hcju/holps huggingface dataset:
FINITE_SEGMENT
|- ((!a. (!b. ((FINITE (closed_segment ((CONS (a , b)) NIL))) = (a = b)))) /\ (!a. (!b. ((FINITE (open_segment (a , b))) = (a = b)))))
REAL_SECOND_MEAN_VALUE_THEOREM_BONNET_FULL
|- (!f. (!g. (!a. (!b. (((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_of_num (NUMERAL _0))) (g x)))) /\ (!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_le x) y))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_real_interval ((CONS (a , b)) NIL))) /\ (((has_real_integral (\x. ((real_mul (g x)) (f x)))) ((real_mul (g b)) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f))) (closed_real_interval ((CONS (a , b)) NIL))))))))))
Fan.hypermap_of_fan_rep
|- (!x. (!V. (!E. (!p. (((FAN (x , (V , E))) /\ (p = (\t. ((res (((t x) V) E)) (d1_fan (x , (V , E))))))) ==> (((dart (hypermap1_of_fanx (x , (V , E)))) = (d_fan (x , (V , E)))) /\ (((edge_map (hypermap1_of_fanx (x , (V , E)))) = (p e_fan)) /\ (((node_map (hypermap1_of_fanx (x , (V , E)))) = (p n_fan)) /\ ((face_map (hypermap1_of_fanx (x , (V , E)))) = (p f1_fan))))))))))
Wrgcvdr_cizmrrh.SPEC_HY_ELEMS
|- (!H. (((tuple_hypermap H) = (D , (e , (n , f)))) = (((dart H) = D) /\ (((edge_map H) = e) /\ (((node_map H) = n) /\ ((face_map H) = f))))))
REAL_LT_LE
|- (!x. (!y. (((real_lt x) y) = (((real_le x) y) /\ (~ (x = y))))))
Seq.mem_filter
|- (!a. (!x. (!s. (((MEM x) ((filter a) s)) = ((a x) /\ ((MEM x) s))))))
DIMINDEX_GE_1
|- (!s. ((NUMERAL (BIT1 _0)) <= (dimindex s)))
Ssrnat.half_add
|- (!m. (!n. ((half (m + n)) = ((((COND ((odd m) /\ (odd n))) (NUMERAL (BIT1 _0))) (NUMERAL _0)) + ((half m) + (half n))))))
SEPARATING_HYPERPLANE_CLOSED_POINT
|- (!s. (!z. (((convex s) /\ ((closed s) /\ (~ (z IN s)))) ==> (?a. (?b. (((real_lt ((dot a) z)) b) /\ (!x. ((x IN s) ==> ((real_gt ((dot a) x)) b)))))))))
IN_NUMSEG
|- (!m. (!n. (!p. ((p IN (m .. n)) = ((m <= p) /\ (p <= n))))))
Seq.perm_eq_sym
|- (!s1. (!s2. (((perm_eq s1) s2) = ((perm_eq s2) s1))))
HAUSDIST_NONTRIVIAL
|- (!s. (!t. (((bounded s) /\ ((bounded t) /\ ((~ (s = EMPTY)) /\ (~ (t = EMPTY))))) ==> ((hausdist (s , t)) = (sup ((GSPEC (\GEN%PVAR%1186. (?x. (((SETSPEC GEN%PVAR%1186) (x IN s)) (setdist ((x INSERT EMPTY) , t)))))) UNION (GSPEC (\GEN%PVAR%1187. (?y. (((SETSPEC GEN%PVAR%1187) (y IN t)) (setdist ((y INSERT EMPTY) , s))))))))))))
dependent
|- (!s. ((dependent s) = (?a. ((a IN s) /\ (a IN (span (s DELETE a)))))))
DET_I
|- ((det (mat (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT1 _0))))
Odxlstc.DEFORMATION_DIST_LE_ALL_LE3
