hcju/holps huggingface dataset:
FACET_OF_HALFSPACE_LE
|- (!f. (!a. (!b. (((facet_of f) (GSPEC (\GEN%PVAR%2216. (?x. (((SETSPEC GEN%PVAR%2216) ((real_le ((dot a) x)) b)) x))))) = ((~ (a = (vec (NUMERAL _0)))) /\ (f = (GSPEC (\GEN%PVAR%2217. (?x. (((SETSPEC GEN%PVAR%2217) (((dot a) x) = b)) x))))))))))
FINITE_RECURSION
|- (!f. (!b. ((!x. (!y. (!s. ((~ (x = y)) ==> (((f x) ((f y) s)) = ((f y) ((f x) s))))))) ==> (((((ITSET f) EMPTY) b) = b) /\ (!x. (!s. ((FINITE s) ==> ((((ITSET f) (x INSERT s)) b) = (((COND (x IN s)) (((ITSET f) s) b)) ((f x) (((ITSET f) s) b)))))))))))
REAL_CONVEX_ON_SECOND_DERIVATIVE
|- (!f. (!f'. (!f''. (!s. (((is_realinterval s) /\ ((~ (?a. (s = (a INSERT EMPTY)))) /\ ((!x. ((x IN s) ==> (((has_real_derivative f) (f' x)) ((within (atreal x)) s)))) /\ (!x. ((x IN s) ==> (((has_real_derivative f') (f'' x)) ((within (atreal x)) s))))))) ==> (((real_convex_on f) s) = (!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f'' x))))))))))
Hypermap.lemma_dart_invariant
|- (!H. (!x. ((x IN (dart H)) ==> ((((edge_map H) x) IN (dart H)) /\ ((((node_map H) x) IN (dart H)) /\ (((face_map H) x) IN (dart H)))))))
Seq.has_predU
|- (!a1. (!a2. (!s. (((has ((predU a1) a2)) s) = (((has a1) s) \/ ((has a2) s))))))
AFF_GT_INJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (!t. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((aff_gt ((IMAGE f) s)) ((IMAGE f) t)) = ((IMAGE f) ((aff_gt s) t)))))))
REAL_LT_IMP_NZ
|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> (~ (x = (real_of_num (NUMERAL _0))))))
log_convex_on
|- (!s. (!f. (((log_convex_on f) s) = (!x. (!y. (!u. (!v. (((x IN s) /\ ((y IN s) /\ (((real_le (real_of_num (NUMERAL _0))) u) /\ (((real_le (real_of_num (NUMERAL _0))) v) /\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> (((real_le (real_of_num (NUMERAL _0))) (f ((vector_add ((% u) x)) ((% v) y)))) /\ ((real_le (f ((vector_add ((% u) x)) ((% v) y)))) ((real_mul ((rpow (f x)) u)) ((rpow (f y)) v))))))))))))
Tame_defs.tame_9a
|- (!H. ((tame_9a H) = (!x. ((x IN (dart H)) ==> (((CARD ((face H) x)) >= (NUMERAL (BIT1 (BIT1 _0)))) /\ ((CARD ((face H) x)) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))))))
CLOSED_IN_CLOSED_TRANS
|- (!s. (!t. ((((closed_in ((subtopology euclidean) t)) s) /\ (closed t)) ==> (closed s))))
COMPLEX_MUL_RZERO
|- (!x. (((complex_mul x) (Cx (real_of_num (NUMERAL _0)))) = (Cx (real_of_num (NUMERAL _0)))))
RE_CCOS
|- (!z. ((Re (ccos z)) = ((real_mul ((real_div ((real_add (exp (Im z))) (exp (real_neg (Im z))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (cos (Re z)))))
HAS_SIZE_FACES_OF_SIMPLEX
|- (!n. (!s. (((simplex n) s) ==> ((GSPEC (\GEN%PVAR%2479. (?f. (((SETSPEC GEN%PVAR%2479) ((face_of f) s)) f)))) HAS_SIZE ((NUMERAL (BIT0 (BIT1 _0))) EXP (num_of_int ((int_add n) (int_of_num (NUMERAL (BIT1 _0))))))))))
segment
