hcju/holps huggingface dataset:
SHIFTPATH_TRANSLATION
|- (!a. (!t. (!g. (((shiftpath t) ((\x. ((vector_add a) x)) o g)) = ((\x. ((vector_add a) x)) o ((shiftpath t) g))))))
NORM_GE_SQUARE
|- (!x. (((real_ge (vector_norm x)) a) = (((real_le a) (real_of_num (NUMERAL _0))) \/ ((real_ge ((dot x) x)) ((real_pow a) (NUMERAL (BIT0 (BIT1 _0))))))))
INDEPENDENT_IMP_AFFINE_DEPENDENT_0
|- (!s. ((independent s) ==> (~ (affine_dependent ((vec (NUMERAL _0)) INSERT s)))))
COMPLEX_MUL_CCOS_CSIN
|- (!w. (!z. (((complex_mul (ccos w)) (csin z)) = ((complex_div ((vector_sub (csin ((vector_add w) z))) (csin ((vector_sub w) z)))) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))
RECTIFIABLE_PATH_IMP_PATH
|- (!g. ((rectifiable_path g) ==> (path g)))
REAL_SGN_EQ
|- ((!x. (((real_sgn x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0))))) /\ ((!x. (((real_sgn x) = (real_of_num (NUMERAL (BIT1 _0)))) = ((real_gt x) (real_of_num (NUMERAL _0))))) /\ (!x. (((real_sgn x) = (real_neg (real_of_num (NUMERAL (BIT1 _0))))) = ((real_lt x) (real_of_num (NUMERAL _0)))))))
ABSOLUTELY_SETCONTINUOUS_ON_DIVISION
|- (!f. (!a. (!b. (!d. ((((operative vector_add) f) /\ ((division_of d) (closed_interval ((CONS (a , b)) NIL)))) ==> ((!k. ((k IN d) ==> ((absolutely_setcontinuous_on f) k))) = ((absolutely_setcontinuous_on f) (closed_interval ((CONS (a , b)) NIL)))))))))
Seq.in_nil
|- (!x. (((MEM x) NIL) = F))
CBALL_MIN_INTER
|- (!x. (!d. (!e. ((cball (x , ((real_min d) e))) = ((cball (x , d)) INTER (cball (x , e)))))))
NEGLIGIBLE_TRANSLATION_EQ
|- (!a. (!s. ((negligible ((IMAGE (\x. ((vector_add a) x))) s)) = (negligible s))))
SQRT_WORKS
|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((real_le (real_of_num (NUMERAL _0))) (sqrt x)) /\ (((real_pow (sqrt x)) (NUMERAL (BIT0 (BIT1 _0)))) = x))))
EXISTS_LIFT
|- ((?x. (P x)) = (?x. (P (lift x))))
CONTINUOUS_AT_CTAN
|- (!z. ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) ==> ((continuous ctan) (at z))))
Ineq.dart_std4
|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!y7. (!y8. (!y9. ((((((((((dart_std4 y1) y2) y3) y4) y5) y6) y7) y8) y9) = ((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 (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0) , (y4 , ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) ((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_NOT_LE
|- (!x. (!y. ((~ ((real_le x) y)) = ((real_lt y) x))))
INT_LE_NEG
|- (!x. (!y. (((int_le (int_neg x)) (int_neg y)) = ((int_le y) x))))
Constants_approx.INTERVAL_INV
|- ((((interval_arith x) (a , b)) /\ (((real_lt (real_of_num (NUMERAL _0))) a) \/ ((real_lt b) (real_of_num (NUMERAL _0))))) ==> ((interval_arith (real_inv x)) ((real_inv b) , (real_inv a))))
RE_MUL_CX
