hcju/holps huggingface dataset:
Ssrnat.expn0
|- (!m. ((m EXP (NUMERAL _0)) = (NUMERAL (BIT1 _0))))
IM_DIV_CX
|- (!z. (!x. ((Im ((complex_div z) (Cx x))) = ((real_div (Im z)) x))))
REAL_MAX_LE
|- (!x. (!y. (!z. (((real_le ((real_max x) y)) z) = (((real_le x) z) /\ ((real_le y) z))))))
MATRIX_OF_MATRIX_VECTOR_MUL
|- (!A. ((matrix (\x. ((matrix_vector_mul A) x))) = A))
BOUNDED_CLOSURE
|- (!s. ((bounded s) ==> (bounded (closure s))))
AFFINE_HULL_TRANSLATION
|- (!a. (!s. (((hull affine) ((IMAGE (\x. ((vector_add a) x))) s)) = ((IMAGE (\x. ((vector_add a) x))) ((hull affine) s)))))
INJECTIVE_ON_ALT
|- (!P. (!f. ((!x. (!y. (((P x) /\ ((P y) /\ ((f x) = (f y)))) ==> (x = y)))) = (!x. (!y. (((P x) /\ (P y)) ==> (((f x) = (f y)) = (x = y))))))))
INTERIOR_COMPLEMENT
|- (!s. ((interior (UNIV DIFF s)) = (UNIV DIFF (closure s))))
Add_triangle.BIJ_IMAGE
|- (!f. (!s. (!t. ((((BIJ f) s) t) ==> (t = ((IMAGE f) s))))))
AFF_DIM_CONVEX_HULL
|- (!s. ((aff_dim ((hull convex) s)) = (aff_dim s)))
HAS_REAL_DERIVATIVE_POW_ATREAL
|- (!f. (!f'. (!x. (!n. ((((has_real_derivative f) f') (atreal x)) ==> (((has_real_derivative (\x. ((real_pow (f x)) n))) ((real_mul (real_of_num n)) ((real_mul ((real_pow (f x)) (n - (NUMERAL (BIT1 _0))))) f'))) (atreal x)))))))
POSITIVE_DEFINITE_TRANSP
|- (!A. ((positive_definite (transp A)) = (positive_definite A)))
Add_triangle.sigma_fan_not_fixed
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\ (fully_surrounded (V , E))) ==> (~ ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) = w))))))))
real_integral
|- (!f. (!i. (((real_integral i) f) = (@y. (((has_real_integral f) y) i)))))
Hypermap.face
|- (!H. (!x. (((face H) x) = ((orbit_map (face_map H)) x))))
CARD_EQ_SYM
|- (!s. (!t. ((s =_c t) = (t =_c s))))
Nonlinear_lemma.lmdih5_0
|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y5) ==> ((((((((y_of_x lmdih5_x_div_sqrtdelta_posbranch) y1) y2) y3) y4) y5) y6) = (real_of_num (NUMERAL _0))))))))))
INTERIOR_BIJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\ (!y. (?x. ((f x) = y))))) ==> ((interior ((IMAGE f) s)) = ((IMAGE f) (interior s))))))
Trigonometry.SCEZKRH1
|- (!x. ((sin ((real_sub ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) x)) = (cos x)))
NORM_BOUND_COMPONENT_LT
|- (!x. (!e. (((real_lt (vector_norm x)) e) ==> (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_lt (real_abs (x $ i))) e))))))
IVT_INCREASING_IM
|- (!f. (!a. (!b. (!y. ((((real_le a) b) /\ ((!x. ((((real_le a) x) /\ ((real_le x) b)) ==> ((continuous f) (at (Cx x))))) /\ (((real_le (Im (f (Cx a)))) y) /\ ((real_le y) (Im (f (Cx b))))))) ==> (?x. (((real_le a) x) /\ (((real_le x) b) /\ ((Im (f (Cx x))) = y)))))))))
Seq2.take
|- (!x. (!s. (!n. ((((take (SUC n)) ((CONS x) s)) = ((CONS x) ((take n) s))) /\ ((((take (NUMERAL _0)) s) = NIL) /\ (((take n) NIL) = NIL))))))
