hcju/holps huggingface dataset:
Lp_ineqs_proofs.INJ_pair |- (!h1. (!h2. (!g. ((!x. ((g x) = ((h1 (FST x)) , (h2 (SND x))))) ==> (!s1. (!t1. (!s2. (!t2. (((((INJ h1) s1) t1) /\ (((INJ h2) s2) t2)) ==> (((INJ g) (s1 CROSS s2)) (t1 CROSS t2)))))))))))
IN_COMPONENTS_UNIONS_COMPLEMENT |- (!s. (!c. ((c IN (components s)) ==> ((s DIFF c) = (UNIONS ((components s) DELETE c))))))
Collect_geom2.EQ_IMP_COPLANAR |- (!a. (!b. (!c. (!d. (((a = b) \/ ((a = c) \/ (a = d))) ==> (coplanar_alt (a INSERT (b INSERT (c INSERT (d INSERT EMPTY))))))))))
HOMOTOPIC_INTO_RETRACT |- (!f. (!g. (!s. (!t. (!u. (((((IMAGE f) s) SUBSET t) /\ ((((IMAGE g) s) SUBSET t) /\ (((retract_of t) u) /\ ((((homotopic_with (\x. T)) (s , u)) f) g)))) ==> ((((homotopic_with (\x. T)) (s , t)) f) g)))))))
HAS_BOUNDED_VARIATION_COMPOSE_INJECTIVE |- (!f. (!g. (!s. (((is_interval s) /\ (((continuous_on f) s) /\ ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) /\ ((has_bounded_variation_on g) ((IMAGE f) s))))) ==> ((has_bounded_variation_on (g o f)) s)))))
SUMS_INTERVALS |- ((!a. (!b. (!c. (!d. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (~ ((closed_interval ((CONS (c , d)) NIL)) = EMPTY))) ==> ((GSPEC (\GEN%PVAR%972. (?x. (?y. (((SETSPEC GEN%PVAR%972) ((x IN (closed_interval ((CONS (a , b)) NIL))) /\ (y IN (closed_interval ((CONS (c , d)) NIL))))) ((vector_add x) y)))))) = (closed_interval ((CONS (((vector_add a) c) , ((vector_add b) d))) NIL)))))))) /\ (!a. (!b. (!c. (!d. (((~ ((open_interval (a , b)) = EMPTY)) /\ (~ ((open_interval (c , d)) = EMPTY))) ==> ((GSPEC (\GEN%PVAR%973. (?x. (?y. (((SETSPEC GEN%PVAR%973) ((x IN (open_interval (a , b))) /\ (y IN (open_interval (c , d))))) ((vector_add x) y)))))) = (open_interval (((vector_add a) c) , ((vector_add b) d))))))))))
Functional_equation.functional_acs_sqrt_x2_d4 |- (acs_sqrt_x2_d4 = (uni (acs , ((scalar6 proj_y2) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))
LOCALLY_COMPACT_INTER_CBALLS |- (!s. (((locally compact) s) = (!x. ((x IN s) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!d. (((real_le d) e) ==> (closed ((cball (x , d)) INTER s))))))))))
REAL_SERIES_DIFFS |- (!f. (!k. ((((---> f) (real_of_num (NUMERAL _0))) sequentially) ==> (((real_sums (\n. ((real_sub (f n)) (f (n + (NUMERAL (BIT1 _0))))))) (f k)) (from k)))))
REAL_POWER_SERIES_CONV_IMP_ABSCONV |- (!a. (!k. (!w. (!z. ((((real_summable k) (\n. ((real_mul (a n)) ((real_pow z) n)))) /\ ((real_lt (real_abs w)) (real_abs z))) ==> ((real_summable k) (\n. (real_abs ((real_mul (a n)) ((real_pow w) n))))))))))
Tame_list.nth_EL |- (!x. (!xs. (!n. (!k. (((n = (NUMERAL _0)) ==> ((((\l. (\n. ((EL n) l))) ((CONS x) xs)) n) = x)) /\ ((n = (SUC k)) ==> ((((\l. (\n. ((EL n) l))) ((CONS x) xs)) n) = (((\l. (\n. ((EL n) l))) xs) k))))))))
