hcju/holps huggingface dataset:
RPOW_1_LE |- (!a. (!x. ((((real_le (real_of_num (NUMERAL _0))) x) /\ (((real_le x) (real_of_num (NUMERAL (BIT1 _0)))) /\ ((real_le (real_of_num (NUMERAL _0))) a))) ==> ((real_le ((rpow x) a)) (real_of_num (NUMERAL (BIT1 _0)))))))
CONNECTED_SUMS |- (!s. (!t. (((connected s) /\ (connected t)) ==> (connected (GSPEC (\GEN%PVAR%917. (?x. (?y. (((SETSPEC GEN%PVAR%917) ((x IN s) /\ (y IN t))) ((vector_add x) y))))))))))
REAL_MUL_COS_SIN |- (!x. (!y. (((real_mul (cos x)) (sin y)) = ((real_div ((real_sub (sin ((real_add x) y))) (sin ((real_sub x) y)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))
Ssrbool.implyNb |- (!a. (!b. (((~ a) ==> b) = (a \/ b))))
Merge_ineq.nonf_mud_135_x_v1 |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a. (!b. (!c. (((real_le (real_of_num (NUMERAL _0))) a) ==> ((((((((((mud_135_x_v1 a) b) c) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((mu6_x ((real_mul a) a)) x1) x3) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0)))) (sqrt (((((((((delta_135_x ((real_mul a) a)) ((real_mul b) b)) ((real_mul c) c)) x1) x2) x3) x4) x5) x6))))))))))))))
LEBESGUE_MEASURABLE_PREIMAGE_OPEN |- (!f. (!t. ((((measurable_on f) UNIV) /\ (open t)) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3550. (?x. (((SETSPEC GEN%PVAR%3550) ((f x) IN t)) x))))))))
Nonlinear_lemma.edge_quadratic |- (!x1. (!x2. (!x3. (!x5. (!x6. ((quadratic_root_plus (x1 , (((real_add ((real_mul x1) x1)) ((real_sub ((real_mul ((real_sub x3) x5)) ((real_sub x2) x6))) ((real_mul x1) ((real_add x2) ((real_add x3) ((real_add x5) x6)))))) , ((real_add ((real_mul x1) ((real_mul x3) x5))) ((real_sub ((real_sub ((real_mul x1) ((real_mul x2) x6))) ((real_mul x3) ((real_mul ((real_add x1) ((real_add ((real_sub x2) x3)) ((real_sub x5) x6)))) x6)))) ((real_mul x2) ((real_mul x5) ((real_add ((real_sub x1) x2)) ((real_add ((real_sub x3) x5)) x6))))))))) = ((real_div ((real_add (real_neg ((real_mul x1) x1))) ((real_add ((real_mul x1) x2)) ((real_add ((real_sub ((real_mul x1) x3)) ((real_mul x2) x3))) ((real_add ((real_mul x1) x5)) ((real_add ((real_mul x2) x5)) ((real_add ((real_mul x1) x6)) ((real_add ((real_sub ((real_mul x3) x6)) ((real_mul x5) x6))) (sqrt ((real_mul (((ups_x x1) x3) x5)) (((ups_x x1) x2) x6))))))))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x1))))))))
Seq.leq_size_perm |- (!s1. (!s2. ((uniq s1) ==> ((!x. (((MEM x) s1) ==> ((MEM x) s2))) ==> (((sizel s2) <= (sizel s1)) ==> ((!x. (((MEM x) s1) = ((MEM x) s2))) /\ ((sizel s1) = (sizel s2))))))))
LIM_CONG_ATREAL |- ((!x. ((~ (x = a)) ==> ((f x) = (g x)))) ==> ((((--> (\x. (f x))) l) (atreal a)) = (((--> g) l) (atreal a))))
HOMEOMORPHIC_SCALING_RIGHT |- (!c. (((real_lt (real_of_num (NUMERAL _0))) c) ==> (!s. (!t. (((homeomorphic s) ((IMAGE (\x. ((% c) x))) t)) = ((homeomorphic s) t))))))
