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))))))))


LOCALLY_CONNECTED_UNIV

|- ((locally connected) UNIV)


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)))


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))))


Vol1.radial_norm_deprecated

|- (radial_norm = radial)


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))))))))))


IN_UNIV

|- (!x. (x IN UNIV))


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)))))))


ANR_EMPTY

|- (ANR EMPTY)


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)))))


FSIGMA_EMPTY

|- (fsigma EMPTY)


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)))))))


TL

|- ((TL ((CONS h) t)) = t)


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))


Local_lemmas.AFF_GE_MONO_TRANS

|- ((S SUBSET X) ==> (((aff_ge (X DIFF S)) (Y UNION S)) SUBSET ((aff_ge X) Y)))


MEASURE_CLOSURE

|- (!s. (((bounded s) /\ (negligible (frontier s))) ==> ((measure (closure s)) = (measure s))))


Hypermap.lemma_two_series_eq

|- (!p. (!q. (!n. ((!i. ((i <= n) ==> ((p i) = (q i)))) ==> ((GSPEC (\GEN%PVAR%5391. (?i. (((SETSPEC GEN%PVAR%5391) (i <= n)) (p i))))) = (GSPEC (\GEN%PVAR%5392. (?i. (((SETSPEC GEN%PVAR%5392) (i <= n)) (q i))))))))))


EMPTY_NOT_UNIV

|- (~ (EMPTY = UNIV))


RE_COMPLEX_DIV_GT_0

|- (!a. (!b. (((real_lt (real_of_num (NUMERAL _0))) (Re ((complex_div a) b))) = ((real_lt (real_of_num (NUMERAL _0))) (Re ((complex_mul a) (cnj b)))))))


real_sums

|- (!s. (!f. (!l. ((((real_sums f) l) s) = (((---> (\n. ((sum (s INTER ((NUMERAL _0) .. n))) f))) l) sequentially)))))


DOT_RSUB

|- (!x. (!y. (!z. (((dot x) ((vector_sub y) z)) = ((real_sub ((dot x) y)) ((dot x) z))))))


UNIONS_COMPONENTS

|- (!u. (u = (UNIONS (components u))))


OPEN_IN_CONTAINS_BALL

|- (!s. (!t. (((open_in ((subtopology euclidean) t)) s) = ((s SUBSET t) /\ (!x. ((x IN s) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (((ball (x , e)) INTER t) SUBSET s)))))))))


REAL_LOG_CONVEX_MUL

|- (!f. (!g. (!s. ((((real_log_convex_on f) s) /\ ((real_log_convex_on g) s)) ==> ((real_log_convex_on (\x. ((real_mul (f x)) (g x)))) s)))))


Nonlin_def.ups234_x

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((ups234_x x1) x2) x3) x4) x5) x6) = (((ups_x x2) x3) x4))))))))


RSTC_REFL

|- (!R. (!x. (((RSTC R) x) x)))


ANGLE_EQ_0_DIST

|- (!A. (!B. (!C. (((angle (A , (B , C))) = (real_of_num (NUMERAL _0))) = ((~ (A = B)) /\ ((~ (C = B)) /\ (((distance (A , B)) = ((real_add (distance (A , C))) (distance (C , B)))) \/ ((distance (B , C)) = ((real_add (distance (A , C))) (distance (A , B)))))))))))


LAMBDA_BETA

|- (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (((lambda g) $ i) = (g i))))


CONTINUOUS_AT

|- (!f. (!x. (((continuous f) (at x)) = (((--> f) (f x)) (at x)))))


CBALL_TRANSLATION

|- (!a. (!x. (!r. ((cball (((vector_add a) x) , r)) = ((IMAGE (\y. ((vector_add a) y))) (cball (x , r)))))))


CONNECTED_EMPTY

|- (connected EMPTY)


ARITH_ADD

|- ((!m. (!n. (((NUMERAL m) + (NUMERAL n)) = (NUMERAL (m + n))))) /\ (((_0 + _0) = _0) /\ ((!n. ((_0 + (BIT0 n)) = (BIT0 n))) /\ ((!n. ((_0 + (BIT1 n)) = (BIT1 n))) /\ ((!n. (((BIT0 n) + _0) = (BIT0 n))) /\ ((!n. (((BIT1 n) + _0) = (BIT1 n))) /\ ((!m. (!n. (((BIT0 m) + (BIT0 n)) = (BIT0 (m + n))))) /\ ((!m. (!n. (((BIT0 m) + (BIT1 n)) = (BIT1 (m + n))))) /\ ((!m. (!n. (((BIT1 m) + (BIT0 n)) = (BIT1 (m + n))))) /\ (!m. (!n. (((BIT1 m) + (BIT1 n)) = (BIT0 (SUC (m + n)))))))))))))))


Tame_defs.heights

|- (!g. ((heights g) = (SND (SND (SND g)))))


ANR_TRIANGULATION

|- (!tr. ((triangulation tr) ==> (ANR (UNIONS tr))))


COMPACT_IMP_BOUNDED

|- (!s. ((compact s) ==> (bounded s)))


CONIC_PCROSS

|- (!s. (!t. (((conic s) /\ (conic t)) ==> (conic (s PCROSS t)))))


Ssrnat.muln_subl

|- (!m. (!n. (!p. (((m - n) * p) = ((m * p) - (n * p))))))


SUMS_EQ

|- (!f. (!g. (!k. (((!x. ((x IN k) ==> ((f x) = (g x)))) /\ (((sums f) l) k)) ==> (((sums g) l) k)))))


LIM_AT_INFINITY

|- (!f. (!l. ((((--> f) l) at_infinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_ge (vector_norm x)) b) ==> ((real_lt (distance ((f x) , l))) e)))))))))


Fan_defs.dart1_of_fan

|- (!V. (!E. ((dart1_of_fan (V , E)) = (GSPEC (\GEN%PVAR%5440. (?v. (?w. (((SETSPEC GEN%PVAR%5440) ((v INSERT (w INSERT EMPTY)) IN E)) (v , w)))))))))


CONVEX_EXPLICIT

|- (!s. ((convex s) = (!t. (!u. (((FINITE t) /\ ((t SUBSET s) /\ ((!x. ((x IN t) ==> ((real_le (real_of_num (NUMERAL _0))) (u x)))) /\ (((sum t) u) = (real_of_num (NUMERAL (BIT1 _0))))))) ==> (((vsum t) (\x. ((% (u x)) x))) IN s))))))


real_lt

|- (!y. (!x. (((real_lt x) y) = (~ ((real_le y) x)))))


SIMPLY_CONNECTED_IMP_PATH_CONNECTED

|- (!s. ((simply_connected s) ==> (path_connected s)))


Add_triangle.fully_surrounded_card_set_of_edge1

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (((fully_surrounded (V , E)) /\ (v IN V)) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))))))


Appendix.BBs_v39

|- (!s. (!vv. (((BBs_v39 s) vv) = ((LET (\V. (LET_END ((LET (\E. (LET_END ((LET (\FF. (LET_END ((V SUBSET ball_annulus) /\ (((periodic vv) (scs_k_v39 s)) /\ ((!i. (!j. (((real_le (((scs_a_v39 s) i) j)) (distance ((vv i) , (vv j)))) /\ ((real_le (distance ((vv i) , (vv j)))) (((scs_b_v39 s) i) j))))) /\ (((scs_k_v39 s) <= (NUMERAL (BIT1 (BIT1 _0)))) \/ (convex_local_fan (V , (E , FF)))))))))) ((IMAGE (\i. ((vv i) , (vv (SUC i))))) UNIV))))) ((IMAGE (\i. ((vv i) INSERT ((vv (SUC i)) INSERT EMPTY)))) UNIV))))) ((IMAGE vv) UNIV)))))


TWO

|- ((NUMERAL (BIT0 (BIT1 _0))) = (SUC (NUMERAL (BIT1 _0))))


INT_NOT_EQ

|- (!x. (!y. ((~ (x = y)) = (((int_lt x) y) \/ ((int_lt y) x)))))


Nonlin_def.compose6

|- (!f. (!p1. (!p2. (!p3. (!p4. (!p5. (!p6. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((((((((compose6 f) p1) p2) p3) p4) p5) p6) x1) x2) x3) x4) x5) x6) = ((((((f ((((((p1 x1) x2) x3) x4) x5) x6)) ((((((p2 x1) x2) x3) x4) x5) x6)) ((((((p3 x1) x2) x3) x4) x5) x6)) ((((((p4 x1) x2) x3) x4) x5) x6)) ((((((p5 x1) x2) x3) x4) x5) x6)) ((((((p6 x1) x2) x3) x4) x5) x6))))))))))))))))


LE

|- ((!m. ((m <= (NUMERAL _0)) = (m = (NUMERAL _0)))) /\ (!m. (!n. ((m <= (SUC n)) = ((m = (SUC n)) \/ (m <= n))))))


LOWDIM_SUBSET_HYPERPLANE

|- (!s. (((dim s) < (dimindex UNIV)) ==> (?a. ((~ (a = (vec (NUMERAL _0)))) /\ ((span s) SUBSET (GSPEC (\GEN%PVAR%183. (?x. (((SETSPEC GEN%PVAR%183) (((dot a) x) = (real_of_num (NUMERAL _0)))) x)))))))))


Tame_lemmas.fully_surrounded_tau_eq_tauq

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((V SUBSET ball_annulus) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ==> ((LET (\w'. (LET_END ((LET (\u. (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 ((LET (\y7. (LET_END ((LET (\y8. (LET_END ((LET (\y9. (LET_END ((tauVEF (V , (E , ((face (hypermap_of_fan (V , E))) (v , w))))) = (((((((((tauq y1) y2) y3) y4) y5) y6) y7) y8) y9))))) (distance (w , u)))))) (distance (w' , u)))))) (vector_norm u))))) (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) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))))))))))


Seq.perm_eq_size

|- (!s1. (!s2. (((perm_eq s1) s2) ==> ((sizel s1) = (sizel s2)))))


NOT_INSERT_EMPTY

|- (!x. (!s. (~ ((x INSERT s) = EMPTY))))


REAL_NEGLIGIBLE_SUBSET

|- (!s. (!t. (((real_negligible s) /\ (t SUBSET s)) ==> (real_negligible t))))


REALLIM_NULL_POW_EQ

|- (!net. (!f. (!n. ((~ (n = (NUMERAL _0))) ==> ((((---> (\x. ((real_pow (f x)) n))) (real_of_num (NUMERAL _0))) net) = (((---> f) (real_of_num (NUMERAL _0))) net))))))


