hcju/holps huggingface dataset:
Fan_misc.dart1_of_fan
|- (!V. (!E. ((dart1_of_fan (V , E)) = (GSPEC (\GEN%PVAR%5545. (?v. (?w. (((SETSPEC GEN%PVAR%5545) ((v INSERT (w INSERT EMPTY)) IN E)) (v , w)))))))))
Vol1.VOLUME_PROPS_MEASURABLE
|- (!C. ((measurable C) ==> ((real_ge (measure C)) (real_of_num (NUMERAL _0)))))
OPEN_IN_RELATIVE_INTERIOR
|- (!s. ((open_in ((subtopology euclidean) ((hull affine) s))) (relative_interior s)))
REAL_MUL_RNEG
|- (!x. (!y. (((real_mul x) (real_neg y)) = (real_neg ((real_mul x) y)))))
Ssrnat.minnA
|- (!m1. (!m2. (!m3. (((minn m1) ((minn m2) m3)) = ((minn ((minn m1) m2)) m3)))))
MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GE
|- (!f. (((measurable_on f) UNIV) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV))) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3514. (?x. (((SETSPEC GEN%PVAR%3514) ((real_ge ((f x) $ k)) a)) x))))))))))
IM_II
|- ((Im ii) = (real_of_num (NUMERAL (BIT1 _0))))
PATHSTART_SUBPATH
|- (!u. (!v. (!g. ((pathstart (((subpath u) v) g)) = (g u)))))
FINITE_INTER_NUMSEG
|- (!s. (!m. (!n. (FINITE (s INTER (m .. n))))))
Appendix.scs_4M5
|- (scs_4M5' = ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) NIL))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , (real_of_num (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , (real_of_num (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))) NIL))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))
Real_ext.REAL_SV_LMUL_0
|- (!x. (((real_mul (real_of_num (NUMERAL _0))) x) = (real_of_num (NUMERAL _0))))
Real_ext.REAL_LE
|- (!m. (!n. (((real_le (real_of_num m)) (real_of_num n)) = (m <= n))))
HAS_BOUNDED_VARIATION_ON_SUB
|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ ((has_bounded_variation_on g) s)) ==> ((has_bounded_variation_on (\x. ((vector_sub (f x)) (g x)))) s)))))
Nonlinear_lemma.halfbump_x
|- (!x. ((halfbump_x x) = (bump ((real_div (sqrt x)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))
AR_DEFORMATION_RETRACT_OF_CONTRACTIBLE
|- (!s. (!t. (((contractible s) /\ ((AR t) /\ ((closed_in ((subtopology euclidean) s)) t))) ==> (?r. (((((homotopic_with (\x. T)) (s , s)) (\x. x)) r) /\ ((retraction (s , t)) r))))))
LIM_NULL
|- (!net. (!f. (!l. ((((--> f) l) net) = (((--> (\x. ((vector_sub (f x)) l))) (vec (NUMERAL _0))) net)))))
REAL_ABS_NEG
|- (!x. ((real_abs (real_neg x)) = (real_abs x)))
Functional_equation.two6
|- (two6 = (constant6 (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))
FRONTIER_FRONTIER
|- (!s. (((open s) \/ (closed s)) ==> ((frontier (frontier s)) = (frontier s))))
OPEN_BALL
|- (!x. (!e. (open (ball (x , e)))))
Fan_misc.inverse_sigma_fan
|- (!x. (!V. (!E. (!v. (((((inverse_sigma_fan x) V) E) v) = (inverse ((((extension_sigma_fan x) V) E) v)))))))
STC_INC
