hcju/holps huggingface dataset:
HAS_REAL_MEASURE_HAS_MEASURE
|- (!s. (!m. (((has_real_measure s) m) = ((has_measure ((IMAGE lift) s)) m))))
ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR
|- (!f. (!u. (!t. (!s. (((ANR s) /\ (((continuous_on f) t) /\ ((((IMAGE f) t) SUBSET s) /\ ((closed_in ((subtopology euclidean) u)) t)))) ==> (?v. (?g. ((t SUBSET v) /\ (((open_in ((subtopology euclidean) u)) v) /\ (((continuous_on g) v) /\ ((((IMAGE g) v) SUBSET s) /\ (!x. ((x IN t) ==> ((g x) = (f x)))))))))))))))
Seq2.EL_nth
|- (!x0. (!s. (!i. ((i < (sizel s)) ==> (((EL i) s) = (((nth x0) s) i))))))
SUBSET_CROSS
|- (!s. (!t. (!s'. (!t'. (((s CROSS t) SUBSET (s' CROSS t')) = ((s = EMPTY) \/ ((t = EMPTY) \/ ((s SUBSET s') /\ (t SUBSET t')))))))))
REAL_ADD_LID
|- (!x. (((real_add (real_of_num (NUMERAL _0))) x) = x))
Fan_misc.EXTENSION_SIGMA_FAN_EQ_RES
|- (!x. (!V. (!E. (!v. (((((extension_sigma_fan x) V) E) v) = ((res ((((sigma_fan x) V) E) v)) (((set_of_edge v) V) E)))))))
AFFINE_INDEPENDENT_IFF_CARD
|- (!s. ((~ (affine_dependent s)) = ((FINITE s) /\ ((aff_dim s) = ((int_sub (int_of_num (CARD s))) (int_of_num (NUMERAL (BIT1 _0))))))))
OPEN_IMP_FSIGMA
|- (!s. ((open s) ==> (fsigma s)))
FRONTIER_TRANSLATION
|- (!a. (!s. ((frontier ((IMAGE (\x. ((vector_add a) x))) s)) = ((IMAGE (\x. ((vector_add a) x))) (frontier s)))))
CX_SIN
|- (!x. ((Cx (sin x)) = (csin (Cx x))))
CENTRE_IN_BALL
|- (!x. (!e. ((x IN (ball (x , e))) = ((real_lt (real_of_num (NUMERAL _0))) e))))
path_image
|- (!g. ((path_image g) = ((IMAGE g) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))))
SUMS_INFSUM
|- (!f. (!s. ((((sums f) ((infsum s) f)) s) = ((summable s) f))))
VECTOR_EQ
|- (!x. (!y. ((x = y) = ((((dot x) x) = ((dot x) y)) /\ (((dot y) y) = ((dot x) x))))))
IMAGE_AFFINITY_INTERVAL
|- (!a. (!b. (!m. (!c. (((IMAGE (\x. ((vector_add ((% m) x)) c))) (closed_interval ((CONS (a , b)) NIL))) = (((COND ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) EMPTY) (((COND ((real_le (real_of_num (NUMERAL _0))) m)) (closed_interval ((CONS (((vector_add ((% m) a)) c) , ((vector_add ((% m) b)) c))) NIL))) (closed_interval ((CONS (((vector_add ((% m) b)) c) , ((vector_add ((% m) a)) c))) NIL)))))))))
CONVEX_BALL
|- (!x. (!e. (convex (ball (x , e)))))
SIN_COS_EQ
|- (!x. (!y. ((((sin y) = (sin x)) /\ ((cos y) = (cos x))) = (?n. ((integer n) /\ (y = ((real_add x) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul n) pi)))))))))
BINOM_LT
|- (!n. (!k. ((n < k) ==> ((binom (n , k)) = (NUMERAL _0)))))
Hypermap.is_node_going
|- (!H. (!L. (!x. (!y. (((((is_node_going H) L) x) y) = (?k. ((y = (((POWER (loop_map L)) k) x)) /\ (!i. ((i <= k) ==> ((((POWER (loop_map L)) i) x) = (((POWER (inverse (node_map H))) i) x)))))))))))
CSQRT_0
|- ((csqrt (Cx (real_of_num (NUMERAL _0)))) = (Cx (real_of_num (NUMERAL _0))))
DIAMETER_BOUNDED_BOUND
|- (!s. (!x. (!y. (((bounded s) /\ ((x IN s) /\ (y IN s))) ==> ((real_le (vector_norm ((vector_sub x) y))) (diameter s))))))