NPRODUCT_ONE |- (!s. (((nproduct s) (\n. (NUMERAL (BIT1 _0)))) = (NUMERAL (BIT1 _0))))
SUP_INSERT |- (!x. (!s. ((bounded ((IMAGE lift) s)) ==> ((sup (x INSERT s)) = (((COND (s = EMPTY)) x) ((real_max x) (sup s)))))))
RELPOW_M |- (!m. (!n. (!x. (!y. (((((RELPOW (m + n)) R) x) y) = (?z. (((((RELPOW m) R) x) z) /\ ((((RELPOW n) R) z) y))))))))
REAL_CONTINUOUS_ON_CASES |- (!P. (!f. (!g. (!s. (!t. (((real_closed s) /\ ((real_closed t) /\ (((real_continuous_on f) s) /\ (((real_continuous_on g) t) /\ (!x. ((((x IN s) /\ (~ (P x))) \/ ((x IN t) /\ (P x))) ==> ((f x) = (g x)))))))) ==> ((real_continuous_on (\x. (((COND (P x)) (f x)) (g x)))) (s UNION t))))))))
List_hypermap_iso.fan_list_nodes_iso_inj |- (!L. (!f. (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((good_list L) ==> (((hyp_iso f) ((hypermap_of_fan (V , E)) , (hypermap_of_list L))) ==> ((good_list_nodes L) ==> (!x. (!y. (((x IN V) /\ ((y IN V) /\ (((fan_list_nodes_iso (f , (V , E))) x) = ((fan_list_nodes_iso (f , (V , E))) y)))) ==> (x = y))))))))))))
List_hypermap_iso.fan_list_nodes_iso_surj |- (!f. (!E. (!V. (!L. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((good_list L) ==> (((hyp_iso f) ((hypermap_of_fan (V , E)) , (hypermap_of_list L))) ==> (((SURJ (fan_list_nodes_iso (f , (V , E)))) V) (elements_of_list L)))))))))
REAL_INTEGRABLE_ON_REFL |- (!f. (!a. ((real_integrable_on f) (closed_real_interval ((CONS (a , a)) NIL)))))
REAL_CONTINUOUS_WITHIN_SQRT |- (!s. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_continuous sqrt) ((within (atreal x)) s)))))
LIM_LOG_OVER_Z |- (((--> (\z. ((complex_div (clog z)) z))) (Cx (real_of_num (NUMERAL _0)))) at_infinity)
CONIC_PCROSS_EQ |- (!s. (!t. ((conic (s PCROSS t)) = ((s = EMPTY) \/ ((t = EMPTY) \/ ((conic s) /\ (conic t)))))))
RSTC_TRANS_R |- (!R. (!x. (!y. (!z. ((((R x) y) /\ (((RSTC R) y) z)) ==> (((RSTC R) x) z))))))
Fatugpd.dot_normalize |- (!v. (((dot v) (normalize v)) = (vector_norm v)))
REAL_CONVEX_ON_LEFT_SECANT |- (!f. (!s. (((real_convex_on f) s) = (!a. (!b. (!x. (((a IN s) /\ ((b IN s) /\ (x IN (open_real_segment (a , b))))) ==> ((real_le ((real_div ((real_sub (f x)) (f a))) (real_abs ((real_sub x) a)))) ((real_div ((real_sub (f b)) (f a))) (real_abs ((real_sub b) a)))))))))))
ANALYTIC_AT_ADD |- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\ ((analytic_on g) (z INSERT EMPTY))) ==> ((analytic_on (\w. ((vector_add (f w)) (g w)))) (z INSERT EMPTY))))))
Lp_gen_theory.hyp_iso_face_set_k |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!k. (((face_set_k k) H) = ((IMAGE (IMAGE g)) ((face_set_k k) G))))))))
INT_SGN_NEG |- (!x. ((int_sgn (int_neg x)) = (int_neg (int_sgn x))))
COMPLEX_EQ_NEG2 |- (!x. (!y. (((vector_neg x) = (vector_neg y)) = (x = y))))
INT_SUB_TRIANGLE |- (!a. (!b. (!c. (((int_add ((int_sub a) b)) ((int_sub b) c)) = ((int_sub a) c)))))