|- (!R. (!x. (!y. (((R x) y) ==> (((STC R) x) y)))))
CARD_EXP_FINITE
|- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> (FINITE ((^_c s) t)))))
RE_CLOG
|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((Re (clog z)) = (log (vector_norm z)))))
Hypermap_iso.hyp_iso_SURJ
|- (!f. (!H. (!G. (((hyp_iso f) (H , G)) ==> (((SURJ f) (dart H)) (dart G))))))
FINITE_RULES
|- ((FINITE EMPTY) /\ (!x. (!s. ((FINITE s) ==> (FINITE (x INSERT s))))))
AR_IMP_CONTRACTIBLE
|- (!s. ((AR s) ==> (contractible s)))
REAL_ENTIRE
|- (!x. (!y. ((((real_mul x) y) = (real_of_num (NUMERAL _0))) = ((x = (real_of_num (NUMERAL _0))) \/ (y = (real_of_num (NUMERAL _0)))))))
vector_neg
|- (!x. ((vector_neg x) = (lambdai. (real_neg (x $ i)))))
Sphere.bis_le
|- (!u. (!v. (((bis_le u) v) = (GSPEC (\GEN%PVAR%5246. (?x. (((SETSPEC GEN%PVAR%5246) ((real_le (distance (x , u))) (distance (x , v)))) x)))))))
Lp_ineqs_proofs.y78_eq
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((LET (\w'. (LET_END ((LET (\u. (LET_END ((((y8_fan (V , E)) (v , w)) = (distance (w' , u))) /\ (((y7_fan (V , E)) (v , w)) = (vector_norm u)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))))
SUBSET_LIFT_IMAGE
|- (!s. (!t. ((((IMAGE lift) s) SUBSET ((IMAGE lift) t)) = (s SUBSET t))))
MAP2
|- (((((MAP2 f) NIL) NIL) = NIL) /\ ((((MAP2 f) ((CONS h1) t1)) ((CONS h2) t2)) = ((CONS ((f h1) h2)) (((MAP2 f) t1) t2))))
DIM_INJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((dim ((IMAGE f) s)) = (dim s)))))
closed_in
|- (!top. (!s. (((closed_in top) s) = ((s SUBSET (topspace top)) /\ ((open_in top) ((topspace top) DIFF s))))))
Appendix.scs_3T3
|- (scs_3T3 = ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT1 _0)))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 NIL) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT1 (BIT1 _0))))) (((funlist_v39 NIL) cstab) (NUMERAL (BIT1 (BIT1 _0))))))
REAL_LT_MAX
|- (!x. (!y. (!z. (((real_lt z) ((real_max x) y)) = (((real_lt z) x) \/ ((real_lt z) y))))))
UNCOUNTABLE_CONTAINS_LIMIT_POINT
|- (!s. ((~ (COUNTABLE s)) ==> (?x. ((x IN s) /\ ((limit_point_of x) s)))))
IM_CX
|- (!x. ((Im (Cx x)) = (real_of_num (NUMERAL _0))))
HAS_SIZE_DIFF
|- (!s. (!t. (!m. (!n. (((s HAS_SIZE m) /\ ((t HAS_SIZE n) /\ (t SUBSET s))) ==> ((s DIFF t) HAS_SIZE (m - n)))))))
REAL_OPEN
|- (!s. ((real_open s) = (open ((IMAGE lift) s))))
real_derivative
|- (!f. (!x. (((real_derivative f) x) = (@f'. (((has_real_derivative f) f') (atreal x))))))
HAUSDIST_TRANSLATION
|- (!a. (!s. (!t. ((hausdist (((IMAGE (\x. ((vector_add a) x))) s) , ((IMAGE (\x. ((vector_add a) x))) t))) = (hausdist (s , t))))))
FRONTIER_INTERIOR_SUBSET
|- (!s. ((frontier (interior s)) SUBSET (frontier s)))
ABSOLUTELY_INTEGRABLE_BOUNDED_SETVARIATION
|- (!f. (!s. (((absolutely_integrable_on f) s) ==> ((has_bounded_setvariation_on (\k. ((integral k) f))) s))))
