let NAN = 0;;

let searchd pat =
  searchl (map (fun t -> ("",t)) (definitions())) pat;;

let ineq_as_thm = 
  map (fun t -> t.idv,ASSUME t.ineq) (!Ineq.ineqs);;

let prep_as_thm = 
  map (fun t -> t.idv,ASSUME t.ineq) (!Prep.prep_ineqs);;

let searchi pat =
      searchl (ineq_as_thm) pat;;

let searchp pat =
      searchl (prep_as_thm) pat;;


Sphere.vol2r;;
Sphere.vol2f;;
Sphere.gamma23f;;
Sphere.a_spine5;;
Sphere.b_spine5;;
Sphere.gamma3f;;
Sphere.vol3f;;
vol3f_palt;;
search[`vol3f`];;
search[`gchi`];;
search[`delta_x`;`eta_x`];;
vol3f_x_sqrt2_lmplus;;
vol3f_x_lfun;;
vol3_x_sqrt;;
vol3f_palt;;
help_grep "GEN";;

Sphere.eta2_135;;
ineq;;
Sphere.vol3r;;
searcht 5 [`lfun`];;
cpp_string_of_term `#3.0`;;
chop_list 3 [`a`;`b`;`c`;`d`];;

Parse_ineq.autogen;;
let ineq = Sphere.ineq;;
let all_forall = Sphere.all_forall;;
split_ineq;;
Sphere.vol3r;;
Sphere.vol_x;;
(*  *)
Sys.command("pwd");;


search[`vol3f`];;
search[`sqrt2`];;
search[`dihRy`];;
search[`solRy`];;
search[`lmfun`;`lfun`];;
search[`lmfun`;`h0`];;

Sphere.dih_x;;
Sphere.atn2;;
Sphere.vol_x;;
Sphere.vol4f;;
Sphere.gamma4f;;

search [`lfun`];;
search [`ly`];;
searcht 3 [`matn`];;
type_of `matan`;;
Sphere.vol4f;;
help "dest_binder";;
help_grep "binder";;
help "is_binder";;
binders();;

Sphere.arclength;;
searcht 5 [`acs`;`atn`];;
searcht 5 [`acs`;`asn`];;

Sphere.a_spine5;;
Sphere.atn2;;
Sphere.dih_x;;
Sphere.vol3r;;
Sphere.vol_x;;
Sphere.vol3f;;
Sphere.sol_y;;
searcht 5 [`sol_y`;`atn`];;
searcht 20 [`delta_x4`];;
search [name "Euler"];;
it;;
Sphere.gamma3f;;
Sphere.vol3f;;
Sphere.gamma23f;;
Sphere.vol2r;;
Sphere.marchal_quartic;;
Sphere.taum;;
Sphere.lnazim;;
Sphere.rhazim;;
Sphere.rho;;
Sphere.rhazim;;
Sphere.ly;;
Sphere.const1;;
Sphere.vol4f;;
Sphere.vol_y;;
Sphere.const1;;
Sphere.lfun;;
Sphere.lmfun;;
Sphere.mm1;;
Sphere.mm2;;
Sphere.rhazim2;;
Sphere.sol0;;
Sphere.edge_flat;;

Sphere.vol2r;;
Sphere.vol2f;;
Sphere.mm1;;
Sphere.mm2;;
Sphere.tau0;;
Sphere.marchal_quartic;;
Sphere.hplus;;
searcht 100 [`asn`;`atn`];;
searcht 100 [`rhazim2`];;
searcht 100 [`sol0`];;
Sphere.arclength;;

REAL_FIELD;;
search[`];;
search[`&0 < sol0`];;

Sphere.edge_flat;;
Sphere.flat_term;;
Sphere.tame_table_d;;


search[`radV`];;
Sphere.gamma4f;;
Sphere.rad2_y;;
Sphere.rad2_x;;
Parse_ineq.y_of_x_e;;
Sphere.vol_x;;
Sphere.dih6_y;;
gchi1_x;;
gchi;;
dih4_x;;
List.length !Ineq.ineqs;;
Sphere.atn2;;
"string";;

curry;;
search [name "Packing"];;
help_grep "list";;
help "chop_list";;

Sphere.gamma4f;;
Sphere.vol4f;;
Sphere.vol3f;;
Sphere.vol2f;;

Sphere.ups_x;;
Sphere.quadratic_root_plus;;
searcht 10 [`edge_flat`];;
searcht 10 [`(sqrt x) pow 2`];;
searcht 4 [`abs`;`sqrt`];;

searcht 5 [`sqrt x pow 2`];; SQRT_POW_2;;
help_grep "REWRITE_TAC";;
(*
Sphere.beta_bump_y;;
Sphere.bump;;
Sphere.lmfun;;
Sphere.beta_bump_force_y;;
Sphere.critical_edge_y;;
Sphere.vol4f;;
Sphere.hminus;;
Sphere.h0;;
Sphere.hplus;;
*)

search[`vol4f`;`lmfun`];;
search[`gcy`;`gchi`];;
search[`gcy`];;
ineq_expand;;


help_grep "REP";;
search [name "cycle_";];;
help "REPEATC";;
help "MATCH_MP";;
help_grep "CHANGED";;
search[name "MVT"];;"MVT_GENERAL";;
search[name "IVT"];;
REAL_IVT_DECREASING;;
REAL_IVT_DECREASING;;
search[`real_continuous_on`];;"REAL_CONTINUOUS_ON_NEG","REAL_CONTINUOUS_ON_MUL","REAL_CONTINUOUS_ON_POW",("REAL_CONTINUOUS_ON_LMUL"),"REAL_CONTINUOUS_ON_SUB","REAL_CONTINUOUS_ON_RMUL","REAL_CONTINUOUS_ON_ID","REAL_ROLLE","REAL_CONTINUOUS_ON_ADD","REAL_CONTINUOUS_ON_CONST";
(* also REAL_CONTINUOUS_MUL, REAL_CONTINUOUS_SUB, etc. *)
REAL_CONTINUOUS_ON_POW;;

searcht 5 [`has_real_derivative`;`u + v`],"HAS_REAL_DERIVATIVE_ADD";;
search [name "HAS_REAL_DERIVATIVE"];;
"HAS_REAL_DERIVATIVE_INCREASING_IMP","HAS_REAL_DERIVATIVE_CONST","HAS_REAL_DERIVATIVE_POW_ATREAL",("HAS_REAL_DERIVATIVE_POW_WITHIN"),"HAS_REAL_DERIVATIVE_INCREASING","HAS_REAL_DERIVATIVE_ADD","HAS_REAL_DERIVATIVE_ID",
search[`has_real_derivative`;name "MVT"];;
REAL_ROLLE;;
HAS_REAL_DERIVATIVE_INCREASING;;
searcht 10 [`has_real_derivative`;`f x < f y`];;


REAL_ROLLE;;

searcht 5 [`&0 <= a /\ &0 <= b ==> &0 <= a * b`];;
searcht 5 [`(a /\ b) /\ c` ; `a /\ b /\ c`];;



searcht 5 [`#1.2`;`hminus`]; hminus_prop;;
searcht 5 [`rhazim_x`];;
rhazim_x;;
rhazim2_x;;


searcht 5 [`(\x. f (x) * g x) real_continuous_on s`];;
searcht 10 [`inv (x * y)`;`inv x * inv y`];;
searcht 10 [`inv(inv x)`];;
searcht 5 [`&0 <= inv (x) <=> &0 <= x`];;

searcht 10 [`x ==> (&0 < u / v)`];;
searcht 10 [`x < t /\ t <= u ==> x < u`];;
searcht 10 [`x pow 2 < y pow 2`];;
searcht 10 [`x ==> (&0 <= u/v)`];;

searcht 5 [`lmfun`;`lfun`];;

Sphere.marchal_quartic;;
Sphere.lmfun;;
Sphere.hminus;;


searcht 5[`FST a,SND a`],PAIR;;
searcht 5 [`sol0`];;GSYM Tame_general.sol0_over_pi_EQ_const1
searcht 5 [`&0 < pi`],PI_POS;

searcht 4 [` (x * y) pow n`];;REAL_POW_MUL;;
searcht 10 [`x * inv y`];;


searcht 10 [`sqrt8`;`sqrt2`];;
searcht 5 [`sqrt ( x pow 2)`];;
searcht 5 [`sqrt (x ) * sqrt(a)`];;


searcht 10 [`&0 <= sqrt (x)`],SQRT_POS_LE;;;

search [name "TAYLOR"];;
search [name "GRAD"];;

(search [`x pow 2 <= y pow 2 `],Collect_gome.POW2_COND);;
searcht 10 [`(sqrt x) pow 2`],Collect_geom.POW2_COND;;
searcht 10 [`&0 <= a pow 2`],Collect_geom.REAL_LE_SQUARE_POW;;
searcht 10 [`abs b pow 2 = b pow 2`];;
searcht 10 [`x <=y /\ y <= z ==> P`],REAL_LE_TRANS;;;
searcht 10 [`&0 <= sqrt x`],Collect_geom.REAL_LE_SQUARE_POW;;
(search [`x pow 2 <= y pow 2 `],SQRT_POW2);;
searcht 10 [`x pow 2 = x * x`], REAL_POW_2;;

searcht 10 [`x pow 2 <= y pow 2`];;
searcht 10 [`sqrt(y pow 2)`];;


search[`sqrt(a*b) = sqrt a * sqrt b`],SQRT_MUL;;
search[`sqrt(a pow 2 * b)`];;
search[`sqrt(x pow 2)`],POW_2_SQRT_ABS;;


search[`acs`;`atn`],ACS_ATN;;
search[`atn`;`pi`],ATN_1;;
search[`pi`;`atn`],ATN_1;;
search[`interval_eps (float a b) r e`];;
search[`real_of_int (& n)`];;
search[`twopow (-- (int_of_num n))`],Float.TWOPOW_NEG;;
search[`x:1`];;
search_thml (term_match []) [(concl Hypermap.lemma_mAdd)] (map (fun t -> ("",t)) (definitions()));;
search!definitions
searcht 10 [`pi`;`atn`];;
searcht 10 [`atn`;`halfatn4`];;
search[`acs`;`atn`];;
searcht 10[`atn`;`pi`];;
searcht 10 [`ssqrt`];;
search[`!i. 1 <= i /\ i <= 3 ==> vec 1$ i = &1`];;
searcht[`x <= y /\ y <= z ==> x <= z`];;
search[`atn`];;
search[`cos`];;
search[`sin`;`FACT`];; 
search [`sum (:num)`];;
search[`has_real_derivative`;`atn`];; (* HAS_REAL_DERIVATIVE_ATN *)
search[`binom`];;
search[`has_real_derivative`;`Re`];;  (* HAS_REAL_COMPLEX_DERIVATIVE_AT *)
search[`has_complex_derivative`];;
search[`atn`;`catn`];; (* yes *)
search[`dihX3`];;
search[`sin x = t`];;
search[`mcell_set`];;
search[`dihV a b c d = dihV a' b' c' d'`];;
search[`facet_of`];;
searcht 5[`aff_ge`];;
searcht 5[`sum X f = &0`], SUM_TRIV_NUMSEG;;
search [`sum (i..SUC j) f`], CONJUNCT2 SUM_CLAUSES_NUMSEG;;
search [`sum (i..i)`],SUM_SING_NUMSEG;;
searcht 5[`sum (i..j) f`;`i< j`];;
search[`sum (SUC i..j) f`];;
search[`fan80 (x,V,E) = y`];;
search[`fan81 (x,V,E) = y`];;
search[`let x = (y:A) in (z:B)`];;
search[`let x = y in z`];;
search[name "CZZH"];;  
fst(List.nth (search[full `sqrt(&x) = y`]) 0);;
search[full `!x y. x + y = y + x`];;
search[`sqrt`;omit `x:num`];;
search[];;
search[exactly `a + b = b + a`];;
search[full `!a b. a + b = b + a`];;
search[`x:hypermap`];;
search[`sqrt(&x) = y`];;

search [`sum (i..SUC j)`];;
Searching.suggest();;
search[`abs(atn u - &16 *x)`];;
search[`abs(x + y) <= abs(x) + abs y`],REAL_ABS_TRIANGLE;;

help_grep "mk_n";;
help "instantiate";;
help "INSTANTIATE";;
help_grep "TERM";;
help "AP_TERM";;
help_grep "DISCH";;
help "UNDISCH";;
help_grep "conj";;
help_grep "mk.*eq";;
help_grep "ty.*var";;
help_grep "ty";;
help_grep "strip";;
help_grep "_TAC$";;
help_grep "^ASM";;
help "PROVE_HYP";;
help "EQ_MP";;
help "AP_TERM";;
Searching.help_grep "AP_";;
help "striplist";;
help"MK_CONJ";;
help "META_EXISTS_TAC";;
help "UNIFY_ACCEPT_TAC";;
help_grep "PURE_";;
help "BETA_CONV";;
help_grep "ARITH";;


(*
g `(P:num-> bool) ( @ t. t >0)`;;
g `(a /\ b) ==> c`;;
g `(a \/ b) ==> c`;;
g `(?x . t) ==> c`;;
g `(f:A->B) = g`;;
g `(x:A->B) = g`;;
g `sqrt(&3) < sqrt(&4)`;;
g `?x. f x`;;
DISCH_TAC;
s 0;;

eh 1;;
eh 0;;
eh 0;;

eh 1;;
suggest();;
g `?x. f x`;;
g `(f:A->B)  =g`;;
*)


(*


termination tactics MESON, REAL_ARITH, SET_TAC, ARITH_TAC;;
REAL_FIELD, REAL_RING, SIMPLE_COMPLEX_ARITH_TAC;
 
 REAL_LET_TRANS;;
FORALL_PAIR_THM;;
REWRITE_TAC[FORALL_PAIR_THM];;
 EXISTS_PAIRED_THM;;
REWRITE_TAC[GSYM EXISTS_PAIRED_THM] ;;
*)


(* IMP_RES_THEN  forward implications of assumption list *)
(* MATCH_ACCEPT_TAC goal is instance of supplied theorem 
    PATH_CONV, find_path, follow_path
    PAT_CONV  (can be used on swapping quantifiers! ) 
  POP_ASSUM, POP_ASSUM_LIST
  REPLICATE_TAC n GEN_TAC
  RULE_ASSUM_TAC (maps an inference rule over assumption list)
     RULE_ASSUM_TAC (TRY (CHOOSE_))
  SUBST1_TAC, SUBST_ALL_TAC.
   *)

(*
NAMED ASSUMPTIONS 
*)

(*
theorem usage.
rewrites, simplifications, introductions, definitions,
assume ispecl,
*)

(*
amplifications,
curtailments,
*)

(*
terminators.
*)

(*
quants.
*)

(searcht 5 [`x pow (a + y) = x pow a * x pow y`],REAL_POW_ADD);;
(searcht 5 [`x pow 1`],REAL_POW_1);;
searcht 5 [`x pow 0`],(CONJUNCT1 real_pow);;

(* some searches that came up in practice. *)

Searching.suggest();;
search[`atn has_real_derivative z`];;
searcht 10[name "WITHIN";`atreal`;`has_real_derivative`],"HAS_REAL_DERIVATIVE_ATREAL_WITHIN";
searcht 3[name "MVT";`has_real_derivative`],REAL_MVT_VERY_SIMPLE;;
searcht 5 [`inv (x * y ) = inv (x ) * inv (y)`],REAL_INV_MUL;;
searcht 5 [`z pow (a + b) = z pow a * z pow b`],REAL_POW_ADD;;
searcht 5 [`(z pow a) pow b = z pow (a * b)`];;
searcht 8 [`halfatn`],halfatn_bounds_abs;;
searcht 8 [`atn(halfatn x)`];;
halfatn_half;
searcht 8 [`&0 <= abs(x)`],REAL_ABS_POS;;
searcht 8 [`inv (x pow k)`],REAL_INV_POW;;
search[`x pow k <= y pow k`],Real_ext.REAL_PROP_LE_POW;
search[`&0 <= x pow 2`];;
search[`inv x <= inv y`];REAL_LE_INV2;;
search[`&0 <= abs(x)`],REAL_ABS_POS;;
search[`abs(inv x)`;`inv(abs x)`],REAL_ABS_INV;;
search[`abs(x * y) = abs(x) * abs y`],REAL_ABS_MUL;;
search[`x * y <= x * z`],REAL_LE_LMUL;;

searcht 5 [`EVEN`;`ODD`],NOT_EVEN;;
searcht 5 [`~(?x. P)`],NOT_EXISTS_THM;;
searcht 10 [`(=) (a..b)`],numseg;;
searcht 3[`sum (IMAGE f X)`],SUM_IMAGE;;
searcht 3[`sum`;`SUBSET`;`&0`],SUM_SUPERSET;;
search[`m/m = &1`],REAL_DIV_REFL;;
search [`&x = &y`;`(x:num) = y`],REAL_OF_NUM_EQ;;
searcht 4 [`& (x * y)`;  `& x * & y`],REAL_OF_NUM_MUL;;
searcht 4 [`ODD`;`2 * k`], ODD_EXISTS;;
searcht 4 [`EVEN`;`ODD`],NOT_EVEN;;
search[`norm (Cx x)`],COMPLEX_NORM_CX;;
search[`Cx x pow i`],CX_POW;;
search[`Cx a - Cx b`],CX_SUB;;
search[`Cx (x) * Cx (y)`],CX_MUL;;
search[`Cx(x)/Cx(y)`],CX_DIV;;
searcht 5 [`f (if x then y else z)`;`(if x then f y else f z)`];; "not found"
search[`Im (Cx x)`];;
search[`catn`;`atn`],GSYM CX_ATN;;
search[`taylor_coeff_catn`;`DIV`],taylor_coeff0;;
(searcht 10 [`Im z = &0`;`Cx`],"not found" (Im z = &0 <=> (?x. z = Cx(x))));;
search[`Im (Cx x) = &0`],IM_CX;;
search[`(x:num) > y <=> y < x`],GT;;
search [`0 < FACT n`],FACT_LT;;
search[`(&x = &y) = (x = y)`],REAL_OF_NUM_EQ;;
(search[`((a:real) * b /a = b)`],REAL_DIV_LMUL);
(search[`taylor_coeff_catn 0`],taylor_coeff_catn0);;
search[`taylor_coeff_catn`;`has_complex_derivative`];;

(* --  hard one to find.  (at x) (at x within y)  *)
([`at x within y`,`has_complex_derivative`],HAS_COMPLEX_DERIVATIVE_AT_WITHIN);
(search[`Im ( v % x )`],IM_CMUL);
(`Im (x + y)`,IM_ADD);
(`&1 pow n`,REAL_POW_ONE);
(`abs( -- x)`,REAL_ABS_NEG);
(`abs(&n)`,REAL_ABS_NUM);
[
(`norm((a:real^A)  + b) <= norm (a) + norm(b)`,"NORM_TRIANGLE",NORM_TRIANGLE);;
];;

MATCH_MPlist:
[
(`((a:real) <= c ) /\ (b <= d) ==> (a + b <= c + d)`,"REAL_LE_ADD2",REAL_LE_ADD2);
];;
num_CASES;;
search[`norm(a-b)`;`dist`];;
search[`0 <= FACT n`];;
search[`norm(inv(a))`];;
search[`&a <= &b`;`(a:num) <= (b:num)`];;
search[`norm (a * b) = norm (a) * norm (b)`];;
search[`norm ii`];;
search[`norm (z pow k)`];;
search[`norm (Cx a)`];;
search[`norm(inv z)`];;
COMPLEX_NORM_POW;;
COMPLEX_NORM_INV;;
search [`(z:real) pow (a + b)`];;
search[`z pow (a + b)`];;
search[`EVEN`;`2*m`];;
search[`atn (&0)`];;
search[`catn`;`atn`];;
search[`EVEN`;`ODD`];;
search[`Cx(&1) pow n`];;
search[`Cx (a *  b)`];;
search[`z pow 0`];;

search[`vsum`;`sum`;`Cx`],VSUM_CX_NUMSEG;;
VSUM_CX_NUMSEG;;
search[`has_complex_derivative`;`has_real_derivative`];;
COMPLEX_TAYLOR;;
HAS_COMPLEX_REAL_DERIVATIVE_AT;;

([`vsum`;`sum`;`Cx`],VSUM_CX_NUMSEG);;
([`vsum:(num->bool) -> (num->complex)->complex`;`sum`],VSUM_CX_NUMSEG);;

type_of `vsum`;;

(* counting_spheres *)


(*
PRESERVES_NORM_INJECTIVE;;
MEASURABLE_LINEAR_IMAGE_EQ;;
RADIAL_NORM_LINEAR_INVARIANT;;
linear_inter_normball;;
search [`solvec0`];;
searcht 10 [`radial`];;
Marchal_cells_2.RADIAL_VS_RADIAL_NORM;;
searcht 1 [`radial_norm`];;
searcht 0 [`linear`;`radial_norm`];;
MEASURABLE_LINEAR_IMAGE_EQ;;
searcht 15 [`measurable`;`linear`];;
Vol1.sol;;
searcht 15  [`sol`];;
searcht 10 [`measurable (a INTER b)`];;
*)


(*
Vol1.sol;;
sol0_linear_r;;
linear_inter_normball;;
MEASURE_LINEAR_IMAGE_SAME;;
search[`vol`;`measure`];;
Marchal_cells_2.RADIAL_VS_RADIAL_NORM;;
search[`linear`;`measurable`];;
Vol1.sol;;
search[`radial_norm`];;
sol;;
SUBGOAL_TAC;;
SUBGOAL_THEN;;
search[`linear`;`t % f u`];;
type_of_goal ();;
help "apropos_types";;
PRESERVES_NORM_INJECTIVE;;
searcht 15 [`norm u = norm v`;`(f x = f  y) ==> (x = y)`];;
searcht 5 [`<=>`;`(A ==> b) /\ (b ==> A)`];;
srch[`dist`;`norm`];;
ball;;
srch[`ball`];;
srch[`X SUBSET ball(u,r)`];;
srch[`vec 0 + x`];;
srch;;
srch [`IMAGE f x SUBSET IMAGE f t`];;
srch [`vec 0 + x`];;
type_of `radial_norm`;;
*)

(* Work in progress 

def_of_regexp "radial$";;
def_of_regexp "radial_norm";;
def_of_regexp "radial$";;
srch [`eventually_radial`];;
srch [`radial_norm`];;
more();;
srch [`normball`;`ball`];;
search [`sol`];; 
Vol1.sol;;
type_of `sol`;;
type_of_thm LINEAR_CMUL;;
type_of_goal();;
linear;;
srch [`f (t % u)`;`linear`];;
help "ISPECL";;
 RELATIVE_INTERIOR_INJECTIVE_LINEAR_IMAGE;;
srch [`relative_interior (IMAGE f c)`];;
Polyhedron.fchanged;;
search_thml (term_match []) [`sol`] (map (fun s -> ("",s)) (!invariant_under_linear));;
srch [`fchanged`];;

*)

(*
add_linear_invariants;;
*)

(* add_linear_invariants [FCHANGED_LINEAR_INVARIANT ; LINEAR_0;RCONE_LINEAR_INVARIANT ];; *)




(*
searcht 15 [`rcone_gt`];;
more();;
def_of_regexp "rconesgn";;
help_grep "without";;
searcht 5 [`dropout`];;
searcht 5 [`azim`;`Arg`];;
type_of `hypermap_of_fanx`;;
def_of_regexp "wedge";;
def_of_regexp "cyclic_set";;
search [`wedge_ge`;];;
search [`azim v1 v2 x y + azim v1 v2 y z`;name "sum"];;
search [`wedge_ge`];;
search [`cwedge`];;
search [`cw_dart_fan`];;
search [`cwedge`];;
search [`fan`];;
search [`FAN`];;
*)

(* a polyhedral bound *)

(* February 2012 searches *)

results[];;
mark 0;;

Calc_derivative.rational_identity `b / a * a = b`;;
searcht 5 [`face_of`;`t % v`];;
searcht 5 [`v dot v`;`&0`];;
mark 3;;
searcht 5 [`a dot (vec 0)`];;
mark 0;;
searcht 15 [`facet_of`;`{}`];;
mark 1;;
searcht 5 [`(a % b) dot c`];;
searcht 15 [`rcone_gt`];;
mark 1;;
searcht 5 [`x:real^3`;`x dot y = &0`];;
EUSOTYP_general;;
1;;
searcht 5 [`pad2d3d x = pad2d3d y`];;
mark 0;;
searcht 5 [`vec 0`;`Cx (&0)`];;
searcht 5 [`norm (facet_rep_a P c)`];;
mark 1;;
searcht 5 [`Cx u = Cx v`];;
searcht 5 [`norm x = r`];;
searcht 5 [`norm (vec 0)`];;
searcht 5 [`-- &1`;`aff_dim p`];;
searcht 5 [`polytope`;`polyhedron`];;
mark 4;;
facet_arg_lt_pi;;
!search_results;;
results[];;
mark 0;;
searcht 10 [`convex`;`polytope`];;
searcht 10 [`(?)`;`extreme_point_of`];;
searcht 10 [def "extreme"];;
searcht 10 [`extreme_point_of`;`face_of`];;
searcht 10 [`?`;`face_of`];;
searcht 15 [`X = {}`;`IN`];;
searcht 5 [`u HAS_SIZE 0`];;
1;;


searcht 15 [`radial_norm`];;
searcht 15 [`fchanged`;`radial_norm`];;
searcht 15 [`fchanged`;`radial`];;
searcht 15 [`fchanged`;`eventually_radial_norm`];;
searcht 15 [`wedge`;`radial_norm`];;
searcht 15 [`wedge`;`radial`];;
searcht 15 [`wedge`;`eventually_radial_norm`];;
searcht 15 [`wedge`;`sol`];;
searchn 0 [`sol`;`UNION`] "Conforming.SOL_DISJOINT_UNION";;
searchn 1 [`sol`;`UNIONS`] "Conforming.SOL_UNIONS";;
searchn 0 [`sol`;`UNIONS`] "Conforming.SUM_SOL_TOPOLOGICAL_COMPONENT_YFAN_EQ_SOL_UNIONS";;
searcht 15 [`sol`];;
searchn 2 [`sol`] "Conforming.SOLID_ANGLE_YFAN";; (* 4 Pi sum *)
searcht 5 [`fchanged`;`sol`];;
searchn 0 [`open`;`measurable`] "MEASURABLE_OPEN";;
searcht 10 [`fchanged`];;
search [name "AMHFNXP"];;
searcht 5 [`sum (n..n)`];;
searchn 2 [`sum X f = sum X g`] "SUM_EQ";;
SUM_ADD_SPLIT;;
SUM_PAIR;;
searcht 15 [`sum`;`(a..( n+m))`];;
searcht 15 [name "SUM_DIFF"];;
searchn 0 [`sum`;`\i. (f (i+1) - f (i))`] "SUM_DIFFS_ALT";;
searcht 15 [name "TELE"];;
searcht 15 [`azim`;`sum`];;
AZIM;;
searcht 5 [`&0 <= azim x y z w`];;
AZIM_BASE_SHIFT_LT;;
X_IN;;
X_GEN_TAC;;
help_grep "GEN";;
search [`p IN {}`];;
REAL_WLOG_LE;;
AZIM_BASE_SHIFT_LT;;
search[`cyclic_set`];;
Fan.sum4_azim_fan;; (* lower ineq *)
Fan.sum5_azim_fan;; (* upper ineq *)
Counting_spheres.EUSOTYP_general;;
searcht 5 [`azim a b c d < azim a b c' d'`];;
searcht 5 [`&0 <= azim a b c d`];;
searchn 1 [`azim`;`&2 * pi`] "Local_lemmas.AZIM_RANGE";;
searchn 2 [`azim`;`&2 * pi`] "Rogers.AZIM_COMPL_EXT";;
searcht 10 [`azim`;`&2 * pi`];;
searcht 25 [`wedge`];;
searcht 5 [`sum`;`azim`];;
searchn 3 [`azim a b c d + azim a b c' d'`] "Fan.sum5_azim_fan";;
searcht 50 [`azim a b c d + azim a b c' d'`;omit `cyclic_set`];;
vol_solid_triangle_ortho;;
vol_solid_triangle;;
searcht 5 [`vol_solid_triangle`];;
searcht 5 [name "solid_triangle"];;
searcht 5 [name "cone0"];;
searcht 5 [name "normball"];;
searcht 5 [`ball`;`( < )`];;
1;;


searcht 5 [`x IN IMAGE (dropout 3) U`];;
pad2d3d_dot;;
VECTOR_ARITH;;
 linear;;
LINEAR_NEG;;
searcht 65 [def "linear"];;
searcht 15 [`linear`;`f (x:real^A) + f y`];;
searcht 15 [`linear`;`pad2d3d`;LINEAR_PAD2D3D];;
searcht 15 [`pad2d3d`;`(x:real^2)`];;
searcht 5 [`pad2d3d`;`dot`];;
types_of (concl pad2d3d_dot);;

EUSOTYP_simple;;

(*
searcht 5 [`(z:complex) * u = u * z`];;
searcht 5 [`Arg (Cx r * u)`];;
searcht 5 [`cnj (x * y)`];;
searcht 5[ `cnj`;`Cx`];;
searcht 5 [`Cx x = Cx y`];;
searcht 5 [`z * u = Cx(&0)`];;
ASM_CASES_TAC;;
searcht 5 [`inv( Cx (&0))`];;
*)
 

asimp[sym Ysskqoy.COS_ARG_VECTOR_ANGLE]
amt[Ysskqoy.COS_ARG_VECTOR_ANGLE;COMPLEX_ENTIRE;CX_INJ;arith `&0 < r ==> ~(r = &0)`]
searcht 5 [`Cx u = Cx v`];;
searcht 15 [`norm x = &0`];;
searcht 15 [`pi = &0`];;
searcht 15 [`cos x = &0`];;
searcht 5 [`inv x = &0`];;
Ysskqoy.COS_ARG_VECTOR_ANGLE;;
searcht 5 [`vector_angle u v = vector_angle v u`];;
GSYM 
Ysskqoy.SEC_DOT;;
searcht 5 [`norm (facet_rep_a P c)`];;
searcht 5 [`norm (Cx r)`];;
searcht 5 [`(x:complex)/ x`];;
facet_arg_lt_pi;;
searcht 5 [` x HAS_SIZE 1`];;
facet_arg_lt_pi;;
POLY_SORT_BIJ;;
searcht 5 [`inv (Cx u)`];;
ARG_MUL_CX;;
searcht 5 [`inv (a * (b:complex))`];;
searcht 5 [`Arg`; `Cx`];;
searcht 5 [`(z:complex) * w`;`r % u`];;
searcht 5 [`facet_rep_a`;`&0`];;
mt[]
art[]

POLYSORT_BIJ2;;
suggest();;
Counting_spheres.facet_rep_nz;;
searcht 5 [`Arg (Cx (&n))`];;
searcht 5 [`(x:complex)/x`];;
POLYTOPE_FACET_EXISTS;;
searcht 5 [`(?)`;`facet_of`];;
searcht 5 [`facet_rep_a P u = Cx (&0)`];;
SPECL [`u:real^2`;`h = \(i:num). facet_rep_a P (if i IN 1..n then f i else f 1)`;`n:num`] ARG_ORDER;;
types_of `h = \(i:num). facet_rep_a P (if i IN 1..n then f i else f 1)`;;
type_of_thm ARG_ORDER;;
ARG_ORDER;;
searcht 5 [`&0 <= Arg z`];;
ARG_INV_ALT;;
REAL;;
complex_frac_cancel;;
SIMPLE_COMPLEX_ARITH;;
help_grep "COMPLEX.*TAC";;
searcht 5 [`Cx (&1) * b = b`];;
searcht 5 [`inv b * b = Cx (&1)`];;
complex_div;;
searcht 0 [`inv (a:complex/b)`] "COMPLEX_INV_DIV" ;;
searcht 5 [`((a:complex)/b)/(c/d)`];;
searcht 5 [`Arg ((x/y)/(z/y))`];;
searcht 5 [`Arg (x/y)`];;
searchn 0 [`a /b = Cx (&0)`] "Ysskqoy.ARG_0_DIV";;
ARG_LE_DIV_SUM;;
searcht 0 [` (inv (z/(u:complex)))`] "COMPLEX_INV_DIV";;
searchn 0 [`&0 <= Arg z`] "ARG";;
searcht 5 [`facet_of`;`polyhedron`;`HAS_SIZE`];;
SEC_DOT;;
searcht 20 [`wedge`];;
searcht 10 [`Arg (z) + Arg (w) `];;
facet_arg_lt_pi;;
searcht 10 [`bisector_point`];;
searcht 10 [`norm (Cx (&0))`];;
searcht 10 [`Arg`;`Cx`];;
searcht 10 [`( * ):real^2->real^2->real^2`;`(%)`];;
searcht 10 [`Arg`;`(%)`];;
    dest_const `(..)`;;
searcht 10 [`(..)`;`IN`];;
searcht 5 [`&0 < r`;`abs r = r`];;
searcht 5 [`norm (a % v)`];;
type_of `facet_rep_a`;;
searchn 3 [`facet_rep_a`] "Counting_spheres.facet_rep_in_poly";;
searchn 9 [`facet_rep_a`] "Counting_spheres.facet_arg_lt_pi";;
searchn 10 [`facet_rep_a`] "Counting_spheres.facet_rep_def";;  (* rep has norm 1 *)
POLYHEDRON_LINEAR_IMAGE;;
Ysskqoy.COS_ARG_VECTOR_ANGLE;;
bisector_point;;
Ysskqoy.SEC_DOT;;
searchn 0 [`BIJ`;`HAS_SIZE`] "Counting_spheres.POLY_SORT_BIJ";;

searcht 10 [`pad2d3d`];;
searcht 10 [`aff_dim`;`plane`];;
searcht 10 [`affine`;`SUBSET`];;
pad_in;;
searcht 15 [`IN`;`SUBSET`];;
searcht 15 [`pad2d3d`];;
searchn 0 [`Cx (&0)`;`vec 0`] "COMPLEX_VEC_0";;
searcht 5 [`x SUBSET x`];;
searcht 5 [`SUBSET`;`facet_of`];;
searcht 5 [`IMAGE h x = x`];;
searcht 5 [`IMAGE f (IMAGE g x)`];;
BIJECTIONS_HAS_SIZE;;
searchn 0 [`linear`;`facet_of`] "FACET_OF_LINEAR_IMAGE" ;;
searcht 5 [`dropout`;`(f x = f y) ==> (x  =y)`] (* none *);; 
searchn 0 [`dropout`;`linear`] "LINEAR_DROPOUT";;
searcht 15 [`dropout`;`face_of`] ;;
PRESERVES_NORM_INJECTIVE;;
searcht 5 [`dropout`;`norm`];; (* none *)
searcht 15  [`(f x = f y) ==> (x = y)`;`HAS_SIZE`];;

searchn 0 [`pad2d3d`;`linear`] "LINEAR_PAD2D3D";;
AZIM_SPECIAL_SCALE;;

PRESERVES_NORM_INJECTIVE;;
NORM_PAD2D3D;;

searchn 0 [`linear`;`polyhedron`] "POLYHEDRON_LINEAR_IMAGE";;
searchn 0 [`linear`;`bounded`] "BOUNDED_LINEAR_IMAGE";;
searchn 0 [`pad2d3d`;`linear`] "LINEAR_PAD2D3D";;
type_of_goal();;

searchn 1 [`norm z = &0`] "COMPLEX_NORM_ZERO"
searchn 1 [`real`;`Re`]"REAL_NORM";;
searcht 5 [`(a = b) = (b = a)`];;
searcht 5 [`Cx (Re (t))`];;
searcht 5 [`Arg (Cx (&0))`];;
searcht 5 [`Arg x = Arg y`];;
searcht 5 [`Arg`;`Re`];;
searcht 5 [`inv (x/(y:complex))`];;
searcht 15 [`Arg`;`real`];;

help "LAND_CONV";;
searchn 6 [`Arg (x)`;`pi`] "ARG_INV";;
type_of_goal();;
type_of `azim`;;
type_of `plane`;;


(*
 GEOM_BASIS_MULTIPLE_TAC 3`v:real^3` 
*)

(* add_linear_invariants [FCHANGED_LINEAR_INVARIANT ; LINEAR_0;RCONE_LINEAR_INVARIANT ];;  *)
(* let backup_invariant_under_linear = !invariant_under_linear;; *)
(*
let restore _ = 
  invariant_under_linear := backup_invariant_under_linear;;
restore();;
*)
(* add_linear_invariants [SOLVEC0_LINEAR_INVARIANT_3];;  *)

(*
searcht 15 [`facet_rep_a`];;
POLY_SORT_BIJ;;
searcht 15 [`plane`];;
*)


(*
let _ = [`ball`;`half_spaces`;`weakly_saturated`;`lmfun_ineq_center`];;

let _ = [ball;Tarjjuw.half_spaces;Tarjjuw.weakly_saturated;Pack_defs.lmfun_ineq_center];;

let _ = [Tarjjuw.TARJJUW;Ysskqoy.YSSKQOY];;
*)





(* TARJJUW *)
(*
searcht 5 [`packing`;`FINITE`];;
searcht 15 [`ball_annulus`];;
searcht 5 [`sup`;`FINITE`];;
weakly_saturated;;
ball_annulus;;
searcht 5 [`weakly_saturated`];;
type_of `ball_annulus`;;
def_of_regexp "ball_annulus";;
constant_of_regexp "ball_annulus";;
def_of_regexp "ball_annul";;
Tarjjuw.TARJJUW;;
search [name "TARJ"];;
apropos();;
constant_of_regexp "TRI";;
def_of_regexp "sol$";;
def_of_regexp "solid_triangle";;
help "apropos_searching";;
constant_of_regexp "satur";; ["weakly_saturated"; "saturated"];;
def_of_regexp "weakly_saturated";;
def_of_regexp "max";;
constant_of_regexp "inter";;
def_of_regexp "closed_real_interval";;
search [`weakly_saturated`];; (* "Tarjjuw.TARJJUW" *)
Tarjjuw.TARJJUW;;
def_of_regexp "ball";; (* open *)
def_of_regexp "half_spaces";;
def_of_regexp "ineq_cen";;
searcht 5 [`weakly_saturated`];;
let ineq693 =  `~lmfun_ineq_center V`;;
type_of `sum`;;
searcht 5 [`..`];;
type_of `lmfun`;;
searcht 5 [`lmfn_ineq_center`];;
*)



searcht 5 [`pad2d3d`;`dot`];;





(* Tue Feb 28 08:27:45 EST 2012 *)

searcht 50 [`facet_of`;`polyhedron`;`IN`;`t % v`];;
mark 0;;
searcht 20 [`rcone_gt`;`vec`];;
searcht 20 [`rconesgn`];;
mark 6;;
mark 7;;
searcht 20 [`x - vec 0`];;
mark 0;;
searcht 5 [`dist (x, vec 0)`];;
mark 0;;
searcht 5 [`(a % v) dot u`];;
mark 0;;
searcht 5 [`norm (a % v)`];;
mark 0;;
searcht 5 [`norm`; `v dot v`];;
mark 4;;
searcht 15 [`(a:real) * c < b * d`];;
searcht 15 [`abs x = x`];;
mark 12;;
searcht 15[`&0 < a / b`];;
mark 1;;
mark 6;;
searcht 15 [`x * y < x * v`];;
mark 4;;
searcht 15[`&0 < a * b`];;
mark 4;;
searcht 15 [`polyhedron`;`face_of`;`a % v`];;
searcht 15 [name "INTERI";name "POLY"];;
mark 0;;
searcht  100 [`relative_interior`;`polyhedron`];;
mark 0;;
FACET_OF_POLYHEDRON_EXPLICIT;;
searcht 50 [`polyhedron`;`fchanged`];;
searcht 5 [`x /\ y <=> ((u:A->bool) = b)`];;
searcht 5 [`x SUBSET y /\ y SUBSET x`];;
searcht 5 [`x SUBSET a INTER b`];;
searcht 5[`relative_interior f SUBSET fchanged f`];;
mark 0;;
searcht 5 [`x SUBSET affine hull x`];;
mark 0;;
searcht 5 [`relative_interior s SUBSET s`];;
mark 0;;
searcht 10 [`fchanged`];;
searcht 10 [`a % v`;` affine hull f`];;
mark 3;;
searcht 10 [`a % v = b % v`];;
mark 0;;
searcht 10 [`affine`;`vec 0`];;
mark 2;;
searcht 10 [`dependent`];;
searcht 5 [`a % vec 0`];;
searcht 5 [`a - vec 0`];;
searcht 5 [`vec 0 + u`];;
searcht 5 [`vec 0 - u`];;
searcht 5 [`u IN (:A)`];;
VECTOR_ARITH;;
searcht 5 [` t % u = vec 0`];;
mark 2;;
searcht 5[`face_of`;`affine`];;

(* Sun Mar  4 10:27:12 EST 2012, counting spheres *)

searcht 15 [`relative_interior`;`face_of`];;
mark 1;;
searcht 10 [`face_of`;`hull`];;
mark 2;;
searcht 10 [`~(X = {})`];;
searcht 10 [`relative_interior f SUBSET y`];;
mark 0;;
searcht 10 [`affine hull A SUBSET affine hull B`];;
mark 0;;
searcht 10 [`&1 % v`];;
mark 0;;
searcht 10 [`u hull A SUBSET u hull B`];;
mark 3;;
searcht 5 [`u hull (u hull A)`];;
mark 1;;
searcht 5 [`u IN {a,b}`];;
research 5;;
searcht 5 [`f SUBSET P hull x`];;
mark 0;;
searcht 5 [`convex`; `polyhedron`];;
mark 0;;
searcht 5 [`face_of`;`facet_of`];;
mark 1;;
searcht 5 [`relative_interior`;`interior`];;
mark 0;;
searcht 15 [`facet_of`];;
searcht 5 [`relative_interior x SUBSET y`];;
mark 0;;
FCHANGED_AFFINE;;
searcht 5 [name "POLY"; name "COLL"; name "FACE"];;
searcht 5 [`affine hull c`;`facet_of`];;
mark 1;;
searcht 5 [`affine hull u = (:real^N)`];;
mark 3;;
searcht 5 [`x dot y = y dot x`];;
mark 0;;
searcht 5 [`v dot (a % w)`];;
searcht 5 [`rcone_gt`];;
searcht 15 [`rconesgn`];;
searcht 15 [`dist(v,vec 0)`];;
searcht 15 [`a < b`;`a pow 2 < b pow 2`];;
mark 3;;
searcht 5 [`(a * b) pow 2`];;
mark 1;;
searcht 5 [`norm p pow 2`];;
mark 1;;
mark 2;;
searcht 5 [`&0 <= a * b`];;
mark 4;;
searcht 5 [`&0 <= norm x`];;
mark 0;;
searcht 15 [`a < b`;`a pow 2 < b pow 2`];;
mark 2;;
searcht 5 [`dist (a,b) pow 2`];;
mark 0;;
searcht 20 [`sqrt(&1 - a pow 2)`];;
searcht 5 [`norm v = &0`];;
mark 3;;
searcht 5 [`sqrt x pow 2`];;
mark 1;;
searcht 5 [`&0 <= &1 - t pow 2`];;
mark 0;;
searcht 15 [`a * x < b * x`];;
mark 2;;
searcht 5 [`&0 < a * b`];;
mark 4;;
searcht 5 [`&0 < t pow 2`];;
mark 0;;
searcht 5 [`cos (arcV a b c)`];;
mark 0;;
searcht 5 [`&0 < a * b`];;
mark 4;;
mark 3;;
searcht 5 [`&0 < &1 - t pow 2`];;
searcht 5 [`t pow 2 < &1`];;
mark 0;;
searcht 5 [`&0 < sqrt u`];;
mark 1;;
searcht 5 [`&0 < norm v`];;
mark 0;;
searcht 5 [`a*b = a * c`];;
mark 0;;
searcht 5 [`a % v = vec 0`];;
mark 2;;
searcht 5 [`a / b = &0`];;
mark 2;;
searcht 5 [`vec 0 dot u = &0`];;
mark 0;;
searcht 5 [`(a/b) pow 2`];;
mark 0;;
searcht 5 [`x pow 2 = y pow 2`];;
mark 4;;
searcht 5 [`a dot b = b dot a`];;
mark 0;;
searchk 5 [`&0 <= a/b`];;
mark 1;;
searcht 5 [`&0 <= a * b`];;
mark 4;;
searcht 5 [`&0 <= norm t`];;
mark 0;;
searcht 5 [`v dot v = &0`];;
mark 0;;
searcht 5 [`norm x = &0`];;
mark 3;;
searchk 5 [`norm x pow 2`];;
searchk 5 [`measure  A <= measure ( B)`];;
mark 0;;
searcht 5 [`measurable`;`wedge`];;
mark 0;;
searcht 5 [`eventually_radial`;`wedge`];;
searcht 14 [`eventually_radial`];;
mark 7;;
vol_solid_triangle_ortho;;
searcht 5 [`vol_solid_triangle`;`solid_triangle`];;
mark 1;;
searcht 3 [`solid_triangle`];;
mark 0;;
searcht 5 [`cone0`];;
mark 1;;
searcht 5 [`measure`;`measurable`];;
searchk 5 [`primitive`];;  (* measurability of solid_triangle *)
mark 0;;
mark 1;;
searchk 50 [`fchanged`;`topological_component_yfan`];;
mark 1;;
searcht 50 [`topological_component_yfan`;`measurable`];;
mark 0;;
searcht 50 [`FAN`;`polyhedron`];;
mark 0;;	
searcht 50 [name "PIIJ"];;
mark 0;;
searcht 5 [`fan80`];;
mark 0;;
searcht 5 [`set_of_edge`;`polyhedron`];;
mark 1;;
searcht 5 [`eventually_radial`;`topological_component_yfan`];;
searchk 5 [`radial`];;
mark 5;;
searchk 5 [`normball`;`ball`];;
mark 1;;
searchk 5 [`fchanged`;`t % v`];;
mark 0;;
searchk 5 [`a % b % c`];;
mark 0;;
searchk 5 [`&0 < a * b`];;
searcht 5 [`radial`;`ball`];;
searcht 5 [`IN`;`ball`];;
mark 3;;
searcht 5 [`dist`;`norm`];;
mark 1;;
searchk 5 [`norm (--u)`];;
mark 0;;
searchk 5 [`norm (t % u)`];;
searchk 10 [`wedge`;`azim`];;
mark 5;;
searcht 5 [`azim`;`azim a b c d + azim a b d e`];;
searcht 5 [`X = {}`;`?`];;
mark 2;;
searchk 5 [name "SHIFT";`azim`];;
mark 0;;
searchk 5 [`azim a b c c `];;
mark 0;;
searchk 5 [`aff_gt`];;
mark 1;;
searchk 5 [`cone0`];;
searcht 15 [`affsign`;`SUBSET`];;
searchk 5 [`affsign`];;
mark 32;;
searcht 15 [`aff_gt`;`convex`];;
mark 0;;
searcht 15 [`convex`;`fchanged`];;
searcht 5 [`convex`;`topological_component_yfan`];;
searcht 15 [`fchanged`;`aff_gt`];;
searcht 15[ `fchanged`;`affsign`];;
searchk 5 [`cone0`];;
searchk 5 [`fchanged`];;
searchk 5 [`convex0`];;
search [`affsign`];;
AFFSIGN;;
searcht 5 [`{} UNION X`];;
mark 0;;
searchk 5 [`x IN {}`];;
mark 0;;
searchk 8 [`vsum`;`INSERT`];;
mark 4;;
mark 6;;
searchk 5 [`collinear`;`~(x = y)`];;
mark 0;;
searcht 15 [`collinear {x,y,z} = collinear {u,v,w}`];;
1;;

