hcju/holps huggingface dataset:
Local_lemmas.AFF_GE_MONO_TRANS
|- ((S SUBSET X) ==> (((aff_ge (X DIFF S)) (Y UNION S)) SUBSET ((aff_ge X) Y)))
MEASURE_CLOSURE
|- (!s. (((bounded s) /\ (negligible (frontier s))) ==> ((measure (closure s)) = (measure s))))
Hypermap.lemma_two_series_eq
|- (!p. (!q. (!n. ((!i. ((i <= n) ==> ((p i) = (q i)))) ==> ((GSPEC (\GEN%PVAR%5391. (?i. (((SETSPEC GEN%PVAR%5391) (i <= n)) (p i))))) = (GSPEC (\GEN%PVAR%5392. (?i. (((SETSPEC GEN%PVAR%5392) (i <= n)) (q i))))))))))
RE_COMPLEX_DIV_GT_0
|- (!a. (!b. (((real_lt (real_of_num (NUMERAL _0))) (Re ((complex_div a) b))) = ((real_lt (real_of_num (NUMERAL _0))) (Re ((complex_mul a) (cnj b)))))))
real_sums
|- (!s. (!f. (!l. ((((real_sums f) l) s) = (((---> (\n. ((sum (s INTER ((NUMERAL _0) .. n))) f))) l) sequentially)))))
DOT_RSUB
|- (!x. (!y. (!z. (((dot x) ((vector_sub y) z)) = ((real_sub ((dot x) y)) ((dot x) z))))))
OPEN_IN_CONTAINS_BALL
|- (!s. (!t. (((open_in ((subtopology euclidean) t)) s) = ((s SUBSET t) /\ (!x. ((x IN s) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (((ball (x , e)) INTER t) SUBSET s)))))))))
REAL_LOG_CONVEX_MUL
|- (!f. (!g. (!s. ((((real_log_convex_on f) s) /\ ((real_log_convex_on g) s)) ==> ((real_log_convex_on (\x. ((real_mul (f x)) (g x)))) s)))))
Nonlin_def.ups234_x
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((ups234_x x1) x2) x3) x4) x5) x6) = (((ups_x x2) x3) x4))))))))
RSTC_REFL
|- (!R. (!x. (((RSTC R) x) x)))
ANGLE_EQ_0_DIST
|- (!A. (!B. (!C. (((angle (A , (B , C))) = (real_of_num (NUMERAL _0))) = ((~ (A = B)) /\ ((~ (C = B)) /\ (((distance (A , B)) = ((real_add (distance (A , C))) (distance (C , B)))) \/ ((distance (B , C)) = ((real_add (distance (A , C))) (distance (A , B)))))))))))
LAMBDA_BETA
|- (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (((lambda g) $ i) = (g i))))
CONTINUOUS_AT
|- (!f. (!x. (((continuous f) (at x)) = (((--> f) (f x)) (at x)))))
CBALL_TRANSLATION
|- (!a. (!x. (!r. ((cball (((vector_add a) x) , r)) = ((IMAGE (\y. ((vector_add a) y))) (cball (x , r)))))))
ARITH_ADD
|- ((!m. (!n. (((NUMERAL m) + (NUMERAL n)) = (NUMERAL (m + n))))) /\ (((_0 + _0) = _0) /\ ((!n. ((_0 + (BIT0 n)) = (BIT0 n))) /\ ((!n. ((_0 + (BIT1 n)) = (BIT1 n))) /\ ((!n. (((BIT0 n) + _0) = (BIT0 n))) /\ ((!n. (((BIT1 n) + _0) = (BIT1 n))) /\ ((!m. (!n. (((BIT0 m) + (BIT0 n)) = (BIT0 (m + n))))) /\ ((!m. (!n. (((BIT0 m) + (BIT1 n)) = (BIT1 (m + n))))) /\ ((!m. (!n. (((BIT1 m) + (BIT0 n)) = (BIT1 (m + n))))) /\ (!m. (!n. (((BIT1 m) + (BIT1 n)) = (BIT0 (SUC (m + n)))))))))))))))
Tame_defs.heights
|- (!g. ((heights g) = (SND (SND (SND g)))))
ANR_TRIANGULATION
|- (!tr. ((triangulation tr) ==> (ANR (UNIONS tr))))
