hcju/holps huggingface dataset:
ADMISSIBLE_MATCH_SEQPATTERN |- (!<<. (!p. (!s. (!c1. (!c2. (!e. ((((((admissible <<) p) s) (\f. (\x. (?y. ((((c1 f) x) ((e f) x)) y))))) /\ (((((admissible <<) (\f. (\x. (((p f) x) /\ (?y. ((((c1 f) x) ((e f) x)) y)))))) s) (\f. (\x. ((_MATCH ((e f) x)) ((c1 f) x))))) /\ ((((admissible <<) (\f. (\x. (((p f) x) /\ (~ (?y. ((((c1 f) x) ((e f) x)) y))))))) s) (\f. (\x. ((_MATCH ((e f) x)) ((c2 f) x))))))) ==> ((((admissible <<) p) s) (\f. (\x. ((_MATCH ((e f) x)) ((_SEQPATTERN ((c1 f) x)) ((c2 f) x)))))))))))))
REAL_DIFFERENTIABLE_WITHIN |- (!f. (!a. (!s. (((real_differentiable f) ((within (atreal x)) s)) = ((differentiable (lift o (f o drop))) ((within (at (lift x))) ((IMAGE lift) s)))))))
REAL_LOG_CONVEX_PRODUCT |- (!f. (!s. (!k. (((FINITE k) /\ (!i. ((i IN k) ==> ((real_log_convex_on (\x. ((f x) i))) s)))) ==> ((real_log_convex_on (\x. ((product k) (f x)))) s)))))
HOLOMORPHIC_ON_CLOG |- (!s. ((!z. (((z IN s) /\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z)))) ==> ((holomorphic_on clog) s)))
REAL_INTEGRABLE_SPIKE_EQ |- (!f. (!g. (!s. (!t. (((real_negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((real_integrable_on f) t) = ((real_integrable_on g) t)))))))
AFFINE_PCROSS_EQ |- (!s. (!t. ((affine (s PCROSS t)) = ((s = EMPTY) \/ ((t = EMPTY) \/ ((affine s) /\ (affine t)))))))
Rdwkarc.SATURATED_TRANS |- (!V. (!x. ((saturated V) ==> (saturated (GSPEC (\GEN%PVAR%6925. (?u. (((SETSPEC GEN%PVAR%6925) (((vector_add u) x) IN V)) u))))))))
EMPTY_INTERIOR_AFF_DIM |- (!s. (((int_lt (aff_dim s)) (int_of_num (dimindex UNIV))) ==> ((interior s) = EMPTY)))
CONVEX_AFF_LT |- (!s. (!t. (convex ((aff_lt s) t))))
Ssrbool.iffRLn |- (!Q. (!P. ((P = Q) ==> ((~ Q) ==> (~ P)))))
REAL_SUMMABLE_ADD |- (!x. (!y. (!s. ((((real_summable s) x) /\ ((real_summable s) y)) ==> ((real_summable s) (\n. ((real_add (x n)) (y n))))))))
Seq.all_predI |- (!a1. (!a2. (!s. (((all ((predI a1) a2)) s) = (((all a1) s) /\ ((all a2) s))))))
REAL_CONTINUOUS_INV_ATREAL |- (!f. (!a. ((((real_continuous f) (atreal a)) /\ (~ ((f a) = (real_of_num (NUMERAL _0))))) ==> ((real_continuous (\x. (real_inv (f x)))) (atreal a)))))
COMPLEX_ADD_SUB |- (!x. (!y. (((vector_sub ((vector_add x) y)) x) = y)))
SUM_PERMUTE_NUMSEG |- (!f. (!p. (!m. (!n. (((permutes p) (m .. n)) ==> (((sum (m .. n)) f) = ((sum (m .. n)) (f o p))))))))
FARKAS_LEMMA_ALT |- (!A. (!b. ((?x. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (((matrix_vector_mul A) x) $ i)) (b $ i))))) = (~ (?y. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (y $ i)))) /\ ((((vector_matrix_mul y) A) = (vec (NUMERAL _0))) /\ ((real_lt ((dot b) y)) (real_of_num (NUMERAL _0))))))))))
LOCALLY_CONNECTED_PATH_IMAGE |- (!p. ((path p) ==> ((locally connected) (path_image p))))
