hcju/holps huggingface dataset:
Functional_equation.dih_all_atn
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\ (((real_lt (real_of_num (NUMERAL _0))) x1) /\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\ ((real_lt (real_of_num (NUMERAL _0))) x6))))))) ==> ((((((((dih_x x1) x2) x3) x4) x5) x6) = ((((((dihatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih2_x x1) x2) x3) x4) x5) x6) = ((((((dih2atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih3_x x1) x2) x3) x4) x5) x6) = ((((((dih3atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih4_x x1) x2) x3) x4) x5) x6) = ((((((dih4atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih5_x x1) x2) x3) x4) x5) x6) = ((((((dih5atn_x x1) x2) x3) x4) x5) x6)) /\ (((((((dih6_x x1) x2) x3) x4) x5) x6) = ((((((dih6atn_x x1) x2) x3) x4) x5) x6))))))))))))))
CX_INJ
|- (!x. (!y. (((Cx x) = (Cx y)) = (x = y))))
AFFINE_IMP_CONVEX
|- (!s. ((affine s) ==> (convex s)))
REAL_LT_RDIV_EQ
|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_lt x) ((real_div y) z)) = ((real_lt ((real_mul x) z)) y))))))
REAL_ABS_0
|- ((real_abs (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))
Ssrnat.leqNgt
|- (!m. (!n. ((m <= n) = (~ (n < m)))))
CLOSED_IN_CLOSED_INTER
|- (!u. (!s. ((closed s) ==> ((closed_in ((subtopology euclidean) u)) (u INTER s)))))
INVERTIBLE_I
|- (invertible (mat (NUMERAL (BIT1 _0))))
Constants_approx.pi_interval
|- ((interval_arith pi) (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0)))))))))))))))))))))))))))))))) , ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))))))))))))))))))))))
real_polynomial_function_INDUCT
|- (!real_polynomial_function'. (((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (real_polynomial_function' (\x. (x $ i))))) /\ ((!c. (real_polynomial_function' (\x. c))) /\ ((!f. (!g. (((real_polynomial_function' f) /\ (real_polynomial_function' g)) ==> (real_polynomial_function' (\x. ((real_add (f x)) (g x))))))) /\ (!f. (!g. (((real_polynomial_function' f) /\ (real_polynomial_function' g)) ==> (real_polynomial_function' (\x. ((real_mul (f x)) (g x)))))))))) ==> (!a. ((real_polynomial_function a) ==> (real_polynomial_function' a)))))
has_complex_derivative
|- (!f. (!f'. (!net. ((((has_complex_derivative f) f') net) = (((has_derivative f) (\x. ((complex_mul f') x))) net)))))
REAL_MEASURE_UNIONS_LE_IMAGE
|- (!f. (!s. (((FINITE f) /\ (!a. ((a IN f) ==> (real_measurable (s a))))) ==> ((real_le (real_measure (UNIONS ((IMAGE s) f)))) ((sum f) (\a. (real_measure (s a))))))))
NSUM_UNION_RZERO
|- (!f. (!u. (!v. (((FINITE u) /\ (!x. (((x IN v) /\ (~ (x IN u))) ==> ((f x) = (NUMERAL _0))))) ==> (((nsum (u UNION v)) f) = ((nsum u) f))))))
HAS_REAL_INTEGRAL_NEG
|- (!f. (!k. (!s. ((((has_real_integral f) k) s) ==> (((has_real_integral (\x. (real_neg (f x)))) (real_neg k)) s)))))
SQRT_POS_LT