|- (((is_scs_v39 s) /\ (((MMs_v39 s) w) /\ (((scs_k_v39 s) = k) /\ (((NUMERAL (BIT1 (BIT1 _0))) <= k) /\ (!i. ((~ ((i MOD k) = (l MOD k))) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance ((w l) , (w i)))))))))) ==> (!i. ((~ ((i MOD k) = (l MOD k))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!t. ((((real_lt (real_of_num (NUMERAL _0))) t) /\ ((real_lt t) e)) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance (((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) t)) (w l)) , (w i)))))))))))
Ssrnat.subnK
|- (!m. (!n. ((m <= n) ==> (((n - m) + m) = n))))
HAS_INTEGRAL_ALT
|- (!f. (!s. (!i. ((((has_integral f) i) s) = ((!a. (!b. ((integrable_on (\x. (((COND (x IN s)) (f x)) (vec (NUMERAL _0))))) (closed_interval ((CONS (a , b)) NIL))))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!a. (!b. (((ball ((vec (NUMERAL _0)) , B)) SUBSET (closed_interval ((CONS (a , b)) NIL))) ==> ((real_lt (vector_norm ((vector_sub ((integral (closed_interval ((CONS (a , b)) NIL))) (\x. (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))))) i))) e)))))))))))))
NORM_LE_L1
|- (!x. ((real_le (vector_norm x)) ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (real_abs (x $ i))))))
REAL_SUB_ARG
|- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> (((real_sub (Arg w)) (Arg z)) = (((COND ((real_le (Arg z)) (Arg w))) (Arg ((complex_div w) z))) ((real_sub (Arg ((complex_div w) z))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))))))
SPAN_SUB
|- (!x. (!y. (!s. (((x IN (span s)) /\ (y IN (span s))) ==> (((vector_sub x) y) IN (span s))))))
EL
|- ((((EL (NUMERAL _0)) l) = (HD l)) /\ (((EL (SUC n)) l) = ((EL n) (TL l))))
DISCRETE_BOUNDED_IMP_FINITE
|- (!s. (!e. ((((real_lt (real_of_num (NUMERAL _0))) e) /\ ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_lt (vector_norm ((vector_sub y) x))) e))) ==> (y = x)))) /\ (bounded s))) ==> (FINITE s))))
HAS_COMPLEX_DERIVATIVE_WITHIN_SUBSET
|- (!f. (!s. (!t. (!x. (((((has_complex_derivative f) f') ((within (at x)) s)) /\ (t SUBSET s)) ==> (((has_complex_derivative f) f') ((within (at x)) t)))))))
RELATIVE_FRONTIER_RETRACT_OF_PUNCTURED_AFFINE_HULL
|- (!s. (!a. (((convex s) /\ ((bounded s) /\ (a IN (relative_interior s)))) ==> ((retract_of (relative_frontier s)) (((hull affine) s) DELETE a)))))
VSUM_DIFF
|- (!f. (!s. (!t. (((FINITE s) /\ (t SUBSET s)) ==> (((vsum (s DIFF t)) f) = ((vector_sub ((vsum s) f)) ((vsum t) f)))))))
HAS_BOUNDED_VARIATION_ON_MAX
|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ ((has_bounded_variation_on g) s)) ==> ((has_bounded_variation_on (\x. (lift ((real_max (drop (f x))) (drop (g x)))))) s)))))
Nonlin_def.constant6
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!c. ((((((((constant6 c) x1) x2) x3) x4) x5) x6) = c))))))))
CONNECTED_CLOSURE
|- (!s. ((connected s) ==> (connected (closure s))))
REAL_LE_SQUARE_ABS
|- (!x. (!y. (((real_le (real_abs x)) (real_abs y)) = ((real_le ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow y) (NUMERAL (BIT0 (BIT1 _0))))))))
