hcju/holps huggingface dataset:
NADD_LE_LMUL
|- (!x. (!y. (!z. (((nadd_le y) z) ==> ((nadd_le ((nadd_mul x) y)) ((nadd_mul x) z))))))
Functional_equation.constant6
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!c. ((((((((constant6 c) x1) x2) x3) x4) x5) x6) = c))))))))
OPEN_IMP_LOCALLY_PATH_CONNECTED
|- (!s. ((open s) ==> ((locally path_connected) s)))
Lp_main_estimate.quad_diags_imp
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (!a. (!b. ((LET (\f. (LET_END ((LET (\w'. (LET_END ((LET (\u. (LET_END ((((v , w) IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) /\ ((~ (a = b)) /\ ((a IN ((v_prime V) f)) /\ ((b IN ((v_prime V) f)) /\ (~ ((a INSERT (b INSERT EMPTY)) IN ((e_prime E) f))))))) ==> ((((a , b) = (w , w')) \/ ((a , b) = (w' , w))) \/ (((a , b) = (v , u)) \/ ((a , b) = (u , v)))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))
SUM_4
|- (!t. (((sum ((NUMERAL (BIT1 _0)) .. (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) t) = ((real_add (t (NUMERAL (BIT1 _0)))) ((real_add (t (NUMERAL (BIT0 (BIT1 _0))))) ((real_add (t (NUMERAL (BIT1 (BIT1 _0))))) (t (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))
ANALYTIC_ON_COMPOSE
|- (!f. (!g. (!s. ((((analytic_on f) s) /\ ((analytic_on g) ((IMAGE f) s))) ==> ((analytic_on (g o f)) s)))))
CLOSED_INJECTIVE_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((closed ((IMAGE f) s)) = (closed s)))))
DIM_UNIQUE
|- (!v. (!b. (((b SUBSET v) /\ ((v SUBSET (span b)) /\ ((independent b) /\ (b HAS_SIZE n)))) ==> ((dim v) = n))))
BOOL_CASES_AX
|- (!t. ((t = T) \/ (t = F)))
TRACE_MATRIX_INV_LMUL
|- (!A. ((trace ((matrix_mul (matrix_inv A)) A)) = (real_of_num (rank A))))
INTEGRABLE_SPIKE
|- (!f. (!g. (!s. (!t. (((negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((integrable_on f) t) ==> ((integrable_on g) t)))))))
IMAGE_EQ_EMPTY
|- (!f. (!s. ((((IMAGE f) s) = EMPTY) = (s = EMPTY))))
MATCH_SEQPATTERN
|- (((_MATCH x) ((_SEQPATTERN r) s)) = (((COND (?y. ((r x) y))) ((_MATCH x) r)) ((_MATCH x) s)))
CONVEX_IMP_SIMPLY_CONNECTED
|- (!s. ((convex s) ==> (simply_connected s)))
ORTHOGONAL_TRANSFORMATION
|- (!f. ((orthogonal_transformation f) = ((linear f) /\ (!v. ((vector_norm (f v)) = (vector_norm v))))))
Lp_ineqs_proofs.y4'_eq_y4
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT1 _0)))) (hypermap_of_fan (V , E)))) ==> (((y4'_fan (V , E)) d) = ((y4_fan (V , E)) d))))))))))
Appendix.scs_am_v39_explicit
|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_am_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = alpha))))))))))
Add_triangle.split_fan_face
|- (!V. (!v. (!w. (!E. (((split_fan_face (V , E)) (v , w)) = (V , ((w INSERT ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) INSERT EMPTY)) INSERT E)))))))
LIM_AT_INFINITY_WITHIN
|- (!f. (!l. (!s. ((((--> f) l) ((within at_infinity) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((x IN s) /\ ((real_ge (vector_norm x)) b)) ==> ((real_lt (distance ((f x) , l))) e))))))))))
