hcju/holps huggingface dataset:
HAS_COMPLEX_DERIVATIVE_CDIV_AT
|- (!f. (!f'. (!c. (!x. (!s. ((((has_complex_derivative f) f') (at x)) ==> (((has_complex_derivative (\x. ((complex_div (f x)) c))) ((complex_div f') c)) (at x))))))))
CONVEX_HULL_FINITE
|- (!s. (((hull convex) s) = (GSPEC (\GEN%PVAR%1491. (?y. (((SETSPEC GEN%PVAR%1491) (?u. ((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (u x)))) /\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum s) (\x. ((% (u x)) x))) = y))))) y))))))
cross
|- (!a. (!b. (((cross a) b) = (vector ((CONS ((real_sub ((real_mul (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_mul (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((CONS ((real_sub ((real_mul (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 _0))))) ((real_mul (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 (BIT1 _0))))))) ((CONS ((real_sub ((real_mul (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT1 _0)))))) NIL)))))))
REAL_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET
|- (!f. (!s. (!t. (((s SUBSET t) /\ (((real_measurable_on f) t) /\ (real_lebesgue_measurable s))) ==> ((real_measurable_on f) s)))))
ADMISSIBLE_COND
|- (!<<. (!p. (!P. (!s. (!h. (!k. ((((((admissible <<) p) s) P) /\ (((((admissible <<) (\f. (\x. (((p f) x) /\ ((P f) x))))) s) h) /\ ((((admissible <<) (\f. (\x. (((p f) x) /\ (~ ((P f) x)))))) s) k))) ==> ((((admissible <<) p) s) (\f. (\x. (((COND ((P f) x)) ((h f) x)) ((k f) x))))))))))))
Lp_ineqs_proofs.dart4_azim_cross_split
|- (!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)))) ==> (((azim_dart (V , E)) d) = ((real_add ((((((dih_y (y1_fan d)) ((y3_fan (V , E)) d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) ((y5_fan (V , E)) d))) ((((((dih_y (y1_fan d)) (y2_fan d)) ((y7_fan (V , E)) d)) ((y9_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) (y6_fan d))))))))))))
REAL_LE_LCANCEL_IMP
|- (!x. (!y. (!z. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_le ((real_mul x) y)) ((real_mul x) z))) ==> ((real_le y) z)))))
INTEGRAL_UNIQUE
|- (!f. (!y. (!k. ((((has_integral f) y) k) ==> (((integral k) f) = y)))))
INTEGRAL_EQ_0
|- (!f. (!s. ((!x. ((x IN s) ==> ((f x) = (vec (NUMERAL _0))))) ==> (((integral s) f) = (vec (NUMERAL _0))))))
CONNECTED_COMPONENT_EQ
|- (!s. (!x. (!y. ((y IN ((connected_component s) x)) ==> (((connected_component s) y) = ((connected_component s) x))))))
REAL_LE_LADD
|- (!x. (!y. (!z. (((real_le ((real_add x) y)) ((real_add x) z)) = ((real_le y) z)))))
REAL_CONTINUOUS_COMPLEX_COMPONENTS_AT
|- (!z. (((real_continuous Re) (at z)) /\ ((real_continuous Im) (at z))))
REAL_MEASURABLE_SCALING
|- (!s. (!c. ((real_measurable s) ==> (real_measurable ((IMAGE (\x. ((real_mul c) x))) s)))))
OPEN_UNION
|- (!s. (!t. (((open s) /\ (open t)) ==> (open (s UNION t)))))