ABSOLUTELY_INTEGRABLE_ADD

|- (!f. (!g. (!s. ((((absolutely_integrable_on f) s) /\ ((absolutely_integrable_on g) s)) ==> ((absolutely_integrable_on (\x. ((vector_add (f x)) (g x)))) s)))))


CARD_EQ_COUNTABLE_SUBSETS_SUBREAL

|- (!s. (((INFINITE s) /\ (s <=_c UNIV)) ==> ((GSPEC (\GEN%PVAR%72. (?t. (((SETSPEC GEN%PVAR%72) ((t SUBSET s) /\ (COUNTABLE t))) t)))) =_c UNIV)))


EDELSTEIN_FIX

|- (!f. (!s. (((compact s) /\ ((~ (s = EMPTY)) /\ ((((IMAGE f) s) SUBSET s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> ((real_lt (distance ((f x) , (f y)))) (distance (x , y))))))))) ==> (?!x. ((x IN s) /\ ((f x) = x))))))


IN_BALL_0

|- (!x. (!e. ((x IN (ball ((vec (NUMERAL _0)) , e))) = ((real_lt (vector_norm x)) e))))


uniformly_continuous_on

|- (!s. (!f. (((uniformly_continuous_on f) s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (!x'. (((x IN s) /\ ((x' IN s) /\ ((real_lt (distance (x' , x))) d))) ==> ((real_lt (distance ((f x') , (f x)))) e)))))))))))


Tame_lemmas.dart_leads_into_eq1

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((((((dart_leads_into (vec (NUMERAL _0))) V) E) v) w) = ((((dartset_leads_into_fan (vec (NUMERAL _0))) V) E) ((IMAGE (ext_dart (V , E))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))))


Lp_details.quad_3862621143_revised

|- (quad_nonlinear_v10 ==> (!y0. (!y1. (!y2. (!y3. (!y5. (!y6. (!y7. (!y8. (!y9. ((ineq ((CONS (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) , (y0 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y5 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y7 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y8 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y9 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL)))))))))) (((real_lt ((real_add ((((((taum y1) y2) y7) y9) y0) y6)) ((((((taum y1) y3) y7) y8) y0) y5))) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) \/ (((real_gt ((real_add ((((((taum y1) y2) y7) y9) y0) y6)) ((real_add ((real_sub ((((((taum y1) y3) y7) y8) y0) y5)) ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((real_add ((((((dih_y y1) y2) y7) y9) y0) y6)) ((((((dih_y y1) y3) y7) y8) y0) y5))))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))) ((DECIMAL (NUMERAL _0)) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) \/ (((real_lt ((((((delta_y y1) y2) y7) y9) y0) y6)) (real_of_num (NUMERAL _0))) \/ ((real_lt ((((((delta_y y1) y3) y7) y8) y0) y5)) (real_of_num (NUMERAL _0)))))))))))))))))


REAL_CONVERGENT_IMP_BOUNDED

|- (!s. (!l. ((((---> s) l) sequentially) ==> (real_bounded ((IMAGE s) UNIV)))))


INSIDE_SUBSET_CONVEX

|- (!s. (!c. (((convex c) /\ (s SUBSET c)) ==> ((inside s) SUBSET c))))


CONTINUOUS_SUB

|- (!f. (!g. (!net. ((((continuous f) net) /\ ((continuous g) net)) ==> ((continuous (\x. ((vector_sub (f x)) (g x)))) net)))))


JOINPATHS_TRANSLATION

|- (!a. (!g1. (!g2. (((++ ((\x. ((vector_add a) x)) o g1)) ((\x. ((vector_add a) x)) o g2)) = ((\x. ((vector_add a) x)) o ((++ g1) g2))))))


subpath

|- (!g. (!v. (!u. ((((subpath u) v) g) = (\x. (g ((vector_add u) ((% (drop ((vector_sub v) u))) x))))))))


FORALL_2

|- (!P. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (NUMERAL (BIT0 (BIT1 _0))))) ==> (P i))) = ((P (NUMERAL (BIT1 _0))) /\ (P (NUMERAL (BIT0 (BIT1 _0)))))))


Hexagons.PROP_EQU_IS_SCS

|- ((is_scs_v39 s) ==> (is_scs_v39 ((scs_prop_equ_v39 s) i)))


REAL_DIFFERENTIABLE_AT_LOG

|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_differentiable log) (atreal x))))


VECTOR_SUB_EQ

|- (!x. (!y. ((((vector_sub x) y) = (vec (NUMERAL _0))) = (x = y))))


Ssrnat.ltn_exp2l

|- (!m. (!n1. (!n2. (((NUMERAL (BIT1 _0)) < m) ==> (((m EXP n1) < (m EXP n2)) = (n1 < n2))))))


REAL_POW_LT

|- (!x. (!n. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_pow x) n)))))


Seq.size_cons

|- (!h. (!t. ((sizel ((CONS h) t)) = (SUC (sizel t)))))


Lp_ineqs_def.rhazim3_fan

|- (!V. (!E. (!x. (((rhazim3_fan (V , E)) x) = ((rhazim_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) x))))))


VECTOR_SUB_RZERO

|- (((vector_sub x) (vec (NUMERAL _0))) = x)


Tame_defs.GRAPH

|- (!fs. (!n. (!f. (!h. (((((Graph fs) n) f) h) = (fs , (n , (f , h))))))))


DROP_NEG

|- (!x. ((drop (vector_neg x)) = (real_neg (drop x))))


COMPLEX_INV_INV

|- (!x. ((complex_inv (complex_inv x)) = x))


CONTINUOUS_ON_EQ

|- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((continuous_on f) s)) ==> ((continuous_on g) s)))))


RIGHT_OR_FORALL_THM

|- (!P. (!Q. ((P \/ (!x. (Q x))) = (!x. (P \/ (Q x))))))


int_gcd

|- (!a. (!b. (((int_le (int_of_num (NUMERAL _0))) (int_gcd (a , b))) /\ (((int_divides (int_gcd (a , b))) a) /\ (((int_divides (int_gcd (a , b))) b) /\ (?x. (?y. ((int_gcd (a , b)) = ((int_add ((int_mul a) x)) ((int_mul b) y))))))))))


PCROSS_EMPTY

|- ((!s. ((s PCROSS EMPTY) = EMPTY)) /\ (!t. ((EMPTY PCROSS t) = EMPTY)))


Add_triangle.dart1_switch

|- (!V. (!E. (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) = ((w , v) IN (dart1_of_fan (V , E))))))))


REAL_LE_SQUARE

|- (!x. ((real_le (real_of_num (NUMERAL _0))) ((real_mul x) x)))


MULT_0

|- (!m. ((m * (NUMERAL _0)) = (NUMERAL _0)))


FRONTIER_UNIONS_SUBSET_CLOSURE

|- (!f. ((frontier (UNIONS f)) SUBSET (closure (UNIONS (GSPEC (\GEN%PVAR%1825. (?t. (((SETSPEC GEN%PVAR%1825) (t IN f)) (frontier t)))))))))


SETDIST_TRANSLATION

|- (!a. (!s. (!t. ((setdist (((IMAGE (\x. ((vector_add a) x))) s) , ((IMAGE (\x. ((vector_add a) x))) t))) = (setdist (s , t))))))


SUM_OVER_PERMUTATIONS_INSERT

|- (!f. (!a. (!s. (((FINITE s) /\ (~ (a IN s))) ==> (((sum (GSPEC (\GEN%PVAR%295. (?p. (((SETSPEC GEN%PVAR%295) ((permutes p) (a INSERT s))) p))))) f) = ((sum (a INSERT s)) (\b. ((sum (GSPEC (\GEN%PVAR%296. (?p. (((SETSPEC GEN%PVAR%296) ((permutes p) s)) p))))) (\q. (f ((swap (a , b)) o q)))))))))))


CONIC_INTERS

|- ((!s. ((s IN f) ==> (conic s))) ==> (conic (INTERS f)))


COMPLEX_VEC_0

|- ((vec (NUMERAL _0)) = (Cx (real_of_num (NUMERAL _0))))


Seq.flatten0

|- ((flatten NIL) = NIL)


REAL_MEASURABLE_ON_UNIV

|- (((real_measurable_on (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) UNIV) = ((real_measurable_on f) s))


OPEN_SET_COCOUNTABLE_COORDINATES

|- (!P. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (COUNTABLE (UNIV DIFF (GSPEC (\GEN%PVAR%1866. (?x. (((SETSPEC GEN%PVAR%1866) ((P i) x)) x)))))))) ==> (!s. (((open s) /\ (~ (s = EMPTY))) ==> (?x. ((x IN s) /\ (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((P i) (x $ i))))))))))


Appendix.periodic2

|- (!k. (!f. (((periodic2 f) k) = (!i. (!j. ((((f (i + k)) j) = ((f i) j)) /\ (((f i) (j + k)) = ((f i) j))))))))


NSUM_SUPPORT

|- (!f. (!s. (((nsum (((support +) f) s)) f) = ((nsum s) f))))


CONVEX_CONE_LINEAR_IMAGE_EQ

|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((convex_cone ((IMAGE f) s)) = (convex_cone s)))))


IS_REALINTERVAL_CONNECTED

|- (!s. ((is_realinterval s) = (connected ((IMAGE lift) s))))


Functional_equation.scalar6

|- (!f. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!r. (((((((((scalar6 f) r) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((f x1) x2) x3) x4) x5) x6)) r))))))))))


LOCALLY_COMPACT_OPEN_INTER_CLOSURE

|- (!s. (((locally compact) s) ==> (?t. ((open t) /\ (s = (t INTER (closure s)))))))


RE_ADD

|- (!x. (!y. ((Re ((vector_add x) y)) = ((real_add (Re x)) (Re y)))))


Conforming.pr23

|- (pr23 = (GABS (\f. (!x. (!y. (!z. (!t. ((GEQ (f (x , (y , (z , t))))) (y , z)))))))))


locally

|- (!s. (!P. (((locally P) s) = (!w. (!x. ((((open_in ((subtopology euclidean) s)) w) /\ (x IN w)) ==> (?u. (?v. (((open_in ((subtopology euclidean) s)) u) /\ ((P v) /\ ((x IN u) /\ ((u SUBSET v) /\ (v SUBSET w)))))))))))))


CLOSURE_SUBSET_EQ

|- (!s. (((closure s) SUBSET s) = (closed s)))