COMPACT_IMP_BOUNDED
|- (!s. ((compact s) ==> (bounded s)))
CONIC_PCROSS
|- (!s. (!t. (((conic s) /\ (conic t)) ==> (conic (s PCROSS t)))))
Ssrnat.muln_subl
|- (!m. (!n. (!p. (((m - n) * p) = ((m * p) - (n * p))))))
SUMS_EQ
|- (!f. (!g. (!k. (((!x. ((x IN k) ==> ((f x) = (g x)))) /\ (((sums f) l) k)) ==> (((sums g) l) k)))))
LIM_AT_INFINITY
|- (!f. (!l. ((((--> f) l) at_infinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_ge (vector_norm x)) b) ==> ((real_lt (distance ((f x) , l))) e)))))))))
Fan_defs.dart1_of_fan
|- (!V. (!E. ((dart1_of_fan (V , E)) = (GSPEC (\GEN%PVAR%5440. (?v. (?w. (((SETSPEC GEN%PVAR%5440) ((v INSERT (w INSERT EMPTY)) IN E)) (v , w)))))))))
CONVEX_EXPLICIT
|- (!s. ((convex s) = (!t. (!u. (((FINITE t) /\ ((t SUBSET s) /\ ((!x. ((x IN t) ==> ((real_le (real_of_num (NUMERAL _0))) (u x)))) /\ (((sum t) u) = (real_of_num (NUMERAL (BIT1 _0))))))) ==> (((vsum t) (\x. ((% (u x)) x))) IN s))))))
real_lt
|- (!y. (!x. (((real_lt x) y) = (~ ((real_le y) x)))))
SIMPLY_CONNECTED_IMP_PATH_CONNECTED
|- (!s. ((simply_connected s) ==> (path_connected s)))
Add_triangle.fully_surrounded_card_set_of_edge1
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (((fully_surrounded (V , E)) /\ (v IN V)) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))))))
Appendix.BBs_v39
|- (!s. (!vv. (((BBs_v39 s) vv) = ((LET (\V. (LET_END ((LET (\E. (LET_END ((LET (\FF. (LET_END ((V SUBSET ball_annulus) /\ (((periodic vv) (scs_k_v39 s)) /\ ((!i. (!j. (((real_le (((scs_a_v39 s) i) j)) (distance ((vv i) , (vv j)))) /\ ((real_le (distance ((vv i) , (vv j)))) (((scs_b_v39 s) i) j))))) /\ (((scs_k_v39 s) <= (NUMERAL (BIT1 (BIT1 _0)))) \/ (convex_local_fan (V , (E , FF)))))))))) ((IMAGE (\i. ((vv i) , (vv (SUC i))))) UNIV))))) ((IMAGE (\i. ((vv i) INSERT ((vv (SUC i)) INSERT EMPTY)))) UNIV))))) ((IMAGE vv) UNIV)))))
TWO
|- ((NUMERAL (BIT0 (BIT1 _0))) = (SUC (NUMERAL (BIT1 _0))))
INT_NOT_EQ
|- (!x. (!y. ((~ (x = y)) = (((int_lt x) y) \/ ((int_lt y) x)))))
Nonlin_def.compose6
|- (!f. (!p1. (!p2. (!p3. (!p4. (!p5. (!p6. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((((((((compose6 f) p1) p2) p3) p4) p5) p6) x1) x2) x3) x4) x5) x6) = ((((((f ((((((p1 x1) x2) x3) x4) x5) x6)) ((((((p2 x1) x2) x3) x4) x5) x6)) ((((((p3 x1) x2) x3) x4) x5) x6)) ((((((p4 x1) x2) x3) x4) x5) x6)) ((((((p5 x1) x2) x3) x4) x5) x6)) ((((((p6 x1) x2) x3) x4) x5) x6))))))))))))))))
LE
|- ((!m. ((m <= (NUMERAL _0)) = (m = (NUMERAL _0)))) /\ (!m. (!n. ((m <= (SUC n)) = ((m = (SUC n)) \/ (m <= n))))))
LOWDIM_SUBSET_HYPERPLANE
|- (!s. (((dim s) < (dimindex UNIV)) ==> (?a. ((~ (a = (vec (NUMERAL _0)))) /\ ((span s) SUBSET (GSPEC (\GEN%PVAR%183. (?x. (((SETSPEC GEN%PVAR%183) (((dot a) x) = (real_of_num (NUMERAL _0)))) x)))))))))