(* Tue Mar  6 06:09:41 EST 2012 *)
searcht 15 [`face_of`;`SUBSET`;`convex`];;
mark 8;;
searchk 5 [`sum`;`INSERT`];;
mark 7;;
mark 2;;
searcht 15 [`convex`;`sum`];; (*117*)
mark 9;;
searchk 5 [`relative_interior a SUBSET relative_interior b`];;
mark 0;;
searcht 15 [`convex`;`facet_of`];;
mark 4;;
searchk 5 [`convex`;`face_of`];; (* ?120 *)
mark 2;;
searcht 5 [`sgn_gt`];;
mark 3;;
searcht 15 [`a <= sum s f `];;
mark 6;;
searcht 5 [`sum (A UNION B) f `];;
mark 0;;
searcht 5 [`x HAS_SIZE n`;`x IN u`];;
searcht 5 [`x HAS_SIZE 0`];;
mark 0;;
searcht 5 [`x HAS_SIZE 1`;`x IN u`];;
mark 0;;
searchk 5 [`sum s f <= sum s' f`];;
mark 0;;
searchk 15 [`CARD`;`INSERT`];;
searchk 15 [`convex hull a SUBSET f`];; (* 128 *)
mark 3;;
searchk 15 [`convex hull s = s`];; (* 129 *)
mark 2;;
searchk 15 [`convex`;`facet_of`];;
searchk 5 [`face_of`;`facet_of`];;
searchk 15 [`affine hull u`;`aff_dim`];;
mark 5;;
searchk 15 [`convex hull u `;`aff_dim`];;
searchk 15 [`aff_dim`;`facet_of`];; (* 42 *)
searchk 15 [`interior`;`aff_dim`];;
mark 0;;
searchk 15 [`collinear`;`aff_dim`];;
searchk 5 [`SUBSET`;`aff_dim`];; (* 132 *)
mark 0;;
searchk 5 [`x SUBSET P hull x`];;
searchk 15 [`aff_dim`;`INSERT`];;
mark 3;;
mark 2;;
searchk 5 [`affine`;`collinear`];;
mark 2;;
searchk 5 [`affsign`;`{}`;`{x}`];;
searchk 5 [`a % vsum s f`];;
mark 0;;
searchk 5 [`a % b % c`];;
searchk 5 [`UNION`;`INSERT`];;
mark 3;;
searchk 15 [`sum`;`INSERT`];;
mark 6;;
searchk 15 [`a % vec 0 `];;
mark 0;;
searchk 5 [`sum s (\v. f v * u)` ];;
mark 1;;
searchk 5 [`x / y`;`inv`];;
mark 1;;
searchk 5 [`face_of`;`interior`];;
mark 0;;
searcht 15 [`a INTER b = {}`];;
mark 14;;
searchk 5 [`aff_dim (convex hull s)`];;
mark 0;;
searchk 5 [`x:real^3`;`?`;`u dot v = &0`;`~(v = vec 0)`];;
searchk 5 [`eventually_radial`;`sol`];;
searcht 50 [`sol`];;
mark 8;;  (* 145, sol def *)
mark 15;; (* 146, SOL_UNIONS *)
searchk 5 [`radial_norm`;`radial`];;
mark 0;; (* 147 *)
vol_solid_triangle_ortho;;
searcht 15 [def "collinear"];;
searchk 15 [`collinear`;`~(a = (b:real^N))`];;
mark 11;;
searchk 15 [`affine hull {x,y}`];;
mark 10;;
searchk 15 [`affine`;`aff`];;
mark 5;;
searchk 15 [`aff_gt`;`wedge`];;
mark 2;;
searchk 15 [`collinear {vec 0 ,a,b}`];;
mark 13;;
mark 4;;
searchk 15 [`v dot v = &0`];;
1;;

(* Wed Mar  7 07:03:30 EST 2012 *)

searchk 5 [`&0 <= sqrt x`];;
mark 2;;
searchk 5 [`(\x. Q x) x = Q x`];;
1;;

(*---^^  command line  ^^ --- *)



(* GG generic commands *)

List.map (fun n -> List.nth !search_results n) (100--120);;
suggest();;
result 200 [];;
mark 0;;
type_of_goal();;
write 0;;

map (fun (_,s,_,_) -> s) !search_results;;

let search_result_dump = 
["REAL_LT_MUL"; "REAL_LT_LMUL"; "REAL_LT_DIV";
   "Local_lemmas.REAL_LT_DIV_NEG"; "Trigonometry2.LT_IMP_ABS_REFL";
   "NORM_POW_2"; "NORM_MUL"; "DOT_LMUL"; "DIST_0"; "VECTOR_SUB_RZERO";
   "rconesgn"; "rcone_gt"; "Polyhedron.REDUCE_POINT_FACET_EXISTS";
   "POLYHEDRON_COLLINEAR_FACES"; "DOT_POS_LT"; "DOT_RZERO"; "facet_of";
   "DOT_LMUL"; "Trigonometry2.EXISTS_OTHOR_VECTOR_DIFFF_VEC0";
   "Trigonometry2.EXISTS_UNIT_OTHOR_VECTOR"; "INJECTIVE_PAD2D3D";
   "COMPLEX_VEC_0"; "COMPLEX_VEC_0"; "Counting_spheres.facet_rep_def";
   "CX_INJ"; "COMPLEX_NORM_CX"; "POLYTOPE_EQ_BOUNDED_POLYHEDRON";
   "POLYTOPE_IMP_CONVEX"; "POLYTOPE_IMP_COMPACT";
   "EXTREME_POINT_EXISTS_CONVEX"; "FACE_OF_SING";
   "FACE_OF_POLYHEDRON_SUBSET_FACET"; "POLYTOPE_FACET_EXISTS"];;

let search_result_dump_march7_8am =   ["Trigonometry2.COLLINEAR_TRANSABLE"; "Trigonometry2.COLLINEAR_TRANSABLE";
   "Trigonometry2.COLLINEAR_TRANSABLE"; "Trigonometry2.COLLINEAR_TRANSABLE";
   "COLLINEAR_LEMMA_ALT"; "WEDGE_LUNE_GT"; "COLLINEAR_3_AFFINE_HULL";
   "Sphere.aff"; "AFFINE_HULL_2_ALT";
   "Marchal_cells_2.RADIAL_VS_RADIAL_NORM"; "Conforming.SOL_UNIONS";
   "Vol1.sol"; "AFF_DIM_CONVEX_HULL"; "Hypermap.lemma_in_disjoint";
   "FACE_OF_DISJOINT_INTERIOR"; "real_div"; "SUM_RMUL"; "VECTOR_MUL_RZERO";
   "SUM_CLAUSES"; "Packing3.SING_UNION_EQ_INSERT"; "VSUM_LMUL";
   "Collect_geom.IN_AFFINE_HULL_IMP_COLLINEAR"; "AFF_DIM_2";
   "AFF_DIM_INSERT"; "AFF_DIM_SUBSET"; "Polyhedron.AFF_DIM_INTERIOR_EQ_3";
   "AFF_DIM_EQ_AFFINE_HULL"; "CONVEX_HULL_EQ";
   "Marchal_cells.CONVEX_HULL_SUBSET";
   "Upfzbzm_support_lemmas.SUM_SET_OF_2_ELEMENTS"; "SUM_SUBSET_SIMPLE";
   "HAS_SIZE_1_EXISTS"; "HAS_SIZE_0"; "SUM_UNION"; "SUM_POS_LE"; "sgn_gt";
   "FACE_OF_IMP_CONVEX"; "POLYTOPE_UNION_CONVEX_HULL_FACETS";
   "SUBSET_RELATIVE_INTERIOR";
   "EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL"; "Geomdetail.SUM_DIS3";
   "face_of"; "FACE_OF_CONVEX_HULL_SUBSET"; "Geomdetail.VSUM_DIS3";
   "Collect_geom.NOT_COLLINEAR_IMP_2_UNEQUAL";
   "Marchal_cells_2.VSUM_CLAUSES_alt"; "NOT_IN_EMPTY"; "UNION_EMPTY";
   "aff_gt_def"; "CONVEX_AFF_GT"; "affsign"; "Sphere.aff_gt_def";
   "AZIM_REFL"; "AZIM_BASE_SHIFT_LT"; "Counting_spheres.EMPTY_NOT_EXISTS_IN";
   "wedge"; "NORM_NEG"; "dist"; "IN_BALL"; "VECTOR_MUL_ASSOC";
   "Polyhedron.fchanged"; "NORMBALL_BALL"; "Sphere.radial";
   "Polyhedron.CARD_SET_OF_EDGE_INEQ_1_POLYHEDRON";
   "Polyhedron.POLYTOPE_FAN80"; "Conforming.PIIJBJK";
   "Polyhedron.POLYHEDRON_FAN";
   "Conforming.MEASURABLE_TOPOLOGICAL_COMPONENT_YFAN_INTER_BALL";
   "Polyhedron.FCHANGED_IN_COMPONENT"; "primitive"; "MEASURABLE_RULES";
   "Sphere.cone0"; "solid_triangle"; "Vol1.volume_prop_fix";
   "Sphere.eventually_radial"; "MEASURABLE_BALL_WEDGE"; "MEASURE_SUBSET";
   "NORM_EQ_0"; "DOT_EQ_0"; "NORM_POS_LE"; "Real_ext.REAL_PROP_NN_MUL2";
   "Calc_derivative.invert_den_le"; "DOT_SYM"; "Collect_geom.EQ_POW2_COND";
   "Trigonometry2.DIV_POW2"; "DOT_LZERO"; "Calc_derivative.invert_den_eq";
   "VECTOR_MUL_EQ_0"; "REAL_EQ_MUL_LCANCEL"; "NORM_POS_LT"; "SQRT_POS_LT";
   "ABS_SQUARE_LT_1"; "Calc_derivative.invert_den_lt"; "REAL_LT_MUL";
   "Trigonometry1.DOT_COS"; "Trigonometry2.NOT_ZERO_EQ_POW2_LT";
   "REAL_LT_MUL"; "REAL_LT_RMUL"; "Trigonometry2.UNIT_BOUNDED_IN_TOW_FORMS";
   "SQRT_POW_2"; "NORM_EQ_0"; "Collect_geom.DIST_POW2_DOT";
   "Tarjjuw.CHANGE_TARJJUW_4"; "NORM_POS_LE"; "Real_ext.REAL_PROP_NN_MUL2";
   "NORM_POW_2"; "DOT_SQUARE_NORM"; "Trigonometry2.MUL_POW2";
   "Tactics_jordan.REAL_POW_2_LT"; "DOT_SYM";
   "Polyhedron.INTERIOR_AFFINIE_HUL_EQ_UNIV";
   "Counting_spheres.affine_facet_hyper"; "RELATIVE_INTERIOR_SUBSET";
   "INTERIOR_SUBSET_RELATIVE_INTERIOR"; "facet_of"; "POLYHEDRON_IMP_CONVEX";
   "Ldurdpn.SUBSET_P_HULL"; "HULL_HULL"; "HULL_MONO"; "VECTOR_MUL_LID";
   "Marchal_cells_2.AFFINE_SUBSET_KY_LEMMA"; "RELATIVE_INTERIOR_SUBSET";
   "FACE_OF_STILLCONVEX"; "FACE_OF_DISJOINT_RELATIVE_INTERIOR";
   "REAL_LT_MUL"; "REAL_LT_LMUL"; "REAL_LT_DIV";
   "Local_lemmas.REAL_LT_DIV_NEG"; "Trigonometry2.LT_IMP_ABS_REFL";
   "NORM_POW_2"; "NORM_MUL"; "DOT_LMUL"; "DIST_0"; "VECTOR_SUB_RZERO";
   "rconesgn"; "rcone_gt"; "Polyhedron.REDUCE_POINT_FACET_EXISTS";
   "POLYHEDRON_COLLINEAR_FACES"; "DOT_POS_LT"; "DOT_RZERO"; "facet_of";
   "DOT_LMUL"; "Trigonometry2.EXISTS_OTHOR_VECTOR_DIFFF_VEC0";
   "Trigonometry2.EXISTS_UNIT_OTHOR_VECTOR"; "INJECTIVE_PAD2D3D";
   "COMPLEX_VEC_0"; "COMPLEX_VEC_0"; "Counting_spheres.facet_rep_def";
   "CX_INJ"; "COMPLEX_NORM_CX"; "POLYTOPE_EQ_BOUNDED_POLYHEDRON";
   "POLYTOPE_IMP_CONVEX"; "POLYTOPE_IMP_COMPACT";
   "EXTREME_POINT_EXISTS_CONVEX"; "FACE_OF_SING";
   "FACE_OF_POLYHEDRON_SUBSET_FACET"; "POLYTOPE_FACET_EXISTS"];;

search_results := [];;
search_results := augment_search_results search_result_dump;;

map retrieve_search_number (21--39);;


(* Oct 24, 2012 searches for EUSOTYP2_general *)
ALL_TAC;;
searc '0' 2 [`azim a b c d = azim a' b' c' d'`];;
searcht 5 [`DISJOINT`;`collinear`];;
EQ_SYM;;
     (GSYM Topology.th1);;;
CHOOSE_TAC;;
VECTOR_ARITH;;
azim_axis;;
searc 'a' 3 [`~(?x. p)`;];;
searc 'b' 2 [`bounded`;`SUBSET`];;
searc 'c' 1 [`collinear`;`vec 0 `;`t % v`];;
searc 'd' 3 [`~(a \/ b)`];;
searc 'e' 0 [`(c % v) dot w`];;
searc 'f' 0 [`face_of`;`facet_of`];;
searc 'g' 0 [`(&0) % v`];;
searc 'h' 3 [`i IN a..b`];;
searc 'i' 3 [`&0 < a / b`];;
searc 'j' 0 [`x % v = y % v`];;
searc 'l' 0 [`(a + b) dot v`];;
searc 'p' 0   [`polyhedron`;`face_of`;`==>`];;

searc 's' 0 [`SUBSET`;`face_of`];;
searc 't'  1 [`collinear {(vec 0),a,b}`;`t % x`];;
searc 'u' 8 [`collinear`;`t % x`];;
searc 'v' 0 [`a % b % c`];;
searc 'w' 6 [`collinear`;`vec 0`];; 
searc 'x' 6 [`&0 < a * b`];;
searc 'y' 0 [`arcV a b c = arcV a c b`];;
searc 'z' 1 [`arcV`;`t % v`];;


Print_types.print_goal_types();;
searcht 5 [`(a - b) dot v`];;


    type_of `rcone`;;
searcht 5[`&0 < u dot u`];;
searcht 5 [`{x,x,y}`];;
RDISK_R;;
Counting_spheres.EUSOTYP_general;;

searc '5' 2 [`x % y = (vec 0)`];;
searcht 5 [`x IN (a INSERT b)`];;
searc '4' 7 [`X INTER Y = {}`];;

(*
searc '2' 1 [`facet_of`;`face_of`];;
searc '1' 3 [`{}`;`?`];;
searc '0' 0 [`interior`;`face_of`];;
*)


(*
WEDGE_LUNE_GT;;
searc '0' 0 [`azim`;`dihV`];;
*)



(* Oct 28, 2012 *)
let searcj c = searcht;;
searc '3' 6 [`&0 < t2/t`];;
searc '2' 0 [`t % u % v`];;
searc '1' 16 [`<=>`;`==>`];;
searcht 25 [`(?t. u) <=> (?t. y)`];;
searc '0' 2 [`aff_gt {x} {u,v,w}`; `t % v`];;
searcht 5 [`collinear`;`DISJOINT`];;
searcht 5 [def "t0"];;
type_of `t0`;;
searcht 5 [`aff_gt`; `t % x`];;
Print_types.print_thm_types CONE0_FCHANGED;;
Print_types.print_goal_types();;
searc '4' 0 [`t % (vec 0)`];;
searc '5' 2 [`x = {}`;`?`];;
searc '6' 2 [`collinear`;`coplanar`];;
searc '7' 0 [`coplanar`;`~(a = (b:real^A))`];;
searc '8' 0 [`coplanar`;`t % v`;`vec 0`];;
searc '9' 0 [`x IN {y}`];;
searc 'a' 0 [`x IN A INTER B`];;

searc 'b' 0 [`coplanar`;`dihV`];;
searc 'c' 1 [`&0 <= dihV a b c d`];;
searc 'd' 6 [`SUBSET`;`INTER`];;
searcht 5 [`wedge`;`aff_gt`];;
searc 'i' 0 [`cone0`;`aff_gt`];;
searc 'e' 0 [`x IN {}`];;
searc '0' 2 [`a SUBSET b INTER c`];;
searcht 5 [`measurable`;`cone0`];;
searc 'f' 0 [`measurable`;`aff_gt`];;
searc 'g' 1 [`ball`;`normball`];;
searc 'h' 0 [`A INTER B = B INTER A`];;
searc 'i' 0 [`cone0`;`aff_gt`];;
searcht 5 [`measurable (A INTER B)`];;
searc 'j' 2 [`radial_norm`;`aff_gt`];;
searcht 5 [`solid_triangle`;`cone0`];;
searcht 5 [`vol`;`sol`];;
searcht 5 [`vol`;`vol_solid_triangle`];;
CONJUNCT2 Vol1.volume_props;;
help "CONJUNCT2";;
searc 'k' 0 [`azim`;`dihV`];;
searcht 5 [`&0 < pi`];;
searc 'l' 2 [`wedge`;`aff_gt`];;
searc 'm' 0 [`measurable (X UNION Y)`];;
searcht 5 [`(A UNION B) INTER C`];;
UNION_OVER_INTER;;
searc 'n' 0 [`radial_norm`;`UNION`];;
searc 'p' 1 [`radial_norm`;`UNION`];;
searc 'q' 1 [`dihV a b c d = dihV a b d c`];;
searc 'r' 2 [`wedge`;`aff_gt`];;
searcht 15 [`aff_gt {x,x'} y SUBSET aff_gt {x,x'} z`];;
searcht 15 [`x IN wedge a b c d`];;
searc 's' 5 [`wedge`;`azim`];;
searc 't' 1 [`fchanged c = x`];;
searcht 5 [`relative_interior x SUBSET y`];;
searc 'u' 0 [`a % v dot w`];;
searcht 5 [`x*b = b`];;
searc 'v' 0 [`sol`;`vol`];;
searc 'w' 8 [`measure`;`sum`];;
type_of `measure`;;
`measure:(real^3->bool)->real`;;
GOTCJAH_convex_sum;;

(* Oct 29, 2012 *)


searc 'a' 0 [`collinear {(vec 0),v,u}`;`t % v`];;
Trigonometry2.EXISTS_OTHOR_VECTOR_DIFFF_VEC0;;
searc 'b' 1 [`~(?t. p)`];;
searc 'c' 2 [`t % v = vec 0`];;

    
searc 'd' 3 [`azim a b c d + azim a b d e `];;


searc 'e' 0 [`measurable {}`];;
searc 'f' 0 [`radial_norm a b {}`];;

help_grep "CAS.*TAC";;
searc 'g' 0 [`{} INTER B`];;
searc 'h' 0 [`sol u {}`];;
searc 'i' 0 [`sin(&0)`];;
searc 'j' 0 [`asn(&0)`];;
SKOLEM_THM;;
searc 'k' 3 [`measure`;`has_measure`];;
searc 'l' 0]\[`measure`;`SUBSET`];;
GOTCJAH_convex_sum;;
WEDGE_ORDER_DISJOINT;;
ORDER_AZIM_SUM2Pi;;
searcht 5 [`sum`;`a..b`;`f o c`];;
searc 'm' 2 [`sum (a..b) f = sum (c..d) g`];;
searc 'n' 1  [`a < (b:num)`;`a < b`];;
searc 'o' 4 [`&0 <= a * b`];;
searc 'p' 0  [`a <= (b:num)`;`a <= b`];;
searc 'q' 1 [`&0 <= azim a b c d`];;
searc 'r' 2 [`sum f x = sum f y`];;
searc 's' 0 [`sum f (\i. t * r i)`];;
searc 't' 0 [`sol`;`vol`];;
searc 'u' 1  [`(UNIONS X) INTER Y`];;
searc 'v' 4 [`measurable (UNIONS s)`];;
searcht 5 [`radial_norm`;`UNIONS`];;

SOL_UNIONS;;
searc 'w' 0 [`radial_norm`;`DIFF`];;
searc 'x' 0 [`sol`;`UNION`];;
searcht 5 [`sol`;`DIFF`];;
searcht 5 [`sol x y <= sol x z`];;
searc 'y' 1  [`sol`;`UNIONS`];;
searc 'z' 0 [`FINITE (IMAGE X u)`];;
searc 'a' 0 [`FINITE (a..b)`];;
searc 'b' 2 [`sum (IMAGE X y)`];;

searc 'c' 3 [`sum (IMAGE X y)`];;
searcht 35 [`(\i. f i = \j. g j) <=> `];;
EXT_AXIOM;;
FUN_EQ_THM;;
searcht 5 [`sol`;`SUBSET`];;
searcht 5 [`sol x y <= sol x z`];;
searc 'd' 0 [`measurable`;`DIFF`];;
searc 'e' 0 [`radial_norm`;`DIFF`];;
searcht 5 [`(t DIFF s) INTER u`];; 
searcht 5 [`sol x y >= &0`];;
searc 'h' 0 [`&0 <= measure u`];;
searc 'f' 4 [`&0 <= x / y`];;
searc 'g' 4 [`&0 <= x * y`];;
searc 'i' 11 [`&0 <= x pow n`];;
searc 'j' 0 [`radial_norm`;`UNIONS`];;
searcht 15 [`FINITE { f s | g s }`];;
searc 'k' 0 [`measurable`;`fchanged`];;
FCHANGED_RADIAL;;
searc 'l' 0 [`radial`;`radial_norm`];;

(* Oct 31, 2012 *)
searcht 5 [`v IN rcone_gt x y z`];;
searcht 15 [`rcone_gt x y z`];;
searc 'a' 0 [`dist(v, vec 0)`;`norm`];;
searcht 5 [`norm v = &0`];;
searc 'b' 4  [`norm`;`dot`];;
searc 'c' 5 [`&0 < a * b`];;
searc 'd' 0 [`&0 < norm x`];;
searc 'e' 0 [`norm x = &0`];;
searc 'f' 0  [`vec 0 dot v`];;
searc 'g' 0 [`v dot vec 0`];;
searc 'h' 0 [`a % v dot u`];;
searcht 2 [`arcV`;`rcone_gt`];;
searc 'i' 0 [`cos`;`a dot b`];;
Print_types.print_thm_types Trigonometry1.DOT_COS;;
searcht 5 [`&0 < cos x`];;
searc 'j' 1 [`cos x < cos y`];;
searc 'k' 0 [`&0 <= arcV a b c`];;
searc 'p' 2 [`x * y < x * z`];;
searc 'q' 0 [`arcV x y z + arcV x y' z'`];;
searc 'r' 1 [`&0 < cos x`];;
searc 's' 0 [`cos x <= &1`];;
searc 't' 4 [`cos x = &1`];;
searc 'u' 2 [`~(?x. p)`];;
searc 'v' 1 [`~(a /\ b) = c`];;
searc 'w' 4 [`a <= b`;`&a <= &b`];;
searc 'x' 4 [`&0 <= x * y`];; 
searcht 5 [`&0 <= arcV x y z`];;
searc  'y' 0 [`arcV x y z = arcV x z y`];;
searc 'z' 12  [`&0 < x * y`];;
searcht 5 [`x * y <= &0`];;


(* START WITH THIS ONE. Nov 1. *)


searc 'a' 0 [name "WBLARHH"];;
searc 'b' 0 [name "AMH"];;
(* need to relat facet_of^2 to edges,edge_of *)

searc 'c' 0  [`edges`;`edge_of`];;
searc 'd' 0 [`edge_of`;`face_of`];; (* def of edge_of=face_of aff_dim 1 *)
searc 'e' 0 [`conforming`];;  (* def of conforming *)

searcht 5 [name "CFYXF"];; (* 0 relates edges<=> EDGES0_FAN *) 
searcht 5 [def "edge_of"];;
searcht 5 [`fully_surrounded`];;
searcht 5 [`plain_hypermap`];;

searcht 5 [def "plain"];;
searcht 50 [`FAN`];;

(* Nov 1, 2012 *)
searcht 5 [`edges`;`face_of`];;
searcht 5 [`hypermap_of_fan`;`hypermap1_of_fanx`];;
searcht 50 [`hypermap_of_fan`];;
searcht 50 [`hypermap1_of_fanx`];;
searcht 5 [`facet_of`;`aff_dim`];;
searc 'a' 1  [`x = y`;`x ==> y`;`x /\ y`];;
searc 'b' 2 [`face_of`;`facet_of`];;
searcht 'f' 18  [`face_of`];;
searc 'c' 3 [`aff_dim`;`{}`];;
type_of `aff_dim`;;
searc 'd' 0 [`aff_dim`;`interior`;`3`];;
searc 'e' 0 [`aff_dim`;`interior`;`2`];;
searc 'f' 18  [`face_of`];;
searc 'g' 1 [`face_of`;`SUBSET`];;
searcht 5 [`facet_of`;`aff_dim`];;
searc 'h' 0 [`edges`;`segment`];;
BIJ_SYM;;
searc 'i' 0 [`r (BIJ f U V) (BIJ g V U)`];;
searcht 15 [`BIJ`;`(a,b)`];;
searcht 5 [def "preimage"];;
searcht 5 [`preimage`;`BIJ`];;
searcht 5 [`IMAGE`;`BIJ`];;
type_of `preimage`;;

searc 'j' 0 [`( p ==> (?x. q x)) <=> (?x. p ==> q x)`];;
SKOLEM_THM;;
searcht 35 [`BIJ`];;
searcht 5 [`BIJ`;`x ==> (u = v)`;`?`];;
searcht 5 [def "BIJ"];;
searcht 10 [def "INJ"];;
searc 'k'  0 [`preimage`];;
searcht 5 [`IMAGE f X SUBSET Y`];;
INST_TYPE;;
Print_types.print_goal_types();;
searcht 5 [`FST (x,y)`];;
searc 'l' 1 [`preimage`];;
searcht 5 [def "hypermap1_of_fanx"];;
searc 'm' 0 [`hypermap1_of_fanx`;`hypermap`;`e_fan`];;
searc 'n' 1 [`face_set`];;
Hypermap.dart;;
searcht 5 [`tuple_hypermap`;`hypermap`];;
searcht 50 [`hypermap_of_fan`;`hypermap`];;
searcht 5 [`hypermap1_of_fanx`;`hypermap`];;
searcht 15 [`HYP`];;
searcht 15 [`d1_fan`];;
searcht 5 [`{a,b} = {a',b'}`];;

 lemmaTGJISOK;;
searcht 50 [`FAN`;`azim`];;
searc 'a' 0 [`conforming`];; (* def *)
searcht 50 [`conforming`];;  
searcht 50 [`conforming_fan`];;
searc 'c' 0 [name "PIIJ"];;
searc 'b' 2 [`fully_surrounded`];; (* dart_of_fan azim_dart *)
searcht 5 [`azim_dart`];; (* azim_dart -> azim_fan *)
searcht 5 [`azim_fan`];; (* sigma_fan *)
searcht 5 [`FAN`;`sigma_fan`];;
List.length (searcht 1000 [`FAN`]);;
searcht 500 [`conforming`];;
searc 'a' 0 [name "BSXA"];;
searc 'b' 0 [`FAN`;`vertices`];; (* POLYHEDRON_FAN *)
searc 'd' 0 [`fan80`;`polyhedron`];;
searc 'e' 1  [`polyhedron`;`set_of_edge`];;
searcht 5 [`polyhedron`;`conforming_fan`];;
searcht 30 [`hypermap1_of_fanx`;`conforming_fan`];;
searc 'f' 5 [`CARD`;`simple_hypermap`];;  (* conditions for 3 <= CARD face *)
searc 'g' 0  [name "TGJ"];;
searc 'h' 0 [name "WGV"];;
searc 'i' 0 [`plain_hypermap`;`FAN`];;
searcht  5 [`plain_hypermap`;`FAN`];;
Fan.plain_hypermap_fan;;

Fan.e_fan_no_fix_point;;
searcht 5 [`edge_map`;`e_fan`];;
Fan.hypermap_of_fan_rep;;
Hypermap.is_restricted;;
searcht 5 [`d1_fan`;`d_fan`];;
searcht 5 [`FAN`;`d20_fan`];;
searc 'j' 7 [`d1_fan`;`d_fan`];;
Print_types.print_goal_types();;
searc 'k'  4 [`res`; `e_fan`];;
searcht 5 [`p e_fan x = e_fan x`];;
searc 'l' 0 [`x IN d1_fan u`;`res`];;  (* not IN *)
searc 'm' 2 [`x IN d1_fan u`;`res`];;
help_grep "GEN";;
GEN_ALL;;
(GEN_ALL Fan.into_domain_e_fan);;
Fan.plain_hypermap_fan;;
searc 'm' 2 [`x IN d1_fan u`;`res`];;
searcht 5 [`is_no_double_joins`];; 
Fan.face_subset_dart_fan;; (* each face is subset of d_fan *)
Fan.FACE_FAN_NOT_EMPTY;;
searc 'n' 4 [`3`;`CARD (face H x)`];;  (* for hypermap_of_fan *)
searc 'o' 0 [`planar_hypermap`;`FAN`];;
searc 'p' 2 [`dart`;`hypermap1_of_fanx`];;
searcht  3 [`dart`;`hypermap1_of_fanx`];;
searc 'q'  0 [name "e_fan_no_fix"];;
searcht 5 [`edge_map`;`e_fan`];;
searcht 5 [`fan80`];;
searcht 5 [`hypermap_of_fan`];;
searcht 50 [`node`;`3`];;
SURROUNDED_IMP_CARD_NODE_GE_3;;
searcht 50 [`node_set`;`hypermap1_of_fanx`];;
Conforming.SUM_AZIM_FAN_OF_NODE_EQ_2PI_I_FAN;;
searcht 5 [`fan80`;`azim`];;
searcht 5 [`n_fan`;`sigma_fan`];;
searcht 5 [def "d1_fan"];;
searcht 5 [def "dart_of_fan"];;
searcht 5 [def "n_fan_pair"];;
searcht 100 [`sigma_fan`;`CARD`];;
searc 'r' 0 [`azim_fan`;`pi`;`sum`];;  (* sum over node set azim_fan is 2pi *)
searcht 5  [def"node_set"];;
searc 'u' 0 [`node`;`node_set`];;
searc 's' 0 [`fan80`;`polyhedron`];;
searc 't' 0 [`polyhedron`;`azim_fan`];; 
searcht 5 [`sum f x < sum f y`];;
searc 'v' 0 [`FINITE`;`node`];;
searcht 5 [`node`;`{}`;omit `INSERT`];;
searc 'w' 0 [`x IN node f x`];;
searc 'x' 14 [`~(x = {})`];;
searc 'y' 0 [`node f x SUBSET dart g`];;
searc 'z' 1 [`sum`;`CARD`];;
searc '0' 3 [`a * x <= b * x`];;
searc '1' 4 [`&x <= &y`];;
searcht 5 [def "edge_set"];;
searcht 25 [`simple_hypermap`];;

(* Nov 3, 2012 *)
searcht 5 [`BIJ`;`o`];;
searc 'a' 3 [`pr2`;`d_fan`];; (* injectivity of pr2,pr3 *)
searc 'b' 2 [`{x,y} = {x',y'}`];;
searc 'c' 7 [`e_fan`];;
searcht 5 [def "e_fan"];;
let  nil = 0;;
searcht nil [`e_fan`;`e_fan_pair`];;
searcht 5 [`?`;`x = (a,b)`];;
searcht 5 [`pr2`];;
searc 'd' 0  [`dart`;`d_fan`];;
searc 'e' 1 [`r e_fan x = e_fan y V E x`];;
searc 'h' 0 [`(x,y) = (x',y')`];;
searc 'f'0  [`d_fan`;`dart`];;
searc 'g' 8 [`d_fan`;`d1_fan`];;

EDGE_PAIR_pr23;;
searc 'h' 10 [`e_fan`;`d1_fan`];;

searc 'n' 14 [`simple_hypermap`];;
searc 'i' 1 [`node_map H o face_map H`];;
  searcht 50 [`node_map H ( face_map H t)`];;
searc 'k' 3 [`inverse f y = z`];;
searc 'l' 0 [`(x = y) <=> (y = x)`];; 
searc 'm' 0 [`edge_map H permutes s`];;
searc 'o' 0 [`node_map H y IN dart H`];;
searc 'p' 0 [`y IN node H y`];;
searcht 5 [`node_map H y IN node H y`];;
searc 'q' 1 [`x IN node H y`];;
searc 'r' 0 [`x POWER 1`];;
searcht 5 [`face H (f x)`];;
searc 's' 0 [`face H x = face H y`];;
searc 't' 2 [`x IN face H y`];;
searcht 15 [`x IN face H y`];;
searcht 15  [`face H x = face H y`];;
searcht 8 [`POWER`;`node`];;
searc 'u' 16 [`(f POWER n) x = x`;`f x = x`];;
searc 'v' 0 [`orbit_map`;`node`];;
searc 'w' 0 [`f POWER 0`];;
searc 'x' 1 [`CARD {x}`];;

POLYHEDRON_CONFORMING_FAN;;
searc 'y' 0 [`simple_hypermap`;`conforming_fan`];;
searc 'a' 0 [`polyhedron`;`FAN`];;
searc 'b' 0 [name "POLYHEDRON_NODE_3"];;
searcht 5 [`d_fan`;`f1_fan`];;
searc 'c' 0 [`d_fan`;`dart`];;
searc 'd' 0 [`face_map H x IN dart H`];;
searc 'e' 2 [`face_set H`;`dart`];;
searcht 5 [`set_of_orbits`;`SUBSET`];;
searc 'f' 0 [`face`;`dart`];;
searc 'g' 6 [`y IN face H x`];;
searc 'h' 0 [`y IN face H x`];;
searcht 5 [name "D1_D"];;
searc 'i' 0 [`set_of_edge`;`CARD`;`1`];;
searcht 5 [name "WBL"];;
searcht 5 [name "AMH"];;
Cfyxfty.WBLARHH_BIJ;;
Polyhedron.AMHFNXP_BIJ;;

type_of `(topological_component_yfan (vec 0,vertices p,edges p))`;;
searcht 5 [name "WGV"];; 
searc 'j' 0 [`edges`;`segment`];;
searcht  5 [`edge_of`;`segment`];;
searc 'k' 0 [`3`;`interior`;`aff_dim`];;
searc 'l' 0 [`2`;`aff_dim`;`facet_of`];;
searc 'm' 3 [`aff_dim`;`{}`];;
searc 'n' 0 [`convex hull x`;`segment`];;
searc 'o' 1 [`segment [a,b] = segment [c,d]`];;
searc 'p' 0 [`{a,b} = {c,d}`];;
searc 'q' 0 [`aff_dim`;`segment`];;
searc 'r' 0 [`edge_of`;`aff_dim`];;
searc 's' 1 [`polytope`;`polyhedron`];;
searcht 5 [`face_of`;`facet_of`];;
PREIMAGE_BIJ;;
searcht 5 [`dartset_leads_into_fan`;`fchanged`];;
Polyhedron.WBLARHH;;
searc 't' 0 [`face H a IN face_set H`];;
searc 'u' 4  [`?!`;`?`];;
searc 'v'  1 [`preimage`];;
searc 'w' 0 [`x IN {y}`];;
searc 'x' 2 [`?`;`facet_of`];;
searcht 5 [`s face_of s' ==> polyhedron s`;`polyhedron`];;
searc 'y' 0 [`face_of`;`polytope`];;
searc 'z' 12 [`facet_of`;`fchanged`];;
searc 'a' 4 [`facet_of`;`fchanged`];;
searc 'b' 0 [`x INTER x`];;
searc 'c'  14 [`~(x = {})`];;
searc 'd' 0 [`(x,y) = (x',y')`];;

(* Nov 5, 2012 *)
let searct _ = searcht;;
searcht 5 [name "eq_sym"];;
searcht 5 [`(x = y) <=> (y = x)`];;
searc 'a' 0 [name "WBL"];;
searc 'b' 2 [`?`;`?!`];;
searc 'c' 0 [name "AMH"];;
searc 'd'  1 [`preimage`];;
searcht 5 [`face_set`;`face`];;
searc 'e' 8 [`==>`;`<=>`];;
searc 'f' 0 [`x IN face H x`];;
Print_types.print_goal_types();;
BIJ_FACET2_EDGE;;
Print_types.print_thm_types BIJ_FACET2_EDGE;;
type_of `edges c`;;
BIJ_EDGES_DART_FACE;;

(* Nov 6, 2012 *)
searcht 15 [`sum f (\x. sum h g)`];;
SUM_SUM_PRODUCT;;
searc 'g' 0 [`CARD`;`sum`];;
searc 'h' 8 [`(FINITE:((A#B)->bool)->bool)`];;
searcht 5 [`SUBSET`;`FINITE`];;
searc 'i' 0 [`face_of`;`facet_of`];;
searc 'i' 0 [`face_of`;`facet_of`];;
FACE_OF_TRANS;;
help_grep "BETA_TAC";;
searc 'j' 0 [`(\(a,b). f a b)`];;
searc 'k' 0 [`FINITE`;`facet_of`];;
searc 'l' 1 [`polytope`;`polyhedron`];;
searc 'm' 0 [`face_of`;`FINITE`];;
searc 'n' 2 [`sum f X = sum f Y`];;
searcht 5 [`&x <= &y`;`(x:num) <= (y:num)`];;
searcht 15 [name "REAL_OF_NUM"];;
searcht 5 [`CARD {f,e | f facet_of P /\ e facet_of f}`];;
searcht 5 [`6 * n - 12`];;
searcht 5 [name "BIJ_FACET"];;
POLYHEDRON_TGJISOK;;
BIJ_DART_POLYEDGE;;
searc 'o' 0 [`number_of_faces`];;
searc 'p' 2 [`BIJ`;`CARD x= CARD y`];;
searcht 5 [`rcone_gt`;`fchanged`];;
searc 'q' 0 [`open`;`INTERS`];;
searc 'r' 0 [`FINITE`;`IMAGE`];;
searc 's' 0 [`FINITE`;`DIFF`];;
searc 't' 0 [`open`;`a dot v < b`];;
searc 'u' 1 [`open`;`ball`];;
searcht 5 [`A INTER B SUBSET D INTER C`];;
searc 'v' 3  [`affine hull x = x`];;
searc 'w' 0 [`affine hull x SUBSET affine hull y`];;
searc 'x' 0 [`affine`;`dot`];;
searc 'y' 0 [`A INTER B SUBSET B`];;
searcht 5 [`(A= B) ==> A SUBSET B `];;

SET_CONV;;    
searcht 15 [`INTERS (IMAGE h f)`];;

search [`relative_interior`;`<`];; 

 Polyhedron.IN_RELATIVE_INTERIOR1;;
RELATIVE_INTERIOR_POLYHEDRON_EXPLICIT;; (* via hyperplane ineqs *)

FACET_OF_POLYHEDRON_EXPLICIT;;
searc 'a' 0 [`( p ==> (?x. q x)) <=> (?x. p ==> q x)`];;
searc 'b' 0 [`interior`;`affine`;`hull`];;
searc 'c' 1 [`(UNIV) INTER A`];;
searcht 5 [`INTERS A SUBSET INTERS B`];;
DE_MORGAN_THM;;
searc 'd' 2 [`~(?x. p)`];;

(* Nov 8 *)
Print_types.print_goal_types();;
searc 'e' 0 [`x dot (vec 0) = &0`];;
searc 'f' 0 [`a dot (t % p)`];; 
searc 'g' 2 [`x * z < y * z`];;
searc 'h' 6 [`&0 < x/y`];;
searc 'i' 0 [`&0 < norm p`];;
searcht 5 [`dist`;`norm`];;
searc 'j' 0 [`norm (t % p)`];;
searc 'k' 3 [`norm p  = &0`];;
searcht 5 [`(a dot x < b) <=> (a' dot x < b')`];;
searc 'l' 0 [`interior`;`a dot x <= b`];;
searc 'm' 0 [`(x = y) <=> (y = x)`];;
avoids;;
searc 'n' 0 [`~(!x. p)`];;
searcht 5 [`polyhedron`;`dot`;`<=`];;
searcht 30 [name "TARJJ"];;
Tarjjuw.TARJJUW;;
searc 'o'  0 [`ball_annulus = y`];;
searc 'p'  0 [`norm (vec 0) = &0`];;
searc 'q' 6  [`&0 < x * y`];;
searc 'r' 1 [`interior`;`ball`];;
searc 's' 0 [`half_spaces`];;
searc 't' 8 [`==>`;`<=>`];;
FACET_OF_POLYHEDRON_EXPLICIT_ALT;;
searc 'u' 0 [`v dot v = &0`];;
searcht 5 [`v IN rcone_gt a v c`];;
searc 'v' 1 [`fchanged`;`relative_interior`];;
searc 'w' 0 [name "rcone_gt_facet"];;
RELATIVE_INTERIOR_POLYHEDRON_EXPLICIT_ALT;;
rcone_gt_facet;;
FACET_OF_POLYHEDRON_EXPLICIT_ALT;;
searc 'x' 0 [`x % y % z`];; 
searc 'y' 0 [`v % x = u % x`];;
searcht 5 [`rcone_gt`;`dot`];;
searc 'z' 0 [`&0 < &1 / x`];;
searcht 5 [`&0 < x/ y `];;
searc 'h' 6 [`&0 < x/y`];;
rcone_gt_facet;;