|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_lt (real_of_num (NUMERAL _0))) (sqrt x))))
COUNTABLE_SING
|- (!x. (COUNTABLE (x INSERT EMPTY)))
Fan_defs.n_fan_pair
|- (!V. (!E. (!v. (!w. (((n_fan_pair (V , E)) (v , w)) = (v , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))
ROTATION_LOWDIM_HORIZONTAL
|- (!s. (((dim s) < (dimindex UNIV)) ==> (?f. ((orthogonal_transformation f) /\ (((det (matrix f)) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((IMAGE f) s) SUBSET (GSPEC (\GEN%PVAR%456. (?z. (((SETSPEC GEN%PVAR%456) ((z $ (dimindex UNIV)) = (real_of_num (NUMERAL _0)))) z))))))))))
matrix_sub
|- (!A. (!B. (((matrix_sub A) B) = (lambdai. (lambdaj. ((real_sub ((A $ i) $ j)) ((B $ i) $ j)))))))
ITERATE_PAIR
|- (!op. ((monoidal op) ==> (!f. (!m. (!n. ((((iterate op) (((NUMERAL (BIT0 (BIT1 _0))) * m) .. (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 _0))))) f) = (((iterate op) (m .. n)) (\i. ((op (f ((NUMERAL (BIT0 (BIT1 _0))) * i))) (f (((NUMERAL (BIT0 (BIT1 _0))) * i) + (NUMERAL (BIT1 _0)))))))))))))
DISJOINT_EMPTY
|- (!s. (((DISJOINT EMPTY) s) /\ ((DISJOINT s) EMPTY)))
NEUTRAL_MUL
|- ((neutral *) = (NUMERAL (BIT1 _0)))
CONTINUOUS_LE_ON_CLOSURE
|- (!f. (!s. (!a. ((((continuous_on (lift o f)) (closure s)) /\ (!x. ((x IN s) ==> ((real_le (f x)) a)))) ==> (!x. ((x IN (closure s)) ==> ((real_le (f x)) a)))))))
HAS_COMPLEX_DERIVATIVE_CONST
|- (!c. (!net. (((has_complex_derivative (\x. c)) (Cx (real_of_num (NUMERAL _0)))) net)))
Conforming.NEGLIGIBLE_AFF_3
|- (!x. (!v. (!u. (negligible (aff (x INSERT (v INSERT (u INSERT EMPTY))))))))
le_c
|- (!t. (!s. ((s <=_c t) = (?f. ((!x. ((x IN s) ==> ((f x) IN t))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))))))))
Seq.has_predC
|- (!a. (!s. (((has (predC a)) s) = (~ ((all a) s)))))
COVERING_SPACE_FINITE_SHEETS
|- (!p. (!c. (!s. (!b. ((((covering_space (c , p)) s) /\ (compact c)) ==> (FINITE (GSPEC (\GEN%PVAR%2084. (?x. (((SETSPEC GEN%PVAR%2084) ((x IN c) /\ ((p x) = b))) x))))))))))
SHIFTPATH_LINEAR_IMAGE
|- (!f. (!t. (!g. ((linear f) ==> (((shiftpath t) (f o g)) = (f o ((shiftpath t) g)))))))
FRONTIER_UNION_SUBSET
|- (!s. (!t. ((frontier (s UNION t)) SUBSET ((frontier s) UNION (frontier t)))))
BOUNDED_SUMS_IMAGE
|- (!f. (!g. (!t. (((bounded (GSPEC (\GEN%PVAR%588. (?x. (((SETSPEC GEN%PVAR%588) (x IN t)) (f x)))))) /\ (bounded (GSPEC (\GEN%PVAR%589. (?x. (((SETSPEC GEN%PVAR%589) (x IN t)) (g x))))))) ==> (bounded (GSPEC (\GEN%PVAR%590. (?x. (((SETSPEC GEN%PVAR%590) (x IN t)) ((vector_add (f x)) (g x)))))))))))
CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP
|- (!f. (!g. (!s. (!t. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET s) /\ (!y. ((y IN t) ==> ((f (g y)) = y))))))) ==> (!u. ((u SUBSET t) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%651. (?x. (((SETSPEC GEN%PVAR%651) ((x IN s) /\ ((f x) IN u))) x))))) = ((open_in ((subtopology euclidean) t)) u)))))))))