|- (!x. (!z. (((Re ((complex_mul (Cx x)) z)) = ((real_mul x) (Re z))) /\ ((Re ((complex_mul z) (Cx x))) = ((real_mul (Re z)) x)))))
Ssrnat.mul2n
|- (!m. (((NUMERAL (BIT0 (BIT1 _0))) * m) = (double m)))
Real_ext.REAL_MK_NZ_EXP
|- (!x. (~ ((exp x) = (real_of_num (NUMERAL _0)))))
HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT
|- (!p. (!f. (!g. (!h. (!W. (!X. (!Y. ((((((homotopic_with (\f. (p (f o h)))) (X , Y)) f) g) /\ (((continuous_on h) W) /\ (((IMAGE h) W) SUBSET X))) ==> ((((homotopic_with p) (W , Y)) (f o h)) (g o h))))))))))
FRECHET_DERIVATIVE_WORKS
|- (!f. (!net. (((differentiable f) net) = (((has_derivative f) ((frechet_derivative f) net)) net))))
SUM_LMUL
|- (!f. (!c. (!s. (((sum s) (\x. ((real_mul c) (f x)))) = ((real_mul c) ((sum s) f))))))
Sphere.dih2_x
|- (!x2. (!x1. (!x3. (!x5. (!x4. (!x6. (((((((dih2_x x1) x2) x3) x4) x5) x6) = ((((((dih_x x2) x1) x3) x5) x4) x6))))))))
CX_EXP
|- (!x. ((Cx (exp x)) = (cexp (Cx x))))
Hypermap.node_map_and_darts
|- (!H. ((FINITE (dart H)) /\ ((permutes (node_map H)) (dart H))))
BASIS_COMPONENT
|- (!k. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (((basis k) $ i) = (((COND (i = k)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))))
COMPLEX_NEG_0
|- ((vector_neg (Cx (real_of_num (NUMERAL _0)))) = (Cx (real_of_num (NUMERAL _0))))
REAL_ABS_POW
|- (!x. (!n. ((real_abs ((real_pow x) n)) = ((real_pow (real_abs x)) n))))
VSUM_SUB
|- (!f. (!g. (!s. ((FINITE s) ==> (((vsum s) (\x. ((vector_sub (f x)) (g x)))) = ((vector_sub ((vsum s) f)) ((vsum s) g)))))))
CLOSED_IMP_LOCALLY_COMPACT
|- (!s. ((closed s) ==> ((locally compact) s)))
LIM_UNION
|- (!f. (!x. (!l. (!s. (!t. (((((--> f) l) ((within (at x)) s)) /\ (((--> f) l) ((within (at x)) t))) ==> (((--> f) l) ((within (at x)) (s UNION t)))))))))
TRANSP_TRANSP
|- (!A. ((transp (transp A)) = A))
MEASURABLE_TRANSLATION_EQ
|- (!a. (!s. ((measurable ((IMAGE (\x. ((vector_add a) x))) s)) = (measurable s))))
Seq2.list_sum_EL
|- (!s. (!f. ((~ (s = NIL)) ==> (((list_sum s) f) = ((sum ((NUMERAL _0) .. ((sizel s) - (NUMERAL (BIT1 _0))))) (\i. (f ((EL i) s))))))))
Hypermap_and_fan.E_N_F_IN_DART1_OF_FAN
|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (x IN (dart1_of_fan (V , E)))) ==> ((((e_fan_pair (V , E)) x) IN (dart1_of_fan (V , E))) /\ ((((n_fan_pair (V , E)) x) IN (dart1_of_fan (V , E))) /\ (((f_fan_pair (V , E)) x) IN (dart1_of_fan (V , E)))))))))
DIST_SYM
|- (!x. (!y. ((distance (x , y)) = (distance (y , x)))))
REAL_MUL_POS_LE
|- (!x. (!y. (((real_le (real_of_num (NUMERAL _0))) ((real_mul x) y)) = ((x = (real_of_num (NUMERAL _0))) \/ ((y = (real_of_num (NUMERAL _0))) \/ ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) \/ (((real_lt x) (real_of_num (NUMERAL _0))) /\ ((real_lt y) (real_of_num (NUMERAL _0))))))))))
Seq.eqS0
|- (!n. (((SUC n) = (NUMERAL _0)) = F))
