hcju/holps huggingface dataset:
AR_IMP_ABSOLUTE_RETRACT
|- (!s. (!u. (!s'. (((AR s) /\ (((homeomorphic s) s') /\ ((closed_in ((subtopology euclidean) u)) s'))) ==> ((retract_of s') u)))))
HAS_BOUNDED_VARIATION_DARBOUX_STRICT
|- (!f. (!a. (!b. (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) = (?g. (?h. ((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_lt (drop x)) (drop y)))) ==> ((real_lt (drop (g x))) (drop (g y)))))) /\ ((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_lt (drop x)) (drop y)))) ==> ((real_lt (drop (h x))) (drop (h y)))))) /\ (!x. ((f x) = ((vector_sub (g x)) (h x))))))))))))
PATH_CONNECTED_PUNCTURED_CONVEX
|- (!s. (!a. (((convex s) /\ (~ ((aff_dim s) = (int_of_num (NUMERAL (BIT1 _0)))))) ==> (path_connected (s DELETE a)))))
CONTINUOUS_AT_CSQRT
|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> ((continuous csqrt) (at z))))
Bump.MCELL2
|- (!V. (!ul. (((mcell2 V) ul) = (((mcell (NUMERAL (BIT0 (BIT1 _0)))) V) ul))))
INT_OF_NUM_MUL
|- (!m. (!n. (((int_mul (int_of_num m)) (int_of_num n)) = (int_of_num (m * n)))))
REAL_LEBESGUE_MEASURABLE_COMPL
|- (!s. ((real_lebesgue_measurable (UNIV DIFF s)) = (real_lebesgue_measurable s)))
PATH_CONNECTED_PCROSS
|- (!s. (!t. (((path_connected s) /\ (path_connected t)) ==> (path_connected (s PCROSS t)))))
DIFFERENTIAL_COMPONENT_ZERO_AT_MAXMIN
|- (!f. (!f'. (!x. (!s. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ ((x IN s) /\ ((open s) /\ ((((has_derivative f) f') (at x)) /\ ((!w. ((w IN s) ==> ((real_le ((f w) $ k)) ((f x) $ k)))) \/ (!w. ((w IN s) ==> ((real_le ((f x) $ k)) ((f w) $ k)))))))))) ==> (!h. (((f' h) $ k) = (real_of_num (NUMERAL _0))))))))))
CONTINUOUS_ON_LIFT_COMPONENT
|- (!i. (!s. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((continuous_on (\x. (lift (x $ i)))) s))))
AR_STRONG_DEFORMATION_RETRACT_OF_AR
|- (!s. (!t. (((AR s) /\ ((AR t) /\ ((closed_in ((subtopology euclidean) s)) t))) ==> (?r. (((((homotopic_with (\h. (!x. ((x IN t) ==> ((h x) = x))))) (s , s)) (\x. x)) r) /\ ((retraction (s , t)) r))))))
MEASURABLE_CBALL
|- (!a. (!r. (measurable (cball (a , r)))))
Lp_ineqs_proofs.split_list_list
|- (!L. (!ds. (((split_list_list L) ds) = (((foldr (\d. (\L. ((split_list_hyp L) d)))) L) ds))))
CARD_EQ_TRANS
|- (!s. (!t. (!u. (((s =_c t) /\ (t =_c u)) ==> (s =_c u)))))
Nonlinear_lemma.lmfun_lfun
|- (!y. (((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) ==> ((lmfun ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (lfun ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))
ITERATE_DELTA
|- (!op. ((monoidal op) ==> (!f. (!a. (!s. ((((iterate op) s) (\x. (((COND (x = a)) (f x)) (neutral op)))) = (((COND (a IN s)) (f a)) (neutral op))))))))
HAS_SIZE_FUNSPACE_UNIV
|- (!m. (!n. (((UNIV HAS_SIZE m) /\ (UNIV HAS_SIZE n)) ==> (UNIV HAS_SIZE (n EXP m)))))
Tame_list.POWER_TO_ITER
|- (!n. (((POWER f) n) = ((ITER n) f)))
Ssrnat.addnn