(* Nov 10, 2012 *)
searc 'i' 0 [`lmfun_ineq_center`];;
searc 'j' 0 [`ball_annulus`;`norm`];;
searc 'k' 0 [`sum V x <= sum V y`];;
searc 'l' 1 [`hl`];;
searc 'm' 1 [`dist(vec 0,x)`];;
searc 'n' 9 [`lmfun`];;
searc 'o' 4 [`h0`];;
searc 'p' 0 [`a / x <= b /x `];;
searc 'q' 0 [`sum V (\v. &1)`];;
searc 'r' 4 [`&x <= &y`];;
searcht 5 [name "weak_sat"];;
searc 's' 0 [`sum w x <= sum v x`];;
searc 't' 0 [`&0 <= lmfun h`];; 
searc 'u' 0 [`sum`;`&4 * pi`];;
searc 'v' 0 [`polyhedron`;`FAN`];;
searc 'w' 0 [`conforming_fan`;`polyhedron`];;
searcht 5 [`SURJ`;`sum`];;
searc 'x' 0 [name "AMH"];;
searc 'y' 1 [`sum`;`topological_component_yfan`];;
searc 'z' 2 [`SURJ`;`IMAGE`];;
searc 'a' 4 [`IMAGE`;`sum`];;
asearcht 5 [`{x | f x } = \y. f y`];;
searcht 5 [name "DLWCHEM"];;
searc 'b' 4 [`3`;`face`];;
searcht 15 [name "CFY"];;
searcht 5 [`facet_of`;`FINITE`];;
searc 'c' 0 [`lfun`;`lmfun`];;
searcht 5 [`regular_spherical_polygon_area`];;
searc 'd' 3 [`cos(x - y)`];;
searc 'e' 0 [`cos (pi/ &6)`];;
searcht 5 [`cos(acs t)`];;
searcht 5 [`BIJ`;`CARD`];;
searc 'f' 0 [`face_of`;`FINITE`];;
searc 'g' 0 [`face_of`;`facet_of`];;
BIJ_FACET2_EDGE;;
BIJ_EDGES_DART_FACE;;
searcht 5 [`facet_of`;`2`];;
searc 'h' 4 [`dartset_leads_into_fan`;`edges`];;
searc 'i' 1 [`fchanged`;`topological_component_yfan`];;
searc 'j' 2 [`?`;`?!`];;
searcht 25 [`fully_surrounded`;];;  
searcht 25 [`conforming_fan`];;
searcht 5  [`facet_of`;`?`];;
searc 'k' 2 [`facet_of`;`?`];;
searc 'l' 1 [`polytope`;`polyhedron`];;
searcht 5 [`facet_of`;`{}`];;
searcht 5 [`face_of`;`SUBSET`];;
searcht 5 [`p facet_of p`];;
searcht 15 [name "KREIN"];;
searc 'm' 0 [`polyhedron`;`convex`];;
searc 'n' 0 [`polytope`;`compact`];;
searc 'o' 4 [`SUBSET`;`convex hull x`];;
searcht 5 [`convex hull x`;`face_of`];;
searc 'p' 2 [`face_of`;`convex`];;
searc 'q' 0 [`convex x`;`convex hull y = y`];;
searc 'r' 0 [`extreme_point_of`;`face_of`];;
searc 's' 0 [`face_of`;`facet_of`;`INTERS`];;
searc 't' 0 [`CARD x = 0`];;
searc 'u' 14 [`~(x = {})`];;
searc 'v' 18 [`CARD x` ;`2`];;

searcht 15 [`CARD`;`3`];;
searcht 5 [`CARD A < CARD B`];;
searc 'w' 12 [`PSUBSET`];;
searc 'x' 0 [`CARD{x,y} = 2`];;
searc 'y' 1 [`INTERS {}`];; 
searcht  15 [`(:A)`;`{x}`];;
searc 'z' 0 [`aff_dim x <= aff_dim y`;`SUBSET`];;
searc 'a' 0 [`aff_dim {y}`];;
searcht 25 [`A = {x}`];;
searc 'b' 0 [`INTERS {x}`];;
searc 'd' 0 [`x ==> polyhedron p`;`face_of`];;
searc 'g' 0 [`face_of`;`facet_of`];;
searcht 5 [`SUBSET`;`bounded`];;
searc 'h' 0 [`face_of`;`SUBSET`];;
searcht 5 [`aff_dim`;`2`;`facet_of`];;
searc 'k' 0 [`3`;`interior`;`aff_dim`];;
searcht 5 [`ball_annulus`;`FINITE`];;
searcht 5 [`x IN {}`];;
searcht 5 [`CARD`;`SUBSET`];;
EXISTS_M_POLYHEDRON;;
searcht 5 [`CARD {}`];;
searc 'l' 0 [`ball_annulus`;`norm`];;
searc 'm' 0 [`arclength`;`arcV`];;
searc 'n' 0 [`arclength a b c <= arclength a b c'`];;
searcht 5 [`arcV a b c <= arcV a' b' c'`];;  
searcht 5 [`arcV a b c < arcV a' b' c'`];;  
searcht 5 [`arclength a b c <= arclength a' b' c'`];;  
searc 'p' 0 [`acs x <= acs y`];;

searc 'o' 3 [`acs`;`arclength`];;
searc 'q' 0 [`dist (x,y) <= dist (x,p) + dist(p,y)`];;
searc 'r' 5 [`(a *a + b *b - c *c)/(&2 *a* b)`];;
searc 's' 0 [`x / y <= z / y`];;
searc 't' 6 [`&0 < x * y`];;
searc 'u' 0 [`x * x <= y * y`];;
searcht 5 [`x < acs y`];;
searc 'v' 0 [`pi / &2 - acs x`];;
COS_PI6;;
searcht 5 [`acs (sqrt(&3)/ &2)`];;
searcht 5 [`acs (x ) = pi/ &6`];;
searcht 5 [`acs (cos x)`];;
searc 'w' 1 [`asn (-- x)`];;
searc 'x' 0 [`acs (&0)`];;
searc 'y' 0 [`acs x < acs y`];;
searc 'z' 0 [`acs x <= acs y`];;
searcht 15 [`sqrt3`];;
searcht 5 [`BIJ`;`a..b`];;
searcht 5 [`BIJ`;`CARD`];;
searcht 5 [`IMAGE`;`CARD`];;
searcht 5 [`HAS_SIZE`;`BIJ`];;

(* Nov 11, 2012 *)
searcht 50 [`CARD`;`f:num->B`;`?`];;
searcht 5 [def "CARD"];;
searcht 5 [`ITSET`];;
searcht 50 [`BIJ`];;
type_of `CHOICE`;;
searc 'a' 0 [`CARD A = 0`];;
searc 'b' 0 [`A SUBSET {}`];;
searcht 5 [`INJ`;`?`;`(f:A->B)`];;
searcht 50 [`(f x = f y) ==> (x = y)`];;
searc 'c' 0 [`x HAS_SIZE 0`];;
searc 'u' 14 [`~(x = {})`];;
searc 'e' 0 [`DELETE`;`HAS_SIZE`];; 
searc 'd' 1 [`DELETE`;`FINITE`];;
type_of `PRE`;;
searcht 5 [`A DIFF B HAS_SIZE c`];;
searcht 5 [`FINITE (A DIFF B) `];;
searc 'f' 1 [`CARD( A DIFF B)`];;
searc 'g' 0 [`CARD (IMAGE j A)`];;
searcht 5 [name "INJ_IFF_SURJ"];;
searcht 5 [` (a..b) HAS_SIZE t`];;
searc 'l' 0 [`ball_annulus`;`norm`];;
searcht 5 [`sum V x = sum V y`];;
searc 'm' 3  [`lmfun_ineq_center`];;
searc 'n' 1 [`hl x`];;
searc 'o' 1 [`lfun`;`lmfun`];;
searcht 5 [`sum(a..n)`;`sum(b..n) f + sum (a..r) f`];;
searc 'p' 1 [`sum(a..a)`];;
SUM_EQ;;
searcht 5 [`3 <= CARD x`;`facet_of`];;
searcht 5 [`sum`;`&6 * &n - &12`];;
searcht  5 [`BIJ`;`HAS_SIZE`];;
searc 'q'  2 [`BIJ`;`CARD`];;
searc 'r' 3 [`BIJ`;`FINITE`];;
GOTCJAH;;
searcht 5 [`&0 <= sol x U`];;
searcht 5 [`fchanged`];;
RADIAL_NORMBALL;;
FCHANGED_RADIAL;;
FCHANGED_MEASURABLE;;
searc 's' 0 [`radial`;`radial_norm`];;
SUM_EQ;;
searc 't' 0 [`regular_spherical_polygon_area`];;
searc 'u' 2 [`FINITE`;`facet_of`];;
searc 'v'  3 [`pack_ineq_def_a`];;
searc 'w' 1 [`acs_sqrt_x1_d4`];;
searc 'x' 2 [`arclength_x_123`];;
searc 'y' 2 [`x pow 2 <= y pow 2`];;
searc 'z' 2 [`sqrt (x pow 2)`];;
searc 'a' 0 [`sqrt(&4)`];;
searcht 5 [`arclength x y z <= arclength x' y' z'`];;
 Trigonometry1.TRI_SQUARES_BOUNDS;;
searc 'b' 0 [`arcV`;`arclength`];;
searc 'c' 0 [`arclength a b c = arclength b a c`];;
searc 'd' 0 [`ball_annulus`;`norm`];;
PACK_INEQ_DEF_A_797;;
searc 'e' 0  [`arcV x y z = arcV x z y`];;
searcht 15 [name "plain_hyper"];;

(* Nov 13, 2012 *)
searcht 5 [`gamma3f_x_div_sqrtdelta`];;
searcht 50 [def "gamma3f"];;
searc 'a' 1 [`delta`;`ups_x`];;
searcht 5 [`delta`;`delta_x`];;
searc 'b' 3 [`x / y < z`];;
searc 'c' 2 [`sqrt x < sqrt y`];;
searc 'd' 4 [`&0 <= x/y`];;
searc 'e' 5 [`&0 <= x * y`];;
searc 'f' 5 [`&0 < x* y`];;
searcht 5 [def "cut"];;
searcht 5 [`lfun`;`h0cut`];;
searcht 5 [`vol_x`;`vol_y`];;
searcht 5 [`vol_y`;`sqrt`];;
searc 'g'  0 [`sqrt( x * x) = x`];;
searc 'h' 0  [`sqrt2 * sqrt2`];;
searcht 5 [`dih_x_div_sqrtdelta_posbranch`;`dih_x`];;
searcht 50 [`dih_x_div_sqrtdelta_posbranch`];;
searcht 15 [name "LET"];;
searc 'i' 1[`sol_y`;`sol_x`];;
searc 'j' 0 [`dih_x x1 x2 x3 x4 x5 x6 = dih_x x1 x2' x3' x4' x5' x6'`];;
searc 'k' 1 [`dih_x x1 x2 x3 x4 x5 x6 = dih_x x1 x2' x3' x4' x5' x6'`];;
searc 'l' 1 [`sol_euler_x_div_sqrtdelta`;`delta_x`];;
searc 'm' 0 [`sol_x`;`sol_euler_x`];;
searc 'n' 0 [`sol_x a b c d e f = sol_x a' b' c' d' e' f'`];;
searc 'o' 1 [`sol_x a b c d e f = sol_x a' b' c' d' e' f'`];;
searc 'p' 0 [`dih_x_div_sqrtdelta_posbranch`;`dih_x`];;
searcht 5 [`delta_x a b c d e f =  delta_x a' b' c' d' e' f'`];;
searc 'q' 1 [`ups_x a b c = ups_x a' b' c'`];;
searc 'r'  4 [`&0 < ups_x x y z`];;
searcht 5 [`sqrt(&2) * sqrt(&2)`];;
searc 's'  0 [`ups_x a b c = ups_x a' b' c'`];;
searcht 5 [`eulerA_x a b c d e f = eulerA_x a' b' c' d' e' f'`];;
searc 't' 12 [`&0 < x * y`];;
searcht 5 [`sqrt x <= sqrt y`];;
searcht 5 [`eta_y`;`&2 * x`];;
searcht 5 [`x * x <= y * y`];;
searcht 5 [`sqrt8 * sqrt8`];;
searc 'u' 13 [`dih_x_div_sqrtdelta_posbranch`];;
searc 'w' 0 [`dih_x`;`atn2`];;
searc 'v' 5 [`atn2`;`pi`;`<=`];;
searc 'w' 5 [`atn`;`<`;`pi`];;
searcht 50 [`&0 <= a * b`];;
searcht 10 [`dih4_x_div_sqrtdelta_posbranch`];;
searcht 5 [`eta_y`;`&0 < delta_x`];;
searcht 5 [`&0 <= ups_x a b c`];;
cell_3_delta_x_pos_x;;  (* &0 < delta_x <=> (eta_x < sqrt2 *)
UPS_X_POS;;
delta_x4_pos;;
searcht 10 [`delta_x`;`ups_x`];;
GSYM delta_delta_x;;
Collect_geom.DELTA_RRR_INTERPRETE;;
searcht 5 [`sqrt8 * sqrt8`];;
searc 'x' 0 [`&0 <= x pow 2`];;
searc 'y' 0 [`eta_y a b c = eta_y a' b' c'`];;
searc 'z' 0 [`sqrt8`;`sqrt2`];;
searcht 5 [`delta_x a b c d e f = delta_x a' b' c' d' e' f'`];;
searc 'a' 0 [`sqrt8 * sqrt8`];;
searcht 5 [`radV`];;

(* Nov 18, 2012 *)
searcht 5 [`gamma3f_x_div_sqrtdelta`];;
searcht 5 [`sol_euler_x_div_sqrtdelta a b c d e f = sol_euler_x_div_sqrtdelta a' b' c' d' e' f'`];; 
searc 'b' 0 [`eta_x a b c = eta_x a' b' c'`];;
searcht 5 [`gamma3f a b c r s = gamma3f a' b' c' r' s'`];;
searc 'h' 0  [`sqrt2 * sqrt2`];;
searc 'i' 0 [`sol_x a b c d e f = sol_x a' b' c' d' e' f'`];;
searc 'j' 1 [`sol_x a b c d e f = sol_x a' b' c' d' e' f'`];;
searc 'k' 1[`sol_y`;`sol_x`];;
searc 'l' 0 [`dih_y a b c d e f = dih_y a' b' c' d' e' f'`];;
searc 'm' 1 [`dih_y a b c d e f = dih_y a' b' c' d' e' f'`];;
searc 'n' 23 [`!y. (p ==> q y)`; `p ==> (!y. q y)`];;
searcht 5 [`&0 <= eta_x a b c`];;
searcht 6  [`&0 <= x*y`];;
searcht 5 [`&0 <= x*x`];;
searcht 5 [`&0 <= y_of_x dih4_x_div_sqrtdelta_posbranch a b c d e f`];;
searc 'o' 0 [`hminus`;`h0`];;
searc 'p' 0 [`h0cut y = &1`];;
searc 'q' 0 [`hplus`;`h0`];;
searc 'r' 0 [`h0cut y = &0`];;
type_of `h0cut`;;
bb 30;;
bb 5;;
bb 12;;
searc 's' 0 [`gamma3f a b c d e = gamma3f a' b' c' d' e'`];;
searc 't' 0 [`eta_y a b c = eta_y a' b' c'`];;
searcht 5 [`ineq u T`];;

(* Nov 22, 2012 *)
searcht 15 [`x * y <  x' * y' `];;
searc 'a' 1 [`gamma2_x1_div_a`;`gamma2_x_div_azim`];;
searc 'b' 0 [`h0cut y = &0`];;
searc 'c' 0 [`h0`;`hplus`];;
searcht 5 [`h0cut`;`lfun`];;
searcht 500 [`rad2_y`];;
searcht 5 [name "RHUFIIB"];;
searcht 5 [`rho_ij`;`rho_x`];;
searc 'd' 11 [`x <= y/ z`];;
searcht 5 [`sqrt x pow 2 `];;
searcht 5 [`x/ y * z`];;
searcht 5 [`x /y <= z`];;
searc 'e' 1 [`ups_x a b c = ups_x a' b' c'`];;
searc 'f' 1 [name "RHUFIIB"];;
searc 'g' 0 [`delta`;`delta_x`];;
searcht 5 [`&0 <= x pow 2`];;
searc 'h' 0 [`sqrt8`;`sqrt2`];;
searcht 5 [`hminus`;`DECIMAL`];;
searcht 1 [`rad2_x`;`rad2_y`];;
searcht 5 [`eta_y a b c < sqrt2`];;
searcht 15 [`x < sqrt y`];;
searcht 5 [`hplus`;`DECIMAL`];;
searc 'i' 0 [`rad2_x x1 x2 x3 x4 x5 x6 = rad2_x a b c d e f`];;
searcht 5 [`delta_x a b c d e f = delta_x a' b' c' d' e' f'`];;
searcht 5 [def "wt"];;
searcht 5 [`wtcount6_y`];;
bb 4;;
searc 'j' 0 [`beta_bump_y`;`beta_bump_lb`;omit `ineq`];;
searcht 5 [`x / y <= x / z`];;
bb 3;;
searcht 5  [`beta_bump_y`;`beta_bump_lb`;omit `ineq`];;
searcht 5 [`if a then b else c = b`];;
searcht 5 [`norm2hh`;`gamma4fgcy`];;
searcht 5 [`gamma23_full8_x`];;
searcht 5 [`gamma3_x`;];;
searcht 5 [`vol3f_456`];;
searcht 5 [`critical_edge_y`];;
searc 'k' 0 [`sqrt2 * sqrt2`];;
searc 'l' 1 [`sol_y`;`sol_x`];;
searcht 5 [`sqrt (y * y)`];;
searc 'm' 0 [`lfun`;`h0cut`];;
searc 'n' 1 [`lmfun x  = lfun x`];;
searc 'o' 0 [`hminus`;`h0`;omit `ineq`];;
searcht 5 [`dih_x a b c d e f = dih_x a' b' c' d' e' f'`];;
searcht 5 [`x * y <= x * z`];;
searcht 5 [`#0.008`;`gamma2_x_div_azim`];;
searcht 5 [`gamma2_x1_div_a`;`gamma2_x_div_azim`];;
searcht 5 [`atn2 (x,y) + atn2 (u,v)`];;
searc 'q' 3 [`tan (x + y)`];;
searc 'p' 2 [`tan x = tan y`];;
searcht 5 [`tan (atn x)`];;
searcht 5 [`abs(atn x) < pi / &2`];;
searcht 15 [`cos x = &0`];;
searc 'r' 7 [`atn2(x,y)`];;
searcht 5 [`&0 < atn x`];;
searc 's' 7 [ `atn2`;`atn`];;
searcht 5 [`atn x < atn y`];;
searcht 35 [`y < u / x`];;
searcht 5 [`delta_x`;`ups_x`];;
Sphere.dih_x;;
searc 't' 0 [`sqrt (x pow 2)`];;
fynlist 3;;
follow_your_nose_string_list();;
searcht 5 [`sqrt x * sqrt y`];;
searcht 5 [`sqrt (x1 pow 2)`];;
searc 'u' 1 [`atn2 (a* x,a*y)`];;
ETA_Y_BOUNDS;;
searc 'v' 4 [`gamma3f_x_div_sqrtdelta`;`gamma3f`];;
searcht 50 [`dih4_x_div_sqrtdelta_posbranch`;omit `ineq`];;
 dih_x_div_sqrtdelta_pos;;
searc 'w' 1 [`&0 < delta_x4 a b c d e f`];;
searcht 5 [`sqrt8 * sqrt8`];;
searcht 5 [`x * x < y * y`];;
searc 'x' 1 [`h0cut y = &1`];;
searcht 5 [`x * y < x * z`];;
searcht 5 [`dih4_x_div_sqrtdelta_posbranch`;`dih_x_div_sqrtdelta_posbranch`];;
bb 22;;
bb 3;;
searcht 5 [`dih_y a b c d e f = dih_y a' b' c' d' e'f 'f`];;
searcht 5 [`eta_y a b c = eta_y a' b' c'`];;
searcht 5 [`gamma3f a b c d e = gamma3f a' b' c' d' e' `];;
searcht 15 [`x <= y/z`];;
searcht 35 [`dih_y_div_sqrtdelta_posbranch a b c d e f = dih_y_div_sqrtdelta_posbranch a' b' c' d' e' f'`];;
searcht 5 [`dih_x_div_sqrtdelta_posbranch`;`dih_x`];;
searc 'y' 0 [`(?x. p /\ q x) <=> (p /\ ?x. q x)`];;
searcht 5 [`gamma2_x1_div_a`;`gamma2_x_div_azim`];;
searcht 5 [name "VICUATE"];;
searcht 5 [def "wedge"];;

(* Nov 29, 2012 *)
bb 3;;
searcht 5 [`gamma4fgcy a b c d e f l = u a' b' c' d' e' f' l`];;
searcht 5 [`rad2_y`;`eta_y`;omit `ineq`];;
searc 'a' 0 [`sqrt8`;`sqrt2`];;
Sphere.delta_y;;
Sphere.bump;;
searcht 5 [`m1`;`pi`];;
searcht 5 [`wedge a b c d INTER wedge a' b' c' d'`];;
searcht 5 [name "WEDGE_ORDER_DIS"];;
searcht 5 [`wedge_ge a b c d INTER wedge_ge a' b' c' d'`];;
searcht 5 [`wedge`;`wedge_ge`];;
searcht 25 [`&2 * pi`;`azim`];;
searcht 15 [`azim a b c d = &0`];;
Local_lemmas.AZIM_EQ_0_GE_ALT2;;
searcht 5 [`x IN Y <=> x IN Z`];;
searcht 5 [`wedge_ge`;`azim`];;
searcht 5 [`&0 <= azim a b c d `];;
searcht 15 [`collinear`;`aff_ge`];;
searc 'b' 3 [`azim a b c d + azim a b d e`];;
searcht 5 [`collinear {a,b,c} ==> (c IN affine hull X)`];;
searcht 5 [`aff_ge X {}`];;
searcht 25 [`collinear {a,b,c}`];;
searc 'c' 4 [`aff_ge`;`aff`];;
searc 'd' 2 [`aff_ge X Y SUBSET aff_ge X' Y'`];;
searcht 35 [`x IN aff_ge X Y`;omit `azim`;omit `%`];;
searc 'e' 22 [`affsign`];;
searcht 35 [`affsign`];;
searcht 5 [`DISJOINT`;`aff_ge S U`];;
searcht 5 [`x IN aff_ge {x} U`];;
AFF_GE_MONO_RIGHT;;
SUBSET;;
searcht 5 [`x IN X /\ X SUBSET Y`];;
searcht 5 [`aff_ge S {}`];;
searcht 5 [`x IN u hull v`];;
searcht 5 [`collinear`;`~(x = y)`];;
searcht 15 [`wedge`;`wedge_ge`];;
searcht 15 [`&0 <= azim a b c d `];;
searcht 15 [`wedge`;`azim`];;
searcht 15 [`collinear`;`affine hull X`];;
AFFINE_HULL_3_IMP_COLLINEAR;;
searcht 15 [`affine hull X`;`aff_ge`];;
searcht 5 [`X INTER {u}`];;
searcht 15 [`X INTER Y = {}`;`!`];;
searcht 5 [`azim`;`aff_ge X Y`];;
searcht 15 [`azim a b c d = azim a b c d'`;omit `norm`];;
searcht 15 [`azim a b c d = &0`];;
searcht 15 [`aff_gt`;`aff_ge`];;
searcht 5 [`azim a b c c`];;

(* Dec 1, 2012 *)
searcht 5 [`&x < &y`];;
bb 6;;
searcht 50 [`beta_bump_y`];;
Sphere.beta_bump_y;;
searcht 5 [`x /y`;`inv`];;
searcht 5 [`inv x <= inv y`];;
searcht 5 [`hminus`;`hplus`];;
searcht 5 [`hplus`;`sqrt8`];;
searc 'a' 0 [`sqrt8`;`sqrt2`];;
searcht 5 [`dih_y a b c d e f = dih_y a' b' c' d' e'f 'f`];;
searcht 3 [`rad2_y`;`y_of_x`];;
searc 'b' 1 [`norm2hh`;omit `ineq`];;

(* Dec 5, 2012 *)
searcht 5 [`sum`;`IMAGE`];;
Counting_spheres.BIJ_SUM;;
searcht 55 [`x MOD n`;`(v = q * x + (b:num))`];;
searcht 5 [`MOD`;`DIV`];;
DIVISION;;
searcht 5 [`x IN a..b`;`?`;`x+(a:num)`];;
searcht 5 [`(a - b) MOD n`];;
searcht 30 [`MOD`];;
searcht 5 [`a <= (b:num)`;`a >= (b:num)`];;
searcht 30 [`DIV`];;
MOD_EQ_0;;
searcht 30 [`(x - y) * (n:num) = x * n - y * n`];;
searc 'c' 0 [`(x * y + a ) MOD n`];;
searcht 5 [`a MOD n = a`];;
searcht 5 [`sum {}`];;
searcht 5 [`CARD X = 0`];;
searcht 5 [`~(X = {})`;`?`];;
searcht 5 [name "STRONG";name "INDUCT"];;
searcht 5 [`{} INTER X`];;
bb 8;;

(* Dec 14, 2012 *)
searcht 5 [`CARD x = 0`];;
searcht 15 [`(X = {})`;`?`];;
searcht 5 [`~(!x. P x)`];;
searcht 15 [`&0 <= sum X f`];;
searcht 15 [`sum {i}`];;
searcht 15 [`i..i`];;
searcht 5 [`x IN Y <=> x IN Z`];;
searcht 15 [`EL 0 x`];;
searcht 5 [`HAS_SIZE`;`0`;`{}`];;
bb 9;;
searcht 5 [`x HAS_SIZE (SUC n)`];;
searcht 5 [`( k * n + i) MOD n`];;
Print_types.print_goal_types();;
searcht 5 [`i MOD n = i`];;
searcht 5 [`0 MOD n`];;
searcht 5 [`periodic`;`cc_qu_v9`;omit `a_spine5`];;
searcht 50 [`sum (a..b) f`;`f a + sum (c..d) f`];;
searcht 5 [`sum (a..b) f`;`(b:num) < a`];;
searcht 5 [`sum (n..n) f`];;
searcht 5 [`sum X f = sum X g`];;

(* Dec 21, 2012 *)
searcht 5 [name "MHFTT"];;
searcht 5 [`collinear`;`aff_dim`];;
searcht 5 [name "IDBEZ"];;
searcht 5 [`voronoi_list`;`omega_list`];;
searcht 15 [`barV`;`aff_dim`];;
searcht 5 [`aff_dim`;`segment`];;
searcht 5 [`segment`;`face_of`];;
searcht 5 [`segment`;`extreme_point_of`];;
searc 'a' 0 [`extreme_point_of`;`face_of`];;
FACE_OF_REFL;;
searcht 5 [`polytope`;`voronoi_list`];;
searcht 5 [`segment`;`hull`];;
searcht 5 [`{x} face_of p`];;
searcht 5 [`aff_dim`;`facet_of`];;
searcht  5 [`segment`;`aff_dim`];;
searc 'b' 8 [`segment`;`~(a = b)`];;
searc 'c' 0 [`aff_dim {c}`];;
searcht 5 [`aff_dim x = &0`];;
searcht 5 [name "YNHY"];;
searcht 5 [name "BARV_3_EXPLICIT"];;
searcht 5 [name "TRUNCATE_SIMPLEX_EXPLICIT_3"];;
searcht 5 [name "LEFT_ACTION_LIST_PROPERTIES"];;
searcht 5 [name "OMEGA_LIST_2_EXPLICIT_NEW"];;
searc 'd' 0 [`permutes`;`hl`;`barV`];;
searcht 5 [`p permutes a..b ==> p permutes c..d`];;
searcht 50 [`u ==> x permutes y`];;
searcht 15 [`truncate_simplex 2 (CONS x y)`];;
searcht 15 [`set_of_list (CONS x y)`];;
searcht 15 [`hl x `;` (left_action_list p q)`];;
searcht 15 [`hl`];;
searcht 15 [`barV V k (truncate_simplex a b)`];;
searcht 15 [`left_action_list`;`swap`];;
searcht 15 [`TABLE`];;
searcht 15 [`EL i x = EL i y`];;
searcht 15 [`LENGTH`;`TABLE`];;
searcht 15 [`inverse (swap(a,b))`];;
searcht 5 [`EL j (truncate_simplex k p)`];;
EL;;

(* Dec 22, 2012 *)
searcht 15 [def "circ"];;
searcht 15 [name "MHFT"];;
searcht 15 [`affine hull (convex hull s)`];;
searc 'e' 14 [`affine hull {a,b}`];;
searcht 15 [`voronoi_list`;`dist`];;
searcht 5 [`norm x pow 2`;`dot`];;
searcht 15 [`voronoi_closed`;`voronoi_list`];;
Packing3.VORONOI_LIST_SUBSET_VORONOI_CLOSED;;
searcht 15 [def "voronoi_closed"];;
searcht 15 [`omega_list`;`voronoi_list`];;
searcht 5 [name "XYOF"];;
searcht 5 [`barV`;`x SUBSET V`];;
bb 4;;
searcht 5 [`affine_dependent`;`barV`];;
searcht 5 [`dist(x,y) < dist (u,v)`];;
searcht 5 [`EL x y IN set_of_list p`];;
searc 'f' 0 [`x IN set_of_list p`];;
searc 'h' 1 [`MEM x u`;`EL`];;
searcht 5 [`aff_dim`;`barV`];;
searcht 15 [`truncate_simplex`;`set_of_list`];;
searc 'g' 0 [`EL`;`truncate_simplex`];;
searc 'h' 0 [`LENGTH (truncate_simplex a b)`];;
searcht 5 [`aff_dim X <= aff_dim Y`];;
searcht 5 [`barV`;`aff_dim`];;
searcht 5 [`voronoi_closed`;`bis`];;
searcht 5 [`voronoi_closed`;`dist`];;
searcht 5 [`x IN s ==> x IN convex hull s`];;
searcht 15 [`barV`;`circumcenter`];;
searcht 10 [`omega_list`;`voronoi_list`;`barV`];;
searcht 10 [`circumcenter`;`dist`];;
searcht 5 [`sqrt x = sqrt y`];;
searcht 5 [`HD (truncate_simplex k u)`];;
searcht 15 [`&0 < x * y`;`&0 < x`];;

(* Dec 23, 2012 *)
searcht 15 [`CONS`;`LENGTH`;`3`];;
searcht 15 [`CONS`;`(x:(A)list) = y`];;
searcht 15 [`LENGTH t = 0`];;
searcht 15 [`set_of_list (CONS x y)`];;
searc 'i' 15 [`affine hull {a,b,c}`];;
searcht 8 [`(a cross b) dot c`];;
searcht 5 [`a cross a`];;
searcht 5 [`(vec 0) dot b`];;
searcht 5 [`coplanar`;`cross`];;
searcht 5 [`aff_dim`;`coplanar`];;
searcht 15 [`aff_dim`;`INSERT`];;
searcht 5 [`coplanar`;`affine_dependent`];;
searcht 50 [`coplanar`];;
searcht 5 [`aff_dim x <= aff_dim y`];;
searcht 5 [`aff_dim`;`{a,b,c}`];;
searcht 5 [`aff_dim (affine hull s)`];;
searcht 5 [`aff_dim {a,b}`];;
searcht 5 [`barV`;`aff_dim`];;
searcht 5 [`x IN y <=> x IN z`];;
searcht 5 [`affine hull (convex hull s)`];;
searcht 5 [`y SUBSET affine hull y`];;
searcht 5 [`t % x = t % y`];;
searcht 5 [`circumcenter x IN affine hull y`];;
searcht 15 [`&0 < a * b`];;
searc 'j' 1 [`extreme_point_of`;`segment`];;
searcht 5 [`aff_dim {a}`];;
searcht 5 [`aff_dim`;` {a,b}`];;
searcht 5 [`aff_dim`;`convex hull x`];;
searcht 5 [`face_of`;`extreme_point_of`];;
searcht 5 [`segment`;`convex`];;
searcht 15 [`X SUBSET aff_gt U V`];;
searcht 5 [`aff_gt {x,y} {z}`];;
searcht 5 [`set_of_list ul`;`LENGTH`];;
searcht 5 [`collinear {a,b}`];;
searcht 5 [`truncate_simplex i ul = ul`];;
searcht 5 [`x IN S ==> x IN (convex hull S)`];;
searcht 5 [`set_of_list (truncate_simplex i ul)`];;
searcht 5 [`EL i ul IN set_of_list ul`];;
searcht 5 [`MEM (EL i ul) ul`];;
searcht 5 [`EL k (truncate_simplex i ul)`];;
searcht 5 [`convex hull X  SUBSET aff_ge X Y`];;
searcht 4 [`aff_ge X Y = aff_ge X Z`];;
searcht 5 [`convex hull X`;`aff_ge`];;
searcht 5 [`aff_ge X Y SUBSET aff_ge X' Y'`];;
searcht 25 [`aff_ge {a,b,c} {d}`];;
searc 'k' 1 [`coplanar`;`DISJOINT`];;
searcht 5 [`coplanar`;`aff_dim`];;
searcht 15 [`aff_dim (r:real^A->bool) <= x`];;
searcht 5 [`dimindex (:3)`];;
searcht 5 [`x IN interior (UNIV)`];;
searcht 5 [`aff_dim`;`(UNIV)`;`affine hull x`];;
searcht 5 [`affine hull {a,b,c,d}`];;
searcht 5 [name "JBD"];;
searcht 5 [`a cross (b+ c)`];;
searcht 5 [`(a + b) dot c`];;
searcht 5 [`vec 0 dot a`];;
searcht 5 [`-- (x dot y)`];;
searcht 5 [`-- (x cross y)`];;
searcht 15 [def "wedge"];;
searcht 15 [`wedge`;`open`];;
searcht 5 [`convex`;`&1`;`%`];;
searcht 5 [`convex hull S = S`];;
searcht 3 [`(a + b) cross d`];;
searcht 5 [`convex hull A SUBSET convex hull b`];;
searcht 5 [`convex`;`omega_list`];;
searcht 5 [`convex hull {a,b,c,d}`;`%`];;
searcht 5 [`vec 0 dot u`];;
searcht 5 [`re_eqvl a (t * b)`];;
searc 'l' 2 [`re_eqvl`];;
searcht 5 [`&0 < inv x`];;
searcht 5 [`barV`;`aff_dim`];;
searcht 5 [`INSERT`;`aff_dim`];;
searcht 5 [`convex hull x`;`mxi`];;
searcht 50 [`mxi`];;
between;;
searcht 5 [`between`;`a % b`];
searcht 5 [`between`;`segment`];;
searcht 5 [`segment`;`a % b`];;
searcht 5 [`omega_list_n`;`omega_list`];;
searcht 5 [`omega_list`;`circumcenter`];;
searcht 5 [`truncate_simplex k ul = ul`];;
searcht 5 [` &0 % x`];;
searcht 5 [`x + vec 0`];;
searcht 5 [`hl`;`dist`];;
searcht 5 [`HD`;`EL 0`];;
searcht 5 [`a % u + b % v IN f hull s`];;
searcht 5 [`a + b IN affine hull s`];;
searcht 25 [`a % u`;`affine`];;
searcht 5 [`affine (affine hull s)`];;
searcht 5 [`circumcenter x IN affine hull s`];;
searcht 5 [`cc_cell`;`convex`];;
searcht 15 [`convex hull y SUBSET x`;`convex u`];;
searcht 5 [`S SUBSET affine hull S`];;
searcht 5 [`convex hull s = s`];;
searcht 5 [`mcell2`;`voronoi_closed`];;
searcht 5 [`vol`;`convex hull {a,b,c,d}`];;
searcht 5 [`coplanar`;`delta`];;
searcht 15 [`coplanar`;`vol`];;
searcht 5 [name  "POLFLZY"];;
searcht 5 [`delta_x`;`delta`];;
searcht 5 [`coplanar`;`coplanar_alt`];;
searcht 5 [`dimindex (:3)`];;
searcht 5 [`&0 <= delta a b c d e f`];;
 Rogers.MHFTTZN1;;
searcht 5 [`mcell4`;`convex hull s`];;
searcht 5 [`mcell4`;`coplanar`];;
searcht 5 [`coplanar`;`aff_dim`];;
searcht 5 [`aff_dim x <= aff_dim y`];;
searcht 5 [`s SUBSET f hull s`];;
searcht 5 [`convex`;`mcell`];;
searc 'a' 0 [`convex hull s SUBSET affine hull s`];;
searc 'b' 0 [`x IN s /\ s SUBSET t`];;
searcht 5 [`set_of_list (truncate_simplex k vl)`];;
searcht 5 [`s SUBSET affine hull s`];;
searcht 5 [`affine hull s SUBSET affine hull t`];;
searcht 5 [`affine hull (affine hull s)`];;
searcht 5 [`coplanar {a,b,c}`];;
searcht 5 [`LENGTH(truncate_simplex k vl)`];;
searcht 25 [`circumcenter`];;
searcht 5 [`affine_dependent`;`collinear`];;
searcht 5 [`truncate_simplex k vl`;` barV V k`];;

AFFINE_DEPENDENT_IMP_COLLINEAR_3;;
searcht 5 [`EL k (truncate_simplex j ul)`];;
searcht 5 [`mxi`;`dist`];;
searc 'c' 3 [`omega_list_n`;`voronoi_list`];;
searcht 5 [`convex`;`voronoi_list`];;
searcht 5 [`between`;`mxi`];;
searcht 5 [`between`;`convex hull s`];;
searc 'd' 0 [`convex hull a SUBSET convex hull b`];;
searcht 5 [`convex hull s = s`];;
searcht 5 [`voronoi_set`;`dist`];;
searcht 15 [`(?v. p v) ==> t`];;
searcht 15 [`~(x = {})`;`?`];;
searcht 15 [`set_of_list ul SUBSET V`];;
searcht 15 [`q IN set_of_list (truncate_simplex k vl)`];;
bb 5;;
searcht 5 [`radV`;`dist`];;
searcht 5 [`collinear`;`leaf`];;
searcht 5 [`coplanar`;`SUBSET`];;
searcht 5 [`s SUBSET f hull s`];;
searcht 5 [`aff_dim`;`coplanar`];;
searcht 5 [`aff_dim`;`collinear`];;
searcht 5 [`x UNION {y} = y INSERT x`];;
type_of `INSERT`;;
searcht 5 [`aff_dim`;`INSERT`];;
type_of `mcell`;;
searcht 5 [def "rog"];;
searcht 5 [`rogers`;`omega_list_n`];;
searcht 5 [`rogers`;`aff_ge`];;
searcht 5 [`aff_ge`;`convex`];;
searcht 15 [`aff_ge a b = aff_ge c d`];;
searcht 15 [`mcell2`;`aff_ge`];;
searcht 5 [`aff_gt`;`aff_ge`];;
searcht 15 [`aff_gt {a,b}`];;
searcht 5 [`aff_ge a b SUBSET aff_ge c d`];;
searcht 15 [`FINITE`;`aff_dim`];;
searcht 5 [`?(x:real^A). p`;`~(x = (a:real^A))`];;
searcht 5 [`DISJOINT`;`collinear`];;
searcht 5 [`{} SUBSET X`];;
searcht 5 [`aff_ge S {}`];;
searcht 5 [`affine`;`aff`];;
searcht 5 [def "aff"];;
searcht 15 [def "aff"];;
searcht 35[`affine_dependent`];;
searcht 5 [`affine hull s`;`aff_ge`];;
searcht 5 [`collinear`;`affine hull {a,b}`];;
searcht 5 [`collinear`;`aff_dim`];;
searcht 5 [`aff_dim {a,b}`];;
searcht 5 [`aff_dim {a,b,c} = &2`];;
searcht 5 [`affine hull s = affine hull t`;`aff_dim`];;
searcht 5 [`x IN s ==> x IN f hull s`];;
searcht 5 [`coplanar`;`aff_dim`];;
searcht 5 [`aff_dim (affine hull s)`];;
searcht 5 [`affine hull (affine hull x)`];;
searcht 5 [`affine hull {a,b,c}`;`%`];;
searcht 5 [`%`;`conv`];;
searcht 5 [`affine hull {a,b}`;`%`];;
searcht 5 [`%`;`aff_ge {a,b} {c,d}`];;
searcht 5 [`conv`;`convex hull s`];;
searcht 5 [def "conv"];;
searcht 5 [`aff_ge`;`conv`];;
searcht 50 [`conv`];;
bb 5;;
searcht 5 [`a % v = a % w`];;
searcht 5 [`a % (b + c)`];;
searcht 5 [`a % b % c`];;
searcht 5 [`collinear`;`DISJOINT`];;
searcht 5 [`affine_dependent`;`~`;`{a,b,c,d}`];;
AFFINE_DEPENDENT_EXPLICIT;;
searcht 8 [`vsum`;`INSERT`];;
searcht 5 [`CARD`;`DISJOINT`];;
searcht 5 [name "CARD4_IMP_DISTINCT"];;
VSUM_CLAUSES;;
CONJUNCT1 VSUM_CLAUSES;;

(* Dec 27, 2012 *)
searc 'a' 0 [`coplanar {a,b,c}`];;
searcht 5 [`s SUBSET f hull s`];;
searc 'b' 1 [`DISJOINT`;`coplanar`];;
searcht 5 [`%`;`aff_gt {a,b} {p}`];;
searcht 5 [`%`;`aff_ge {a,b} {p}`];;
searcht 5 [`DISJOINT`;`collinear`];;
searcht 5 [`%`;`affine hull {a,b,p}`];;
searcht 5 [`affine_dependent`;`coplanar`];;
searcht 5 [`coplanar`;`CARD`];;
searcht 5 [`CARD {a,b,c,d}`];;
searcht 5 [`coplanar`;`coplanar_alt`];;
searcht 5 [`dimindex (:3)`];;
searcht 15 [`affine_dependent {a,b,c,d}`];;
searcht 5 [`aff_dim`;`coplanar`];;
searc 'c' 0 [`(&0 <= w * t3) <=> p`];;
searcht 5 [`affine hull {a,b}`;`%`];;
searcht 15 [`mxi`];;
searcht 15 [`convex hull {a,b,c}`];;
searcht 5 [`&0 <= inv x `];;
searcht 5 [`abs(a * b)`];;
searcht 5 [`%`;`aff_gt {a,b,c} {p}`];;
searcht 5 [`DISJOINT`;`coplanar`];;
searcht 5 [`x INTER x`];;
searcht 5 [`convex hull {a,b,c,d}`;`%`];;
searcht 5 [`a/b < c`];;
searcht 15 [`coplanar {a,b,c,d}`];;
searcht 5 [`chi_msb [a;b;c]`;`%`];;
searcht 5 [`aff_gt`;`affine hull s`];;
searcht 5 [`x = [a;b;c]`;`3`];;
searcht 5 [`coplanar {a,b,c}`];;
searcht 5 [`coplanar (affine hull s)`];;
searcht 5 [`coplanar`;`SUBSET`];;
searcht 5 [`convex`;`mcell_set`];;
searc 'd' 1 [`convex`;`rcone_ge`];;
searcht 5 [`&0 <= hl ul`];;
searcht 5 [`convex`;`aff_ge`];;
searc 'e' 0 [`hl [a;b]`];;
searcht 5 [`truncate_simplex 1`];;
searcht 5 [`convex {}`];;
searc 'f' 0 [`convex (convex hull s)`];;
searcht 50 [`edgeX`];;
searcht 50 [`VX`];;
searcht 5 [`rogers`;`voronoi_closed`];;
searcht 5 [`rogers`;`VX`];;
searcht 5 [`VX`;`mcell`];;
searcht 15 [`voronoi_closed V v INTER x`];;
searcht 15 [`voronoi_set V {u,v}`];;
searcht 5 [`dist(x,w) = &0`];;
searcht 5 [`&0 <= dist (x,y)`];;
Hdtfnfz.HDTFNFZ;;
searcht 5 [`cell_params`];;
searcht 5 [`(\(a,b). f a b) t`];;
searcht 5 [`NULLSET`;`vol`];;
searcht 15 [`cell_params`];;
searcht 15 [`barV`;`HD vl IN V`];;
searcht 5 [`barV`;`set_of_list ul SUBSET V`];;
searcht 5 [`sqrt(&4)`];;
searcht 50 [`NULLSET`];;
COPLANAR_IMP_NEGLIGIBLE;;
CHI_MSB_COPLANAR;;
searcht 5 [`aff_dim`;`barV`];;
searcht 5 [`convex hull X SUBSET convex hull Y`];;
searcht 15 [`edgeX`];;
searcht 5 [`convex hull X = X`];;
CC_CELL34;;
ZASUVOR;;
searcht 5 [`x SUBSET f hull y`];;
searcht 5 [`p IN aff_gt s {p}`];;
searcht 5 [`chi_msb`;`coplanar`];;
searcht 10 [`coplanar {a,b,c,d}`;`%`];;
searcht 15 [`affine_dependent`;`{a,b,c,(d:real^3)}`];;
searcht 5 [`affine_dependent`;`aff_dim`];;
searcht 5 [`aff_dim x <= aff_dim y`;`x = y`];;
searcht 5 [`aff_dim (:real^A)`];;
searcht 5 [`aff_dim`;`coplanar`];;
searcht 5 [`affine hull {a,b,c,d}`];;
searcht 15 [`(&0 < z) ==> u`;`&0 < x * y`];;
searcht 5 [`aff_gt {a,b,c} {d}`;`%`];;
searcht 5 [`collinear`;`DISJOINT`];;
searcht 5 [`x IN convex hull (x INSERT y)`];;

(* Dec 29, 2012 *)
searcht 5 [`%`;`aff_gt {a,b} {c}`];;
searcht 5 [name "chi_msb_add"];;
Fan.th3a;;

GBEWYFX;;
searcht 5 [`coplanar`;`chi_msb`];;
bb 15;;
searcht 5 [name "AJRI"];;
searcht 5 [`coplanar`;`NULLSET`];;
searcht 5 [`convex hull x SUBSET convex hull y`];;
HULL_INC;;
searcht 15 [`a hull X = X`];;
searcht 5 [`convex`;`mcell`];;
searcht 5 [`barV`;`cc_uh`];;
searcht 5 [`convex hull s SUBSET convex hull y`];;
searcht 5 [`f hull (f hull s)`];;
HULL_INC;;
searcht 5 [`NULLSET`;`SUBSET`];;
searcht 5 [`NULLSET`;`vol`];;
searcht 5 [`SUBSET`;`measure`];;
searcht 25 [ `negligible`;`SUBSET`];;
searcht 5 [`X INTER Y = Y INTER X`];;
cc_cell;;
bb 4;;
searcht 5 [`NULLSET`;`coplanar`];;
searcht 5 [`coplanar`;`vol`];;
searcht 5 [name "ZWV"];;
searcht 5 [name "RVF"];;
Pack_defs.mcell_set;;
searcht 5 [`mcell_set`;`cc_cell`];;
searcht 30 [`VX`];;
searcht 15 [`convex hull j`;`rogers`];;
searcht 15 [`rogers`;`voronoi_closed`];;
searcht 15 [`EL`;`left_action_list`];;
Packing3.EL_LEFT_ACTION_LIST;;
searcht 15 [`set_of_list`;`left_action_list`];;
Packing3.SET_OF_LIST_LEFT_ACTION_LIST;;
searcht 5 [`set_of_list [a;b;c;d]`];;
set_of_list4;;
searcht 5 [`CARD {a,b,c,d}`];;
searcht 5 [name "WQP"];;
searcht 5 [name "AJR"];;
searcht 15 [`convex hull s = convex hull t`];;
EXTREME_POINTS_OF_CONVEX_HULL_EQ;;
Packing3.CONVEX_HULL_EQ_EQ_SET_EQ;;
Rogers.WQPRRD4;;
searcht 5 [`coplanar (convex hull s)`];;
searcht 5 [`coplanar`;`convex hull s`];;
searcht 5 [`coplanar`;`affine hull s`];;
searcht 5 [`coplanar s`;`s SUBSET t`];;
searcht 5 [`convex hull s SUBSET affine hull s`];;
MCELL3_NONPLANAR;;
searcht 5 [`truncate_simplex 2 (cc_uh V ul)`];;
cc_uh;;
searcht 5 [`X INTER X`];;