NUMSEG_LREC
|- (!m. (!n. ((m <= n) ==> ((m INSERT ((m + (NUMERAL (BIT1 _0))) .. n)) = (m .. n)))))
EQ_EXT
|- (!f. (!g. ((!x. ((f x) = (g x))) ==> (f = g))))
REAL_ABS_BETWEEN1
|- (!x. (!y. (!z. ((((real_lt x) z) /\ ((real_lt (real_abs ((real_sub y) x))) ((real_sub z) x))) ==> ((real_lt y) z)))))
ITERATE_BIJECTION
|- (!op. ((monoidal op) ==> (!f. (!p. (!s. (((!x. ((x IN s) ==> ((p x) IN s))) /\ (!y. ((y IN s) ==> (?!x. ((x IN s) /\ ((p x) = y)))))) ==> ((((iterate op) s) f) = (((iterate op) s) (f o p)))))))))
Ssrbool.andFb
|- (!b. ((F /\ b) = F))
REAL_CONTINUOUS_ON_RPOW
|- (!s. (!y. ((((real_of_num (NUMERAL _0)) IN s) ==> ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_continuous_on (\x. ((rpow x) y))) s))))
NSUM_CLAUSES_LEFT
|- (!f. (!m. (!n. ((m <= n) ==> (((nsum (m .. n)) f) = ((f m) + ((nsum ((m + (NUMERAL (BIT1 _0))) .. n)) f)))))))
DIM_EQ_CARD
|- (!s. ((independent s) ==> ((dim s) = (CARD s))))
DIFFERENTIABLE_IMP_CONTINUOUS_AT
|- (!f. (!x. (((differentiable f) (at x)) ==> ((continuous f) (at x)))))
SUM_INJECTION
|- (!f. (!p. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> ((p x) IN s))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((p x) = (p y)))) ==> (x = y)))))) ==> (((sum s) (f o p)) = ((sum s) f))))))
Hypermap_and_fan.CARD_FACE_GT_1
|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((CARD ((face (hypermap_of_fan (V , E))) x)) > (NUMERAL (BIT1 _0)))) ==> (x IN (dart1_of_fan (V , E)))))))
COND_RAND
|- (!b. (!f. (!x. (!y. ((f (((COND b) x) y)) = (((COND b) (f x)) (f y)))))))
Nonlin_def.promote3_to_6
|- (!x4. (!x5. (!x6. (!f. (!x1. (!x2. (!x3. ((((((((promote3_to_6 f) x1) x2) x3) x4) x5) x6) = (((f x1) x2) x3)))))))))
HAS_DERIVATIVE_NEG_EQ
|- (!f. (!f'. (!net. ((((has_derivative (\x. (vector_neg (f x)))) (\h. (vector_neg (f' h)))) net) = (((has_derivative f) f') net)))))
Hypermap_iso.bij_disjoint_union
|- (!f. (!g. (!s. (!t. (!s'. (!t'. ((((DISJOINT s) t) /\ (((DISJOINT s') t') /\ ((((BIJ f) s) s') /\ (((BIJ g) t) t')))) ==> (((BIJ (\x. (((COND (x IN s)) (f x)) (g x)))) (s UNION t)) (s' UNION t')))))))))
ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV
|- (!f. (!s. (((absolutely_integrable_on (\x. (((COND (x IN s)) (f x)) (vec (NUMERAL _0))))) UNIV) = ((absolutely_integrable_on f) s))))
Fan_misc.INVERSE_SIGMA_FAN_EQ_INVERSE1_SIGMA_FAN
|- (!x. (!V. (!E. (!v. (!w. (((FAN (x , (V , E))) /\ ((v INSERT (w INSERT EMPTY)) IN E)) ==> ((((((inverse1_sigma_fan x) V) E) v) w) = (((((inverse_sigma_fan x) V) E) v) w))))))))
DROP_VEC
|- (!n. ((drop (vec n)) = (real_of_num n)))