REAL_MIN_MAX
|- (!x. (!y. (((real_min x) y) = (real_neg ((real_max (real_neg x)) (real_neg y))))))
Nonlin_def.flat_term2_234_x
|- (!d. (!x5. (!x6. ((((flat_term2_234_x d) x5) x6) = (uni (flat_term_x , (((edge2_234_x d) x5) x6)))))))
RSTC_INC
|- (!R. (!x. (!y. (((R x) y) ==> (((RSTC R) x) y)))))
AFF_DIM_NONEMPTY_INTERIOR
|- (!s. ((~ ((interior s) = EMPTY)) ==> ((aff_dim s) = (int_of_num (dimindex UNIV)))))
REAL_MEASURABLE_MEASURABLE
|- (!s. ((real_measurable s) = (measurable ((IMAGE lift) s))))
Seq2.perm_eq_cons_delete1
|- (!x. (!s. (((MEM x) s) ==> ((perm_eq ((CONS x) ((delete1 x) s))) s))))
complex_mul
|- (!w. (!z. (((complex_mul w) z) = (complex (((real_sub ((real_mul (Re w)) (Re z))) ((real_mul (Im w)) (Im z))) , ((real_add ((real_mul (Re w)) (Im z))) ((real_mul (Im w)) (Re z))))))))
COPLANAR_TRANSLATION_EQ
|- (!a. (!s. ((coplanar ((IMAGE (\x. ((vector_add a) x))) s)) = (coplanar s))))
SUBSPACE_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((subspace ((IMAGE f) s)) = (subspace s)))))
Constants_approx.INTERVAL_MUL
|- ((((interval_arith x) (a , b)) /\ ((interval_arith y) (c , d))) ==> ((interval_arith ((real_mul x) y)) (((real_min ((real_min ((real_mul a) c)) ((real_mul a) d))) ((real_min ((real_mul b) c)) ((real_mul b) d))) , ((real_max ((real_max ((real_mul a) c)) ((real_mul a) d))) ((real_max ((real_mul b) c)) ((real_mul b) d))))))
WF_FINITE
|- (!<<. (((!x. (~ ((<< x) x))) /\ ((!x. (!y. (!z. ((((<< x) y) /\ ((<< y) z)) ==> ((<< x) z))))) /\ (!x. (FINITE (GSPEC (\GEN%PVAR%149. (?y. (((SETSPEC GEN%PVAR%149) ((<< y) x)) y)))))))) ==> (WF <<)))
REAL_LT_ADD1
|- (!x. (!y. (((real_le x) y) ==> ((real_lt x) ((real_add y) (real_of_num (NUMERAL (BIT1 _0))))))))
BOUNDED_CLOSURE_EQ
|- (!s. ((bounded (closure s)) = (bounded s)))
DENSE_IMP_PERFECT
|- (!s. (((closure s) = UNIV) ==> (!x. ((x IN s) ==> ((limit_point_of x) s)))))
OPEN_IN_SUBSET_TRANS
|- (!s. (!t. (!u. ((((open_in ((subtopology euclidean) u)) s) /\ ((s SUBSET t) /\ (t SUBSET u))) ==> ((open_in ((subtopology euclidean) t)) s)))))
Lp_ineqs_proofs.delta_y_not_neg
|- (!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_lt ((((((delta_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))) (real_of_num (NUMERAL _0))))))))))))
CONTINUOUS_ON_COMPONENTS_OPEN
|- (!f. (!s. (((open s) /\ (!c. ((c IN (components s)) ==> ((continuous_on f) c)))) ==> ((continuous_on f) s))))
int_gt
|- (!x. (!y. (((int_gt x) y) = ((real_gt (real_of_int x)) (real_of_int y)))))
REAL_POW_MUL
|- (!x. (!y. (!n. (((real_pow ((real_mul x) y)) n) = ((real_mul ((real_pow x) n)) ((real_pow y) n))))))
REAL_INF_UNIQUE
|- (!s. (!b. (((!x. ((x IN s) ==> ((real_le b) x))) /\ (!b'. (((real_lt b) b') ==> (?x. ((x IN s) /\ ((real_lt x) b')))))) ==> ((inf s) = b))))