|- (((closed_segment ((CONS (a , b)) NIL)) = (GSPEC (\GEN%PVAR%559. (?u. (((SETSPEC GEN%PVAR%559) (((real_le (real_of_num (NUMERAL _0))) u) /\ ((real_le u) (real_of_num (NUMERAL (BIT1 _0)))))) ((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) u)) a)) ((% u) b))))))) /\ ((open_segment (a , b)) = ((closed_segment ((CONS (a , b)) NIL)) DIFF (a INSERT (b INSERT EMPTY)))))
bilinear
|- (!f. ((bilinear f) = ((!x. (linear (\y. ((f x) y)))) /\ (!y. (linear (\x. ((f x) y)))))))
LIPSCHITZ_IMP_HAS_BOUNDED_VARIATION
|- (!f. (!s. (!B. (((bounded 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)))))))) ==> ((has_bounded_variation_on f) s)))))
WITHINREAL_UNIV
|- (!x. (((within (atreal x)) UNIV) = (atreal x)))
IN_COMPONENTS_CONNECTED
|- (!s. (!c. ((c IN (components s)) ==> (connected c))))
NUM_OF_INT
|- (!x. (((int_le (int_of_num (NUMERAL _0))) x) = ((int_of_num (num_of_int x)) = x)))
HOMOTOPIC_WITH_IMP_SUBSET
|- (!P. (!X. (!Y. (!f. (!g. (((((homotopic_with P) (X , Y)) f) g) ==> ((((IMAGE f) X) SUBSET Y) /\ (((IMAGE g) X) SUBSET Y))))))))
Hypermap_and_fan.N_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((permutes (n_fan_pair_ext (V , E))) (dart_of_fan (V , E))))))
ORTHOGONAL_TRANSFORMATION_ID
|- (orthogonal_transformation (\x. x))
BASIS_EXISTS
|- (!v. (?b. ((b SUBSET v) /\ ((independent b) /\ ((v SUBSET (span b)) /\ (b HAS_SIZE (dim v)))))))
RE_DEF
|- (!z. ((Re z) = (z $ (NUMERAL (BIT1 _0)))))
ORTHOGONAL_SUBSPACE_DECOMP_UNIQUE
|- (!s. (!t. (!x. (!y. (!x'. (!y'. (((!a. (!b. (((a IN s) /\ (b IN t)) ==> ((orthogonal a) b)))) /\ ((x IN (span s)) /\ ((x' IN (span s)) /\ ((y IN (span t)) /\ ((y' IN (span t)) /\ (((vector_add x) y) = ((vector_add x') y'))))))) ==> ((x = x') /\ (y = y')))))))))
FACE_OF_HALFSPACE_LE
|- (!f. (!a. (!b. (((face_of f) (GSPEC (\GEN%PVAR%2153. (?x. (((SETSPEC GEN%PVAR%2153) ((real_le ((dot a) x)) b)) x))))) = ((f = EMPTY) \/ ((f = (GSPEC (\GEN%PVAR%2154. (?x. (((SETSPEC GEN%PVAR%2154) (((dot a) x) = b)) x))))) \/ (f = (GSPEC (\GEN%PVAR%2155. (?x. (((SETSPEC GEN%PVAR%2155) ((real_le ((dot a) x)) b)) x)))))))))))
DIAMETER_SUBSET
|- (!s. (!t. (((s SUBSET t) /\ (bounded t)) ==> ((real_le (diameter s)) (diameter t)))))
HAS_MEASURE_0
|- (!s. (((has_measure s) (real_of_num (NUMERAL _0))) = (negligible s)))
INTEGRAL_COMPONENT
|- (!f. (((integrable_on f) s) ==> ((((integral s) f) $ k) = (drop ((integral s) (\x. (lift ((f x) $ k))))))))
LIFT_NEG
|- (!x. ((lift (real_neg x)) = (vector_neg (lift x))))
SETDIST_EMPTY
|- ((!t. ((setdist (EMPTY , t)) = (real_of_num (NUMERAL _0)))) /\ (!s. ((setdist (s , EMPTY)) = (real_of_num (NUMERAL _0)))))
REAL_INV_POW
|- (!x. (!n. ((real_inv ((real_pow x) n)) = ((real_pow (real_inv x)) n))))
real_infsum
|- (!f. (!s. (((real_infsum s) f) = (@l. (((real_sums f) l) s)))))
CLOSED_IN_EMPTY
|- (!top. ((closed_in top) EMPTY))
CONTINUOUS_ON_ADD