HAS_BOUNDED_VARIATION_ON_LINEAR_IMAGE |- (!f. (!g. (!a. (!b. (((linear f) /\ ((has_bounded_variation_on g) ((IMAGE f) (closed_interval ((CONS (a , b)) NIL))))) ==> ((has_bounded_variation_on (g o f)) (closed_interval ((CONS (a , b)) NIL))))))))
COMPACT_CONTINUOUS_IMAGE_EQ |- (!f. (!s. ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) ==> (((continuous_on f) s) = (!t. (((compact t) /\ (t SUBSET s)) ==> (compact ((IMAGE f) t))))))))
EMPTY_INSIDE_PSUBSET_CONVEX_FRONTIER |- (!s. (!t. (((convex s) /\ (t PSUBSET (frontier s))) ==> ((inside t) = EMPTY))))
Geomdetail.IN_SET3 |- ((x IN (a INSERT (b INSERT (c INSERT EMPTY)))) = ((x = a) \/ ((x = b) \/ (x = c))))
MBOUNDED_INSERT |- (!m. (!a. (!s. (((mbounded m) (a INSERT s)) = ((a IN (mspace m)) /\ ((mbounded m) s))))))
INSIDE_SUBSET_CONVEX_HULL |- (!s. ((inside s) SUBSET ((hull convex) s)))
Hdplygy.INDEX_J1_SY |- (!s. ((x IN (J1_SY s)) ==> (((NUMERAL (BIT1 _0)) <= (FST x)) /\ ((FST x) <= (k_sy s)))))
Fan.remark_fan1 |- (!v. (!w. (!V. (!E. (((v IN V) /\ (w IN V)) ==> ((w IN (((set_of_edge v) V) E)) = (v IN (((set_of_edge w) V) E))))))))
Float.REAL_EQ_RCANCEL_IMP' |- (!x. (!y. (!z. ((((real_mul x) z) = ((real_mul y) z)) ==> ((~ (z = (real_of_num (NUMERAL _0)))) ==> (x = y))))))
COMPLEX_POW_DIV |- (!x. (!y. (!n. (((complex_pow ((complex_div x) y)) n) = ((complex_div ((complex_pow x) n)) ((complex_pow y) n))))))
Fan.FAN5_LINEAR_IMAGE_EQ |- (!f. (!x. (!V. (!E. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((fan5 ((f x) , (((IMAGE f) V) , ((IMAGE (IMAGE f)) E)))) = (fan5 (x , (V , E)))))))))
REAL_CONVEX_DISTANCE |- (!s. (!a. ((real_convex_on (\x. (real_abs ((real_sub a) x)))) s)))
CONIC_HALFSPACE_LE |- (!a. (conic (GSPEC (\GEN%PVAR%1464. (?x. (((SETSPEC GEN%PVAR%1464) ((real_le ((dot a) x)) (real_of_num (NUMERAL _0)))) x))))))
HAS_REAL_DERIVATIVE_DIFFERENTIABLE |- (!f. (!x. ((((has_real_derivative f) ((real_derivative f) x)) (atreal x)) = ((real_differentiable f) (atreal x)))))
NEGATIONS_BALL |- (!r. (((IMAGE vector_neg) (ball ((vec (NUMERAL _0)) , r))) = (ball ((vec (NUMERAL _0)) , r))))
SUMMABLE_COMPONENT |- (!f. (!s. (!k. ((((summable s) f) /\ (((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV)))) ==> ((summable s) (\i. (lift ((f i) $ k))))))))
REAL_VARIATION_AFFINITY2 |- (!m. (!c. (!f. (!s. (((real_variation ((IMAGE (\x. ((real_add ((real_mul (real_inv m)) x)) (real_neg ((real_mul (real_inv m)) c))))) s)) (\x. (f ((real_add ((real_mul m) x)) c)))) = (((COND (m = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL _0))) ((real_variation s) f)))))))
NSUM_RESTRICT |- (!f. (!s. ((FINITE s) ==> (((nsum s) (\x. (((COND (x IN s)) (f x)) (NUMERAL _0)))) = ((nsum s) f)))))
FORALL_SUBSET_IMAGE_INJ |- (!P. (!f. (!s. ((!t. ((t SUBSET ((IMAGE f) s)) ==> (P t))) = (!t. (((t SUBSET s) /\ (!x. (!y. (((x IN t) /\ (y IN t)) ==> (((f x) = (f y)) = (x = y)))))) ==> (P ((IMAGE f) t))))))))
CONTINUOUS_ON_COMPONENTS_OPEN_EQ |- (!f. (!s. ((open s) ==> (((continuous_on f) s) = (!c. ((c IN (components s)) ==> ((continuous_on f) c)))))))