List_hypermap.components_hypermap_of_list

|- (!L. ((good_list L) ==> (((dart (hypermap_of_list L)) = (darts_of_list L)) /\ (((edge_map (hypermap_of_list L)) = (e_list_ext L)) /\ (((node_map (hypermap_of_list L)) = (n_list_ext L)) /\ ((face_map (hypermap_of_list L)) = (f_list_ext L)))))))


Sphere.x1_delta_x

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((x1_delta_x x1) x2) x3) x4) x5) x6) = ((real_mul x1) ((((((delta_x x1) x2) x3) x4) x5) x6)))))))))


COVERING_SPACE_FIBRE_NO_LIMPT

|- (!p. (!c. (!s. (!a. (!b. ((((covering_space (c , p)) s) /\ (a IN c)) ==> (~ ((limit_point_of a) (GSPEC (\GEN%PVAR%2072. (?x. (((SETSPEC GEN%PVAR%2072) ((x IN c) /\ ((p x) = b))) x))))))))))))


fl

|- (!l. (!x. (((fl l) x) = (?y. ((l (x , y)) \/ (l (y , x)))))))


SUMMABLE_EQ

|- (!f. (!g. (!k. (((!x. ((x IN k) ==> ((f x) = (g x)))) /\ ((summable k) f)) ==> ((summable k) g)))))


MATRIX_INV_COVARIANCE_LMUL

|- (!A. (((matrix_mul (transp A)) (matrix_inv ((matrix_mul A) (transp A)))) = (matrix_inv A)))


Hypermap_iso.bij_alt

|- (!f. (!s. (!t. ((((BIJ f) s) t) = ((!x. ((x IN s) ==> ((f x) IN t))) /\ ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) /\ (!y. ((y IN t) ==> (?x. ((x IN s) /\ ((f x) = y)))))))))))


IS_INTERVAL_CONNECTED_1

|- (!s. ((is_interval s) = (connected s)))


REAL_SERIES_ADD

|- (!x. (!x0. (!y. (!y0. (!s. (((((real_sums x) x0) s) /\ (((real_sums y) y0) s)) ==> (((real_sums (\n. ((real_add (x n)) (y n)))) ((real_add x0) y0)) s)))))))


Hypermap_iso.darts_of_fan_disj

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((DISJOINT (dart1_of_fan (V , E))) (GSPEC (\GEN%PVAR%7629. (?v. (((SETSPEC GEN%PVAR%7629) ((v IN V) /\ ((((set_of_edge v) V) E) = EMPTY))) (v , v)))))))))


FINITE_DELETE

|- (!s. (!x. ((FINITE (s DELETE x)) = (FINITE s))))


HOMEOMORPHISM_INTO_1D

|- (!f. (!s. (!t. (((path_connected s) /\ (((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)))))))


BOUNDED_CONVEX_HULL

|- (!s. ((bounded s) ==> (bounded ((hull convex) s))))


ABSOLUTELY_REAL_INTEGRABLE_CONST

|- (!a. (!b. (!c. ((absolutely_real_integrable_on (\x. c)) (closed_real_interval ((CONS (a , b)) NIL))))))


Tame_general.IN_ESTD

|- (!V. (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN (ESTD V)) = ((v IN V) /\ ((w IN V) /\ ((~ (v = w)) /\ ((real_le (distance (v , w))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))


AFF_DIM_SEGMENT

|- ((!a. (!b. ((aff_dim (closed_segment ((CONS (a , b)) NIL))) = (((COND (a = b)) (int_of_num (NUMERAL _0))) (int_of_num (NUMERAL (BIT1 _0))))))) /\ (!a. (!b. ((aff_dim (open_segment (a , b))) = (((COND (a = b)) (int_neg (int_of_num (NUMERAL (BIT1 _0))))) (int_of_num (NUMERAL (BIT1 _0))))))))


Seq2.list_sum_nil

|- (!f. (((list_sum NIL) f) = (real_of_num (NUMERAL _0))))


Conforming.SIGMA_FAN_OF_FANADD_AT_POINT4

|- (!x. (!V. (!E. (!v. (!u. (!w. (!w1. (((FAN (x , (V , E))) /\ ((FAN (x , (V , E1))) /\ ((fan80 (x , (V , E))) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\ ((~ (u = w1)) /\ (((v INSERT (w1 INSERT EMPTY)) IN E) /\ (((((((sigma_fan x) V) E) u) w) = v) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))))) ==> ((((((sigma_fan x) V) E1) v) w1) = (((((sigma_fan x) V) E) v) w1))))))))))


COUNTABLE_INTEGER

|- (COUNTABLE integer)


ANR_PCROSS

|- (!s. (!t. (((ANR s) /\ (ANR t)) ==> (ANR (s PCROSS t)))))


Sphere.y_of_x

|- (!fx. (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((((((((y_of_x fx) y1) y2) y3) y4) y5) y6) = ((((((fx ((real_mul y1) y1)) ((real_mul y2) y2)) ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5)) ((real_mul y6) y6))))))))))


Tame_general.sol0_over_pi_EQ_const1

|- (((real_div sol0) pi) = const1)


LIM_AT_ZERO

|- (!f. (!l. (!a. ((((--> f) l) (at a)) = (((--> (\x. (f ((vector_add a) x)))) l) (at (vec (NUMERAL _0))))))))


NOT_ON_PATH_BALL

|- (!g. (!z. (((path g) /\ (~ (z IN (path_image g)))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (((ball (z , e)) INTER (path_image g)) = EMPTY))))))


Miqmcsn.K_SCS_3M1_prime

|- ((scs_k_v39 scs_3M1_prime) = (NUMERAL (BIT1 (BIT1 _0))))


Real_ext.REAL_MK_POS_EXP

|- (!x. ((real_lt (real_of_num (NUMERAL _0))) (exp x)))


REAL_INTERVAL_INTERVAL

|- (((closed_real_interval ((CONS (a , b)) NIL)) = ((IMAGE drop) (closed_interval ((CONS ((lift a) , (lift b))) NIL)))) /\ ((open_real_interval (a , b)) = ((IMAGE drop) (open_interval ((lift a) , (lift b))))))


REAL_EQ_IMP_LE

|- (!x. (!y. ((x = y) ==> ((real_le x) y))))


DIFF_EMPTY

|- (!s. ((s DIFF EMPTY) = s))


SUBSET_TRANS

|- (!s. (!t. (!u. (((s SUBSET t) /\ (t SUBSET u)) ==> (s SUBSET u)))))


Seq.mem_subseq

|- (!s1. (!s2. (((subseq s1) s2) ==> (!x. (((MEM x) s1) ==> ((MEM x) s2))))))


Functional_equation.ldih_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))))))) ==> ((((((((ldih_x x1) x2) x3) x4) x5) x6) = ((((((ldihatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih2_x x1) x2) x3) x4) x5) x6) = ((((((ldih2atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((ldih3_x x1) x2) x3) x4) x5) x6) = ((((((ldih3atn_x x1) x2) x3) x4) x5) x6)) /\ (((((((ldih6_x x1) x2) x3) x4) x5) x6) = ((((((ldih6atn_x x1) x2) x3) x4) x5) x6))))))))))))


SERIES_COMPARISON

|- (!f. (!g. (!s. (((?l. (((sums (lift o g)) l) s)) /\ (?N. (!n. (((n >= N) /\ (n IN s)) ==> ((real_le (vector_norm (f n))) (g n)))))) ==> (?l. (((sums f) l) s))))))


REAL_MUL_LNEG

|- (!x. (!y. (((real_mul (real_neg x)) y) = (real_neg ((real_mul x) y)))))


BOUNDED_PARTIAL_SUMS

|- (!f. (!k. ((bounded (GSPEC (\GEN%PVAR%1123. (?n. (((SETSPEC GEN%PVAR%1123) (n IN UNIV)) ((vsum (k .. n)) f)))))) ==> (bounded (GSPEC (\GEN%PVAR%1124. (?m. (?n. (((SETSPEC GEN%PVAR%1124) ((m IN UNIV) /\ (n IN UNIV))) ((vsum (m .. n)) f))))))))))


HOMOTOPIC_PATHS_REFL

|- (!s. (!p. ((((homotopic_paths s) p) p) = ((path p) /\ ((path_image p) SUBSET s)))))


TC_TRANS

|- (!R. (!x. (!y. (!z. (((((TC R) x) y) /\ (((TC R) y) z)) ==> (((TC R) x) z))))))


SUBSET_HULL

|- (!P. (!s. (!t. ((P t) ==> ((((hull P) s) SUBSET t) = (s SUBSET t))))))


REAL_POLYNOMIAL_FUNCTION_EXPLICIT_UNIV

|- (!f. ((real_polynomial_function f) = (?a. ((FINITE (GSPEC (\GEN%PVAR%5014. (?k. (((SETSPEC GEN%PVAR%5014) (~ ((a k) = (real_of_num (NUMERAL _0))))) k))))) /\ (f = (\x. ((sum UNIV) (\k. ((real_mul (a k)) ((product ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. ((real_pow (x $ i)) (k $ i)))))))))))))


POLYTOPE_IMP_CLOSED

|- (!s. ((polytope s) ==> (closed s)))


LEBESGUE_MEASURABLE_UNIV

|- (lebesgue_measurable UNIV)


has_vector_derivative

|- (!f. (!f'. (!net. ((((has_vector_derivative f) f') net) = (((has_derivative f) (\x. ((% (drop x)) f'))) net)))))


AFF_DIM_EMPTY

|- ((aff_dim EMPTY) = (int_neg (int_of_num (NUMERAL (BIT1 _0)))))


DET_2

|- (!A. ((det A) = ((real_sub ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))))))


CONTINUOUS_GE_ON_CLOSURE

|- (!f. (!s. (!a. ((((continuous_on (lift o f)) (closure s)) /\ (!x. ((x IN s) ==> ((real_le a) (f x))))) ==> (!x. ((x IN (closure s)) ==> ((real_le a) (f x))))))))


VITALI_COVERING_LEMMA_CBALLS_BALLS

|- (!a. (!r. (!k. (!B. ((!i. ((i IN k) ==> (((real_lt (real_of_num (NUMERAL _0))) (r i)) /\ ((real_le (r i)) B)))) ==> (?c. ((COUNTABLE c) /\ ((c SUBSET k) /\ (((pairwise (\i. (\j. ((DISJOINT (cball ((a i) , (r i)))) (cball ((a j) , (r j))))))) c) /\ (!i. ((i IN k) ==> (?j. ((j IN c) /\ ((~ ((DISJOINT (cball ((a i) , (r i)))) (cball ((a j) , (r j))))) /\ ((cball ((a i) , (r i))) SUBSET (ball ((a j) , ((real_mul (real_of_num (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) (r j)))))))))))))))))))


