hcju/holps huggingface dataset:
Lp_ineqs_proofs.y7_hi
|- (!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 ((y7_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))
Fan_defs.n_fan_pair_ext
|- (!V. (!E. (!x. (((n_fan_pair_ext (V , E)) x) = (((COND (x IN (dart1_of_fan (V , E)))) ((n_fan_pair (V , E)) x)) x)))))
Seq2.list_sum_add
|- (!s. (!f1. (!f2. (((list_sum s) (\x. ((real_add (f1 x)) (f2 x)))) = ((real_add ((list_sum s) f1)) ((list_sum s) f2))))))
Tame_lemmas.tauVEF_alt3
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((LET (\f. (LET_END ((tauVEF (V , (E , f))) = ((real_add ((sum f) (\x. ((real_sub ((real_sub ((real_mul ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul ((real_div sol0) pi)) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (lmfun (h_dart x)))))) ((azim_dart (V , E)) x))) pi)) sol0)))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_add pi) sol0))))))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))
REAL_SUP_LE
|- (!b. (((~ (s = EMPTY)) /\ (!x. ((x IN s) ==> ((real_le x) b)))) ==> ((real_le (sup s)) b)))
DOT_SYM
|- (!x. (!y. (((dot x) y) = ((dot y) x))))
REAL_ABS_ABS
|- (!x. ((real_abs (real_abs x)) = (real_abs x)))
CONNECTED_COMPONENT_EQ_EMPTY
|- (!s. (!x. ((((connected_component s) x) = EMPTY) = (~ (x IN s)))))
REAL_BOUNDED
|- ((real_bounded s) = (bounded ((IMAGE lift) s)))
SPAN_CARD_GE_DIM
|- (!v. (!b. (((v SUBSET (span b)) /\ (FINITE b)) ==> ((dim v) <= (CARD b)))))
REAL_DIFFERENTIABLE_AT_EXP
|- (!x. ((real_differentiable exp) (atreal x)))
LIM_COMPLEX_LMUL
|- (!f. (!l. (!c. ((((--> f) l) net) ==> (((--> (\x. ((complex_mul c) (f x)))) ((complex_mul c) l)) net)))))
REAL_LT_INF_FINITE
|- (!s. (!a. (((FINITE s) /\ (~ (s = EMPTY))) ==> (((real_lt a) (inf s)) = (!x. ((x IN s) ==> ((real_lt a) x)))))))
Fan.properties_of_set_of_edge_fan
|- (!x. (!V. (!E. (!v. (!u. ((FAN (x , (V , E))) ==> (((v INSERT (u INSERT EMPTY)) IN E) = (u IN (((set_of_edge v) V) E)))))))))
LIFT_CMUL
|- (!x. (!c. ((lift ((real_mul c) x)) = ((% c) (lift x)))))
Hypermap.NON_ZERO
|- (!n. (~ ((SUC n) = (NUMERAL _0))))
ARITH_LE
|- ((!m. (!n. (((NUMERAL m) <= (NUMERAL n)) = (m <= n)))) /\ (((_0 <= _0) = T) /\ ((!n. (((BIT0 n) <= _0) = (n <= _0))) /\ ((!n. (((BIT1 n) <= _0) = F)) /\ ((!n. ((_0 <= (BIT0 n)) = T)) /\ ((!n. ((_0 <= (BIT1 n)) = T)) /\ ((!m. (!n. (((BIT0 m) <= (BIT0 n)) = (m <= n)))) /\ ((!m. (!n. (((BIT0 m) <= (BIT1 n)) = (m <= n)))) /\ ((!m. (!n. (((BIT1 m) <= (BIT0 n)) = (m < n)))) /\ (!m. (!n. (((BIT1 m) <= (BIT1 n)) = (m <= n)))))))))))))
HOMEOMORPHISM_INJECTIVE_CLOSED_MAP_EQ
|- (!f. (!s. (!t. ((((continuous_on f) s) /\ ((((IMAGE f) s) = t) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))))) ==> ((?g. ((homeomorphism (s , t)) (f , g))) = (!u. (((closed_in ((subtopology euclidean) s)) u) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) u)))))))))
Appendix.scs_k_v39_explicit