ARZELA_ASCOLI
|- (!f. (!s. (!M. (((compact s) /\ ((!n. (!x. ((x IN s) ==> ((real_le (vector_norm ((f n) x))) M)))) /\ (!x. (!e. (((x IN s) /\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!n. (!y. (((y IN s) /\ ((real_lt (vector_norm ((vector_sub x) y))) d)) ==> ((real_lt (vector_norm ((vector_sub ((f n) x)) ((f n) y)))) e))))))))))) ==> (?g. (((continuous_on g) s) /\ (?r. ((!m. (!n. ((m < n) ==> ((r m) < (r n))))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\ (x IN s)) ==> ((real_lt (vector_norm ((vector_sub ((f (r n)) x)) (g x)))) e)))))))))))))))
real_min
|- (!m. (!n. (((real_min m) n) = (((COND ((real_le m) n)) m) n))))
Nonlin_def.add6
|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((add6 f) g) x1) x2) x3) x4) x5) x6) = ((real_add ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))
REAL_LE_ADD2
|- (!w. (!x. (!y. (!z. ((((real_le w) x) /\ ((real_le y) z)) ==> ((real_le ((real_add w) y)) ((real_add x) z)))))))
REAL_SGN_NEG
|- (!x. ((real_sgn (real_neg x)) = (real_neg (real_sgn x))))
COND_ELIM_THM
|- ((P (((COND c) x) y)) = ((c ==> (P x)) /\ ((~ c) ==> (P y))))
Lp_ineqs_proofs.y9_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))))) ((y9_fan (V , E)) d))))))))))
CONTINUOUS_ON_ID
|- (!s. ((continuous_on (\x. x)) s))
transp
|- (!A. ((transp A) = (lambdai. (lambdaj. ((A $ j) $ i)))))
ARC_SIMPLE_PATH_SUBPATH
|- (!g. (!u. (!v. (((simple_path g) /\ ((u IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ ((v IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ (~ ((g u) = (g v)))))) ==> (arc (((subpath u) v) g))))))
CONTINUOUS_INJECTIVE_IMAGE_SEGMENT_1
|- (!f. (!a. (!b. ((((continuous_on f) (closed_segment ((CONS (a , b)) NIL))) /\ (!x. (!y. (((x IN (closed_segment ((CONS (a , b)) NIL))) /\ ((y IN (closed_segment ((CONS (a , b)) NIL))) /\ ((f x) = (f y)))) ==> (x = y))))) ==> (((IMAGE f) (closed_segment ((CONS (a , b)) NIL))) = (closed_segment ((CONS ((f a) , (f b))) NIL)))))))
Hypermap.face_map_and_darts
|- (!H. ((FINITE (dart H)) /\ ((permutes (face_map H)) (dart H))))
bounded
|- (!s. ((bounded s) = (?a. (!x. ((x IN s) ==> ((real_le (vector_norm x)) a))))))
linear
|- (!f. ((linear f) = ((!x. (!y. ((f ((vector_add x) y)) = ((vector_add (f x)) (f y))))) /\ (!c. (!x. ((f ((% c) x)) = ((% c) (f x))))))))
RSTC
|- (!R. ((RSTC R) = (RC (TC (SC R)))))
ABSOLUTELY_REAL_INTEGRABLE_ADD
|- (!f. (!g. (!s. ((((absolutely_real_integrable_on f) s) /\ ((absolutely_real_integrable_on g) s)) ==> ((absolutely_real_integrable_on (\x. ((real_add (f x)) (g x)))) s)))))
EDGE_OF_TRANSLATION_EQ
|- (!a. (!f. (!s. (((edge_of ((IMAGE (\x. ((vector_add a) x))) f)) ((IMAGE (\x. ((vector_add a) x))) s)) = ((edge_of f) s)))))
ITERATE_SUPPORT
|- (!op. (!f. (!s. ((((iterate op) (((support op) f) s)) f) = (((iterate op) s) f)))))