Sphere.azim_cycle
|- (!W. (!p. (!w. (!v. (((((azim_cycle W) v) w) p) = (((COND (W SUBSET (p INSERT EMPTY))) p) (@u. ((~ (u = p)) /\ ((W u) /\ (!q. (((~ (q = p)) /\ (W q)) ==> (((real_lt ((((azim v) w) p) u)) ((((azim v) w) p) q)) \/ ((((((azim v) w) p) u) = ((((azim v) w) p) q)) /\ ((real_le (vector_norm ((projection ((vector_sub w) v)) ((vector_sub u) v)))) (vector_norm ((projection ((vector_sub w) v)) ((vector_sub q) v)))))))))))))))))
INT_LE_LNEG
|- (!x. (!y. (((int_le (int_neg x)) y) = ((int_le (int_of_num (NUMERAL _0))) ((int_add x) y)))))
Lp_ineqs_proofs.const1_ge0
|- ((real_le (real_of_num (NUMERAL _0))) const1)
HAS_REAL_INTEGRAL_SUBSET_LE
|- (!f. (!s. (!t. (!i. (!j. (((s SUBSET t) /\ ((((has_real_integral f) i) s) /\ ((((has_real_integral f) j) t) /\ (!x. ((x IN t) ==> ((real_le (real_of_num (NUMERAL _0))) (f x))))))) ==> ((real_le i) j)))))))
LIM_CONST_EQ
|- (!net. (!c. (!d. ((((--> (\x. c)) d) net) = ((trivial_limit net) \/ (c = d))))))
SIN_VECTOR_ANGLE_LMUL
|- (!a. (!x. (!y. ((sin ((vector_angle ((% a) x)) y)) = (((COND (a = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL (BIT1 _0)))) (sin ((vector_angle x) y)))))))
REAL_ABS_NORM
|- (!x. ((real_abs (vector_norm x)) = (vector_norm x)))
CONVEX_ON_COUNTABLE_NONDIFFERENTIABLE
|- (!f. (!s. ((((convex_on f) s) /\ (COUNTABLE (components s))) ==> (COUNTABLE (GSPEC (\GEN%PVAR%3315. (?x. (((SETSPEC GEN%PVAR%3315) ((x IN s) /\ (~ ((differentiable (lift o f)) (at x))))) x))))))))
SPAN_CLAUSES
|- ((!a. (!s. ((a IN s) ==> (a IN (span s))))) /\ (((vec (NUMERAL _0)) IN (span s)) /\ ((!x. (!y. (!s. (((x IN (span s)) /\ (y IN (span s))) ==> (((vector_add x) y) IN (span s)))))) /\ (!x. (!c. (!s. ((x IN (span s)) ==> (((% c) x) IN (span s)))))))))
HAS_MEASURE_TRANSLATION
|- (!s. (!m. (!a. (((has_measure s) m) ==> ((has_measure ((IMAGE (\x. ((vector_add a) x))) s)) m)))))
SUBPATH_LINEAR_IMAGE
|- (!f. (!g. (!u. (!v. ((linear f) ==> ((((subpath u) v) (f o g)) = (f o (((subpath u) v) g))))))))
REAL_SGN_INEQS
|- ((!x. (((real_le (real_of_num (NUMERAL _0))) (real_sgn x)) = ((real_le (real_of_num (NUMERAL _0))) x))) /\ ((!x. (((real_lt (real_of_num (NUMERAL _0))) (real_sgn x)) = ((real_lt (real_of_num (NUMERAL _0))) x))) /\ ((!x. (((real_ge (real_of_num (NUMERAL _0))) (real_sgn x)) = ((real_ge (real_of_num (NUMERAL _0))) x))) /\ ((!x. (((real_gt (real_of_num (NUMERAL _0))) (real_sgn x)) = ((real_gt (real_of_num (NUMERAL _0))) x))) /\ ((!x. (((real_of_num (NUMERAL _0)) = (real_sgn x)) = ((real_of_num (NUMERAL _0)) = x))) /\ ((!x. (((real_le (real_sgn x)) (real_of_num (NUMERAL _0))) = ((real_le x) (real_of_num (NUMERAL _0))))) /\ ((!x. (((real_lt (real_sgn x)) (real_of_num (NUMERAL _0))) = ((real_lt x) (real_of_num (NUMERAL _0))))) /\ ((!x. (((real_ge (real_sgn x)) (real_of_num (NUMERAL _0))) = ((real_ge x) (real_of_num (NUMERAL _0))))) /\ ((!x. (((real_gt (real_sgn x)) (real_of_num (NUMERAL _0))) = ((real_gt x) (real_of_num (NUMERAL _0))))) /\ (!x. (((real_sgn x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0))))))))))))))