INTEGRABLE_ON_LITTLE_SUBINTERVALS
|- (!f. (!a. (!b. ((!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!u. (!v. (((x IN (closed_interval ((CONS (u , v)) NIL))) /\ (((closed_interval ((CONS (u , v)) NIL)) SUBSET (ball (x , d))) /\ ((closed_interval ((CONS (u , v)) NIL)) SUBSET (closed_interval ((CONS (a , b)) NIL))))) ==> ((integrable_on f) (closed_interval ((CONS (u , v)) NIL)))))))))) ==> ((integrable_on f) (closed_interval ((CONS (a , b)) NIL)))))))
STC_SYM
|- (!R. (!x. (!y. ((((STC R) x) y) ==> (((STC R) y) x)))))
TENDSTO_REAL
|- (((---> s) l) = ((--> (lift o s)) (lift l)))
CONTINUOUS_WITHIN_ID
|- (!a. (!s. ((continuous (\x. x)) ((within (at a)) s))))
EXTENSION_INTO_AR_LOCAL
|- (!f. (!c. (!s. (!t. ((((continuous_on f) c) /\ ((((IMAGE f) c) SUBSET t) /\ (((retract_of t) UNIV) /\ ((closed_in ((subtopology euclidean) s)) c)))) ==> (?g. (((continuous_on g) s) /\ ((((IMAGE g) UNIV) SUBSET t) /\ (!x. ((x IN c) ==> ((g x) = (f x))))))))))))
EXISTS_PASTECART
|- ((?p. (P p)) = (?x. (?y. (P ((pastecart x) y)))))
MEASURABLE_ON_PREIMAGE_OPEN
|- (!f. (((measurable_on f) UNIV) = (!t. ((open t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3534. (?x. (((SETSPEC GEN%PVAR%3534) ((f x) IN t)) x)))))))))
OUTSIDE_COMPACT_IN_OPEN
|- (!s. (!t. (((compact s) /\ ((open t) /\ ((s SUBSET t) /\ (~ (t = EMPTY))))) ==> (~ (((outside s) INTER t) = EMPTY)))))
MEM
|- ((((MEM x) NIL) = F) /\ (((MEM x) ((CONS h) t)) = ((x = h) \/ ((MEM x) t))))
VECTOR_ADD_RID
|- (!x. (((vector_add x) (vec (NUMERAL _0))) = x))
DIST_REFL
|- (!x. ((distance (x , x)) = (real_of_num (NUMERAL _0))))
CONNECTED_PATH_IMAGE
|- (!g. ((path g) ==> (connected (path_image g))))
IM_COMPLEX_INV_GE_0
|- (!z. (((real_le (real_of_num (NUMERAL _0))) (Im (complex_inv z))) = ((real_le (Im z)) (real_of_num (NUMERAL _0)))))
EXP_EQ_0
|- (!m. (!n. (((m EXP n) = (NUMERAL _0)) = ((m = (NUMERAL _0)) /\ (~ (n = (NUMERAL _0)))))))
infsum
|- (!f. (!s. (((infsum s) f) = (@l. (((sums f) l) s)))))
INTEGRABLE_0
|- (!s. ((integrable_on (\x. (vec (NUMERAL _0)))) s))
LAMBDA_SKOLEM
|- ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (?x. ((P i) x)))) = (?x. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((P i) (x $ i))))))
AFF_DIM_INJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((aff_dim ((IMAGE f) s)) = (aff_dim s)))))
SURJECTIVE_IMAGE
|- (!f. ((!t. (?s. (((IMAGE f) s) = t))) = (!y. (?x. ((f x) = y)))))
Real_ext.REAL_SV_SQRT_0
|- ((sqrt (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))
BILINEAR_COMPLEX_MUL
|- (bilinear complex_mul)
COMPACT_UNION
|- (!s. (!t. (((compact s) /\ (compact t)) ==> (compact (s UNION t)))))
Seq.pred1
|- (!a. ((pred1 a) = (\x. (x = a))))
REAL_SUP_UNIQUE
|- (!s. (!b. (((!x. ((x IN s) ==> ((real_le x) b))) /\ (!b'. (((real_lt b') b) ==> (?x. ((x IN s) /\ ((real_lt b') x)))))) ==> ((sup s) = b))))
Misc_defs_and_lemmas.REAL_ADD
|- (!m. (!n. (((real_add (real_of_num m)) (real_of_num n)) = (real_of_num (m + n)))))
Functional_equation.uni
