hcju/holps huggingface dataset:
SUBSET_CONVEX_HULL_RELATIVE_FRONTIER |- (!s. (((bounded s) /\ (~ (?a. (s = (a INSERT EMPTY))))) ==> (s SUBSET ((hull convex) (relative_frontier s)))))
Tactics_jordan.expand_prod_le |- (!B1. (!B2. (!C. ((((real_le (real_of_num (NUMERAL _0))) ((real_mul B1) B2)) ==> C) ==> (((real_le (real_of_num (NUMERAL _0))) B1) ==> (((real_le (real_of_num (NUMERAL _0))) B2) ==> C))))))
Ssrnat.half_gt0 |- (!n. (((NUMERAL _0) < (half n)) = ((NUMERAL (BIT1 _0)) < n)))
Hypermap_iso.finite_inj_eq_bij |- (!s. (!t. ((FINITE s) ==> ((FINITE t) ==> (((CARD s) = (CARD t)) ==> (!f. ((((INJ f) s) t) = (((BIJ f) s) t))))))))
FLOOR_DIV_DIV |- (!m. (!n. ((~ (m = (NUMERAL _0))) ==> ((floor ((real_div (real_of_num n)) (real_of_num m))) = (real_of_num (n DIV m))))))
CARD_EQ_FULLSIZE_POWERSET |- (!s. ((INFINITE s) ==> ((GSPEC (\GEN%PVAR%51. (?t. (((SETSPEC GEN%PVAR%51) ((t SUBSET s) /\ (t =_c s))) t)))) =_c (GSPEC (\GEN%PVAR%52. (?t. (((SETSPEC GEN%PVAR%52) (t SUBSET s)) t)))))))
ABSOLUTELY_REAL_INTEGRABLE_INCREASING |- (!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))))) ==> ((absolutely_real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL)))))))
Nonlinear_lemma.c2001 |- (!y. (((real_le y) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ==> ((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))))
REAL_ABS_SIN_BOUND_LE |- (!x. ((real_le (real_abs (sin x))) (real_abs x)))
FACES_OF_LINEAR_IMAGE |- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((GSPEC (\GEN%PVAR%2138. (?t. (((SETSPEC GEN%PVAR%2138) ((face_of t) ((IMAGE f) s))) t)))) = ((IMAGE (IMAGE f)) (GSPEC (\GEN%PVAR%2139. (?t. (((SETSPEC GEN%PVAR%2139) ((face_of t) s)) t)))))))))
VECTOR_DIFF_CHAIN_AT |- (!f. (!g. (!f'. (!g'. (!x. (((((has_vector_derivative f) f') (at x)) /\ (((has_vector_derivative g) g') (at (f x)))) ==> (((has_vector_derivative (g o f)) ((% (drop f')) g')) (at x))))))))
GRASSMANN_PLUCKER_3 |- (!x1. (!x2. (!x3. (!y1. (!y2. (!y3. (((real_mul (det (vector ((CONS x1) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS y1) ((CONS y2) ((CONS y3) NIL)))))) = ((real_add ((real_mul (det (vector ((CONS y1) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS x1) ((CONS y2) ((CONS y3) NIL))))))) ((real_add ((real_mul (det (vector ((CONS y2) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS y1) ((CONS x1) ((CONS y3) NIL))))))) ((real_mul (det (vector ((CONS y3) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS y1) ((CONS y2) ((CONS x1) NIL)))))))))))))))
SET_VARIATION_LBOUND_ON_INTERVAL |- (!f. (!a. (!b. (!B. ((((has_bounded_setvariation_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (?d. (((division_of d) (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le B) ((sum d) (\k. (vector_norm (f k)))))))) ==> ((real_le B) ((set_variation (closed_interval ((CONS (a , b)) NIL))) f)))))))
REAL_ADD_COS |- (!x. (!y. (((real_add (cos x)) (cos y)) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (cos ((real_div ((real_add x) y)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) (cos ((real_div ((real_sub x) y)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))
