hcju/holps huggingface dataset:
Nonlinear_lemma.vol3f_x_lfun_alt
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL _0))) x1) /\ (((real_le (real_of_num (NUMERAL _0))) x2) /\ ((real_le (real_of_num (NUMERAL _0))) x6))) ==> (((((((vol3f_x_lfun x1) x2) x3) x4) x5) x6) = ((real_sub ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div mm1) pi))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih2_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih6_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6))) ((real_add ((((((dih3_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((real_add ((((((dih4_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((real_sub ((((((dih5_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) pi))))))))) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) ((real_div mm2) pi))) ((real_add ((((((ldih_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((real_add ((((((ldih2_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((((((ldih6_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)))))))))))))
Miqmcsn.scs_4M8_13
|- (scs_4M8_13 = ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) NIL))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , (real_of_num (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))) NIL))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))
NORM_VSUM_PYTHAGOREAN
|- (!k. (!u. (((FINITE k) /\ ((pairwise (\i. (\j. ((orthogonal (u i)) (u j))))) k)) ==> (((real_pow (vector_norm ((vsum k) u))) (NUMERAL (BIT0 (BIT1 _0)))) = ((sum k) (\i. ((real_pow (vector_norm (u i))) (NUMERAL (BIT0 (BIT1 _0))))))))))
Merge_ineq.nonfunctional_mu6_x_y
|- (!x1. (!x2. (!x3. (!a. (!b. (!c. (((((((mu6_x x1) x2) x3) a) b) c) = (((mu_y (sqrt x1)) (sqrt x2)) (sqrt x3)))))))))
VSUM_CLAUSES_NUMSEG
|- ((!m. (((vsum (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (vec (NUMERAL _0))))) /\ (!m. (!n. (((vsum (m .. (SUC n))) f) = (((COND (m <= (SUC n))) ((vector_add ((vsum (m .. n)) f)) (f (SUC n)))) ((vsum (m .. n)) f))))))
HOMOTOPIC_WITH_IMP_CONTINUOUS
|- (!P. (!X. (!Y. (!f. (!g. (((((homotopic_with P) (X , Y)) f) g) ==> (((continuous_on f) X) /\ ((continuous_on g) X))))))))
SIMPLE_PATH_TRANSLATION_EQ
|- (!a. (!g. ((simple_path ((\x. ((vector_add a) x)) o g)) = (simple_path g))))
Constants_approx.acs3_interval
|- ((interval_arith (acs ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))) (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0)))))))))))))))))))))))))))))))) , ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))))))))))))))))))))))
COMPLEX_NORM_MUL
|- (!w. (!z. ((vector_norm ((complex_mul w) z)) = ((real_mul (vector_norm w)) (vector_norm z)))))
Ssrbool.andbAC
|- (!b. (!c. (!p. (((b /\ c) /\ p) = ((b /\ p) /\ c)))))
TAN_PI4
|- ((tan ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))) = (real_of_num (NUMERAL (BIT1 _0))))
