needs "../formal_lp/hypermap/ineqs/lp_ineqs_proofs-compiled.hl";;
needs "local/RNSYJXM-compiled.hl";;
needs "local/polar_fan.hl";;

(* Module Lp_main_estimate*)
module Lp_main_estimate = struct

open Sphere;;
open Localization;;
open Tame_defs;;
open Ssrbool;;
open Ssrnat;;
open Lp_ineqs_def;;
open Lp_ineqs_proofs;;
open Hypermap_and_fan;;
open Fan_defs;;
open Hypermap;;
open Add_triangle;;
open Tame_general;;
open Hypermap_iso;;
open Rnsyjxm;;
open List_hypermap;;
let rho_fun = new_definition
	`rho_fun y =  &1 + (inv (&2 * h0 - &2)) * (inv pi) * sol0 * (y - &2)`;;
let tau_fun = new_definition
	`tau_fun V E f =  sum (f) (\e. rho_fun(norm(FST e)) * (azim_in_fan e E)) 
			- (pi + sol0) * &(CARD f -2)`;;
let tau3 = new_definition `tau3 (v1:real^3) v2 v3 = 
  rho (norm v1) * dihV (vec 0) v1 v2 v3 + rho(norm v2) * dihV (vec 0) v2 v3 v1 +
    rho(norm v3) * dihV (vec 0) v3 v1 v2 - (pi + sol0)`;;
let JEJTVGB_std_concl = 
  `!V E FF. 
    convex_local_fan (V,E,FF) /\
    packing V /\
    V SUBSET ball_annulus /\
    4 <= CARD V /\ CARD V <= 6 /\
    (!v w. ~(v = w) /\ v IN V /\ w IN V /\ ~({v,w} IN E) ==> &2 * h0 <= dist(v,w)) /\
    (!v w. {v,w} IN E ==> &2 <= dist(v,w) /\ dist(v,w) <= &2 * h0) ==>
    d_tame (CARD V) <= tau_fun V E FF`;;
let JEJTVGB_std3_concl = 
  `!v1 v2 v3. 
    &2 <= norm v1 /\
    &2 <= norm v2 /\
    &2 <= norm v3 /\
    norm v1 <= &2 * h0 /\
    norm v2 <= &2 * h0 /\
    norm v3 <= &2 * h0 /\
    &2 <= dist(v1,v2) /\
    &2 <= dist(v1,v3) /\
    &2 <= dist(v2,v3) /\
    dist(v1,v2) <= &2 * h0 /\
    dist(v1,v3) <= &2 * h0 /\
    dist(v2,v3) <= &2 * h0 ==>
    &0 <= tau3 v1 v2 v3`;;
let JEJTVGB_pent_diag_concl = 
  `!V E FF. 
    convex_local_fan (V,E,FF) /\
    packing V /\
    V SUBSET ball_annulus /\
    CARD V = 5 /\
    (!v w. ~(v = w) /\ v IN V /\ w IN V /\ ~({v,w} IN E) ==> sqrt8 <= dist(v,w)) /\
    (!v w. {v,w} IN E ==> &2 <= dist(v,w) /\ dist(v,w) <= &2 * h0) ==>
    #0.616 <= tau_fun V E FF`;;
let JEJTVGB_pent_pro_concl = 
  `!V E FF. 
    convex_local_fan (V,E,FF) /\
    packing V /\
    V SUBSET ball_annulus /\
    CARD V = 5 /\ 
    (!v w. ~(v = w) /\ v IN V /\ w IN V /\ ~({v,w} IN E) ==> &2 * h0 <= dist(v,w)) /\
    (?v0 w0.
       (!v w. {v,w} IN E /\ ~({v,w} = {v0,w0}) ==> &2 <= dist(v,w) /\ dist(v,w) <= &2 * h0) /\
       {v0,w0} IN E /\
       &2 *h0 <= dist(v0,w0) /\ dist(v0,w0) <= sqrt8)
     ==>
    #0.616 <= tau_fun V E FF`;;
let JEJTVGB_quad_pro_concl = 
  `!V E FF. 
    convex_local_fan (V,E,FF) /\
    packing V /\
    V SUBSET ball_annulus /\
    CARD V = 4 /\ 
    (!v w. ~(v = w) /\ v IN V /\ w IN V /\ ~({v,w} IN E) ==> sqrt8 <= dist(v,w)) /\
    (?v0 w0.
       (!v w. {v,w} IN E /\ ~({v,w} = {v0,w0}) ==> &2 <= dist(v,w) /\ dist(v,w) <= &2 * h0) /\
       {v0,w0} IN E /\
       &2 *h0 <= dist(v0,w0) /\ dist(v0,w0) <= sqrt8)
     ==>
    #0.477 <= tau_fun V E FF`;;
let JEJTVGB_quad_diag_concl = 
  `!V E FF. 
    convex_local_fan (V,E,FF) /\
    packing V /\
    V SUBSET ball_annulus /\
    CARD V = 4 /\
    (!v w. ~(v = w) /\ v IN V /\ w IN V /\ ~({v,w} IN E) ==> &3 <= dist(v,w)) /\
    (!v w. {v,w} IN E ==> &2 <= dist(v,w) /\ dist(v,w) <= &2 * h0) ==>
    #0.467 <= tau_fun V E FF`;;
let JEJTVGB_concl = 
  let co = [JEJTVGB_std_concl;JEJTVGB_std3_concl;JEJTVGB_pent_diag_concl;JEJTVGB_pent_pro_concl;
				      JEJTVGB_quad_pro_concl;JEJTVGB_quad_diag_concl] in
    list_mk_conj co;;
let lp_main_estimate = new_definition (mk_eq (`lp_main_estimate:bool`,JEJTVGB_concl));;
(* Section Misc *)
Sections.begin_section "Misc";;
(Sections.add_section_type (mk_var ("f", (`:real^3#real^3->bool`))));;

(* Lemma rho_rho_fun *)
let rho_rho_fun = Sections.section_proof ["y"]
`rho_fun y = rho y`
[
   ((((use_arg_then2 ("Sphere.rho", [Sphere.rho]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("rho_fun", [rho_fun]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("Sphere.const1", [Sphere.const1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("Sphere.ly", [Sphere.ly]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("Sphere.interp", [Sphere.interp]))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("Nonlinear_lemma.sol0_EQ_sol_y", [Nonlinear_lemma.sol0_EQ_sol_y]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("Sphere.h0", [Sphere.h0]))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("PI_POS", [PI_POS])) (disch_tac [])) THEN (clear_assumption "PI_POS") THEN BETA_TAC) THEN (CONV_TAC REAL_FIELD) THEN (done_tac));
];;

(* Lemma packing_v_prime *)
let packing_v_prime = Sections.section_proof ["V";"f"]
`packing V ==> packing (v_prime V f)`
[
   (((DISCH_THEN (fun snd_th -> (use_arg_then2 ("Rdwkarc.PACKING_SUBSET", [Rdwkarc.PACKING_SUBSET])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then2 ("V_PRIME_SUBSET_V", [V_PRIME_SUBSET_V]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma v_prime_subset *)
let v_prime_subset = Sections.section_proof ["V";"X";"f"]
`V SUBSET X ==> v_prime V f SUBSET X`
[
   (((DISCH_THEN (fun snd_th -> (fun arg_tac -> (use_arg_then2 ("SUBSET_TRANS", [SUBSET_TRANS])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`v_prime V f`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then2 ("V_PRIME_SUBSET_V", [V_PRIME_SUBSET_V]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma in_e_prime *)
let in_e_prime = Sections.section_proof ["E";"f";"v";"w"]
`{v,w} IN e_prime E f <=> {v,w} IN E /\ (v,w IN f \/ w,v IN f)`
[
   ((((use_arg_then2 ("e_prime", [e_prime]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_ELIM_THM", [IN_ELIM_THM]))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((THENL_ROT (-1)) ((THENL) (split_tac) [((case THEN (move ["a"])) THEN (case THEN (move ["b"])) THEN (case THEN (move ["ab_in"]))); (case THEN (move ["vw_inE"]))]));
   (((THENL) (case THEN (move ["in_f"])) [(((use_arg_then2 ("v", [])) (term_tac exists_tac)) THEN ((use_arg_then2 ("w", [])) (term_tac exists_tac))); ((((use_arg_then2 ("w", [])) (term_tac exists_tac)) THEN ((use_arg_then2 ("v", [])) (term_tac exists_tac))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Collect_geom.PER_SET2)))(thm_tac (new_rewrite [] [])))))]) THEN (done_tac));
   (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Geomdetail.PAIR_EQ_EXPAND)))(thm_tac (new_rewrite [] []))));
   ((case THEN ALL_TAC THEN (case THEN ((((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Collect_geom.PER_SET2)))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma set_vv_eq_v *)
let set_vv_eq_v = Sections.section_proof ["v"]
`{v,v} = {v}`
[
   (((((use_arg_then2 ("EXTENSION", [EXTENSION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_SING", [IN_SING]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section Misc *)
let rho_rho_fun = Sections.finalize_theorem rho_rho_fun;;
let packing_v_prime = Sections.finalize_theorem packing_v_prime;;
let v_prime_subset = Sections.finalize_theorem v_prime_subset;;
let in_e_prime = Sections.finalize_theorem in_e_prime;;
let set_vv_eq_v = Sections.finalize_theorem set_vv_eq_v;;
Sections.end_section "Misc";;

(* Section Fan *)
Sections.begin_section "Fan";;
(Sections.add_section_var (mk_var ("V", (`:real^3->bool`))));;
(Sections.add_section_var (mk_var ("E", (`:(real^3->bool)->bool`))));;
(Sections.add_section_hyp "fanV" (`FAN (vec 0,V,E)`));;

(* Let dartH *)
Sections.add_section_lemma "dartH" (Sections.section_proof []
`dart (hypermap_of_fan (V,E)) = dart_of_fan (V,E)`
[
   ((((use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Let in_dart_inV *)
Sections.add_section_lemma "in_dart_inV" (Sections.section_proof ["d"]
`d IN dart_of_fan (V,E) ==> FST d IN V /\ SND d IN V`
[
   (((((use_arg_then2 ("pair_expand", [pair_expand]))(thm_tac (new_rewrite [1] [])))) THEN (DISCH_THEN (fun snd_th -> (fun arg_tac -> (use_arg_then2 ("PAIR_IN_DART_OF_FAN", [PAIR_IN_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac (match_mp_then snd_th MP_TAC))))) THEN (done_tac));
]);;

(* Lemma fan_aff_gt_1_2_not_empty *)
let fan_aff_gt_1_2_not_empty = Sections.section_proof ["v";"w"]
`v,w IN dart_of_fan (V,E)
	==> ~(aff_gt {vec 0} {v,w} = {})`
[
   (BETA_TAC THEN (move ["vw_in"]));
   ((fun arg_tac -> arg_tac (Arg_term (`~(v = vec 0) /\ ~(w = vec 0)`))) (term_tac (have_gen_tac [](move ["n0"]))));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("PAIR_IN_DART_OF_FAN", [PAIR_IN_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["inV"])));
   ((((use_arg_then2 ("fanV", [])) (disch_tac [])) THEN (clear_assumption "fanV") THEN BETA_TAC) THEN (((((use_arg_then2 ("FAN", [FAN]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("fan2", [fan2]))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["n0V"])) THEN (move ["_"])));
   ((split_tac) THEN (((use_arg_then2 ("n0V", [])) (disch_tac [])) THEN (clear_assumption "n0V") THEN ((use_arg_then2 ("contra", [contra])) (disch_tac [])) THEN (clear_assumption "contra") THEN (DISCH_THEN apply_tac) THEN (((conv_thm_tac DISCH_THEN)(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("vw_in", [])) (disch_tac [])) THEN (clear_assumption "vw_in") THEN BETA_TAC) THEN ((((use_arg_then2 ("dart_of_fan_eq", [dart_of_fan_eq]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] []))))) THEN ((THENL) case [(move ["vw_in1"]); ALL_TAC]));
   (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("Planarity.POINT_IN_CLOSURE_AFF_GT_1_2", [Planarity.POINT_IN_CLOSURE_AFF_GT_1_2])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`vec 0`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("v", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("w", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   ((((fun arg_tac -> (use_arg_then2 ("PAIR_IN_DART1_OF_FAN_IMP_NOT_EQ", [PAIR_IN_DART1_OF_FAN_IMP_NOT_EQ])) (fun fst_arg -> (use_arg_then2 ("vw_in1", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("n0", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (simp_tac));
   ((((use_arg_then2 ("contraL", [contraL])) (disch_tac [])) THEN (clear_assumption "contraL") THEN (DISCH_THEN apply_tac) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN ((((use_arg_then2 ("CLOSURE_EMPTY", [CLOSURE_EMPTY]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("NOT_IN_EMPTY", [NOT_IN_EMPTY]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then2 ("IN_ELIM_THM", [IN_ELIM_THM]))(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then2 ("PAIR_EQ", [PAIR_EQ]))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["x"])) THEN (case THEN (move ["_"])) THEN (case THEN ((((conv_thm_tac DISCH_THEN)(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   ((((use_arg_then2 ("set_vv_eq_v", [set_vv_eq_v]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("AFF_GT_1_1", [AFF_GT_1_1]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Local_lemmas.DISJOINT_DOUBLE_SING)))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("n0", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("MEMBER_NOT_EMPTY", [MEMBER_NOT_EMPTY]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))));
   (((fun arg_tac -> arg_tac (Arg_term (`#0.5 % v`))) (term_tac exists_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.5`))) (term_tac exists_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.5`))) (term_tac exists_tac)));
   ((THENL_FIRST) ((((use_arg_then2 ("andbA", [andbA]))(thm_tac (new_rewrite [] [])))) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((VECTOR_ARITH_TAC) THEN (done_tac));
];;

(* Lemma tau_fun_local *)
let tau_fun_local = Sections.section_proof ["f"]
`f IN face_set (hypermap_of_fan (V,E))
	==> tau_fun (v_prime V f) (e_prime E f) f = tau_fun V E f`
[
   ((BETA_TAC THEN (move ["f_in"])) THEN (((use_arg_then2 ("f_in", [])) (disch_tac [])) THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("lemma_face_representation", [lemma_face_representation])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (case THEN (case THEN ((move ["v"]) THEN (move ["w"])))) THEN (case THEN (move ["vw_in"])) THEN (move ["f_eq"])));
   ((repeat_tactic 1 9 (((use_arg_then2 ("tau_fun", [tau_fun]))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("real_sub", [real_sub]))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("REAL_EQ_ADD_RCANCEL", [REAL_EQ_ADD_RCANCEL]))(thm_tac (new_rewrite [] [])))));
   (((use_arg_then2 ("SUM_EQ", [SUM_EQ])) (thm_tac apply_tac)) THEN (move ["d"]) THEN (move ["d_in"]) THEN (simp_tac));
   ((((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Lvducxu.AZIM_IN_FAN_EQ_IZIM_E_PRIME))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`v,w`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("hypermap_HYP_elim", [hypermap_HYP_elim]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma v_prime_eq *)
let v_prime_eq = Sections.section_proof ["d";"k"]
`d IN darts_k k (hypermap_of_fan (V,E))
	==> v_prime V (face (hypermap_of_fan (V,E)) d)
		= {FST ((f_fan_pair_ext (V,E) POWER i) d) | i < k}`
[
   ((((use_arg_then2 ("EXTENSION", [EXTENSION]))(thm_tac (new_rewrite [] [])))) THEN (move ["d_in_k"]) THEN (move ["x"]));
   ((((use_arg_then2 ("d_in_k", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["d_in"])) THEN (move ["card_f"])));
   ((((use_arg_then2 ("v_prime", [v_prime]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("face_eq_power", [face_eq_power])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("d_in_k", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("IN_ELIM_THM", [IN_ELIM_THM]))(thm_tac (new_rewrite [] []))))) THEN (simp_tac));
   ((THENL) (split_tac) [((case THEN (move ["xV"])) THEN (case THEN (move ["w"])) THEN (case THEN (move ["i"])) THEN (case THEN (move ["i_lt"])) THEN (move ["xw_eq"])); ((case THEN (move ["i"])) THEN (case THEN (move ["i_lt"])) THEN (move ["x_eq"]))]);
   (((use_arg_then2 ("i", [])) (term_tac exists_tac)) THEN (((use_arg_then2 ("xw_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_in_dart", [f_fan_pair_ext_power_in_dart])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("d_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("i", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("pair_expand", [pair_expand]))(thm_tac (new_rewrite [1] [])))) THEN (DISCH_THEN (fun snd_th -> (fun arg_tac -> (use_arg_then2 ("PAIR_IN_DART_OF_FAN", [PAIR_IN_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac (match_mp_then snd_th MP_TAC)))));
   ((((use_arg_then2 ("x_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   (((fun arg_tac -> arg_tac (Arg_term (`SND ((f_fan_pair_ext (V,E) POWER i) d)`))) (term_tac exists_tac)) THEN ((use_arg_then2 ("i", [])) (term_tac exists_tac)));
   (((((use_arg_then2 ("x_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("pair_expand", [pair_expand]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma v_prime4_eq *)
let v_prime4_eq = Sections.section_proof ["v";"w"]
`let w' = sigma_fan (vec 0) V E v w in
		      let u = sigma_fan (vec 0) V E w' v in
	v,w IN darts_k 4 (hypermap_of_fan (V,E))
	==> v_prime V (face (hypermap_of_fan (V,E)) (v,w))
		= {v, w, u, w'}`
[
   ((repeat_tactic 1 9 ((CONV_TAC let_CONV))) THEN (BETA_TAC THEN (move ["vw_in4"])));
   ((((fun arg_tac -> (use_arg_then2 ("v_prime_eq", [v_prime_eq])) (fun fst_arg -> (use_arg_then2 ("vw_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (gen_FINITE_SET 4)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("POWER_0", [POWER_0]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))));
   ((repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face4_eq))) (fun fst_arg -> (use_arg_then2 ("vw_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma v_prime5_eq *)
let v_prime5_eq = Sections.section_proof ["v";"w"]
`let w' = sigma_fan (vec 0) V E v w in
		   let u = sigma_fan (vec 0) V E w' v in
		   let z = sigma_fan (vec 0) V E u w' in
	v,w IN darts_k 5 (hypermap_of_fan (V,E))
	==> v_prime V (face (hypermap_of_fan (V,E)) (v,w))
		= {v, w, z, u, w'}`
[
   ((repeat_tactic 1 9 ((CONV_TAC let_CONV))) THEN (BETA_TAC THEN (move ["vw_in5"])));
   ((((fun arg_tac -> (use_arg_then2 ("v_prime_eq", [v_prime_eq])) (fun fst_arg -> (use_arg_then2 ("vw_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (gen_FINITE_SET 5)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("POWER_0", [POWER_0]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))));
   ((repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face5_eq))) (fun fst_arg -> (use_arg_then2 ("vw_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma quad_diags_imp *)
let quad_diags_imp = Sections.section_proof ["v";"w";"a";"b"]
`let f = face (hypermap_of_fan (V,E)) (v,w) in
	 let w' = sigma_fan (vec 0) V E v w in
	 let u = sigma_fan (vec 0) V E w' v in
	v,w IN darts_k 4 (hypermap_of_fan (V,E))
	/\ ~(a = b) /\ a IN v_prime V f /\ b IN v_prime V f
	/\ ~({a,b} IN e_prime E f)
	==> ((a,b) = (w, w') \/ (a,b) = (w',w)) \/ ((a,b) = (v,u) \/ (a,b) = (u,v))`
[
   ((repeat_tactic 1 9 ((CONV_TAC let_CONV))) THEN (BETA_TAC THEN (case THEN (move ["vw_in4"])) THEN (case THEN (move ["a_neq_b"]))));
   ((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE v_prime4_eq))) (fun fst_arg -> (use_arg_then2 ("vw_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac ->(use_arg_then2 ("IN_SING", [IN_SING]))(fun tmp_arg1 -> (use_arg_then2 ("IN_INSERT", [IN_INSERT]))(fun tmp_arg2 -> arg_tac (Arg_theorem (CONJ (get_arg_thm tmp_arg1) (get_arg_thm tmp_arg2))))))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("in_e_prime", [in_e_prime]))(thm_tac (new_rewrite [] [])))));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face4_eq))) (fun fst_arg -> (use_arg_then2 ("vw_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["f_eqs"]) THEN (case THEN (move ["in_darts"])) THEN (move ["face_eq"])))));
   ((fun arg_tac -> arg_tac (Arg_term (`face _ (v,w)`))) (term_tac (set_tac "f")));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E w' v`))) (term_tac (set_tac "u")));
   (BETA_TAC THEN (case THEN (move ["a_conds"])) THEN (case THEN (move ["b_conds"])));
   (((use_arg_then2 ("a_neq_b", [])) (disch_tac [])) THEN (clear_assumption "a_neq_b") THEN ((use_arg_then2 ("b_conds", [])) (disch_tac [])) THEN (clear_assumption "b_conds") THEN ((use_arg_then2 ("a_conds", [])) (disch_tac [])) THEN (clear_assumption "a_conds") THEN BETA_TAC);
   ((((fun arg_tac -> (use_arg_then2 ("inE_eq_in_dart1", [inE_eq_in_dart1])) (fun fst_arg -> (use_arg_then2 ("V", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("face_eq", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac ->(use_arg_then2 ("IN_SING", [IN_SING]))(fun tmp_arg1 -> (use_arg_then2 ("IN_INSERT", [IN_INSERT]))(fun tmp_arg2 -> arg_tac (Arg_theorem (CONJ (get_arg_thm tmp_arg1) (get_arg_thm tmp_arg2))))))(thm_tac (new_rewrite [] []))))));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN ((simp_tac THEN TRY done_tac))) THEN (((use_arg_then2 ("dart1_switch", [dart1_switch]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma pent_diags_imp *)
let pent_diags_imp = Sections.section_proof ["v";"w";"a";"b"]
`let f = face (hypermap_of_fan (V,E)) (v,w) in
	 let w' = sigma_fan (vec 0) V E v w in
	 let u = sigma_fan (vec 0) V E w' v in
	 let z = sigma_fan (vec 0) V E u w' in
	v,w IN darts_k 5 (hypermap_of_fan (V,E))
	/\ ~(a = b) /\ a IN v_prime V f /\ b IN v_prime V f
	/\ ~({a,b} IN e_prime E f)
	==> ((a,b) = (w, w') \/ (a,b) = (w',w)) 
		\/ ((a,b) = (z,v) \/ (a,b) = (v,z))
		\/ ((a,b) = (u,w) \/ (a,b) = (w,u))
		\/ ((a,b) = (w',z) \/ (a,b) = (z,w'))
		\/ ((a,b) = (v,u) \/ (a,b) = (u,v))`
[
   ((repeat_tactic 1 9 ((CONV_TAC let_CONV))) THEN (BETA_TAC THEN (case THEN (move ["vw_in5"])) THEN (case THEN (move ["a_neq_b"]))));
   ((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE v_prime5_eq))) (fun fst_arg -> (use_arg_then2 ("vw_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac ->(use_arg_then2 ("IN_SING", [IN_SING]))(fun tmp_arg1 -> (use_arg_then2 ("IN_INSERT", [IN_INSERT]))(fun tmp_arg2 -> arg_tac (Arg_theorem (CONJ (get_arg_thm tmp_arg1) (get_arg_thm tmp_arg2))))))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("in_e_prime", [in_e_prime]))(thm_tac (new_rewrite [] [])))));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face5_eq))) (fun fst_arg -> (use_arg_then2 ("vw_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["f_eqs"]) THEN (case THEN (move ["in_darts"])) THEN (move ["face_eq"])))));
   ((fun arg_tac -> arg_tac (Arg_term (`face _ (v,w)`))) (term_tac (set_tac "f")));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E w' v`))) (term_tac (set_tac "u")));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E u w'`))) (term_tac (set_tac "z")));
   (BETA_TAC THEN (case THEN (move ["a_conds"])) THEN (case THEN (move ["b_conds"])));
   (((use_arg_then2 ("a_neq_b", [])) (disch_tac [])) THEN (clear_assumption "a_neq_b") THEN ((use_arg_then2 ("b_conds", [])) (disch_tac [])) THEN (clear_assumption "b_conds") THEN ((use_arg_then2 ("a_conds", [])) (disch_tac [])) THEN (clear_assumption "a_conds") THEN BETA_TAC);
   ((((fun arg_tac -> (use_arg_then2 ("inE_eq_in_dart1", [inE_eq_in_dart1])) (fun fst_arg -> (use_arg_then2 ("V", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("face_eq", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac ->(use_arg_then2 ("IN_SING", [IN_SING]))(fun tmp_arg1 -> (use_arg_then2 ("IN_INSERT", [IN_INSERT]))(fun tmp_arg2 -> arg_tac (Arg_theorem (CONJ (get_arg_thm tmp_arg1) (get_arg_thm tmp_arg2))))))(thm_tac (new_rewrite [] []))))));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN ((simp_tac THEN TRY done_tac))) THEN (((use_arg_then2 ("dart1_switch", [dart1_switch]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma e_fan_pair_eq_f_fan_pair *)
let e_fan_pair_eq_f_fan_pair = Sections.section_proof ["v";"w"]
`simple_hypermap (hypermap_of_fan (V,E))
	/\ v, w IN dart_of_fan (V,E)
	/\ w, v IN face (hypermap_of_fan (V,E)) (v,w)
	==> f_fan_pair_ext (V,E) (v,w) = (w,v)`
[
   (BETA_TAC THEN (case THEN (move ["simpleH"])) THEN (case THEN (move ["vw_in"])) THEN (move ["wv_in_f"]));
   (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("HYPERMAP_OF_FAN_FACE_NODE_INJ", [HYPERMAP_OF_FAN_FACE_NODE_INJ])) (fun fst_arg -> (use_arg_then2 ("simpleH", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`face (hypermap_of_fan (V,E)) (v,w)`))) (term_tac exists_tac));
   ((((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("vw_in", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("wv_in_f", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [1] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("POWER_1", [POWER_1])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`face_map _`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face", [lemma_in_face]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("vw_in", [])) (disch_tac [])) THEN (clear_assumption "vw_in") THEN BETA_TAC) THEN ((((use_arg_then2 ("dart_of_fan_eq", [dart_of_fan_eq]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_fan_pair_ext", [f_fan_pair_ext]))(thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) (((fun arg_tac -> arg_tac (Arg_term (`v,w IN _1`))) (disch_eq_tac "vw_in1" [])) THEN case THEN (simp_tac) THEN (process_fst_eq_tac)) ((((use_arg_then2 ("f_fan_pair", [f_fan_pair]))(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   (((((use_arg_then2 ("IN_ELIM_THM", [IN_ELIM_THM]))(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (case THEN (move ["u"]))) THEN ((((use_arg_then2 ("PAIR_EQ", [PAIR_EQ]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["_"])) THEN (case THEN ((((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) THEN (done_tac));
];;