FRONTIER_UNIONS_SUBSET |- (!f. ((FINITE f) ==> ((frontier (UNIONS f)) SUBSET (UNIONS (GSPEC (\GEN%PVAR%1826. (?t. (((SETSPEC GEN%PVAR%1826) (t IN f)) (frontier t)))))))))
REAL_LOG_CONVEX_IMP_CONVEX |- (!f. (!s. (((real_log_convex_on f) s) ==> ((real_convex_on f) s))))
Lp_main_estimate.ineq_tau3 |- (!V. (!E. ((lp_fan (V , E)) ==> ((!v1. (!v2. (!v3. ((((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v1)) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v2)) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v3)) /\ (((real_le (vector_norm v1)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ (((real_le (vector_norm v2)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ (((real_le (vector_norm v3)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v1 , v2))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v1 , v3))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v2 , v3))) /\ (((real_le (distance (v1 , v2))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ (((real_le (distance (v1 , v3))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\ ((real_le (distance (v2 , v3))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))))))))))))) ==> ((real_le (real_of_num (NUMERAL _0))) (((tau3 v1) v2) v3)))))) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT1 _0)))) (hypermap_of_fan (V , E)))) ==> ((real_le (real_of_num (NUMERAL _0))) ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)))))))))
REAL_INTEGRAL_NEG |- (!f. (!s. (((real_integrable_on f) s) ==> (((real_integral s) (\x. (real_neg (f x)))) = (real_neg ((real_integral s) f))))))
Lp_ineqs_proofs.RHB |- (!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 ((rhazim_fan (V , E)) d)) ((real_mul ((azim_dart (V , E)) d)) ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_div sol0) pi))))))))))))
PATH_LENGTH_EQ_LINE_SEGMENT |- (!g. (((rectifiable_path g) /\ ((path_length g) = (distance ((pathstart g) , (pathfinish g))))) ==> ((path_image g) = (closed_segment ((CONS ((pathstart g) , (pathfinish g))) NIL)))))
VSUM_CMUL_NUMSEG |- (!f. (!c. (!m. (!n. (((vsum (m .. n)) (\x. ((% c) (f x)))) = ((% c) ((vsum (m .. n)) f)))))))
CONNECTED_OPEN_VECTOR_POLYNOMIAL_CONNECTED |- (!s. (((open s) /\ (connected s)) ==> (!x. (!y. (((x IN s) /\ (y IN s)) ==> (?g. ((vector_polynomial_function g) /\ (((path_image g) SUBSET s) /\ (((pathstart g) = x) /\ ((pathfinish g) = y))))))))))
INT_MAX_ACI |- ((((int_max x) y) = ((int_max y) x)) /\ ((((int_max ((int_max x) y)) z) = ((int_max x) ((int_max y) z))) /\ ((((int_max x) ((int_max y) z)) = ((int_max y) ((int_max x) z))) /\ ((((int_max x) x) = x) /\ (((int_max x) ((int_max x) y)) = ((int_max x) y))))))
Functional_equation.functional_proj_y4 |- (proj_y4 = (rotate4 proj_y1))
COMPACT_IN_IFF_COMPACT |- (!s. (((compact_in s) euclidean) = (compact s)))
