. (!ds. (((lp_cond (L , (g , h))) (V , E)) ==> ((lp_cond (((split_list_list L) ds) , (g , h))) (V , ((add_diag_list (g , (V , E))) ds))))))))))
AFFINE_INDEPENDENT_STDBASIS |- (~ (affine_dependent ((vec (NUMERAL _0)) INSERT (GSPEC (\GEN%PVAR%1517. (?i. (((SETSPEC GEN%PVAR%1517) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (basis i))))))))
INTEGER_POW |- (!x. (!n. ((integer x) ==> (integer ((real_pow x) n)))))
RECTIFIABLE_PATH_VECTOR_POLYNOMIAL_FUNCTION |- (!p. ((vector_polynomial_function p) ==> (rectifiable_path p)))
PATH_CONNECTED_NEGATIONS |- (!s. ((path_connected s) ==> (path_connected ((IMAGE vector_neg) s))))
HOMOTOPY_EQUIVALENT |- (!s. (!t. (((homotopy_equivalent s) t) = (?f. (?g. (?h. (((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET s) /\ (((continuous_on h) t) /\ ((((IMAGE h) t) SUBSET s) /\ (((((homotopic_with (\x. T)) (s , s)) (g o f)) I) /\ ((((homotopic_with (\x. T)) (t , t)) (f o h)) I))))))))))))))
MAXIMAL_AFFINE_INDEPENDENT_SUBSET_AFFINE |- (!s. (!b. (((affine s) /\ ((b SUBSET s) /\ ((~ (affine_dependent b)) /\ (!b'. (((b SUBSET b') /\ ((b' SUBSET s) /\ (~ (affine_dependent b')))) ==> (b' = b)))))) ==> (((hull affine) b) = s))))
Lp_ineqs_proofs.rhazim3_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 ((rhazim3_fan (V , E)) d)) ((real_add pi) sol0))))))))))
NONNEGATIVE_ABSOLUTELY_REAL_INTEGRABLE |- (!f. (!s. (((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))) /\ ((real_integrable_on f) s)) ==> ((absolutely_real_integrable_on f) s))))
LIM_NULL_COMPLEX_POW_EQ |- (!net. (!f. (!n. ((~ (n = (NUMERAL _0))) ==> ((((--> (\x. ((complex_pow (f x)) n))) (Cx (real_of_num (NUMERAL _0)))) net) = (((--> f) (Cx (real_of_num (NUMERAL _0)))) net))))))
Trigonometry1.COS_PERIODIC_N2PI |- (!x. (!n. ((cos ((real_add x) ((real_mul (real_of_num n)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))) = (cos x))))
ARC_SUBPATH_EQ |- (!g. (!u. (!v. ((arc (((subpath u) v) g)) = ((path (((subpath u) v) g)) /\ ((~ (u = v)) /\ (!x. (!y. (((x IN (closed_segment ((CONS (u , v)) NIL))) /\ ((y IN (closed_segment ((CONS (u , v)) NIL))) /\ ((g x) = (g y)))) ==> (x = y))))))))))
Tame_lemmas.fully_surrounded_sum_sol |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (((sum (face_set (hypermap_of_fan (V , E)))) (\f. ((sol (vec (NUMERAL _0))) ((((dartset_leads_into_fan (vec (NUMERAL _0))) V) E) ((IMAGE (ext_dart (V , E))) f))))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi))))))
LIFT_IN_INTERVAL |- ((!a. (!b. (!x. (((lift x) IN (closed_interval ((CONS (a , b)) NIL))) = (x IN (closed_real_interval ((CONS ((drop a) , (drop b))) NIL))))))) /\ (!a. (!b. (!x. (((lift x) IN (open_interval (a , b))) = (x IN (open_real_interval ((drop a) , (drop b)))))))))
CONTRACTIBLE_CONVEX_TWEAK_BOUNDARY_POINTS |- (!s. (!t. (((convex s) /\ (((relative_interior s) SUBSET t) /\ (t SUBSET (closure s)))) ==> (contractible t))))