|- (!f. (!x. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((uni (f , x)) x1) x2) x3) x4) x5) x6) = (f ((((((x x1) x2) x3) x4) x5) x6)))))))))))
SYMMETRIC_MATRIX_INV_RMUL
|- (!A. ((transp ((matrix_mul A) (matrix_inv A))) = ((matrix_mul A) (matrix_inv A))))
AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR
|- (!s. (!t. (((convex s) /\ (~ ((s INTER (interior t)) = EMPTY))) ==> (((hull affine) (s INTER t)) = ((hull affine) s)))))
BETWEEN_MIDPOINT
|- (!a. (!b. (((between (midpoint (a , b))) (a , b)) /\ ((between (midpoint (a , b))) (b , a)))))
Seq.uniq_size_uniq
|- (!s1. (!s2. ((uniq s1) ==> ((!x. (((MEM x) s1) = ((MEM x) s2))) ==> ((uniq s2) = ((sizel s2) = (sizel s1)))))))
CLOSED_IN_TRANSLATION_EQ
|- (!a. (!s. (!t. (((closed_in ((subtopology euclidean) ((IMAGE (\x. ((vector_add a) x))) t))) ((IMAGE (\x. ((vector_add a) x))) s)) = ((closed_in ((subtopology euclidean) t)) s)))))
FINITE_IMAGE_INJ_EQ
|- (!f. (!s. ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) ==> ((FINITE ((IMAGE f) s)) = (FINITE s)))))
INSERT
|- ((x INSERT s) = (GSPEC (\GEN%PVAR%5. (?y. (((SETSPEC GEN%PVAR%5) ((y IN s) \/ (y = x))) y)))))
Nonlinear_lemma.gcy_low
|- (!y. (((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) ==> ((gcy y) = (gchi y))))
ANTIDERIVATIVE_CONTINUOUS
|- (!f. (!a. (!b. (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) ==> (?g. (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> (((has_vector_derivative g) (f x)) ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))))))))))
PATH_COMPONENT_LINEAR_IMAGE
|- (!f. (!s. (!x. (((linear f) /\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\ (!y. (?x. ((f x) = y))))) ==> (((path_component ((IMAGE f) s)) (f x)) = ((IMAGE f) ((path_component s) x)))))))
VECTOR_MUL_LCANCEL
|- (!a. (!x. (!y. ((((% a) x) = ((% a) y)) = ((a = (real_of_num (NUMERAL _0))) \/ (x = y))))))
BETWEEN_DIST_LT
|- (!r. (!a. (!b. (!c. ((((real_lt (distance (c , a))) r) /\ (((real_lt (distance (c , b))) r) /\ ((between x) (a , b)))) ==> ((real_lt (distance (c , x))) r))))))
Sphere.sol0
|- (sol0 = ((real_sub ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) (acs ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))) pi))
ANALYTIC_AT
|- (!f. (!z. (((analytic_on f) (z INSERT EMPTY)) = (?s. ((open s) /\ ((z IN s) /\ ((holomorphic_on f) s)))))))
ABSOLUTELY_INTEGRABLE_CMUL_EQ
|- (!f. (!s. (!c. (((absolutely_integrable_on (\x. ((% c) (f x)))) s) = ((c = (real_of_num (NUMERAL _0))) \/ ((absolutely_integrable_on f) s))))))
IMAGE_LIFT_UNIV
|- (((IMAGE lift) UNIV) = UNIV)
CLOSEST_POINT_UNIQUE
|- (!s. (!a. (!x. (((convex s) /\ ((closed s) /\ ((x IN s) /\ (!z. ((z IN s) ==> ((real_le (distance (a , x))) (distance (a , z)))))))) ==> (x = ((closest_point s) a))))))
FORALL_IN_CLAUSES
|- ((!P. ((!x. ((x IN EMPTY) ==> (P x))) = T)) /\ (!P. (!a. (!s. ((!x. ((x IN (a INSERT s)) ==> (P x))) = ((P a) /\ (!x. ((x IN s) ==> (P x)))))))))
VECTOR_VARIATION_GE_NORM_FUNCTION