|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_k_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = k))))))))))
ORTHOGONAL_LINEAR_IMAGE_EQ
|- (!f. (!x. (!y. (((linear f) /\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> (((orthogonal (f x)) (f y)) = ((orthogonal x) y))))))
OUTSIDE_CONNECTED_COMPONENT_LT
|- (!s. ((outside s) = (GSPEC (\GEN%PVAR%2011. (?x. (((SETSPEC GEN%PVAR%2011) (!B. (?y. (((real_lt B) (vector_norm y)) /\ (((connected_component (UNIV DIFF s)) x) y))))) x))))))
TRACE_TRANSP
|- (!A. ((trace (transp A)) = (trace A)))
REALLIM_COMPLEX
|- (((---> s) l) = ((--> (Cx o s)) (Cx l)))
DIFFERENTIABLE_BOUND
|- (!f. (!f'. (!s. (!B. (((convex s) /\ ((!x. ((x IN s) ==> (((has_derivative f) (f' x)) ((within (at x)) s)))) /\ (!x. ((x IN s) ==> ((real_le (onorm (f' x))) B))))) ==> (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y))))))))))))
CLOSED_LIMPT
|- (!s. ((closed s) = (!x. (((limit_point_of x) s) ==> (x IN s)))))
Ssrbool.implybF
|- (!b. ((b ==> F) = (~ b)))
Functional_equation.domain6
|- (!h. (!f. (!g. ((((domain6 h) f) g) = (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((h x1) x2) x3) x4) x5) x6) ==> (((((((f x1) x2) x3) x4) x5) x6) = ((((((g x1) x2) x3) x4) x5) x6)))))))))))))
FINITE_INDEX_INRANGE
|- (!i. (?k. (((NUMERAL (BIT1 _0)) <= k) /\ ((k <= (dimindex UNIV)) /\ (!x. ((x $ i) = (x $ k)))))))
INTEGRABLE_CONST
|- (!a. (!b. (!c. ((integrable_on (\x. c)) (closed_interval ((CONS (a , b)) NIL))))))
SIN_0
|- ((sin (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))
REAL_SGN_REAL_SGN
|- (!x. ((real_sgn (real_sgn x)) = (real_sgn x)))
CENTRE_IN_CBALL
|- (!x. (!e. ((x IN (cball (x , e))) = ((real_le (real_of_num (NUMERAL _0))) e))))
REAL_LEBESGUE_MEASURABLE_INTER
|- (!s. (!t. (((real_lebesgue_measurable s) /\ (real_lebesgue_measurable t)) ==> (real_lebesgue_measurable (s INTER t)))))
DOT_LSUB
|- (!x. (!y. (!z. (((dot ((vector_sub x) y)) z) = ((real_sub ((dot x) z)) ((dot y) z))))))
EXP
|- ((!m. ((m EXP (NUMERAL _0)) = (NUMERAL (BIT1 _0)))) /\ (!m. (!n. ((m EXP (SUC n)) = (m * (m EXP n))))))
TAGGED_PARTIAL_DIVISION_SUBSET
|- (!s. (!t. (!i. ((((tagged_partial_division_of s) i) /\ (t SUBSET s)) ==> ((tagged_partial_division_of t) i)))))
CARD_EQ_IMP_LE
|- (!s. (!t. ((s =_c t) ==> (s <=_c t))))
MOD_MOD
|- (!m. (!n. (!p. ((~ ((n * p) = (NUMERAL _0))) ==> (((m MOD (n * p)) MOD n) = (m MOD n))))))
IN_INTERIOR
|- (!x. (!s. ((x IN (interior s)) = (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ ((ball (x , e)) SUBSET s))))))
Lp_ineqs_def.sol_fan
|- (!f. (!V. (!E. (((sol_fan (V , E)) f) = (real_abs ((real_add ((sum f) (\x. ((real_sub ((azim_dart (V , E)) x)) pi)))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))))))
INTERIOR_INTER
|- (!s. (!t. ((interior (s INTER t)) = ((interior s) INTER (interior t)))))
CONTINUOUS_AT_TRANSLATION