(* Dec 30, 2012 *)
CC_CELL34;;
searcht 5 [`affine_dependent`;`coplanar`];;
searcht 5 [`affine_dependent`;`aff_dim`];;
searcht 5 [`CARD`;`coplanar`];;
searcht 5 [`CARD {a,b,c,d}`];;
searcht 5 [`coplanar`;`coplanar_alt`];;
searcht 5 [`coplanar`;`affine hull s`];;
searcht 5 [`coplanar`;`SUBSET`];;
ZWVCBMN;;
searcht 5 [`FINITE`;`INSERT`];;
searcht 5 [`aff_dim {a,b,c,d}`];;
searcht 5 [`coplanar`;`aff_dim`];;
searcht 5 [`aff_dim`;`dimindex`];;
searcht 5 [`cc_uh`;`barV`];;
searcht 5 [`set_of_length`;`INSERT`];;
EL_CC_UH;;
K4_CHI_MSB_POS;;
set_of_list2;;
searcht 5 [`set_of_list`;`truncate_simplex 1`];;
searcht 5 [`EL`;`truncate_simplex`];;
searcht 20 [`LENGTH`;`truncate_simplex`];;
LENGTH3;;
LENGTH4_SET2;;
searcht 5 [`CARD`;`4`;`cc_uh`];;
set_of_list4_explicit;;
LIST_OF_CC_UH;;
searcht 5 [`truncate_simplex 2 (cc_uh V ul)`];;
searcht 5[`sqrt(&4)`];;
searcht 5 [`set_of_list`;`barV`];;
searcht 5 [`HD ul`;`packing V`];;
Packing3.BARV_SUBSET;;
Packing3.BARV_IMP_HD_IN_SET_OF_LIST;;
searcht 5 [`x IN Y /\ Y SUBSET Z`];;
searcht 5 [`barV`;`set_of_list ul SUBSET V`];;
searcht 5 [`A SUBSET B /\ B SUBSET A`];;
searcht 15 [`cc_ke V ul = 4`];;
searcht 5 [name "CARD4"];;
searcht 5 [`CARD {EL 0 ul,b,c,d}`];;
searcht 5 [`{a,b} = {c,d}`];;
bb 6;;
searcht 5 [`{a,b,c} = {a',b',c'}`];;
searcht 5 [`(A =B) <=> (B  = A)`];;
searcht 5 [``;`barV`];;
searcht 50 [`left_action_list`];;
searcht 5 [`CARD`;`DIFF`];;

(* Dec 31, 2012 *)
Pack_defs.HL;;
searcht 5 [name "RDW"];;
Rvfxzbu.RVFXZBU;;
searcht 5 [`TSKAJXY_statement`];;

Marchal_cells_3.LEFT_ACTION_LIST_3_EXISTS;;
Marchal_cells_3.LEFT_ACTION_LIST_2_EXISTS;;
Rvfxzbu.RVFXZBU;;
searcht 5 [`left_action_list`;`mcell`];;
CARD_DIFF;;
searcht 5 [`left_action_list`;`barV`];;
Rogers.YIFVQDV_1;;
searcht 5 [`cell_cluster_estimate`];;
searcht 50 [`left_action_list`;`mcell`];;
Marchal_cells_2_new.LEFT_ACTION_LIST_PROPERTIES;;

searcht 5 [`convex hull s`;`mcell3`];;
searcht 5 [name "REUH"];;
searcht 5 [`{a,b} = {c,d}`];;
searcht 5 [`VX`];;
searcht 5 [`VX`;`set_of_list`];;
searcht 5 [`cell_params`];;
searcht 15 [`mcell4 V ul = mcell4 V vl`];;
searcht 25 [` x IN mcell i V vl`];;
searcht 5 [`x IN mcell2 V vl`];;
Urrphbz1.MCELL_2_PROPERTIES_lemma1;;
Rogers.XYOFCGX;;
searcht 5 [name "CGX"];;
searcht 5 [`VX`;`mcell2`];;
searcht 5 [name "AJR"];;
searcht 5 [`VX V X SUBSET y`];;
searcht 5 [`midpoint`;`circumcenter {a,b}`];;
searcht 5 [`X SUBSET set_of_list ul`];;
searcht 5 [`set_of_list V SUBSET Q`];;
searcht 5 [`affine_dependent {a,b}`];;
searcht 15 [`set_of_list (truncate_simplex k ul)`];;
searcht 15 [`truncate_simplex 1 ul`];;
searcht 5 [`stem x = {a,b}`];;
searcht 5 [`set_of_list [a;b]`];;
searcht 5 [`radV`;`midpoint`];;
searcht 5 [`radV`;`dist`];;
searcht 5 [`hl [a;b]`];;
searcht 5 [`midpoint`;`dist`];;
searcht 50 [`VX`];;
searcht 5 [`gammaX`;`gamma4fgcy`];;
searcht 15 [`permutes`;`hl`];;
Pack_defs.mcell3;;
Pack_defs.VX;;
searcht 15 [`(\(k,u). f k u) (a,b)`];;
searcht 15 [name "BETA_PAIR"];;
searcht 15 [`(FST x, SND x)`];;
Pack_defs.cell_params;;
searcht 5 [`mcell4`;`mcell 4`];;
searcht 5 [`x INTER x`];;
searcht 25 [`(@)`];;
searcht 5 [`(?x. P x) ==> (P (@x. P x))`];;
searcht 5 [`(A = B) ==> A SUBSET B`];;
searcht 5 [`truncate_simplex k X = X`];;

(* Jan 1, 2013 *)  
searc 'a' 0  [`set_of_list`;`FINITE`];;
searc 'b' 1 [`CARD`;`LENGTH`;`set_of_list`];;
searcht 5 [`CARD B <= CARD A`;`A SUBSET B`;`FINITE`];;
searcht 5 [`mcell4`;`convex hull s`];;
searcht 5 [`set_of_list ul SUBSET V`;`barV`];;
searcht 5 [`set_of_set ul SUBSET mcell4 V ul`];;
searcht 5 [`s SUBSET f hull s`];;
searcht 5 [`neglibible`;`{}`];;
searcht 5 [`VX V X`;`INTER`];;
searcht 5 [`CARD`;`barV`];;
searc 'c' 2 [`{a,b} = {c,d}`];;
 Ajripqn.AJRIPQN;;
searc 'd' 0 [`set_of_list`;`truncate_simplex`;`SUBSET`];;
searc 'e' 5 [`LENGTH`;`truncate_simplex`];;
Geomdetail.CARD_SET_OF_LIST_LE;;
Marchal_cells_3.BARV_CARD_LEMMA;;
searcht 5 [`INSERT`;`set_of_list`];;
searcht 5 [`truncate_simplex 2 (CONS x y)`];;
searcht 5 [`set_of_list (CONS x y)`];;
CARD_CLAUSES;;
searcht 5 [`CARD {a,b,c}`];;
searcht 5 [`{ x| F}`];;
GSPEC_EMPTY;;
conj_list;;
searcht 5 [`\x. F`];;
CONJ;;
searcht 5 [`sum {}`];;
searcht 5 [`edgeX`;`mcell2`];;
searcht 15 [`truncate_simplex 2 ul`];;
Marchal_cells.TRUNCATE_SIMPLEX_EXPLICIT_2;;
searcht 5 [`LENGTH (truncate_simplex k ul)`];;
searcht 5 [`EL k (truncate_simplex r ul)`];;
searcht 5 [`edgeX`;`mcell`];;
searcht 5 [`CARD {u,v}`];;
set_of_list3;;
searcht 15 [`truncate_simplex k ul`];;
set_of_list3;;
searcht 5 [`CARD (X UNION Y)`];;
searcht 5 [`CARD {}`];;
searcht 5 [`CARD x  <= CARD y`;`SUBSET`];;
searcht 5 [name "HDT"];;
searcht 5 [`set_of_list ul = A INTER B`];;
set_of_list4;;
searcht 5 [`SUBSET`;`CARD`;`a=b`];;
searcht 5 [`CARD`;`edgeX`];;
searcht 5[ `CARD {a,b} <= 2`];;
searcht 5 [`edgeX`;`mcell4`];;
searcht 5 [`CARD`;`barV`];;
searcht 5 [`CARD`;`DIFF`];;
searcht;;
HDTFNFZ_ALT;;
Hdtfnfz.HDTFNFZ;;
searcht 15 [`BIJ`;`f(g(x))`];;
searcht 15 [`sum X f = sum Y f`;`f x = &0`];;
searcht 15 [`sum X f = sum X g`];;
searcht 5 [`sum s (\x. -- f x)`];;
searcht 5 [name"BIJ_SUM"];;

(* Jan 2, 2013 *)
searcht 5 [`abs x <= abs a + abs b`];;
searcht 5 [`abs (x - y) <= abs x + abs y`];;
searcht 5 [`abs (x/ y) = abs x / abs y`];;
searcht 5 [`x / y <= z / y`];;
searcht 5 [`x/y = x * inv y`];;
searcht 15 [`x * c  <= y * c`];;
searcht 5 [`x pow 2 <= y pow 2`];;
searcht 5 [`abs (x pow n)`];; 
searcht 5 [name "HL_2"];;
searcht 5 [`dihu2`];;
searcht 5 [`dihu3`];;
searcht 15 [`gammaX`];;
searcht 5 [`permutes`;`barV`;`mcell`];;
searcht 5 [`mcell2`;`edgeX`];;
searcht 5 [`{a,b} = {c,d}`];;
searcht 5 [`permutes`;`INSERT`];;
Marchal_cells_3.LEFT_ACTION_LIST_3_EXISTS;;
Marchal_cells_3.LEFT_ACTION_LIST_2_EXISTS;;
searcht 5 [`p permutes 0..1`;`left_action_list`];;
Qzksykg.TWO_REARRANGEMENT_LEMMA;;
Rvfxzbu.RVFXZBU;;
Qzksykg.QZKSYKG1;;
searcht 5 [`edgeX`;`{}`];;
searcht 5 [`negligible {}`];;
RIJRIED;;
searcht 5 [`edgeX`;`mcell3`];;
searcht 5 [`CARD {a,b,c}`];;
searcht 5 [`CARD`;`DIFF`];;
searcht 5 [`CARD {a,b}`];;
searcht 5 [`CARD X = 0`];;
searcht 5 [`CARD X = 1`];;
searcht 5 [`set_of_list (truncate_simplex 2 ul) = x INSERT  y`];;
searcht 5 [`barV`;`CARD`];;
LENGTH4;;
set_of_list4;;
searcht 15 [`X DIFF e`;`edgeX`];;
searcht 5 [`CARD {x,y} = 2`];;
searcht 15 [`CARD x = 2`];;
searcht 5 [`edgeX`;`mcell k`];;
searcht 5 [`hl a < hl b`];;
searcht 5 [`truncate_simplex k ul = ul`];;
searcht 5 [`truncate_simplex`;`barV`];;
searcht 5 [name "CFFO"];;
searcht 5 [`EL k (truncate_simplex r ul)`];;
searcht 5 [`stem`;`EL`];;
searcht 5 [`cc_A0`;`EL`];;
searcht 5 [`~(x = {})`;`?`];;
searcht 5 [`V INTER X  = set_of_list a`];;
searcht 5 [`x IN aff_gt u {x}`];;
searcht 5 [`leaf V (CONS x y)`];;
searcht 15 [`truncate_simplex k vl`;`CONS`];;
LENGTH3;;
searcht 5 [`LENGTH (truncate_simplex k vl)`];;
Marchal_cells.TRUNCATE_SIMPLEX_EXPLICIT_2;;
STEM_OF_LEAF;;
STEM_OF_LEAF;;
searcht 5 [name "CFF"];;
searcht 5 [`edgeX`;`cc_cell`];;
searcht 5 [`mcell3`;`edgeX`];;
searcht 5 [`cc_cell`;`NULLSET`];;
searcht 5 [`barV V 3`;`cc_uh`];;
searcht 5 [`EL`;`cc_uh`];;
searcht 5 [`set_of_list (truncate_simplex k (ul))`];;
searcht 15 [`EL k (cc_uh V ul)`];;
searcht 15 [`LENGTH (cc_uh V ul)`];;
searcht 5 [`CARD`;`barV`];;
cc_uh;;
Bump.SET_OF_LIST_TRUNCATE_2;;
LENGTH3;;
set_of_list3;;
searcht 5 [`CARD`;`4`;`~(a  =b)`];;
searcht 5 [`truncate_simplex k ul SUBSET ul`];;
searcht 5 [`mcell_set`;`cc_cell`];;
searcht 5 [`cc_cell`;`cc_A0`];;
searcht 5 [name "FUE"];;
searcht 5 [`CONS x y = CONS x' y'`];;
ZASUVOR;;
 FUEIMOV_4;;
searcht 5 [`cc_uh V ul = CONS x y`];;
searcht 5 [`{a,b} = {c,d}`];;
searcht 5 [`cc_uh V ul = CONS x y`];;
searcht 5 [`coplanar`;`cc_uh`];;
searcht 5 [`coplanar`;`affine hull {a,b,c}`];;
searcht 5 [`aff_gt`;`affine hull s`];;
searcht 15 [`s IN aff_gt u v`];;
SUBSET_HULL;;
searcht 15 [`s SUBSET f hull s`];;
searcht 15 [`s IN S /\ S SUBSET U`];;

(* Jan 3, 2013 *)
searc 'a' 1 [`coplanar`;`affine hull s`];;
searc 'b' 0 [`aff_dim`;`collinear`];;
searcht 5 [`aff_dim`;`INSERT`];;
searcht 5 [`aff_dim x <= aff_dim y`;`x = y`];;
searcht 5 [`affine hull (affine hull h)`];;
searcht 5 [`s SUBSET f hull s`];;
searcht 5 [`aff_dim (affine hull s)`];;
coplanar;;
searc 'c' 1 [`connected`;`segment`];;
searcht 25 [`connected`;`open`];;
searcht 5 [`a IN segment [a,b]`];;
searcht 5 [`wedge_ge`;`wedge`];;
searcht 5 [`wedge_ge`];;
searcht 35 [`wedge`];;

Local_lemmas1.AFF_GE_WEDGE_DISJOINTION;;
OPEN_WEDGE;;
Local_lemmas1.EDGE_NOT_INTER_WITH_WEDGE;;
WEDGE_ALT;;
searcht 5 [name "WEDGE_GE_ALMOST"];;
WEDGE_GE_ALMOST_DISJOINT ;; (* not loaded in work_in_progress_merge.hl *)
COPLANAR_UNION;;
CONNECTED_SEGMENT_NOT_COVERED;;
searcht NAN [`closed`;`wedge_ge`];;
searcht 5 [`mcell`;`convex`];;

searcht 5 [def "wedge_ge"];;
searcht 55 [def "azim"];;
Sphere.AZIM;;
azim;;
searc 'e' 0 [`wedge_ge`;`aff_ge`];;
searcht NAN [`wedge_ge`;name "split"];;
searcht 5 [`wedge`;`UNION`];;
searc 'd' 1 [`closed`;`aff_ge`];;
searcht NAN [`?`;`azim a b c d = e`];;
searcht 5 [`wedge_ge`;`azim a b c d = pi`];;
searcht 5 [`convex`;`closed`];;
searc 'f' [`polyhedron`;`closed`];;
searcht 5 [`~(x = {})`;`?`];;
searcht 5 [`wedge_ge`;`azim`];;
searcht 50 [`collinear`;`azim`];;
searcht 5 [`azim a b c d + azim a b d e`];;
searc 'g' 0 [`collinear`;`azim`;omit `~`];;
searcht 5 [`&0 <= azim a b c d`];;
searcht 5 [`open`;`closed`];;
searcht 5 [`azim`;`-- x`];;
searcht 5 [`azim`;`pi`];;
searcht 25 [`conv0`];;
searcht 5 [`aff_ge {} {a,b}`];;
searcht 5 [`aff_sgn`;`aff_gt`];;
searcht 15 [`aff_gt`;`%`];;
searcht 15[ `azim`;`&2 * pi`];;
searcht 5 [`closed`;` a dot b`];;
searcht 5 [`aff_ge`;`closed`];;
searcht 5 [`closed (A UNION B)`];;
searcht 5 [`NULLSET`;`edgeX`];;
searcht 5 [`edgeX V X = {}`];;
searcht 5 [`convex`;`mcell`];;
searcht 5 [`coplanar`;`SUBSET`];;
searcht 5 [`coplanar`;`NULLSET`];;
WEDGE_WEDGE_GE;;
searcht 5 [`NULLSET `;`UNION`];;
searcht 5 [`NULLSET`;`aff_ge`];;
Conforming.NEGLIGIBLE_AFF_GE_2_1;;
searcht 5 [`collinear`;`leaf`];;
searcht 5 [`set_of_list (CONS x y)`];;
searcht 5 [`SUBSET`;`negligible`];;
searcht 5 [`coplanar`;`SUBSET`];;
searcht 5 [`azim`;`aff_ge`];;
AZIM_DEGENERATE;;
searcht 5 [`wedge_ge`;`azim`];;

(* Jan 4, 2012 *)
searcht 5 [`((!x. p x) /\ (!x. q x)) <=> R`];;
Local_lemmas.AZIM_EQ_0_GE_ALT2;;
searc 'r' 0 [`&0 <= azim a b c d`];;
searcht 5 [`azim a b c d + azim a b d e`];;
searcht 5 [`collinear`;`affine hull {a,b}`];;
help_grep "DISJ";;
searcht 5 [`collinear`;`~(a = b)`];;
searcht 5 [`affine hull s SUBSET aff_ge a b`];;
searcht 5 [`DISJOINT`;`collinear`];;
bb 15;;
COPLANAR_INSERT;;
searcht 5 [name "COPLANAR_INSERT"];;
searcht 5 [`open`;`wedge`];;
searcht 5 [`closed`;`wedge_ge`];;
searcht 5 [`open`;`closed`];;
searcht 5 [`wedge a b c d SUBSET wedge_ge a b c d`];;
searcht 5 [`wedge`;`wedge_ge`];;
searcht 5 [`wedge_ge`];;
searcht 5 [`segment`;`convex`];;
searcht 15 [`aff_ge`;`aff_gt`];;
 DISJOINT {u, v} {w}
Collect_geom.IN_AFF_GE_INTERPRET_TO_AFF_GT_AND_AFF;;
searcht 5 [`segment`;`%`];;
searcht 5 [`wedge`;`wedge_ge`];;
searcht 5 [name "CFF"];;
searcht 20 [`coplanar {a,b,c,d}`];;
searcht 5 [`coplanar`;`INTER`];;
searcht 5 [`s SUBSET f hull s`];;
searcht 15 [`aff_dim s <= &2`];;
Print_types.print_thm_types Njiutiu.AFF_DEPENDENT_AFF_DIM_4;;
searcht 5 [`coplanar`;`aff_dim`];;
searcht 5 [`coplanar`;`affine_dependent`];;
searcht 15 [`affine_dependent  {a,b,c,d}`];;
searcht 15 [`affine_dependent`;`sum`];;
searcht 15 [`sum (x INSERT y)`];;
searcht 5 [`affine hull {a,b}`;`%`];;
searcht 5 [`affine hull {a,b,c}`;`%`];;
searcht 5 [`a % v = a % u`];;
searcht 5 [`a % (b + c)`];;
searcht 5 [`a % (b % c)`];;
searcht 5 [`aff_ge`;`aff_gt`];;
searcht 5 [`collinear`;`DISJOINT`];;
searcht 5 [`stem [a;b;c]`];;
searcht 5 [`stem`];;
CELL_NN;;
searcht 5 [`set_of_list [a;b;c]`];;
searcht 5 [`aff_gt x y SUBSET affine hull u`];;
searcht 5 [`chi_msb ul`;`%`];;
searcht 5 [`affine hull s`;`chi_msb`];;
CHI_MSB_IMP_COPLANAR;;
searcht 15 [`coplanar`;`affine hull s`];;
searcht 5 [`aff_dim {a,b,c} <= &2`];;
searcht 5 [`aff_dim`;`collinear`];;
searcht 5 [`collinear`;`leaf`];;
bb 20;;
chi_msb_coplanar;;
COPLANAR_INSERT;;
searcht 5 [`set_of_list ul = {a,b,c}`];;
searcht 5 [`azim a b c d = &0`];;
searcht 50 [`wedge_ge`;`wedge`;];;
searcht 5 [`collinear`;`leaf`];;
searcht 5 [`wedge_ge`;`(:real^3)`];;
searcht 5 [`aff_ge`;`azim a b c d = &0`];;
searcht 5 [`SUBSET`;`negligible`];;
searcht 5 [`wedge_ge`];;
searcht 15 [`collinear`;`affine hull s`];;
searcht 15 [`affine hull s SUBSET aff_ge u v`];;
searcht 5 [`DISJOINT`;`collinear`];;
Local_lemmas.AZIM_EQ_0_GE_ALT2;;
AZIM_BASE_SHIFT_LE;;
searcht 5 [`azim a b c c`];;
 Local_lemmas.AZIM_EQ_0_GE_ALT2;;
searcht 5 [`azim a b c d`;`&2 * pi - azim a b c d`];;
Rogers.AZIM_COMPL_EXT;;

(* Jan 5, 2012 *)
searcht 5 [`wedge_ge`];;
AZIM_DEGENERATE;;
searcht 5 [`&0 <= azim a b c d `];;
searcht 5 [`azim a b c d + azim a b d e`];;
searcht 5 [`azim a b c d = &0`];;
searcht 15 [`wedge_ge`;`wedge`;`aff_ge`];;

(* link in Ky's work *)
searcht 5 [name "REU"];;
searcht 15 [name "Tsk"];;
Tskajxy_lemmas.SOL_SOL_Y_EXPLICIT;;
Tskajxy_lemmas.DIHX_DIH_Y_lemma;;
searcht 5 [name "REU"];;
Reuhady.REUHADY1;;
searcht 5 [`hl x < hl y`];;
searcht 5 [`truncate_simplex k (CONS x y)`];;
searcht 5 [`set_of_list [a;b]`];;
searcht 5 [`CONS a b = CONS a' b'`];;
searcht 5 [`collinear`;`leaf`];;
searcht 5 [`set_of_list [u0;u1;u2]`];;
searcht 5 [`wedge_ge`;`INTER`;`aff_ge`];;
searcht 5 [`collinear`;`~(a = b)`];;
searcht 5 [`set_of_list x SUBSET V`;`barV`];;
searcht 5 [`azim`;`aff_ge`];;
Leaf_cell.FCHKUGT;;
TL;;
searcht 5 [`aff_gt s t = aff_gt s t'`];;
searcht 15 [`aff_gt`;`aff_ge`];;
searcht 5 [`DISJOINT`;`collinear`];;
searcht 5 [`affine hull {a,b}`;`collinear`];;
Pack_defs.mcell_set;;
Print_types.print_thm_types;; 
Tskajxy_lemmas.gammaX_gamma3f;;
bb 5;;
searcht 5 [`radV`;`eta_y`];;
Sphere.packing;;
searcht 5 [`dist(a,b) + dist(b,c)`];;
bb 4;;
searcht 5 [`coplanar`;`collinear`];;
searcht 5 [`set_of_list (truncate_simplex 2 ul)`];;
searcht 5 [`{}`;`negligible`];;
print_proof();;
Print_types.print_goal_types();;
print_goalstack;;
print_proof();;


(* Jan 6, 2013 TSKAJXY 1 and 2 cells *)
searcht 15 [`mcell1`];;
searc 'a' 1 [`mcell1`;`HD`];;
searc 'b' 0 [`mcell`;`HD`];;
searc 'c' 2 [`edgeX`;`mcell`];;
searcht 10 [`gammaX`];;
searcht 5 [`gammaX`;`{}`];;
searcht 5 [`total_solid`];;
searcht 5 [`sum {x}`];;
searc 'h' 1 [`a SUBSET b /\ b SUBSET a`];;
searcht 25 [`VX V`];;
searc 'd' 0 [`{}`;`negligible`];; 
searc 'e' 0 [`VX V X = {}`];;
searcht 5 [`edgeX V X = {}`];;
searcht 5 [`NULLSET`;`vol`];;
searc 'f' 0 [`set_of_list ul SUBSET V`;`barV`];;
searcht 15 [`set_of_list ul = x INSERT y `];;
searcht 5 [`mcell2`;`VX`];;
searcht 5 [`mcell2`;`edgeX`];;
searcht 15 [`mcell`;`barV`;`V INTER X`];;
searcht 15 [`set_of_list (truncate_simplex 1 ul)`];;
update_database();;
Sphere.BARV;;
searcht 4 [`CARD`;`barV`];;
searc 'g' 1 [`CARD X = 4`;`~(x = y)`];;
searcht 5 [`set_of_list ul`;`4`];;
searcht 5 [`sum {a,b}`];;
searc 'i' 0 [name "BETA_PAIR"];;
searcht 5 [`set_of_list [a;b]`];;
searcht 5 [`dihX`];;
searcht 5 [`mcell2`;`VX`];;
searcht 15 [`radial`];;
searcht 15 [`radial_norm`];;
searcht 15 [`eventually_radial`];;
searcht 15 [def "eventually"];;
searc 'j' 0 [`radial_norm`;`DIFF`];;
searcht 5 [`radial_norm`;`INTER`];;
searcht 5 [def "rcone"];;
searcht 5 [`radial_norm`;`rcone_lt`];;
searcht 500 [name "Vol1.";`sol`];;
searcht 5 [`solid_triangle`];;
searcht 15 [`conic_cap`];;
searc 'k' 0 [`eventually_radial`;`mcell`];;
Urrphbz2.URRPHBZ2;;
Urrphbz2.EVENTUALLY_RADIAL_RCONE_GE_ABC_A;;
searcht 5 [name "Count";`rcone_gt`];;
Sphere.rcone_gt;;
searcht 5 [`rcone_ge`;`dot`];;
searcht 5 [`&0 < dist (p,q)`];;
Print_types.print_thm_types DIST_NZ;;
DIST_NZ;;
type_of `radial_norm`;;
searcht 15 [`convex hull s`;`INTERS`];;
searcht 15 [`aff_ge s t INTER aff_ge s' t'`];;
Cfyxfty.inter_aff_ge_3_1_is_aff_ge_1_3;; (* ! *)
Collect_geom2.ARIKWRQ;;
searcht 5 [`&0 <= dist(a,b)`];;
searcht 5 [`dist (p,x ) <= dist (p,y)`;`dot`];;
searcht 5 [`dist (x,y) pow 2`;`dot`];;
searcht 5 [`norm = r`;`NULLSET`];;
searcht 5 [name "OWCZKJR"];;
type_of `radial_norm`;;
radial_norm;;
searcht 5 [`normball`;`dist (y,x) < u`];;
searcht 5 [def "normball"];;
searcht 5 [`sphere`;`NULLSET`];;
searcht 5 [`eventually_radial_norm`;`eventually_radial`];;
List.length(searcht 500 [`radial_norm`]);; (* 37 *)
hd (definitions());;
searchl (map (fun t -> ("",t)) (definitions())) [`radial_norm`];;
type_of `eventually_radial_norm`;;
Vol1.eventually_radial_norm;;
searcht 15 [def "^sol$"];;
searcht 5 [`sol`;`measurable`;`INTER`;`normball`];;
Vol1.pre_def_4_3b;;
open Vol1;;
radial_normball;;
Vol1.sol;;
searcht 50 [`rect`];;
searcht 50 [`volume_prop_fix`];;
searcht 50 [`primitive`];;
searcht 50 [`ellipsoid`];;
searcht 50 [`hinhcau`];;
Vol1.hinhcau;;
searcht 50 [`map0`];;
DISJOINT;;
INTER_ACI;;
nonempty_cube;;
Vol1.sol_spec;;

(* Jan 7, 2013 *)
searcht 5 [`vol X = vol Y`];;
searcht 5 [`measurable`;`mcell`];;
volume_props;;
searcht 5 [`sphere`];;
searcht 5 [`measurable (X INTER Y)`];;
searcht 5 [`measurable (ball (x,r))`];;
searcht 1 [`SDIFF`;`vol`];;
searcht 5 [`NULLSET`;`UNION`];;
searcht 5 [`negligible {}`];;
print_proof();;
type_of `sphere`;;
searcht 5 [`sphere`];;
NEGLIGIBLE_SPHERE;;
searc 'a' 0 [`negligible`;`SUBSET`];;
searcht NAN [`sol`;`INTER`;`ball`];;
searcht 50 [`sol`];;
searcht 5 [`mcell`;`eventually_radial`];;
searcht 5 [`set_of_list ul SUBSET V`;`barV`];;
searcht 20 [`set_of_list ul = x INSERT y`];;
searc 'b' 0 [`radial_norm`;`x < t`;`==>`];;
searcht 5 [`radial_norm`;`radial`];;
Vol1.sol;;
searcht 5 [`(X INTER Y) INTER Z`;`X INTER Y INTER Z`];;
Cfyxfty.inter_aff_ge_3_1_is_aff_ge_1_3;; (* ! *)
Collect_geom2.ARIKWRQ;;
searcht 5 [`aff_ge`;`convex hull s`];;
searcht 25 [def "^conv"];;
searcht 5 [`conv`;`affsign`];;
searcht 5 [`aff_ge`;`affsign`];;
searcht 5 [`coplanar`;`coplanar_alt`];;
searcht 5 [name "MCELL2_VX_PROPS"];;
searcht 5 [`coplanar`;`CARD`];;
searcht 5 [`CARD {a,b,c,d}`;`~(a = b)`];;
print_proof();;
searcht 50 [`rogers`];;
searcht 5 [`aff `;`rogers`];;
searcht 5 [`omega_list_n`;`aff`];;
searcht 50 [`aff`];;

searcht 5 [name "COMPATIBLE_BW_TWO_LEMMAS2"];;
type_of `hypermap`;;
type_of `HYP`;;
 searcht 5 [`coplanar`;`coplanar_alt`];;

(* Jan 18, 2013 *)
(* TSK_in_progress *)
searcht 5 [`norm`;`dist`];;
searcht 5 [`norm (a % b)`];;
searcht 5 [`aff_ge {a} {b,c,d}`;`%`];;
searcht 5 [`DISJOINT`;`coplanar`];;
searcht 15 [`convex`;`%`;`INSERT`;`{a,b,c,d}`];;
searcht 5 [def "bis"];;
searcht 100 [`omega_list_n`];;
searcht 100 [`omega_list`];;
searcht 5 [`coplanar`;`omega_list_n`];;
searcht 5 [`omega_list_n`;`aff_dim`];;
searcht 5 [`negligible`;`aff_dim`];;
searcht 5 [`mcell1`;`rogers`];;
searcht 5 [`SUBSET`;`negligible`];;
searcht 5 [`convex`;`rogers`;`omega_list_n`];;
Marchal_cells.OMEGA_LIST_0_EXPLICIT;;
searcht 15 [`omega_list`;`HD`];;
searcht 5 [`omega_list_n`];;
searcht 5 [`omega_list_n V ul 0`];;
searcht 5 [`NULLSET`;`affine hull s`];;
searcht 5 [`coplanar`;`affine hull s`];;
searcht 5 [`convex hull s SUBSET affine hull s`];;
searcht 5 [`f hull s SUBSET f hull t`];;
searcht 5 [`affine hull (affine hull x)`];;
searcht 5 [`aff_ge {a,b,c} {d}`;`%`];;
searcht 5 [`DISJOINT a b = DISJOINT b a`];;
searcht 5 [`dot`;`dist(a,b) <= dist(c,d)`];;
searc 'a' 1 [`convex`;`a dot b`];;
searcht 8 [`convex`;`%`;`vsum`];;
searcht 5 [`vsum`;`{a,b,c,d}`];;
searcht 15 [`sum (i..k)`;`SUC`];;
searcht 5 [`convex`;`%`;`&1`;omit `vsum`];;
Leaf_cell.DIST_LE_HALF_PLANE;;
searcht 5 [`voronoi_list`;`omega_list_n`];;
searcht 5 [`voronoi_list`;`dist (a,b)`];;
searcht 5 [`set_of_list(truncate_simplex k ul)`];;
searcht 5 [`barV`;`truncate_simplex`];;
searcht 5 [`dist(a,b) = dist(c,d)`;`dot`];;
searcht 5 [`&0 <= dist(x,y)`];;
searcht 25 [`~coplanar {(a:real^3),b,c,d}`;`%`;`&1`];;
searcht 15 [`affine hull s`;`(:real^A)`];;
searcht 5 [`coplanar`;`aff_dim`];;
searcht 5 [`aff_dim`;`dimindex`];;
searcht 5 [`affine hull {a,b,c,d}`;`%`];;
searcht 15 [`&0 <= a * b`;`&0 < x`];;
searcht 15 [`dot`;`dist(a,b) < dist(c,d)`];;
searcht 5 [`&0 < dist (x,y)`];;
searcht 15 [`barV`;`~(x = y)`];;
Sphere.BARV;;
searcht 5 [def "voronoi_nondg"];;
searcht 5 [`voronoi_nondg`];;
searcht 5 [`voronoi_list`;`aff_dim`];;
Sphere.VORONOI_LIST;;
searcht 5 [`initial_sublist ul ul`];;
searcht 5 [`barV`;`truncate_simplex`];;
searcht 5 [`truncate_simplex 0`];;
searcht 15 [`set_of_list`;`LENGTH`];;
searcht 15 [`set_of_list`;`CONS x y`];;

searcht 5 [`barV`;`truncate_simplex`];;
searcht 5 [`EL`;`truncate_simplex`];;
searcht 5 [`mcell1`;`cball`];;
searcht 5 [`ball`;`cball`;`SUBSET`];;
searcht 5 [`(X DIFF Y) INTER Z`;`(X INTER Y) DIFF Z`];;
Pack_defs.mcell1;;
searcht 5 [`rcone_gt`;`dot`];;
searcht 30 [`rcone_gt`];;
RCONE_GT_HYPERPLANE;;
searcht 5 [`a % b dot c`];;
searcht 5 [`a % b % c`];;
searcht 5 [`rogers`;`{a,b,c,d}`;`convex`];;
searcht 5 [`hl [a;b]`];;
searcht 15 [`truncate_simplex k`;`CONS`];;
searcht 5 [`barV`;`CONS`;`EL`];;
searcht 5 [name "LENGTH4"];;
searcht 5 [`radial`;`sol`];;
searcht 5 [` A INTER A`];;
searcht 5 [`measurable`;`ball`];;
searcht 5 [`measurable`;`mcell`];;
searcht 5 [`mcell`;`mcell1`];;
searcht 5 [`sqrt x = &0`];;
searcht 5 [`gammaX`];;
searcht 5 [`gammaX`];;
searcht 5 [`&0 <= sol a b`];;
searcht 5 [`pow`;`mm1`];;
searcht 5 [`&0 <= vol X`];;
searcht 5 [`vol X >= &0`];;
searcht 15 [`&0 <= x * y`];;
searcht 5 [`sqrt x pow n`];;
searcht 5 [`sqrt x pow 2`];;
searcht 5 [def "mxi"];;

(* Jan 20 *)
print_proof();;
GAMMAX_MCELL2;;
Pack_defs.dihX;;
Pack_defs.dihu2;;
searcht 5 [`cell_params_d`];;
searcht 5 [`cell_params`];;
searcht 25 [`initial_sublist`];;
searcht 15 [`CONS`;`LENGTH`];;
LENGTH4;;
searcht 30 [`APPEND`];;
searcht 5 [name "AJR"];;
searcht 5 [`omega_list_n`;`permutes`];;
Marchal_cells_2_new.LEFT_ACTION_LIST_1_PROPERTIES;;
Bump.BETA_ORDERED_PAIR_THM;;
searcht 5 [`mxi V ul = mxi V vl`];;
Marchal_cells_3.MCELL_ID_MXI_2;;
searcht 15 [`omega_list_n V ul k = omega_list_n V vl k`];;
Marchal_cells_3.MCELL_ID_OMEGA_LIST_N;;
searcht 5 [`CONS x y  = CONS x' y'`];;
searcht 5 [`measure X + measure Y`];;
searcht 5 [`bis_le`;`bis`];;
searc 'a' 0 [`negligible`;`dot`];;
DOT_RMUL;;
searcht 5 [`measurable`;`dist`];;
searc 'e' 1 [`measurable`;`convex`;`bounded`];;
searc 'b' 0 [`convex`;`mcell`];;
searc 'c' 0 [`bounded`;`mcell`];;
searc 'd' 0 [`convex`;`bis_le`];;
searcht 5 [`negligible`;`SUBSET`];;
searcht 5 [`a % v = vec 0`];;
searcht 5 [`mcell2`;`rcone_ge`];;
searcht 5 [`negligible {}`];;
searcht 5 [`~ ~ x`];;
searcht 5 [`frustum`];;
searcht 15 [`frustt`];;
VOLUME_FRUSTT_WEDGE;;
searcht 5 [def "^wedge"];;
RCONE_GT_HYPERPLANE;;
searcht 8 [`rcone_ge`;`dot`];;
RCONE_GE;;
searcht 5 [`dot`;`dist(a,b) <= dist(c,d)`];;
searcht 5 [`dist(x,y) = &0`];;
searcht 5 [name "Cou";`ball`];;
DIST_NZ;;
searcht 5 [`norm`;`dot`];;
searcht 15 [`x pow 2 <= y pow 2`];;
searcht 15 [`atn2`;`ups_x`];;
Sphere.arclength;;
searcht 5 [`arcV`;`dot`];;
Trigonometry1.DOT_COS;;
searcht 5 [`arclength`;`arcV`];;
Trigonometry1.arcVarc;;
searcht 5 [`cos`;`rcone_ge`];;
searcht 5 [`arcV a b c = arcV a c b`];;
searcht 5 [`arcV u u v`];;
searcht 5 [`(v:real^A) - v`];;
searcht 5 [`norm (vec 0)`];;
searcht 5 [`norm x = &0`];;
searcht 15 [`&0 <= x * y <=> &0 <= y`];;
searcht 5 [`&0 < norm x`];;
searcht 5 [`arcV a b c = arcV a' b' c'`];;
searcht 5 [`arcV v u u`];;
searcht 5 [`cos (&0)`];;
searcht 5 [`arcV`;`pi`];;
Local_lemmas1.ARCV_BOUNDS;;
searcht 5 [`cos x <= cos y`];;
searcht 5 [`atn2 (&0,y)`];;
searcht 5 [`ups_x a b c = ups_x a' b' c'`];;
searcht 5 [`atn2 (x,y) <= atn2(x',y')`];;
searcht 17 [`arcV`;`cos`];;
searcht 5 [`&0 < dist(a,b)`];;
searcht 5 [`&0 <= ups_x a b c`];;
searcht 5 [`atn2(x, -- y)`];;
searcht 5 [`atn`;`atn2`];;
searcht 15 [`atn(x / y)`];;
searcht 5 [`x / (--u)`];;
searcht 5 [`inv (--x)`];;


searcht 5 [`a pow 2 <= b pow 2`];;
searcht 5 [`atn x <= atn y`];; 
searcht 5 [`x pow 2`;`x * x`];;
searcht 5 [`frustt`;`vol`];;
searcht 5 [`frustum`];;
type_of `dihV`;;
searcht 5 [`bis_le`;`dot`];;
searcht 5 [`SDIFF`];;
searcht 5 [`NULLSET`;`sphere`];;
searcht 5 [`circular_cone`];;
searcht 5 [`c_cone`];;
searcht 5 [`rcone_ge`;`rcone_gt`];;
searcht 5 [`SDIFF`];;
searcht 5 [`negligible {}`];;
searcht 15 [`rcone_ge`;`rcone_gt`];;
searcht 5 [`?(u,v). p`];;
searc 'k' 0 [`negligible`;`dot`];;
searcht 5 [`vol`;`rcone_ge`];;
searcht 5 [`SDIFF`;`measurable`];;
searcht 15 [`measure X = measure Y`];;
searcht 5 [`measurable`;`rcone_ge`];;
searcht 5 [`wedge`;`wedge_ge`];;
searcht 5 [`affine`;`NULLSET`];;
searcht 5 [`aff_ge s t SUBSET affine hull t'`];;
searcht 5 [`(A UNION B) SUBSET (A' UNION B')`];;
searcht 5 [`aff_ge`;`wedge_ge`];;
searcht 5 [`dihV a b c d <= pi`];;
searcht 5 [`dihV a b c d = pi`];;
searcht 5 [`coplanar`;`mxi`];;
searcht 2 [`coplanar`;`NULLSET`];;
searcht 5 [`coplanar`;`affine hull s`];;
searcht 5 [`collinear`;`coplanar`];;
SET_RULE;;
searcht 15 [`azim`;`dihV`];;
searcht 5 [`azim a b c d`;`&2 * pi`];;
searc 'l' 2 [`{a,b} = {c,d}`];;
searc 'm' 1 [`&0 < hl u`];;
searcht 5 [`sqrt2 * sqrt2`];;
searcht 1 [`SDIFF`;`vol X = vol Y`];;
searcht 5 [`measurable`;`mcell2`];;
searcht 1 [`measurable`;`frustt`;`wedge`];;
searcht 5 [`bis_le`;`dot`];;
searcht 5 [`norm`;`pow`;`dot`];;
searcht 5 [`a dot a - b dot b`];;
searcht 5 [`(a + b) dot c`];;
searcht 5 [`hl`;`dist`];;
searcht 5 [`SDIFF`;`mcell2`];;
FRUSTT_WEDGE_RCONE_GE;;
MCELL2_SPLIT;;
searcht 5 [`wedge_ge`;`aff_ge`];;
searcht 5 [`mcell2`;`rcone_ge`];;
searcht 5 [`SDIFF X Y = SDIFF Y X`];;
searcht 5 [`&0 < hl(truncate_simplex 1 ul)`];;

searcht 5 [def "gamma2_x_div_azim"];;
searcht 5 [def "^lmfun"];;
searcht 5 [`gamma2_x_div_azim`;`lfun`;`sqrt`];;
Pack_defs.mcell2;;



(* Jan 22, 2013 *)
searcht 5 [`vol X = vol Y + vol Z`;`mcell2`];;
searcht 5 [`p permutes 0..1`];;
Marchal_cells_2_new.LEFT_ACTION_LIST_1_PROPERTIES;;
searcht 5 [`permutes`;`left_action_list`;`mcell i j k = mcell i' j' k'`];;
searcht 3 [`truncate_simplex`;`permutes`];;
searcht 5 [`set_of_list`;`truncate_simplex`];;
searcht 5 [`dihV a b c d = dihV b a c d`];;
searcht 5 [`sol`;`mcell2`];;
searc 't' 0 [`eventually_radial`;`mcell`];;
searcht 5 [def"^radial"];;
searcht 5 [`radial`;`sol`];;
searcht 15 [`dist`;`hl`];;
VOLUME_FRUSTT_WEDGE;;
searcht 15 [`conic_cap`;`vol`];;

(* Jan 23, 2013 *)
MCELL2_VX_PROPS;;
searcht 5 [`VX`;`V INTER X`];;
searcht 5 [`eventually_radial`;`sol`];;
searcht 5 [`radial_norm`;`r' <= r`];;
searcht 5 [`radial`;`radial_norm`];;
searcht 5 [`normball`;`ball`];;
Marchal_cells_2_new.RADIAL_VS_RADIAL_NORM;;
RADIAL_LE;;
searcht 5 [`sol`;`radial`];;
searcht 5 [`measurable`;`mcell`];;
Sphere.bis_le;;
Reuhady.VOLUME_CONIC_CAP_WEDGE_GE_VS_CONIC_CAP;;
searcht 5 [`vol X = vol Y`];;
searcht 5 [`measurable`;`wedge`];;
searcht 5 [`SDIFF`;`rcone_ge`];;
searcht 5 [`h / a < t`];;
searcht 5 [`SDIFF`;`X INTER Y`];;
bb 10;;
searcht 5 [`dist a < dist b`;`dot`];;
searcht  5 [`mcell2`;`bis_le`];;
searcht 5 [`wedge_ge`;`aff_ge`];;
searcht 5 [`vol`;`mcell2`];;
searcht 5 [`hl(truncate_simplex 1 ul) = hl(truncate_simplex 1 vl)`];;
searcht 5 [`dihV a b c d = dihV b a c d`];;
searcht 5 [`dihX V X`;`dihV`];;
searcht 5 [`dihV`;`dih_y`];;
searcht 5 [`gamma2_x_div_azim`];;
searcht 5 [`truncate_simplex`;`CONS`];;
searcht 2 [`lmfun`;`h0cut`];;
searcht 5 [`sqrt8`;`sqrt2`];;
searcht 5 [`sqrt(x) = &0`];;
searcht 15 [`gamma2_x_div_azim`];;
searcht 5 [name "BALL_DIFF_RCONE_GE"];;

(* Jan 24, 2013 *)
searcht 5 [`&0 <= a`;`dih_x`];;
searcht 5 [`&0 <= dihV a b c d`];;
searcht 5 [`gamma2_x_div_azim_v2`;`ineq`];;
searcht 5 [`sqrt8`;`sqrt2`];;
searcht 5 [`mcell1`;name "TSK"];;