Functional_equation.gchi_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))))))) ==> ((((((((gchi1_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x1))) ((((((dihatn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi2_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x2))) ((((((dih2atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi3_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x3))) ((((((dih3atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi4_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x4))) ((((((dih4atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi5_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x5))) ((((((dih5atn_x x1) x2) x3) x4) x5) x6))) /\ (((((((gchi6_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x6))) ((((((dih6atn_x x1) x2) x3) x4) x5) x6)))))))))))))))
Tame_defs.contracted_dart
|- (!x. (!w1. (!v. (!w. ((contracted_dart (x , (v , (w , w1)))) = (v , w))))))
CONDENSATION_POINTS_EQ_EMPTY
|- (!s. (((GSPEC (\GEN%PVAR%1861. (?x. (((SETSPEC GEN%PVAR%1861) ((condensation_point_of x) s)) x)))) = EMPTY) = (COUNTABLE s)))
SPAN_SPAN
|- (!s. ((span (span s)) = (span s)))
Planarity.properties_inside_collinear_fan
|- (!x. (!u. (!w. (!a. ((((real_lt (real_of_num (NUMERAL _0))) a) /\ (((real_lt a) (real_of_num (NUMERAL (BIT1 _0)))) /\ (~ (collinear (x INSERT (u INSERT (w INSERT EMPTY))))))) ==> ((~ (collinear (x INSERT (((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) a)) u)) ((% a) w)) INSERT (u INSERT EMPTY))))) /\ (~ (collinear (x INSERT (((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) a)) u)) ((% a) w)) INSERT (w INSERT EMPTY)))))))))))
Fan.pr3
|- (pr3 = (GABS (\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) w))))))))
RELATIVE_INTERIOR_EMPTY
|- ((relative_interior EMPTY) = EMPTY)
Miqmcsn.K_SCS_3T4_prime2
|- ((scs_k_v39 scs_3T4_prime2) = (NUMERAL (BIT1 (BIT1 _0))))
Lp_ineqs_proofs.y3_hi
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((y3_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))
SIMPLICIAL_COMPLEX_IMP_TRIANGULATION
|- (!tr. ((simplicial_complex tr) ==> (triangulation tr)))
UNIFORM_LIM_COMPLEX_MUL
|- (!net. (!P. (!f. (!g. (!l. (!m. (!b1. (!b2. ((((eventually (\x. (!n. ((P n) ==> ((real_le (vector_norm (l n))) b1))))) net) /\ (((eventually (\x. (!n. ((P n) ==> ((real_le (vector_norm (m n))) b2))))) net) /\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((f n) x)) (l n)))) e))))) net))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((g n) x)) (m n)))) e))))) net)))))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((complex_mul ((f n) x)) ((g n) x))) ((complex_mul (l n)) (m n))))) e))))) net))))))))))))
ISO
|- (!g. (!f. (((ISO f) g) = ((!x. ((f (g x)) = x)) /\ (!y. ((g (f y)) = y))))))
MEM_EXISTS_EL
|- (!l. (!x. (((MEM x) l) = (?i. ((i < (LENGTH l)) /\ (x = ((EL i) l)))))))
CONVEX_SIMPLEX
|- (!n. (!s. (((simplex n) s) ==> (convex s))))
Hypermap.orbit_map
|- (!f. (!x. (((orbit_map f) x) = (GSPEC (\GEN%PVAR%5390. (?n. (((SETSPEC GEN%PVAR%5390) (n >= (NUMERAL _0))) (((POWER f) n) x))))))))
