module Jotswix = struct

  open Hales_tactic;;


let LET_THM = CONJ LET_DEF LET_END_DEF;;
let UNDISCH2 = repeat UNDISCH;;


(*
let get_work = 
  let is_main = function 
    | Main_estimate -> true
    | _ -> false in
  let has_main ind = 
    exists (is_main) ind.tags in
  let main_ineq_data1 = 
    filter has_main (!Ineq.ineqs) in
  let id = map (fun t-> t.idv) main_ineq_data1 in
    (* debug 
  let id = nonlinear_ineq_terminal in *)
  let main_ineq_data = map (fun t -> hd(Ineq.getexact t)) id in
  let ineql = map (fun ind -> ind.ineq) main_ineq_data in
  let sl = map (fun ind -> ind.idv) main_ineq_data in
  let main_ineq_conj = end_itlist (curry mk_conj) ineql in
  
let th = new_definition (mk_eq (`main_work4:bool`,main_ineq_conj)) in
  let th1 = UNDISCH (MATCH_MP (TAUT `(a <=> b) ==> (a ==> b)`) th) in
  let co1 thm = if (is_conj (concl thm)) then CONJUNCT1 thm else thm in
  let tryindex s sl = try index s sl with  _ -> report s; failwith s in
    fun s ->
      let i = tryindex s sl in
      let th2 = funpow i CONJUNCT2 th1 in
	co1 th2;;
*)

let taum_attains_inf = prove_by_refinement(
  (* main_work4 ==> *) `main_nonlinear_terminal_v11 ==> (!y1 y2 y3 y4 y5 y6. &2 <= y1 /\ y1 <= &2 * h0 /\
	&2 <= y2 /\ y2 <= &2 * h0 /\
	&2 <= y3 /\ y3 <= &2 * h0 /\
	&2 <= y4 /\ y4 <= &2 * h0 /\
	&2 <= y5 /\ y5 <= &2 * h0 /\
	&2 <= y6 /\ y6 <= &2 * h0 
	==>
	(?y4'. &2 <= y4' /\ y4' <= &2 * h0 /\ (!y4. &2 <= y4 /\ y4 <= &2 * h0 ==>
	   taum y1 y2 y3 y4' y5 y6 <= taum y1 y2 y3 y4 y5 y6)))`,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC REAL_CONTINUOUS_ATTAINS_INF [`\q. taum y1 y2 y3 q y5 y6`;`real_interval [&2, &2 * h0]`];
  ANTS_TAC;
    REWRITE_TAC[REAL_COMPACT_INTERVAL];
    CONJ_TAC;
      REWRITE_TAC[EXTENSION;NOT_IN_EMPTY;IN_REAL_INTERVAL;NOT_FORALL_THM;Sphere.h0];
      TYPIFY `&2` EXISTS_TAC;
      BY(REAL_ARITH_TAC);
    REWRITE_TAC[REAL_CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN];
    REWRITE_TAC[IN_REAL_INTERVAL];
    REPEAT WEAKER_STRIP_TAC;
    MATCH_MP_TAC REAL_CONTINUOUS_ATREAL_WITHINREAL;
    MATCH_MP_TAC Bkossge.real_continuous_taum;
    ASM_SIMP_TAC[arith `&2 <= y ==> &0 < y`];
    CONJ_TAC;
      INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "4717061266") [`y1`;`y2`;`y3`;`x`;`y5`;`y6`];
      REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
      REWRITE_TAC[arith `x > &0 <=> &0 < x`] THEN DISCH_THEN MATCH_MP_TAC;
      BY(ASM_REWRITE_TAC[]);
    BY(REPEAT CONJ_TAC THEN MATCH_MP_TAC Ysskqoy.TRI_UPS_X_STRICT_POS THEN ASM_TAC THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
  REWRITE_TAC[IN_REAL_INTERVAL];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `x` EXISTS_TAC;
  BY(ASM_MESON_TAC[])
  ]);;
  (* }}} *)

let TAUM_EXTREMAL = prove_by_refinement(
  (* main_work4 ==> *) ` main_nonlinear_terminal_v11 ==> (!y1 y2 y3 y4 y5 y6.
	&2 <= y1 /\ y1 <= &2 * h0 /\
	&2 <= y2 /\ y2 <= &2 * h0 /\
	&2 <= y3 /\ y3 <= &2 * h0 /\
	&2 <= y4 /\ y4 <= &2 * h0 /\
	&2 <= y5 /\ y5 <= &2 * h0 /\
	&2 <= y6 /\ y6 <= &2 * h0 
	==>
	(?y4'. (y4' = &2 * h0 \/ y4' = &2) /\
	   taum y1 y2 y3 y4' y5 y6 <= taum y1 y2 y3 y4 y5 y6)
				  )`,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC (UNDISCH2 taum_attains_inf) [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `y4'` EXISTS_TAC;
  CONJ2_TAC;
    FIRST_X_ASSUM MATCH_MP_TAC;
    BY(ASM_REWRITE_TAC[]);
  PROOF_BY_CONTR_TAC;
  TYPIFY `&2 < y4' /\ y4' < &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_TAC THEN REAL_ARITH_TAC);
  INTRO_TAC (UNDISCH Ocbicby.LEMMA_1834976363) [`y1`;`y2`;`y3`;`y4'`;`y5`;`y6`;`&2`;`&2 * h0`];
  ANTS_TAC;
    ASM_REWRITE_TAC[SUBSET;IN_REAL_INTERVAL;IN_ELIM_THM];
    ASM_SIMP_TAC [arith `&2 <= y ==> &0 < y`];
    INTRO_TAC Ocbicby.xrr_simple_lower_bound [`y2`;`y3`;`y4'`;`&2`];
    INTRO_TAC Ocbicby.xrr_simple_lower_bound [`y1`;`y3`;`y5`;`&2`];
    INTRO_TAC Ocbicby.xrr_simple_lower_bound [`y1`;`y2`;`y6`;`&2`];
    ASM_REWRITE_TAC[arith `x <= x`];
    REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[];
    CONJ_TAC;
      INTRO_TAC Ocbicby.xrr_simple_upper_bound [`y2`;`y3`;`y4'`;`&2 * h0`];
      ANTS_TAC;
        BY(ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
      BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
    CONJ2_TAC;
      BY(CONJ_TAC THEN MATCH_MP_TAC Bkossge.UPS_X_STD_POS THEN ASM_REWRITE_TAC[]);
    REPEAT WEAKER_STRIP_TAC;
    ASM_SIMP_TAC[arith `&2 < x ==> &0 < x`];
    CONJ2_TAC;
      MATCH_MP_TAC Bkossge.UPS_X_STD_POS THEN ASM_REWRITE_TAC[];
      BY(ASM_TAC THEN REAL_ARITH_TAC);
    INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "4717061266") [`y1`;`y2`;`y3`;`x`;`y5`;`y6`];
    REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
    REWRITE_TAC[arith `x > &0 <=> &0 < x`] THEN DISCH_THEN MATCH_MP_TAC;
    (ASM_REWRITE_TAC[]);
    BY(REPLICATE_TAC 2 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  REWRITE_TAC[IN_REAL_INTERVAL];
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM (C INTRO_TAC [`y4''`]);
  ANTS_TAC;
    BY(ASM_TAC THEN REAL_ARITH_TAC);
  BY(ASM_REWRITE_TAC[arith `~(x <= y) <=> (y < x)`])
  ]);;
  (* }}} *)

let TAUM_STD_POS = prove_by_refinement(
  (* main_work4 ==> *) ` main_nonlinear_terminal_v11 ==> (!y1 y2 y3 y4 y5 y6.
	&2 <= y1 /\ y1 <= &2 * h0 /\
	&2 <= y2 /\ y2 <= &2 * h0 /\
	&2 <= y3 /\ y3 <= &2 * h0 /\
	&2 <= y4 /\ y4 <= &2 * h0 /\
	&2 <= y5 /\ y5 <= &2 * h0 /\
	&2 <= y6 /\ y6 <= &2 * h0 
	==>
	(&0 <= taum y1 y2 y3 y4 y5 y6))`,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[arith `x <= y <=> ~(y < x)`];
  DISCH_TAC;
  TYPIFY `&2 * h0 <= sqrt8 /\ #2.52 = &2 * h0 /\ &2 <= &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REWRITE_TAC[Sphere.h0] THEN MP_TAC Flyspeck_constants.bounds THEN REAL_ARITH_TAC);
  COMMENT "y4";
  INTRO_TAC (UNDISCH2 TAUM_EXTREMAL) [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM (DISJ_CASES_TAC);
    REPEAT (FIRST_X_ASSUM_ST `taum` MP_TAC) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
    INTRO_TAC ( Terminal.get_main_nonlinear "5541487347") [`y2`;`y3`;`y1`;`y5`;`y6`;`y4'`];
    REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
    DISCH_THEN MP_TAC THEN ANTS_TAC;
      BY(ASM_REWRITE_TAC[arith `x <= x`]);
    TYPIFY `taum y2 y3 y1 y5 y6 y4' = taum y1 y2 y3 y4' y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
      BY(MESON_TAC[Terminal.taum_sym]);
    ASM_REWRITE_TAC[];
    BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  TYPIFY `taum y1 y2 y3 (&2) y5 y6 < &0` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM_ST `taum` MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_TAC THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM MP_TAC;
  POP_ASSUM kill;
  REPEAT (FIRST_X_ASSUM_ST `taum` kill);
  DISCH_TAC;
  COMMENT "y5";
  INTRO_TAC (UNDISCH2 TAUM_EXTREMAL) [`y2`;`y3`;`y1`;`y5`;`y6`;`(&2)`];
  ASM_REWRITE_TAC[arith `x <= x`];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `taum y1 y2 y3 (&2) y5 y6 = taum y2 y3 y1 y5 y6 (&2) /\ taum y3 y1 y2 y6 (&2) y4' = taum y2 y3 y1 y4' y6 (&2)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[Terminal.taum_sym]);
  FIRST_X_ASSUM (DISJ_CASES_TAC);
    REPEAT (FIRST_X_ASSUM_ST `taum` MP_TAC) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
    INTRO_TAC ( Terminal.get_main_nonlinear "5541487347") [`y3`;`y1`;`y2`;`y6`;`&2`;`y4'`];
    REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
    DISCH_THEN MP_TAC THEN ANTS_TAC;
      BY(ASM_REWRITE_TAC[arith `x <= x`]);
    ASM_REWRITE_TAC[];
    BY(REPLICATE_TAC 4 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  TYPIFY `taum y1 y2 y3 (&2) (&2) y6 < &0` (C SUBGOAL_THEN ASSUME_TAC);
    TYPIFY `taum y1 y2 y3 (&2) (&2) y6 = taum  y2 y3 y1 (&2) y6 (&2)` (C SUBGOAL_THEN SUBST1_TAC);
      BY(MESON_TAC[Terminal.taum_sym]);
    BY(REPEAT (FIRST_X_ASSUM_ST `taum` MP_TAC) THEN ASM_REWRITE_TAC[] THEN ASM_TAC THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM MP_TAC;
  POP_ASSUM kill;
  REPEAT (FIRST_X_ASSUM_ST `taum` kill);
  DISCH_TAC;
  COMMENT "y6";
  INTRO_TAC (UNDISCH2 TAUM_EXTREMAL) [`y3`;`y1`;`y2`;`y6`;`(&2)`;`(&2)`];
  ASM_REWRITE_TAC[arith `x <= x`];
  REPEAT WEAKER_STRIP_TAC;
  POP_ASSUM MP_TAC;
  TYPIFY_GOAL_THEN `!z1 z2 z3. taum y3 y1 y2 z3 z1 z2 = taum y1 y2 y3 z1 z2 z3` (unlist REWRITE_TAC);
    BY(MESON_TAC[Terminal.taum_sym]);
  DISCH_TAC;
  FIRST_X_ASSUM (DISJ_CASES_TAC);
    REPEAT (FIRST_X_ASSUM_ST `taum` MP_TAC) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
    INTRO_TAC ( Terminal.get_main_nonlinear "5541487347") [`y1`;`y2`;`y3`;`&2`;`&2`;`&2 * h0`];
    REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
    DISCH_THEN MP_TAC THEN ANTS_TAC;
      BY(ASM_REWRITE_TAC[arith `x <= x`]);
    BY(REPLICATE_TAC 4 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  TYPIFY `taum y1 y2 y3 (&2) (&2) (&2) < &0` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (FIRST_X_ASSUM_ST `taum` MP_TAC) THEN ASM_REWRITE_TAC[] THEN ASM_TAC THEN REAL_ARITH_TAC);
  INTRO_TAC ( Terminal.get_main_nonlinear "OMKYNLT 3336871894") [`y1`;`y2`;`y3`;`&2`;`&2`;`&2`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  ASM_REWRITE_TAC[arith `#2.0 = &2`];
  BY(POP_ASSUM MP_TAC THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let LFLACKU = prove_by_refinement(
  (* main_work4 ==> *) ` main_nonlinear_terminal_v11 ==>
   scs_arrow_v39 {scs_3I1} {}`,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REPEAT (GMATCH_SIMP_TAC Ocbicby.pos_imp_scs_arrow_empty);
  REWRITE_TAC[Appendix.scs_3I1;Appendix.mk_unadorned_v39];
  MATCH_MP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] Terminal.taustar_taum_dfun);
  REWRITE_TAC[Terminal.FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.is_ear_v39;Appendix.scs_v39_explicit;EMPTY_GSPEC;NOT_INSERT_EMPTY];
  REWRITE_TAC[Appendix.d_tame];
  REWRITE_TAC[Appendix.cs_adj;Terminal.FUNLIST_EXPLICIT];
  NUM_REDUCE_TAC;
  REWRITE_TAC[arith `x <= x`];
  TYPIFY_GOAL_THEN `&2 * h0 <= #3.62` (unlist REWRITE_TAC);
    BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[arith `&0 +  #0.1 * -- &1 * (&0 + &0 + &0 + &0) = &0`];
  MATCH_MP_TAC (UNDISCH2 TAUM_STD_POS);
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)


let s_init_list_alt = 
prove_by_refinement(
  `s_init_list_v39 = [scs_6I1;scs_5I1;scs_4I1;scs_3I1;scs_5I2;scs_4I2;scs_5I3;scs_4I3]`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.s_init_list_v39];
  REPEAT LET_TAC;
  ASM_REWRITE_TAC[CONS_11;Appendix.scs_6I1;Appendix.scs_5I1;Appendix.scs_4I1;Appendix.scs_3T1;Appendix.scs_5I2;Appendix.scs_5I3;Appendix.scs_4I3;Appendix.scs_3I1;Appendix.scs_4I2];
  TYPIFY `(!a1 a2 a3. (a_pro 5 a1 a2 a3 = (funlist_v39   [(0,1),a1; (0,2),a3; (0,3),a3; (1,3),a3; (1,4),a3;   (2,4),  a3]  a2 5))) /\ (!a1 a2 a3. a_pro 4 a1 a2 a3  = (funlist_v39 [(0,1),a1; (0,2),a3; (1,3),a3] a2 4))` ENOUGH_TO_SHOW_TAC;
    BY(DISCH_THEN (unlist REWRITE_TAC));
  CONJ_TAC;
    REPEAT WEAKER_STRIP_TAC;
    REWRITE_TAC[FUN_EQ_THM];
    MATCH_MP_TAC Terminal.periodic2_mod_reduce;
    TYPIFY `5` EXISTS_TAC;
    REWRITE_TAC[arith `~(5 =0)`];
    CONJ_TAC;
      REWRITE_TAC[Appendix.periodic2];
      EXPAND_TAC "a_pro";
      ASM_REWRITE_TAC[Oxl_2012.MOD_ADD_CANCEL;arith `SUC (j+5) = (SUC j) + 5`];
      BY(MESON_TAC[Terminal.periodic2_funlist;Appendix.periodic2]);
    REWRITE_TAC[arith `x < 5 <=> x = 0 \/ x = 1 \/ x = 2 \/ x = 3 \/ x = 4`];
    REWRITE_TAC[TAUT ` ((a \/ b) /\ c) <=> ((a /\ c) \/ (b /\ c))`;TAUT `(a /\ (b \/ c)) <=> ((a /\ b) \/ (a /\ c))`];
    EXPAND_TAC "a_pro";
    REWRITE_TAC[Geomdetail.PAIR_EQ_EXPAND];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[Terminal.FUNLIST_EXPLICIT;Hexagons.PSORT_5_EXPLICIT] THEN NUM_REDUCE_TAC);
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[FUN_EQ_THM];
  MATCH_MP_TAC Terminal.periodic2_mod_reduce;
  TYPIFY `4` EXISTS_TAC;
  REWRITE_TAC[arith `~(4 =0)`];
  CONJ_TAC;
    REWRITE_TAC[Appendix.periodic2];
    EXPAND_TAC "a_pro";
    ASM_REWRITE_TAC[Oxl_2012.MOD_ADD_CANCEL;arith `SUC (j+4) = (SUC j) + 4`];
    BY(MESON_TAC[Terminal.periodic2_funlist;Appendix.periodic2]);
  REWRITE_TAC[arith `x < 4 <=> x = 0 \/ x = 1 \/ x = 2 \/ x = 3`];
  REWRITE_TAC[TAUT ` ((a \/ b) /\ c) <=> ((a /\ c) \/ (b /\ c))`;TAUT `(a /\ (b \/ c)) <=> ((a /\ b) \/ (a /\ c))`];
  EXPAND_TAC "a_pro";
  REWRITE_TAC[Geomdetail.PAIR_EQ_EXPAND];
  BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[Terminal.FUNLIST_EXPLICIT] THEN NUM_REDUCE_TAC)
  ]);;
  (* }}} *)

(* moved to JEJTVGB.hl
let JEJTVGB_case_breakdown = prove_by_refinement(
  (* main_work4 ==> *) ` main_nonlinear_terminal_v11 ==> (
  scs_arrow_v39 { scs_6I1 } { scs_6T1, scs_5M1, scs_4M2, scs_3T1 } /\ // OEHDBEN
    scs_arrow_v39 { scs_5I1 } { scs_stab_diag_v39 scs_5I1 0 2 , scs_5M2 } /\ // OTMTOTJ1
scs_arrow_v39 { scs_5I2 } { scs_stab_diag_v39 scs_5I2 0 2 , scs_5M2 } /\ // 

scs_arrow_v39 { scs_5I3 } { scs_stab_diag_v39 scs_5M1 0 2 , 
						    scs_stab_diag_v39 scs_5M1 0 3, scs_stab_diag_v39 scs_5M1 2 4, scs_5M2 } /\ //  OTMTOTJ3

scs_arrow_v39 { scs_5M1 } { scs_stab_diag_v39 scs_5M1 0 2 , 
						    scs_stab_diag_v39 scs_5M1 0 3, scs_stab_diag_v39 scs_5M1 2 4, scs_5M2 } /\ // OTMTOTJ4

scs_arrow_v39 { scs_5M2 } { scs_3T1,scs_3T4,scs_4M6',scs_4M7,scs_4M8,
						   scs_5T1, scs_stab_diag_v39 scs_5I2 0 2 , 
						   scs_stab_diag_v39 scs_5M1 0 2 ,     scs_stab_diag_v39 scs_5M1 0 3, scs_stab_diag_v39 scs_5M1 2 4 } /\ // HIJQAHA

scs_arrow_v39 { scs_stab_diag_v39 scs_5I1 0 2 }
    {scs_4M2, scs_3M1 } /\ //  CNICGSF1 ..

scs_arrow_v39 { scs_stab_diag_v39 scs_5I2 0 2 }
    {scs_4M3', scs_3T1 } /\ //  CNICGSF2

scs_arrow_v39 { scs_stab_diag_v39 scs_5M1 0 2 }
    {scs_4M2, scs_3T4 } /\ //  CNICGSF3

scs_arrow_v39 { scs_stab_diag_v39 scs_5M1 0 3 }
    {scs_4M4', scs_3M1 } /\ //  CNICGSF4

scs_arrow_v39 { scs_stab_diag_v39 scs_5M1 2 4 }
    {scs_4M5', scs_3M1 } /\ //  CNICGSF5 ..

  scs_arrow_v39 { scs_4I1 } {scs_4I2, scs_stab_diag_v39 scs_4I1 0 2 } /\ // FYSSVEV


  scs_arrow_v39 { scs_4I2 } { scs_4T1, scs_4T2 } /\ // ARDBZYE


  scs_arrow_v39 { scs_stab_diag_v39 scs_4I1 0 2 } { scs_3M1 } /\ // AUEAHEH

  // scs_arrow_v39 { scs_stab_diag_v39 scs_4I3 0 2 } { scs_4T4 } /\ // ZNLLLDL (internal)

  scs_arrow_v39 { scs_4I3 } { scs_4T4, scs_4M6' } /\ // VQFYMZY ..