(* Jan 25, 2013 *)
Counting_spheres.POLY_SORT_BIJ;;
Counting_spheres.WEDGE_ORDER_DISJOINT;;
Counting_spheres.PREIMAGE_BIJ;;
Counting_spheres.ORDER_AZIM_SUM2Pi;;
TOPOLOGICAL_SORT;;
searcht 25 [name "SORT"];;
searcht 5 [name "EL_EXPLICIT"];;
searcht 5 [name "aff"];;
searcht 5 [name "FCHK"];;
searcht 5 [`azim`;`aff_ge`];;
searcht 5 [`s_leaf`];;
searcht 5 [`leaf`;`collinear`];;
searcht 5 [`set_of_list (CONS x y)`];;
searcht 15 [`azim a b c d = azim a b c d'`];;
searcht 5 [`aff_gt s t = aff_gt s {y}`];;
searcht 5 [`(i + j) MOD n`];;
searcht 5 [`i MOD n = i`];;
searcht 5 [`(x MOD n) < n`];;
DIVISION;;
type_of `PRE`;;
type_of `min:num->num->bool`;;
searcht 25 [`?(x:num). P x`;`x <= (y:num)`];;
Print_types.print_thm_types UPPER_BOUND_FINITE_SET;;
searcht 50 [name "MAX";`x:num`];;
;`(x:num) IN s`];;
searcht 15 [`x:(num->bool)->num`];;
type_of`min_num`;;
searcht 15 [`min_num`];;
searcht 15 [`wedge`;`wedge_ge`;`mcell_set`];;
Leaf_cell.BDXKHTW;;
Leaf_cell.EWYBJUA;;
searcht 25 [`azim a b c d = &0`;omit `wedge`;omit `wedge_ge`;omit `aff_ge`];;
searcht 5 [`azim a b c d + azim a b c' d'`];;
searcht 5 [`wedge`;`azim`];;
searcht 5 [`azim a b c c`];;
searcht 5 [`&0 <= azim a b c d`];;
searcht 5 [`azim a b c d = &0`;`aff_ge`];;
searcht 5 [`aff_ge s t SUBSET affine hull t'`];;
searcht 5 [`negligible`;`affine hull {a,b,c}`];;
searcht 5 [`a /\ (?x. P x)`];;


(* Jan 26, 2013 *)
searcht 5 [`barV`;`FINITE`];;
searcht 5 [`packing`;`FINITE`];;
searcht 5 [`barV`;`ball`];;
searcht 5 [`set_of_list ul SUBSET V`;`barV`];;
Packing3.BARV_SUBSET;;
searcht 5 [`set_of_list`;`CONS`];;
searcht 5 [`hl`;`dist pr`];;
DIST_SYM;;
Qzyzmjc.BARV_3_IMP_FINITE_lemma2;;
SUBSET_UNIONS;;
IN_UNIONS;;
searcht 5 [`negligible`;`UNIONS`];;
searcht 5 [`NULLSET`;`aff_ge`];;
searcht 5 [`NULLSET`;`affine`];;
searcht 5 [`aff_ge`;`affine hull (s UNION t)`];;
searcht 5 [`aff_ge {a,b} {c}`;`collinear`];;
Local_lemmas.AZIM_EQ_0_GE_ALT2;;
searcht 5 [`wedge_ge a b c d SUBSET X`;`aff_ge`];;
searcht 5 [`collinear`;`s_leaf`];;
searcht 5 [name "OXL"];;
searcht 5 [`sum X f = &0`];;
searcht 5 [name "TSK"];;
searcht 5 [`beta_bump_v1`];;
searcht 5 [`critical_weight`;`&0 <= x`];;
searcht 5 [`0 <= CARD X`];;
searcht 5 [`;;
Leaf_cell.BDXKHTW;;
searcht 5 [`s_leaf`;`leaf`];;
searcht 5 [`collinear`;`leaf`];;
searcht 5 [`azim a b c d = &0`;`aff_ge`];;
searcht 5 [`periodic`];;
searcht 50 [`azim a b w w' = &0`];;
searcht 15 [`azim a b c d + azim a b d e`];;
searcht 5 [`&0 <= azim a b c d`];;
searcht 5 [`periodic `;`MOD`];;
searcht 5 [`((a ==> b ) /\ (b ==> a))`];;
searcht 5 [`i MOD n < n`];;
searcht 15 [`(a + b) MOD n`];;
searcht  5[`wedeg`;`wedge_ge`;`DIFF`];;
searcht 5 [`a MOD n = b MOD n`;`a < n`];;
searcht 5 [`0 MOD n`];;
DIVISION;;
searcht 15 [`a MOD n = a`];;
searcht 5 [`wedge_ge`;`azim`];;
Local_lemmas.AZIM_EQ_0_GE_ALT2;;
searcht 5 [`wedge`;`azim`];;
searcht 5 [`azim a b c d - azim a' b' c' d' = azim a'' b'' c'' d'' - azim a''' b''' c''' d'''`];;
searcht 15 [`collinear`;`aff`];;
searcht 15 [`collinear`;`affine hull s`];;
COLLINEAR_3_AFFINE_HULL;;
searcht 5 [`collinear`;`~(a = b)`];;
searcht 5 [`affine hull s SUBSET aff_ge s' t'`];;
searcht 5 [`DISJOINT`;`collinear {a,b,c}`];;
bb 3;;
searcht 5 [`azim a b c d = &0 <=> p`];;
searcht 5 [`azim a b c d = &0`;`aff_ge`];;
searcht 5 [name "REUH"];;
Oxl_def.periodic_mod;;
searcht 5 [`azim a b c c `];;
searcht 5 [`sum`;`azim`;`&2 * pi`];;
searcht 5 [name "GRUT"];;
searcht 15 [`sum X (\i. sum Y f)`];;
searcht 5 [`FINITE {i | (i:num) < n}`];;
searcht 5 [`FINITE`;`mcell_set`];;
searcht 5 [`preimage`;`BIJ`];;
searcht 5 [`a,b = a',b'`];;
MCELL_WEDGE_UNIQUE;;
searcht 5 [`edgeX`;`NULLSET`];;
Bump.RIJRIED;;
searcht  5 [`BIJ`;`sum`];;
searcht 5 [name "BETA"];;
searcht 5 [`collinear`;`~(a = b)`];;
searcht 5[`hplus`;`hminus`];;
searcht 5 [`periodic`;`leaf_rank`];;


(* Jan 28, 2013 *)
searcht 5 [`beta_bumpA_y`];;
Counting_spheres.ORDER_AZIM_SUM2Pi;;
searcht 5 [`sum x f = sum x f'`];;
searcht 5 [`collinear`;`leaf_rank`];;

(* Jan 30, 2013 *)
searcht 5 [`mcell4`;`leaf`];;
searcht 5 [`mcell`;`leaf`];;
searcht 15 [`cc_cell`;`leaf`];;
searcht 5 [`hl`;`radV`];;
searcht 15 [`radV`;`eta_y`];;
searcht 5 [`dist`;`d3`];;
searcht 15 [`hl`;`dist`];;
searcht 15 [`hl a < hl b`];;
searcht 15 [`hl a <= hl b`];;
searc 'a' 0 [`radV a <= radV b`];;
searcht 5 [`affine_dependent`;`collinear`];;
searcht 5 [`radV {a,b}`];;
searcht 5 [`hl [a;b]`];;
Pack_defs.HL;;
Marchal_cells_3.HL_2;;
searcht 5 [`eta_y a b c = eta_y a b' c'`];;
searcht 5 [`&0 <= ups_x a b c`];;
searcht 5 [`x pow 2`;`x*x`];;
searcht 5 [`&0 <= eta_y a b c`];;
searcht 5 [`ups_x a b c = ups_x a c b`];;
searcht 5 [`x pow 2 <= y pow 2`];;
Sphere.BARV;;
searcht 5 [`barV`;`X SUBSET V`];;
searcht 5 [`collinear`;`leaf`];;
searcht 5 [`collinear `;`~(a = b)`];;
searcht 5 [`NULLSET`;`edgeX`];;
searcht 15 [`dihX`;`NULLSET`];;
searcht 15 [`dihX`;`dihV`];;
searcht 50 [`dihX`];;
searcht 50 [`dihu4`];;
searcht 50 [`mcell2`;`aff_ge`];;
searcht 5 [`mcell2`;`dihX`];;
searcht 5 [`&0 <= dihV a b c d`];;
searcht 5 [`dihV a b c d = &0`];;
searcht 5 [`collinear`;`mxi`];;
searcht 5 [`collinear`;`omega_list_n`];;
searcht 5 [`coplanar`;`collinear`];;
searcht 5 [`NULLSET`;`coplanar`];;
searcht 5 [`coplanar`;`affine hull s`];;
searcht 5 [`aff_ge s t SUBSET affine hull s'`];;
searcht 5 [`dihX`;`COND`];;
searcht 5 [`dihu3`];;
searcht 5 [`FST`;`cell_params_d`];;
searcht 5 [`mcell i V ul = mcell i V ul'`];;
searcht 15 [`initial_sublist`];;
searcht 5 [`barV`;`LENGTH`];;
searcht 5 [`SND`;`cell_params_d`];;
searcht 5 [`mcell3`;`convex`];;
Pack_defs.mcell3;;
searcht 15 [`initial_sublist`];;
searcht 5 [`APPEND (CONS x y)`];;
searcht 5 [`((a ==> b) /\ (b ==> a))`];;
searcht 5 [`convex hull s = convex hull y`];;
searcht 5 [`set_of_list (truncate_simplex 2 ul)`];;
searcht 5 [`affine_dependent`;`NULLSET`];;
searcht 5 [`affine_dependent`;`aff_dim`];;
searcht 5 [`CARD`;`mxi`];;
searcht 5 [`mcell 3`;`cc_cell`];;
searcht 15 [`coplanar`;`hull`];;
searcht 5 [`affine hull (convex hull s)`];;
searcht 5 [`FINITE`;`INSERT`];;
searcht 5 [`aff_dim`;`coplanar`];;
searcht 5 [`aff_dim`;`DIMINDEX`];;
searcht 5 [`coplanar (convex hull s)`];;
searcht 25 [`CARD`;`4`];;
searcht 5 [`dihV a b c d = dihV a b d c`];;
DIHV_SYM;;
MCELL2_DIHX_POS;;
searcht 5 [`mcell4`;`edgeX`];;


(* Jan 31, 2013 *)
searcht 5 [`barV`;`mcell4`;`leaf`];;
Leaf_cell.RBUTTCS;;
searcht 5 [`mcell4`;`convex`];;
searcht 5 [`barV V`;`truncate_simplex`];;
searcht 5 [`truncate_simplex 2`];;
searcht 5 [`hl s <= hl t`];;
searcht 15 [`permutes`;`left_action_list`;`?`];;
Marchal_cells_3.LEFT_ACTION_LIST_3_EXISTS;;
searcht 5 [`CARD`;`mcell4`];;
Collect_geom2.NOT_COPLANAR_IMP_CARD4;;
Bump.LENGTH4;;
searcht 15 [`set_of_list`;`LENGTH`];;
searcht 5 [`barV`;`LENGTH`];;
searcht 5 [`mcell i V ul = mcell i' V ul'`];;
Rvfxzbu.RVFXZBU;;
searcht 5 [`barV`;`left_action_list`];;
searcht 15 [`CARD`;`2`];;
Hypermap.CARD_TWO_ELEMENTS;;
searcht 5 [`CARD x <= CARD y`;`SUBSET`];;
searcht 5 [`CARD`;`4`;`~(x = y)`];;
Marchal_cells_2_new.CARD4_IMP_DISTINCT;;
searcht 5 [`a = b ==> b = a`];;
searcht 5 [`FINITE`;`s_leaf`];;
searcht 3 [`edgeX`;`cc_cell`];;
searcht 5 [`cc_cell`;`cc_ke`];;
searcht 5 [`mcell4`;`edgeX`];;
searcht 5 [`u IN p hull s`];;
searcht 5 [`u IN aff_gt s t`];;
searcht 10 [`mcell i V ul = mcell i' V ul'`];;
searcht 5 [`barV`;`cc_uh`];;
searcht 5 [`cc_ke`;`4`];;
searcht 5 [`leaf V a ==> leaf V b`];;
Leaf_cell.cc_uh;;
searcht 5 [`wedge_ge`;`wedge`;`aff_ge`];;
searcht 5 [`collinear`;`leaf_rank`];;
Print_types.print_goal_types();;
searcht 50 [`leaf_rank`;`periodic`];;
searcht 5 [`j MOD n < n`];;
searcht 5 [`periodic`;`MOD`];;
searcht 5 [`wedge`;`azim`];;
searcht 5 [`azim a b c c `];;
searcht 5 [name "SHIFT";`azim a b c d - azim a' b' c' d'`];;
searcht 8 [`azim a b c d + azim a b d e`;omit `pi`];;

print_proof();;
(* Feb 1, 2013 *)
searcht 5 [`azim`;`wedge_ge`];;
searcht 5 [`wedge_ge`;`DIFF`];;
GSYM Leaf_cell.WEDGE_COMPLEMENT;;
searcht 5 [`azim a b c d = &0`];;
searcht 5 [`azim a b c d`;`&2 * pi`];;
searcht 5 [`mcell4`;`convex`];;
searcht 5 [`{a,b} = {c,d}`;`~(a = b)`];;
searcht 5 [`CARD`;`mcell4`];;
searcht 5 [`CARD`;`4`;`~(a = b)`];;
MCELL4_LEAF2;;
searcht 5 [`x IN convex hull s`];;
searcht 5 [`chi_msb`;`sin`];;
searcht 5 [`sin`;`azim`];;
searcht 5 [`aff_ge`;`wedge_ge`];;
searcht 5 [`convex hull s SUBSET aff_ge a b`];;
searcht 5 [`azim a b c d = pi`;`coplanar`];;
searcht 5 [`leaf_rank`;`collinear`];;
searcht 5 [`coplanar (convex hull s)`];;
searcht 5 [`coplanar`;`NULLSET`];;
searcht 5 [`azim a b c d + azim a b d c`];;
searcht 5 [`wedge_ge a b c d INTER wedge_ge a b c' d'`];;
searcht 5 [`NULLSET`;`aff_ge`];;
searcht 5 [`aff_ge s t SUBSET affine hull s'`];;
searcht 5 [`azim`;`dihV`];;
searcht 5 [`dihX`;`dihV`];;
searcht 5 [`dihX`;`mcell 4`];;
Tskajxy_lemmas.DIHX_DIH_Y_lemma;;
searcht 5 [`dihV`;`dih_y`];;
searcht 5 [`CARD`;`4`;`mcell4`];;
searcht 5 [`barV`;`left_action_list`];;
searcht 5 [`mcell j V ul = mcell i V ul'`];;
searcht 5 [`barV`;`LENGTH`];;
MCELL4_BARV_FI;;
searcht 5 [name "S_LEAF_COLLINEAR"];;
searcht 5 [`packing V /\ saturated V /\ leaf_rank V ul w0 n f
            ==> ~collinear {EL 0 ul, EL 1 ul, f k}`];;
searcht 5 [`mcell`;`?`;`4`];;
searcht 5 [`edgeX V X = {}`];;
searcht 5 [`edgeX`;`NULLSET`];;
Bump.RIJRIED;;
searcht 5 [`leaf_rank`;`CARD`];;
searcht 5 [`s_leaf`;`FINITE`];;
searcht 5 [`leaf_rank`;name "INJ"];;
searcht 5 [`periodic`;`IMAGE`];;
searcht 5 [`periodic`;`MOD`];;
searcht 5 [`BIJ`;`CARD`];;
searcht 5 [`{i | i < n}`;`HAS_SIZE`];;
searcht 5 [`i<(n:num)`;`FINITE`];;
searcht 5 [`u MOD n = u`];;
GSYM MOD_LT;;
searcht 5 [`sum s f <= sum s' f`;`SUBSET`];;
searcht 5 [`mcell_set`;`FINITE`];;
searcht 5 [`mcell4`;`edgeX`];;
searcht 5 [`CARD`;`4`;`barV`];;
searcht 5 [`set_of_list`;`LENGTH`;`4`];;
searcht 5 [`&0 <= dihX V x u`];;
searcht 5 [`sum {a,b}`];;
searcht 5 [`mcell2`;`edgeX`];;
type_of `gg_mcell`;;
searcht 5 [`gg_mcell`];;
searcht 5 [`sum {X}`];;

print_proof();;
update_database();;
(* Feb 2, 2013 *)
searcht 10 [`EL`;`cc_uh`];;
searcht 5 [`leaf`;`leaf_rank`];;
searcht 5 [`leaf`;`s_leaf`];;
searcht 15 [`s_leaf`;`leaf_rank`];;
searcht 5 [`{a,b} = {c,d}`];;
searcht 5 [`leaf V a ==> leaf V b`];;
Leaf_cell.ZASUVOR;;
MCELL_IN_WEDGE;;
searcht 5 [`s_leaf`;`HAS_SIZE`];;
searcht 5 [`mcell_set`;`cc_cell`];;
searcht 5 [`NULLSET`;`cc_cell`];;
searcht 5 [`edgeX`;`cc_cell`];;
searcht 5 [`x IN cc_cell V ul`];;
bb 3;;
 S_LEAF_IN_WEDGE_GE ;;
searcht 5 [`leaf_rank`;name "INJ"];;
searcht 5 [`azim a b c c = &0`];;
searcht 5 [`periodic`;`leaf_rank`];;
searcht 5 [`MOD`;`SUC`];;
searcht 5 [`periodic`;`f (i MOD n)`];;
Oxl_def.periodic_mod;;
searcht 15 [`((i MOD n) + j) MOD n`;];;
searcht 5 [`(a*n + i) MOD n`];;
Leaf_cell.cc_cell;;
searcht 5 [`cc_cell V a = cc_cell V b`];;
searcht 5 [`stem [a;b;c]`];;
searcht 5 [`stem`;`{a,b}`];;

print_proof();;
update_database();;
(* Feb 3, 2013 *)
Leaf_cell.ZASUVOR;;
Leaf_cell.STEM_OF_LEAF;;
searcht 50 [`hl`;`sqrt`;`barV`];;
(* want barV as a conclusion if hl < sqrt2 *)
Marchal_cells_3.HL_LE_SQRT2_IMP_BARV_1;;
Leaf_cell.leaf;;
MCELL4_FULL_WEDGE;;
searcht 5 [`mcell4`;name "FIRST"];;
searcht 5 [`cc_uh`;`barV`];;
searcht 2 [`cc_cell`;`edgeX`];;
searcht 2 [`NULLSET`;`cc_cell`];;
EQ_SYM_EQ;;
searcht 5 [`mcell_set`;`cc_cell`];;
searcht 5 [`stem ul = {a,b}`];;
searcht 5 [`cc_cell`;`convex hull s`];;
searcht 8 [`mcell i V ul = mcell j V ul'`];;
searcht 5 [`barV`;`cc_uh`];;
searcht 5 [`cc_ke`;`3`;`4`];;
INTER_IDEMPOT;;
searcht 5 [`NULLSET`;`affine hull s`];;
searcht 5 [`convex hull s SUBSET convex hull t`];;
searcht 5 [`convex hull (affine hull s)`];;
searcht 5 [`convex hull s = s`];;
searcht 5 [`affine s ==> convex s`];;
searcht 5 [`s SUBSET P hull t`];;
searcht 5 [`set_of_list`;`LENGTH`;`4`];;
searcht 5 [`barV`;`cc_uh`];;
searcht 5 [`&0 < sin x`];;
searcht 15 [`azim (vec 0) b c d`];;
searcht 5 [`azim (a + x)`];;
searcht 5 [`&0 <= azim a b c d`];;
searcht 5 [`azim a b c d = &0`;`leaf_rank`];;
searcht 5 [`i MOD n = i`];;
searcht 5 [`n MOD n`];;
Oxl_def.periodic_mod;;
searcht 5 [`cc_cell`;`convex hull s`];;
searcht 15 [`convex hull s = convex hull s'`];;
searcht 5 [`affine_dependent`;`aff_dim`];;
bb 5;;
searcht 5 [`affine hull (convex hull s)`];;
searcht 5 [`aff_dim`;`affine hull s`];;
searcht 5 [`set_of_list`;`cc_uh`];;
searcht 15 [`CARD`;`4`;`{a,b}`];;
Leaf_cell.LENGTH4_SET2;;
searcht 15 [`CARD`;;
MCELL4_CARD4;;
searcht 5 [`aff_dim`;`coplanar`];;
searcht 5 [`coplanar`;`NULLSET`];;
searcht 5 [`coplanar`;`convex hull s`];;
searcht 5 [`int_of_num x = int_of_num y`];;
searcht 5 [`mcell4`;`set_of_list`];;
searcht 5 [`CARD`;`aff_dim`];;
AFF_DIM_LE_CARD;;
searcht 5 [`CARD {a,b,c,d} <= 4`];;
searcht 5 [`cc_ke`;`&0 < chi_msb V ul`];;
searcht 5 [`chi_msb a b = -- chi_msb a' b'`];;
CC_CELL_WEDGE_MATCH_UH;;
searcht 5 [`MOD`;`periodic`;`f i = f j`];;
searcht 5[`(x MOD n) MOD n`];;
searcht 5 [name "INJ";`azim`];;
searcht 5 [`azim a b c c = &0`];;
searcht 5 [`i MOD n = j MOD n`];;
searcht 5 [`~(i MOD n = j MOD n)`];;
LEAF_RANK4_CHI_MSB_ALT;;
searcht 5 [`mcell4`;`set_of_list`];;
searcht 5 [`set_of_list ul = {EL 0 ul, EL 1 ul, EL 2 ul, EL 3 ul}`];;
searcht 5[`barV`;`LENGTH x = k + 1`];;
searcht 15 [`left_action_list`;`barV`];;
searcht 15 [`cc_cell V a = cc_cell V b`];;
LEAF_RANK4_CHI_MSB_ALT;;
searcht 5 [`CARD`;`4`;`cc_uh`];;
searcht 5 [`barV`;`cc_uh`];;
searcht 5 [`NULLSET`;`cc_cell`];;
 CC_CELL_WEDGE_MATCH_UH_ALT;;
LEAF_RANK_AZIM_INJ;;
Oxl_2012.MOD_INJ1;;
searcht 5 [name "WEDGE_UNIQUE"];;
searcht 5 [`i MOD n MOD n`];;
searcht 8 [`(a + b) MOD n`];;
searcht 5 [`i MOD n = i`];;

print_proof();;
update_database();;
(* Feb 5, 2013 *)
 WEDGE_UNIQUE_CC_CELL;;
searcht 5 [`cc_ke`;`3`;`4`];;
searcht 5 [`barV`;`cc_uh`];;
searcht 1 [`edgeX`;`cc_cell`];;
Leaf_cell.cc_cell;;
searcht 5 [`SUC i + n - 1`;`MOD`];;
searcht 5 [`cc_cell V a = cc_cell V b`];;
searcht 5 [`e IN critical_edgeX V X`];;
searcht 5[`sum {}`];;
searcht 5 [`mcell4`;name "FIRST"];;
searcht 5 [`NULLSET`;`edgeX`];;
searcht 5 [`sum`;`edgeX`];;
searcht 2 [`cc_cell`;`edgeX`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
(* Feb 6, 2013 *)
cc_prep_model_data;;
searcht 5 [`leaf_rank`;`cc_ke`];;
searcht 5 [`leaf_rank`;`s_leaf`];;
S_LEAF_SYM;;
searcht 5 [`barV`;`cc_uh`];;
searcht 5 [`gammaX`;`&0 <= x`];;
Tskajxy.GAMMAX_GAMMA2_X;;
searcht 5 [`gammaX`;`x >= &0`];;
Tskajxy.TSKAJXY_1;;
Tskajxy.TSKAJXY_2;;
searcht 5 [`mcell`;`edgeX V X = {}`];;
Bump.EDGE_IMP_K2;;
searcht 5 [`&0 <= sum X f`];;
searcht 5 [`FINITE`;`mcell_set`];;
searcht 5 [`mcell_set`;`4`];;
searcht 5 [`NULLSET`;`edgeX`];;
searcht 5 [`critical_edgeX`;`hl`];;
searcht 5 [`gammaX`;name "034"];;
searcht 5 [`gammaX`;`mcell 3`];;
searcht 15 [`gamma3f`];;
searcht 5 [`eta_y`;`radV`];;
searcht 5 [`eta_y a b c = eta_y a' b' c'`];;
searcht 5 [`radV`;`hl`];;
bb 3;;
searcht 5 [`truncate_simplex 2 ul`];;
searcht 5 [`collinear`;`ups_x`];;
searcht 5 [`ups_x (x pow 2)`;`a * b * c`];;
searcht 5 [`hl x <= hl y`];;
searc 'a' 0 [`radV x <= radV y`];;
searcht 5 [`aff_dim`;`affine_dependent`];;
searcht 5 [`collinear`;`coplanar`];;
searcht 5 [`coplanar (convex hull s)`];;
searcht 5 [`convex hull s`;`mcell3`];;
searcht 3 [`coplanar`;`NULLSET`];;
searcht 5 [`affine_dependent`;`collinear`];;
searcht 5 [`radV {x,y}`;`dist`];;
searcht 5 [`barV`;`set_of_list ul SUBSET V`];;
searcht 5 [`set_of_list ul = x INSERT y`];;
searcht 5 [`collinear`;`~(x = y)`];;
searcht 5 [`initial_sublist`;`CONS`];;
searcht 15 [`truncate_simplex 2 ul`];;
Marchal_cells.TRUNCATE_SIMPLEX_EXPLICIT_2;;
Print_types.print_thm_types Marchal_cells.TRUNCATE_SIMPLEX_EXPLICIT_2;;
searcht 5 [`CONS`;`[]`];;
searcht 5 [`x = []`;`x = CONS u v`];;
help_grep "CASE";;
help "STRUCT_CASES_TAC";;
searcht 5 [`initial_sublist []`];;
searcht 5 [`initial_sublist`;`APPEND`];;
searcht 5 [`aff_dim x = &0`];;
searcht 5 [def "circum"];;
searcht 5 [`circumcenter`;`voronoi_list`];;
searcht 5 [name "XYOF"];;
searcht 5 [`coplanar`;`affine_dependent`];;
searcht 5 [`coplanar`;`aff_dim`];;
searcht 5 [`aff_dim`;`coplanar`];;
searcht 5 [`CARD`;`coplanar`];;
searcht 5 [`CARD {a,b,c,d}`];;
searcht 5 [`coplanar`;`coplanar_alt`];;
searcht 5 [`voronoi_list V ul`;`voronoi_set`];;
searcht 15 [`voronoi_set`];;
Sphere.VORONOI_SET;;
searcht 5 [`voronoi_set`;`circumcenter`];;
searcht 50 [`circumcenter`];;
searcht 5 [name "OAPV"];;
searcht 5 [`affine hull s `;`coplanar`;`(:real^3)`];;
searcht 5 [`coplanar`;`azim`;`pi`];;
searcht 5 [`leaf_rank`;`azim`;name "INJ"];;
searcht 5 [`SUC i MOD n`];;
searcht 5 [`barV`;`x SUBSET V`];;
searcht 5 [`truncate_simplex 2 (CONS x y)`];;
searcht 5 [`mcell4`;`edgeX`];;
searcht 5 [`vol`;`convex hull s`];;
searcht 5 [`vol`;`NULLSET`];;
searcht 5 [`collinear`;`~(a = b)`];;
searcht 5 [`mcell4`;`convex hull s`];;
searcht 5 [`wedge_ge`;`aff_ge`];;
searcht 5 [`leaf`;`collinear`];;
searcht 5 [`collinear`;`leaf_rank`];;
searcht 5 [`convex hull s`;`aff_ge`];;
MCELL3_DOMAIN;;
searcht 5 [`truncate_simplex 2 ul = ul`];;
searcht 5 [`leaf`;`collinear`];;
searcht 5 [`delta_x`;`delta_y`];;
searcht 5 [`sqrt8`;`sqrt2`];;
searcht 15 [`rad2_x`];;
searcht 5 [`dih_x a b c d e f < dih_x a' b' c' d' e' f'`];;
Tame_inequalities.DIH_X_MONO_LT_4;;
searcht 5 [`eta_x`;`rad2_x`];;
searcht 5 [`x <= f(eta_x a b c)`;];;
searcht 50 [`eta_y`;omit `ineq`];;
Merge_ineq.rad2_x_eta_x;;
Collect_geom.HVXIKHW;;
SQRT_MUL;;
searcht 5 [`sqrt x * sqrt x`];;
searcht 5 [`x pow 2 <= y pow 2`];;
searcht 5 [`max_real3`];;
searcht 5 [`x <= max_real u v`];;
searcht 5 [`ups_x`;`collinear`];;
Collect_geom.COL_EQ_UPS_0;;

print_proof();;
update_database();;
Print_types.print_goal_types();;
WEDGE3_Y4;;
(* Feb 7, 2013 *)
searcht 5 [`rad2_y`;`rad2_x`];;
searcht 5 [`sqrt8 * sqrt8`];;
searcht 15 [`ups_x (y * y) ( u * u) (v * v)`];;
searcht 5 [`dih_y`;` pi`];;
searcht 5 [`atn2 (x,y)`;`pi / &2`];;
searcht 5 [`atn (x) < pi/ &2`];;
searcht 5 [`atn2`;`atn`];;
searcht 5 [`&0 < atn x`];;
searcht 5 [`atn u < c`];;
ATN_BOUNDS;;
Flyspeck_constants.bounds;;
searcht 5 [`hminus`;`DECIMAL`];;
searcht 5 [`sqrt2`;`sqrt8`];;
searcht 5 [`sqrt8 < y`];;
bb 4;;
searcht 5 [`azim`;`dihV`];;
searcht 5 [`dih_y`;`dihV`];;
searcht 5 [`delta_x (x  *x )`;`delta_y`];;
searcht 5 [`azim`;`delta_x`];;
searcht 35 [def "^delta"];; (* no vector version *)

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Merge_ineq.GDRQXLGv2 ;;
WEDGE3_Y4;;
(* Feb 8, 2013 *)
searcht 5 [`coplanar_alt`;`delta`];;
Collect_geom2.POS_EQ_NOT_COPLANANR;;
searcht 5 [`coplanar`;`coplanar_alt`];;
searcht 5 [`delta`;`delta_x`];;
searcht 5 [`delta_x (x *x)`;`delta_y`];;
searcht 5 [`azim a b c c `];;
searcht 5 [`barV`;`set_of_list ul SUBSET V`];;
searcht 5 [`azim`;`dihV`];;
searcht 5 [`dihV`;`dih_y`];;
searcht 5 [`rad2_x`;`radV`];;
Merge_ineq.GDRQXLGv2;;
searcht 5 [name "GDRQ"];;
searcht 5 [`coplanar`;`CARD`];;
	    type_of `coplanar`;;
searcht 5 [`CARD {a,b,c,d}`];;
searcht 5 [`rad2_y`;`rad2_x`];;
searcht 5 [`&0 <= radV s`];;
Rogers.AFFINE_HULL_RADV;;
RAD_PI_IMP_WEDGE4;;
searcht 5 [`sqrt2 * sqrt2`];;
searcht 5 [`x * x <= y * y`];;
searcht 15 [`abs x = x`];;
searcht 5 [`gg_mcell`];;
searcht 5 [`azim_mcell`];;
searcht 5 [`cell_cluster_estimate_v1`;`edgeX`];;
searcht 5 [`?`;`~collinear {a,b,c}`];;
searcht 5 [`edgeX`;`~(u = v)`];;
searcht 5 [`{a,b} = {c,d}`];;
searcht 5 [`~a ==> ~b`];;
searcht 5 [`FINITE`;`s_leaf`];;
searcht 5 [`gg_mcell`;`gammaX`];;
searcht 5 [`leaf_rank`;`periodic`];;
searcht 5 [`sum`;`azim_mcell`];;
searcht 5 [`hplus`;`sqrt2`];;
searcht 5 [`azim_mcell`;`azim`];;
searcht 5 [`bool_model_data_v10`;`cc_data_v4`];;
searcht 5 [`hl [a;b]`];;
cc_bool_model_data;;
searcht 5 [`cc_data_v4`;`leaf_rank`];;
searcht 5 [`cc_card_v10`;`cc_data_v4`];;
searcht 5 [`cc_hassmall_v10`;`cc_data_v4`];;
Oxl_def.cc_qx_v10;;
	    type_of `(\x. azim_mcell V x u0 u1)`;;
searcht 5 [`0..r`;`{i  | i < j}`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Merge_ineq.GDRQXLGv2 ;;
(* Feb 9, 2013 *)
searcht 5 [`gamma4fgcy`;`gammaX`];;  (* gamma4fgcy <=> gammaX *)
Tskajxy_lemmas.gammaX_gamm4fgcy;;
Merge_ineq.DIHV_DIH_X;; (* dihV <=> dih_x *)
Merge_ineq.UPS_X_POS;;
cell3_008_from_ineq;;
Merge_ineq.ox3q1h_merge;;
searcht 5 [`gg_mcell`;`gammaX`];;
searcht 25 [`beta_bump_v1`];;
searcht 5 [`beta_bump_force_y`];;
Sphere.beta_bump_force_y;;
Sphere.beta_bump_y;;
searcht 5 [`cc_cell`;`wedge_ge`];;
searcht 5 [`cc_ke`;`3`;`4`];;
searcht 5 [`j < (n:num)`;`f (i:num) = f j`;`f (SUC i) = f (SUC j)`];;
Bump.RIJRIED;;
searcht 5 [`mcell_set`;`cc_cell`];;
searcht 5 [`edgeX`;`cc_cell`];;
searcht 15 [`mcell i V ul = mcell j V ul'`];;
searcht 5 [`barV`;`cc_uh`];;
searcht 5 [`mcell`;`cc_uh`;`cc_cell`];;
Bump.RIJRIED;;
searcht 8 [`EL`;`cc_uh`];;
searcht 5 [`NULLSET`;`cc_cell`];;
DIV_LE;;
searcht 5 [`truncate_simplex (SUC n) (CONS x y)`];;
searcht 5 [`set_of_list`;`CONS`];;
searcht 5 [`truncate_simplex 0`];;
searcht 5 [`inv (&0)`];;
searcht 5 [`&0 <= x pow 2`];;
searcht 5 [`truncate_simplex 0`];;
searcht 15 [`truncate_simplex`;`initial_sublist`];;
searcht 15 [`initial_sublist (CONS x y) (CONS x' y')`];;
searcht 5 [`LENGTH (CONS x y)`];;
searcht 5 [`initial_sublist []`];;
searcht 5 [`LENGTH (truncate_simplex k ul)`];;
searcht 5 [`initial_sublist(truncate_simplex k z) y`];;