FORALL_IN_UNION
|- (!P. (!s. (!t. ((!x. ((x IN (s UNION t)) ==> (P x))) = ((!x. ((x IN s) ==> (P x))) /\ (!x. ((x IN t) ==> (P x))))))))
REAL_INTEGRAL_EQ
|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((real_integral s) f) = ((real_integral s) g))))))
CLOSED_IN_COMPACT
|- (!s. (!t. (((compact s) /\ ((closed_in ((subtopology euclidean) s)) t)) ==> (compact t))))
matrix_add
|- (!A. (!B. (((matrix_add A) B) = (lambdai. (lambdaj. ((real_add ((A $ i) $ j)) ((B $ i) $ j)))))))
HAS_SIZE_1_EXISTS
|- (!s. ((s HAS_SIZE (NUMERAL (BIT1 _0))) = (?!x. (x IN s))))
Collect_geom.PER_SET2
|- ((a INSERT (b INSERT EMPTY)) = (b INSERT (a INSERT EMPTY)))
Float.INT_POW_NEG1
|- (!x. (!n. (((int_pow (int_neg (int_of_num x))) n) = ((int_mul ((int_pow (int_neg (int_of_num (NUMERAL (BIT1 _0))))) n)) ((int_pow (int_of_num x)) n)))))
Local_lemmas.CONDS_IN_HAFL_LINE
|- ((((real_le (real_of_num (NUMERAL _0))) t) /\ (((vector_sub a) x) = ((% t) ((vector_sub b) x)))) ==> (a IN ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))))
SUM_IMAGE
|- (!f. (!g. (!s. ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) ==> (((sum ((IMAGE f) s)) g) = ((sum s) (g o f)))))))
LE_1
|- ((!n. ((~ (n = (NUMERAL _0))) ==> ((NUMERAL _0) < n))) /\ ((!n. ((~ (n = (NUMERAL _0))) ==> ((NUMERAL (BIT1 _0)) <= n))) /\ ((!n. (((NUMERAL _0) < n) ==> (~ (n = (NUMERAL _0))))) /\ ((!n. (((NUMERAL _0) < n) ==> ((NUMERAL (BIT1 _0)) <= n))) /\ ((!n. (((NUMERAL (BIT1 _0)) <= n) ==> ((NUMERAL _0) < n))) /\ (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (~ (n = (NUMERAL _0))))))))))
COMPACT_CONVEX_HULL
|- (!s. ((compact s) ==> (compact ((hull convex) s))))
Lp_ineqs_proofs.y1_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))))) (y1_fan d))))))))))
LE_SUC
|- (!m. (!n. (((SUC m) <= (SUC n)) = (m <= n))))
frontier
|- (!s. ((frontier s) = ((closure s) DIFF (interior s))))
PATH_CONNECTED_CONVEX_DIFF_CARD_LT
|- (!u. (!s. (((convex u) /\ ((~ (collinear u)) /\ (s <_c UNIV))) ==> (path_connected (u DIFF s)))))
FINITE_SUBSET
|- (!s. (!t. (((FINITE t) /\ (s SUBSET t)) ==> (FINITE s))))
o_ASSOC
|- (!f. (!g. (!h. ((f o (g o h)) = ((f o g) o h)))))
CONVEX_LINEAR_IMAGE
|- (!f. (!s. (((convex s) /\ (linear f)) ==> (convex ((IMAGE f) s)))))
Fan.POWER_2
|- (!f. (((POWER f) (NUMERAL (BIT0 (BIT1 _0)))) = (f o f)))
Topology.FINITE_ORBITS_SIGMA_FAN
|- (!x. (!V. (!E. (!v. (!u. (((FAN (x , (V , E))) /\ ((v INSERT (u INSERT EMPTY)) IN E)) ==> (FINITE (((((set_of_orbits_points_fan x) V) E) v) u))))))))
COMPLEX_DIFFERENTIABLE_AT_CLOG
|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> ((complex_differentiable clog) (at z))))
MEASURE_NEGLIGIBLE_UNIONS_IMAGE