COMPACT_FRONTIER
|- (!s. ((compact s) ==> (compact (frontier s))))
Fan_misc.FAN_IN_SET_OF_EDGE
|- (!x. (!V. (!E. (!v. (!w. (((FAN (x , (V , E))) /\ ((v INSERT (w INSERT EMPTY)) IN E)) ==> ((v IN V) /\ ((w IN V) /\ ((w IN (((set_of_edge v) V) E)) /\ (v IN (((set_of_edge w) V) E)))))))))))
SETDIST_CLOSED_COMPACT
|- (!s. (!t. (((closed s) /\ ((compact t) /\ ((~ (s = EMPTY)) /\ (~ (t = EMPTY))))) ==> (?x. (?y. ((x IN s) /\ ((y IN t) /\ ((distance (x , y)) = (setdist (s , t))))))))))
Appendix.scs_hi_v39_explicit
|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_hi_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = hi))))))))))
ANALYTIC_ON_NEG
|- (!f. (!s. (((analytic_on f) s) ==> ((analytic_on (\z. (vector_neg (f z)))) s))))
FORALL_PASTECART
|- ((!p. (P p)) = (!x. (!y. (P ((pastecart x) y)))))
REAL_ABS_BETWEEN2
|- (!x0. (!x. (!y0. (!y. ((((real_lt x0) y0) /\ (((real_lt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_abs ((real_sub x) x0)))) ((real_sub y0) x0)) /\ ((real_lt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_abs ((real_sub y) y0)))) ((real_sub y0) x0)))) ==> ((real_lt x) y))))))
REAL_DIFFERENTIABLE_IMP_CONTINUOUS_ATREAL
|- (!f. (!x. (((real_differentiable f) (atreal x)) ==> ((real_continuous f) (atreal x)))))
REAL_LE_NEGL
|- (!x. (((real_le (real_neg x)) x) = ((real_le (real_of_num (NUMERAL _0))) x)))
VECTOR_MUL_RNEG
|- (((% c) (vector_neg x)) = (vector_neg ((% c) x)))
BAIRE
|- (!g. (!s. ((((locally compact) s) /\ ((COUNTABLE g) /\ (!t. ((t IN g) ==> (((open_in ((subtopology euclidean) s)) t) /\ (s SUBSET (closure t))))))) ==> (s SUBSET (closure (INTERS g))))))
LIM_AT
|- (!f. (!l. (!a. ((((--> f) l) (at a)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. ((((real_lt (real_of_num (NUMERAL _0))) (distance (x , a))) /\ ((real_lt (distance (x , a))) d)) ==> ((real_lt (distance ((f x) , l))) e)))))))))))
MATRIX_TRANSP_MUL
|- (!A. (!B. ((transp ((matrix_mul A) B)) = ((matrix_mul (transp B)) (transp A)))))
SUM_INTEGRAL_UBOUND_INCREASING
|- (!f. (!g. (!m. (!n. (((m <= n) /\ ((!x. ((x IN (closed_segment ((CONS ((Cx (real_of_num m)) , (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_of_num m)) 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 ((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)))))))))))
HAS_REAL_DERIVATIVE_SEQUENCE
|- (!s. (!f. (!f'. (!g'. (((is_realinterval s) /\ ((!n. (!x. ((x IN s) ==> (((has_real_derivative (f n)) ((f' n) x)) ((within (atreal x)) s))))) /\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\ (x IN s)) ==> ((real_le (real_abs ((real_sub ((f' n) x)) (g' x)))) e))))))) /\ (?x. (?l. ((x IN s) /\ (((---> (\n. ((f n) x))) l) sequentially))))))) ==> (?g. (!x. ((x IN s) ==> ((((---> (\n. ((f n) x))) (g x)) sequentially) /\ (((has_real_derivative g) (g' x)) ((within (atreal x)) s)))))))))))