Float.ABS_NUM |- (!m. (!n. ((real_abs ((real_sub (real_of_num n)) (real_of_num m))) = (real_of_num ((m - n) + (n - m))))))
COMPLEX_DIFFERENTIABLE_WITHIN_SUBSET |- (!f. (!s. (!t. ((((complex_differentiable f) ((within (at x)) s)) /\ (t SUBSET s)) ==> ((complex_differentiable f) ((within (at x)) t))))))
GDELTA_PCROSS_EQ |- (!s. (!t. ((gdelta (s PCROSS t)) = ((s = EMPTY) \/ ((t = EMPTY) \/ ((gdelta s) /\ (gdelta t)))))))
COMPLEX_NEG_EQ |- (!x. (!y. (((vector_neg x) = y) = (x = (vector_neg y)))))
NPRODUCT_CLAUSES_LEFT |- (!f. (!m. (!n. ((m <= n) ==> (((nproduct (m .. n)) f) = ((f m) * ((nproduct ((m + (NUMERAL (BIT1 _0))) .. n)) f)))))))
Add_triangle.fully_surrounded_card_set_of_edge |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (((fully_surrounded (V , E)) /\ (v IN V)) ==> ((CARD (((set_of_edge v) V) E)) >= (NUMERAL (BIT1 (BIT1 _0)))))))))
DIM_SPECIAL_SUBSPACE |- (!k. ((dim (GSPEC (\GEN%PVAR%216. (?x. (((SETSPEC GEN%PVAR%216) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (i IN k))) ==> ((x $ i) = (real_of_num (NUMERAL _0)))))) x))))) = (CARD (((NUMERAL (BIT1 _0)) .. (dimindex UNIV)) DIFF k))))
HOMEOMORPHIC_COMPACT_ARNESS |- (!s. (!s'. (((homeomorphic s) s') ==> (((compact s) /\ ((retract_of s) UNIV)) = ((compact s') /\ ((retract_of s') UNIV))))))
REAL_SUMS_SUMMABLE |- (!f. (!l. (!s. ((((real_sums f) l) s) ==> ((real_summable s) f)))))
INTERIOR_SURJECTIVE_LINEAR_IMAGE |- (!f. (!s. (((linear f) /\ (!y. (?x. ((f x) = y)))) ==> ((interior ((IMAGE f) s)) = ((IMAGE f) (interior s))))))
INT_POW_LT2 |- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ (((int_le (int_of_num (NUMERAL _0))) x) /\ ((int_lt x) y))) ==> ((int_lt ((int_pow x) n)) ((int_pow y) n))))))
HOLOMORPHIC_ON_CASN |- (!s. ((!z. (((z IN s) /\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((holomorphic_on casn) s)))
CART_EQ_FULL |- (!x. (!y. ((x = y) = (!i. ((x $ i) = (y $ i))))))
MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE_AE |- (!f. (!g. (!s. (!t. ((((measurable_on f) s) /\ (((integrable_on g) s) /\ ((negligible t) /\ (!x. ((x IN (s DIFF t)) ==> ((real_le (vector_norm (f x))) (drop (g x)))))))) ==> ((absolutely_integrable_on f) s))))))
DOT_4 |- (((dot x) y) = ((real_add ((real_mul (x $ (NUMERAL (BIT1 _0)))) (y $ (NUMERAL (BIT1 _0))))) ((real_add ((real_mul (x $ (NUMERAL (BIT0 (BIT1 _0))))) (y $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add ((real_mul (x $ (NUMERAL (BIT1 (BIT1 _0))))) (y $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_mul (x $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (y $ (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))
AFFINE_HULL_AFFINE_INTER_OPEN_IN |- (!s. (!t. (((affine s) /\ (((open_in ((subtopology euclidean) s)) t) /\ (~ ((s INTER t) = EMPTY)))) ==> (((hull affine) (s INTER t)) = s))))
Nonlinear_lemma.rad2_y_e |- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((((((rad2_y y1) y2) y3) y4) y5) y6) = (((((((y_of_x rad2_x) y1) y2) y3) y4) y5) y6))))))))