|- (!f. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> (measurable (f x)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> (negligible ((f x) INTER (f y)))))))) ==> ((measure (UNIONS ((IMAGE f) s))) = ((sum s) (\x. (measure (f x))))))))
Seq.gtS0
|- (!n. ((NUMERAL _0) < (SUC n)))
Rnsyjxm.f_surr_localization_convex_local
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((LET (\f. (LET_END (convex_local_fan (((v_prime V) f) , (((e_prime E) f) , f)))))) ((face (hypermap_of_fan (V , E))) d))))))))
IM_CLOG_POS_LT
|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((((real_lt (real_of_num (NUMERAL _0))) (Im (clog z))) /\ ((real_lt (Im (clog z))) pi)) = ((real_lt (real_of_num (NUMERAL _0))) (Im z)))))
CONNECTED_PCROSS
|- (!s. (!t. (((connected s) /\ (connected t)) ==> (connected (s PCROSS t)))))
HOMEOMORPHISM_SYM
|- (!f. (!g. (!s. (!t. (((homeomorphism (s , t)) (f , g)) = ((homeomorphism (t , s)) (g , f)))))))
CONVEX_HALFSPACE_GT
|- (!a. (!b. (convex (GSPEC (\GEN%PVAR%1441. (?x. (((SETSPEC GEN%PVAR%1441) ((real_gt ((dot a) x)) b)) x)))))))
COMPLEX_DERIVATIVE_UNIQUE_AT
|- (!f. (!z. (!f'. (!f''. (((((has_complex_derivative f) f') (at z)) /\ (((has_complex_derivative f) f'') (at z))) ==> (f' = f''))))))
VECTOR_POLYNOMIAL_FUNCTION_ADD
|- (!f. (!g. (((vector_polynomial_function f) /\ (vector_polynomial_function g)) ==> (vector_polynomial_function (\x. ((vector_add (f x)) (g x)))))))
Ssrnat.uphalf0
|- ((uphalf (NUMERAL _0)) = (NUMERAL _0))
POSITIVE_SEMIDEFINITE_2_DET
|- (!A. ((positive_semidefinite A) = (((transp A) = A) /\ (((real_le (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) /\ (((real_le (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\ ((real_le (real_of_num (NUMERAL _0))) (det A)))))))
HOMEOMORPHISM_OF_SUBSETS
|- (!f. (!g. (!s. (!t. (!s'. (!t'. ((((homeomorphism (s , t)) (f , g)) /\ ((s' SUBSET s) /\ ((t' SUBSET t) /\ (((IMAGE f) s') = t')))) ==> ((homeomorphism (s' , t')) (f , g)))))))))
COS_DOUBLE
|- (!x. ((cos ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x)) = ((real_sub ((real_pow (cos x)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (sin x)) (NUMERAL (BIT0 (BIT1 _0)))))))
LINEPATH_TRANSLATION
|- (!a. (!b. (!c. ((linepath (((vector_add a) b) , ((vector_add a) c))) = ((\x. ((vector_add a) x)) o (linepath (b , c)))))))
CONTRACTIBLE_IMP_SIMPLY_CONNECTED
|- (!s. ((contractible s) ==> (simply_connected s)))
REAL_NEG_SUB
|- (!x. (!y. ((real_neg ((real_sub x) y)) = ((real_sub y) x))))
INTEGRABLE_INCREASING_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 x)) (g y)))))) ==> ((integrable_on (\x. ((% (g x)) (f x)))) (closed_interval ((CONS (a , b)) NIL))))))))
Ssrnat.ltnNge
|- (!m. (!n. ((m < n) = (~ (n <= m)))))
CLOSED_CLOSED_PREIMAGES_IMP_CONTINUOUS_ON