|- (!f. (!g. (!s. ((((continuous_on f) s) /\ ((continuous_on g) s)) ==> ((continuous_on (\x. ((vector_add (f x)) (g x)))) s)))))
SUM_SUB
|- (!f. (!g. (!s. ((FINITE s) ==> (((sum s) (\x. ((real_sub (f x)) (g x)))) = ((real_sub ((sum s) f)) ((sum s) g)))))))
NOT_LE
|- (!m. (!n. ((~ (m <= n)) = (n < m))))
Planarity.exists_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)))))))) ==> (?s. (!y. ((y IN ds) ==> (s = (((((dart_leads_into x) V) E) (pr2 y)) (pr3 y)))))))))))
real_uniformly_continuous_on
|- (!s. (!f. (((real_uniformly_continuous_on f) s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (!x'. (((x IN s) /\ ((x' IN s) /\ ((real_lt (real_abs ((real_sub x') x))) d))) ==> ((real_lt (real_abs ((real_sub (f x')) (f x)))) e)))))))))))
SERIES_BOUND
|- (!f. (!g. (!s. (!a. (!b. (((((sums f) a) s) /\ ((((sums (lift o g)) (lift b)) s) /\ (!i. ((i IN s) ==> ((real_le (vector_norm (f i))) (g i)))))) ==> ((real_le (vector_norm a)) b)))))))
DIST_IN_CLOSED_SEGMENT
|- (!a. (!b. (!x. ((x IN (closed_segment ((CONS (a , b)) NIL))) ==> (((real_le (distance (x , a))) (distance (a , b))) /\ ((real_le (distance (x , b))) (distance (a , b))))))))
POLYHEDRON_NEGATIONS
|- (!s. ((polyhedron s) ==> (polyhedron ((IMAGE vector_neg) s))))
HAS_COMPLEX_DERIVATIVE_CSIN
|- (!z. (((has_complex_derivative csin) (ccos z)) (at z)))
SKOLEM_THM
|- (!P. ((!x. (?y. ((P x) y))) = (?y. (!x. ((P x) (y x))))))
COMPLEX_POLY_CLAUSES
|- ((!x. (!y. (!z. (((vector_add x) ((vector_add y) z)) = ((vector_add ((vector_add x) y)) z))))) /\ ((!x. (!y. (((vector_add x) y) = ((vector_add y) x)))) /\ ((!x. (((vector_add (Cx (real_of_num (NUMERAL _0)))) x) = x)) /\ ((!x. (!y. (!z. (((complex_mul x) ((complex_mul y) z)) = ((complex_mul ((complex_mul x) y)) z))))) /\ ((!x. (!y. (((complex_mul x) y) = ((complex_mul y) x)))) /\ ((!x. (((complex_mul (Cx (real_of_num (NUMERAL (BIT1 _0))))) x) = x)) /\ ((!x. (((complex_mul (Cx (real_of_num (NUMERAL _0)))) x) = (Cx (real_of_num (NUMERAL _0))))) /\ ((!x. (!y. (!z. (((complex_mul x) ((vector_add y) z)) = ((vector_add ((complex_mul x) y)) ((complex_mul x) z)))))) /\ ((!x. (((complex_pow x) (NUMERAL _0)) = (Cx (real_of_num (NUMERAL (BIT1 _0)))))) /\ (!x. (!n. (((complex_pow x) (SUC n)) = ((complex_mul x) ((complex_pow x) n))))))))))))))
CONVEX_ON_LEFT_SECANT
|- (!f. (!s. (((convex_on f) s) = (!a. (!b. (!x. (((a IN s) /\ ((b IN s) /\ (x IN (open_segment (a , b))))) ==> ((real_le ((real_div ((real_sub (f x)) (f a))) (vector_norm ((vector_sub x) a)))) ((real_div ((real_sub (f b)) (f a))) (vector_norm ((vector_sub b) a)))))))))))
Tame_defs2.list_update
|- (((((list_update NIL) i) v) = NIL) /\ ((((list_update ((CONS x) xs)) i) v) = (((COND (i = (NUMERAL _0))) ((CONS v) xs)) ((CONS x) (((list_update xs) (i - (NUMERAL (BIT1 _0)))) v)))))
Trigonometry2.TWO_PI_POS
|- ((real_lt (real_of_num (NUMERAL _0))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))