MIDPOINTS_IN_CONVEX_HULL |- (!x. (!s. (((x IN ((hull convex) s)) /\ (y IN ((hull convex) s))) ==> ((midpoint (x , y)) IN ((hull convex) s)))))
Asfutbf.INTER_NEG |- (!A. (!B. ((((IMAGE vector_neg) A) INTER ((IMAGE vector_neg) B)) = ((IMAGE vector_neg) (A INTER B)))))
ABSOLUTELY_INTEGRABLE_ON_MUL_BERNOULLI_FRAC |- (!f. (!s. (!n. (((absolutely_integrable_on f) s) ==> ((absolutely_integrable_on (\x. ((% ((bernoulli n) (frac (drop x)))) (f x)))) s)))))
REAL_SUMMABLE_LMUL |- (!s. (!x. (!c. (((real_summable s) x) ==> ((real_summable s) (\n. ((real_mul c) (x n))))))))
HAS_BOUNDED_REAL_VARIATION_REFLECT_EQ_INTERVAL |- (!f. (!u. (!v. (((has_bounded_real_variation_on (\x. (f (real_neg x)))) (closed_real_interval ((CONS (u , v)) NIL))) = ((has_bounded_real_variation_on f) (closed_real_interval ((CONS ((real_neg v) , (real_neg u))) NIL)))))))
REAL_INTEGRABLE_SPIKE_INTERIOR |- (!f. (!g. (!a. (!b. ((!x. ((x IN (open_real_interval (a , b))) ==> ((g x) = (f x)))) ==> (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_integrable_on g) (closed_real_interval ((CONS (a , b)) NIL)))))))))
Float.FLOAT_ADD_PPv2 |- (!a. (!b. (!m. (!n. ((m < n) ==> (((real_add ((float a) (int_of_num m))) ((float b) (int_of_num n))) = ((float ((int_add ((int_mul (int_of_num ((NUMERAL (BIT0 (BIT1 _0))) EXP (n - m)))) b)) a)) (int_of_num m))))))))
More_list_hypermap.nodes_hypermap_of_list_all |- (!L. ((good_list_nodes L) ==> ((ALL (\n. ((ALL (\d. (((node (hypermap_of_list L)) d) = (set_of_list n)))) n))) (list_of_nodes L))))
SUP_INSERT_INSERT |- (!a. (!b. (!s. ((sup (b INSERT (a INSERT s))) = (sup (((real_max a) b) INSERT s))))))
AFF_LT_EQ_AFFINE_HULL |- (!s. (!t. (((aff_lt s) EMPTY) = ((hull affine) s))))
Nonlinear_lemma.lmdih5_ldih5' |- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((((real_le (real_of_num (NUMERAL _0))) y5) /\ ((real_le y5) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ==> (((((((lmdih5_x_div_sqrtdelta_posbranch ((real_mul y1) y1)) ((real_mul y2) y2)) ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5)) ((real_mul y6) y6)) = ((((((ldih5_x_div_sqrtdelta_posbranch ((real_mul y1) y1)) ((real_mul y2) y2)) ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5)) ((real_mul y6) y6))))))))))
TC_RTC_CASES_R |- (!R. (!x. (!z. ((((TC R) x) z) = (?y. (((R x) y) /\ (((RTC R) y) z)))))))
LIM_AT_INFINITY_WITHIN_POS |- (!f. (!l. (!s. ((((--> f) l) ((within at_infinity) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (((real_lt (real_of_num (NUMERAL _0))) b) /\ (!x. (((x IN s) /\ ((real_ge (vector_norm x)) b)) ==> ((real_lt (distance ((f x) , l))) e)))))))))))
COMPLEX_RNEG_UNIQ |- (!x. (!y. ((((vector_add x) y) = (Cx (real_of_num (NUMERAL _0)))) = (y = (vector_neg x)))))
BOUNDED_RELATIVE_INTERIOR |- (!s. ((bounded s) ==> (bounded (relative_interior s))))
Ardbzye.J_SCS_3T3 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_3T3) i)) i1) j) = F)
HOMOTOPY_EQUIVALENT_SING |- (!s. (!a. (((homotopy_equivalent s) (a INSERT EMPTY)) = ((~ (s = EMPTY)) /\ (contractible s)))))
