hcju/holps huggingface dataset:
REAL_INTEGRAL_LBOUND |- (!f. (!a. (!b. ((((real_le a) b) /\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ (!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_le B) (f x)))))) ==> ((real_le ((real_mul B) ((real_sub b) a))) ((real_integral (closed_real_interval ((CONS (a , b)) NIL))) f))))))
REAL_OPEN_RATIONAL |- (!P. (((real_open (GSPEC (\GEN%PVAR%4978. (?x. (((SETSPEC GEN%PVAR%4978) (P x)) x))))) /\ (?x. (P x))) ==> (?x. ((rational x) /\ (P x)))))
REAL_INTEGRABLE_BY_PARTS_EQ |- (!f. (!g. (!f'. (!g'. (!a. (!b. (!c. (((COUNTABLE c) /\ (((real_continuous_on (\x. ((real_mul (f x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL))) /\ (!x. ((x IN ((open_real_interval (a , b)) DIFF c)) ==> ((((has_real_derivative f) (f' x)) (atreal x)) /\ (((has_real_derivative g) (g' x)) (atreal x))))))) ==> (((real_integrable_on (\x. ((real_mul (f x)) (g' x)))) (closed_real_interval ((CONS (a , b)) NIL))) = ((real_integrable_on (\x. ((real_mul (f' x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))))))
Packing3.INTERS_INTER_INTERS_ALT |- (!f. (!g. (!P. (((INTERS (GSPEC (\GEN%PVAR%6019. (?x. (((SETSPEC GEN%PVAR%6019) (x IN f)) (P x)))))) INTER (INTERS (GSPEC (\GEN%PVAR%6020. (?y. (((SETSPEC GEN%PVAR%6020) (y IN g)) (P y))))))) = (INTERS (GSPEC (\GEN%PVAR%6021. (?u. (((SETSPEC GEN%PVAR%6021) (u IN (f UNION g))) (P u))))))))))
Trigonometry.ISRTTNZ |- (((FINITE W) /\ (((CARD W) = n) /\ ((!x. ((W x) ==> (~ (x = (vec (NUMERAL _0)))))) /\ (((polar_cycle_on xicm) W) /\ ((W p) /\ (?p. (?q. ((W p) /\ ((W q) /\ (~ (p = q))))))))))) ==> ((((sum ((NUMERAL _0) .. (n - (NUMERAL (BIT1 _0))))) (\i. ((arg_diff (((ITER i) xicm) p)) (((ITER (i + (NUMERAL (BIT1 _0)))) xicm) p)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ (!p. (!i. (!j. (((W p) /\ (((NUMERAL _0) <= i) /\ ((i <= j) /\ (j < n)))) ==> (((real_add ((arg_diff p) (((ITER i) xicm) p))) ((arg_diff (((ITER i) xicm) p)) (((ITER j) xicm) p))) = ((arg_diff p) (((ITER j) xicm) p)))))))))
ANR_COVERING_SPACE |- (!p. (!s. (!c. ((((covering_space (c , p)) s) /\ (ANR c)) ==> (ANR s)))))
Miqmcsn.mod_prep |- (((i + (NUMERAL (BIT1 (BIT1 _0)))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT1 (BIT1 _0)))) + i)) /\ (((i + (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) + i)) /\ (((i + (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) + i)) /\ (((i + (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) + i)) /\ ((SUC (((NUMERAL (BIT1 _0)) * k) + i)) = (((NUMERAL (BIT1 _0)) * k) + (SUC i)))))))
DIFFERENTIAL_ZERO_MAXMIN_COMPONENT |- (!f. (!x. (!e. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ (((real_lt (real_of_num (NUMERAL _0))) e) /\ (((!y. ((y IN (ball (x , e))) ==> ((real_le ((f y) $ k)) ((f x) $ k)))) \/ (!y. ((y IN (ball (x , e))) ==> ((real_le ((f x) $ k)) ((f y) $ k))))) /\ ((differentiable f) (at x)))))) ==> ((((jacobian f) (at x)) $ k) = (vec (NUMERAL _0))))))))