Leaf_cell.AFF_GE_MONO_TRANS |- (!X. (!Y. (!S. ((S SUBSET X) ==> (((aff_ge (X DIFF S)) (Y UNION S)) SUBSET ((aff_ge X) Y))))))
REALLIM_1_OVER_LOG |- (((---> (\n. (real_inv (log (real_of_num n))))) (real_of_num (NUMERAL _0))) sequentially)
CONTINUOUS_CONTINUOUS_ATREAL |- (!f. (!x. (((continuous f) (atreal x)) = ((continuous (f o drop)) (at (lift x))))))
DIAMETER_ATTAINED_FRONTIER |- (!s. (((bounded s) /\ (~ ((diameter s) = (real_of_num (NUMERAL _0))))) ==> (?x. (?y. ((x IN (frontier s)) /\ ((y IN (frontier s)) /\ ((vector_norm ((vector_sub x) y)) = (diameter s))))))))
CARD_EQ_ARC_IMAGE |- (!g. ((arc g) ==> ((path_image g) =_c UNIV)))
COMPLEX_DERIVATIVE_LMUL_AT |- (!f. (!c. (!z. (((analytic_on f) (z INSERT EMPTY)) ==> (((complex_derivative (\w. ((complex_mul c) (f w)))) z) = ((complex_mul c) ((complex_derivative f) z)))))))
INT_EQ_ADD_LCANCEL |- (!x. (!y. (!z. ((((int_add x) y) = ((int_add x) z)) = (y = z)))))
Collect_geom2.INSERT_SUBSET |- ((EMPTY SUBSET s) /\ (((a INSERT s) SUBSET ss) = ((a IN ss) /\ (s SUBSET ss))))
EXISTS_PATH_SUBPATH_TO_FRONTIER_CLOSED |- (!g. (!s. (((closed s) /\ ((path g) /\ (((pathstart g) IN s) /\ (~ ((pathfinish g) IN s))))) ==> (?h. ((path h) /\ (((pathstart h) = (pathstart g)) /\ (((path_image h) SUBSET ((path_image g) INTER s)) /\ ((pathfinish h) IN (frontier s)))))))))
OPEN_CONTAINS_CBALL_EQ |- (!s. ((open s) ==> (!x. ((x IN s) = (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ ((cball (x , e)) SUBSET s)))))))
TAN_BOUND_PI2 |- (!x. (((real_lt (real_abs x)) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))) ==> ((real_lt (real_abs (tan x))) (real_of_num (NUMERAL (BIT1 _0))))))
REAL_MEASURABLE_REAL_NEGLIGIBLE_SYMDIFF |- (!s. (!t. (((real_measurable s) /\ (real_negligible ((s DIFF t) UNION (t DIFF s)))) ==> (real_measurable t))))
OPEN_MAP_INTERIORS |- (!f. ((!s. ((open s) ==> (open ((IMAGE f) s)))) = (!s. (((IMAGE f) (interior s)) SUBSET (interior ((IMAGE f) s))))))
INTERIOR_BALL |- (!a. (!r. ((interior (ball (a , r))) = (ball (a , r)))))
INT_LE_TRANS_LT |- (!x. (!y. (((int_le x) y) = (!z. (((int_lt y) z) ==> ((int_lt x) z))))))
LOCALLY_NEGLIGIBLE |- (!s. (((locally negligible) s) = (negligible s)))
SUM_INTEGRAL_BOUNDS_INCREASING |- (!f. (!g. (!m. (!n. (((m <= n) /\ ((!x. ((x IN (closed_segment ((CONS ((Cx ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))) , (Cx ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) NIL))) ==> (((has_complex_derivative g) (f x)) (at x)))) /\ (!x. (!y. ((((real_le ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))) x) /\ (((real_le x) y) /\ ((real_le y) ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) ==> ((real_le (Re (f (Cx x)))) (Re (f (Cx y))))))))) ==> (((real_le (Re ((vector_sub (g (Cx (real_of_num n)))) (g (Cx ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))))))) ((sum (m .. n)) (\k. (Re (f (Cx (real_of_num k))))))) /\ ((real_le ((sum (m .. n)) (\k. (Re (f (Cx (real_of_num k))))))) (Re ((vector_sub (g (Cx ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) (g (Cx (real_of_num m))))))))))))