HOMOTOPY_EQUIVALENT_CONTRACTIBLE_SETS
|- (!s. (!t. (((contractible s) /\ ((contractible t) /\ ((s = EMPTY) = (t = EMPTY)))) ==> ((homotopy_equivalent s) t))))
Lp_ineqs_def.y8_fan
|- (!V. (!E. (!x. (((y8_fan (V , E)) x) = ((y5_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) x))))))
PAIR_SURJECTIVE
|- (!p. (?x. (?y. (p = (x , y)))))
SEPARATE_COMPACT_CLOSED
|- (!s. (!t. (((compact s) /\ ((closed t) /\ ((s INTER t) = EMPTY))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (!y. (((x IN s) /\ (y IN t)) ==> ((real_le d) (distance (x , y)))))))))))
Nonlinear_lemma.h0_lt_hplus
|- ((real_lt h0) hplus)
Tame_defs2.hideDups
|- (((hideDups NIL) = NIL) /\ ((hideDups ((CONS b) bs)) = ((CONS (SOME b)) ((hideDupsRec b) bs))))
SUBSET_INTERS
|- (!s. (!f. ((s SUBSET (INTERS f)) = (!t. ((t IN f) ==> (s SUBSET t))))))
DROP_IN_REAL_INTERVAL
|- ((!a. (!b. (!x. (((drop x) IN (closed_real_interval ((CONS (a , b)) NIL))) = (x IN (closed_interval ((CONS ((lift a) , (lift b))) NIL))))))) /\ (!a. (!b. (!x. (((drop x) IN (open_real_interval (a , b))) = (x IN (open_interval ((lift a) , (lift b)))))))))
Lp_ineqs_def.y1_fan
|- (!x. ((y1_fan x) = (vector_norm (FST x))))
PATHFINISH_SUBPATH
|- (!u. (!v. (!g. ((pathfinish (((subpath u) v) g)) = (g v)))))
REAL_LE_RMUL_EQ
|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_le ((real_mul x) z)) ((real_mul y) z)) = ((real_le x) y))))))
WOSET_WF
|- (!l. ((woset l) = ((WF (\x. (\y. ((l (x , y)) /\ (~ (x = y)))))) /\ (!x. (!y. ((((fl l) x) /\ ((fl l) y)) ==> ((l (x , y)) \/ (l (y , x)))))))))
option_RECURSION
|- (!NONE'. (!SOME'. (?fn. (((fn NONE) = NONE') /\ (!a. ((fn (SOME a)) = (SOME' a)))))))
REAL_INTEGRAL_UNIQUE
|- (!f. (!y. (!k. ((((has_real_integral f) y) k) ==> (((real_integral k) f) = y)))))
BILINEAR_DIFFERENTIABLE_ON_COMPOSE
|- (!f. (!g. (!h. (!s. ((((differentiable_on f) s) /\ (((differentiable_on g) s) /\ (bilinear h))) ==> ((differentiable_on (\x. ((h (f x)) (g x)))) s))))))
AR_ANR
|- (!s. ((AR s) = ((ANR s) /\ ((contractible s) /\ (~ (s = EMPTY))))))
Hypermap_iso.bij_contracted_dart
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (((BIJ contracted_dart) (d_fan ((vec (NUMERAL _0)) , (V , E)))) (dart_of_fan (V , E))))))
HULL_EQ
|- (!P. (!s. ((!f. ((!s. ((s IN f) ==> (P s))) ==> (P (INTERS f)))) ==> ((((hull P) s) = s) = (P s)))))
REAL_NEG_LT0
|- (!x. (((real_lt (real_neg x)) (real_of_num (NUMERAL _0))) = ((real_lt (real_of_num (NUMERAL _0))) x)))
Seq.cons_uniq
|- (!x. (!s. ((uniq ((CONS x) s)) = ((~ ((MEM x) s)) /\ (uniq s)))))
HAS_BOUNDED_VARIATION_ON_NEG
|- (!f. (!s. (((has_bounded_variation_on f) s) ==> ((has_bounded_variation_on (\x. (vector_neg (f x)))) s))))
Sphere.node3_y
|- (!f. (!y3. (!y1. (!y2. (!y6. (!y4. (!y5. ((((((((node3_y f) y1) y2) y3) y4) y5) y6) = ((((((f y3) y1) y2) y6) y4) y5)))))))))