|- (!a. (!z. (!f. (((continuous f) (at ((vector_add a) z))) = ((continuous (\x. (f ((vector_add a) x)))) (at z))))))
Seq.cat
|- (!x. (!t. (!s2. ((((cat NIL) s2) = s2) /\ (((cat ((CONS x) t)) s2) = ((CONS x) ((cat t) s2)))))))
LE_SUC_LT
|- (!m. (!n. (((SUC m) <= n) = (m < n))))
Hypermap_iso.hyp_iso
|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) = ((((BIJ f) (dart H)) (dart G)) /\ (!x. ((x IN (dart H)) ==> ((((edge_map G) (f x)) = (f ((edge_map H) x))) /\ ((((node_map G) (f x)) = (f ((node_map H) x))) /\ (((face_map G) (f x)) = (f ((face_map H) x))))))))))))
sign
|- (!p. ((sign p) = (((COND (evenperm p)) (real_of_num (NUMERAL (BIT1 _0)))) (real_neg (real_of_num (NUMERAL (BIT1 _0)))))))
CLOSED_IN_UNION_COMPLEMENT_COMPONENTS
|- (!u. (!s. (!c. ((((locally connected) u) /\ (((closed_in ((subtopology euclidean) u)) s) /\ (c SUBSET (components (u DIFF s))))) ==> ((closed_in ((subtopology euclidean) u)) (s UNION (UNIONS c)))))))
Pack1.ineq_lm5_3_step3
|- (!S. (!p. (!r. (!A. ((((real_le (real_of_num (NUMERAL _0))) r) /\ ((packing S) /\ ((saturated S) /\ ((fcc_compatible A) S)))) ==> ((real_le ((real_mul (sqrt (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (CARD (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0))))))))))) ((sum (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) (\v. ((real_add (A v)) (measure ((voronoi_open S) v)))))))))))
HAS_REAL_INTEGRAL_LMUL
|- (!f. (!k. (!s. (!c. ((((has_real_integral f) k) s) ==> (((has_real_integral (\x. ((real_mul c) (f x)))) ((real_mul c) k)) s))))))
HAS_COMPLEX_DERIVATIVE_POW_AT
|- (!f. (!f'. (!x. (!n. ((((has_complex_derivative f) f') (at x)) ==> (((has_complex_derivative (\x. ((complex_pow (f x)) n))) ((complex_mul (Cx (real_of_num n))) ((complex_mul ((complex_pow (f x)) (n - (NUMERAL (BIT1 _0))))) f'))) (at x)))))))
CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN
|- (!f. (!s. (((continuous_on f) s) = (!x. ((x IN s) ==> ((continuous f) ((within (at x)) s)))))))
SUM_GP
|- (!x. (!m. (!n. (((sum (m .. n)) (\i. ((real_pow x) i))) = (((COND (n < m)) (real_of_num (NUMERAL _0))) (((COND (x = (real_of_num (NUMERAL (BIT1 _0))))) (real_of_num ((n + (NUMERAL (BIT1 _0))) - m))) ((real_div ((real_sub ((real_pow x) m)) ((real_pow x) (SUC n)))) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) x))))))))
REAL_LT_IMP_LE
|- (!x. (!y. (((real_lt x) y) ==> ((real_le x) y))))
CONTENT_1
|- (!a. (!b. (((real_le (drop a)) (drop b)) ==> ((content (closed_interval ((CONS (a , b)) NIL))) = ((real_sub (drop b)) (drop a))))))
TAN_NEG
|- (!x. ((tan (real_neg x)) = (real_neg (tan x))))
REAL_LTE_TRANS
|- (!x. (!y. (!z. ((((real_lt x) y) /\ ((real_le y) z)) ==> ((real_lt x) z)))))
SIMPLE_IMAGE
|- (!f. (!s. ((GSPEC (\GEN%PVAR%11. (?x. (((SETSPEC GEN%PVAR%11) (x IN s)) (f x))))) = ((IMAGE f) s))))
BOUNDED_POS
|- (!s. ((bounded s) = (?b. (((real_lt (real_of_num (NUMERAL _0))) b) /\ (!x. ((x IN s) ==> ((real_le (vector_norm x)) b)))))))