COND_ABS |- (!b. (!f. (!g. ((\x. (((COND b) (f x)) (g x))) = (((COND b) f) g)))))
ABSOLUTELY_CONTINUOUS_ON_NEG |- (!f. (!s. (((absolutely_continuous_on f) s) ==> ((absolutely_continuous_on (\x. (vector_neg (f x)))) s))))
Ssrnat.distn_eq0 |- (!m. (!n. ((((distn m) n) = (NUMERAL _0)) = (m = n))))
HOLOMORPHIC_ON_COMPOSE_GEN |- (!f. (!g. (!s. (!t. ((((holomorphic_on f) s) /\ (((holomorphic_on g) t) /\ (!z. ((z IN s) ==> ((f z) IN t))))) ==> ((holomorphic_on (g o f)) s))))))
REAL_DIFFERENTIABLE_WITHIN_SIN |- (!s. (!x. ((real_differentiable sin) ((within (atreal x)) s))))
REAL_MEASURABLE_ON_CASES |- (!P. (!f. (!g. (!s. (((real_lebesgue_measurable (GSPEC (\GEN%PVAR%5045. (?x. (((SETSPEC GEN%PVAR%5045) (P x)) x))))) /\ (((real_measurable_on f) s) /\ ((real_measurable_on g) s))) ==> ((real_measurable_on (\x. (((COND (P x)) (f x)) (g x)))) s))))))
REAL_INTEGRABLE_DECREASING_PRODUCT |- (!f. (!g. (!a. (!b. ((((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ (!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 y)) (g x)))))) ==> ((real_integrable_on (\x. ((real_mul (g x)) (f x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))
Ckqowsa_3_points.in_aff_ge_0_2_imp_dot_pos |- (!v1. (!v2. (!w. ((((real_lt (real_of_num (NUMERAL _0))) ((dot v1) v2)) /\ ((w IN ((aff_ge ((vec (NUMERAL _0)) INSERT EMPTY)) (v1 INSERT (v2 INSERT EMPTY)))) /\ (~ (w = (vec (NUMERAL _0)))))) ==> (((real_lt (real_of_num (NUMERAL _0))) ((dot v1) w)) /\ ((real_lt (real_of_num (NUMERAL _0))) ((dot v2) w)))))))
REAL_LE_ROOT |- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ ((real_le (real_of_num (NUMERAL _0))) x)) ==> (((real_le x) ((root n) y)) = ((real_le ((real_pow x) n)) y))))))
CROSS_MONO |- (!s. (!t. (!s'. (!t'. (((s SUBSET s') /\ (t SUBSET t')) ==> ((s CROSS t) SUBSET (s' CROSS t')))))))
REAL_INTEGRAL_SUBSET_LE |- (!f. (!s. (!t. (((s SUBSET t) /\ (((real_integrable_on f) s) /\ (((real_integrable_on f) t) /\ (!x. ((x IN t) ==> ((real_le (real_of_num (NUMERAL _0))) (f x))))))) ==> ((real_le ((real_integral s) f)) ((real_integral t) f))))))
Audit_formal_proof.chk_basic_definitions |- ((((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_abs x) = x)) /\ ((((real_lt x) (real_of_num (NUMERAL _0))) ==> ((real_abs x) = (real_neg x))) /\ ((((real_pow x) (NUMERAL _0)) = (real_of_num (NUMERAL (BIT1 _0)))) /\ ((((real_pow x) (SUC n)) = ((real_mul x) ((real_pow x) n))) /\ ((((real_le (real_of_num (NUMERAL _0))) x) ==> (((real_le (real_of_num (NUMERAL _0))) (sqrt x)) /\ (((real_pow (sqrt x)) (NUMERAL (BIT0 (BIT1 _0)))) = x))) /\ ((((sum ((NUMERAL _0) .. (NUMERAL _0))) f) = (f (NUMERAL _0))) /\ ((((sum ((NUMERAL _0) .. (SUC n))) f) = ((real_add ((sum ((NUMERAL _0) .. n)) f)) (f (SUC n)))) /\ (((real_le (real_abs ((real_sub ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))) ((sum ((NUMERAL _0) .. n)) (\i. ((real_div ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) i)) (real_of_num (((NUMERAL (BIT0 (BIT1 _0))) * i) + (NUMERAL (BIT1 _0)))))))))) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 (BIT1 _0))))))) /\ (((A HAS_SIZE n) = ((FINITE A) /\ ((CARD A) = n))) /\ ((EMPTY HAS_SIZE (NUMERAL _0)) /\ (((a INSERT EMPTY) HAS_SIZE (NUMERAL (BIT1 _0))) /\ ((((A HAS_SIZE m) /\ ((B HAS_SIZE n) /\ ((A INTER B) HAS_SIZE p))) ==> ((A UNION B) HAS_SIZE (m + (n - p)))) /\ (((triple_of_real3 o real3_of_triple) = (\w. w)) /\ (((real3_of_triple o triple_of_real3) = (\v. v)) /\ (((vec (NUMERAL _0)) = (real3_of_triple ((real_of_num (NUMERAL _0)) , ((real_of_num (NUMERAL _0)) , (real_of_num (NUMERAL _0)))))) /\ (((distance ((real3_of_triple (x , (y , z))) , (real3_of_triple (x' , (y' , z'))))) = (sqrt ((real_add ((real_pow ((real_sub x) x')) (NUMERAL (BIT0 (BIT1 _0))))) ((real_add ((real_pow ((real_sub y) y')) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow ((real_sub z) z')) (NUMERAL (BIT0 (BIT1 _0)))))))) /\ ((packing V) = (!u. (!v. (((u IN V) /\ ((v IN V) /\ ((real_lt (distance (u , v))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> (u = v)))))))))))))))))))))
REAL_POLYNOMIAL_FUNCTION_EXPLICIT_NZ |- (!f. ((real_polynomial_function f) = (?s. (?a. ((FINITE s) /\ ((!k. ((k IN s) ==> (~ ((a k) = (real_of_num (NUMERAL _0)))))) /\ (f = (\x. ((sum s) (\k. ((real_mul (a k)) ((product ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. ((real_pow (x $ i)) (k $ i)))))))))))))))
HAS_BOUNDED_REAL_VARIATION_ON_RMUL |- (!f. (!c. (!s. (((has_bounded_real_variation_on f) s) ==> ((has_bounded_real_variation_on (\x. ((real_mul (f x)) c))) s)))))
MATRIX_INV_EQ_0 |- (!A. (((matrix_inv A) = (mat (NUMERAL _0))) = (A = (mat (NUMERAL _0)))))
SIMPLICIAL_COMPLEX_DISJOINT_RELATIVE_INTERIORS |- (!t. (!c. (!d. (((simplicial_complex t) /\ ((c IN t) /\ ((d IN t) /\ (~ (c = d))))) ==> (((relative_interior c) INTER (relative_interior d)) = EMPTY)))))
HOMEOMORPHIC_CBALLS |- (!a. (!b. (!d. (!e. ((((real_lt (real_of_num (NUMERAL _0))) d) /\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> ((homeomorphic (cball (a , d))) (cball (b , e))))))))
CTAN_DOUBLE |- (!z. (((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) /\ (~ ((ccos ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) z)) = (Cx (real_of_num (NUMERAL _0)))))) ==> ((ctan ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) z)) = ((complex_div ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (ctan z))) ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow (ctan z)) (NUMERAL (BIT0 (BIT1 _0)))))))))
EVENTUALLY_AT_INFINITY_POS |- (!p. (((eventually p) at_infinity) = (?b. (((real_lt (real_of_num (NUMERAL _0))) b) /\ (!x. (((real_ge (vector_norm x)) b) ==> (p x)))))))