Hypermap.hypermap_lemma
|- (!H. ((FINITE (dart H)) /\ (((permutes (edge_map H)) (dart H)) /\ (((permutes (node_map H)) (dart H)) /\ (((permutes (face_map H)) (dart H)) /\ (((edge_map H) o ((node_map H) o (face_map H))) = I))))))
COVERING_SPACE_IMP_SURJECTIVE
|- (!p. (!c. (!s. (((covering_space (c , p)) s) ==> (((IMAGE p) c) = s)))))
INJECTIVE_IMAGE
|- (!f. ((!s. (!t. ((((IMAGE f) s) = ((IMAGE f) t)) ==> (s = t)))) = (!x. (!y. (((f x) = (f y)) ==> (x = y))))))
ROOT_MONO_LT
|- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ ((real_lt x) y)) ==> ((real_lt ((root n) x)) ((root n) y))))))
INSIDE_CONVEX
|- (!s. ((convex s) ==> ((inside s) = EMPTY)))
Seq.size_eq0
|- (!s. (((sizel s) = (NUMERAL _0)) = (s = NIL)))
Real_ext.REAL_MK_POS_PI
|- ((real_lt (real_of_num (NUMERAL _0))) pi)
Seq.index_mem
|- (!x. (!s. ((((indexl x) s) < (sizel s)) = ((MEM x) s))))
DROP_SUB
|- (!x. (!y. ((drop ((vector_sub x) y)) = ((real_sub (drop x)) (drop y)))))
CARD_UNION_OVERLAP_EQ
|- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> (((CARD (s UNION t)) = ((CARD s) + (CARD t))) = ((s INTER t) = EMPTY)))))
Conforming.SIGMA_FAN_OF_FANADD_AT_POINT1
|- (!x. (!V. (!E. (!E1. (!v. (!u. (!w. (((FAN (x , (V , E))) /\ ((FAN (x , (V , E1))) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\ (((((((sigma_fan x) V) E) u) w) = v) /\ ((fan80 (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))) ==> ((((((sigma_fan x) V) E1) v) w) = (((((sigma_fan x) V) E) v) u))))))))))
CLOSURE_CLOSED
|- (!s. ((closed s) ==> ((closure s) = s)))
HULL_MONO
|- (!P. (!s. (!t. ((s SUBSET t) ==> (((hull P) s) SUBSET ((hull P) t))))))
IS_INTERVAL_TRANSLATION_EQ
|- (!a. (!s. ((is_interval ((IMAGE (\x. ((vector_add a) x))) s)) = (is_interval s))))
sums
|- (!s. (!f. (!l. ((((sums f) l) s) = (((--> (\n. ((vsum (s INTER ((NUMERAL _0) .. n))) f))) l) sequentially)))))
REAL_LE_NEG
|- (!x. (!y. (((real_le (real_neg x)) (real_neg y)) = ((real_le y) x))))
PATH_COMPONENT_SYM
|- (!s. (!x. (!y. ((((path_component s) x) y) ==> (((path_component s) y) x)))))
BOREL_TRANSLATION
|- (!a. (!s. ((borel ((IMAGE (\x. ((vector_add a) x))) s)) = (borel s))))
Tame_list.size_list_update
|- (!xs. (!i. (!v. ((sizel (((list_update xs) i) v)) = (sizel xs)))))
CONVEX_RELATIVE_INTERIOR
|- (!s. ((convex s) ==> (convex (relative_interior s))))
Optimize.ineq
|- ((!c. (((ineq NIL) c) = c)) /\ (!a. (!x. (!b. (!xs. (!c. (((ineq ((CONS (a , (x , b))) xs)) c) = ((((real_le a) x) /\ ((real_le x) b)) ==> ((ineq xs) c)))))))))
CONTINUOUS_CONST
|- (!net. (!c. ((continuous (\x. c)) net)))
LIM_SUB
|- (!net. (!f. (!g. (!l. (!m. (((((--> f) l) net) /\ (((--> g) m) net)) ==> (((--> (\x. ((vector_sub (f x)) (g x)))) ((vector_sub l) m)) net)))))))
INTERVAL_EQ_EMPTY_1
