hcju/holps huggingface dataset:
EXTENSION
|- (!s. (!t. ((s = t) = (!x. ((x IN s) = (x IN t))))))
holomorphic_on
|- (!f. (!s. (((holomorphic_on f) s) = (!x. ((x IN s) ==> (?f'. (((has_complex_derivative f) f') ((within (at x)) s))))))))
Planarity.DARTSET_LEADS_INTO_FAN
|- (!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)))))))) ==> (!y. ((y IN ds) ==> (((((dartset_leads_into_fan x) V) E) ds) = (((((dart_leads_into x) V) E) (pr2 y)) (pr3 y))))))))))
Bump.MCELL_PARAM_UL
|- (!V. (!ul. (!vl. (!X. (!k. (((k <= (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\ ((packing V) /\ ((saturated V) /\ ((X = (((mcell k) V) ul)) /\ ((ul IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))) /\ ((~ (negligible X)) /\ (vl = (SND ((cell_params V) X))))))))) ==> ((X = (((mcell k) V) vl)) /\ (vl IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))))))))))
Lp_gen_theory.mem_list_dart_pairs
|- (!L. (!p. (((MEM p) (list_dart_pairs L)) = (((MEM (FST p)) (list_of_darts L)) /\ ((MEM (SND p)) (list_of_darts L))))))
COMPLEX_POW_MUL
|- (!x. (!y. (!n. (((complex_pow ((complex_mul x) y)) n) = ((complex_mul ((complex_pow x) n)) ((complex_pow y) n))))))
VSUM_EQ
|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((vsum s) f) = ((vsum s) g))))))
CLOSED_HALFSPACE_COMPONENT_GE
|- (!a. (!k. (closed (GSPEC (\GEN%PVAR%775. (?x. (((SETSPEC GEN%PVAR%775) ((real_ge (x $ k)) a)) x)))))))
CLOSED_IN_INJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (!t. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((closed_in ((subtopology euclidean) ((IMAGE f) t))) ((IMAGE f) s)) = ((closed_in ((subtopology euclidean) t)) s))))))
COLLINEAR_LEMMA_ALT
|- (!x. (!y. ((collinear ((vec (NUMERAL _0)) INSERT (x INSERT (y INSERT EMPTY)))) = ((x = (vec (NUMERAL _0))) \/ (?c. (y = ((% c) x)))))))
CLOSED_IN_SING
|- (!u. (!x. (((closed_in ((subtopology euclidean) u)) (x INSERT EMPTY)) = (x IN u))))
Lp_ineqs_def.rhazim_fan
|- (!V. (!E. (!x. (((rhazim_fan (V , E)) x) = ((real_mul (real_abs ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul const1) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (lmfun (h_dart x))))))) ((azim_dart (V , E)) x))))))
NPRODUCT_CONST
|- (!c. (!s. ((FINITE s) ==> (((nproduct s) (\x. c)) = (c EXP (CARD s))))))
Seq.constant
|- (((constant NIL) = T) /\ ((constant ((CONS x) s')) = ((all (pred1 x)) s')))
LIM_TRANSFORM_WITHIN_SET_IMP
|- (!f. (!l. (!a. (!s. (!t. ((((eventually (\x. ((x IN t) ==> (x IN s)))) (at a)) /\ (((--> f) l) ((within (at a)) s))) ==> (((--> f) l) ((within (at a)) t))))))))
RSTC_TRANS
|- (!R. (!x. (!y. (!z. (((((RSTC R) x) y) /\ (((RSTC R) y) z)) ==> (((RSTC R) x) z))))))
CONIC_HULL_AS_IMAGE
|- (!s. (((hull conic) s) = ((IMAGE (\z. ((% (drop (fstcart z))) (sndcart z)))) ((GSPEC (\GEN%PVAR%1458. (?t. (((SETSPEC GEN%PVAR%1458) ((real_le (real_of_num (NUMERAL _0))) (drop t))) t)))) PCROSS s))))
MTOPOLOGY_REAL_EUCLIDEAN_METRIC