REAL_INTEGRAL_REFLECT |- (!f. (!a. (!b. (((real_integral (closed_real_interval ((CONS ((real_neg b) , (real_neg a))) NIL))) (\x. (f (real_neg x)))) = ((real_integral (closed_real_interval ((CONS (a , b)) NIL))) f)))))
Ssrnat.ltn_pexp2l |- (!m. (!n1. (!n2. (((NUMERAL _0) < m) ==> (((m EXP n1) < (m EXP n2)) ==> (n1 < n2))))))
STC_INC_TC |- (!R. (!x. (!y. ((((TC R) x) y) ==> (((STC R) x) y)))))
OPEN_MAP_IFF_CLOSED_MAP_BIJECTIVE |- (!f. (!s. (!t. (((((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)))) = (!c. (((closed_in ((subtopology euclidean) s)) c) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) c)))))))))
Nonlinear_lemma.vol2r_y |- (!y. (!r. (((vol2r y) r) = ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul pi) ((real_mul r) ((real_mul r) (real_inv (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))) ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) ((real_mul pi) ((real_mul (real_inv (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))) ((real_pow y) (NUMERAL (BIT0 (BIT1 _0)))))))))))
LOCALLY_CONNECTED_SPHERE_GEN |- (!s. (((bounded s) /\ (convex s)) ==> ((locally connected) (relative_frontier s))))
Wrgcvdr_cizmrrh.hypermap |- (!H. (((tuple_hypermap H) = (D , (e , (n , f)))) ==> (((permutes e) D) /\ (((permutes n) D) /\ (((permutes f) D) /\ ((e o (n o f)) = I))))))
SUPERADMISSIBLE_MATCH_SEQPATTERN |- (!<<. (!p. (!s. (!c1. (!c2. (!e. ((((((admissible <<) p) s) (\f. (\x. (?y. ((((c1 f) x) ((e f) x)) y))))) /\ (((((superadmissible <<) (\f. (\x. (((p f) x) /\ (?y. ((((c1 f) x) ((e f) x)) y)))))) s) (\f. (\x. ((_MATCH ((e f) x)) ((c1 f) x))))) /\ ((((superadmissible <<) (\f. (\x. (((p f) x) /\ (~ (?y. ((((c1 f) x) ((e f) x)) y))))))) s) (\f. (\x. ((_MATCH ((e f) x)) ((c2 f) x))))))) ==> ((((superadmissible <<) p) s) (\f. (\x. ((_MATCH ((e f) x)) ((_SEQPATTERN ((c1 f) x)) ((c2 f) x)))))))))))))
SEPARATION_NORMAL_COMPACT |- (!s. (!t. (((compact s) /\ ((closed t) /\ ((s INTER t) = EMPTY))) ==> (?u. (?v. ((open u) /\ ((compact (closure u)) /\ ((open v) /\ ((s SUBSET u) /\ ((t SUBSET v) /\ ((u INTER v) = EMPTY)))))))))))
Upfzbzm_support_lemmas.john_harrison_lemma1 |- ((GABS (\f. (!x. (!y. ((GEQ (f (x , y))) ((P x) y)))))) = (\p. ((P (FST p)) (SND p))))
INT_LT_ADDL |- (!x. (!y. (((int_lt y) ((int_add x) y)) = ((int_lt (int_of_num (NUMERAL _0))) x))))
ORTHOGONAL_BASIS_EXISTS |- (!v. (?b. ((independent b) /\ ((b SUBSET (span v)) /\ ((v SUBSET (span b)) /\ ((b HAS_SIZE (dim v)) /\ ((pairwise orthogonal) b)))))))
FSIGMA_IMP_LEBESGUE_MEASURABLE |- (!s. ((fsigma s) ==> (lebesgue_measurable s)))
NPRODUCT_DELTA |- (!s. (!a. (((nproduct s) (\x. (((COND (x = a)) b) (NUMERAL (BIT1 _0))))) = (((COND (a IN s)) b) (NUMERAL (BIT1 _0))))))
CLOSED_MAP_CLOSURES |- (!f. ((!s. ((closed s) ==> (closed ((IMAGE f) s)))) = (!s. ((closure ((IMAGE f) s)) SUBSET ((IMAGE f) (closure s))))))