|- (!a. (!b. ((((closed_interval ((CONS (a , b)) NIL)) = EMPTY) = ((real_lt (drop b)) (drop a))) /\ (((open_interval (a , b)) = EMPTY) = ((real_le (drop b)) (drop a))))))
IN_DIFF
|- (!s. (!t. (!x. ((x IN (s DIFF t)) = ((x IN s) /\ (~ (x IN t)))))))
Add_triangle.dart_of_fan_eq_image
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((dart_of_fan (V , E)) = ((IMAGE contracted_dart) (d_fan ((vec (NUMERAL _0)) , (V , E))))))))
PATH_CONNECTED_OPEN_DIFF_CARD_LT
|- (!s. (!t. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ ((open s) /\ ((connected s) /\ (t <_c UNIV)))) ==> (path_connected (s DIFF t)))))
int_min_th
|- (!x. (!y. ((real_of_int ((int_min x) y)) = ((real_min (real_of_int x)) (real_of_int y)))))
SUB_SUC
|- (!m. (!n. (((SUC m) - (SUC n)) = (m - n))))
SUB_0
|- (!m. ((((NUMERAL _0) - m) = (NUMERAL _0)) /\ ((m - (NUMERAL _0)) = m)))
CARD_NUMSEG
|- (!m. (!n. ((CARD (m .. n)) = ((n + (NUMERAL (BIT1 _0))) - m))))
BOUNDED_TRANSLATION_EQ
|- (!a. (!s. ((bounded ((IMAGE (\x. ((vector_add a) x))) s)) = (bounded s))))
pairwise
|- (!s. (!r. (((pairwise r) s) = (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> ((r x) y)))))))
HAS_BOUNDED_SETVARIATION_ON_EQ
|- (!f. (!g. (!s. (((!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ ((closed_interval ((CONS (a , b)) NIL)) SUBSET s)) ==> ((f (closed_interval ((CONS (a , b)) NIL))) = (g (closed_interval ((CONS (a , b)) NIL))))))) /\ ((has_bounded_setvariation_on f) s)) ==> ((has_bounded_setvariation_on g) s)))))
COMPLEX_SUB_0
|- (!x. (!y. ((((vector_sub x) y) = (Cx (real_of_num (NUMERAL _0)))) = (x = y))))
HAS_INTEGRAL_COMBINE_TAGGED_DIVISION
|- (!f. (!s. (!p. (!i. ((((tagged_division_of p) s) /\ (!x. (!k. (((x , k) IN p) ==> (((has_integral f) (i k)) k))))) ==> (((has_integral f) ((vsum p) (GABS (\f. (!x. (!k. ((GEQ (f (x , k))) (i k)))))))) s))))))
SUBSET_UNION_ABSORPTION
|- (!s. (!t. ((s SUBSET t) = ((s UNION t) = t))))
RELATIVE_FRONTIER_CLOSURE
|- (!s. ((convex s) ==> ((relative_frontier (closure s)) = (relative_frontier s))))
COS_NPI
|- (!n. ((cos ((real_mul (real_of_num n)) pi)) = ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) n)))
REAL_CONTINUOUS_ON_BERNOULLI
|- (!n. (!s. ((real_continuous_on (bernoulli n)) s)))
CONVEX_CONNECTED_1
|- (!s. ((convex s) = (connected s)))
CEXP_N
|- (!n. (!x. ((cexp ((complex_mul (Cx (real_of_num n))) x)) = ((complex_pow (cexp x)) n))))
SUMMABLE_ZETA_INTEGER
|- (!n. (!m. (((NUMERAL (BIT0 (BIT1 _0))) <= m) ==> ((summable (from n)) (\k. (complex_inv ((complex_pow (Cx (real_of_num k))) m)))))))
Lp_ineqs_proofs.y9_hi_std
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((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)))))))))))))))))))
COMPLETE_INJECTIVE_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((complete ((IMAGE f) s)) = (complete s)))))
LE_RDIV_EQ