IN_EPIGRAPH |- (!x. (!y. ((((pastecart x) (lift y)) IN ((epigraph s) f)) = ((x IN s) /\ ((real_le (f x)) y)))))
HAS_INTEGRAL_SPIKE_INTERIOR_EQ |- (!f. (!g. (!a. (!b. (!y. ((!x. ((x IN (open_interval (a , b))) ==> ((g x) = (f x)))) ==> ((((has_integral f) y) (closed_interval ((CONS (a , b)) NIL))) = (((has_integral g) y) (closed_interval ((CONS (a , b)) NIL))))))))))
Trigonometry2.WHEN_K_DIFF0_ARCV |- ((~ (k = (real_of_num (NUMERAL _0)))) ==> ((((arcV (vec (NUMERAL _0))) u) v) = (((arcV (vec (NUMERAL _0))) u) ((% (real_abs k)) v))))
TC_RC |- (!R. ((TC (RC R)) = (RC (TC R))))
INT_ABS_SUB |- (!x. (!y. ((int_abs ((int_sub x) y)) = (int_abs ((int_sub y) x)))))
UNION_FRONTIER |- (!s. (!t. (((frontier s) UNION (frontier t)) = ((frontier (s UNION t)) UNION ((frontier (s INTER t)) UNION ((frontier s) INTER (frontier t)))))))
REAL_CONVEX_ON_CONTINUOUS |- (!f. (!s. (((real_open s) /\ ((real_convex_on f) s)) ==> ((real_continuous_on f) s))))
Ssrnat.max0n |- (!n. (((maxn (NUMERAL _0)) n) = n))
REAL_SECOND_MEAN_VALUE_THEOREM_BONNET |- (!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))) /\ (((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\x. ((real_mul (g x)) (f x)))) = ((real_mul (g b)) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f))))))))))
INT_MUL_AC |- ((((int_mul m) n) = ((int_mul n) m)) /\ ((((int_mul ((int_mul m) n)) p) = ((int_mul m) ((int_mul n) p))) /\ (((int_mul m) ((int_mul n) p)) = ((int_mul n) ((int_mul m) p)))))
Conforming.NEGLIGIBLE_AFF_3_INTER_BALL |- (!x. (!v. (!u. (!r. (negligible ((aff (x INSERT (v INSERT (u INSERT EMPTY)))) INTER ((normball x) r)))))))
REAL_INTEGRABLE_ON_SUPERSET |- (!f. (!s. (!t. (((!x. ((~ (x IN s)) ==> ((f x) = (real_of_num (NUMERAL _0))))) /\ ((s SUBSET t) /\ ((real_integrable_on f) s))) ==> ((real_integrable_on f) t)))))
REAL_COMPACT_IMP_CLOSED |- (!s. ((real_compact s) ==> (real_closed s)))
ARZELA_ASCOLI_LIPSCHITZ |- (!f. (!s. (!t. (!b. (((compact s) /\ ((bounded t) /\ ((!n. (!x. ((x IN s) ==> (~ ((((IMAGE (f n)) s) INTER t) = EMPTY))))) /\ (!n. (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub ((f n) x)) ((f n) y)))) ((real_mul b) (vector_norm ((vector_sub x) y))))))))))) ==> (?g. ((!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (g x)) (g y)))) ((real_mul b) (vector_norm ((vector_sub x) y))))))) /\ (?r. ((!m. (!n. ((m < n) ==> ((r m) < (r n))))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\ (x IN s)) ==> ((real_lt (vector_norm ((vector_sub ((f (r n)) x)) (g x)))) e))))))))))))))))
REALLIM_CONST_EQ |- (!net. (!c. (!d. ((((---> (\x. c)) d) net) = ((trivial_limit net) \/ (c = d))))))
MOD_MOD_EXP_MIN |- (!x. (!p. (!m. (!n. ((~ (p = (NUMERAL _0))) ==> (((x MOD (p EXP m)) MOD (p EXP n)) = (x MOD (p EXP ((MIN m) n)))))))))