RSC

|- (!R. ((RSC R) = (RC (SC R))))


ARG_EQ_0

|- (!z. (((Arg z) = (real_of_num (NUMERAL _0))) = ((real z) /\ ((real_le (real_of_num (NUMERAL _0))) (Re z)))))


Fan_misc.dart1_of_fan

|- (!V. (!E. ((dart1_of_fan (V , E)) = (GSPEC (\GEN%PVAR%5545. (?v. (?w. (((SETSPEC GEN%PVAR%5545) ((v INSERT (w INSERT EMPTY)) IN E)) (v , w)))))))))


Vol1.VOLUME_PROPS_MEASURABLE

|- (!C. ((measurable C) ==> ((real_ge (measure C)) (real_of_num (NUMERAL _0)))))


OPEN_IN_RELATIVE_INTERIOR

|- (!s. ((open_in ((subtopology euclidean) ((hull affine) s))) (relative_interior s)))


REAL_MUL_RNEG

|- (!x. (!y. (((real_mul x) (real_neg y)) = (real_neg ((real_mul x) y)))))


Ssrnat.minnA

|- (!m1. (!m2. (!m3. (((minn m1) ((minn m2) m3)) = ((minn ((minn m1) m2)) m3)))))


MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GE

|- (!f. (((measurable_on f) UNIV) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV))) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3514. (?x. (((SETSPEC GEN%PVAR%3514) ((real_ge ((f x) $ k)) a)) x))))))))))


IM_II

|- ((Im ii) = (real_of_num (NUMERAL (BIT1 _0))))


PATHSTART_SUBPATH

|- (!u. (!v. (!g. ((pathstart (((subpath u) v) g)) = (g u)))))


FINITE_INTER_NUMSEG

|- (!s. (!m. (!n. (FINITE (s INTER (m .. n))))))


Appendix.scs_4M5

|- (scs_4M5' = ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) NIL))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , (real_of_num (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , (real_of_num (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))) NIL))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))


Real_ext.REAL_SV_LMUL_0

|- (!x. (((real_mul (real_of_num (NUMERAL _0))) x) = (real_of_num (NUMERAL _0))))


Real_ext.REAL_LE

|- (!m. (!n. (((real_le (real_of_num m)) (real_of_num n)) = (m <= n))))


HAS_BOUNDED_VARIATION_ON_SUB

|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ ((has_bounded_variation_on g) s)) ==> ((has_bounded_variation_on (\x. ((vector_sub (f x)) (g x)))) s)))))


Nonlinear_lemma.halfbump_x

|- (!x. ((halfbump_x x) = (bump ((real_div (sqrt x)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))


AR_DEFORMATION_RETRACT_OF_CONTRACTIBLE

|- (!s. (!t. (((contractible s) /\ ((AR t) /\ ((closed_in ((subtopology euclidean) s)) t))) ==> (?r. (((((homotopic_with (\x. T)) (s , s)) (\x. x)) r) /\ ((retraction (s , t)) r))))))


LIM_NULL

|- (!net. (!f. (!l. ((((--> f) l) net) = (((--> (\x. ((vector_sub (f x)) l))) (vec (NUMERAL _0))) net)))))


REAL_ABS_NEG

|- (!x. ((real_abs (real_neg x)) = (real_abs x)))


Functional_equation.two6

|- (two6 = (constant6 (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))


FRONTIER_FRONTIER

|- (!s. (((open s) \/ (closed s)) ==> ((frontier (frontier s)) = (frontier s))))


OPEN_BALL

|- (!x. (!e. (open (ball (x , e)))))


Fan_misc.inverse_sigma_fan

|- (!x. (!V. (!E. (!v. (((((inverse_sigma_fan x) V) E) v) = (inverse ((((extension_sigma_fan x) V) E) v)))))))


STC_INC

|- (!R. (!x. (!y. (((R x) y) ==> (((STC R) x) y)))))


CARD_EXP_FINITE

|- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> (FINITE ((^_c s) t)))))


RE_CLOG

|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((Re (clog z)) = (log (vector_norm z)))))


Hypermap_iso.hyp_iso_SURJ

|- (!f. (!H. (!G. (((hyp_iso f) (H , G)) ==> (((SURJ f) (dart H)) (dart G))))))


FINITE_RULES

|- ((FINITE EMPTY) /\ (!x. (!s. ((FINITE s) ==> (FINITE (x INSERT s))))))


AR_IMP_CONTRACTIBLE

|- (!s. ((AR s) ==> (contractible s)))


REAL_ENTIRE

|- (!x. (!y. ((((real_mul x) y) = (real_of_num (NUMERAL _0))) = ((x = (real_of_num (NUMERAL _0))) \/ (y = (real_of_num (NUMERAL _0)))))))


vector_neg

|- (!x. ((vector_neg x) = (lambdai. (real_neg (x $ i)))))


Sphere.bis_le

|- (!u. (!v. (((bis_le u) v) = (GSPEC (\GEN%PVAR%5246. (?x. (((SETSPEC GEN%PVAR%5246) ((real_le (distance (x , u))) (distance (x , v)))) x)))))))


Lp_ineqs_proofs.y78_eq

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((LET (\w'. (LET_END ((LET (\u. (LET_END ((((y8_fan (V , E)) (v , w)) = (distance (w' , u))) /\ (((y7_fan (V , E)) (v , w)) = (vector_norm u)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))))


SUBSET_LIFT_IMAGE

|- (!s. (!t. ((((IMAGE lift) s) SUBSET ((IMAGE lift) t)) = (s SUBSET t))))


MAP2

|- (((((MAP2 f) NIL) NIL) = NIL) /\ ((((MAP2 f) ((CONS h1) t1)) ((CONS h2) t2)) = ((CONS ((f h1) h2)) (((MAP2 f) t1) t2))))


DIM_INJECTIVE_LINEAR_IMAGE

|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((dim ((IMAGE f) s)) = (dim s)))))


closed_in

|- (!top. (!s. (((closed_in top) s) = ((s SUBSET (topspace top)) /\ ((open_in top) ((topspace top) DIFF s))))))


Appendix.scs_3T3

|- (scs_3T3 = ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT1 _0)))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 NIL) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT1 (BIT1 _0))))) (((funlist_v39 NIL) cstab) (NUMERAL (BIT1 (BIT1 _0))))))


REAL_LT_MAX

|- (!x. (!y. (!z. (((real_lt z) ((real_max x) y)) = (((real_lt z) x) \/ ((real_lt z) y))))))


UNCOUNTABLE_CONTAINS_LIMIT_POINT

|- (!s. ((~ (COUNTABLE s)) ==> (?x. ((x IN s) /\ ((limit_point_of x) s)))))


IM_CX

|- (!x. ((Im (Cx x)) = (real_of_num (NUMERAL _0))))


HAS_SIZE_DIFF

|- (!s. (!t. (!m. (!n. (((s HAS_SIZE m) /\ ((t HAS_SIZE n) /\ (t SUBSET s))) ==> ((s DIFF t) HAS_SIZE (m - n)))))))


REAL_OPEN

|- (!s. ((real_open s) = (open ((IMAGE lift) s))))


real_derivative

|- (!f. (!x. (((real_derivative f) x) = (@f'. (((has_real_derivative f) f') (atreal x))))))


HAUSDIST_TRANSLATION

|- (!a. (!s. (!t. ((hausdist (((IMAGE (\x. ((vector_add a) x))) s) , ((IMAGE (\x. ((vector_add a) x))) t))) = (hausdist (s , t))))))


FRONTIER_INTERIOR_SUBSET

|- (!s. ((frontier (interior s)) SUBSET (frontier s)))


ABSOLUTELY_INTEGRABLE_BOUNDED_SETVARIATION

|- (!f. (!s. (((absolutely_integrable_on f) s) ==> ((has_bounded_setvariation_on (\k. ((integral k) f))) s))))


REAL_MIN_MAX

|- (!x. (!y. (((real_min x) y) = (real_neg ((real_max (real_neg x)) (real_neg y))))))


Nonlin_def.flat_term2_234_x

|- (!d. (!x5. (!x6. ((((flat_term2_234_x d) x5) x6) = (uni (flat_term_x , (((edge2_234_x d) x5) x6)))))))


RSTC_INC

|- (!R. (!x. (!y. (((R x) y) ==> (((RSTC R) x) y)))))


AFF_DIM_NONEMPTY_INTERIOR

|- (!s. ((~ ((interior s) = EMPTY)) ==> ((aff_dim s) = (int_of_num (dimindex UNIV)))))


REAL_MEASURABLE_MEASURABLE

|- (!s. ((real_measurable s) = (measurable ((IMAGE lift) s))))


Seq2.perm_eq_cons_delete1

|- (!x. (!s. (((MEM x) s) ==> ((perm_eq ((CONS x) ((delete1 x) s))) s))))


complex_mul

|- (!w. (!z. (((complex_mul w) z) = (complex (((real_sub ((real_mul (Re w)) (Re z))) ((real_mul (Im w)) (Im z))) , ((real_add ((real_mul (Re w)) (Im z))) ((real_mul (Im w)) (Re z))))))))


COPLANAR_TRANSLATION_EQ

|- (!a. (!s. ((coplanar ((IMAGE (\x. ((vector_add a) x))) s)) = (coplanar s))))


SUBSPACE_LINEAR_IMAGE_EQ

|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((subspace ((IMAGE f) s)) = (subspace s)))))


Constants_approx.INTERVAL_MUL

|- ((((interval_arith x) (a , b)) /\ ((interval_arith y) (c , d))) ==> ((interval_arith ((real_mul x) y)) (((real_min ((real_min ((real_mul a) c)) ((real_mul a) d))) ((real_min ((real_mul b) c)) ((real_mul b) d))) , ((real_max ((real_max ((real_mul a) c)) ((real_mul a) d))) ((real_max ((real_mul b) c)) ((real_mul b) d))))))


WF_FINITE

|- (!<<. (((!x. (~ ((<< x) x))) /\ ((!x. (!y. (!z. ((((<< x) y) /\ ((<< y) z)) ==> ((<< x) z))))) /\ (!x. (FINITE (GSPEC (\GEN%PVAR%149. (?y. (((SETSPEC GEN%PVAR%149) ((<< y) x)) y)))))))) ==> (WF <<)))