Ssrnat.sqrn_add_sub |- (!m. (!n. ((n <= m) ==> ((((m + n) EXP (NUMERAL (BIT0 (BIT1 _0)))) - ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) * (m * n))) = ((m - n) EXP (NUMERAL (BIT0 (BIT1 _0))))))))
COMPLEX_IN_BALL_0 |- (!v. (!r. ((v IN (ball ((Cx (real_of_num (NUMERAL _0))) , r))) = ((real_lt (vector_norm v)) r))))
NORM_CCOS_POW_2 |- (!z. (((real_pow (vector_norm (ccos z))) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_add ((real_pow (cos (Re z))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_div ((real_pow ((real_sub (exp (Im z))) (real_inv (exp (Im z))))) (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))
Seq.unzip2_zip |- (!s. (!t. (((sizel t) <= (sizel s)) ==> ((unzip2 ((zip s) t)) = t))))
EXPOSED_POINT_OF_FURTHEST_POINT |- (!s. (!a. (!b. (((b IN s) /\ (!x. ((x IN s) ==> ((real_le (distance (a , x))) (distance (a , b)))))) ==> ((exposed_face_of (b INSERT EMPTY)) s)))))
CONVEX_NORM |- (!s. ((convex_on vector_norm) s))
Tame_defs2.good_list_nodes |- (!L. ((good_list_nodes L) = ((node_set (hypermap_of_list L)) = (set_of_list (nodes_of_list L)))))
DEPENDENT_LINEAR_IMAGE |- (!f. (!s. (((linear f) /\ ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) /\ (dependent s))) ==> (dependent ((IMAGE f) s)))))
ANR_INSERT |- (!s. (!a. (((closed s) /\ (ANR s)) ==> (ANR (a INSERT s)))))
HAS_VECTOR_DERIVATIVE_ID |- (!net. (((has_vector_derivative (\x. x)) (vec (NUMERAL (BIT1 _0)))) net))
REAL_INTEGRABLE_REAL_BOUNDED_VARIATION_PRODUCT |- (!f. (!g. (!a. (!b. ((((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ ((has_bounded_real_variation_on g) (closed_real_interval ((CONS (a , b)) NIL)))) ==> ((real_integrable_on (\x. ((real_mul (g x)) (f x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))
HAS_REAL_INTEGRAL_NEGLIGIBLE_EQ |- (!f. (!s. ((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))) ==> ((((has_real_integral f) (real_of_num (NUMERAL _0))) s) = (real_negligible (GSPEC (\GEN%PVAR%4964. (?x. (((SETSPEC GEN%PVAR%4964) ((x IN s) /\ (~ ((f x) = (real_of_num (NUMERAL _0)))))) x)))))))))
CARD_EXP_C |- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> ((CARD ((^_c s) t)) = ((CARD s) EXP (CARD t))))))
Nonlinear_lemma.gcy_high_hplus |- (!y. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) hplus)) y) ==> ((gcy y) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_div mm1) pi)))))
NPRODUCT_FACT |- (!n. (((nproduct ((NUMERAL (BIT1 _0)) .. n)) (\m. m)) = (FACT n)))
INT_NEG_LT0 |- (!x. (((int_lt (int_neg x)) (int_of_num (NUMERAL _0))) = ((int_lt (int_of_num (NUMERAL _0))) x)))
CONVEX_CONE_PCROSS_EQ |- (!s. (!t. ((convex_cone (s PCROSS t)) = ((convex_cone s) /\ (convex_cone t)))))
REAL_UNIFORMLY_CONTINUOUS_ON_COMPOSE |- (!f. (!g. (!s. ((((real_uniformly_continuous_on f) s) /\ ((real_uniformly_continuous_on g) ((IMAGE f) s))) ==> ((real_uniformly_continuous_on (g o f)) s)))))