ANR_CLOSED_UNION
|- (!s. (!t. (((closed s) /\ ((closed t) /\ ((ANR s) /\ ((ANR t) /\ (ANR (s INTER t)))))) ==> (ANR (s UNION t)))))
Ssrnat.expnS
|- (!m. (!n. ((m EXP (SUC n)) = (m * (m EXP n)))))
gt_c
|- (!t. (!s. ((s >_c t) = (t <_c s))))
WITHIN_UNIV
|- (!x. (((within (at x)) UNIV) = (at x)))
CONTINUOUS_CLOSED_GRAPH_EQ
|- (!f. (!s. (!t. (((compact t) /\ (((IMAGE f) s) SUBSET t)) ==> (((continuous_on f) s) = ((closed_in ((subtopology euclidean) (s PCROSS t))) (GSPEC (\GEN%PVAR%965. (?x. (((SETSPEC GEN%PVAR%965) (x IN s)) ((pastecart x) (f x))))))))))))
AFFINE_HULL_2_ALT
|- (!a. (!b. (((hull affine) (a INSERT (b INSERT EMPTY))) = (GSPEC (\GEN%PVAR%1416. (?u. (((SETSPEC GEN%PVAR%1416) (u IN UNIV)) ((vector_add a) ((% u) ((vector_sub b) a))))))))))
path_connected
|- (!s. ((path_connected s) = (!x. (!y. (((x IN s) /\ (y IN s)) ==> (?g. ((path g) /\ (((path_image g) SUBSET s) /\ (((pathstart g) = x) /\ ((pathfinish g) = y))))))))))
CART_EQ
|- (!x. (!y. ((x = y) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((x $ i) = (y $ i)))))))
DIM_SUBSET
|- (!s. (!t. ((s SUBSET t) ==> ((dim s) <= (dim t)))))
REAL_MEASURABLE_ON_ADD
|- (!f. (!g. (!s. ((((real_measurable_on f) s) /\ ((real_measurable_on g) s)) ==> ((real_measurable_on (\x. ((real_add (f x)) (g x)))) s)))))
MEASURABLE_ON_COMPOSE_CONTINUOUS
|- (!f. (!g. ((((measurable_on f) UNIV) /\ ((continuous_on g) UNIV)) ==> ((measurable_on (g o f)) UNIV))))
PASTECART_VEC
|- (!n. (((pastecart (vec n)) (vec n)) = (vec n)))
PATH_CONNECTED_IMP_CONNECTED
|- (!s. ((path_connected s) ==> (connected s)))
Real_ext.REAL_PROP_ZERO_ABS
|- (!x. (((real_abs x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0)))))
Trigonometry2.POW2_COND
|- (!a. (!b. ((((real_le (real_of_num (NUMERAL _0))) a) /\ ((real_le (real_of_num (NUMERAL _0))) b)) ==> (((real_le a) b) = ((real_le ((real_pow a) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow b) (NUMERAL (BIT0 (BIT1 _0)))))))))
ABSOLUTELY_CONTINUOUS_ON_VMUL_EQ
|- (!f. (!s. (!v. (((absolutely_continuous_on (\x. ((% (f x)) v))) s) = ((v = (vec (NUMERAL _0))) \/ ((absolutely_continuous_on (\x. (lift (f x)))) s))))))
LET_TRANS
|- (!m. (!n. (!p. (((m <= n) /\ (n < p)) ==> (m < p)))))
IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT
|- (!s. (!x. (!a. (((convex s) /\ ((x IN (relative_interior s)) /\ ((a IN ((hull affine) s)) /\ (~ (x = a))))) ==> (?b. ((b IN s) /\ (x IN (open_segment (a , b)))))))))
LOG_RPOW
|- (!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((log ((rpow x) y)) = ((real_mul y) (log x))))))
LINEAR_SUB
|- (!f. (!x. (!y. ((linear f) ==> ((f ((vector_sub x) y)) = ((vector_sub (f x)) (f y)))))))