HAS_COMPLEX_DERIVATIVE_AT_WITHIN
|- (!f. (!f'. (!x. (!s. ((((has_complex_derivative f) f') (at x)) ==> (((has_complex_derivative f) f') ((within (at x)) s)))))))
RETRACT_OF_TRANSLATION_EQ
|- (!a. (!t. (!s. (((retract_of ((IMAGE (\x. ((vector_add a) x))) t)) ((IMAGE (\x. ((vector_add a) x))) s)) = ((retract_of t) s)))))
CONTRACTIBLE_INJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((contractible ((IMAGE f) s)) = (contractible s)))))
BASIS_EXPANSION
|- (!x. (((vsum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. ((% (x $ i)) (basis i)))) = x))
MEASURE_INTERVAL_2
|- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ ((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL _0)))))) /\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ ((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL _0)))))))
HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS_IMP
|- (!f. (!g. (((orthogonal_transformation f) /\ ((orthogonal_transformation g) /\ ((((homotopic_with (\x. T)) ((sphere ((vec (NUMERAL _0)) , (real_of_num (NUMERAL (BIT1 _0))))) , (sphere ((vec (NUMERAL _0)) , (real_of_num (NUMERAL (BIT1 _0))))))) f) g))) ==> ((det (matrix f)) = (det (matrix g))))))
int_eq
|- (!x. (!y. ((x = y) = ((real_of_int x) = (real_of_int y)))))
connected_component
|- (!s. (!x. (!y. ((((connected_component s) x) y) = (?t. ((connected t) /\ ((t SUBSET s) /\ ((x IN t) /\ (y IN t)))))))))
HAS_REAL_INTEGRAL_SPIKE_INTERIOR
|- (!f. (!g. (!a. (!b. (!y. (((!x. ((x IN (open_real_interval (a , b))) ==> ((g x) = (f x)))) /\ (((has_real_integral f) y) (closed_real_interval ((CONS (a , b)) NIL)))) ==> (((has_real_integral g) y) (closed_real_interval ((CONS (a , b)) NIL)))))))))
TOPSPACE_EUCLIDEAN
|- ((topspace euclidean) = UNIV)
Hypermap.number_of_faces
|- (!H. ((number_of_faces H) = (CARD (face_set H))))
OPEN_UNIONS
|- ((!s. ((s IN f) ==> (open s))) ==> (open (UNIONS f)))
Lp_ineqs_proofs.add_diag_list
|- (!V. (!g. (!E. (!diags. (((add_diag_list (g , (V , E))) diags) = (((foldr (\d. (\E'. ((add_diag (V , E')) (g d))))) E) diags))))))
HULL_INC
|- (!P. (!s. (!x. ((x IN s) ==> (x IN ((hull P) s))))))
FACE_OF_CONVEX_HULLS
|- (!f. (!s. (((FINITE s) /\ ((f SUBSET s) /\ ((DISJOINT ((hull affine) f)) ((hull convex) (s DIFF f))))) ==> ((face_of ((hull convex) f)) ((hull convex) s)))))
REAL_LT_REFL
|- (!x. (~ ((real_lt x) x)))
REALLIM_SEQUENTIALLY
|- (!s. (!l. ((((---> s) l) sequentially) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (real_abs ((real_sub (s n)) l))) e)))))))))
COUNTABLE_ALT
|- (!s. ((COUNTABLE s) = (s <=_c UNIV)))
Lp_ineqs_proofs.y1_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 (y1_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))
lt_c
|- (!t. (!s. ((s <_c t) = ((s <=_c t) /\ (~ (t <=_c s))))))
WLOG_LE
|- (((!m. (!n. (((P m) n) = ((P n) m)))) /\ (!m. (!n. ((m <= n) ==> ((P m) n))))) ==> (!m. (!n. ((P m) n))))