|- (!n. ((n + n) = (double n)))
INT_LE_SUB_LADD
|- (!x. (!y. (!z. (((int_le x) ((int_sub y) z)) = ((int_le ((int_add x) z)) y)))))
INT_OF_NUM_EQ
|- (!m. (!n. (((int_of_num m) = (int_of_num n)) = (m = n))))
Tame_defs2.core
|- (!L. (!r. (((core L) r) = ((MAP (FST o HD)) ((parts L) r)))))
MEASURABLE_IMP_LEBESGUE_MEASURABLE
|- (!s. ((measurable s) ==> (lebesgue_measurable s)))
Tame_defs.scriptL
|- (!V. ((scriptL V) = ((sum V) (\v. (lmfun ((real_div (vector_norm v)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))
POLYTOPE_SEGMENT
|- (!a. (!b. (polytope (closed_segment ((CONS (a , b)) NIL)))))
COMPLEX_MUL_SYM
|- (!x. (!y. (((complex_mul x) y) = ((complex_mul y) x))))
Tame_defs.tame_1
|- (!H. ((tame_1 H) = ((plain_hypermap H) /\ (planar_hypermap H))))
CONVEX_HALFSPACE_LT
|- (!a. (!b. (convex (GSPEC (\GEN%PVAR%1439. (?x. (((SETSPEC GEN%PVAR%1439) ((real_lt ((dot a) x)) b)) x)))))))
BOUNDED_SUBSET
|- (!s. (!t. (((bounded t) /\ (s SUBSET t)) ==> (bounded s))))
LET_END_DEF
|- (!t. ((LET_END t) = t))
LIM_NULL_COMPARISON
|- (!net. (!f. (!g. ((((eventually (\x. ((real_le (vector_norm (f x))) (g x)))) net) /\ (((--> (\x. (lift (g x)))) (vec (NUMERAL _0))) net)) ==> (((--> f) (vec (NUMERAL _0))) net)))))
ABSOLUTELY_REAL_INTEGRABLE_0
|- (!s. ((absolutely_real_integrable_on (\x. (real_of_num (NUMERAL _0)))) s))
ANR_TRANSLATION
|- (!a. (!s. ((ANR ((IMAGE (\x. ((vector_add a) x))) s)) = (ANR s))))
GABS_DEF
|- (!P. ((GABS P) = (@ P)))
extreme_point_of
|- (!s. (!x. (((extreme_point_of x) s) = ((x IN s) /\ (!a. (!b. (((a IN s) /\ (b IN s)) ==> (~ (x IN (open_segment (a , b)))))))))))
RC_SC
|- (!R. ((RC (SC R)) = (SC (RC R))))
HAS_INTEGRAL_COMPLEX_LMUL
|- (!f. (!y. (!i. (!c. ((((has_integral f) y) i) ==> (((has_integral (\x. ((complex_mul c) (f x)))) ((complex_mul c) y)) i))))))
INTEGRABLE_DECREASING_PRODUCT
|- (!f. (!g. (!a. (!b. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le (drop x)) (drop y)))) ==> ((real_le (g y)) (g x)))))) ==> ((integrable_on (\x. ((% (g x)) (f x)))) (closed_interval ((CONS (a , b)) NIL))))))))
List_hypermap_iso.fan_list_iso
|- (!v. (!f. (!V. (!E. (!w. (((fan_list_iso (f , (V , E))) (v , w)) = (((fan_list_nodes_iso (f , (V , E))) v) , ((fan_list_nodes_iso (f , (V , E))) w))))))))
HAS_BOUNDED_SETVARIATION_WORKS_ON_INTERVAL
|- (!f. (!a. (!b. (((has_bounded_setvariation_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((!d. (((division_of d) (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le ((sum d) (\k. (vector_norm (f k))))) ((set_variation (closed_interval ((CONS (a , b)) NIL))) f)))) /\ (!B. ((!d. (((division_of d) (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le ((sum d) (\k. (vector_norm (f k))))) B))) ==> ((real_le ((set_variation (closed_interval ((CONS (a , b)) NIL))) f)) B))))))))
LIM_INV_N_OFFSET
|- (!z. (((--> (\n. (complex_inv ((vector_add (Cx (real_of_num n))) z)))) (Cx (real_of_num (NUMERAL _0)))) sequentially))
LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF
|- (!s. (!t. (((lebesgue_measurable s) /\ (negligible ((s DIFF t) UNION (t DIFF s)))) ==> (lebesgue_measurable t))))
COMPLEX_SUB_RZERO
|- (!x. (((vector_sub x) (Cx (real_of_num (NUMERAL _0)))) = x))
OPEN_IN_OPEN
|- (!s. (!u. (((open_in ((subtopology euclidean) u)) s) = (?t. ((open t) /\ (s = (u INTER t)))))))
Calc_derivative.invert_den_eq
|- (!a. (!b. ((((real_div a) b) = (real_of_num (NUMERAL _0))) = (((real_mul a) b) = (real_of_num (NUMERAL _0))))))
CONVEX_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((convex ((IMAGE f) s)) = (convex s)))))
HAS_SIZE_0
|- (!s. ((s HAS_SIZE (NUMERAL _0)) = (s = EMPTY)))
Fan.sigma_fan_in_set_of_edge
|- (!x. (!V. (!E. (!v. (!u. (((FAN (x , (V , E))) /\ (u IN (((set_of_edge v) V) E))) ==> ((((((sigma_fan x) V) E) v) u) IN (((set_of_edge v) V) E))))))))
Localization.has_orders
|- (!k. (!f. (((has_orders f) k) = ((!i. ((((NUMERAL _0) < i) /\ (i < k)) ==> (~ (((ITER i) f) = I)))) /\ (((ITER k) f) = I)))))
SETDIST_SUBSETS_EQ
|- (!s. (!t. (!s'. (!t'. (((s' SUBSET s) /\ ((t' SUBSET t) /\ (!x. (!y. (((x IN s) /\ (y IN t)) ==> (?x'. (?y'. ((x' IN s') /\ ((y' IN t') /\ ((real_le (distance (x' , y'))) (distance (x , y)))))))))))) ==> ((setdist (s' , t')) = (setdist (s , t))))))))
SUBSET_INTERIOR_EQ
|- (!s. ((s SUBSET (interior s)) = (open s)))
REAL_LE_INV2
|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_le x) y)) ==> ((real_le (real_inv y)) (real_inv x)))))
Add_triangle.azim_dart1_eq_azim
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((((v , w) IN (dart1_of_fan (V , E))) /\ ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0)))) ==> (((azim_dart (V , E)) (v , w)) = ((((azim (vec (NUMERAL _0))) v) w) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))
FSTCART_SUB
|- (!x. (!y. ((fstcart ((vector_sub x) y)) = ((vector_sub (fstcart x)) (fstcart y)))))
OPEN_SEGMENT_LINEAR_IMAGE
|- (!f. (!a. (!b. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((open_segment ((f a) , (f b))) = ((IMAGE f) (open_segment (a , b))))))))
AFF_DIM_OPEN
|- (!s. (((open s) /\ (~ (s = EMPTY))) ==> ((aff_dim s) = (int_of_num (dimindex UNIV)))))
SUM_DEGENERATE
|- (!f. (!s. ((~ (FINITE (GSPEC (\GEN%PVAR%194. (?x. (((SETSPEC GEN%PVAR%194) ((x IN s) /\ (~ ((f x) = (real_of_num (NUMERAL _0)))))) x)))))) ==> (((sum s) f) = (real_of_num (NUMERAL _0))))))
UNIONS_IMAGE
|- (!f. (!s. ((UNIONS ((IMAGE f) s)) = (GSPEC (\GEN%PVAR%29. (?y. (((SETSPEC GEN%PVAR%29) (?x. ((x IN s) /\ (y IN (f x))))) y)))))))
PAIRWISE_DISJOINT_COMPONENTS
|- (!u. ((pairwise DISJOINT) (components u)))
Localization.v_prime
|- (!V. (!FF. (((v_prime V) FF) = (GSPEC (\GEN%PVAR%7534. (?v. (((SETSPEC GEN%PVAR%7534) ((v IN V) /\ (?w. ((v , w) IN FF)))) v)))))))
matrix_neg
|- (!A. ((matrix_neg A) = (lambdai. (lambdaj. (real_neg ((A $ i) $ j))))))
DIMINDEX_1