VECTOR_MUL_RCANCEL
|- (!a. (!b. (!x. ((((% a) x) = ((% b) x)) = ((a = b) \/ (x = (vec (NUMERAL _0))))))))
Real_ext.REAL_PROP_POS_LMUL
|- ((!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) x) ==> (((real_lt (real_of_num (NUMERAL _0))) ((real_mul x) y)) = ((real_lt (real_of_num (NUMERAL _0))) y))))) /\ (!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) y) ==> (((real_lt (real_of_num (NUMERAL _0))) ((real_mul x) y)) = ((real_lt (real_of_num (NUMERAL _0))) x))))))
COMPLEX_ADD_LINV
|- (!x. (((vector_add (vector_neg x)) x) = (Cx (real_of_num (NUMERAL _0)))))
TRIANGULATION_TRANSLATION
|- (!a. (!tr. ((triangulation ((IMAGE (IMAGE (\x. ((vector_add a) x)))) tr)) = (triangulation tr))))
COMPLEX_ADD_CSIN
|- (!w. (!z. (((vector_add (csin w)) (csin z)) = ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((complex_mul (csin ((complex_div ((vector_add w) z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) (ccos ((complex_div ((vector_sub w) z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))))
REAL_LE_SETDIST_EQ
|- (!d. (!s. (!t. (((real_le d) (setdist (s , t))) = ((!x. (!y. (((x IN s) /\ (y IN t)) ==> ((real_le d) (distance (x , y)))))) /\ (((s = EMPTY) \/ (t = EMPTY)) ==> ((real_le d) (real_of_num (NUMERAL _0)))))))))
Ssrnat.leqSS
|- (!n. (!m. (((SUC n) <= (SUC m)) = (n <= m))))
Lp_ineqs_quads.invf_in_dart
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((inverse (f_fan_pair_ext (V , E))) d) IN (dart_of_fan (V , E))))))))
SUM_SUPERSET
|- (!f. (!u. (!v. (((u SUBSET v) /\ (!x. (((x IN v) /\ (~ (x IN u))) ==> ((f x) = (real_of_num (NUMERAL _0)))))) ==> (((sum v) f) = ((sum u) f))))))
homotopic_with
|- (!X. (!Y. (!p. (!q. (!P. (((((homotopic_with P) (X , Y)) p) q) = (?h. (((continuous_on h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) /\ ((((IMAGE h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) SUBSET Y) /\ ((!x. ((h ((pastecart (vec (NUMERAL _0))) x)) = (p x))) /\ ((!x. ((h ((pastecart (vec (NUMERAL (BIT1 _0)))) x)) = (q x))) /\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> (P (\x. (h ((pastecart t) x)))))))))))))))))
Seq.zip
|- ((((zip ((CONS x) s)) ((CONS y) t)) = ((CONS (x , y)) ((zip s) t))) /\ ((((zip NIL) t) = NIL) /\ (((zip s) NIL) = NIL)))
HAS_BOUNDED_VARIATION_ON_EMPTY
|- (!f. ((has_bounded_variation_on f) EMPTY))
COLLINEAR_DIST_BETWEEN
|- (!a. (!b. (!x. (((collinear (x INSERT (a INSERT (b INSERT EMPTY)))) /\ (((real_le (distance (x , a))) (distance (a , b))) /\ ((real_le (distance (x , b))) (distance (a , b))))) ==> ((between x) (a , b))))))
FORALL_BOOL_THM
|- ((!b. (P b)) = ((P T) /\ (P F)))
Ssrbool.contraL
|- (!c. (!b. ((c ==> (~ b)) ==> (b ==> (~ c)))))
numseg
|- (!m. (!n. ((m .. n) = (GSPEC (\GEN%PVAR%150. (?x. (((SETSPEC GEN%PVAR%150) ((m <= x) /\ (x <= n))) x)))))))