(* Lemma linear_face_simple_hyp *)
let linear_face_simple_hyp = Sections.section_proof ["v";"w"]
`simple_hypermap (hypermap_of_fan (V,E))
	/\ v,w IN dart_of_fan (V,E)
	/\ w,v IN face (hypermap_of_fan (V,E)) (v,w)
	==> CARD (face (hypermap_of_fan (V,E)) (v,w)) <= 2`
[
   (BETA_TAC THEN (case THEN (move ["simpleH"])) THEN (case THEN (move ["vw_in"])) THEN (move ["wv_in_f"]));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["_"]) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["fH"])))));
   ((THENL_FIRST) (((((use_arg_then2 ("face", [face]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_orbit_le", [card_orbit_le]))(thm_tac (new_rewrite [] []))))) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("fH", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("e_fan_pair_eq_f_fan_pair", [e_fan_pair_eq_f_fan_pair]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("e_fan_pair_eq_f_fan_pair", [e_fan_pair_eq_f_fan_pair]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("simpleH", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("in_face_imp_in_dart", [in_face_imp_in_dart])) (fun fst_arg -> (use_arg_then2 ("wv_in_f", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((fun arg_tac -> (use_arg_then2 ("lemma_face_identity", [lemma_face_identity])) (fun fst_arg -> (use_arg_then2 ("wv_in_f", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("face_refl", [face_refl]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Section BallAnnulus *)
Sections.begin_section "BallAnnulus";;
(Sections.add_section_hyp "subV" (`V SUBSET ball_annulus`));;

(* Let norm_inV *)
Sections.add_section_lemma "norm_inV" (Sections.section_proof ["v"]
`v IN V ==> &2 <= norm v /\ norm v <= &2 * h0`
[
   ((in_tac ["subV"] false (((use_arg_then2 ("SUBSET", [SUBSET]))(thm_tac (new_rewrite [] []))))) THEN ((DISCH_THEN (fun snd_th -> (use_arg_then2 ("subV", [])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC) THEN (((use_arg_then2 ("Fnjlbxs.in_ball_annulus", [Fnjlbxs.in_ball_annulus]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Lemma tau_fun_eq_tauVEF *)
let tau_fun_eq_tauVEF = Sections.section_proof ["f"]
`f IN face_set (hypermap_of_fan (V,E)) /\ 2 <= CARD f
	==> tau_fun V E f = tauVEF (V,E,f)`
[
   (((((use_arg_then2 ("tau_fun", [tau_fun]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("tauVEF", [tauVEF]))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN ((move ["f_in"]) THEN (move ["card_f"]))));
   ((((use_arg_then2 ("REAL_OF_NUM_SUB", [REAL_OF_NUM_SUB]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("real_sub", [real_sub]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("REAL_MUL_RNEG", [REAL_MUL_RNEG]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("REAL_NEG_SUB", [REAL_NEG_SUB]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("REAL_EQ_ADD_RCANCEL", [REAL_EQ_ADD_RCANCEL]))(thm_tac (new_rewrite [] [])))));
   (((use_arg_then2 ("SUM_EQ", [SUM_EQ])) (thm_tac apply_tac)) THEN (move ["d"]) THEN (move ["d_in_f"]) THEN (simp_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`d IN dart_of_fan (V,E)`))) (term_tac (have_gen_tac [](move ["d_in"])))) (((((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("in_face_imp_in_dart_alt", [in_face_imp_in_dart_alt])) (fun fst_arg -> (use_arg_then2 ("d_in_f", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac)));
   ((((use_arg_then2 ("rho_rho_fun", [rho_rho_fun]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("azim_in_fan_eq_azim_dart", [azim_in_fan_eq_azim_dart])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("Sphere.rho", [Sphere.rho]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("ly_EQ_lmfun", [ly_EQ_lmfun]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("norm_inV", []))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((fun arg_tac -> (use_arg_then2 ("in_dart_inV", [])) (fun fst_arg -> (use_arg_then2 ("d_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("sol0_over_pi_EQ_const1", [sol0_over_pi_EQ_const1]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
];;

(* Finalization of the section BallAnnulus *)
let tau_fun_eq_tauVEF = Sections.finalize_theorem tau_fun_eq_tauVEF;;
Sections.end_section "BallAnnulus";;

(* Finalization of the section Fan *)
let fan_aff_gt_1_2_not_empty = Sections.finalize_theorem fan_aff_gt_1_2_not_empty;;
let tau_fun_local = Sections.finalize_theorem tau_fun_local;;
let v_prime_eq = Sections.finalize_theorem v_prime_eq;;
let v_prime4_eq = Sections.finalize_theorem v_prime4_eq;;
let v_prime5_eq = Sections.finalize_theorem v_prime5_eq;;
let quad_diags_imp = Sections.finalize_theorem quad_diags_imp;;
let pent_diags_imp = Sections.finalize_theorem pent_diags_imp;;
let e_fan_pair_eq_f_fan_pair = Sections.finalize_theorem e_fan_pair_eq_f_fan_pair;;
let linear_face_simple_hyp = Sections.finalize_theorem linear_face_simple_hyp;;
let tau_fun_eq_tauVEF = Sections.finalize_theorem tau_fun_eq_tauVEF;;
Sections.end_section "Fan";;

(* Section FullySurrounded *)
Sections.begin_section "FullySurrounded";;
(Sections.add_section_var (mk_var ("V", (`:real^3->bool`))));;
(Sections.add_section_var (mk_var ("E", (`:(real^3->bool)->bool`))));;
(Sections.add_section_hyp "fanV" (`FAN (vec 0,V,E)`));;
(Sections.add_section_hyp "f_surr" (`fully_surrounded (V,E)`));;

(* Let dartH *)
Sections.add_section_lemma "dartH" (Sections.section_proof []
`dart (hypermap_of_fan (V,E)) = dart_of_fan (V,E)`
[
   ((((use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Let dart1_eq *)
Sections.add_section_lemma "dart1_eq" (Sections.section_proof []
`dart1_of_fan (V,E) = dart_of_fan (V,E)`
[
   ((((use_arg_then2 ("fully_surrounded_dart_of_fan_eq", [fully_surrounded_dart_of_fan_eq]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Lemma f_surr_in_e_prime *)
let f_surr_in_e_prime = Sections.section_proof ["f";"v";"w"]
`f IN face_set (hypermap_of_fan (V,E))
	==> ({v,w} IN e_prime E f <=> (v,w IN f \/ w,v IN f))`
[
   (((((use_arg_then2 ("in_e_prime", [in_e_prime]))(thm_tac (new_rewrite [] [])))) THEN (move ["f_in"])) THEN (((THENL) (split_tac) [(((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))); ALL_TAC]) THEN ((TRY done_tac))));
   ((((fun arg_tac -> (use_arg_then2 ("inE_eq_in_dart1", [inE_eq_in_dart1])) (fun fst_arg -> (use_arg_then2 ("V", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))));
   ((BETA_TAC THEN (move ["h"])) THEN ((((use_arg_then2 ("h", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("andbT", [andbT]))(thm_tac (new_rewrite [] []))))));
   (((use_arg_then2 ("f_in", [])) (disch_tac [])) THEN (clear_assumption "f_in") THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("lemma_face_representation", [lemma_face_representation])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (case THEN (move ["d"])) THEN (case THEN (move ["d_in"])) THEN (move ["f_eq"]));
   (((THENL) (((use_arg_then2 ("h", [])) (disch_tac [])) THEN (clear_assumption "h") THEN case) [(ALL_TAC); (((use_arg_then2 ("dart_switch", [dart_switch]))(thm_tac (new_rewrite [] []))))]) THEN (((((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] []))))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("in_face_imp_in_dart", [in_face_imp_in_dart])) (thm_tac (match_mp_then snd_th MP_TAC))))) THEN (DISCH_THEN apply_tac) THEN (done_tac));
];;

(* Lemma diag_not_in_dart *)
let diag_not_in_dart = Sections.section_proof ["a";"b";"f"]
`f IN face_set (hypermap_of_fan (V,E))
	/\ a IN v_prime V f /\ b IN v_prime V f /\
	~(a,b IN f) /\ ~(b,a IN f)
	==> ~(a,b IN dart_of_fan (V,E))`
[
   (BETA_TAC THEN (case THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("lemma_face_representation", [lemma_face_representation])) (thm_tac (match_mp_then snd_th MP_TAC))))) THEN (case THEN (case THEN ((move ["v"]) THEN (move ["w"])))));
   ((((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["f_eq"]));
   (((((use_arg_then2 ("v_prime", [v_prime]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))))) THEN ALL_TAC THEN (case THEN (move ["a_in"])) THEN (case THEN (move ["b_in"])) THEN (move ["not_in_f"]));
   ((fun arg_tac -> (use_arg_then2 ("fan_hypermaps_iso_explicit2", [fan_hypermaps_iso_explicit2])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((fun arg_tac -> arg_tac (Arg_term (`ext_dart _`))) (term_tac (set_tac "h")));
   ((((use_arg_then2 ("implybF", [implybF]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (move ["h_iso"]) THEN (move ["ab_in"]));
   ((fun arg_tac -> arg_tac (Arg_term (`~(aff_gt {vec 0} {a,b} INTER xfan (vec 0,V,E) = {})`))) (term_tac (have_gen_tac []ALL_TAC)));
   ((((use_arg_then2 ("Vol1.subset_inter", [Vol1.subset_inter]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((fun arg_tac -> (use_arg_then2 ("fan_aff_gt_1_2_not_empty", [fan_aff_gt_1_2_not_empty])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((fun arg_tac -> (use_arg_then2 ("SUBSET_TRANS", [SUBSET_TRANS])) (fun fst_arg -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("AFF_GT_SUBSET_AFF_GE", [AFF_GT_SUBSET_AFF_GE])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`{vec 0}`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`{a,b}`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac apply_tac));
   (((((use_arg_then2 ("Planarity.AFF_GE_SUBSET_XFAN", [Planarity.AFF_GE_SUBSET_XFAN]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("inE_eq_in_dart1", [inE_eq_in_dart1])) (fun fst_arg -> (use_arg_then2 ("V", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("implybF", [implybF]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("negbK", [negbK]))(thm_tac (new_rewrite [] [])))));
   (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`aff_gt {vec 0} {a,b} SUBSET dart_leads_into (vec 0) V E v w`))) (term_tac (have_gen_tac []ALL_TAC))));
   ((DISCH_THEN (fun snd_th -> (use_arg_then2 ("SUBSET_TRANS", [SUBSET_TRANS])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN ((fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`yfan (vec 0,V,E)`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC)));
   ((((use_arg_then2 ("Planarity.topological_component_subset_yfan", [Planarity.topological_component_subset_yfan]))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   (((use_arg_then2 ("Topology.dart_leads_into_fan_in_topological_component_yfan", [Topology.dart_leads_into_fan_in_topological_component_yfan]))(thm_tac (new_rewrite [] []))));
   (((((fun arg_tac -> (use_arg_then2 ("inE_eq_in_dart1", [inE_eq_in_dart1])) (fun fst_arg -> (use_arg_then2 ("V", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then2 ("yfan", [yfan]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("SUBSET", [SUBSET]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_DIFF", [IN_DIFF]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNIV", [IN_UNIV]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("EXTENSION", [EXTENSION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("NOT_IN_EMPTY", [NOT_IN_EMPTY]))(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (move ["h"]) THEN (move ["x"]));
   ((((use_arg_then2 ("IN_INTER", [IN_INTER]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("negb_and", [negb_and]))(thm_tac (new_rewrite [] [])))));
   ((((fun arg_tac -> arg_tac (Arg_term (`x IN _`))) (disch_eq_tac "x_in" [])) THEN case THEN (simp_tac) THEN (process_fst_eq_tac)) THEN (((use_arg_then2 ("h", [])) (disch_tac [])) THEN (clear_assumption "h") THEN (exact_tac)) THEN (done_tac));
   (((use_arg_then2 ("a_in", [])) (disch_tac [])) THEN (clear_assumption "a_in") THEN ((use_arg_then2 ("b_in", [])) (disch_tac [])) THEN (clear_assumption "b_in") THEN BETA_TAC THEN (case THEN (move ["bV"])) THEN (case THEN (move ["b'"])) THEN (move ["bb'_in_f"]) THEN (case THEN (move ["aV"])) THEN (case THEN (move ["a'"])) THEN (move ["aa'_in_f"]));
   ((fun arg_tac -> arg_tac (Arg_term (`a,a' IN dart_of_fan (V,E) /\ b,b' IN dart_of_fan (V,E)`))) (term_tac (have_gen_tac [](case THEN ((move ["aa'_in"]) THEN (move ["bb'_in"]))))));
   ((((use_arg_then2 ("vw_in", [])) (disch_tac [])) THEN (clear_assumption "vw_in") THEN BETA_TAC) THEN ((((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (move ["vw_in"])));
   (((repeat_tactic 1 9 (((fun arg_tac -> (use_arg_then2 ("in_face_imp_in_dart", [in_face_imp_in_dart])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("f_eq", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fully_surrounded_imp_conforming", [fully_surrounded_imp_conforming])) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   ((((use_arg_then2 ("Conforming.conforming_fan", [Conforming.conforming_fan]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("Conforming.conforming_diagonal_fan", [Conforming.conforming_diagonal_fan]))(thm_tac (new_rewrite [] [])))));
   ((BETA_TAC THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN ((fun arg_tac -> (fun arg_tac -> (fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`IMAGE h f`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`h (a,a')`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`h (b,b')`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC))) THEN (ANTS_TAC));
   ((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [1] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("iso_components", [iso_components])) (fun fst_arg -> (use_arg_then2 ("h_iso", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then2 ("hyp_iso_dart", [hyp_iso_dart])) (fun fst_arg -> (use_arg_then2 ("h_iso", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("IN_IMAGE", [IN_IMAGE]))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) (split_tac) (((fun arg_tac -> arg_tac (Arg_term (`a,a'`))) (term_tac exists_tac)) THEN (done_tac)));
   ((THENL_FIRST) (split_tac) (((fun arg_tac -> arg_tac (Arg_term (`b,b'`))) (term_tac exists_tac)) THEN (done_tac)));
   ((((use_arg_then2 ("ab_in", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("dart1_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (DISCH_THEN (fun snd_th -> (fun arg_tac -> (use_arg_then2 ("PAIR_IN_DART1_OF_FAN_IMP_NOT_EQ", [PAIR_IN_DART1_OF_FAN_IMP_NOT_EQ])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac (match_mp_then snd_th MP_TAC))))));
   ((((use_arg_then2 ("contra", [contra])) (disch_tac [])) THEN (clear_assumption "contra") THEN (DISCH_THEN apply_tac) THEN (DISCH_THEN (fun snd_th -> (fun arg_tac -> (use_arg_then2 ("hyp_iso_inj", [hyp_iso_inj])) (fun fst_arg -> (use_arg_then2 ("h_iso", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac (match_mp_then snd_th MP_TAC))))) THEN ((((use_arg_then2 ("PAIR_EQ", [PAIR_EQ]))(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (BETA_TAC THEN (case THEN (move ["_"])));
   ((((use_arg_then2 ("h_def", []))(gsym_then (thm_tac (new_rewrite [2; 4] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("fan_map_ext", [fan_map_ext]))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("h_def", []))(thm_tac (new_rewrite [] [])))));
   case;
   (((DISCH_THEN (fun snd_th -> (fun arg_tac -> (use_arg_then2 ("hyp_iso_inj", [hyp_iso_inj])) (fun fst_arg -> (use_arg_then2 ("h_iso", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC) THEN (ANTS_TAC));
   (((((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("aa'_in", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("E_N_F_IN_DART1_OF_FAN", [E_N_F_IN_DART1_OF_FAN]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then2 ("f_fan_pair", [f_fan_pair]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("PAIR_EQ", [PAIR_EQ]))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["eq"])) THEN (move ["_"]));
   ((((use_arg_then2 ("not_in_f", [])) (disch_tac [])) THEN (clear_assumption "not_in_f") THEN ((use_arg_then2 ("bb'_in_f", [])) (disch_tac [])) THEN (clear_assumption "bb'_in_f") THEN BETA_TAC) THEN ((((use_arg_then2 ("eq", []))(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   case;
   (((DISCH_THEN (fun snd_th -> (fun arg_tac -> (use_arg_then2 ("hyp_iso_inj", [hyp_iso_inj])) (fun fst_arg -> (use_arg_then2 ("h_iso", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC) THEN (ANTS_TAC));
   (((((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("bb'_in", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("E_N_F_IN_DART1_OF_FAN", [E_N_F_IN_DART1_OF_FAN]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then2 ("f_fan_pair", [f_fan_pair]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("PAIR_EQ", [PAIR_EQ]))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["eq"])) THEN (move ["_"]));
   ((((use_arg_then2 ("not_in_f", [])) (disch_tac [])) THEN (clear_assumption "not_in_f") THEN ((use_arg_then2 ("aa'_in_f", [])) (disch_tac [])) THEN (clear_assumption "aa'_in_f") THEN BETA_TAC) THEN ((((use_arg_then2 ("eq", []))(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then2 ("h_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("ext_dart", [ext_dart]))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("Fan.pr2", [Fan.pr2]))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("Tame_lemmas.dart_leads_into_eq1", [Tame_lemmas.dart_leads_into_eq1]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma tauVEF_add_diag_eq *)
let tauVEF_add_diag_eq = Sections.section_proof ["d";"f"]
`d IN dart_of_fan (V,E)
	/\ f IN face_set (hypermap_of_fan (V, add_diag (V,E) d))
	/\ f IN face_set (hypermap_of_fan (V,E))
	==> tauVEF (V, add_diag (V,E) d, f) = tauVEF (V, E, f)`
[
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (case THEN (move ["vw_in"])) THEN (case THEN (move ["f_in2"])) THEN (move ["f_in"]));
   (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fully_surrounded_card_face_alt", [fully_surrounded_card_face_alt])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN case THEN (move ["card_f"]));
   ((((use_arg_then2 ("add_diag_eq_E", [add_diag_eq_E]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((((use_arg_then2 ("add_diag", [add_diag]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("vw_in", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("tau_split_fan_face_eq", [tau_split_fan_face_eq])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("IN_DELETE", [IN_DELETE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_in", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("implybF", [implybF]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (move ["f_eq"]));
   ((((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN (clear_assumption "f_in2") THEN BETA_TAC) THEN ((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (let_RULE faces_add_diag)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))));
   ((((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_DELETE", [IN_DELETE]))(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_SING", [IN_SING]))(thm_tac (new_rewrite [] [])))));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face_split_fan_face_explicit))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("card_f", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   (((((use_arg_then2 ("split_fan_face_eq_add_diag", [split_fan_face_eq_add_diag]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN ALL_TAC THEN (case THEN (move ["fs_eq"])) THEN (case THEN (move ["_"])) THEN (case THEN ALL_TAC) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["n_in"]) THEN (move ["_"]));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((THENL_ROT (-1)) (((repeat_tactic 1 9 (((use_arg_then2 ("fs_eq", []))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("negb_or", [negb_or]))(thm_tac (new_rewrite [] []))))) THEN (split_tac)));
   ((((use_arg_then2 ("n_in", [])) (disch_tac [])) THEN (clear_assumption "n_in") THEN ((use_arg_then2 ("contra", [contra])) (disch_tac [])) THEN (clear_assumption "contra") THEN (DISCH_THEN apply_tac) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`~(w,w' IN face (hypermap_of_fan (V,E)) (v,w))`))) (term_tac (have_gen_tac []ALL_TAC)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fully_surrounded_diag_not_in_dart", [fully_surrounded_diag_not_in_dart])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("card_f", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("contra", [contra])) (disch_tac [])) THEN (clear_assumption "contra") THEN (DISCH_THEN apply_tac)) THEN (((((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("in_face_imp_in_dart", [in_face_imp_in_dart])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("contra", [contra])) (disch_tac [])) THEN (clear_assumption "contra") THEN (DISCH_THEN apply_tac) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma tau3_eq_tau_fun *)
let tau3_eq_tau_fun = Sections.section_proof ["v";"w"]
`(v,w) IN dart_of_fan (V,E) 
	/\ CARD (face (hypermap_of_fan (V,E)) (v,w)) = 3
	==> tau3 (sigma_fan (vec 0) V E v w) v w 
		= tau_fun V E (face (hypermap_of_fan (V,E)) (v,w))`
[
   (BETA_TAC THEN (case THEN ((move ["vw_in"]) THEN (move ["card3"]))));
   ((((use_arg_then2 ("vw_in", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("dart1_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (move ["vw_in1"])));
   (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE TRIANGULAR_FACE))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in1", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("card3", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC)) THEN (move ["f_eq"]));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((((use_arg_then2 ("tau_fun", [tau_fun]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card3", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))));
   ((fun arg_tac -> arg_tac (Arg_term (`~(v = w) /\ ~(w = w')`))) (term_tac (have_gen_tac [](move ["neqs"]))));
   (((repeat_tactic 1 9 (((fun arg_tac -> (use_arg_then2 ("PAIR_IN_DART1_OF_FAN_IMP_NOT_EQ", [PAIR_IN_DART1_OF_FAN_IMP_NOT_EQ])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_fan_pair3", [f_fan_pair3]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((repeat_tactic 1 9 (((use_arg_then2 ("SUM_CLAUSES", [SUM_CLAUSES]))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((fun arg_tac ->(use_arg_then2 ("FINITE_INSERT", [FINITE_INSERT]))(fun tmp_arg1 -> (use_arg_then2 ("FINITE_EMPTY", [FINITE_EMPTY]))(fun tmp_arg2 -> arg_tac (Arg_theorem (CONJ (get_arg_thm tmp_arg1) (get_arg_thm tmp_arg2))))))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((repeat_tactic 1 9 (((fun arg_tac ->(use_arg_then2 ("IN_INSERT", [IN_INSERT]))(fun tmp_arg1 -> (use_arg_then2 ("NOT_IN_EMPTY", [NOT_IN_EMPTY]))(fun tmp_arg2 -> arg_tac (Arg_theorem (CONJ (get_arg_thm tmp_arg1) (get_arg_thm tmp_arg2))))))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("PAIR_EQ", [PAIR_EQ]))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("neqs", []))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then2 ("neqs", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (simp_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`x * &(3 - 2) = x`))) (term_tac (have_gen_tac ["x"](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`w,w' IN dart_of_fan (V,E) /\ w',v IN dart_of_fan (V,E)`))) (term_tac (have_gen_tac [](move ["in_dart"]))));
   (((((use_arg_then2 ("dart_of_fan_eq", [dart_of_fan_eq]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_fan_pair3", [f_fan_pair3]))(thm_tac (new_rewrite [] [])))) THEN ((simp_tac THEN TRY done_tac))) THEN (DISJ1_TAC));
   (((((use_arg_then2 ("dart1_switch", [dart1_switch]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("sigma_in_dart1", [sigma_in_dart1]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((repeat_tactic 1 9 (((use_arg_then2 ("rho_rho_fun", [rho_rho_fun]))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (use_arg_then2 ("azim_in_fan_eq_azim_dart", [azim_in_fan_eq_azim_dart])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("tau3", [tau3]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("azim_dart_eq_dihV", [azim_dart_eq_dihV]))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))));
   ((arith_tac) THEN (done_tac));
];;