Tame_lemmas.fully_surrounded_tau_eq_tauq
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((V SUBSET ball_annulus) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ==> ((LET (\w'. (LET_END ((LET (\u. (LET_END ((LET (\y1. (LET_END ((LET (\y2. (LET_END ((LET (\y3. (LET_END ((LET (\y4. (LET_END ((LET (\y5. (LET_END ((LET (\y6. (LET_END ((LET (\y7. (LET_END ((LET (\y8. (LET_END ((LET (\y9. (LET_END ((tauVEF (V , (E , ((face (hypermap_of_fan (V , E))) (v , w))))) = (((((((((tauq y1) y2) y3) y4) y5) y6) y7) y8) y9))))) (distance (w , u)))))) (distance (w' , u)))))) (vector_norm u))))) (distance (v , w)))))) (distance (v , w')))))) (distance (w , w')))))) (vector_norm w'))))) (vector_norm w))))) (vector_norm v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))))))))))
Seq.perm_eq_size
|- (!s1. (!s2. (((perm_eq s1) s2) ==> ((sizel s1) = (sizel s2)))))
NOT_INSERT_EMPTY
|- (!x. (!s. (~ ((x INSERT s) = EMPTY))))
REAL_NEGLIGIBLE_SUBSET
|- (!s. (!t. (((real_negligible s) /\ (t SUBSET s)) ==> (real_negligible t))))
REALLIM_NULL_POW_EQ
|- (!net. (!f. (!n. ((~ (n = (NUMERAL _0))) ==> ((((---> (\x. ((real_pow (f x)) n))) (real_of_num (NUMERAL _0))) net) = (((---> f) (real_of_num (NUMERAL _0))) net))))))
ABSOLUTELY_INTEGRABLE_ADD
|- (!f. (!g. (!s. ((((absolutely_integrable_on f) s) /\ ((absolutely_integrable_on g) s)) ==> ((absolutely_integrable_on (\x. ((vector_add (f x)) (g x)))) s)))))
CARD_EQ_COUNTABLE_SUBSETS_SUBREAL
|- (!s. (((INFINITE s) /\ (s <=_c UNIV)) ==> ((GSPEC (\GEN%PVAR%72. (?t. (((SETSPEC GEN%PVAR%72) ((t SUBSET s) /\ (COUNTABLE t))) t)))) =_c UNIV)))
EDELSTEIN_FIX
|- (!f. (!s. (((compact s) /\ ((~ (s = EMPTY)) /\ ((((IMAGE f) s) SUBSET s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> ((real_lt (distance ((f x) , (f y)))) (distance (x , y))))))))) ==> (?!x. ((x IN s) /\ ((f x) = x))))))
IN_BALL_0
|- (!x. (!e. ((x IN (ball ((vec (NUMERAL _0)) , e))) = ((real_lt (vector_norm x)) e))))
uniformly_continuous_on
|- (!s. (!f. (((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 (distance (x' , x))) d))) ==> ((real_lt (distance ((f x') , (f x)))) e)))))))))))
Tame_lemmas.dart_leads_into_eq1
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((((((dart_leads_into (vec (NUMERAL _0))) V) E) v) w) = ((((dartset_leads_into_fan (vec (NUMERAL _0))) V) E) ((IMAGE (ext_dart (V , E))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))))
Lp_details.quad_3862621143_revised