REAL_LT_ADD1

|- (!x. (!y. (((real_le x) y) ==> ((real_lt x) ((real_add y) (real_of_num (NUMERAL (BIT1 _0))))))))


BOUNDED_CLOSURE_EQ

|- (!s. ((bounded (closure s)) = (bounded s)))


DENSE_IMP_PERFECT

|- (!s. (((closure s) = UNIV) ==> (!x. ((x IN s) ==> ((limit_point_of x) s)))))


OPEN_IN_SUBSET_TRANS

|- (!s. (!t. (!u. ((((open_in ((subtopology euclidean) u)) s) /\ ((s SUBSET t) /\ (t SUBSET u))) ==> ((open_in ((subtopology euclidean) t)) s)))))


Lp_ineqs_proofs.delta_y_not_neg

|- (!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_lt ((((((delta_y (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d))) (real_of_num (NUMERAL _0))))))))))))


CONTINUOUS_ON_COMPONENTS_OPEN

|- (!f. (!s. (((open s) /\ (!c. ((c IN (components s)) ==> ((continuous_on f) c)))) ==> ((continuous_on f) s))))


int_gt

|- (!x. (!y. (((int_gt x) y) = ((real_gt (real_of_int x)) (real_of_int y)))))


REAL_POW_MUL

|- (!x. (!y. (!n. (((real_pow ((real_mul x) y)) n) = ((real_mul ((real_pow x) n)) ((real_pow y) n))))))


REAL_INF_UNIQUE

|- (!s. (!b. (((!x. ((x IN s) ==> ((real_le b) x))) /\ (!b'. (((real_lt b) b') ==> (?x. ((x IN s) /\ ((real_lt x) b')))))) ==> ((inf s) = b))))


HOMOTOPY_EQUIVALENT_CONTRACTIBLE_SETS

|- (!s. (!t. (((contractible s) /\ ((contractible t) /\ ((s = EMPTY) = (t = EMPTY)))) ==> ((homotopy_equivalent s) t))))


Lp_ineqs_def.y8_fan

|- (!V. (!E. (!x. (((y8_fan (V , E)) x) = ((y5_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) x))))))


PAIR_SURJECTIVE

|- (!p. (?x. (?y. (p = (x , y)))))


SEPARATE_COMPACT_CLOSED

|- (!s. (!t. (((compact s) /\ ((closed t) /\ ((s INTER t) = EMPTY))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (!y. (((x IN s) /\ (y IN t)) ==> ((real_le d) (distance (x , y)))))))))))


Nonlinear_lemma.h0_lt_hplus

|- ((real_lt h0) hplus)


Tame_defs2.hideDups

|- (((hideDups NIL) = NIL) /\ ((hideDups ((CONS b) bs)) = ((CONS (SOME b)) ((hideDupsRec b) bs))))


SUBSET_INTERS

|- (!s. (!f. ((s SUBSET (INTERS f)) = (!t. ((t IN f) ==> (s SUBSET t))))))


DROP_IN_REAL_INTERVAL

|- ((!a. (!b. (!x. (((drop x) IN (closed_real_interval ((CONS (a , b)) NIL))) = (x IN (closed_interval ((CONS ((lift a) , (lift b))) NIL))))))) /\ (!a. (!b. (!x. (((drop x) IN (open_real_interval (a , b))) = (x IN (open_interval ((lift a) , (lift b)))))))))


Lp_ineqs_def.y1_fan

|- (!x. ((y1_fan x) = (vector_norm (FST x))))


PATHFINISH_SUBPATH

|- (!u. (!v. (!g. ((pathfinish (((subpath u) v) g)) = (g v)))))


REAL_LE_RMUL_EQ

|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_le ((real_mul x) z)) ((real_mul y) z)) = ((real_le x) y))))))


WOSET_WF

|- (!l. ((woset l) = ((WF (\x. (\y. ((l (x , y)) /\ (~ (x = y)))))) /\ (!x. (!y. ((((fl l) x) /\ ((fl l) y)) ==> ((l (x , y)) \/ (l (y , x)))))))))


option_RECURSION

|- (!NONE'. (!SOME'. (?fn. (((fn NONE) = NONE') /\ (!a. ((fn (SOME a)) = (SOME' a)))))))


REAL_INTEGRAL_UNIQUE

|- (!f. (!y. (!k. ((((has_real_integral f) y) k) ==> (((real_integral k) f) = y)))))


BILINEAR_DIFFERENTIABLE_ON_COMPOSE

|- (!f. (!g. (!h. (!s. ((((differentiable_on f) s) /\ (((differentiable_on g) s) /\ (bilinear h))) ==> ((differentiable_on (\x. ((h (f x)) (g x)))) s))))))


AR_ANR

|- (!s. ((AR s) = ((ANR s) /\ ((contractible s) /\ (~ (s = EMPTY))))))


Hypermap_iso.bij_contracted_dart

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (((BIJ contracted_dart) (d_fan ((vec (NUMERAL _0)) , (V , E)))) (dart_of_fan (V , E))))))


HULL_EQ

|- (!P. (!s. ((!f. ((!s. ((s IN f) ==> (P s))) ==> (P (INTERS f)))) ==> ((((hull P) s) = s) = (P s)))))


REAL_NEG_LT0

|- (!x. (((real_lt (real_neg x)) (real_of_num (NUMERAL _0))) = ((real_lt (real_of_num (NUMERAL _0))) x)))


Seq.cons_uniq

|- (!x. (!s. ((uniq ((CONS x) s)) = ((~ ((MEM x) s)) /\ (uniq s)))))


HAS_BOUNDED_VARIATION_ON_NEG

|- (!f. (!s. (((has_bounded_variation_on f) s) ==> ((has_bounded_variation_on (\x. (vector_neg (f x)))) s))))


Sphere.node3_y

|- (!f. (!y3. (!y1. (!y2. (!y6. (!y4. (!y5. ((((((((node3_y f) y1) y2) y3) y4) y5) y6) = ((((((f y3) y1) y2) y6) y4) y5)))))))))


Hypermap.hypermap_lemma

|- (!H. ((FINITE (dart H)) /\ (((permutes (edge_map H)) (dart H)) /\ (((permutes (node_map H)) (dart H)) /\ (((permutes (face_map H)) (dart H)) /\ (((edge_map H) o ((node_map H) o (face_map H))) = I))))))


COVERING_SPACE_IMP_SURJECTIVE

|- (!p. (!c. (!s. (((covering_space (c , p)) s) ==> (((IMAGE p) c) = s)))))


INJECTIVE_IMAGE

|- (!f. ((!s. (!t. ((((IMAGE f) s) = ((IMAGE f) t)) ==> (s = t)))) = (!x. (!y. (((f x) = (f y)) ==> (x = y))))))


ROOT_MONO_LT

|- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ ((real_lt x) y)) ==> ((real_lt ((root n) x)) ((root n) y))))))


INSIDE_CONVEX

|- (!s. ((convex s) ==> ((inside s) = EMPTY)))


Seq.size_eq0

|- (!s. (((sizel s) = (NUMERAL _0)) = (s = NIL)))


Real_ext.REAL_MK_POS_PI

|- ((real_lt (real_of_num (NUMERAL _0))) pi)


Seq.index_mem

|- (!x. (!s. ((((indexl x) s) < (sizel s)) = ((MEM x) s))))


COMPACT_SING

|- (!a. (compact (a INSERT EMPTY)))


DROP_SUB

|- (!x. (!y. ((drop ((vector_sub x) y)) = ((real_sub (drop x)) (drop y)))))


CARD_UNION_OVERLAP_EQ

|- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> (((CARD (s UNION t)) = ((CARD s) + (CARD t))) = ((s INTER t) = EMPTY)))))


Conforming.SIGMA_FAN_OF_FANADD_AT_POINT1

|- (!x. (!V. (!E. (!E1. (!v. (!u. (!w. (((FAN (x , (V , E))) /\ ((FAN (x , (V , E1))) /\ (((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) /\ ((fan80 (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))) ==> ((((((sigma_fan x) V) E1) v) w) = (((((sigma_fan x) V) E) v) u))))))))))


CLOSURE_CLOSED

|- (!s. ((closed s) ==> ((closure s) = s)))


HULL_MONO

|- (!P. (!s. (!t. ((s SUBSET t) ==> (((hull P) s) SUBSET ((hull P) t))))))


IS_INTERVAL_TRANSLATION_EQ

|- (!a. (!s. ((is_interval ((IMAGE (\x. ((vector_add a) x))) s)) = (is_interval s))))


sums

|- (!s. (!f. (!l. ((((sums f) l) s) = (((--> (\n. ((vsum (s INTER ((NUMERAL _0) .. n))) f))) l) sequentially)))))


REAL_LE_NEG

|- (!x. (!y. (((real_le (real_neg x)) (real_neg y)) = ((real_le y) x))))


PATH_COMPONENT_SYM

|- (!s. (!x. (!y. ((((path_component s) x) y) ==> (((path_component s) y) x)))))


BOREL_TRANSLATION

|- (!a. (!s. ((borel ((IMAGE (\x. ((vector_add a) x))) s)) = (borel s))))


Tame_list.size_list_update

|- (!xs. (!i. (!v. ((sizel (((list_update xs) i) v)) = (sizel xs)))))


CONVEX_RELATIVE_INTERIOR

|- (!s. ((convex s) ==> (convex (relative_interior s))))


Optimize.ineq

|- ((!c. (((ineq NIL) c) = c)) /\ (!a. (!x. (!b. (!xs. (!c. (((ineq ((CONS (a , (x , b))) xs)) c) = ((((real_le a) x) /\ ((real_le x) b)) ==> ((ineq xs) c)))))))))


CONTINUOUS_CONST

|- (!net. (!c. ((continuous (\x. c)) net)))


LIM_SUB

|- (!net. (!f. (!g. (!l. (!m. (((((--> f) l) net) /\ (((--> g) m) net)) ==> (((--> (\x. ((vector_sub (f x)) (g x)))) ((vector_sub l) m)) net)))))))


INTERVAL_EQ_EMPTY_1

|- (!a. (!b. ((((closed_interval ((CONS (a , b)) NIL)) = EMPTY) = ((real_lt (drop b)) (drop a))) /\ (((open_interval (a , b)) = EMPTY) = ((real_le (drop b)) (drop a))))))


IN_DIFF

|- (!s. (!t. (!x. ((x IN (s DIFF t)) = ((x IN s) /\ (~ (x IN t)))))))