INFSUM_NEG |- (!s. (!x. (((summable s) x) ==> (((infsum s) (\n. (vector_neg (x n)))) = (vector_neg ((infsum s) x))))))
KERNEL_MATRIX_INV |- (!A. ((GSPEC (\GEN%PVAR%227. (?x. (((SETSPEC GEN%PVAR%227) (((matrix_vector_mul (matrix_inv A)) x) = (vec (NUMERAL _0)))) x)))) = (GSPEC (\GEN%PVAR%228. (?x. (((SETSPEC GEN%PVAR%228) (((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0)))) x))))))
Ssrbool.orbK |- (!a. (!b. (((a \/ b) /\ a) = a)))
GATEAUX_DERIVATIVE |- (!f. (!f'. (!x. (!y. ((((has_derivative f) f') (at x)) ==> (((--> (\t. ((% (real_inv (drop t))) ((vector_sub (f ((vector_add x) ((% (drop t)) y)))) (f x))))) (f' y)) (at (vec (NUMERAL _0)))))))))
CONNECTED_PUNCTURED_CONVEX |- (!s. (!a. (((convex s) /\ (~ ((aff_dim s) = (int_of_num (NUMERAL (BIT1 _0)))))) ==> (connected (s DELETE a)))))
Hypermap_iso.surj_ext |- (!f. (!g. (!s. (!t. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> ((((SURJ f) s) t) ==> (((SURJ g) s) t)))))))
SYMMETRIX_MATRIX_CONJUGATE |- (!A. (!B. (((transp B) = B) ==> ((transp ((matrix_mul (transp A)) ((matrix_mul B) A))) = ((matrix_mul (transp A)) ((matrix_mul B) A))))))
NSUM_BIJECTION |- (!f. (!p. (!s. (((!x. ((x IN s) ==> ((p x) IN s))) /\ (!y. ((y IN s) ==> (?!x. ((x IN s) /\ ((p x) = y)))))) ==> (((nsum s) f) = ((nsum s) (f o p)))))))
CONTINUOUS_ON_CSQRT |- (!s. ((!z. (((z IN s) /\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z)))) ==> ((continuous_on csqrt) s)))
Tame_list.sub_nth |- (((sub a) n) = ((#!# a) n))
Appendix.scs_unadorned |- (!s. ((is_scs_v39 s) ==> (is_scs_v39 (scs_v39 ((scs_k_v39 s) , ((scs_d_v39 s) , ((scs_a_v39 s) , ((scs_a_v39 s) , ((scs_b_v39 s) , ((scs_b_v39 s) , ((scs_J_v39 s) , (EMPTY , (EMPTY , EMPTY)))))))))))))
EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL |- (!s. (((FINITE s) /\ (((hull affine) s) = UNIV)) ==> ((GSPEC (\GEN%PVAR%1794. (?y. (((SETSPEC GEN%PVAR%1794) (?u. ((!x. ((x IN s) ==> (((real_lt (real_of_num (NUMERAL _0))) (u x)) /\ ((real_lt (u x)) (real_of_num (NUMERAL (BIT1 _0))))))) /\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum s) (\x. ((% (u x)) x))) = y))))) y)))) SUBSET (interior ((hull convex) s)))))
CONTINUOUS_INJECTIVE_IMAGE_OPEN_SEGMENT_1 |- (!f. (!a. (!b. ((((continuous_on f) (closed_segment ((CONS (a , b)) NIL))) /\ (!x. (!y. (((x IN (closed_segment ((CONS (a , b)) NIL))) /\ ((y IN (closed_segment ((CONS (a , b)) NIL))) /\ ((f x) = (f y)))) ==> (x = y))))) ==> (((IMAGE f) (open_segment (a , b))) = (open_segment ((f a) , (f b))))))))
HAS_MEASURE_ALMOST_EQ |- (!s. (!s'. (!t. (((negligible t) /\ ((s UNION t) = (s' UNION t))) ==> (((has_measure s) m) = ((has_measure s') m))))))
OPEN_IN_SEGMENT |- (!s. (!a. (!b. ((((open_segment (a , b)) SUBSET s) /\ (s SUBSET ((hull affine) (open_segment (a , b))))) ==> ((open_in ((subtopology euclidean) s)) (open_segment (a , b)))))))