Packing3.MINIMAL_INTERS_EXISTS |- (!s. (!f. (((FINITE f) /\ (s = (INTERS f))) ==> (?g. ((g SUBSET f) /\ ((s = (INTERS g)) /\ (!g'. ((g' PSUBSET g) ==> (s PSUBSET (INTERS g'))))))))))
FORALL_IN_INTERMEDIATE_CLOSURE_EQ |- (!f. (!s. (!s'. (!t. (((closed t) /\ (((continuous_on f) s') /\ ((s SUBSET s') /\ (s' SUBSET (closure s))))) ==> ((!x. ((x IN s') ==> ((f x) IN t))) = (!x. ((x IN s) ==> ((f x) IN t)))))))))
DIVISION_OF_UNIONS |- (!f. (((FINITE f) /\ ((!p. ((p IN f) ==> ((division_of p) (UNIONS p)))) /\ (!k1. (!k2. (((k1 IN (UNIONS f)) /\ ((k2 IN (UNIONS f)) /\ (~ (k1 = k2)))) ==> (((interior k1) INTER (interior k2)) = EMPTY)))))) ==> ((division_of (UNIONS f)) (UNIONS (UNIONS f)))))
SUM_GP_OFFSET |- (!x. (!m. (!n. (((sum (m .. (m + n))) (\i. ((real_pow x) i))) = (((COND (x = (real_of_num (NUMERAL (BIT1 _0))))) ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))) ((real_mul ((real_pow x) m)) ((real_div ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) ((real_pow x) (SUC n)))) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) x))))))))
REAL_CONTINUOUS_AT_RPOW_RIGHT |- (!a. (!x. (((real_lt (real_of_num (NUMERAL _0))) a) ==> ((real_continuous (\x. ((rpow a) x))) (atreal x)))))
CONNECTED_FROM_OPEN_UNION_AND_INTER |- (!s. (!t. (((open s) /\ ((open t) /\ ((connected (s UNION t)) /\ (connected (s INTER t))))) ==> ((connected s) /\ (connected t)))))
CONTINUOUS_IMP_REAL_MEASURABLE_ON |- (!f. (((real_continuous_on f) UNIV) ==> ((real_measurable_on f) UNIV)))
DIM_NONEMPTY_INTERIOR |- (!s. ((~ ((interior s) = EMPTY)) ==> ((dim s) = (dimindex UNIV))))
SIMPLE_PATH_CONTINUOUS_IMAGE |- (!f. (!g. (((simple_path g) /\ (((continuous_on f) (path_image g)) /\ (!x. (!y. (((x IN (path_image g)) /\ ((y IN (path_image g)) /\ ((f x) = (f y)))) ==> (x = y)))))) ==> (simple_path (f o g)))))
CONVEX_ON_EPIGRAPH_SLICE_LT |- (!f. (!s. (!a. ((((convex_on f) s) /\ (convex s)) ==> (convex (GSPEC (\GEN%PVAR%1705. (?x. (((SETSPEC GEN%PVAR%1705) ((x IN s) /\ ((real_lt (f x)) a))) x)))))))))
TRANSP_ROWVECTOR |- (!v. ((transp (rowvector v)) = (columnvector v)))
ALL_IMP |- (!P. (!Q. (!l. (((!x. ((((MEM x) l) /\ (P x)) ==> (Q x))) /\ ((ALL P) l)) ==> ((ALL Q) l)))))
Ssrbool.andKb |- (!a. (!b. ((a \/ (b /\ a)) = a)))
COMMUTING_MATRIX_INV_NORMAL |- (!A. ((((matrix_mul (transp A)) A) = ((matrix_mul A) (transp A))) ==> (((matrix_mul (matrix_inv A)) A) = ((matrix_mul A) (matrix_inv A)))))
SUBSPACE_EQ_AFFINE |- (!s. ((subspace s) = ((affine s) /\ ((vec (NUMERAL _0)) IN s))))
EXTREME_POINT_OF_CONVEX_HULL_INSERT_EQ |- (!s. (!a. (!x. (((FINITE s) /\ (~ (a IN ((hull affine) s)))) ==> (((extreme_point_of x) ((hull convex) (a INSERT s))) = ((x = a) \/ ((extreme_point_of x) ((hull convex) s))))))))
REAL_INTEGRAL_EQ_0 |- (!f. (!s. ((!x. ((x IN s) ==> ((f x) = (real_of_num (NUMERAL _0))))) ==> (((real_integral s) f) = (real_of_num (NUMERAL _0))))))
Miqmcsn.STAB_4M8_13_SCS |- ((((scs_diag (scs_k_v39 scs_4M8_13)) i) j) ==> ((is_scs_v39 (((scs_stab_diag_v39 scs_4M8_13) i) j)) /\ (scs_basic_v39 (((scs_stab_diag_v39 scs_4M8_13) i) j))))