Dih2k_hypermap.EXISTS_POINT_DART_OF_HYP
|- (((FAN ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l))))) /\ (((NUMERAL (BIT0 (BIT1 _0))) < (dimindex UNIV)) /\ (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ ((j <= (dimindex UNIV)) /\ (((row i) (vecmats l)) = ((row j) (vecmats l))))))) ==> (i = j)))))) ==> (?x. (x IN ((darts_of_hyp (E_SY (vecmats l))) (V_SY (vecmats l))))))
FSIGMA_TRANSLATION
|- (!a. (!s. ((fsigma ((IMAGE (\x. ((vector_add a) x))) s)) = (fsigma s))))
COBOUNDED_UNIQUE_UNBOUNDED_COMPONENTS
|- (!s. (!c. (!c'. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ ((bounded (UNIV DIFF s)) /\ ((c IN (components s)) /\ ((~ (bounded c)) /\ ((c' IN (components s)) /\ (~ (bounded c'))))))) ==> (c' = c)))))
Ssrbool.orbF
|- (!b. ((b \/ F) = b))
real_variation
|- (!s. (!f. (((real_variation s) f) = ((vector_variation ((IMAGE lift) s)) (lift o (f o drop))))))
Fan.d1_fan
|- (!x. (!V. (!E. ((d1_fan (x , (V , E))) = (GSPEC (\GEN%PVAR%5465. (?x'. (?v. (?w. (?w1. (((SETSPEC GEN%PVAR%5465) ((x' = x) /\ (((v INSERT (w INSERT EMPTY)) IN E) /\ (w1 = (((((sigma_fan x) V) E) v) w))))) (x' , (v , (w , w1))))))))))))))
REAL_ABS_NZ
|- (!x. ((~ (x = (real_of_num (NUMERAL _0)))) = ((real_lt (real_of_num (NUMERAL _0))) (real_abs x))))
UNIONS_GSPEC
|- ((!P. (!f. ((UNIONS (GSPEC (\GEN%PVAR%31. (?x. (((SETSPEC GEN%PVAR%31) (P x)) (f x)))))) = (GSPEC (\GEN%PVAR%32. (?a. (((SETSPEC GEN%PVAR%32) (?x. ((P x) /\ (a IN (f x))))) a))))))) /\ ((!P. (!f. ((UNIONS (GSPEC (\GEN%PVAR%33. (?x. (?y. (((SETSPEC GEN%PVAR%33) ((P x) y)) ((f x) y))))))) = (GSPEC (\GEN%PVAR%34. (?a. (((SETSPEC GEN%PVAR%34) (?x. (?y. (((P x) y) /\ (a IN ((f x) y)))))) a))))))) /\ (!P. (!f. ((UNIONS (GSPEC (\GEN%PVAR%35. (?x. (?y. (?z. (((SETSPEC GEN%PVAR%35) (((P x) y) z)) (((f x) y) z)))))))) = (GSPEC (\GEN%PVAR%36. (?a. (((SETSPEC GEN%PVAR%36) (?x. (?y. (?z. ((((P x) y) z) /\ (a IN (((f x) y) z))))))) a)))))))))
COMPLEX_ADD_ASSOC
|- (!x. (!y. (!z. (((vector_add x) ((vector_add y) z)) = ((vector_add ((vector_add x) y)) z)))))
SNDCART_PASTECART
|- (!x. (!y. ((sndcart ((pastecart x) y)) = y)))
INTEGRABLE_INTEGRAL
|- (!f. (!i. (((integrable_on f) i) ==> (((has_integral f) ((integral i) f)) i))))
RETRACTION
|- (!s. (!t. (!r. (((retraction (s , t)) r) = ((t SUBSET s) /\ (((continuous_on r) s) /\ ((((IMAGE r) s) = t) /\ (!x. ((x IN t) ==> ((r x) = x))))))))))
HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT
|- (!p. (!f. (!g. (!h. (!X. (!Y. (!Z. ((((((homotopic_with (\f. (p (h o f)))) (X , Y)) f) g) /\ (((continuous_on h) Y) /\ (((IMAGE h) Y) SUBSET Z))) ==> ((((homotopic_with p) (X , Z)) (h o f)) (h o g))))))))))
REAL_OF_NUM_SUB
|- (!m. (!n. ((m <= n) ==> (((real_sub (real_of_num n)) (real_of_num m)) = (real_of_num (n - m))))))
UNION_SUBSET
|- (!s. (!t. (!u. (((s UNION t) SUBSET u) = ((s SUBSET u) /\ (t SUBSET u))))))