TAGGED_DIVISION_UNIONS_EXISTS |- (!d. (!iset. (!i. (((FINITE iset) /\ ((!i. ((i IN iset) ==> (?p. (((tagged_division_of p) i) /\ ((fine d) p))))) /\ ((!i1. (!i2. (((i1 IN iset) /\ ((i2 IN iset) /\ (~ (i1 = i2)))) ==> (((interior i1) INTER (interior i2)) = EMPTY)))) /\ ((UNIONS iset) = i)))) ==> (?p. (((tagged_division_of p) i) /\ ((fine d) p)))))))
Betwn_corek_z_x.core_rot |- (((good_list L) /\ (((all (\x. ((MEM x) (list_of_darts L)))) r) /\ ((is_edge_nondegenerate (hypermap_of_list L)) /\ ((uniq r) /\ ((final_list L) r))))) ==> (((core L) ((rot n) r)) = ((rot n) ((core L) r))))
COMPONENTS_OPEN_UNIQUE |- (!f. (!s. (((!c. ((c IN f) ==> ((open c) /\ ((connected c) /\ (~ (c = EMPTY)))))) /\ (((pairwise DISJOINT) f) /\ ((UNIONS f) = s))) ==> ((components s) = f))))
Ssrbool.andbK |- (!a. (!b. (((a /\ b) \/ a) = a)))
INT_MIN |- (!x. (!y. (((int_min x) y) = (((COND ((int_le x) y)) x) y))))
NORM_CPOW |- (!w. (!z. (((real w) /\ ((real_lt (real_of_num (NUMERAL _0))) (Re w))) ==> ((vector_norm ((cpow w) z)) = ((rpow (vector_norm w)) (Re z))))))
COMPLEX_DIFFSQ |- (!x. (!y. (((complex_mul ((vector_add x) y)) ((vector_sub x) y)) = ((vector_sub ((complex_mul x) x)) ((complex_mul y) y)))))
LIM_NULL_COMPLEX_MUL |- (!net. (!f. (!g. (((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) /\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) net)) ==> (((--> (\x. ((complex_mul (f x)) (g x)))) (Cx (real_of_num (NUMERAL _0)))) net)))))
REAL_NEGLIGIBLE_TRANSLATION_EQ |- (!s. (!a. ((real_negligible ((IMAGE (\x. ((real_add a) x))) s)) = (real_negligible s))))
NSUM_OFFSET_0 |- (!f. (!m. (!n. ((m <= n) ==> (((nsum (m .. n)) f) = ((nsum ((NUMERAL _0) .. (n - m))) (\i. (f (i + m)))))))))
BOUNDED_HYPERPLANE_EQ_TRIVIAL |- (!a. (!b. ((bounded (GSPEC (\GEN%PVAR%1578. (?x. (((SETSPEC GEN%PVAR%1578) (((dot a) x) = b)) x))))) = (((COND (a = (vec (NUMERAL _0)))) (~ (b = (real_of_num (NUMERAL _0))))) ((dimindex UNIV) = (NUMERAL (BIT1 _0)))))))
FRONTIER_INTER_SUBSET_INTER |- (!s. (!t. ((frontier (s INTER t)) SUBSET (((closure s) INTER (frontier t)) UNION ((frontier s) INTER (closure t))))))
INTEGRABLE_SUBINTERVALS_IMP_REAL_MEASURABLE |- (!f. ((!a. (!b. ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))))) ==> ((real_measurable_on f) UNIV)))
Functional_equation.functional_proj_y3 |- (proj_y3 = (rotate3 proj_y1))
REAL_MEASURABLE_ON_PREIMAGE_HALFSPACE_GE |- (!f. (((real_measurable_on f) UNIV) = (!a. (real_lebesgue_measurable (GSPEC (\GEN%PVAR%5049. (?x. (((SETSPEC GEN%PVAR%5049) ((real_ge (f x)) a)) x))))))))
HAS_REAL_INTEGRAL_EQ_EQ |- (!f. (!g. (!k. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> ((((has_real_integral f) k) s) = (((has_real_integral g) k) s)))))))
PATH_IMAGE_SYM |- (!p. (!q. ((((pathfinish p) = (pathstart q)) /\ ((pathfinish q) = (pathstart p))) ==> ((path_image ((++ p) q)) = (path_image ((++ q) p))))))