|- ((dimindex UNIV) = (NUMERAL (BIT1 _0)))
DISJOINT_RELATIVE_INTERIOR_CONVEX_HULL
|- (!s. (((~ (affine_dependent s)) /\ (~ (?a. (s = (a INSERT EMPTY))))) ==> (((relative_interior ((hull convex) s)) INTER s) = EMPTY)))
Collect_geom2.NOT_COL_EQ_UPS_X_POS
|- (!v1. (!v2. (!v3. ((~ (collinear (v1 INSERT (v2 INSERT (v3 INSERT EMPTY))))) = ((real_lt (real_of_num (NUMERAL _0))) (((ups_x ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0))))))))))
COUNTABLE_DELETE
|- (!x. (!s. ((COUNTABLE (s DELETE x)) = (COUNTABLE s))))
PATH_SUBPATH
|- (!u. (!v. (!g. (((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))))) ==> (path (((subpath u) v) g))))))
RC_CLOSED
|- (!R. (((RC R) = R) = (!x. ((R x) x))))
CROSS_RADD
|- (!x. (!y. (!z. (((cross x) ((vector_add y) z)) = ((vector_add ((cross x) y)) ((cross x) z))))))
Tame_defs2.transform_assumption
|- (!L. ((transform_assumption_v2 L) = (!N. (!r. (!x. (((good_list L) /\ ((good_list_nodes L) /\ ((((marked_list L) N) r) x))) ==> ((((marked_list L) (((((ntrans L) N) r) x) (NUMERAL (BIT1 _0)))) (((((rtrans L) N) r) x) (NUMERAL (BIT1 _0)))) x)))))))
EXTREME_POINTS_OF_CONVEX_HULL_AFFINE_INDEPENDENT
|- (!s. ((~ (affine_dependent s)) ==> ((GSPEC (\GEN%PVAR%2247. (?x. (((SETSPEC GEN%PVAR%2247) ((extreme_point_of x) ((hull convex) s))) x)))) = s)))
INSEG_SUBSET_FL
|- (!l. (!m. (((inseg m) l) ==> (!x. (((fl m) x) ==> ((fl l) x))))))
CARD_ADD_C
|- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> ((CARD ((+_c s) t)) = ((CARD s) + (CARD t))))))
LINEAR_1
|- (!f. ((linear f) = (?c. (f = (\x. ((% c) x))))))
REAL_DIFFSQ
|- (!x. (!y. (((real_mul ((real_add x) y)) ((real_sub x) y)) = ((real_sub ((real_mul x) x)) ((real_mul y) y)))))
CSIN_ADD
|- (!w. (!z. ((csin ((vector_add w) z)) = ((vector_add ((complex_mul (csin w)) (ccos z))) ((complex_mul (ccos w)) (csin z))))))
CARD_PSUBSET
|- (!a. (!b. (((a PSUBSET b) /\ (FINITE b)) ==> ((CARD a) < (CARD b)))))
OPEN_IN_TRANS
|- (!s. (!t. (!u. ((((open_in ((subtopology euclidean) t)) s) /\ ((open_in ((subtopology euclidean) u)) t)) ==> ((open_in ((subtopology euclidean) u)) s)))))
Add_triangle.dart_switch
|- (!V. (!E. (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) = ((w , v) IN (dart_of_fan (V , E))))))))
DET_DEPENDENT_ROWS
|- (!A. ((dependent (rows A)) ==> ((det A) = (real_of_num (NUMERAL _0)))))
ADD_SUB
|- (!m. (!n. (((m + n) - n) = m)))
poset
|- (!l. ((poset l) = ((!x. (((fl l) x) ==> (l (x , x)))) /\ ((!x. (!y. (!z. (((l (x , y)) /\ (l (y , z))) ==> (l (x , z)))))) /\ (!x. (!y. (((l (x , y)) /\ (l (y , x))) ==> (x = y))))))))
HAS_REAL_MEASURE_EMPTY
|- ((has_real_measure EMPTY) (real_of_num (NUMERAL _0)))
MEM_LIST_OF_SET
|- (!s. ((FINITE s) ==> (!x. (((MEM x) (list_of_set s)) = (x IN s)))))
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))))))
Fan.d_fan
|- (!x. (!V. (!E. ((d_fan (x , (V , E))) = ((d1_fan (x , (V , E))) UNION (d20_fan (x , (V , E))))))))