TRANSP_MATRIX_INV
|- (!A. ((transp (matrix_inv A)) = (matrix_inv (transp A))))
REAL_EQ_SUB_RADD
|- (!x. (!y. (!z. ((((real_sub x) y) = z) = (x = ((real_add z) y))))))
BOUNDED_INTERVAL
|- ((!a. (!b. (bounded (closed_interval ((CONS (a , b)) NIL))))) /\ (!a. (!b. (bounded (open_interval (a , b))))))
COVERING_SPACE_LIFT_STRONG
|- (!p. (!c. (!s. (!f. (!u. (!a. (!z. ((((covering_space (c , p)) s) /\ ((a IN c) /\ ((z IN u) /\ ((simply_connected u) /\ (((locally path_connected) u) /\ (((continuous_on f) u) /\ ((((IMAGE f) u) SUBSET s) /\ ((f z) = (p a))))))))) ==> (?g. (((continuous_on g) u) /\ ((((IMAGE g) u) SUBSET c) /\ (((g z) = a) /\ (!y. ((y IN u) ==> ((p (g y)) = (f y))))))))))))))))
MEASURABLE_MEASURE_POS_LT
|- (!s. ((measurable s) ==> (((real_lt (real_of_num (NUMERAL _0))) (measure s)) = (~ (negligible s)))))
FSIGMA_CONTINUOUS_IMAGE
|- (!f. (!s. ((((continuous_on f) s) /\ (fsigma s)) ==> (fsigma ((IMAGE f) s)))))
MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE
|- (!s. (!t. (((measurable s) /\ (lebesgue_measurable t)) ==> (measurable (s INTER t)))))
Nonlin_def.div6
|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((div6 f) g) x1) x2) x3) x4) x5) x6) = ((real_div ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))
STARLIKE_CONVEX_TWEAK_BOUNDARY_POINTS
|- (!s. (!t. (((convex s) /\ ((~ (s = EMPTY)) /\ (((relative_interior s) SUBSET t) /\ (t SUBSET (closure s))))) ==> (starlike t))))
Hypermap.lemma_face_subset
|- (!H. (!x. ((x IN (dart H)) ==> (((face H) x) SUBSET (dart H)))))
SURJECTIVE_IMAGE_EQ
|- (!s. (!t. (((!y. ((y IN t) ==> (?x. ((f x) = y)))) /\ (!x. (((f x) IN t) = (x IN s)))) ==> (((IMAGE f) s) = t))))
ARG_ROTATE2D_UNIQUE
|- (!t. (!a. (!z. (((~ (z = (Cx (real_of_num (NUMERAL _0))))) /\ ((Arg ((rotate2d t) z)) = a)) ==> (?n. ((integer n) /\ (t = ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul n) pi))) ((real_sub a) (Arg z))))))))))
HYPERPLANE_EQ_UNIV
|- (!a. (!b. (((GSPEC (\GEN%PVAR%172. (?x. (((SETSPEC GEN%PVAR%172) (((dot a) x) = b)) x)))) = UNIV) = ((a = (vec (NUMERAL _0))) /\ (b = (real_of_num (NUMERAL _0)))))))
BOUNDED_HAS_INF
|- (!s. (((bounded ((IMAGE lift) s)) /\ (~ (s = EMPTY))) ==> ((!x. ((x IN s) ==> ((real_le (inf s)) x))) /\ (!b. ((!x. ((x IN s) ==> ((real_le b) x))) ==> ((real_le b) (inf s)))))))
Nonlinear_lemma.lmfun0
|- (!y. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y) ==> ((lmfun ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (real_of_num (NUMERAL _0)))))
Collect_geom.pow_g
|- (!x. ((real_le (real_of_num (NUMERAL _0))) ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))))
COMPLEX_NORM_DIV
|- (!w. (!z. ((vector_norm ((complex_div w) z)) = ((real_div (vector_norm w)) (vector_norm z)))))
SUM_DIFFS
|- (!m. (!n. (((sum (m .. n)) (\k. ((real_sub (f k)) (f (k + (NUMERAL (BIT1 _0))))))) = (((COND (m <= n)) ((real_sub (f m)) (f (n + (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL _0))))))