COUNTABLE_ANR_PATH_COMPONENTS
|- (!s. (!t. ((ANR s) ==> (COUNTABLE (GSPEC (\GEN%PVAR%2677. (?x. (((SETSPEC GEN%PVAR%2677) (x IN t)) ((path_component s) x)))))))))
CROSS_LMUL
|- (!c. (!x. (!y. (((cross ((% c) x)) y) = ((% c) ((cross x) y))))))
EXP_ONE
|- (!n. (((NUMERAL (BIT1 _0)) EXP n) = (NUMERAL (BIT1 _0))))
EMPTY_INTERIOR_FINITE
|- (!s. ((FINITE s) ==> ((interior s) = EMPTY)))
ENDS_IN_INTERVAL
|- ((!a. (!b. ((a IN (closed_interval ((CONS (a , b)) NIL))) = (~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\ ((!a. (!b. ((b IN (closed_interval ((CONS (a , b)) NIL))) = (~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\ ((!a. (!b. (~ (a IN (open_interval (a , b)))))) /\ (!a. (!b. (~ (b IN (open_interval (a , b)))))))))
Trigonometry.KGLLRQT2
|- (!x. ((sin (real_neg x)) = (real_neg (sin x))))
sphere
|- (!x. (!e. ((sphere (x , e)) = (GSPEC (\GEN%PVAR%555. (?y. (((SETSPEC GEN%PVAR%555) ((distance (x , y)) = e)) y)))))))
CONNECTED_CONNECTED_IMP_CLOSED_POINTIMAGES
|- (!f. (!s. (((!c. (((c SUBSET s) /\ (connected c)) ==> (connected ((IMAGE f) c)))) /\ (!y. (connected (GSPEC (\GEN%PVAR%1938. (?x. (((SETSPEC GEN%PVAR%1938) ((x IN s) /\ ((f x) = y))) x))))))) ==> (!y. ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%1939. (?x. (((SETSPEC GEN%PVAR%1939) ((x IN s) /\ ((f x) = y))) x)))))))))
HAS_BOUNDED_VARIATION_ON_CONST
|- (!s. (!c. ((has_bounded_variation_on (\x. c)) s)))
COLLINEAR_TRANSLATION_EQ
|- (!a. (!s. ((collinear ((IMAGE (\x. ((vector_add a) x))) s)) = (collinear s))))
ORTHOGONAL_MATRIX_TRANSFORMATION
|- (!A. ((orthogonal_matrix A) = (orthogonal_transformation (\x. ((matrix_vector_mul A) x)))))
retract_of
|- (!s. (!t. (((retract_of t) s) = (?r. ((retraction (s , t)) r)))))
PRODUCT_CLAUSES
|- ((!f. (((product EMPTY) f) = (real_of_num (NUMERAL (BIT1 _0))))) /\ (!x. (!f. (!s. ((FINITE s) ==> (((product (x INSERT s)) f) = (((COND (x IN s)) ((product s) f)) ((real_mul (f x)) ((product s) f)))))))))
Nonlin_def.sqrt_x6
|- (((((((sqrt_x6 x1) x2) x3) x4) x5) x6) = (sqrt x6))
Tame_defs2.mapAt
|- (((((mapAt NIL) f) xs) = xs) /\ ((((mapAt ((CONS n) ns)) f) xs) = (((COND (n < (LENGTH xs))) (((mapAt ns) f) (((list_update xs) n) (f ((EL n) xs))))) (((mapAt ns) f) xs))))
Packing3.EL_TABLE
|- (!f. (!n. (!i. ((i < n) ==> (((EL i) ((TABLE f) n)) = (f i))))))
COMPLEX_DIFFERENTIABLE_AT_CSIN
|- (!z. ((complex_differentiable csin) (at z)))
CONVEX_IMP_BOUNDED_ON_INTERVAL
|- (!f. (!a. (!b. (((convex_on f) (closed_interval ((CONS (a , b)) NIL))) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_abs (f x))) B)))))))))
SUM_CLAUSES
|- ((!f. (((sum EMPTY) f) = (real_of_num (NUMERAL _0)))) /\ (!x. (!f. (!s. ((FINITE s) ==> (((sum (x INSERT s)) f) = (((COND (x IN s)) ((sum s) f)) ((real_add (f x)) ((sum s) f)))))))))
CONVEX_CONVEX_HULL
|- (!s. (convex ((hull convex) s)))
REAL_INV_NEG
|- (!x. ((real_inv (real_neg x)) = (real_neg (real_inv x))))