Nonlin_def.ldih_x_div_sqrtdelta_posbranch
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((ldih_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((real_mul (lfun ((real_div (sqrt x1)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6)))))))))
REAL_ADD_SYM
|- (!x. (!y. (((real_add x) y) = ((real_add y) x))))
Add_triangle.fan_map_ext
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> ((((((f1_fan (vec (NUMERAL _0))) V) E) ((ext_dart (V , E)) d)) = ((ext_dart (V , E)) ((f_fan_pair (V , E)) d))) /\ ((((((n_fan (vec (NUMERAL _0))) V) E) ((ext_dart (V , E)) d)) = ((ext_dart (V , E)) ((n_fan_pair (V , E)) d))) /\ (((((e_fan (vec (NUMERAL _0))) V) E) ((ext_dart (V , E)) d)) = ((ext_dart (V , E)) ((e_fan_pair (V , E)) d))))))))))
RPOW_RPOW
|- (!x. (!y. (!z. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((rpow ((rpow x) y)) z) = ((rpow x) ((real_mul y) z)))))))
Seq2.list_sum_set_of_list
|- (!s. (!f. ((uniq s) ==> (((list_sum s) f) = ((sum (set_of_list s)) f)))))
AFFSIGN_SPECIAL_SCALE
|- (!sgn. (!s. (!t. (!a. (!v. (((FINITE s) /\ ((FINITE t) /\ ((~ ((vec (NUMERAL _0)) IN t)) /\ ((~ (v IN t)) /\ ((~ (((% a) v) IN t)) /\ ((!x. ((sgn x) ==> (sgn ((real_div x) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) /\ ((!x. (!y. (((sgn x) /\ (sgn y)) ==> (sgn ((real_add x) y))))) /\ ((real_lt (real_of_num (NUMERAL _0))) a)))))))) ==> ((((affsign sgn) ((vec (NUMERAL _0)) INSERT (((% a) v) INSERT s))) t) = (((affsign sgn) ((vec (NUMERAL _0)) INSERT (v INSERT s))) t))))))))
CLOSEST_POINT_IN_RELATIVE_FRONTIER
|- (!s. (!x. (((closed s) /\ ((~ (s = EMPTY)) /\ (x IN (((hull affine) s) DIFF (relative_interior s))))) ==> (((closest_point s) x) IN (relative_frontier s)))))
Betwn_corek_z_x.parts_rot
|- (((good_list L) /\ (((all (\x. ((MEM x) (list_of_darts L)))) r) /\ ((is_edge_nondegenerate (hypermap_of_list L)) /\ ((uniq r) /\ ((final_list L) r))))) ==> (((parts L) ((rot n) r)) = ((rot n) ((parts L) r))))
Seq.uniq_leq_size
|- (!s1. (!s2. ((uniq s1) ==> ((!x. (((MEM x) s1) ==> ((MEM x) s2))) ==> ((sizel s1) <= (sizel s2))))))
LIM_NULL_CMUL
|- (!net. (!f. (!c. ((((--> f) (vec (NUMERAL _0))) net) ==> (((--> (\x. ((% c) (f x)))) (vec (NUMERAL _0))) net)))))
INT_OF_NUM_GE
|- (!m. (!n. (((int_ge (int_of_num m)) (int_of_num n)) = (m >= n))))
REAL_CONTINUOUS_CONTINUOUS1
|- (((real_continuous f) net) = ((continuous (lift o f)) net))
REAL_CONTINUOUS_CONTINUOUS_ATREAL
|- (((real_continuous f) (atreal x)) = ((continuous (lift o (f o drop))) (at (lift x))))
SECOND_MEAN_VALUE_THEOREM_GEN_FULL
|- (!f. (!g. (!a. (!b. (!u. (!v. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (open_interval (a , b))) ==> (((real_le u) (g x)) /\ ((real_le (g x)) v)))) /\ (!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le (drop x)) (drop y)))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_interval ((CONS (a , b)) NIL))) /\ (((has_integral (\x. ((% (g x)) (f x)))) ((vector_add ((% u) ((integral (closed_interval ((CONS (a , c)) NIL))) f))) ((% v) ((integral (closed_interval ((CONS (c , b)) NIL))) f)))) (closed_interval ((CONS (a , b)) NIL))))))))))))