TAGGED_DIVISION_OF_NONTRIVIAL |- (!s. (!a. (!b. ((((tagged_division_of s) (closed_interval ((CONS (a , b)) NIL))) /\ (~ ((content (closed_interval ((CONS (a , b)) NIL))) = (real_of_num (NUMERAL _0))))) ==> ((tagged_division_of (GSPEC (\GEN%PVAR%2765. (?x. (?k. (((SETSPEC GEN%PVAR%2765) (((x , k) IN s) /\ (~ ((content k) = (real_of_num (NUMERAL _0)))))) (x , k))))))) (closed_interval ((CONS (a , b)) NIL)))))))
Lp_gen_theory.iso_face_trans |- (!H. (!G. (!g. (((hyp_iso g) (G , H)) ==> (!P. ((!f. ((f IN (face_set H)) ==> (P f))) ==> (!f. ((f IN (face_set G)) ==> (P ((IMAGE g) f))))))))))
Seq.mem_seq4 |- (!x. (!y1. (!y2. (!y3. (!y4. (((MEM x) ((CONS y1) ((CONS y2) ((CONS y3) ((CONS y4) NIL))))) = ((x = y1) \/ ((x = y2) \/ ((x = y3) \/ (x = y4))))))))))
REAL_OPEN_REAL_CLOSED |- (!s. ((real_open s) = (real_closed (UNIV DIFF s))))
Ssrbool.implybE |- (!a. (!b. ((a ==> b) = ((~ a) \/ b))))
IMAGE_MATRIX_INV |- (!A. (((IMAGE (\x. ((matrix_vector_mul (matrix_inv A)) x))) UNIV) = ((IMAGE (\x. ((matrix_vector_mul (transp A)) x))) UNIV)))
ANR_DELETE |- (!s. (!a. ((ANR s) ==> (ANR (s DELETE a)))))
RECTIFIABLE_PATH_DIFFERENTIABLE |- (!g. (!s. (((COUNTABLE s) /\ ((path g) /\ (!t. ((t IN ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) DIFF s)) ==> ((differentiable g) (at t)))))) ==> ((rectifiable_path g) = ((absolutely_integrable_on (\t. ((vector_derivative g) (at t)))) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))))))
COVERING_SPACE_COUNTABLE_SHEETS |- (!p. (!c. (!s. (!b. (((covering_space (c , p)) s) ==> (COUNTABLE (GSPEC (\GEN%PVAR%2073. (?x. (((SETSPEC GEN%PVAR%2073) ((x IN c) /\ ((p x) = b))) x))))))))))
COUNTABLE_DIFF_FINITE |- (!s. (!t. ((FINITE s) ==> ((COUNTABLE (t DIFF s)) = (COUNTABLE t)))))
Hypermap_and_fan.F_HAS_NO_FIXED_POINTS_IN_D1 |- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (x IN (dart1_of_fan (V , E)))) ==> (~ (((f_fan_pair (V , E)) x) = x))))))
NOT_EMPTY_INSERT |- (!x. (!s. (~ (EMPTY = (x INSERT s)))))
Lp_ineqs_proofs.azim3_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 ((azim3_fan (V , E)) d)) pi)))))))))
Functional_equation.functional_ups234_x |- (ups234_x = (((((((compose6 ups_126) proj_x2) proj_x3) dummy6) dummy6) dummy6) proj_x4))
Add_triangle.fan_hypermap_map2 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (d1_fan ((vec (NUMERAL _0)) , (V , E)))) ==> ((((face_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) d) = ((((f1_fan (vec (NUMERAL _0))) V) E) d)) /\ ((((node_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) d) = ((((n_fan (vec (NUMERAL _0))) V) E) d)) /\ (((edge_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) d) = ((((e_fan (vec (NUMERAL _0))) V) E) d)))))))))
Float.INT_POS_NEG |- (!n. (((int_lt (int_of_num (NUMERAL _0))) (int_neg (int_of_num n))) = F))
Ssrnat.distn0 |- (!n. (((distn n) (NUMERAL _0)) = n))