searcht 5 [`leaf`;`barV`];;
searcht 5 [`x ==> (?u. p u)`];;
searcht 5 [`(?x. P x) ==> q`;`!`];;
searcht 5 [`P u ==> (?x. P x)`];;
searcht 5 [`(?x. P x) ==> (?x. Q x)`];;
searcht 25 [`p ==> (?x. Q x)`];;
searcht 5 [`x <= (x:num)`];;
searcht 5 [`DROP`];;
searcht 5 [`truncate_simplex`];;
searcht 50 [`APPEND`];;
searcht 50 [`BUTLAST`];;
searcht 5 [`initial_sublist a b /\ initial_sublist b c`];;
searcht 5 [`1 <= LENGTH v`;`CONS`];;
searcht 5 [`initial_sublist`;`BUTLAST`];;
searcht 15 [`initial_sublist`];;
searcht 5 [`APPEND a b = APPEND a' b'`];;
searcht 5 [`initial_sublist x []`];;
searcht 5 [`APPEND x y = []`];;
searcht 5 [`LENGTH x = 0`];;
searcht 5 [`?x y. u = CONS x y`];;
searcht 5 [`CONS x y = []`];;
searcht 5 [`SUC x = SUC y`];;
searcht 5 [`initial_sublist`;`LENGTH`];;
searcht 5 [`APPEND`;`LENGTH`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Merge_ineq.GDRQXLGv2 ;;
(* Feb 11, 2013 *)
searcht 5 [`CONS x (CONS y z)`;`LENGTH`];;
searcht 5 [name "LENGTH_CONS"];;
LENGTH0;;
searcht 5 [`beta_bump_v1`;`sum`];;
searcht 5 [`beta_bump_force_y`];;
searcht 15 [`cc_4`];;
searcht 5 [`leaf_rank`;`leaf`];;
cc_4_cc_cell;;
Oxl_merge.MCELL4_GG;;
searcht  15 [`cc_uh`;`EL`];;
searcht 15 [`beta_bump_v1`];;
searcht 5[`mcell_set`;`cc_cell`];;
searcht 5 [`NULLSET`;`cc_cell`];;
searcht 5 [`barV`;`leaf`];;
searcht 5 [`barV V 3`;`NULLSET`];;
searcht 5 [`barV V 3`;`[u0;u1;f i;f (SUC i)]`];;
searcht 5 [`barV`;`cc_uh`];;
searcht 5 [`radV {a,b}`];;
searcht 15 [`critical_edgeX`];;
searcht 5 [`subcritical_edgeX`];;
searcht 15 [`edgeX`;`mcell4`];;
searcht 15 [`!x. (p ==> Q x)`;`(p ==> (!x. Q x)) <=> r`];;
RIGHT_FORALL_IMP_THM;;
searcht 5 [`edgeX`];;
searcht 5 [`CARD e = 2`;`INSERT`];;
searcht 5 [`edgeX`;`CARD`];;
searcht 5 [`CARD {a,b,c,d}= 4`;`mcell4`;`4`];;
searcht 5 [`CARD`;`~(a = b)`;`4`];;
searcht 5 [`{a,b} = {c,d}`];;
searcht 5 [`gammaX`;`gamma4fgcy`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Merge_ineq.GDRQXLGv2 ;;
(* Feb 12, 2013 *)
searcht 15 [`critical_weight`];;
searcht 5 [`wtcount6_y`;`critical_weight`];;
searcht 5 [`inv x = inv y`];;
searcht 5 [`&x = &y`];;
searcht 5 [`critical_edgeX`;`FINITE`];;
searcht 5 [`CARD( x INSERT y)`];;
searcht 5 [`CARD (Z INTER ((x INSERT y)))`];;
CARD_CLAUSES;;
searcht 5 [`(x INSERT y) INTER Z`];;
searcht 5 [`FINITE (A INTER B)`];;
 MCELL4_EDGE_EXPLICIT;;
searcht 5 [`{} INTER X`];;
searcht 5 [`wtcount3_y`];;
Pack_defs.critical_edgeX;;
searcht 5 [`critical_edgeX`;`critical_edge_y`];;
searcht 5 [`hl [a;b]`];;
searcht 5[`{} INTER x`];;
searcht 5 [`azim_mcell`;`azim`];;
searcht 5 [`azim a b c d < pi`;`mcell4`];;
searcht 15 [`azim`;`dihV`];;
searcht 5 [`leaf`;`collinear`];;
searcht 5 [`dihV`;`dih_y`];;
EDGE_LE_2RAD;;
searcht 5 [`radV`;`rad2_x`];;
 Collect_geom2.NOT_COPLANAR_IMP_CARD4;;
searcht 5 [`coplanar (convex hull s)`];;
searcht 5 [`mcell4`;`convex hull s`];;
searcht 5 [`delta_y`;`coplanar`];;
searcht 5 [`&0 < ups_x (dist(x,y) pow 2) b c`];;
searcht 5 [`collinear`;`coplanar`];;
searcht 5 [`barV`;`set_of_list ul SUBSET V`];;
searcht 5 [`rad2_x a b c d e f = rad2_f a' b' c' d' e' f'`];;
searcht 5 [`&0 <= radV s`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Collect_geom2.NOT_COPLANAR_IMP_CARD4;;
(* Feb 13, 2013 *)
searcht 5 [`x <= radV s`];;
searcht 5 [`affine_dependent`;`coplanar`];;
searcht 5 [`CARD`;`coplanar`];;
searcht 8 [`CARD {a,b,c,d}`;`4`];;
searcht 5 [`radV {a,b}`];;
searcht 15 [`d pow 2 <= r pow 2`];;
searcht 5 [`rad2_x`;`radV`];;
searcht 5 [`{a,b} SUBSET {a',b',c,d}`];;
searcht 5 [`CARD`;`~(a=b)`;`4`];;
searcht 5 [`sqrt2 * sqrt2`];;
searcht 5 [`x pow 2 <= r pow 2`];;
searcht 5 [`&0 <= dist (x,y)`];;
searcht 5 [`sqrt8 = &2 * x`];;
searcht 5 [`cc_eps`];;
searcht 5 [`gg_mcell`];;
searcht 5 [`gammaX`;`gamma4fgcy`];;
searcht 5 [`cc_4`;`beta_bump_v1 V e x = &0`];;
searcht 5 [`hminus`;`sqrt8`];;
searcht 5 [`radV`;`eta_y`];;
searcht 5 [`d3`;`&0 <= eta_y a b c`];;
searcht 5 [`dist`;`d3`];;
searcht 5 [`collinear`;`leaf`];;
searcht 5 [`gamma4fgcy a b c d e f r = gamma4fgcy a' b' c' d' e' f' r`];;
searcht 5 [`rad2_y a b c d e f = rad2_y a' b' c' d' e' f'`];;
searcht 5 [`azim_mcell`;`azim`];;
WEDGE3_Y4;;
searcht 5 [`azim_mcell`;`azim`];;
searcht 5 [`leaf_rank`;`leaf`];;
searcht 5 [`leaf V ul = leaf V ul'`];;
searcht 5 [`s_leaf V ul = s_leaf V ul'`];;
searcht 5 [`s_leaf`;`leaf`];;
searcht 5 [`leaf_rank`;`s_leaf`];;
LEAF_RANK_TRUNCATE;;
searcht 5 [`leaf`;`s_leaf`];;
searcht 5 [`NULLSET`;`cc_cell`];;
searcht 5 [`collinear`;`leaf`];;
searcht 5 [`eta_y a b c = eta_y a' b' c'`];;
searcht 5 [`azim_mcell`;`sum`;`pi`];;
searcht 5 [`hl [u;v]`];;
searcht 5 [`hplus`;`sqrt2`];;
searcht 3 [`sum`;`periodic`];;
searcht 3 [`beta_bump_v1`;`bump`];;
searcht 5 [`periodic`;`gg_mcell`];;
searcht 15 [`sum (a..b)`;`SUC`];;
SUM_CLAUSES_NUMSEG;;
searcht 5 [`sum (a..a)`];;
searcht 5 [`periodic`;`f i = f j`];;
searcht 5 [`azim_mcell`;`sum`;`pi`];;
searcht 5 [`cc_4`;`azim_mcell`;`dih_y`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Collect_geom2.NOT_COPLANAR_IMP_CARD4;;
(* Feb 14, 2013 *)
searcht 5 [`gammaX`;`gamma3f`];;
searcht 5 [`mxi`;`sqrt2`];;
searcht 5 [`barV`;`cc_uh`];;
searcht 5 [`cc_uh`;`EL`];;
searcht 5 [`gamma3f a b c r f =gamma3f a' b' c' r f`];;
searcht 5 [`dih_y a b c d e f = dih_y a' b' c' d' e' f'`];;
searcht 5 [`dihX V X (u0,u1) = dihX V X (u0',u1')`];;
searcht 5 [`mcell_set`;`cc_cell`];;
searcht 5 [`edgeX`;`cc_cell`];;
searcht 5 [`gg_mcell`;`gamma4fgcy`];;
searcht 5 [`gamma4fgcy a b c d e f r = gamma4fgcy a c b d f e r`];;
searcht 5 [`gamma4fgcy a b c d e f r = gamma4fgcy a' b' c' d' e' f' r`];;
searcht 5 [`rad2_y a b c d e f = rad2_y a' b' c' d' e' f'`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Collect_geom2.NOT_COPLANAR_IMP_CARD4;;
(* Feb 15, 2013 *)
searcht 5 [def "beta_bumpA_y"];;
searcht 5 [`beta_bumpA_y`];;
searcht 5 [`cc_4`;`azim_mcell`];;
Merge_ineq.gamma_qx;;
Merge_ineq.gamma10_gamma11;;
searcht 5 [`hminus`;`DECIMAL`];;
searcht 5 [`hplus`;`sqrt`];;
searcht 5 [`wtcount6_y a b c d e f = wtcount6_y a' b' c' d' e' f'`];;
Merge_ineq.cell3_from_ineq_thm;;
searcht 3 [`radV`;`eta_y`];;
searcht 5 [`collinear`;`leaf`];;
searcht 7 [`x pow 2 < y pow 2`];;
searcht 5 [`&0 <= sqrt2`];;
searcht 5 [`&0 <= eta_y a b c`];;
searcht 5 [`sqrt2 * sqrt2`];;
searcht 5 [` gamma4fgcy a b c d e f r > &0`;omit `ineq`];;
searcht 5 [`beta_bumpA_y a b c d e f = &0`];;
searcht 5 [`norm2hh`;omit `ineq`];;
searcht 15 [`hminus`;`sqrt8`;omit `ineq`];;
searcht 5 [`beta_bumpA_y`;`beta_bump_lb`];;
Merge_ineq.beta_bumpA_lb1 ;;
searcht 15 [`#0.008`];;
NO_4CELL_IMP_K3;;
searcht 5 [`sum`;`gg_mcell`];;
gg_mcell;; (*def*)
searcht 5 [`sum`;`azim_mcell`];;
searcht 5 [`sum (x INSERT y)`];;
searcht 5 [`sum `;`DISJOINT`];;
searcht 5 [`cc_cell`;`edgeX`];;
searcht 5 [`mcell_set`;`cc_cell`];;
searcht 5 [`FINITE`;`mcell_set`;`edgeX`];;
searcht 5 [`sum{a,b}`];;
Merge_ineq.cell3_008_from_ineq_ALT;;
searcht 15 [`mcell3`;`gammaX`];;
searcht 15 [`edgeX`;`mcell`];;
searcht 5 [`mcell_set`;`mcell`];;
searcht 5 [`mcell`;name "FIRST"];;
Bump.LENGTH4;;
Pack_defs.critical_weight;;
searcht 5 [`critical_edgeX`;`edgeX`];;
searcht 5 [`CARD x = 1`];;
searcht 5 [`SING`];;
searcht 8 [`edgeX`;`?`];;
searcht 5 [`edgeX`;`mcell3`];;
searcht 5 [`edgeX`;`mcell4`;`{a,b,c,d}`];;
searcht 5 [`mcell3`;`CARD`];;
searcht 5 [`truncate_simplex 2 [a;b;c;d]`];;
searcht 5 [`cc_uh V ul = CONS x y`];;
searcht 5 [`{a,b}= {c,d} ==> dist u = dist v`];;
searcht 5[`leaf`;`barV`;`cc_uh`];;
searcht 5 [`beta_bump_v1`;`mcell`];;
Bump.MCELL_BUMP_0;;
searcht 5 [`sum A f <= sum A g`];;
searcht 5 [`edgeX`;`NULLSET`];;
searcht 5 [`mcell2`;`#0.008`];;
searcht 50 [`gamma2_x_div_azim_v2`];;
Tskajxy.GAMMAX_GAMMA2_X;;
searcht 5 [`dihX`;`mcell2`];;
Tskajxy.MCELL2_DIHX;;
searcht 5 [`mcell2`;name "FIRST"];;
searcht 5 [`a * x < b * x`];;
searcht 5 [`gamma2_x_div_azim_v2`;`gamma2_x_div_a_v2`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Collect_geom2.NOT_COPLANAR_IMP_CARD4;;
(* Feb 17, 2013 *)
searcht 5 [`hminus`;`<`];;
 Tskajxy.GAMMAX_GAMMA2_X;;
searcht 5 [`dihV`;`dih_y`];;
searcht 5 [`collinear`;`mxi`];;
searcht 5 [`coplanar`;`mcell2`];;
searcht 5 [`coplanar`;`collinear`];;
critical_weight_wtcount6_y;;
searcht 5 [`inv(x) = y`];;
searcht 5 [`CARD x = 1`];;
searcht 5 [`x IN critical_edgeX V X`];;
searcht 5 [`edgeX`;`mcell2`];;
searcht 5 [`gammaX`;`gamma3f`];;
searcht 5 [`critical_weight`;`cc_cell`];;
critical_weight_wtcount6_y;;
critical_edgeX_critical_edge_y;;
searcht 5 [`critical_edgeX`;`edgeX`];;
searcht 5 [`CARD ((x INSERT y) INTER b)`];;
searcht 5 [`FINITE`;`critical_edgeX`];;
searcht 5 [`{} INTER B`];;
searcht 5 [`critical_edgeX`;`edgeX`];;
searcht 5 [`CARD`;`4`;`mcell3`];;
searcht 5 [`CARD s = 4`;`~(a = b)`];;
searcht 5 [`gamma3f a b c r f = gamma3f a' b' c' r f`];;
searcht 5 [`eta_y a b c = eta_y a' b' c'`];;
searcht 5 [`&0 <= sum s f`];;
searcht 5 [`&0 <= dihX V x e`];;
searcht 5 [`critical_weight`;`cc_cell`];;
searcht 5 [`&0 <= critical_weight V X`];;
searcht 15 [`&0`;`gammaX V X f`];;
searcht 15 [`beta_bump_v1`;`cc_cell`];;
searcht 5 [`sum s f = sum s g`];;
searcht 5 [`beta_bump_v1 V e X = &0`];;
searcht 5 [`a * sum s f`];;
real_model_008;;
searcht 5 [`mcell3`;name "FIRST"];;
Bump.MCELL_BUMP_0;;
searcht 5 [`cc_cell`;`edgeX`];;
searcht 5 [`dihX V X e = dihX V X e'`];;
searcht 5 [`mcell_set`;`cc_cell`];;
searcht 5 [`azim_mcell`;`azim`];;
Sphere.b_spine5;;
searcht 5 [`gamma23_keep135_x`;name "Merge_ineq"];;
TSKAJXY_3;;
searcht 5 [`gammaX`;`gamma3f`];;
searcht 5 [`gamma23_keep135_x`;`gamma3f`];;
searcht 5 [`gamma23_keep135_x`;omit `ineq`];;
searcht 5 [`gamma3_x`;`gamma3f`];;
Merge_ineq.gamma3_x_gamma3f;;
searcht 5 [`dih_y`;`dih_x`];;
searcht 5 [`hl`;`eta_y`];;
Collect_geom.POW2_COND_LT;;
searcht 5 [`&0 <= eta_y a b c`];;
searcht 5 [`sqrt2 * sqrt2`];;
bb 4;;
Nonlinear_lemma.dih_y_sym;;
searcht 5 [`dihX`;`cc_cell`;`dih_y`];;
mcell3_dihX_dih_y;;
searcht 5 [`gamma23_full8_x`;`gamma3f`];;
searcht 15 [`delta_x4`;`dih_x`];;
searcht 5 [`&0 < atn2 (x,y)`];;
searcht 5 [`&0 < atn x`];;
searcht 8 [`&0 < x / y`];;
searcht 5 [`hminus`;`DECIMAL`];;
searcht 5[`abs (atn x)`];;
searcht 5 [`tan(atn x)`];;
searcht 5 [`sqrt s pow 2 = s`];;
searcht 6 [`x pow 2 < y pow 2`];;
searcht 5 [`x pow 2`;`eta_y`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Collect_geom2.NOT_COPLANAR_IMP_CARD4;;
(* Feb 18, 2013 *)
critical_weight1;;
searcht 5 [`azim a b c d`;`dih_y`];;
searcht 10 [`azim`;`dihV`];;
searcht 5 [`dih_y`;`dihV`];;
searcht 5 [`&2 <= dist(f i,f (SUC i))`];;
searcht 5 [`radV`;`azim a b c d < pi`];;
searcht 5 [`radV`;`rad2_x`];;
searcht 15 [`coplanar`;`azim`];;
searcht 5 [`coplanar`;`f i`;`f (SUC i)`];;
searcht 15 [`azim a b c d = &0`];;
searcht 5 [`leaf`;`cc_A0`];;
searcht 3 [`aff_gt a s = aff_gt a t`];;
searcht 5 [`aff_gt`;`aff_ge {a,b} {c}`];;
searcht 5 [`collinear`;`DISJOINT`];;
searcht 5 [`aff`;`collinear`];;
searcht 5 [`dist(x,x)`];;
searcht 5 [`delta_y`;`coplanar`];;
GSYM (REWRITE_RULE[LET_DEF;LET_END_DEF] GDRQXLGv3);;
searcht 5 [`&2`;`sqrt2 * sqrt2`];;
searcht 10 [`x pow 2 <= y pow 2`];;
searcht 5 [`eta_y`;`x pow 2 <= y pow 2`];;
searcht 5 [name "TRUNCATE_SIMPLEX_EXPLICIT"];;
searcht 5 [name "Local_lemmas1"];;
searcht 5 [name "Local_lemmas"];;
searcht 5 [name "MOD_REFL"];;
MOD_REFL;;
searcht 5 [name "EJR"];;
REBIND_TAC;;
searcht 5 [`x IN {(a,b) | f a b}`];;
searcht 15 [`!(x:A#B). f x`;`!a b. g a b`];;
FORALL_PAIR_THM;;
  IN_ELIM_PAIR_THM;;
searcht 5 [`(a,b) = (c,d)`];;
searcht 5 [`?v. P v /\ (p = v)`];;
searcht 50 [`fan`];;
searcht 50 [`base_point_fan`];;
searcht  15 [`FAN`;`SUBSET`];;
searcht 5 [`dart1_of_fan`;`dart_of_fan`];;
	    type_of `nn_of_hyp`;;
searcht 5 [`azim_cycle`;`sigma_fan`];;
searcht 5 [`azim_cycle`];;
searcht 5 [`ivs_azim_cycle`;`inverse_sigma_fan`];;
searcht 50 [`dih2k`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Collect_geom2.NOT_COPLANAR_IMP_CARD4;;
(* Feb 21, 2013 *)
searcht 5 [`gamma2_x1_div_a`];;
searcht 5 [`gamma2_x1_div_a`];;
searcht 5 [`beta`];;
searcht 5 [`radius`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.ALL_TO_THE_NONPARALLEL_PART;;
(* Feb 22, 2013 *)
searcht 5 [`azim_cycle`];;
searcht 5 [`sigma_fan`];;
searcht 15 [`inverse1_sigma_fan`;`inverse_sigma_fan`];;
searcht 15 [`ivs_azim_cycle`];;
searcht 25 [`dart1_of_fan`];;
searcht 5 [name "PAIR";name "ELIM"];;
searcht 5 [`{a,b} = {b,a}`];;
searcht 5 [`ivs_azim_cycle {}`];;
searcht 5 [`wedge_ge`];;
searcht 15 [`hypermap`];;
	    type_of `face_set`;;
searcht 5 [`face`;`face_set`];;
searcht 5 [name "WRGC"];;
	    type_of `hro`;;
dest_const `hro`;;
	    type_of `rho`;;
searcht 5 [def "rho_node"];;
	    type_of `rho_node1`;;
searcht 50 [`rho_node1`;`FST`];;
searcht 5 [`rho_node1`];;
searcht 5 [name "KOMW"];;
searcht 15 [`rho_node`];;
searcht 5 [`rho_node1`;`rho_node`];;
searcht 5 [`per`];;
searcht 5 [`perimeterbound`];;
searcht 5 [`tauVEF`];;
searcht 50 [`w_node_fan`];;
searcht 50 [`wedge_in_fan_ge`];;
searcht 50 [name "LVD"];;
searcht 5 [name "RNS"];;
searcht 5 [def "fan80"];;
searcht 5 [`fan80`;`convex_local_fan`];;
searcht 5 [name "CIZM"];;
searcht 25 [def "flat"];;
searcht 5 [name "LDU"];;
searcht 5 [def "straight"];;
searcht 5 [name "HKIR"];;
searcht 5 [name "interior_angle1"];;
searcht 50 [`interior_angle1`];;
searcht 5 [name "KOM"];;
searcht 5 [name "OZQ"];;
searcht 5 [name "KCH"];;
searcht 5 [name "HKIR"];;
searcht 5 [name "EGH"];;
searcht 5 [name "deform"];;
searcht 5 [name "XRECQNS"];;
searcht 5 [`deformation`];;
searcht 5 [def "^FAN$"];;
searcht 5 [`FAN`;`fan1`];;
searcht 5 [`slicef`;`f_slice`];;
searcht 15 [def "slice"];;
searcht 5 [`slice`];;
searcht 50 [`f_slice`];;
List.length (searcht 50 [`e_slice`]);;
searcht 50 [`slicef`];;
searcht 50 [name "Wjs"];;
searcht 100 [name "Dih2k_";omit `vecmat`;omit `dimindex`];;
searcht 5 [name "COMPATIBLE_BW_TWO_LEMMAS2"];;
searcht 5 [name "EJRC"];;
searcht 5 [`sol_local_fan`];;
searcht 5 [`azim_in_fan`;`interior_angle1`];;
searcht 5 [`v_prime`];;
searcht 5 [`v_prime`];;
searcht 8 [`aff_gt {x} {y}`];;
searcht 5 [`v IN wedge a v c d`];;
searcht 5 [`collinear {a,b}`];;
searcht 7 [`v,rho_node1 FF v`];;
searcht 5 [`convex_local_fan`;`local_fan`];;
searcht 5 [`{v,v} = {v}`];;
searcht 5 [`CARD`;`EE`];;
searcht 1 [`v IN v_prime V U`];;
searcht 5 [`face HS a INTER face HS b`];;
searcht 5 [`mdtau_y_LC`];;
searcht 1 [`safesqrt`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.ALL_TO_THE_NONPARALLEL_PART;;
(* Feb 23, 2013 *)
searcht 50 [name "Arc_properties"];;
searcht 50 [name "Ckqowsa."];;
searcht 50 [`tame_13a`];;
	    type_of `deformation`;;
	    type_of `localization`;;
	    type_of `taum`;;
searcht 5 [def "stable"];;
searcht 5 [`lunar_deform`];;
searcht 5 [`spherical_map`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types SUM_CONST;;
(* Feb 24, 2013 *)
searcht 5 [`slicev`;`CARD`];;
searcht 15 [`v_prime`;`CARD`];;
Local_lemmas1.CONDS_IN_V_PRIME_NUM2;;
Local_lemmas1.CONDS_IN_V_PRIME_NUM;;
searcht 5 [`rho_node1`;`v_prime`];;
Local_lemmas1.POINTS_IN_HAFL_CIRCLE;;
searcht 5 [`ITER i (rho_node1 FF) v IN V`];;
Local_lemmas.LOCAL_FAN_ORBIT_MAP_VITER;;
searcht 5 [`ITER n f v = v`;`rho_node1`];;
Local_lemmas1.CARD_IS_LEAST_CYCLE;;
Local_lemmas.LOFA_IMP_ITER_RHO_NODE_ID;;
searcht 5 [`POWER`;`ITER`];;
searcht 5 [`ITER`;`?`;`rho_node1`;`v IN V`];;
searcht 5 [`coplanar`;`plane`];;
searcht 50 [`plane`];;
searcht 5 [`local_fan`;`CARD`;`EE`];;
searcht 5 [`aff_gt`;`wedge`];;
searcht 5 [`wedge`;`wedge_ge`];;
searcht 5 [`wedge_in_fan_gt`;`{}`];;
searcht 5 [`wedge a b c d = {}`];;
searcht 5 [`collinear`;`rho_node1`];;
searcht 5 [`collinear`;`azim_cycle`];;
searcht 5 [`collinear`;`ivs_rho_node1`];;
searcht 5 [`azim_cycle`;`ivs_rho_node1`];;
searcht 5 [`x IN wedge a b c d `];;
searcht 5 [`azim a b c c`];;
searcht 5 [`ITER (a + b)`];;
searcht 5 [`slicev`;`v_prime`];;
searcht 5 [`local_fan`;`convex_local_fan`];;
searcht 15 [`CARD`;`local_fan`];;
Local_lemmas.ITER_CARD_MINUS1_EQ_IVS_RN1;;
searcht 5 [`ITER 0`];;
searcht 5 [`{a,b}={b,a}`];;
searcht 5 [`(w,rho_node1 FF w) IN FF`];;
searcht 5 [`rho_node1 FF (ivs_rho_node1 FF v)`];;
REWRITE_RULE[TAUT `(a ==> b ==> c) <=> (a /\ b ==> c)`] Local_lemmas1.CONDS_IN_V_PRIME_NUM;;
searcht 5 [`A SUBSET B /\ B SUBSET A`];;
searcht 5 [`v_prime`;`ITER`];;
searcht 5 [`v_prime V f SUBSET E`];;
help_grep "GEN";;
help "X_GEN_TAC";;
searcht 5 [`ITER i (rho_node1 FF) v = ITER j (rho_node1 FF) v`];;
Local_lemmas1.LT_CARD_MONO_LOFA;;
searcht 5 [`IMAGE`;`CARD`];;
searcht 5 [`FINITE`;`{i | (i:num) < j}`];;
searcht 5[ `CARD`;`{i | (i:num) < j}`];;
searcht 5 [`{a,b,c} = {a,c,b}`];;
searcht 50 [`generic`];;
searcht 5 [`e_prime`;`SUBSET`];;
searcht 5 [`aff_ge {a} {b,c}`;`aff_gt`];;
searcht 5 [`(A UNION B) INTER C`];;
searcht 5 [`A SUBSET {}`];;
searcht 5 [`A UNION B SUBSET C`];;
searcht 5 [`aff_ge`;`aff_lt`;`collinear`];;
searcht 5 [`aff_ge {a} {b}`;`aff_lt`];;
searcht 5 [def "aff_ge"];;
searcht 5 [`aff_ge`;`affsign`];;
searcht 5 [`aff_ge {a} {b}`;`%`];;
searcht 5 [`aff_lt {a} {b}`;`%`];;
searcht 5 [`a % (vec 0)`];;
searcht 5 [`a % y = vec 0`];;
searcht 5 [`v_prime V fw SUBSET V`];;
searcht 5 [`wedge_in_fan_gt`;`wedge`];;
searcht 5 [`vec 0 + a`];;
searcht 5 [`collinear`;`%`];;
searcht 5 [`interior_angle1`;`azim_in_fan`];;
searcht 5 [`convex_local_fan`;`slicev`];;
Local_lemmas1.CONDS_IN_V_PRIME_NUM2;;
Polar_fan.LOCAL_FAN_ORBIT_MAP_EXPLICIT;;
searcht 15 [`IMAGE`;`?`];;
searcht 5 [`SURJ`;`IMAGE`];;
searcht 5 [`CARD V = 0`];;
searcht 5 [`FINITE`;`local_fan`];;
searcht 5 [`BIJ`;`CARD`];;
searcht 5 [`FINITE {i | i < (j:num)}`];;
searcht 5 [`FINITE(A UNION B)`];;
searcht 5 [`CARD`;`{x  | f (x:num)}`;`(a:num) <= b`];;
searcht 5 [`CARD (A UNION B)`];;
searcht 5 [`ITER 0`];;
searcht 5 [`CARD {}`];;
searcht 5 [`CARD {a}`];;
searcht 5 [`CARD (a..b)`];;
searcht 5 [`a..b`;`(a:num) <= x`];;
searcht 15 [`local_fan`;`CARD:(real^3#real^3->bool) -> num`];;
searcht 25 [`local_fan`;`CARD:(real^3->bool) -> num`];;
Local_lemmas1.LOFA_HYP_UNION_CARD_GT2;;
searcht 15 [`ITER`;`rho_node1`];;
Local_lemmas.LOFA_IMP_LT_CARD_SET_V;;
searcht 5 [`BIJ`;`sum`];;
searcht 5 [`BIJ`;`local_fan`];;
searcht 5 [`sum V f = sum V g`];;
searcht 5 [`interior_angle1`;`azim_in_fan`];;
searcht 15 [`v,rho_node1 FF v`];;
searcht 5 [`local_fan`;`FST x IN V`];;
searcht 5[`local_fan`;`v_prime V fv,E,f`];;
Local_lemmas1.HAFL_CIRCLE_FORM_LOCAL_FAN;;
Local_lemmas1.HAFL_CIRCLE_FORM_LOCAL_FAN2;;
searcht 5[`&a + &b`];;
searcht 5 [`&a = &b`];;
X_CHOOSE_TAC;;
searcht 5 [`BIJ`;`CARD`];; 
searcht 5 [`FINITE`;`local_fan`];;
bb 4;;
Local_lemmas.LOFA_IMP_CARD_FF_V_EQ;;
searcht 5 [`2 <= CARD x`];;
searcht 5 [`FINITE`;`local_fan`];;
searcht 5 [`sum V (\e. f e - c)`];;
searcht 5 [`sum V (\e. f)`;`CARD`];;
searcht 5 [`x:(A->B)->(C->D)->(A#C) -> (B#D)`];;
searcht 15 [`x:(A->B->C)->(B->A->C)`];;
searcht 10 [def "rho"];;
searcht 5 [`sol_y`;`sol0`];;
searcht 5 [`taum`;`rho_fun`];;
searcht 5 [`rhazim`];;
searcht 5 [`const1`];;
searcht 5 [`ly`;`lnfun`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types SUM_CONST;;
(* Feb 26, 2013 *)
searcht 5 [`tau_fun`];;
searcht 5 [`main_estimate`];;
searcht 5 [`#0.206`];;
searcht 5 [`d_tame`];;
searcht 5 [`diagonal1`];;
searcht 5 [`protracted`];;
searcht 5 [name "AZIM_SUM"];;
searcht 5 [`arcV`;`rho_node`];;
searcht 20 [`azim a b c d + azim a b d e`];;
searcht 5 [name "KCH"];;
searcht 5 [`azim a b c d = pi`];;
searcht 5 [def "power_map_points"];;
searcht 5 [`power_map_points`];;
searcht 5 [`sigma_fan`;`rho_node1`];;
searcht 5 [def "if_azims_fan"];;
searcht 23 [`azim_cycle`];;
searcht 5 [`azim_cycle`;`cyclic_set`];;
TOPOLOGICAL_SORT;;


print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types SUM_CONST;;
(* March 2, 2013 *)
searcht 5 [`(a:num) * i + b`;`MOD`];;
searcht 5 [`k - (a MOD k)`];;
searcht 5 [`MOD`;`(a:num) - b`];;
searcht 5 [`FST (SND (SND(SND x)))`];;
searcht 10 [def "^part"];;
	    type_of `taustar`;;
searcht 5 [`{i | F}`];;

searcht 5 [`part0`];;

(* March 18, 2013 *)
help "MK_CONJ";;
searcht 5 [`periodic {}`];;
searcht 50 [def "min"];;
searcht 5 [`min_num`];;
	    type_of `MEM`;;
searcht 5 [`sol_local`];;
searcht 5 [`circular`];;
searcht 5 [`azim_in_fan`];;
	    type_of `min_num`;;
	    type_of `max_num`;;
searcht 5 [`max_num`];;
	    type_of `max`;;
	    type_of `scs_J_v4 s`;;

(* March 31, 2013 *)
searcht 5 [def "assoc"];;
searcht 5 [`ASSOC`];;
searcht 5 [`MAP`];;

(* April 1, 2013 *)
help_grep "list";;
listof;;
mk_list;;
help "mk_flist";;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types SUM_CONST;;

(* April 17, 2013 *)
searcht 15 [`taum`];;
searcht 0 [`&0 <= sol_x y1 y2 y3 y4 y5 y6`];;
searcht 0 [`sol_x`;`&0`];;
searcht 0 [`&0`;`sol_euler_x_div_sqrtdelta`];;
searcht 0 [`sol_x`;`x:real^3`];;
searcht 5 [`sol`;`dihV`];;
searcht 2 [`&0 <= sol a b`];;
Counting_spheres.SOL_NN;;
searcht 5 [`sol`;`dihV`];;
Tskajxy_lemmas.SOL_SOLID_TRIANGLE;;
searcht 5 [`dihV`;`dih_y`];;
Merge_ineq.DIHV_EQ_DIH_Y;;
searcht 5 [`dih_y`;`sol_y`];;
Sphere.sol_y;;
searcht 5 [`sol_x`;`sol_y`];;
searcht 5 [`collinear`;`ups_x`];;
searcht 5 [`ups_x a b c = ups_x a' b' c'`];;
searcht 5 [`radial`;`convex hull S`];;
MEASURABLE_RULES;;
searcht 5 [`primitive`];;
searcht 5 [`solid_triangle`];;
searcht 5 [`convex hull S`;` normball`];;
searcht 5 [name "NKEZ"];;
searcht 5 [`sol_local`];;
Nkezbfc_local.SOL_LOCAL_FAN_POS_CASE3;;
searcht 15 [`convex_local_fan`];;
searcht 5 [`local_fan`;`convex_local_fan`;`FAN`];;
searcht 5 [def "sol_eu"];;
searcht 5 [`sol_euler_x`;`delta_x`];;
searcht 15 [`eulerA_x`;omit `compose6`];;
searcht 5 [`cstab`;`DECIMAL`];;
Merge_ineq.sol_x_sol_euler_x;;
searcht 5 [`eulerA_x`;`sqrt`];;
Sphere.eulerA_x;;
searcht 5 [`sqrt(x  *y) = sqrt x * sqrt y`];;
searcht 5 [`&0 < atn2 (x,y)`];;
Euler_main_theorem.EULER_TRIANGLE;;
searcht 5 [`dihV`;`coplanar`];;
DIHV_EQ_0_PI_EQ_COPLANAR;;
searcht 5 [`dih_x a b c d e f = pi`];;
searcht 15 [`azim a b c d = pi`];;
searcht 5 [`atn2`;`dih_x`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types SUM_CONST;;
(* April 17, 2013 *)
searcht 5 [`cstab`;`DECIMAL`];;
searcht 5 [`sol0`;`DECIMAL`];;
searcht 5 [`taum`;`rhazim`];;
searcht 5 [`&0 <= dih_x y1 y2 y3 y4 y5 y6`];;
searcht 5 [`dih_y`;`rhazim`];;
searcht 5 [`a <= rho y`];;
searcht 5 [`ly y`;`<=`];;
searcht 5 [`rho`;`lfun`];;
searcht 5 [`rho y = x`];;
searcht 5 [`const1`;`DECIMAL`];;
searcht 5 [`const1`;`sol0`];;
searcht 5 [`atn2(&0,x)`];;
searcht 5 [`delta_y y1 y2 y3 y4 y5 y6 = delta_y y'1 y'2 y'3 y'4 y'5 y'6`];;
searcht 5 [`delta_x y1 y2 y3 y4 y5 y6 = delta_x y'1 y'2 y'3 y'4 y'5 y'6`];;
searcht 5 [`delta_x4 y1 y2 y3 y4 y5 y6 = delta_x4 y'1 y'2 y'3 y'4 y'5 y'6`];;
UNDISCH;;
searcht 5 [`dih_x y1 y2 y3 y4 y5 y6`;`pi`];;
searcht 5 [`sol_y`;`sol_x`];;
searcht 5 [`ups_x (a*a) (b*b) (c*c) = u`];;
searcht 5 [`tau_residual_x`];;
searcht 15 [`taum_x`;`rhazim_x`];;
searcht 5 [`taum`;`rhazim`];;
searcht 5 [`sqrt(x * x)`];;
searcht 5 [`scs_k_v14 s = y`];;
searcht 5 [`{i | F}`];;
searcht 5 [`sum {}`];;
searcht 5 [`ball_annulus`;`h0`];;
searcht 5 [`IMAGE v S SUBSET U`];;
searcht 5 [`taum`;`rhazim`];;
searcht 5 [`rhazim`;`rho`];;
searcht 5 [`dihV`;`dih_y`];;
searcht 5 [`collinear`;`ball_annulus`];;
Dih2k_hypermap.NONPARALLEL_BALL_ANNULUS;;
searcht 5 [`norm`;`dist`;`vec 0`];;
searcht 5 [`norm(v-w)`;`dist`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types SUM_CONST;;
(* April 19, 2013 *)
searcht 5 [`&0`;`delta`;`dist`];;
searcht 5 [`delta`;`delta_x`];;
searcht 5 [`norm`;`dist`];;
searcht 5 [`MAP f []`];;
searcht 5 [`i MOD k = i`];;
searcht 5 [`d_tame2`];;
Sphere.tame_table_d;;
help_grep "recurs";;
help "new_recursive_definition";;
searcht 5 [`(a,b) = (c,d)`];;
	    type_of `ASSOCD_v18`;;
searcht 5 [`taum y1 y2 y3 y4 y5 y6 = taum y'1 y'2 y'3 y'4 y'5 y'6`];;
searcht 5 [`taum`;`rhazim`];;
searcht 5 [`&2 <= sqrt8`];;
0.103 +. 2.0 *. 0.2759;;
NONPARALLEL_BALL_ANNULUS_ALT;;
searcht 5 [`tau_star`];;
searcht 5 [`tauVEF`;`tau_fun`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types SUM_CONST;;
(* April 21, 2013 *)
searcht 5 [name "EJRCF"];;
searcht 2 [`tauVEF`];;
searcht 5 [`tau_fun`;`sum`];;
searcht 5 [`azim_dart`;`azim_in_fan`];;
searcht 5 [def "azim_dart"];;
searcht 5 [`azim_dart`;`azim_fan`];;
searcht 5 [`azim_in_fan`;`azim_fan`];;
searcht 5 [`azim_fan`];;
searcht 5 [`periodic {}`];;
searcht 5 [`a * k + b:num`;`MOD`];;
searcht 5 [`periodic`;`MOD`];;
searcht 5 [`i MOD k < k`];;
searcht 5 [`(a ==> b) /\ (b ==> a)`];;
searcht 15 [`(!x. (a ==> b x)) <=> (a ==> (!x. b x))`];;
[GSYM (REWRITE_RULE[Appendix.periodic2] periodic2_funlist);GSYM (REWRITE_RULE[Appendix.periodic2] periodic2_funlistA)];;
searcht 15 [`sum (A INTER B)`;`FINITE`];;
searcht 5 [`sum`;`SUBSET`;`&0`];;
searcht 5 [`sum A f = sum A g`];;
SUM_CLAUSES;;
searcht 5 [`FINITE`;`INSERT`];;
searcht 5 [`tau_fun`];;
searcht 5 [`tau_fun`;`tauVEF`];;
searcht 15 [`tauVEF`];;
searcht 5 [`&(x - y) `];;
searcht 5 [`rho_fun`;`lmfun`];;
searcht 5 [`azim_dart`;`azim_in_fan`];;
searcht 5 [`rho_fun`;`rho`];;
searcht 5 [`rho`;`lfun`];;
searcht 5 [`rho_fun`];;
searcht 5 [`rho`];;
searcht 5 [`const1`;`sol0`];;
searcht 5 [`ly`;`lmfun`];;
searcht 5 [`azim_dart`;`azim_fan`];;
searcht 0 [`azim_in_fan`;`azim_fan`];;
searcht 0 [`azim_in_fan`;`azim_dart`];;
Tame_general.ly_EQ_lmfun;;
searcht 5 [`ly`;`lfun`];;
searcht 5 [def "azim_in_fan"];;
searcht 5 [`azim_cycle`;`sigma_fan`];;
searcht 5 [`azim_cycle`;`azim_in_fan`];;
searcht 5 [`azim`;`azim_fan`];;
searcht 5 [def "EE"];;
searcht 5 [def "set_of_edge"];;
searcht 5 [`set_of_edge`;`EE`];;
searcht 5 [`FAN`;`CARD`;`2`];;
searcht 15 [def "FAN"];;
searcht 5 [`FAN`;`fan2`];;
searcht 5 [`graph`];;
searcht 5 [`HAS_SIZE`;`{x}`];;
searcht 5 [`FAN`;`local_fan`];;
searcht 5 [`local_fan`;`{x,y} IN E`];;
searcht 5 [`(i + k) MOD k`];;
searcht 5 [`(a * k + b) MOD k`];;
searcht 5 [`k MOD k`];;
searcht 5 [`0 MOD k`];;
searcht 5 [`CARD x <= CARD y`];;
searcht 5 [`CARD {i | i <= k }`];;
searcht 5 [`CARD (i..j)`];;
searcht 5 [`(i..j)`;`{i | a <= i /\  i <= j}`];;
IN_NUMSEG;;
searcht 5 [`FINITE`;`{i | i < (j:num)}`];;
Appendix.is_ear_v25;;
searcht 5 [`unadorned_v25`];;
searcht 5 [`i < j /\ j < 3`];;
searcht 15 [`sqrt8`;`cstab`];;
searcht 5 [`funlist_v25 a a0 k i i`];;
searcht 5 [`k MOD k`];;


print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types SUM_CONST;;
(* May 3, 2013 *)
searcht 5 [`{i | F}`];;
searcht 5 [`a INSERT  b = {}`];;
Appendix.get_main_nonlinear "5691615370";;
searcht 5 [`IMAGE f X SUBSET Y`];;
searcht 5 [`x IN ball_annulus`];;
searcht 5 [`i MOD k = i`];;
searcht 5 [`k MOD k`];;
searcht 5 [`&0 <= delta_y a b c d e f`;`dist`];;
bb 3;;
Appendix.taustar_v25;;
Merge_ineq.REAL_FINITE_MAX_EXISTS;;
searcht 5 [name "WLOG";`P x1 x2 = P x2 x1`];;
searcht 5 [`enclosed`];;
searcht 5 [name "dart_leads_into_eq1"];;
Appendix.tau_fun;;
searcht 5 [`IMAGE`;`periodic`;`(:A)`];;
searcht 5 [`CARD (IMAGE f X)`];;
searcht 5[`FINITE`;`{i | (i:num) < j}`];;
searcht 5 [`CARD`;`{i | (i:num) < j}`];;
searcht 5 [`&(a - b)`];;
searcht 5 [`sum (IMAGE f X) g`];;
searcht  15 [`azim_in_fan`;`azim`];;
searcht 5 [`azim_cycle`];;
searcht 5 [def "EE"];;
searcht 5 [`EE`;`IN`];;
help_grep "GEN";;
searcht 5 [`x IN IMAGE f X`];;
searcht 5 [`{a,b} = {c,d}`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types SUM_CONST;;
(* May 4, 2013 *)
searcht 5 [`periodic`;`vv (i:num) = ww (i:num)`];;
searcht 5 [name "FORALL";name "IMP"];;
DIVISION;;
searcht 5 [`(i + j) MOD k`];;
searcht 5 [`(a * k + v) MOD k`];;
searcht 5 [`azim_cycle`;`{a,b}`];;
searcht 5 [`CARD {a,b}`];;
searcht 5 [`~(a MOD k = b MOD k)`];;
searcht 5 [`dist(x,x)`];;
searcht 5 [`i MOD k = i`];;
searcht 5 [`azim a b c d = azim a b c e + azim a b e d`];;
searcht 7 [`sum ( a INSERT  b)`];;
searcht 5 [`convex_local_fan`;`azim_in_fan`];;
Local_lemmas.PRIOR_TO_LESS_THAN_PI_LEMMA;;
searcht 5 [`azim_in_fan`];;
Localization.azim_in_fan;;
searcht 5 [`rho_node1`];;
searcht 15 [`rho_node1`;name "Localization."];;
searcht 5 [`periodic`;`MOD`];;
searcht 5 [`(a + j) MOD k`];;
searcht 5 [`collinear`;`ball_annulus`];;
searcht 5 [`ball_annulus = x`];;
searcht 5 [`x pow 2 <= y pow 2 `];;
searcht 5 [`IMAGE f X SUBSET Y`];;
searcht 5 [name "EGH"];;
Local_lemmas.EGHNAVX;;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
(* May 5, 2013 *)
help_grep "SPEC";;
help "GEN";;
searcht 5 [`ITER 0`];;
searcht 5 [`ITER (SUC j)`];;
searcht 5 [`periodic`;`IMAGE`];;
searcht 5 [`CARD (IMAGE f X)`];;
 NONPARALLEL_BALL_ANNULUS40_ALT;;
searcht 5 [def "generic"];;
searcht 5 [`generic`];;
searcht 5 [`local_fan`;`convex_local_fan`];;
Appendix.cstab;;
searcht 15 [name "Tame_lemmas";`reflection`;`dist`];;
searcht 15 [name "enclosed";name "Tame_lemmas."];;
searcht 15 [name "quadratic";name "Tame_lemmas."];;
searcht 50 [name "Tame_lemmas."];;
searcht 5 [`enclosed`];;
Collect_geom2.CAYLEYR_5POINTS;;
 POW_2_SQRT ;;

DIST_POS_LE;;
searcht 5 [`muR`];;
Mur.muR;;
searcht 5 [`cayleyR`];;
searcht 5 [`abc_of_quadratic`];;
searcht 5 [`delta`;`delta_x`];;
searcht 3 [`&0 <= delta_y a b c d e f`;`dist`];;
DELTA_Y_POS_4POINTS;;
searcht 50 [`chi_msb`];;
Tame_lemmas.delta_y_pos;;
searcht 5 [`vec 0`;`cross`];;
searcht 5 [`dot`;`cross`;`&0`];;
searcht 5 [`cross`;`ups_x`];;
searcht 5 [`&0 < norm x`];;
searcht 5 [`&0 < x pow 2`];;
searcht 5 [`&0 <= norm x`];;
searcht 5 [`&0 < x dot x`];;
searcht 5 [`chi_msb`;`coplanar`];;
searcht 15 [`chi_msb`;`cross`];;
searcht  5 [`coplanar`;`delta_y`];;
REWRITE_RULE[LET_DEF;LET_END_DEF] enclosed4_lemma;;
searcht 5 [`ups_x`;`collinear`];;
GSYM Collect_geom2.NOT_COL_EQ_UPS_X_POS;;
searcht 5 [`collinear`;`ball_annulus`];;
searcht 5 [`chi_msb`;`coplanar`];;
searcht 15 [`coplanar`;`collinear`];;
searcht 5 [`convex_local_fan`;`pi`];;
Local_lemmas.IN_V_IMP_AZIM_LESS_PI;;
searcht 5 [name "Terminal";`rho_node1`];;
searcht 5 [`wedge_ge`;`aff_ge`];;
searcht 5 [`wedge_ge`;`azim`];;
bb 3;;
vv_azim_le;;
REWRITE_RULE[LET_THM] vv_azim_le;;
searcht 15 [`vv i = vv j`;`periodic`;`MOD`];;
Local_lemmas.WEDGE_GE_EQ_AFF_GE;;
searcht 15 [`aff_ge {a,b} {c,d}`;`%`];;
searcht 5 [`DISJOINT`;`collinear`];;
searcht 5 [`DISJOINT {a,b} {c,d}`];;
searcht 5 [`chi_msb`;`%`];;
searcht 5 [`chi_msb [a;b;c] d = -- chi_msb [a;b;c'] d'`];;
searcht 5 [`azim a b c d = pi`;`aff`;`INTER`];;
searcht 5 [`conv0`;`%`];;
searcht 5 [`chi_msb u d = &0`];;
searcht 5 [`set_of_list [a;b;c]`];;
searcht 15 [`aff {u,v}`];;
searcht 5 [`affine (P hull s)`];;
searcht 5 [`S SUBSET P hull S`];;
searcht 5 [`chi_msb u (a + b)`];;
searcht 5 [`a * b <= &0`];;
searcht 15 [`a * x = a * y`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
(* May 7, 2013 *)
searcht 5 [`azim`;`dihV`];;
Local_lemmas.AZIM_LE_PI_EQ_DIHV;;
	    type_of `part8`;;
searcht 5 [`drop3`];;
searcht 5 [`part5`];;
	    type_of `h0`;;
frees (mk_flist terminal_cs);;
map frees terminal_cs;;
searcht 5 [`(a * k + b) MOD k`];;
searcht 50 [name "Zithlqn."];;
searcht 50 [name "Xwitccn.";`scs_lo_v28`];;
searcht  5 [`pi`;`convex_locaL_fan`];;
searcht 5 [name "convex_local"];;
searcht 5 [`azim`;`azim_in_fan`];;
searcht 5 [`azim_cycle {a,b}`];;
searcht 5 [`rho`;`rho_fun`];;
Local_lemmas.AZIM_LE_PI_EQ_DIHV;;
searcht 5 [`&0 <= azim a b c d`];;
searcht 5 [`collinear`;`ball_annulus`];;
searcht 5 [`IMAGE f X SUBSET B`;name "Terminal"];;
periodic_vv_inj;;
searcht 5 [`{a,b,c} = {a,c,b}`];;
UNDISCH;;
UNDISCH terminal_quad_lemma;;
vv_quad_split012;;
searcht 5 [`vv i = vv (i MOD k)`];;
periodic2_mod_reduce;;
bb 5;;
Appendix.cs_adj;;
searcht 5 [`SUC 3 = 4`];;
searcht 5 [`x IN ball_annulus`];;
searcht 5 [`taum a b c d e f = taum a' b' c' d' e' f'`];;
Merge_ineq.tau_x_tau_residual_x;;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
(* May 7, 2013 *)
searcht 5 [`sqrt8 * sqrt8`];;
searcht 5 [`dist`;`enclosed`];;
searcht 5 [`tauq`;`taum`];;
Appendix.get_main_nonlinear "4680581274 delta top issue";;
searcht 5 [`delta_x4 a b c d e f = delta_x4 a' b' c' d' e' f'`];;
searcht 5 [def "cayleyR"];;
searcht 10 [`cayleyR x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 = y`];;
searcht 5 [`tau3`;`taum`];;
searcht 5 [`tau3 a b c = tau3 a' b' c'`];;
searcht 5 [`dihV a b c d = dihV a b c' d'`];;
bb 5;;
searcht 5 [`INSERT`;`SUBSET`];;
I_THM;;
searcht 5 [`sqrt8`;`DECIMAL`;`pi`];;
searcht 5 [name "safe"];;
searcht 5 [`sin(azim a b c d)`];;
searcht 8 [`dihV`;`azim`];;
searcht 15 [`azim a b c d + azim a b d e`;omit `cyclic_set`];;
searcht 15 [`cyclic_set`];;
	    open Hales_tactic;;
searcht 5 [`sqrt(y * y)`];;
searcht 5 [`safesqrt x = sqrt x`];;
searcht 5 [`safesqrt x`];;
searcht 5 [`sqrt x * sqrt x `];;
Functional_equation.functional_edge_flat2_x;;
Functional_equation.functional_euler_3flat_x;;
Functional_equation.functional_euler_2flat_x;;
Functional_equation.functional_euler_1flat_x;;
searcht 5 [def "ups_"];;
searcht 10 [`abc_of_quadratic`];;
searcht 5 [`quadratic_root_plus`];;
Sphere.flat_term_x;;
searcht 100 [def "min"];;
searcht 8 [`min_num`];;

State_manager.neutralize_state();;
print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
(* May 21, 2013 *)
searcht 5 [`(x + y) * (x - y)`];;
searcht 5 [`sqrt x * sqrt x`];;
searcht 5 [`sqrt x = &0`];;
searcht 5 [`&0 <= sqrt x1`];;
searcht 5 [`&0 < sqrt x1`];;
searcht 5 [`a* b <= c * b`];;
searcht 5[ `flat_term x <= &0`];;
searcht 5 [`a <= sqrt y`];;
searcht 15 [`a <= b / c`];;
searcht 50 [`has_real_derivative`;name "MAX"];;
REAL_DERIVATIVE_ZERO_MAXMIN;;
searcht 50 [`has_real_derivative`;` f (y:real) <= f x`];; (* good stuff *)
Counting_spheres.REAL_CONVEX_ON_SECOND_SECANT;;
REAL_CONVEX_ON_SECOND_DERIVATIVE;;
REAL_CONVEX_ON_SECANT_DERIVATIVE;;
searcht 5 [`real_continuous`];;
searcht 5 [def "real_continuous"];;
searcht 5 [`real_continuous`;`real_open`];;
searcht 50 [`real_continuous`;`abs`];;

print_proof();;
update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
(* May 22, 2013 *)
searcht 5 [`has_real_derivative`;`delta_x`];;
strip_comb `derived_form p f f' x s`;;
Calc_derivative.derived_form_sin;;
help_grep "strip";;
strip_forall;;
GEN_LIST;;
help_grep "GEN";;
GENL;;
searcht 5 [`sqrt x = &0`];;
searcht 5 [`derived_form F`];;
Functional_equation.nonfunctional_taud_D1;;
searcht 5 [`sqrt(x * x)`];;
has_real_derivative;;
searcht 5 [`(f has_real_derivative f') net  ==> (g has_real_derivative f') net`];;
snd(chop_list 50 (searcht 150 [`has_real_derivative`]));;
drop;;
Arc_properties.HAS_REAL_DERIVATIVE_LOCAL;;
searcht 5 [`atreal`;`within`;`(:real)`];;
WITHIN_UNIV;;
searcht 5 [`real_open`;`>`];;
derived_form_delta_y;;
searcht 5 [`sqrt x = &0`];;
searcht 5 [`taum`;`delta_y`];;
	    open Terminal;;
"2314572187";;
searcht 5 [`taud`;`taud_x`];;
Merge_ineq.tau_x_tau_residual_x;;
searcht 5 [`tau_residual_x`;`taum_x`];;
searcht 5 [`const`;`sol0`];;
searcht 5 [`has_real_derivative`;`real_continuous`];;
searcht 5 [`abs (&0)`];;
searcht 5 [`real_open`;`<`];;
searcht 5 [`real_interval`;`real_open`];;
searcht 5 [`real_interval`];;
searcht 5 [`real_interval (a,b)`];;
searcht 5 [`z IN real_interval s`];;
searcht 5 [`is_real_interval(real_interval (a,b))`];;
searcht 5 [`real_interval(a,b) = {c}`];;
searcht 5 [`(\x. -- f x)`;`has_real_derivative`];;
searcht 5 [`atreal x within s`;`real_open`];;
searcht 5 [`real_open(real_interval(a,b))`];;
searcht 5 [`(\x. c) has_real_derivative u`];;
searcht 5 [`has_real_derivative`;name "UNI"];;
REAL_DERIVATIVE_UNIQUE_ATREAL;;

searcht 15 [`real_open`;`real_continuous_on`];;
searcht 15 [`real_closed`;`real_continuous_on`];;
searcht 15 [`real_closed`;`x <= y`];;
searcht 5 [def "compact"];;
searcht 5 [`real_compact`;`real_closed`];;
REAL_COMPACT_EQ_BOUNDED_CLOSED;;
REAL_OPEN_DIFF;;
REAL_CONTINUOUS_ATTAINS_INF;;
searcht 15 [`real_open`;`PREIMAGE`];;
searcht 50 [`open_in`;`real_continuous_on`];;
searcht 5 [`real_continuous_on`;`real_compact`];;
searcht 500 [`real_open`];;
searcht 15 [`real_continuous`;`has_real_derivative`];;
searcht 5 [`real_continuous_on`;`real_continuous`];;
searcht 5 [`atreal x within (:real)`];;
searcht 5 [`(\x. f x + g x) real_continuous z`];;
searcht 5 [`(\x. c * f x) real_continuous z`];;
searcht 15 [`(\x. g x * f x) real_continuous z`];;
searcht 5 [`(\x. c) real_continuous z`];;
searcht 5 [`(\x. (sqrt (f x))) real_continuous z`];;
searcht 5 [`sqrt`;`real_continuous`];;
searcht 5 [`(f o g) real_continuous z`];;
searcht 5 [`real_continuous`;`SUBSET`];;
searcht 5 [`real_continuous_on`;`SUBSET`];;
searcht 5 [`real_bounded (X)`;`SUBSET`];;
searcht 5 [`real_bounded (real_interval s)`];;
searcht 5 [`real_closed`;`d <= t`];;
searcht 5 [`real_open`;`d < t`];;
searcht 5 [`min`];;
searcht 5 [name "EXISTS";`?f. p ==> Q f`];;
SPEC `t:real` (ASSUME `!(x:real). &3 > &4 * x`);;
SPECL;;
strip_forall;;
help_grep "EXIST";;
help "EXISTS";;

help_grep "comb";;
help_grep "exist";;
searcht 5 [`x / y = &0`];;
searcht 5 [`&0 <= x / y`];;
searcht 5 [`&0 < x pow n`];;
searcht 5 [name "taum_taum_x"];;

print_proof();;

update_database();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
(* May 24, 2013 *)
searcht 15 [`mud_126_x_v1`];;
searcht 5 [`delta_x y1 y2 y3 y4 y5 y6 = delta_x a b c d e f`];;
searcht 5 [`taum y1 y2 y3 y4 y5 y6 = taum a b c d e f`];;
searcht 5 [`taud`;`taum`];;
searcht 5 [`flat_term`;`sol0`];;
searcht 5 [`flat_term2_126_x`];;
bb 3;;
searcht 15 [`abc_of_quadratic`;`edge_flat2_x`];;
searcht 50 [`abc_of_quadratic`];;
searcht 15 [`quadratic_root_plus`];;
searcht 5 [`a pow 2 < b pow 2`;`abs`];;
searcht 5 [`flat_term`;`flat_term_x`];;
searcht 5 [name "AFFSIGN_MONO_SHUFFLE"];;
searcht 5 [`taud`;`flat_term_x`];;
searcht 5 [`taud`;`taud_x`];;
UNDISCH taud_minimizer_terminal_pent_cases;;
UNDISCH OWZLKVY4;;
help_grep "REP";;
replicate;;
REPEATC;;
UNDISCH;;
searcht 5 [`taum`;`taud`];;
searcht 5 [`mud_126_x_v1`;`taud`];;
   (UNDISCH2 edge2_flatD_x1_delta_lemma3);;