CONIC_POSITIVE_ORTHANT |- (conic (GSPEC (\GEN%PVAR%1467. (?x. (((SETSPEC GEN%PVAR%1467) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (x $ i))))) x)))))
INTEGRABLE_BOUNDED_VARIATION_COMPLEX_RMUL |- (!f. (!g. (!a. (!b. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((has_bounded_variation_on g) (closed_interval ((CONS (a , b)) NIL)))) ==> ((integrable_on (\x. ((complex_mul (f x)) (g x)))) (closed_interval ((CONS (a , b)) NIL))))))))
FRONTIER_UNION |- (!s. (!t. ((((closure s) INTER (closure t)) = EMPTY) ==> ((frontier (s UNION t)) = ((frontier s) UNION (frontier t))))))
LOG_CONVEX_ON_SUBSET |- (!f. (!s. (!t. ((((log_convex_on f) t) /\ (s SUBSET t)) ==> ((log_convex_on f) s)))))
Ssrnat.distn_eq1 |- (!m. (!n. ((((distn m) n) = (NUMERAL (BIT1 _0))) = (((COND (m < n)) ((SUC m) = n)) (m = (SUC n))))))
LOCALLY_CONNECTED_PCROSS_EQ |- (!s. (!t. (((locally connected) (s PCROSS t)) = ((s = EMPTY) \/ ((t = EMPTY) \/ (((locally connected) s) /\ ((locally connected) t)))))))
Seq.size_behead |- (!s. ((sizel (behead s)) = ((sizel s) - (NUMERAL (BIT1 _0)))))
LIM_CSIN_OVER_X |- (((--> (\z. ((complex_div (csin z)) z))) (Cx (real_of_num (NUMERAL (BIT1 _0))))) (at (Cx (real_of_num (NUMERAL _0)))))
Functional_equation.functional_proj_x3 |- (proj_x3 = (rotate3 proj_x1))
Ssrbool.addbI |- (!x. (!y1. (!y2. ((((XOR x) y1) = ((XOR x) y2)) ==> (y1 = y2)))))
CONVEX_ON_IMP_MIDPOINT_CONVEX |- (!f. (!s. (!x. (!y. ((((convex_on f) s) /\ ((x IN s) /\ (y IN s))) ==> ((real_le (f (midpoint (x , y)))) ((real_div ((real_add (f x)) (f y))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))
CARD_EQ_COMPACT_SETS |- ((GSPEC (\GEN%PVAR%1256. (?s. (((SETSPEC GEN%PVAR%1256) (compact s)) s)))) =_c UNIV)
HAUSDIST_ALT |- (!s. (!t. (((bounded s) /\ ((bounded t) /\ ((~ (s = EMPTY)) /\ (~ (t = EMPTY))))) ==> ((hausdist (s , t)) = (sup (GSPEC (\GEN%PVAR%1203. (?x. (((SETSPEC GEN%PVAR%1203) (x IN UNIV)) (real_abs ((real_sub (setdist ((x INSERT EMPTY) , s))) (setdist ((x INSERT EMPTY) , t)))))))))))))
RELATIVE_BOUNDARY_RETRACT_OF_PUNCTURED_AFFINE_HULL |- (!s. (!a. (((convex s) /\ ((compact s) /\ (a IN (relative_interior s)))) ==> ((retract_of (s DIFF (relative_interior s))) (((hull affine) s) DELETE a)))))
FCCOVERABLE_INTERMEDIATE_CLOSURE |- (!s. (!t. (((s SUBSET t) /\ ((t SUBSET (closure s)) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?c. ((FINITE c) /\ (((UNIONS c) = s) /\ (!t. ((t IN c) ==> ((connected t) /\ ((bounded t) /\ ((real_le (diameter t)) e)))))))))))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?c. ((FINITE c) /\ (((UNIONS c) = t) /\ (!t. ((t IN c) ==> ((connected t) /\ ((bounded t) /\ ((real_le (diameter t)) e)))))))))))))
DENSE_LIMIT_POINTS |- (!x. (((GSPEC (\GEN%PVAR%583. (?x. (((SETSPEC GEN%PVAR%583) ((limit_point_of x) s)) x)))) = UNIV) = ((closure s) = UNIV)))