|- (!f. (!s. (!a. (!b. ((((has_bounded_variation_on f) s) /\ ((closed_segment ((CONS (a , b)) NIL)) SUBSET s)) ==> ((real_le (vector_norm ((vector_sub (f b)) (f a)))) ((vector_variation s) f)))))))
LIM_WITHIN_OPEN
|- (!f. (!l. (!a. (!s. (((a IN s) /\ (open s)) ==> ((((--> f) l) ((within (at a)) s)) = (((--> f) l) (at a))))))))
Float.POW_1
|- (!x. (((real_pow x) (NUMERAL (BIT1 _0))) = x))
NORM_NEG
|- (!x. ((vector_norm (vector_neg x)) = (vector_norm x)))
REAL_CONTINUOUS_AT_WITHIN
|- (!f. (!s. (!x. (((real_continuous f) (at x)) ==> ((real_continuous f) ((within (at x)) s))))))
DOT_2
|- (((dot x) y) = ((real_add ((real_mul (x $ (NUMERAL (BIT1 _0)))) (y $ (NUMERAL (BIT1 _0))))) ((real_mul (x $ (NUMERAL (BIT0 (BIT1 _0))))) (y $ (NUMERAL (BIT0 (BIT1 _0)))))))
Seq.cat1s
|- (!x. (!s. (((cat ((CONS x) NIL)) s) = ((CONS x) s))))
LEBESGUE_MEASURABLE_ALMOST_FSIGMA
|- (!s. ((lebesgue_measurable s) ==> (?c. (?t. ((fsigma c) /\ ((negligible t) /\ (((c UNION t) = s) /\ ((DISJOINT c) t))))))))
Trigonometry1.NORM_CAUCHY_SCHWARZ_FRAC
|- (!u. (!v. (((~ (u = (vec (NUMERAL _0)))) /\ (~ (v = (vec (NUMERAL _0))))) ==> (((real_le (real_neg (real_of_num (NUMERAL (BIT1 _0))))) ((real_div ((dot u) v)) ((real_mul (vector_norm u)) (vector_norm v)))) /\ ((real_le ((real_div ((dot u) v)) ((real_mul (vector_norm u)) (vector_norm v)))) (real_of_num (NUMERAL (BIT1 _0))))))))
VECTOR_DIFFERENTIABLE
|- (!f. (!net. (((differentiable f) net) = (?f'. (((has_vector_derivative f) f') net)))))
Sphere.h0
|- (h0 = ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))
CARD_EXP_CONG
|- (!s. (!s'. (!t. (!t'. (((s =_c s') /\ (t =_c t')) ==> (((^_c s) t) =_c ((^_c s') t')))))))
FORALL_IN_UNIONS
|- (!P. (!s. ((!x. ((x IN (UNIONS s)) ==> (P x))) = (!t. (!x. (((t IN s) /\ (x IN t)) ==> (P x)))))))
Bump.MCELL_CELL_PARAMETERS_EXIST
|- (!V. (!ul. (!k. (!X. (((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))))))) ==> ((FST ((cell_params V) X)) = k))))))
Cuxvzoz.vv_azim_le_alt
|- (!vv. (!k. (!i. (!j. ((LET (\V. (LET_END ((LET (\E. (LET_END ((LET (\f. (LET_END ((((periodic vv) k) /\ (((NUMERAL (BIT1 (BIT1 _0))) <= k) /\ ((convex_local_fan (V , (E , f))) /\ (!i. (!j. (((i < k) /\ ((j < k) /\ ((vv i) = (vv j)))) ==> (i = j))))))) ==> ((real_le ((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv j))) ((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv (i + (k - (NUMERAL (BIT1 _0))))))))))) ((IMAGE (\i. ((vv i) , (vv (SUC i))))) UNIV))))) ((IMAGE (\i. ((vv i) INSERT ((vv (SUC i)) INSERT EMPTY)))) UNIV))))) ((IMAGE vv) UNIV))))))
Ssrnat.minnCA
|- (!m1. (!m2. (!m3. (((minn m1) ((minn m2) m3)) = ((minn m2) ((minn m1) m3))))))
IN_CBALL_0
|- (!x. (!e. ((x IN (cball ((vec (NUMERAL _0)) , e))) = ((real_le (vector_norm x)) e))))
Float.ABS_NUM_P
|- (!m. (!n. ((m <= n) ==> ((real_abs ((real_sub (real_of_num n)) (real_of_num m))) = (real_of_num ((m - n) + (n - m)))))))
Real_ext.REAL_PROP_NN_DOUBLE
|- (!x. (((real_le (real_of_num (NUMERAL _0))) ((real_add x) x)) = ((real_le (real_of_num (NUMERAL _0))) x)))