UNDISCH2 delta_126_x_2h0_le_d;;
searcht 5 [`flat_term`;`taud`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
bb 3;;
(* May 24, 2013 *)
searcht 5 [`a <= sqrt b`];;
searcht 25 [`a <= c /\ b <= d`;`a * b <= c * d`];;
REAL_LE_MUL2;;
delta_126_135;;
searcht 5 [`&0 <= taud a b c d e f`];;
hd(Ineq.getexact "1008824382");;
hd(Ineq.getexact "8875146520");;
Ineq.getexact "1586903463";;
searcht 5 [`#0.705`];;
searcht 5 [`tau3 v0 v1 v2 + tau3 v0' v1' v2'`];;
searcht 5 [`taud`;`taum`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
bb 3;;
(* May 27, 2013 *)
searcht 5 [`taum`;`taud`];;
appendix_get_main_nonlinear "6877738680";;
searcht 5 [`taud_x`;`taud`];;
searcht 50 [name "WLOG"];;

REAL_WLOG_LE;;
Merge_ineq.REAL_WLOG_SIMPLEX_3d;;
searcht 5 [`delta_234_x`];;
searcht 5 [`mud_126_x_v1`];;
searcht 10 [`flat_term2_126_x`];;
searcht 5 [`eulerA`];;
searcht 5 [`ineq`;omit `CONS`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
bb 3;;
(* May 28, 2013 *)
searcht 5 [`x + y = y + x`;`(a + b) + c = a + b +c `];;
3860.0/. 60.0;;
searcht 5 [`taum`;`taud`];;
searcht 15 [`v ==> (a = sqrt b)`];;
help_grep "CONJ";;
basic_rewrites();;
searcht 5 [name "BETA"];;
searcht 25 [`eulerA_x`];;
searcht 25 [def "euler"];;
searcht 15 [`sol_euler_x`];;
searcht 5 [`dihV`;`sol`];;
searcht 5 [`eulerA_x`;`euler_p`];;
searcht 5 [`sqrt(x pow 2)`];;
searcht 5 [`&0 <= dist(x,y)`];;
searcht 5 [`dist(v2,v3)`;`dot`];;
searcht 15 [`(u + v) dot w`];;
searcht 5 [`x dot y = y dot x`];;

searcht 5 [`delta_x4`;`dihV`];;
searcht 5 [name "EULER";`euler_p`];;
searcht 5 [`dihV v0 v1 v2 v3 = &0`];;
searcht 5 [`azim v0 v1 v2 v3 = pi`];;
searcht 15 [`delta_x`;`v:real^3`];;
searcht 5 [`&0 < atn2(d,p)`];;
searcht 5[`&0 <= delta_y a b c d e f `;`dist`];;
searcht 15 [`atn2`;`atn`];;
Pent_hex.derived_form_delta_x;;
searcht 5 [`dihV`;`delta_x4`];;
searcht 5 [`atn2 (-- x, -- y)`];;
searcht 5 [`atn x < pi / &2`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
bb 5;;
(* May 31, 2013 *)
searcht 5 [`coplanar`;`delta_y`];;
searcht 5 [`eulerA_x a b c d e f = eulerA_x a' b' c' d' e' f'`];;
searcht 15 [`delta_x`;`pow `;`delta_x4`];;
searcht 5 [`delta_x`;`dih_x`];;
searcht 5 [`ups_x`;`dot`];;
derived_form_delta_x_wrt_x5;;
searcht 5 [`delta_x a b c d e f = delta_x a b' c' d' e' f'`];;
searcht 5 [name "Terminal";`main_nonlinear_terminal_v6`];;
searcht 5 [`sqrt(&4)`];;
searcht 5 [`sqrt(x)  = &0`];;
searcht 5 [`sqrt(x pow 2)`];;
searcht 5 [`&0 < x pow 2`];;
searcht 5[`atn2 (t * x, t * y)`];;
searcht 5 [name "taustar_taum"];;
	    open Terminal;;
taustar_taum;;
searcht 5 [name "AZIM_DIHV_SAME_STRONG"];;
searcht 5 [name "Lunar_deform"];;
searcht 5 [name "AZIM_EQ_IMP"];;
searcht 5 [`convex_local_fan`];;
List.length Build.build_sequence;;
searcht 50 [def "_eq"];;
searcht 50 [def "continuous"];;
searcht 15 [`real_continuous`];;

searcht 5 [def "generic"];;
searcht 5 [`generic`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
Print_types.print_term_types AXJRPNC_concl;;
bb 5;;
(* Jun 17, 2013 *)
searcht 5 [`lunar`];;
searcht 55 [def "BB"];;
searcht 5 [`arcV`;`dihV`];;
searcht 5 [name "XWN"];;
searcht 5 [def "d_fun3"];;
searcht 5 [def "edge2_flatD_x1"];;
searcht 5 [`edge2_flatD_x1`;`quadratic_root_plus`];;
searcht 5 [name "IV"];;
searcht 5 [`?x.  f (x:real) = &0 /\ P x`];;
searcht 15 [`real_continuous_on`;`f x = a`];;
Sphere.abc_of_quadratic;;
searcht 5 [`abc_of_quadratic (\z. f z)`];;
searcht 50 [`derived_form`;`?`];;
searcht 15 [`real_continuous`;`has_real_derivative`];;
HAS_REAL_DERIVATIVE_IMP_CONTINUOUS_ATREAL;;
searcht 5 [`real_continuous_on`;`real_continuous`];;
searcht 5 [`x within (:real)`];;
REAL_IVT_INCREASING;;
searcht 5 [`edge2_flatD_x1 d x2 x3 x4 x5 x6 = x1`];;
searcht 5 [`f real_continuous x within s`;`f real_continuous atreal x`];;
searcht 5 [`delta_x`;`edge2_flatD_x1`];;
searcht 5 [`quadratic_root_plus`;name "work"];;
searcht 5 [`ups_x x1 x2 x3 = ups_x a b c`];;
searcht 5 [`delta_x a b c d e f = delta_x a b' c' d' e' f'`];;
searcht 5 [`delta_x1 a b c d e f = delta_x1 a b' c' d' e' f'`];;
searcht 5 [`delta_y a b c d e f = delta_y a b' c' d' e' f'`];;
searcht 5 [name "WLOG";name "SQ"];;
Terminal.REAL_WLOG_SQUARE_LEMMA;;
Pent_hex.delta_diff;;
searcht 5 [`&0 < ups_x (a*a) (b*b) (c*c)`];;
Merge_ineq.UPS_X_POS;;
Pent_hex.delta_diff;;
searcht 5 [`&0 <= abs(x)`];;
searcht 5 [`abs(a / b)`];;
searcht 5 [`a * x pow 2 + b * x + c`;`quadratic_root_plus`];;
searcht 5 [`&0 <= x pow 2`];;
searcht 5 [`sqrt(x pow 2 ) = abs x`];;
searcht 5 [`x * x < y * y`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
Print_types.print_term_types AXJRPNC_concl;;
bb 5;;
(* Jun 19, 2013 *)
searcht 5 [`x * x <= y * y`];;
searcht 5 [`x pow 2 <= y pow 2`];;
searcht 5 [`sqrt8 * sqrt8`];;
searcht 5 [`&0 <= ups_x (a*a) (b*b) (c*c)`];;
searcht 5 [`ups_x a b c = ups_x a' b' c'`];;
searcht 50 [`delta_x`;`ups_x`];;
searcht 5 [`xrr`];;
searcht 5 [`derived_form F`];;
searcht 5 [`derived_form b f f' x s = derived_form b' g g' x' s'`];;
searcht 5 [`(f has_real_derivative f' x) s /\ (f has_real_derivative g' x') s'`];;
searcht 5 [`f has_real_derivative f' x s`];;
searcht 5 [def "has_real_derivative"];;
searcht 5 [`has_real_derivative`;name "UNI"];;
searcht 5 [`(atreal x within (:real))`];;
searcht 5 [`re_eqvl`;`?`];;
searcht 50 [`(f has_real_derivative (f x)) s`];;
Vukhacky_tactics.HAS_REAL_DERIVATIVE_CHAIN2;;
searcht 5 [`real_open`;`has_real_derivative`];;
searcht 500 [`derived_form`];;
Arc_properties.HAS_REAL_DERIVATIVE_LOCAL;;
searcht 5 [`re_eqvl a b <=> re_eqvl b a`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
Print_types.print_term_types AXJRPNC_concl;;
bb 5;;
(* Jun 20, 2013 *)
searcht 5 [`taum`;`rhofn`];;
searcht 5 [`taum_x`];;
searcht 5 [`taum`;`rhazim`];;
searcht 5 [`dih_x a b c d e f = dih_x a' b' c' d' e' f'`];;
searcht 50 [`real_open`];;
searcht 5 [`real_open`;`<`];;
searcht 5 [`real_continuous_on`;`!`;`?`];;
Pent_hex.continuous_preimage_open;;
searcht 5 [`real_continuous`;`has_real_derivative`];;
   HAS_REAL_DERIVATIVE_IMP_CONTINUOUS_ATREAL;;
searcht 5 [`real_continuous`;`real_continuous_on`];;
derived_form_xrr;;
GEN_ALL Calc_derivative.derived_form_sub;;
searcht 15[`ups_x (y1*y1) (y2*y2) (y6*y6)`];;
searcht 15 [`&0 < x * y`];;
searcht 15 [`x * y < &0`];;
searcht 5 [`arclength`;`acs`];;
searcht 5 [`sqrt x = &0`];;
searcht 5 [`(f x - f y) = f' * (x - y)`;`has_real_derivative`];;
searcht 15 [`&0 <= a * b`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
Print_types.print_term_types AXJRPNC_concl;;
bb 3;;
(* Jun 21, 2013 *)
searcht 5 [`xrr y2 y3 y4 < &16`];;
searcht 5 [` &0 < x pow 2`];;
searcht 5 [`real_closed`;`real_interval`];;
searcht 5 [`&0 < x/y`];;
searcht 5 [`ups_x`;`xrr`];;
searcht 5 [`&0 < x * y`;`\/`];;
searcht 5 [`real_open`;`delta_y`];;
searcht 5 [`taum`;`dih_x`];;
searcht 5 [`x / y = &0`];;
searcht 5 [`sqrt x = &0`];;
searcht 5 [`&0 <= x / y`];;
REAL_MVT_SIMPLE;;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
Print_types.print_term_types AXJRPNC_concl;;
bb 3;;
(* Jun 22, 2013 *)
searcht 5 [`derived_form`;`(\q. g' (f q) * f' q)`];;
searcht 5 [`&0 < x`;`delta_x`;`delta_y`;`xrr`];;
derived_form_xrr;;
searcht 5 [`real_open`;`real_interval`];;
REAL_MVT_SIMPLE;;
searcht 5 [`(atreal x within s)`;`f has_real_derivative f' atreal x`];;

HAS_REAL_DERIVATIVE_ATREAL_WITHIN;;
searcht 5 [`(f has_real_derivative f') (atreal x) ==> (f has_real_derivative f') (atreal x within s)`];;
searcht 5 [`derived_form b (g o f)`];;
GSYM Calc_derivative.derived_form;;
derived_form_sum_dih444sub;;
searcht 5 [`real_interval(a,b) SUBSET real_interval[a,b]`];;
searcht 5 [`taum`;`dih_x`];;
searcht 5 [`rho_fun y <= c`];;
searcht 5 [def "rho"];;
searcht 5 [`rho y = x`];;
searcht 5 [`const1`;`sol0/pi`];;
searcht 5 [`&0 < const1`];;
searcht 5 [`xrr y1 y3 y5`;` &16`];;
searcht 5 [`x * x < y * y`];;
searcht 5 [`&0 < ups_x (a*a) (b*b) (c*c)`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types vv_azim_le;;
Print_types.print_term_types t';;
bb 3;;
(* Jun 22, 2013 *)
searcht 5 [`taustar_v39`;`mk_unadorned`];;
searcht 5 [`scs_k_v39`;`mk_unadorned_v39`];;
searcht 5 [`dsv_v39`;`mk_unadorned`];;
Terminal.tau_fun_azim;;
searcht 5 [`dist(x,x)`];;
searcht 5 [`i MOD k = i`];;
searcht 5 [`azim`;`tau_fun`];;
searcht 5 [name "VPW";name "PRIME"];;
searcht 5 [`SUC i MOD k`];;
searcht 15 [`(i + a) MOD k`];;
GSYM Oxl_2012.MOD_INJ1;;
searcht 5 [`ball_annulus`;`norm`];;
Fnjlbxs.in_ball_annulus;;
searcht 5 [`IMAGE`;`SUBSET`];;
searcht 5 [`dist(a,b)+dist(b,c)`];;
searcht 5[`#2.52`;`h0`];;
searcht 5 [`delta_y`;`norm`];;
REWRITE_RULE[LET_THM] Tame_lemmas.delta_y_pos;;
searcht 15 [name "JKQ"];;
searcht 15 [`is_scs_v39`;`mk_unadorned_v39`];;
searcht 50 [`mk_unadorned_v39`;`funlist_v39`];;
Terminal.is_scs_funlist;;
searcht 5 [`funlistA_v39`;`F`];;
searcht 5 [`periodic {}`];;
searcht 5 [`funlist_v39`;`cs_adj`];;
Terminal.periodic2_funlist;;
searcht 5 [`X SUBSET Y`;`CARD`];;
searcht 5 [`{i | i < (k:num)}`];;
searcht 5 [`CARD {}`];;
searcht 5 [`x MOD k = SUC x MOD k`];;
searcht 5 [`d_tame`;`#0.712`];;
searcht 5 [`sol_local`;`azim_in_fan`];;
searcht 5 [`generic`;`circular`;`lunar`];;
searcht 5 [`convex_local_fan`;`local_fan`];;
searcht 15 [`lunar`];;
searcht 15 [`sol_local`];;
searcht 15 [`interior_angle1`;`azim`];;
searcht  5 [def "interior_an"];;
searcht 5 [`interior_angle1`];;
searcht 5 [`azim`;`rho_node1`;`ivs_rho_node1`];;
searcht 5 [`rho_node1`;`i+1`];;
SUM_SUB;;
searcht 5 [`FINITE {i | i < (k:num)}`];;
searcht 5 [`sum X (\x. c)`;`CARD`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  Local_lemmas.AZIM_CYCLE_TWO_POINT_SET;;
Print_types.print_term_types t';;
bb 6;;
(* Jun 24, 2013 *)
Local_lemmas.CVX_LO_IMP_LO;;
 Wrgcvdr_cizmrrh.CIZMRRH;;
searcht 5 [`interior_angle1`;`azim_in_fan`];;
Wrgcvdr_cizmrrh.lunar;;
searcht 5 [name "vv_rho_node1"];;
Terminal.EE_vv;;
Local_lemmas.AZIM_CYCLE_TWO_POINT_SET;;
searcht 15 [`azim_in_fan`;`azim`];;
DIVISION;;
searcht 5 [name "SYN"];;
searcht 10 [`dihV`;`azim`];;
GSYM Polar_fan.AZIM_DIHV_SAME_STRONG;;
searcht 5 [`collinear`;`BBs_v39`];;
Appendix.BBs_v39;;
searcht 5 [`periodic`;`f (i MOD k)`];;
searcht 5 [`dih_x`;`delta_x4`];;
searcht 5 [`dih_x a b c d e f < dih_x a' b' c' d' e' f'`];;
searcht 5 [`interior_angle1`;`&0`];;
searcht 5 [`(!i j. i < k /\ j < k /\ v i = v j ==> i = j)`];;
searcht 5 [`azim_in_fan`;`interior_angle1`];;
DIVISION;;
searcht 5 [`a *(k:num) + b `;`MOD`];;
searcht 5 [`&0 < atn2(x,y)`];;
searcht 5 [`atn2(&0,x)`];;
searcht 5 [`dih_y`;`dih_x`];;
searcht 5 [`dihV`;`dih_y`];;
searcht 5 [`periodic`;`MOD`];;
searcht 5 [`dist`;`norm`];;
searcht 5 [`delta_y`;`norm`];;
searcht 5 [`i MOD k = SUC i MOD k`];;
searcht 15 [`(i + a) MOD k`;`~ x`];;
Oxl_2012.MOD_INJ1;;
searcht 5 [name "dec";name "Lp_d"];;
searcht 5 [`delta_x4 a b c d e f = delta_x4 a' b' c' d' e' f'`];;
Wrgcvdr_cizmrrh.CIZMRRH;;
searcht 5 [`scs_generic`];;
Local_lemmas.INTERIOR_ANGLE1_POS;;
GSYM INTERIOR_ANGLE1_AZIM;;
(REWRITE_RULE[LET_THM] (GSYM INTERIOR_ANGLE1_AZIM));;
scs_6T1;;
searcht 5 [`scs_basic_v39`];;
searcht 5 [`mk_unadorned_v39`;`unadorned_v39`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  Local_lemmas.AZIM_CYCLE_TWO_POINT_SET;;
Print_types.print_term_types t';;
bb 11;;
(* Jun 25, 2013 *)
searcht 5 [`cs_adj`;`scs_diag`];;
searcht 5 [`SUC i MOD k`];;
is_scs_6T1;;
searcht 5 [`periodic`];;
Pent_hex.terminal_pent_taum;;
searcht 5 [`eulerA_x`;`dihV`];;
searcht 5 [`azim_in_fan`;`convex_local_fan`];;
azim_in_fan_azim;;
searcht 5 [`generic`;`collinear`];;
Nkezbfc_local.PROPERTIES_GENERIC_LOCAL_FAN;;
searcht 5 [`sol`;`dihV`];;
searcht 5 [`taum`;`sol0`];;
searcht 5 [`taum`;`dih_x`];;
Sphere.taum;;
searcht 5 [`taum y1 y2 y3 y4 y5 y6 = x`];;
Sphere.rhazim;;
searcht 5 [`&0 <= sol_y y1 y2 y3 y4 y5 y6`];;
Sphere.sol_y;;
searcht 5 [`generic`;`azim`;`+`];;
DIH_IMP_EULER_A_POS;;
searcht 5 [`sol_euler_x_div_sqrtdelta`];;
searcht 5 [`&0 < sol x y`];;
searcht 5 [`taum`;`sol_y`];;
searcht 5 [`sol`;`vol`];;
searcht 5 [`convex hull x`;`measurable`];;
searcht 5 [`radial_norm`;`convex hull x`];;
searcht 5 [`&0 <= dih_y y1 y2 y3 y4 y5 y6`];;
searcht 5 [def "sol_euler_x"];;
searcht 5 [`sol_euler_x`];;
Localization.convex_local_fan;;
azim_in_fan_azim;;
searcht 5 [`sol_local`;`azim`];;
Terminal.vv_split_azim_generic;;
searcht 5 [`sum (a INSERT b)`];;
searcht 5 [`FINITE {}`];;
searcht 5 [`FINITE {a,b,c}`];;
searcht 5 [`&0 <= x`; `azim`];;
searcht 15 [`azim`;`dihV`];;
Polar_fan.AZIM_DIHV_SAME_STRONG;;
searcht 5 [`collinear`;`generic`];;
searcht 5 [`IN`;`IMAGE`;`UNIV`];;
searcht 5 [`periodic`;`x+(b:num)`];;
Appendix.tau3;;
`(3 + 4)  - 1`;;
searcht 5 [`rho_fun`;`rho`];;
Terminal.tau3_taum_40;;
Pent_hex.terminal_pent_taum;;
searcht 5 [`taum a b c d e f = taum a' b' c' d' e' f'`];;
searcht 5 [`delta_y`;`norm`];;
DIH_IMP_EULER_A_POS;;
searcht 5 [`dist(x, vec 0)`;`norm`];;
searcht 5 [`sol_x`;`dihV`];;
searcht 5 [`dihV a b c d = dihV a b c' d'`];;
searcht 5 [`dihV`;`dih_y`];;
searcht 2 [`dih_y`;`sol_y`];;
searcht 5 [`&0 <= sol_y a b c d e f`];;
searcht 5 [name "VPW"];;
searcht 5 [`norm (x - y) `;`dist`];;
Sphere.packing;;
searcht 5 [`packing`;`IN`];;
IN_ELIM_THM;;
searcht 15 [`(s INSERT b) c`];;
searcht 5 [`periodic`;`MOD`];;
searcht 5 [`periodic`;`sum`];;
Oxl_def.periodic_sum;;
searcht 5 [`sum`;`BIJ`];;
searcht 5 [`sum X f = sum X g`];;
help_grep "ARITH";;
help_grep "NUM";;
Local_lemmas.CVX_LO_IMP_LO;;
DIVISION;;
searcht 25 [`(j + i) MOD k`];;
MOD_ADD_MOD;;
searcht 5 [`i MOD k = i`];;
searcht 15 [`(a + b) MOD k = (a'+ b') MOD k`];;
Polar_fan.AZIM_DIHV_SAME_STRONG;;
searcht 5 [`sqrt(x pow 2)`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  Local_lemmas.AZIM_CYCLE_TWO_POINT_SET;;
Print_types.print_term_types t';;
bb 11;;
(* Jun 26, 2013 *)
searcht 5 [`delta4_squared_y`];;
searcht 5 [`dih_y`;`dih_x`];;
searcht 5 [`x pow 2 < y pow 2`];;
searcht 5 [`&0 <= tan x`];;
searcht 5 [`x * y < x' * y`];;
searcht 15 [`a < x pow 2`];;
searcht 5 [`tan`;`atn`];;
searcht 5 [`atn x < atn y`];;
searcht 5 [`dih_y a b c d e f = dih_y a' b' c' d' e' f'`];;
searcht 5 [`delta_y a b c d e f = delta_y a' b' c' d' e' f'`;`&2`];;
searcht 5 [`taum a b c d e f = taum a' b' c' d' e' f'`];;
searcht 5 [`dih_x a b c d e f < dih_x a' b' c' d' e' f`];;
Tame_inequalities.DIH_X_MONO_LT_4;;
searcht 5 [`delta_y y1 y2 y3 y4 y5 y6 = &0`;`dih_y`];;
Sphere.dih_x;;
searcht 5 [`atn2(&0,x)`];;
searcht 5 [`dih_x`;`pi`];;
Oxlzlez.DIH_X_LT_PI;;
searcht 15 [`delta_x`;`pow`];;
searcht 5 [name "diff";`delta_x`];;
searcht 5 [`delta_x`;`delta_x1`;`pow`];;
searcht 5 [`dihV`;`dih_y`];;
searcht 5 [`cs_diag`;`cs_adj`];;
searcht 15 [`(a+B) MOD n`;omit `5`];;
searcht 5 [`1 MOD 5`];;
searcht 5 [`i < 4`;`\/`];;
searcht 5 [name "funlist";name "expl"];;
searcht 5 [`1 MOD 4`;`2 MOD 4`];;
searcht 5 [`psort 4`];;
searcht 5 [`CARD x <= 2`];;
searcht 5 [`CARD x = 1`];;
searcht 15 [`CARD`;`3`];;
searcht 5 [`CARD X <= CARD Y`];;
searcht 5 [`CARD`;`{i | (i:num) < k}`];;
searcht 5 [`CARD s <= 2`];;
searcht 5 [`CARD {a,b}`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  Local_lemmas.AZIM_CYCLE_TWO_POINT_SET;;
Print_types.print_term_types t';;
bb 25;;
(* Jun 27, 2013 *)
searcht 2 [`delta_y`;`norm`];;
get_main_nonlinear "5405130650";;
searcht 5 [def "deform"];;
searcht 5 [`deformation`];;
searcht 15 [def "continuous"];;
	    type_of `real_continuous_on`;;
searcht 5 [`continuous_on`;`azim`];;
searcht 5 [`real_continuous`;`dihV`];;
searcht 5 [`continuous_on`;`dist`];;
searcht 5 [`generic`];;
searcht 15 [name "HYPERPL";name "SEP"];;
SEPARATING_HYPERPLANE_CLOSED_POINT;;
REAL_CONTINUOUS_AT_AZIM;;
searcht 5 [`azim`;`real_continuous`];;
searcht 5 [`real_continuous_on`;`real_uniformly_continuous_on`];;
	    type_of `compact`;;
searcht 15 [`compact`;`closed`];;
searcht 15 [`compact`;`real_interval`];;
searcht 5 [`real_continuous_on`;`dot`];;
searcht 25 [`azim a b c d = pi`];;
searcht 50 [`wedge_ge`;name "Cou"];;
searcht 5 [def "wedge"];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  Local_lemmas.AZIM_CYCLE_TWO_POINT_SET;;
Print_types.print_term_types t';;
	    let named s = searcht 50 [name s];;
bb 25;;
(* Jun 28, 2013 *)
searcht 50 [`(A:(V)net)`;omit `atreal`;omit `at`];;
searcht 50 [`mk_net`];;
searcht 50 [`real_continuous`;`dist`];;
Local_lemmas1.CON_ATREAL_REAL_CON2_REDO;;
searcht 15 [`continuous`;name "COMPO"];;
BILINEAR_CONTINUOUS_AT_COMPOSE;;
searcht 15 [`continuous`;`h (f x) (g x) (g2 x)`];;
Local_lemmas1.UPS_X_CONTS_FUNC;;
Local_lemmas1.CONTINUOUS_PRESERVE_COLLINEAR;;
searcht 5 [`continuous`;`collinear`];;
searcht 5 [`delta_y`;`coplanar`];;
searcht 50 [`real_continuous`;`<`;`?`];;
searcht 5 [def "generic"];;
searcht 50 [def "wedge"];;
searcht 5 [`wedge`;`azim`];;
searcht 5 [`aff_gt`;`wedge`];;
searcht 5 [`wedge`;`DIFF`];;
searcht 5 [`azim a b c d = &0`];;
searcht 5 [`coplanar`;`cross`];;
Local_lemmas1.AZIM_COND_FOR_COPLANAR;;
Local_lemmas.NOT_COLL_IMP_COPL;;
searcht 5 [`wedge_ge`;`aff_ge`];;
searcht 15 [`collinear`;`cross`];;
Local_lemmas.COLL_IFF_COLL_CROSS;;
searcht 50 [`collinear`;`t % (u:real^A)`];;
COLLINEAR_SPECIAL_SCALE;;
searcht 5 [`azim`;`sin`;`pi`];;
searcht 5 [`sin`;`azim`;`cross`];;
Local_lemmas.SIN_AZIM_MUTUAL_SROSS;;
searcht 5 [`sin`;`pi`;`&0 < sin x`];;
searcht 5 [`azim a b c d <= pi`];;
searcht 5 [`sin pi`];;
searcht 5 [`&0 <= sin (azim a b c d)`];;
searcht 5 [`(a cross b) dot c = (a' cross b') dot c'`];;
searcht 5 [`&0 < x dot x`];;
searcht 5 [`a cross b = vec 0`;`collinear`];;
searcht 5 [`{a,b,c} = {a',b',c'}`];;
searcht 5 [`-- (v cross w)`];;
searcht 15 [`aff_ge {a} {c,d}`;`%`];;
searcht 8 [`DISJOINT`;`collinear`];;
searcht 5 [`v cross w = v`];;
searcht 5 [`collinear {a,b}`];;
searcht 5 [`aff_ge {a} {b,c}`;`%`];;
searcht 5 [`coplanar{a,b,c}`];;
searcht 5[`u IN aff_ge {u} {v,w}`];;
searcht 5[`vec 0 IN aff_lt a b`];;
searcht 50 [`aff_lt {a} {b}`];;
searcht 15 [`coplanar`;`%`];;
searcht 5 [`coplanar`;`cross`];;
searcht 5 [`x dot (-- v)`];;
searcht 5 [`x dot (u + v)`];;
searcht 5 [`x dot (vec 0)`];;
searcht 5 [`(v cross w) dot v`];;
searcht 15 [`coplanar`;`%`;omit `~coplanar s`];;
searcht 5 [`t % x IN wedge a b c d`];;
searcht 5 [`wedge`;`azim`];;
searcht 5 [`azim`;`%`];;
searcht 5 [`aff_ge {a,b} {c,d}`;`%`];;
searcht 5 [name "LDUR"];;
searcht 50 [name "Lunar_de"];;
searcht 5 [`azim a w v w`];;
searcht 20 [`wedge_ge`;`SUBSET`];;
searcht 20 [`wedge`;`afffine hull {a,c}`];;
Local_lemmas.INTERIOR_ANGLE1_POS;;
searcht 50 [name "Counting_";name "SPL"];;
searcht 5 [`dihV`;`azim`];;
searcht 5 [`azim`;`x < &2 * pi`];;
searcht 5 [`sin`;`azim`;`cross`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  Local_lemmas.AZIM_CYCLE_TWO_POINT_SET;;
Print_types.print_term_types t';;
bb 25;;
(* Jun 28, 2013 *)
searcht 15 [`aff_ge {a,b} {c}`;`%`];;
searcht 5 [`DISJOINT`;`collinear`];;
searcht 5 [`collinear`;`%`];;
CROSS_EQ_SELF;;
searcht 5 [`collinear`;`cross`];;
searcht 5 [`v dot v = &0`];;
searcht 5 [`collinear {a,b}`];;
searcht 5 [`collinear {a,b,a}`];;
searcht 5 [`cross`;`azim`];;
searcht 5 [def "re_eqvl"];;
searcht 5 [`re_eqvl`;`?`;`&0`];;
searcht 5 [`re_eqvl a b <=> re_eqvl b a`];;
Local_lemmas.SIN_AZIM_POS_PI_LT;;
searcht 5 [`sin pi`];;
searcht 5 [`&0 < sin x`];;
searcht 5 [`&0 <= azim a b c d`];;
CROSS_LMUL;;
searcht 5 [`x cross x`];;
searcht 5 [`&0 < x dot x`];;
searcht 5 [`collinear`;`cross`];;
searcht 5 [def "cycle"];;
searcht 50 [`azim_cycle`;`azim`];;
Wrgcvdr_cizmrrh.IDENTIFY_AZIM_CYCLE;;
Wrgcvdr_cizmrrh.AZIM_CYCLE_PROPERTIES;;
searcht 15 [`azim a b c d = azim a b c' d'`];;
searcht 5 [`collinear`;`cross`];;
searcht 5 [`DISJOINT`;`collinear`];;
searcht 5 [`(a cross (b cross c))`;`dot`];;
searcht 5 [`a cross b = -- (b cross a)`];;
searcht 5 [`t  % (u % v)`];;
searcht 5 [`t % (a + b)`];;
searcht 15 [`azim a b c d= &0`];;
CROSS_EQ_SELF;;
searcht 5 [`(a cross b) dot a`];;
searcht 5 [`azim a b c c`];;
EQ_SYM;;


print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  Local_lemmas.AZIM_CYCLE_TWO_POINT_SET;;
Print_types.print_term_types t';;
bb 7;;
(* Jun 30, 2013 *)
searcht 5 [`aff_ge`;`aff_gt`];;
searcht 5 [`FINITE`;`IMAGE`];;
searcht 5 [`FINITE`;`{i | (i:num) <= m}`];;
searcht 5 [`azim`;name "SHIFT"];;
searcht 5 [`xrr`];;
searcht 5 [`arclength a b c < arclength a' b' c'`];;
searcht 5 [`u IN aff_ge {a,u} {c}`];;
searcht 5 [`collinear`;`cross`];;
Leaf_cell.AZIM_BASE_SHIFT_LE;;
searcht 6 [`azim a b c d + azim a b d e`;omit `cyclic_set`];;
searcht 5 [`&0 <= azim a b c d`];;
searcht 5 [name "LDUR"];;
searcht 5 [`plane`;`coplanar`];;
Trigonometry2.coplanar1;;
searcht 5 [name "GE_2_2"];;
searcht 5 [name "EGH"];;
searcht 5 [`affine hull s`;`aff_gt`];;
searcht 5 [`aff_gt a b SUBSET aff_gt a' b'`];;
searcht 5 [`x IN S ==> x IN g hull S`];;
searcht 5 [`coplanar`;`affine hull s`];;
searcht 5 [`coplanar`;`azim`];;
searcht 5 [`conv0`;`conv`];;
searcht 5 [`conv0 {a,b} SUBSET X`];;
searcht 5 [`conv0`;`%`];;
searcht 5 [`convex`;`aff_gt`];;
searcht 5 [`aff {a,b}`;`%`];;
searcht 10 [`collinear`;`%`];;
searcht 5 [`a % u = a % v`];;
searcht 5 [`a % b % c`];;
Trigonometry2.AFF2_VEC0;;;
searcht 5 [`&1 % v`];;
searcht 5 [`aff_gt`;`conv0`];;
searcht 5 [`aff_gt s t SUBSET aff_gt s' t'`];;

searcht 5 [`wedge`;`aff_gt`];;
searcht 15 [`collinear`;`cross`];;
searcht 15 [`azim a b c d = &0`];;
searcht 5 [`a IN aff_ge {b,a} s`];;
searcht 5 [`wedge`;`azim`];;
searcht 5[`azim`;`a % c`];;
searcht 10 [`azim`;`&2 * pi`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  Local_lemmas.AZIM_CYCLE_TWO_POINT_SET;;
Print_types.print_term_types t';;
bb 7;;
(* July 1, 2013 *)
searcht 5 [`circular`];;
searcht 5 [def "deformation"];;
searcht 5 [`deformation`;`real_interval`];;
searcht 5 [`deformation`];;
searcht 50 [`wedge_in_fan_gt`];;
searcht 5 [name "AZIM_EQ_IMP"];;


print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  Local_lemmas.AZIM_CYCLE_TWO_POINT_SET;;
Print_types.print_term_types t';;
bb 7;;
(* July 3, 2013 *)
searcht 5 [name "MHAE"];;
searcht 5 [`lunar`;`convex_local_fan`;`IMAGE`;`interior_angle1`];;
searcht 15 [`FINITE`;`FAN`];;
searcht 100 [def "FAN"];;
searcht 5 [`fan3`;`FAN`];;
searcht 50 [`FAN u = y`;`graph`];;
searcht 5 [`graph E`];;
searcht 5 [`local_fan`;`FINITE`];;
searcht 5 [`local_fan`;`convex_local_fan`];;
searcht 5 [`wedge`;`azim`];;
searcht 25 [`a < b /\ b < c`;`&2`];;
searcht 5 [`real_open`;`real_interval`];;
searcht 5 [name "xrr_pos"];;
	    type_of `real_continuous_on`;;
searcht 5 [`azim`;`&0`;`aff_ge`];;
searcht 5 [`aff_ge`;`coplanar`;`wedge`];;
searcht 5 [`open`;`wedge`];;
searcht 30 [`azim a b c d = pi`];;
searcht 15 [`aff_lt {a,b,c}`];;
searcht 5 [name "LDU"];;
searcht 5 [`coplanar`;`azim`;`&0`];;
searcht 15 [`coplanar`;`collinear`];;
searcht 15 [`affine hull {a,b,c}`;`%`];;
Ckqowsa_4_points.in_affine_hull_lemma;;
searcht 5 [`collinear`;`aff_dim`];;
searcht 5 [def "DIM"];;
searcht 5 [`aff_dim`;`INSERT`];;
searcht 5 [`(a cross b) dot c = (a' cross b') dot c'`];;
searcht 5 [`a cross a`];;
searcht 5 [`&0 < a dot a`];;
searcht 15 [`&0 <= x * y`];;
searcht 5 [`cross`;`collinear`];;
searcht 5 [`azim a b c d = pi`;`coplanar`;`&0`];;
coplanar;;
plane;;
searcht 5 [`S SUBSET affine hull S`];;
searcht 5 [`aff {a,b}`;`%`];;
searcht 5 [`conv0`;`%`];;
searcht 5 [`collinear {vec 0,a,b}`;`%`];;
searcht 5 [`a % v = a % w`];;
searcht 5 [`a % (b + c)`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  CONTINUOUS_AT_LIFT_DOT2;;
Print_types.print_term_types t';;
bb 8;;
(* July 4, 2013 *)
searcht 5 [`&0 <= azim a b c d`];;
    NONPLANAR_OPEN;;
searcht 5 [`(\t. c) continuous atz`];;
searcht 5 [`open`;`dot`;`>`];;
searcht 5 [def "open"];;
searcht 5 [`open`;`!`;`?`;`dist`];;
searcht 15 [`dot`;`continuous`];;
searcht 5 [`continuous`;`lift`;`real_continuous`];;
searcht 5 [`lift`;`drop`;`continuous`];;
searcht 5 [`continuous`;`cross`];;
searcht 5 [`real_open`;`real_interval`];;
searcht 5 [`real_open`;`{x | x > c}`];;
searcht 5 [`continuous`;`(-- (v:real^A))`];;
searcht 5 [`azim`;`real_continuous`];;
searcht 5 [`azim a b c d = &0`;`aff_ge`];;
searcht 5 [name "MIN";name "FINITE";omit `dimindex`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  CONTINUOUS_AT_LIFT_DOT2;;
Print_types.print_term_types t';;
bb 5;;
	    let s = searcht 5;;
RENAME_FREE_VAR (`f:real^3#real^3->real^3#real^3`,"Q")
(* July 4, 2013 *)
s [`local_fan`;`IMAGE`];;
searcht 5 [name "XREC"];;
s [`local_fan`;`convex_local_fan`];;
rename;;
s [`?(x:A#B). P x`];;
SKOLEM_THM;;
s [`local_fan`;`FINITE`];;
searcht 5 [`local_fan`;`~(FF = {})`];;
s [`IMAGE f X = {}`];;
searcht 25 [`local_fan`;`(u,v)`];;
searcht 5 [`x = p1,p2`];;
searcht 8 [`?e. (p ==> q)`];;
searcht 8 [`((?x. p) ==> q)`];;
searcht 5 [`azim_in_fan`;`interior_angle1`];;
Nkezbfc_local.CONVEX_LOFA_IMP_INANGLE_EQ_AZIM;;
	    let named x =  (searcht 3 [name x]);;
searcht 5 [`azim_in_fan`;`azim`];;
Local_lemmas1.AZIM_IN_FAN_RHOND_IVS_RHOND;;
searcht 15 [`rho_node1`];;
searcht 5 [`v,rho_node1 FF v IN FF`];;
searcht 5 [`p1 IN (V:real^3->bool)`;`local_fan`;`(x:real^3#real^3) IN FF`];;
searcht 10 [`rho_node1 FF v = w`];;
searcht 5 [`azim`;`azim_in_fan`];;
Local_lemmas1.AZIM_IN_FAN_RHOND_IVS_RHOND;;
searcht 5 [`rho_node1 FF p1 IN V`];;
searcht 5 [`~collinear s`;`local_fan`];;
searcht 5 [`ivs_rho_node1 FF v = w`];;
searcht 5 [`ivs_rho_node1 FF v,v IN FF`];;
searcht 5 [`collinear`;`ivs_rho_node1`];;
searcht 5 [`FST x,SND x`];;
FST;;
searcht 5 [`&0 < interior_angle1 v f p`];;
searcht 25[`(x,y) IN U`];;
searcht 5 [`FINITE`;`CROSS`];;
searcht 5 [`local_fan`;`~(V = {})`];;
searcht 5 [`V CROSS FF = {}`];;
help_grep "BETA";;
searcht 5 [`local_fan`;`{u,v} IN E`;`(u,v) IN FF`];;
searcht 5 [`wedge_in_fan_ge`;`wedge_ge`];;
searcht 15 [`local_fan`;`x = {u,v}`];;
searcht 5 [`graph`;`FAN`];;
s  [`FAN`;`local_fan`];;
searcht 5 [`graph `;`{u,v}`];;
searcht 5 [`graph`;`HAS_SIZE`];;
searcht 5 [`x HAS_SIZE 2`;`{a,b}`];;
searcht 25 [`IMAGE`;`INSERT`];;
searcht 5 [`{u,v} IN (E)`;`local_fan`;`(x:real^3#real^3) IN FF`];;
searcht 5[`{a,b} = {c,d}`];;
Local_lemmas.LOCAL_FAN_IMP_IN_V;;
searcht 5 [`azim`;`azim_in_fan`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  CONTINUOUS_AT_LIFT_DOT2;;
Print_types.print_term_types t';;
bb 8;;
(* July 7, 2013 *)
searcht 5 [`{u,v} IN E`;`(u,v) IN FF`;`local_fan`];;
searcht 5 [`convex_local_fan`;`local_fan`];;
searcht 5 [`local_fan`;`collinear`];;
searcht 5 [`v IN V`;`(u,v) IN FF`;`local_fan`];;
searcht 5 [`~(u = vec 0)`;`local_fan`];;
searcht 5 [`generic`;`collinear`];;
searcht 5 [`collinear`;`wedge`];;
searcht 5 [`collinear`;`cross`];;
searcht 5 [`real_continuous`;`norm`];;
searcht 5 [`norm`;`(x = vec 0)`];;
searcht 5 [`real_continuous_on`;`real_continuous`];;
searcht 15 [`f o g real_continuous net`];;
searcht 5 [`(\t. c) continuous net`];;
searcht 5 [`funlist`];;
searcht 5 [`v cross w = v`];;
searcht 5 [`collinear {a,b}`];;
searcht 5 [`azim`;`wedge`];;
searcht 5 [`azim a b c c `];;
searcht 5 [`cross`;`continuous`];;
searcht 5 [`(\t. f t) = f`];;
searcht 5 [`continuous`;`(\(t:real). -- (f:real->real^A) t)`];;
searcht 5 [`continuous`;`(--):real^A->real^A`];;
searcht 5 [`x IN wedge_ge a b c d`];;
searcht 5 [`wedge_ge`;`wedge_in_fan_ge`];;
searcht 5 [`x IN wedge_ge a x b d`];;
searcht 5 [`wedge_ge`;`azim`];;
searcht 5 [`azim a b c b`];;
NHCXLRV;;
Oxl_2012.CHQSQEY;;
Oxl_2012.MTMLSRF;;
Oxl_2012.LXDEYBO;;
Oxl_2012.UNPNFVW;;
Oxl_2012.DHCVTVE;;
Oxl_2012.PMZTATI;;
Oxl_2012.IPVICGW;;
searcht 5 [`ivs_rho_node1 FF w IN V`];;
searcht  5[`collinear`;`rho_node1`];;
searcht 5 [`collinear`;`local_fan`;`generic`];;
searcht 5 [`wedge a b c d SUBSET wedge_ge a b c d`];;
searcht 5 [name "EGHNAVX"];;
searcht 10 [`local_fan`;`(x:real^3#real^3) IN FF`];;
searcht 5 [`x = a,b`;`FST`;`SND`];;
searcht 5 [`a,b = c,d`];;
searcht 5 [`wedge_ge a b c d`;`wedge a b c d`];;
searcht 5 [`convex_local_fan`;`local_fan`];;
searcht 1 [`ITER`;`rho_node1`;`ivs_rho_node1`];;
searcht 5 [`ITER 1`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  SKOLEM_EPSILON;;
Print_types.print_term_types t';;
bb 5;;
(* July 8, 2013 *)
searcht 5 [name "KOMW"];;
searcht 5 [`x IN P hull s`];;
searcht 5 [`aff_gt s t SUBSET affine hull u`];;
searcht 5 [`ITER i (rho_node1 FF) w = ITER j (rho_node1 FF) w`];;
searcht 5 [`local_fan`;`CARD V = CARD FF`];;
searcht 5 [`ITER i (rho_node1 FF) v IN V`];;
searcht 5 [`ITER (i+j)`];;
searcht 5 [name "XIV"];;
searcht 5 [`azim v0 v1 v2 v1`];;
Local_lemmas.AZIM_SPEC_DEGENERATE;;
searcht 5 [`azim v0 v1 v1 v2`];;
searcht 5 [`azim v0 v1 v2 v3 + azim v0 v1 v3 v2`];;
azim;;
searcht 5 [`v IN wedge_ge a b c v`];;
searcht 10 [`real_continuous`;`?`;`atreal`];;
real_continuous_at;;
searcht 5 [`dihV`;`real_continuous`];;
searcht 5 [`(\t. c) continuous net`];;
GSYM SKOLEM_EPSILON;;
searcht 5 [`?d. (a /\ b d ==> c d)`];;
searcht 5 [`azim`;`a IN aff_ge u v`];;
searcht 5 [`azim a b c d = &0`;`&0 < x`];;
searcht 5 [`SUC i = SUC j`];;
searcht 15 [`azim`;`dihV`];;
searcht 5 [`azim a b c d = &0`];;
searcht 15 [`cyclic_set`;`%`];;
searcht 5 [`t % e = vec 0`];;
searcht 10 [`affine hull s`;`%`];;
searcht 25 [`azim_cycle`;omit `FAN`;omit `azim_in_fan`;omit `local_fan`;omit `convex_local_fan`;omit `wedge_in_fan_gt`;`@`];;
searcht 5 [`azim (vec 0) (t % e)`];;
searcht 15 [`projection`;`%`];;
Sphere.cyclic_set;;
searcht 5 [`azim_cycle {x}`];;
Sphere.azim_cycle;;
searcht 5 [`collinear`;`cyclic_set`];;
Wrgcvdr_cizmrrh.IDENTIFY_AZIM_CYCLE;;
searcht 5 [`azim a (-- b) c d`];;
searcht 15 [`%`;`continuous`];;
searcht 5 [`wedge_in_fan_ge`];;
Local_lemmas.LOCAL_FAN_CHARACTER_OF_RHO_NODE2;;
deformation_rho_node1_equivariant1;;
searcht 5 [`&0 < interior_angle1 u FF v`];;
Local_lemmas.INTERIOR_ANGLE1_POS;;
searcht 5 [`interior_angle1`;`azim`];;
Nkezbfc_local.CONVEX_LOFA_IMP_INANGLE_EQ_AZIM_IVS;;
searcht 5 [`rho_node1`;`ivs_rho_node1`];;
searcht 5 [`ITER (a+b)`];;
XIV_ECAU;;
searcht 5 [`ivs_rho_node1 FF w IN V`];;
searcht 5 [`ITER i (rho_node1 FF) w IN V`];;
ITER;;
searcht 5 [`ITER`;`rho_node1`;`ivs_rho_node1`];;