VOLUME_FRUSTT |- (!v0. (!v1. (!h. (!a. (((real_lt (real_of_num (NUMERAL _0))) a) ==> ((measurable ((((frustt v0) v1) h) a)) /\ ((measure ((((frustt v0) v1) h) a)) = (((COND ((v1 = v0) \/ (((real_le (real_of_num (NUMERAL (BIT1 _0)))) a) \/ ((real_lt h) (real_of_num (NUMERAL _0)))))) (real_of_num (NUMERAL _0))) ((real_mul pi) ((real_mul ((real_sub ((real_pow ((real_div h) a)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow h) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_div h) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))))))))
LIMPT_OF_OPEN |- (!s. (!x. (((open s) /\ (x IN s)) ==> ((limit_point_of x) s))))
COMPLEX_DIFFERENTIABLE_CARATHEODORY_WITHIN |- (!f. (!z. (!s. (((complex_differentiable f) ((within (at z)) s)) = (?g. ((!w. (((vector_sub (f w)) (f z)) = ((complex_mul (g w)) ((vector_sub w) z)))) /\ ((continuous g) ((within (at z)) s))))))))
Pack2.POLYTOPE_VORONOI_CLOSED |- (!s. (!v. (((packing s) /\ ((saturated s) /\ (v IN s))) ==> (polytope ((voronoi_closed s) v)))))
PROPER_MAP_FROM_COMPOSITION_LEFT |- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\ ((((IMAGE f) s) = t) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET u) /\ (!k. (((k SUBSET u) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%738. (?x. (((SETSPEC GEN%PVAR%738) ((x IN s) /\ (((g o f) x) IN k))) x))))))))))) ==> (!k. (((k SUBSET u) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%739. (?x. (((SETSPEC GEN%PVAR%739) ((x IN t) /\ ((g x) IN k))) x)))))))))))))
UNIFORMLY_CONTINUOUS_ON_COMPLEX_MUL |- (!f. (!g. (!s. ((((uniformly_continuous_on f) s) /\ (((uniformly_continuous_on g) s) /\ ((bounded ((IMAGE f) s)) /\ (bounded ((IMAGE g) s))))) ==> ((uniformly_continuous_on (\x. ((complex_mul (f x)) (g x)))) s)))))
NSUM_EQ_GENERAL_INVERSES |- (!s. (!t. (!f. (!g. (!h. (!k. (((!y. ((y IN t) ==> (((k y) IN s) /\ ((h (k y)) = y)))) /\ (!x. ((x IN s) ==> (((h x) IN t) /\ (((k (h x)) = x) /\ ((g (h x)) = (f x))))))) ==> (((nsum s) f) = ((nsum t) g)))))))))
HAS_BOUNDED_REAL_VARIATION_ON_SUB |- (!f. (!g. (!s. ((((has_bounded_real_variation_on f) s) /\ ((has_bounded_real_variation_on g) s)) ==> ((has_bounded_real_variation_on (\x. ((real_sub (f x)) (g x)))) s)))))
RECTIFIABLE_PATH_REPARAMETRIZATION |- (!g. (!h. (!h'. (((rectifiable_path g) /\ ((homeomorphism ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) , (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))) (h , h'))) ==> (rectifiable_path (g o h))))))
Tame_list.hideDups_eq_nil |- (!s. (((hideDups s) = NIL) = (s = NIL)))
Nonlinear_lemma.sqrt_x2 |- (((((((sqrt_x2 x1) x2) x3) x4) x5) x6) = (sqrt x2))
HAS_REAL_DERIVATIVE_DERIVATIVE |- (!f. (!f'. (!x. ((((has_real_derivative f) f') (atreal x)) ==> (((real_derivative f) x) = f')))))
CONVEX_CONE_HULL_CONVEX_HULL |- (!s. (((hull convex_cone) s) = ((vec (NUMERAL _0)) INSERT (GSPEC (\GEN%PVAR%1701. (?c. (?x. (((SETSPEC GEN%PVAR%1701) (((real_le (real_of_num (NUMERAL _0))) c) /\ (x IN ((hull convex) s)))) ((% c) x)))))))))