affine
|- (!s. ((affine s) = (!x. (!y. (!u. (!v. (((x IN s) /\ ((y IN s) /\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))) ==> (((vector_add ((% u) x)) ((% v) y)) IN s))))))))
COMPLEX_SUB_RDISTRIB
|- (!x. (!y. (!z. (((complex_mul ((vector_sub x) y)) z) = ((vector_sub ((complex_mul x) z)) ((complex_mul y) z))))))
LIMPT_INJECTIVE_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((limit_point_of (f x)) ((IMAGE f) s)) = ((limit_point_of x) s)))))
MOD_MOD_REFL
|- (!m. (!n. ((~ (n = (NUMERAL _0))) ==> (((m MOD n) MOD n) = (m MOD n)))))
REAL_AFFINITY_EQ
|- (!m. (!c. (!x. (!y. ((~ (m = (real_of_num (NUMERAL _0)))) ==> ((((real_add ((real_mul m) x)) c) = y) = (x = ((real_add ((real_mul (real_inv m)) y)) (real_neg ((real_div c) m))))))))))
NOT_CLAUSES
|- ((!t. ((~ (~ t)) = t)) /\ (((~ T) = F) /\ ((~ F) = T)))
CTAN_SUB
|- (!w. (!z. (((~ ((ccos w) = (Cx (real_of_num (NUMERAL _0))))) /\ ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) /\ (~ ((ccos ((vector_sub w) z)) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((ctan ((vector_sub w) z)) = ((complex_div ((vector_sub (ctan w)) (ctan z))) ((vector_add (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_mul (ctan w)) (ctan z))))))))
ENDS_NOT_IN_SEGMENT
|- (!a. (!b. ((~ (a IN (open_segment (a , b)))) /\ (~ (b IN (open_segment (a , b)))))))
TRIANGULATION_DISJOINT_RELATIVE_INTERIORS
|- (!t. (!c. (!d. (((triangulation t) /\ ((c IN t) /\ ((d IN t) /\ (~ (c = d))))) ==> (((relative_interior c) INTER (relative_interior d)) = EMPTY)))))
eq_c
|- (!t. (!s. ((s =_c t) = (?f. ((!x. ((x IN s) ==> ((f x) IN t))) /\ (!y. ((y IN t) ==> (?!x. ((x IN s) /\ ((f x) = y))))))))))
MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LE_DENSE
|- (!f. (!r. (((closure ((IMAGE lift) r)) = UNIV) ==> (((measurable_on f) UNIV) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ (a IN r))) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3594. (?x. (((SETSPEC GEN%PVAR%3594) ((real_le ((f x) $ k)) a)) x))))))))))))
Planarity.properties1_inside_fan
|- (!x. (!u. (!w. ((((DISJOINT (x INSERT EMPTY)) (u INSERT (w INSERT EMPTY))) /\ (((real_lt (real_of_num (NUMERAL _0))) a) /\ ((real_lt a) (real_of_num (NUMERAL (BIT1 _0)))))) ==> (((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) a)) u)) ((% a) w)) IN ((aff_ge (x INSERT EMPTY)) (u INSERT (w INSERT EMPTY))))))))
CONVEX_HULL_INSERT_SEGMENTS
|- (!s. (!a. (((hull convex) (a INSERT s)) = (((COND (s = EMPTY)) (a INSERT EMPTY)) (UNIONS (GSPEC (\GEN%PVAR%1485. (?x. (((SETSPEC GEN%PVAR%1485) (x IN ((hull convex) s))) (closed_segment ((CONS (a , x)) NIL)))))))))))
REALLIM_SUB