|- (!f. (!s. (!t. (((((IMAGE f) s) SUBSET t) /\ ((compact t) /\ ((!y. ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%1941. (?x. (((SETSPEC GEN%PVAR%1941) ((x IN s) /\ ((f x) = y))) x)))))) /\ (!c. (((closed_in ((subtopology euclidean) s)) c) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) c))))))) ==> ((continuous_on f) s)))))
FINITE_INSERT
|- (!s. (!x. ((FINITE (x INSERT s)) = (FINITE s))))
CONTINUOUS_COMPLEX_MUL
|- (!net. (!f. (!g. ((((continuous f) net) /\ ((continuous g) net)) ==> ((continuous (\x. ((complex_mul (f x)) (g x)))) net)))))
CARD_MUL_C
|- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> ((CARD ((*_c s) t)) = ((CARD s) * (CARD t))))))
ANALYTIC_ON_SUBSET
|- (!f. (!s. (!t. ((((analytic_on f) s) /\ (t SUBSET s)) ==> ((analytic_on f) t)))))
POLYTOPE_TRANSLATION_EQ
|- (!a. (!s. ((polytope ((IMAGE (\x. ((vector_add a) x))) s)) = (polytope s))))
MATRIX_VECTOR_MUL_RZERO
|- (!A. (((matrix_vector_mul A) (vec (NUMERAL _0))) = (vec (NUMERAL _0))))
Ardbzye.K_SCS_3T3
|- ((scs_k_v39 scs_3T3) = (NUMERAL (BIT1 (BIT1 _0))))
Fan.subset_d_fan
|- (!x. (!V. (!E. (((d1_fan (x , (V , E))) SUBSET (d_fan (x , (V , E)))) /\ ((d20_fan (x , (V , E))) SUBSET (d_fan (x , (V , E))))))))
UNIFORMLY_CONTINUOUS_ON_COMPOSE
|- (!f. (!g. (!s. ((((uniformly_continuous_on f) s) /\ ((uniformly_continuous_on g) ((IMAGE f) s))) ==> ((uniformly_continuous_on (g o f)) s)))))
HAS_MEASURE_UNION_NEGLIGIBLE_EQ
|- (!s. (!t. (!m. ((negligible t) ==> (((has_measure (s UNION t)) m) = ((has_measure s) m))))))
Hypermap_iso.res_inv_left
|- (!f. (!s. (!x. (((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) /\ (x IN s)) ==> ((((res_inv f) s) (f x)) = x)))))
ARG_CNJ
|- (!z. ((Arg (cnj z)) = (((COND ((real z) /\ ((real_le (real_of_num (NUMERAL _0))) (Re z)))) (Arg z)) ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) (Arg z)))))
INTEGRAL_SPIKE_SET
|- (!f. (!g. (!s. (!t. ((negligible ((s DIFF t) UNION (t DIFF s))) ==> (((integral s) f) = ((integral t) f)))))))
HAS_MEASURE_DISJOINT_UNIONS
|- (!m. (!f. (((FINITE f) /\ ((!s. ((s IN f) ==> ((has_measure s) (m s)))) /\ (!s. (!t. (((s IN f) /\ ((t IN f) /\ (~ (s = t)))) ==> ((DISJOINT s) t)))))) ==> ((has_measure (UNIONS f)) ((sum f) m)))))
FORALL_LIFT
|- ((!x. (P x)) = (!x. (P (lift x))))
CONTINUOUS_MUL
|- (!net. (!f. (!c. ((((continuous (lift o c)) net) /\ ((continuous f) net)) ==> ((continuous (\x. ((% (c x)) (f x)))) net)))))
Real_ext.REAL_SV_EXP_0
|- ((exp (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL (BIT1 _0))))
NSUM_LMUL
|- (!f. (!c. (!s. (((nsum s) (\x. (c * (f x)))) = (c * ((nsum s) f))))))
HAS_BOUNDED_VARIATION_ON_VMUL
|- (!f. (!s. (!v. (((has_bounded_variation_on (\x. (lift (f x)))) s) ==> ((has_bounded_variation_on (\x. ((% (f x)) v))) s)))))
IM_NEG
|- (!x. ((Im (vector_neg x)) = (real_neg (Im x))))
CARD_LE_SUBSET
|- (!s. (!t. ((s SUBSET t) ==> (s <=_c t))))
SPHERE_LINEAR_IMAGE