Pqcsxwg.REAL_CONTINUOUS_WITHIN_SQRT_COMPOSE |- (!f. (!s. (!a. ((((real_continuous f) ((within (at a)) s)) /\ (((real_lt (real_of_num (NUMERAL _0))) (f a)) \/ (!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))))) ==> ((real_continuous (\x. (sqrt (f x)))) ((within (at a)) s))))))
Lp_ineqs_proofs.rhazim_lo |- (!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 (real_of_num (NUMERAL _0))) ((rhazim_fan (V , E)) d))))))))))
REAL_INFSUM_RESTRICT |- (!k. (!a. (((real_infsum UNIV) (\n. (((COND (n IN k)) (a n)) (real_of_num (NUMERAL _0))))) = ((real_infsum k) a))))
Tame_lemmas.reflection_eq2 |- (!n. (!x. ((((reflection n) x) = x) = (((dot x) n) = (real_of_num (NUMERAL _0))))))
Tame_list.nextElem_ALT |- (((((nextElem NIL) b) x) = b) /\ ((((nextElem ((CONS a) as)) b) x) = (((COND (x = a)) (((COND (as = NIL)) b) ((LET (\a_. (LET_END ((LET (\as_. (LET_END a_))) (TL as))))) (HD as)))) (((nextElem as) b) x))))
Seq.has_mask_cons |- (!a. (!b. (!m. (!x. (!s. (((has a) ((mask ((CONS b) m)) ((CONS x) s))) = ((b /\ (a x)) \/ ((has a) ((mask m) s)))))))))
ORTHOGONAL_MATRIX_NORM_EQ |- (!A. ((orthogonal_matrix A) = (!x. ((vector_norm ((matrix_vector_mul A) x)) = (vector_norm x)))))
Tame_general.RUNOQPQ |- (!H. ((tame_planar_hypermap H) ==> (restricted_hypermap H)))
Bump.MCELL2_PARAM_UL |- (!V. (!ul. (!vl. (!X. (((packing V) /\ ((saturated V) /\ ((X = ((mcell2 V) ul)) /\ ((ul IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))) /\ ((~ (negligible X)) /\ (vl = (SND ((cell_params V) X)))))))) ==> ((X = ((mcell2 V) vl)) /\ (vl IN ((barV V) (NUMERAL (BIT1 (BIT1 _0)))))))))))
DECREASING_LEFT_LIMIT |- (!f. (!a. (!b. (!c. (((!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_le x) y))) ==> ((real_le (f y)) (f x))))) /\ (c IN (closed_real_interval ((CONS (a , b)) NIL)))) ==> (?l. (((---> f) l) ((within (atreal c)) (closed_real_interval ((CONS (a , c)) NIL))))))))))
MEASURE_LIPSCHITZ_IMAGE |- (!f. (!s. (!B. ((((dimindex UNIV) <= (dimindex UNIV)) /\ ((measurable s) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y))))))))) ==> ((real_le (measure ((IMAGE f) s))) ((real_mul ((real_pow B) (dimindex UNIV))) (measure s)))))))
IM_COMPLEX_INV_EQ_0 |- (!z. (((Im (complex_inv z)) = (real_of_num (NUMERAL _0))) = ((Im z) = (real_of_num (NUMERAL _0)))))
CARD_EQ_COUNTABLE_SUBSETS_REAL |- ((GSPEC (\GEN%PVAR%73. (?s. (((SETSPEC GEN%PVAR%73) (COUNTABLE s)) s)))) =_c UNIV)
QUOTIENT_MAP_FROM_SUBSET |- (!f. (!s. (!t. (!u. ((((continuous_on f) t) /\ ((((IMAGE f) t) SUBSET u) /\ ((s SUBSET t) /\ ((((IMAGE f) s) = u) /\ (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%669. (?x. (((SETSPEC GEN%PVAR%669) ((x IN s) /\ ((f x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v)))))))) ==> (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) t)) (GSPEC (\GEN%PVAR%670. (?x. (((SETSPEC GEN%PVAR%670) ((x IN t) /\ ((f x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v)))))))))