|- (!net. (!f. (!g. (!l. (!m. (((((---> f) l) net) /\ (((---> g) m) net)) ==> (((---> (\x. ((real_sub (f x)) (g x)))) ((real_sub l) m)) net)))))))
POLYHEDRON_POSITIVE_ORTHANT
|- (polyhedron (GSPEC (\GEN%PVAR%2275. (?x. (((SETSPEC GEN%PVAR%2275) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (x $ i))))) x)))))
NSUM_PERMUTE
|- (!f. (!p. (!s. (((permutes p) s) ==> (((nsum s) f) = ((nsum s) (f o p)))))))
DIFFERENTIABLE_TRANSFORM_WITHIN
|- (!f. (!g. (!x. (!s. (!d. ((((real_lt (real_of_num (NUMERAL _0))) d) /\ ((x IN s) /\ ((!x'. (((x' IN s) /\ ((real_lt (distance (x' , x))) d)) ==> ((f x') = (g x')))) /\ ((differentiable f) ((within (at x)) s))))) ==> ((differentiable g) ((within (at x)) s))))))))
real_negligible
|- (!s. ((real_negligible s) = (negligible ((IMAGE lift) s))))
REAL_LE_LDIV_EQ
|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_le ((real_div x) z)) y) = ((real_le x) ((real_mul y) z)))))))
LOCALLY_PATH_CONNECTED_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((locally path_connected) ((IMAGE f) s)) = ((locally path_connected) s)))))
HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT
|- (!f. (!s. (!x. ((((holomorphic_on f) s) /\ ((open s) /\ (x IN s))) ==> ((complex_differentiable f) (at x))))))
CLOSED_SUBSET
|- (!s. (!t. (((s SUBSET t) /\ (closed s)) ==> ((closed_in ((subtopology euclidean) t)) s))))
CLOSED_IN_IMP_SUBSET
|- (!top. (!s. (!t. (((closed_in ((subtopology top) s)) t) ==> (t SUBSET s)))))
Functional_equation.proj_y4
|- (!x1. (!x2. (!x3. (!x5. (!x6. (!x4. (((((((proj_y4 x1) x2) x3) x4) x5) x6) = (sqrt x4))))))))
INTER_SUBSET
|- ((!s. (!t. ((s INTER t) SUBSET s))) /\ (!s. (!t. ((t INTER s) SUBSET s))))
ROOT_POS_LT
|- (!n. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_lt (real_of_num (NUMERAL _0))) ((root n) x)))))
VSUM_RESTRICT_SET
|- (!P. (!s. (!f. (((vsum (GSPEC (\GEN%PVAR%111. (?x. (((SETSPEC GEN%PVAR%111) ((x IN s) /\ (P x))) x))))) f) = ((vsum s) (\x. (((COND (P x)) (f x)) (vec (NUMERAL _0)))))))))
REAL_POLY_CLAUSES
|- ((!x. (!y. (!z. (((real_add x) ((real_add y) z)) = ((real_add ((real_add x) y)) z))))) /\ ((!x. (!y. (((real_add x) y) = ((real_add y) x)))) /\ ((!x. (((real_add (real_of_num (NUMERAL _0))) x) = x)) /\ ((!x. (!y. (!z. (((real_mul x) ((real_mul y) z)) = ((real_mul ((real_mul x) y)) z))))) /\ ((!x. (!y. (((real_mul x) y) = ((real_mul y) x)))) /\ ((!x. (((real_mul (real_of_num (NUMERAL (BIT1 _0)))) x) = x)) /\ ((!x. (((real_mul (real_of_num (NUMERAL _0))) x) = (real_of_num (NUMERAL _0)))) /\ ((!x. (!y. (!z. (((real_mul x) ((real_add y) z)) = ((real_add ((real_mul x) y)) ((real_mul x) z)))))) /\ ((!x. (((real_pow x) (NUMERAL _0)) = (real_of_num (NUMERAL (BIT1 _0))))) /\ (!x. (!n. (((real_pow x) (SUC n)) = ((real_mul x) ((real_pow x) n))))))))))))))
CARD_NUMSEG_1
|- (!n. ((CARD ((NUMERAL (BIT1 _0)) .. n)) = n))
MDIST_REVERSE_TRIANGLE
|- (!m. (!x. (!y. (!z. (((x IN (mspace m)) /\ ((y IN (mspace m)) /\ (z IN (mspace m)))) ==> ((real_le (real_abs ((real_sub ((mdist m) (x , y))) ((mdist m) (y , z))))) ((mdist m) (x , z))))))))