Rogers.QXSKIIT |- (!vf. (!b. (((FINITE ((IMAGE vf) UNIV)) /\ ((~ (affine_dependent ((IMAGE vf) UNIV))) /\ (!i. (!j. (((vf i) = (vf j)) ==> ((b i) = (b j))))))) ==> (?!p. ((p IN ((hull affine) ((IMAGE vf) UNIV))) /\ (!i. (!j. (((dot p) ((vector_sub (vf i)) (vf j))) = ((real_sub (b i)) (b j))))))))))
SECOND_MEAN_VALUE_THEOREM_GEN |- (!f. (!g. (!a. (!b. (!u. (!v. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (open_interval (a , b))) ==> (((real_le u) (g x)) /\ ((real_le (g x)) v)))) /\ (!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le (drop x)) (drop y)))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_interval ((CONS (a , b)) NIL))) /\ (((integral (closed_interval ((CONS (a , b)) NIL))) (\x. ((% (g x)) (f x)))) = ((vector_add ((% u) ((integral (closed_interval ((CONS (a , c)) NIL))) f))) ((% v) ((integral (closed_interval ((CONS (c , b)) NIL))) f)))))))))))))
EXTREME_POINT_OF_CONIC |- (!s. (!x. (((conic s) /\ ((extreme_point_of x) s)) ==> (x = (vec (NUMERAL _0))))))
REAL_DIFFERENTIABLE_WITHIN_SQRT |- (!s. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_differentiable sqrt) ((within (atreal x)) s)))))
IS_REALINTERVAL_CONTINUOUS_IMAGE |- (!s. ((((real_continuous_on f) s) /\ (is_realinterval s)) ==> (is_realinterval ((IMAGE f) s))))
Seq.count_predUI |- (!a1. (!a2. (!s. ((((count ((predU a1) a2)) s) + ((count ((predI a1) a2)) s)) = (((count a1) s) + ((count a2) s))))))
CAUCHY_CONTINUOUS_IMP_CONTINUOUS |- (!f. (!s. ((!x. (((cauchy x) /\ (!n. ((x n) IN s))) ==> (cauchy (f o x)))) ==> ((continuous_on f) s))))
ANR_FROM_UNION_AND_INTER |- (!s. (!t. (((closed s) /\ ((closed t) /\ ((ANR (s UNION t)) /\ (ANR (s INTER t))))) ==> ((ANR s) /\ (ANR t)))))
LIEB |- (!f. (!g. (!s. (!t. (((!n. ((absolutely_integrable_on (f n)) s)) /\ (((absolutely_integrable_on g) s) /\ ((negligible t) /\ (!x. ((x IN (s DIFF t)) ==> (((--> (\n. ((f n) x))) (g x)) sequentially)))))) ==> (((--> (\n. ((vector_sub ((integral s) (\x. (lift (vector_norm ((vector_sub ((f n) x)) (g x))))))) ((vector_sub ((integral s) (\x. (lift (vector_norm ((f n) x)))))) ((integral s) (\x. (lift (vector_norm (g x))))))))) (vec (NUMERAL _0))) sequentially))))))
RADON |- (!c. ((affine_dependent c) ==> (?m. (?p. ((m SUBSET c) /\ ((p SUBSET c) /\ (((DISJOINT m) p) /\ (~ ((DISJOINT ((hull convex) m)) ((hull convex) p))))))))))
Ssrbool.andb_orl |- (!b. (!c. (!p. (((b \/ c) /\ p) = ((b /\ p) \/ (c /\ p))))))
Ssrbool.orb_idl |- (!a. (!b. ((a ==> b) ==> ((a \/ b) = b))))
REAL_ADD_SIN |- (!x. (!y. (((real_add (sin x)) (sin y)) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (sin ((real_div ((real_add x) y)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) (cos ((real_div ((real_sub x) y)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))
Seq.reshapeKr |- (!sh. (!s. (((sizel s) <= (sumn sh)) ==> ((flatten ((reshape sh) s)) = s))))
WF_REFL |- (!x. ((WF <<) ==> (~ ((<< x) x))))