CARD_EQ_SIMPLE_PATH_IMAGE
|- (!g. ((simple_path g) ==> ((path_image g) =_c UNIV)))
INTEGRABLE_SUBINTERVALS_IMP_MEASURABLE
|- (!f. ((!a. (!b. ((integrable_on f) (closed_interval ((CONS (a , b)) NIL))))) ==> ((measurable_on f) UNIV)))
CLOSED_IN_SUBTOPOLOGY_EMPTY
|- (!top. (!s. (((closed_in ((subtopology top) EMPTY)) s) = (s = EMPTY))))
ITERATE_INJECTION
|- (!op. ((monoidal op) ==> (!f. (!p. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> ((p x) IN s))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((p x) = (p y)))) ==> (x = y)))))) ==> ((((iterate op) s) (f o p)) = (((iterate op) s) f))))))))
PERMUTES_INSERT_LEMMA
|- (!p. (!a. (!s. (((permutes p) (a INSERT s)) ==> ((permutes ((swap (a , (p a))) o p)) s)))))
REAL_LTE_TOTAL
|- (!x. (!y. (((real_lt x) y) \/ ((real_le y) x))))
ORTHOGONAL_MATRIX_IMP_INVERTIBLE
|- (!A. ((orthogonal_matrix A) ==> (invertible A)))
Seq.all_filterP
|- (!a. (!s. ((((filter a) s) = s) = ((all a) s))))
INTEGRAL_0
|- (!s. (((integral s) (\x. (vec (NUMERAL _0)))) = (vec (NUMERAL _0))))
CONDENSATION_POINT_IMP_LIMPT
|- (!x. (!s. (((condensation_point_of x) s) ==> ((limit_point_of x) s))))
Real_ext.REAL_PROP_POS_ADD2
|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_add x) y)))))
VECTOR_VARIATION_AFFINITY2
|- (!m. (!c. (!f. (!s. (((vector_variation ((IMAGE (\x. ((vector_add ((% (real_inv m)) x)) (vector_neg ((% (real_inv m)) c))))) s)) (\x. (f ((vector_add ((% m) x)) c)))) = (((COND (m = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL _0))) ((vector_variation s) f)))))))
Rogers.KHEJKCI_GEN
|- (!V. (!k. (!r. (!ul. (!vl. (((saturated V) /\ ((packing V) /\ ((((barV V) k) ul) /\ ((((barV V) r) vl) /\ ((initial_sublist ul) vl))))) ==> ((face_of ((voronoi_list V) vl)) ((voronoi_list V) ul))))))))
REAL_LT_ADDNEG
|- (!x. (!y. (!z. (((real_lt y) ((real_add x) (real_neg z))) = ((real_lt ((real_add y) z)) x)))))
LIM_ATREAL
|- (!f. (!l. (!a. ((((--> f) l) (atreal 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))) (real_abs ((real_sub x) a))) /\ ((real_lt (real_abs ((real_sub x) a))) d)) ==> ((real_lt (distance ((f x) , l))) e)))))))))))
FINITE_CARD_LT
|- (!s. ((FINITE s) = (s <_c UNIV)))
Ssrbool.negb_and
|- (!a. (!b. ((~ (a /\ b)) = ((~ a) \/ (~ b)))))
Fan_defs.e_fan_pair_ext
|- (!V. (!E. (!x. (((e_fan_pair_ext (V , E)) x) = (((COND (x IN (dart1_of_fan (V , E)))) ((e_fan_pair (V , E)) x)) x)))))
FUN_IN_IMAGE
|- (!f. (!s. (!x. ((x IN s) ==> ((f x) IN ((IMAGE f) s))))))
Trigonometry2.ARG_DIFF_SUCCESSIBLE_IN_FIRST_PERIOD
|- (!W. (!xicm. (((FINITE W) /\ (((CARD W) = n) /\ ((!x. ((W x) ==> (~ (x = (vec (NUMERAL _0)))))) /\ ((polar_cycle_on xicm) W)))) ==> (!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))))))))))
Lp_ineqs_proofs.y3_lo
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((y3_fan (V , E)) d))))))))))