//  scs_arrow_v39 { scs_4M1 } { scs_4M2, scs_4M6' } /\ // unused.

  scs_arrow_v39 { scs_4M2 } { scs_3M1, scs_3T4, scs_4M6' } /\ //  BNAWVNH

  scs_arrow_v39 { scs_4M3' } { scs_3T1, scs_3T6', scs_4M6' } /\ // RAWZDIB

  scs_arrow_v39 { scs_4M4' } { scs_3M1, scs_3T4, scs_3T3, scs_4M7 } /\ //  MFKLVDK

  scs_arrow_v39 { scs_4M5' } { scs_3T4, scs_4M8 } /\ // RYPDIXT

scs_arrow_v39 {scs_4M6'} {scs_4T3,scs_4T5} /\ // NWDGKXH

scs_arrow_v39 {scs_4M7} {scs_3M1,scs_3T3,scs_3T4} /\ // YOBIMPP

scs_arrow_v39 {scs_4M8} {scs_4M6',scs_3T7,scs_3T4} /\ // MIQMCSN

    scs_arrow_v39 {scs_3M1} {scs_3T1,scs_3T5} // BKOSSGE

==> 
JEJTVGB_assume_v39)
`,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC (UNDISCH Ocbicby.OCBICBY) [];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `scs_arrow_v39 {scs_3M1} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_3T1, scs_3T5}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_4M6'} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_4T3, scs_4T5}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_4M7} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_3M1, scs_3T3, scs_3T4}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_4M8} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_4M6', scs_3T7, scs_3T4}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_4M5'} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_3T4, scs_4M8}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_4M4'} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{ scs_3M1, scs_3T4, scs_3T3, scs_4M7 }` EXISTS_TAC; (* revised *)
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_4M3'} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_3T1, scs_3T6', scs_4M6'}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_4M2} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{ scs_3M1, scs_3T4, scs_4M6' }` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_4I3} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_4T4, scs_4M6'}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_stab_diag_v39 scs_4I1 0 2} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_3M1}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_4I2}  {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_4T1, scs_4T2}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_4I1}  {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_4I2, scs_stab_diag_v39 scs_4I1 0 2}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_stab_diag_v39 scs_5M1 2 4}   {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_4M5', scs_3M1}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_stab_diag_v39 scs_5M1 0 3}   {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_4M4', scs_3M1}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_stab_diag_v39 scs_5M1 0 2} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY ` {scs_4M2, scs_3T4}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_stab_diag_v39 scs_5I2 0 2} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_4M3', scs_3T1}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_stab_diag_v39 scs_5I1 0 2} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY ` {scs_4M2, scs_3M1}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_5M2} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `      {scs_3T1, scs_3T4, scs_4M6', scs_4M7, scs_4M8, scs_5T1, scs_stab_diag_v39                                                              scs_5I2                                                              0                                                              2, scs_stab_diag_v39                                                                 scs_5M1                                                                 0                                                                 2, scs_stab_diag_v39                                                                    scs_5M1                                                                    0                                                                    3,       scs_stab_diag_v39 scs_5M1 2      4}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_5M1} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_stab_diag_v39 scs_5M1 0 2, scs_stab_diag_v39 scs_5M1 0 3,       scs_stab_diag_v39 scs_5M1 2      4, scs_5M2}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_5I3} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_stab_diag_v39 scs_5M1 0 2, scs_stab_diag_v39 scs_5M1 0 3,       scs_stab_diag_v39 scs_5M1 2      4, scs_5M2}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_5I2} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_stab_diag_v39 scs_5I2 0 2, scs_5M2}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_5I1} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY ` {scs_stab_diag_v39 scs_5I1 0 2, scs_5M2}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  TYPIFY `scs_arrow_v39 {scs_6I1} {}` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC SCS_ARROW_TRANS_SING;
    TYPIFY `{scs_6T1, scs_5M1, scs_4M2, scs_3T1}` EXISTS_TAC;
    ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
    BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
  COMMENT "final kill";
  MATCH_MP_TAC Ayqjtmd.EAPGLE;
  REWRITE_TAC[GSYM Ocbicby.scs_arrow_sing_empty];
  REWRITE_TAC[s_init_list_alt;MEM];
  REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[];
  BY(REWRITE_TAC[UNDISCH2 LFLACKU])
  ]);;
  (* }}} *)
*)

let MOD_4_CASES = 
prove_by_refinement(
  `!j p. j < 4 ==> j = (p+0) MOD 4 \/ j = (p+1) MOD 4 \/ j = (p+2) MOD 4 \/ j = (p+3) MOD 4`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `p MOD 4 < 4` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[DIVISION;arith `~(4 = 0)`]);
  TYPED_ABBREV_TAC `i = p MOD 4`;
  TYPIFY `(j = 0 \/ j = 1 \/ j = 2 \/ j = 3) /\ (i = 0 \/ i = 1 \/ i = 2 \/ i = 3)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_TAC THEN ARITH_TAC);
  TYPIFY_GOAL_THEN `!k. (p + k) MOD 4 = (i + k) MOD 4` (unlist REWRITE_TAC);
    GEN_TAC;
    ONCE_REWRITE_TAC[arith `(i:num) + j = j + i`];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    EXPAND_TAC "i";
    GMATCH_SIMP_TAC MOD_MOD_REFL;
    BY(ARITH_TAC);
  POP_ASSUM MP_TAC THEN REWRITE_TAC[TAUT ` ((a \/ b) /\ c) <=> ((a /\ c) \/ (b /\ c))`;TAUT `(a /\ (b \/ c)) <=> ((a /\ b) \/ (a /\ c))`];
  BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[Terminal.MOD_4_EXPLICIT])
  ]);;
  (* }}} *)

let ab4_assumption_reduction = 
prove_by_refinement(
  `!s p f v e.
    is_scs_v39 s /\
    scs_k_v39 s  = 4 /\
    BBs_v39 s v /\
    (!i j. scs_diag 4 i j  ==> scs_a_v39 s i j < dist(v i,v j) /\ &4 * h0 < scs_b_v39 s i j) /\
    deformation f (IMAGE v (:num)) (--e,e) /\
    &0 < e /\
    (!w t. ~(w = v p)  ==> (f w t = w)) /\
    (!t. abs t < e ==> dist (v (p + 3),f (v p) t) = dist (v (p + 3),v p)) /\
    scs_a_v39 s p (p + 1) < dist (v p,v (p + 1)) /\ dist (v p,v (p + 1)) < scs_b_v39 s p (p + 1) 
    ==>
    (!i j. ?e1. &0 < e1 /\ (scs_a_v39 s i j = dist(v i,v j) ==> 
	(!t. abs t < e1 ==> scs_a_v39 s i j <= dist(f (v i) t,f (v j) t)))) /\
   (!i j.
      ?e2. &0 < e2 /\
           (dist (v i,v j) = scs_b_v39 s i j
            ==> (!t. abs t < e2
                     ==> dist (f (v i) t,f (v j) t) <= scs_b_v39 s i j)))
    `,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  ASSUME_TAC (arith `~(4 = 0)`);
  TYPIFY `!i j.scs_a_v39 s (i MOD 4) (j MOD 4) = scs_a_v39 s i j  /\ scs_b_v39 s (i MOD 4) (j MOD 4) = scs_b_v39 s i j ` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC;
    ASM_REWRITE_TAC[Appendix.is_scs_v39];
    BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[Cuxvzoz.periodic2_MOD]);
  TYPIFY `!i. i MOD 4 < 4` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[DIVISION]);
  TYPIFY `periodic v 4` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN MESON_TAC[]);
  TYPIFY `!i. v (i MOD 4) = v i` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC (GSYM Oxl_def.periodic_mod);
    BY(ASM_REWRITE_TAC[]);
  INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`4`];
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  TYPIFY `!j. j < 4 ==> (j MOD 4 = j)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REWRITE_TAC[MOD_LT]);
  COMMENT "scs_a_v39";
  CONJ_TAC;
    MATCH_MP_TAC Terminal.periodic2_mod_reduce;
    TYPIFY `4` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      MATCH_MP_TAC Cuxvzoz.MOD_periodic2;
      BY(REPLICATE_TAC 7 (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[] THEN MESON_TAC[]);
    TYPIFY `!j. j < 4 ==> (?e1. &0 < e1 /\ (scs_a_v39 s p j = dist (v p,v j)           ==> (!t. abs t < e1                    ==> scs_a_v39 s p j <= dist (f (v p) t,f (v j) t))))` ENOUGH_TO_SHOW_TAC;
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `i = ( p MOD 4)` ASM_CASES_TAC;
        FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`j`]);
        ASM_REWRITE_TAC[];
        REPEAT WEAKER_STRIP_TAC;
        TYPIFY `e1` EXISTS_TAC;
        ASM_REWRITE_TAC[];
        TYPIFY_GOAL_THEN `scs_a_v39 s (p MOD 4) j = scs_a_v39 s p j ` (unlist REWRITE_TAC);
          BY(REPLICATE_TAC 12 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
        BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[]);
      TYPIFY `j = p MOD 4` ASM_CASES_TAC;
        FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`i`]);
        ASM_REWRITE_TAC[DIST_SYM];
        REPEAT WEAKER_STRIP_TAC;
        TYPIFY `e1` EXISTS_TAC;
        ASM_REWRITE_TAC[];
        TYPIFY_GOAL_THEN `scs_a_v39 s i (p MOD 4)  = scs_a_v39 s i p ` (unlist REWRITE_TAC);
          BY(REPLICATE_TAC 12 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
        BY(FIRST_X_ASSUM MP_TAC THEN FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN MESON_TAC[]);
      TYPIFY `&1` EXISTS_TAC;
      REWRITE_TAC[arith `&0 < &1`];
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `!w t. ~(w = v p) ==> f w t = w` ( (C FIRST_ASSUM_ST GMATCH_SIMP_TAC));
      TYPIFY `!w t. ~(w = v p) ==> f w t = w` ( (C FIRST_X_ASSUM_ST GMATCH_SIMP_TAC));
      FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
      nCONJ_TAC 2;
        BY(FIRST_X_ASSUM (C INTRO_TAC [`i`;`j`]) THEN MESON_TAC[]);
      ASM_REWRITE_TAC[];
      BY(REPLICATE_TAC 20 (POP_ASSUM MP_TAC) THEN MESON_TAC[]);
    REPEAT WEAKER_STRIP_TAC;
    INTRO_TAC MOD_4_CASES [`j`;`p`];
    ASM_REWRITE_TAC[arith `p + 0 = p`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[DIST_REFL];
          BY(TYPIFY `&1` EXISTS_TAC THEN MESON_TAC[arith `&0 <= &0 /\ &0 < &1`]);
        TYPIFY `&1` EXISTS_TAC THEN REWRITE_TAC[arith `&0 < &1`];
        BY(ASM_MESON_TAC[arith `x < y ==> ~(x = y)`]);
      FIRST_X_ASSUM_ST `scs_diag` (C INTRO_TAC [`p`;`p+2`]);
      TYPIFY `scs_diag 4 p (p+2)` ENOUGH_TO_SHOW_TAC;
        DISCH_THEN (unlist REWRITE_TAC);
        REPEAT WEAKER_STRIP_TAC;
        TYPIFY `&1` EXISTS_TAC THEN REWRITE_TAC[arith `&0 < &1`];
        BY(ASM_MESON_TAC[arith `x < y ==> ~(x = y)`]);
      REWRITE_TAC[arith `p+2 = SUC(SUC p)`];
      MATCH_MP_TAC Tfitskc.SCS_DIAG_2;
      BY(NUM_REDUCE_TAC);
    TYPIFY `e` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    DISCH_TAC;
    REPEAT WEAKER_STRIP_TAC;
    ASM_REWRITE_TAC[];
    REPEAT (FIRST_X_ASSUM_ST `abs` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[DIST_SYM];
    DISCH_THEN (SUBST1_TAC o GSYM);
    MATCH_MP_TAC (arith `(x = y) ==> x <= y`);
    REPEAT (AP_TERM_TAC ORELSE AP_THM_TAC);
    ONCE_REWRITE_TAC[EQ_SYM_EQ];
    FIRST_X_ASSUM MATCH_MP_TAC;
    FIRST_X_ASSUM_ST `v (i MOD 4) = v i` ((unlist ONCE_REWRITE_TAC) o GSYM);
    TYPIFY_GOAL_THEN `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` MATCH_MP_TAC;
      BY(ASM_MESON_TAC[]);
    nCONJ_TAC 2;
      TYPIFY `~((p + 3) MOD 4 = (p+0) MOD 4)` ENOUGH_TO_SHOW_TAC;
        BY(REWRITE_TAC[arith `p+0=p`]);
      GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
      BY(NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[Terminal.MOD_4_EXPLICIT]);
    BY(ASM_REWRITE_TAC[]);
  COMMENT "now the b end";
  MATCH_MP_TAC Terminal.periodic2_mod_reduce;
  TYPIFY `4` EXISTS_TAC;
  ASM_REWRITE_TAC[];
  CONJ_TAC;
    MATCH_MP_TAC Cuxvzoz.MOD_periodic2;
    BY(REPLICATE_TAC 7 (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[] THEN MESON_TAC[]);
  TYPIFY `!j. j < 4 ==> (?e1. &0 < e1 /\ (dist (v p,v j) = scs_b_v39 s p j           ==> (!t. abs t < e1                    ==> dist (f (v p) t,f (v j) t) <= scs_b_v39 s p j)))` ENOUGH_TO_SHOW_TAC;
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `i = ( p MOD 4)` ASM_CASES_TAC;
      FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`j`]);
      ASM_REWRITE_TAC[];
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `e1` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      TYPIFY_GOAL_THEN `scs_b_v39 s (p MOD 4) j = scs_b_v39 s p j ` (unlist REWRITE_TAC);
        BY(REPLICATE_TAC 12 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
      BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[]);
    TYPIFY `j = p MOD 4` ASM_CASES_TAC;
      FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`i`]);
      ASM_REWRITE_TAC[DIST_SYM];
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `e1` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      TYPIFY_GOAL_THEN `scs_b_v39 s i (p MOD 4)  = scs_b_v39 s i p ` (unlist REWRITE_TAC);
        BY(REPLICATE_TAC 12 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
      BY(FIRST_X_ASSUM MP_TAC THEN FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN MESON_TAC[]);
    TYPIFY `&1` EXISTS_TAC;
    REWRITE_TAC[arith `&0 < &1`];
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `!w t. ~(w = v p) ==> f w t = w` ( (C FIRST_ASSUM_ST GMATCH_SIMP_TAC));
    TYPIFY `!w t. ~(w = v p) ==> f w t = w` ( (C FIRST_X_ASSUM_ST GMATCH_SIMP_TAC));
    FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
    nCONJ_TAC 2;
      BY(FIRST_X_ASSUM (C INTRO_TAC [`i`;`j`]) THEN MESON_TAC[]);
    ASM_REWRITE_TAC[];
    BY(REPLICATE_TAC 20 (POP_ASSUM MP_TAC) THEN MESON_TAC[]);
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC MOD_4_CASES [`j`;`p`];
  ASM_REWRITE_TAC[arith `p + 0 = p`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[DIST_REFL];
        BY(TYPIFY `&1` EXISTS_TAC THEN MESON_TAC[arith `&0 <= &0 /\ &0 < &1`]);
      TYPIFY `&1` EXISTS_TAC THEN REWRITE_TAC[arith `&0 < &1`];
      BY(ASM_MESON_TAC[arith `x < y ==> ~(x = y)`]);
    FIRST_X_ASSUM_ST `scs_diag` (C INTRO_TAC [`p`;`p+2`]);
    TYPIFY_GOAL_THEN `scs_diag 4 p (p+2)` (unlist REWRITE_TAC);
      REWRITE_TAC[arith `p+2 = SUC(SUC p)`];
      MATCH_MP_TAC Tfitskc.SCS_DIAG_2;
      BY(NUM_REDUCE_TAC);
    TYPIFY `dist(v p,v (p+2)) <= &4 * h0` ENOUGH_TO_SHOW_TAC;
      BY(ASM_MESON_TAC[arith `x <= y /\ y < z ==> ~(x = z)`]);
    INTRO_TAC DIST_TRIANGLE [`v p`;`(vec 0):real^3`;`v (p+2)`];
    REWRITE_TAC[DIST_0];
    TYPIFY `!i. norm (v i) <= &2 * h0` ENOUGH_TO_SHOW_TAC;
      BY(DISCH_TAC THEN FIRST_ASSUM (C INTRO_TAC [`p`]) THEN FIRST_X_ASSUM (C INTRO_TAC [`p+2`]) THEN REAL_ARITH_TAC);
    FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
    BY(FIRST_X_ASSUM_ST `ball_annulus` MP_TAC THEN REWRITE_TAC[Terminal.IMAGE_SUBSET_IN;IN_UNIV] THEN MESON_TAC[Fnjlbxs.in_ball_annulus]);
  TYPIFY `e` EXISTS_TAC;
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  REPEAT WEAKER_STRIP_TAC;
  ASM_REWRITE_TAC[];
  REPEAT (FIRST_X_ASSUM_ST `abs` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[DIST_SYM];
  DISCH_THEN (SUBST1_TAC o GSYM);
  MATCH_MP_TAC (arith `(x = y) ==> x <= y`);
  REPEAT (AP_TERM_TAC ORELSE AP_THM_TAC);
  FIRST_X_ASSUM MATCH_MP_TAC;
  FIRST_X_ASSUM_ST `v (i MOD 4) = v i` ((unlist ONCE_REWRITE_TAC) o GSYM);
  TYPIFY_GOAL_THEN `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` MATCH_MP_TAC;
    BY(ASM_MESON_TAC[]);
  nCONJ_TAC 2;
    TYPIFY `~((p + 3) MOD 4 = (p+0) MOD 4)` ENOUGH_TO_SHOW_TAC;
      BY(REWRITE_TAC[arith `p+0=p`]);
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    BY(NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[Terminal.MOD_4_EXPLICIT]);
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let PERIODIC_INJ_MOD = 
prove_by_refinement(
  `!v k i j. ~(k= 0) /\ periodic (v:num->A) k /\ 
    (!i j. i < k /\ j < k /\ v i = v j ==> i = j)
	==>
    ((v i = v j) <=> (i MOD k = j MOD k))`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `!i. v (i MOD k) = v i` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC (GSYM Oxl_def.periodic_mod);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `    (!i j. i < (k:num) /\ j < k /\ ~(i = j) ==> ~(v i = v j))` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[]);
  MATCH_MP_TAC (TAUT `((~b ==> ~a ) /\ (b ==> a)) ==> (a = b)`);
  CONJ_TAC;
    DISCH_TAC;
    FIRST_X_ASSUM_ST `v (i MOD k) = v i` ((unlist ONCE_REWRITE_TAC) o GSYM);
    FIRST_X_ASSUM MATCH_MP_TAC;
    ASM_REWRITE_TAC[];
    BY(ASM_MESON_TAC[DIVISION]);
  BY(ASM_MESON_TAC[])
  ]);;
  (* }}} *)