BROUWER_REDUCTION_THEOREM
|- (!P. (!s. (((!f. (((!n. ((compact (f n)) /\ ((~ ((f n) = EMPTY)) /\ (P (f n))))) /\ (!n. ((f (SUC n)) SUBSET (f n)))) ==> (P (INTERS (GSPEC (\GEN%PVAR%1277. (?n. (((SETSPEC GEN%PVAR%1277) (n IN UNIV)) (f n))))))))) /\ ((compact s) /\ ((~ (s = EMPTY)) /\ (P s)))) ==> (?t. ((t SUBSET s) /\ ((compact t) /\ ((~ (t = EMPTY)) /\ ((P t) /\ (!u. (((u SUBSET s) /\ ((closed u) /\ ((~ (u = EMPTY)) /\ (P u)))) ==> (~ (u PSUBSET t))))))))))))
DIAMETER_TRANSLATION
|- (!a. (!s. ((diameter ((IMAGE (\x. ((vector_add a) x))) s)) = (diameter s))))
Lp_main_estimate.f_surr_tau_fun_eq_tauVEF
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((V SUBSET ball_annulus) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((((tau_fun V) E) f) = (tauVEF (V , (E , f)))))))))))
REAL_EQ_RDIV_EQ
|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> ((x = ((real_div y) z)) = (((real_mul x) z) = y))))))
SUMMABLE_IMP_BOUNDED
|- (!f. (!k. (((summable k) f) ==> (bounded ((IMAGE f) k)))))
NEGLIGIBLE_CONVEX_HULL
|- (!s. (((FINITE s) /\ ((CARD s) <= (dimindex UNIV))) ==> (negligible ((hull convex) s))))
HAS_BOUNDED_REAL_VARIATION_TRANSLATION_EQ
|- (!a. (!f. (!s. (((has_bounded_real_variation_on (\x. (f ((real_add a) x)))) s) = ((has_bounded_real_variation_on f) ((IMAGE (\x. ((real_add a) x))) s))))))
ROWS_MAPROWS
|- (!f. (!A. ((rows ((maprows f) A)) = ((IMAGE f) (rows A)))))
HAS_DERIVATIVE_SEQUENCE
|- (!s. (!f. (!f'. (!g'. (((convex s) /\ ((!n. (!x. ((x IN s) ==> (((has_derivative (f n)) ((f' n) x)) ((within (at x)) s))))) /\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (!h. (((n >= N) /\ (x IN s)) ==> ((real_le (vector_norm ((vector_sub (((f' n) x) h)) ((g' x) h)))) ((real_mul e) (vector_norm h)))))))))) /\ (?x. (?l. ((x IN s) /\ (((--> (\n. ((f n) x))) l) sequentially))))))) ==> (?g. (!x. ((x IN s) ==> ((((--> (\n. ((f n) x))) (g x)) sequentially) /\ (((has_derivative g) (g' x)) ((within (at x)) s)))))))))))
POSITIVE_SEMIDEFINITE_ADD
|- (!A. (!B. (((positive_semidefinite A) /\ (positive_semidefinite B)) ==> (positive_semidefinite ((matrix_add A) B)))))
REAL_CONTINUOUS_AT_ACS
|- (!x. (((real_lt (real_abs x)) (real_of_num (NUMERAL (BIT1 _0)))) ==> ((real_continuous acs) (atreal x))))
lim
|- (!f. (!net. (((lim net) f) = (@l. (((--> f) l) net)))))
MOD_LT
|- (!m. (!n. ((m < n) ==> ((m MOD n) = m))))
STC_TRANS
|- (!R. (!x. (!y. (!z. (((((STC R) x) y) /\ (((STC R) y) z)) ==> (((STC R) x) z))))))
AFFINE_INDEPENDENT_IMP_FINITE
|- (!s. ((~ (affine_dependent s)) ==> (FINITE s)))
Miqmcsn.K_SCS_4M8_13
|- ((scs_k_v39 scs_4M8_13) = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))
SEGMENT_TO_FRONTIER_SIMPLE
|- (!s. (!x. (((bounded s) /\ (x IN s)) ==> (?a. (?b. ((a IN (frontier s)) /\ ((b IN (frontier s)) /\ (x IN (closed_segment ((CONS (a , b)) NIL))))))))))
RPOW_ONE