EXP_LOG
|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((exp (log x)) = x)))
PATH_CONNECTED_TRANSLATION_EQ
|- (!a. (!s. ((path_connected ((IMAGE (\x. ((vector_add a) x))) s)) = (path_connected s))))
REVERSEPATH_TRANSLATION
|- (!a. (!g. ((reversepath ((\x. ((vector_add a) x)) o g)) = ((\x. ((vector_add a) x)) o (reversepath g)))))
MEASURE_LOCALLY_LIPSCHITZ_IMAGE
|- (!f. (!s. (!B. ((((dimindex UNIV) <= (dimindex UNIV)) /\ ((measurable s) /\ (!x. ((x IN s) ==> (?t. ((open t) /\ ((x IN t) /\ (!y. ((y IN (s INTER t)) ==> ((real_le (vector_norm ((vector_sub (f y)) (f x)))) ((real_mul B) (vector_norm ((vector_sub y) x))))))))))))) ==> ((real_le (measure ((IMAGE f) s))) ((real_mul ((real_pow B) (dimindex UNIV))) (measure s)))))))
Conforming.conforming_solid_angle_fan
|- (!x. (!V. (!E. ((conforming_solid_angle_fan (x , (V , E))) = (!f. ((f IN (face_set (hypermap1_of_fanx (x , (V , E))))) ==> ((LET (\U. (LET_END ((!r. (measurable ((ball (x , r)) INTER U))) /\ (((eventually_radial x) U) /\ (((sol x) U) = ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) ((sum f) (\y. ((real_sub (((((azim_fan x) V) E) (pr2 y)) (pr3 y))) pi)))))))))) ((((dartset_leads_into_fan x) V) E) f))))))))
HYPERPLANE_EQ_EMPTY
|- (!a. (!b. (((GSPEC (\GEN%PVAR%171. (?x. (((SETSPEC GEN%PVAR%171) (((dot a) x) = b)) x)))) = EMPTY) = ((a = (vec (NUMERAL _0))) /\ (~ (b = (real_of_num (NUMERAL _0))))))))
IS_REALINTERVAL_CONVEX
|- (!s. ((is_realinterval s) = (convex ((IMAGE lift) s))))
SUMMABLE_ZETA
|- (!n. (!z. (((real_lt (real_of_num (NUMERAL (BIT1 _0)))) (Re z)) ==> ((summable (from n)) (\k. (complex_inv ((cpow (Cx (real_of_num k))) z)))))))
REAL_SUB_TRIANGLE
|- (!a. (!b. (!c. (((real_add ((real_sub a) b)) ((real_sub b) c)) = ((real_sub a) c)))))
IN_CARD_MUL
|- (!s. (!t. (!x. (!y. (((x , y) IN ((*_c s) t)) = ((x IN s) /\ (y IN t)))))))
REAL_ABS_NUM
|- (!n. ((real_abs (real_of_num n)) = (real_of_num n)))
Tame_defs.restricted_hypermap
|- (!H. ((restricted_hypermap H) = ((is_no_double_joins H) /\ ((~ ((dart H) = EMPTY)) /\ ((planar_hypermap H) /\ ((connected_hypermap H) /\ ((plain_hypermap H) /\ ((simple_hypermap H) /\ ((is_edge_nondegenerate H) /\ ((is_node_nondegenerate H) /\ (!f. ((f IN (face_set H)) ==> ((CARD f) >= (NUMERAL (BIT1 (BIT1 _0))))))))))))))))
APPEND_NIL
|- (!l. (((APPEND l) NIL) = l))
BILINEAR_SWAP
|- (!op. ((bilinear (\x. (\y. ((op y) x)))) = (bilinear op)))
triangulation
|- (!tr. ((triangulation tr) = ((FINITE tr) /\ ((!t. ((t IN tr) ==> (?n. ((simplex n) t)))) /\ (!t. (!t'. (((t IN tr) /\ (t' IN tr)) ==> (((face_of (t INTER t')) t) /\ ((face_of (t INTER t')) t')))))))))
Hypermap_and_fan.DART_EXISTS
|- (!V. (!E. (!v. ((v IN V) ==> (?w. ((v , w) IN (dart_of_fan (V , E))))))))
HAS_REAL_INTEGRAL_LBOUND
|- (!f. (!a. (!b. (!i. ((((real_le a) b) /\ ((((has_real_integral f) i) (closed_real_interval ((CONS (a , b)) NIL))) /\ (!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_le B) (f x)))))) ==> ((real_le ((real_mul B) ((real_sub b) a))) i))))))