let ab4_assumption_reduction2 = 
prove_by_refinement(
  `!s p f v e.
    is_scs_v39 s /\
    scs_k_v39 s  = 4 /\
    BBs_v39 s v /\
    (!i j. scs_diag 4 i j  ==> scs_a_v39 s i j < dist(v i,v j) /\ &4 * h0 < scs_b_v39 s i j) /\
    deformation f (IMAGE v (:num)) (--e,e) /\
    &0 < e /\
    (!w t. ~(w = v p)  ==> (f w t = w)) /\
    (!t. abs t < e ==> dist (v (p + 3),f (v p) t) = dist (v (p + 3),v p)) /\
    scs_a_v39 s p (p + 1) < dist (v p,v (p + 1)) /\
    (?e3. &0 < e3 /\ (!t. abs t < e3 ==> dist (f (v p) t,v (p + 1)) <= scs_b_v39 s p (p + 1)))
    ==>
    (!i j. ?e1. &0 < e1 /\ (scs_a_v39 s i j = dist(v i,v j) ==> 
	(!t. abs t < e1 ==> scs_a_v39 s i j <= dist(f (v i) t,f (v j) t)))) /\
   (!i j.
      ?e2. &0 < e2 /\
           (dist (v i,v j) = scs_b_v39 s i j
            ==> (!t. abs t < e2
                     ==> dist (f (v i) t,f (v j) t) <= scs_b_v39 s i j)))
    `,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM MP_TAC THEN BURY_MP_TAC;
  ASSUME_TAC (arith `~(4 = 0)`);
  TYPIFY `!i j.scs_a_v39 s (i MOD 4) (j MOD 4) = scs_a_v39 s i j  /\ scs_b_v39 s (i MOD 4) (j MOD 4) = scs_b_v39 s i j ` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC;
    ASM_REWRITE_TAC[Appendix.is_scs_v39];
    BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[Cuxvzoz.periodic2_MOD]);
  TYPIFY `!i. i MOD 4 < 4` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[DIVISION]);
  TYPIFY `periodic v 4` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN MESON_TAC[]);
  TYPIFY `!i. v (i MOD 4) = v i` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC (GSYM Oxl_def.periodic_mod);
    BY(ASM_REWRITE_TAC[]);
  INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`4`];
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  TYPIFY `!j. j < 4 ==> (j MOD 4 = j)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REWRITE_TAC[MOD_LT]);
  COMMENT "scs_a_v39";
  CONJ_TAC;
    MATCH_MP_TAC Terminal.periodic2_mod_reduce;
    TYPIFY `4` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      MATCH_MP_TAC Cuxvzoz.MOD_periodic2;
      BY(REPLICATE_TAC 7 (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[] THEN MESON_TAC[]);
    TYPIFY `!j. j < 4 ==> (?e1. &0 < e1 /\ (scs_a_v39 s p j = dist (v p,v j)           ==> (!t. abs t < e1                    ==> scs_a_v39 s p j <= dist (f (v p) t,f (v j) t))))` ENOUGH_TO_SHOW_TAC;
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `i = ( p MOD 4)` ASM_CASES_TAC;
        FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`j`]);
        ASM_REWRITE_TAC[];
        REPEAT WEAKER_STRIP_TAC;
        TYPIFY `e1` EXISTS_TAC;
        ASM_REWRITE_TAC[];
        TYPIFY_GOAL_THEN `scs_a_v39 s (p MOD 4) j = scs_a_v39 s p j ` (unlist REWRITE_TAC);
          BY(REPLICATE_TAC 12 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
        BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[]);
      TYPIFY `j = p MOD 4` ASM_CASES_TAC;
        FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`i`]);
        ASM_REWRITE_TAC[DIST_SYM];
        REPEAT WEAKER_STRIP_TAC;
        TYPIFY `e1` EXISTS_TAC;
        ASM_REWRITE_TAC[];
        TYPIFY_GOAL_THEN `scs_a_v39 s i (p MOD 4)  = scs_a_v39 s i p ` (unlist REWRITE_TAC);
          BY(REPLICATE_TAC 12 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
        BY(FIRST_X_ASSUM MP_TAC THEN FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN MESON_TAC[]);
      TYPIFY `&1` EXISTS_TAC;
      REWRITE_TAC[arith `&0 < &1`];
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `!w t. ~(w = v p) ==> f w t = w` ( (C FIRST_ASSUM_ST GMATCH_SIMP_TAC));
      TYPIFY `!w t. ~(w = v p) ==> f w t = w` ( (C FIRST_X_ASSUM_ST GMATCH_SIMP_TAC));
      FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
      nCONJ_TAC 2;
        BY(FIRST_X_ASSUM (C INTRO_TAC [`i`;`j`]) THEN MESON_TAC[]);
      ASM_REWRITE_TAC[];
      BY(REPLICATE_TAC 20 (POP_ASSUM MP_TAC) THEN MESON_TAC[]);
    REPEAT WEAKER_STRIP_TAC;
    INTRO_TAC MOD_4_CASES [`j`;`p`];
    ASM_REWRITE_TAC[arith `p + 0 = p`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[DIST_REFL];
          BY(TYPIFY `&1` EXISTS_TAC THEN MESON_TAC[arith `&0 <= &0 /\ &0 < &1`]);
        TYPIFY `&1` EXISTS_TAC THEN REWRITE_TAC[arith `&0 < &1`];
        BY(ASM_MESON_TAC[arith `x < y ==> ~(x = y)`]);
      FIRST_X_ASSUM_ST `scs_diag` (C INTRO_TAC [`p`;`p+2`]);
      TYPIFY `scs_diag 4 p (p+2)` ENOUGH_TO_SHOW_TAC;
        DISCH_THEN (unlist REWRITE_TAC);
        REPEAT WEAKER_STRIP_TAC;
        TYPIFY `&1` EXISTS_TAC THEN REWRITE_TAC[arith `&0 < &1`];
        BY(ASM_MESON_TAC[arith `x < y ==> ~(x = y)`]);
      REWRITE_TAC[arith `p+2 = SUC(SUC p)`];
      MATCH_MP_TAC Tfitskc.SCS_DIAG_2;
      BY(NUM_REDUCE_TAC);
    TYPIFY `e` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    DISCH_TAC;
    REPEAT WEAKER_STRIP_TAC;
    ASM_REWRITE_TAC[];
    REPEAT (FIRST_X_ASSUM_ST `abs` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[DIST_SYM];
    DISCH_TAC;
    DISCH_THEN (SUBST1_TAC o GSYM);
    MATCH_MP_TAC (arith `(x = y) ==> x <= y`);
    REPEAT (AP_TERM_TAC ORELSE AP_THM_TAC);
    ONCE_REWRITE_TAC[EQ_SYM_EQ];
    FIRST_X_ASSUM MATCH_MP_TAC;
    FIRST_X_ASSUM_ST `v (i MOD 4) = v i` ((unlist ONCE_REWRITE_TAC) o GSYM);
    TYPIFY_GOAL_THEN `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` MATCH_MP_TAC;
      BY(ASM_MESON_TAC[]);
    nCONJ_TAC 2;
      TYPIFY `~((p + 3) MOD 4 = (p+0) MOD 4)` ENOUGH_TO_SHOW_TAC;
        BY(REWRITE_TAC[arith `p+0=p`]);
      GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
      BY(NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[Terminal.MOD_4_EXPLICIT]);
    BY(ASM_REWRITE_TAC[]);
  COMMENT "now the b end";
  MATCH_MP_TAC Terminal.periodic2_mod_reduce;
  TYPIFY `4` EXISTS_TAC;
  ASM_REWRITE_TAC[];
  CONJ_TAC;
    MATCH_MP_TAC Cuxvzoz.MOD_periodic2;
    BY(REPLICATE_TAC 7 (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[] THEN MESON_TAC[]);
  TYPIFY `!j. j < 4 ==> (?e1. &0 < e1 /\ (dist (v p,v j) = scs_b_v39 s p j           ==> (!t. abs t < e1                    ==> dist (f (v p) t,f (v j) t) <= scs_b_v39 s p j)))` ENOUGH_TO_SHOW_TAC;
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `i = ( p MOD 4)` ASM_CASES_TAC;
      FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`j`]);
      ASM_REWRITE_TAC[];
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `e1` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      TYPIFY_GOAL_THEN `scs_b_v39 s (p MOD 4) j = scs_b_v39 s p j ` (unlist REWRITE_TAC);
        BY(REPLICATE_TAC 12 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
      BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[]);
    TYPIFY `j = p MOD 4` ASM_CASES_TAC;
      FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`i`]);
      ASM_REWRITE_TAC[DIST_SYM];
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `e1` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      TYPIFY_GOAL_THEN `scs_b_v39 s i (p MOD 4)  = scs_b_v39 s i p ` (unlist REWRITE_TAC);
        BY(REPLICATE_TAC 12 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
      BY(FIRST_X_ASSUM MP_TAC THEN FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN MESON_TAC[]);
    TYPIFY `&1` EXISTS_TAC;
    REWRITE_TAC[arith `&0 < &1`];
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `!w t. ~(w = v p) ==> f w t = w` ( (C FIRST_ASSUM_ST GMATCH_SIMP_TAC));
    TYPIFY `!w t. ~(w = v p) ==> f w t = w` ( (C FIRST_X_ASSUM_ST GMATCH_SIMP_TAC));
    FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
    nCONJ_TAC 2;
      BY(FIRST_X_ASSUM (C INTRO_TAC [`i`;`j`]) THEN MESON_TAC[]);
    ASM_REWRITE_TAC[];
    BY(REPLICATE_TAC 20 (POP_ASSUM MP_TAC) THEN MESON_TAC[]);
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC MOD_4_CASES [`j`;`p`];
  ASM_REWRITE_TAC[arith `p + 0 = p`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[DIST_REFL];
        BY(TYPIFY `&1` EXISTS_TAC THEN MESON_TAC[arith `&0 <= &0 /\ &0 < &1`]);
      TYPIFY `e3` EXISTS_TAC THEN REWRITE_TAC[arith `&0 < &1`];
      ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
      TYPIFY_GOAL_THEN `scs_b_v39 s p ((p + 1) MOD 4) = scs_b_v39 s p ((p + 1))` (unlist REWRITE_TAC);
        BY(ASM_MESON_TAC[]);
      TYPIFY `f (v(p+1)) t = v (p+1)` (C SUBGOAL_THEN SUBST1_TAC);
        FIRST_X_ASSUM MATCH_MP_TAC;
        GMATCH_SIMP_TAC PERIODIC_INJ_MOD;
        TYPIFY `4` EXISTS_TAC THEN ASM_REWRITE_TAC[];
        GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p + 0`];
        GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
        BY(NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[Terminal.MOD_4_EXPLICIT]);
      FIRST_X_ASSUM MATCH_MP_TAC;
      BY(ASM_REWRITE_TAC[]);
    FIRST_X_ASSUM_ST `scs_diag` (C INTRO_TAC [`p`;`p+2`]);
    TYPIFY_GOAL_THEN `scs_diag 4 p (p+2)` (unlist REWRITE_TAC);
      REWRITE_TAC[arith `p+2 = SUC(SUC p)`];
      MATCH_MP_TAC Tfitskc.SCS_DIAG_2;
      BY(NUM_REDUCE_TAC);
    TYPIFY `dist(v p,v (p+2)) <= &4 * h0` ENOUGH_TO_SHOW_TAC;
      BY(ASM_MESON_TAC[arith `x <= y /\ y < z ==> ~(x = z)`]);
    INTRO_TAC DIST_TRIANGLE [`v p`;`(vec 0):real^3`;`v (p+2)`];
    REWRITE_TAC[DIST_0];
    TYPIFY `!i. norm (v i) <= &2 * h0` ENOUGH_TO_SHOW_TAC;
      BY(DISCH_TAC THEN FIRST_ASSUM (C INTRO_TAC [`p`]) THEN FIRST_X_ASSUM (C INTRO_TAC [`p+2`]) THEN REAL_ARITH_TAC);
    FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
    BY(FIRST_X_ASSUM_ST `ball_annulus` MP_TAC THEN REWRITE_TAC[Terminal.IMAGE_SUBSET_IN;IN_UNIV] THEN MESON_TAC[Fnjlbxs.in_ball_annulus]);
  TYPIFY `e` EXISTS_TAC;
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  REPEAT WEAKER_STRIP_TAC;
  ASM_REWRITE_TAC[];
  REPEAT (FIRST_X_ASSUM_ST `abs` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[DIST_SYM];
  DISCH_TAC;
  DISCH_THEN (SUBST1_TAC o GSYM);
  MATCH_MP_TAC (arith `(x = y) ==> x <= y`);
  REPEAT (AP_TERM_TAC ORELSE AP_THM_TAC);
  FIRST_X_ASSUM MATCH_MP_TAC;
  FIRST_X_ASSUM_ST `v (i MOD 4) = v i` ((unlist ONCE_REWRITE_TAC) o GSYM);
  TYPIFY_GOAL_THEN `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` MATCH_MP_TAC;
    BY(ASM_MESON_TAC[]);
  nCONJ_TAC 2;
    TYPIFY `~((p + 3) MOD 4 = (p+0) MOD 4)` ENOUGH_TO_SHOW_TAC;
      BY(REWRITE_TAC[arith `p+0=p`]);
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    BY(NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[Terminal.MOD_4_EXPLICIT]);
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let ab4_assumption_reduction_sym = 
prove_by_refinement(
  `!s p f v e.
    is_scs_v39 s /\
    scs_k_v39 s  = 4 /\
    BBs_v39 s v /\
    (!i j. scs_diag 4 i j  ==> scs_a_v39 s i j < dist(v i,v j) /\ &4 * h0 < scs_b_v39 s i j) /\
    deformation f (IMAGE v (:num)) (--e,e) /\
    &0 < e /\
    (!w t. ~(w = v (p+1))  ==> (f w t = w)) /\
    (!t. abs t < e ==> dist (f (v (p + 1)) t ,(v (p+2))) = dist (v (p + 1),v (p+2))) /\
    scs_a_v39 s p (p + 1) < dist (v p,v (p + 1)) /\ dist (v p,v (p + 1)) < scs_b_v39 s p (p + 1) 
    ==>
    (!i j. ?e1. &0 < e1 /\ (scs_a_v39 s i j = dist(v i,v j) ==> 
	(!t. abs t < e1 ==> scs_a_v39 s i j <= dist(f (v i) t,f (v j) t)))) /\
   (!i j.
      ?e2. &0 < e2 /\
           (dist (v i,v j) = scs_b_v39 s i j
            ==> (!t. abs t < e2
                     ==> dist (f (v i) t,f (v j) t) <= scs_b_v39 s i j)))
    `,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  ASSUME_TAC (arith `~(4 = 0)`);
  TYPIFY `!i j.scs_a_v39 s (i MOD 4) (j MOD 4) = scs_a_v39 s i j  /\ scs_b_v39 s (i MOD 4) (j MOD 4) = scs_b_v39 s i j ` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC;
    ASM_REWRITE_TAC[Appendix.is_scs_v39];
    BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[Cuxvzoz.periodic2_MOD]);
  TYPIFY `!i. i MOD 4 < 4` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[DIVISION]);
  TYPIFY `periodic v 4` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN MESON_TAC[]);
  TYPIFY `!i. v (i MOD 4) = v i` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC (GSYM Oxl_def.periodic_mod);
    BY(ASM_REWRITE_TAC[]);
  INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`4`];
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[]);
  TYPIFY `!j. j < 4 ==> (j MOD 4 = j)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REWRITE_TAC[MOD_LT]);
  TYPIFY `!t. f (v (p+0)) t = v(p+0) /\ f (v (p+2)) t = v(p+2) /\ f (v (p+3)) t = v(p+3)` (C SUBGOAL_THEN MP_TAC);
    GEN_TAC THEN (FIRST_X_ASSUM_ST `~(w = v (p+1)) ==> a` (REPEAT o GMATCH_SIMP_TAC));
    INTRO_TAC PERIODIC_INJ_MOD [`v`;`4`];
    ASM_REWRITE_TAC[arith `~(4=0)`];
    DISCH_TAC THEN ASM_REWRITE_TAC[];
    REPEAT ( GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT);
    BY(NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[Terminal.MOD_4_EXPLICIT]);
  DISCH_TAC;
  COMMENT "scs_a_v39";
  CONJ_TAC;
    MATCH_MP_TAC Terminal.periodic2_mod_reduce;
    TYPIFY `4` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      MATCH_MP_TAC Cuxvzoz.MOD_periodic2;
      BY(ASM_REWRITE_TAC[]);
    TYPIFY `!j. j < 4 ==> (?e1. &0 < e1 /\ (scs_a_v39 s (p+1) j = dist (v (p+1),v j)           ==> (!t. abs t < e1                    ==> scs_a_v39 s (p+1) j <= dist (f (v (p+1)) t,f (v j) t))))` ENOUGH_TO_SHOW_TAC;
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `i = ( (p+1) MOD 4)` ASM_CASES_TAC;
        FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`j`]);
        ASM_REWRITE_TAC[];
        REPEAT WEAKER_STRIP_TAC;
        TYPIFY `e1` EXISTS_TAC;
        ASM_REWRITE_TAC[];
        TYPIFY_GOAL_THEN `scs_a_v39 s ((p+1) MOD 4) j = scs_a_v39 s (p+1) j ` (unlist REWRITE_TAC);
          BY(REPLICATE_TAC 14 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
        BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[]);
      TYPIFY `j = (p+1) MOD 4` ASM_CASES_TAC;
        FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`i`]);
        ASM_REWRITE_TAC[DIST_SYM];
        REPEAT WEAKER_STRIP_TAC;
        TYPIFY `e1` EXISTS_TAC;
        ASM_REWRITE_TAC[];
        TYPIFY_GOAL_THEN `scs_a_v39 s i ((p+1) MOD 4)  = scs_a_v39 s i (p+1) ` (unlist REWRITE_TAC);
          BY(REPLICATE_TAC 14 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
        BY(FIRST_X_ASSUM MP_TAC THEN FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN MESON_TAC[]);
      TYPIFY `&1` EXISTS_TAC;
      REWRITE_TAC[arith `&0 < &1`];
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `!w t. ~(w = v (p+1)) ==> f w t = w` ( (C FIRST_ASSUM_ST GMATCH_SIMP_TAC));
      TYPIFY `!w t. ~(w = v (p+1)) ==> f w t = w` ( (C FIRST_X_ASSUM_ST GMATCH_SIMP_TAC));
      FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
      nCONJ_TAC 2;
        BY(FIRST_X_ASSUM (C INTRO_TAC [`i`;`j`]) THEN MESON_TAC[]);
      ASM_REWRITE_TAC[];
      BY(REPLICATE_TAC 20 (POP_ASSUM MP_TAC) THEN MESON_TAC[]);
    REPEAT WEAKER_STRIP_TAC;
    INTRO_TAC MOD_4_CASES [`j`;`(p+1)`];
    ASM_REWRITE_TAC[arith `p + 0 = p /\ (p+1)+j = p+(1+j)`] THEN NUM_REDUCE_TAC;
    TYPIFY `(p+4) MOD 4 = p MOD 4` (C SUBGOAL_THEN SUBST1_TAC);
      BY(REWRITE_TAC[Oxl_2012.MOD_ADD_CANCEL]);
    REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[DIST_REFL];
          BY(TYPIFY `&1` EXISTS_TAC THEN MESON_TAC[arith `&0 <= &0 /\ &0 < &1`]);
        TYPIFY `e` EXISTS_TAC THEN REWRITE_TAC[];
        ASM_REWRITE_TAC[];
        DISCH_THEN SUBST1_TAC;
        REPEAT WEAKER_STRIP_TAC;
        FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`t`]);
        BY(FIRST_X_ASSUM MP_TAC THEN REAL_ARITH_TAC);
      FIRST_X_ASSUM_ST `scs_diag` (C INTRO_TAC [`p+1`;`p+3`]);
      TYPIFY `scs_diag 4 (p+1) (p+3)` ENOUGH_TO_SHOW_TAC;
        DISCH_THEN (unlist REWRITE_TAC);
        REPEAT WEAKER_STRIP_TAC;
        TYPIFY `&1` EXISTS_TAC THEN REWRITE_TAC[arith `&0 < &1`];
        BY(ASM_MESON_TAC[arith `x < y ==> ~(x = y)`]);
      REWRITE_TAC[arith `p+3 = SUC(SUC (p+1))`];
      MATCH_MP_TAC Tfitskc.SCS_DIAG_2;
      BY(NUM_REDUCE_TAC);
    TYPIFY `&1` EXISTS_TAC;
    TYPIFY `scs_a_v39 s (p+1) (p MOD 4) = scs_a_v39 s p (p+1)` ENOUGH_TO_SHOW_TAC;
      DISCH_THEN (unlist REWRITE_TAC);
      REWRITE_TAC[arith `&0 < &1`];
      BY(FIRST_X_ASSUM_ST `scs_a_v39 s p (p+1) < d` MP_TAC THEN REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC);
    BY(FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN ASM_MESON_TAC[]);
  COMMENT "now the b end";
  MATCH_MP_TAC Terminal.periodic2_mod_reduce;
  TYPIFY `4` EXISTS_TAC;
  ASM_REWRITE_TAC[];
  CONJ_TAC;
    MATCH_MP_TAC Cuxvzoz.MOD_periodic2;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!j. j < 4 ==> (?e1. &0 < e1 /\ (dist (v (p+1),v j) = scs_b_v39 s (p+1) j           ==> (!t. abs t < e1                    ==> dist (f (v (p+1)) t,f (v j) t) <= scs_b_v39 s (p+1) j)))` ENOUGH_TO_SHOW_TAC;
    (REPEAT WEAKER_STRIP_TAC);
    TYPIFY `i = ( (p+1) MOD 4)` ASM_CASES_TAC;
      FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`j`]);
      ASM_REWRITE_TAC[];
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `e1` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      TYPIFY_GOAL_THEN `scs_b_v39 s ((p+1) MOD 4) j = scs_b_v39 s (p+1) j ` (unlist REWRITE_TAC);
        BY(REPLICATE_TAC 14 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
      BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[]);
    TYPIFY `j = (p+1) MOD 4` ASM_CASES_TAC;
      FIRST_X_ASSUM_ST `dist` (C INTRO_TAC [`i`]);
      ASM_REWRITE_TAC[DIST_SYM];
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `e1` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      TYPIFY_GOAL_THEN `scs_b_v39 s i ((p+1) MOD 4)  = scs_b_v39 s i (p+1) ` (unlist REWRITE_TAC);
        BY(REPLICATE_TAC 14 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
      BY(FIRST_X_ASSUM MP_TAC THEN FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN MESON_TAC[]);
    TYPIFY `&1` EXISTS_TAC;
    REWRITE_TAC[arith `&0 < &1`];
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `!w t. ~(w = v (p+1)) ==> f w t = w` ( (C FIRST_ASSUM_ST GMATCH_SIMP_TAC));
    TYPIFY `!w t. ~(w = v (p+1)) ==> f w t = w` ( (C FIRST_X_ASSUM_ST GMATCH_SIMP_TAC));
    FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
    nCONJ_TAC 2;
      BY(FIRST_X_ASSUM (C INTRO_TAC [`i`;`j`]) THEN MESON_TAC[]);
    ASM_REWRITE_TAC[];
    BY(REPLICATE_TAC 20 (POP_ASSUM MP_TAC) THEN MESON_TAC[]);
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC MOD_4_CASES [`j`;`p+1`];
  ASM_REWRITE_TAC[arith `p + 0 = p /\ (p+1)+j = p+(1+j)`] THEN NUM_REDUCE_TAC;
  TYPIFY `(p+4) MOD 4 = p MOD 4` (C SUBGOAL_THEN SUBST1_TAC);
    BY(REWRITE_TAC[Oxl_2012.MOD_ADD_CANCEL]);
  ASM_REWRITE_TAC[arith `p + 0 = p`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[DIST_REFL];
        BY(TYPIFY `&1` EXISTS_TAC THEN MESON_TAC[arith `&0 <= &0 /\ &0 < &1`]);
      TYPIFY `e` EXISTS_TAC THEN ASM_REWRITE_TAC[];
      BY(ASM_MESON_TAC[arith `x = y ==> x <= y`]);
    FIRST_X_ASSUM_ST `scs_diag` (C INTRO_TAC [`p+1`;`p+3`]);
    TYPIFY_GOAL_THEN `scs_diag 4 (p+1) (p+3)` (unlist REWRITE_TAC);
      REWRITE_TAC[arith `p+3 = SUC(SUC (p+1))`];
      MATCH_MP_TAC Tfitskc.SCS_DIAG_2;
      BY(NUM_REDUCE_TAC);
    TYPIFY `dist(v (p+1),v (p+3)) <= &4 * h0` ENOUGH_TO_SHOW_TAC;
      BY(ASM_MESON_TAC[arith `x <= y /\ y < z ==> ~(x = z)`]);
    INTRO_TAC DIST_TRIANGLE [`v (p+1)`;`(vec 0):real^3`;`v (p+3)`];
    REWRITE_TAC[DIST_0];
    TYPIFY `!i. norm (v i) <= &2 * h0` ENOUGH_TO_SHOW_TAC;
      BY(DISCH_TAC THEN FIRST_ASSUM (C INTRO_TAC [`p+1`]) THEN FIRST_X_ASSUM (C INTRO_TAC [`p+3`]) THEN REAL_ARITH_TAC);
    FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
    BY(FIRST_X_ASSUM_ST `ball_annulus` MP_TAC THEN REWRITE_TAC[Terminal.IMAGE_SUBSET_IN;IN_UNIV] THEN MESON_TAC[Fnjlbxs.in_ball_annulus]);
  TYPIFY `&1` EXISTS_TAC THEN REWRITE_TAC[arith `&0 < &1`];
  FIRST_X_ASSUM_ST `dist (v p,v (p + 1)) < scs_b_v39 s p (p + 1)` MP_TAC THEN REWRITE_TAC[DIST_SYM];
  TYPIFY `scs_b_v39 s (p + 1) (p MOD 4) = scs_b_v39 s p (p+1)` ENOUGH_TO_SHOW_TAC;
    BY(REAL_ARITH_TAC);
  BY(FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN ASM_MESON_TAC[])
  ]);;
  (* }}} *)