MEASURABLE_ON_CONVOLUTION
|- (!bop. (!f. (!g. (!x. (((bilinear bop) /\ (((measurable_on f) UNIV) /\ ((measurable_on g) UNIV))) ==> ((measurable_on (\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))
VECTOR_ADD_COMPONENT
|- (!x. (!y. (!i. ((((vector_add x) y) $ i) = ((real_add (x $ i)) (y $ i))))))
real_bounded
|- (!s. ((real_bounded s) = (?B. (!x. ((x IN s) ==> ((real_le (real_abs x)) B))))))
NEGLIGIBLE_COUNTABLE
|- (!s. ((COUNTABLE s) ==> (negligible s)))
CSIN_CIRCLE
|- (!z. (((vector_add ((complex_pow (csin z)) (NUMERAL (BIT0 (BIT1 _0))))) ((complex_pow (ccos z)) (NUMERAL (BIT0 (BIT1 _0))))) = (Cx (real_of_num (NUMERAL (BIT1 _0))))))
SURJECTIVE_IFF_INJECTIVE_GEN
|- (!s. (!t. (!f. (((FINITE s) /\ ((FINITE t) /\ (((CARD s) = (CARD t)) /\ (((IMAGE f) s) SUBSET t)))) ==> ((!y. ((y IN t) ==> (?x. ((x IN s) /\ ((f x) = y))))) = (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))))))))
OPEN_TRANSLATION_EQ
|- (!a. (!s. ((open ((IMAGE (\x. ((vector_add a) x))) s)) = (open s))))
real_gt
|- (!y. (!x. (((real_gt x) y) = ((real_lt y) x))))
CONVEX_CONE_PCROSS
|- (!s. (!t. (((convex_cone s) /\ (convex_cone t)) ==> (convex_cone (s PCROSS t)))))
HAS_DERIVATIVE_INVERSE_STRONG
|- (!f. (!g. (!f'. (!g'. (!s. (!x. (((open s) /\ ((x IN s) /\ (((continuous_on f) s) /\ ((!x. ((x IN s) ==> ((g (f x)) = x))) /\ ((((has_derivative f) f') (at x)) /\ ((f' o g') = I)))))) ==> (((has_derivative g) g') (at (f x))))))))))
ALTERNATING_SUM_BOUND
|- (!a. (((!n. ((real_le (real_abs (a (SUC n)))) (real_abs (a n)))) /\ ((!n. ((EVEN n) ==> ((real_le (real_of_num (NUMERAL _0))) (a n)))) /\ (!n. ((ODD n) ==> ((real_le (a n)) (real_of_num (NUMERAL _0))))))) ==> (!m. (!n. ((real_le (real_abs ((sum (m .. n)) a))) (real_abs (a m)))))))
Hypermap_iso.bij_res_inv
|- (!f. (!s. (!t. ((((BIJ f) s) t) ==> (((BIJ ((res_inv f) s)) t) s)))))
Seq.map_rot
|- (!n0. (!f. (!s. (((map f) ((rot n0) s)) = ((rot n0) ((map f) s))))))
inverse
|- (!f. ((inverse f) = (\y. (@x. ((f x) = y)))))
AFFINE_HULL_FINITE
|- (!s. (((hull affine) s) = (GSPEC (\GEN%PVAR%1412. (?y. (((SETSPEC GEN%PVAR%1412) (?u. ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum s) (\v. ((% (u v)) v))) = y)))) y))))))
CEXP_EULER
|- (!z. ((cexp ((complex_mul ii) z)) = ((vector_add (ccos z)) ((complex_mul ii) (csin z)))))
Sphere.rhazim3
|- (rhazim3 = (node3_y rhazim))
Fan.POWER_1
|- (!f. (((POWER f) (NUMERAL (BIT1 _0))) = f))
COMPLEX_DERIVATIVE_LMUL
|- (!f. (!c. (!z. (((complex_differentiable f) (at z)) ==> (((complex_derivative (\w. ((complex_mul c) (f w)))) z) = ((complex_mul c) ((complex_derivative f) z)))))))
COVERING_SPACE_HOMEOMORPHISM
|- (!p. (!c. (!s. ((((covering_space (c , p)) s) /\ ((path_connected c) /\ (simply_connected s))) ==> (?q. ((homeomorphism (c , s)) (p , q)))))))
SUBSET_INTER
|- (!s. (!t. (!u. ((s SUBSET (t INTER u)) = ((s SUBSET t) /\ (s SUBSET u))))))
CFUNSPACE_IMP_BOUNDED2