FSIGMA_IMP_BOREL
|- (!s. ((fsigma s) ==> (borel s)))
REAL_LE_01
|- ((real_le (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0))))
Appendix.scs_bm_v39_explicit
|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_bm_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = beta))))))))))
NSUM_CLAUSES_NUMSEG
|- ((!m. (((nsum (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (NUMERAL _0)))) /\ (!m. (!n. (((nsum (m .. (SUC n))) f) = (((COND (m <= (SUC n))) (((nsum (m .. n)) f) + (f (SUC n)))) ((nsum (m .. n)) f))))))
DEPENDENT_2
|- (!a. (!b. ((dependent (a INSERT (b INSERT EMPTY))) = (((COND (a = b)) (a = (vec (NUMERAL _0)))) (?x. (?y. ((((vector_add ((% x) a)) ((% y) b)) = (vec (NUMERAL _0))) /\ (~ ((x = (real_of_num (NUMERAL _0))) /\ (y = (real_of_num (NUMERAL _0))))))))))))
Seq2.uniq_nthP
|- (!x0. (!s. ((uniq s) = (!i. (!j. (((i < j) /\ (j < (sizel s))) ==> (~ ((((nth x0) s) i) = (((nth x0) s) j)))))))))
Oxl_2012.CARD_INSERT
|- (!x. (!s. ((FINITE s) ==> ((CARD (x INSERT s)) = (((COND (x IN s)) (CARD s)) (SUC (CARD s)))))))
NOT_FORALL_THM
|- (!P. ((~ (!x. (P x))) = (?x. (~ (P x)))))
COMPLEX_MUL_RNEG
|- (!x. (!y. (((complex_mul x) (vector_neg y)) = (vector_neg ((complex_mul x) y)))))
CONNECTED_INTERMEDIATE_CLOSURE
|- (!s. (!t. (((connected s) /\ ((s SUBSET t) /\ (t SUBSET (closure s)))) ==> (connected t))))
Ssrnat.leq_trans
|- (!n. (!m. (!p. ((m <= n) ==> ((n <= p) ==> (m <= p))))))
COUNTABLE_OPEN_COMPONENTS
|- (!s. ((open s) ==> (COUNTABLE (components s))))
submetric
|- (!s. (!m. (((submetric m) s) = (metric ((s INTER (mspace m)) , (mdist m))))))
BIJECTIONS_CARD_EQ
|- (!s. (!t. (!f. (!g. ((((FINITE s) \/ (FINITE t)) /\ ((!x. ((x IN s) ==> (((f x) IN t) /\ ((g (f x)) = x)))) /\ (!y. ((y IN t) ==> (((g y) IN s) /\ ((f (g y)) = y)))))) ==> ((CARD s) = (CARD t)))))))
CONTINUOUS_ABS
|- (!f. (!net. (((continuous f) net) ==> ((continuous (\x. (lambdai. (real_abs ((f x) $ i))))) net))))
CONTINUOUS_WITHINREAL
|- (!f. (!x. (!s. (((continuous f) ((within (atreal x)) s)) = (((--> f) (f x)) ((within (atreal x)) s))))))
CRAMER_MATRIX_LEFT
|- (!A. (!X. (!B. ((~ ((det A) = (real_of_num (NUMERAL _0)))) ==> ((((matrix_mul X) A) = B) = (X = (lambdak. (lambdal. ((real_div (det (lambdai. (lambdaj. (((COND (j = l)) ((B $ k) $ i)) ((A $ j) $ i)))))) (det A))))))))))
lin_combo
|- (!V. (!f. (((lin_combo V) f) = ((vsum V) (\v. ((% (f v)) v))))))
SUB_ELIM_THM
|- ((P (a - b)) = (!d. (((a = (b + d)) \/ ((a < b) /\ (d = (NUMERAL _0)))) ==> (P d))))
REAL_MEASURABLE_COUNTABLE_UNIONS_STRONG
|- (!s. (!B. (((!n. (real_measurable (s n))) /\ (!n. ((real_le (real_measure (UNIONS (GSPEC (\GEN%PVAR%4945. (?k. (((SETSPEC GEN%PVAR%4945) (k <= n)) (s k)))))))) B))) ==> (real_measurable (UNIONS (GSPEC (\GEN%PVAR%4946. (?n. (((SETSPEC GEN%PVAR%4946) (n IN UNIV)) (s n))))))))))
Ssrnat.addn0
|- (!n. ((n + (NUMERAL _0)) = n))
IM_MUL_II
|- (!z. (((Im ((complex_mul z) ii)) = (Re z)) /\ ((Im ((complex_mul ii) z)) = (Re z))))