REAL_LNEG_UNIQ
|- (!x. (!y. ((((real_add x) y) = (real_of_num (NUMERAL _0))) = (x = (real_neg y)))))
LIM_CONG_AT
|- ((!x. ((~ (x = a)) ==> ((f x) = (g x)))) ==> ((((--> (\x. (f x))) l) (at a)) = (((--> g) l) (at a))))
POWER_SERIES_CONV_IMP_ABSCONV
|- (!a. (!k. (!w. (!z. ((((summable k) (\n. ((complex_mul (a n)) ((complex_pow z) n)))) /\ ((real_lt (vector_norm w)) (vector_norm z))) ==> ((summable k) (\n. (Cx (vector_norm ((complex_mul (a n)) ((complex_pow w) n)))))))))))
INVERTIBLE_TRANSP
|- (!A. ((invertible (transp A)) = (invertible A)))
Lp_ineqs_proofs.f_fan_pair_ext_in_dart
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((f_fan_pair_ext (V , E)) d) IN (dart_of_fan (V , E))))))))
Ssrnat.ltn0
|- (!n. ((n < (NUMERAL _0)) = F))
CLOSED_IN_CLOSED_EQ
|- (!s. (!t. ((closed s) ==> (((closed_in ((subtopology euclidean) s)) t) = ((closed t) /\ (t SUBSET s))))))
AFF_DIM_EQ_AFFINE_HULL
|- (!s. (!t. (((s SUBSET t) /\ ((int_le (aff_dim t)) (aff_dim s))) ==> (((hull affine) s) = ((hull affine) t)))))
IMAGE_SUBSET
|- (!f. (!s. (!t. ((s SUBSET t) ==> (((IMAGE f) s) SUBSET ((IMAGE f) t))))))
AFFSIGN_EQ_AFFINE_HULL
|- (!sgn. (!s. (!t. ((((affsign sgn) s) EMPTY) = ((hull affine) s)))))
PATH_IMAGE_TRANSLATION
|- (!a. (!g. ((path_image ((\x. ((vector_add a) x)) o g)) = ((IMAGE (\x. ((vector_add a) x))) (path_image g)))))
REAL_LE_RSQRT
|- (!x. (!y. (((real_le ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))) y) ==> ((real_le x) (sqrt y)))))
REAL_ABS_MUL
|- (!x. (!y. ((real_abs ((real_mul x) y)) = ((real_mul (real_abs x)) (real_abs y)))))
SEGMENT_SYM
|- ((!a. (!b. ((closed_segment ((CONS (a , b)) NIL)) = (closed_segment ((CONS (b , a)) NIL))))) /\ (!a. (!b. ((open_segment (a , b)) = (open_segment (b , a))))))
REAL_EUCLIDEAN_METRIC
|- (((mspace real_euclidean_metric) = UNIV) /\ (!x. (!y. (((mdist real_euclidean_metric) (x , y)) = (real_abs ((real_sub y) x))))))
Appendix.BBprime_v39
|- (!s. (!vv. (((BBprime_v39 s) vv) = (((BBs_v39 s) vv) /\ ((!ww. (((BBs_v39 s) ww) ==> ((real_le ((taustar_v39 s) vv)) ((taustar_v39 s) ww)))) /\ ((real_lt ((taustar_v39 s) vv)) (real_of_num (NUMERAL _0))))))))
MATRIX_VECTOR_MUL_RMUL
|- (!A. (!x. (!c. (((matrix_vector_mul A) ((% c) x)) = ((% c) ((matrix_vector_mul A) x))))))
Nonlinear_lemma.gcy_high
|- (!y. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y) ==> ((gcy y) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_div mm1) pi)))))
REAL_SUB_SUB2
|- (!x. (!y. (((real_sub x) ((real_sub x) y)) = y)))
COMPACT_IMP_TOTALLY_BOUNDED
|- (!s. ((compact s) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?k. ((FINITE k) /\ ((k SUBSET s) /\ (s SUBSET (UNIONS ((IMAGE (\x. (ball (x , e)))) k))))))))))
Constants_approx.interval_arith
|- (!lo. (!x. (!hi. (((interval_arith x) (lo , hi)) = (((real_le lo) x) /\ ((real_le x) hi))))))
SETDIST_CLOSURE