Functional_equation.functional_vol3f_x_lfun |- (((domain6 (\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) ((sub6 ((mul6 (constant6 ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div mm1) pi)))) ((add6 ((mul6 two6) (mk_126 dih_x))) ((add6 ((mul6 two6) (mk_126 dih2_x))) ((add6 ((mul6 two6) (mk_126 dih6_x))) ((add6 (mk_126 dih3_x)) ((add6 (mk_126 dih4_x)) ((sub6 (mk_126 dih5_x)) (constant6 ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) pi)))))))))) ((mul6 (constant6 ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) ((real_div mm2) pi)))) ((add6 (mk_126 ldih_x)) ((add6 (mk_126 ldih2_x)) (mk_126 ldih6_x))))))
BOUNDED_PARTIAL_REAL_SUMS |- (!f. (!k. ((real_bounded (GSPEC (\GEN%PVAR%4868. (?n. (((SETSPEC GEN%PVAR%4868) (n IN UNIV)) ((sum (k .. n)) f)))))) ==> (real_bounded (GSPEC (\GEN%PVAR%4869. (?m. (?n. (((SETSPEC GEN%PVAR%4869) ((m IN UNIV) /\ (n IN UNIV))) ((sum (m .. n)) f))))))))))
TRANS_ALT |- (!R. (!S. (!U. ((!x. (!z. ((?y. (((R x) y) /\ ((S y) z))) ==> ((U x) z)))) = (!x. (!y. (!z. ((((R x) y) /\ ((S y) z)) ==> ((U x) z)))))))))
Ssrnat.maxn_mull |- (!m1. (!m2. (!n. ((((maxn m1) m2) * n) = ((maxn (m1 * n)) (m2 * n))))))
ISO_FUN |- ((((ISO f) f') /\ ((ISO g) g')) ==> ((ISO (\h. (\a'. (g (h (f' a')))))) (\h. (\a. (g' (h (f a)))))))
FRECHET_DERIVATIVE_CONST_AT |- (!c. (!a. (((frechet_derivative (\x. c)) (at a)) = (\h. (vec (NUMERAL _0))))))
COMPLEX_SUB_POW_L1 |- (!x. (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow x) n)) = ((complex_mul ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) x)) ((vsum ((NUMERAL _0) .. (n - (NUMERAL (BIT1 _0))))) (\i. ((complex_pow x) i))))))))
Trigonometry.LLOYXRK2 |- (!x. ((sin ((real_add ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) x)) = (cos x)))
REAL_INTEGRABLE_REFLECT |- (!f. (!a. (!b. (((real_integrable_on (\x. (f (real_neg x)))) (closed_real_interval ((CONS ((real_neg b) , (real_neg a))) NIL))) = ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL)))))))
POLYTOPE_PCROSS_EQ |- (!s. (!t. ((polytope (s PCROSS t)) = ((s = EMPTY) \/ ((t = EMPTY) \/ ((polytope s) /\ (polytope t)))))))
COLLINEAR_CONVEX_HULL_COLLINEAR |- (!s. ((collinear ((hull convex) s)) = (collinear s)))
SETDIST_LE_HAUSDIST |- (!s. (!t. (((bounded s) /\ (bounded t)) ==> ((real_le (setdist (s , t))) (hausdist (s , t))))))
REAL_DIFFERENTIABLE_ON_COMPOSE |- (!f. (!g. (!s. ((((real_differentiable_on f) s) /\ ((real_differentiable_on g) ((IMAGE f) s))) ==> ((real_differentiable_on (g o f)) s)))))
Sphere.vol_solid_triangle |- (!r. (!v0. (!v1. (!v2. (!v3. ((((((vol_solid_triangle v0) v1) v2) v3) r) = ((LET (\a123. (LET_END ((LET (\a231. (LET_END ((LET (\a312. (LET_END ((real_mul ((real_add a123) ((real_add a231) ((real_sub a312) pi)))) ((real_div ((real_pow r) (NUMERAL (BIT1 (BIT1 _0))))) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))))) ((((dihV v0) v3) v1) v2))))) ((((dihV v0) v2) v3) v1))))) ((((dihV v0) v1) v2) v3))))))))