(* Section BallAnnulus *)
Sections.begin_section "BallAnnulus";;
(Sections.add_section_hyp "subV" (`V SUBSET ball_annulus`));;

(* Lemma f_surr_tau_fun_eq_tauVEF *)
let f_surr_tau_fun_eq_tauVEF = Sections.section_proof ["f"]
`f IN face_set (hypermap_of_fan (V,E))
	==> tau_fun V E f = tauVEF (V,E,f)`
[
   ((BETA_TAC THEN (move ["f_in"])) THEN ((((fun arg_tac -> (use_arg_then2 ("tau_fun_eq_tauVEF", [tau_fun_eq_tauVEF])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("subV", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_in", []))(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then2 ("f_in", [])) (disch_tac [])) THEN (clear_assumption "f_in") THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("lemma_face_representation", [lemma_face_representation])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (case THEN (move ["x"]))) THEN ((((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["x_in"])) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("FULLY_SURROUNDED_IMP_CARD_FACE_GE_3", [FULLY_SURROUNDED_IMP_CARD_FACE_GE_3])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("x_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
];;

(* Lemma f_surr_tau_fun_eq_tauVEF_alt *)
let f_surr_tau_fun_eq_tauVEF_alt = Sections.section_proof ["d"]
`d IN dart_of_fan (V,E)
	==> tau_fun V E (face (hypermap_of_fan (V,E)) d) 
		= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(thm_tac (new_rewrite [] []))))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("f_surr_tau_fun_eq_tauVEF", [f_surr_tau_fun_eq_tauVEF])) (thm_tac (match_mp_then snd_th MP_TAC))))) THEN (done_tac));
];;

(* Lemma f_surr_tau3_eq_tauVEF *)
let f_surr_tau3_eq_tauVEF = Sections.section_proof ["v";"w"]
`(v,w) IN dart_of_fan (V,E) 
	/\ CARD (face (hypermap_of_fan (V,E)) (v,w)) = 3
	==> tau3 (sigma_fan (vec 0) V E v w) v w 
		= tauVEF (V, E, face (hypermap_of_fan (V,E)) (v,w))`
[
   ((BETA_TAC THEN (move ["h"])) THEN ((((use_arg_then2 ("tau3_eq_tau_fun", [tau3_eq_tau_fun]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("f_surr_tau_fun_eq_tauVEF_alt", [f_surr_tau_fun_eq_tauVEF_alt]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section BallAnnulus *)
let f_surr_tau_fun_eq_tauVEF = Sections.finalize_theorem f_surr_tau_fun_eq_tauVEF;;
let f_surr_tau_fun_eq_tauVEF_alt = Sections.finalize_theorem f_surr_tau_fun_eq_tauVEF_alt;;
let f_surr_tau3_eq_tauVEF = Sections.finalize_theorem f_surr_tau3_eq_tauVEF;;
Sections.end_section "BallAnnulus";;

(* Finalization of the section FullySurrounded *)
let f_surr_in_e_prime = Sections.finalize_theorem f_surr_in_e_prime;;
let diag_not_in_dart = Sections.finalize_theorem diag_not_in_dart;;
let tauVEF_add_diag_eq = Sections.finalize_theorem tauVEF_add_diag_eq;;
let tau3_eq_tau_fun = Sections.finalize_theorem tau3_eq_tau_fun;;
let f_surr_tau_fun_eq_tauVEF = Sections.finalize_theorem f_surr_tau_fun_eq_tauVEF;;
let f_surr_tau_fun_eq_tauVEF_alt = Sections.finalize_theorem f_surr_tau_fun_eq_tauVEF_alt;;
let f_surr_tau3_eq_tauVEF = Sections.finalize_theorem f_surr_tau3_eq_tauVEF;;
Sections.end_section "FullySurrounded";;

(* Section MainEstimate *)
Sections.begin_section "MainEstimate";;
(Sections.add_section_var (mk_var ("V", (`:real^3->bool`))));;
(Sections.add_section_var (mk_var ("E", (`:(real^3->bool)->bool`))));;
(Sections.add_section_hyp "h_fan" (`lp_fan (V,E)`));;

(* Let dart1_eq *)
Sections.add_section_lemma "dart1_eq" (Sections.section_proof []
`dart1_of_fan (V,E) = dart_of_fan (V,E)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN (move ["h"])) THEN (((use_arg_then2 ("fully_surrounded_dart_of_fan_eq", [fully_surrounded_dart_of_fan_eq]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Lemma local_diag *)
let local_diag = Sections.section_proof ["f";"v";"w"]
`f IN face_set (hypermap_of_fan (V,E))
	/\ ~(v = w) /\ v IN v_prime V f /\ w IN v_prime V f
	/\ ~({v,w} IN e_prime E f) ==> &2 * h0 <= dist (v,w)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (move ["_"])));
   (((((use_arg_then2 ("in_e_prime", [in_e_prime]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("negb_and", [negb_and]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("negb_or", [negb_or]))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["f_in"])) THEN (case THEN (move ["v_n_w"])) THEN (case THEN (move ["v_in"])) THEN (case THEN (move ["w_in"])) THEN (move ["h"]));
   (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`~({v,w} IN E)`))) (term_tac (have_gen_tac [](move ["nE"])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`&2 * h0 = #2.52`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((((use_arg_then2 ("h0", [h0]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("diag", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("nE", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("v_n_w", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((((use_arg_then2 ("w_in", [])) (disch_tac [])) THEN (clear_assumption "w_in") THEN ((use_arg_then2 ("v_in", [])) (disch_tac [])) THEN (clear_assumption "v_in") THEN BETA_TAC) THEN (((((use_arg_then2 ("v_prime", [v_prime]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((use_arg_then2 ("h", [])) (disch_tac [])) THEN (clear_assumption "h") THEN case THEN ((TRY done_tac)) THEN (move ["not_in_f"]));
   ((((fun arg_tac -> (use_arg_then2 ("inE_eq_in_dart1", [inE_eq_in_dart1])) (fun fst_arg -> (use_arg_then2 ("V", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))));
   (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("diag_not_in_dart", [diag_not_in_dart])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (exact_tac));
];;

(* Lemma local_std *)
let local_std = Sections.section_proof ["f";"d"]
`f = face (hypermap_of_fan (V,E)) d /\ d IN dart_of_fan (V,E)
	==> ((!v w. {v,w} IN e_prime E f ==> &2 <= dist(v,w) /\ dist(v,w) <= &2 * h0)
		<=> (!k. k < CARD f ==> dist ((f_fan_pair_ext (V,E) POWER k) d) <= #2.52))`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["packV"]) THEN (case THEN ((move ["f_eq"]) THEN (move ["d_in"])))));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["dartH"]) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["fH"])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`&2 * h0 = #2.52`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((((use_arg_then2 ("h0", [h0]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`f IN face_set (hypermap_of_fan (V,E))`))) (term_tac (have_gen_tac [](move ["f_in"]))));
   (((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`x IN f <=> ?k. k < CARD f /\ x = (f_fan_pair_ext (V,E) POWER k) d`))) (term_tac (have_gen_tac ["x"](move ["x_in_f"]))));
   ((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then2 ("face", [face]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("FINITE_ORBIT_MAP", [FINITE_ORBIT_MAP])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`dart_of_fan (V,E)`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`CARD f`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("face_map_and_darts", [face_map_and_darts]))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("face", [face]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_eq", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((((use_arg_then2 ("fH", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_ELIM_THM", [IN_ELIM_THM]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL) (split_tac) [((move ["h"]) THEN (move ["k"]) THEN (move ["k_lt"])); ((move ["h"]) THEN (move ["v"]) THEN (move ["w"]))]);
   ((fun arg_tac -> arg_tac (Arg_term (`(_1 POWER k) d`))) (term_tac (set_tac "x")));
   ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("h", [])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`FST x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`SND x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("pair_expand", [pair_expand]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_surr_in_e_prime", [f_surr_in_e_prime])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("pair_expand", [pair_expand]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (DISJ1_TAC));
   ((((use_arg_then2 ("x_in_f", []))(thm_tac (new_rewrite [] [])))) THEN ((use_arg_then2 ("k", [])) (term_tac exists_tac)) THEN (done_tac));
   (((((fun arg_tac -> (use_arg_then2 ("f_surr_in_e_prime", [f_surr_in_e_prime])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (move ["h_vw"]));
   ((fun arg_tac -> arg_tac (Arg_term (`v,w IN dart_of_fan (V,E)`))) (term_tac (have_gen_tac [](move ["vw_in"]))));
   (((THENL) (((use_arg_then2 ("h_vw", [])) (disch_tac [])) THEN (clear_assumption "h_vw") THEN case) [(ALL_TAC); (((use_arg_then2 ("dart_switch", [dart_switch]))(thm_tac (new_rewrite [] []))))]) THEN (((((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] []))))) THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("in_face_imp_in_dart", [in_face_imp_in_dart])) (thm_tac (match_mp_then snd_th MP_TAC))))) THEN (DISCH_THEN apply_tac) THEN (done_tac));
   ((((fun arg_tac -> (use_arg_then2 ("pack_dist", [pack_dist])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] [])))));
   (((THENL) (((use_arg_then2 ("h_vw", [])) (disch_tac [])) THEN (clear_assumption "h_vw") THEN case) [(ALL_TAC); (((use_arg_then2 ("DIST_SYM", [DIST_SYM]))(thm_tac (new_rewrite [] []))))]) THEN ((((use_arg_then2 ("x_in_f", []))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["k"])) THEN (case THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("h", [])) (thm_tac (match_mp_then snd_th MP_TAC))))) THEN (move ["ineq"]) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma local_pro *)
let local_pro = Sections.section_proof ["f";"d"]
`f = face (hypermap_of_fan (V,E)) d /\ d IN dart_of_fan (V,E)
	==> ((?v0 w0. (!v w. {v,w} IN e_prime E f /\ ~({v,w} = {v0,w0}) 
		==> &2 <= dist(v,w) /\ dist(v,w) <= &2 * h0) 
		/\ {v0,w0} IN e_prime E f /\ &2 *h0 <= dist(v0,w0) /\ dist(v0,w0) <= sqrt8)
	<=> (?k0. (!k. k < CARD f /\ ~(k = k0) ==> dist ((f_fan_pair_ext (V,E) POWER k) d) <= #2.52)
		/\ k0 < CARD f /\ #2.52 <= dist ((f_fan_pair_ext (V,E) POWER k0) d)
				/\ dist ((f_fan_pair_ext (V,E) POWER k0) d) <= sqrt8))`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["packV"]) THEN (case THEN ((move ["f_eq"]) THEN (move ["d_in"])))));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["dartH"]) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["fH"])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`&2 * h0 = #2.52`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((((use_arg_then2 ("h0", [h0]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`f IN face_set (hypermap_of_fan (V,E))`))) (term_tac (have_gen_tac [](move ["f_in"]))));
   (((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`x IN f <=> ?k. k < CARD f /\ x = (f_fan_pair_ext (V,E) POWER k) d`))) (term_tac (have_gen_tac ["x"](move ["x_in_f"]))));
   ((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then2 ("face", [face]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("FINITE_ORBIT_MAP", [FINITE_ORBIT_MAP])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`dart_of_fan (V,E)`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`CARD f`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("face_map_and_darts", [face_map_and_darts]))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("face", [face]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_eq", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((((use_arg_then2 ("fH", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_ELIM_THM", [IN_ELIM_THM]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_ROT (-1)) ((THENL) (split_tac) [((case THEN (move ["v0"])) THEN (case THEN (move ["w0"])) THEN (case THEN (move ["h_all"])) THEN (case THEN (move ["v0w0_inE"])) THEN (move ["dist0"])); ((case THEN (move ["k0"])) THEN (case THEN (move ["h_all"])) THEN (case THEN (move ["k0_lt"])))]));
   ((fun arg_tac -> arg_tac (Arg_term (`(_1 POWER k0) d`))) (term_tac (set_tac "x")));
   ((BETA_TAC THEN (move ["dist_x"])) THEN (((fun arg_tac -> arg_tac (Arg_term (`FST x`))) (term_tac exists_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`SND x`))) (term_tac exists_tac))) THEN ((((use_arg_then2 ("pair_expand", [pair_expand]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("dist_x", []))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andbT", [andbT]))(thm_tac (new_rewrite [] [])))))));
   ((THENL_LAST) (split_tac) (((((fun arg_tac -> (use_arg_then2 ("f_surr_in_e_prime", [f_surr_in_e_prime])) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("pair_expand", [pair_expand]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("x_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("fH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face", [lemma_in_face]))(thm_tac (new_rewrite [] []))))) THEN (done_tac)));
   (((((use_arg_then2 ("in_e_prime", [in_e_prime]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("inE_eq_in_dart1", [inE_eq_in_dart1])) (fun fst_arg -> (use_arg_then2 ("V", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] []))))) THEN (move ["v"]) THEN (move ["w"]) THEN (case THEN ALL_TAC) THEN (case THEN (move ["vw_in"])) THEN (move ["vw_in_f"]) THEN (move ["neq"]));
   ((((fun arg_tac -> (use_arg_then2 ("pack_dist", [pack_dist])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] [])))));
   (((THENL) (((use_arg_then2 ("vw_in_f", [])) (disch_tac [])) THEN (clear_assumption "vw_in_f") THEN case) [(ALL_TAC); (((use_arg_then2 ("DIST_SYM", [DIST_SYM]))(thm_tac (new_rewrite [] []))))]) THEN ((((use_arg_then2 ("x_in_f", []))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["k"])) THEN (case THEN (move ["k_lt"])) THEN (move ["vw_eq"])) THEN (((use_arg_then2 ("vw_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("h_all", [])) (fun fst_arg -> (use_arg_then2 ("k_lt", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac)) THEN (((use_arg_then2 ("neq", [])) (disch_tac [])) THEN (clear_assumption "neq") THEN ((use_arg_then2 ("contra", [contra])) (disch_tac [])) THEN (clear_assumption "contra") THEN (DISCH_THEN apply_tac) THEN (move ["k_eq"])) THEN (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Geomdetail.PAIR_EQ_EXPAND)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("vw_eq", [])) (disch_tac [])) THEN (clear_assumption "vw_eq") THEN BETA_TAC) THEN (((((use_arg_then2 ("k_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("x_def", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("pair_expand", [pair_expand])) (fun fst_arg -> (use_arg_then2 ("x", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then2 ("PAIR_EQ", [PAIR_EQ]))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN ((((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`?k0. k0 < CARD f /\ (v0,w0 = (f_fan_pair_ext (V,E) POWER k0) d
				\/ w0,v0 = (f_fan_pair_ext (V,E) POWER k0) d)`))) (term_tac (have_gen_tac [](case THEN ((move ["k0"]) THEN (case THEN (move ["k0_lt"])) THEN (move ["v0w0_eq"]))))));
   ((((use_arg_then2 ("v0w0_inE", [])) (disch_tac [])) THEN (clear_assumption "v0w0_inE") THEN BETA_TAC) THEN ((((use_arg_then2 ("in_e_prime", [in_e_prime]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("x_in_f", []))(thm_tac (new_rewrite [] [])))))));
   ((BETA_TAC THEN (case THEN (move ["_"])) THEN (case THEN ALL_TAC) THEN (case THEN (move ["k0"])) THEN (move ["h"])) THEN ((use_arg_then2 ("k0", [])) (term_tac exists_tac)) THEN (done_tac));
   (((use_arg_then2 ("k0", [])) (term_tac exists_tac)) THEN ((((use_arg_then2 ("k0_lt", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((THENL_LAST) ((THENL) (split_tac) [((move ["k"]) THEN (case THEN (move ["k_lt"])) THEN (move ["k_neq"])); ALL_TAC]) ((((use_arg_then2 ("v0w0_eq", [])) (disch_tac [])) THEN (clear_assumption "v0w0_eq") THEN BETA_TAC THEN (case THEN (((conv_thm_tac DISCH_THEN)(gsym_then (thm_tac (new_rewrite [] []))))))) THEN ((repeat_tactic 0 10 (((fun arg_tac -> (use_arg_then2 ("DIST_SYM", [DIST_SYM])) (fun fst_arg -> (use_arg_then2 ("w0", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("dist0", []))(thm_tac (new_rewrite [] [])))))) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`(_1 POWER k) d`))) (term_tac (set_tac "x")));
   ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("h_all", [])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`FST x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`SND x`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("pair_expand", [pair_expand]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_surr_in_e_prime", [f_surr_in_e_prime])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("pair_expand", [pair_expand]))(gsym_then (thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) (split_tac) ((DISJ1_TAC) THEN ((((use_arg_then2 ("x_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("fH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face", [lemma_in_face]))(thm_tac (new_rewrite [] []))))) THEN (done_tac)));
   ((((use_arg_then2 ("k_neq", [])) (disch_tac [])) THEN (clear_assumption "k_neq") THEN ((use_arg_then2 ("contra", [contra])) (disch_tac [])) THEN (clear_assumption "contra") THEN (DISCH_THEN apply_tac)) THEN (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Geomdetail.PAIR_EQ_EXPAND)))(thm_tac (new_rewrite [] [])))));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("FULLY_SURROUNDED_IMP_CARD_FACE_GE_3", [FULLY_SURROUNDED_IMP_CARD_FACE_GE_3])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("d_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("f_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (move ["card_f_ge3"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fully_surrounded_simple_hypermap", [fully_surrounded_simple_hypermap])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["simpleH"])));
   ((((use_arg_then2 ("v0w0_eq", [])) (disch_tac [])) THEN (clear_assumption "v0w0_eq") THEN BETA_TAC THEN (case THEN (move ["eq"])) THEN (case THEN ALL_TAC)) THEN ((((use_arg_then2 ("PAIR_EQ", [PAIR_EQ]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("pair_expand", [pair_expand]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("x_def", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (((repeat_tactic 0 10 (((fun arg_tac ->(use_arg_then2 ("eq", []))(fun tmp_arg1 -> (use_arg_then2 ("f_fan_pair_ext_power_inj", [f_fan_pair_ext_power_inj]))(fun tmp_arg2 -> arg_tac (Arg_theorem (CONJ (get_arg_thm tmp_arg1) (get_arg_thm tmp_arg2))))))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_eq", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac))) THEN (move ["eq2"])) THEN ((THENL_FIRST) (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`CARD f <= 2`))) (term_tac (have_gen_tac []ALL_TAC)))) ((((use_arg_then2 ("card_f_ge3", [])) (disch_tac [])) THEN (clear_assumption "card_f_ge3") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac))));
   ((fun arg_tac -> arg_tac (Arg_term (`f = face (hypermap_of_fan (V,E)) (v0,w0)`))) (term_tac (have_gen_tac [](move ["f_eq2"]))));
   ((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN ((use_arg_then2 ("lemma_face_identity", [lemma_face_identity])) (thm_tac apply_tac)) THEN ((((use_arg_then2 ("eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("fH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face", [lemma_in_face]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("f_eq2", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("linear_face_simple_hyp", [linear_face_simple_hyp])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("simpleH", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_eq2", []))(gsym_then (thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("eq2", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("fH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face", [lemma_in_face]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_fan_pair_ext_power_in_dart", [f_fan_pair_ext_power_in_dart]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`f = face (hypermap_of_fan (V,E)) (w0,v0)`))) (term_tac (have_gen_tac [](move ["f_eq2"]))));
   ((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN ((use_arg_then2 ("lemma_face_identity", [lemma_face_identity])) (thm_tac apply_tac)) THEN ((((use_arg_then2 ("eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("fH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face", [lemma_in_face]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("f_eq2", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("linear_face_simple_hyp", [linear_face_simple_hyp])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("simpleH", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_eq2", []))(gsym_then (thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("eq2", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("fH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face", [lemma_in_face]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_fan_pair_ext_power_in_dart", [f_fan_pair_ext_power_in_dart]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Section Face3 *)
Sections.begin_section "Face3";;
(Sections.add_section_hyp "ineq" (JEJTVGB_std3_concl));;