|- (!top. (!m. (!f. (!g. (((f IN (mspace ((cfunspace top) m))) /\ (g IN (mspace ((cfunspace top) m)))) ==> (?b. (!x. ((x IN (topspace top)) ==> ((real_le ((mdist m) ((f x) , (g x)))) b)))))))))
LIPSCHITZ_REAL_POLYNOMIAL_FUNCTION
|- (!f. (!s. (((real_polynomial_function f) /\ (bounded s)) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (real_abs ((real_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y))))))))))))
PRESERVES_NORM_PRESERVES_DOT
|- (!f. (!x. (!y. (((linear f) /\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> (((dot (f x)) (f y)) = ((dot x) y))))))
Ssrnat.factS
|- (!n. ((FACT (SUC n)) = ((SUC n) * (FACT n))))
REAL_SUB_ADD
|- (!x. (!y. (((real_add ((real_sub x) y)) y) = x)))
EVENTUALLY_AT_NEGINFINITY
|- (!p. (((eventually p) at_neginfinity) = (?b. (!x. (((real_le x) b) ==> (p x))))))
CONNECTED_INTER_FRONTIER
|- (!s. (!t. (((connected s) /\ ((~ ((s INTER t) = EMPTY)) /\ (~ ((s DIFF t) = EMPTY)))) ==> (~ ((s INTER (frontier t)) = EMPTY)))))
PROPER_LOCALLY_INJECTIVE_OPEN_IMP_COVERING_MAP
|- (!p. (!c. (!s. ((((continuous_on p) c) /\ ((((IMAGE p) c) = s) /\ ((!k. (((k SUBSET s) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%2101. (?x. (((SETSPEC GEN%PVAR%2101) ((x IN c) /\ ((p x) IN k))) x))))))) /\ ((!u. (((open_in ((subtopology euclidean) c)) u) ==> ((open_in ((subtopology euclidean) s)) ((IMAGE p) u)))) /\ (!x. ((x IN c) ==> (?t. ((x IN t) /\ (((open_in ((subtopology euclidean) c)) t) /\ (!y. (!z. (((y IN t) /\ ((z IN t) /\ ((p y) = (p z)))) ==> (y = z))))))))))))) ==> ((covering_space (c , p)) s)))))
REAL_CONVEX_ON_SUBSET
|- (!f. (!s. (!t. ((((real_convex_on f) t) /\ (s SUBSET t)) ==> ((real_convex_on f) s)))))
Fan.FAN
|- (!x. (!V. (!E. ((FAN (x , (V , E))) = (((UNIONS E) SUBSET V) /\ ((graph E) /\ ((fan1 (x , (V , E))) /\ ((fan2 (x , (V , E))) /\ ((fan6 (x , (V , E))) /\ (fan7 (x , (V , E))))))))))))
Appendix.is_scs_v39
|- (!s. ((is_scs_v39 s) = (((real_lt (scs_d_v39 s)) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 _0)))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) /\ (((NUMERAL (BIT1 (BIT1 _0))) <= (scs_k_v39 s)) /\ (((scs_k_v39 s) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) /\ (((periodic (scs_lo_v39 s)) (scs_k_v39 s)) /\ (((periodic (scs_hi_v39 s)) (scs_k_v39 s)) /\ (((periodic (scs_str_v39 s)) (scs_k_v39 s)) /\ (((periodic (scs_str_v39 s)) (scs_k_v39 s)) /\ (((periodic2 (scs_a_v39 s)) (scs_k_v39 s)) /\ (((periodic2 (scs_am_v39 s)) (scs_k_v39 s)) /\ (((periodic2 (scs_bm_v39 s)) (scs_k_v39 s)) /\ (((periodic2 (scs_b_v39 s)) (scs_k_v39 s)) /\ (((periodic2 (scs_J_v39 s)) (scs_k_v39 s)) /\ ((!i. (!j. (((((scs_a_v39 s) i) j) = (((scs_a_v39 s) j) i)) /\ (((((scs_am_v39 s) i) j) = (((scs_am_v39 s) j) i)) /\ (((((scs_bm_v39 s) i) j) = (((scs_bm_v39 s) j) i)) /\ (((((scs_b_v39 s) i) j) = (((scs_b_v39 s) j) i)) /\ ((((scs_J_v39 s) i) j) = (((scs_J_v39 s) j) i)))))))) /\ ((!i. (!j. (((real_le (((scs_a_v39 s) i) j)) (((scs_am_v39 s) i) j)) /\ (((real_le (((scs_am_v39 s) i) j)) (((scs_bm_v39 s) i) j)) /\ ((real_le (((scs_bm_v39 s) i) j)) (((scs_b_v39 s) i) j)))))) /\ ((!i. ((((scs_a_v39 s) i) i) = (real_of_num (NUMERAL _0)))) /\ ((!i. (!j. (((i < (scs_k_v39 s)) /\ ((j < (scs_k_v39 s)) /\ (~ (i = j)))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (((scs_a_v39 s) i) j))))) /\ ((!i. (((scs_k_v39 s) = (NUMERAL (BIT1 (BIT1 _0)))) ==> ((real_lt (((scs_b_v39 s) i) (SUC i))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))) /\ ((!i. (((NUMERAL (BIT1 (BIT1 _0))) < (scs_k_v39 s)) ==> ((real_le (((scs_b_v39 s) i) (SUC i))) cstab))) /\ ((!i. (!j. ((((scs_J_v39 s) i) j) ==> (((j MOD (scs_k_v39 s)) = ((SUC i) MOD (scs_k_v39 s))) \/ ((i MOD (scs_k_v39 s)) = ((SUC j) MOD (scs_k_v39 s))))))) /\ ((!i. (!j. ((((scs_J_v39 s) i) j) ==> (((((scs_a_v39 s) i) j) = sqrt8) /\ ((((scs_b_v39 s) i) j) = cstab))))) /\ (((CARD (GSPEC (\GEN%PVAR%7668. (?i. (((SETSPEC GEN%PVAR%7668) ((i < (scs_k_v39 s)) /\ (((real_lt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (((scs_b_v39 s) i) (SUC i))) \/ ((real_lt (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (((scs_a_v39 s) i) (SUC i)))))) i))))) + (scs_k_v39 s)) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))))))))))
REAL_POW_EQ_ABS
|- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ (((real_pow x) n) = ((real_pow y) n))) ==> ((real_abs x) = (real_abs y))))))
CONNECTED_SUBSET_SEGMENT
|- (!s. (!a. (!b. (((connected s) /\ ((s SUBSET (closed_segment ((CONS (a , b)) NIL))) /\ ((a IN s) /\ (b IN s)))) ==> (s = (closed_segment ((CONS (a , b)) NIL)))))))
Tame_defs.contravening
|- (!V. ((contravening V) = ((packing V) /\ ((V SUBSET ball_annulus) /\ (((real_gt (scriptL V)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) /\ ((!W. (((packing W) /\ (W SUBSET ball_annulus)) ==> ((real_le (scriptL W)) (scriptL V)))) /\ ((((CARD V) = (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 _0)))))) \/ (((CARD V) = (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 _0)))))) \/ ((CARD V) = (NUMERAL (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))) /\ ((!v. ((v IN V) ==> ((surrounded_node (V , (ESTD V))) v))) /\ (!v. ((v IN V) ==> (((surrounded_node (V , (ECTC V))) v) \/ ((vector_norm v) = (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))))))))
Calc_derivative.ratform
|- (!p. (!r. (!a. (!b. (((((ratform p) r) a) b) = (p ==> ((~ (b = (real_of_num (NUMERAL _0)))) /\ (r = ((real_div a) b)))))))))
EVENTUALLY_MONO
|- (!net. (!p. (!q. (((!x. ((p x) ==> (q x))) /\ ((eventually p) net)) ==> ((eventually q) net)))))
IN_CLOSURE_DELETE
|- (!s. (!x. ((x IN (closure (s DELETE x))) = ((limit_point_of x) s))))
Trigonometry.UTNKIAC
|- (!x. (!y. ((((real_lt (real_neg ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) x) /\ (((real_lt x) y) /\ ((real_lt y) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) ==> ((real_lt (tan x)) (tan y)))))