|- (quad_nonlinear_v10 ==> (!y0. (!y1. (!y2. (!y3. (!y5. (!y6. (!y7. (!y8. (!y9. ((ineq ((CONS (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) , (y0 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y5 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y7 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y8 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y9 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL)))))))))) (((real_lt ((real_add ((((((taum y1) y2) y7) y9) y0) y6)) ((((((taum y1) y3) y7) y8) y0) y5))) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) \/ (((real_gt ((real_add ((((((taum y1) y2) y7) y9) y0) y6)) ((real_add ((real_sub ((((((taum y1) y3) y7) y8) y0) y5)) ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((real_add ((((((dih_y y1) y2) y7) y9) y0) y6)) ((((((dih_y y1) y3) y7) y8) y0) y5))))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (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) y7) y9) y0) y6)) (real_of_num (NUMERAL _0))) \/ ((real_lt ((((((delta_y y1) y3) y7) y8) y0) y5)) (real_of_num (NUMERAL _0)))))))))))))))))
REAL_CONVERGENT_IMP_BOUNDED
|- (!s. (!l. ((((---> s) l) sequentially) ==> (real_bounded ((IMAGE s) UNIV)))))
INSIDE_SUBSET_CONVEX
|- (!s. (!c. (((convex c) /\ (s SUBSET c)) ==> ((inside s) SUBSET c))))
CONTINUOUS_SUB
|- (!f. (!g. (!net. ((((continuous f) net) /\ ((continuous g) net)) ==> ((continuous (\x. ((vector_sub (f x)) (g x)))) net)))))
JOINPATHS_TRANSLATION
|- (!a. (!g1. (!g2. (((++ ((\x. ((vector_add a) x)) o g1)) ((\x. ((vector_add a) x)) o g2)) = ((\x. ((vector_add a) x)) o ((++ g1) g2))))))
subpath
|- (!g. (!v. (!u. ((((subpath u) v) g) = (\x. (g ((vector_add u) ((% (drop ((vector_sub v) u))) x))))))))
FORALL_2
|- (!P. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (NUMERAL (BIT0 (BIT1 _0))))) ==> (P i))) = ((P (NUMERAL (BIT1 _0))) /\ (P (NUMERAL (BIT0 (BIT1 _0)))))))
Hexagons.PROP_EQU_IS_SCS
|- ((is_scs_v39 s) ==> (is_scs_v39 ((scs_prop_equ_v39 s) i)))
REAL_DIFFERENTIABLE_AT_LOG
|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_differentiable log) (atreal x))))
VECTOR_SUB_EQ
|- (!x. (!y. ((((vector_sub x) y) = (vec (NUMERAL _0))) = (x = y))))
Ssrnat.ltn_exp2l
|- (!m. (!n1. (!n2. (((NUMERAL (BIT1 _0)) < m) ==> (((m EXP n1) < (m EXP n2)) = (n1 < n2))))))
REAL_POW_LT
|- (!x. (!n. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_pow x) n)))))
Seq.size_cons
|- (!h. (!t. ((sizel ((CONS h) t)) = (SUC (sizel t)))))
Lp_ineqs_def.rhazim3_fan
|- (!V. (!E. (!x. (((rhazim3_fan (V , E)) x) = ((rhazim_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) x))))))
VECTOR_SUB_RZERO
|- (((vector_sub x) (vec (NUMERAL _0))) = x)
Tame_defs.GRAPH
|- (!fs. (!n. (!f. (!h. (((((Graph fs) n) f) h) = (fs , (n , (f , h))))))))
DROP_NEG
|- (!x. ((drop (vector_neg x)) = (real_neg (drop x))))
COMPLEX_INV_INV
|- (!x. ((complex_inv (complex_inv x)) = x))