LINEAR_OPEN_MAPPING |- (!f. (!g. (((linear f) /\ ((linear g) /\ ((f o g) = I))) ==> (!s. ((open s) ==> (open ((IMAGE f) s)))))))
Tactics_jordan.strip_lt_lemma |- (!B1. (!B2. (!C. ((((real_lt (real_of_num (NUMERAL _0))) ((real_add B1) B2)) ==> C) ==> (((real_lt (real_of_num (NUMERAL _0))) B2) ==> (((real_le (real_of_num (NUMERAL _0))) B1) ==> C))))))
Lp_ineqs_proofs.sol_y_sym_reflect |- (!y1. (!y3. (!y2. (!y4. (!y6. (!y5. (((((((sol_y y1) y2) y3) y4) y5) y6) = ((((((sol_y y1) y3) y2) y4) y6) y5))))))))
Ssrnat.half_leq |- (!m. (!n. ((m <= n) ==> ((half m) <= (half n)))))
CARD_FINITE_IMAGE |- (!s. ((CARD UNIV) = (dimindex s)))
TAGGED_DIVISION_OF_TRIVIAL |- (!p. (((tagged_division_of p) EMPTY) = (p = EMPTY)))
ROTATE2D_ADD_VECTORS |- (!t. (!w. (!z. (((rotate2d t) ((vector_add w) z)) = ((vector_add ((rotate2d t) w)) ((rotate2d t) z))))))
INTERVAL_IMAGE_STRETCH_INTERVAL |- (!a. (!b. (!m. (?u. (?v. (((IMAGE (\x. (lambdak. ((real_mul (m k)) (x $ k))))) (closed_interval ((CONS (a , b)) NIL))) = (closed_interval ((CONS (u , v)) NIL))))))))
SIMPLEX_IMP_CLOSED |- (!s. (!n. (((simplex n) s) ==> (closed s))))
DIV_MONO_LT |- (!m. (!n. (!p. (((~ (p = (NUMERAL _0))) /\ ((m + p) <= n)) ==> ((m DIV p) < (n DIV p))))))
LOG_SQRT |- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((log (sqrt x)) = ((real_div (log x)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))
Vol1.RADIAL_VS_RADIAL_NORM |- (!x. (!r. (!C. ((((radial r) x) C) = (((radial_norm r) x) C)))))
Seq.mask1 |- (!b. (!x. (((mask ((CONS b) NIL)) ((CONS x) NIL)) = ((nseq (((COND b) (NUMERAL (BIT1 _0))) (NUMERAL _0))) x))))
ALL2_MAP |- (!P. (!f. (!l. ((((ALL2 P) ((MAP f) l)) l) = ((ALL (\a. ((P (f a)) a))) l)))))
Lp_main_estimate.ineq_tau6_tauVEF |- (!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))))))))))))) (tauVEF (V , (E , ((face (hypermap_of_fan (V , E))) d))))))))))))
Trigonometry1.VECTOR_NEG_COMPONENT_3 |- (!x. ((((vector_neg x) $ (NUMERAL (BIT1 _0))) = (real_neg (x $ (NUMERAL (BIT1 _0))))) /\ ((((vector_neg x) $ (NUMERAL (BIT0 (BIT1 _0)))) = (real_neg (x $ (NUMERAL (BIT0 (BIT1 _0)))))) /\ (((vector_neg x) $ (NUMERAL (BIT1 (BIT1 _0)))) = (real_neg (x $ (NUMERAL (BIT1 (BIT1 _0)))))))))
REAL_MUL_SIN_COS |- (!x. (!y. (((real_mul (sin x)) (cos y)) = ((real_div ((real_add (sin ((real_add x) y))) (sin ((real_sub x) y)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))
POLYTOPE_VERTEX_LOWER_BOUND |- (!p. ((polytope p) ==> ((int_le ((int_add (aff_dim p)) (int_of_num (NUMERAL (BIT1 _0))))) (int_of_num (CARD (GSPEC (\GEN%PVAR%2456. (?v. (((SETSPEC GEN%PVAR%2456) ((extreme_point_of v) p)) v)))))))))
INT_LT_ADDNEG2 |- (!x. (!y. (!z. (((int_lt ((int_add x) (int_neg y))) z) = ((int_lt x) ((int_add z) y))))))
Hypermap_iso.image_res |- (!f. (!s. (!u. ((s SUBSET u) ==> (((IMAGE f) s) = ((IMAGE ((res f) u)) s))))))
Lp_ineqs_proofs.rhazim3c |- (!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))) ==> (((rhazim3_fan (V , E)) d) = ((rhazim_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))