INCREASING_RIGHT_LIMIT |- (!f. (!a. (!b. (!c. (((!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 (f x)) (f y))))) /\ (c IN (closed_real_interval ((CONS (a , b)) NIL)))) ==> (?l. (((---> f) l) ((within (atreal c)) (closed_real_interval ((CONS (c , b)) NIL))))))))))
SETDIST_ZERO_STRONG |- (!s. (!t. ((~ ((DISJOINT (closure s)) (closure t))) ==> ((setdist (s , t)) = (real_of_num (NUMERAL _0))))))
REAL_CONTINUOUS_TRANSFORM_WITHIN_SET_IMP |- (!f. (!a. (!s. (!t. ((((eventually (\x. ((x IN t) ==> (x IN s)))) (at a)) /\ ((real_continuous f) ((within (at a)) s))) ==> ((real_continuous f) ((within (at a)) t)))))))
Nonlinear_lemma.quartic_has_real_derivative |- (!x. (!c0. (!c1. (!c2. (!c3. (!c4. (((has_real_derivative (\x. ((real_add ((real_mul c0) (real_of_num (NUMERAL (BIT1 _0))))) ((real_add ((real_mul c1) ((real_pow x) (NUMERAL (BIT1 _0))))) ((real_add ((real_mul c2) ((real_pow x) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add ((real_mul c3) ((real_pow x) (NUMERAL (BIT1 (BIT1 _0)))))) ((real_mul c4) ((real_pow x) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))))) ((real_add ((real_mul c0) (real_of_num (NUMERAL _0)))) ((real_add ((real_mul c1) ((real_mul (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul ((real_pow x) ((NUMERAL (BIT1 _0)) - (NUMERAL (BIT1 _0))))) (real_of_num (NUMERAL (BIT1 _0))))))) ((real_add ((real_mul c2) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow x) ((NUMERAL (BIT0 (BIT1 _0))) - (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 _0)))))) ((real_add ((real_mul c3) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((real_pow x) ((NUMERAL (BIT1 (BIT1 _0))) - (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 _0)))))) ((real_mul c4) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_pow x) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) - (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 _0)))))))))) (atreal x))))))))
LIM_RE_UBOUND |- (!net. (!f. (!l. (!b. (((~ (trivial_limit net)) /\ ((((--> f) l) net) /\ ((eventually (\x. ((real_le (Re (f x))) b))) net))) ==> ((real_le (Re l)) b))))))
CONVEX_HALFSPACE_IM_LE |- (!b. (convex (GSPEC (\GEN%PVAR%4774. (?z. (((SETSPEC GEN%PVAR%4774) ((real_le (Im z)) b)) z))))))
TRACE_MUL_POSITIVE_SEMIDEFINITE_LE |- (!A. (!B. (((positive_semidefinite A) /\ (positive_semidefinite B)) ==> ((real_le (trace ((matrix_mul A) B))) ((real_mul (trace A)) (trace B))))))
REAL_INTEGRABLE_ON_LITTLE_SUBINTERVALS |- (!f. (!a. (!b. ((!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!u. (!v. (((x IN (closed_real_interval ((CONS (u , v)) NIL))) /\ (!y. ((y IN (closed_real_interval ((CONS (u , v)) NIL))) ==> (((real_lt (real_abs ((real_sub y) x))) d) /\ (y IN (closed_real_interval ((CONS (a , b)) NIL))))))) ==> ((real_integrable_on f) (closed_real_interval ((CONS (u , v)) NIL)))))))))) ==> ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL)))))))