print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  SKOLEM_EPSILON;;
Print_types.print_term_types t';;
bb 4;;
(* July 10, 2013 *)
searcht 5 [`azim_in_fan`;`azim`];;
searcht 5 [`v,rho_node1 FF v IN FF`];;
PROPERTIES_GENERIC_LOCAL_FAN_ALT;;
searcht 5 [`ITER i f w = ITER j f w`];;
Local_lemmas1.LT_CARD_MONO_LOFA;;
searcht 5[`ITER (CARD V - 1)`];;
searcht 5[`ITER (CARD V)`];;
searcht 5 [`ITER i (ivs_rho_node1 FF) v`;`ITER j (rho_node1 FF) v`];;
searcht 5 [`collinear`;`ivs_rho_node1`];;
searcht 5 [`rho_node1 FF v IN VV`];;
searcht  5[`ITER i (ivs_rho_node1  FF) v IN V`];;
searcht 5 [`ivs_rho_node1 FF (rho_node1 FF w)`];;
searcht 5 [`rho_node1 FF (ivs_rho_node1 FF w)`];;
XIV_ECAU;;
KCZXLLE_SYM;;
searcht 5 [`collinear`;`cross`];;
searcht 15[`coplanar`;`affine hull s`];;
searcht 15[`v2 IN affine hull {a,b,c}`;`%`;`vec 0`];;
searcht 5 [`u1 cross u2 = -- (u2 cross u1)`];;
searcht 15 [`&0 < x * y`];;
searcht 2 [`&0 < x dot x`];;
searcht 5 [`&0 < azim a b c d`;`azm a b c d = &0`];;
searcht 5 [`azim a b c d = &0`;`aff_gt`];;
searcht 5 [`collinear`;`%`];;
searcht 5 [`e dot (e cross f)`];;
searcht 5 [`azim a b c d <= azim a' b' c' d'`;`azim_cycle`];;
searcht 5 [`azim_cycle U a b v IN U`];;
searcht 5 [`aff_gt {a,b} {c}`;`%`];;
searcht 5 [`DISJOINT`;`collinear`];;
Polar_fan.AZIM_CYCLE_BASIC_PROPERTIES;;
searcht 5 [`azim_cycle U o e  v = v`];;
searcht 5 [`azim a (-- e)`];;
searcht 8 [`azim a b c d + azim a b d e`;omit `cyclic_set`];;
searcht 5 [`&0 <= azim a b c d`];;
searcht 50 [`azim`;`azim_cycle`];;
Wrgcvdr_cizmrrh.AZIM_CYCLE_PROPERTIES;;
searcht 15 [`s SUBSET affine hull t`];;
searcht 5 [`aff_gt {a,b} {c}`;`%`];;
AFF_GT_SUBSET_AFFINE_HULL;;
searcht 5 [`periodic`;`ITER`];;
searcht 5 [`local_fan`;`CARD FF = CARD V`];;


print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  SKOLEM_EPSILON;;
Print_types.print_term_types t';;
bb 4;;
(* July 12, 2013 *)
searcht 5 [name "A_482"];;
searcht 5 [`MMs_v39`;`BBs_prime_v39`];;
searcht 5 [`BBprime2_v39`;`BBprime_v39`];;
Appendix.BBs_v39;;
searcht 15 [`IMAGE f X SUBSET Y`];;
searcht 5 [`local_fan`;`convex_local_fan`];;
searcht 5 [`&0 < x`;`interior_angle1`];;
searcht 5 [`IMAGE f (:A)`];;
searcht 5 [`interior_angle1`;`azim`];;
searcht 5 [`coplanar`;`azim_in_fan`];;
searcht 5 [`rho_node1`;`is_scs_v39`];;
searcht 5 [`ivs_rho_node1`;`is_scs_v39`];;
searcht 5 [`ivs_rho_node1 FF u IN V`];;
searcht 5 [`rho_node1 FF v = rho_node1 FF w`];;
Qknvmlb.VV_SUC_EQ_RHO_NODE_PRIME;;
searcht 5 [`rho_node1 FF (ivs_rho_node1 FF v)`];;
searcht 5 [`generic`;`collinear`];;
searcht  5 [`generic`;`scs_generic`];;
searcht 5 [`is_scs_v39`;`&2 <= dist (v,u)`];;
searcht 10 [`(p + a) MOD k = (p + b) MOD k`];;
searcht 10 [`(a MOD k = a)`];;
searcht 5 [`coplanar`;`azim`];;
searcht 5 [`norm`;`ball_annulus`];;
searcht 5 [`&0`;`delta_y`;`norm`];;
Tame_lemmas.delta_y_pos;;
rr[LET_DEF;LET_END_DEF] Tame_lemmas.delta_y_pos];;
searcht 5 [`coplanar`;`delta_y`];;
searcht 5 [`dist(vec 0,v)`;`norm`];;
GSYM Oxlzlez.coplanar_delta_y;;
searcht 5 [`&0 < x`;`ups_x`;`dist`];;
Localization.deformation;;
searcht 5 [`dist`;`~collinear {a,b,c}`;`x ==> ((u:real^3) = v)`;omit `cos`;omit `radV`];;
searcht 5 [def "eqvl"];;
searcht 5 [`coplanar`;`cross`];;
Local_lemmas.NOT_COLL_IMP_COPL;;
searcht 15 [`coplanar`;`%`];;
Trigonometry2.NONCOPLANAR_3_BASIS;;
searcht 5 [`v1 dot (v1 cross v2)`];;
searcht 5 [`re_eqvl`];;
searcht 5 [`cross`;`collinear`];;
searcht 5 [`a dot a = &0`];;
searcht 5 [`dist`;`norm`];;
NORM_ADD_PYTHAGOREAN;;
searcht 5 [`norm (t % v)`];;
searcht 5 [`(norm x) pow 2`;`dot`];;
searcht 5 [`t pow 2 = &1`];;
searcht 5 [`t2 pow 2 = &0`];;
searcht 5 [`(v1 cross v2) dot (v1 cross v2)`];;
searcht 5 [name "CONTINUOUS_CONTINUOUS_WITHINREAL"];;
Localization.deformation;;
searcht 5 [`dist(a,b)= &0`];;
searcht 5 [`collinear`;`coplanar`];;
searcht 5 [`coplanar`;`~((v:real^A)  = w)`];;
searcht 5 [`coplanar`;`delta_y`;`dist`];;
searcht 5 [`x pow 2 = y pow 2`];;
searcht 5 [`&0 <= dist(x,y)`];;
searcht 5 [`continuous`;`coplanar`];;
searcht 5 [`dist(vec 0,v)`];;
mk_simplex_uniq;;
searcht 5 [`cross`;`azim`];;
searcht 5 [`delta_y`;`norm`];;
searcht 5 [`norm`;`dist`];;
 Oxlzlez.coplanar_delta_y;;
REAL_CONTINUOUS_SUB;;
REAL_CONTINUOUS_CONST;;
) orelse (mmp REAL_CONTINUOUS_NEG) orelse mmp REAL_CONTINUOUS_ABS)
searcht 5 [`real_abs`;`real_continuous`];;
ETA_AX;;
REAL_CONTINUOUS_ABS;;
searcht 5 [`I`;`real_continuous`];;
searcht 5 [`(\x. x)`;` real_continuous`];;
searcht 15 [`real_continuous`;`?`];;
searcht 5 [`real_continuous`;`delta_x`];;
searcht 5 [name "OPEN";name "REAL_INTERVAL"];;
searcht 5 [`real_continuous`;`pow`];;
DIST_0;;
searcht 5 [`&0 < dist(x,y)`];;
searcht 5 [`x pow 2 = y pow 2`];;
searcht 5 [`&0 <= norm x`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  SKOLEM_EPSILON;;
Print_types.print_term_types t';;
bb 4;;
(* July 12, 2013 *)
searcht 5 [`a dot a`;`~(a = vec 0)`];;
searcht 5 [`collinear {a,b}`];;
searcht 5 [`coplanar`;`cross`];;
searcht 5 [`collinear`;`cross`];;
searcht 15 [`~coplanar {a,b,c,d}`;`%`];;
VECTOR_SUB_RZERO;;
searcht 15 [`coplanar`;`collinear`];;
searcht 5 [`u IN affine hull {a,b,c}`;`%`];;
searcht 5 [`x dot x = &0`];;
searcht 5 [`norm (t % a)`];;
searcht 5 [`norm v1`;`dot`];;
searcht 10 [`a * b = c * b`];;
searcht 5 [`n dot n = &0`];;
searcht 5 [`a pow 2 = b pow 2`];;
searcht 5 [`a % w = b % w`];;
searcht 5 [`a * b = c * b`];;
searcht 5 [`sqrt`;`real_continuous`];;
searcht 5 [`continuous`;`cross`];;
searcht 5 [`lift`;`continuous`];;
searcht 5 [`real_continuous`;`(/)`];;
searcht 5 [`collinear`;`ups_x`];;
mk_planar2_continuous;;
searcht 5 [`norm x = &0`];;
searcht 5 [`&0 < ups_x a b c`;`dist`];;
searcht 5 [`norm`;`dist`];;
searcht 2 [`coplanar`;`~((u:real^3)=v)`];;
mk_planar_unique;;
searcht 5 [`~coplanar s`;`affine hull s`];;
searcht 5 [`affine hull {a,u,v,w}`;`%`];;
searcht 15 [`coplanar`;`%`;omit `~coplanar s`];;
searcht 15 [`coplanar`];;
searcht 20 [`coplanar`;`{a,b,c}`];;
searcht 5 [`collinear`;`aff_dim`];;
searcht 5 [`coplanar`;`aff_dim`];;
searcht 5 [`aff_dim`;`s SUBSET t`];;
searcht 5 [`x IN P hull s`];;
searcht 5 [`affine hull {a,b,c}`;`%`];;
searcht 2 [`&0 < v dot v`];;
searcht 5 [`&0 < t * y <=> &0 < t`];;
searcht 5 [`a % v = a % w`];;
searcht 5 [`ups_x a b c = ups_x a' b' c'`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  SKOLEM_EPSILON;;
Print_types.print_term_types t';;
bb 5;;
(* July 16, 2013 *)

searcht 5 [`is_scs_v39 s`;`&2 <= dist(a,b)`];;
searcht 5 [`(a + b) MOD k = (a + c) MOD k`];;
searcht 15 [`(a MOD k = b MOD k)`;omit `5`;omit `3`;omit `is_scs_v39`];;
searcht 5 [`a MOD k = a`];;
searcht 5 [`{a,b}={c,d}`];;
searcht 5 [`deformation`;`is_scs_v39`];;
searcht 5 [def "ww_defor"];;
searcht 5 [`convex_local_fan`;`deformation`;name ".ZLZ"];;
frees_of_goal ([],`a b c`);;
searcht 5 [name "SKOLEM_EPSILON"];;
searcht 0 [`periodic`;`{(i:num) | i < k}`];;
searcht 15 [`periodic2`;`0..u`];;
searcht 15 [`periodic2`];;
Oxl_def.periodic_numseg;;
Terminal.periodic2_mod_reduce;;
searcht 15 [`periodic2`];;
SKOLEM_PERIODIC2;;
searcht 5 [`is_scs_v39`;`3`];;
help_grep "conj";;
conjuncts `a /\ b /\ c`;;
REWRITE_CONV [GSYM CONJ_ASSOC] `(a /\ b) /\ c`;;
searcht 5 [name "Terminal";name "IMAGE"];;
searcht 5 [`ball_annulus`;`norm`];;
searcht 2 [`b_spine5`;`DECIMAL`];;
searcht 5 [`dist`;`real_continuous`];;
searcht 5 [`IMAGE f (IMAGE g X)`];;
searcht 5 [`IMAGE f (a INSERT s)`];;
searcht 5 [`generic`;`scs_generic`];;
searcht 50 [`interior_angle1`;`azim`];;
searcht 5 [`deformation`;`interior_angle1`];;
searcht 5 [`azim_in_fan`;`azim`];;
searcht 5 [`azim_in_fan`;`azim`];;
searcht 5 [`azim_in_fan`;`interior_angle1`];;
searcht 5 [`interior_angle1`;`azim`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  SKOLEM_EPSILON;;
Print_types.print_term_types t';;
bb 8;;
(* July 17, 2013 *)
searcht 5 [`deformation`;`local_fan`];;
searcht 5 [`is_scs_v39`;`i + (k-1)`];;
searcht 5 [`convex_local_fan`;`local_fan`];;
 Ocbicby.INTERIOR_ANGLE1_AZIM;;
named "azim_in_fan2";;
named "EXISTS_INVERSE_OF_V";;
map named[  "LOFA_IMP_EE_TWO_ELMS";
  "LOFA_CARD_EE_V_1";
  "LOCAL_FAN_RHO_NODE_PROS2";];;
Local_lemmas.azim_in_fan2;;
searcht 5 [`rho_node1`;`vv (SUC i)`];;
searcht 5 [`is_scs_v39`;`(!i j. i < k /\ j < k /\ v i = v j ==> i = j)`];;
searcht 5 [`is_scs_v39`;`&2 <= dist(a,b)`];;
searcht 5 [`i MOD k = i`];;
searcht 5 [`ivs_rho_node1`;`vv (i + (k-1))`];;
searcht 5 [`interior_angle1`;`azim`;`is_scs_v39`];;
 Ocbicby.INTERIOR_ANGLE1_AZIM;;
GSYM (REWRITE_RULE[LET_DEF;LET_END_DEF;arith `SUC i = i+1`] Ocbicby.INTERIOR_ANGLE1_AZIM);;
searcht 5 [`cross`;`azim `;`&0 < x`];;
deform_simplex_edge_exists;;
searcht 5 [`{a,b} = {c,d}`];;
searcht 5 [`(\t .t) real_continuous net`];;
searcht 5 [name "periodic2";name "reduce"];;
searcht 5 [`periodic2`;`MOD`];;
searcht 5 [`periodic`;`MOD`];;
searcht 5 [`i MOD k < k`];;
searcht 0 [`(i + k) MOD k`];;
searcht 5 [`(a*b+c) MOD k`];;
searcht 5 [`j MOD k = j`];;
searcht 5 [name "split";name "generic"];;
searcht 15 [`azim a b c d + azim a' b' c' d'`;name "3"];;
searcht 5 [name "tau_fun_azim"];;
searcht 5 [`scs_generic`;`generic`];;
searcht 5 [`azim a b c d < &2 * pi`];;
searcht 5 [`collinear`;`azim`;omit `~collinear s`];;
searcht 5 [`I x = x`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  SKOLEM_EPSILON;;
Print_types.print_term_types t';;
bb 6;;
(* July 18, 2013 *)
searcht 5 [`psort`];;
searcht 5 [`(a,b) = (c,d)`];;
searcht 5 [`(a*b + c) MOD k`];;
searcht 5 [`dist(a,b) + dist(b,c)`];;
searcht 5 [`ball_annulus`;`norm`];;
searcht 5 [name "IMAGE_SUBSET";name "IN"];;
searcht 5 [`(a + b) MOD k`;`(a MOD k + b MOD k) MOD k`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types  SKOLEM_EPSILON;;
Print_types.print_term_types t';;
bb 10;;
(* July 19, 2013 *)
searcht 10 [`azim`;`real_continuous`];;
searcht 5 [`generic`;`collinear`];;
searcht 5 [`local_fan`;`convex_local_fan`];;
searcht 5 [`(a + c) MOD k = (a + c') MOD k`];;
searcht 5 [`deformation`;`collinear`];;
searcht 5 [name "continuous_nbd_pos"];;
searcht 5 [`azim`;`aff_ge`];;
searcht 5 [`&0 < azim a b c d`;`~(azim a b c d = &0)`];;
searcht 5 [`azim a b c d = &0`;`coplanar`];;
searcht 5 [`azim`;`convex_local_fan`;`i + k - 1`];;
searcht 5 [`atn x < atn y`];;
searcht 10 [`coplanar`;`collinear`;`( /\ )`];;
searcht 5 [`sin x`;`&0 < x /\ x < pi`];;
searcht 5 [`azim a b c d < &2 * pi`];;
searcht 5 [`re_eqvl`;`sin`];;
searcht 5 [`re_eqvl`;`&0 < x`];;
searcht 15 [`azim`;`dihV`];;
AZIM_DIHV_SAME;;
searcht 5 [`pi / &2 `;`dih_x `];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types CONTINUOUS_LIFT_DOT2 ;;
Print_types.print_term_types t';;
bb 10;;
back_to "CVX_LO_IMP";;
(* July 20, 2013 *)
searcht 5 [`dihV`;`dih_x`];;
searcht 8 [name "REAL_MVT"];;
searcht 5 [`derived_form`;`dih_x`;`delta_x6`];;
HAS_REAL_DERIVATIVE_TRANSFORM_WITHIN;;
searcht 5 [`f'`];;
searcht 5 [`net within (:real)`];;
searcht 5 [`net within s`;`(f has_real_derivative f') (atreal x)`];;
searcht 5 [`net within s`;`f real_continuous (atreal x)`];;
searcht 5 [`(f o g) has_real_derivative f'`];;
searcht 5 [`dih_x`;`delta_x4`];;
searcht 5 [`real_continuous`;`delta_x`];;
searcht 5 [`(\t. -- f t) real_continuous net`];;
REAL_CONTINUOUS_SUB;;
searcht 5 [`(\t. t) real_continuous net`];;
searcht 5 [`dih_x a b c d e f <= dih_x a b c d' e' f'`];;
Tame_inequalities.DIH_X_MONO_LT_4;;
searcht 5 [`delta_x5`;`delta_x6`];;
searcht 5 [`dih_x a b c d e f = dih_x a c b d' e' f'`];;
searcht 5 [`&0 < ups_x a b c`;`collinear`];;
searcht 5 [`pi / &2 < dih_x a b c d e f`;`delta_x4`];;
searcht 5 [`coplanar {a,b,c}`];;
searcht 3 [`&0 < delta_y a b c d e f`;`dist`];;
searcht 5 [`real_min`];;
searcht 5 [`abs`;`real_continuous`];;
dihV_obtuse_mono_a;;
searcht 5 [name "CONTINUOUS_AT_LIFT_DOT2"];;
searcht 5 [`dihV a b c d = dihV a b d c`];;
searcht 15 [`abs`;`real_continuous`];;
searcht 5 [`(\u. u)`;`real_continuous`];;
Searching.searcht 5 [name "CONTINUOUS_LIFT_DOT"];;
g `!net f i.
            f continuous net /\ g continuous net
            ==> (\x. lift (f x dot g x)) continuous net`;;
searcht 5 [name "SYNQ"];;
Searching.searcht 5 [name "REAL_CONTINUOUS_ATREAL_DIHV_COMPOSE"];;
Searching.searcht 5 [`lift`;`continuous`;`real_continuous`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types CONTINUOUS_LIFT_DOT2 ;;
Print_types.print_term_types t';;
bb 4;;
back_to "kill";;
(* July 22, 2013 *)
searcht 5 [`interior_angle1`;`azim`;`i + k - 1`];;
searcht 5 [`psort k (a,b) = psort k (b,a)`];;
searcht 5 [`psort k (a MOD k,b)`];;
searcht 5 [`(a MOD k) MOD k`];;
searcht 5 [`scs_a_v39`;`MOD`];;
searcht 50 [def "tau"];;
searcht 15 [`taustar_v39`;`tau_fun`];;
searcht 5 [`tau_fun`;`azim`];;
searcht 5 [`BBs_v39`;`periodic`];;
searcht 5 [`dsv_v39`];;
Appendix.is_basic;;
searcht 50 [`is_basic`];;
searcht 5 [`sum (A DIFF B)`];;
searcht 5 [`tau3`;`taum`];;
searcht 5 [`tau3`;`dihV`];;
searcht 5 [`FINITE {i | (i:num) < k}`];;
searcht 5 [`sum A f = sum A g`];;
searcht 5 [`sum {a,b,c} f`];;
searcht 15 [`(a + b) MOD k`;omit `5`];;
MOD_ADD_MOD;;
DIVISION;;
searcht 5 [`0 MOD k`;omit `a MOD 4`];;
searcht 5 [`x MOD k = x`];;
MOD_SHIFT;;
searcht 5 [`(a + c) MOD k = (a + b) MOD k`];;
searcht 5 [`taum`;`tau3`];;
searcht 15 [`azim`;`dihV`];;
searcht 5 [name "FSQ"];;
searcht 5 [`real_interval (a,b)`;`abs t`];;
Zlzthic.real_interval_contains_0_ball;;
searcht 5 [`rho`;`rho_fun`];;
MOD_SHIFT;;s
searcht 5 [`taum a b c d e f = taum a' b' c' d' e' f'`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types CONTINUOUS_LIFT_DOT2 ;;
Print_types.print_term_types t';;
bb 4;;
back_to "kill";;
(* July 23, 2013 *)
searcht  5 [name "deform_simplex_684"];;
searcht 5 [`coplanar`;`cross`;`dot`];;
searcht 5 [`azim`;`cross`;`dot`];;
searcht 5 [`re_eqvl`;`&0 < x`];;
searcht 5 [`re_eqvl`;name "SYM"];;
searcht 5 [def "quadratic"];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types CONTINUOUS_LIFT_DOT2 ;;
Print_types.print_term_types t';;
bb 4;;
back_to "kill";;
(* July 24, 2013 *)
searcht 15 [`quadratic_root_plus`];;
searcht 5 [name "2x2"];;
searcht 25 [`real_sgn`];;
searcht 5 [`real_sgn(x * y)`];;
searcht 5 [`real_sgn x = &1`];;
GSYM REAL_SGN_EQ;;
searcht 5 [`real_sgn`;`abs`];;
searcht 5 [`y / x  = &0`];;
searcht 15 [`&0 < x * y`];;
searcht 5 [`re_eqvl`;`&0 <= x`];;
searcht 5 [`sin`;`azim`];;
searcht 5 [`coplanar`;`delta_y`];;
Collect_geom2.POS_EQ_NOT_COPLANANR;;
searcht 5 [`&0 <= delta_y a b c d e f`;`dist`];;
searcht 5 [`delta_x`;`abc_of_quadratic`];;
searcht 5 [`delta_x`;`a * u pow 2 + b * u + c`];;
searcht 5 [`convex`;`real_interval`];;
searcht 5 [`FINITE`;`connected`];;
CONTINUOUS_FINITE_RANGE_CONSTANT;;
Localization.deformation;;
searcht 5 [`convex`;`connected`];;
searcht 5 [`real_interval`;`interval`];;
searcht 5 [`real_interval`;`real_connected`];;
searcht 5 [`real_continuous`;`FINITE`];;
searcht 5 [def "kepler"];;
searcht 5 [`kepler_conjecture`;`saturated`;`packing`];;
searcht 5 [`drop (lift x)`];;
searcht 15 [`f x IN IMAGE f X`];;
searcht 5 [`lift`;`real_interval`];;
searcht 5 [`connected`;`interval`];;
searcht 5 [`continuous_on`;`real_continuous_on`];;
ETA_AX;;
searcht 5  [`IMAGE f (IMAGE g X)`];;
searcht 5 [`drop o lift`];;
searcht 5 [`real_continuous`;`dist`];;
searcht 5 [`real_continuous`;`real_continuous_on`];;
searcht 5 [`real_open`;`real_interval(a,b)`];;
FINITE_SUBSET;;
searcht 5 [`FINITE`;`INSERT`];;
searcht 15 [`IMAGE f X`;`HAS_SIZE`];;
searcht 5 [`azim a b c d = &0`;`cross`];;
searcht 5 [`interior_angle1`;`azim_in_fan`];;
searcht 15 [`azim_in_fan`;`azim`];;
searcht 5 [`interior_angle1`;`BBs_v39`];;
Ocbicby.INTERIOR_ANGLE1_AZIM;;
searcht 5 [`&0 < azim a b c d`;`convex_local_fan`];;
searcht 5 [`5 MOD 5`];;
Appendix.scs_diag;;
searcht 5 [`(a /\ c) \/ (a /\  b)`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas.EGHNAVX;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 5;;
back_to "azim 0";;
(* July 25, 2013 *)
searcht 5 [name "EGH"];;
searcht 5 [`  IMAGE (\i. (v i,v (SUC i))) (:num)`;`rho_node1`];;
searcht 5 [name "FEKT"];;
searcht 5 [`coplanar`;`&0`;`pi`;`azim`];;
type_of `scs_basic_v39`;;
searcht 15 [`CARD`;`is_scs_v39`];;
searcht 5 [`collinear`;`generic`];;
searcht 5 [`scs_generic`;`generic`];;
help "GEN_ALL";;
help "GENL";;
searcht 15 [`(!i. a ==> P i)`;`a ==> (!i. P i)`;name "FORALL"];;
searcht 5 [name "FORALL";name "AND"];;
 (GSYM Ocbicby.INTERIOR_ANGLE1_AZIM);;
searcht 5 [`aff_gt {a,b} {c}`;`%`];;
searcht 5 [`i MOD k < k`];;
searcht 5 [name "SHIFT";`(a + b) MOD k = (a + c) MOD k`];;
searcht 5 [`i MOD k = i`];;
searcht 5 [`cross`;`collinear`];;
searcht 5 [`coplanar`;`affine hull s`];;
searcht 5 [`x IN P hull s`];;
searcht 5 [`periodic`;`IMAGE`;`a..b`];;
searcht 5 [`periodic`;`a..b`];;
searcht 5 [`aff_gt s t SUBSET affine hull s'`];;
searcht 5 [`collinear`;`affine hull s`;`SUBSET`];;
AFFINE_HULL_3_GENERATED;;
searcht 5 [`coplanar`;`affine hull s`];;
AFFINE_HULL_3_GENERATED;;
Terminal.vv_split_azim_generic;;
searcht 5 [`&0 <= azim a b c d`];;
Local_lemmas1.AZIM_COND_FOR_COPLANAR;;
searcht 5 [`real_continuous`;`azim a b c d < x`];;
WNWSHJT_ALT;;
searcht 5 [`coplanar`;`collinear`;`/\`];;
searcht 5 [`azim`;name "fan3"];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.CON_ATREAL_REAL_CON;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 3;;
back_to "deform coll";;
(* July 27, 2013 *)
searcht 10 [`dihV`;`azim`;name "STRONG"];;
searcht 5 [`dihV`;`dih_x`];;
searcht 5 [`collinear`;`ups_x`];;
searcht 5 [name "SYN"];;
searcht 5 [def "xrr"];;
searcht 5 [`xrr`;`&1`];;
Appendix.scs_v39;;

searcht 5 [`dest_scs_v39`];;
searcht 5 [`FST x = FST x'`];;
searcht 5 [`part1`];;
searcht 5 [name "EGH"];;
searcht 5 [`s SUBSET P hull s`];;
searcht 5 [`aff_gt s t SUBSET affine hull t'`];;
searcht 5 [`convex_local_fan`;`interior_angle1`];;
searcht 5 [`azim`;`dot`;`cross`];;
searcht 5 [`re_eqvl`;`&0 < x`;`==>`];;
searcht 5 [`&0 < sin x`;`azim`];;
searcht 5 [`abs`;`real_continuous`];;
a5_assumption_reduction;;
searcht 5 [name "nbd_pos"];;
searcht 5 [`is_scs_v39`;`&2 <= scs_a_v39 s i j`];;
searcht 5 [`dist`;`real_continuous`];;
 Local_lemmas1.CON_ATREAL_REAL_CON;;
type_of `t0`;;
searcht 50 [`t0`];;
searcht 50 [`quarter`];;
searcht 50 [`quasi_tri`];;
searcht 15 [`collinear`;`continuous`];;
searcht 15 [`collinear`;`deformation`];;
searcht 5 [`azim`;`dih_y`];;
Cuxvzoz.deform_684_pent_exists;;
searcht 5 [name "FSQK"];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.CON_ATREAL_REAL_CON;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 3;;
back_to "epsilon_hex";;
(* July 27, 2013 *)
searcht 3 [`azim`;`dih_y`];;
searcht 5 [`coplanar`;`azim`;`pi`;`&0`];;
searcht 5 [`&0 < azim a b c d`;`abs t < e`];;
searcht 5 [`deformation`;`subset`];;
searcht 5 [name "WNW"];;
searcht 5 [`dih_y a b c d e f <= dih_y a b c d' e f`];;
Ocbicby.dih_y_mono;;
searcht 5 [`continuous`;`coplanar`];;
azim_dih_y;;
searcht 5 [`&0 < norm x`];;
searcht 5 [`~coplanar s`;`~(x = y)`;`/\ `];;
searcht 5 [`delta_y a b c d e f`;`coplanar`];;
searcht 5 [`delta_y a b c d e f = delta_y a' b' c' d' e' f'`];;
searcht 5 [`&0 <= delta_y a b c d e f`];;
searcht 25 [`sum (a..b)`];;
SUM_4;;
searcht 5 [`sum (0..4)`];;
searcht 5 [`sum (a..b)`;`sum ((SUC n)..b)`];;
searcht 5 [`sum (0..3)`];;
searcht 5 [name "sym";`xrr`];;
searcht 5 [`ball_annulus`;`norm`];;
searcht 5 [`taum`;`tau3`];;
Cuxvzoz.tau3_taum_nonplanar;;
searcht 5 [`tau3`;`azim`];;
Cuxvzoz.tau3_azim;;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.CON_ATREAL_REAL_CON;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 3;;
back_to "COMMENT";;
(* July 29, 2013 *)
searcht 5 [`taum a b c d e f = taum a' b' c' d' e' f'`];;
searcht 5 [`arclength a b c + arclength a' b' c'`];;
searcht 12 [`arcV  u v w + arcV u' v' w'`];;
searcht 15 [`arclength`;`aff_gt`];;
searcht 5 [`angle s + angle s'`];;
ANGLES_ADD_AFF_GE;;
searcht 5 [`angle`;`arcV`];;
Nkezbfc_local.PROPERTIES_GENERIC_LOCAL_FAN;;
searcht 5 [`DISJOINT`;`collinear`];;
searcht 5 [`DISJOINT`;name "SYM"];;
Qknvmlb.VV_SUC_EQ_RHO_NODE_PRIME;;
Local_lemmas.LUNAR_IMP_HALF_CIRCLE_SUBSET_AFF_GT;;
Local_lemmas.LOCAL_FAN_CHARACTER_OF_RHO_NODE2;;
Planarity.decomposition_planar_by_angle_fan;;
Planarity.POINT_IN_AFF_GE_IMP_IN_EDGE;;
searcht 5 [name "WL_IN_E"];;
searcht 5 [`FAN`;`aff_ge`];;
searcht 5 [`FAN`;`local_fan`];;
searcht 5 [`FAN`;`{a,b} IN E`;`~(a = b)`];;
searcht 5 [`arclength`;`atn`];;
searcht 5 [`xrr`;`arclength`];;
searcht 5 [`acs`;`pi`;`cos`];;
searcht 5 [`cos z < &1`];;
searcht 5 [`cos (z+pi)`];;
searcht 5 [`cos (-- z)`];;
searcht 5 [`sqrt x * sqrt x = x`];;
seracht 5 [`xrr a b c <= d`];;
searcht 5 [`arclength`;`acs`];;
searcht 5 [`(a * a + b * b - c*c)/(&2 * a*b)`;`<=`];;
Oxl_2012.GRHIDFA;;
searcht 5 [`cc_card_v11`;`CARD`];;
searcht 5 [`gg_mcell`;`cc_gg_v11`];;
searcht 5 [`cell_cluster_estimate_v1`];;
searcht 5 [`TSKAJXY_statement`];;
searcht 10 [name "TSKAJXY"];;
searcht 5 [name "Zorn"];;
searcht 5 [`delta4_squared_y`;`x1_delta_y`];;
searcht 15 [`hminus`;`DECIMAL`];;
searcht 5 [`const1`;`pi`];;
searcht 5 [`sol0`;`DECIMAL`];;
Flyspeck_constants.bounds;;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.CON_ATREAL_REAL_CON;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 3;;
back_to "RIGHT_FORALL";;
(* Aug 4, 2013 *)
searcht 5 [`delta_x4`;`dih_x`];;
searcht 5 [`ups_x (a*a)`;`a*b*c`];;
searcht 5 [`&0 < x * y <=> &0 < y`];;
Sphere.delta4_y;;
Sphere.delta_y;;
searcht 5 [`atn2 (x,y)`;`x < y`];;
searcht 5 [`atn x < pi / &2`];;
searcht 5 [`atn (-- x)`];;
DIST_TRIANGLE;;
searcht 5 [`&0 <= delta_y y1 y2 y3 y4 y5 y6`;`dist`];;
searcht 5 [`cos`;`acs`;`pi`];;
searcht 5 [`sqrt x * sqrt x`];;
searcht 5 [`x pow 2 <= y pow 2`];;
searcht 5 [`abs x pow 2`];;
searcht 5 [name "1834"];;
searchl ineq_as_thm [`[(&2,y1,&2*h0);(&2,y2,&2*h0);(&2,y3,&2*h0);(&2,y4,&2*h0);(&2,y5,&2*h0);(&2,y6,&2*h0)]`];;
searcht 5 [`compact`;`real_continuous_on`];;
searcht 5 [`derived_form`;`dih_x`];;
searcht 5 [`taum`;`dih_x`];;
searcht 5 [`taum`;`dih_y`];;
Sphere.taum;;
searcht 5 [`taum`;`rhazim`];;
Sphere.rhazim;;
searcht 5 [`arclength`;`xrr`];;


print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.CON_ATREAL_REAL_CON;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 4;;
back_to "periodic2_mod_reduce";;
(* Aug 4, 2013 *)
searcht 5 [`rho`;`ly`];;
searcht 5 [`sol0`;`const1`];;
searcht 5 [`real_continuous`;`dih_y`];;
searcht 15 [`f o g`;`real_continuous`];;
searcht 5 [`(\t. t) real_continuous net`];;
searcht 5 [`real_continuous`;`has_real_derivative`];;
searcht 5 [`ups_x a b c = ups_x b a c`];;
searcht 5 [`real_compact`;`real_interval [a,b]`];;
searcht 5 [`real_continuous_on`;`real_continuous`];;
searcht 15 [`&0 < ups_x a b c`];;
searcht 5 [`c <= xrr a b d`];;
searcht 5 [`xrr a b c <= d`];;
searcht 5 [`&0 <= taum y1 y2 y3 y4 y5 y6`];;
searcht 5 [`c <= sqrt8`;`h0`];;
searcht 5 [`d_tame i = j`];;
searcht 5 [`scs_arrow_v39`;`scs_6I1`];;
searcht 5 [name "EAPG"];;
searcht 5 [`CONS a b = CONS a' b'`];;
searcht 26 [`s_init_list_v39`;`scs_6I1`];;
searcht 5 [name "periodic2";name "red"];;
searcht 5 [`funlist_v39`;`periodic2`];;
searcht 5 [`(i+k) MOD k`];;
searcht 5 [`{a,b} = {c,d}`];;
searcht 5 [`psort 5 (0,2)`];;
Ocbicby.psort_5;;
searcht 5 [`scs_arrow_v39`;`MMs_v39`];;
searcht 5 [name "OTMT"];;
searcht 5 [`is_scs_v39`;`scs_3T1`];;
searcht 5 [`BBs_v39`;`MMs_v39`];;
searcht 5 [`~(x = {})`;`?`];;
searcht 5 [`scs_basic_v39`;`scs_3T5`];;
searcht 5 [`mk_unadorned_v39`;name "explicit"];;
searcht 5 [`BBs_v39`;`MMs_v39`];;
searcht 5 [`psort 3 (0,1)`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.CON_ATREAL_REAL_CON;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 7;;
back_to "Cuxv";;
(* Aug 6, 2013 *)
searcht 5 [`5 MOD 4`];;
Cuxvzoz.deform_simplex_edge_exists;;
searcht 5 [`coplanar`;`azim`;`pi`;`&0`];;
searcht 5 [name "coplanar_cross_reduction"];;
searcht 20 [`azim`;`cross`;`dot`];;
searcht 15 [`coplanar`;`collinear`;`/\`];;
searcht 15 [`{a,b,c,d} = {a',b',c',d'}`];;
searcht 5 [`p MOD k < k`];;
searcht 5 [`((a MOD k) + b) MOD k`;omit `5`];;
searcht 5 [`(a + b) MOD k = (a + b') MOD k`];;
searcht 5 [`p MOD k MOD k`];;
searcht 5 [`3 MOD 4`];;
searcht 5 [`scs_diag k p q`];;

searcht 5 [name "SKOLEM_EPSILON"];;
searcht 5 [name "SKOLEM";name "PERIODIC"];;
searcht 5 [`&0 < azim a b c d`;`deformation`];;
searcht 5 [`deformation`;`SUBSET`];;
searcht 5 [name "epsilon_quad"];;
searcht 5 [`collinear`;`ups_x`];;
help "RAND_CONV";;
help "RATOR_CONV";;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.CON_ATREAL_REAL_CON;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 5;;
back_to "STRONG";;
(* Aug 8, 2013 *)
searcht 5 [`convex_local_fan`;`interior_angle1`;`pi`];;
Cuxvzoz.WNWSHJT_ALT;;
UNDISCH Ocbicby.LEMMA_1834976363;;
searcht 50 [`azim`;`+`;`deformation`];;
Localization.deformation;;
searcht 5 [name "WNW"];;
searcht 15 [`~collinear s`;`continuous`];;
searcht 15 [`azim a b c d + azim a b d e`;`pi`;name "fan"];;
searcht 50 [`(azim a b c d)  < pi `];;
searcht 5 [`d <= xrr a b c`];;
searcht 5 [`xrr a b c <= d`];;
searcht 5 [`&2 <= dist s`;`BBs_v39`];;
searcht 5 [`sqrt(x pow 2)`];;
searcht 15 [`x <= sqrt y`];;
searcht 5 [`coplanar`;`delta_y`];;
searcht 5 [`coplanar`;`collinear`;`/\`];;
searcht 5 [name "SUM_NUMSEG3"];;
Cuxvzoz.MMs_minimize_tau_fun;;
searcht 5 [`periodic`;`sum`];;
searcht 15 [`sum (a..b)`;`SUC`];;
searcht 5 [`sum (p..p)`];;
searcht 5 [`interior_angle1`;`azim`;`is_scs_v39`];;
REWRITE_RULE[LET_THM] Ocbicby.INTERIOR_ANGLE1_AZIM;;
searcht 5 [`(p + k) MOD k`];;
searcht 5 [name "LEMMA_684"];;
searcht 5 [`(p+k) MOD k`];;
searcht 5 [`i MOD k = i`];;
searcht 5 [`delta4_y`;`dih_y`];;
searcht 5 [`dihV`;`dih_y`];;
searcht 3 [`&0 <= delta_y a b c d e f`;`dist`];;
searcht 5 [`azim`;`dihV`;name "STRONG"];;
searcht 5 [`dihV`;`dih_y`];;
Merge_ineq.DIHV_EQ_DIH_Y;;
searcht 5 [`dih_y a b c d e f <= dih_y a b c d' e f`];;
searcht 5 [`arclength`;`atn`];;
searcht 15 [`sqrt x < y`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.CON_ATREAL_REAL_CON;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 3;;
back_to "edge lengths";;
(* Aug 11, 2013 *)
searcht 5 [`coplanar`;`azim`];;
searcht 3 [`is_scs_v39`;`&0 < azim a b c d`];;
searcht 5 [`&0 < delta_y a b c d e f`;`coplanar`];;
searcht 5 [`dih_y a b c d e f = dih_y a' b' c' d' e' f'`];;
searcht 5 [`x1_delta_y`];;
Trigonometry2.NOT_ZERO_EQ_POW2_LT;;
searcht 5 [`&0 <= delta_y a b c d e f`;`dist`];;
searcht 5 [`hypermap_of_list`];;
searcht 15 [`hypermap_of_list`];;
searcht 5 [name "lemma";name "merge"];;
searcht 5 [name "planar_hypermap"];;
searcht 5 [name "TGJ"];;
searcht 5 [name "samsara"];;
type_of `is_inj_list`;;
searcht 5 [name "ThreeDarts"];;
searcht 5 [name "is_loop"];;
searcht 5 [`x:(A) loop`];;
searcht 5 [name "iso";`iso`];;
searcht 5 [name "Quotient"];;
searcht 15 [name "lemmaNodalFixedPoint"];;
searcht 5 [`is_no_double_joins`];;
Hypermap.cycle;;
searcht 5 [name "dih2k"];;
Hypermap.lemma_mInside;;
Hypermap.lemmaHQYMRTX;;
searcht 50 [def "tame"];;
type_of `bn_cong_iso`;;
searcht 5[name "hypermap_of_list"];;
searcht 15 [`hypermap_of_list`];;
List_hypermap.good_list;;
searcht 5 [`MAP I`];;
searcht 5 [`IMAGE I`];;
searcht 15 [`good_list`;`hypermap_of_list`];;
searcht 5 [name "perm_eq"];;
searcht 15 [`perm_eq`];;
searcht 5 [`dropl 0`];;
searcht 5 [`cat s []`];;
searcht 5 [`acs`;`atn`];;
searcht 5 [`kepler_conjecture`;`packing`];;


searcht 5 [name "OEHDBEN"];;
let nd s = searcht 5 [name s];;
map nd ["OEHDBEN";"OTMTOTJ1";"OTMTOTJ3"];;
map nd ["CNICGSF"];;
map nd ["FYSSVEV";"ARDBZYE";"AUEAHEH";"VQFYMZY"];;
map nd ["VQFYMZY"];;
map nd ["BNAWVNH";"RAWZDIB";"MFKLVDK";"RYPDIXT"];;
map nd ["NWDGKXH";"YOBIMPP";"MIQMCSN";"BKOSSGE"];;
searcht 5 [`sin (x  + y) = sin x`];;
open Counting_spheres;;
open Tactics_jordan;;
open Ysskqoy;;
open Hales_tactic;;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.CON_ATREAL_REAL_CON;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 3;;
back_to "edge lengths";;
(* Aug 15, 2013 *)



searcht 5 [`delta`;`delta_x`];;
searchd [`delta`];;
searcht 5 [`b_coef`];;
searchd [`d3`];;
searcht 5 [`voronoi_trg`];;
searcht 5 [`voro2`];;
searcht 5 [`voronoi_open`];;
searchd [`cm3_ups_x`];;
searcht 5 [`cm3_ups_x`];;
searcht 5 [`condA`];;
List.length (definitions());;
searchd [`rho_ij`];;
searcht 5 [`rho_ij`;`rho_x`];;
searcht 15 [`chi`];;
searcht 5 [`the`];;
searcht 20 [`SOME x`];;
searcht 5 [`concat`];;
searcht 5 [`filter`;`FILTER`];;
searcht 5 [`filter`;`CONS`];;
searcht 5 [`f (CONS (x:A list) xs) = APPEND x (concat xs)`];;
searcht 5 [`MAP`;`map`];;
type_of `minimal`;;
searcht 50 [def "MIN"];;
searchd [`MIN`];;
searchd [`minn`];;
searcht 50 [`minn`];;
searcht 5 [`set_of_list []`];;
searcht 5 [`POWER`;`ITER`];;
let s1 t = searcht 5 [t];;
s1 `min_num`;;
searcht 5 [`min_num`;`INSERT`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.CON_ATREAL_REAL_CON;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 3;;
back_to "edge lengths";;
(* Aug 15, 2013 *)
Misc_defs_and_lemmas.min_least;;
searcht 5 [`MIN`];;
searcht 5 [`minn x y <= x`];;
searcht 5 [`set_of_list`;`{}`];;
Seq.filter;;
searcht 5 [`dropl (SUC n)`];;
Seq.take;;
SPEC_TAC;;
searcht 5 [`x ++ []`];;
Seq.rot;;
searcht 5 [`sol_euler_x`;`delta_x`];;
searchd [`dih4_x`];;
searcht 50 [`mardih6_x`];;
searcht 50 [`dih4_x`];;
searcht 50 [`domain6`];;
searcht 10 [`gamma3f_x_div_sqrtdelta`];;

print_proof();;
Print_types.print_goal_types();;
Print_types.print_thm_types Local_lemmas1.CON_ATREAL_REAL_CON;;
Print_types.print_term_types t';;
frees (concl Local_lemmas.EGHNAVX);;
bb 3;;
back_to "edge lengths";;
(* Aug 19, 2013 *)
searcht 10 [`ldih6_x`];;
searchp [`vol3f_x_sqrt2_lmplus`];;
searchp [`arclength_x_123`];;
Nonlin_def.vol3f_x_sqrt2_lmplus;;
Sphere.rhazim_x;;
searcht 5 [name "sq_pow2"];;