REAL_INTEGRABLE_INCREASING_PRODUCT |- (!f. (!g. (!a. (!b. ((((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)))))) ==> ((real_integrable_on (\x. ((real_mul (g x)) (f x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))
FACE_OF_SLICE |- (!f. (!s. (!t. ((((face_of f) s) /\ (convex t)) ==> ((face_of (f INTER t)) (s INTER t))))))
INT_MIN_LT |- (!x. (!y. (!z. (((int_lt ((int_min x) y)) z) = (((int_lt x) z) \/ ((int_lt y) z))))))
INT_EXISTS_ABS |- (!P. ((?n. (P (int_of_num n))) = (?x. (P (int_abs x)))))
SUMMABLE_ADD |- (!x. (!y. (!s. ((((summable s) x) /\ ((summable s) y)) ==> ((summable s) (\n. ((vector_add (x n)) (y n))))))))
LIM_NULL_COMPLEX_RMUL |- (!net. (!f. (!c. ((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) ==> (((--> (\x. ((complex_mul (f x)) c))) (Cx (real_of_num (NUMERAL _0)))) net)))))
Ssrnat.succn_inj |- (!n. (!m. (((SUC n) = (SUC m)) ==> (n = m))))
Seq.zip_unzip |- (!s. (((zip (unzip1 s)) (unzip2 s)) = s))
HAS_DERIVATIVE_INVERSE_ON |- (!f. (!s. (((open s) /\ (!x. ((x IN s) ==> ((((has_derivative f) (f' x)) (at x)) /\ (((g (f x)) = x) /\ (((f' x) o (g' x)) = I)))))) ==> (!x. ((x IN s) ==> (((has_derivative g) (g' x)) (at (f x))))))))
REAL_UNIFORMLY_CONTINUOUS_ON_ADD |- (!f. (!g. (!s. ((((real_uniformly_continuous_on f) s) /\ ((real_uniformly_continuous_on g) s)) ==> ((real_uniformly_continuous_on (\x. ((real_add (f x)) (g x)))) s)))))
OPEN_INTERVAL_EQ |- ((!a. (!b. ((open (closed_interval ((CONS (a , b)) NIL))) = ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)))) /\ (!a. (!b. (open (open_interval (a , b))))))
FINITE_FUNSPACE_UNIV |- (((FINITE UNIV) /\ (FINITE UNIV)) ==> (FINITE UNIV))
INT_MAX |- (!x. (!y. (((int_max x) y) = (((COND ((int_le x) y)) y) x))))
Pack2.ineq_lm5_3_step3 |- (!S. (!p. (!r. (!A. ((((real_le (real_of_num (NUMERAL _0))) r) /\ ((packing S) /\ ((saturated S) /\ ((fcc_compatible A) S)))) ==> ((real_le ((real_mul (sqrt (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (CARD (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0))))))))))) ((sum (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) (\v. ((real_add (A v)) (measure ((voronoi_closed S) v)))))))))))
INSIDE_INSIDE_EQ_EMPTY |- (!s. (((closed s) /\ (connected s)) ==> ((inside (inside s)) = EMPTY)))
INT_MIN_MAX |- (!x. (!y. (((int_min x) y) = (int_neg ((int_max (int_neg x)) (int_neg y))))))
WOSET_FINITE_TOSET |- (!l. (((toset l) /\ (FINITE (fl l))) ==> (woset l)))
EXISTS_DOUBLE_ARC |- (!g. (!a. (!b. (((simple_path g) /\ (((pathfinish g) = (pathstart g)) /\ ((a IN (path_image g)) /\ ((b IN (path_image g)) /\ (~ (a = b)))))) ==> (?u. (?d. ((arc u) /\ ((arc d) /\ (((pathstart u) = a) /\ (((pathfinish u) = b) /\ (((pathstart d) = b) /\ (((pathfinish d) = a) /\ ((((path_image u) INTER (path_image d)) = (a INSERT (b INSERT EMPTY))) /\ (((path_image u) UNION (path_image d)) = (path_image g)))))))))))))))
Lp_ineqs_proofs.y4'_list_def |- (!h. (!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)))) ==> ((!x. ((g x) = ((h (FST x)) , (h (SND x))))) ==> ((ALL (\d. (((y4'_list (g , (V , E))) d) = ((ye_list g) ((SND d) , (FST ((inverse (f_list_ext L)) d))))))) (list_of_darts L)))))))))))