|- ((mtopology real_euclidean_metric) = euclideanreal)
COMPONENTS_EQ_EMPTY
|- (!s. (((components s) = EMPTY) = (s = EMPTY)))
LOCALLY_COMPACT_TRANSLATION_EQ
|- (!a. (!s. (((locally compact) ((IMAGE (\x. ((vector_add a) x))) s)) = ((locally compact) s))))
REAL_LT_INV_EQ
|- (!x. (((real_lt (real_of_num (NUMERAL _0))) (real_inv x)) = ((real_lt (real_of_num (NUMERAL _0))) x)))
DOT_POS_LE
|- (!x. ((real_le (real_of_num (NUMERAL _0))) ((dot x) x)))
AFFSIGN_ALT
|- ((((affsign sgn) s) t) = (GSPEC (\GEN%PVAR%5090. (?y. (((SETSPEC GEN%PVAR%5090) (?f. ((!w. ((w IN (s UNION t)) ==> ((w IN t) ==> (sgn (f w))))) /\ ((((sum (s UNION t)) f) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum (s UNION t)) (\v. ((% (f v)) v))) = y))))) y)))))
POLYTOPE_LINEAR_IMAGE
|- (!f. (!p. (((linear f) /\ (polytope p)) ==> (polytope ((IMAGE f) p)))))
HAS_REAL_MEASURE_0
|- (!s. (((has_real_measure s) (real_of_num (NUMERAL _0))) = (real_negligible s)))
List_hypermap.face_set_k
|- (!H. (!k. (((face_set_k k) H) = (GSPEC (\GEN%PVAR%8596. (?f. (((SETSPEC GEN%PVAR%8596) ((f IN (face_set H)) /\ ((CARD f) = k))) f)))))))
LIM_NORM_LBOUND
|- (!net. (!f. (!l. (!b. (((~ (trivial_limit net)) /\ ((((--> f) l) net) /\ ((eventually (\x. ((real_le b) (vector_norm (f x))))) net))) ==> ((real_le b) (vector_norm l)))))))
REAL_SUMMABLE_FROM_ELSEWHERE
|- (!f. (!m. (!n. (((real_summable (from m)) f) ==> ((real_summable (from n)) f)))))
RE_CACS_BOUNDS
|- (!z. (((real_lt (real_neg pi)) (Re (cacs z))) /\ ((real_le (Re (cacs z))) pi)))
SPHERE_EQ_EMPTY
|- (!a. (!r. (((sphere (a , r)) = EMPTY) = ((real_lt r) (real_of_num (NUMERAL _0))))))
HAS_REAL_INTEGRAL_NEGLIGIBLE
|- (!f. (!s. (!t. (((real_negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((f x) = (real_of_num (NUMERAL _0)))))) ==> (((has_real_integral f) (real_of_num (NUMERAL _0))) t)))))
MEASURE_COUNTABLE_UNIONS_LE
|- (!d. (!B. (((!n. (measurable (d n))) /\ (!n. ((real_le ((sum ((NUMERAL _0) .. n)) (\k. (measure (d k))))) B))) ==> ((measurable (UNIONS (GSPEC (\GEN%PVAR%3383. (?n. (((SETSPEC GEN%PVAR%3383) (n IN UNIV)) (d n))))))) /\ ((real_le (measure (UNIONS (GSPEC (\GEN%PVAR%3384. (?n. (((SETSPEC GEN%PVAR%3384) (n IN UNIV)) (d n)))))))) B)))))
IN_CBALL
|- (!x. (!y. (!e. ((y IN (cball (x , e))) = ((real_le (distance (x , y))) e)))))
Seq.cat0s
|- (!s. (((cat NIL) s) = s))
Tame_defs.tame_8
|- (!H. ((tame_8 H) = ((number_of_faces H) >= (NUMERAL (BIT1 (BIT1 _0))))))
MEASURABLE_LEBESGUE_MEASURABLE_SUBSET
|- (!s. (!t. (((lebesgue_measurable s) /\ ((measurable t) /\ (s SUBSET t))) ==> (measurable s))))
Hypermap_iso.finite_surj_eq_bij
|- (!s. (!t. ((FINITE s) ==> ((FINITE t) ==> (((CARD s) = (CARD t)) ==> (!f. ((((SURJ f) s) t) = (((BIJ f) s) t))))))))
INTEGRABLE_COMPLEX_LMUL_EQ
|- (!f. (!s. (!c. (((integrable_on (\x. ((complex_mul c) (f x)))) s) = ((c = (Cx (real_of_num (NUMERAL _0)))) \/ ((integrable_on f) s))))))
Fan.remark1_fan