EVEN_MULT
|- (!m. (!n. ((EVEN (m * n)) = ((EVEN m) \/ (EVEN n)))))
COMPLEX_POW_2
|- (!x. (((complex_pow x) (NUMERAL (BIT0 (BIT1 _0)))) = ((complex_mul x) x)))
CLOSURE_IMAGE_CLOSURE
|- (!f. (!s. (((continuous_on f) (closure s)) ==> ((closure ((IMAGE f) (closure s))) = (closure ((IMAGE f) s))))))
Conforming.SIGMA_FAN_OF_FANADD1
|- (!x. (!V. (!E. (!E1. (!v. (!w. (((FAN (x , (V , E))) /\ ((FAN (x , (V , E1))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((~ ((v INSERT (w INSERT EMPTY)) IN E)) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))) ==> (!v1. (!w1. ((((v1 INSERT (w1 INSERT EMPTY)) IN E) /\ (~ (v1 IN (v INSERT (w INSERT EMPTY))))) ==> ((((((sigma_fan x) V) E1) v1) w1) = (((((sigma_fan x) V) E) v1) w1))))))))))))
Lp_ineqs_def.ln_fan
|- (!v. ((ln_fan v) = (lmfun ((real_div (yn_fan v)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))
Local_lemmas1.HAFL_CIRCLE_FORM_LOCAL_FAN
|- ((((local_fan (V , (E , FF))) /\ ((v IN V) /\ ((w IN V) /\ ((~ (v = w)) /\ ((!z. (!t. (((z IN (v INSERT (w INSERT EMPTY))) /\ (t IN (V DIFF (z INSERT EMPTY)))) ==> (~ (collinear ((vec (NUMERAL _0)) INSERT (z INSERT (t INSERT EMPTY)))))))) /\ (!x. ((x IN FF) ==> (((aff_gt ((vec (NUMERAL _0)) INSERT EMPTY)) (v INSERT (w INSERT EMPTY))) SUBSET ((wedge_in_fan_gt x) E))))))))) /\ ((HS = (hypermap (HYP ((vec (NUMERAL _0)) , (V , (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))))))) /\ (fv = ((face HS) (v , ((rho_node1 FF) v)))))) ==> (local_fan (((v_prime V) fv) , (((e_prime (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))) fv) , fv))))
HAUSDIST_STILL_INSIDE_INTERIOR
|- (!s. (!t. (!x. (((bounded s) /\ ((bounded t) /\ ((convex s) /\ ((convex t) /\ ((~ (t = EMPTY)) /\ ((real_lt (hausdist (s , t))) (setdist ((x INSERT EMPTY) , (UNIV DIFF s))))))))) ==> (x IN (interior t))))))
CONNECTED_LINEAR_IMAGE
|- (!f. (!s. (((connected s) /\ (linear f)) ==> (connected ((IMAGE f) s)))))
HAS_REAL_DERIVATIVE_ATREAL_WITHIN
|- (!f. (!f'. (!x. (!s. ((((has_real_derivative f) f') (atreal x)) ==> (((has_real_derivative f) f') ((within (atreal x)) s)))))))
SIN_COS_INJ
|- (!x. (!y. ((((sin x) = (sin y)) /\ (((cos x) = (cos y)) /\ ((real_lt (real_abs ((real_sub x) y))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))) ==> (x = y))))
Miqmcsn.SCS_3T7_OPP_IS_SCS
|- (is_scs_v39 (scs_opp_v39 scs_3T7))
MEASURABLE_ON_MAX
|- (!f. (!g. (!s. ((((measurable_on f) s) /\ ((measurable_on g) s)) ==> ((measurable_on (\x. (lambdai. ((real_max ((f x) $ i)) ((g x) $ i))))) s)))))
REAL_SUB_LNEG
|- (!x. (!y. (((real_sub (real_neg x)) y) = (real_neg ((real_add x) y)))))
HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_LEFT_EQ
|- (!f. (!s. (!t. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((homotopy_equivalent ((IMAGE f) s)) t) = ((homotopy_equivalent s) t))))))
Hypermap_iso.iso_components