REALLIM_AT_POSINFINITY |- (!f. (!l. ((((---> f) l) at_posinfinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_ge x) b) ==> ((real_lt (real_abs ((real_sub (f x)) l))) e)))))))))
FRAC_NEG |- (!x. ((frac (real_neg x)) = (((COND (integer x)) (real_of_num (NUMERAL _0))) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (frac x)))))
Lp_ineqs_proofs.ln_hi |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (ln_fan v)) (real_of_num (NUMERAL (BIT1 _0))))))))))))
REAL_INTEGRABLE_AFFINITY |- (!f. (!a. (!b. (!m. (!c. ((((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ (~ (m = (real_of_num (NUMERAL _0))))) ==> ((real_integrable_on (\x. (f ((real_add ((real_mul m) x)) c)))) ((IMAGE (\x. ((real_mul (real_inv m)) ((real_sub x) c)))) (closed_real_interval ((CONS (a , b)) NIL))))))))))
Hypermap_iso.iso_number_of_edges |- (!f. (!G. (!H. (((hyp_iso f) (H , G)) ==> ((number_of_edges G) = (number_of_edges H))))))
Ssrbool.andb_idl |- (!a. (!b. ((b ==> a) ==> ((a /\ b) = b))))
CONTINUOUS_TRANSFORM_WITHIN_OPEN |- (!f. (!g. (!s. (!a. (((open s) /\ ((a IN s) /\ ((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((continuous f) (at a))))) ==> ((continuous g) (at a)))))))
EXISTS_SUBPATH_OF_ARC_NOENDS |- (!g. (!a. (!b. (((arc g) /\ ((a IN (path_image g)) /\ ((b IN (path_image g)) /\ (((a INSERT (b INSERT EMPTY)) INTER ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))) = EMPTY)))) ==> (?h. ((path h) /\ (((pathstart h) = a) /\ (((pathfinish h) = b) /\ ((path_image h) SUBSET ((path_image g) DIFF ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))))))))))))
REAL_DIFFERENTIABLE_FROM_COMPLEX_AT |- (!f. (!x. ((((complex_differentiable f) (at (Cx x))) /\ (!z. ((real z) ==> (real (f z))))) ==> ((real_differentiable (Re o (f o Cx))) (atreal x)))))
Lp_ineqs_proofs.dart4_tau_fan_eq_tauq |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)) = (((((((((tauq (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y9_fan (V , E)) d))))))))))))
YOUNG_INEQUALITY |- (!a. (!b. (!p. (!q. ((((real_le (real_of_num (NUMERAL _0))) a) /\ (((real_le (real_of_num (NUMERAL _0))) b) /\ (((real_lt (real_of_num (NUMERAL _0))) p) /\ (((real_lt (real_of_num (NUMERAL _0))) q) /\ (((real_add (real_inv p)) (real_inv q)) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> ((real_le ((real_mul a) b)) ((real_add ((real_div ((rpow a) p)) p)) ((real_div ((rpow b) q)) q))))))))
Lp_ineqs_proofs.yy10_std |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (ye_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))
Nonlinear_lemma.pathL_bound |- (!y. (!a. ((((real_le (FST (pathL a))) y) /\ ((real_le y) (SND (pathL a)))) ==> (((real_le (FST a)) y) /\ ((real_le y) (SND a))))))