|- (!z. (((rpow (real_of_num (NUMERAL (BIT1 _0)))) z) = (real_of_num (NUMERAL (BIT1 _0)))))
Hypermap_and_fan.PAIR_IN_DART1_OF_FAN
|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((v , w) IN (dart1_of_fan (V , E)))) ==> ((v IN V) /\ ((w IN V) /\ ((w IN (((set_of_edge v) V) E)) /\ (v IN (((set_of_edge w) V) E))))))))))
SEGMENT_TO_RELATIVE_FRONTIER_SIMPLE
|- (!s. (!x. (((bounded s) /\ ((x IN s) /\ (~ (s = (x INSERT EMPTY))))) ==> (?a. (?b. ((a IN (relative_frontier s)) /\ ((b IN (relative_frontier s)) /\ (x IN (closed_segment ((CONS (a , b)) NIL))))))))))
HAS_REAL_MEASURE_REAL_NEGLIGIBLE_UNIONS
|- (!m. (!f. (((FINITE f) /\ ((!s. ((s IN f) ==> ((has_real_measure s) (m s)))) /\ (!s. (!t. (((s IN f) /\ ((t IN f) /\ (~ (s = t)))) ==> (real_negligible (s INTER t))))))) ==> ((has_real_measure (UNIONS f)) ((sum f) m)))))
LE_MULT_RCANCEL
|- (!m. (!n. (!p. (((m * p) <= (n * p)) = ((m <= n) \/ (p = (NUMERAL _0)))))))
NPRODUCT_POS_LT
|- (!f. (!s. (((FINITE s) /\ (!x. ((x IN s) ==> ((NUMERAL _0) < (f x))))) ==> ((NUMERAL _0) < ((nproduct s) f)))))
CONVEX_HULL_SUBSET_AFFINE_HULL
|- (!s. (((hull convex) s) SUBSET ((hull affine) s)))
Nonlin_def.sol_euler156_x_div_sqrtdelta
|- (sol_euler156_x_div_sqrtdelta = (rotate6 sol_euler_x_div_sqrtdelta))
IM_CNJ
|- (!z. ((Im (cnj z)) = (real_neg (Im z))))
Seq2.list_sum_eq0
|- (!s. (!f. ((!x. (((MEM x) s) ==> ((f x) = (real_of_num (NUMERAL _0))))) ==> (((list_sum s) f) = (real_of_num (NUMERAL _0))))))
IMAGE_FSTCART_PCROSS
|- (!s. (!t. (((IMAGE fstcart) (s PCROSS t)) = (((COND (t = EMPTY)) EMPTY) s))))
LTE_TRANS
|- (!m. (!n. (!p. (((m < n) /\ (n <= p)) ==> (m < p)))))
has_bounded_real_variation_on
|- (!f. (!s. (((has_bounded_real_variation_on f) s) = ((has_bounded_variation_on (lift o (f o drop))) ((IMAGE lift) s)))))
INSIDE_OUTSIDE_INTERSECT_CONNECTED
|- (!s. (!t. (((connected t) /\ ((~ (((inside s) INTER t) = EMPTY)) /\ (~ (((outside s) INTER t) = EMPTY)))) ==> (~ ((s INTER t) = EMPTY)))))
OPEN_IN_INJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (!t. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((open_in ((subtopology euclidean) ((IMAGE f) t))) ((IMAGE f) s)) = ((open_in ((subtopology euclidean) t)) s))))))
UNBOUNDED_COMPLEMENT_COMPONENT_CONVEX
|- (!s. (!c. (((convex s) /\ (c IN (components (UNIV DIFF s)))) ==> (~ (bounded c)))))
REAL_NEGLIGIBLE_COUNTABLE_UNIONS
|- (!s. ((!n. (real_negligible (s n))) ==> (real_negligible (UNIONS (GSPEC (\GEN%PVAR%4943. (?n. (((SETSPEC GEN%PVAR%4943) (n IN UNIV)) (s n)))))))))
sgn_ge
|- (sgn_ge = (\t. ((real_le (real_of_num (NUMERAL _0))) t)))
Merge_ineq.delta_y_sym
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((delta_y x1) x2) x3) x4) x5) x6) = ((((((delta_y x2) x1) x3) x5) x4) x6)) /\ ((((((((delta_y x1) x2) x3) x4) x5) x6) = ((((((delta_y x1) x3) x2) x4) x6) x5)) /\ (((((((delta_y x1) x2) x3) x4) x5) x6) = ((((((delta_y x5) x4) x3) x2) x1) x6))))))))))