MEASURE_INTERVAL_4
|- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ (((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_le (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\ ((real_le (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 (BIT1 _0))))) (a $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (NUMERAL _0)))))) /\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ (((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_le (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\ ((real_le (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 (BIT1 _0))))) (a $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (NUMERAL _0)))))))
CARD_SING
|- (!a. ((CARD (a INSERT EMPTY)) = (NUMERAL (BIT1 _0))))
FRONTIER_STRADDLE
|- (!a. (!s. ((a IN (frontier s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((?x. ((x IN s) /\ ((real_lt (distance (a , x))) e))) /\ (?x. ((~ (x IN s)) /\ ((real_lt (distance (a , x))) e)))))))))
EVENTUALLY_WITHINREAL_LE
|- (!s. (!a. (!p. (((eventually p) ((within (atreal a)) s)) = (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (((x IN s) /\ (((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\ ((real_le (real_abs ((real_sub x) a))) d))) ==> (p x)))))))))
EXTREME_POINT_NOT_IN_INTERIOR
|- (!s. (!x. (((extreme_point_of x) s) ==> (~ (x IN (interior s))))))
CBALL_LINEAR_IMAGE
|- (!f. (!x. (!r. (((linear f) /\ ((!y. (?x. ((f x) = y))) /\ (!x. ((vector_norm (f x)) = (vector_norm x))))) ==> ((cball ((f x) , r)) = ((IMAGE f) (cball (x , r))))))))
INT_LT_LE
|- (!x. (!y. (((int_lt x) y) = (((int_le x) y) /\ (~ (x = y))))))
Hijqaha.K_SCS_4M5
|- ((scs_k_v39 scs_4M5') = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))
REAL_OF_NUM_POW
|- (!x. (!n. (((real_pow (real_of_num x)) n) = (real_of_num (x EXP n)))))
BALL_1
|- (!x. (!r. (((cball (x , r)) = (closed_interval ((CONS (((vector_sub x) (lift r)) , ((vector_add x) (lift r)))) NIL))) /\ ((ball (x , r)) = (open_interval (((vector_sub x) (lift r)) , ((vector_add x) (lift r))))))))
COMPLEX_DIFFERENTIABLE_AT_WITHIN
|- (!f. (!s. (!z. (((complex_differentiable f) (at z)) ==> ((complex_differentiable f) ((within (at z)) s))))))
complex_sub
|- (((vector_sub w) z) = ((vector_add w) (vector_neg z)))
COMMUTING_MATRIX_INV_COVARIANCE
|- (!A. (((matrix_mul (matrix_inv ((matrix_mul (transp A)) A))) ((matrix_mul (transp A)) A)) = ((matrix_mul ((matrix_mul (transp A)) A)) (matrix_inv ((matrix_mul (transp A)) A)))))
MATRIX_ADD
|- (!f. (!g. (((linear f) /\ (linear g)) ==> ((matrix (\x. ((vector_add (f x)) (g x)))) = ((matrix_add (matrix f)) (matrix g))))))
REAL_LT_POW2
|- (!n. ((real_lt (real_of_num (NUMERAL _0))) ((real_pow (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)))