CONIC_RELATIVE_INTERIOR |- (!s. (((conic s) /\ ((vec (NUMERAL _0)) IN (relative_interior s))) ==> (conic (relative_interior s))))
Merge_ineq.nonf_mud_135_x |- (!a. (!b. (!c. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((real_le (real_of_num (NUMERAL _0))) a) ==> ((((((((((mud_135_x_v1 a) b) c) x1) x2) x3) x4) x5) x6) = ((real_mul (((mu_y a) (sqrt x1)) (sqrt x3))) (sqrt (((((((((delta_135_x ((real_mul a) a)) ((real_mul b) b)) ((real_mul c) c)) x1) x2) x3) x4) x5) x6))))))))))))))
CONTINUOUS_ON_COMPONENTS_FINITE |- (!f. (!s. (((FINITE (components s)) /\ (!c. ((c IN (components s)) ==> ((continuous_on f) c)))) ==> ((continuous_on f) s))))
Seq2.perm_eq_set_of_list |- (!l. (!s. (((FINITE s) /\ ((perm_eq l) (list_of_set s))) ==> ((set_of_list l) = s))))
REAL_SUMMABLE_NEG |- (!x. (!s. (((real_summable s) x) ==> ((real_summable s) (\n. (real_neg (x n)))))))
COMPLEX_SUB_SUB |- (!x. (!y. (((vector_sub ((vector_sub x) y)) x) = (vector_neg y))))
Seq.mem_seq3 |- (!x. (!y1. (!y2. (!y3. (((MEM x) ((CONS y1) ((CONS y2) ((CONS y3) NIL)))) = ((x = y1) \/ ((x = y2) \/ (x = y3))))))))
EDELSTEIN_FIX_ITER |- (!f. (!s. (!n. (((compact s) /\ ((~ (s = EMPTY)) /\ ((((IMAGE f) s) SUBSET s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> ((real_lt (distance ((((ITER n) f) x) , (((ITER n) f) y)))) (distance (x , y))))))))) ==> (?!x. ((x IN s) /\ ((f x) = x)))))))
ULC_IMP_LOCALLY_CONNECTED |- (!s. ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_lt (distance (x , y))) d))) ==> (?c. ((x IN c) /\ ((y IN c) /\ ((c SUBSET s) /\ ((connected c) /\ ((bounded c) /\ ((real_le (diameter c)) e)))))))))))))) ==> ((locally connected) s)))
MINIMAL_CONTINUUM |- (!t. (!s. (((t SUBSET s) /\ ((compact s) /\ (connected s))) ==> (?u. ((t SUBSET u) /\ ((u SUBSET s) /\ ((compact u) /\ ((connected u) /\ (!v. (((v SUBSET u) /\ ((t SUBSET v) /\ ((compact v) /\ (connected v)))) ==> (v = u)))))))))))
INSIDE_INSIDE_SUBSET |- (!s. ((inside (inside s)) SUBSET s))
REAL_MEASURABLE_ON_RPOW |- (!f. (!r. (!s. ((((real_measurable_on f) s) /\ ((real_lt (real_of_num (NUMERAL _0))) r)) ==> ((real_measurable_on (\x. ((rpow (f x)) r))) s)))))
CONNECTED_COMPONENT_EMPTY |- (!x. (((connected_component EMPTY) x) = EMPTY))
POLYTOPE_AFF_DIM_1 |- (!p. (((polytope p) /\ ((aff_dim p) = (int_of_num (NUMERAL (BIT1 _0))))) = (?a. (?b. ((~ (a = b)) /\ (p = (closed_segment ((CONS (a , b)) NIL))))))))
EXISTS_VECTOR_3 |- ((?v. (P v)) = (?x. (?y. (?z. (P (vector ((CONS x) ((CONS y) ((CONS z) NIL)))))))))
REAL_LE_TRANS_LE |- (!x. (!y. (((real_le x) y) = (!z. (((real_le y) z) ==> ((real_le x) z))))))
CONIC_HULL_EQ |- (!s. ((((hull conic) s) = s) = (conic s)))
AFF_GE_1_1 |- (!x. (!v. (!w. (((DISJOINT (x INSERT EMPTY)) (v INSERT EMPTY)) ==> (((aff_ge (x INSERT EMPTY)) (v INSERT EMPTY)) = (GSPEC (\GEN%PVAR%5098. (?y. (((SETSPEC GEN%PVAR%5098) (?t1. (?t2. (((real_le (real_of_num (NUMERAL _0))) t2) /\ ((((real_add t1) t2) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (y = ((vector_add ((% t1) x)) ((% t2) v)))))))) y)))))))))
IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT_STRONG |- (!s. (!x. (!a. (((convex s) /\ ((x IN (relative_interior s)) /\ ((a IN ((hull affine) s)) /\ (~ (x = a))))) ==> (?b. ((b IN (relative_interior s)) /\ (x IN (open_segment (a , b)))))))))
HOLOMORPHIC_ON_LMUL |- (!f. (!c. (!s. (((holomorphic_on f) s) ==> ((holomorphic_on (\x. ((complex_mul c) (f x)))) s)))))
SUM_BIJECTION |- (!f. (!p. (!s. (((!x. ((x IN s) ==> ((p x) IN s))) /\ (!y. ((y IN s) ==> (?!x. ((x IN s) /\ ((p x) = y)))))) ==> (((sum s) f) = ((sum s) (f o p)))))))
RETRACT_FIXPOINT_PROPERTY |- (!s. (!t. ((((retract_of t) s) /\ (!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))))))))
DOUBLE_LEBESGUE_MEASURABLE |- (!f. ((!t. ((lebesgue_measurable t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3690. (?x. (((SETSPEC GEN%PVAR%3690) ((f x) IN t)) x))))))) = (((measurable_on f) UNIV) /\ (!t. ((negligible t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3691. (?x. (((SETSPEC GEN%PVAR%3691) ((f x) IN t)) x))))))))))
FRONTIER_SURJECTIVE_LINEAR_IMAGE |- (!f. (((linear f) /\ (!y. (?x. ((f x) = y)))) ==> ((frontier ((IMAGE f) s)) = ((IMAGE f) (frontier s)))))
Lp_ineqs_proofs.yn_hi |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (yn_fan v)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))
OPEN_EXISTS |- (!Q. ((!a. (open (GSPEC (\GEN%PVAR%544. (?x. (((SETSPEC GEN%PVAR%544) ((Q a) x)) x)))))) ==> (open (GSPEC (\GEN%PVAR%545. (?x. (((SETSPEC GEN%PVAR%545) (?a. ((Q a) x))) x)))))))
REAL_INTEGRAL_RESTRICT_INTER |- (!f. (!s. (!t. (((real_integral t) (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) = ((real_integral (s INTER t)) f)))))
REAL_LEBESGUE_MEASURABLE_COUNTABLE_INTERS_EXPLICIT |- (!s. ((!n. (real_lebesgue_measurable (s n))) ==> (real_lebesgue_measurable (INTERS (GSPEC (\GEN%PVAR%5042. (?n. (((SETSPEC GEN%PVAR%5042) (n IN UNIV)) (s n)))))))))
Tactics.WELLDEFINED_FUNCTION_1P |- (!P. ((?f. (!x. ((P x) ==> ((f (s x)) = (t x))))) = (!x. (!x'. (((P x) /\ ((P x') /\ ((s x) = (s x')))) ==> ((t x) = (t x')))))))
Merge_ineq.nonf_dih4_x_div_sqrtdelta_posbranch |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dih4_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((real_mul ((real_div (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x4))) ((((((delta_x4 x4) x2) x6) x1) x5) x3))) (matan ((real_div ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x4) ((((((delta_x x1) x2) x3) x4) x5) x6)))) ((real_pow ((((((delta_x4 x4) x2) x6) x1) x5) x3)) (NUMERAL (BIT0 (BIT1 _0))))))))))))))
REALLIM_POSINFINITY_SEQUENTIALLY |- (!f. (!l. ((((---> f) l) at_posinfinity) ==> (((---> (\n. (f (real_of_num n)))) l) sequentially))))
COMPLEX_DERIVATIVE_MUL_AT |- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\ ((analytic_on g) (z INSERT EMPTY))) ==> (((complex_derivative (\w. ((complex_mul (f w)) (g w)))) z) = ((vector_add ((complex_mul (f z)) ((complex_derivative g) z))) ((complex_mul ((complex_derivative f) z)) (g z))))))))