let deform_azim_sum = 
prove_by_refinement(
  `!v1 v2 v3 v4 f e. deformation f {v1,v2,v3,v4} (--e,e) /\
   &0 < azim (vec 0) v1 v2 v4 /\  azim (vec 0) v1 v2 v4 < pi /\
    ~collinear {vec 0,v1,v2} /\ ~collinear {vec 0,v1,v3} /\ ~collinear {vec 0,v1,v4} /\
    azim (vec 0) v1 v2 v3 + azim (vec 0) v1 v3 v4 = azim (vec 0) v1 v2 v4 /\
    (((!t.  (azim (vec 0) (f v1 t) (f v2 t) (f v3 t) = azim (vec 0) v1 v2 v3)) /\
        (&0 < azim (vec 0) v1 v3 v4 /\ azim (vec 0) v1 v3 v4 < pi)) \/       
    ((!t.  (azim (vec 0) (f v1 t) (f v3 t) (f v4 t) = azim (vec 0) v1 v3 v4)) /\  
        (&0 < azim (vec 0) v1 v2 v3 /\ azim (vec 0) v1 v2 v3 < pi))) ==>
    (?e1. &0 < e1 /\
       (!t. abs t < e1 ==> 
	     ~collinear {vec 0,(f v1 t),(f v2 t)} /\
	     ~collinear {vec 0,(f v1 t),(f v3 t)} /\
	     ~collinear {vec 0,(f v1 t),(f v4 t)} /\
	      azim (vec 0) (f v1 t) (f v2 t) (f v3 t) + 
	      azim (vec 0) (f v1 t) (f v3 t) (f v4 t) = azim (vec 0) (f v1 t) (f v2 t) (f v4 t)))
	 `,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC (GEN_ALL Local_lemmas1.CONTINUOUS_PRESERVE_COLLINEAR) [`&0`;`(vec 0):real^3`;`f v1`];
  DISCH_TAC;
  FIRST_ASSUM (C INTRO_TAC [`f v2`]);
  FIRST_ASSUM (C INTRO_TAC [`f v3`]);
  FIRST_X_ASSUM (C INTRO_TAC [`f v4`]);
  FIRST_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation;IN_INSERT;NOT_IN_EMPTY];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `f v1 continuous atreal (&0) /\ f v2 continuous atreal (&0) /\ f v3 continuous atreal (&0) /\ f v4 continuous atreal (&0)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[]);
  TYPIFY `f v1 (&0) = v1 /\ f v2 (&0) = v2 /\ f v4 (&0) = v4 /\ f v3 (&0) = v3` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[]);
  REPLICATE_TAC 3 (FIRST_X_ASSUM_ST `collinear` MP_TAC) THEN ASM_REWRITE_TAC[];
  REWRITE_TAC[arith `abs(&0 - r) = abs r`];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC Cuxvzoz.epsilon_triple [`e'`;`e''`;`e'''`];
  ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC Cuxvzoz.WNWSHJT_ALT [`v4`;`v1`;`v2`;`f`;`-- e`;`e`;`pi`];
  ASM_REWRITE_TAC[];
  ANTS_TAC;
    MATCH_MP_TAC Zlzthic.deformation_subset;
    BY(TYPIFY `{v1,v2,v3,v4}` EXISTS_TAC THEN ASM_REWRITE_TAC[] THEN SET_TAC[]);
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `?e1. &0 < e1 /\ (!t. abs t < e1 ==>			azim (vec 0) (f v1 t) (f v2 t) (f v3 t) +               azim (vec 0) (f v1 t) (f v3 t) (f v4 t) < &2 * pi)` (C SUBGOAL_THEN MP_TAC);
    TYPIFY `(!t. azim (vec 0) (f v1 t) (f v2 t) (f v3 t) = azim (vec 0) v1 v2 v3) /\       &0 < azim (vec 0) v1 v3 v4 /\      azim (vec 0) v1 v3 v4 < pi` ASM_CASES_TAC;
      INTRO_TAC Cuxvzoz.WNWSHJT_ALT [`v4`;`v1`;`v3`;`f`;`-- e`;`e`;`pi - azim (vec 0) v1 v2 v3`];
      ASM_REWRITE_TAC[];
      ANTS_TAC;
        CONJ_TAC;
          MATCH_MP_TAC Zlzthic.deformation_subset;
          BY(TYPIFY `{v1,v2,v3,v4}` EXISTS_TAC THEN ASM_REWRITE_TAC[] THEN SET_TAC[]);
        BY(ASM_REWRITE_TAC[arith `a < p - b <=> b + a < p`]);
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `e''''''` EXISTS_TAC;
      ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
      FIRST_X_ASSUM (C INTRO_TAC [`t`]);
      ASM_REWRITE_TAC[];
      BY(MP_TAC PI_POS THEN REAL_ARITH_TAC);
    TYPIFY `(!t. azim (vec 0) (f v1 t) (f v3 t) (f v4 t) = azim (vec 0) v1 v3 v4) /\      &0 < azim (vec 0) v1 v2 v3 /\      azim (vec 0) v1 v2 v3 < pi` (C SUBGOAL_THEN MP_TAC);
      BY(ASM_MESON_TAC[]);
    REPEAT WEAKER_STRIP_TAC;
    INTRO_TAC Cuxvzoz.WNWSHJT_ALT [`v3`;`v1`;`v2`;`f`;`-- e`;`e`;`pi - azim (vec 0) v1 v3 v4`];
    ASM_REWRITE_TAC[];
    ANTS_TAC;
      CONJ_TAC;
        MATCH_MP_TAC Zlzthic.deformation_subset;
        BY(TYPIFY `{v1,v2,v3,v4}` EXISTS_TAC THEN ASM_REWRITE_TAC[] THEN SET_TAC[]);
      BY(ASM_REWRITE_TAC[arith `a < p - b <=> a + b < p`]);
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `e''''''` EXISTS_TAC;
    ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM (C INTRO_TAC [`t`]);
    ASM_REWRITE_TAC[];
    BY(MP_TAC PI_POS THEN REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC Cuxvzoz.epsilon_triple [`e''''`;`e'''''`;`e1`];
  ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  TYPIFY `e''''''` EXISTS_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  ONCE_REWRITE_TAC[EQ_SYM_EQ];
  REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC;
    BY(REPEAT CONJ_TAC THEN (REPEAT (FIRST_X_ASSUM MATCH_MP_TAC)) THEN ASM_REWRITE_TAC[]);
  MATCH_MP_TAC Fan.sum3_azim_fan;
  CONJ_TAC;
    BY(REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]);
  BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let NEHXMWH = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
  (!s FF v p.
  IMAGE (\i. (v i,v (SUC i))) (:num) = FF /\
  is_scs_v39 s /\
  scs_k_v39 s = 4 /\
  MMs_v39 s v /\
  scs_basic_v39 s /\
  scs_generic v /\
  (!i j. scs_diag 4 i j ==> scs_a_v39 s i j <= cstab /\ cstab < dist(v i,v j) /\ &4 * h0 < scs_b_v39 s i j) /\
  (!i. ~(i MOD 4 = (p+2) MOD 4) ==> interior_angle1 (vec 0) FF (v i) < pi)
  ==> (dist(v p,v(p+1)) = scs_a_v39 s p (p+1) \/ dist(v p,v(p+1)) = scs_b_v39 s p (p+1)))`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `BBprime2_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.scs_basic;Ayqjtmd.unadorned_MMs]);
  TYPIFY `BBprime_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime2_v39]);
  TYPIFY `BBs_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime_v39]);
  INTRO_TAC Appendix.BBs_v39 [`s`;`v`];
  ASM_REWRITE_TAC[LET_DEF;LET_END_DEF;Terminal.IMAGE_SUBSET_IN;IN_UNIV;arith `~(4 <= 3)`];
  REPEAT WEAKER_STRIP_TAC;
  TYPED_ABBREV_TAC `V = IMAGE v (:num)`;
  TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`;
  TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    EXPAND_TAC "V";
    REWRITE_TAC[IN_IMAGE;IN_UNIV];
    BY(MESON_TAC[]);
  INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`4`];
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  TYPIFY `!i. interior_angle1 (vec 0) FF (v i) = azim (vec 0) (v i) (v (i+1)) (v (i+3))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    EXPAND_TAC "FF";
    GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] (GSYM Ocbicby.INTERIOR_ANGLE1_AZIM));
    TYPIFY `s` EXISTS_TAC;
    BY(ASM_REWRITE_TAC[IN;arith `4 - 1 = 3`;arith `3 < 4`;arith `SUC i = i+1`]);
  INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`];
  ASM_REWRITE_TAC[LET_THM;IN;arith `4-1 = 3`;arith `SUC i = i+1`];
  DISCH_TAC;
  TYPIFY `!i. v (i MOD 4) = v i` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC (GSYM Oxl_def.periodic_mod);
    BY(ASM_REWRITE_TAC[arith `~(0=4)`]);
  TYPIFY `v 4 = v 0 /\ v 5 = v 1 /\ v 6 = v 2 /\ v 7 = v 3 /\ v 8  = v 4` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[Terminal.MOD_4_EXPLICIT;arith `7 = 1*4 + 3`;arith `8 = 2*4 + 0`;MOD_MULT_ADD]);
  TYPIFY `generic V E` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM_ST `scs_generic` MP_TAC THEN ASM_REWRITE_TAC[Appendix.scs_generic]);
  TYPIFY `v (p+4) = v p /\ v (p+5) = v (p+1) /\ v (p+6) = v (p+2) /\ v (p+7) = v (p+3) /\ v (p+8)  = v (p)` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `MOD` ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[arith `p+4 = 1*4 + p /\ p+5 = 1*4+(p+1) /\ p+6=1*4+(p+2) /\ p+7 = 1*4 + (p+3) /\ p+8 = 2*4 + p`;MOD_MULT_ADD]);
  INTRO_TAC Terminal.vv_split_azim_generic [`v`;`4`;`p+3`;`2`;`3`];
  REWRITE_TAC[arith `SUC (x+k) = (x+k)+1`;arith `((p:num)+i)+j = p+(i+j)`];
  NUM_REDUCE_TAC;
  ASM_REWRITE_TAC[LET_THM];
  DISCH_TAC;
  INTRO_TAC Terminal.vv_split_azim_generic [`v`;`4`;`p+1`;`2`;`3`];
  REWRITE_TAC[arith `SUC (x+k) = (x+k)+1`;arith `((p:num)+i)+j = p+(i+j)`];
  NUM_REDUCE_TAC;
  ASM_REWRITE_TAC[LET_THM];
  DISCH_TAC;
  COMMENT "collinearity";
  TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0,v i, v j}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT GEN_TAC THEN DISCH_TAC;
    MATCH_MP_TAC Zlzthic.PROPERTIES_GENERIC_LOCAL_FAN_ALT;
    GEXISTL_TAC [`V`;`E`;`FF`];
    BY(ASM_REWRITE_TAC[IN_IMAGE;IN_UNIV]);
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
  INTRO_TAC PERIODIC_INJ_MOD [`v`;`4`];
  ASM_REWRITE_TAC[arith `~(4 = 0)`];
  DISCH_TAC;
  TYPIFY `azim (vec 0) (v p) (v (p+1)) (v (p+3)) < pi` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
    FIRST_X_ASSUM MATCH_MP_TAC;
    GEN_REWRITE_TAC(RAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[arith`p = p+0`];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    BY(NUM_REDUCE_TAC);
  TYPIFY `~coplanar {vec 0,v p,v (p+1),v (p+3)}` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR];
    MATCH_MP_TAC (arith `&0 < x /\ x < pi ==> ~ ((x = &0) \/ (x = pi))`);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i a b. a < 4 /\ b < 4 /\ ~(a = b) ==> ~collinear {vec 0,v(i+a),v(i+b)}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[];
    FIRST_X_ASSUM_ST `v (i MOD 4) = v i` ((unlist ONCE_REWRITE_TAC) o GSYM);
    REPEAT (FIRST_X_ASSUM MATCH_MP_TAC);
    nCONJ_TAC 2;
      GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
      NUM_REDUCE_TAC;
      BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[DIVISION;MOD_LT;arith `~(4=0)`]);
    ASM_REWRITE_TAC[];
    BY(MESON_TAC[DIVISION;arith `~(4=0)`]);
  COMMENT "introduce deformation";
  INTRO_TAC Cuxvzoz.deform_simplex_edge_exists [`V`;`(\ (t:real). &0)`;`(\(t:real).  t)`;`v (p +3)`;`v p`;`v (p +1)`;`&1`];
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM MP_TAC;
  ANTS_TAC;
    REWRITE_TAC[arith `&0 < &1`];
    SUBCONJ_TAC;
      ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d}={a,c,d,b}`];
      BY(ASM_REWRITE_TAC[]);
    DISCH_TAC;
    CONJ_TAC;
      INTRO_TAC Planarity.cross_dot_fully_surrounded_fan [`(vec 0):real^3`;`v p`;`v (p+3)`;`v(p+1)`];
      ASM_REWRITE_TAC[];
      ANTS_TAC;
        INTRO_TAC Planarity.notcoplanar_imp_notcollinear_fan [`(vec 0):real^3`;`v p`;`v (p+1)`;`v(p+3)`];
        BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
      MATCH_MP_TAC (arith `x = y ==> (&0 < x ==> y > &0)`);
      BY(VEC3_TAC);
    REPEAT ( GMATCH_SIMP_TAC REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT);
    REWRITE_TAC[Ocbicby.REAL_OPEN_REAL_INTERVAL];
    BY(REWRITE_TAC[REAL_CONTINUOUS_CONST;REAL_CONTINUOUS_AT_ID]);
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `~(scs_a_v39 s p (p+1) < dist(v p,v (p+1)) /\ dist(v p,v (p+1)) < scs_b_v39 s p (p+1))` ENOUGH_TO_SHOW_TAC;
    FIRST_X_ASSUM_ST `scs_a_v39` (C INTRO_TAC [`p`;`(p+1)`]);
    BY(REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `(!i t. abs t < e' ==> norm (f (v i) t) = norm (v i))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `v i = v p` ASM_CASES_TAC;
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM_ST `norm` (C INTRO_TAC [`t`]);
      BY(ASM_REWRITE_TAC[] THEN MESON_TAC[]);
    FIRST_X_ASSUM GMATCH_SIMP_TAC;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. 0 < i /\ i < 4 ==> ~(v (p + i) = v p)` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM MP_TAC;
    ASM_REWRITE_TAC[];
    GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p+0`];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    NUM_REDUCE_TAC;
    BY(REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[DIVISION;MOD_LT;arith `~(4=0)`;arith `0 < i ==> ~(i=0)`]);
  TYPIFY `!i j (k:num). v i = v j ==> v (i+k) = v (j+k)` (C SUBGOAL_THEN ASSUME_TAC);
    ASM_REWRITE_TAC[];
    REPEAT WEAKER_STRIP_TAC;
    ONCE_REWRITE_TAC[arith `(i:num) + k = k + i`];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    BY(ASM_REWRITE_TAC[arith `~(4 = 0)`]);
  TYPIFY `!t. azim (vec 0) (f (v (p+2)) t) (f (v (p+3)) t) (f (v (p+1)) t) = azim(vec 0) (v(p+2)) (v(p+3)) (v(p+1))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    FIRST_X_ASSUM_ST `~(w = v p) ==> a` (REPEAT o GMATCH_SIMP_TAC);
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  TYPIFY `!i. azim (vec 0) (v i) (v (i+1)) (v (i+3)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
    MATCH_MP_TAC Local_lemmas.CONVEX_LOFA_IMP_INANGLE_LE_PI;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `(!i. ?e0. &0 < e0 /\           (azim (vec 0) (v i) (v (i + 1)) (v (i + 3)) = pi            ==> (!t. abs t < e0                     ==> azim (vec 0) (f (v i) t) (f (v (i + 1)) t)                         (f (v (i + 3)) t) <=                         pi)))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    TYPIFY `v i = v (p+2)` ASM_CASES_TAC;
      FIRST_ASSUM (fun t -> FIRST_X_ASSUM (ASSUME_TAC o (C MATCH_MP t)));
      TYPIFY `&1` EXISTS_TAC;
      ASM_REWRITE_TAC[arith `&0 < &1`;arith `(p+2)+k = p + (2+k)`];
      NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[];
      BY(REAL_ARITH_TAC);
    TYPIFY `azim (vec 0) (v i) (v (i + 1)) (v (i + 3)) < pi` ENOUGH_TO_SHOW_TAC;
      BY(MESON_TAC[arith `x < pi ==> ~(x = pi)`;arith `&0 < &1`]);
    BY(ASM_MESON_TAC[]);
  (COMMENT "deformation in BBs");
  INTRO_TAC Cuxvzoz.deformation_BBs_ALT [`s`;`4`;`f`;`v`;`e'`];
  ANTS_TAC;
    NUM_REDUCE_TAC;
    ASM_REWRITE_TAC[];
    MATCH_MP_TAC ab4_assumption_reduction;
    GEXISTL_TAC [`p`;`e'`];
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      REPEAT WEAKER_STRIP_TAC;
      FIRST_X_ASSUM_ST `scs_diag` (C INTRO_TAC [`i`;`j`]);
      BY(ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);
    REPEAT WEAKER_STRIP_TAC;
    (FIRST_X_ASSUM_ST `norm` (C INTRO_TAC [`t`]));
    BY(ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);
  (REPEAT WEAKER_STRIP_TAC);
  COMMENT "azim sums";
  INTRO_TAC deform_azim_sum [`v (p+3)`;`v p`;`v (p+1)`;`v(p+2)`;`f`;`e'`];
  ANTS_TAC;
    REPEAT (FIRST_X_ASSUM_ST `c = a + b` MP_TAC) THEN ASM_REWRITE_TAC[];
    REPLICATE_TAC 3 DISCH_TAC;
    CONJ_TAC;
      MATCH_MP_TAC Zlzthic.deformation_subset;
      (TYPIFY `V` EXISTS_TAC THEN ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]);
      BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
    TYPIFY `~collinear {vec 0, v (p + 3), v (p+0)} /\ ~collinear {vec 0, v (p + 3), v (p + 1)} /\ ~collinear {vec 0, v (p + 3), v (p + 2)}` (C SUBGOAL_THEN MP_TAC);
      BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN (unlist REWRITE_TAC);
    nCONJ_TAC 2;
      BY(ASM_REWRITE_TAC[]);
    FIRST_X_ASSUM (SUBST1_TAC) THEN REWRITE_TAC[];
    REPLICATE_TAC 4 (FIRST_X_ASSUM kill);
    ASM_REWRITE_TAC[];
    TYPIFY_GOAL_THEN `&0 < azim (vec 0) (v (p + 3)) (v p) (v (p + 2)) /\ azim (vec 0) (v (p + 3)) (v p) (v (p + 2)) < pi` (unlist REWRITE_TAC);
      FIRST_X_ASSUM_ST `&0 < azim a b c d` (C INTRO_TAC [`p+3`]);
      TYPIFY `azim (vec 0) (v (p+3)) (v p) (v (p+2)) = azim (vec 0) (v (p + 3)) (v ((p + 3) + 1)) (v ((p + 3) + 3))` (C SUBGOAL_THEN ASSUME_TAC);
        BY(REWRITE_TAC[arith `((p:num) + i) + j = p + i+j`] THEN NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[]);
      ASM_REWRITE_TAC[];
      DISCH_THEN (unlist REWRITE_TAC);
      FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM MATCH_MP_TAC;
      GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
      BY(NUM_REDUCE_TAC);
    DISJ2_TAC;
    CONJ_TAC;
      GEN_TAC;
      FIRST_X_ASSUM_ST `~(w = v p) ==> a` (REPEAT o GMATCH_SIMP_TAC);
      ASM_REWRITE_TAC[];