Sphere.rotate6
|- (!f. (!x6. (!x1. (!x5. (!x3. (!x4. (!x2. ((((((((rotate6 f) x1) x2) x3) x4) x5) x6) = ((((((f x6) x1) x5) x3) x4) x2)))))))))
Fnjlbxs.length_list_of_vector
|- (!d. (!n. (!v. ((LENGTH (((list_of_vector d) n) v)) = n))))
Lp_main_estimate.ineq_tau4_tauVEF
|- (!V. (!E. ((lp_fan (V , E)) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\ ((packing V) /\ ((V SUBSET ball_annulus) /\ (((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) <= (CARD V)) /\ (((CARD V) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) /\ ((!v. (!w. (((~ (v = w)) /\ ((v IN V) /\ ((w IN V) /\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (distance (v , w)))))) /\ (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v , w))) /\ ((real_le (distance (v , w))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))))))))))) ==> ((real_le (d_tame (CARD V))) (((tau_fun V) E) FF)))))) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le ((y5_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (y6_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le ((y8_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ ((real_le ((y9_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (tauVEF (V , (E , ((face (hypermap_of_fan (V , E))) d))))))))))))
CONNECTED_CONVEX_1_GEN
|- (!s. (((dimindex UNIV) = (NUMERAL (BIT1 _0))) ==> ((convex s) = (connected s))))
INT_OF_NUM_GT
|- (!m. (!n. (((int_gt (int_of_num m)) (int_of_num n)) = (m > n))))
REAL_EXP_MONO_LT
|- (!x. (!y. (((real_lt (exp x)) (exp y)) = ((real_lt x) y))))
REAL_SUB_LE
|- (!x. (!y. (((real_le (real_of_num (NUMERAL _0))) ((real_sub x) y)) = ((real_le y) x))))
COND_RATOR
|- (!b. (!f. (!g. (!x. (((((COND b) f) g) x) = (((COND b) (f x)) (g x)))))))
Functional_equation.add6
|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((add6 f) g) x1) x2) x3) x4) x5) x6) = ((real_add ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))
BASIS_NONZERO
|- (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV))) ==> (~ ((basis k) = (vec (NUMERAL _0))))))
Hdplygy.sigma_sy
|- (!s. ((sigma_sy s) = (((COND (ear_sy s)) (real_of_num (NUMERAL (BIT1 _0)))) (real_neg (real_of_num (NUMERAL (BIT1 _0)))))))
MEASURABLE_ON_MIN
|- (!f. (!g. (!s. ((((measurable_on f) s) /\ ((measurable_on g) s)) ==> ((measurable_on (\x. (lambdai. ((real_min ((f x) $ i)) ((g x) $ i))))) s)))))
columns
|- (!A. ((columns A) = (GSPEC (\GEN%PVAR%126. (?i. (((SETSPEC GEN%PVAR%126) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) ((column i) A)))))))
COMPLEX_ADD_CCOS
|- (!w. (!z. (((vector_add (ccos w)) (ccos z)) = ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((complex_mul (ccos ((complex_div ((vector_add w) z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) (ccos ((complex_div ((vector_sub w) z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))))