REAL_DIFFERENTIABLE_BOUND |- (!f. (!f'. (!s. (!B. (((is_realinterval s) /\ (!x. ((x IN s) ==> ((((has_real_derivative f) (f' x)) ((within (atreal x)) s)) /\ ((real_le (real_abs (f' x))) B))))) ==> (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (real_abs ((real_sub (f x)) (f y)))) ((real_mul B) (real_abs ((real_sub x) y))))))))))))
RELATIVE_ORIENTATION_COMPOSE |- (!f. (!g. (!c. ((((simplex ((int_sub (int_of_num (dimindex UNIV))) (int_of_num (NUMERAL (BIT1 _0))))) c) /\ (~ (((relative_orientation f) c) = (real_of_num (NUMERAL _0))))) ==> (((relative_orientation (g o f)) c) = ((real_mul ((relative_orientation g) ((vertex_image f) c))) ((relative_orientation f) c)))))))
DIFFERENTIABLE_ON_MUL |- (!f. (!g. (!s. ((((differentiable_on (lift o f)) s) /\ ((differentiable_on g) s)) ==> ((differentiable_on (\x. ((% (f x)) (g x)))) s)))))
CONTINUOUS_ON_CX_IM |- (!s. ((continuous_on (Cx o Im)) s))
ORTHOGONAL_TRANSFORMATION_BALL |- (!f. (!r. ((orthogonal_transformation f) ==> (((IMAGE f) (ball ((vec (NUMERAL _0)) , r))) = (ball ((vec (NUMERAL _0)) , r))))))
REAL_SUMMABLE_IMP_REAL_SUMS_BOUNDED |- (!f. (!k. (((real_summable (from k)) f) ==> (real_bounded (GSPEC (\GEN%PVAR%4863. (?n. (((SETSPEC GEN%PVAR%4863) (n IN UNIV)) ((sum (k .. n)) f)))))))))
PATH_CONNECTED_SCALING |- (!s. (!c. ((path_connected s) ==> (path_connected ((IMAGE (\x. ((% c) x))) s)))))
INT_LE_SQUARE_ABS |- (!x. (!y. (((int_le (int_abs x)) (int_abs y)) = ((int_le ((int_pow x) (NUMERAL (BIT0 (BIT1 _0))))) ((int_pow y) (NUMERAL (BIT0 (BIT1 _0))))))))
INT_LTE_TOTAL |- (!x. (!y. (((int_lt x) y) \/ ((int_le y) x))))
CLOG_NEG |- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((clog (vector_neg z)) = (((COND (((real_le (Im z)) (real_of_num (NUMERAL _0))) /\ (~ (((real_lt (Re z)) (real_of_num (NUMERAL _0))) /\ ((Im z) = (real_of_num (NUMERAL _0))))))) ((vector_add (clog z)) ((complex_mul ii) (Cx pi)))) ((vector_sub (clog z)) ((complex_mul ii) (Cx pi)))))))
Seq2.next_el_hd_belast |- (!h. (!t. (((NUMERAL (BIT1 _0)) < (sizel t)) ==> (((next_el ((belast h) t)) h) = (HD t)))))
HOMOTOPIC_NEARBY_PATHS |- (!g. (!s. (((path g) /\ ((open s) /\ ((path_image g) SUBSET s))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!h. (((path h) /\ (((pathstart h) = (pathstart g)) /\ (((pathfinish h) = (pathfinish g)) /\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> ((real_lt (vector_norm ((vector_sub (h t)) (g t)))) e)))))) ==> (((homotopic_paths s) g) h))))))))
REAL_INTEGRABLE_RESTRICT_INTER |- (!f. (!s. (!t. (((real_integrable_on (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) t) = ((real_integrable_on f) (s INTER t))))))
PSUBSET_INSERT_SUBSET |- (!s. (!t. ((s PSUBSET t) = (?x. ((~ (x IN s)) /\ ((x INSERT s) SUBSET t))))))
LIM_UNION_UNIV |- (!f. (!x. (!l. (!s. (!t. (((((--> f) l) ((within (at x)) s)) /\ ((((--> f) l) ((within (at x)) t)) /\ ((s UNION t) = UNIV))) ==> (((--> f) l) (at x))))))))