|- (!f. (!x. (!r. (((linear f) /\ ((!y. (?x. ((f x) = y))) /\ (!x. ((vector_norm (f x)) = (vector_norm x))))) ==> ((sphere ((f x) , r)) = ((IMAGE f) (sphere (x , r))))))))
RANK_TRANSP
|- (!A. ((rank (transp A)) = (rank A)))
HAS_REAL_DERIVATIVE_CONST
|- (!c. (!net. (((has_real_derivative (\x. c)) (real_of_num (NUMERAL _0))) net)))
Fan.INVERSE1_SIGMA_FAN
|- (!x. (!V. (!E. (!v. ((FAN (x , (V , E))) ==> ((!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> ((v INSERT ((((((inverse1_sigma_fan x) V) E) v) w) INSERT EMPTY)) IN E))) /\ ((!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> ((((((sigma_fan x) V) E) v) (((((inverse1_sigma_fan x) V) E) v) w)) = w))) /\ (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> ((((((inverse1_sigma_fan x) V) E) v) (((((sigma_fan x) V) E) v) w)) = w))))))))))
SC_INC
|- (!R. (!x. (!y. (((R x) y) ==> (((SC R) x) y)))))
CPRODUCT_CLAUSES
|- ((!f. (((cproduct EMPTY) f) = (Cx (real_of_num (NUMERAL (BIT1 _0)))))) /\ (!x. (!f. (!s. ((FINITE s) ==> (((cproduct (x INSERT s)) f) = (((COND (x IN s)) ((cproduct s) f)) ((complex_mul (f x)) ((cproduct s) f)))))))))
DOT_LNEG
|- (!x. (!y. (((dot (vector_neg x)) y) = (real_neg ((dot x) y)))))
ARC_IMP_SIMPLE_PATH
|- (!g. ((arc g) ==> (simple_path g)))
Lp_ineqs_def.ye_fan
|- (!x. ((ye_fan x) = (distance x)))
HAS_REAL_MEASURE_REAL_NEGLIGIBLE_UNION
|- (!s1. (!s2. (!m1. (!m2. ((((has_real_measure s1) m1) /\ (((has_real_measure s2) m2) /\ (real_negligible (s1 INTER s2)))) ==> ((has_real_measure (s1 UNION s2)) ((real_add m1) m2)))))))
Pack1.saturated
|- (!S. ((saturated S) = (!x. (?y. ((y IN S) /\ ((real_lt (distance (x , y))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))
MATRIX_TRANSP_MULTIPLE_INV_LEFT
|- (!A. ((transp A) = ((matrix_mul (matrix_inv A)) ((matrix_mul A) (transp A)))))
SELF_ADJOINT_ORTHOGONAL_EIGENVECTORS
|- (!f. (!v. (!w. (!a. (!b. (((linear f) /\ (((adjoint f) = f) /\ (((f v) = ((% a) v)) /\ (((f w) = ((% b) w)) /\ (~ (a = b)))))) ==> ((orthogonal v) w)))))))
Seq.lastI
|- (!x. (!s. (((CONS x) s) = ((rcons ((belast x) s)) ((last x) s)))))
REAL_CONTINUOUS_WITHIN_ACS_STRONG
|- (!x. ((real_continuous acs) ((within (atreal x)) (GSPEC (\GEN%PVAR%4911. (?t. (((SETSPEC GEN%PVAR%4911) ((real_le (real_abs t)) (real_of_num (NUMERAL (BIT1 _0))))) t)))))))
STEINHAUS
|- (!s. (((measurable s) /\ ((real_lt (real_of_num (NUMERAL _0))) (measure s))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ ((ball ((vec (NUMERAL _0)) , d)) SUBSET (GSPEC (\GEN%PVAR%3461. (?x. (?y. (((SETSPEC GEN%PVAR%3461) ((x IN s) /\ (y IN s))) ((vector_sub x) y)))))))))))
INTERIOR_MAXIMAL
|- (!s. (!t. (((t SUBSET s) /\ (open t)) ==> (t SUBSET (interior s)))))
RELATIVE_INTERIOR_INJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((relative_interior ((IMAGE f) s)) = ((IMAGE f) (relative_interior s))))))