|- (!x. (!V. (!E. (!u. (!v. ((FAN (x , (V , E))) ==> ((FINITE (((set_of_edge v) V) E)) /\ ((((v INSERT (u INSERT EMPTY)) IN E) = (u IN (((set_of_edge v) V) E))) /\ (((v INSERT (u INSERT EMPTY)) IN E) ==> ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) /\ ((~ (x = v)) /\ ((~ (x = u)) /\ ((~ (v = u)) /\ (((DISJOINT (x INSERT (v INSERT EMPTY))) (u INSERT EMPTY)) /\ (((DISJOINT (x INSERT (u INSERT EMPTY))) (v INSERT EMPTY)) /\ (((DISJOINT (x INSERT EMPTY)) (v INSERT (u INSERT EMPTY))) /\ ((~ (u IN (aff (x INSERT (v INSERT EMPTY))))) /\ ((v IN V) /\ ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) = (~ (u IN (aff (x INSERT (v INSERT EMPTY))))))))))))))))))))))))
LOCALLY_PATH_CONNECTED_SPHERE_GEN
|- (!s. (((bounded s) /\ (convex s)) ==> ((locally path_connected) (relative_frontier s))))
INFSUM_UNIQUE
|- (!f. (!l. (!s. ((((sums f) l) s) ==> (((infsum s) f) = l)))))
closure
|- (!s. ((closure s) = (s UNION (GSPEC (\GEN%PVAR%568. (?x. (((SETSPEC GEN%PVAR%568) ((limit_point_of x) s)) x)))))))
CONNECTED_COMPONENT_MAXIMAL
|- (!s. (!t. (!x. (((x IN t) /\ ((connected t) /\ (t SUBSET s))) ==> (t SUBSET ((connected_component s) x))))))
REAL_DIFFERENTIABLE_COMPOSE_WITHIN
|- (!f. (!g. (!x. (!s. ((((real_differentiable f) ((within (atreal x)) s)) /\ ((real_differentiable g) ((within (atreal (f x))) ((IMAGE f) s)))) ==> ((real_differentiable (g o f)) ((within (atreal x)) s)))))))
VECTOR_VARIATION_CONTINUOUS
|- (!f. (!a. (!b. (!c. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (((continuous (\x. (lift ((vector_variation (closed_interval ((CONS (a , x)) NIL))) f)))) ((within (at c)) (closed_interval ((CONS (a , b)) NIL)))) = ((continuous f) ((within (at c)) (closed_interval ((CONS (a , b)) NIL))))))))))
Oxl_2012.CARD_EMPTY
|- ((CARD EMPTY) = (NUMERAL _0))
Tame_lemmas.reflection
|- (!x. (!n. (((reflection n) x) = ((vector_sub x) ((% ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div ((dot x) n)) ((dot n) n)))) n)))))
Tame_defs.rotate_to
|- (!v. (!vs. (((rotate_to vs) v) = ((CONS v) ((APPEND (SND ((split_at v) vs))) (FST ((split_at v) vs)))))))
ORTHOGONAL_MATRIX_TRANSP
|- (!A. ((orthogonal_matrix (transp A)) = (orthogonal_matrix A)))
POLYHEDRON_SUMS
|- (!s. (!t. (((polyhedron s) /\ (polyhedron t)) ==> (polyhedron (GSPEC (\GEN%PVAR%2421. (?x. (?y. (((SETSPEC GEN%PVAR%2421) ((x IN s) /\ (y IN t))) ((vector_add x) y))))))))))
int_mul_th
|- (!x. (!y. ((real_of_int ((int_mul x) y)) = ((real_mul (real_of_int x)) (real_of_int y)))))
CONNECTED_COMPONENT_EQ_SELF
|- (!s. (!x. (((connected s) /\ (x IN s)) ==> (((connected_component s) x) = s))))
support
|- (!s. (!f. (!op. ((((support op) f) s) = (GSPEC (\GEN%PVAR%158. (?x. (((SETSPEC GEN%PVAR%158) ((x IN s) /\ (~ ((f x) = (neutral op))))) x))))))))
Functional_equation.proj_x6
|- (((((((proj_x6 x1) x2) x3) x4) x5) x6) = x6)
DIAMETER_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> ((diameter ((IMAGE f) s)) = (diameter s)))))