CONTINUOUS_ON_EQ
|- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((continuous_on f) s)) ==> ((continuous_on g) s)))))
RIGHT_OR_FORALL_THM
|- (!P. (!Q. ((P \/ (!x. (Q x))) = (!x. (P \/ (Q x))))))
int_gcd
|- (!a. (!b. (((int_le (int_of_num (NUMERAL _0))) (int_gcd (a , b))) /\ (((int_divides (int_gcd (a , b))) a) /\ (((int_divides (int_gcd (a , b))) b) /\ (?x. (?y. ((int_gcd (a , b)) = ((int_add ((int_mul a) x)) ((int_mul b) y))))))))))
PCROSS_EMPTY
|- ((!s. ((s PCROSS EMPTY) = EMPTY)) /\ (!t. ((EMPTY PCROSS t) = EMPTY)))
Add_triangle.dart1_switch
|- (!V. (!E. (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) = ((w , v) IN (dart1_of_fan (V , E))))))))
REAL_LE_SQUARE
|- (!x. ((real_le (real_of_num (NUMERAL _0))) ((real_mul x) x)))
MULT_0
|- (!m. ((m * (NUMERAL _0)) = (NUMERAL _0)))
FRONTIER_UNIONS_SUBSET_CLOSURE
|- (!f. ((frontier (UNIONS f)) SUBSET (closure (UNIONS (GSPEC (\GEN%PVAR%1825. (?t. (((SETSPEC GEN%PVAR%1825) (t IN f)) (frontier t)))))))))
SETDIST_TRANSLATION
|- (!a. (!s. (!t. ((setdist (((IMAGE (\x. ((vector_add a) x))) s) , ((IMAGE (\x. ((vector_add a) x))) t))) = (setdist (s , t))))))
SUM_OVER_PERMUTATIONS_INSERT
|- (!f. (!a. (!s. (((FINITE s) /\ (~ (a IN s))) ==> (((sum (GSPEC (\GEN%PVAR%295. (?p. (((SETSPEC GEN%PVAR%295) ((permutes p) (a INSERT s))) p))))) f) = ((sum (a INSERT s)) (\b. ((sum (GSPEC (\GEN%PVAR%296. (?p. (((SETSPEC GEN%PVAR%296) ((permutes p) s)) p))))) (\q. (f ((swap (a , b)) o q)))))))))))
CONIC_INTERS
|- ((!s. ((s IN f) ==> (conic s))) ==> (conic (INTERS f)))
COMPLEX_VEC_0
|- ((vec (NUMERAL _0)) = (Cx (real_of_num (NUMERAL _0))))
REAL_MEASURABLE_ON_UNIV
|- (((real_measurable_on (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) UNIV) = ((real_measurable_on f) s))
OPEN_SET_COCOUNTABLE_COORDINATES
|- (!P. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (COUNTABLE (UNIV DIFF (GSPEC (\GEN%PVAR%1866. (?x. (((SETSPEC GEN%PVAR%1866) ((P i) x)) x)))))))) ==> (!s. (((open s) /\ (~ (s = EMPTY))) ==> (?x. ((x IN s) /\ (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((P i) (x $ i))))))))))
Appendix.periodic2
|- (!k. (!f. (((periodic2 f) k) = (!i. (!j. ((((f (i + k)) j) = ((f i) j)) /\ (((f i) (j + k)) = ((f i) j))))))))
NSUM_SUPPORT
|- (!f. (!s. (((nsum (((support +) f) s)) f) = ((nsum s) f))))
CONVEX_CONE_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((convex_cone ((IMAGE f) s)) = (convex_cone s)))))
IS_REALINTERVAL_CONNECTED
|- (!s. ((is_realinterval s) = (connected ((IMAGE lift) s))))
Functional_equation.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))))))))))
LOCALLY_COMPACT_OPEN_INTER_CLOSURE
|- (!s. (((locally compact) s) ==> (?t. ((open t) /\ (s = (t INTER (closure s)))))))