COMPLEX_DERIVATIVE_MUL
|- (!f. (!g. (!z. ((((complex_differentiable f) (at z)) /\ ((complex_differentiable g) (at z))) ==> (((complex_derivative (\w. ((complex_mul (f w)) (g w)))) z) = ((vector_add ((complex_mul (f z)) ((complex_derivative g) z))) ((complex_mul ((complex_derivative f) z)) (g z))))))))
AFF_DIM_LE_DIM
|- (!s. ((int_le (aff_dim s)) (int_of_num (dim s))))
VECTOR_VARIATION_MINUS_FUNCTION_MONOTONE
|- (!f. (!a. (!b. (!c. (!d. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (((closed_interval ((CONS (c , d)) NIL)) SUBSET (closed_interval ((CONS (a , b)) NIL))) /\ (~ ((closed_interval ((CONS (c , d)) NIL)) = EMPTY)))) ==> ((real_le ((real_sub ((vector_variation (closed_interval ((CONS (c , d)) NIL))) f)) (drop ((vector_sub (f d)) (f c))))) ((real_sub ((vector_variation (closed_interval ((CONS (a , b)) NIL))) f)) (drop ((vector_sub (f b)) (f a)))))))))))
TC_SYM
|- (!R. ((!x. (!y. (((R x) y) ==> ((R y) x)))) ==> (!x. (!y. ((((TC R) x) y) ==> (((TC R) y) x))))))
NORM_LE
|- (!x. (!y. (((real_le (vector_norm x)) (vector_norm y)) = ((real_le ((dot x) x)) ((dot y) y)))))
Ssrnat.distnC
|- (!m. (!n. (((distn m) n) = ((distn n) m))))
MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL
|- (!f. (!s. ((lebesgue_measurable s) ==> (((measurable_on f) s) = (!t. ((borel t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3639. (?x. (((SETSPEC GEN%PVAR%3639) ((x IN s) /\ ((f x) IN t))) x)))))))))))
SUPPORT_CLAUSES
|- ((!f. ((((support op) f) EMPTY) = EMPTY)) /\ ((!f. (!x. (!s. ((((support op) f) (x INSERT s)) = (((COND ((f x) = (neutral op))) (((support op) f) s)) (x INSERT (((support op) f) s))))))) /\ ((!f. (!x. (!s. ((((support op) f) (s DELETE x)) = ((((support op) f) s) DELETE x))))) /\ ((!f. (!s. (!t. ((((support op) f) (s UNION t)) = ((((support op) f) s) UNION (((support op) f) t)))))) /\ ((!f. (!s. (!t. ((((support op) f) (s INTER t)) = ((((support op) f) s) INTER (((support op) f) t)))))) /\ ((!f. (!s. (!t. ((((support op) f) (s DIFF t)) = ((((support op) f) s) DIFF (((support op) f) t)))))) /\ (!f. (!g. (!s. ((((support op) g) ((IMAGE f) s)) = ((IMAGE f) (((support op) (g o f)) s))))))))))))
SETDIST_SING_IN_SET
|- (!x. (!s. ((x IN s) ==> ((setdist ((x INSERT EMPTY) , s)) = (real_of_num (NUMERAL _0))))))
HAS_COMPLEX_DERIVATIVE_VSUM
|- (!f. (!net. (!s. (((FINITE s) /\ (!a. ((a IN s) ==> (((has_complex_derivative (f a)) (f' a)) net)))) ==> (((has_complex_derivative (\x. ((vsum s) (\a. ((f a) x))))) ((vsum s) f')) net)))))
real_div
|- (!x. (!y. (((real_div x) y) = ((real_mul x) (real_inv y)))))
REAL_BOUNDS_LT
|- (!x. (!k. ((((real_lt (real_neg k)) x) /\ ((real_lt x) k)) = ((real_lt (real_abs x)) k))))
WF_SUBSET
|- (((!x. (!y. (((<< x) y) ==> ((<<< x) y)))) /\ (WF <<<)) ==> (WF <<))