INT_POW_NEG |- (!x. (!n. (((int_pow (int_neg x)) n) = (((COND (EVEN n)) ((int_pow x) n)) (int_neg ((int_pow x) n))))))
CLOSURE_IMAGE_BOUNDED |- (!f. (!s. ((((continuous_on f) (closure s)) /\ (bounded s)) ==> ((closure ((IMAGE f) s)) = ((IMAGE f) (closure s))))))
PATH_COMPOSE_JOIN |- (!f. (!p. (!q. ((f o ((++ p) q)) = ((++ (f o p)) (f o q))))))
REAL_INFSUM_SUB |- (!x. (!y. (!s. ((((real_summable s) x) /\ ((real_summable s) y)) ==> (((real_infsum s) (\i. ((real_sub (x i)) (y i)))) = ((real_sub ((real_infsum s) x)) ((real_infsum s) y)))))))
NADD_LE_RMUL |- (!x. (!y. (!z. (((nadd_le x) y) ==> ((nadd_le ((nadd_mul x) z)) ((nadd_mul y) z))))))
NEGLIGIBLE_TRANSLATION |- (!s. (!a. ((negligible s) ==> (negligible ((IMAGE (\x. ((vector_add a) x))) s)))))
Seq2.list_sum_EL_alt |- (!s. (!f. (((list_sum s) f) = ((sum ((NUMERAL (BIT1 _0)) .. (sizel s))) (\i. (f ((EL (i - (NUMERAL (BIT1 _0)))) s)))))))
Functional_equation.functional_proj_y6 |- (proj_y6 = (rotate6 proj_y1))
AFFINE_INDEPENDENT_CONVEX_AFFINE_HULL |- (!s. (!t. (((~ (affine_dependent s)) /\ (t SUBSET s)) ==> (((hull convex) t) = (((hull affine) t) INTER ((hull convex) s))))))
RECTIFIABLE_PATH_SYM |- (!p. (!q. ((((pathfinish p) = (pathstart q)) /\ ((pathfinish q) = (pathstart p))) ==> ((rectifiable_path ((++ p) q)) = (rectifiable_path ((++ q) p))))))
Lp_ineqs_proofs.tau_sum4 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((CARD f) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ==> (((sum f) (rhazim_fan (V , E))) = ((real_add ((tau_fan (V , E)) f)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_add pi) sol0))))))))))))))
LINEAR_IMP_HAS_BOUNDED_VARIATION |- (!f. (((linear f) /\ (bounded s)) ==> ((has_bounded_variation_on f) s)))
Fnjlbxs.scriptL_eq |- (!S. (!k. ((((NUMERAL (BIT1 (BIT1 _0))) * k) <= (dimindex UNIV)) ==> ((!l. ((l IN S) = (((set_of_list l) SUBSET ball_annulus) /\ ((packing (set_of_list l)) /\ (((LENGTH l) = k) /\ (uniq l)))))) ==> (!v. ((v IN ((IMAGE (vector_of_list (NUMERAL (BIT1 (BIT1 _0))))) S)) ==> ((scriptL (set_of_list (((list_of_vector (NUMERAL (BIT1 (BIT1 _0)))) k) v))) = ((list_sum (((list_of_vector (NUMERAL (BIT1 (BIT1 _0)))) k) v)) (\v. (lmfun ((real_div (vector_norm v)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))))
RCONE_GT_NEG |- (!v0. (!v1. (!a. ((((rcone_gt v0) v1) (real_neg a)) = ((IMAGE (\x. ((vector_sub ((% (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) v0)) x))) (UNIV DIFF (((rcone_ge v0) v1) a)))))))
RE_CACS_BOUND |- (!z. ((real_le (real_abs (Re (cacs z)))) pi))
COMPLEX_NEG_MINUS1 |- (!x. ((vector_neg x) = ((complex_mul (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) x)))
AFFINE_NEGATIONS |- (!s. ((affine s) ==> (affine ((IMAGE vector_neg) s))))
DIAMETER_SCALING |- (!a. (!s. ((bounded s) ==> ((diameter ((IMAGE (\x. ((% a) x))) s)) = ((real_mul (real_abs a)) (diameter s))))))