|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) ==> (!d. ((d IN (dart H)) ==> ((((node G) (f d)) = ((IMAGE f) ((node H) d))) /\ ((((face G) (f d)) = ((IMAGE f) ((face H) d))) /\ (((edge G) (f d)) = ((IMAGE f) ((edge H) d)))))))))))
INT_GE
|- (!x. (!y. (((int_ge x) y) = ((int_le y) x))))
CONTINUOUS_ON_SUB
|- (!f. (!g. (!s. ((((continuous_on f) s) /\ ((continuous_on g) s)) ==> ((continuous_on (\x. ((vector_sub (f x)) (g x)))) s)))))
LIMPT_OF_OPEN_IN
|- (!s. (!t. (!x. ((((open_in ((subtopology euclidean) s)) t) /\ (((limit_point_of x) s) /\ (x IN t))) ==> ((limit_point_of x) t)))))
BOUNDED_PATH_IMAGE
|- (!g. ((path g) ==> (bounded (path_image g))))
ABSOLUTELY_INTEGRABLE_SUB
|- (!f. (!g. (!s. ((((absolutely_integrable_on f) s) /\ ((absolutely_integrable_on g) s)) ==> ((absolutely_integrable_on (\x. ((vector_sub (f x)) (g x)))) s)))))
REAL_FLOOR_NEG
|- (!x. ((floor (real_neg x)) = (((COND (integer x)) (real_neg x)) (real_neg ((real_add (floor x)) (real_of_num (NUMERAL (BIT1 _0))))))))
CONVEX_ON_DIRECTIONAL_DERIVATIVE_FUNCTIONS
|- (!f. (!s. ((((convex_on f) s) /\ ((is_interval s) /\ (open s))) ==> (?l. (?r. ((!x. ((x IN s) ==> (((has_vector_derivative (lift o f)) (l x)) ((within (at x)) (GSPEC (\GEN%PVAR%3307. (?t. (((SETSPEC GEN%PVAR%3307) ((real_le (drop t)) (drop x))) t)))))))) /\ ((!x. ((x IN s) ==> (((has_vector_derivative (lift o f)) (r x)) ((within (at x)) (GSPEC (\GEN%PVAR%3308. (?t. (((SETSPEC GEN%PVAR%3308) ((real_le (drop x)) (drop t))) t)))))))) /\ ((!x. ((x IN s) ==> ((real_le (drop (l x))) (drop (r x))))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_lt (drop x)) (drop y)))) ==> ((real_le (drop (r x))) (drop (l y))))))))))))))
VSUM_IMAGE
|- (!f. (!g. (!s. (((FINITE s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y))))) ==> (((vsum ((IMAGE f) s)) g) = ((vsum s) (g o f)))))))
SWAP_IDEMPOTENT
|- (!a. (!b. (((swap (a , b)) o (swap (a , b))) = I)))
convex_on
|- (!s. (!f. (((convex_on f) s) = (!x. (!y. (!u. (!v. (((x IN s) /\ ((y IN s) /\ (((real_le (real_of_num (NUMERAL _0))) u) /\ (((real_le (real_of_num (NUMERAL _0))) v) /\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> ((real_le (f ((vector_add ((% u) x)) ((% v) y)))) ((real_add ((real_mul u) (f x))) ((real_mul v) (f y))))))))))))
CONIC_LINEAR_IMAGE
|- (!f. (!s. (((conic s) /\ (linear f)) ==> (conic ((IMAGE f) s)))))
SEPARATING_HYPERPLANE_CLOSED_0
|- (!s. (((convex s) /\ ((closed s) /\ (~ ((vec (NUMERAL _0)) IN s)))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ (((real_lt (real_of_num (NUMERAL _0))) b) /\ (!x. ((x IN s) ==> ((real_gt ((dot a) x)) b)))))))))
Lp_main_estimate.diag_not_in_dart
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!a. (!b. (!f. (((f IN (face_set (hypermap_of_fan (V , E)))) /\ ((a IN ((v_prime V) f)) /\ ((b IN ((v_prime V) f)) /\ ((~ ((a , b) IN f)) /\ (~ ((b , a) IN f)))))) ==> (~ ((a , b) IN (dart_of_fan (V , E))))))))))))