|- ((!s. (!t. ((setdist ((closure s) , t)) = (setdist (s , t))))) /\ (!s. (!t. ((setdist (s , (closure t))) = (setdist (s , t))))))
CONVEX_HULL_SING
|- (!a. (((hull convex) (a INSERT EMPTY)) = (a INSERT EMPTY)))
Hypermap_iso.hyp_iso_inv
|- (!f. (!G. (!H. (((hyp_iso f) (H , G)) ==> ((hyp_iso ((res_inv f) (dart H))) (G , H))))))
OPEN_CONTAINS_CBALL
|- (!s. ((open s) = (!x. ((x IN s) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ ((cball (x , e)) SUBSET s)))))))
COMPLEX_MUL_LNEG
|- (!x. (!y. (((complex_mul (vector_neg x)) y) = (vector_neg ((complex_mul x) y)))))
EVENTUALLY_AT_POSINFINITY
|- (!p. (((eventually p) at_posinfinity) = (?b. (!x. (((real_ge x) b) ==> (p x))))))
SERIES_NEG
|- (!x. (!x0. (!s. ((((sums x) x0) s) ==> (((sums (\n. (vector_neg (x n)))) (vector_neg x0)) s)))))
DECIMAL
|- (!x. (!y. (((DECIMAL x) y) = ((real_div (real_of_num x)) (real_of_num y)))))
DIAGONAL_MATRIX_MUL_COMPONENT
|- (!A. (!B. (!i. (!j. (((diagonal_matrix A) /\ ((diagonal_matrix B) /\ (((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ (j <= (dimindex UNIV))))))) ==> (((((matrix_mul A) B) $ i) $ j) = ((real_mul ((A $ i) $ j)) ((B $ i) $ j))))))))
LIM_ATREAL_AT
|- ((((--> f) a) (atreal x)) = (((--> (f o drop)) a) (at (lift x))))
ABSOLUTELY_REAL_INTEGRABLE_ON
|- (((absolutely_real_integrable_on f) s) = ((absolutely_integrable_on (lift o (f o drop))) ((IMAGE lift) s)))
HAS_INTEGRAL_SPIKE_FINITE
|- (!f. (!g. (!s. (!t. (!y. (((FINITE s) /\ ((!x. ((x IN (t DIFF s)) ==> ((g x) = (f x)))) /\ (((has_integral f) y) t))) ==> (((has_integral g) y) t)))))))
COLLINEAR_AFFINE_HULL_COLLINEAR
|- (!s. ((collinear ((hull affine) s)) = (collinear s)))
CBALL_SCALING
|- (!c. (((real_lt (real_of_num (NUMERAL _0))) c) ==> (!x. (!r. ((cball (((% c) x) , ((real_mul c) r))) = ((IMAGE (\x. ((% c) x))) (cball (x , r))))))))
EXTREME_POINT_OF_LINEAR_IMAGE
|- (!f. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((extreme_point_of (f x)) ((IMAGE f) s)) = ((extreme_point_of x) s))))
OPEN_IN_TOPSPACE
|- (!top. ((open_in top) (topspace top)))
SUBSET_RESTRICT
|- (!s. (!P. ((GSPEC (\GEN%PVAR%8. (?x. (((SETSPEC GEN%PVAR%8) ((x IN s) /\ (P x))) x)))) SUBSET s)))
TAYLOR_CSIN
|- (!n. (!z. ((real_le (vector_norm ((vector_sub (csin z)) ((vsum ((NUMERAL _0) .. n)) (\k. ((complex_mul ((complex_pow (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) k)) ((complex_div ((complex_pow z) (((NUMERAL (BIT0 (BIT1 _0))) * k) + (NUMERAL (BIT1 _0))))) (Cx (real_of_num (FACT (((NUMERAL (BIT0 (BIT1 _0))) * k) + (NUMERAL (BIT1 _0))))))))))))) ((real_mul (exp (real_abs (Im z)))) ((real_div ((real_pow (vector_norm z)) (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 (BIT1 _0)))))) (real_of_num (FACT (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT0 (BIT1 _0)))))))))))
VECTOR_ANGLE_SYM
|- (!x. (!y. (((vector_angle x) y) = ((vector_angle y) x))))
CX_LOG