CBALL_INTERVAL
|- (!x. (!e. ((cball (x , e)) = (closed_interval ((CONS (((vector_sub x) (lift e)) , ((vector_add x) (lift e)))) NIL)))))
Real_ext.REAL_PROP_POS_MUL2
|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_mul x) y)))))
Collect_geom.C_COEF_FORMULA
|- (!x12. (!x13. (!x14. (!x23. (!x24. ((((((c_coef x12) x13) x14) x23) x24) = ((real_add ((real_mul ((real_mul x14) x23)) x12)) ((real_add ((real_mul ((real_mul x14) x23)) x13)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x14)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x23)) ((real_add ((real_mul ((real_mul x14) x23)) x24)) ((real_add ((real_mul ((real_mul (real_neg x12)) x13)) x23)) ((real_add ((real_mul ((real_mul (real_neg x12)) x14)) x24)) ((real_add ((real_mul ((real_mul x13) x24)) x12)) ((real_add ((real_mul ((real_mul (real_neg x13)) x24)) x13)) ((real_add ((real_mul ((real_mul x13) x24)) x14)) ((real_add ((real_mul ((real_mul x13) x24)) x23)) ((real_mul ((real_mul (real_neg x13)) x24)) x24))))))))))))))))))
CASN_CSIN
|- (!z. ((((real_lt (real_abs (Re z))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) \/ (((real_abs (Re z)) = ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) /\ ((Im z) = (real_of_num (NUMERAL _0))))) ==> ((casn (csin z)) = z)))
Dih2k_hypermap.SUC_NOT
|- (((NUMERAL (BIT1 _0)) < k) ==> (((NUMERAL (BIT1 _0)) <= (SUC (i MOD k))) /\ (((SUC (i MOD k)) <= k) /\ (~ (i = (SUC (i MOD k)))))))
LEBESGUE_DENSITY_THEOREM_INTEGRAL_CBALL
|- (!f. (!s. ((!a. (!b. ((absolutely_integrable_on f) (s INTER (closed_interval ((CONS (a , b)) NIL)))))) ==> (?k. ((negligible k) /\ (!x. ((~ (x IN k)) ==> (((--> (\e. ((% (real_inv (measure (cball (x , (drop e)))))) ((integral (s INTER (cball (x , (drop e))))) f)))) (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))) ((within (at (vec (NUMERAL _0)))) (GSPEC (\GEN%PVAR%3724. (?t. (((SETSPEC GEN%PVAR%3724) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t)))))))))))))
Nonlinear_lemma.rho_alt
|- (!y. ((rho y) = ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul const1) ((real_div ((real_sub y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 _0)))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))
Ssrnat.subn_gt0
|- (!m. (!n. (((NUMERAL _0) < (n - m)) = (m < n))))
Fan.f1_fan
|- (!x. (!V. (!E. ((((f1_fan x) V) E) = (GABS (\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) (x , (w , ((((((inverse1_sigma_fan x) V) E) w) v) , v))))))))))))))
Lp_ineqs_def.y3_fan
|- (!V. (!E. (!x. (((y3_fan (V , E)) x) = (vector_norm (FST ((inverse (f_fan_pair_ext (V , E))) x)))))))
ANR_BALL
|- (!a. (!r. (ANR (ball (a , r)))))
CLOSED_IN_TRANS
|- (!s. (!t. (!u. ((((closed_in ((subtopology euclidean) t)) s) /\ ((closed_in ((subtopology euclidean) u)) t)) ==> ((closed_in ((subtopology euclidean) u)) s)))))
Nonlin_def.scalar6