Ssrnat.eqn_addr
|- (!p. (!m. (!n. (((m + p) = (n + p)) = (m = n)))))
LENGTH_EQ_NIL
|- (!l. (((LENGTH l) = (NUMERAL _0)) = (l = NIL)))
INTERVAL_NE_EMPTY
|- (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (a $ i)) (b $ i))))) /\ ((~ ((open_interval (a , b)) = EMPTY)) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_lt (a $ i)) (b $ i))))))
division_of
|- (!s. (!i. (((division_of s) i) = ((FINITE s) /\ ((!k. ((k IN s) ==> ((k SUBSET i) /\ ((~ (k = EMPTY)) /\ (?a. (?b. (k = (closed_interval ((CONS (a , b)) NIL))))))))) /\ ((!k1. (!k2. (((k1 IN s) /\ ((k2 IN s) /\ (~ (k1 = k2)))) ==> (((interior k1) INTER (interior k2)) = EMPTY)))) /\ ((UNIONS s) = i)))))))
IN_CROSS
|- (!x. (!y. (!s. (!t. (((x , y) IN (s CROSS t)) = ((x IN s) /\ (y IN t)))))))
Real_ext.REAL_PROP_ZERO_POW
|- (!x. (!n. ((((real_pow x) n) = (real_of_num (NUMERAL _0))) = ((x = (real_of_num (NUMERAL _0))) /\ (~ (n = (NUMERAL _0)))))))
VECTOR_VARIATION_MONOTONE
|- (!f. (!s. (!t. ((((has_bounded_variation_on f) s) /\ (t SUBSET s)) ==> ((real_le ((vector_variation t) f)) ((vector_variation s) f))))))
Hypermap.edge_set
|- (!H. ((edge_set H) = ((set_of_orbits (dart H)) (edge_map H))))
ADD1
|- (!m. ((SUC m) = (m + (NUMERAL (BIT1 _0)))))
VECTOR_MUL_EQ_0
|- (!a. (!x. ((((% a) x) = (vec (NUMERAL _0))) = ((a = (real_of_num (NUMERAL _0))) \/ (x = (vec (NUMERAL _0)))))))
Lp_ineqs_proofs.y2_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 (BIT0 (BIT1 _0))))) (y2_fan d))))))))))
CARD_EXP_0
|- (!s. (!c. (((^_c s) EMPTY) =_c (c INSERT EMPTY))))
SUM_EQ_0_NUMSEG
|- (!f. (!m. (!n. ((!i. (((m <= i) /\ (i <= n)) ==> ((f i) = (real_of_num (NUMERAL _0))))) ==> (((sum (m .. n)) f) = (real_of_num (NUMERAL _0)))))))
REAL_ABS_POS
|- (!x. ((real_le (real_of_num (NUMERAL _0))) (real_abs x)))
INSIDE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\ (!y. (?x. ((f x) = y))))) ==> ((inside ((IMAGE f) s)) = ((IMAGE f) (inside s))))))
VSUM_CASES
|- (!s. (!P. (!f. (!g. ((FINITE s) ==> (((vsum s) (\x. (((COND (P x)) (f x)) (g x)))) = ((vector_add ((vsum (GSPEC (\GEN%PVAR%112. (?x. (((SETSPEC GEN%PVAR%112) ((x IN s) /\ (P x))) x))))) f)) ((vsum (GSPEC (\GEN%PVAR%113. (?x. (((SETSPEC GEN%PVAR%113) ((x IN s) /\ (~ (P x)))) x))))) g))))))))
REAL_CONTINUOUS_ON_EQ
|- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((real_continuous_on f) s)) ==> ((real_continuous_on g) s)))))
CLOSED_PATH_IMAGE
|- (!g. ((path g) ==> (closed (path_image g))))
STARLIKE_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((starlike ((IMAGE f) s)) = (starlike s)))))
HOMOTOPIC_ON_COMPONENTS
|- (!s. (!t. (!f. (!g. ((((locally connected) s) /\ (!c. ((c IN (components s)) ==> ((((homotopic_with (\x. T)) (c , t)) f) g)))) ==> ((((homotopic_with (\x. T)) (s , t)) f) g))))))
VECTOR_NEG_NEG
|- ((vector_neg (vector_neg x)) = x)
REAL_ABS_DIV