(* Lemma ineq_tau3_tauVEF *)
let ineq_tau3_tauVEF = Sections.section_proof ["d"]
`d IN darts_k 3 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y4_fan (V,E) d <= #2.52
	==> &0 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["dartH"]) THEN (move ["_"])))));
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in3"]));
   ((((use_arg_then2 ("vw_in3", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"]) THEN (move ["bounds"])));
   ((((use_arg_then2 ("f_surr_tau3_eq_tauVEF", [f_surr_tau3_eq_tauVEF]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("ineq", [ineq]))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`&2 * h0 = #2.52`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((((use_arg_then2 ("h0", [h0]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((repeat_tactic 1 9 (((fun arg_tac -> (use_arg_then2 ("DIST_SYM", [DIST_SYM])) (fun fst_arg -> (use_arg_then2 ("w'", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))));
   ((repeat_tactic 1 9 (((fun arg_tac -> (use_arg_then2 ("ys_eq", [ys_eq])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y4'_eq_y4", [y4'_eq_y4]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then2 ("y1_lo", [y1_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y3_lo", [y3_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   ((((fun arg_tac -> (use_arg_then2 ("y1_hi", [y1_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y3_hi", [y3_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((fun arg_tac -> (use_arg_then2 ("y4_lo", [y4_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y5_lo", [y5_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_lo", [y6_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma ineq_tau3 *)
let ineq_tau3 = Sections.section_proof ["d"]
`d IN darts_k 3 (hypermap_of_fan (V,E))
	==> &0 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   (((((use_arg_then2 ("tau_fan", [tau_fan]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("REAL_ABS_POS", [REAL_ABS_POS]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Section Std *)
Sections.begin_section "Std";;
(Sections.add_section_hyp "std" (`E = ESTD V`));;

(* Lemma ineq_tau3_tauVEF_std *)
let ineq_tau3_tauVEF_std = Sections.section_proof ["d"]
`d IN darts_k 3 (hypermap_of_fan (V,E))
	==> &0 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((BETA_TAC THEN (move ["d_in3"])) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau3_tauVEF", [ineq_tau3_tauVEF])) (fun fst_arg -> (use_arg_then2 ("d_in3", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac)));
   ((((use_arg_then2 ("d_in3", [])) (disch_tac [])) THEN (clear_assumption "d_in3") THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["d_in"])) THEN (move ["_"])));
   (((((use_arg_then2 ("y5_hi_std2", [y5_hi_std2]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_hi_std2", [y6_hi_std2])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y4_hi_std2", [y4_hi_std2]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section Std *)
let ineq_tau3_tauVEF_std = Sections.finalize_theorem ineq_tau3_tauVEF_std;;
Sections.end_section "Std";;

(* Finalization of the section Face3 *)
let ineq_tau3_tauVEF = Sections.finalize_theorem ineq_tau3_tauVEF;;
let ineq_tau3 = Sections.finalize_theorem ineq_tau3;;
let ineq_tau3_tauVEF_std = Sections.finalize_theorem ineq_tau3_tauVEF_std;;
Sections.end_section "Face3";;

(* Section Face4 *)
Sections.begin_section "Face4";;
(Sections.add_section_hyp "ineq" (JEJTVGB_std_concl));;

(* Lemma ineq_tau4_tauVEF *)
let ineq_tau4_tauVEF = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	==> #0.206 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["dartH"]) THEN (move ["_"])))));
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in4"]));
   ((((use_arg_then2 ("vw_in4", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"]) THEN (move ["bounds"])));
   ((((use_arg_then2 ("f_surr_tau_fun_eq_tauVEF", [f_surr_tau_fun_eq_tauVEF]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_surr_localization_convex_local))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["c_local"])));
   ((fun arg_tac -> arg_tac (Arg_term (`face _ (v,w)`))) (term_tac (set_tac "f")));
   ((fun arg_tac -> arg_tac (Arg_term (`f IN face_set (hypermap_of_fan (V,E))`))) (term_tac (have_gen_tac [](move ["f_in"]))));
   (((((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("c_local", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("convex_local_fan", [convex_local_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["local"])) THEN (move ["_"])));
   (((fun arg_tac -> (use_arg_then2 ("ineq", [ineq])) (fun fst_arg -> (use_arg_then2 ("c_local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   ((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ))) (fun fst_arg -> (use_arg_then2 ("local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("d_tame", [d_tame]))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`~(4 = 3)`))) (term_tac (have_gen_tac []((((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (simp_tac))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("tau_fun_local", [tau_fun_local]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac))) THEN (DISCH_THEN apply_tac));
   ((((use_arg_then2 ("packing_v_prime", [packing_v_prime]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("v_prime_subset", [v_prime_subset]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("leqnn", [leqnn]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) (split_tac) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((THENL) (split_tac) [((move ["a"]) THEN (move ["b"]) THEN (move ["h"])); ALL_TAC]) ((((fun arg_tac -> (use_arg_then2 ("local_diag", [local_diag])) (fun fst_arg -> (use_arg_then2 ("f_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   (((((fun arg_tac -> (use_arg_then2 ("local_std", [local_std])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) THEN (move ["k"]) THEN (move ["k_lt"]));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`k = 0 \/ k = 1 \/ k = 2 \/ k = 3`))) (term_tac (have_gen_tac []ALL_TAC))) ((((use_arg_then2 ("k_lt", [])) (disch_tac [])) THEN (clear_assumption "k_lt") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("POWER_0", [POWER_0]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y6_fan", [y6_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y9_fan", [y9_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`4 - 2 = 2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("y8_fan", [y8_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`4 - 3 = 1`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma ineq_tau4 *)
let ineq_tau4 = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	==> #0.206 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   (BETA_TAC THEN (move ["d_in4"]) THEN (move ["bounds"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("ineq_tau4_tauVEF", [ineq_tau4_tauVEF])) (fun fst_arg -> (use_arg_then2 ("d_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("bounds", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("tau_fan", [tau_fan]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
];;

(* Section Std *)
Sections.begin_section "Std";;
(Sections.add_section_hyp "std" (`E = ESTD V`));;

(* Lemma ineq_tau4_tauVEF_std *)
let ineq_tau4_tauVEF_std = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> #0.206 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((BETA_TAC THEN (move ["d_in4"])) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau4_tauVEF", [ineq_tau4_tauVEF])) (fun fst_arg -> (use_arg_then2 ("d_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac)));
   ((((use_arg_then2 ("d_in4", [])) (disch_tac [])) THEN (clear_assumption "d_in4") THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["d_in"])) THEN (move ["_"])));
   (((((use_arg_then2 ("y5_hi_std2", [y5_hi_std2]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_hi_std2", [y6_hi_std2])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y8_hi_std", [y8_hi_std]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_hi_std", [y9_hi_std]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma ineq_tau4_std *)
let ineq_tau4_std = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> #0.206 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   (((DISCH_THEN (fun snd_th -> (use_arg_then2 ("ineq_tau4_tauVEF_std", [ineq_tau4_tauVEF_std])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC) THEN (((use_arg_then2 ("tau_fan", [tau_fan]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
];;

(* Finalization of the section Std *)
let ineq_tau4_tauVEF_std = Sections.finalize_theorem ineq_tau4_tauVEF_std;;
let ineq_tau4_std = Sections.finalize_theorem ineq_tau4_std;;
Sections.end_section "Std";;

(* Finalization of the section Face4 *)
let ineq_tau4_tauVEF = Sections.finalize_theorem ineq_tau4_tauVEF;;
let ineq_tau4 = Sections.finalize_theorem ineq_tau4;;
let ineq_tau4_tauVEF_std = Sections.finalize_theorem ineq_tau4_tauVEF_std;;
let ineq_tau4_std = Sections.finalize_theorem ineq_tau4_std;;
Sections.end_section "Face4";;

(* Section Face5 *)
Sections.begin_section "Face5";;
(Sections.add_section_hyp "ineq" (JEJTVGB_std_concl));;

(* Lemma ineq_tau5_tauVEF *)
let ineq_tau5_tauVEF = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	==> #0.4819 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["dartH"]) THEN (move ["_"])))));
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in_k"]));
   ((((use_arg_then2 ("vw_in_k", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"]) THEN (move ["bounds"])));
   ((((use_arg_then2 ("f_surr_tau_fun_eq_tauVEF", [f_surr_tau_fun_eq_tauVEF]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_surr_localization_convex_local))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["c_local"])));
   ((fun arg_tac -> arg_tac (Arg_term (`face _ (v,w)`))) (term_tac (set_tac "f")));
   ((fun arg_tac -> arg_tac (Arg_term (`f IN face_set (hypermap_of_fan (V,E))`))) (term_tac (have_gen_tac [](move ["f_in"]))));
   (((((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("c_local", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("convex_local_fan", [convex_local_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["local"])) THEN (move ["_"])));
   (((fun arg_tac -> (use_arg_then2 ("ineq", [ineq])) (fun fst_arg -> (use_arg_then2 ("c_local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   ((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ))) (fun fst_arg -> (use_arg_then2 ("local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("d_tame", [d_tame]))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`~(5 = 3)`))) (term_tac (have_gen_tac []((((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (simp_tac))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`~(5 = 4)`))) (term_tac (have_gen_tac []((((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN (simp_tac))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("tau_fun_local", [tau_fun_local]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac))) THEN (DISCH_THEN apply_tac));
   ((((use_arg_then2 ("packing_v_prime", [packing_v_prime]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("v_prime_subset", [v_prime_subset]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) (split_tac) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) (split_tac) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((THENL) (split_tac) [((move ["a"]) THEN (move ["b"]) THEN (move ["h"])); ALL_TAC]) ((((fun arg_tac -> (use_arg_then2 ("local_diag", [local_diag])) (fun fst_arg -> (use_arg_then2 ("f_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   (((((fun arg_tac -> (use_arg_then2 ("local_std", [local_std])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) THEN (move ["k"]) THEN (move ["k_lt"]));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`k = 0 \/ k = 1 \/ k = 2 \/ k = 3 \/ k = 4`))) (term_tac (have_gen_tac []ALL_TAC))) ((((use_arg_then2 ("k_lt", [])) (disch_tac [])) THEN (clear_assumption "k_lt") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("POWER_0", [POWER_0]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y6_fan", [y6_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y9_fan", [y9_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("ye_fan", [ye_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 3 = 2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("y8_fan", [y8_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 4 = 1`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma ineq_tau5 *)
let ineq_tau5 = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	==> #0.4819 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   (BETA_TAC THEN (move ["d_in"]) THEN (move ["bounds"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("ineq_tau5_tauVEF", [ineq_tau5_tauVEF])) (fun fst_arg -> (use_arg_then2 ("d_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("bounds", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("tau_fan", [tau_fan]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
];;

(* Section Std *)
Sections.begin_section "Std";;
(Sections.add_section_hyp "std" (`E = ESTD V`));;

(* Lemma ineq_tau5_tauVEF_std *)
let ineq_tau5_tauVEF_std = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> #0.4819 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((BETA_TAC THEN (move ["d_in5"])) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau5_tauVEF", [ineq_tau5_tauVEF])) (fun fst_arg -> (use_arg_then2 ("d_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac)));
   ((((use_arg_then2 ("d_in5", [])) (disch_tac [])) THEN (clear_assumption "d_in5") THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["d_in"])) THEN (move ["_"])));
   ((((use_arg_then2 ("y5_hi_std2", [y5_hi_std2]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_hi_std2", [y6_hi_std2])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y8_hi_std", [y8_hi_std]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_hi_std", [y9_hi_std]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y6_def", [y6_def]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((fun arg_tac -> (use_arg_then2 ("y6_hi_std2", [y6_hi_std2])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_fan_pair_ext_power_in_dart", [f_fan_pair_ext_power_in_dart]))(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma ineq_tau5_std *)
let ineq_tau5_std = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> #0.4819 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   (((DISCH_THEN (fun snd_th -> (use_arg_then2 ("ineq_tau5_tauVEF_std", [ineq_tau5_tauVEF_std])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC) THEN (((use_arg_then2 ("tau_fan", [tau_fan]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
];;

(* Finalization of the section Std *)
let ineq_tau5_tauVEF_std = Sections.finalize_theorem ineq_tau5_tauVEF_std;;
let ineq_tau5_std = Sections.finalize_theorem ineq_tau5_std;;
Sections.end_section "Std";;

(* Finalization of the section Face5 *)
let ineq_tau5_tauVEF = Sections.finalize_theorem ineq_tau5_tauVEF;;
let ineq_tau5 = Sections.finalize_theorem ineq_tau5;;
let ineq_tau5_tauVEF_std = Sections.finalize_theorem ineq_tau5_tauVEF_std;;
let ineq_tau5_std = Sections.finalize_theorem ineq_tau5_std;;
Sections.end_section "Face5";;

(* Section Face6 *)
Sections.begin_section "Face6";;
(Sections.add_section_hyp "ineq" (JEJTVGB_std_concl));;

(* Lemma ineq_tau6_tauVEF *)
let ineq_tau6_tauVEF = Sections.section_proof ["d"]
`d IN darts_k 6 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 3) d) <= #2.52
	==> #0.712 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["dartH"]) THEN (move ["_"])))));
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in_k"]));
   ((((use_arg_then2 ("vw_in_k", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"]) THEN (move ["bounds"])));
   ((((use_arg_then2 ("f_surr_tau_fun_eq_tauVEF", [f_surr_tau_fun_eq_tauVEF]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_surr_localization_convex_local))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["c_local"])));
   ((fun arg_tac -> arg_tac (Arg_term (`face _ (v,w)`))) (term_tac (set_tac "f")));
   ((fun arg_tac -> arg_tac (Arg_term (`f IN face_set (hypermap_of_fan (V,E))`))) (term_tac (have_gen_tac [](move ["f_in"]))));
   (((((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("c_local", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("convex_local_fan", [convex_local_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["local"])) THEN (move ["_"])));
   (((fun arg_tac -> (use_arg_then2 ("ineq", [ineq])) (fun fst_arg -> (use_arg_then2 ("c_local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   ((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ))) (fun fst_arg -> (use_arg_then2 ("local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("d_tame", [d_tame]))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`~(6 = 3) /\ ~(6 = 4) /\ ~(6 = 5)`))) (term_tac (have_gen_tac [](move ["neqs"])))) ((arith_tac) THEN (done_tac)));
   (((repeat_tactic 1 9 (((use_arg_then2 ("neqs", []))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (((use_arg_then2 ("tau_fun_local", [tau_fun_local]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac))) THEN (DISCH_THEN apply_tac));
   ((((use_arg_then2 ("packing_v_prime", [packing_v_prime]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("v_prime_subset", [v_prime_subset]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("leqnn", [leqnn]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) (split_tac) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((THENL) (split_tac) [((move ["a"]) THEN (move ["b"]) THEN (move ["h"])); ALL_TAC]) ((((fun arg_tac -> (use_arg_then2 ("local_diag", [local_diag])) (fun fst_arg -> (use_arg_then2 ("f_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   (((((fun arg_tac -> (use_arg_then2 ("local_std", [local_std])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) THEN (move ["k"]) THEN (move ["k_lt"]));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`k = 0 \/ k = 1 \/ k = 2 \/ k = 3 \/ k = 4 \/ k = 5`))) (term_tac (have_gen_tac []ALL_TAC))) ((((use_arg_then2 ("k_lt", [])) (disch_tac [])) THEN (clear_assumption "k_lt") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("POWER_0", [POWER_0]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y6_fan", [y6_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y9_fan", [y9_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("ye_fan", [ye_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("ye_fan", [ye_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`6 - 4 = 2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("y8_fan", [y8_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`6 - 5 = 1`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma ineq_tau6 *)
let ineq_tau6 = Sections.section_proof ["d"]
`d IN darts_k 6 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 3) d) <= #2.52
	==> #0.712 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   (BETA_TAC THEN (move ["d_in"]) THEN (move ["bounds"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("ineq_tau6_tauVEF", [ineq_tau6_tauVEF])) (fun fst_arg -> (use_arg_then2 ("d_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("bounds", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("tau_fan", [tau_fan]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
];;

(* Section Std *)
Sections.begin_section "Std";;
(Sections.add_section_hyp "std" (`E = ESTD V`));;

(* Lemma ineq_tau6_tauVEF_std *)
let ineq_tau6_tauVEF_std = Sections.section_proof ["d"]
`d IN darts_k 6 (hypermap_of_fan (V,E))
	==> #0.712 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((BETA_TAC THEN (move ["d_in6"])) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau6_tauVEF", [ineq_tau6_tauVEF])) (fun fst_arg -> (use_arg_then2 ("d_in6", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac)));
   ((((use_arg_then2 ("d_in6", [])) (disch_tac [])) THEN (clear_assumption "d_in6") THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["d_in"])) THEN (move ["_"])));
   ((((use_arg_then2 ("y5_hi_std2", [y5_hi_std2]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_hi_std2", [y6_hi_std2])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y8_hi_std", [y8_hi_std]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_hi_std", [y9_hi_std]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("y6_def", [y6_def]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((fun arg_tac -> (use_arg_then2 ("y6_hi_std2", [y6_hi_std2])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_fan_pair_ext_power_in_dart", [f_fan_pair_ext_power_in_dart]))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y6_def", [y6_def]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((fun arg_tac -> (use_arg_then2 ("y6_hi_std2", [y6_hi_std2])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_fan_pair_ext_power_in_dart", [f_fan_pair_ext_power_in_dart]))(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma ineq_tau6_std *)
let ineq_tau6_std = Sections.section_proof ["d"]
`d IN darts_k 6 (hypermap_of_fan (V,E))
	==> #0.712 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   (((DISCH_THEN (fun snd_th -> (use_arg_then2 ("ineq_tau6_tauVEF_std", [ineq_tau6_tauVEF_std])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC) THEN (((use_arg_then2 ("tau_fan", [tau_fan]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
];;

(* Finalization of the section Std *)
let ineq_tau6_tauVEF_std = Sections.finalize_theorem ineq_tau6_tauVEF_std;;
let ineq_tau6_std = Sections.finalize_theorem ineq_tau6_std;;
Sections.end_section "Std";;

(* Finalization of the section Face6 *)
let ineq_tau6_tauVEF = Sections.finalize_theorem ineq_tau6_tauVEF;;
let ineq_tau6 = Sections.finalize_theorem ineq_tau6;;
let ineq_tau6_tauVEF_std = Sections.finalize_theorem ineq_tau6_tauVEF_std;;
let ineq_tau6_std = Sections.finalize_theorem ineq_tau6_std;;
Sections.end_section "Face6";;

(* Section Face5_pro *)
Sections.begin_section "Face5_pro";;
(Sections.add_section_hyp "ineq" (JEJTVGB_pent_pro_concl));;

(* Lemma ineq_tau5_pro_tauVEF_aux *)
let ineq_tau5_pro_tauVEF_aux = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	/\ #2.52 <= y6_fan d /\ y6_fan d <= sqrt8
	==> #0.616 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["dartH"]) THEN (move ["_"])))));
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in_k"]));
   ((((use_arg_then2 ("vw_in_k", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"]) THEN (move ["bounds"])));
   ((((use_arg_then2 ("f_surr_tau_fun_eq_tauVEF", [f_surr_tau_fun_eq_tauVEF]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_surr_localization_convex_local))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["c_local"])));
   ((fun arg_tac -> arg_tac (Arg_term (`face _ (v,w)`))) (term_tac (set_tac "f")));
   ((fun arg_tac -> arg_tac (Arg_term (`f IN face_set (hypermap_of_fan (V,E))`))) (term_tac (have_gen_tac [](move ["f_in"]))));
   (((((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("c_local", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("convex_local_fan", [convex_local_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["local"])) THEN (move ["_"])));
   (((fun arg_tac -> (use_arg_then2 ("ineq", [ineq])) (fun fst_arg -> (use_arg_then2 ("c_local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   ((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ))) (fun fst_arg -> (use_arg_then2 ("local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   (((((use_arg_then2 ("tau_fun_local", [tau_fun_local]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac))) THEN (DISCH_THEN apply_tac));
   ((((use_arg_then2 ("packing_v_prime", [packing_v_prime]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("v_prime_subset", [v_prime_subset]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) ((THENL) (split_tac) [((move ["a"]) THEN (move ["b"]) THEN (move ["h"])); ALL_TAC]) ((((fun arg_tac -> (use_arg_then2 ("local_diag", [local_diag])) (fun fst_arg -> (use_arg_then2 ("f_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   ((THENL_ROT (-1)) (((((fun arg_tac -> (use_arg_then2 ("local_pro", [local_pro])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) THEN ((fun arg_tac -> arg_tac (Arg_term (`0`))) (term_tac exists_tac)) THEN (split_tac)));
   (((((use_arg_then2 ("POWER_0", [POWER_0]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y6_fan", [y6_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] [])))))) THEN (arith_tac) THEN (done_tac));
   (BETA_TAC THEN (move ["k"]) THEN (move ["k_ineq"]));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`k = 1 \/ k = 2 \/ k = 3 \/ k = 4`))) (term_tac (have_gen_tac []ALL_TAC))) ((((use_arg_then2 ("k_ineq", [])) (disch_tac [])) THEN (clear_assumption "k_ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y9_fan", [y9_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("ye_fan", [ye_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 3 = 2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("y8_fan", [y8_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 4 = 1`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma ineq_tau5_pro_tauVEF *)
let ineq_tau5_pro_tauVEF = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	/\ #2.52 <= y9_fan (V,E) d /\ y9_fan (V,E) d <= sqrt8
	==> #0.616 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["dartH"]) THEN (move ["_"])))));
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in_k"]));
   ((((use_arg_then2 ("vw_in_k", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"]) THEN (move ["bounds"])));
   ((((use_arg_then2 ("f_surr_tau_fun_eq_tauVEF", [f_surr_tau_fun_eq_tauVEF]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_surr_localization_convex_local))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["c_local"])));
   ((fun arg_tac -> arg_tac (Arg_term (`face _ (v,w)`))) (term_tac (set_tac "f")));
   ((fun arg_tac -> arg_tac (Arg_term (`f IN face_set (hypermap_of_fan (V,E))`))) (term_tac (have_gen_tac [](move ["f_in"]))));
   (((((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("c_local", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("convex_local_fan", [convex_local_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["local"])) THEN (move ["_"])));
   (((fun arg_tac -> (use_arg_then2 ("ineq", [ineq])) (fun fst_arg -> (use_arg_then2 ("c_local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   ((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ))) (fun fst_arg -> (use_arg_then2 ("local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   (((((use_arg_then2 ("tau_fun_local", [tau_fun_local]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac))) THEN (DISCH_THEN apply_tac));
   ((((use_arg_then2 ("packing_v_prime", [packing_v_prime]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("v_prime_subset", [v_prime_subset]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) ((THENL) (split_tac) [((move ["a"]) THEN (move ["b"]) THEN (move ["h"])); ALL_TAC]) ((((fun arg_tac -> (use_arg_then2 ("local_diag", [local_diag])) (fun fst_arg -> (use_arg_then2 ("f_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   ((THENL_ROT (-1)) (((((fun arg_tac -> (use_arg_then2 ("local_pro", [local_pro])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) THEN ((fun arg_tac -> arg_tac (Arg_term (`1`))) (term_tac exists_tac)) THEN (split_tac)));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y9_fan", [y9_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] [])))))) THEN (arith_tac) THEN (done_tac));
   (BETA_TAC THEN (move ["k"]) THEN (move ["k_ineq"]));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`k = 0 \/ k = 2 \/ k = 3 \/ k = 4`))) (term_tac (have_gen_tac []ALL_TAC))) ((((use_arg_then2 ("k_ineq", [])) (disch_tac [])) THEN (clear_assumption "k_ineq") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("POWER_0", [POWER_0]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y6_fan", [y6_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("ye_fan", [ye_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 3 = 2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("y8_fan", [y8_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 4 = 1`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma ineq_tau5_pro *)
let ineq_tau5_pro = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	/\ #2.52 <= y9_fan (V,E) d /\ y9_fan (V,E) d <= sqrt8
	==> #0.616 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   (BETA_TAC THEN (move ["d_in"]) THEN (move ["bounds"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("ineq_tau5_pro_tauVEF", [ineq_tau5_pro_tauVEF])) (fun fst_arg -> (use_arg_then2 ("d_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("bounds", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("tau_fan", [tau_fan]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
];;