REAL_EXP_NEG_MUL |- (!x. (((real_mul (exp x)) (exp (real_neg x))) = (real_of_num (NUMERAL (BIT1 _0)))))
VECTOR_DERIVATIVE_CONST_AT |- (!c. (!a. (((vector_derivative (\x. c)) (at a)) = (vec (NUMERAL _0)))))
HAS_BOUNDED_VARIATION_DARBOUX_STRONG |- (!f. (!a. (!b. (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) ==> (?g. (?h. ((!x. ((f x) = ((vector_sub (g x)) (h 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 (drop (g x))) (drop (g y)))))) /\ ((!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 (h x))) (drop (h y)))))) /\ ((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_lt (drop x)) (drop y)))) ==> ((real_lt (drop (g x))) (drop (g y)))))) /\ ((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_lt (drop x)) (drop y)))) ==> ((real_lt (drop (h x))) (drop (h y)))))) /\ ((!x. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((continuous f) ((within (at x)) (closed_interval ((CONS (a , x)) NIL))))) ==> (((continuous g) ((within (at x)) (closed_interval ((CONS (a , x)) NIL)))) /\ ((continuous h) ((within (at x)) (closed_interval ((CONS (a , x)) NIL))))))) /\ ((!x. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((continuous f) ((within (at x)) (closed_interval ((CONS (x , b)) NIL))))) ==> (((continuous g) ((within (at x)) (closed_interval ((CONS (x , b)) NIL)))) /\ ((continuous h) ((within (at x)) (closed_interval ((CONS (x , b)) NIL))))))) /\ (!x. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((continuous f) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))) ==> (((continuous g) ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))) /\ ((continuous h) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))))))))))))))))
Lp_ineqs_proofs.rho_hi |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (rho_fan v)) ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_div sol0) pi)))))))))))
SUMS_OFFSET_REV |- (!f. (!l. (!m. (!n. (((((sums f) l) (from m)) /\ (((NUMERAL _0) < m) /\ (n <= m))) ==> (((sums f) ((vector_add l) ((vsum (n .. (m - (NUMERAL (BIT1 _0))))) f))) (from n)))))))
Ssrnat.maxKn |- (!m. (!n. (((minn n) ((maxn m) n)) = n)))
REAL_INFSUM_RMUL |- (!s. (!x. (!c. (((real_summable s) x) ==> (((real_infsum s) (\n. ((real_mul (x n)) c))) = ((real_mul ((real_infsum s) x)) c))))))
Tactics_jordan.REAL_DOUBLE |- (!x. (((real_add x) x) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x)))
Tame_list.HEIGHTS |- ((heights ((((Graph fs) n) f) h)) = h)
Rnsyjxm.f_fan_pair_ext_in_dart1 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> (((f_fan_pair_ext (V , E)) d) IN (dart1_of_fan (V , E))))))))
Gbycpxs.SIGMA_SY_LE1 |- (!s. ((real_le (sigma_sy s)) (real_of_num (NUMERAL (BIT1 _0)))))
CONNECTED_LIMIT_POINTS |- (!s. ((connected s) ==> (connected (GSPEC (\GEN%PVAR%585. (?x. (((SETSPEC GEN%PVAR%585) ((limit_point_of x) s)) x)))))))
HOLOMORPHIC_ON_CCOS |- (!s. ((holomorphic_on ccos) s))