int_abs_th
|- (!x. ((real_of_int (int_abs x)) = (real_abs (real_of_int x))))
Hypermap_iso.orbit_comm
|- (!s. (!g2. (!f. (!g1. ((!x. ((x IN s) ==> ((g1 x) IN s))) ==> ((!x. ((x IN s) ==> ((f (g1 x)) = (g2 (f x))))) ==> (!x. ((x IN s) ==> (((orbit_map g2) (f x)) = ((IMAGE f) ((orbit_map g1) x)))))))))))
ABSOLUTELY_REAL_INTEGRABLE_MIN
|- (!f. (!g. (!s. ((((absolutely_real_integrable_on f) s) /\ ((absolutely_real_integrable_on g) s)) ==> ((absolutely_real_integrable_on (\x. ((real_min (f x)) (g x)))) s)))))
Lp_ineqs_def.y6_fan
|- (!x. ((y6_fan x) = (distance x)))
REAL_DIV_RMUL
|- (!x. (!y. ((~ (y = (real_of_num (NUMERAL _0)))) ==> (((real_mul ((real_div x) y)) y) = x))))
int_le
|- (!x. (!y. (((int_le x) y) = ((real_le (real_of_int x)) (real_of_int y)))))
Hypermap.FACE_NOT_EMPTY
|- (!H. (!x. ((NUMERAL (BIT1 _0)) <= (CARD ((face H) x)))))
CONVEX_IMP_CONTRACTIBLE
|- (!s. ((convex s) ==> (contractible s)))
MIDPOINT_EQ_ENDPOINT
|- (!a. (!b. ((((midpoint (a , b)) = a) = (a = b)) /\ ((((midpoint (a , b)) = b) = (a = b)) /\ (((a = (midpoint (a , b))) = (a = b)) /\ ((b = (midpoint (a , b))) = (a = b)))))))
MEASURABLE_MEASURABLE_PREIMAGE_CLOSED
|- (!f. (!s. (!t. ((((measurable_on f) s) /\ ((measurable s) /\ (closed t))) ==> (measurable (GSPEC (\GEN%PVAR%3655. (?x. (((SETSPEC GEN%PVAR%3655) ((x IN s) /\ ((f x) IN t))) x)))))))))
PATH_IMAGE_JOIN
|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((path_image ((++ g1) g2)) = ((path_image g1) UNION (path_image g2))))))
POWER_SERIES_CONV_IMP_ABSCONV_WEAK
|- (!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. ((complex_mul (Cx (vector_norm (a n)))) ((complex_pow w) n)))))))))
MATRIX_VECTOR_MUL_LINEAR
|- (!A. (linear (\x. ((matrix_vector_mul A) x))))
HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ
|- (!f. (!s. (!t. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((homotopy_equivalent s) ((IMAGE f) t)) = ((homotopy_equivalent s) t))))))
CONVEX_HULL_TRANSLATION
|- (!a. (!s. (((hull convex) ((IMAGE (\x. ((vector_add a) x))) s)) = ((IMAGE (\x. ((vector_add a) x))) ((hull convex) s)))))
REAL_LEBESGUE_MEASURABLE
|- (!s. ((real_lebesgue_measurable s) = (lebesgue_measurable ((IMAGE lift) s))))
Yxionxl.YXIONXL3
|- (!s. (!i. ((is_scs_v39 s) ==> ((scs_arrow_v39 (s INSERT EMPTY)) (((scs_prop_equ_v39 s) i) INSERT EMPTY)))))
RIGHT_POLAR_DECOMPOSITION_UNIQUE
|- (!A. (?!P. (?U. ((orthogonal_matrix U) /\ ((positive_semidefinite P) /\ (((matrix_mul U) P) = A))))))
INTEGRABLE_CMUL
|- (!f. (!c. (!s. (((integrable_on f) s) ==> ((integrable_on (\x. ((% c) (f x)))) s)))))
ABSOLUTELY_CONTINUOUS_DIFFERENTIABLE_BV_GEN
|- (!f. (!s. (!t. (((is_interval s) /\ ((bounded s) /\ (((continuous_on f) s) /\ (((has_bounded_variation_on f) s) /\ ((COUNTABLE t) /\ (!x. ((x IN (s DIFF t)) ==> ((differentiable f) ((within (at x)) s))))))))) ==> ((absolutely_continuous_on f) s)))))
MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GT
|- (!f. (!s. ((lebesgue_measurable s) ==> (((measurable_on f) s) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV))) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3635. (?x. (((SETSPEC GEN%PVAR%3635) ((x IN s) /\ ((real_gt ((f x) $ k)) a))) x))))))))))))
List_hypermap_iso.fan_imp_finite_set
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (FINITE V))))
VECTOR_MUL_ASSOC
|- (!a. (!b. (!x. (((% a) ((% b) x)) = ((% ((real_mul a) b)) x)))))
HAS_BOUNDED_VARIATION_ON_ID
|- (!s. ((bounded s) ==> ((has_bounded_variation_on (\x. x)) s)))
Ssrbool.andb_idr
|- (!a. (!b. ((a ==> b) ==> ((a /\ b) = a))))
PATH_COMPONENT_OF_SUBSET
|- (!s. (!t. (!x. (((s SUBSET t) /\ (((path_component s) x) y)) ==> (((path_component t) x) y)))))
Lp_ineqs_proofs.dart4_yf_eq
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((y1_fan ((f_fan_pair_ext (V , E)) d)) = (y2_fan d)) /\ (((y2_fan ((f_fan_pair_ext (V , E)) d)) = ((y7_fan (V , E)) d)) /\ ((((y3_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = (y1_fan d)) /\ ((((y5_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = (y6_fan d)) /\ (((y6_fan ((f_fan_pair_ext (V , E)) d)) = ((y9_fan (V , E)) d)) /\ ((((y7_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y3_fan (V , E)) d)) /\ ((((y8_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y5_fan (V , E)) d)) /\ (((y9_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y8_fan (V , E)) d)))))))))))))))))
REAL_LE_ANTISYM
|- (!x. (!y. ((((real_le x) y) /\ ((real_le y) x)) = (x = y))))
Lp_ineqs_def.yn_fan
|- (!v. ((yn_fan v) = (vector_norm v)))
POLYTOPE_IMP_CONVEX
|- (!s. ((polytope s) ==> (convex s)))
COHOMOTOPICALLY_TRIVIAL_ON_COMPONENTS
|- (!s. (!t. ((((locally connected) s) \/ ((compact s) /\ (ANR t))) ==> ((!f. (!g. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) s) /\ (((IMAGE g) s) SUBSET t)))) ==> ((((homotopic_with (\x. T)) (s , t)) f) g)))) = (!c. ((c IN (components s)) ==> (!f. (!g. ((((continuous_on f) c) /\ ((((IMAGE f) c) SUBSET t) /\ (((continuous_on g) c) /\ (((IMAGE g) c) SUBSET t)))) ==> ((((homotopic_with (\x. T)) (c , t)) f) g))))))))))
BILINEAR_BOUNDED_POS
|- (!h. ((bilinear h) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!x. (!y. ((real_le (vector_norm ((h x) y))) ((real_mul B) ((real_mul (vector_norm x)) (vector_norm y))))))))))
Functional_equation.proj_x1
|- (((((((proj_x1 x1) x2) x3) x4) x5) x6) = x1)
DIAGONAL_POSITIVE_SEMIDEFINITE
|- (!A. (!i. (((positive_semidefinite A) /\ (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) ==> ((real_le (real_of_num (NUMERAL _0))) ((A $ i) $ i)))))
azim_def