(* Finalization of the section Face5_pro *)
let ineq_tau5_pro_tauVEF_aux = Sections.finalize_theorem ineq_tau5_pro_tauVEF_aux;;
let ineq_tau5_pro_tauVEF = Sections.finalize_theorem ineq_tau5_pro_tauVEF;;
let ineq_tau5_pro = Sections.finalize_theorem ineq_tau5_pro;;
Sections.end_section "Face5_pro";;

(* Section Face4_pro *)
Sections.begin_section "Face4_pro";;
(Sections.add_section_hyp "ineq" (JEJTVGB_quad_pro_concl));;

(* Lemma ineq_tau4_pro_tauVEF_aux *)
let ineq_tau4_pro_tauVEF_aux = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ #2.52 <= y6_fan d /\ y6_fan d <= sqrt8
	/\ sqrt8 <= y4'_fan (V,E) d
	/\ sqrt8 <= y4'_fan (V,E) (f_fan_pair_ext (V,E) d)
	==> #0.477 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["dartH"]) THEN (move ["_"])))));
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in4"]));
   ((((use_arg_then2 ("vw_in4", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"]) THEN (move ["bounds"])));
   ((((use_arg_then2 ("f_surr_tau_fun_eq_tauVEF", [f_surr_tau_fun_eq_tauVEF]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_surr_localization_convex_local))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["c_local"])));
   ((fun arg_tac -> arg_tac (Arg_term (`face _ (v,w)`))) (term_tac (set_tac "f")));
   ((fun arg_tac -> arg_tac (Arg_term (`f IN face_set (hypermap_of_fan (V,E))`))) (term_tac (have_gen_tac [](move ["f_in"]))));
   (((((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("c_local", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("convex_local_fan", [convex_local_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["local"])) THEN (move ["_"])));
   (((fun arg_tac -> (use_arg_then2 ("ineq", [ineq])) (fun fst_arg -> (use_arg_then2 ("c_local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   ((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ))) (fun fst_arg -> (use_arg_then2 ("local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   (((((use_arg_then2 ("tau_fun_local", [tau_fun_local]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac))) THEN (DISCH_THEN apply_tac));
   ((((use_arg_then2 ("packing_v_prime", [packing_v_prime]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("v_prime_subset", [v_prime_subset]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((THENL) (split_tac) [((move ["a"]) THEN (move ["b"]) THEN (case THEN (move ["a_neq_b"])) THEN (case THEN (move ["a_in"])) THEN (case THEN (move ["b_in"])) THEN (move ["ab_not_in"])); ALL_TAC]);
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE quad_diags_imp))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("a_neq_b", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN ((fun arg_tac -> (fun arg_tac -> (fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (use_arg_then2 ("a_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("b_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("ab_not_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC)));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E w' v`))) (term_tac (set_tac "u")));
   (((use_arg_then2 ("bounds", [])) (disch_tac [])) THEN (clear_assumption "bounds") THEN BETA_TAC THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])));
   ((repeat_tactic 1 9 (((use_arg_then2 ("y4'_fan", [y4'_fan]))(thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("PERMUTES_INVERSES", [PERMUTES_INVERSES])) (fun fst_arg -> (fun arg_tac -> (use_arg_then2 ("F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", [F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("INVERSE_F_FAN_PAIR_EXT_EXPLICIT", [INVERSE_F_FAN_PAIR_EXT_EXPLICIT]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("res", [res]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("vw_in", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   (((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face4_eq))) (fun fst_arg -> (use_arg_then2 ("vw_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("SND", [SND]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("u_def", []))(thm_tac (new_rewrite [] []))))) THEN (move ["diag_ineqs"]));
   (case THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (((use_arg_then2 ("DIST_SYM", [DIST_SYM]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((((fun arg_tac -> (use_arg_then2 ("local_pro", [local_pro])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))));
   ((THENL_ROT (-1)) (((fun arg_tac -> arg_tac (Arg_term (`0`))) (term_tac exists_tac)) THEN (split_tac)));
   (((((use_arg_then2 ("POWER_0", [POWER_0]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y6_fan", [y6_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] [])))))) THEN (arith_tac) THEN (done_tac));
   (BETA_TAC THEN (move ["k"]) THEN (move ["k_ineqs"]));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`k = 1 \/ k = 2 \/ k = 3`))) (term_tac (have_gen_tac []ALL_TAC))) ((((use_arg_then2 ("k_ineqs", [])) (disch_tac [])) THEN (clear_assumption "k_ineqs") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y9_fan", [y9_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] [])))))) THEN (arith_tac) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`4 - 2 = 2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("y8_fan", [y8_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`4 - 3 = 1`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma ineq_tau4_pro_tauVEF *)
let ineq_tau4_pro_tauVEF = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ #2.52 <= y9_fan (V,E) d /\ y9_fan (V,E) d <= sqrt8
	/\ sqrt8 <= y4'_fan (V,E) d
	/\ sqrt8 <= y4'_fan (V,E) (f_fan_pair_ext (V,E) d)
	==> #0.477 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   (BETA_TAC THEN (move ["d_in4"]) THEN (move ["bounds"]));
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((fun arg_tac -> arg_tac (Arg_term (`face (hypermap_of_fan (V,E)) d = face (hypermap_of_fan (V,E)) (f_fan_pair_ext (V,E) d)`))) (term_tac (have_gen_tac [](move ["f_eq"]))));
   (((use_arg_then2 ("lemma_face_identity", [lemma_face_identity])) (thm_tac apply_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face", [lemma_in_face]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN ((use_arg_then2 ("ineq_tau4_pro_tauVEF_aux", [ineq_tau4_pro_tauVEF_aux])) (thm_tac apply_tac)));
   ((repeat_tactic 1 9 (((use_arg_then2 ("dart4_yf_eq", [dart4_yf_eq]))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`_1 IN _2`))) (term_tac (have_gen_tac [](move ["in4"])))));
   (((((use_arg_then2 ("dart4_cross_eq_y4'", [dart4_cross_eq_y4']))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("PERMUTES_INVERSES", [PERMUTES_INVERSES])) (fun fst_arg -> (fun arg_tac -> (use_arg_then2 ("F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", [F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("d_in4", [])) (disch_tac [])) THEN (clear_assumption "d_in4") THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ALL_TAC THEN (case THEN (move ["d_in"])) THEN (move ["card_f"])));
   ((((use_arg_then2 ("lemma_dart_invariant", [lemma_dart_invariant]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma ineq_tau4_pro *)
let ineq_tau4_pro = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ #2.52 <= y9_fan (V,E) d /\ y9_fan (V,E) d <= sqrt8
	/\ sqrt8 <= y4'_fan (V,E) d
	/\ sqrt8 <= y4'_fan (V,E) (f_fan_pair_ext (V,E) d)
	==> #0.477 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   (BETA_TAC THEN (move ["d_in4"]) THEN (move ["bounds"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("ineq_tau4_pro_tauVEF", [ineq_tau4_pro_tauVEF])) (fun fst_arg -> (use_arg_then2 ("d_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("bounds", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("tau_fan", [tau_fan]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
];;

(* Finalization of the section Face4_pro *)
let ineq_tau4_pro_tauVEF_aux = Sections.finalize_theorem ineq_tau4_pro_tauVEF_aux;;
let ineq_tau4_pro_tauVEF = Sections.finalize_theorem ineq_tau4_pro_tauVEF;;
let ineq_tau4_pro = Sections.finalize_theorem ineq_tau4_pro;;
Sections.end_section "Face4_pro";;

(* Section Face4_diag3 *)
Sections.begin_section "Face4_diag3";;
(Sections.add_section_hyp "ineq" (JEJTVGB_quad_diag_concl));;

(* Lemma ineq_tau4_diag3_tauVEF *)
let ineq_tau4_diag3_tauVEF = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ &3 <= y4'_fan (V,E) d
	/\ &3 <= y4'_fan (V,E) (f_fan_pair_ext (V,E) d)
	==> #0.467 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["dartH"]) THEN (move ["_"])))));
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in4"]));
   ((((use_arg_then2 ("vw_in4", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"]) THEN (move ["bounds"])));
   ((((use_arg_then2 ("f_surr_tau_fun_eq_tauVEF", [f_surr_tau_fun_eq_tauVEF]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_surr_localization_convex_local))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["c_local"])));
   ((fun arg_tac -> arg_tac (Arg_term (`face _ (v,w)`))) (term_tac (set_tac "f")));
   ((fun arg_tac -> arg_tac (Arg_term (`f IN face_set (hypermap_of_fan (V,E))`))) (term_tac (have_gen_tac [](move ["f_in"]))));
   (((((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("c_local", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("convex_local_fan", [convex_local_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["local"])) THEN (move ["_"])));
   (((fun arg_tac -> (use_arg_then2 ("ineq", [ineq])) (fun fst_arg -> (use_arg_then2 ("c_local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   ((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ))) (fun fst_arg -> (use_arg_then2 ("local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   (((((use_arg_then2 ("tau_fun_local", [tau_fun_local]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac))) THEN (DISCH_THEN apply_tac));
   ((((use_arg_then2 ("packing_v_prime", [packing_v_prime]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("v_prime_subset", [v_prime_subset]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((THENL) (split_tac) [((move ["a"]) THEN (move ["b"]) THEN (case THEN (move ["a_neq_b"])) THEN (case THEN (move ["a_in"])) THEN (case THEN (move ["b_in"])) THEN (move ["ab_not_in"])); ALL_TAC]);
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE quad_diags_imp))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("a_neq_b", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN ((fun arg_tac -> (fun arg_tac -> (fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (use_arg_then2 ("a_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("b_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("ab_not_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC)));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E w' v`))) (term_tac (set_tac "u")));
   (((use_arg_then2 ("bounds", [])) (disch_tac [])) THEN (clear_assumption "bounds") THEN BETA_TAC THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])));
   ((repeat_tactic 1 9 (((use_arg_then2 ("y4'_fan", [y4'_fan]))(thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("PERMUTES_INVERSES", [PERMUTES_INVERSES])) (fun fst_arg -> (fun arg_tac -> (use_arg_then2 ("F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", [F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("INVERSE_F_FAN_PAIR_EXT_EXPLICIT", [INVERSE_F_FAN_PAIR_EXT_EXPLICIT]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("res", [res]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("vw_in", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   (((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face4_eq))) (fun fst_arg -> (use_arg_then2 ("vw_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("SND", [SND]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("u_def", []))(thm_tac (new_rewrite [] []))))) THEN (move ["diag_ineqs"]));
   (case THEN (case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (((use_arg_then2 ("DIST_SYM", [DIST_SYM]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   (((((fun arg_tac -> (use_arg_then2 ("local_std", [local_std])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) THEN (move ["k"]) THEN (move ["k_lt"]));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`k = 0 \/ k = 1 \/ k = 2 \/ k = 3`))) (term_tac (have_gen_tac []ALL_TAC))) ((((use_arg_then2 ("k_lt", [])) (disch_tac [])) THEN (clear_assumption "k_lt") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("POWER_0", [POWER_0]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y6_fan", [y6_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y9_fan", [y9_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`4 - 2 = 2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("y8_fan", [y8_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`4 - 3 = 1`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Lemma ineq_tau4_diag3 *)
let ineq_tau4_diag3 = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ &3 <= y4'_fan (V,E) d
	/\ &3 <= y4'_fan (V,E) (f_fan_pair_ext (V,E) d)
	==> #0.467 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   (BETA_TAC THEN (move ["d_in4"]) THEN (move ["bounds"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("ineq_tau4_diag3_tauVEF", [ineq_tau4_diag3_tauVEF])) (fun fst_arg -> (use_arg_then2 ("d_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("bounds", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("tau_fan", [tau_fan]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
];;

(* Section Std *)
Sections.begin_section "Std";;
(Sections.add_section_hyp "std" (`E = ESTD V`));;

(* Lemma ineq_tau4_diag3_std *)
let ineq_tau4_diag3_std = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> &3 <= y4'_fan (V,E) d 
	/\ &3 <= y4'_fan (V,E) (f_fan_pair_ext (V,E) d)
	==> #0.467 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((BETA_TAC THEN (move ["d_in4"]) THEN (move ["bounds"])) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau4_diag3", [ineq_tau4_diag3])) (fun fst_arg -> (use_arg_then2 ("d_in4", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac)));
   ((((use_arg_then2 ("d_in4", [])) (disch_tac [])) THEN (clear_assumption "d_in4") THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["d_in"])) THEN (move ["_"])));
   (((((use_arg_then2 ("y5_hi_std2", [y5_hi_std2]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_hi_std2", [y6_hi_std2])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y8_hi_std", [y8_hi_std]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_hi_std", [y9_hi_std]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section Std *)
let ineq_tau4_diag3_std = Sections.finalize_theorem ineq_tau4_diag3_std;;
Sections.end_section "Std";;

(* Finalization of the section Face4_diag3 *)
let ineq_tau4_diag3_tauVEF = Sections.finalize_theorem ineq_tau4_diag3_tauVEF;;
let ineq_tau4_diag3 = Sections.finalize_theorem ineq_tau4_diag3;;
let ineq_tau4_diag3_std = Sections.finalize_theorem ineq_tau4_diag3_std;;
Sections.end_section "Face4_diag3";;

(* Section Face5_diags *)
Sections.begin_section "Face5_diags";;
(Sections.add_section_hyp "ineq" (JEJTVGB_pent_diag_concl));;

(* Lemma ineq_tau5_diags_tauVEF *)
let ineq_tau5_diags_tauVEF = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	/\ sqrt8 <= y4'_fan (V,E) d
	/\ sqrt8 <= y4'_fan (V,E) (f_fan_pair_ext (V,E) d)
	/\ sqrt8 <= y4'_fan (V,E) ((f_fan_pair_ext (V,E) POWER 2) d)
	/\ sqrt8 <= y4'_fan (V,E) ((f_fan_pair_ext (V,E) POWER 3) d)
	/\ sqrt8 <= y4'_fan (V,E) (inverse (f_fan_pair_ext (V,E)) d)
	==> #0.616 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (case THEN ((move ["dartH"]) THEN (move ["_"])))));
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in5"]));
   ((((use_arg_then2 ("vw_in5", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"]) THEN (move ["bounds"])));
   ((((use_arg_then2 ("f_surr_tau_fun_eq_tauVEF", [f_surr_tau_fun_eq_tauVEF]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_surr_localization_convex_local))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["c_local"])));
   ((fun arg_tac -> arg_tac (Arg_term (`face _ (v,w)`))) (term_tac (set_tac "f")));
   ((fun arg_tac -> arg_tac (Arg_term (`f IN face_set (hypermap_of_fan (V,E))`))) (term_tac (have_gen_tac [](move ["f_in"]))));
   (((((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("c_local", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("convex_local_fan", [convex_local_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["local"])) THEN (move ["_"])));
   (((fun arg_tac -> (use_arg_then2 ("ineq", [ineq])) (fun fst_arg -> (use_arg_then2 ("c_local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC);
   ((((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ))) (fun fst_arg -> (use_arg_then2 ("local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   (((((use_arg_then2 ("tau_fun_local", [tau_fun_local]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_def", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("lemma_in_face_set", [lemma_in_face_set]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac))) THEN (DISCH_THEN apply_tac));
   ((((use_arg_then2 ("packing_v_prime", [packing_v_prime]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("v_prime_subset", [v_prime_subset]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((THENL_ROT (-1)) ((THENL) (split_tac) [((move ["a"]) THEN (move ["b"]) THEN (case THEN (move ["a_neq_b"])) THEN (case THEN (move ["a_in"])) THEN (case THEN (move ["b_in"])) THEN (move ["ab_not_in"])); ALL_TAC]));
   (((((fun arg_tac -> (use_arg_then2 ("local_std", [local_std])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) THEN (move ["k"]) THEN (move ["k_lt"]));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`k = 0 \/ k = 1 \/ k = 2 \/ k = 3 \/ k = 4`))) (term_tac (have_gen_tac []ALL_TAC))) ((((use_arg_then2 ("k_lt", [])) (disch_tac [])) THEN (clear_assumption "k_lt") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac)));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("POWER_0", [POWER_0]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y6_fan", [y6_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y9_fan", [y9_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("ye_fan", [ye_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 3 = 2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("y8_fan", [y8_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 4 = 1`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE pent_diags_imp))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("a_neq_b", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN ((fun arg_tac -> (fun arg_tac -> (fun arg_tac ->  (conv_thm_tac DISCH_THEN)  (fun fst_arg -> (use_arg_then2 ("a_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("b_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("ab_not_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (thm_tac MP_TAC)));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E w' v`))) (term_tac (set_tac "u")));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E u w'`))) (term_tac (set_tac "z")));
   (((use_arg_then2 ("bounds", [])) (disch_tac [])) THEN (clear_assumption "bounds") THEN BETA_TAC THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])));
   ((fun arg_tac -> arg_tac (Arg_term (`inverse (f_fan_pair_ext (V,E)) (v,w) = (f_fan_pair_ext (V,E) POWER 4) (v,w)`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] []))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 4 = 1`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((fun arg_tac -> (use_arg_then2 ("PERMUTES_INVERSES", [PERMUTES_INVERSES])) (fun fst_arg -> (fun arg_tac -> (use_arg_then2 ("F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", [F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["inv_f_f"])));
   ((repeat_tactic 1 9 (((use_arg_then2 ("y4'_fan", [y4'_fan]))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("THREE", [THREE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("POWER", [POWER]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("inv_f_f", []))(thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`4 = SUC 3`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [2] []))))))) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("COM_POWER", [COM_POWER]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inv_f_f", []))(thm_tac (new_rewrite [] [])))));
   (((repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face5_eq))) (fun fst_arg -> (use_arg_then2 ("vw_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("u_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("z_def", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (move ["diag_ineqs"]));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (((use_arg_then2 ("DIST_SYM", [DIST_SYM]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Lemma ineq_tau5_diags *)
let ineq_tau5_diags = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	/\ sqrt8 <= y4'_fan (V,E) d
	/\ sqrt8 <= y4'_fan (V,E) (f_fan_pair_ext (V,E) d)
	/\ sqrt8 <= y4'_fan (V,E) ((f_fan_pair_ext (V,E) POWER 2) d)
	/\ sqrt8 <= y4'_fan (V,E) ((f_fan_pair_ext (V,E) POWER 3) d)
	/\ sqrt8 <= y4'_fan (V,E) (inverse (f_fan_pair_ext (V,E)) d)
	==> #0.616 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   (BETA_TAC THEN (move ["d_in"]) THEN (move ["bounds"]));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("ineq_tau5_diags_tauVEF", [ineq_tau5_diags_tauVEF])) (fun fst_arg -> (use_arg_then2 ("d_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("bounds", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("tau_fan", [tau_fan]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
];;

(* Section Std *)
Sections.begin_section "Std";;
(Sections.add_section_hyp "std" (`E = ESTD V`));;

(* Lemma ineq_tau5_diags_std *)
let ineq_tau5_diags_std = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> sqrt8 <= y4'_fan (V,E) d
	/\ sqrt8 <= y4'_fan (V,E) (f_fan_pair_ext (V,E) d)
	/\ sqrt8 <= y4'_fan (V,E) ((f_fan_pair_ext (V,E) POWER 2) d)
	/\ sqrt8 <= y4'_fan (V,E) ((f_fan_pair_ext (V,E) POWER 3) d)
	/\ sqrt8 <= y4'_fan (V,E) (inverse (f_fan_pair_ext (V,E)) d)
	==> #0.616 <= tau_fan (V,E) (face (hypermap_of_fan (V,E)) d)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((BETA_TAC THEN (move ["d_in5"]) THEN (move ["bounds"])) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau5_diags", [ineq_tau5_diags])) (fun fst_arg -> (use_arg_then2 ("d_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac)));
   ((((use_arg_then2 ("d_in5", [])) (disch_tac [])) THEN (clear_assumption "d_in5") THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["d_in"])) THEN (move ["_"])));
   ((((use_arg_then2 ("y5_hi_std2", [y5_hi_std2]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_hi_std2", [y6_hi_std2])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y8_hi_std", [y8_hi_std]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_hi_std", [y9_hi_std]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("y6_def", [y6_def]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((fun arg_tac -> (use_arg_then2 ("y6_hi_std2", [y6_hi_std2])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("f_fan_pair_ext_power_in_dart", [f_fan_pair_ext_power_in_dart]))(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Finalization of the section Std *)
let ineq_tau5_diags_std = Sections.finalize_theorem ineq_tau5_diags_std;;
Sections.end_section "Std";;

(* Finalization of the section Face5_diags *)
let ineq_tau5_diags_tauVEF = Sections.finalize_theorem ineq_tau5_diags_tauVEF;;
let ineq_tau5_diags = Sections.finalize_theorem ineq_tau5_diags;;
let ineq_tau5_diags_std = Sections.finalize_theorem ineq_tau5_diags_std;;
Sections.end_section "Face5_diags";;