INTEGRAL_PASTECART_CONTINUOUS |- (!f. (!a. (!b. (!c. (!d. (((continuous_on f) (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) ==> (((integral (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) f) = ((integral (closed_interval ((CONS (a , b)) NIL))) (\x. ((integral (closed_interval ((CONS (c , d)) NIL))) (\y. (f ((pastecart x) y)))))))))))))
PROPER_MAP_FROM_COMPOSITION_RIGHT |- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET u) /\ (!k. (((k SUBSET u) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%740. (?x. (((SETSPEC GEN%PVAR%740) ((x IN s) /\ (((g o f) x) IN k))) x))))))))))) ==> (!k. (((k SUBSET t) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%741. (?x. (((SETSPEC GEN%PVAR%741) ((x IN s) /\ ((f x) IN k))) x)))))))))))))
Localization.wedge_in_fan_ge |- (!E. (!v. (!w. (((wedge_in_fan_ge (v , w)) E) = (((COND ((CARD ((EE v) E)) > (NUMERAL (BIT1 _0)))) ((((wedge_ge (vec (NUMERAL _0))) v) w) ((((azim_cycle ((EE v) E)) (vec (NUMERAL _0))) v) w))) (GSPEC (\GEN%PVAR%7533. (?x. (((SETSPEC GEN%PVAR%7533) T) x)))))))))
LEBESGUE_MEASURABLE_ALMOST_GDELTA |- (!s. ((lebesgue_measurable s) ==> (?c. (?t. ((gdelta c) /\ ((negligible t) /\ (((s UNION t) = c) /\ ((DISJOINT t) s))))))))
AR_INTERVAL |- ((!a. (!b. ((AR (closed_interval ((CONS (a , b)) NIL))) = (~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\ (!a. (!b. ((AR (open_interval (a , b))) = (~ ((open_interval (a , b)) = EMPTY))))))
RELATIVE_INTERIOR_RELATIVE_INTERIOR |- (!s. ((relative_interior (relative_interior s)) = (relative_interior s)))
MEASURABLE_SMALL_IMP_NEGLIGIBLE |- (!s. (((measurable s) /\ (s <_c UNIV)) ==> (negligible s)))
LEBESGUE_DENSITY_THEOREM_INTEGRAL_BALL |- (!f. (!s. ((!a. (!b. ((absolutely_integrable_on f) (s INTER (closed_interval ((CONS (a , b)) NIL)))))) ==> (?k. ((negligible k) /\ (!x. ((~ (x IN k)) ==> (((--> (\e. ((% (real_inv (measure (ball (x , (drop e)))))) ((integral (s INTER (ball (x , (drop e))))) f)))) (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))) ((within (at (vec (NUMERAL _0)))) (GSPEC (\GEN%PVAR%3725. (?t. (((SETSPEC GEN%PVAR%3725) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t)))))))))))))
CCOS_DOUBLE_CCOS |- (!z. ((ccos ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) z)) = ((vector_sub ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((complex_pow (ccos z)) (NUMERAL (BIT0 (BIT1 _0)))))) (Cx (real_of_num (NUMERAL (BIT1 _0)))))))
LINEAR_FRECHET_DERIVATIVE |- (!f. (!net. (((differentiable f) net) ==> (linear ((frechet_derivative f) net)))))
HOLOMORPHIC_ON_CACS |- (!s. ((!z. (((z IN s) /\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((holomorphic_on cacs) s)))
DIFFERENTIABLE_REAL_COMPLEX |- (!f. (!a. (((complex_differentiable f) (at (Cx (drop a)))) ==> ((differentiable (\x. (f (Cx (drop x))))) (at a)))))