Add_triangle.dart_of_fan_eq_image

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((dart_of_fan (V , E)) = ((IMAGE contracted_dart) (d_fan ((vec (NUMERAL _0)) , (V , E))))))))


PATH_CONNECTED_OPEN_DIFF_CARD_LT

|- (!s. (!t. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ ((open s) /\ ((connected s) /\ (t <_c UNIV)))) ==> (path_connected (s DIFF t)))))


MONOIDAL_REAL_ADD

|- (monoidal real_add)


int_min_th

|- (!x. (!y. ((real_of_int ((int_min x) y)) = ((real_min (real_of_int x)) (real_of_int y)))))


SUB_SUC

|- (!m. (!n. (((SUC m) - (SUC n)) = (m - n))))


SUB_0

|- (!m. ((((NUMERAL _0) - m) = (NUMERAL _0)) /\ ((m - (NUMERAL _0)) = m)))


CARD_NUMSEG

|- (!m. (!n. ((CARD (m .. n)) = ((n + (NUMERAL (BIT1 _0))) - m))))


BOUNDED_TRANSLATION_EQ

|- (!a. (!s. ((bounded ((IMAGE (\x. ((vector_add a) x))) s)) = (bounded s))))


pairwise

|- (!s. (!r. (((pairwise r) s) = (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> ((r x) y)))))))


HAS_BOUNDED_SETVARIATION_ON_EQ

|- (!f. (!g. (!s. (((!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ ((closed_interval ((CONS (a , b)) NIL)) SUBSET s)) ==> ((f (closed_interval ((CONS (a , b)) NIL))) = (g (closed_interval ((CONS (a , b)) NIL))))))) /\ ((has_bounded_setvariation_on f) s)) ==> ((has_bounded_setvariation_on g) s)))))


COMPLEX_SUB_0

|- (!x. (!y. ((((vector_sub x) y) = (Cx (real_of_num (NUMERAL _0)))) = (x = y))))


HAS_INTEGRAL_COMBINE_TAGGED_DIVISION

|- (!f. (!s. (!p. (!i. ((((tagged_division_of p) s) /\ (!x. (!k. (((x , k) IN p) ==> (((has_integral f) (i k)) k))))) ==> (((has_integral f) ((vsum p) (GABS (\f. (!x. (!k. ((GEQ (f (x , k))) (i k)))))))) s))))))


SUBSET_UNION_ABSORPTION

|- (!s. (!t. ((s SUBSET t) = ((s UNION t) = t))))


RELATIVE_FRONTIER_CLOSURE

|- (!s. ((convex s) ==> ((relative_frontier (closure s)) = (relative_frontier s))))


COS_NPI

|- (!n. ((cos ((real_mul (real_of_num n)) pi)) = ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) n)))


REAL_CONTINUOUS_ON_BERNOULLI

|- (!n. (!s. ((real_continuous_on (bernoulli n)) s)))


CONVEX_CONNECTED_1

|- (!s. ((convex s) = (connected s)))


CEXP_N

|- (!n. (!x. ((cexp ((complex_mul (Cx (real_of_num n))) x)) = ((complex_pow (cexp x)) n))))


SUMMABLE_ZETA_INTEGER

|- (!n. (!m. (((NUMERAL (BIT0 (BIT1 _0))) <= m) ==> ((summable (from n)) (\k. (complex_inv ((complex_pow (Cx (real_of_num k))) m)))))))


Lp_ineqs_proofs.y9_hi_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 ((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)))))))))))))))))))


COMPLETE_INJECTIVE_LINEAR_IMAGE_EQ

|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((complete ((IMAGE f) s)) = (complete s)))))


LE_RDIV_EQ

|- (!a. (!b. (!n. ((~ (a = (NUMERAL _0))) ==> ((n <= (b DIV a)) = ((a * n) <= b))))))


CARD_EQ_IMAGE

|- (!f. (!s. ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) ==> (((IMAGE f) s) =_c s))))


CONVEX_TRANSLATION_EQ

|- (!a. (!s. ((convex ((IMAGE (\x. ((vector_add a) x))) s)) = (convex s))))


ENR_OPEN_IN

|- (!s. (!t. ((((open_in ((subtopology euclidean) t)) s) /\ (ENR t)) ==> (ENR s))))


REAL_MUL_RZERO

|- (!x. (((real_mul x) (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0))))


OPEN_IN_TRANSLATION_EQ

|- (!a. (!s. (!t. (((open_in ((subtopology euclidean) ((IMAGE (\x. ((vector_add a) x))) t))) ((IMAGE (\x. ((vector_add a) x))) s)) = ((open_in ((subtopology euclidean) t)) s)))))


INTERIOR_OPEN

|- (!s. ((open s) ==> ((interior s) = s)))


DIAMETER_SING

|- (!a. ((diameter (a INSERT EMPTY)) = (real_of_num (NUMERAL _0))))


URYSOHN_LOCAL_STRONG

|- (!s. (!t. (!u. (!a. (!b. ((((closed_in ((subtopology euclidean) u)) s) /\ (((closed_in ((subtopology euclidean) u)) t) /\ (((s INTER t) = EMPTY) /\ (~ (a = b))))) ==> (?f. (((continuous_on f) u) /\ ((!x. ((x IN u) ==> ((f x) IN (closed_segment ((CONS (a , b)) NIL))))) /\ ((!x. ((x IN u) ==> (((f x) = a) = (x IN s)))) /\ (!x. ((x IN u) ==> (((f x) = b) = (x IN t))))))))))))))


convex_cone

|- (!s. ((convex_cone s) = ((~ (s = EMPTY)) /\ ((convex s) /\ (conic s)))))


CNJ_CX

|- (!x. ((cnj (Cx x)) = (Cx x)))


DEPENDENT_EQ_DIM_LT_CARD

|- (!s. ((dependent s) = ((FINITE s) ==> ((dim s) < (CARD s)))))


DET_MUL

|- (!A. (!B. ((det ((matrix_mul A) B)) = ((real_mul (det A)) (det B)))))


SIMPLICIAL_COMPLEX_TRANSLATION

|- (!a. (!tr. ((simplicial_complex ((IMAGE (IMAGE (\x. ((vector_add a) x)))) tr)) = (simplicial_complex tr))))


HOMOTOPIC_WITH_IMP_PROPERTY

|- (!P. (!X. (!Y. (!f. (!g. (((((homotopic_with P) (X , Y)) f) g) ==> ((P f) /\ (P g))))))))


COUNTABLE_IMAGE

|- (!f. (!s. ((COUNTABLE s) ==> (COUNTABLE ((IMAGE f) s)))))


FINITE_BOOL

|- (FINITE UNIV)


Nonlin_def.mudLs_126_x

|- (!d2s. (!y3. (!y4. (!y5. (!d1s. ((((((mudLs_126_x d1s) d2s) y3) y4) y5) = ((mul6 (((((((compose6 mu6_x) (constant6 ((real_mul y3) y3))) proj_x1) proj_x2) dummy6) dummy6) dummy6)) ((add6 ((mul6 (constant6 ((real_div (real_of_num (NUMERAL (BIT1 _0)))) ((real_add d1s) d2s)))) ((sub6 (((delta_126_x ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5))) (constant6 ((real_mul d1s) d1s))))) (constant6 d1s)))))))))


INT_MUL_LNEG

|- (!x. (!y. (((int_mul (int_neg x)) y) = (int_neg ((int_mul x) y)))))


ARITH_MULT

|- ((!m. (!n. (((NUMERAL m) * (NUMERAL n)) = (NUMERAL (m * n))))) /\ (((_0 * _0) = _0) /\ ((!n. ((_0 * (BIT0 n)) = _0)) /\ ((!n. ((_0 * (BIT1 n)) = _0)) /\ ((!n. (((BIT0 n) * _0) = _0)) /\ ((!n. (((BIT1 n) * _0) = _0)) /\ ((!m. (!n. (((BIT0 m) * (BIT0 n)) = (BIT0 (BIT0 (m * n)))))) /\ ((!m. (!n. (((BIT0 m) * (BIT1 n)) = ((BIT0 m) + (BIT0 (BIT0 (m * n))))))) /\ ((!m. (!n. (((BIT1 m) * (BIT0 n)) = ((BIT0 n) + (BIT0 (BIT0 (m * n))))))) /\ (!m. (!n. (((BIT1 m) * (BIT1 n)) = ((BIT1 m) + ((BIT0 n) + (BIT0 (BIT0 (m * n)))))))))))))))))


Ckqowsa_3_points.aff_ge_0_2

|- (!u. (!v. (((~ (u = (vec (NUMERAL _0)))) /\ (~ (v = (vec (NUMERAL _0))))) ==> (((aff_ge ((vec (NUMERAL _0)) INSERT EMPTY)) (u INSERT (v INSERT EMPTY))) = (GSPEC (\GEN%PVAR%7632. (?y. (((SETSPEC GEN%PVAR%7632) (?t1. (?t2. (((real_le (real_of_num (NUMERAL _0))) t1) /\ (((real_le (real_of_num (NUMERAL _0))) t2) /\ (y = ((vector_add ((% t1) u)) ((% t2) v)))))))) y))))))))


CONVEX_CONE_HULL_LINEAR_IMAGE

|- (!f. (!s. ((linear f) ==> (((hull convex_cone) ((IMAGE f) s)) = ((IMAGE f) ((hull convex_cone) s))))))


LEBESGUE_MEASURABLE_CLOSED

|- (!s. ((closed s) ==> (lebesgue_measurable s)))


BALL_LINEAR_IMAGE

|- (!f. (!x. (!r. (((linear f) /\ ((!y. (?x. ((f x) = y))) /\ (!x. ((vector_norm (f x)) = (vector_norm x))))) ==> ((ball ((f x) , r)) = ((IMAGE f) (ball (x , r))))))))


EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL

|- (!s. ((FINITE s) ==> ((GSPEC (\GEN%PVAR%1789. (?y. (((SETSPEC GEN%PVAR%1789) (?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 (relative_interior ((hull convex) s)))))


AFF_DIM_AFFINE_HULL

|- (!s. ((aff_dim ((hull affine) s)) = (aff_dim s)))


RAT_LEMMA2

|- ((((real_lt (real_of_num (NUMERAL _0))) y1) /\ ((real_lt (real_of_num (NUMERAL _0))) y2)) ==> (((real_add ((real_div x1) y1)) ((real_div x2) y2)) = ((real_mul ((real_add ((real_mul x1) y2)) ((real_mul x2) y1))) ((real_mul (real_inv y1)) (real_inv y2)))))