(* Finalization of the section MainEstimate *)
let local_diag = Sections.finalize_theorem local_diag;;
let local_std = Sections.finalize_theorem local_std;;
let local_pro = Sections.finalize_theorem local_pro;;
let ineq_tau3_tauVEF = Sections.finalize_theorem ineq_tau3_tauVEF;;
let ineq_tau3 = Sections.finalize_theorem ineq_tau3;;
let ineq_tau3_tauVEF_std = Sections.finalize_theorem ineq_tau3_tauVEF_std;;
let ineq_tau4_tauVEF = Sections.finalize_theorem ineq_tau4_tauVEF;;
let ineq_tau4 = Sections.finalize_theorem ineq_tau4;;
let ineq_tau4_tauVEF_std = Sections.finalize_theorem ineq_tau4_tauVEF_std;;
let ineq_tau4_std = Sections.finalize_theorem ineq_tau4_std;;
let ineq_tau5_tauVEF = Sections.finalize_theorem ineq_tau5_tauVEF;;
let ineq_tau5 = Sections.finalize_theorem ineq_tau5;;
let ineq_tau5_tauVEF_std = Sections.finalize_theorem ineq_tau5_tauVEF_std;;
let ineq_tau5_std = Sections.finalize_theorem ineq_tau5_std;;
let ineq_tau6_tauVEF = Sections.finalize_theorem ineq_tau6_tauVEF;;
let ineq_tau6 = Sections.finalize_theorem ineq_tau6;;
let ineq_tau6_tauVEF_std = Sections.finalize_theorem ineq_tau6_tauVEF_std;;
let ineq_tau6_std = Sections.finalize_theorem ineq_tau6_std;;
let ineq_tau5_pro_tauVEF_aux = Sections.finalize_theorem ineq_tau5_pro_tauVEF_aux;;
let ineq_tau5_pro_tauVEF = Sections.finalize_theorem ineq_tau5_pro_tauVEF;;
let ineq_tau5_pro = Sections.finalize_theorem ineq_tau5_pro;;
let ineq_tau4_pro_tauVEF_aux = Sections.finalize_theorem ineq_tau4_pro_tauVEF_aux;;
let ineq_tau4_pro_tauVEF = Sections.finalize_theorem ineq_tau4_pro_tauVEF;;
let ineq_tau4_pro = Sections.finalize_theorem ineq_tau4_pro;;
let ineq_tau4_diag3_tauVEF = Sections.finalize_theorem ineq_tau4_diag3_tauVEF;;
let ineq_tau4_diag3 = Sections.finalize_theorem ineq_tau4_diag3;;
let ineq_tau4_diag3_std = Sections.finalize_theorem ineq_tau4_diag3_std;;
let ineq_tau5_diags_tauVEF = Sections.finalize_theorem ineq_tau5_diags_tauVEF;;
let ineq_tau5_diags = Sections.finalize_theorem ineq_tau5_diags;;
let ineq_tau5_diags_std = Sections.finalize_theorem ineq_tau5_diags_std;;
Sections.end_section "MainEstimate";;

(* Section PerimeterBound *)
Sections.begin_section "PerimeterBound";;
(Sections.add_section_var (mk_var ("V", (`:real^3->bool`))));;
(Sections.add_section_var (mk_var ("E", (`:(real^3->bool)->bool`))));;
(Sections.add_section_hyp "fanV" (`FAN (vec 0,V,E)`));;
(Sections.add_section_hyp "f_surr" (`fully_surrounded (V,E)`));;

(* Let dartH *)
Sections.add_section_lemma "dartH" (Sections.section_proof []
`dart (hypermap_of_fan (V,E)) = dart_of_fan (V,E)`
[
   ((((use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Section OneFace *)
Sections.begin_section "OneFace";;
(Sections.add_section_var (mk_var ("f", (`:real^3#real^3->bool`))));;
(Sections.add_section_hyp "f_in" (`f IN face_set (hypermap_of_fan (V,E))`));;

(* Let local *)
Sections.add_section_lemma "local" (Sections.section_proof []
`local_fan (v_prime V f, e_prime E f, f)`
[
   (((use_arg_then2 ("f_in", [])) (disch_tac [])) THEN (clear_assumption "f_in") THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("lemma_face_representation", [lemma_face_representation])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (case THEN (move ["d"])) THEN (case THEN (move ["d_in"])) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))));
   (((((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_surr_localization))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
]);;

(* Lemma face_eq_image_rho_node1 *)
let face_eq_image_rho_node1 = Sections.section_proof []
`f = IMAGE (\v. v, rho_node1 f v) (v_prime V f)`
[
   ((((((use_arg_then2 ("EXTENSION", [EXTENSION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_IMAGE", [IN_IMAGE]))(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN ALL_TAC THEN (case THEN ((move ["a"]) THEN (move ["b"])))) THEN (((use_arg_then2 ("PAIR_EQ", [PAIR_EQ]))(thm_tac (new_rewrite [] [])))));
   ((THENL_ROT (-1)) ((THENL) (split_tac) [(move ["ab_in"]); ((case THEN (move ["x"])) THEN (case THEN (move ["ab_eq"])) THEN (move ["in_v_prime"]))]));
   (((repeat_tactic 1 9 (((use_arg_then2 ("ab_eq", []))(thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("Local_lemmas.LOCAL_FAN_RHO_NODE_PROS", [Local_lemmas.LOCAL_FAN_RHO_NODE_PROS])) (fun fst_arg -> (use_arg_then2 ("local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((use_arg_then2 ("a", [])) (term_tac exists_tac));
   ((((use_arg_then2 ("v_prime", [v_prime]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Local_lemmas.DETER_RHO_NODE))) (fun fst_arg -> (use_arg_then2 ("local", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("ab_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((THENL_LAST) (split_tac) (((use_arg_then2 ("b", [])) (term_tac exists_tac)) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`a,b IN dart_of_fan (V,E)`))) (term_tac (have_gen_tac []ALL_TAC)));
   (((((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("in_face_imp_in_dart_alt", [in_face_imp_in_dart_alt])) (fun fst_arg -> (use_arg_then2 ("f_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((DISCH_THEN (fun snd_th -> (use_arg_then2 ("PAIR_IN_DART_OF_FAN", [PAIR_IN_DART_OF_FAN])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma sum_face_rho_node1 *)
let sum_face_rho_node1 = Sections.section_proof ["h"]
`sum f h = sum (v_prime V f) (\v. h (v, rho_node1 f v))`
[
   ((((use_arg_then2 ("face_eq_image_rho_node1", [face_eq_image_rho_node1]))(thm_tac (new_rewrite [1] [])))) THEN (((use_arg_then2 ("SUM_IMAGE", [SUM_IMAGE]))(thm_tac (new_rewrite [] [])))));
   ((BETA_TAC THEN (move ["x"]) THEN (move ["y"]) THEN (simp_tac) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"]))) THEN ((((use_arg_then2 ("PAIR_EQ", [PAIR_EQ]))(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`h o (\v. v, rho_node1 f v) = \v. h (v, rho_node1 f v)`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))))) ((ALL_TAC) THEN (done_tac)));
   (((((use_arg_then2 ("FUN_EQ_THM", [FUN_EQ_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section OneFace *)
let face_eq_image_rho_node1 = Sections.finalize_theorem face_eq_image_rho_node1;;
let sum_face_rho_node1 = Sections.finalize_theorem sum_face_rho_node1;;
Sections.end_section "OneFace";;

(* Lemma fully_surrounded_perimeter_bound *)
let fully_surrounded_perimeter_bound = Sections.section_proof []
`perimeterbound (V,E)`
[
   (((((use_arg_then2 ("perimeterbound", [perimeterbound]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("face_set_of_fan", [face_set_of_fan]))(thm_tac (new_rewrite [] []))))) THEN (move ["f"]) THEN (move ["f_in"]));
   (((use_arg_then2 ("f_in", [])) (disch_tac [])) THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("lemma_face_representation", [lemma_face_representation])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN BETA_TAC THEN (case THEN (move ["d"])) THEN (case THEN (move ["d_in"])) THEN (move ["f_eq"]));
   ((((use_arg_then2 ("sum_face_rho_node1", [sum_face_rho_node1]))(thm_tac (new_rewrite [] [])))) THEN ((simp_tac THEN TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("Polar_fan.FAN_PERIMETER", [Polar_fan.FAN_PERIMETER])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`e_prime E f`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_surr_localization))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((((use_arg_then2 ("Polar_fan.WSEWPCH", [Polar_fan.WSEWPCH]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_surr_localization_convex_local)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (done_tac));
];;

(* Finalization of the section PerimeterBound *)
let face_eq_image_rho_node1 = Sections.finalize_theorem face_eq_image_rho_node1;;
let sum_face_rho_node1 = Sections.finalize_theorem sum_face_rho_node1;;
let fully_surrounded_perimeter_bound = Sections.finalize_theorem fully_surrounded_perimeter_bound;;
Sections.end_section "PerimeterBound";;

(* Section ExtraPentSplit *)
Sections.begin_section "ExtraPentSplit";;
(Sections.add_section_var (mk_var ("V", (`:real^3->bool`))));;
(Sections.add_section_var (mk_var ("E", (`:(real^3->bool)->bool`))));;
(Sections.add_section_hyp "h_fan" (`lp_fan (V,E)`));;

(* Let dart1_eq *)
Sections.add_section_lemma "dart1_eq" (Sections.section_proof []
`dart1_of_fan (V,E) = dart_of_fan (V,E)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN (move ["h"])) THEN (((use_arg_then2 ("fully_surrounded_dart_of_fan_eq", [fully_surrounded_dart_of_fan_eq]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Let dartH *)
Sections.add_section_lemma "dartH" (Sections.section_proof []
`dart (hypermap_of_fan (V,E)) = dart_of_fan (V,E)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN (move ["h"])) THEN (((use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Let split_eq *)
Sections.add_section_lemma "split_eq" (Sections.section_proof ["d"]
`d IN dart_of_fan (V,E) ==> split_fan_face (V,E) d = V, add_diag (V,E) d`
[
   ((((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in"])) THEN ((((use_arg_then2 ("split_fan_face", [split_fan_face]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("add_diag", [add_diag]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("vw_in", []))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
]);;
(Sections.add_section_hyp "ineq_apex_flat" ((hd (Ineq.getexact "6988401556")).ineq));;
(Sections.add_section_hyp "ineq_apex_A" ((hd (Ineq.getexact "8082208587")).ineq));;

(* Lemma ineq_tau4_extra *)
let ineq_tau4_extra = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ #2.52 <= y6_fan d /\ y6_fan d <= sqrt8
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ y4'_fan (V,E) d <= sqrt8
	==> &0 <= tauVEF (V,E,face (hypermap_of_fan (V,E)) d)`
[
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in4"]) THEN (move ["bounds"]));
   ((((use_arg_then2 ("vw_in4", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"])));
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((((use_arg_then2 ("dart4_tauVEF_eq_tauq", [dart4_tauVEF_eq_tauq]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("tauq", [tauq]))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("taum_sym_reflect", [taum_sym_reflect]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("Tame_lemmas.taum_sym", [Tame_lemmas.taum_sym]))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`#2.0 = &2 /\ &2 * h0 = #2.52`))) (term_tac (have_gen_tac [](move ["eqs"])))) ((((use_arg_then2 ("h0", [h0]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   (in_tac ["ineq_apex_flat"; "ineq_apex_A"] false ((repeat_tactic 1 9 (((use_arg_then2 ("real_gt", [real_gt]))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac ->(use_arg_then2 ("Ineq.apex_flat", [Ineq.apex_flat]))(fun tmp_arg1 -> (use_arg_then2 ("Ineq.apex_A", [Ineq.apex_A]))(fun tmp_arg2 -> arg_tac (Arg_theorem (CONJ (get_arg_thm tmp_arg1) (get_arg_thm tmp_arg2))))))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("INEQ_ALT", [INEQ_ALT]))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("eqs", []))(thm_tac (new_rewrite [] [])))))));
   ((((use_arg_then2 ("REAL_LE_ADD", [REAL_LE_ADD]))(thm_tac (new_rewrite [] [])))) THEN (split_tac) THEN (((use_arg_then2 ("REAL_LT_IMP_LE", [REAL_LT_IMP_LE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("REAL_LT_TRANS", [REAL_LT_TRANS])) (disch_tac [])) THEN (clear_assumption "REAL_LT_TRANS") THEN (DISCH_THEN apply_tac)));
   ((THENL_FIRST) (((fun arg_tac -> arg_tac (Arg_term (`#0.2759`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("ineq_apex_A", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then2 ("y1_lo", [y1_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y1_hi", [y1_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y3_lo", [y3_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_hi", [y3_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y5_lo", [y5_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("dart4_y4'_lo", [dart4_y4'_lo]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) (((fun arg_tac -> arg_tac (Arg_term (`#0.103`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("ineq_apex_flat", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then2 ("y7_lo", [y7_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y7_hi", [y7_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_lo", [y3_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_hi", [y3_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y8_lo", [y8_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_lo", [y9_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("dart4_y4'_lo", [dart4_y4'_lo]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Lemma lp_tau_split4_extra *)
let lp_tau_split4_extra = Sections.section_proof ["d"]
`lp_tau (V,E) /\ d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ #2.52 <= y6_fan d /\ y6_fan d <= sqrt8
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ y4'_fan (V,E) d <= sqrt8
	==> lp_tau (V, add_diag (V,E) d)`
[
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (case THEN (move ["h_tau"])) THEN (move ["vw_in4"]) THEN (move ["bounds"]));
   ((((use_arg_then2 ("vw_in4", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"])));
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] [])))) THEN (move ["f"]) THEN (move ["f_in2"]));
   ((((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((fun arg_tac -> arg_tac (Arg_theorem (let_RULE faces_add_diag)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_DELETE", [IN_DELETE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_SING", [IN_SING]))(thm_tac (new_rewrite [] []))))));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((THENL_ROT (-1)) ((THENL) case [(move ["f_eqs"]); ((case THEN (move ["f_in"])) THEN (move ["_"]))]));
   (((((use_arg_then2 ("tauVEF_add_diag_eq", [tauVEF_add_diag_eq]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (((use_arg_then2 ("h_tau", [])) (disch_tac [])) THEN (clear_assumption "h_tau") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`3 < CARD (face (hypermap_of_fan (V,E)) (v,w))`))) (term_tac (have_gen_tac [](move ["card_f_gt3"])))) ((((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face_split_fan_face_explicit))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("card_f_gt3", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fan_split_fan_face", [fan_split_fan_face])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("split_eq", []))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`4 - 1 = 3`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`hypermap_of_fan _`))) (term_tac (set_tac "H2")));
   (((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN (clear_assumption "f_in2") THEN BETA_TAC THEN (move ["_"]) THEN (case THEN (move ["fan2"])) THEN (case THEN (move ["dart2"])) THEN (move ["f_surr2"]) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["f2_vw"]) THEN (move ["f2_w'w"]))) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["card2_vw"]) THEN (move ["card2_w'w"]))) THEN (move ["_"]));
   ((((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["dartH2"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["fH2"])));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`#2.0 = &2 /\ &2 * h0 = #2.52`))) (term_tac (have_gen_tac [](move ["eqs"])))) ((((use_arg_then2 ("h0", [h0]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   (in_tac ["ineq_apex_flat"; "ineq_apex_A"] false ((repeat_tactic 1 9 (((use_arg_then2 ("real_gt", [real_gt]))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac ->(use_arg_then2 ("Ineq.apex_flat", [Ineq.apex_flat]))(fun tmp_arg1 -> (use_arg_then2 ("Ineq.apex_A", [Ineq.apex_A]))(fun tmp_arg2 -> arg_tac (Arg_theorem (CONJ (get_arg_thm tmp_arg1) (get_arg_thm tmp_arg2))))))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("INEQ_ALT", [INEQ_ALT]))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("eqs", []))(thm_tac (new_rewrite [] [])))))));
   (((use_arg_then2 ("f_eqs", [])) (disch_tac [])) THEN (clear_assumption "f_eqs") THEN case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((((use_arg_then2 ("REAL_LT_IMP_LE", [REAL_LT_IMP_LE]))(thm_tac (new_rewrite [] [])))) THEN ((use_arg_then2 ("REAL_LT_TRANS", [REAL_LT_TRANS])) (thm_tac apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.2759`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`v,w IN darts_k 3 H2`))) (term_tac (have_gen_tac [](move ["vw_in2_3"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("face_face_map_eq", [face_face_map_eq]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dart3_tauVEF_eq_taum", [dart3_tauVEF_eq_taum]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("face_map_in_darts_k", [face_map_in_darts_k]))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("fH2", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (use_arg_then2 ("dart3_yf_eq", [dart3_yf_eq])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y4'_eq_y4", [y4'_eq_y4]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("ys_split_eq", [ys_split_eq]))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("ineq_apex_A", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then2 ("y1_lo", [y1_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y1_hi", [y1_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y3_lo", [y3_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_hi", [y3_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y5_lo", [y5_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("dart4_y4'_lo", [dart4_y4'_lo]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((((use_arg_then2 ("REAL_LT_IMP_LE", [REAL_LT_IMP_LE]))(thm_tac (new_rewrite [] [])))) THEN ((use_arg_then2 ("REAL_LT_TRANS", [REAL_LT_TRANS])) (thm_tac apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.103`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`w',w IN darts_k 3 H2`))) (term_tac (have_gen_tac [](move ["w'w_in2_3"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("w'w_in2_3", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["w'w_in2"])) THEN (move ["_"])));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dart3_tauVEF_eq_taum", [dart3_tauVEF_eq_taum]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y4_def", [y4_def]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_def", [y9_def]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE ys_split_eq2))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("Tame_lemmas.taum_sym", [Tame_lemmas.taum_sym]))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("ineq_apex_flat", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then2 ("y7_lo", [y7_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y7_hi", [y7_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_lo", [y3_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_hi", [y3_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y8_lo", [y8_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_lo", [y9_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("dart4_y4'_lo", [dart4_y4'_lo]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section ExtraPentSplit *)
let ineq_tau4_extra = Sections.finalize_theorem ineq_tau4_extra;;
let lp_tau_split4_extra = Sections.finalize_theorem lp_tau_split4_extra;;
Sections.end_section "ExtraPentSplit";;

(* Section LpTau *)
Sections.begin_section "LpTau";;
(Sections.add_section_var (mk_var ("V", (`:real^3->bool`))));;
(Sections.add_section_var (mk_var ("E", (`:(real^3->bool)->bool`))));;

(* Section Contravening *)
Sections.begin_section "Contravening";;
(Sections.add_section_hyp "main_estimate" (`lp_main_estimate`));;
(Sections.add_section_hyp "contrV" (`contravening V`));;

(* Let h_fan *)
Sections.add_section_lemma "h_fan" (Sections.section_proof []
`lp_fan (V,ESTD V)`
[
   ((((use_arg_then2 ("contravening_lp_fan", [contravening_lp_fan]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Let p_bound *)
Sections.add_section_lemma "p_bound" (Sections.section_proof []
`perimeterbound (V,ESTD V)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN (move ["h"])));
   ((((use_arg_then2 ("fully_surrounded_perimeter_bound", [fully_surrounded_perimeter_bound]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Lemma contravening_lp_tau *)
let contravening_lp_tau = Sections.section_proof []
`lp_tau (V,ESTD V)`
[
   (((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] []))));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("Crttxat_tame.CRTTXAT", [Crttxat_tame.CRTTXAT])) (fun fst_arg -> (use_arg_then2 ("contrV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("p_bound", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("Jgtdebu.JGTDEBU4", [Jgtdebu.JGTDEBU4])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("tame_9a", [tame_9a]))(thm_tac (new_rewrite [] [])))) THEN (move ["card_f_ineq"]));
   (BETA_TAC THEN (move ["f"]) THEN (DISCH_THEN (fun snd_th -> (use_arg_then2 ("lemma_face_representation", [lemma_face_representation])) (thm_tac (match_mp_then snd_th MP_TAC)))) THEN (case THEN (move ["d"])) THEN (case THEN (move ["d_in"])) THEN (move ["f_eq"]));
   ((fun arg_tac -> arg_tac (Arg_term (`CARD f = 3 \/ CARD f = 4 \/ CARD f = 5 \/ CARD f = 6`))) (term_tac (have_gen_tac []ALL_TAC)));
   ((((fun arg_tac -> (use_arg_then2 ("card_f_ineq", [])) (fun fst_arg -> (use_arg_then2 ("d_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN (((use_arg_then2 ("f_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   ((repeat_tactic 1 9 (case)) THEN (BETA_TAC THEN (move ["card_f"])));
   ((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau3_tauVEF_std", [ineq_tau3_tauVEF_std])) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("d_in", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((((use_arg_then2 ("main_estimate", [])) (disch_tac [])) THEN (clear_assumption "main_estimate") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_main_estimate", [lp_main_estimate]))(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (done_tac));
   ((THENL_FIRST) ((((use_arg_then2 ("REAL_LE_TRANS", [REAL_LE_TRANS])) (disch_tac [])) THEN (clear_assumption "REAL_LE_TRANS") THEN (DISCH_THEN apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.206`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau4_tauVEF_std", [ineq_tau4_tauVEF_std])) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("d_in", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((((use_arg_then2 ("main_estimate", [])) (disch_tac [])) THEN (clear_assumption "main_estimate") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_main_estimate", [lp_main_estimate]))(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (done_tac));
   ((THENL_FIRST) ((((use_arg_then2 ("REAL_LE_TRANS", [REAL_LE_TRANS])) (disch_tac [])) THEN (clear_assumption "REAL_LE_TRANS") THEN (DISCH_THEN apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.4819`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau5_tauVEF_std", [ineq_tau5_tauVEF_std])) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("d_in", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((((use_arg_then2 ("main_estimate", [])) (disch_tac [])) THEN (clear_assumption "main_estimate") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_main_estimate", [lp_main_estimate]))(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (done_tac));
   ((THENL_FIRST) ((((use_arg_then2 ("REAL_LE_TRANS", [REAL_LE_TRANS])) (disch_tac [])) THEN (clear_assumption "REAL_LE_TRANS") THEN (DISCH_THEN apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.712`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("f_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau6_tauVEF_std", [ineq_tau6_tauVEF_std])) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("d_in", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((((use_arg_then2 ("main_estimate", [])) (disch_tac [])) THEN (clear_assumption "main_estimate") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_main_estimate", [lp_main_estimate]))(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) THEN (done_tac));
];;

(* Finalization of the section Contravening *)
let contravening_lp_tau = Sections.finalize_theorem contravening_lp_tau;;
Sections.end_section "Contravening";;

(* Section Split *)
Sections.begin_section "Split";;
(Sections.add_section_hyp "h_fan" (`lp_fan (V,E)`));;

(* Let dart1_eq *)
Sections.add_section_lemma "dart1_eq" (Sections.section_proof []
`dart1_of_fan (V,E) = dart_of_fan (V,E)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN (move ["h"])) THEN (((use_arg_then2 ("fully_surrounded_dart_of_fan_eq", [fully_surrounded_dart_of_fan_eq]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Let dartH *)
Sections.add_section_lemma "dartH" (Sections.section_proof []
`dart (hypermap_of_fan (V,E)) = dart_of_fan (V,E)`
[
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN (clear_assumption "h_fan") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN (move ["h"])) THEN (((use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
]);;

(* Let split_eq *)
Sections.add_section_lemma "split_eq" (Sections.section_proof ["d"]
`d IN dart_of_fan (V,E) ==> split_fan_face (V,E) d = V, add_diag (V,E) d`
[
   ((((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in"])) THEN ((((use_arg_then2 ("split_fan_face", [split_fan_face]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("add_diag", [add_diag]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("vw_in", []))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
]);;

(* Section Split4 *)
Sections.begin_section "Split4";;
(Sections.add_section_hyp "h_tau" (`lp_tau (V,E)`));;

(* Section Apex_flat *)
Sections.begin_section "Apex_flat";;
(Sections.add_section_hyp "ineq103" ((hd (Ineq.getexact "6988401556")).ineq));;