      BY(REPEAT CONJ_TAC THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p+0`] THEN GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC);
    REWRITE_TAC[Zlzthic.azim_lt_pi_cross];
    TYPIFY `(v (p + 3) cross v p) dot v (p + 1) = (v (p) cross (v (p+1))) dot v (p + 3)` (C SUBGOAL_THEN SUBST1_TAC);
      BY(VEC3_TAC);
    REWRITE_TAC[GSYM Zlzthic.azim_lt_pi_cross];
    BY(ASM_REWRITE_TAC[]);
  (REPEAT WEAKER_STRIP_TAC);
  COMMENT "second azim sum";
  INTRO_TAC deform_azim_sum [`v (p+1)`;`v (p+2)`;`v (p+3)`;`v(p)`;`f`;`e'`];
  ANTS_TAC;
    REPEAT (FIRST_X_ASSUM_ST `c = a + b` MP_TAC) THEN ASM_REWRITE_TAC[];
    REPLICATE_TAC 3 DISCH_TAC;
    CONJ_TAC;
      MATCH_MP_TAC Zlzthic.deformation_subset;
      (TYPIFY `V` EXISTS_TAC THEN ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]);
      BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
    TYPIFY `~collinear {vec 0, v (p + 1), v (p + 2)} /\ ~collinear {vec 0, v (p + 1), v (p + 3)} /\ ~collinear {vec 0, v (p + 1), v (p + 0)}` (C SUBGOAL_THEN MP_TAC);
      BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN (unlist REWRITE_TAC);
    nCONJ_TAC 2;
      BY(ASM_REWRITE_TAC[]);
    FIRST_X_ASSUM (SUBST1_TAC) THEN REWRITE_TAC[];
    REPLICATE_TAC 4 (FIRST_X_ASSUM kill);
    ASM_REWRITE_TAC[];
    TYPIFY_GOAL_THEN `&0 < azim (vec 0) (v (p + 1)) (v (p + 2)) (v p) /\ azim (vec 0) (v (p + 1)) (v (p + 2)) (v p) < pi` (unlist REWRITE_TAC);
      FIRST_X_ASSUM_ST `&0 < azim a b c d` (C INTRO_TAC [`p+1`]);
      TYPIFY `azim (vec 0) (v (p+1)) (v (p+2)) (v (p)) = azim (vec 0) (v (p + 1)) (v ((p + 1) + 1)) (v ((p + 1) + 3))` (C SUBGOAL_THEN ASSUME_TAC);
        BY(REWRITE_TAC[arith `((p:num) + i) + j = p + i+j`] THEN NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[]);
      ASM_REWRITE_TAC[];
      DISCH_THEN (unlist REWRITE_TAC);
      FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM MATCH_MP_TAC;
      GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
      BY(NUM_REDUCE_TAC);
    DISJ1_TAC;
    CONJ_TAC;
      GEN_TAC;
      FIRST_X_ASSUM_ST `~(w = v p) ==> a` (REPEAT o GMATCH_SIMP_TAC);
      ASM_REWRITE_TAC[];
      BY(REPEAT CONJ_TAC THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p+0`] THEN GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC);
    REWRITE_TAC[Zlzthic.azim_lt_pi_cross];
    TYPIFY `(v (p + 1) cross v (p+3)) dot v (p) = (v (p) cross (v (p+1))) dot v (p + 3)` (C SUBGOAL_THEN SUBST1_TAC);
      BY(VEC3_TAC);
    REWRITE_TAC[GSYM Zlzthic.azim_lt_pi_cross];
    BY(ASM_REWRITE_TAC[]);
  (REPEAT WEAKER_STRIP_TAC);
  TYPIFY `?e3. &0 < e3 /\ (!t. abs t < e3 ==> dist(v p,v (p+1)) + t IN real_interval(scs_a_v39 s p (p+1),scs_b_v39 s p (p+1)))` (C SUBGOAL_THEN MP_TAC);
    INTRO_TAC Zlzthic.real_interval_contains_0_ball [`scs_a_v39 s p (p+1)-dist(v p,v (p+1))`;`scs_b_v39 s p (p+1) - dist(v p,v (p+1))`;`&1`];
    ASM_REWRITE_TAC[arith `&0 < &1`;arith `a - d < &0 <=> a < d`;arith `&0 < b - d <=> d < b`];
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `e` EXISTS_TAC;
    ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM (C INTRO_TAC [`t`]);
    ASM_REWRITE_TAC[IN_REAL_INTERVAL];
    BY(REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "coplanar";
  INTRO_TAC Zlzthic.NONPLANAR_OPEN [`(\ (t:real). (vec 0):real^3)`;`f (v p)`;`\ (t:real). v (p+1)`;`\ (t:real). v (p+3)`;`&0`];
  ASM_REWRITE_TAC[CONTINUOUS_CONST];
  DISCH_THEN MP_TAC THEN ANTS_TAC;
    FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation] THEN REPEAT WEAKER_STRIP_TAC;
    CONJ2_TAC;
      BY(FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]);
    POP_ASSUM GMATCH_SIMP_TAC;
    BY(ASM_REWRITE_TAC[]);
  REWRITE_TAC[arith `abs (&0 - t') = abs t'`] THEN REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC Zlzthic.WNWSHJT [`v(p+3)`;`v p`;`v (p+1)`;`f`;`-- e'`;`e'`];
  ANTS_TAC;
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      MATCH_MP_TAC Zlzthic.deformation_subset;
      (TYPIFY `V` EXISTS_TAC THEN ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]);
      BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
    FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation];
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM (REPEAT o GMATCH_SIMP_TAC) THEN ASM_REWRITE_TAC[];
    TYPIFY `!i. ~(i=0) /\ i < 4 ==> ~collinear {vec 0,v (p+0),v(p+i)}` ENOUGH_TO_SHOW_TAC;
      BY(MESON_TAC[arith `p+0=p` ;arith `~(1=0) /\ 1 < 4 /\ ~(3=0) /\ 3 < 4`]);
    GEN_TAC THEN DISCH_TAC;
    FIRST_X_ASSUM MATCH_MP_TAC;
    BY(ASM_REWRITE_TAC[arith `0 < 4`]);
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "introduce 1834";
  INTRO_TAC Iunbuig.epsilon_hept [`e`;`e'`;`e''`;`e'''`;`e3`;`e1`;`e1'`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  FIRST_ASSUM_ST `!i. 0 < i /\ i < 4 ==> ~(v (p + i) = v p)` MP_TAC;
  FIRST_ASSUM_ST `i MOD 4 = j MOD 4` (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC (UNDISCH Ocbicby.LEMMA_1834976363) [`norm (v (p+3))`;`norm (v p)`;`norm (v (p+1))`;`dist(v p,v(p+1))`;`dist (v(p+1),v(p+3))`;`dist(v p,v (p+3))`;`dist(v p,v (p+1)) - e7`;`dist(v p,v(p+1)) + e7`];
  ANTS_TAC;
    ASM_REWRITE_TAC[];
    (COMMENT "1834 ants");
    COMMENT "restart";
    TYPIFY `!i. &2 <= norm (v i) /\ norm (v i) <= &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
      GEN_TAC;
      REWRITE_TAC[GSYM Fnjlbxs.in_ball_annulus];
      FIRST_X_ASSUM_ST `BBs_v39` kill THEN FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN EXPAND_TAC "V" THEN REPEAT WEAKER_STRIP_TAC;
      FIRST_X_ASSUM_ST `ball_annulus` MP_TAC;
      REWRITE_TAC[SUBSET;IN_IMAGE;IN_UNIV];
      BY(MESON_TAC[]);
    ASM_REWRITE_TAC[];
    FIRST_ASSUM_ST `collinear` (C INTRO_TAC [`p`;`3`;`0`]);
    FIRST_X_ASSUM_ST `collinear` (C INTRO_TAC [`p`;`3`;`1`]);
    NUM_REDUCE_TAC;
    REWRITE_TAC[arith `p + 0 = p`];
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY_GOAL_THEN `&0 < ups_x (norm (v (p + 3)) * norm (v (p + 3))) (norm (v p) * norm (v p)) (dist (v p,v (p + 3)) * dist (v p,v (p + 3))) /\ &0 < ups_x (norm (v (p + 3)) * norm (v (p + 3))) (norm (v (p + 1)) * norm (v (p + 1))) (dist (v (p + 1),v (p + 3)) * dist (v (p + 1),v (p + 3)))` (unlist REWRITE_TAC);
      REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC);
      REWRITE_TAC[Collect_geom2.NOT_COL_EQ_UPS_X_POS;DIST_0;DIST_SYM;arith`x pow 2 = x*x`];
      BY(MESON_TAC[]);
    TYPIFY `!i j. &2 <= dist(v i,v j) ==> (&2 / h0) pow 2 <= xrr (norm(v i)) (norm(v (j))) (dist (v i,v j))` ((C SUBGOAL_THEN ASSUME_TAC));
      REPEAT WEAKER_STRIP_TAC;
      MATCH_MP_TAC Ocbicby.xrr_simple_lower_bound;
      BY(ASM_REWRITE_TAC[arith `x <= x`]);
    CONJ_TAC;
      POP_ASSUM MATCH_MP_TAC;
      MATCH_MP_TAC Ocbicby.scs_lb_2;
      TYPIFY `s` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      ONCE_REWRITE_TAC[EQ_SYM_EQ];
      FIRST_X_ASSUM MATCH_MP_TAC;
      BY(NUM_REDUCE_TAC);
    CONJ_TAC;
      TYPIFY `#15.53 = sqrt(#15.53) pow 2` (C SUBGOAL_THEN SUBST1_TAC);
        GMATCH_SIMP_TAC SQRT_POW_2;
        BY(REAL_ARITH_TAC);
      MATCH_MP_TAC Ocbicby.xrr_simple_upper_bound;
      ASM_REWRITE_TAC[];
      TYPIFY `&4 = sqrt(&16)` (C SUBGOAL_THEN SUBST1_TAC);
        REWRITE_TAC[arith `&16 = &4 pow 2`];
        REWRITE_TAC[POW_2_SQRT_ABS];
        BY(REAL_ARITH_TAC);
      GMATCH_SIMP_TAC SQRT_MONO_LT_EQ;
      TYPIFY_GOAL_THEN `&0 <=  #15.53 /\ &0 < &16 /\  #15.53 < &16` (unlist REWRITE_TAC);
        BY(REAL_ARITH_TAC);
      CONJ_TAC;
        BY(REAL_ARITH_TAC);
      CONJ_TAC;
        MATCH_MP_TAC Ocbicby.scs_lb_2;
        TYPIFY `s` EXISTS_TAC;
        ASM_REWRITE_TAC[];
        ONCE_REWRITE_TAC[EQ_SYM_EQ];
        FIRST_X_ASSUM MATCH_MP_TAC;
        BY(NUM_REDUCE_TAC);
      MATCH_MP_TAC REAL_LE_TRANS;
      TYPIFY `cstab` EXISTS_TAC;
      CONJ_TAC;
        MATCH_MP_TAC (arith `d <= scs_b_v39 s p (p+1) /\ scs_b_v39 s p (p+1) <= cstab ==> d <= cstab`);
        CONJ_TAC;
          FIRST_X_ASSUM_ST `BBs_v39` kill;
          (FIRST_X_ASSUM_ST `BBs_v39` MP_TAC) THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM];
          BY(MESON_TAC[]);
        FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39];
        ASM_REWRITE_TAC[arith `3 < 4`;arith `SUC i = i+1`];
        BY(MESON_TAC[]);
      REWRITE_TAC[Sphere.cstab];
      MATCH_MP_TAC REAL_LE_RSQRT;
      BY(REAL_ARITH_TAC);
    CONJ_TAC;
      ONCE_REWRITE_TAC[DIST_SYM];
      FIRST_X_ASSUM MATCH_MP_TAC;
      MATCH_MP_TAC Ocbicby.scs_lb_2;
      TYPIFY `s` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      BY(GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC);
    CONJ_TAC;
      ONCE_REWRITE_TAC[DIST_SYM];
      FIRST_X_ASSUM MATCH_MP_TAC;
      MATCH_MP_TAC Ocbicby.scs_lb_2;
      TYPIFY `s` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      BY(FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    CONJ_TAC;
      REWRITE_TAC[IN_REAL_INTERVAL];
      BY(FIRST_X_ASSUM_ST `&0 < e` MP_TAC THEN REAL_ARITH_TAC);
    CONJ_TAC;
      REWRITE_TAC[SUBSET;IN_REAL_INTERVAL;IN_ELIM_THM];
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `?t. abs t < e7 /\ x = dist(v p,v(p+1)) + t` (C SUBGOAL_THEN MP_TAC);
        TYPIFY `x - dist(v p,v(p+1))` EXISTS_TAC;
        CONJ2_TAC;
          BY(REAL_ARITH_TAC);
        BY(REPLICATE_TAC 2 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
      REPEAT WEAKER_STRIP_TAC;
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM_ST `norm (f (v p) t) = norm (v p)` (C INTRO_TAC [`t`]);
      ANTS_TAC;
        BY(REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]);
      REWRITE_TAC[DIST_SYM] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[];
      FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`t`]);
      ANTS_TAC;
        BY(REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]);
      DISCH_TAC;
      FIRST_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,d,b,c}`] THEN REWRITE_TAC[Oxlzlez.coplanar_delta_y;DIST_0;DIST_SYM];
      RULE_ASSUM_TAC (REWRITE_RULE[arith `d + &0 = d`]);
      ASM_REWRITE_TAC[];
      DISCH_THEN (unlist REWRITE_TAC);
      REWRITE_TAC[GSYM DIST_NZ];
      CONJ_TAC;
        DISCH_TAC;
        FIRST_X_ASSUM_ST `coplanar` MP_TAC;
        BY(ASM_REWRITE_TAC[SET_RULE `{a,b,b,c} = {a,b,c}`;COPLANAR_3]);
      TYPIFY `~collinear {vec 0,f (v p) t,v (p+1)}` (C SUBGOAL_THEN MP_TAC);
        BY(POP_ASSUM MP_TAC THEN MESON_TAC[Planarity.notcoplanar_imp_notcollinear_fan]);
      REWRITE_TAC[Collect_geom2.NOT_COL_EQ_UPS_X_POS];
      BY(REWRITE_TAC[DIST_0;arith `x pow 2 = x*x`;DIST_SYM]);
    TYPIFY `!i.  i < 3 ==> &0 < dist (v (p+i), v (p+3))` ENOUGH_TO_SHOW_TAC;
      BY(MESON_TAC[arith `p + 0 = (p:num)`;arith `0 < 3`;arith `1 < 3`]);
    REPEAT WEAKER_STRIP_TAC;
    MATCH_MP_TAC (arith `&2 <= x ==> &0 < x`);
    MATCH_MP_TAC Ocbicby.scs_lb_2;
    TYPIFY `s` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC;
    BY(FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[arith `i < 3 <=> i = 0 \/ i = 1 \/ i = 2`] THEN DISCH_TAC THEN REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[] THEN NUM_REDUCE_TAC);
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `?t. abs t < e7 /\ y4' = dist(v p,v(p+1)) + t` (C SUBGOAL_THEN MP_TAC);
    TYPIFY `y4' - dist(v p,v(p+1))` EXISTS_TAC;
    CONJ2_TAC;
      BY(REAL_ARITH_TAC);
    BY(REPLICATE_TAC 2 (POP_ASSUM MP_TAC) THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM_ST `taum` MP_TAC;
  ASM_REWRITE_TAC[];
  REWRITE_TAC[arith `~(x < y) <=> (y <= x)`];
  FIRST_X_ASSUM_ST `norm (f (v p) t) = norm (v p)` (C INTRO_TAC [`t`]);
  ANTS_TAC;
    BY(REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]);
  (REWRITE_TAC[DIST_SYM;arith `d + &0 = d`] THEN REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[]);
  COMMENT "back to root, next tau";
  TYPIFY `sum {i | i < 4} (\i. rho_fun (norm (v i)) * azim (vec 0) (v i) (v (i + 1)) (v (i + 4 - 1))) <=    sum {i | i < 4} (\i. rho_fun (norm (f (v i) t)) * azim (vec 0) (f (v i) t) (f(v(i+1)) t) (f (v (i + 4 - 1)) t))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    MATCH_MP_TAC Cuxvzoz.MMs_minimize_tau_fun;
    TYPIFY `s` EXISTS_TAC;
    ASM_REWRITE_TAC[arith `3 < 4`];
    FIRST_X_ASSUM MATCH_MP_TAC;
    BY(FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]);
  POP_ASSUM MP_TAC;
  TYPIFY `{i | i < 4} = 0..(4-1)` (C SUBGOAL_THEN SUBST1_TAC);
    BY(REWRITE_TAC[EXTENSION;IN_ELIM_THM;IN_NUMSEG] THEN ARITH_TAC);
  REPEAT ( GMATCH_SIMP_TAC (GSYM Oxl_def.periodic_sum));
  TYPIFY `p` EXISTS_TAC;
  CONJ_TAC;
    NUM_REDUCE_TAC;
    FIRST_ASSUM_ST `periodic` MP_TAC THEN REWRITE_TAC[Oxl_def.periodic];
    REWRITE_TAC[arith `(i+4)+k = (i+k)+4`];
    BY(DISCH_THEN (unlist REWRITE_TAC));
  TYPIFY `p` EXISTS_TAC;
  CONJ_TAC;
    NUM_REDUCE_TAC;
    FIRST_ASSUM_ST `periodic` MP_TAC THEN REWRITE_TAC[Oxl_def.periodic];
    REWRITE_TAC[arith `(i+4)+k = (i+k)+4`];
    BY(DISCH_THEN (unlist REWRITE_TAC));
  NUM_REDUCE_TAC;
  REWRITE_TAC[arith `3+p = SUC(SUC(SUC p))`;SUM_CLAUSES_NUMSEG;SUM_SING_NUMSEG;arith `p <= SUC p /\ p <= SUC (SUC p) /\ p <= SUC(SUC(SUC p))`];
  REWRITE_TAC[arith `SUC i = i+1`;arith `(i + j) + (k:num) = i+(j+k)`];
  NUM_REDUCE_TAC;
  ASM_REWRITE_TAC[];
  REPLICATE_TAC 7 (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  REPEAT (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM_ST `rho_fun` MP_TAC THEN ASM_REWRITE_TAC[Appendix.rho_rho_fun];
  TYPIFY `f (v(p+1)) t = v(p+1) /\ f (v (p+2)) t = v(p+2) /\ f(v (p+3)) t = v(p+3)` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `~(w = v p) ==> a` (REPEAT o GMATCH_SIMP_TAC);
    BY(ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  POP_ASSUM (fun t -> ASM_TAC THEN unlist REWRITE_TAC t THEN REPEAT DISCH_TAC THEN ASSUME_TAC t);
  FIRST_X_ASSUM_ST `azim` MP_TAC;
  FIRST_X_ASSUM_ST `azim` MP_TAC THEN BURY_MP_TAC;
  FIRST_ASSUM_ST `azim` ((unlist REWRITE_TAC) o GSYM);
  FIRST_ASSUM_ST `azim (vec 0) (v (p + 3)) (f (v p) t) (v (p + 2))` ((unlist REWRITE_TAC) o GSYM);
  REWRITE_TAC[arith `(a+b)+c = a + b + c`];
  REWRITE_TAC[arith `r0 * (a013) + r1 * (a123 + a130) + r2 * a231 + r3 * (a301+a312)  <=  r0 * (f013) + r1 * (a123 + f130) + r2 * a231 + r3 *(f301 +a312) <=> r0 * a013 + r1*a130 + r3* a301 <=  r0 * f013 + r1 * f130 + r3 * f301`];
  MATCH_MP_TAC (arith `a - (pi+sol0) = a' /\ b - (pi+sol0) = b' ==> (a <= b ==> a' <= b')`);
  REWRITE_TAC[arith `(a + b+c) - (pi + sol0) = a + b + c - (pi + sol0)`];
  CONJ_TAC;
    GMATCH_SIMP_TAC (GSYM Cuxvzoz.tau3_azim);
    GMATCH_SIMP_TAC Cuxvzoz.tau3_taum_nonplanar;
    ASM_REWRITE_TAC[];
    REWRITE_TAC[DIST_SYM];
    BY(MESON_TAC[Terminal.taum_sym]);
  FIRST_X_ASSUM_ST `norm a  = norm b` (SUBST1_TAC o GSYM);
  FIRST_X_ASSUM_ST `dist s + t` (SUBST1_TAC o GSYM);
  GMATCH_SIMP_TAC (GSYM Cuxvzoz.tau3_azim);
  GMATCH_SIMP_TAC Cuxvzoz.tau3_taum_nonplanar;
  REWRITE_TAC[DIST_SYM];
  REWRITE_TAC[CONJ_ASSOC] THEN CONJ2_TAC;
    ASM_REWRITE_TAC[DIST_SYM];
    BY(MESON_TAC[Terminal.taum_sym]);
  ASM_REWRITE_TAC[];
  BY(FIRST_X_ASSUM_ST `azim (vec 0) (f (v p) t) (v (p + 1)) (v (p + 3)) < pi` MP_TAC THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let BZQNDMN = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
  (!s FF v p.
  IMAGE (\i. (v i,v (SUC i))) (:num) = FF /\
  is_scs_v39 s /\
  scs_k_v39 s = 4 /\