RE_ADD
|- (!x. (!y. ((Re ((vector_add x) y)) = ((real_add (Re x)) (Re y)))))
Conforming.pr23
|- (pr23 = (GABS (\f. (!x. (!y. (!z. (!t. ((GEQ (f (x , (y , (z , t))))) (y , z)))))))))
locally
|- (!s. (!P. (((locally P) s) = (!w. (!x. ((((open_in ((subtopology euclidean) s)) w) /\ (x IN w)) ==> (?u. (?v. (((open_in ((subtopology euclidean) s)) u) /\ ((P v) /\ ((x IN u) /\ ((u SUBSET v) /\ (v SUBSET w)))))))))))))
CLOSURE_SUBSET_EQ
|- (!s. (((closure s) SUBSET s) = (closed s)))
List_hypermap.components_hypermap_of_list
|- (!L. ((good_list L) ==> (((dart (hypermap_of_list L)) = (darts_of_list L)) /\ (((edge_map (hypermap_of_list L)) = (e_list_ext L)) /\ (((node_map (hypermap_of_list L)) = (n_list_ext L)) /\ ((face_map (hypermap_of_list L)) = (f_list_ext L)))))))
Sphere.x1_delta_x
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((x1_delta_x x1) x2) x3) x4) x5) x6) = ((real_mul x1) ((((((delta_x x1) x2) x3) x4) x5) x6)))))))))
COVERING_SPACE_FIBRE_NO_LIMPT
|- (!p. (!c. (!s. (!a. (!b. ((((covering_space (c , p)) s) /\ (a IN c)) ==> (~ ((limit_point_of a) (GSPEC (\GEN%PVAR%2072. (?x. (((SETSPEC GEN%PVAR%2072) ((x IN c) /\ ((p x) = b))) x))))))))))))
fl
|- (!l. (!x. (((fl l) x) = (?y. ((l (x , y)) \/ (l (y , x)))))))
SUMMABLE_EQ
|- (!f. (!g. (!k. (((!x. ((x IN k) ==> ((f x) = (g x)))) /\ ((summable k) f)) ==> ((summable k) g)))))
MATRIX_INV_COVARIANCE_LMUL
|- (!A. (((matrix_mul (transp A)) (matrix_inv ((matrix_mul A) (transp A)))) = (matrix_inv A)))
Hypermap_iso.bij_alt
|- (!f. (!s. (!t. ((((BIJ f) s) t) = ((!x. ((x IN s) ==> ((f x) IN t))) /\ ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) /\ (!y. ((y IN t) ==> (?x. ((x IN s) /\ ((f x) = y)))))))))))
IS_INTERVAL_CONNECTED_1
|- (!s. ((is_interval s) = (connected s)))
REAL_SERIES_ADD
|- (!x. (!x0. (!y. (!y0. (!s. (((((real_sums x) x0) s) /\ (((real_sums y) y0) s)) ==> (((real_sums (\n. ((real_add (x n)) (y n)))) ((real_add x0) y0)) s)))))))
Hypermap_iso.darts_of_fan_disj
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((DISJOINT (dart1_of_fan (V , E))) (GSPEC (\GEN%PVAR%7629. (?v. (((SETSPEC GEN%PVAR%7629) ((v IN V) /\ ((((set_of_edge v) V) E) = EMPTY))) (v , v)))))))))
FINITE_DELETE
|- (!s. (!x. ((FINITE (s DELETE x)) = (FINITE s))))
HOMEOMORPHISM_INTO_1D
|- (!f. (!s. (!t. (((path_connected s) /\ (((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)))))))
BOUNDED_CONVEX_HULL
|- (!s. ((bounded s) ==> (bounded ((hull convex) s))))
ABSOLUTELY_REAL_INTEGRABLE_CONST
|- (!a. (!b. (!c. ((absolutely_real_integrable_on (\x. c)) (closed_real_interval ((CONS (a , b)) NIL))))))
Tame_general.IN_ESTD