Tame_list.list_update_ALT |- (((((list_update NIL) i) v) = NIL) /\ ((((list_update ((CONS x) xs)) i) v) = (((COND (i = (NUMERAL _0))) ((CONS v) xs)) ((LET (\j. (LET_END ((CONS x) (((list_update xs) j) v))))) (PRE i)))))
COMPLEX_DIFFERENTIABLE_POW_AT |- (!f. (!n. (!z. (((complex_differentiable f) (at z)) ==> ((complex_differentiable (\z. ((complex_pow (f z)) n))) (at z))))))
COS_EQ_MINUS1 |- (!x. (((cos x) = (real_neg (real_of_num (NUMERAL (BIT1 _0))))) = (?n. ((integer n) /\ (x = ((real_mul ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)) (real_of_num (NUMERAL (BIT1 _0))))) pi))))))
TAN_ABS_GE_X |- (!x. (((real_lt (real_abs x)) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ==> ((real_le (real_abs x)) (real_abs (tan x)))))
REAL_INFSUM_0 |- (((real_infsum s) (\i. (real_of_num (NUMERAL _0)))) = (real_of_num (NUMERAL _0)))
INT_POW_EQ_ABS |- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ (((int_pow x) n) = ((int_pow y) n))) ==> ((int_abs x) = (int_abs y))))))
INT_LE_SQUARE |- (!x. ((int_le (int_of_num (NUMERAL _0))) ((int_mul x) x)))
OPEN_SET_IRRATIONAL_COORDINATES |- (!s. (((open s) /\ (~ (s = EMPTY))) ==> (?x. ((x IN s) /\ (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (~ (rational (x $ i)))))))))
MOD_EXISTS |- (!m. (!n. ((?q. (m = (n * q))) = (((COND (n = (NUMERAL _0))) (m = (NUMERAL _0))) ((m MOD n) = (NUMERAL _0))))))
COBOUNDED_HAS_BOUNDED_COMPONENT |- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ ((bounded (UNIV DIFF s)) /\ (~ (connected s)))) ==> (?c. ((c IN (components s)) /\ (bounded c)))))
Nonlinear_lemma.delta_x_eq0 |- ((((((((delta_x (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = (real_of_num (NUMERAL _0))) /\ (((((((delta_x (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = (real_of_num (NUMERAL _0))))
REAL_MEASURABLE_ON_COMPOSE_CONTINUOUS_CLOSED_SET_0 |- (!f. (!g. (!s. (!t. (((real_closed s) /\ (((real_measurable_on f) t) /\ ((!x. ((f x) IN s)) /\ (((real_continuous_on g) s) /\ (((real_of_num (NUMERAL _0)) IN s) /\ ((g (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))))))) ==> ((real_measurable_on (g o f)) t))))))
UNIFORMLY_CONTINUOUS_ON_ID |- (!s. ((uniformly_continuous_on (\x. x)) s))
ORTHOGONAL_SUBSPACE_DECOMP |- (!s. (!x. (?! (GABS (\f. (!y. (!z. ((GEQ (f (y , z))) ((y IN (span s)) /\ ((z IN (GSPEC (\GEN%PVAR%185. (?z. (((SETSPEC GEN%PVAR%185) (!x. ((x IN (span s)) ==> ((orthogonal z) x)))) z))))) /\ (x = ((vector_add y) z))))))))))))
Ssrnat.leq_pexp2l |- (!m. (!n1. (!n2. (((NUMERAL _0) < m) ==> ((n1 <= n2) ==> ((m EXP n1) <= (m EXP n2)))))))
Lp_ineqs_proofs.dart3_rhazim3_fan_eq_rhazim3 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT1 _0)))) (hypermap_of_fan (V , E)))) ==> (((rhazim3_fan (V , E)) d) = ((((((rhazim3 (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)))))))))))