BROUWER_INESSENTIAL_ANR
|- (!f. (!s. (((compact s) /\ ((~ (s = EMPTY)) /\ ((ANR s) /\ (((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET s) /\ (?a. ((((homotopic_with (\x. T)) (s , s)) f) (\x. a)))))))) ==> (?x. ((x IN s) /\ ((f x) = x))))))
HOMEOMORPHIC_SIMPLE_PATH_ARC
|- (!g. (!h. (((arc g) /\ ((simple_path h) /\ ((homeomorphic (path_image g)) (path_image h)))) ==> (arc h))))
List_hypermap.good_list_nodes_condition
|- (!L. ((good_list L) ==> ((good_list_nodes L) = ((CARD (node_set (hypermap_of_list L))) = (sizel (list_of_elements L))))))
CLOSURE_INTER_SUBSET
|- (!s. (!t. ((closure (s INTER t)) SUBSET ((closure s) INTER (closure t)))))
aff_gt_def
|- (aff_gt = (affsign sgn_gt))
AFFINE_EQ_SUBSPACE
|- (!s. (((vec (NUMERAL _0)) IN s) ==> ((affine s) = (subspace s))))
LINSEG_INSEG
|- (!l. (!a. ((woset l) ==> ((inseg ((linseg l) a)) l))))
List_hypermap.mem_list_pairs
|- (!x. (!y. (!l. (((MEM (x , y)) (list_pairs l)) ==> (((MEM x) l) /\ ((MEM y) l))))))
INT_MUL_LID
|- (!x. (((int_mul (int_of_num (NUMERAL (BIT1 _0)))) x) = x))
Lp_ineqs_proofs.lp_fan
|- (!E. (!V. ((lp_fan (V , E)) = ((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((fully_surrounded (V , E)) /\ ((!v. (!w. (((v IN V) /\ ((w IN V) /\ ((~ (v = w)) /\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (distance (v , w)))))) /\ ((V SUBSET ball_annulus) /\ (packing V))))))))
REAL_ABS_REFL
|- (!x. (((real_abs x) = x) = ((real_le (real_of_num (NUMERAL _0))) x)))
aff_lt_def
|- (aff_lt = (affsign sgn_lt))
COMPLEX_SUB_POW_R1
|- (!x. (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (((vector_sub ((complex_pow x) n)) (Cx (real_of_num (NUMERAL (BIT1 _0))))) = ((complex_mul ((vector_sub x) (Cx (real_of_num (NUMERAL (BIT1 _0)))))) ((vsum ((NUMERAL _0) .. (n - (NUMERAL (BIT1 _0))))) (\i. ((complex_pow x) i))))))))
Ssrnat.leqif_imp_eq
|- (!m. (!n. (!c. ((((leqif m) n) c) ==> ((m = n) = c)))))
LOG_INV
|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((log (real_inv x)) = (real_neg (log x)))))
REAL_POW_ADD
|- (!x. (!m. (!n. (((real_pow x) (m + n)) = ((real_mul ((real_pow x) m)) ((real_pow x) n))))))
num_gcd
|- (!a. (!b. ((num_gcd (a , b)) = (num_of_int (int_gcd ((int_of_num a) , (int_of_num b)))))))
Appendix.mk_unadorned_v39
|- (!k. (!d. (!a. (!b. (((((mk_unadorned_v39 k) d) a) b) = (scs_v39 (k , (d , (a , (a , (b , (b , ((\i. (\j. F)) , (EMPTY , (EMPTY , EMPTY)))))))))))))))
PASTECART_FST_SND
|- (!z. (((pastecart (fstcart z)) (sndcart z)) = z))
CARD_EQ_OPEN_IN
|- (!u. (!s. ((((locally connected) u) /\ (((open_in ((subtopology euclidean) u)) s) /\ (?x. ((x IN s) /\ ((limit_point_of x) u))))) ==> (s =_c UNIV))))
HOMEOMORPHISM_INJECTIVE_OPEN_MAP_EQ
|- (!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)))))) ==> ((?g. ((homeomorphism (s , t)) (f , g))) = (!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))))))))
HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES
|- (!f. (!g. (!g'. (!s. (!b. (((lebesgue_measurable s) /\ ((!x. ((x IN s) ==> (((has_derivative g) (g' x)) ((within (at x)) s)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((g x) = (g y)))) ==> (x = y)))))) ==> ((((absolutely_integrable_on (\x. ((% (real_abs (det (matrix (g' x))))) (f (g x))))) s) /\ (((integral s) (\x. ((% (real_abs (det (matrix (g' x))))) (f (g x))))) = b)) = (((absolutely_integrable_on f) ((IMAGE g) s)) /\ (((integral ((IMAGE g) s)) f) = b)))))))))
vector_angle
|- (!x. (!y. (((vector_angle x) y) = (((COND ((x = (vec (NUMERAL _0))) \/ (y = (vec (NUMERAL _0))))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (acs ((real_div ((dot x) y)) ((real_mul (vector_norm x)) (vector_norm y))))))))
STARLIKE_IMP_CONTRACTIBLE
|- (!s. ((starlike s) ==> (contractible s)))
WF_ANTISYM
|- (!<<. (!x. (!y. ((WF <<) ==> (~ (((<< x) y) /\ ((<< y) x)))))))
SC_INDUCT
|- (!R. (!SC'. (((!x. (!y. (((R x) y) ==> ((SC' x) y)))) /\ (!x. (!y. (((SC' x) y) ==> ((SC' y) x))))) ==> (!a0. (!a1. ((((SC R) a0) a1) ==> ((SC' a0) a1)))))))
SUM_CONST
|- (!c. (!s. ((FINITE s) ==> (((sum s) (\n. c)) = ((real_mul (real_of_num (CARD s))) c)))))
IMAGE_AFFINITY_REAL_INTERVAL
|- (!a. (!b. (!m. (!c. (((IMAGE (\x. ((real_add ((real_mul m) x)) c))) (closed_real_interval ((CONS (a , b)) NIL))) = (((COND ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) EMPTY) (((COND ((real_le (real_of_num (NUMERAL _0))) m)) (closed_real_interval ((CONS (((real_add ((real_mul m) a)) c) , ((real_add ((real_mul m) b)) c))) NIL))) (closed_real_interval ((CONS (((real_add ((real_mul m) b)) c) , ((real_add ((real_mul m) a)) c))) NIL)))))))))
INTERVAL_SUBSET_IS_INTERVAL
|- (!s. (!a. (!b. ((is_interval s) ==> (((closed_interval ((CONS (a , b)) NIL)) SUBSET s) = (((closed_interval ((CONS (a , b)) NIL)) = EMPTY) \/ ((a IN s) /\ (b IN s))))))))
SIMPLE_PATH_LINEAR_IMAGE_EQ
|- (!f. (!g. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((simple_path (f o g)) = (simple_path g)))))
EXPOSED_POINT_OF_INTER_SUPPORTING_HYPERPLANE_LE
|- (!s. (!a. (!b. (!c. (((!x. ((x IN s) ==> ((real_le ((dot a) x)) b))) /\ ((s INTER (GSPEC (\GEN%PVAR%2202. (?x. (((SETSPEC GEN%PVAR%2202) (((dot a) x) = b)) x))))) = (c INSERT EMPTY))) ==> ((exposed_face_of (c INSERT EMPTY)) s))))))
NOT_COPLANAR_NOT_COLLINEAR
|- (!v1. (!v2. (!v3. (!w. ((~ (coplanar (v1 INSERT (v2 INSERT (v3 INSERT (w INSERT EMPTY)))))) ==> (~ (collinear (v1 INSERT (v2 INSERT (v3 INSERT EMPTY))))))))))
CONNECTED_CHAIN_GEN
|- (!f. (((!s. ((s IN f) ==> ((closed s) /\ (connected s)))) /\ ((?s. ((s IN f) /\ (compact s))) /\ (!s. (!t. (((s IN f) /\ (t IN f)) ==> ((s SUBSET t) \/ (t SUBSET s))))))) ==> (connected (INTERS f))))
CONTENT_POS_LT
|- (!a. (!b. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_lt (a $ i)) (b $ i)))) ==> ((real_lt (real_of_num (NUMERAL _0))) (content (closed_interval ((CONS (a , b)) NIL)))))))
LINEAR_INJECTIVE_ISOMORPHISM
|- (!f. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (?f'. ((linear f') /\ ((!x. ((f' (f x)) = x)) /\ (!x. ((f (f' x)) = x)))))))