Lp_ineqs_proofs.y6_def
|- (!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))) ==> ((y6_fan d) = (ye_fan d))))))))))
CONVEX_CLOSURE
|- (!s. ((convex s) ==> (convex (closure s))))
DIFFERENTIABLE_ADD
|- (!f. (!g. (!net. ((((differentiable f) net) /\ ((differentiable g) net)) ==> ((differentiable (\z. ((vector_add (f z)) (g z)))) net)))))
CARD_LE_FINITE_INFINITE
|- (!s. (!t. (((FINITE s) /\ (INFINITE t)) ==> (s <=_c t))))
OPEN_IN_CLOSED_IN_EQ
|- (!top. (!s. (((open_in top) s) = ((s SUBSET (topspace top)) /\ ((closed_in top) ((topspace top) DIFF s))))))
CARD_EQ_REFL
|- (!s. (s =_c s))
REAL_EQ_MUL_RCANCEL
|- (!x. (!y. (!z. ((((real_mul x) z) = ((real_mul y) z)) = ((x = y) \/ (z = (real_of_num (NUMERAL _0))))))))
LOCALLY_COMPACT_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((locally compact) ((IMAGE f) s)) = ((locally compact) s)))))
COMPLEX_SQNORM
|- (((real_pow (vector_norm z)) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_add ((real_pow (Re z)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (Im z)) (NUMERAL (BIT0 (BIT1 _0))))))
Hypermap_iso.dart1_of_fan_eq_image
|- (!V. (!E. ((dart1_of_fan (V , E)) = ((IMAGE contracted_dart) (d1_fan ((vec (NUMERAL _0)) , (V , E)))))))
CONVEX_ON_CONTINUOUS
|- (!f. (!s. (((open s) /\ ((convex_on f) s)) ==> ((continuous_on (lift o f)) s))))
LINEAR_VMUL_DROP
|- (!f. (!v. ((linear f) ==> (linear (\x. ((% (drop (f x))) v))))))
Lp_ineqs_proofs.dart4_y4'_lo
|- (!V. (!E. ((lp_fan (V , E)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) ((y4'_fan (V , E)) d)))))))
Rogers.UNIQUE_SOLUTION_AFFINE_INDEPENDENT
|- (!S. (!b. (((~ (S = EMPTY)) /\ (~ (affine_dependent S))) ==> (?!p. ((p IN ((hull affine) S)) /\ (!x. (!y. (((x IN S) /\ (y IN S)) ==> (((dot p) ((vector_sub x) y)) = ((real_sub (b x)) (b y)))))))))))
ORTHONORMAL_BASIS_SUBSPACE
|- (!s. ((subspace s) ==> (?b. ((b SUBSET s) /\ (((pairwise orthogonal) b) /\ ((!x. ((x IN b) ==> ((vector_norm x) = (real_of_num (NUMERAL (BIT1 _0)))))) /\ ((independent b) /\ ((b HAS_SIZE (dim s)) /\ ((span b) = s)))))))))
Seq2.prev_el
|- (!x. (!s. (((prev_el s) x) = (((COND (~ ((MEM x) s))) x) (((COND (((indexl x) s) = (NUMERAL _0))) ((last x) s)) (((nth x) s) (((indexl x) s) - (NUMERAL (BIT1 _0)))))))))
homotopy_equivalent
|- (!s. (!t. (((homotopy_equivalent s) t) = (?f. (?g. (((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET s) /\ (((((homotopic_with (\x. T)) (s , s)) (g o f)) I) /\ ((((homotopic_with (\x. T)) (t , t)) (f o g)) I)))))))))))
list_cases
|- (!P. (((P NIL) /\ (!h. (!t. (P ((CONS h) t))))) ==> (!l. (P l))))
SIMPLE_PATH_IMP_PATH
|- (!g. ((simple_path g) ==> (path g)))
Ssrnat.maxnK
|- (!m. (!n. (((minn ((maxn m) n)) m) = m)))
COMPLEX_INV_MUL
|- (!w. (!z. ((complex_inv ((complex_mul w) z)) = ((complex_mul (complex_inv w)) (complex_inv z)))))
NORM_POW_2