Tame_defs.tame_2
|- (!H. ((tame_2 H) = ((connected_hypermap H) /\ (simple_hypermap H))))
LENGTH
|- (((LENGTH NIL) = (NUMERAL _0)) /\ (!h. (!t. ((LENGTH ((CONS h) t)) = (SUC (LENGTH t))))))
VECTOR_4
|- ((((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT1 _0))) = w) /\ ((((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT0 (BIT1 _0)))) = x) /\ ((((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT1 (BIT1 _0)))) = y) /\ (((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) = z))))
REAL_SERIES_RMUL
|- (!x. (!x0. (!c. (!s. ((((real_sums x) x0) s) ==> (((real_sums (\n. ((real_mul (x n)) c))) ((real_mul x0) c)) s))))))
CANTOR_BENDIXSON_GEN
|- (!s. (?t. (?u. (((closed_in ((subtopology euclidean) s)) t) /\ ((!x. ((x IN t) ==> ((condensation_point_of x) t))) /\ ((COUNTABLE u) /\ (s = (t UNION u))))))))
SIN_NPI
|- (!n. ((sin ((real_mul (real_of_num n)) pi)) = (real_of_num (NUMERAL _0))))
FLOOR_FRAC
|- (!x. ((integer (floor x)) /\ (((real_le (real_of_num (NUMERAL _0))) (frac x)) /\ (((real_lt (frac x)) (real_of_num (NUMERAL (BIT1 _0)))) /\ (x = ((real_add (floor x)) (frac x)))))))
covering_space
|- (!s. (!c. (!p. (((covering_space (c , p)) s) = (((continuous_on p) c) /\ ((((IMAGE p) c) = s) /\ (!x. ((x IN s) ==> (?t. ((x IN t) /\ (((open_in ((subtopology euclidean) s)) t) /\ (?v. (((UNIONS v) = (GSPEC (\GEN%PVAR%2050. (?x. (((SETSPEC GEN%PVAR%2050) ((x IN c) /\ ((p x) IN t))) x))))) /\ ((!u. ((u IN v) ==> ((open_in ((subtopology euclidean) c)) u))) /\ (((pairwise DISJOINT) v) /\ (!u. ((u IN v) ==> (?q. ((homeomorphism (u , t)) (p , q))))))))))))))))))))
TAN_ATN
|- (!x. ((((real_lt (real_neg ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) x) /\ ((real_lt x) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> ((atn (tan x)) = x)))
Hypermap_iso.hyp_iso_dart
|- (!H. (!f. (!G. (((hyp_iso f) (H , G)) ==> (!x. ((x IN (dart H)) ==> ((f x) IN (dart G))))))))
CARD_IMAGE_INJ
|- (!f. (!s. (((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) /\ (FINITE s)) ==> ((CARD ((IMAGE f) s)) = (CARD s)))))
integer
|- (!x. ((integer x) = (?n. ((real_abs x) = (real_of_num n)))))
Miqmcsn.SCS_4M8_13_BASIC
|- (scs_basic_v39 scs_4M8_13)
SEPARATING_HYPERPLANE_SET_POINT_INAFF
|- (!s. (!z. (((convex s) /\ ((~ (s = EMPTY)) /\ (~ (z IN s)))) ==> (?a. (?b. ((((vector_add z) a) IN ((hull affine) (z INSERT s))) /\ ((~ (a = (vec (NUMERAL _0)))) /\ (((real_le ((dot a) z)) b) /\ (!x. ((x IN s) ==> ((real_ge ((dot a) x)) b)))))))))))
Constants_approx.INTERVAL_SUB
|- ((((interval_arith x) (a , b)) /\ ((interval_arith y) (c , d))) ==> ((interval_arith ((real_sub x) y)) (((real_sub a) d) , ((real_sub b) c))))
Nonlin_def.vol3f_sqrt2_lmplus