(* Lemma lp_tau_split4_apex_flat *)
let lp_tau_split4_apex_flat = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ y4'_fan (V,E) d <= sqrt8
	==> lp_tau (V, add_diag (V,E) d)`
[
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in4"]) THEN (move ["bounds"]));
   ((((use_arg_then2 ("vw_in4", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"])));
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] [])))) THEN (move ["f"]) THEN (move ["f_in2"]));
   ((((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((fun arg_tac -> arg_tac (Arg_theorem (let_RULE faces_add_diag)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_DELETE", [IN_DELETE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_SING", [IN_SING]))(thm_tac (new_rewrite [] []))))));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((THENL_ROT (-1)) ((THENL) case [(move ["f_eqs"]); ((case THEN (move ["f_in"])) THEN (move ["_"]))]));
   (((((use_arg_then2 ("tauVEF_add_diag_eq", [tauVEF_add_diag_eq]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (((use_arg_then2 ("h_tau", [])) (disch_tac [])) THEN (clear_assumption "h_tau") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`3 < CARD (face (hypermap_of_fan (V,E)) (v,w))`))) (term_tac (have_gen_tac [](move ["card_f_gt3"])))) ((((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face_split_fan_face_explicit))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("card_f_gt3", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fan_split_fan_face", [fan_split_fan_face])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("split_fan_face_eq_add_diag", [split_fan_face_eq_add_diag]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`4 - 1 = 3`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`hypermap_of_fan _`))) (term_tac (set_tac "H2")));
   (((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN (clear_assumption "f_in2") THEN BETA_TAC THEN (move ["_"]) THEN (case THEN (move ["fan2"])) THEN (case THEN (move ["dart2"])) THEN (move ["f_surr2"]) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["f2_vw"]) THEN (move ["f2_w'w"]))) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["card2_vw"]) THEN (move ["card2_w'w"]))) THEN (move ["_"]));
   ((((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["dartH2"])) THEN (move ["_"])));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`#2.0 = &2 /\ &2 * h0 = #2.52`))) (term_tac (have_gen_tac [](move ["eqs"])))) ((((use_arg_then2 ("h0", [h0]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   (in_tac ["ineq103"] false ((((use_arg_then2 ("real_gt", [real_gt]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("Ineq.apex_flat", [Ineq.apex_flat]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("INEQ_ALT", [INEQ_ALT]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("eqs", []))(thm_tac (new_rewrite [] [])))))));
   ((THENL_FIRST) ((((use_arg_then2 ("REAL_LT_IMP_LE", [REAL_LT_IMP_LE]))(thm_tac (new_rewrite [] [])))) THEN ((use_arg_then2 ("REAL_LT_TRANS", [REAL_LT_TRANS])) (thm_tac apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.103`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   (((use_arg_then2 ("f_eqs", [])) (disch_tac [])) THEN (clear_assumption "f_eqs") THEN case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))));
   ((fun arg_tac -> arg_tac (Arg_term (`v,w IN darts_k 3 H2`))) (term_tac (have_gen_tac [](move ["vw_in2_3"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dart3_tauVEF_eq_taum", [dart3_tauVEF_eq_taum]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y4'_eq_y4", [y4'_eq_y4]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("ys_split_eq", [ys_split_eq]))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("ineq103", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then2 ("y1_lo", [y1_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y1_hi", [y1_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y3_lo", [y3_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_hi", [y3_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y5_lo", [y5_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_lo", [y6_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("dart4_y4'_lo", [dart4_y4'_lo]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`w',w IN darts_k 3 H2`))) (term_tac (have_gen_tac [](move ["w'w_in2_3"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("w'w_in2_3", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["w'w_in2"])) THEN (move ["_"])));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dart3_tauVEF_eq_taum", [dart3_tauVEF_eq_taum]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y4_def", [y4_def]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_def", [y9_def]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE ys_split_eq2))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("Tame_lemmas.taum_sym", [Tame_lemmas.taum_sym]))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("ineq103", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then2 ("y7_lo", [y7_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y7_hi", [y7_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_lo", [y3_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_hi", [y3_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y8_lo", [y8_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_lo", [y9_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("dart4_y4'_lo", [dart4_y4'_lo]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section Apex_flat *)
let lp_tau_split4_apex_flat = Sections.finalize_theorem lp_tau_split4_apex_flat;;
Sections.end_section "Apex_flat";;

(* Section Apex_sup_flat *)
Sections.begin_section "Apex_sup_flat";;
(Sections.add_section_hyp "ineq86" ((hd (Ineq.getexact "1642527039")).ineq));;

(* Let ineq_tau *)
Sections.add_section_lemma "ineq_tau" (Sections.section_proof ["y1";"y2";"y3";"y4";"y5";"y6"]
`ineq (apex_sup_flat y1 y2 y3 y4 y5 y6)
		(&0 <= taum y1 y2 y3 y4 y5 y6)`
[
   ((in_tac ["ineq86"] true ((repeat_tactic 1 9 (((use_arg_then2 ("INEQ_ALT", [INEQ_ALT]))(thm_tac (new_rewrite [] []))))) THEN (simp_tac))) THEN (BETA_TAC THEN (move ["ys_all"])));
   ((((use_arg_then2 ("REAL_LT_IMP_LE", [REAL_LT_IMP_LE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("REAL_LT_TRANS", [REAL_LT_TRANS])) (disch_tac [])) THEN (clear_assumption "REAL_LT_TRANS") THEN (DISCH_THEN apply_tac)));
   (((fun arg_tac -> arg_tac (Arg_term (`#0.128 + #0.053 * ((y5 + y6 - &4) - #2.75 / &2 * (y4 - sqrt8))`))) (term_tac exists_tac)) THEN (split_tac));
   ((((use_arg_then2 ("ys_all", [])) (disch_tac [])) THEN (clear_assumption "ys_all") THEN BETA_TAC) THEN ((((use_arg_then2 ("Ineq.apex_sup_flat", [Ineq.apex_sup_flat]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac)));
   ((((use_arg_then2 ("Flyspeck_constants.bounds", [Flyspeck_constants.bounds])) (disch_tac [])) THEN (clear_assumption "Flyspeck_constants.bounds") THEN BETA_TAC) THEN (arith_tac) THEN (done_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`a + c < b <=> &0 < (b - a) - c`))) (term_tac (have_gen_tac ["a"; "b"; "c"](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   (((((use_arg_then2 ("real_gt", [real_gt]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("ineq86", []))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
]);;

(* Lemma lp_tau_split4_apex_sup_flat *)
let lp_tau_split4_apex_sup_flat = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ sqrt8 <= y4'_fan (V,E) d
	/\ y4'_fan (V,E) d <= &3
	==> lp_tau (V, add_diag (V,E) d)`
[
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in4"]) THEN (move ["bounds"]));
   ((((use_arg_then2 ("vw_in4", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"])));
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] [])))) THEN (move ["f"]) THEN (move ["f_in2"]));
   ((((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((fun arg_tac -> arg_tac (Arg_theorem (let_RULE faces_add_diag)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_DELETE", [IN_DELETE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_SING", [IN_SING]))(thm_tac (new_rewrite [] []))))));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((THENL_ROT (-1)) ((THENL) case [(move ["f_eqs"]); ((case THEN (move ["f_in"])) THEN (move ["_"]))]));
   (((((use_arg_then2 ("tauVEF_add_diag_eq", [tauVEF_add_diag_eq]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (((use_arg_then2 ("h_tau", [])) (disch_tac [])) THEN (clear_assumption "h_tau") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`3 < CARD (face (hypermap_of_fan (V,E)) (v,w))`))) (term_tac (have_gen_tac [](move ["card_f_gt3"])))) ((((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face_split_fan_face_explicit))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("card_f_gt3", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fan_split_fan_face", [fan_split_fan_face])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("split_fan_face_eq_add_diag", [split_fan_face_eq_add_diag]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`4 - 1 = 3`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`hypermap_of_fan _`))) (term_tac (set_tac "H2")));
   (((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN (clear_assumption "f_in2") THEN BETA_TAC THEN (move ["_"]) THEN (case THEN (move ["fan2"])) THEN (case THEN (move ["dart2"])) THEN (move ["f_surr2"]) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["f2_vw"]) THEN (move ["f2_w'w"]))) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["card2_vw"]) THEN (move ["card2_w'w"]))) THEN (move ["_"]));
   ((((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["dartH2"])) THEN (move ["_"])));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`#2.0 = &2 /\ #3.0 = &3`))) (term_tac (have_gen_tac [](move ["eqs"])))) ((arith_tac) THEN (done_tac)));
   (in_tac ["ineq_tau"] false ((((use_arg_then2 ("INEQ_ALT", [INEQ_ALT]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("Ineq.apex_sup_flat", [Ineq.apex_sup_flat]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("eqs", []))(thm_tac (new_rewrite [] [])))))));
   (((use_arg_then2 ("f_eqs", [])) (disch_tac [])) THEN (clear_assumption "f_eqs") THEN case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))));
   ((fun arg_tac -> arg_tac (Arg_term (`v,w IN darts_k 3 H2`))) (term_tac (have_gen_tac [](move ["vw_in2_3"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dart3_tauVEF_eq_taum", [dart3_tauVEF_eq_taum]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y4'_eq_y4", [y4'_eq_y4]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("ys_split_eq", [ys_split_eq]))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("ineq_tau", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then2 ("y1_lo", [y1_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y1_hi", [y1_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y3_lo", [y3_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_hi", [y3_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y5_lo", [y5_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_lo", [y6_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`w',w IN darts_k 3 H2`))) (term_tac (have_gen_tac [](move ["w'w_in2_3"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("w'w_in2_3", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["w'w_in2"])) THEN (move ["_"])));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dart3_tauVEF_eq_taum", [dart3_tauVEF_eq_taum]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y4_def", [y4_def]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_def", [y9_def]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE ys_split_eq2))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("Tame_lemmas.taum_sym", [Tame_lemmas.taum_sym]))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("ineq_tau", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then2 ("y7_lo", [y7_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y7_hi", [y7_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_lo", [y3_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_hi", [y3_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y8_lo", [y8_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y9_lo", [y9_lo]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section Apex_sup_flat *)
let lp_tau_split4_apex_sup_flat = Sections.finalize_theorem lp_tau_split4_apex_sup_flat;;
Sections.end_section "Apex_sup_flat";;
(Sections.add_section_hyp "ineq103" ((hd (Ineq.getexact "6988401556")).ineq));;
(Sections.add_section_hyp "ineq86" ((hd (Ineq.getexact "1642527039")).ineq));;

(* Lemma lp_tau_split4 *)
let lp_tau_split4 = Sections.section_proof ["d"]
`d IN darts_k 4 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ y4'_fan (V,E) d <= &3
	==> lp_tau (V, add_diag (V,E) d)`
[
   (BETA_TAC THEN (move ["d_in4"]) THEN (move ["bounds"]));
   (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("REAL_LE_TOTAL", [REAL_LE_TOTAL])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`y4'_fan (V,E) d`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`sqrt8`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN case THEN (move ["y4'_ineq"]));
   ((((use_arg_then2 ("lp_tau_split4_apex_flat", [lp_tau_split4_apex_flat]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((((use_arg_then2 ("lp_tau_split4_apex_sup_flat", [lp_tau_split4_apex_sup_flat]))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
];;

(* Finalization of the section Split4 *)
let lp_tau_split4_apex_flat = Sections.finalize_theorem lp_tau_split4_apex_flat;;
let lp_tau_split4_apex_sup_flat = Sections.finalize_theorem lp_tau_split4_apex_sup_flat;;
let lp_tau_split4 = Sections.finalize_theorem lp_tau_split4;;
Sections.end_section "Split4";;

(* Section Split5 *)
Sections.begin_section "Split5";;
(Sections.add_section_hyp "h_tau" (`lp_tau (V,E)`));;

(* Section One_diag *)
Sections.begin_section "One_diag";;
(Sections.add_section_hyp "quad_pro" (JEJTVGB_quad_pro_concl));;
(Sections.add_section_hyp "ineq103" ((hd (Ineq.getexact "6988401556")).ineq));;

(* Lemma lp_tau_split5_one_diag *)
let lp_tau_split5_one_diag = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	/\ y4'_fan (V,E) d <= sqrt8
	/\ sqrt8 <= y4'_fan (V,E) ((f_fan_pair_ext (V,E) POWER 2) d)
	/\ sqrt8 <= y4'_fan (V,E) ((f_fan_pair_ext (V,E) POWER 3) d)
	==> lp_tau (V, add_diag (V,E) d)`
[
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in5"]) THEN (move ["bounds"]));
   ((((use_arg_then2 ("vw_in5", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"])));
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] [])))) THEN (move ["f"]) THEN (move ["f_in2"]));
   ((((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((fun arg_tac -> arg_tac (Arg_theorem (let_RULE faces_add_diag)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_DELETE", [IN_DELETE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_SING", [IN_SING]))(thm_tac (new_rewrite [] []))))));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((THENL_ROT (-1)) ((THENL) case [(move ["f_eqs"]); ((case THEN (move ["f_in"])) THEN (move ["_"]))]));
   (((((use_arg_then2 ("tauVEF_add_diag_eq", [tauVEF_add_diag_eq]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (((use_arg_then2 ("h_tau", [])) (disch_tac [])) THEN (clear_assumption "h_tau") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`3 < CARD (face (hypermap_of_fan (V,E)) (v,w))`))) (term_tac (have_gen_tac [](move ["card_f_gt3"])))) ((((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face_split_fan_face_explicit))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("card_f_gt3", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fan_split_fan_face", [fan_split_fan_face])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_split_fan_face))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("card_f_gt3", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("split_fan_face_eq_add_diag", [split_fan_face_eq_add_diag]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 1 = 4`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`hypermap_of_fan _`))) (term_tac (set_tac "H2")));
   (((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN (clear_assumption "f_in2") THEN BETA_TAC THEN (move ["_"]) THEN (move ["f2_eqs"]) THEN (case THEN (move ["fan2"])) THEN (case THEN (move ["dart2"])) THEN (move ["f_surr2"]) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["f2_vw"]) THEN (move ["f2_w'w"]))) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["card2_vw"]) THEN (move ["card2_w'w"]))) THEN (move ["_"]));
   ((((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["dartH2"])) THEN (move ["_"])));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`#2.0 = &2 /\ &2 * h0 = #2.52`))) (term_tac (have_gen_tac [](move ["eqs"])))) ((((use_arg_then2 ("h0", [h0]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   (in_tac ["ineq103"] false ((((use_arg_then2 ("real_gt", [real_gt]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("Ineq.apex_flat", [Ineq.apex_flat]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("INEQ_ALT", [INEQ_ALT]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("eqs", []))(thm_tac (new_rewrite [] [])))))));
   (((use_arg_then2 ("f_eqs", [])) (disch_tac [])) THEN (clear_assumption "f_eqs") THEN case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((((use_arg_then2 ("REAL_LT_IMP_LE", [REAL_LT_IMP_LE]))(thm_tac (new_rewrite [] [])))) THEN ((use_arg_then2 ("REAL_LT_TRANS", [REAL_LT_TRANS])) (thm_tac apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.103`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`v,w IN darts_k 3 H2`))) (term_tac (have_gen_tac [](move ["vw_in2_3"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dart3_tauVEF_eq_taum", [dart3_tauVEF_eq_taum]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y4'_eq_y4", [y4'_eq_y4]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("ys_split_eq", [ys_split_eq]))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("ineq103", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then2 ("y1_lo", [y1_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y1_hi", [y1_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y3_lo", [y3_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_hi", [y3_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y5_lo", [y5_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_lo", [y6_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("y4'_lo_2h0", [y4'_lo_2h0])) (fun fst_arg -> (use_arg_then2 ("vw_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`w',w IN darts_k 4 H2`))) (term_tac (have_gen_tac [](move ["w'w_in2_5"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((((use_arg_then2 ("REAL_LE_TRANS", [REAL_LE_TRANS])) (disch_tac [])) THEN (clear_assumption "REAL_LE_TRANS") THEN (DISCH_THEN apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.477`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau4_pro_tauVEF_aux", [ineq_tau4_pro_tauVEF_aux])) (fun fst_arg -> (use_arg_then2 ("quad_pro", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("w'w_in2_5", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("lp_fan_trans1", [lp_fan_trans1]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE ys_split_eq2))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("y4'_lo_2h0", [y4'_lo_2h0])) (fun fst_arg -> (use_arg_then2 ("vw_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fully_surrounded_dart_of_fan_eq", [fully_surrounded_dart_of_fan_eq])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr2", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["dart1_2_eq"])));
   ((fun arg_tac -> arg_tac (Arg_term (`w', w IN dart_of_fan (V, add_diag (V,E) (v,w))`))) (term_tac (have_gen_tac [](move ["w'w_in2"]))));
   (((((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (split_tac);
   ((((use_arg_then2 ("y8_fan", [y8_fan]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("ye_fan", [ye_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("o_THM", [o_THM])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`inverse _`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("POWER_2", [POWER_2]))(gsym_then (thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`2 = CARD (face H2 (w',w)) - 2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [1] []))))))) ((((use_arg_then2 ("card2_w'w", []))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse]))(gsym_then (thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("fan2", [fan2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card2_w'w", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("w'w_in2", []))(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   (((((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_fan_pair_ext_power_add_diag))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))))) THEN (TRY ((arith_tac))));
   ((repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_ROT (-1)) (split_tac));
   ((((use_arg_then2 ("y4'_fan", [y4'_fan]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("PERMUTES_INVERSES", [PERMUTES_INVERSES])) (fun fst_arg -> (fun arg_tac -> (use_arg_then2 ("F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", [F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("FST", [FST]))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("POWER_1", [POWER_1]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_fan_pair_power", [f_fan_pair_power]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("dart1_2_eq", []))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f2_eqs", []))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`sqrt8 <= y4'_fan (V,E) ((f_fan_pair_ext (V,E) POWER 3) (v,w))`))) (term_tac (have_gen_tac []ALL_TAC))) ((ALL_TAC) THEN (done_tac)));
   (((use_arg_then2 ("y4'_fan", [y4'_fan]))(thm_tac (new_rewrite [] []))));
   (((fun arg_tac -> arg_tac (Arg_term (`SND _`))) (term_tac (set_tac "x1"))) THEN ((fun arg_tac -> arg_tac (Arg_term (`FST _`))) (term_tac (set_tac "x2"))));
   ((fun arg_tac -> arg_tac (Arg_term (`x1 = w'`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   ((((use_arg_then2 ("x1_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 3 = 2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   ((((fun arg_tac -> arg_tac (Arg_theorem (let_RULE inverse_f_fan_pair_ext_power2)))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((THENL_FIRST) (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`x2 = SND (f_fan_pair (V,E) (v,w))`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))))) ((((use_arg_then2 ("DIST_SYM", [DIST_SYM]))(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   ((((use_arg_then2 ("x2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("THREE", [THREE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("COM_POWER", [COM_POWER]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("o_THM", [o_THM])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`inverse _`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("o_ASSOC", [o_ASSOC]))(thm_tac (new_rewrite [] [])))));
   ((((fun arg_tac -> (use_arg_then2 ("PERMUTES_INVERSES_o", [PERMUTES_INVERSES_o])) (fun fst_arg -> (fun arg_tac -> (use_arg_then2 ("F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", [F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then2 ("f_fan_pair_power", [f_fan_pair_power]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("f_fan_pair", [f_fan_pair]))(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   (((use_arg_then2 ("y4'_fan", [y4'_fan]))(thm_tac (new_rewrite [] []))));
   ((fun arg_tac -> arg_tac (Arg_term (`inverse _1 _2`))) (term_tac (set_tac "x1")));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan (vec 0) V E w' v`))) (term_tac (set_tac "u")));
   ((fun arg_tac -> arg_tac (Arg_term (`u,w' IN dart_of_fan (V,E)`))) (term_tac (have_gen_tac [](move ["uw'_in"]))));
   ((((use_arg_then2 ("u_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (let_RULE inverse_f_fan_pair_ext_power2)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`2 = CARD (face (hypermap_of_fan (V,E)) (v,w)) - 3`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   (((repeat_tactic 1 9 (((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] [])))))) THEN (((use_arg_then2 ("lemma_dart_invariant_power_face", [lemma_dart_invariant_power_face]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`x1 = u, w'`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   ((((use_arg_then2 ("x1_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("PERMUTES_INVERSE_EQ", [PERMUTES_INVERSE_EQ])) (fun fst_arg -> (fun arg_tac -> (use_arg_then2 ("F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", [F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))));
   (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`u,w' IN dart1_of_fan (V,add_diag (V,E) (v,w))`))) (term_tac (have_gen_tac [](move ["uw'_in2"])))));
   (((((use_arg_then2 ("POWER_1", [POWER_1]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_fan_pair_power", [f_fan_pair_power]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("POWER_1", [POWER_1]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f2_eqs", []))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((((use_arg_then2 ("dart1_2_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`sqrt8 <= y4'_fan (V,E) ((f_fan_pair_ext (V,E) POWER 2) (v,w))`))) (term_tac (have_gen_tac []ALL_TAC))) ((ALL_TAC) THEN (done_tac)));
   (((use_arg_then2 ("y4'_fan", [y4'_fan]))(thm_tac (new_rewrite [] []))));
   (((fun arg_tac -> arg_tac (Arg_term (`SND _`))) (term_tac (set_tac "x2"))) THEN ((fun arg_tac -> arg_tac (Arg_term (`FST _`))) (term_tac (set_tac "x3"))));
   ((fun arg_tac -> arg_tac (Arg_term (`x3 = w`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   ((((use_arg_then2 ("x3_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("o_THM", [o_THM])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`inverse _`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("o_ASSOC", [o_ASSOC]))(thm_tac (new_rewrite [] [])))));
   ((((fun arg_tac -> (use_arg_then2 ("PERMUTES_INVERSES_o", [PERMUTES_INVERSES_o])) (fun fst_arg -> (fun arg_tac -> (use_arg_then2 ("F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", [F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then2 ("f_fan_pair_ext", [f_fan_pair_ext]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("vw_in", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then2 ("f_fan_pair", [f_fan_pair]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`x2 = u`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))))) ((((use_arg_then2 ("DIST_SYM", [DIST_SYM]))(thm_tac (new_rewrite [] [])))) THEN (done_tac)));
   ((THENL_FIRST) ((((use_arg_then2 ("x2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))))) ((arith_tac) THEN (done_tac)));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 2 = 3`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("THREE", [THREE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("COM_POWER", [COM_POWER]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (let_RULE inverse_f_fan_pair_ext_power2)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("u_def", []))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then2 ("INVERSE_F_FAN_PAIR_EXT_EXPLICIT", [INVERSE_F_FAN_PAIR_EXT_EXPLICIT]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("res", [res]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("uw'_in", []))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section One_diag *)
let lp_tau_split5_one_diag = Sections.finalize_theorem lp_tau_split5_one_diag;;
Sections.end_section "One_diag";;

(* Section Two_diags *)
Sections.begin_section "Two_diags";;
(Sections.add_section_hyp "ineq_apex_flat" ((hd (Ineq.getexact "6988401556")).ineq));;
(Sections.add_section_hyp "ineq_apex_A" ((hd (Ineq.getexact "8082208587")).ineq));;