|- (!x. (((real_pow (vector_norm x)) (NUMERAL (BIT0 (BIT1 _0)))) = ((dot x) x)))
VECTOR_ADD_LDISTRIB
|- (((% c) ((vector_add x) y)) = ((vector_add ((% c) x)) ((% c) y)))
LIM_TRANSFORM_EVENTUALLY
|- (!net. (!f. (!g. (!l. ((((eventually (\x. ((f x) = (g x)))) net) /\ (((--> f) l) net)) ==> (((--> g) l) net))))))
HAS_INTEGRAL_COMPONENT_LE_AE
|- (!f. (!g. (!s. (!i. (!j. (!k. (!t. ((((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ ((negligible t) /\ ((((has_integral f) i) s) /\ ((((has_integral g) j) s) /\ (!x. ((x IN (s DIFF t)) ==> ((real_le ((f x) $ k)) ((g x) $ k))))))))) ==> ((real_le (i $ k)) (j $ k))))))))))
Vol1.sol_spec
|- (!x. (!C. (!r. ((((real_gt r) (real_of_num (NUMERAL _0))) /\ ((measurable (C INTER (ball (x , r)))) /\ (((radial r) x) (C INTER (ball (x , r)))))) ==> (((sol x) C) = ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((real_div (measure (C INTER (ball (x , r))))) ((real_pow r) (NUMERAL (BIT1 (BIT1 _0)))))))))))
AFFINE_HULL_CONVEX_HULL
|- (!s. (((hull affine) ((hull convex) s)) = ((hull affine) s)))
EXISTS_PAIRED_THM
|- (!P. ((? (GABS (\f. (!x. (!y. ((GEQ (f (x , y))) ((P x) y))))))) = (?x. (?y. ((P x) y)))))
GDELTA_COMPLEMENT
|- (!s. ((gdelta (UNIV DIFF s)) = (fsigma s)))
COND_CLAUSES
|- (!t1. (!t2. (((((COND T) t1) t2) = t1) /\ ((((COND F) t1) t2) = t2))))
HAS_BOUNDED_SETVARIATION_ON_NEG
|- (!f. (!s. (((has_bounded_setvariation_on (\x. (vector_neg (f x)))) s) = ((has_bounded_setvariation_on f) s))))
IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT
|- (!s. (!a. (!b. (((convex s) /\ ((a IN (relative_interior s)) /\ (b IN (closure s)))) ==> ((open_segment (a , b)) SUBSET (relative_interior s))))))
CONTINUOUS_ON_CMUL
|- (!f. (!c. (!s. (((continuous_on f) s) ==> ((continuous_on (\x. ((% c) (f x)))) s)))))
LIM_COMPONENT_UBOUND
|- (!net. (!f. (!l. (!b. (!k. (((~ (trivial_limit net)) /\ ((((--> f) l) net) /\ (((eventually (\x. ((real_le ((f x) $ k)) b))) net) /\ (((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV)))))) ==> ((real_le (l $ k)) b)))))))
ITER
|- ((!f. ((((ITER (NUMERAL _0)) f) x) = x)) /\ (!f. (!n. ((((ITER (SUC n)) f) x) = (f (((ITER n) f) x))))))
DIM_SPAN
|- (!s. ((dim (span s)) = (dim s)))
Tame_lemmas.fully_surrounded_azim_eq_dih_y
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((LET (\w'. (LET_END ((LET (\y1. (LET_END ((LET (\y2. (LET_END ((LET (\y3. (LET_END ((LET (\y4. (LET_END ((LET (\y5. (LET_END ((LET (\y6. (LET_END (((azim_dart (V , E)) (v , w)) = ((((((dih_y y1) y2) y3) y4) y5) y6))))) (distance (v , w)))))) (distance (v , w')))))) (distance (w , w')))))) (vector_norm w'))))) (vector_norm w))))) (vector_norm v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))
CONVEX_POSITIVE_ORTHANT
|- (convex (GSPEC (\GEN%PVAR%1454. (?x. (((SETSPEC GEN%PVAR%1454) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (x $ i))))) x)))))