DIST_FSTCART |- (!x. (!y. ((real_le (distance ((fstcart x) , (fstcart y)))) (distance (x , y)))))
INT_DIFFSQ |- (!x. (!y. (((int_mul ((int_add x) y)) ((int_sub x) y)) = ((int_sub ((int_mul x) x)) ((int_mul y) y)))))
INT_NEG_SUB |- (!x. (!y. ((int_neg ((int_sub x) y)) = ((int_sub y) x))))
COMPLEX_EQ_ADD_RCANCEL |- (!x. (!y. (!z. ((((vector_add x) z) = ((vector_add y) z)) = (x = y)))))
PRODUCT_NEG_NUMSEG_1 |- (!f. (!n. (((product ((NUMERAL (BIT1 _0)) .. n)) (\i. (real_neg (f i)))) = ((real_mul ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) n)) ((product ((NUMERAL (BIT1 _0)) .. n)) f)))))
REAL_MEASURE_DISJOINT_UNIONS_IMAGE_STRONG |- (!f. (!s. (((FINITE (GSPEC (\GEN%PVAR%4934. (?x. (((SETSPEC GEN%PVAR%4934) ((x IN s) /\ (~ ((f x) = EMPTY)))) x))))) /\ ((!x. ((x IN s) ==> (real_measurable (f x)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> ((DISJOINT (f x)) (f y))))))) ==> ((real_measure (UNIONS ((IMAGE f) s))) = ((sum s) (\x. (real_measure (f x))))))))
HELLY_COMPACT |- (!f. ((((FINITE f) ==> ((CARD f) >= ((dimindex UNIV) + (NUMERAL (BIT1 _0))))) /\ ((!s. ((s IN f) ==> ((convex s) /\ (compact s)))) /\ (!t. (((t SUBSET f) /\ ((FINITE t) /\ ((CARD t) = ((dimindex UNIV) + (NUMERAL (BIT1 _0)))))) ==> (~ ((INTERS t) = EMPTY)))))) ==> (~ ((INTERS f) = EMPTY))))
Tame_list.BUTLAST_NIL |- (!xs. (((NUMERAL (BIT1 _0)) < (sizel xs)) ==> (~ ((BUTLAST xs) = NIL))))
REAL_SUMMABLE_ZETA_INTEGER |- (!n. (!m. (((NUMERAL (BIT0 (BIT1 _0))) <= m) ==> ((real_summable (from n)) (\k. (real_inv ((real_pow (real_of_num k)) m)))))))
REAL_CLOSED_OPEN_INTERVAL |- (!a. (!b. (((real_le a) b) ==> ((closed_real_interval ((CONS (a , b)) NIL)) = ((open_real_interval (a , b)) UNION (a INSERT (b INSERT EMPTY)))))))
Packing3.HULL_INTER_EQ_INTER |- (!P. (!s. (!t. (((P s) /\ (P t)) ==> (((hull P) (s INTER t)) = (s INTER t))))))
Constants_approx.tgt_interval |- ((interval_arith tgt) (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))))) , ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))
Lp_main_estimate.ineq_tau4_diag3_std |- (!V. (!E. ((lp_fan (V , E)) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\ ((packing V) /\ ((V SUBSET ball_annulus) /\ (((CARD V) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\ ((!v. (!w. (((~ (v = w)) /\ ((v IN V) /\ ((w IN V) /\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) (distance (v , w)))))) /\ (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v , w))) /\ ((real_le (distance (v , w))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))))))))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((tau_fun V) E) FF)))))) ==> ((E = (ESTD V)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((y4'_fan (V , E)) d)) /\ ((real_le (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d)))) ==> ((real_le ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (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)))))))))))