|- (!a. (!b. (!n. ((~ (a = (NUMERAL _0))) ==> ((n <= (b DIV a)) = ((a * n) <= b))))))
CARD_EQ_IMAGE
|- (!f. (!s. ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) ==> (((IMAGE f) s) =_c s))))
CONVEX_TRANSLATION_EQ
|- (!a. (!s. ((convex ((IMAGE (\x. ((vector_add a) x))) s)) = (convex s))))
ENR_OPEN_IN
|- (!s. (!t. ((((open_in ((subtopology euclidean) t)) s) /\ (ENR t)) ==> (ENR s))))
REAL_MUL_RZERO
|- (!x. (((real_mul x) (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0))))
OPEN_IN_TRANSLATION_EQ
|- (!a. (!s. (!t. (((open_in ((subtopology euclidean) ((IMAGE (\x. ((vector_add a) x))) t))) ((IMAGE (\x. ((vector_add a) x))) s)) = ((open_in ((subtopology euclidean) t)) s)))))
INTERIOR_OPEN
|- (!s. ((open s) ==> ((interior s) = s)))
DIAMETER_SING
|- (!a. ((diameter (a INSERT EMPTY)) = (real_of_num (NUMERAL _0))))
URYSOHN_LOCAL_STRONG
|- (!s. (!t. (!u. (!a. (!b. ((((closed_in ((subtopology euclidean) u)) s) /\ (((closed_in ((subtopology euclidean) u)) t) /\ (((s INTER t) = EMPTY) /\ (~ (a = b))))) ==> (?f. (((continuous_on f) u) /\ ((!x. ((x IN u) ==> ((f x) IN (closed_segment ((CONS (a , b)) NIL))))) /\ ((!x. ((x IN u) ==> (((f x) = a) = (x IN s)))) /\ (!x. ((x IN u) ==> (((f x) = b) = (x IN t))))))))))))))
convex_cone
|- (!s. ((convex_cone s) = ((~ (s = EMPTY)) /\ ((convex s) /\ (conic s)))))
CNJ_CX
|- (!x. ((cnj (Cx x)) = (Cx x)))
DEPENDENT_EQ_DIM_LT_CARD
|- (!s. ((dependent s) = ((FINITE s) ==> ((dim s) < (CARD s)))))
DET_MUL
|- (!A. (!B. ((det ((matrix_mul A) B)) = ((real_mul (det A)) (det B)))))
SIMPLICIAL_COMPLEX_TRANSLATION
|- (!a. (!tr. ((simplicial_complex ((IMAGE (IMAGE (\x. ((vector_add a) x)))) tr)) = (simplicial_complex tr))))
HOMOTOPIC_WITH_IMP_PROPERTY
|- (!P. (!X. (!Y. (!f. (!g. (((((homotopic_with P) (X , Y)) f) g) ==> ((P f) /\ (P g))))))))
COUNTABLE_IMAGE
|- (!f. (!s. ((COUNTABLE s) ==> (COUNTABLE ((IMAGE f) s)))))
Nonlin_def.mudLs_126_x
|- (!d2s. (!y3. (!y4. (!y5. (!d1s. ((((((mudLs_126_x d1s) d2s) y3) y4) y5) = ((mul6 (((((((compose6 mu6_x) (constant6 ((real_mul y3) y3))) proj_x1) proj_x2) dummy6) dummy6) dummy6)) ((add6 ((mul6 (constant6 ((real_div (real_of_num (NUMERAL (BIT1 _0)))) ((real_add d1s) d2s)))) ((sub6 (((delta_126_x ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5))) (constant6 ((real_mul d1s) d1s))))) (constant6 d1s)))))))))
INT_MUL_LNEG
|- (!x. (!y. (((int_mul (int_neg x)) y) = (int_neg ((int_mul x) y)))))
ARITH_MULT
|- ((!m. (!n. (((NUMERAL m) * (NUMERAL n)) = (NUMERAL (m * n))))) /\ (((_0 * _0) = _0) /\ ((!n. ((_0 * (BIT0 n)) = _0)) /\ ((!n. ((_0 * (BIT1 n)) = _0)) /\ ((!n. (((BIT0 n) * _0) = _0)) /\ ((!n. (((BIT1 n) * _0) = _0)) /\ ((!m. (!n. (((BIT0 m) * (BIT0 n)) = (BIT0 (BIT0 (m * n)))))) /\ ((!m. (!n. (((BIT0 m) * (BIT1 n)) = ((BIT0 m) + (BIT0 (BIT0 (m * n))))))) /\ ((!m. (!n. (((BIT1 m) * (BIT0 n)) = ((BIT0 n) + (BIT0 (BIT0 (m * n))))))) /\ (!m. (!n. (((BIT1 m) * (BIT1 n)) = ((BIT1 m) + ((BIT0 n) + (BIT0 (BIT0 (m * n)))))))))))))))))