LEBESGUE_DENSITY_THEOREM_LIFT_CBALL
|- (!s. ((lebesgue_measurable s) ==> (?k. ((negligible k) /\ (!x. ((~ (x IN k)) ==> (((--> (\e. (lift ((real_div (measure (s INTER (cball (x , (drop e)))))) (measure (cball (x , (drop e)))))))) (((COND (x IN s)) (vec (NUMERAL (BIT1 _0)))) (vec (NUMERAL _0)))) ((within (at (vec (NUMERAL _0)))) (GSPEC (\GEN%PVAR%3726. (?t. (((SETSPEC GEN%PVAR%3726) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t))))))))))))
Conforming.SUM_SOL_IN_FACE_SET_EQ_4PI
|- (!x. (!V. (!E. (((FAN (x , (V , E))) /\ (conforming_fan (x , (V , E)))) ==> (((sum (face_set (hypermap1_of_fanx (x , (V , E))))) (\f. ((sol x) ((((dartset_leads_into_fan x) V) E) f)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi))))))
VECTOR_SUB_REFL
|- (!x. (((vector_sub x) x) = (vec (NUMERAL _0))))
IMAGE_LIFT_DROP
|- ((!s. (((IMAGE (lift o drop)) s) = s)) /\ (!s. (((IMAGE (drop o lift)) s) = s)))
IN_RELATIVE_INTERIOR
|- (!x. (!s. ((x IN (relative_interior s)) = ((x IN s) /\ (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (((ball (x , e)) INTER ((hull affine) s)) SUBSET s)))))))
REAL_CONTINUOUS_CONST
|- (!net. (!c. ((real_continuous (\x. c)) net)))
COMPONENTS_EQ_SING
|- (!s. (((components s) = (s INSERT EMPTY)) = ((connected s) /\ (~ (s = EMPTY)))))
MULT_ASSOC
|- (!m. (!n. (!p. ((m * (n * p)) = ((m * n) * p)))))
RANK_CMUL
|- (!A. (!c. ((rank ((%% c) A)) = (((COND (c = (real_of_num (NUMERAL _0)))) (NUMERAL _0)) (rank A)))))
Planarity.dartset_leads_into_is_topological_component_yfan
|- (!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)))))))) ==> (((((dartset_leads_into_fan x) V) E) ds) IN (topological_component_yfan (x , (V , E)))))))))
Dih2k_hypermap.INDEX_VECMAT
|- (((i < (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= i') /\ (i' <= (dimindex UNIV)))) ==> (((NUMERAL (BIT1 _0)) <= ((i * (dimindex UNIV)) + i')) /\ (((i * (dimindex UNIV)) + i') <= (dimindex UNIV))))
LIMPT_TRANSLATION_EQ
|- (!a. (!s. (!x. (((limit_point_of ((vector_add a) x)) ((IMAGE (\y. ((vector_add a) y))) s)) = ((limit_point_of x) s)))))
minimal
|- (!P. ((minimal P) = (@n. ((P n) /\ (!m. ((m < n) ==> (~ (P m))))))))
HAS_REAL_MEASURE_REAL_NEGLIGIBLE_UNIONS_IMAGE
|- (!f. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> (real_measurable (f x)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> (real_negligible ((f x) INTER (f y)))))))) ==> ((has_real_measure (UNIONS ((IMAGE f) s))) ((sum s) (\x. (real_measure (f x))))))))
LINEAR_INDEPENDENT_EXTEND
|- (!f. (!b. ((independent b) ==> (?g. ((linear g) /\ (!x. ((x IN b) ==> ((g x) = (f x)))))))))
IN_UNIONS
|- (!s. (!x. ((x IN (UNIONS s)) = (?t. ((t IN s) /\ (x IN t))))))
OPEN_CONVEX_HULL
|- (!s. ((open s) ==> (open ((hull convex) s))))
MAT_CMUL
|- (!a. ((mat a) = ((%% (real_of_num a)) (mat (NUMERAL (BIT1 _0))))))