|- (!f. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!r. (((((((((scalar6 f) r) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((f x1) x2) x3) x4) x5) x6)) r))))))))))
vertex_image
|- (!f. (!s. (((vertex_image f) s) = ((hull convex) ((IMAGE f) (GSPEC (\GEN%PVAR%2552. (?v. (((SETSPEC GEN%PVAR%2552) ((extreme_point_of v) s)) v)))))))))
WF_INSEG_WOSET
|- (WF (\x. (\y. ((woset x) /\ ((woset y) /\ (((inseg x) y) /\ (~ (x = y))))))))
List_hypermap.list_of_elements
|- (!L. ((list_of_elements L) = (undup (flatten L))))
CHOICE_DEF
|- (!s. ((~ (s = EMPTY)) ==> ((CHOICE s) IN s)))
SUM_SUPPORT
|- (!f. (!s. (((sum (((support real_add) f) s)) f) = ((sum s) f))))
Fnjlbxs.infnorm_lt
|- (!e. (!v. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_lt (real_abs (v $ i))) e))) ==> ((real_lt (infnorm v)) e))))
I_O_ID
|- (!f. (((I o f) = f) /\ ((f o I) = f)))
Fan.collinear_imp_azim_is_rezo_fan
|- (!x. (!v. (!u. (!y. (!h1. (!h2. (((~ (v = x)) /\ ((~ (u = x)) /\ ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) /\ (((real_le (real_of_num (NUMERAL _0))) y) /\ (((real_lt y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ (!e1. (!e2. (!e3. (((((orthonormal e1) e2) e3) /\ (((% (distance (v , x))) e3) = ((vector_sub v) x))) ==> (?psi. (?r1. (?r2. ((((vector_sub u) x) = ((vector_add ((% ((real_mul r1) (cos psi))) e1)) ((vector_add ((% ((real_mul r1) (sin psi))) e2)) ((% h1) ((vector_sub v) x))))) /\ ((((vector_sub u) x) = ((vector_add ((% ((real_mul r2) (cos ((real_add psi) y)))) e1)) ((vector_add ((% ((real_mul r2) (sin ((real_add psi) y)))) e2)) ((% h2) ((vector_sub v) x))))) /\ (((real_lt (real_of_num (NUMERAL _0))) r1) /\ ((real_lt (real_of_num (NUMERAL _0))) r2)))))))))))))))) ==> (y = (real_of_num (NUMERAL _0))))))))))
Collect_geom.ETA_Y_SYYM
|- (!x. (!y. (!z. (((((eta_y x) y) z) = (((eta_y y) x) z)) /\ ((((eta_y x) y) z) = (((eta_y z) y) x))))))
rcone_ge
|- (rcone_ge = (rconesgn real_ge))
Hypermap.face_set
|- (!H. ((face_set H) = ((set_of_orbits (dart H)) (face_map H))))
IN_DELETE
|- (!s. (!x. (!y. ((x IN (s DELETE y)) = ((x IN s) /\ (~ (x = y)))))))
TRACE_COVARIANCE_CAUCHY_SCHWARZ
|- (!A. (!B. ((real_le (trace ((matrix_mul (transp A)) B))) ((real_mul (sqrt (trace ((matrix_mul (transp A)) A)))) (sqrt (trace ((matrix_mul (transp B)) B)))))))
Planarity.AFFINE_HULL_1
|- (!a. (((hull affine) (a INSERT EMPTY)) = (GSPEC (\GEN%PVAR%5568. (?u. (((SETSPEC GEN%PVAR%5568) (u = (real_of_num (NUMERAL (BIT1 _0))))) ((% u) a)))))))
Ssrnat.subn_sub
|- (!m. (!n. (!p. (((n - m) - p) = (n - (m + p))))))
EMPTY_INTERIOR_LOWDIM
|- (!s. (((dim s) < (dimindex UNIV)) ==> ((interior s) = EMPTY)))
INTERVAL_TRANSLATION
|- ((!c. (!a. (!b. ((closed_interval ((CONS (((vector_add c) a) , ((vector_add c) b))) NIL)) = ((IMAGE (\x. ((vector_add c) x))) (closed_interval ((CONS (a , b)) NIL))))))) /\ (!c. (!a. (!b. ((open_interval (((vector_add c) a) , ((vector_add c) b))) = ((IMAGE (\x. ((vector_add c) x))) (open_interval (a , b))))))))