REAL_ADD_RINV

|- (!x. (((real_add x) (real_neg x)) = (real_of_num (NUMERAL _0))))


HAS_REAL_INTEGRAL_RESTRICT_UNIV

|- (!f. (!s. (!i. ((((has_real_integral (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) i) UNIV) = (((has_real_integral f) i) s)))))


REAL_DERIVATIVE_UNIQUE_ATREAL

|- (!f. (!z. (!f'. (!f''. (((((has_real_derivative f) f') (atreal z)) /\ (((has_real_derivative f) f'') (atreal z))) ==> (f' = f''))))))


Lp_ineqs_proofs.y2_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 (y2_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))


SUBSET_CLOSURE

|- (!s. (!t. ((s SUBSET t) ==> ((closure s) SUBSET (closure t)))))


DIAMETER_ATTAINED_RELATIVE_FRONTIER

|- (!s. (((bounded s) /\ (~ ((diameter s) = (real_of_num (NUMERAL _0))))) ==> (?x. (?y. ((x IN (relative_frontier s)) /\ ((y IN (relative_frontier s)) /\ ((vector_norm ((vector_sub x) y)) = (diameter s))))))))


face_of

|- (!s. (!t. (((face_of t) s) = ((t SUBSET s) /\ ((convex t) /\ (!a. (!b. (!x. (((a IN s) /\ ((b IN s) /\ ((x IN t) /\ (x IN (open_segment (a , b)))))) ==> ((a IN t) /\ (b IN t)))))))))))


FINITE_SEGMENT

|- ((!a. (!b. ((FINITE (closed_segment ((CONS (a , b)) NIL))) = (a = b)))) /\ (!a. (!b. ((FINITE (open_segment (a , b))) = (a = b)))))


REAL_SECOND_MEAN_VALUE_THEOREM_BONNET_FULL

|- (!f. (!g. (!a. (!b. (((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_of_num (NUMERAL _0))) (g x)))) /\ (!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_le x) y))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_real_interval ((CONS (a , b)) NIL))) /\ (((has_real_integral (\x. ((real_mul (g x)) (f x)))) ((real_mul (g b)) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f))) (closed_real_interval ((CONS (a , b)) NIL))))))))))


Fan.hypermap_of_fan_rep

|- (!x. (!V. (!E. (!p. (((FAN (x , (V , E))) /\ (p = (\t. ((res (((t x) V) E)) (d1_fan (x , (V , E))))))) ==> (((dart (hypermap1_of_fanx (x , (V , E)))) = (d_fan (x , (V , E)))) /\ (((edge_map (hypermap1_of_fanx (x , (V , E)))) = (p e_fan)) /\ (((node_map (hypermap1_of_fanx (x , (V , E)))) = (p n_fan)) /\ ((face_map (hypermap1_of_fanx (x , (V , E)))) = (p f1_fan))))))))))


Wrgcvdr_cizmrrh.SPEC_HY_ELEMS

|- (!H. (((tuple_hypermap H) = (D , (e , (n , f)))) = (((dart H) = D) /\ (((edge_map H) = e) /\ (((node_map H) = n) /\ ((face_map H) = f))))))


REAL_LT_LE

|- (!x. (!y. (((real_lt x) y) = (((real_le x) y) /\ (~ (x = y))))))


Seq.mem_filter

|- (!a. (!x. (!s. (((MEM x) ((filter a) s)) = ((a x) /\ ((MEM x) s))))))


DIMINDEX_GE_1

|- (!s. ((NUMERAL (BIT1 _0)) <= (dimindex s)))


Ssrnat.half_add

|- (!m. (!n. ((half (m + n)) = ((((COND ((odd m) /\ (odd n))) (NUMERAL (BIT1 _0))) (NUMERAL _0)) + ((half m) + (half n))))))


SEPARATING_HYPERPLANE_CLOSED_POINT

|- (!s. (!z. (((convex s) /\ ((closed s) /\ (~ (z IN s)))) ==> (?a. (?b. (((real_lt ((dot a) z)) b) /\ (!x. ((x IN s) ==> ((real_gt ((dot a) x)) b)))))))))


IN_NUMSEG

|- (!m. (!n. (!p. ((p IN (m .. n)) = ((m <= p) /\ (p <= n))))))


Seq.perm_eq_sym

|- (!s1. (!s2. (((perm_eq s1) s2) = ((perm_eq s2) s1))))


HAUSDIST_NONTRIVIAL

|- (!s. (!t. (((bounded s) /\ ((bounded t) /\ ((~ (s = EMPTY)) /\ (~ (t = EMPTY))))) ==> ((hausdist (s , t)) = (sup ((GSPEC (\GEN%PVAR%1186. (?x. (((SETSPEC GEN%PVAR%1186) (x IN s)) (setdist ((x INSERT EMPTY) , t)))))) UNION (GSPEC (\GEN%PVAR%1187. (?y. (((SETSPEC GEN%PVAR%1187) (y IN t)) (setdist ((y INSERT EMPTY) , s))))))))))))


dependent

|- (!s. ((dependent s) = (?a. ((a IN s) /\ (a IN (span (s DELETE a)))))))


DET_I

|- ((det (mat (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT1 _0))))


Odxlstc.DEFORMATION_DIST_LE_ALL_LE3

|- (((is_scs_v39 s) /\ (((MMs_v39 s) w) /\ (((scs_k_v39 s) = k) /\ (((NUMERAL (BIT1 (BIT1 _0))) <= k) /\ (!i. ((~ ((i MOD k) = (l MOD k))) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance ((w l) , (w i)))))))))) ==> (!i. ((~ ((i MOD k) = (l MOD k))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!t. ((((real_lt (real_of_num (NUMERAL _0))) t) /\ ((real_lt t) e)) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance (((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) t)) (w l)) , (w i)))))))))))


Ssrnat.subnK

|- (!m. (!n. ((m <= n) ==> (((n - m) + m) = n))))


HAS_INTEGRAL_ALT

|- (!f. (!s. (!i. ((((has_integral f) i) s) = ((!a. (!b. ((integrable_on (\x. (((COND (x IN s)) (f x)) (vec (NUMERAL _0))))) (closed_interval ((CONS (a , b)) NIL))))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!a. (!b. (((ball ((vec (NUMERAL _0)) , B)) SUBSET (closed_interval ((CONS (a , b)) NIL))) ==> ((real_lt (vector_norm ((vector_sub ((integral (closed_interval ((CONS (a , b)) NIL))) (\x. (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))))) i))) e)))))))))))))


NORM_LE_L1

|- (!x. ((real_le (vector_norm x)) ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (real_abs (x $ i))))))


REAL_SUB_ARG

|- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> (((real_sub (Arg w)) (Arg z)) = (((COND ((real_le (Arg z)) (Arg w))) (Arg ((complex_div w) z))) ((real_sub (Arg ((complex_div w) z))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))))))


SPAN_SUB

|- (!x. (!y. (!s. (((x IN (span s)) /\ (y IN (span s))) ==> (((vector_sub x) y) IN (span s))))))


EL

|- ((((EL (NUMERAL _0)) l) = (HD l)) /\ (((EL (SUC n)) l) = ((EL n) (TL l))))


DISCRETE_BOUNDED_IMP_FINITE

|- (!s. (!e. ((((real_lt (real_of_num (NUMERAL _0))) e) /\ ((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_lt (vector_norm ((vector_sub y) x))) e))) ==> (y = x)))) /\ (bounded s))) ==> (FINITE s))))


HAS_COMPLEX_DERIVATIVE_WITHIN_SUBSET

|- (!f. (!s. (!t. (!x. (((((has_complex_derivative f) f') ((within (at x)) s)) /\ (t SUBSET s)) ==> (((has_complex_derivative f) f') ((within (at x)) t)))))))


RELATIVE_FRONTIER_RETRACT_OF_PUNCTURED_AFFINE_HULL

|- (!s. (!a. (((convex s) /\ ((bounded s) /\ (a IN (relative_interior s)))) ==> ((retract_of (relative_frontier s)) (((hull affine) s) DELETE a)))))


VSUM_DIFF

|- (!f. (!s. (!t. (((FINITE s) /\ (t SUBSET s)) ==> (((vsum (s DIFF t)) f) = ((vector_sub ((vsum s) f)) ((vsum t) f)))))))


HAS_BOUNDED_VARIATION_ON_MAX

|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ ((has_bounded_variation_on g) s)) ==> ((has_bounded_variation_on (\x. (lift ((real_max (drop (f x))) (drop (g x)))))) s)))))


Nonlin_def.constant6

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!c. ((((((((constant6 c) x1) x2) x3) x4) x5) x6) = c))))))))


CONNECTED_CLOSURE

|- (!s. ((connected s) ==> (connected (closure s))))


REAL_LE_SQUARE_ABS

|- (!x. (!y. (((real_le (real_abs x)) (real_abs y)) = ((real_le ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow y) (NUMERAL (BIT0 (BIT1 _0))))))))


MEASURE_INTERVAL_4

|- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ (((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_le (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\ ((real_le (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 (BIT1 _0))))) (a $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (NUMERAL _0)))))) /\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ (((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))) /\ (((real_le (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\ ((real_le (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 (BIT1 _0))))) (a $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (NUMERAL _0)))))))


CARD_SING

|- (!a. ((CARD (a INSERT EMPTY)) = (NUMERAL (BIT1 _0))))


FRONTIER_STRADDLE

|- (!a. (!s. ((a IN (frontier s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((?x. ((x IN s) /\ ((real_lt (distance (a , x))) e))) /\ (?x. ((~ (x IN s)) /\ ((real_lt (distance (a , x))) e)))))))))


EVENTUALLY_WITHINREAL_LE

|- (!s. (!a. (!p. (((eventually p) ((within (atreal a)) s)) = (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (((x IN s) /\ (((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\ ((real_le (real_abs ((real_sub x) a))) d))) ==> (p x)))))))))


EXTREME_POINT_NOT_IN_INTERIOR

|- (!s. (!x. (((extreme_point_of x) s) ==> (~ (x IN (interior s))))))


CBALL_LINEAR_IMAGE

|- (!f. (!x. (!r. (((linear f) /\ ((!y. (?x. ((f x) = y))) /\ (!x. ((vector_norm (f x)) = (vector_norm x))))) ==> ((cball ((f x) , r)) = ((IMAGE f) (cball (x , r))))))))