|- (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> ((Cx (log z)) = (clog (Cx z)))))
Fan_defs.f_fan_pair
|- (!V. (!E. (!w. (!v. (((f_fan_pair (V , E)) (v , w)) = (w , (((((inverse_sigma_fan (vec (NUMERAL _0))) V) E) w) v)))))))
EXISTS_IN_IMAGE
|- (!f. (!s. ((?y. ((y IN ((IMAGE f) s)) /\ (P y))) = (?x. ((x IN s) /\ (P (f x)))))))
REAL_DIV_EQ_0
|- (!x. (!y. ((((real_div x) y) = (real_of_num (NUMERAL _0))) = ((x = (real_of_num (NUMERAL _0))) \/ (y = (real_of_num (NUMERAL _0)))))))
Fnjlbxs.vector_of_list_comp0
|- (!d. (!list. (((dimindex UNIV) = d) ==> (((d * (LENGTH list)) <= (dimindex UNIV)) ==> (!i. (((((d * (LENGTH list)) + (NUMERAL (BIT1 _0))) <= i) /\ (i <= (dimindex UNIV))) ==> ((((vector_of_list d) list) $ i) = (real_of_num (NUMERAL _0)))))))))
Trigonometry.OWECYNV1
|- (!s. ((convex s) = (!x. (!y. (!u. (!v. (((x IN s) /\ ((y IN s) /\ (((real_le (real_of_num (NUMERAL _0))) u) /\ (((real_le (real_of_num (NUMERAL _0))) v) /\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> (((vector_add ((% u) x)) ((% v) y)) IN s))))))))
ROTATE2D_ADD
|- (!s. (!t. (!x. (((rotate2d ((real_add s) t)) x) = ((rotate2d s) ((rotate2d t) x))))))
pair_INDUCT
|- (!P. ((!x. (!y. (P (x , y)))) ==> (!p. (P p))))
Tame_list.rev1
|- (!h. ((rev ((CONS h) NIL)) = ((CONS h) NIL)))
IM_CCOS
|- (!z. ((Im (ccos z)) = ((real_mul ((real_div ((real_sub (exp (real_neg (Im z)))) (exp (Im z)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (sin (Re z)))))
NEGLIGIBLE_IFF_LEBESGUE_MEASURABLE_SUBSETS
|- (!s. ((negligible s) = (!t. ((t SUBSET s) ==> (lebesgue_measurable t)))))
CNJ_CLOG
|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> ((cnj (clog z)) = (clog (cnj z)))))
CONTINUOUS_WITHIN_CX_NORM
|- (!z. (!s. ((continuous (\z. (Cx (vector_norm z)))) ((within (at z)) s))))
BALL_INTERVAL
|- (!x. (!e. ((ball (x , e)) = (open_interval (((vector_sub x) (lift e)) , ((vector_add x) (lift e)))))))
REAL_LE_INF
|- (!b. (((~ (s = EMPTY)) /\ (!x. ((x IN s) ==> ((real_le b) x)))) ==> ((real_le b) (inf s))))
Hdplygy.J1_SY
|- (!s. ((J1_SY s) = (GSPEC (\GEN%PVAR%7598. (?x. (((SETSPEC GEN%PVAR%7598) (?i. ((((i MOD (k_sy s)) INSERT (((f_sy s) (i MOD (k_sy s))) INSERT EMPTY)) IN (J_SY s)) /\ ((i IN ((NUMERAL (BIT1 _0)) .. (k_sy s))) /\ (x = (i , (SUC (i MOD (k_sy s))))))))) x))))))
ABSOLUTE_RETRACT_IMP_AR
|- (!s. (!s'. ((((retract_of s) UNIV) /\ (((homeomorphic s) s') /\ (closed s'))) ==> ((retract_of s') UNIV))))
SUM_2
|- (!t. (((sum ((NUMERAL (BIT1 _0)) .. (NUMERAL (BIT0 (BIT1 _0))))) t) = ((real_add (t (NUMERAL (BIT1 _0)))) (t (NUMERAL (BIT0 (BIT1 _0)))))))
REAL_MAX_LT
|- (!x. (!y. (!z. (((real_lt ((real_max x) y)) z) = (((real_lt x) z) /\ ((real_lt y) z))))))
Wrgcvdr_cizmrrh.ee_of_hyp2
|- (((ee_of_hyp (x , (V , E))) u) = (((COND (u IN ((darts_of_hyp E) V))) ((SND u) , (FST u))) u))
MULT_2
|- (!n. (((NUMERAL (BIT0 (BIT1 _0))) * n) = (n + n)))