  MMs_v39 s v /\
  scs_basic_v39 s /\
  scs_generic v /\
  (!i j. scs_diag 4 i j ==> scs_a_v39 s i j <= cstab /\ cstab < dist(v i,v j) /\ &4 * h0 < scs_b_v39 s i j) /\
  (!i. ~(i MOD 4 = (p+3) MOD 4) ==> interior_angle1 (vec 0) FF (v i) < pi)
  ==> (dist(v p,v(p+1)) = scs_a_v39 s p (p+1) \/ dist(v p,v(p+1)) = scs_b_v39 s p (p+1)))`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `BBprime2_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.scs_basic;Ayqjtmd.unadorned_MMs]);
  TYPIFY `BBprime_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime2_v39]);
  TYPIFY `BBs_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime_v39]);
  INTRO_TAC Appendix.BBs_v39 [`s`;`v`];
  ASM_REWRITE_TAC[LET_DEF;LET_END_DEF;Terminal.IMAGE_SUBSET_IN;IN_UNIV;arith `~(4 <= 3)`];
  REPEAT WEAKER_STRIP_TAC;
  TYPED_ABBREV_TAC `V = IMAGE v (:num)`;
  TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`;
  TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    EXPAND_TAC "V";
    REWRITE_TAC[IN_IMAGE;IN_UNIV];
    BY(MESON_TAC[]);
  INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`4`];
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  TYPIFY `!i. interior_angle1 (vec 0) FF (v i) = azim (vec 0) (v i) (v (i+1)) (v (i+3))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    EXPAND_TAC "FF";
    GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] (GSYM Ocbicby.INTERIOR_ANGLE1_AZIM));
    TYPIFY `s` EXISTS_TAC;
    BY(ASM_REWRITE_TAC[IN;arith `4 - 1 = 3`;arith `3 < 4`;arith `SUC i = i+1`]);
  INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`];
  ASM_REWRITE_TAC[LET_THM;IN;arith `4-1 = 3`;arith `SUC i = i+1`];
  DISCH_TAC;
  TYPIFY `!i. v (i MOD 4) = v i` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC (GSYM Oxl_def.periodic_mod);
    BY(ASM_REWRITE_TAC[arith `~(0=4)`]);
  TYPIFY `v 4 = v 0 /\ v 5 = v 1 /\ v 6 = v 2 /\ v 7 = v 3 /\ v 8  = v 4` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[Terminal.MOD_4_EXPLICIT;arith `7 = 1*4 + 3`;arith `8 = 2*4 + 0`;MOD_MULT_ADD]);
  TYPIFY `generic V E` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM_ST `scs_generic` MP_TAC THEN ASM_REWRITE_TAC[Appendix.scs_generic]);
  TYPIFY `v (p+4) = v p /\ v (p+5) = v (p+1) /\ v (p+6) = v (p+2) /\ v (p+7) = v (p+3) /\ v (p+8)  = v (p)` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `MOD` ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[arith `p+4 = 1*4 + p /\ p+5 = 1*4+(p+1) /\ p+6=1*4+(p+2) /\ p+7 = 1*4 + (p+3) /\ p+8 = 2*4 + p`;MOD_MULT_ADD]);
  COMMENT "change here";
  INTRO_TAC Terminal.vv_split_azim_generic [`v`;`4`;`p`;`2`;`3`];
  REWRITE_TAC[arith `SUC (x+k) = (x+k)+1`;arith `((p:num)+i)+j = p+(i+j)`];
  NUM_REDUCE_TAC;
  ASM_REWRITE_TAC[LET_THM];
  DISCH_TAC;
  INTRO_TAC Terminal.vv_split_azim_generic [`v`;`4`;`p+2`;`2`;`3`];
  REWRITE_TAC[arith `SUC (x+k) = (x+k)+1`;arith `((p:num)+i)+j = p+(i+j)`];
  NUM_REDUCE_TAC;
  ASM_REWRITE_TAC[LET_THM];
  DISCH_TAC;
  COMMENT "collinearity";
  TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0,v i, v j}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT GEN_TAC THEN DISCH_TAC;
    MATCH_MP_TAC Zlzthic.PROPERTIES_GENERIC_LOCAL_FAN_ALT;
    GEXISTL_TAC [`V`;`E`;`FF`];
    BY(ASM_REWRITE_TAC[IN_IMAGE;IN_UNIV]);
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
  INTRO_TAC PERIODIC_INJ_MOD [`v`;`4`];
  ASM_REWRITE_TAC[arith `~(4 = 0)`];
  DISCH_TAC;
  TYPIFY `azim (vec 0) (v (p+1)) (v((p+1)+1)) (v ((p+1)+3)) = azim (vec 0) (v (p+1)) (v (p+2)) (v p)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REWRITE_TAC[arith `(p+1)+i = p+(1+i)`] THEN NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC);
  TYPIFY `azim (vec 0) (v (p+1)) (v ((p+1)+1)) (v ((p+1)+3)) < pi` (C SUBGOAL_THEN MP_TAC);
    FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
    FIRST_X_ASSUM MATCH_MP_TAC;
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    BY(NUM_REDUCE_TAC);
  ASM_REWRITE_TAC[] THEN DISCH_TAC;
  TYPIFY `&0 < azim (vec 0) (v (p+1)) (v ((p+1)+1)) (v ((p+1)+3))` (C SUBGOAL_THEN MP_TAC);
    BY(ASM_REWRITE_TAC[]);
  FIRST_X_ASSUM SUBST1_TAC THEN DISCH_TAC;
  TYPIFY `~coplanar {vec 0,v p,v (p+1),v (p+2)}` (C SUBGOAL_THEN ASSUME_TAC);
    ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,c,d,b}`];
    REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR];
    MATCH_MP_TAC (arith `&0 < x /\ x < pi ==> ~ ((x = &0) \/ (x = pi))`);
    BY((ASM_REWRITE_TAC[]));
  TYPIFY `!i a b. a < 4 /\ b < 4 /\ ~(a = b) ==> ~collinear {vec 0,v(i+a),v(i+b)}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[];
    FIRST_X_ASSUM_ST `v (i MOD 4) = v i` ((unlist ONCE_REWRITE_TAC) o GSYM);
    REPEAT (FIRST_X_ASSUM MATCH_MP_TAC);
    nCONJ_TAC 2;
      GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
      NUM_REDUCE_TAC;
      BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[DIVISION;MOD_LT;arith `~(4=0)`]);
    ASM_REWRITE_TAC[];
    BY(MESON_TAC[DIVISION;arith `~(4=0)`]);
  COMMENT "introduce deformation";
  INTRO_TAC Cuxvzoz.deform_simplex_edge_exists [`V`;`(\ (t:real). t)`;`(\(t:real).  &0)`;`v (p)`;`v (p+1)`;`v (p +2)`;`&1`];
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM MP_TAC;
  ANTS_TAC;
    REWRITE_TAC[arith `&0 < &1`];
    CONJ_TAC;
      INTRO_TAC Planarity.cross_dot_fully_surrounded_fan [`(vec 0):real^3`;`v (p+1)`;`v (p)`;`v(p+2)`];
      ASM_REWRITE_TAC[];
      ANTS_TAC;
        INTRO_TAC Planarity.notcoplanar_imp_notcollinear_fan [`(vec 0):real^3`;`v p`;`v (p+1)`;`v(p+2)`];
        BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[SET_RULE `{a,b,c}={a,c,b}`]);
      MATCH_MP_TAC (arith `x = y ==> (&0 < x ==> y > &0)`);
      BY(VEC3_TAC);
    REPEAT ( GMATCH_SIMP_TAC REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT);
    REWRITE_TAC[Ocbicby.REAL_OPEN_REAL_INTERVAL];
    BY(REWRITE_TAC[REAL_CONTINUOUS_CONST;REAL_CONTINUOUS_AT_ID]);
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "A";
  TYPIFY `~(scs_a_v39 s p (p+1) < dist(v p,v (p+1)) /\ dist(v p,v (p+1)) < scs_b_v39 s p (p+1))` ENOUGH_TO_SHOW_TAC;
    FIRST_X_ASSUM_ST `scs_a_v39` (C INTRO_TAC [`p`;`(p+1)`]);
    BY(REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `(!i t. abs t < e' ==> norm (f (v i) t) = norm (v i))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `v i = v (p+1)` ASM_CASES_TAC;
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM_ST `norm` (C INTRO_TAC [`t`]);
      BY(ASM_REWRITE_TAC[] THEN MESON_TAC[]);
    FIRST_X_ASSUM GMATCH_SIMP_TAC;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. 0 < i /\ i < 4 ==> ~(v (p + i) = v p)` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM MP_TAC;
    ASM_REWRITE_TAC[];
    GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p+0`];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    NUM_REDUCE_TAC;
    BY(REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[DIVISION;MOD_LT;arith `~(4=0)`;arith `0 < i ==> ~(i=0)`]);
  TYPIFY `!i. i=0 \/ i = 2 \/ i = 3 ==> ~(v (p + i) = v (p+1))` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT STRIP_TAC THEN FIRST_X_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC);
  TYPIFY `!i j (k:num). v i = v j ==> v (i+k) = v (j+k)` (C SUBGOAL_THEN ASSUME_TAC);
    ASM_REWRITE_TAC[];
    REPEAT WEAKER_STRIP_TAC;
    ONCE_REWRITE_TAC[arith `(i:num) + k = k + i`];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    BY(ASM_REWRITE_TAC[arith `~(4 = 0)`]);
  TYPIFY `!t. azim (vec 0) (f (v (p+3)) t) (f (v (p)) t) (f (v (p+2)) t) = azim(vec 0) (v(p+3)) (v(p)) (v(p+2))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    FIRST_X_ASSUM_ST `~(w = v (p+1)) ==> a` (REPEAT o GMATCH_SIMP_TAC);
    ONCE_REWRITE_TAC[arith `(v p = (v:num->real^3) (p+1)) <=> (v(p+1) = v p)`];
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  TYPIFY `!i. azim (vec 0) (v i) (v (i+1)) (v (i+3)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
    MATCH_MP_TAC Local_lemmas.CONVEX_LOFA_IMP_INANGLE_LE_PI;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `(!i. ?e0. &0 < e0 /\           (azim (vec 0) (v i) (v (i + 1)) (v (i + 3)) = pi            ==> (!t. abs t < e0                     ==> azim (vec 0) (f (v i) t) (f (v (i + 1)) t)                         (f (v (i + 3)) t) <=                         pi)))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    TYPIFY `v i = v (p+3)` ASM_CASES_TAC;
      FIRST_ASSUM (fun t -> FIRST_X_ASSUM (ASSUME_TAC o (C MATCH_MP t)));
      TYPIFY `&1` EXISTS_TAC;
      ASM_REWRITE_TAC[arith `&0 < &1`;arith `(p+3)+k = p + (3+k)`];
      NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[];
      BY(REAL_ARITH_TAC);
    TYPIFY `azim (vec 0) (v i) (v (i + 1)) (v (i + 3)) < pi` ENOUGH_TO_SHOW_TAC;
      BY(MESON_TAC[arith `x < pi ==> ~(x = pi)`;arith `&0 < &1`]);
    BY(ASM_MESON_TAC[]);
  (COMMENT "deformation in BBs");
  INTRO_TAC Cuxvzoz.deformation_BBs_ALT [`s`;`4`;`f`;`v`;`e'`];
  ANTS_TAC;
    NUM_REDUCE_TAC;
    ASM_REWRITE_TAC[];
    MATCH_MP_TAC ab4_assumption_reduction_sym;
    GEXISTL_TAC [`p`;`e'`];
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      REPEAT WEAKER_STRIP_TAC;
      FIRST_X_ASSUM_ST `scs_diag` (C INTRO_TAC [`i`;`j`]);
      BY(ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);
    REPEAT WEAKER_STRIP_TAC;
    (FIRST_X_ASSUM_ST `norm` (C INTRO_TAC [`t`]));
    BY(ASM_REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC);
  (REPEAT WEAKER_STRIP_TAC);
  COMMENT "azim sums";
  INTRO_TAC deform_azim_sum [`v (p)`;`v (p+1)`;`v (p+2)`;`v(p+3)`;`f`;`e'`];
  ANTS_TAC;
    REPEAT (FIRST_X_ASSUM_ST `c = a + b` MP_TAC) THEN ASM_REWRITE_TAC[];
    REPLICATE_TAC 3 DISCH_TAC;
    CONJ_TAC;
      MATCH_MP_TAC Zlzthic.deformation_subset;
      (TYPIFY `V` EXISTS_TAC THEN ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]);
      BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
    TYPIFY `~collinear {vec 0, v (p + 0), v (p+1)} /\ ~collinear {vec 0, v (p + 0), v (p + 2)} /\ ~collinear {vec 0, v (p + 0), v (p + 3)}` (C SUBGOAL_THEN MP_TAC);
      BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN (unlist REWRITE_TAC);
    nCONJ_TAC 1;
      BY(FIRST_X_ASSUM_ST `SUC` MP_TAC THEN REWRITE_TAC[arith `SUC p = p+1`] THEN DISCH_THEN (unlist REWRITE_TAC));
    FIRST_X_ASSUM (SUBST1_TAC) THEN REWRITE_TAC[];
    REPLICATE_TAC 3 (FIRST_X_ASSUM kill);
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM MATCH_MP_TAC;
      GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p+0`];
      GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
      BY(NUM_REDUCE_TAC);
    DISJ2_TAC;
    CONJ_TAC;
      GEN_TAC;
      FIRST_X_ASSUM_ST `~(w = v p) ==> a` (REPEAT o GMATCH_SIMP_TAC);
      ASM_REWRITE_TAC[];
      TYPIFY `p MOD 4 = (p+0) MOD 4` (C SUBGOAL_THEN SUBST1_TAC);
        BY(REWRITE_TAC[arith `p+0 = p`]);
      BY(REPEAT CONJ_TAC THEN GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC);
    REWRITE_TAC[Zlzthic.azim_lt_pi_cross];
    TYPIFY `(v (p) cross v (p+1)) dot v (p + 2) = (v (p+1) cross (v (p+2))) dot v (p)` (C SUBGOAL_THEN SUBST1_TAC);
      BY(VEC3_TAC);
    REWRITE_TAC[GSYM Zlzthic.azim_lt_pi_cross];
    BY(ASM_REWRITE_TAC[]);
  (REPEAT WEAKER_STRIP_TAC);
  INTRO_TAC deform_azim_sum [`v (p+2)`;`v (p+3)`;`v (p)`;`v(p+1)`;`f`;`e'`];
  ANTS_TAC;
    REPEAT (FIRST_X_ASSUM_ST `c = a + b` MP_TAC) THEN ASM_REWRITE_TAC[];
    REPLICATE_TAC 3 DISCH_TAC;
    CONJ_TAC;
      MATCH_MP_TAC Zlzthic.deformation_subset;
      (TYPIFY `V` EXISTS_TAC THEN ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]);
      BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
    TYPIFY `~collinear {vec 0, v (p + 2), v (p + 3)} /\ ~collinear {vec 0, v (p + 2), v (p + 0)} /\ ~collinear {vec 0, v (p + 2), v (p + 1)}` (C SUBGOAL_THEN MP_TAC);
      BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN (unlist REWRITE_TAC);
    nCONJ_TAC 2;
      BY(ASM_REWRITE_TAC[]);
    FIRST_X_ASSUM (SUBST1_TAC) THEN REWRITE_TAC[];
    REPLICATE_TAC 4 (FIRST_X_ASSUM kill);
    ASM_REWRITE_TAC[];
    TYPIFY_GOAL_THEN `&0 < azim (vec 0) (v (p + 2)) (v (p + 3)) (v (p+1)) /\ azim (vec 0) (v (p + 2)) (v (p + 3)) (v (p+1)) < pi` (unlist REWRITE_TAC);
      FIRST_X_ASSUM_ST `&0 < azim a b c d` (C INTRO_TAC [`p+2`]);
      TYPIFY `azim (vec 0) (v (p+2)) (v (p+3)) (v (p+1)) = azim (vec 0) (v (p + 2)) (v ((p + 2) + 1)) (v ((p + 2) + 3))` (C SUBGOAL_THEN ASSUME_TAC);
        BY(REWRITE_TAC[arith `((p:num) + i) + j = p + i+j`] THEN NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[]);
      ASM_REWRITE_TAC[];
      DISCH_THEN (unlist REWRITE_TAC);
      FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM MATCH_MP_TAC;
      GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
      BY(NUM_REDUCE_TAC);
    DISJ1_TAC;
    CONJ_TAC;
      GEN_TAC;
      FIRST_X_ASSUM_ST `~(w = v (p+1)) ==> a` (REPEAT o GMATCH_SIMP_TAC);
      ASM_REWRITE_TAC[];
      TYPIFY `p MOD 4 = (p+0) MOD 4` (C SUBGOAL_THEN SUBST1_TAC);
        BY(REWRITE_TAC[arith `p + 0 = p`]);
      BY(REPEAT CONJ_TAC THEN GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC);
    REWRITE_TAC[Zlzthic.azim_lt_pi_cross];
    TYPIFY `(v (p + 2) cross v (p)) dot v (p+1) = (v (p+1) cross (v (p+2))) dot v (p)` (C SUBGOAL_THEN SUBST1_TAC);
      BY(VEC3_TAC);
    REWRITE_TAC[GSYM Zlzthic.azim_lt_pi_cross];
    BY(ASM_REWRITE_TAC[]);
  (REPEAT WEAKER_STRIP_TAC);
  TYPIFY `?e3. &0 < e3 /\ (!t. abs t < e3 ==> dist(v p,v (p+1)) + t IN real_interval(scs_a_v39 s p (p+1),scs_b_v39 s p (p+1)))` (C SUBGOAL_THEN MP_TAC);
    INTRO_TAC Zlzthic.real_interval_contains_0_ball [`scs_a_v39 s p (p+1)-dist(v p,v (p+1))`;`scs_b_v39 s p (p+1) - dist(v p,v (p+1))`;`&1`];
    ASM_REWRITE_TAC[arith `&0 < &1`;arith `a - d < &0 <=> a < d`;arith `&0 < b - d <=> d < b`];
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `e` EXISTS_TAC;
    ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM (C INTRO_TAC [`t`]);
    ASM_REWRITE_TAC[IN_REAL_INTERVAL];
    BY(REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  (COMMENT "coplanar");
  INTRO_TAC Zlzthic.NONPLANAR_OPEN [`(\ (t:real). (vec 0):real^3)`;`\ (t:real). (v p)`;`f( v (p+1))`;`\ (t:real). v (p+2)`;`&0`];
  ASM_REWRITE_TAC[CONTINUOUS_CONST];
  DISCH_THEN MP_TAC THEN ANTS_TAC;
    FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation] THEN REPEAT WEAKER_STRIP_TAC;
    CONJ2_TAC;
      BY(FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]);
    POP_ASSUM GMATCH_SIMP_TAC;
    BY(ASM_REWRITE_TAC[]);
  REWRITE_TAC[arith `abs (&0 - t') = abs t'`] THEN REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC Zlzthic.WNWSHJT [`v(p)`;`v (p+1)`;`v (p+2)`;`f`;`-- e'`;`e'`];
  ANTS_TAC;
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      MATCH_MP_TAC Zlzthic.deformation_subset;
      (TYPIFY `V` EXISTS_TAC THEN ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]);
      BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
    FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation];
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM (REPEAT o GMATCH_SIMP_TAC) THEN ASM_REWRITE_TAC[];
    TYPIFY `!i. ~(i=1) /\ i < 3 ==> ~collinear {vec 0,v (p+1),v(p+i)}` ENOUGH_TO_SHOW_TAC;
      BY(MESON_TAC[arith `p+0=p` ;arith `~(0=1) /\ 0 < 3 /\ ~(2=1) /\ 2 < 3`]);