REAL_ABS_SIGN2
|- (!x. (!y. (((real_lt (real_abs ((real_sub x) y))) (real_neg y)) ==> ((real_lt x) (real_of_num (NUMERAL _0))))))
CONVEX_CONE_CONVEX_CONE_HULL
|- (!s. (convex_cone ((hull convex_cone) s)))
LIM_WITHINREAL_WITHIN
|- ((((--> f) a) ((within (atreal x)) s)) = (((--> (f o drop)) a) ((within (at (lift x))) ((IMAGE lift) s))))
COMPLEX_NEG_SUB
|- (!x. (!y. ((vector_neg ((vector_sub x) y)) = ((vector_sub y) x))))
CONTINUOUS_WITHIN_SQRT_COMPOSE
|- (!f. (!s. (!a. ((((continuous (\x. (lift (f x)))) ((within (at a)) s)) /\ (((real_lt (real_of_num (NUMERAL _0))) (f a)) \/ (!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))))) ==> ((continuous (\x. (lift (sqrt (f x))))) ((within (at a)) s))))))
COMPACT_EQ_BOUNDED_CLOSED
|- (!s. ((compact s) = ((bounded s) /\ (closed s))))
Collect_geom.COPLANAR_3
|- (!a. (!b. (!c. (((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) ==> (coplanar_alt (a INSERT (b INSERT (c INSERT EMPTY))))))))
SUBSET_IMAGE
|- (!f. (!s. (!t. ((s SUBSET ((IMAGE f) t)) = (?u. ((u SUBSET t) /\ (s = ((IMAGE f) u))))))))
Functional_equation.functional_taum_x
|- (taum_x = ((add6 rhazim_x) ((add6 rhazim2_x) ((sub6 rhazim3_x) (constant6 ((real_mul ((real_add (real_of_num (NUMERAL (BIT1 _0)))) const1)) pi))))))
Collect_geom.RADV_FORMULAR
|- (!va. (!vb. (!vc. ((~ (collinear (va INSERT (vb INSERT (vc INSERT EMPTY))))) ==> ((radV (va INSERT (vb INSERT (vc INSERT EMPTY)))) = (((eta_y ((dist3 vb) vc)) ((dist3 va) vc)) ((dist3 va) vb)))))))
Pack1.fcc_compatible
|- (!S. (!f. (((fcc_compatible f) S) = (!v. ((v IN S) ==> ((real_le (sqrt (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_add (measure ((voronoi_open S) v))) (f v))))))))
REAL_OPEN_SET_RATIONAL
|- (!s. (((real_open s) /\ (~ (s = EMPTY))) ==> (?x. ((rational x) /\ (x IN s)))))
FROM_INTER_NUMSEG_MAX
|- (!m. (!n. (!p. (((from p) INTER (m .. n)) = (((MAX p) m) .. n)))))
Lp_ineqs_proofs.dart4_y_invf_eq
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((y1_fan ((inverse (f_fan_pair_ext (V , E))) d)) = ((y3_fan (V , E)) d)) /\ (((y2_fan ((inverse (f_fan_pair_ext (V , E))) d)) = (y1_fan d)) /\ ((((y3_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = ((y7_fan (V , E)) d)) /\ ((((y5_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = ((y8_fan (V , E)) d)) /\ (((y6_fan ((inverse (f_fan_pair_ext (V , E))) d)) = ((y5_fan (V , E)) d)) /\ ((((y7_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = (y2_fan d)) /\ ((((y8_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = ((y9_fan (V , E)) d)) /\ (((y9_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = (y6_fan d)))))))))))))))))
FINITE_INDUCT_STRONG
|- (!P. (((P EMPTY) /\ (!x. (!s. (((P s) /\ ((~ (x IN s)) /\ (FINITE s))) ==> (P (x INSERT s)))))) ==> (!s. ((FINITE s) ==> (P s)))))
Lp_details.quad_4240815464_a