LIM_NORM
|- (!net. (!f. (!l. ((((--> f) l) net) ==> (((--> (\x. (lift (vector_norm (f x))))) (lift (vector_norm l))) net)))))
HAUSDIST_CLOSURE
|- ((!s. (!t. ((hausdist ((closure s) , t)) = (hausdist (s , t))))) /\ (!s. (!t. ((hausdist (s , (closure t))) = (hausdist (s , t))))))
REAL_CONVEX_ON_DERIVATIVES_IMP
|- (!f. (!f'x. (!f'y. (!s. (!x. (!y. ((((real_convex_on f) s) /\ (((closed_real_segment ((CONS (x , y)) NIL)) SUBSET s) /\ ((((has_real_derivative f) f'x) ((within (atreal x)) s)) /\ (((has_real_derivative f) f'y) ((within (atreal y)) s))))) ==> ((real_le ((real_mul f'x) ((real_sub y) x))) ((real_mul f'y) ((real_sub y) x))))))))))
PCROSS_INTERVAL
|- (!a. (!b. (!c. (!d. (((closed_interval ((CONS (a , b)) NIL)) PCROSS (closed_interval ((CONS (c , d)) NIL))) = (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL)))))))
FRONTIER_SUBSET_RETRACTION
|- (!s. (!t. (!r. (((bounded s) /\ (((frontier s) SUBSET t) /\ (((continuous_on r) (closure s)) /\ ((((IMAGE r) s) SUBSET t) /\ (!x. ((x IN t) ==> ((r x) = x))))))) ==> (s SUBSET t)))))
REALLIM_CONST
|- (!net. (!a. (((---> (\x. a)) a) net)))
LOG_MONO_LE
|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> (((real_le (log x)) (log y)) = ((real_le x) y)))))
MATRIX_INV_MULTIPLE_TRANP_LEFT
|- (!A. ((matrix_inv A) = ((matrix_mul (transp A)) ((matrix_mul (transp (matrix_inv A))) (matrix_inv A)))))
SUMS_FINITE_UNION
|- (!f. (!s. (!t. (!l. (((FINITE t) /\ (((sums f) l) s)) ==> (((sums f) ((vector_add l) ((vsum (t DIFF s)) f))) (s UNION t)))))))
binom
|- ((!n. ((binom (n , (NUMERAL _0))) = (NUMERAL (BIT1 _0)))) /\ ((!k. ((binom ((NUMERAL _0) , (SUC k))) = (NUMERAL _0))) /\ (!n. (!k. ((binom ((SUC n) , (SUC k))) = ((binom (n , (SUC k))) + (binom (n , k))))))))
ROTATE2D_COMPLEX
|- (!t. (!z. (((rotate2d t) z) = ((complex_mul (cexp ((complex_mul ii) (Cx t)))) z))))
PATH_LINEAR_IMAGE_EQ
|- (!f. (!g. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((path (f o g)) = (path g)))))
SELECT_AX
|- (!P. (!x. ((P x) ==> (P (@ P)))))
HOMEOMORPHIC_TRANSLATION_RIGHT_EQ
|- (!a. (!s. (!t. (((homeomorphic s) ((IMAGE (\x. ((vector_add a) x))) t)) = ((homeomorphic s) t)))))
SERIES_TERMS_TOZERO
|- (!f. (!l. (!n. ((((sums f) l) (from n)) ==> (((--> f) (vec (NUMERAL _0))) sequentially)))))
CONIC_HULL_LINEAR_IMAGE
|- (!f. (!s. ((linear f) ==> (((hull conic) ((IMAGE f) s)) = ((IMAGE f) ((hull conic) s))))))
AFF_DIM_DIM_0
|- (!s. (((vec (NUMERAL _0)) IN ((hull affine) s)) ==> ((aff_dim s) = (int_of_num (dim s)))))
LOCALLY_COMPACT_INTER_CBALL
|- (!s. (((locally compact) s) = (!x. ((x IN s) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (closed ((cball (x , e)) INTER s))))))))
REAL_MAX_MAX
|- (!x. (!y. (((real_le x) ((real_max x) y)) /\ ((real_le y) ((real_max x) y)))))