|- (!y3. (!y4. (!y5. (!y1. (!y2. (!y6. (((((((vol3f_sqrt2_lmplus y1) y2) y3) y4) y5) y6) = ((real_sub ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div mm1) pi))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih_y y1) y2) sqrt2) sqrt2) sqrt2) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih2_y y1) y2) sqrt2) sqrt2) sqrt2) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih6_y y1) y2) sqrt2) sqrt2) sqrt2) y6))) ((real_add ((((((dih3_y y1) y2) sqrt2) sqrt2) sqrt2) y6)) ((real_add ((((((dih4_y y1) y2) sqrt2) sqrt2) sqrt2) y6)) ((real_sub ((((((dih5_y y1) y2) sqrt2) sqrt2) sqrt2) y6)) ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) pi))))))))) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) ((real_div mm2) pi))) ((real_add ((real_mul (lfun ((real_div y2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih2_y y1) y2) sqrt2) sqrt2) sqrt2) y6))) ((real_mul (lfun ((real_div y6) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih6_y y1) y2) sqrt2) sqrt2) sqrt2) y6))))))))))))
SUM_EQ
|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((sum s) f) = ((sum s) g))))))
Hypermap_iso.dart_of_fan_eq
|- (!V. (!E. ((dart_of_fan (V , E)) = ((dart1_of_fan (V , E)) UNION (GSPEC (\GEN%PVAR%7628. (?v. (((SETSPEC GEN%PVAR%7628) ((v IN V) /\ ((((set_of_edge v) V) E) = EMPTY))) (v , v)))))))))
REAL_ABS_INV
|- (!x. ((real_abs (real_inv x)) = (real_inv (real_abs x))))
RC_CASES
|- (!R. (!a0. (!a1. ((((RC R) a0) a1) = (((R a0) a1) \/ (a1 = a0))))))
AFF_DIM_TRANSLATION_EQ
|- (!a. (!s. ((aff_dim ((IMAGE (\x. ((vector_add a) x))) s)) = (aff_dim s))))
SUM_RMUL
|- (!f. (!c. (!s. (((sum s) (\x. ((real_mul (f x)) c))) = ((real_mul ((sum s) f)) c)))))
Pack2.MEASURABLE_VORONOI_CLOSED
|- (!s. (!v. ((saturated s) ==> (measurable ((voronoi_closed s) v)))))
continuous_atreal
|- (((continuous f) (atreal x)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x'. (((real_lt (real_abs ((real_sub x') x))) d) ==> ((real_lt (distance ((f x') , (f x)))) e))))))))
CLOSEST_POINT_IN_SET
|- (!s. (!a. (((closed s) /\ (~ (s = EMPTY))) ==> (((closest_point s) a) IN s))))
FINITE_POWERSET
|- (!s. ((FINITE s) ==> (FINITE (GSPEC (\GEN%PVAR%109. (?t. (((SETSPEC GEN%PVAR%109) (t SUBSET s)) t)))))))
Collect_geom2.COLLINEAR3_IMP_COPLANAR
|- (!v1. (!v2. (!v3. (!v. ((collinear (v1 INSERT (v2 INSERT (v3 INSERT EMPTY)))) ==> (coplanar_alt (v1 INSERT (v2 INSERT (v3 INSERT (v INSERT EMPTY))))))))))
CONTINUOUS_TRANSFORM_WITHIN_SET_IMP
|- (!f. (!a. (!s. (!t. ((((eventually (\x. ((x IN t) ==> (x IN s)))) (at a)) /\ ((continuous f) ((within (at a)) s))) ==> ((continuous f) ((within (at a)) t)))))))
REAL_LE_DIV
|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\ ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_le (real_of_num (NUMERAL _0))) ((real_div x) y)))))
IM_CLOG_EQ_PI
|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((Im (clog z)) = pi) = (((real_lt (Re z)) (real_of_num (NUMERAL _0))) /\ ((Im z) = (real_of_num (NUMERAL _0)))))))
Float.POW_MINUS1