mspace
|- (!m. ((mspace m) = (FST (dest_metric m))))
REAL_SERIES_RESTRICT
|- (!f. (!k. (!l. ((((real_sums (\n. (((COND (n IN k)) (f n)) (real_of_num (NUMERAL _0))))) l) UNIV) = (((real_sums f) l) k)))))
POLYTOPE_IMP_COMPACT
|- (!s. ((polytope s) ==> (compact s)))
CONVEX_CONE_HULL_SEPARATE
|- (!s. (((hull convex_cone) s) = ((vec (NUMERAL _0)) INSERT ((hull conic) ((hull convex) s)))))
SND
|- (!x. (!y. ((SND (x , y)) = y)))
CONTRAPOS_THM
|- (!t1. (!t2. (((~ t1) ==> (~ t2)) = (t2 ==> t1))))
RE_MUL_II
|- (!z. (((Re ((complex_mul z) ii)) = (real_neg (Im z))) /\ ((Re ((complex_mul ii) z)) = (real_neg (Im z)))))
Hypermap.atom
|- (!H. (!L. (!x. ((((atom H) L) x) = (GSPEC (\GEN%PVAR%5421. (?y. (((SETSPEC GEN%PVAR%5421) (((((is_node_going H) L) x) y) \/ ((((is_node_going H) L) y) x))) y))))))))
Nonlinear_lemma.halfbump_x1
|- (!x2. (!x3. (!x4. (!x5. (!x6. (!x1. (((((((halfbump_x1 x1) x2) x3) x4) x5) x6) = (halfbump_x x1))))))))
WF_REC
|- ((WF <<) ==> (!H. ((!f. (!g. (!x. ((!z. (((<< z) x) ==> ((f z) = (g z)))) ==> (((H f) x) = ((H g) x)))))) ==> (?f. (!x. ((f x) = ((H f) x)))))))
STARLIKE_TRANSLATION_EQ
|- (!a. (!s. ((starlike ((IMAGE (\x. ((vector_add a) x))) s)) = (starlike s))))
PATH_COMPONENT_EQ_CONNECTED_COMPONENT
|- (!s. (!x. (((locally path_connected) s) ==> (((path_component s) x) = ((connected_component s) x)))))
Pqcsxwg.PQCSXWG2_ATREAL
|- (!v0. (!v1. (!v2. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a. (((~ (collinear ((v0 a) INSERT ((v1 a) INSERT ((v2 a) INSERT EMPTY))))) /\ (((x1 a) = ((real_pow (distance ((v1 a) , (v0 a)))) (NUMERAL (BIT0 (BIT1 _0))))) /\ (((x2 a) = ((real_pow (distance ((v2 a) , (v0 a)))) (NUMERAL (BIT0 (BIT1 _0))))) /\ (((x6 a) = ((real_pow (distance ((v1 a) , (v2 a)))) (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x (x1 a)) (x2 a)) (x3 a)) (x4 a)) (x5 a)) (x6 a))) /\ (((continuous v0) (atreal a)) /\ (((continuous v1) (atreal a)) /\ (((continuous v2) (atreal a)) /\ (((real_continuous x1) (atreal a)) /\ (((real_continuous x2) (atreal a)) /\ (((real_continuous x3) (atreal a)) /\ (((real_continuous x4) (atreal a)) /\ (((real_continuous x5) (atreal a)) /\ ((real_continuous x6) (atreal a))))))))))))))) ==> ((continuous (\t. (((((((((mk_simplex1 (v0 t)) (v1 t)) (v2 t)) (x1 t)) (x2 t)) (x3 t)) (x4 t)) (x5 t)) (x6 t)))) (atreal a)))))))))))))
Add_triangle.fully_surrounded_imp_conforming
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (conforming_fan ((vec (NUMERAL _0)) , (V , E)))))))
CARD_EQ_CLOSED
|- (!s. ((closed s) ==> ((s <=_c UNIV) \/ (s =_c UNIV))))
CONTRACTIBLE_IMP_PATH_CONNECTED
|- (!s. ((contractible s) ==> (path_connected s)))
INTERIOR_CLOSURE
|- (!s. ((interior s) = (UNIV DIFF (closure (UNIV DIFF s)))))
Lp_ineqs_def.AZIM_DART_POS
|- ((real_le (real_of_num (NUMERAL _0))) ((azim_dart (V , E)) x))