CLOSURE_HALFSPACE_COMPONENT_GT
|- (!a. (!k. ((closure (GSPEC (\GEN%PVAR%812. (?x. (((SETSPEC GEN%PVAR%812) ((real_gt (x $ k)) a)) x))))) = (GSPEC (\GEN%PVAR%813. (?x. (((SETSPEC GEN%PVAR%813) ((real_ge (x $ k)) a)) x)))))))
FINITE_PERMUTATIONS
|- (!s. ((FINITE s) ==> (FINITE (GSPEC (\GEN%PVAR%249. (?p. (((SETSPEC GEN%PVAR%249) ((permutes p) s)) p)))))))
MEASURE_SING
|- (!a. ((measure (a INSERT EMPTY)) = (real_of_num (NUMERAL _0))))
POLYTOPE_IMP_POLYHEDRON
|- (!p. ((polytope p) ==> (polyhedron p)))
Hypermap.orbit_cyclic
|- (!f. (!m. (!x. (((~ (m = (NUMERAL _0))) /\ ((((POWER f) m) x) = x)) ==> (((orbit_map f) x) = (GSPEC (\GEN%PVAR%5396. (?k. (((SETSPEC GEN%PVAR%5396) (k < m)) (((POWER f) k) x))))))))))
REALLIM_NULL_COMPARISON
|- (!net. (!f. (!g. ((((eventually (\x. ((real_le (real_abs (f x))) (g x)))) net) /\ (((---> g) (real_of_num (NUMERAL _0))) net)) ==> (((---> f) (real_of_num (NUMERAL _0))) net)))))
Real_ext.REAL_PROP_NN_INV
|- (!x. (((real_le (real_of_num (NUMERAL _0))) (real_inv x)) = ((real_le (real_of_num (NUMERAL _0))) x)))
QUANTIFY_SURJECTION_HIGHER_THM
|- (!f. ((!y. (?x. ((f x) = y))) ==> (((!P. ((!x. (P x)) = (!x. (P (f x))))) /\ ((!P. ((?x. (P x)) = (?x. (P (f x))))) /\ ((!Q. ((!s. (Q s)) = (!s. (Q ((IMAGE f) s))))) /\ ((!Q. ((?s. (Q s)) = (?s. (Q ((IMAGE f) s))))) /\ ((!Q. ((!s. (Q s)) = (!s. (Q ((IMAGE (IMAGE f)) s))))) /\ ((!Q. ((?s. (Q s)) = (?s. (Q ((IMAGE (IMAGE f)) s))))) /\ ((!P. ((!g. (P g)) = (!g. (P (f o g))))) /\ ((!P. ((?g. (P g)) = (?g. (P (f o g))))) /\ ((!P. ((!g. (P g)) = (!g. (P (f o g))))) /\ ((!P. ((?g. (P g)) = (?g. (P (f o g))))) /\ ((!Q. ((!l. (Q l)) = (!l. (Q ((MAP f) l))))) /\ (!Q. ((?l. (Q l)) = (?l. (Q ((MAP f) l)))))))))))))))) /\ ((!P. ((GSPEC (\GEN%PVAR%238. (?x. (((SETSPEC GEN%PVAR%238) (P x)) x)))) = ((IMAGE f) (GSPEC (\GEN%PVAR%239. (?x. (((SETSPEC GEN%PVAR%239) (P (f x))) x))))))) /\ ((!Q. ((GSPEC (\GEN%PVAR%240. (?s. (((SETSPEC GEN%PVAR%240) (Q s)) s)))) = ((IMAGE (IMAGE f)) (GSPEC (\GEN%PVAR%241. (?s. (((SETSPEC GEN%PVAR%241) (Q ((IMAGE f) s))) s))))))) /\ (!R. ((GSPEC (\GEN%PVAR%242. (?l. (((SETSPEC GEN%PVAR%242) (R l)) l)))) = ((IMAGE (MAP f)) (GSPEC (\GEN%PVAR%243. (?l. (((SETSPEC GEN%PVAR%243) (R ((MAP f) l))) l))))))))))))
Ssrnat.maxn
|- (!n. (!m. (((maxn m) n) = (((COND (m < n)) n) m))))
SIMPLEX_ORDERING_EXISTS
|- (!s. (((simplex ((int_sub (int_of_num (dimindex UNIV))) (int_of_num (NUMERAL (BIT1 _0))))) s) ==> (?A. ((rows A) = (GSPEC (\GEN%PVAR%2546. (?v. (((SETSPEC GEN%PVAR%2546) ((extreme_point_of v) s)) v))))))))
IS_INTERVAL_IMP_ENR
|- (!s. ((is_interval s) ==> (ENR s)))
REAL_MUL_ASSOC
|- (!x. (!y. (!z. (((real_mul x) ((real_mul y) z)) = ((real_mul ((real_mul x) y)) z)))))
VECTOR_VARIATION_REFLECT2