|- (!n. (((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) * n)) = (real_of_num (NUMERAL (BIT1 _0)))))
DIST_POS_LE
|- (!x. (!y. ((real_le (real_of_num (NUMERAL _0))) (distance (x , y)))))
LEBESGUE_MEASURABLE_UNION
|- (!s. (!t. (((lebesgue_measurable s) /\ (lebesgue_measurable t)) ==> (lebesgue_measurable (s UNION t)))))
rpow
|- (!y. (!x. (((rpow x) y) = (((COND ((real_lt (real_of_num (NUMERAL _0))) x)) (exp ((real_mul y) (log x)))) (((COND (x = (real_of_num (NUMERAL _0)))) (((COND (y = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))) (((COND (?m. (?n. ((ODD m) /\ ((ODD n) /\ ((real_abs y) = ((real_div (real_of_num m)) (real_of_num n)))))))) (real_neg (exp ((real_mul y) (log (real_neg x)))))) (exp ((real_mul y) (log (real_neg x))))))))))
CONTINUOUS_ON_CNJ
|- (!s. ((continuous_on cnj) s))
REAL_ADD_LINV
|- (!x. (((real_add (real_neg x)) x) = (real_of_num (NUMERAL _0))))
int_lt
|- (!x. (!y. (((int_lt x) y) = ((real_lt (real_of_int x)) (real_of_int y)))))
VSUM_CLAUSES
|- ((!f. (((vsum EMPTY) f) = (vec (NUMERAL _0)))) /\ (!x. (!f. (!s. ((FINITE s) ==> (((vsum (x INSERT s)) f) = (((COND (x IN s)) ((vsum s) f)) ((vector_add (f x)) ((vsum s) f)))))))))
Ssrnat.add1n
|- (!n. (((NUMERAL (BIT1 _0)) + n) = (SUC n)))
Lp_ineqs_proofs.y5_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))))) ((y5_fan (V , E)) d))))))))))
INVERTIBLE_CMUL
|- (!A. (!c. ((invertible ((%% c) A)) = ((~ (c = (real_of_num (NUMERAL _0)))) /\ (invertible A)))))
REAL_SUB_0
|- (!x. (!y. ((((real_sub x) y) = (real_of_num (NUMERAL _0))) = (x = y))))
REAL_EQ_ADD_LCANCEL
|- (!x. (!y. (!z. ((((real_add x) y) = ((real_add x) z)) = (y = z)))))
REAL_POW_DIV
|- (!x. (!y. (!n. (((real_pow ((real_div x) y)) n) = ((real_div ((real_pow x) n)) ((real_pow y) n))))))
INTEGRABLE_RESTRICT_UNIV
|- (!f. (!s. (((integrable_on (\x. (((COND (x IN s)) (f x)) (vec (NUMERAL _0))))) UNIV) = ((integrable_on f) s))))
CSIN_EQ_1
|- (!z. (((csin z) = (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_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) pi)))))))
Ssrbool.andbb
|- (!b. ((b /\ b) = b))
DET_NEG
|- (!A. ((det (matrix_neg A)) = ((real_mul ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) (dimindex UNIV))) (det A))))
LINEAR_VSUM
|- (!f. (!g. (!s. (((linear f) /\ (FINITE s)) ==> ((f ((vsum s) g)) = ((vsum s) (f o g)))))))
BOUNDED_HAS_SUP
|- (!s. (((bounded ((IMAGE lift) s)) /\ (~ (s = EMPTY))) ==> ((!x. ((x IN s) ==> ((real_le x) (sup s)))) /\ (!b. ((!x. ((x IN s) ==> ((real_le x) b))) ==> ((real_le (sup s)) b))))))
tagged_partial_division_of
|- (!i. (!s. (((tagged_partial_division_of s) i) = ((FINITE s) /\ ((!x. (!k. (((x , k) IN s) ==> ((x IN k) /\ ((k SUBSET i) /\ (?a. (?b. (k = (closed_interval ((CONS (a , b)) NIL)))))))))) /\ (!x1. (!k1. (!x2. (!k2. ((((x1 , k1) IN s) /\ (((x2 , k2) IN s) /\ (~ ((x1 , k1) = (x2 , k2))))) ==> (((interior k1) INTER (interior k2)) = EMPTY)))))))))))