COMPLEX_L1_LE_NORM |- (!z. ((real_le ((real_mul ((real_div (sqrt (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add (real_abs (Re z))) (real_abs (Im z))))) (vector_norm z)))
Merge_ineq.nonf_sol_euler156_x_div_sqrtdelta |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((sol_euler156_x_div_sqrtdelta x1) x2) x3) x4) x5) x6) = ((LET (\a. (LET_END ((real_div (matan ((real_div ((((((delta_x x1) x2) x3) x4) x5) x6)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_pow a) (NUMERAL (BIT0 (BIT1 _0)))))))) a)))) ((real_add (sqrt ((real_mul x6) ((real_mul x1) x5)))) ((real_add ((real_mul (sqrt x6)) ((real_div ((real_add x1) ((real_sub x5) x3))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_add ((real_mul (sqrt x1)) ((real_div ((real_add x6) ((real_sub x5) x4))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul (sqrt x5)) ((real_div ((real_add x6) ((real_sub x1) x2))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))))))
INDEFINITE_INTEGRAL_UNIFORMLY_CONTINUOUS_EXPLICIT |- (!f. (!a. (!b. (!e. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?k. (((real_lt (real_of_num (NUMERAL _0))) k) /\ (!c. (!d. (!c'. (!d'. (((c IN (closed_interval ((CONS (a , b)) NIL))) /\ ((d IN (closed_interval ((CONS (a , b)) NIL))) /\ ((c' IN (closed_interval ((CONS (a , b)) NIL))) /\ ((d' IN (closed_interval ((CONS (a , b)) NIL))) /\ (((real_le (vector_norm ((vector_sub c') c))) k) /\ ((real_le (vector_norm ((vector_sub d') d))) k)))))) ==> ((real_lt (vector_norm ((vector_sub ((integral (closed_interval ((CONS (c' , d')) NIL))) f)) ((integral (closed_interval ((CONS (c , d)) NIL))) f)))) e)))))))))))))
MULT_DIV_LE |- (!m. (!n. (!p. ((~ (p = (NUMERAL _0))) ==> ((m * (n DIV p)) <= ((m * n) DIV p))))))
ABSOLUTELY_REAL_INTEGRABLE_INF |- (!fs. (!s. (!k. (((FINITE k) /\ ((~ (k = EMPTY)) /\ (!i. ((i IN k) ==> ((absolutely_real_integrable_on (\x. ((fs x) i))) s))))) ==> ((absolutely_real_integrable_on (\x. (inf ((IMAGE (fs x)) k)))) s)))))
REAL_CONVEX_ON_EQ |- (!f. (!g. (!s. (((is_realinterval s) /\ ((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((real_convex_on f) s))) ==> ((real_convex_on g) s)))))
HAS_BOUNDED_REAL_VARIATION_DARBOUX_STRICT |- (!f. (!a. (!b. (((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , b)) NIL))) = (?g. (?h. ((!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_lt x) y))) ==> ((real_lt (g x)) (g y))))) /\ ((!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_lt x) y))) ==> ((real_lt (h x)) (h y))))) /\ (!x. ((f x) = ((real_sub (g x)) (h x))))))))))))
ORTHOGONAL_TO_ORTHOGONAL_2D |- (!x. (!y. (!z. (((~ (x = (vec (NUMERAL _0)))) /\ (((orthogonal x) y) /\ ((orthogonal x) z))) ==> (collinear ((vec (NUMERAL _0)) INSERT (y INSERT (z INSERT EMPTY))))))))
PERMUTES_INDUCT |- (!P. (!s. (((FINITE s) /\ ((P I) /\ (!a. (!b. (!p. (((a IN s) /\ ((b IN s) /\ ((P p) /\ (permutation p)))) ==> (P ((swap (a , b)) o p)))))))) ==> (!p. (((permutes p) s) ==> (P p))))))