Constants_approx.INTERVAL_SQRT |- ((((interval_arith x) (a , b)) /\ (((real_le ((real_mul c) c)) a) /\ ((real_le b) ((real_mul d) d)))) ==> ((interval_arith (sqrt x)) ((real_abs c) , (real_abs d))))
INT_LT_MUL |- (!x. (!y. ((((int_lt (int_of_num (NUMERAL _0))) x) /\ ((int_lt (int_of_num (NUMERAL _0))) y)) ==> ((int_lt (int_of_num (NUMERAL _0))) ((int_mul x) y)))))
MEASURE_INTERVAL_2_ALT |- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND (((real_lt (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ ((real_lt (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL _0)))))) /\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND (((real_lt (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ ((real_lt (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL _0)))))))
ALL2_MAP2 |- (!l. (!m. ((((ALL2 P) ((MAP f) l)) ((MAP g) m)) = (((ALL2 (\x. (\y. ((P (f x)) (g y))))) l) m))))
ENR_PCROSS |- (!s. (!t. (((ENR s) /\ (ENR t)) ==> (ENR (s PCROSS t)))))
REAL_CONTINUOUS_WITHIN_SUBSET |- (!f. (!s. (!t. ((((real_continuous f) ((within (at x)) s)) /\ (t SUBSET s)) ==> ((real_continuous f) ((within (at x)) t))))))
ANGLE_EQ_0_DIST_ABS |- (!A. (!B. (!C. (((angle (A , (B , C))) = (real_of_num (NUMERAL _0))) = ((~ (A = B)) /\ ((~ (C = B)) /\ ((distance (A , C)) = (real_abs ((real_sub (distance (A , B))) (distance (C , B)))))))))))
Pack2.finite_voronoi2 |- (!v. (!S. ((packing S) ==> (FINITE (GSPEC (\GEN%PVAR%5977. (?x. (?y. (((SETSPEC GEN%PVAR%5977) ((y IN (S DELETE v)) /\ ((x = v) /\ (y IN (ball (v , (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))) ((bis_le x) y))))))))))
INT_LE_DOUBLE |- (!x. (((int_le (int_of_num (NUMERAL _0))) ((int_add x) x)) = ((int_le (int_of_num (NUMERAL _0))) x)))
Functional_equation.domain6_ineq |- (!c1. (!c2. (!P. (!F1. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a1. (!a2. (!a3. (!a4. (!a5. (!a6. (!b1. (!b2. (!b3. (!b4. (!b5. (!b6. ((((domain6 P) c2) c1) ==> (((ineq ((CONS (a1 , (x1 , b1))) ((CONS (a2 , (x2 , b2))) ((CONS (a3 , (x3 , b3))) ((CONS (a4 , (x4 , b4))) ((CONS (a5 , (x5 , b5))) ((CONS (a6 , (x6 , b6))) NIL))))))) ((((((P x1) x2) x3) x4) x5) x6)) ==> (((ineq ((CONS (a1 , (x1 , b1))) ((CONS (a2 , (x2 , b2))) ((CONS (a3 , (x3 , b3))) ((CONS (a4 , (x4 , b4))) ((CONS (a5 , (x5 , b5))) ((CONS (a6 , (x6 , b6))) NIL))))))) (F1 ((((((c2 x1) x2) x3) x4) x5) x6))) = ((ineq ((CONS (a1 , (x1 , b1))) ((CONS (a2 , (x2 , b2))) ((CONS (a3 , (x3 , b3))) ((CONS (a4 , (x4 , b4))) ((CONS (a5 , (x5 , b5))) ((CONS (a6 , (x6 , b6))) NIL))))))) (F1 ((((((c1 x1) x2) x3) x4) x5) x6))))))))))))))))))))))))))))
HAS_BOUNDED_REAL_VARIATION_ON_MAX |- (!f. (!g. (!s. ((((has_bounded_real_variation_on f) s) /\ ((has_bounded_real_variation_on g) s)) ==> ((has_bounded_real_variation_on (\x. ((real_max (f x)) (g x)))) s)))))
HOLOMORPHIC_ON_CSIN |- (!s. ((holomorphic_on csin) s))