INT_LT_LE

|- (!x. (!y. (((int_lt x) y) = (((int_le x) y) /\ (~ (x = y))))))


Hijqaha.K_SCS_4M5

|- ((scs_k_v39 scs_4M5') = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))


REAL_OF_NUM_POW

|- (!x. (!n. (((real_pow (real_of_num x)) n) = (real_of_num (x EXP n)))))


BALL_1

|- (!x. (!r. (((cball (x , r)) = (closed_interval ((CONS (((vector_sub x) (lift r)) , ((vector_add x) (lift r)))) NIL))) /\ ((ball (x , r)) = (open_interval (((vector_sub x) (lift r)) , ((vector_add x) (lift r))))))))


COMPLEX_DIFFERENTIABLE_AT_WITHIN

|- (!f. (!s. (!z. (((complex_differentiable f) (at z)) ==> ((complex_differentiable f) ((within (at z)) s))))))


complex_sub

|- (((vector_sub w) z) = ((vector_add w) (vector_neg z)))


COMMUTING_MATRIX_INV_COVARIANCE

|- (!A. (((matrix_mul (matrix_inv ((matrix_mul (transp A)) A))) ((matrix_mul (transp A)) A)) = ((matrix_mul ((matrix_mul (transp A)) A)) (matrix_inv ((matrix_mul (transp A)) A)))))


MATRIX_ADD

|- (!f. (!g. (((linear f) /\ (linear g)) ==> ((matrix (\x. ((vector_add (f x)) (g x)))) = ((matrix_add (matrix f)) (matrix g))))))


REAL_LT_POW2

|- (!n. ((real_lt (real_of_num (NUMERAL _0))) ((real_pow (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)))


COMPLEX_DERIVATIVE_MUL

|- (!f. (!g. (!z. ((((complex_differentiable f) (at z)) /\ ((complex_differentiable g) (at z))) ==> (((complex_derivative (\w. ((complex_mul (f w)) (g w)))) z) = ((vector_add ((complex_mul (f z)) ((complex_derivative g) z))) ((complex_mul ((complex_derivative f) z)) (g z))))))))


AFF_DIM_LE_DIM

|- (!s. ((int_le (aff_dim s)) (int_of_num (dim s))))


VECTOR_VARIATION_MINUS_FUNCTION_MONOTONE

|- (!f. (!a. (!b. (!c. (!d. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (((closed_interval ((CONS (c , d)) NIL)) SUBSET (closed_interval ((CONS (a , b)) NIL))) /\ (~ ((closed_interval ((CONS (c , d)) NIL)) = EMPTY)))) ==> ((real_le ((real_sub ((vector_variation (closed_interval ((CONS (c , d)) NIL))) f)) (drop ((vector_sub (f d)) (f c))))) ((real_sub ((vector_variation (closed_interval ((CONS (a , b)) NIL))) f)) (drop ((vector_sub (f b)) (f a)))))))))))


TC_SYM

|- (!R. ((!x. (!y. (((R x) y) ==> ((R y) x)))) ==> (!x. (!y. ((((TC R) x) y) ==> (((TC R) y) x))))))


NORM_LE

|- (!x. (!y. (((real_le (vector_norm x)) (vector_norm y)) = ((real_le ((dot x) x)) ((dot y) y)))))


one

|- (!v. (v = one))


Ssrnat.distnC

|- (!m. (!n. (((distn m) n) = ((distn n) m))))


MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL

|- (!f. (!s. ((lebesgue_measurable s) ==> (((measurable_on f) s) = (!t. ((borel t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3639. (?x. (((SETSPEC GEN%PVAR%3639) ((x IN s) /\ ((f x) IN t))) x)))))))))))


SUPPORT_CLAUSES

|- ((!f. ((((support op) f) EMPTY) = EMPTY)) /\ ((!f. (!x. (!s. ((((support op) f) (x INSERT s)) = (((COND ((f x) = (neutral op))) (((support op) f) s)) (x INSERT (((support op) f) s))))))) /\ ((!f. (!x. (!s. ((((support op) f) (s DELETE x)) = ((((support op) f) s) DELETE x))))) /\ ((!f. (!s. (!t. ((((support op) f) (s UNION t)) = ((((support op) f) s) UNION (((support op) f) t)))))) /\ ((!f. (!s. (!t. ((((support op) f) (s INTER t)) = ((((support op) f) s) INTER (((support op) f) t)))))) /\ ((!f. (!s. (!t. ((((support op) f) (s DIFF t)) = ((((support op) f) s) DIFF (((support op) f) t)))))) /\ (!f. (!g. (!s. ((((support op) g) ((IMAGE f) s)) = ((IMAGE f) (((support op) (g o f)) s))))))))))))


SETDIST_SING_IN_SET

|- (!x. (!s. ((x IN s) ==> ((setdist ((x INSERT EMPTY) , s)) = (real_of_num (NUMERAL _0))))))


HAS_COMPLEX_DERIVATIVE_VSUM

|- (!f. (!net. (!s. (((FINITE s) /\ (!a. ((a IN s) ==> (((has_complex_derivative (f a)) (f' a)) net)))) ==> (((has_complex_derivative (\x. ((vsum s) (\a. ((f a) x))))) ((vsum s) f')) net)))))


real_div

|- (!x. (!y. (((real_div x) y) = ((real_mul x) (real_inv y)))))


REAL_BOUNDS_LT

|- (!x. (!k. ((((real_lt (real_neg k)) x) /\ ((real_lt x) k)) = ((real_lt (real_abs x)) k))))


WF_SUBSET

|- (((!x. (!y. (((<< x) y) ==> ((<<< x) y)))) /\ (WF <<<)) ==> (WF <<))


LIM_NORM

|- (!net. (!f. (!l. ((((--> f) l) net) ==> (((--> (\x. (lift (vector_norm (f x))))) (lift (vector_norm l))) net)))))


HAUSDIST_CLOSURE

|- ((!s. (!t. ((hausdist ((closure s) , t)) = (hausdist (s , t))))) /\ (!s. (!t. ((hausdist (s , (closure t))) = (hausdist (s , t))))))


REAL_CONVEX_ON_DERIVATIVES_IMP

|- (!f. (!f'x. (!f'y. (!s. (!x. (!y. ((((real_convex_on f) s) /\ (((closed_real_segment ((CONS (x , y)) NIL)) SUBSET s) /\ ((((has_real_derivative f) f'x) ((within (atreal x)) s)) /\ (((has_real_derivative f) f'y) ((within (atreal y)) s))))) ==> ((real_le ((real_mul f'x) ((real_sub y) x))) ((real_mul f'y) ((real_sub y) x))))))))))


PCROSS_INTERVAL

|- (!a. (!b. (!c. (!d. (((closed_interval ((CONS (a , b)) NIL)) PCROSS (closed_interval ((CONS (c , d)) NIL))) = (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL)))))))


FRONTIER_SUBSET_RETRACTION

|- (!s. (!t. (!r. (((bounded s) /\ (((frontier s) SUBSET t) /\ (((continuous_on r) (closure s)) /\ ((((IMAGE r) s) SUBSET t) /\ (!x. ((x IN t) ==> ((r x) = x))))))) ==> (s SUBSET t)))))


REALLIM_CONST

|- (!net. (!a. (((---> (\x. a)) a) net)))


LOG_MONO_LE

|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> (((real_le (log x)) (log y)) = ((real_le x) y)))))


MATRIX_INV_MULTIPLE_TRANP_LEFT

|- (!A. ((matrix_inv A) = ((matrix_mul (transp A)) ((matrix_mul (transp (matrix_inv A))) (matrix_inv A)))))


SUMS_FINITE_UNION

|- (!f. (!s. (!t. (!l. (((FINITE t) /\ (((sums f) l) s)) ==> (((sums f) ((vector_add l) ((vsum (t DIFF s)) f))) (s UNION t)))))))


binom

|- ((!n. ((binom (n , (NUMERAL _0))) = (NUMERAL (BIT1 _0)))) /\ ((!k. ((binom ((NUMERAL _0) , (SUC k))) = (NUMERAL _0))) /\ (!n. (!k. ((binom ((SUC n) , (SUC k))) = ((binom (n , (SUC k))) + (binom (n , k))))))))


ROTATE2D_COMPLEX

|- (!t. (!z. (((rotate2d t) z) = ((complex_mul (cexp ((complex_mul ii) (Cx t)))) z))))


PATH_LINEAR_IMAGE_EQ

|- (!f. (!g. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((path (f o g)) = (path g)))))


SPAN_INC

|- (!s. (s SUBSET (span s)))


SELECT_AX

|- (!P. (!x. ((P x) ==> (P (@ P)))))


HOMEOMORPHIC_TRANSLATION_RIGHT_EQ

|- (!a. (!s. (!t. (((homeomorphic s) ((IMAGE (\x. ((vector_add a) x))) t)) = ((homeomorphic s) t)))))


SERIES_TERMS_TOZERO

|- (!f. (!l. (!n. ((((sums f) l) (from n)) ==> (((--> f) (vec (NUMERAL _0))) sequentially)))))


CONIC_HULL_LINEAR_IMAGE

|- (!f. (!s. ((linear f) ==> (((hull conic) ((IMAGE f) s)) = ((IMAGE f) ((hull conic) s))))))


AFF_DIM_DIM_0

|- (!s. (((vec (NUMERAL _0)) IN ((hull affine) s)) ==> ((aff_dim s) = (int_of_num (dim s)))))


LOCALLY_COMPACT_INTER_CBALL

|- (!s. (((locally compact) s) = (!x. ((x IN s) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (closed ((cball (x , e)) INTER s))))))))


REAL_MAX_MAX

|- (!x. (!y. (((real_le x) ((real_max x) y)) /\ ((real_le y) ((real_max x) y)))))


POLYHEDRON_IMP_CLOSED

|- (!s. ((polyhedron s) ==> (closed s)))


IN_INTERS

|- (!s. (!x. ((x IN (INTERS s)) = (!t. ((t IN s) ==> (x IN t))))))


REAL_INTEGRAL_0

|- (!s. (((real_integral s) (\x. (real_of_num (NUMERAL _0)))) = (real_of_num (NUMERAL _0))))


HAS_MEASURE_TRANSLATION_EQ

|- (!a. (!s. (!m. (((has_measure ((IMAGE (\x. ((vector_add a) x))) s)) m) = ((has_measure s) m)))))


CONVEX_INTERIOR

|- (!s. ((convex s) ==> (convex (interior s))))