COUNTABLE_CLOPEN_IN |- (!u. ((compact u) ==> (COUNTABLE (GSPEC (\GEN%PVAR%1247. (?c. (((SETSPEC GEN%PVAR%1247) (((closed_in ((subtopology euclidean) u)) c) /\ ((open_in ((subtopology euclidean) u)) c))) c)))))))
OPEN_IN_GDELTA |- (!s. (!t. ((((open_in ((subtopology euclidean) t)) s) /\ (gdelta t)) ==> (gdelta s))))
REAL_MEASURE_REAL_NEGLIGIBLE_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)))) ==> (real_negligible (s INTER t))))))) ==> ((real_measure (UNIONS f)) = ((sum f) m)))))
CROSS_EQ |- (!s. (!s'. (!t. (!t'. (((s CROSS t) = (s' CROSS t')) = ((((s = EMPTY) \/ (t = EMPTY)) /\ ((s' = EMPTY) \/ (t' = EMPTY))) \/ ((s = s') /\ (t = t'))))))))
HOMEOMORPHIC_OPEN_INTERVALS |- (!a. (!b. (!c. (!d. ((((open_interval (a , b)) = EMPTY) = ((open_interval (c , d)) = EMPTY)) ==> ((homeomorphic (open_interval (a , b))) (open_interval (c , d))))))))
SC_RC |- (!R. ((SC (RC R)) = (RC (SC R))))
INT_LT_MAX |- (!x. (!y. (!z. (((int_lt z) ((int_max x) y)) = (((int_lt z) x) \/ ((int_lt z) y))))))
REAL_SUMMABLE_IMP_BOUNDED |- (!f. (!k. (((real_summable k) f) ==> (real_bounded ((IMAGE f) k)))))
Optimize.split_2h0 |- (!a. (!b. (!y. ((((real_le a) y) /\ ((real_le y) b)) ==> ((((real_le a) y) /\ ((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) \/ (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y) /\ ((real_le y) b)))))))
MEASURABLE_COUNTABLE_UNIONS |- (!s. (!B. (((!n. (measurable (s n))) /\ (!n. ((real_le ((sum ((NUMERAL _0) .. n)) (\k. (measure (s k))))) B))) ==> (measurable (UNIONS (GSPEC (\GEN%PVAR%3387. (?n. (((SETSPEC GEN%PVAR%3387) (n IN UNIV)) (s n))))))))))
Tame_lemmas.dart_leads_into_same |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!f. (!v. (!w. (!x. (!y. (((f IN (face_set (hypermap_of_fan (V , E)))) /\ (((v , w) IN f) /\ ((x , y) IN f))) ==> ((((((dart_leads_into (vec (NUMERAL _0))) V) E) v) w) = (((((dart_leads_into (vec (NUMERAL _0))) V) E) x) y))))))))))))
MEASURE_INTERVAL_1_ALT |- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND ((real_lt (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))) /\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND ((real_lt (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))))
NSUM_PAIR |- (!f. (!m. (!n. (((nsum (((NUMERAL (BIT0 (BIT1 _0))) * m) .. (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 _0))))) f) = ((nsum (m .. n)) (\i. ((f ((NUMERAL (BIT0 (BIT1 _0))) * i)) + (f (((NUMERAL (BIT0 (BIT1 _0))) * i) + (NUMERAL (BIT1 _0)))))))))))
ABSOLUTELY_CONTINUOUS_ON_LIFT_ABS |- (!f. (((absolutely_continuous_on (\x. (lift (f x)))) s) ==> ((absolutely_continuous_on (\x. (lift (real_abs (f x))))) s)))
CONTINUOUS_CARD_LT_RANGE_CONSTANT |- (!f. (!s. (((connected s) /\ (((continuous_on f) s) /\ (((IMAGE f) s) <_c UNIV))) ==> (?a. (!x. ((x IN s) ==> ((f x) = a)))))))
IN_CBALL_IM |- (!x. (!z. (!e. ((x IN (cball (z , e))) ==> ((real_le (real_abs ((real_sub (Im x)) (Im z)))) e)))))
NUM_GCD |- (!a. (!b. ((int_of_num (num_gcd (a , b))) = (int_gcd ((int_of_num a) , (int_of_num b))))))