ANALYTIC_ON_MUL
|- (!f. (!g. (!s. ((((analytic_on f) s) /\ ((analytic_on g) s)) ==> ((analytic_on (\z. ((complex_mul (f z)) (g z)))) s)))))
ORTHOGONAL_CLAUSES
|- ((!a. ((orthogonal a) (vec (NUMERAL _0)))) /\ ((!a. (!x. (!c. (((orthogonal a) x) ==> ((orthogonal a) ((% c) x)))))) /\ ((!a. (!x. (((orthogonal a) x) ==> ((orthogonal a) (vector_neg x))))) /\ ((!a. (!x. (!y. ((((orthogonal a) x) /\ ((orthogonal a) y)) ==> ((orthogonal a) ((vector_add x) y)))))) /\ ((!a. (!x. (!y. ((((orthogonal a) x) /\ ((orthogonal a) y)) ==> ((orthogonal a) ((vector_sub x) y)))))) /\ ((!a. ((orthogonal (vec (NUMERAL _0))) a)) /\ ((!a. (!x. (!c. (((orthogonal x) a) ==> ((orthogonal ((% c) x)) a))))) /\ ((!a. (!x. (((orthogonal x) a) ==> ((orthogonal (vector_neg x)) a)))) /\ ((!a. (!x. (!y. ((((orthogonal x) a) /\ ((orthogonal y) a)) ==> ((orthogonal ((vector_add x) y)) a))))) /\ (!a. (!x. (!y. ((((orthogonal x) a) /\ ((orthogonal y) a)) ==> ((orthogonal ((vector_sub x) y)) a))))))))))))))
SQRT_WORKS_GEN
|- (!x. (((real_sgn (sqrt x)) = (real_sgn x)) /\ (((real_pow (sqrt x)) (NUMERAL (BIT0 (BIT1 _0)))) = (real_abs x))))
REAL_EQ_LCANCEL_IMP
|- (!x. (!y. (!z. (((~ (z = (real_of_num (NUMERAL _0)))) /\ (((real_mul z) x) = ((real_mul z) y))) ==> (x = y)))))
Nkezbfc_local.slicef
|- (!E. (!FF. (!w. (!v. (((((slicef E) FF) v) w) = ((GSPEC (\GEN%PVAR%7012. (?f. (((SETSPEC GEN%PVAR%7012) (?u. ((u IN (((((slicev E) FF) v) w) DELETE w)) /\ (f = (u , ((rho_node1 FF) u)))))) f)))) UNION ((w , v) INSERT EMPTY)))))))
SUP
|- (!s. (((~ (s = EMPTY)) /\ (?b. (!x. ((x IN s) ==> ((real_le x) b))))) ==> ((!x. ((x IN s) ==> ((real_le x) (sup s)))) /\ (!b. ((!x. ((x IN s) ==> ((real_le x) b))) ==> ((real_le (sup s)) b))))))
REALLIM_LMUL
|- (!f. (!l. (!c. ((((---> f) l) net) ==> (((---> (\x. ((real_mul c) (f x)))) ((real_mul c) l)) net)))))
aff_le_def
|- (aff_le = (affsign sgn_le))
LIMPT_OF_UNIV
|- (!x. ((limit_point_of x) UNIV))
MATRIX_MUL_ASSOC
|- (!A. (!B. (!C. (((matrix_mul A) ((matrix_mul B) C)) = ((matrix_mul ((matrix_mul A) B)) C)))))
CCOS_EQ_MINUS1
|- (!z. (((ccos z) = (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) = (?n. ((integer n) /\ (z = (Cx ((real_mul ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)) (real_of_num (NUMERAL (BIT1 _0))))) pi)))))))
RAT_LEMMA4
|- ((((real_lt (real_of_num (NUMERAL _0))) y1) /\ ((real_lt (real_of_num (NUMERAL _0))) y2)) ==> (((real_le ((real_div x1) y1)) ((real_div x2) y2)) = ((real_le ((real_mul x1) y2)) ((real_mul x2) y1))))
CONIC_IMP_PATH_CONNECTED
|- (!s. ((conic s) ==> (path_connected s)))
Ssrbool.if_same
|- (!b. (!vT. ((((COND b) vT) vT) = vT)))
Conforming.SIGMA_FAN_OF_FANADD_AT_POINT6