|- (!V. (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN (ESTD V)) = ((v IN V) /\ ((w IN V) /\ ((~ (v = w)) /\ ((real_le (distance (v , w))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))
AFF_DIM_SEGMENT
|- ((!a. (!b. ((aff_dim (closed_segment ((CONS (a , b)) NIL))) = (((COND (a = b)) (int_of_num (NUMERAL _0))) (int_of_num (NUMERAL (BIT1 _0))))))) /\ (!a. (!b. ((aff_dim (open_segment (a , b))) = (((COND (a = b)) (int_neg (int_of_num (NUMERAL (BIT1 _0))))) (int_of_num (NUMERAL (BIT1 _0))))))))
Seq2.list_sum_nil
|- (!f. (((list_sum NIL) f) = (real_of_num (NUMERAL _0))))
Conforming.SIGMA_FAN_OF_FANADD_AT_POINT4
|- (!x. (!V. (!E. (!v. (!u. (!w. (!w1. (((FAN (x , (V , E))) /\ ((FAN (x , (V , E1))) /\ ((fan80 (x , (V , E))) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\ ((~ (u = w1)) /\ (((v INSERT (w1 INSERT EMPTY)) IN E) /\ (((((((sigma_fan x) V) E) u) w) = v) /\ ((!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) w1) = (((((sigma_fan x) V) E) v) w1))))))))))
ANR_PCROSS
|- (!s. (!t. (((ANR s) /\ (ANR t)) ==> (ANR (s PCROSS t)))))
Sphere.y_of_x
|- (!fx. (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((((((((y_of_x fx) y1) y2) y3) y4) y5) y6) = ((((((fx ((real_mul y1) y1)) ((real_mul y2) y2)) ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5)) ((real_mul y6) y6))))))))))
Tame_general.sol0_over_pi_EQ_const1
|- (((real_div sol0) pi) = const1)
LIM_AT_ZERO
|- (!f. (!l. (!a. ((((--> f) l) (at a)) = (((--> (\x. (f ((vector_add a) x)))) l) (at (vec (NUMERAL _0))))))))
NOT_ON_PATH_BALL
|- (!g. (!z. (((path g) /\ (~ (z IN (path_image g)))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (((ball (z , e)) INTER (path_image g)) = EMPTY))))))
Miqmcsn.K_SCS_3M1_prime
|- ((scs_k_v39 scs_3M1_prime) = (NUMERAL (BIT1 (BIT1 _0))))
Real_ext.REAL_MK_POS_EXP
|- (!x. ((real_lt (real_of_num (NUMERAL _0))) (exp x)))
REAL_INTERVAL_INTERVAL
|- (((closed_real_interval ((CONS (a , b)) NIL)) = ((IMAGE drop) (closed_interval ((CONS ((lift a) , (lift b))) NIL)))) /\ ((open_real_interval (a , b)) = ((IMAGE drop) (open_interval ((lift a) , (lift b))))))
REAL_EQ_IMP_LE
|- (!x. (!y. ((x = y) ==> ((real_le x) y))))
SUBSET_TRANS
|- (!s. (!t. (!u. (((s SUBSET t) /\ (t SUBSET u)) ==> (s SUBSET u)))))
Seq.mem_subseq
|- (!s1. (!s2. (((subseq s1) s2) ==> (!x. (((MEM x) s1) ==> ((MEM x) s2))))))
Functional_equation.ldih_all_atn
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\ (((real_lt (real_of_num (NUMERAL _0))) x1) /\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\ ((real_lt (real_of_num (NUMERAL _0))) x6))))))) ==> ((((((((ldih_x x1) x2) x3) x4) x5) x6) = ((((((ldihatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih2_x x1) x2) x3) x4) x5) x6) = ((((((ldih2atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih3_x x1) x2) x3) x4) x5) x6) = ((((((ldih3atn_x x1) x2) x3) x4) x5) x6)) /\ (((((((ldih6_x x1) x2) x3) x4) x5) x6) = ((((((ldih6atn_x x1) x2) x3) x4) x5) x6))))))))))))