|- (!w1. (!w2. (!w. (!v. (((((azim v) w) w1) w2) = (((COND ((collinear (v INSERT (w INSERT (w1 INSERT EMPTY)))) \/ (collinear (v INSERT (w INSERT (w2 INSERT EMPTY)))))) (real_of_num (NUMERAL _0))) (@theta. (((real_le (real_of_num (NUMERAL _0))) theta) /\ (((real_lt theta) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ (?h1. (?h2. (!e1. (!e2. (!e3. (((((orthonormal e1) e2) e3) /\ ((((% (distance (w , v))) e3) = ((vector_sub w) v)) /\ (~ (w = v)))) ==> (?psi. (?r1. (?r2. ((((vector_sub w1) v) = ((vector_add ((% ((real_mul r1) (cos psi))) e1)) ((vector_add ((% ((real_mul r1) (sin psi))) e2)) ((% h1) ((vector_sub w) v))))) /\ ((((vector_sub w2) v) = ((vector_add ((% ((real_mul r2) (cos ((real_add psi) theta)))) e1)) ((vector_add ((% ((real_mul r2) (sin ((real_add psi) theta)))) e2)) ((% h2) ((vector_sub w) v))))) /\ (((real_lt (real_of_num (NUMERAL _0))) r1) /\ ((real_lt (real_of_num (NUMERAL _0))) r2))))))))))))))))))))))
HOMEOMORPHIC_REFL
|- (!s. ((homeomorphic s) s))
Functional_equation.functional_dih4_x_div_sqrtdelta_posbranch
|- (dih4_x_div_sqrtdelta_posbranch = (rotate4 dih_x_div_sqrtdelta_posbranch))
Nonlinear_lemma.hminus_lt_h0
|- ((real_lt hminus) h0)
MBOUNDED_SUBSET
|- (!m. (!s. (!t. ((((mbounded m) t) /\ (s SUBSET t)) ==> ((mbounded m) s)))))
ENR_TRANSLATION
|- (!a. (!s. ((ENR ((IMAGE (\x. ((vector_add a) x))) s)) = (ENR s))))
Seq.leq_size_uniq
|- (!s1. (!s2. ((uniq s1) ==> ((!x. (((MEM x) s1) ==> ((MEM x) s2))) ==> (((sizel s2) <= (sizel s1)) ==> (uniq s2))))))
BOUNDED_BALL
|- (!x. (!e. (bounded (ball (x , e)))))
Seq.all_pred1P
|- (!x. (!s. ((s = ((nseq (sizel s)) x)) = ((all (pred1 x)) s))))
FORALL_1
|- ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (NUMERAL (BIT1 _0)))) ==> (P i))) = (P (NUMERAL (BIT1 _0))))
ALL_MEM
|- (!P. (!l. ((!x. (((MEM x) l) ==> (P x))) = ((ALL P) l))))
OPEN_CLOSED
|- (!s. ((open s) = (closed (UNIV DIFF s))))
PASTECART_IN_PCROSS
|- (!s. (!t. (!x. (!y. ((((pastecart x) y) IN (s PCROSS t)) = ((x IN s) /\ (y IN t)))))))
CLOSURE_UNION
|- (!s. (!t. ((closure (s UNION t)) = ((closure s) UNION (closure t)))))
EULER
|- (!z. ((cexp z) = ((complex_mul (Cx (exp (Re z)))) ((vector_add (Cx (cos (Im z)))) ((complex_mul ii) (Cx (sin (Im z))))))))
COMPLEX_DIFFERENTIABLE_AT_CASN
|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> ((complex_differentiable casn) (at z))))
HAS_REAL_INTEGRAL_STRADDLE_NULL
|- (!f. (!g. (!s. (((!x. ((x IN s) ==> (((real_le (real_of_num (NUMERAL _0))) (f x)) /\ ((real_le (f x)) (g x))))) /\ (((has_real_integral g) (real_of_num (NUMERAL _0))) s)) ==> (((has_real_integral f) (real_of_num (NUMERAL _0))) s)))))
Conforming.STEP3_REDUCE_FAN
|- (!x. (!V. (!E. (!E1. (!ds. (!f1. (!f2. (!f3. (!v. (!u. (!w. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((fan80 (x , (V , E))) /\ ((ds IN (face_set (hypermap1_of_fanx (x , (V , E))))) /\ (((CARD ds) > (NUMERAL (BIT1 (BIT1 _0)))) /\ (((f1 INSERT (f2 INSERT (f3 INSERT EMPTY))) SUBSET ds) /\ ((((((f1_fan x) V) E) f1) = f2) /\ ((((((f1_fan x) V) E) f2) = f3) /\ ((~ (((((f1_fan x) V) E) f3) = f1)) /\ (((pr2 f1) = v) /\ (((pr2 f2) = u) /\ (((pr2 f3) = w) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\ (((((((sigma_fan x) V) E) u) w) = v) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))))))))))) ==> (FAN (x , (V , E1)))))))))))))))