|- (!x. (!y. ((real_abs ((real_div x) y)) = ((real_div (real_abs x)) (real_abs y)))))
DISJOINT
|- (!s. (!t. (((DISJOINT s) t) = ((s INTER t) = EMPTY))))
ABSOLUTELY_SETCONTINUOUS_INDEFINITE_INTEGRAL
|- (!f. (!s. ((((absolutely_integrable_on f) s) /\ (lebesgue_measurable s)) ==> ((absolutely_setcontinuous_on (\k. ((integral k) f))) s))))
Hypermap.lemmaZHQCZLX
|- (!H. (((simple_hypermap H) /\ ((plain_hypermap H) /\ (!x. ((x IN (dart H)) ==> ((NUMERAL (BIT1 (BIT1 _0))) <= (CARD ((face H) x))))))) ==> (!x. ((x IN (dart H)) ==> (~ (((node_map H) x) = x))))))
APPEND
|- ((!l. (((APPEND NIL) l) = l)) /\ (!h. (!t. (!l. (((APPEND ((CONS h) t)) l) = ((CONS h) ((APPEND t) l)))))))
BALL_TRANSLATION
|- (!a. (!x. (!r. ((ball (((vector_add a) x) , r)) = ((IMAGE (\y. ((vector_add a) y))) (ball (x , r)))))))
MAT_0_COMPONENT
|- (!i. (!j. ((((mat (NUMERAL _0)) $ i) $ j) = (real_of_num (NUMERAL _0)))))
drop
|- (!x. ((drop x) = (x $ (NUMERAL (BIT1 _0)))))
SUBSPACE_0
|- ((subspace s) ==> ((vec (NUMERAL _0)) IN s))
real_continuous_atreal
|- (((real_continuous f) (atreal x)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x'. (((real_lt (real_abs ((real_sub x') x))) d) ==> ((real_lt (real_abs ((real_sub (f x')) (f x)))) e))))))))
real_summable
|- (!f. (!s. (((real_summable s) f) = (?l. (((real_sums f) l) s)))))
ROOT_MONO_LE_EQ
|- (!n. (!x. (!y. ((~ (n = (NUMERAL _0))) ==> (((real_le ((root n) x)) ((root n) y)) = ((real_le x) y))))))
VSUM_0
|- (((vsum s) (\x. (vec (NUMERAL _0)))) = (vec (NUMERAL _0)))
Add_triangle.fully_surrounded_imp_fan80
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (fan80 ((vec (NUMERAL _0)) , (V , E)))))))
RELPOW_R
|- ((((((RELPOW (NUMERAL _0)) R) x) y) = (x = y)) /\ (((((RELPOW (SUC n)) R) x) y) = (?z. (((R x) z) /\ ((((RELPOW n) R) z) y)))))
NORM_REAL
|- (!x. ((vector_norm x) = (real_abs (x $ (NUMERAL (BIT1 _0))))))
Seq.sumn0
|- ((sumn NIL) = (NUMERAL _0))
CONVEX_CLOSURE_RELATIVE_INTERIOR
|- (!s. ((convex s) ==> ((closure (relative_interior s)) = (closure s))))
LE_ANTISYM
|- (!m. (!n. (((m <= n) /\ (n <= m)) = (m = n))))
COMPLEX_POLY_NEG_CLAUSES
|- ((!x. ((vector_neg x) = ((complex_mul (Cx (real_neg (real_of_num (NUMERAL (BIT1 _0)))))) x))) /\ (!x. (!y. (((vector_sub x) y) = ((vector_add x) ((complex_mul (Cx (real_neg (real_of_num (NUMERAL (BIT1 _0)))))) y))))))
MEM_LINEAR_IMAGE
|- (!f. (!x. (!l. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((MEM (f x)) ((MAP f) l)) = ((MEM x) l))))))
PATH_LENGTH_TRANSLATION
|- (!a. (!g. ((path_length ((\x. ((vector_add a) x)) o g)) = (path_length g))))
INT_LE_TRANS
|- (!x. (!y. (!z. ((((int_le x) y) /\ ((int_le y) z)) ==> ((int_le x) z)))))
HAS_REAL_INTEGRAL_REFL
|- (!f. (!a. (((has_real_integral f) (real_of_num (NUMERAL _0))) (closed_real_interval ((CONS (a , a)) NIL)))))
EXISTS_SUBSET_IMAGE_INJ
|- (!P. (!f. (!s. ((?t. ((t SUBSET ((IMAGE f) s)) /\ (P t))) = (?t. ((t SUBSET s) /\ ((!x. (!y. (((x IN t) /\ (y IN t)) ==> (((f x) = (f y)) = (x = y))))) /\ (P ((IMAGE f) t)))))))))