|- (!f. (!s. (((vector_variation ((IMAGE vector_neg) s)) (\x. (f (vector_neg x)))) = ((vector_variation s) f))))
BALL_EQ_EMPTY
|- (!x. (!e. (((ball (x , e)) = EMPTY) = ((real_le e) (real_of_num (NUMERAL _0))))))
IN_SING
|- (!x. (!y. ((x IN (y INSERT EMPTY)) = (x = y))))
AFF_DIM_UNIV
|- ((aff_dim UNIV) = (int_of_num (dimindex UNIV)))
AFFINE_DEPENDENT_IMP_DEPENDENT
|- (!s. ((affine_dependent s) ==> (dependent s)))
BOUNDED_HALFSPACE_LT
|- (!a. (!b. ((bounded (GSPEC (\GEN%PVAR%844. (?x. (((SETSPEC GEN%PVAR%844) ((real_lt ((dot a) x)) b)) x))))) = ((a = (vec (NUMERAL _0))) /\ ((real_le b) (real_of_num (NUMERAL _0)))))))
ONE
|- ((NUMERAL (BIT1 _0)) = (SUC (NUMERAL _0)))
BARYCENTRE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((barycentre ((IMAGE f) s)) = (f (barycentre s))))))
CLOSURE_SEGMENT
|- ((!a. (!b. ((closure (closed_segment ((CONS (a , b)) NIL))) = (closed_segment ((CONS (a , b)) NIL))))) /\ (!a. (!b. ((closure (open_segment (a , b))) = (((COND (a = b)) EMPTY) (closed_segment ((CONS (a , b)) NIL)))))))
SECOND_MEAN_VALUE_THEOREM_BONNET_FULL
|- (!f. (!g. (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_of_num (NUMERAL _0))) (g x)))) /\ (!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)))) ((% (g b)) ((integral (closed_interval ((CONS (c , b)) NIL))) f))) (closed_interval ((CONS (a , b)) NIL))))))))))
SERIES_LINEAR
|- (!f. (!h. (!l. (!s. (((((sums f) l) s) /\ (linear h)) ==> (((sums (\n. (h (f n)))) (h l)) s))))))
Tame_lemmas.fully_surrounded_sol_eq_sol_y
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT1 (BIT1 _0))))) ==> ((LET (\w'. (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 (((sol (vec (NUMERAL _0))) (((((dart_leads_into (vec (NUMERAL _0))) V) E) v) w)) = ((((((sol_y y1) y2) y3) y4) y5) y6))))) (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) v) w)))))))))
COMPACT_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((compact ((IMAGE f) s)) = (compact s)))))
Functional_equation.promote1_to_6
|- (!x2. (!x3. (!x4. (!x5. (!x6. (!f. (!x1. ((((((((promote1_to_6 f) x1) x2) x3) x4) x5) x6) = (f x1)))))))))
Hypermap_and_fan.FULLY_SURROUNDED_IMP_CARD_FACE_GE_3
|- (!V. (!E. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (fully_surrounded (V , E))) ==> (!x. ((x IN (dart_of_fan (V , E))) ==> ((CARD ((face (hypermap_of_fan (V , E))) x)) >= (NUMERAL (BIT1 (BIT1 _0)))))))))
real_sub
|- (!x. (!y. (((real_sub x) y) = ((real_add x) (real_neg y)))))
NOT_LT
|- (!m. (!n. ((~ (m < n)) = (n <= m))))
ARITH_ZERO
|- (((NUMERAL (NUMERAL _0)) = (NUMERAL _0)) /\ ((BIT0 _0) = _0))
COMPLEX_DIV_1
|- (!z. (((complex_div z) (Cx (real_of_num (NUMERAL (BIT1 _0))))) = z))
Appendix.scs_str_v39_explicit
|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_str_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = str))))))))))