|- (quad_nonlinear_v10 ==> (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!y7. (!y8. (!y9. ((ineq (((((((((dart_std4 y1) y2) y3) y4) y5) y6) y7) y8) y9)) (((real_lt (((((((((tauq y1) y2) y3) y4) y5) y6) y7) y8) y9)) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) \/ (((real_lt ((real_add y2) y3)) y4) \/ (((real_gt ((real_add (((((((((tauq y1) y2) y3) y4) y5) y6) y7) y8) y9)) ((real_sub ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) ((((((dih_y y1) y2) y3) y4) y5) y6))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))) ((DECIMAL (NUMERAL _0)) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) \/ ((real_lt ((((((delta_y y1) y2) y3) y4) y5) y6)) (real_of_num (NUMERAL _0)))))))))))))))))
LTE_CASES
|- (!m. (!n. ((m < n) \/ (n <= m))))
POSITIVE_SEMIDEFINITE_SIMILAR
|- (!A. (!B. ((positive_semidefinite A) ==> (positive_semidefinite ((matrix_mul (transp B)) ((matrix_mul A) B))))))
REAL_LE_NEGR
|- (!x. (((real_le x) (real_neg x)) = ((real_le x) (real_of_num (NUMERAL _0)))))
POLYTOPE_PCROSS
|- (!s. (!t. (((polytope s) /\ (polytope t)) ==> (polytope (s PCROSS t)))))
INTEGRABLE_SUBINTERVAL
|- (!f. (!a. (!b. (!c. (!d. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((closed_interval ((CONS (c , d)) NIL)) SUBSET (closed_interval ((CONS (a , b)) NIL)))) ==> ((integrable_on f) (closed_interval ((CONS (c , d)) NIL)))))))))
Ssrbool.negb_or
|- (!a. (!b. ((~ (a \/ b)) = ((~ a) /\ (~ b)))))
Miqmcsn.K_SCS_OPP_3T7
|- ((scs_k_v39 (scs_opp_v39 scs_3T7)) = (NUMERAL (BIT1 (BIT1 _0))))
DROP_EQ
|- (!x. (!y. (((drop x) = (drop y)) = (x = y))))
STONE_WEIERSTRASS_VECTOR_POLYNOMIAL_FUNCTION_SUBSPACE
|- (!f. (!s. (!e. (!t. (((compact s) /\ (((continuous_on f) s) /\ (((real_lt (real_of_num (NUMERAL _0))) e) /\ ((subspace t) /\ (((IMAGE f) s) SUBSET t))))) ==> (?g. ((vector_polynomial_function g) /\ ((((IMAGE g) s) SUBSET t) /\ (!x. ((x IN s) ==> ((real_lt (vector_norm ((vector_sub (f x)) (g x)))) e)))))))))))
pathfinish
|- (!g. ((pathfinish g) = (g (vec (NUMERAL (BIT1 _0))))))
MAXIMAL_AFFINE_INDEPENDENT_SUBSET
|- (!s. (!b. (((b SUBSET s) /\ ((~ (affine_dependent b)) /\ (!b'. (((b SUBSET b') /\ ((b' SUBSET s) /\ (~ (affine_dependent b')))) ==> (b' = b))))) ==> (s SUBSET ((hull affine) b)))))
Yrtafyh.YRTAFYH
|- (!s. (!i. (!j. (((is_scs_v39 s) /\ ((scs_basic_v39 s) /\ (((NUMERAL (BIT1 (BIT1 _0))) < (scs_k_v39 s)) /\ ((((scs_diag (scs_k_v39 s)) i) j) /\ ((real_le (((scs_a_v39 s) i) j)) cstab))))) ==> ((is_scs_v39 (((scs_stab_diag_v39 s) i) j)) /\ (scs_basic_v39 (((scs_stab_diag_v39 s) i) j)))))))
REAL_CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN
|- (!f. (!s. (((real_continuous_on f) s) = (!x. ((x IN s) ==> ((real_continuous f) ((within (atreal x)) s)))))))
REAL_CONVEX_BOUND_LT
|- (!x. (!y. (!a. (!u. (!v. ((((real_lt x) a) /\ (((real_lt y) a) /\ (((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_lt ((real_add ((real_mul u) x)) ((real_mul v) y))) a)))))))