SERIES_COMPARISON
|- (!f. (!g. (!s. (((?l. (((sums (lift o g)) l) s)) /\ (?N. (!n. (((n >= N) /\ (n IN s)) ==> ((real_le (vector_norm (f n))) (g n)))))) ==> (?l. (((sums f) l) s))))))
REAL_MUL_LNEG
|- (!x. (!y. (((real_mul (real_neg x)) y) = (real_neg ((real_mul x) y)))))
BOUNDED_PARTIAL_SUMS
|- (!f. (!k. ((bounded (GSPEC (\GEN%PVAR%1123. (?n. (((SETSPEC GEN%PVAR%1123) (n IN UNIV)) ((vsum (k .. n)) f)))))) ==> (bounded (GSPEC (\GEN%PVAR%1124. (?m. (?n. (((SETSPEC GEN%PVAR%1124) ((m IN UNIV) /\ (n IN UNIV))) ((vsum (m .. n)) f))))))))))
HOMOTOPIC_PATHS_REFL
|- (!s. (!p. ((((homotopic_paths s) p) p) = ((path p) /\ ((path_image p) SUBSET s)))))
TC_TRANS
|- (!R. (!x. (!y. (!z. (((((TC R) x) y) /\ (((TC R) y) z)) ==> (((TC R) x) z))))))
SUBSET_HULL
|- (!P. (!s. (!t. ((P t) ==> ((((hull P) s) SUBSET t) = (s SUBSET t))))))
REAL_POLYNOMIAL_FUNCTION_EXPLICIT_UNIV
|- (!f. ((real_polynomial_function f) = (?a. ((FINITE (GSPEC (\GEN%PVAR%5014. (?k. (((SETSPEC GEN%PVAR%5014) (~ ((a k) = (real_of_num (NUMERAL _0))))) k))))) /\ (f = (\x. ((sum UNIV) (\k. ((real_mul (a k)) ((product ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. ((real_pow (x $ i)) (k $ i)))))))))))))
POLYTOPE_IMP_CLOSED
|- (!s. ((polytope s) ==> (closed s)))
LEBESGUE_MEASURABLE_UNIV
|- (lebesgue_measurable UNIV)
has_vector_derivative
|- (!f. (!f'. (!net. ((((has_vector_derivative f) f') net) = (((has_derivative f) (\x. ((% (drop x)) f'))) net)))))
AFF_DIM_EMPTY
|- ((aff_dim EMPTY) = (int_neg (int_of_num (NUMERAL (BIT1 _0)))))
DET_2
|- (!A. ((det A) = ((real_sub ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))))))
CONTINUOUS_GE_ON_CLOSURE
|- (!f. (!s. (!a. ((((continuous_on (lift o f)) (closure s)) /\ (!x. ((x IN s) ==> ((real_le a) (f x))))) ==> (!x. ((x IN (closure s)) ==> ((real_le a) (f x))))))))
VITALI_COVERING_LEMMA_CBALLS_BALLS
|- (!a. (!r. (!k. (!B. ((!i. ((i IN k) ==> (((real_lt (real_of_num (NUMERAL _0))) (r i)) /\ ((real_le (r i)) B)))) ==> (?c. ((COUNTABLE c) /\ ((c SUBSET k) /\ (((pairwise (\i. (\j. ((DISJOINT (cball ((a i) , (r i)))) (cball ((a j) , (r j))))))) c) /\ (!i. ((i IN k) ==> (?j. ((j IN c) /\ ((~ ((DISJOINT (cball ((a i) , (r i)))) (cball ((a j) , (r j))))) /\ ((cball ((a i) , (r i))) SUBSET (ball ((a j) , ((real_mul (real_of_num (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) (r j)))))))))))))))))))
ARG_EQ_0
|- (!z. (((Arg z) = (real_of_num (NUMERAL _0))) = ((real z) /\ ((real_le (real_of_num (NUMERAL _0))) (Re z)))))