HAS_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)))))) ==> ((has_real_measure (UNIONS f)) ((sum f) m)))))
DIMINDEX_2
|- ((dimindex UNIV) = (NUMERAL (BIT0 (BIT1 _0))))
Miqmcsn.scs_3M1_prime
|- (scs_3M1_prime = ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT1 _0)))) ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) NIL)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT1 (BIT1 _0))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) NIL)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT1 (BIT1 _0))))))
HOLOMORPHIC_ON_ADD
|- (!f. (!g. (!s. ((((holomorphic_on f) s) /\ ((holomorphic_on g) s)) ==> ((holomorphic_on (\z. ((vector_add (f z)) (g z)))) s)))))
INT_OF_NUM_LT
|- (!m. (!n. (((int_lt (int_of_num m)) (int_of_num n)) = (m < n))))
Ssrbool.andTb
|- (!b. ((T /\ b) = b))
REAL_ARCH_SIMPLE
|- (!x. (?n. ((real_le x) (real_of_num n))))
real_max
|- (!n. (!m. (((real_max m) n) = (((COND ((real_le m) n)) n) m))))
REAL_UNIFORMLY_CONTINUOUS_ON
|- (!f. (!s. (((real_uniformly_continuous_on f) s) = ((uniformly_continuous_on (lift o (f o drop))) ((IMAGE lift) s)))))
Asfutbf.IN_NEG
|- (!v. ((v IN ((IMAGE vector_neg) V)) = ((vector_neg v) IN V)))
ITER_ALT
|- ((!f. (!x. ((((ITER (NUMERAL _0)) f) x) = x))) /\ (!f. (!n. (!x. ((((ITER (SUC n)) f) x) = (((ITER n) f) (f x)))))))
SPAN_SUPERSET
|- (!x. ((x IN s) ==> (x IN (span s))))
CARD_LT_CONG
|- (!s. (!s'. (!t. (!t'. (((s =_c s') /\ (t =_c t')) ==> ((s <_c t) = (s' <_c t')))))))
SUBSET_EMPTY
|- (!s. ((s SUBSET EMPTY) = (s = EMPTY)))
ANR_INTERVAL
|- ((!a. (!b. (ANR (closed_interval ((CONS (a , b)) NIL))))) /\ (!a. (!b. (ANR (open_interval (a , b))))))
Ssrbool.contraNN
|- (!c. (!b. ((c ==> b) ==> ((~ b) ==> (~ c)))))
Ssrnat.mulnn
|- (!m. ((m * m) = (m EXP (NUMERAL (BIT0 (BIT1 _0))))))
TAGGED_PARTIAL_DIVISION_OF_TRIVIAL
|- (!p. (((tagged_partial_division_of p) EMPTY) = (p = EMPTY)))
LIM_SEQUENTIALLY
|- (!s. (!l. ((((--> s) l) sequentially) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (distance ((s n) , l))) e)))))))))
REAL_SERIES_0
|- (!s. (((real_sums (\n. (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL _0))) s))
COMPONENTS_EQ_SING_EXISTS
|- (!s. ((?a. ((components s) = (a INSERT EMPTY))) = ((connected s) /\ (~ (s = EMPTY)))))
SEGMENT_1
|- ((!a. (!b. ((closed_segment ((CONS (a , b)) NIL)) = (((COND ((real_le (drop a)) (drop b))) (closed_interval ((CONS (a , b)) NIL))) (closed_interval ((CONS (b , a)) NIL)))))) /\ (!a. (!b. ((open_segment (a , b)) = (((COND ((real_le (drop a)) (drop b))) (open_interval (a , b))) (open_interval (b , a)))))))
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON
|- (!f. (!s. (((holomorphic_on f) s) ==> ((continuous_on f) s))))
summable
|- (!f. (!s. (((summable s) f) = (?l. (((sums f) l) s)))))
Sphere.dihatn_x