|- (!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)) /\ ((w1 = (((((inverse1_sigma_fan x) V) E) w) u)) /\ (((w 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) w) w1) = v)))))))))
ORTHOGONAL_SUBSPACE_DECOMP_EXISTS
|- (!s. (!x. (?y. (?z. ((y IN (span s)) /\ ((!w. ((w IN (span s)) ==> ((orthogonal z) w))) /\ (x = ((vector_add y) z))))))))
DROP_CMUL
|- (!x. (!c. ((drop ((% c) x)) = ((real_mul c) (drop x)))))
AFFSIGN_INJECTIVE_LINEAR_IMAGE
|- (!f. (!sgn. (!s. (!t. (!v. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((((affsign sgn) ((IMAGE f) s)) ((IMAGE f) t)) = ((IMAGE f) (((affsign sgn) s) t)))))))))
DOT_LZERO
|- (!x. (((dot (vec (NUMERAL _0))) x) = (real_of_num (NUMERAL _0))))
SUBSET_INTERIOR
|- (!s. (!t. ((s SUBSET t) ==> ((interior s) SUBSET (interior t)))))
LIM_NULL_NORM
|- (!net. (!f. ((((--> f) (vec (NUMERAL _0))) net) = (((--> (\x. (lift (vector_norm (f x))))) (vec (NUMERAL _0))) net))))
Hypermap_iso.hypermap_of_fan_rep_alt
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (((dart (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = (d_fan ((vec (NUMERAL _0)) , (V , E)))) /\ (((edge_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = ((res (((e_fan (vec (NUMERAL _0))) V) E)) (d1_fan ((vec (NUMERAL _0)) , (V , E))))) /\ (((node_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = ((res (((n_fan (vec (NUMERAL _0))) V) E)) (d1_fan ((vec (NUMERAL _0)) , (V , E))))) /\ ((face_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = ((res (((f1_fan (vec (NUMERAL _0))) V) E)) (d1_fan ((vec (NUMERAL _0)) , (V , E)))))))))))
CLOPEN
|- (!s. (((closed s) /\ (open s)) = ((s = EMPTY) \/ (s = UNIV))))
ORTHOGONAL_TRANSFORMATION_INTO_SUBSPACE
|- (!s. (!t. (((subspace s) /\ ((subspace t) /\ ((dim s) <= (dim t)))) ==> (?f. ((orthogonal_transformation f) /\ (((IMAGE f) s) SUBSET t))))))
HAS_COMPLEX_DERIVATIVE_TRANSFORM_WITHIN_OPEN
|- (!f. (!g. (!f'. (!s. (!z. (((open s) /\ ((z IN s) /\ ((!w. ((w IN s) ==> ((f w) = (g w)))) /\ (((has_complex_derivative f) f') (at z))))) ==> (((has_complex_derivative g) f') (at z))))))))
COMPLEX_MUL_CSIN_CSIN
|- (!w. (!z. (((complex_mul (csin w)) (csin z)) = ((complex_div ((vector_sub (ccos ((vector_sub w) z))) (ccos ((vector_add w) z)))) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))
FORALL_IN_IMAGE
|- (!f. (!s. ((!y. ((y IN ((IMAGE f) s)) ==> (P y))) = (!x. ((x IN s) ==> (P (f x)))))))
column
|- (!A. (!j. (((column j) A) = (lambdai. ((A $ i) $ j)))))
VECTOR_ADD_LID
|- (!x. (((vector_add (vec (NUMERAL _0))) x) = x))
INTER_UNIV
|- ((!s. ((UNIV INTER s) = s)) /\ (!s. ((s INTER UNIV) = s)))
cpow
|- (!z. (!w. (((cpow w) z) = (((COND (w = (Cx (real_of_num (NUMERAL _0))))) (Cx (real_of_num (NUMERAL _0)))) (cexp ((complex_mul z) (clog w)))))))
HOMOTOPIC_ON_EMPTY
|- (!t. (!f. (!g. ((((homotopic_with (\x. T)) (EMPTY , t)) f) g))))