PATH_COMPONENT_EQ_EMPTY
|- (!s. (!x. ((((path_component s) x) = EMPTY) = (~ (x IN s)))))
NORM_POS_LE
|- (!x. ((real_le (real_of_num (NUMERAL _0))) (vector_norm x)))
SEPARATING_HYPERPLANE_AFFINE_AFFINE
|- (!s. (!t. (((affine s) /\ ((affine t) /\ ((~ (s = EMPTY)) /\ ((~ (t = EMPTY)) /\ ((DISJOINT s) t))))) ==> (?a. (?b. (?c. ((~ (a = (vec (NUMERAL _0)))) /\ (((real_lt b) c) /\ ((!x. ((x IN s) ==> (((dot a) x) = b))) /\ (!x. ((x IN t) ==> (((dot a) x) = c))))))))))))
AFFINE_HULL_SING
|- (!a. (((hull affine) (a INSERT EMPTY)) = (a INSERT EMPTY)))
LIM_COMPLEX_REAL
|- (!f. (!g. (!l. (!m. ((((eventually (\n. ((Re (g n)) = (f n)))) sequentially) /\ (((Re m) = l) /\ (((--> g) m) sequentially))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (real_abs ((real_sub (f n)) l))) e)))))))))))
UNIONS_INSERT
|- ((UNIONS (s INSERT u)) = (s UNION (UNIONS u)))
CONNECTED_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((connected ((IMAGE f) s)) = (connected s)))))
CLOSED_IN_COMPONENT
|- (!s. (!c. ((c IN (components s)) ==> ((closed_in ((subtopology euclidean) s)) c))))
REAL_ADD_LDISTRIB
|- (!x. (!y. (!z. (((real_mul x) ((real_add y) z)) = ((real_add ((real_mul x) y)) ((real_mul x) z))))))
RANK_DIM_IM
|- (!A. ((rank A) = (dim ((IMAGE (\x. ((matrix_vector_mul A) x))) UNIV))))
Trigonometry2.SIMPLIZE_COS_IF_OTHOR
|- (!v0. (!v1. (!w. (!p. (((~ (v0 = w)) /\ ((~ (v0 = v1)) /\ ((((vector_sub p) v0) = ((% k) ((vector_sub v1) v0))) /\ (((dot ((vector_sub w) p)) ((vector_sub v1) v0)) = (real_of_num (NUMERAL _0)))))) ==> ((cos (((arcV v0) v1) w)) = ((real_mul k) ((real_div (vector_norm ((vector_sub v1) v0))) (vector_norm ((vector_sub w) v0))))))))))
Hypermap.lemma_dart_invariant_power_face
|- (!H. (!x. (!n. ((x IN (dart H)) ==> ((((POWER (face_map H)) n) x) IN (dart H))))))
INTERVAL_UPPERBOUND_NONEMPTY
|- (!a. (!b. ((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) ==> ((interval_upperbound (closed_interval ((CONS (a , b)) NIL))) = b))))
REAL_COMPACT_INTERVAL
|- (!a. (!b. (real_compact (closed_real_interval ((CONS (a , b)) NIL)))))
LIM_WITHINREAL
|- (!f. (!l. (!a. (!s. ((((--> f) l) ((within (atreal a)) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (((x IN s) /\ (((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\ ((real_lt (real_abs ((real_sub x) a))) d))) ==> ((real_lt (distance ((f x) , l))) e))))))))))))
LINEAR_FROM_1
|- (!f. ((linear f) = (?c. (f = (\x. ((% (drop x)) c))))))
Sphere.eventually_radial
|- (!C. (!x. (((eventually_radial x) C) = (?r. (((real_gt r) (real_of_num (NUMERAL _0))) /\ (((radial r) x) (C INTER (ball (x , r)))))))))
ITER_POINTLESS
|- ((!f. (((ITER (NUMERAL _0)) f) = I)) /\ (!f. (!n. (((ITER (SUC n)) f) = (f o ((ITER n) f))))))
LINEAR_CONTINUOUS_ON
|- (!f. (!s. ((linear f) ==> ((continuous_on f) s))))
LIM_CONST
|- (!net. (!a. (((--> (\x. a)) a) net)))