ALL
|- ((((ALL P) NIL) = T) /\ (((ALL P) ((CONS h) t)) = ((P h) /\ ((ALL P) t))))
LET_DEF
|- (!f. (!x. (((LET f) x) = (f x))))
HAUSDIST_EMPTY
|- ((!t. ((hausdist (EMPTY , t)) = (real_of_num (NUMERAL _0)))) /\ (!s. ((hausdist (s , EMPTY)) = (real_of_num (NUMERAL _0)))))
int_ge
|- (!x. (!y. (((int_ge x) y) = ((real_ge (real_of_int x)) (real_of_int y)))))
Lp_ineqs_def.azim3_fan
|- (!V. (!E. (!x. (((azim3_fan (V , E)) x) = ((azim_dart (V , E)) ((inverse (f_fan_pair_ext (V , E))) x))))))
Lp_main_estimate.in_e_prime
|- (!E. (!f. (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN ((e_prime E) f)) = (((v INSERT (w INSERT EMPTY)) IN E) /\ (((v , w) IN f) \/ ((w , v) IN f))))))))
REAL_CLOSED
|- (!s. ((real_closed s) = (closed ((IMAGE lift) s))))
CONNECTED_WITH_OUTSIDE
|- (!s. (((closed s) /\ (connected s)) ==> (connected (s UNION (outside s)))))
SUPERADMISSIBLE_COND
|- (!<<. (!p. (!P. (!s. (!h. (!k. ((((((admissible <<) p) s) P) /\ (((((superadmissible <<) (\f. (\x. (((p f) x) /\ ((P f) x))))) s) h) /\ ((((superadmissible <<) (\f. (\x. (((p f) x) /\ (~ ((P f) x)))))) s) k))) ==> ((((superadmissible <<) p) s) (\f. (\x. (((COND ((P f) x)) ((h f) x)) ((k f) x))))))))))))
PATH_CONNECTED_OPEN_DELETE
|- (!s. (!a. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ ((open s) /\ (connected s))) ==> (path_connected (s DELETE a)))))
CLOSURE_TRANSLATION
|- (!a. (!s. ((closure ((IMAGE (\x. ((vector_add a) x))) s)) = ((IMAGE (\x. ((vector_add a) x))) (closure s)))))
COMPACT_SPHERE
|- (!a. (!r. (compact (GSPEC (\GEN%PVAR%555. (?y. (((SETSPEC GEN%PVAR%555) ((vector_norm ((vector_sub y) a)) = r)) y)))))))
NPRODUCT_CLAUSES_NUMSEG
|- ((!m. (((nproduct (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (NUMERAL (BIT1 _0))))) /\ (!m. (!n. (((nproduct (m .. (SUC n))) f) = (((COND (m <= (SUC n))) (((nproduct (m .. n)) f) * (f (SUC n)))) ((nproduct (m .. n)) f))))))
REAL_MEASURE_UNIQUE
|- (!s. (!m. (((has_real_measure s) m) ==> ((real_measure s) = m))))
LINEAR_COMPOSE_ADD
|- (!f. (!g. (((linear f) /\ (linear g)) ==> (linear (\x. ((vector_add (f x)) (g x)))))))
orthogonal
|- (!x. (!y. (((orthogonal x) y) = (((dot x) y) = (real_of_num (NUMERAL _0))))))
REAL_SEGMENT_SEGMENT
|- ((!a. (!b. ((closed_real_segment ((CONS (a , b)) NIL)) = ((IMAGE drop) (closed_segment ((CONS ((lift a) , (lift b))) NIL)))))) /\ (!a. (!b. ((open_real_segment (a , b)) = ((IMAGE drop) (open_segment ((lift a) , (lift b))))))))
COMPLEX_MUL_RINV
|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((complex_mul z) (complex_inv z)) = (Cx (real_of_num (NUMERAL (BIT1 _0)))))))
PATH_COMPONENT_TRANS
|- (!s. (!x. (!y. (((((path_component s) x) y) /\ (((path_component s) y) z)) ==> (((path_component s) x) z)))))