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dihatn_x x1) x2) x3) x4) x5) x6) = ((LET (\d_x4. (LET_END ((LET (\d. (LET_END ((real_add ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (atn ((real_div (real_neg d_x4)) (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x1) d))))))))) ((((((delta_x x1) x2) x3) x4) x5) x6))))) ((((((delta_x4 x1) x2) x3) x4) x5) x6)))))))))
less
|- (!l. (!x. (!y. (((less l) (x , y)) = ((l (x , y)) /\ (~ (x = y)))))))
Real_ext.REAL_PROP_POS_RADD
|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_add x) y)))))
IS_REALINTERVAL_UNIV
|- (is_realinterval UNIV)
INTEGRAL_INTERVALS_INCLUSION_EXCLUSION
|- (!f. (!a. (!b. (!c. (!d. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((c IN (closed_interval ((CONS (a , b)) NIL))) /\ (d IN (closed_interval ((CONS (a , b)) NIL))))) ==> (((integral (closed_interval ((CONS (a , d)) NIL))) f) = ((vsum (GSPEC (\GEN%PVAR%2926. (?s. (((SETSPEC GEN%PVAR%2926) (s SUBSET ((NUMERAL (BIT1 _0)) .. (dimindex UNIV)))) s))))) (\s. ((% ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) (CARD (GSPEC (\GEN%PVAR%2927. (?i. (((SETSPEC GEN%PVAR%2927) ((i IN s) /\ ((real_lt (d $ i)) (c $ i)))) i))))))) ((integral (closed_interval ((CONS ((lambdai. (((COND (i IN s)) ((real_min (c $ i)) (d $ i))) (a $ i))) , (lambdai. (((COND (i IN s)) ((real_max (c $ i)) (d $ i))) (c $ i))))) NIL))) f)))))))))))
Seq.undup
|- (((undup NIL) = NIL) /\ ((undup ((CONS x) s')) = (((COND ((MEM x) s')) (undup s')) ((CONS x) (undup s')))))
Hypermap.number_of_edges
|- (!H. ((number_of_edges H) = (CARD (edge_set H))))
Functional_equation.gamma2_x1_div_a
|- (!m. ((gamma2_x1_div_a m) = (promote1_to_6 (gamma2_x_div_azim m))))
REAL_SERIES_NEG
|- (!x. (!x0. (!s. ((((real_sums x) x0) s) ==> (((real_sums (\n. (real_neg (x n)))) (real_neg x0)) s)))))
FINITE_IMP_BOUNDED
|- (!s. ((FINITE s) ==> (bounded s)))
DISJOINT_INSERT
|- (!x. (!s. (!t. (((DISJOINT (x INSERT s)) t) = (((DISJOINT s) t) /\ (~ (x IN t)))))))
ABSOLUTELY_REAL_INTEGRABLE_DECREASING_PRODUCT
|- (!f. (!g. (!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 y)) (f x))))) /\ ((absolutely_real_integrable_on g) (closed_real_interval ((CONS (a , b)) NIL)))) ==> ((absolutely_real_integrable_on (\x. ((real_mul (f x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))
HAS_COMPLEX_DERIVATIVE_CCOS
|- (!z. (((has_complex_derivative ccos) (vector_neg (csin z))) (at z)))
ABSOLUTELY_INTEGRABLE_CONVOLUTION_SYM
|- (!bop. (!f. (!g. (!x. (((absolutely_integrable_on (\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV) = ((absolutely_integrable_on (\y. ((bop (f y)) (g ((vector_sub x) y))))) UNIV))))))
INDEPENDENT_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((independent ((IMAGE f) s)) = (independent s)))))
CONNECTED_INSERT
|- (!s. (!a. ((connected s) ==> ((connected (a INSERT s)) = ((s = EMPTY) \/ (a IN (closure s)))))))
LT_IMP_LE
|- (!m. (!n. ((m < n) ==> (m <= n))))
Sphere.sqrt8