POLYHEDRON_IMP_CLOSED
|- (!s. ((polyhedron s) ==> (closed s)))
IN_INTERS
|- (!s. (!x. ((x IN (INTERS s)) = (!t. ((t IN s) ==> (x IN t))))))
REAL_INTEGRAL_0
|- (!s. (((real_integral s) (\x. (real_of_num (NUMERAL _0)))) = (real_of_num (NUMERAL _0))))
HAS_MEASURE_TRANSLATION_EQ
|- (!a. (!s. (!m. (((has_measure ((IMAGE (\x. ((vector_add a) x))) s)) m) = ((has_measure s) m)))))
CONVEX_INTERIOR
|- (!s. ((convex s) ==> (convex (interior s))))
DET_3
|- (!A. ((det A) = ((real_add ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0))))))) ((real_add ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))))) ((real_sub ((real_sub ((real_sub ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0)))))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 _0))))))))))
EXTREME_POINT_OF_TRANSLATION_EQ
|- (!a. (!x. (!s. (((extreme_point_of ((vector_add a) x)) ((IMAGE (\x. ((vector_add a) x))) s)) = ((extreme_point_of x) s)))))
SPHERE_EMPTY
|- (!a. (!r. (((real_lt r) (real_of_num (NUMERAL _0))) ==> ((sphere (a , r)) = EMPTY))))
HULL_P
|- (!P. (!s. ((P s) ==> (((hull P) s) = s))))
COUNTABLE_NON_LIMIT_POINTS
|- (!s. (COUNTABLE (s DIFF (GSPEC (\GEN%PVAR%1262. (?x. (((SETSPEC GEN%PVAR%1262) ((limit_point_of x) s)) x)))))))
DOT_RMUL
|- (!c. (!x. (!y. (((dot x) ((% c) y)) = ((real_mul c) ((dot x) y))))))
Lp_ineqs_proofs.y6_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))))) (y6_fan d))))))))))
SPHERE_SING
|- (!x. (!e. ((e = (real_of_num (NUMERAL _0))) ==> ((sphere (x , e)) = (x INSERT EMPTY)))))
PATH_COMPONENT_TRANSLATION
|- (!a. (!s. (!x. (((path_component ((IMAGE (\x. ((vector_add a) x))) s)) ((vector_add a) x)) = ((IMAGE (\x. ((vector_add a) x))) ((path_component s) x))))))
COUNTABLE_IMAGE_INJ_EQ
|- (!f. (!s. ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) ==> ((COUNTABLE ((IMAGE f) s)) = (COUNTABLE s)))))
FINITE_UNIONS
|- (!s. ((FINITE (UNIONS s)) = ((FINITE s) /\ (!t. ((t IN s) ==> (FINITE t))))))
Kepler_spec.ZERO_SPEC
|- ((vec (NUMERAL _0)) = (real3_of_triple ((real_of_num (NUMERAL _0)) , ((real_of_num (NUMERAL _0)) , (real_of_num (NUMERAL _0))))))
real_measurable_on
|- (!f. (!s. (((real_measurable_on f) s) = ((measurable_on (lift o (f o drop))) ((IMAGE lift) s)))))
REAL_SUMMABLE_CAUCHY
|- (!f. (!s. (((real_summable s) f) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!m. (!n. ((m >= N) ==> ((real_lt (real_abs ((sum (s INTER (m .. n))) f))) e))))))))))
Ssrnat.addnS
|- (!m. (!n. ((m + (SUC n)) = (SUC (m + n)))))
COMPLEX_EXPAND
|- (!z. (z = ((vector_add (Cx (Re z))) ((complex_mul ii) (Cx (Im z))))))
INTEGER_POS
|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((integer x) = (?n. (x = (real_of_num n))))))
CARD_EXP_SING
|- (!s. (!b. (((^_c s) (b INSERT EMPTY)) =_c s)))