Ckqowsa_3_points.aff_ge_0_2
|- (!u. (!v. (((~ (u = (vec (NUMERAL _0)))) /\ (~ (v = (vec (NUMERAL _0))))) ==> (((aff_ge ((vec (NUMERAL _0)) INSERT EMPTY)) (u INSERT (v INSERT EMPTY))) = (GSPEC (\GEN%PVAR%7632. (?y. (((SETSPEC GEN%PVAR%7632) (?t1. (?t2. (((real_le (real_of_num (NUMERAL _0))) t1) /\ (((real_le (real_of_num (NUMERAL _0))) t2) /\ (y = ((vector_add ((% t1) u)) ((% t2) v)))))))) y))))))))
CONVEX_CONE_HULL_LINEAR_IMAGE
|- (!f. (!s. ((linear f) ==> (((hull convex_cone) ((IMAGE f) s)) = ((IMAGE f) ((hull convex_cone) s))))))
LEBESGUE_MEASURABLE_CLOSED
|- (!s. ((closed s) ==> (lebesgue_measurable s)))
BALL_LINEAR_IMAGE
|- (!f. (!x. (!r. (((linear f) /\ ((!y. (?x. ((f x) = y))) /\ (!x. ((vector_norm (f x)) = (vector_norm x))))) ==> ((ball ((f x) , r)) = ((IMAGE f) (ball (x , r))))))))
EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL
|- (!s. ((FINITE s) ==> ((GSPEC (\GEN%PVAR%1789. (?y. (((SETSPEC GEN%PVAR%1789) (?u. ((!x. ((x IN s) ==> (((real_lt (real_of_num (NUMERAL _0))) (u x)) /\ ((real_lt (u x)) (real_of_num (NUMERAL (BIT1 _0))))))) /\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum s) (\x. ((% (u x)) x))) = y))))) y)))) SUBSET (relative_interior ((hull convex) s)))))
AFF_DIM_AFFINE_HULL
|- (!s. ((aff_dim ((hull affine) s)) = (aff_dim s)))
RAT_LEMMA2
|- ((((real_lt (real_of_num (NUMERAL _0))) y1) /\ ((real_lt (real_of_num (NUMERAL _0))) y2)) ==> (((real_add ((real_div x1) y1)) ((real_div x2) y2)) = ((real_mul ((real_add ((real_mul x1) y2)) ((real_mul x2) y1))) ((real_mul (real_inv y1)) (real_inv y2)))))
REAL_ADD_RINV
|- (!x. (((real_add x) (real_neg x)) = (real_of_num (NUMERAL _0))))
HAS_REAL_INTEGRAL_RESTRICT_UNIV
|- (!f. (!s. (!i. ((((has_real_integral (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) i) UNIV) = (((has_real_integral f) i) s)))))
REAL_DERIVATIVE_UNIQUE_ATREAL
|- (!f. (!z. (!f'. (!f''. (((((has_real_derivative f) f') (atreal z)) /\ (((has_real_derivative f) f'') (atreal z))) ==> (f' = f''))))))
Lp_ineqs_proofs.y2_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 (y2_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))
SUBSET_CLOSURE
|- (!s. (!t. ((s SUBSET t) ==> ((closure s) SUBSET (closure t)))))
DIAMETER_ATTAINED_RELATIVE_FRONTIER
|- (!s. (((bounded s) /\ (~ ((diameter s) = (real_of_num (NUMERAL _0))))) ==> (?x. (?y. ((x IN (relative_frontier s)) /\ ((y IN (relative_frontier s)) /\ ((vector_norm ((vector_sub x) y)) = (diameter s))))))))
face_of
|- (!s. (!t. (((face_of t) s) = ((t SUBSET s) /\ ((convex t) /\ (!a. (!b. (!x. (((a IN s) /\ ((b IN s) /\ ((x IN t) /\ (x IN (open_segment (a , b)))))) ==> ((a IN t) /\ (b IN t)))))))))))