Seq.undup_uniq
|- (!s. (uniq (undup s)))
DEPENDENT_IMP_AFFINE_DEPENDENT
|- (!a. (!s. (((dependent (GSPEC (\GEN%PVAR%1511. (?x. (((SETSPEC GEN%PVAR%1511) (x IN s)) ((vector_sub x) a)))))) /\ (~ (a IN s))) ==> (affine_dependent (a INSERT s)))))
BOUNDED_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((bounded ((IMAGE f) s)) = (bounded s)))))
EVENTUALLY_WITHIN
|- (!s. (!a. (!p. (((eventually p) ((within (at a)) s)) = (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (((x IN s) /\ (((real_lt (real_of_num (NUMERAL _0))) (distance (x , a))) /\ ((real_lt (distance (x , a))) d))) ==> (p x)))))))))
HULL_ANTIMONO
|- (!P. (!Q. (!s. ((P SUBSET Q) ==> (((hull Q) s) SUBSET ((hull P) s))))))
Sphere.vol2r
|- (!r. (!y. (((vol2r y) r) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul pi) ((real_div ((real_sub ((real_mul r) r)) ((real_pow ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (NUMERAL (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))))))
Ssrnat.uphalfS
|- (!n. ((uphalf (SUC n)) = (SUC (half n))))
Functional_equation.functional_gchi4_x
|- (gchi4_x = ((mul6 (uni (gchi , proj_y4))) dih4_x))
Lp_ineqs_proofs.f_fan_pair_ext_power_inverse
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. (((d IN (dart_of_fan (V , E))) /\ (k < (CARD ((face (hypermap_of_fan (V , E))) d)))) ==> ((((POWER (f_fan_pair_ext (V , E))) k) d) = (((POWER (inverse (f_fan_pair_ext (V , E)))) ((CARD ((face (hypermap_of_fan (V , E))) d)) - k)) d))))))))
Seq.cats1
|- (!s. (!z. (((cat s) ((CONS z) NIL)) = ((rcons s) z))))
RELATIVE_INTERIOR_INTERIOR
|- (!s. ((((hull affine) s) = UNIV) ==> ((relative_interior s) = (interior s))))
SUB_ADD
|- (!m. (!n. ((n <= m) ==> (((m - n) + n) = m))))
HAS_BOUNDED_VARIATION_COMPOSE_HOMEOMORPHISM
|- (!f. (!f'. (!g. (!s. (!t. (((is_interval s) /\ (((homeomorphism (s , t)) (f , f')) /\ ((has_bounded_variation_on g) t))) ==> ((has_bounded_variation_on (g o f)) s)))))))
CONNECTED_IMP_PERFECT_AFF_DIM
|- (!s. (!x. (((connected s) /\ ((~ ((aff_dim s) = (int_of_num (NUMERAL _0)))) /\ (x IN s))) ==> ((limit_point_of x) s))))
LENGTH_MAP
|- (!l. (!f. ((LENGTH ((MAP f) l)) = (LENGTH l))))
Hypermap_and_fan.FACE_LAST_POINT
|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((v , w) IN (dart1_of_fan (V , E)))) ==> ((LET (\w'. (LET_END ((w' , v) IN ((face (hypermap_of_fan (V , E))) (v , w)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))
Hypermap.LT1_NZ
|- (!n. (((NUMERAL (BIT1 _0)) <= n) = ((NUMERAL _0) < n)))
SETDIST_ZERO
|- (!s. (!t. ((~ ((DISJOINT s) t)) ==> ((setdist (s , t)) = (real_of_num (NUMERAL _0))))))
CONVEX_DISTANCE
|- (!s. (!a. ((convex_on (\x. (distance (a , x)))) s)))
POLYHEDRON_TRANSLATION_EQ
|- (!a. (!s. ((polyhedron ((IMAGE (\x. ((vector_add a) x))) s)) = (polyhedron s))))
IS_HULL
|- (!P. (!s. ((!f. ((!s. ((s IN f) ==> (P s))) ==> (P (INTERS f)))) ==> ((P s) = (?t. (s = ((hull P) t)))))))