    GEN_TAC THEN DISCH_TAC;
    FIRST_X_ASSUM MATCH_MP_TAC;
    BY(POP_ASSUM MP_TAC THEN ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "introduce 1834";
  INTRO_TAC Iunbuig.epsilon_hept [`e`;`e'`;`e''`;`e'''`;`e3`;`e1`;`e1'`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  FIRST_ASSUM_ST `i = 0 \/ i = 2 \/ i = 3` MP_TAC;
  FIRST_ASSUM_ST `i MOD 4 = j MOD 4` (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC (UNDISCH Ocbicby.LEMMA_1834976363) [`norm (v (p+2))`;`norm (v (p))`;`norm (v (p+1))`;`dist(v (p),v(p+1))`;`dist (v(p+2),v(p+1))`;`dist(v (p+2),v (p))`;`dist(v (p),v (p+1)) - e7`;`dist(v (p),v(p+1)) + e7`];
  ANTS_TAC;
    ASM_REWRITE_TAC[];
    (COMMENT "1834 ants");
    TYPIFY `!i. &2 <= norm (v i) /\ norm (v i) <= &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
      GEN_TAC;
      REWRITE_TAC[GSYM Fnjlbxs.in_ball_annulus];
      FIRST_X_ASSUM_ST `BBs_v39` kill THEN FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN EXPAND_TAC "V" THEN REPEAT WEAKER_STRIP_TAC;
      FIRST_X_ASSUM_ST `ball_annulus` MP_TAC;
      REWRITE_TAC[SUBSET;IN_IMAGE;IN_UNIV];
      BY(MESON_TAC[]);
    ASM_REWRITE_TAC[];
    FIRST_ASSUM_ST `collinear` (C INTRO_TAC [`p`;`2`;`0`]);
    FIRST_X_ASSUM_ST `collinear` (C INTRO_TAC [`p`;`2`;`1`]);
    NUM_REDUCE_TAC;
    REWRITE_TAC[arith `p + 0 = p`];
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY_GOAL_THEN `&0 < ups_x (norm (v (p + 2)) * norm (v (p + 2))) (norm (v p) * norm (v p)) (dist (v (p + 2),v p) * dist (v (p + 2),v p)) /\ &0 < ups_x (norm (v (p + 2)) * norm (v (p + 2))) (norm (v (p + 1)) * norm (v (p + 1))) (dist (v (p + 2),v (p + 1)) * dist (v (p + 2),v (p + 1)))` (unlist REWRITE_TAC);
      REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC);
      REWRITE_TAC[Collect_geom2.NOT_COL_EQ_UPS_X_POS;DIST_0;DIST_SYM;arith`x pow 2 = x*x`];
      BY(MESON_TAC[]);
    TYPIFY `!i j. &2 <= dist(v i,v j) ==> (&2 / h0) pow 2 <= xrr (norm(v i)) (norm(v (j))) (dist (v i,v j))` ((C SUBGOAL_THEN ASSUME_TAC));
      REPEAT WEAKER_STRIP_TAC;
      MATCH_MP_TAC Ocbicby.xrr_simple_lower_bound;
      BY(ASM_REWRITE_TAC[arith `x <= x`]);
    CONJ_TAC;
      POP_ASSUM MATCH_MP_TAC;
      MATCH_MP_TAC Ocbicby.scs_lb_2;
      TYPIFY `s` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p+0`];
      FIRST_X_ASSUM MATCH_MP_TAC;
      BY(NUM_REDUCE_TAC);
    CONJ_TAC;
      TYPIFY `#15.53 = sqrt(#15.53) pow 2` (C SUBGOAL_THEN SUBST1_TAC);
        GMATCH_SIMP_TAC SQRT_POW_2;
        BY(REAL_ARITH_TAC);
      MATCH_MP_TAC Ocbicby.xrr_simple_upper_bound;
      ASM_REWRITE_TAC[];
      TYPIFY `&4 = sqrt(&16)` (C SUBGOAL_THEN SUBST1_TAC);
        REWRITE_TAC[arith `&16 = &4 pow 2`];
        REWRITE_TAC[POW_2_SQRT_ABS];
        BY(REAL_ARITH_TAC);
      GMATCH_SIMP_TAC SQRT_MONO_LT_EQ;
      TYPIFY_GOAL_THEN `&0 <=  #15.53 /\ &0 < &16 /\  #15.53 < &16` (unlist REWRITE_TAC);
        BY(REAL_ARITH_TAC);
      CONJ_TAC;
        BY(REAL_ARITH_TAC);
      CONJ_TAC;
        MATCH_MP_TAC Ocbicby.scs_lb_2;
        TYPIFY `s` EXISTS_TAC;
        ASM_REWRITE_TAC[];
        GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p+0`];
        FIRST_X_ASSUM MATCH_MP_TAC;
        BY(NUM_REDUCE_TAC);
      MATCH_MP_TAC REAL_LE_TRANS;
      TYPIFY `cstab` EXISTS_TAC;
      CONJ_TAC;
        MATCH_MP_TAC (arith `d <= scs_b_v39 s p (p+1) /\ scs_b_v39 s p (p+1) <= cstab ==> d <= cstab`);
        CONJ_TAC;
          FIRST_X_ASSUM_ST `BBs_v39` kill;
          (FIRST_X_ASSUM_ST `BBs_v39` MP_TAC) THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM];
          BY(MESON_TAC[]);
        FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39];
        ASM_REWRITE_TAC[arith `3 < 4`;arith `SUC i = i+1`];
        BY(MESON_TAC[]);
      REWRITE_TAC[Sphere.cstab];
      MATCH_MP_TAC REAL_LE_RSQRT;
      BY(REAL_ARITH_TAC);
    CONJ_TAC;
      FIRST_X_ASSUM MATCH_MP_TAC;
      MATCH_MP_TAC Ocbicby.scs_lb_2;
      TYPIFY `s` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      BY(GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC);
    CONJ_TAC;
      FIRST_X_ASSUM MATCH_MP_TAC;
      MATCH_MP_TAC Ocbicby.scs_lb_2;
      TYPIFY `s` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p+0`];
      BY(GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC);
    CONJ_TAC;
      REWRITE_TAC[IN_REAL_INTERVAL];
      BY(FIRST_X_ASSUM_ST `&0 < e` MP_TAC THEN REAL_ARITH_TAC);
    COMMENT "subset constraints";
    CONJ_TAC;
      REWRITE_TAC[SUBSET;IN_REAL_INTERVAL;IN_ELIM_THM];
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `?t. abs t < e7 /\ x = dist(v p,v(p+1)) + t` (C SUBGOAL_THEN MP_TAC);
        TYPIFY `x - dist(v p,v(p+1))` EXISTS_TAC;
        CONJ2_TAC;
          BY(REAL_ARITH_TAC);
        BY(REPLICATE_TAC 2 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
      REPEAT WEAKER_STRIP_TAC;
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM_ST `norm (f (v (p+1)) t) = norm (v (p+1))` (C INTRO_TAC [`t`]);
      ANTS_TAC;
        BY(REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]);
      REWRITE_TAC[DIST_SYM] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[];
      FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`t`]);
      ANTS_TAC;
        BY(REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]);
      DISCH_TAC;
      FIRST_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,d,b,c}`] THEN REWRITE_TAC[Oxlzlez.coplanar_delta_y;DIST_0;DIST_SYM];
      RULE_ASSUM_TAC (REWRITE_RULE[arith `d + &0 = d`]);
      ASM_REWRITE_TAC[];
      DISCH_THEN (unlist REWRITE_TAC);
      REWRITE_TAC[GSYM DIST_NZ];
      CONJ_TAC;
        DISCH_TAC;
        FIRST_X_ASSUM_ST `coplanar` MP_TAC;
        BY(ASM_REWRITE_TAC[SET_RULE `{a,b,b,c} = {a,b,c}`;COPLANAR_3]);
      TYPIFY `~collinear {vec 0, (v p) ,f (v (p+1)) t}` (C SUBGOAL_THEN MP_TAC);
        BY(POP_ASSUM MP_TAC THEN MESON_TAC[Planarity.notcoplanar_imp_notcollinear_fan]);
      REWRITE_TAC[Collect_geom2.NOT_COL_EQ_UPS_X_POS];
      BY(REWRITE_TAC[DIST_0;arith `x pow 2 = x*x`;DIST_SYM]);
    TYPIFY `!i.  i < 2 ==> &0 < dist ( v (p+2),v (p+i))` ENOUGH_TO_SHOW_TAC;
      BY(MESON_TAC[arith `p + 0 = (p:num)`;arith `0 < 2`;arith `1 < 2`]);
    REPEAT WEAKER_STRIP_TAC;
    MATCH_MP_TAC (arith `&2 <= x ==> &0 < x`);
    MATCH_MP_TAC Ocbicby.scs_lb_2;
    TYPIFY `s` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC;
    BY(FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[arith `i < 2 <=> i = 0 \/ i = 1`] THEN DISCH_TAC THEN REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[] THEN NUM_REDUCE_TAC);
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "root";
  TYPIFY `?t. abs t < e7 /\ y4' = dist(v p,v(p+1)) + t` (C SUBGOAL_THEN MP_TAC);
    TYPIFY `y4' - dist(v p,v(p+1))` EXISTS_TAC;
    CONJ2_TAC;
      BY(REAL_ARITH_TAC);
    BY(REPLICATE_TAC 2 (POP_ASSUM MP_TAC) THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM_ST `taum` MP_TAC;
  ASM_REWRITE_TAC[];
  REWRITE_TAC[arith `~(x < y) <=> (y <= x)`];
  FIRST_X_ASSUM_ST `norm (f (v p) t) = norm (v p)` (C INTRO_TAC [`t`]);
  ANTS_TAC;
    BY(REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]);
  (REWRITE_TAC[DIST_SYM;arith `d + &0 = d`] THEN REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[]);
  COMMENT "back to root, next tau";
  TYPIFY `sum {i | i < 4} (\i. rho_fun (norm (v i)) * azim (vec 0) (v i) (v (i + 1)) (v (i + 4 - 1))) <=    sum {i | i < 4} (\i. rho_fun (norm (f (v i) t)) * azim (vec 0) (f (v i) t) (f(v(i+1)) t) (f (v (i + 4 - 1)) t))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    MATCH_MP_TAC Cuxvzoz.MMs_minimize_tau_fun;
    TYPIFY `s` EXISTS_TAC;
    ASM_REWRITE_TAC[arith `3 < 4`];
    FIRST_X_ASSUM MATCH_MP_TAC;
    BY(FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]);
  POP_ASSUM MP_TAC;
  TYPIFY `{i | i < 4} = 0..(4-1)` (C SUBGOAL_THEN SUBST1_TAC);
    BY(REWRITE_TAC[EXTENSION;IN_ELIM_THM;IN_NUMSEG] THEN ARITH_TAC);
  REPEAT ( GMATCH_SIMP_TAC (GSYM Oxl_def.periodic_sum));
  TYPIFY `p` EXISTS_TAC;
  CONJ_TAC;
    NUM_REDUCE_TAC;
    FIRST_ASSUM_ST `periodic` MP_TAC THEN REWRITE_TAC[Oxl_def.periodic];
    REWRITE_TAC[arith `(i+4)+k = (i+k)+4`];
    BY(DISCH_THEN (unlist REWRITE_TAC));
  TYPIFY `p` EXISTS_TAC;
  CONJ_TAC;
    NUM_REDUCE_TAC;
    FIRST_ASSUM_ST `periodic` MP_TAC THEN REWRITE_TAC[Oxl_def.periodic];
    REWRITE_TAC[arith `(i+4)+k = (i+k)+4`];
    BY(DISCH_THEN (unlist REWRITE_TAC));
  NUM_REDUCE_TAC;
  REWRITE_TAC[arith `3+p = SUC(SUC(SUC p))`;SUM_CLAUSES_NUMSEG;SUM_SING_NUMSEG;arith `p <= SUC p /\ p <= SUC (SUC p) /\ p <= SUC(SUC(SUC p))`];
  REWRITE_TAC[arith `SUC i = i+1`;arith `(i + j) + (k:num) = i+(j+k)`];
  NUM_REDUCE_TAC;
  ASM_REWRITE_TAC[];
  REPLICATE_TAC 7 (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  REPEAT (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM_ST `rho_fun` MP_TAC THEN ASM_REWRITE_TAC[Appendix.rho_rho_fun];
  TYPIFY `f (v(p+0)) t = v(p+0) /\ f (v (p+2)) t = v(p+2) /\ f(v (p+3)) t = v(p+3)` (C SUBGOAL_THEN MP_TAC);
    FIRST_X_ASSUM_ST `~(w = v p) ==> a` (REPEAT o GMATCH_SIMP_TAC);
    BY(ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  REWRITE_TAC[arith `p + 0 = p`] THEN DISCH_TAC;
  POP_ASSUM (fun t -> ASM_TAC THEN unlist REWRITE_TAC t THEN REPEAT DISCH_TAC THEN ASSUME_TAC t);
  FIRST_X_ASSUM_ST `azim` MP_TAC;
  FIRST_X_ASSUM_ST `azim` MP_TAC THEN BURY_MP_TAC;
  FIRST_ASSUM_ST `azim` ((unlist REWRITE_TAC) o GSYM);
  FIRST_ASSUM_ST ` azim (vec 0) (v p) (f (v (p + 1)) t) (v (p + 3))` ((unlist REWRITE_TAC) o GSYM);
  REWRITE_TAC[arith `(a+b)+c = a + b + c`];
  FIRST_X_ASSUM_ST `SUC` MP_TAC THEN REWRITE_TAC[arith `SUC p = p+1`];
  DISCH_THEN (SUBST1_TAC );
  REWRITE_TAC[arith `r0 * (a012+a023) + r1 * (a120) + r2 * (a230+a201) + r3 * (a302)  <=  r0 * (f012+a023) + r1 * (f120) + r2 * (a230+f201) + r3 *(a302) <=> r0 * a012 + r1*a120 + r2* a201 <=  r0 * f012 + r1 * f120 + r2 * f201`];
  MATCH_MP_TAC (arith `a - (pi+sol0) = a' /\ b - (pi+sol0) = b' ==> (a <= b ==> a' <= b')`);
  REWRITE_TAC[arith `(a + b+c) - (pi + sol0) = a + b + c - (pi + sol0)`];
  CONJ_TAC;
    ONCE_REWRITE_TAC[arith `a + b + c - p = b + c + a - p`];
    GMATCH_SIMP_TAC (GSYM Cuxvzoz.tau3_azim);
    GMATCH_SIMP_TAC Cuxvzoz.tau3_taum_nonplanar;
    ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d}  = {a,d,b,c}`];
    ASM_REWRITE_TAC[];
    CONJ2_TAC;
      REWRITE_TAC[DIST_SYM];
      BY(MESON_TAC[Terminal.taum_sym]);
    FIRST_X_ASSUM_ST `a <= pi` kill;
    FIRST_X_ASSUM_ST `a <= pi` (C INTRO_TAC [`p+1`]);
    BY(REWRITE_TAC[arith `(p+1)+k = p + (1 +k)`] THEN NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[]);
  FIRST_X_ASSUM_ST `norm a  = norm b` (SUBST1_TAC o GSYM);
  FIRST_X_ASSUM_ST `dist s + t` (SUBST1_TAC o GSYM);
  ONCE_REWRITE_TAC[arith `a + b + c - p = b + c + a - p`];
  GMATCH_SIMP_TAC (GSYM Cuxvzoz.tau3_azim);
  GMATCH_SIMP_TAC Cuxvzoz.tau3_taum_nonplanar;
  REWRITE_TAC[DIST_SYM];
  REWRITE_TAC[CONJ_ASSOC] THEN CONJ2_TAC;
    ASM_REWRITE_TAC[DIST_SYM];
    BY(MESON_TAC[Terminal.taum_sym]);
  ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,d,b,c}`];
  ASM_REWRITE_TAC[];
  BY(FIRST_X_ASSUM_ST `azim (vec 0) (f (v (p+1)) t) (v (p + 2)) (v (p)) < pi` MP_TAC THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let JOTSWIX = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
  (!s FF v p.
  IMAGE (\i. (v i,v (SUC i))) (:num) = FF /\
  is_scs_v39 s /\
  scs_k_v39 s = 4 /\
  MMs_v39 s v /\
  scs_basic_v39 s /\
  scs_generic v /\
  scs_a_v39 s p (p+1) < cstab /\
  (!i j. scs_diag 4 i j ==> scs_a_v39 s i j <= cstab /\ cstab < dist(v i,v j) /\ &4 * h0 < scs_b_v39 s i j) /\
  (!i. ~(i MOD 4 = (p+2) MOD 4) ==> interior_angle1 (vec 0) FF (v i) < pi) /\
  (xrr (norm (v (p+3))) (norm (v (p+1))) (dist(v (p+3),v(p+1))) <= #15.53)
  ==> (dist(v p,v(p+1)) < cstab \/ dist(v p,v(p+3)) < cstab))`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC (arith `~((cstab <= d) /\ (cstab <= d')) ==> (d < cstab \/ d' < cstab)`);
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `BBprime2_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.scs_basic;Ayqjtmd.unadorned_MMs]);
  TYPIFY `BBprime_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime2_v39]);
  TYPIFY `BBs_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime_v39]);
  INTRO_TAC Appendix.BBs_v39 [`s`;`v`];
  ASM_REWRITE_TAC[LET_DEF;LET_END_DEF;Terminal.IMAGE_SUBSET_IN;IN_UNIV;arith `~(4 <= 3)`];
  REPEAT WEAKER_STRIP_TAC;
  TYPED_ABBREV_TAC `V = IMAGE v (:num)`;
  TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`;
  TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    EXPAND_TAC "V";
    REWRITE_TAC[IN_IMAGE;IN_UNIV];
    BY(MESON_TAC[]);
  INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`4`];
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  TYPIFY `!i. interior_angle1 (vec 0) FF (v i) = azim (vec 0) (v i) (v (i+1)) (v (i+3))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    EXPAND_TAC "FF";
    GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] (GSYM Ocbicby.INTERIOR_ANGLE1_AZIM));
    TYPIFY `s` EXISTS_TAC;
    BY(ASM_REWRITE_TAC[IN;arith `4 - 1 = 3`;arith `3 < 4`;arith `SUC i = i+1`]);
  INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`];
  ASM_REWRITE_TAC[LET_THM;IN;arith `4-1 = 3`;arith `SUC i = i+1`];
  DISCH_TAC;
  TYPIFY `!i. v (i MOD 4) = v i` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC (GSYM Oxl_def.periodic_mod);
    BY(ASM_REWRITE_TAC[arith `~(0=4)`]);
  TYPIFY `v 4 = v 0 /\ v 5 = v 1 /\ v 6 = v 2 /\ v 7 = v 3 /\ v 8  = v 4` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[Terminal.MOD_4_EXPLICIT;arith `7 = 1*4 + 3`;arith `8 = 2*4 + 0`;MOD_MULT_ADD]);
  TYPIFY `generic V E` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM_ST `scs_generic` MP_TAC THEN ASM_REWRITE_TAC[Appendix.scs_generic]);
  TYPIFY `v (p+4) = v p /\ v (p+5) = v (p+1) /\ v (p+6) = v (p+2) /\ v (p+7) = v (p+3) /\ v (p+8)  = v (p)` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `MOD` ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[arith `p+4 = 1*4 + p /\ p+5 = 1*4+(p+1) /\ p+6=1*4+(p+2) /\ p+7 = 1*4 + (p+3) /\ p+8 = 2*4 + p`;MOD_MULT_ADD]);
  INTRO_TAC Terminal.vv_split_azim_generic [`v`;`4`;`p+3`;`2`;`3`];
  REWRITE_TAC[arith `SUC (x+k) = (x+k)+1`;arith `((p:num)+i)+j = p+(i+j)`];
  NUM_REDUCE_TAC;
  ASM_REWRITE_TAC[LET_THM];
  DISCH_TAC;
  INTRO_TAC Terminal.vv_split_azim_generic [`v`;`4`;`p+1`;`2`;`3`];
  REWRITE_TAC[arith `SUC (x+k) = (x+k)+1`;arith `((p:num)+i)+j = p+(i+j)`];
  NUM_REDUCE_TAC;
  ASM_REWRITE_TAC[LET_THM];
  DISCH_TAC;
  COMMENT "collinearity";
  TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0,v i, v j}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT GEN_TAC THEN DISCH_TAC;
    MATCH_MP_TAC Zlzthic.PROPERTIES_GENERIC_LOCAL_FAN_ALT;
    GEXISTL_TAC [`V`;`E`;`FF`];
    BY(ASM_REWRITE_TAC[IN_IMAGE;IN_UNIV]);
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
  INTRO_TAC PERIODIC_INJ_MOD [`v`;`4`];
  ASM_REWRITE_TAC[arith `~(4 = 0)`];
  DISCH_TAC;
  TYPIFY `azim (vec 0) (v p) (v (p+1)) (v (p+3)) < pi` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
    FIRST_X_ASSUM MATCH_MP_TAC;