(* Lemma lp_tau_split5_two_diags *)
let lp_tau_split5_two_diags = Sections.section_proof ["d"]
`d IN darts_k 5 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	/\ y4'_fan (V,E) d <= sqrt8
	/\ y4'_fan (V,E) ((f_fan_pair_ext (V,E) POWER 2) d) <= sqrt8
	==> lp_tau (V, add_diag (V, (add_diag (V,E) d)) ((f_fan_pair_ext (V,E) POWER 2) d))`
[
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in5"]) THEN (move ["bounds"]));
   ((((use_arg_then2 ("vw_in5", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"])));
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("lp_fan_trans1", [lp_fan_trans1])) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`v,w`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["h_fan2"])));
   ((fun arg_tac -> arg_tac (Arg_term (`add_diag (V,E) (v,w)`))) (term_tac (set_tac "E2")));
   ((fun arg_tac -> arg_tac (Arg_term (`(_1 POWER 2) (v,w)`))) (term_tac (set_tac "d2")));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`3 < CARD (face (hypermap_of_fan (V,E)) (v,w))`))) (term_tac (have_gen_tac [](move ["card_f_gt3"])))) ((((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face_split_fan_face_explicit))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("card_f_gt3", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fan_split_fan_face", [fan_split_fan_face])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_split_fan_face))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("card_f_gt3", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("split_eq", []))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("E2_def", []))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`5 - 1 = 4`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`hypermap_of_fan _`))) (term_tac (set_tac "H2")));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   (BETA_TAC THEN (move ["f2_eqs"]) THEN (case THEN (move ["fan2"])) THEN (case THEN (move ["dart2"])) THEN (move ["f_surr2"]) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["f2_vw"]) THEN (move ["f2_w'w"]))) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["card2_vw"]) THEN (move ["card2_w'w"]))) THEN (move ["_"]));
   ((fun arg_tac -> arg_tac (Arg_term (`d2 = (f_fan_pair_ext (V,E2) POWER 2) (w',w)`))) (term_tac (have_gen_tac [](move ["d2_eq"]))));
   ((((use_arg_then2 ("E2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_fan_pair_ext_power_add_diag)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("fanV", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("f_surr", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("vw_in", []))(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   ((((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["dartH2"])) THEN (case THEN (move ["_"])) THEN (case THEN (move ["_"])) THEN (move ["fH2"])));
   ((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fully_surrounded_dart_of_fan_eq", [fully_surrounded_dart_of_fan_eq])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr2", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN (move ["dart1_2_eq"])));
   ((fun arg_tac -> arg_tac (Arg_term (`w',w IN darts_k 4 H2`))) (term_tac (have_gen_tac [](move ["w'w_in4"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card2_w'w", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("w'w_in4", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["w'w_in2"])) THEN (move ["card_w'w"])));
   ((fun arg_tac -> arg_tac (Arg_term (`d2 IN darts_k 4 H2`))) (term_tac (have_gen_tac [](move ["d2_in4"]))));
   (((((use_arg_then2 ("d2_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("fH2", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("face_map_in_darts_k", [face_map_in_darts_k]))(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((((use_arg_then2 ("d2_in4", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["d2_in2"])) THEN (move ["_"])));
   ((fun arg_tac -> arg_tac (Arg_term (`FST (inverse (f_fan_pair_ext (V,E2)) d2) = w`))) (term_tac (have_gen_tac [](move ["eq1"]))));
   ((((use_arg_then2 ("d2_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("o_THM", [o_THM])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`inverse _`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("o_ASSOC", [o_ASSOC]))(thm_tac (new_rewrite [] [])))));
   ((((fun arg_tac -> (use_arg_then2 ("PERMUTES_INVERSES_o", [PERMUTES_INVERSES_o])) (fun fst_arg -> (fun arg_tac -> (use_arg_then2 ("F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", [F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then2 ("f_fan_pair_ext", [f_fan_pair_ext]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_2_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("w'w_in2", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then2 ("f_fan_pair", [f_fan_pair]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("FST", [FST]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan (vec 0) V E2 w' w = SND d2`))) (term_tac (have_gen_tac [](move ["eq2"]))));
   ((THENL_FIRST) ((((use_arg_then2 ("d2_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart4_f_fan_pair_ext_power2", [dart4_f_fan_pair_ext_power2]))(thm_tac (new_rewrite [] []))))) (((((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (done_tac)));
   ((((fun arg_tac -> arg_tac (Arg_theorem (let_RULE inverse_f_fan_pair_ext_power2)))(thm_tac (new_rewrite [] [])))) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`add_diag (V,E2) d2 = add_diag (V,E2) (w',w)`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   ((((fun arg_tac -> (use_arg_then2 ("pair_expand", [pair_expand])) (fun fst_arg -> (use_arg_then2 ("d2", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("add_diag", [add_diag]))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("pair_expand", [pair_expand]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dart1_2_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("d2_in2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("w'w_in2", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan (vec 0) V E2 (FST d2) (SND d2) = FST (inverse (f_fan_pair_ext (V,E2)) d2)`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))))));
   ((((use_arg_then2 ("INVERSE_F_FAN_PAIR_EXT_EXPLICIT", [INVERSE_F_FAN_PAIR_EXT_EXPLICIT]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("res", [res]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_2_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("d2_in2", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac));
   ((((fun arg_tac -> (use_arg_then2 ("pair_expand", [pair_expand])) (fun fst_arg -> (use_arg_then2 ("d2", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [3] [])))) THEN (done_tac));
   (((((use_arg_then2 ("eq1", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("eq2", []))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL Collect_geom.PER_SET2)))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("lp_tau_split4_extra", [lp_tau_split4_extra])) (fun fst_arg -> (use_arg_then2 ("h_fan2", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("ineq_apex_flat", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("ineq_apex_A", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN (DISCH_THEN apply_tac));
   ((((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("w'w_in4", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("andbT", [andbT]))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("E2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE ys_split_eq2))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("E2_def", []))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("y4'_lo_2h0", [y4'_lo_2h0])) (fun fst_arg -> (use_arg_then2 ("vw_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) ((arith_tac) THEN (done_tac)));
   (((THENL_ROT 1)) ((fun arg_tac -> arg_tac (Arg_term (`y8_fan (V,E2) (w',w) <= #2.52 /\ y4'_fan (V,E2) (w',w) <= sqrt8`))) (term_tac (have_gen_tac [](move ["ineqs"])))));
   (((repeat_tactic 1 9 (((use_arg_then2 ("ineqs", []))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andbT", [andbT]))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("E2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] []))))) THEN (move ["f"]) THEN (move ["f_in2"]));
   ((((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((fun arg_tac -> arg_tac (Arg_theorem (let_RULE faces_add_diag)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_DELETE", [IN_DELETE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_SING", [IN_SING]))(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("E2_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))));
   ((THENL_ROT (-1)) ((THENL) case [(move ["f_eqs"]); ((case THEN (move ["f_in"])) THEN (move ["_"]))]));
   (((((use_arg_then2 ("E2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("tauVEF_add_diag_eq", [tauVEF_add_diag_eq]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (((use_arg_then2 ("h_tau", [])) (disch_tac [])) THEN (clear_assumption "h_tau") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`#2.0 = &2 /\ &2 * h0 = #2.52`))) (term_tac (have_gen_tac [](move ["eqs"])))) ((((use_arg_then2 ("h0", [h0]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   (((use_arg_then2 ("f_eqs", [])) (disch_tac [])) THEN (clear_assumption "f_eqs") THEN case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))));
   (in_tac ["ineq_apex_flat"] false ((((use_arg_then2 ("real_gt", [real_gt]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("Ineq.apex_flat", [Ineq.apex_flat]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("INEQ_ALT", [INEQ_ALT]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("eqs", []))(thm_tac (new_rewrite [] [])))))));
   ((THENL_FIRST) ((((use_arg_then2 ("REAL_LT_IMP_LE", [REAL_LT_IMP_LE]))(thm_tac (new_rewrite [] [])))) THEN ((use_arg_then2 ("REAL_LT_TRANS", [REAL_LT_TRANS])) (thm_tac apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.103`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`v,w IN darts_k 3 H2`))) (term_tac (have_gen_tac [](move ["vw_in2_3"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("E2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dart3_tauVEF_eq_taum", [dart3_tauVEF_eq_taum]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y4'_eq_y4", [y4'_eq_y4]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("ys_split_eq", [ys_split_eq]))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("ineq_apex_flat", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then2 ("y1_lo", [y1_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y1_hi", [y1_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y3_lo", [y3_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_hi", [y3_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y5_lo", [y5_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_lo", [y6_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("y4'_lo_2h0", [y4'_lo_2h0])) (fun fst_arg -> (use_arg_then2 ("vw_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("ineq_tau4_extra", [ineq_tau4_extra])) (fun fst_arg -> (use_arg_then2 ("h_fan2", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("ineq_apex_flat", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("ineq_apex_A", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))));
   ((((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("w'w_in4", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("E2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE ys_split_eq2))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("E2_def", []))(thm_tac (new_rewrite [] [])))));
   (((repeat_tactic 1 9 (((use_arg_then2 ("ineqs", []))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("y4'_lo_2h0", [y4'_lo_2h0])) (fun fst_arg -> (use_arg_then2 ("vw_in5", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   (split_tac);
   ((((use_arg_then2 ("y8_fan", [y8_fan]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("ye_fan", [ye_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("o_THM", [o_THM])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`inverse _`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("POWER_2", [POWER_2]))(gsym_then (thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`2 = CARD (face H2 (w',w)) - 2`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [1] []))))))) ((((use_arg_then2 ("card2_w'w", []))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse]))(gsym_then (thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("fan2", [fan2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card2_w'w", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("w'w_in2", []))(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   ((((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("E2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_fan_pair_ext_power_add_diag))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac));
   (((((use_arg_then2 ("d2_def", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] [])))))) THEN (done_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`y4'_fan (V,E) d2 <= sqrt8`))) (term_tac (have_gen_tac []ALL_TAC))) ((ALL_TAC) THEN (done_tac)));
   ((repeat_tactic 1 9 (((use_arg_then2 ("y4'_fan", [y4'_fan]))(thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("d2_def", []))(gsym_then (thm_tac (new_rewrite [2] []))))) THEN (((use_arg_then2 ("POWER_2", [POWER_2]))(thm_tac (new_rewrite [] [])))) THEN (((fun arg_tac -> (use_arg_then2 ("o_THM", [o_THM])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`inverse _`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("o_ASSOC", [o_ASSOC]))(thm_tac (new_rewrite [] [])))));
   ((((fun arg_tac -> (use_arg_then2 ("PERMUTES_INVERSES_o", [PERMUTES_INVERSES_o])) (fun fst_arg -> (fun arg_tac -> (use_arg_then2 ("F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", [F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("o_THM", [o_THM]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("I_THM", [I_THM]))(thm_tac (new_rewrite [] [])))));
   ((((fun arg_tac -> (use_arg_then2 ("INVERSE_F_FAN_PAIR_EXT_EXPLICIT", [INVERSE_F_FAN_PAIR_EXT_EXPLICIT])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("res", [res]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_2_eq", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("w'w_in2", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then2 ("eq2", []))(thm_tac (new_rewrite [] [])))));
   (((((use_arg_then2 ("f_fan_pair_ext", [f_fan_pair_ext]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("vw_in", []))(thm_tac (new_rewrite [] [])))) THEN (simp_tac) THEN (((use_arg_then2 ("f_fan_pair", [f_fan_pair]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("FST", [FST]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("DIST_SYM", [DIST_SYM]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section Two_diags *)
let lp_tau_split5_two_diags = Sections.finalize_theorem lp_tau_split5_two_diags;;
Sections.end_section "Two_diags";;

(* Finalization of the section Split5 *)
let lp_tau_split5_one_diag = Sections.finalize_theorem lp_tau_split5_one_diag;;
let lp_tau_split5_two_diags = Sections.finalize_theorem lp_tau_split5_two_diags;;
Sections.end_section "Split5";;

(* Section Split6 *)
Sections.begin_section "Split6";;
(Sections.add_section_hyp "h_tau" (`lp_tau (V,E)`));;
(Sections.add_section_hyp "pent_pro" (JEJTVGB_pent_pro_concl));;
(Sections.add_section_hyp "ineq103" ((hd (Ineq.getexact "6988401556")).ineq));;

(* Lemma lp_tau_split6 *)
let lp_tau_split6 = Sections.section_proof ["d"]
`d IN darts_k 6 (hypermap_of_fan (V,E))
	==> y5_fan (V,E) d <= #2.52
	/\ y6_fan d <= #2.52
	/\ y8_fan (V,E) d <= #2.52
	/\ y9_fan (V,E) d <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 2) d) <= #2.52
	/\ ye_fan ((f_fan_pair_ext (V,E) POWER 3) d) <= #2.52
	/\ y4'_fan (V,E) d <= sqrt8
	==> lp_tau (V, add_diag (V,E) d)`
[
   (((use_arg_then2 ("d", [])) (disch_tac [])) THEN (clear_assumption "d") THEN case THEN (move ["v"]) THEN (move ["w"]) THEN (move ["vw_in6"]) THEN (move ["bounds"]));
   ((((use_arg_then2 ("vw_in6", [])) (disch_tac [])) THEN BETA_TAC) THEN (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH", []))(thm_tac (new_rewrite [] []))))) THEN ALL_TAC THEN (case THEN (move ["vw_in"])) THEN (move ["card_f"])));
   ((((use_arg_then2 ("h_fan", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_fan", [lp_fan]))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["fanV"])) THEN (case THEN (move ["f_surr"])) THEN (case THEN (move ["diag"])) THEN (case THEN (move ["subV"])) THEN (move ["packV"])));
   ((((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] [])))) THEN (move ["f"]) THEN (move ["f_in2"]));
   ((((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN BETA_TAC) THEN ((((fun arg_tac -> arg_tac (Arg_theorem (let_RULE faces_add_diag)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_DELETE", [IN_DELETE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_SING", [IN_SING]))(thm_tac (new_rewrite [] []))))));
   ((fun arg_tac -> arg_tac (Arg_term (`sigma_fan _ V E v w`))) (term_tac (set_tac "w'")));
   ((THENL_ROT (-1)) ((THENL) case [(move ["f_eqs"]); ((case THEN (move ["f_in"])) THEN (move ["_"]))]));
   (((((use_arg_then2 ("tauVEF_add_diag_eq", [tauVEF_add_diag_eq]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (((use_arg_then2 ("h_tau", [])) (disch_tac [])) THEN (clear_assumption "h_tau") THEN BETA_TAC) THEN ((((use_arg_then2 ("lp_tau", [lp_tau]))(thm_tac (new_rewrite [] [])))) THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`3 < CARD (face (hypermap_of_fan (V,E)) (v,w))`))) (term_tac (have_gen_tac [](move ["card_f_gt3"])))) ((((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE face_split_fan_face_explicit))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("card_f_gt3", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("fan_split_fan_face", [fan_split_fan_face])) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun arg -> thm_tac MP_TAC arg THEN ALL_TAC));
   ((((use_arg_then2 ("split_fan_face_eq_add_diag", [split_fan_face_eq_add_diag]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`6 - 1 = 5`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] []))))))) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`hypermap_of_fan _`))) (term_tac (set_tac "H2")));
   (((use_arg_then2 ("f_in2", [])) (disch_tac [])) THEN (clear_assumption "f_in2") THEN BETA_TAC THEN (move ["_"]) THEN (case THEN (move ["fan2"])) THEN (case THEN (move ["dart2"])) THEN (move ["f_surr2"]) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["f2_vw"]) THEN (move ["f2_w'w"]))) THEN (case THEN ALL_TAC) THEN (case THEN ((move ["card2_vw"]) THEN (move ["card2_w'w"]))) THEN (move ["_"]));
   ((((fun arg_tac -> (use_arg_then2 ("COMPONENTS_HYPERMAP_OF_FAN", [COMPONENTS_HYPERMAP_OF_FAN])) (fun fst_arg -> (use_arg_then2 ("fan2", [fan2])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (disch_tac [])) THEN BETA_TAC) THEN ((((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN ALL_TAC THEN (case THEN (move ["dartH2"])) THEN (move ["_"])));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`#2.0 = &2 /\ &2 * h0 = #2.52`))) (term_tac (have_gen_tac [](move ["eqs"])))) ((((use_arg_then2 ("h0", [h0]))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   (in_tac ["ineq103"] false ((((use_arg_then2 ("real_gt", [real_gt]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("Ineq.apex_flat", [Ineq.apex_flat]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("INEQ_ALT", [INEQ_ALT]))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("eqs", []))(thm_tac (new_rewrite [] [])))))));
   (((use_arg_then2 ("f_eqs", [])) (disch_tac [])) THEN (clear_assumption "f_eqs") THEN case THEN (((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [] [])))));
   ((THENL_FIRST) ((((use_arg_then2 ("REAL_LT_IMP_LE", [REAL_LT_IMP_LE]))(thm_tac (new_rewrite [] [])))) THEN ((use_arg_then2 ("REAL_LT_TRANS", [REAL_LT_TRANS])) (thm_tac apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.103`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((fun arg_tac -> arg_tac (Arg_term (`v,w IN darts_k 3 H2`))) (term_tac (have_gen_tac [](move ["vw_in2_3"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("dart3_tauVEF_eq_taum", [dart3_tauVEF_eq_taum]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y4'_eq_y4", [y4'_eq_y4]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("ys_split_eq", [ys_split_eq]))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)));
   ((((use_arg_then2 ("ineq103", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((fun arg_tac -> arg_tac (Arg_theorem (GEN_ALL ALL)))(thm_tac (new_rewrite [] []))))) THEN (simp_tac) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((((fun arg_tac -> (use_arg_then2 ("y1_lo", [y1_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y1_hi", [y1_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_lo", [y2_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y2_hi", [y2_hi])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)));
   (((((use_arg_then2 ("y3_lo", [y3_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y3_hi", [y3_hi]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("y5_lo", [y5_lo]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (use_arg_then2 ("y6_lo", [y6_lo])) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("y4'_lo_2h0", [y4'_lo_2h0])) (fun fst_arg -> (use_arg_then2 ("vw_in6", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac))) THEN (arith_tac) THEN (done_tac));
   ((fun arg_tac -> arg_tac (Arg_term (`w',w IN darts_k 5 H2`))) (term_tac (have_gen_tac [](move ["w'w_in2_5"]))));
   (((((use_arg_then2 ("darts_k", [darts_k]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("inE", [inE]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dartH2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart1_eq", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
   ((THENL_FIRST) ((((use_arg_then2 ("REAL_LE_TRANS", [REAL_LE_TRANS])) (disch_tac [])) THEN (clear_assumption "REAL_LE_TRANS") THEN (DISCH_THEN apply_tac)) THEN ((fun arg_tac -> arg_tac (Arg_term (`#0.616`))) (term_tac exists_tac)) THEN (split_tac)) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("ineq_tau5_pro_tauVEF_aux", [ineq_tau5_pro_tauVEF_aux])) (fun fst_arg -> (use_arg_then2 ("pent_pro", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("w'w_in2_5", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("lp_fan_trans1", [lp_fan_trans1]))(thm_tac (new_rewrite [] [])))) THEN ((TRY done_tac)) THEN (repeat_tactic 1 9 (((use_arg_then2 ("andTb", [andTb]))(thm_tac (new_rewrite [] []))))));
   ((((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE ys_split_eq2))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN ((TRY done_tac)) THEN (((use_arg_then2 ("w'_def", []))(thm_tac (new_rewrite [] [])))) THEN (repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) ((((fun arg_tac -> (fun arg_tac -> (use_arg_then2 ("y4'_lo_2h0", [y4'_lo_2h0])) (fun fst_arg -> (use_arg_then2 ("vw_in6", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("h_fan", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] [])))) THEN (simp_tac)) ((arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("y8_fan", [y8_fan]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("y5_fan", [y5_fan]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("ye_fan", [ye_fan]))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((fun arg_tac -> (use_arg_then2 ("o_THM", [o_THM])) (fun fst_arg -> (fun arg_tac -> arg_tac (Arg_term (`inverse _`))) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("POWER_2", [POWER_2]))(gsym_then (thm_tac (new_rewrite [] []))))));
   ((THENL_FIRST) ((fun arg_tac -> arg_tac (Arg_term (`2 = CARD (face H2 (w',w)) - 3`))) (term_tac (have_gen_tac [](((conv_thm_tac DISCH_THEN)(thm_tac (new_rewrite [1] []))))))) ((((use_arg_then2 ("card2_w'w", []))(thm_tac (new_rewrite [] [])))) THEN (arith_tac) THEN (done_tac)));
   ((((use_arg_then2 ("H2_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (((use_arg_then2 ("f_fan_pair_ext_power_inverse", [f_fan_pair_ext_power_inverse]))(gsym_then (thm_tac (new_rewrite [] []))))));
   (((((use_arg_then2 ("fan2", [fan2]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("H2_def", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("card2_w'w", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("dart2", []))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_UNION", [IN_UNION]))(thm_tac (new_rewrite [] [])))) THEN (((use_arg_then2 ("IN_INSERT", [IN_INSERT]))(thm_tac (new_rewrite [] []))))) THEN (arith_tac) THEN (done_tac));
   (((((use_arg_then2 ("w'_def", []))(gsym_then (thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 1 9 (((fun arg_tac -> (fun arg_tac -> (fun arg_tac -> (fun arg_tac -> arg_tac (Arg_theorem (let_RULE f_fan_pair_ext_power_add_diag))) (fun fst_arg -> (use_arg_then2 ("fanV", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("f_surr", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg))) (fun fst_arg -> (use_arg_then2 ("vw_in", [])) (fun snd_arg -> combine_args_then arg_tac fst_arg snd_arg)))(thm_tac (new_rewrite [] []))))) THEN (repeat_tactic 0 10 (((use_arg_then2 ("card_f", []))(thm_tac (new_rewrite [] [])))))) THEN (TRY ((arith_tac))));
   ((repeat_tactic 1 9 (((use_arg_then2 ("bounds", []))(thm_tac (new_rewrite [] []))))) THEN (done_tac));
];;

(* Finalization of the section Split6 *)
let lp_tau_split6 = Sections.finalize_theorem lp_tau_split6;;
Sections.end_section "Split6";;

(* Finalization of the section Split *)
let lp_tau_split4_apex_flat = Sections.finalize_theorem lp_tau_split4_apex_flat;;
let lp_tau_split4_apex_sup_flat = Sections.finalize_theorem lp_tau_split4_apex_sup_flat;;
let lp_tau_split4 = Sections.finalize_theorem lp_tau_split4;;
let lp_tau_split5_one_diag = Sections.finalize_theorem lp_tau_split5_one_diag;;
let lp_tau_split5_two_diags = Sections.finalize_theorem lp_tau_split5_two_diags;;
let lp_tau_split6 = Sections.finalize_theorem lp_tau_split6;;
Sections.end_section "Split";;

(* Finalization of the section LpTau *)
let contravening_lp_tau = Sections.finalize_theorem contravening_lp_tau;;
let lp_tau_split4_apex_flat = Sections.finalize_theorem lp_tau_split4_apex_flat;;
let lp_tau_split4_apex_sup_flat = Sections.finalize_theorem lp_tau_split4_apex_sup_flat;;
let lp_tau_split4 = Sections.finalize_theorem lp_tau_split4;;
let lp_tau_split5_one_diag = Sections.finalize_theorem lp_tau_split5_one_diag;;
let lp_tau_split5_two_diags = Sections.finalize_theorem lp_tau_split5_two_diags;;
let lp_tau_split6 = Sections.finalize_theorem lp_tau_split6;;
Sections.end_section "LpTau";;

(* Close the module *)
end;;