Hypermap_and_fan.FINITE_ORBIT_MAP_INVERSE
|- (!f. (!s. (!x. (!n. (!k. (((FINITE s) /\ (((permutes f) s) /\ (((CARD ((orbit_map f) x)) = n) /\ (k <= n)))) ==> ((((POWER (inverse f)) k) x) = (((POWER f) (n - k)) x))))))))
Tame_defs2.marked_list
|- (!L. (!N. (!r. (!x. (((((marked_list L) N) r) x) = ((is_restricted (hypermap_of_list L)) /\ (((normal_list L) N) /\ (((MEM r) N) /\ (((MEM x) r) /\ (((LET (\qs. (LET_END (((all uniq) qs) /\ ((all (\v. ((sizel ((filter (MEM v)) qs)) > (NUMERAL (BIT1 _0))))) (flatten qs)))))) ((quotient_list L) N)) /\ ((((next_el r) x) = ((f_list L) x)) /\ (((LET (\r'. (LET_END ((final_list L) r')))) ((find_list N) (e_list x))) /\ ((((final_list L) r) ==> ((flag_list L) N)) /\ ((~ ((final_list L) r)) ==> ((((s_flag_list L) N) r) x)))))))))))))))
CONVERGENT_BOUNDED_MONOTONE_1
|- (!s. (((bounded ((IMAGE s) UNIV)) /\ ((!n. ((real_le (drop (s n))) (drop (s (SUC n))))) \/ (!n. ((real_le (drop (s (SUC n)))) (drop (s n)))))) ==> (?l. (((--> s) l) sequentially))))
Local_lemmas.LOFA_IMP_CARD_FF_V_EQ
|- ((local_fan (V , (E , FF))) ==> ((CARD FF) = (CARD V)))
REAL_SUMMABLE_COMPLEX
|- (((real_summable s) f) = ((summable s) (Cx o f)))
NORM_SUB
|- (!x. (!y. ((vector_norm ((vector_sub x) y)) = (vector_norm ((vector_sub y) x)))))
PRODUCT_CLAUSES_NUMSEG
|- ((!m. (((product (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0)))))) /\ (!m. (!n. (((product (m .. (SUC n))) f) = (((COND (m <= (SUC n))) ((real_mul ((product (m .. n)) f)) (f (SUC n)))) ((product (m .. n)) f))))))
EIGENVALUES_CHARACTERISTIC_ALT
|- (!A. (!c. ((?v. ((~ (v = (vec (NUMERAL _0)))) /\ (((matrix_vector_mul A) v) = ((% c) v)))) = ((det ((matrix_sub A) ((%% c) (mat (NUMERAL (BIT1 _0)))))) = (real_of_num (NUMERAL _0))))))
ccos
|- (!z. ((ccos z) = ((complex_div ((vector_add (cexp ((complex_mul ii) z))) (cexp ((complex_mul (vector_neg ii)) z)))) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))
Seq.behead
|- (!h. (!t. (((behead NIL) = NIL) /\ ((behead ((CONS h) t)) = t))))
MATRIX_INV_COVARIANCE_RMUL
|- (!A. (((matrix_mul (matrix_inv ((matrix_mul (transp A)) A))) (transp A)) = (matrix_inv A)))
HAS_VECTOR_DERIVATIVE_UNIQUE_AT
|- (!f. (!f'. (!x. ((((has_vector_derivative f) f') (at x)) ==> (((vector_derivative f) (at x)) = f')))))
CNJ_CEXP
|- (!z. ((cnj (cexp z)) = (cexp (cnj z))))
LE_EXP
|- (!x. (!m. (!n. (((x EXP m) <= (x EXP n)) = (((COND (x = (NUMERAL _0))) ((m = (NUMERAL _0)) ==> (n = (NUMERAL _0)))) ((x = (NUMERAL (BIT1 _0))) \/ (m <= n)))))))
homeomorphism
|- (!f. (!s. (!g. (!t. (((homeomorphism (s , t)) (f , g)) = ((!x. ((x IN s) ==> ((g (f x)) = x))) /\ ((((IMAGE f) s) = t) /\ (((continuous_on f) s) /\ ((!y. ((y IN t) ==> ((f (g y)) = y))) /\ ((((IMAGE g) t) = s) /\ ((continuous_on g) t)))))))))))
tendsto_real