SAME_EIGENVECTORS_MATRIX_INV |- (!A. (!c. (!v. (((transp A) = A) ==> ((((matrix_vector_mul (matrix_inv A)) v) = ((% c) v)) = (((matrix_vector_mul A) v) = ((% (real_inv c)) v)))))))
REAL_VARIATION_REFLECT2 |- (!f. (!s. (((real_variation ((IMAGE real_neg) s)) (\x. (f (real_neg x)))) = ((real_variation s) f))))
CONTINUOUS_MAP_CLOSURES |- (!f. (((continuous_on f) UNIV) = (!s. (((IMAGE f) (closure s)) SUBSET (closure ((IMAGE f) s))))))
Functional_equation.gchi4_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)))))) ==> (((((((gchi4_x x1) x2) x3) x4) x5) x6) = ((((((gchi4_x' x1) x2) x3) x4) x5) x6)))))))))
OPEN_MAP_IFF_LOWER_HEMICONTINUOUS_PREIMAGE |- (!f. (!s. (!t. ((((IMAGE f) s) SUBSET t) ==> ((!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))) = (!u. (((closed_in ((subtopology euclidean) s)) u) ==> ((closed_in ((subtopology euclidean) t)) (GSPEC (\GEN%PVAR%885. (?y. (((SETSPEC GEN%PVAR%885) ((y IN t) /\ ((GSPEC (\GEN%PVAR%884. (?x. (((SETSPEC GEN%PVAR%884) ((x IN s) /\ ((f x) = y))) x)))) SUBSET u))) y))))))))))))
Localization.HAFL_CIRCLE_FORM_LOCAL_FAN_ALT |- (!V. (!E. (!FF. (!v. (!w. (!HS. (!fv. ((((local_fan (V , (E , FF))) /\ ((v IN V) /\ ((w IN V) /\ ((~ (v = w)) /\ ((!z. (!t. (((z IN (v INSERT (w INSERT EMPTY))) /\ (t IN (V DIFF (z INSERT EMPTY)))) ==> (~ (collinear ((vec (NUMERAL _0)) INSERT (z INSERT (t INSERT EMPTY)))))))) /\ (!x. ((x IN FF) ==> (((aff_gt ((vec (NUMERAL _0)) INSERT EMPTY)) (v INSERT (w INSERT EMPTY))) SUBSET ((wedge_in_fan_gt x) E))))))))) /\ ((HS = (hypermap (HYP ((vec (NUMERAL _0)) , (V , (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))))))) /\ (fv = ((face HS) (v , ((rho_node1 FF) v)))))) ==> (local_fan (((v_prime V) fv) , (((e_prime (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))) fv) , fv)))))))))))
Planarity.RWXUYZZ |- (!x. (!V. (!E. (!ds. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((fan80 (x , (V , E))) /\ (ds IN (face_set (hypermap1_of_fanx (x , (V , E)))))))) ==> ((?s. (!y. ((y IN ds) ==> (s = (((((dart_leads_into x) V) E) (pr2 y)) (pr3 y)))))) /\ (((((dartset_leads_into_fan x) V) E) ds) IN (topological_component_yfan (x , (V , E))))))))))
IN_CBALL_RE |- (!x. (!z. (!e. ((x IN (cball (z , e))) ==> ((real_le (real_abs ((real_sub (Re x)) (Re z)))) e)))))
Lp_ineqs_proofs.f_fan_pair_ext_power_in_dart1 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. ((d IN (dart1_of_fan (V , E))) ==> ((((POWER (f_fan_pair_ext (V , E))) k) d) IN (dart1_of_fan (V , E)))))))))
Ssrnat.distn_add2l |- (!d. (!m. (!n. (((distn (d + m)) (d + n)) = ((distn m) n)))))
Ssrbool.negbNE |- (!b. ((~ (~ b)) ==> b))
LEBESGUE_DENSITY_THEOREM |- (!s. ((lebesgue_measurable s) ==> (?k. ((negligible k) /\ (!x. ((~ (x IN k)) ==> (((---> (\e. ((real_div (measure (s INTER (cball (x , e))))) (measure (cball (x , e)))))) (((COND (x IN s)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))) ((within (atreal (real_of_num (NUMERAL _0)))) (GSPEC (\GEN%PVAR%5068. (?e. (((SETSPEC GEN%PVAR%5068) ((real_lt (real_of_num (NUMERAL _0))) e)) e))))))))))))