|- (sqrt8 = (sqrt (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))))
REAL_SUM_INTEGRAL_UBOUND_INCREASING
|- (!f. (!g. (!m. (!n. (((m <= n) /\ ((!x. ((x IN (closed_real_interval ((CONS ((real_of_num m) , ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) NIL))) ==> (((has_real_derivative g) (f x)) ((within (atreal x)) (closed_real_interval ((CONS ((real_of_num m) , ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) NIL)))))) /\ (!x. (!y. ((((real_le (real_of_num m)) x) /\ (((real_le x) y) /\ ((real_le y) ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) ==> ((real_le (f x)) (f y))))))) ==> ((real_le ((sum (m .. n)) (\k. (f (real_of_num k))))) ((real_sub (g ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) (g (real_of_num m)))))))))
COMPACT_SIMPLEX
|- (!n. (!s. (((simplex n) s) ==> (compact s))))
Hypermap_iso.bij_contracted_dart2
|- (!V. (!E. (((BIJ contracted_dart) (d20_fan ((vec (NUMERAL _0)) , (V , E)))) (GSPEC (\GEN%PVAR%7631. (?v. (((SETSPEC GEN%PVAR%7631) ((v IN V) /\ ((((set_of_edge v) V) E) = EMPTY))) (v , v))))))))
Lp_ineqs_proofs.azim_fan_eq_dih_y
|- (!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))) ==> (((azim_dart (V , E)) d) = ((((((dih_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)))))))))))
Sphere.packing_lt
|- ((packing V) = (!u. (!v. (((u IN V) /\ ((v IN V) /\ ((real_lt (distance (u , v))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> (u = v)))))
DIFF
|- (!s. (!t. ((s DIFF t) = (GSPEC (\GEN%PVAR%4. (?x. (((SETSPEC GEN%PVAR%4) ((x IN s) /\ (~ (x IN t)))) x)))))))
CX_POW
|- (!x. (!n. ((Cx ((real_pow x) n)) = ((complex_pow (Cx x)) n))))
CONTINUOUS_ON_CLOSED_GEN
|- (!f. (!s. (!t. ((((IMAGE f) s) SUBSET t) ==> (((continuous_on f) s) = (!u. (((closed_in ((subtopology euclidean) t)) u) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%616. (?x. (((SETSPEC GEN%PVAR%616) ((x IN s) /\ ((f x) IN u))) x))))))))))))
REAL_CONTINUOUS_ATREAL_WITHINREAL
|- (!f. (!s. (!x. (((real_continuous f) (atreal x)) ==> ((real_continuous f) ((within (atreal x)) s))))))
LOCALLY_CONNECTED_PCROSS
|- (!s. (!t. ((((locally connected) s) /\ ((locally connected) t)) ==> ((locally connected) (s PCROSS t)))))
REAL_SUMS
|- (((real_sums f) l) = ((sums (lift o f)) (lift l)))
REAL_LT_MIN
|- (!x. (!y. (!z. (((real_lt z) ((real_min x) y)) = (((real_lt z) x) /\ ((real_lt z) y))))))
FACE_OF_LINEAR_IMAGE
|- (!f. (!c. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((face_of ((IMAGE f) c)) ((IMAGE f) s)) = ((face_of c) s))))))
NEGLIGIBLE_SUBSET
|- (!s. (!t. (((negligible s) /\ (t SUBSET s)) ==> (negligible t))))
IN_IMAGE
|- (!y. (!s. (!f. ((y IN ((IMAGE f) s)) = (?x. ((y = (f x)) /\ (x IN s)))))))
OPEN_EXISTS_IN
|- (!P. (!Q. ((!a. ((P a) ==> (open (GSPEC (\GEN%PVAR%542. (?x. (((SETSPEC GEN%PVAR%542) ((Q a) x)) x))))))) ==> (open (GSPEC (\GEN%PVAR%543. (?x. (((SETSPEC GEN%PVAR%543) (?a. ((P a) /\ ((Q a) x)))) x))))))))