REAL_MEASURABLE_IMP_REAL_LEBESGUE_MEASURABLE
|- (!s. ((real_measurable s) ==> (real_lebesgue_measurable s)))
INT_LE_REFL
|- (!x. ((int_le x) x))
RE_DIV_CX
|- (!z. (!x. ((Re ((complex_div z) (Cx x))) = ((real_div (Re z)) x))))
HAS_DERIVATIVE_ID
|- (!net. (((has_derivative (\x. x)) (\h. h)) net))
Hypermap_and_fan.F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((permutes (f_fan_pair_ext (V , E))) (dart_of_fan (V , E))))))
REAL_POS
|- (!n. ((real_le (real_of_num (NUMERAL _0))) (real_of_num n)))
DROP_DIFFERENTIAL_NEG_AT_MAXIMUM
|- (!f. (!f'. (!x. (!s. (!e. (((x IN s) /\ ((convex s) /\ ((((has_derivative f) f') ((within (at x)) s)) /\ (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!w. ((w IN (s INTER (ball (x , e)))) ==> ((real_le (drop (f w))) (drop (f x))))))))) ==> (!y. ((y IN s) ==> ((real_le (drop (f' ((vector_sub y) x)))) (real_of_num (NUMERAL _0)))))))))))
path_length
|- (!g. ((path_length g) = ((vector_variation (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) g)))
PERMUTES_IN_IMAGE
|- (!p. (!s. (!x. (((permutes p) s) ==> (((p x) IN s) = (x IN s))))))
Appendix.FZIOTEF_TRANS
|- (!S1. (!S2. (!S3. ((((scs_arrow_v39 S1) S2) /\ ((scs_arrow_v39 S2) S3)) ==> ((scs_arrow_v39 S1) S3)))))
NORM_CPOW_REAL
|- (!w. (!z. (((real w) /\ ((real_lt (real_of_num (NUMERAL _0))) (Re w))) ==> ((vector_norm ((cpow w) z)) = (exp ((real_mul (Re z)) (log (Re w))))))))
REAL_INTEGRABLE_ON
|- (((real_integrable_on f) s) = ((integrable_on (lift o (f o drop))) ((IMAGE lift) s)))
LIM_NULL_COMPARISON_COMPLEX
|- (!net. (!f. (!g. ((((eventually (\x. ((real_le (vector_norm (f x))) (vector_norm (g x))))) net) /\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) net)) ==> (((--> f) (Cx (real_of_num (NUMERAL _0)))) net)))))
EXTREME_POINT_OF_CONVEX_HULL
|- (!x. (!s. (((extreme_point_of x) ((hull convex) s)) ==> (x IN s))))
OPEN_MAP_CLOSED_SUPERSET_PREIMAGE
|- (!f. (!s. (!t. (!u. (!w. (((!k. (((open_in ((subtopology euclidean) s)) k) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) k)))) /\ (((closed_in ((subtopology euclidean) s)) u) /\ ((w SUBSET t) /\ ((GSPEC (\GEN%PVAR%697. (?x. (((SETSPEC GEN%PVAR%697) ((x IN s) /\ ((f x) IN w))) x)))) SUBSET u)))) ==> (?v. (((closed_in ((subtopology euclidean) t)) v) /\ ((w SUBSET v) /\ ((GSPEC (\GEN%PVAR%698. (?x. (((SETSPEC GEN%PVAR%698) ((x IN s) /\ ((f x) IN v))) x)))) SUBSET u))))))))))
Tame_lemmas.reflection_eq
|- (!n. (!x. ((((dot x) n) = (real_of_num (NUMERAL _0))) ==> (((reflection n) x) = x))))
differentiable
|- (!f. (!net. (((differentiable f) net) = (?f'. (((has_derivative f) f') net)))))
HOLOMORPHIC_ON_MUL
|- (!f. (!g. (!s. ((((holomorphic_on f) s) /\ ((holomorphic_on g) s)) ==> ((holomorphic_on (\z. ((complex_mul (f z)) (g z)))) s)))))
REAL_ADD_ASSOC
|- (!x. (!y. (!z. (((real_add x) ((real_add y) z)) = ((real_add ((real_add x) y)) z)))))