MEASURABLE_ON_COMPOSE_CONTINUOUS_OPEN_INTERVAL
|- (!f. (!g. (!a. (!b. ((((measurable_on f) UNIV) /\ ((!x. ((f x) IN (open_interval (a , b)))) /\ ((continuous_on g) (open_interval (a , b))))) ==> ((measurable_on (g o f)) UNIV))))))
II_NZ
|- (~ (ii = (Cx (real_of_num (NUMERAL _0)))))
REAL_CONTINUOUS_AT_AZIM_SHARP
|- (!v. (!w. (!w1. (!w2. (((~ (collinear (v INSERT (w INSERT (w1 INSERT EMPTY))))) /\ (~ (w2 IN ((aff_ge (v INSERT (w INSERT EMPTY))) (w1 INSERT EMPTY))))) ==> ((real_continuous (((azim v) w) w1)) (at w2)))))))
Misc_defs_and_lemmas.min_num
|- (!X. ((min_num X) = (@m. ((m IN X) /\ (!n. ((n IN X) ==> (m <= n)))))))
real
|- (!z. ((real z) = ((Im z) = (real_of_num (NUMERAL _0)))))
CSQRT
|- (!z. (((complex_pow (csqrt z)) (NUMERAL (BIT0 (BIT1 _0)))) = z))
CONVEX_HAS_BOUNDED_VARIATION_EQ
|- (!f. (!s. ((((convex_on (drop o f)) s) /\ (is_interval s)) ==> (((has_bounded_variation_on f) s) = (bounded ((IMAGE f) s))))))
DIFFERENTIABLE_CHAIN_WITHIN
|- (!f. (!g. (!x. (!s. ((((differentiable f) ((within (at x)) s)) /\ ((differentiable g) ((within (at (f x))) ((IMAGE f) s)))) ==> ((differentiable (g o f)) ((within (at x)) s)))))))
LE_TRANS
|- (!m. (!n. (!p. (((m <= n) /\ (n <= p)) ==> (m <= p)))))
REAL_LE_TOTAL
|- (!x. (!y. (((real_le x) y) \/ ((real_le y) x))))
REAL_CLOSED_IN
|- (!s. ((real_closed s) = ((closed_in euclideanreal) s)))
Dih2k_hypermap.IMAGE_NN_OF_HYP_F_SY
|- (((FAN ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l))))) /\ (((NUMERAL (BIT1 _0)) < (dimindex UNIV)) /\ (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ ((j <= (dimindex UNIV)) /\ (((row i) (vecmats l)) = ((row j) (vecmats l))))))) ==> (i = j)))))) ==> (((IMAGE (nn_of_hyp ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l)))))) (F_SY (vecmats l))) = (GSPEC (\GEN%PVAR%7498. (?i. (((SETSPEC GEN%PVAR%7498) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (((row (SUC (i MOD (dimindex UNIV)))) (vecmats l)) , ((row i) (vecmats l)))))))))
FROM_INTER_NUMSEG
|- (!k. (!n. (((from k) INTER ((NUMERAL _0) .. n)) = (k .. n))))
REAL_INV_INV
|- (!x. ((real_inv (real_inv x)) = x))
COMPLEX_EQ_SUB_LADD
|- (!x. (!y. (!z. ((x = ((vector_sub y) z)) = (((vector_add x) z) = y)))))
POLYHEDRON_IMP_CONVEX
|- (!s. ((polyhedron s) ==> (convex s)))
Hypermap.lemma_loop_outside_node
|- (!H. (!NF. (!L. (!x. ((((is_normal H) NF) /\ (L IN NF)) ==> (~ ((dart_of_loop L) SUBSET ((node H) x))))))))
orthogonal_matrix
|- (!Q. ((orthogonal_matrix Q) = ((((matrix_mul (transp Q)) Q) = (mat (NUMERAL (BIT1 _0)))) /\ (((matrix_mul Q) (transp Q)) = (mat (NUMERAL (BIT1 _0)))))))
ITERATE_OP_GEN
|- (!op. ((monoidal op) ==> (!f. (!g. (!s. (((FINITE (((support op) f) s)) /\ (FINITE (((support op) g) s))) ==> ((((iterate op) s) (\x. ((op (f x)) (g x)))) = ((op (((iterate op) s) f)) (((iterate op) s) g)))))))))