ORTHOGONAL_TRANSFORMATION_LOWDIM_HORIZONTAL |- (!s. (((dim s) < (dimindex UNIV)) ==> (?f. ((orthogonal_transformation f) /\ (((IMAGE f) s) SUBSET (GSPEC (\GEN%PVAR%457. (?z. (((SETSPEC GEN%PVAR%457) ((z $ (dimindex UNIV)) = (real_of_num (NUMERAL _0)))) z)))))))))
DIV_REFL |- (!n. ((~ (n = (NUMERAL _0))) ==> ((n DIV n) = (NUMERAL (BIT1 _0)))))
Asfutbf.azim_cycle_mono |- (!V. (!E. (!v. (!x. (!x'. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((x IN (((set_of_edge v) V) E)) /\ ((x' IN (((set_of_edge v) V) E)) /\ (((((azim_cycle (((set_of_edge v) V) E)) (vec (NUMERAL _0))) v) x) = ((((azim_cycle (((set_of_edge v) V) E)) (vec (NUMERAL _0))) v) x'))))) ==> (x = x')))))))
SUM_OVER_PERMUTATIONS_NUMSEG |- (!f. (!m. (!n. ((m <= n) ==> (((sum (GSPEC (\GEN%PVAR%297. (?p. (((SETSPEC GEN%PVAR%297) ((permutes p) (m .. n))) p))))) f) = ((sum (m .. n)) (\i. ((sum (GSPEC (\GEN%PVAR%298. (?p. (((SETSPEC GEN%PVAR%298) ((permutes p) ((m + (NUMERAL (BIT1 _0))) .. n))) p))))) (\q. (f ((swap (m , i)) o q)))))))))))
SPHERE_EQ_SING |- (!a. (!r. (!x. (((sphere (a , r)) = (x INSERT EMPTY)) = ((x = a) /\ (r = (real_of_num (NUMERAL _0))))))))
ABSOLUTELY_INTEGRABLE_ON_LINEAR_IMAGE |- (!f. (!g. (!s. ((linear g) ==> (((absolutely_integrable_on f) ((IMAGE g) s)) = (((absolutely_integrable_on (f o g)) s) \/ ((det (matrix g)) = (real_of_num (NUMERAL _0)))))))))
NON_MEASURABLE_SET |- (?s. (~ (lebesgue_measurable s)))
INT_ABS_SIGN2 |- (!x. (!y. (((int_lt (int_abs ((int_sub x) y))) (int_neg y)) ==> ((int_lt x) (int_of_num (NUMERAL _0))))))
Collect_geom2.ZJEWPAP |- (!v1. (!v2. (!v3. (!v4. ((LET (\s. (LET_END ((((CARD s) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\ (~ (coplanar_alt s))) ==> ((real_le (radV (v1 INSERT (v2 INSERT (v3 INSERT EMPTY))))) (radV s)))))) (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))))))
URYSOHN_STRONG |- (!s. (!t. (!a. (!b. (((closed s) /\ ((closed t) /\ (((s INTER t) = EMPTY) /\ (~ (a = b))))) ==> (?f. (((continuous_on f) UNIV) /\ ((!x. ((f x) IN (closed_segment ((CONS (a , b)) NIL)))) /\ ((!x. (((f x) = a) = (x IN s))) /\ (!x. (((f x) = b) = (x IN t))))))))))))
INT_BOUNDS_LT |- (!x. (!k. ((((int_lt (int_neg k)) x) /\ ((int_lt x) k)) = ((int_lt (int_abs x)) k))))
INT_MAX_LT |- (!x. (!y. (!z. (((int_lt ((int_max x) y)) z) = (((int_lt x) z) /\ ((int_lt y) z))))))
CEXP_MUL_CPOW |- (!w. (!z. ((((real_lt (real_neg pi)) (Im w)) /\ ((real_le (Im w)) pi)) ==> ((cexp ((complex_mul w) z)) = ((cpow (cexp w)) z)))))
Lp_gen_theory.hyp_iso_comm_special |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!d. ((d IN (dart G)) ==> ((((face_map H) (g d)) = (g ((face_map G) d))) /\ ((((node_map H) (g d)) = (g ((node_map G) d))) /\ ((((edge_map H) (g d)) = (g ((edge_map G) d))) /\ (!k. ((((POWER (face_map H)) k) (g d)) = (g (((POWER (face_map G)) k) d)))))))))))))