LOCALLY_PATH_CONNECTED_RIGHT_INVERTIBLE_IMAGE |- (!f. (!g. (!s. ((((continuous_on f) s) /\ (((continuous_on g) ((IMAGE f) s)) /\ ((((IMAGE g) ((IMAGE f) s)) SUBSET s) /\ ((!x. ((x IN ((IMAGE f) s)) ==> ((f (g x)) = x))) /\ ((locally path_connected) s))))) ==> ((locally path_connected) ((IMAGE f) s))))))
ABSOLUTELY_REAL_INTEGRABLE_SUM |- (!f. (!s. (!t. (((FINITE t) /\ (!a. ((a IN t) ==> ((absolutely_real_integrable_on (f a)) s)))) ==> ((absolutely_real_integrable_on (\x. ((sum t) (\a. ((f a) x))))) s)))))
CPOW_REAL_REAL |- (!w. (!z. (((real w) /\ ((real z) /\ ((real_lt (real_of_num (NUMERAL _0))) (Re w)))) ==> (((cpow w) z) = (Cx (exp ((real_mul (Re z)) (log (Re w)))))))))
EXISTS_CNJ |- ((?z. (P (cnj z))) = (?z. (P z)))
LEFT_POLAR_DECOMPOSITION_INVERTIBLE |- (!A. ((invertible A) = (?U. (?P. ((orthogonal_matrix U) /\ ((positive_definite P) /\ (((matrix_mul P) U) = A)))))))
BANACH_FIX_ITER |- (!f. (!s. (!c. (!n. (((complete s) /\ ((~ (s = EMPTY)) /\ (((real_le (real_of_num (NUMERAL _0))) c) /\ (((real_lt c) (real_of_num (NUMERAL (BIT1 _0)))) /\ ((((IMAGE f) s) SUBSET s) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (distance ((((ITER n) f) x) , (((ITER n) f) y)))) ((real_mul c) (distance (x , y)))))))))))) ==> (?!x. ((x IN s) /\ ((f x) = x))))))))
Bump.MCELL2_CELL_PARAMETERS_EXIST |- (!V. (!ul. (!X. (((packing V) /\ ((saturated V) /\ ((X = ((mcell2 V) ul)) /\ ((ul IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))) /\ (~ (negligible X)))))) ==> ((FST ((cell_params V) X)) = (NUMERAL (BIT0 (BIT1 _0))))))))
NEGLIGIBLE_AFFINE_HULL_2 |- (!a. (!b. (negligible ((hull affine) (a INSERT (b INSERT EMPTY))))))
Functional_equation.functional_rotate2 |- (!f. ((rotate2 f) = (((((((compose6 f) proj_x2) proj_x3) proj_x1) proj_x5) proj_x6) proj_x4)))
ROOT_EQ_0 |- (!n. (!x. ((((root n) x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0))))))
ABSOLUTELY_REAL_INTEGRABLE_ABSOLUTELY_REAL_INTEGRABLE_UBOUND |- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((real_le (f x)) (g x)))) /\ (((real_integrable_on f) s) /\ ((absolutely_real_integrable_on g) s))) ==> ((absolutely_real_integrable_on g) s)))))
PROPER_LOCALLY_INJECTIVE_OPEN_IMP_HOMEOMORPHISM |- (!f. (!s. (!t. (((path_connected s) /\ ((simply_connected t) /\ (((continuous_on f) s) /\ ((((IMAGE f) s) = t) /\ ((!k. (((k SUBSET t) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%2105. (?x. (((SETSPEC GEN%PVAR%2105) ((x IN s) /\ ((f x) IN k))) x))))))) /\ ((!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))) /\ (!x. ((x IN s) ==> (?u. ((x IN u) /\ (((open_in ((subtopology euclidean) s)) u) /\ (!y. (!z. (((y IN u) /\ ((z IN u) /\ ((f y) = (f z)))) ==> (y = z))))))))))))))) ==> (?g. ((homeomorphism (s , t)) (f , g)))))))
REAL_MEASURE_REAL_NEGLIGIBLE_UNION |- (!s. (!t. (((real_measurable s) /\ ((real_measurable t) /\ (real_negligible (s INTER t)))) ==> ((real_measure (s UNION t)) = ((real_add (real_measure s)) (real_measure t))))))