|- (!f. (!l. (!net. ((((---> f) l) net) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. ((real_lt (real_abs ((real_sub (f x)) l))) e))) net)))))))
ADD_ASSOC
|- (!m. (!n. (!p. ((m + (n + p)) = ((m + n) + p)))))
IS_INTERVAL_CONVEX_1
|- (!s. ((is_interval s) = (convex s)))
REAL_INV_MUL
|- (!x. (!y. ((real_inv ((real_mul x) y)) = ((real_mul (real_inv x)) (real_inv y)))))
COMPONENTS_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\ (!y. (?x. ((f x) = y))))) ==> ((components ((IMAGE f) s)) = ((IMAGE (IMAGE f)) (components s))))))
Hypermap_iso.bij_contracted_dart1
|- (!V. (!E. (((BIJ contracted_dart) (d1_fan ((vec (NUMERAL _0)) , (V , E)))) (dart1_of_fan (V , E)))))
INTERVAL_LOWERBOUND_NONEMPTY
|- (!a. (!b. ((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) ==> ((interval_lowerbound (closed_interval ((CONS (a , b)) NIL))) = a))))
Fan.set_of_edge
|- (!v. (!E. (!V. ((((set_of_edge v) V) E) = (GSPEC (\GEN%PVAR%5448. (?w. (((SETSPEC GEN%PVAR%5448) (((v INSERT (w INSERT EMPTY)) IN E) /\ (w IN V))) w))))))))
CARD_LE_DIM_SPANNING
|- (!v. (!b. (((v SUBSET (span b)) /\ ((FINITE b) /\ ((CARD b) <= (dim v)))) ==> (independent b))))
Ssrnat.addnC
|- (!m. (!n. ((m + n) = (n + m))))
COMPLEX_EQ
|- (!w. (!z. ((w = z) = (((Re w) = (Re z)) /\ ((Im w) = (Im z))))))
PATH_TRANSLATION_EQ
|- (!a. (!g. ((path ((\x. ((vector_add a) x)) o g)) = (path g))))
path
|- (!g. ((path g) = ((continuous_on g) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))))
CONNECTED_OUTSIDE
|- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ (bounded s)) ==> (connected (outside s))))
CARD_LE_CONG
|- (!s. (!s'. (!t. (!t'. (((s =_c s') /\ (t =_c t')) ==> ((s <=_c t) = (s' <=_c t')))))))
vec
|- (!n. ((vec n) = (lambdai. (real_of_num n))))
HAS_REAL_VECTOR_DERIVATIVE_WITHIN
|- ((((has_real_derivative f) f') ((within (atreal x)) s)) = (((has_vector_derivative (lift o (f o drop))) (lift f')) ((within (at (lift x))) ((IMAGE lift) s))))
HOLOMORPHIC_ON_DIFFERENTIABLE
|- (!f. (!s. (((holomorphic_on f) s) = (!x. ((x IN s) ==> ((complex_differentiable f) ((within (at x)) s)))))))
CROSS_ORTHOGONAL_MATRIX
|- (!A. (!x. (!y. ((orthogonal_matrix A) ==> (((cross ((matrix_vector_mul A) x)) ((matrix_vector_mul A) y)) = ((% (det A)) ((matrix_vector_mul A) ((cross x) y))))))))
CNJ_CNJ
|- (!z. ((cnj (cnj z)) = z))
Seq.size_subseq_leqif
|- (!s1. (!s2. (((subseq s1) s2) ==> (((leqif (sizel s1)) (sizel s2)) (s1 = s2)))))
LIM_CONTINUOUS
|- (!net. (!f. (!l. ((((continuous f) net) /\ ((f (netlimit net)) = l)) ==> (((--> f) l) net)))))
EQ_ADD_LCANCEL
|- (!m. (!n. (!p. (((m + n) = (m + p)) = (n = p)))))
Hypermap.set_of_orbits
|- (!D. (!f. (((set_of_orbits D) f) = (GSPEC (\GEN%PVAR%5395. (?x. (((SETSPEC GEN%PVAR%5395) (x IN D)) ((orbit_map f) x))))))))