INT_LT_SUB_LADD |- (!x. (!y. (!z. (((int_lt x) ((int_sub y) z)) = ((int_lt ((int_add x) z)) y)))))
REAL_CONTINUOUS_COMPLEX_COMPONENTS_WITHIN |- (!s. (!z. (((real_continuous Re) ((within (at z)) s)) /\ ((real_continuous Im) ((within (at z)) s)))))
INTEGRAL_COMPONENT_LE_AE |- (!f. (!g. (!s. (!k. (!t. ((((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ ((negligible t) /\ (((integrable_on f) s) /\ (((integrable_on g) s) /\ (!x. ((x IN (s DIFF t)) ==> ((real_le ((f x) $ k)) ((g x) $ k))))))))) ==> ((real_le (((integral s) f) $ k)) (((integral s) g) $ k))))))))
REAL_INTEGRABLE_COMBINE |- (!f. (!a. (!b. (!c. ((((real_le a) c) /\ (((real_le c) b) /\ (((real_integrable_on f) (closed_real_interval ((CONS (a , c)) NIL))) /\ ((real_integrable_on f) (closed_real_interval ((CONS (c , b)) NIL)))))) ==> ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))))))))
Lp_ineqs_proofs.dart3_sol_fan_eq_sol |- (!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)))) ==> (((sol_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)) = ((((((sol_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)))))))))))
AFF_DIM_BALL |- (!a. (!r. ((aff_dim (ball (a , r))) = (((COND ((real_lt (real_of_num (NUMERAL _0))) r)) (int_of_num (dimindex UNIV))) (int_neg (int_of_num (NUMERAL (BIT1 _0))))))))
INJECTIVE_MAP_OPEN_IFF_CLOSED |- (!f. (!s. (!t. ((((continuous_on f) s) /\ ((((IMAGE f) s) = t) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))))) ==> ((!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))) = (!u. (((closed_in ((subtopology euclidean) s)) u) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) u)))))))))
CHOICE_UNPAIR_THM |- (!P. ((@ (GABS (\f. (!x. (!y. ((GEQ (f (x , y))) ((P x) y))))))) = (@p. ((P (FST p)) (SND p)))))
SERIES_DIRICHLET_COMPLEX |- (!f. (!g. (!N. (!k. (!m. (((bounded (GSPEC (\GEN%PVAR%4782. (?n. (((SETSPEC GEN%PVAR%4782) (n IN UNIV)) ((vsum (m .. n)) f)))))) /\ ((!n. (real (g n))) /\ ((!n. ((N <= n) ==> ((real_le (Re (g (n + (NUMERAL (BIT1 _0)))))) (Re (g n))))) /\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) sequentially)))) ==> ((summable (from k)) (\n. ((complex_mul (f n)) (g n))))))))))
CONVEX_AFF_LE |- (!s. (!t. (convex ((aff_le s) t))))
IVT_DECREASING_IM |- (!f. (!a. (!b. (!y. ((((real_le a) b) /\ ((!x. ((((real_le a) x) /\ ((real_le x) b)) ==> ((continuous f) (at (Cx x))))) /\ (((real_le (Im (f (Cx b)))) y) /\ ((real_le y) (Im (f (Cx a))))))) ==> (?x. (((real_le a) x) /\ (((real_le x) b) /\ ((Im (f (Cx x))) = y)))))))))
REAL_MEASURABLE_ON_COMPOSE_CONTINUOUS_OPEN_INTERVAL |- (!f. (!g. (!a. (!b. ((((real_measurable_on f) UNIV) /\ ((!x. ((f x) IN (open_real_interval (a , b)))) /\ ((real_continuous_on g) (open_real_interval (a , b))))) ==> ((real_measurable_on (g o f)) UNIV))))))
LOCALLY_CONVEX |- (!s. (((locally convex) s) = (!x. ((x IN s) ==> (?u. (?v. ((x IN u) /\ ((u SUBSET v) /\ ((v SUBSET s) /\ (((open_in ((subtopology euclidean) s)) u) /\ (convex v)))))))))))
MEASURABLE_ON_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GT |- (!f. (!s. ((measurable s) ==> (((measurable_on f) s) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV))) ==> (measurable (GSPEC (\GEN%PVAR%3650. (?x. (((SETSPEC GEN%PVAR%3650) ((x IN s) /\ ((real_gt ((f x) $ k)) a))) x))))))))))))
ONORM_TRIANGLE_LT |- (!f. (!g. (((linear f) /\ ((linear g) /\ ((real_lt ((real_add (onorm f)) (onorm g))) e))) ==> ((real_lt (onorm (\x. ((vector_add (f x)) (g x))))) e))))
NSUM_DEGENERATE |- (!f. (!s. ((~ (FINITE (GSPEC (\GEN%PVAR%171. (?x. (((SETSPEC GEN%PVAR%171) ((x IN s) /\ (~ ((f x) = (NUMERAL _0))))) x)))))) ==> (((nsum s) f) = (NUMERAL _0)))))
QUOTIENT_MAP_IMP_CONTINUOUS_CLOSED |- (!f. (!s. (!t. (((((IMAGE f) s) SUBSET t) /\ (!u. ((u SUBSET t) ==> (((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%645. (?x. (((SETSPEC GEN%PVAR%645) ((x IN s) /\ ((f x) IN u))) x))))) = ((closed_in ((subtopology euclidean) t)) u))))) ==> ((continuous_on f) s)))))
LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ |- (!s. (!t. ((negligible ((s DIFF t) UNION (t DIFF s))) ==> ((lebesgue_measurable s) = (lebesgue_measurable t)))))
INT_LE_NEGL |- (!x. (((int_le (int_neg x)) x) = ((int_le (int_of_num (NUMERAL _0))) x)))
EXISTS_SUBARC_OF_ARC_NOENDS |- (!g. (!a. (!b. (((arc g) /\ ((a IN (path_image g)) /\ ((b IN (path_image g)) /\ ((~ (a = b)) /\ (((a INSERT (b INSERT EMPTY)) INTER ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))) = EMPTY))))) ==> (?h. ((arc h) /\ (((pathstart h) = a) /\ (((pathfinish h) = b) /\ ((path_image h) SUBSET ((path_image g) DIFF ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))))))))))))
Tame_list.AQIUNPP2 |- (!L. (!N. (!r. (!x. (((transform_assumption_v2 L) /\ ((good_list L) /\ ((good_list_nodes L) /\ ((((marked_list L) N) r) x)))) ==> ((((marked_list L) (((((ntrans L) N) r) x) (NUMERAL (BIT1 _0)))) (((((rtrans L) N) r) x) (NUMERAL (BIT1 _0)))) x))))))
URYSOHN |- (!s. (!t. (!a. (!b. (((closed s) /\ ((closed t) /\ ((s INTER t) = EMPTY))) ==> (?f. (((continuous_on f) UNIV) /\ ((!x. ((f x) IN (closed_segment ((CONS (a , b)) NIL)))) /\ ((!x. ((x IN s) ==> ((f x) = a))) /\ (!x. ((x IN t) ==> ((f x) = b))))))))))))
CARD_MUL_ASSOC |- (!s. (!t. (!u. (((*_c s) ((*_c t) u)) =_c ((*_c ((*_c s) t)) u)))))
LINEAR_SUBSPACE_GRAPH |- (!f. ((linear f) = (subspace (GSPEC (\GEN%PVAR%153. (?x. (((SETSPEC GEN%PVAR%153) (x IN UNIV)) ((pastecart x) (f x)))))))))
Lp_ineqs_proofs.rhazim2_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 ((rhazim2_fan (V , E)) d)) ((real_add pi) sol0))))))))))
MEASURABLE_ON_TRANSLATION_EQ |- (!f. (!s. (!a. (((measurable_on (\x. (f ((vector_add a) x)))) s) = ((measurable_on f) ((IMAGE (\x. ((vector_add a) x))) s))))))
Lp_gen_theory.sum_face_iso |- (!H. (!G. (!g. (((hyp_iso g) (G , H)) ==> (!r. (!f. ((f IN (face_set G)) ==> (((sum ((IMAGE g) f)) r) = ((sum f) (r o g))))))))))