    GEN_REWRITE_TAC(RAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[arith`p = p+0`];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    BY(NUM_REDUCE_TAC);
  TYPIFY `~coplanar {vec 0,v p,v (p+1),v (p+3)}` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR];
    MATCH_MP_TAC (arith `&0 < x /\ x < pi ==> ~ ((x = &0) \/ (x = pi))`);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i a b. a < 4 /\ b < 4 /\ ~(a = b) ==> ~collinear {vec 0,v(i+a),v(i+b)}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[];
    FIRST_X_ASSUM_ST `v (i MOD 4) = v i` ((unlist ONCE_REWRITE_TAC) o GSYM);
    REPEAT (FIRST_X_ASSUM MATCH_MP_TAC);
    nCONJ_TAC 2;
      GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
      NUM_REDUCE_TAC;
      BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[DIVISION;MOD_LT;arith `~(4=0)`]);
    ASM_REWRITE_TAC[];
    BY(MESON_TAC[DIVISION;arith `~(4=0)`]);
  COMMENT "introduce deformation";
  INTRO_TAC Cuxvzoz.deform_simplex_edge_exists [`V`;`(\ (t:real). &0)`;`(\(t:real). -- abs t)`;`v (p +3)`;`v p`;`v (p +1)`;`&1`];
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM MP_TAC;
  ANTS_TAC;
    REWRITE_TAC[arith `&0 < &1`];
    SUBCONJ_TAC;
      ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d}={a,c,d,b}`];
      BY(ASM_REWRITE_TAC[]);
    DISCH_TAC;
    CONJ_TAC;
      INTRO_TAC Planarity.cross_dot_fully_surrounded_fan [`(vec 0):real^3`;`v p`;`v (p+3)`;`v(p+1)`];
      ASM_REWRITE_TAC[];
      ANTS_TAC;
        INTRO_TAC Planarity.notcoplanar_imp_notcollinear_fan [`(vec 0):real^3`;`v p`;`v (p+1)`;`v(p+3)`];
        BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
      MATCH_MP_TAC (arith `x = y ==> (&0 < x ==> y > &0)`);
      BY(VEC3_TAC);
    REPEAT ( GMATCH_SIMP_TAC REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT);
    REWRITE_TAC[Ocbicby.REAL_OPEN_REAL_INTERVAL];
    (REWRITE_TAC[REAL_CONTINUOUS_CONST;REAL_CONTINUOUS_AT_ID;arith `-- abs(&0) = &0`]);
    REPEAT WEAKER_STRIP_TAC;
    MATCH_MP_TAC REAL_CONTINUOUS_NEG;
    BY(REWRITE_TAC[Cuxvzoz.real_continuous_abs]);
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "A";
  TYPIFY `!i. dist (v i,v (i+1)) <= cstab` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC REAL_LE_TRANS;
    TYPIFY `scs_b_v39 s i (i+1)` EXISTS_TAC;
    CONJ_TAC;
      BY(ASM_MESON_TAC[]);
    FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM_ST `scs_b_v39 s i j <= cstab` (C INTRO_TAC [`i`]);
    BY(ASM_REWRITE_TAC[arith `3 < 4`;arith `SUC i =  i+1`]);
  TYPIFY `(!i t. abs t < e' ==> norm (f (v i) t) = norm (v i))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `v i = v p` ASM_CASES_TAC;
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM_ST `norm` (C INTRO_TAC [`t`]);
      BY(ASM_REWRITE_TAC[] THEN MESON_TAC[]);
    FIRST_X_ASSUM GMATCH_SIMP_TAC;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. 0 < i /\ i < 4 ==> ~(v (p + i) = v p)` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM MP_TAC;
    ASM_REWRITE_TAC[];
    GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p+0`];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    NUM_REDUCE_TAC;
    BY(REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[DIVISION;MOD_LT;arith `~(4=0)`;arith `0 < i ==> ~(i=0)`]);
  TYPIFY `!i j (k:num). v i = v j ==> v (i+k) = v (j+k)` (C SUBGOAL_THEN ASSUME_TAC);
    ASM_REWRITE_TAC[];
    REPEAT WEAKER_STRIP_TAC;
    ONCE_REWRITE_TAC[arith `(i:num) + k = k + i`];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    BY(ASM_REWRITE_TAC[arith `~(4 = 0)`]);
  TYPIFY `!t. azim (vec 0) (f (v (p+2)) t) (f (v (p+3)) t) (f (v (p+1)) t) = azim(vec 0) (v(p+2)) (v(p+3)) (v(p+1))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    FIRST_X_ASSUM_ST `~(w = v p) ==> a` (REPEAT o GMATCH_SIMP_TAC);
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  TYPIFY `!i. azim (vec 0) (v i) (v (i+1)) (v (i+3)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
    MATCH_MP_TAC Local_lemmas.CONVEX_LOFA_IMP_INANGLE_LE_PI;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `(!i. ?e0. &0 < e0 /\           (azim (vec 0) (v i) (v (i + 1)) (v (i + 3)) = pi            ==> (!t. abs t < e0                     ==> azim (vec 0) (f (v i) t) (f (v (i + 1)) t)                         (f (v (i + 3)) t) <=                         pi)))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    TYPIFY `v i = v (p+2)` ASM_CASES_TAC;
      FIRST_ASSUM (fun t -> FIRST_X_ASSUM (ASSUME_TAC o (C MATCH_MP t)));
      TYPIFY `&1` EXISTS_TAC;
      ASM_REWRITE_TAC[arith `&0 < &1`;arith `(p+2)+k = p + (2+k)`];
      NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[];
      BY(REAL_ARITH_TAC);
    TYPIFY `azim (vec 0) (v i) (v (i + 1)) (v (i + 3)) < pi` ENOUGH_TO_SHOW_TAC;
      BY(MESON_TAC[arith `x < pi ==> ~(x = pi)`;arith `&0 < &1`]);
    BY(ASM_MESON_TAC[]);
  (COMMENT "deformation in BBs");
  INTRO_TAC Cuxvzoz.deformation_BBs_ALT [`s`;`4`;`f`;`v`;`e'`];
  ANTS_TAC;
    NUM_REDUCE_TAC;
    ASM_REWRITE_TAC[];
    MATCH_MP_TAC ab4_assumption_reduction2;
    GEXISTL_TAC [`p`;`e'`];
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      REPEAT WEAKER_STRIP_TAC;
      FIRST_X_ASSUM_ST `scs_diag` (C INTRO_TAC [`i`;`j`]);
      BY(ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);
    REWRITE_TAC[CONJ_ASSOC] THEN CONJ2_TAC;
      TYPIFY `e'` EXISTS_TAC;
      ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
      (FIRST_X_ASSUM_ST `norm` (C INTRO_TAC [`t`]));
      TYPIFY `dist (v p,v (p+1)) <= scs_b_v39 s p (p+1)` ENOUGH_TO_SHOW_TAC;
        BY(POP_ASSUM (unlist REWRITE_TAC) THEN REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC);
      BY(ASM_REWRITE_TAC[]);
    CONJ_TAC;
      REPEAT WEAKER_STRIP_TAC;
      (FIRST_X_ASSUM_ST `norm` (C INTRO_TAC [`t`]));
      BY(ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);
    MATCH_MP_TAC REAL_LTE_TRANS;
    TYPIFY `cstab` EXISTS_TAC;
    BY(ASM_REWRITE_TAC[]);
  (REPEAT WEAKER_STRIP_TAC);
  (COMMENT "azim sums");
  INTRO_TAC deform_azim_sum [`v (p+3)`;`v p`;`v (p+1)`;`v(p+2)`;`f`;`e'`];
  ANTS_TAC;
    REPEAT (FIRST_X_ASSUM_ST `c = a + b` MP_TAC) THEN ASM_REWRITE_TAC[];
    REPLICATE_TAC 3 DISCH_TAC;
    CONJ_TAC;
      MATCH_MP_TAC Zlzthic.deformation_subset;
      (TYPIFY `V` EXISTS_TAC THEN ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]);
      BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
    TYPIFY `~collinear {vec 0, v (p + 3), v (p+0)} /\ ~collinear {vec 0, v (p + 3), v (p + 1)} /\ ~collinear {vec 0, v (p + 3), v (p + 2)}` (C SUBGOAL_THEN MP_TAC);
      BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN (unlist REWRITE_TAC);
    nCONJ_TAC 2;
      BY(ASM_REWRITE_TAC[]);
    FIRST_X_ASSUM (SUBST1_TAC) THEN REWRITE_TAC[];
    REPLICATE_TAC 4 (FIRST_X_ASSUM kill);
    ASM_REWRITE_TAC[];
    TYPIFY_GOAL_THEN `&0 < azim (vec 0) (v (p + 3)) (v p) (v (p + 2)) /\ azim (vec 0) (v (p + 3)) (v p) (v (p + 2)) < pi` (unlist REWRITE_TAC);
      FIRST_X_ASSUM_ST `&0 < azim a b c d` (C INTRO_TAC [`p+3`]);
      TYPIFY `azim (vec 0) (v (p+3)) (v p) (v (p+2)) = azim (vec 0) (v (p + 3)) (v ((p + 3) + 1)) (v ((p + 3) + 3))` (C SUBGOAL_THEN ASSUME_TAC);
        BY(REWRITE_TAC[arith `((p:num) + i) + j = p + i+j`] THEN NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[]);
      ASM_REWRITE_TAC[];
      DISCH_THEN (unlist REWRITE_TAC);
      FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM MATCH_MP_TAC;
      GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
      BY(NUM_REDUCE_TAC);
    DISJ2_TAC;
    CONJ_TAC;
      GEN_TAC;
      FIRST_X_ASSUM_ST `~(w = v p) ==> a` (REPEAT o GMATCH_SIMP_TAC);
      ASM_REWRITE_TAC[];
      BY(REPEAT CONJ_TAC THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p+0`] THEN GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC);
    REWRITE_TAC[Zlzthic.azim_lt_pi_cross];
    TYPIFY `(v (p + 3) cross v p) dot v (p + 1) = (v (p) cross (v (p+1))) dot v (p + 3)` (C SUBGOAL_THEN SUBST1_TAC);
      BY(VEC3_TAC);
    REWRITE_TAC[GSYM Zlzthic.azim_lt_pi_cross];
    BY(ASM_REWRITE_TAC[]);
  (REPEAT WEAKER_STRIP_TAC);
  COMMENT "second azim sum";
  INTRO_TAC deform_azim_sum [`v (p+1)`;`v (p+2)`;`v (p+3)`;`v(p)`;`f`;`e'`];
  ANTS_TAC;
    REPEAT (FIRST_X_ASSUM_ST `c = a + b` MP_TAC) THEN ASM_REWRITE_TAC[];
    REPLICATE_TAC 3 DISCH_TAC;
    CONJ_TAC;
      MATCH_MP_TAC Zlzthic.deformation_subset;
      (TYPIFY `V` EXISTS_TAC THEN ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]);
      BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
    TYPIFY `~collinear {vec 0, v (p + 1), v (p + 2)} /\ ~collinear {vec 0, v (p + 1), v (p + 3)} /\ ~collinear {vec 0, v (p + 1), v (p + 0)}` (C SUBGOAL_THEN MP_TAC);
      BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN (unlist REWRITE_TAC);
    nCONJ_TAC 2;
      BY(ASM_REWRITE_TAC[]);
    FIRST_X_ASSUM (SUBST1_TAC) THEN REWRITE_TAC[];
    REPLICATE_TAC 4 (FIRST_X_ASSUM kill);
    ASM_REWRITE_TAC[];
    TYPIFY_GOAL_THEN `&0 < azim (vec 0) (v (p + 1)) (v (p + 2)) (v p) /\ azim (vec 0) (v (p + 1)) (v (p + 2)) (v p) < pi` (unlist REWRITE_TAC);
      FIRST_X_ASSUM_ST `&0 < azim a b c d` (C INTRO_TAC [`p+1`]);
      TYPIFY `azim (vec 0) (v (p+1)) (v (p+2)) (v (p)) = azim (vec 0) (v (p + 1)) (v ((p + 1) + 1)) (v ((p + 1) + 3))` (C SUBGOAL_THEN ASSUME_TAC);
        BY(REWRITE_TAC[arith `((p:num) + i) + j = p + i+j`] THEN NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[]);
      ASM_REWRITE_TAC[];
      DISCH_THEN (unlist REWRITE_TAC);
      FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM MATCH_MP_TAC;
      GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
      BY(NUM_REDUCE_TAC);
    DISJ1_TAC;
    CONJ_TAC;
      GEN_TAC;
      FIRST_X_ASSUM_ST `~(w = v p) ==> a` (REPEAT o GMATCH_SIMP_TAC);
      ASM_REWRITE_TAC[];
      BY(REPEAT CONJ_TAC THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [arith `p = p+0`] THEN GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT THEN NUM_REDUCE_TAC);
    REWRITE_TAC[Zlzthic.azim_lt_pi_cross];
    TYPIFY `(v (p + 1) cross v (p+3)) dot v (p) = (v (p) cross (v (p+1))) dot v (p + 3)` (C SUBGOAL_THEN SUBST1_TAC);
      BY(VEC3_TAC);
    REWRITE_TAC[GSYM Zlzthic.azim_lt_pi_cross];
    BY(ASM_REWRITE_TAC[]);
  (REPEAT WEAKER_STRIP_TAC);
  TYPIFY `?e3. &0 < e3 /\ (!t. abs t < e3 /\ t < &0 ==> dist(v p,v (p+1)) + t IN real_interval(scs_a_v39 s p (p+1),scs_b_v39 s p (p+1)))` (C SUBGOAL_THEN MP_TAC);
    TYPIFY `dist(v p,v(p+1)) - scs_a_v39 s p (p+1)` EXISTS_TAC;
    ASM_REWRITE_TAC[arith `&0 < &1`;arith `a - d < &0 <=> a < d`;arith `&0 < b - d <=> d < b`];
    SUBCONJ_TAC;
      MATCH_MP_TAC REAL_LTE_TRANS;
      TYPIFY `cstab` EXISTS_TAC;
      BY(ASM_REWRITE_TAC[]);
    DISCH_TAC;
    REPEAT WEAKER_STRIP_TAC;
    REWRITE_TAC[IN_REAL_INTERVAL];
    CONJ_TAC;
      BY(REPLICATE_TAC 3 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
    MATCH_MP_TAC (arith `t < &0 /\ d <= b ==> d + t < b`);
    BY(ASM_REWRITE_TAC[]);
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "coplanar";
  INTRO_TAC Zlzthic.NONPLANAR_OPEN [`(\ (t:real). (vec 0):real^3)`;`f (v p)`;`\ (t:real). v (p+1)`;`\ (t:real). v (p+3)`;`&0`];
  ASM_REWRITE_TAC[CONTINUOUS_CONST];
  DISCH_THEN MP_TAC THEN ANTS_TAC;
    FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation] THEN REPEAT WEAKER_STRIP_TAC;
    CONJ2_TAC;
      BY(FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]);
    POP_ASSUM GMATCH_SIMP_TAC;
    BY(ASM_REWRITE_TAC[]);
  REWRITE_TAC[arith `abs (&0 - t') = abs t'`] THEN REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC Zlzthic.WNWSHJT [`v(p+3)`;`v p`;`v (p+1)`;`f`;`-- e'`;`e'`];
  ANTS_TAC;
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      MATCH_MP_TAC Zlzthic.deformation_subset;
      (TYPIFY `V` EXISTS_TAC THEN ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]);
      BY(REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]);
    FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation];
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM (REPEAT o GMATCH_SIMP_TAC) THEN ASM_REWRITE_TAC[];
    TYPIFY `!i. ~(i=0) /\ i < 4 ==> ~collinear {vec 0,v (p+0),v(p+i)}` ENOUGH_TO_SHOW_TAC;
      BY(MESON_TAC[arith `p+0=p` ;arith `~(1=0) /\ 1 < 4 /\ ~(3=0) /\ 3 < 4`]);
    GEN_TAC THEN DISCH_TAC;
    FIRST_X_ASSUM MATCH_MP_TAC;
    BY(ASM_REWRITE_TAC[arith `0 < 4`]);
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC Iunbuig.epsilon_hept [`e`;`e'`;`e''`;`e'''`;`e3`;`e1`;`e1'`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  FIRST_ASSUM_ST `!i. 0 < i /\ i < 4 ==> ~(v (p + i) = v p)` MP_TAC;
  FIRST_ASSUM_ST `i MOD 4 = j MOD 4` (unlist REWRITE_TAC);
  DISCH_TAC;
  COMMENT "introduce 684";
  INTRO_TAC (UNDISCH Ocbicby.LEMMA_6843920790) [`norm (v (p+3))`;`norm (v p)`;`norm (v (p+1))`;`dist(v p,v(p+1))`;`dist (v(p+1),v(p+3))`;`dist(v p,v (p+3))`];
  ANTS_TAC;
    ASM_REWRITE_TAC[];
    (COMMENT "684 ants");
    TYPIFY `!i. &2 <= norm (v i) /\ norm (v i) <= &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
      GEN_TAC;
      REWRITE_TAC[GSYM Fnjlbxs.in_ball_annulus];
      FIRST_X_ASSUM_ST `BBs_v39` kill THEN FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN EXPAND_TAC "V" THEN REPEAT WEAKER_STRIP_TAC;
      FIRST_X_ASSUM_ST `ball_annulus` MP_TAC;
      REWRITE_TAC[SUBSET;IN_IMAGE;IN_UNIV];
      BY(MESON_TAC[]);
    ASM_REWRITE_TAC[];
    REWRITE_TAC[CONJ_ASSOC] THEN CONJ2_TAC;
      FIRST_X_ASSUM_ST `coplanar` kill;
      FIRST_X_ASSUM_ST `coplanar` MP_TAC;
      ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d}= {a,d,b,c}`];
      BY(REWRITE_TAC[Oxlzlez.coplanar_delta_y;DIST_0;DIST_SYM]);
    ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[];
    TYPIFY `!i j. &2 <= dist(v i,v j) /\ dist(v i,v j) <= cstab ==> xrr (norm (v i)) (norm (v j)) (dist (v i,v j)) <= #15.53` (C SUBGOAL_THEN ASSUME_TAC);
      REPEAT WEAKER_STRIP_TAC;
      TYPIFY `#15.53 = sqrt(#15.53) pow 2` (C SUBGOAL_THEN SUBST1_TAC);
        GMATCH_SIMP_TAC SQRT_POW_2;
        BY(REAL_ARITH_TAC);
      MATCH_MP_TAC Ocbicby.xrr_simple_upper_bound;
      ASM_REWRITE_TAC[];
      TYPIFY `&4 = sqrt(&16)` (C SUBGOAL_THEN SUBST1_TAC);
        REWRITE_TAC[arith `&16 = &4 pow 2`];
        REWRITE_TAC[POW_2_SQRT_ABS];
        BY(REAL_ARITH_TAC);
      GMATCH_SIMP_TAC SQRT_MONO_LT_EQ;
      TYPIFY_GOAL_THEN `&0 <=  #15.53 /\ &0 <= &16 /\  #15.53 < &16` (unlist REWRITE_TAC);
        BY(REAL_ARITH_TAC);
      MATCH_MP_TAC REAL_LE_TRANS;
      TYPIFY `cstab` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      REWRITE_TAC[Sphere.cstab];
      MATCH_MP_TAC REAL_LE_RSQRT;
      BY(REAL_ARITH_TAC);
    FIRST_X_ASSUM ( GMATCH_SIMP_TAC);
    TYPIFY ` &2 <= dist(v p,v(p+1)) /\ dist(v p,v (p+3)) <= cstab /\ cstab <= dist(v p,v(p+3)) /\ dist(v p,v(p+1)) <= cstab /\ cstab <= dist (v (p+1),v(p+3))` ENOUGH_TO_SHOW_TAC;
      BY(REWRITE_TAC[Sphere.cstab;DIST_SYM] THEN REAL_ARITH_TAC);
    CONJ_TAC;
      MATCH_MP_TAC Ocbicby.scs_lb_2;
      TYPIFY `s` EXISTS_TAC;
      ASM_REWRITE_TAC[];
      ONCE_REWRITE_TAC[EQ_SYM_EQ];
      FIRST_X_ASSUM MATCH_MP_TAC;
      BY(NUM_REDUCE_TAC);
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      MATCH_MP_TAC REAL_LE_TRANS;
      TYPIFY `scs_b_v39 s (p+3) (SUC (p+3))` EXISTS_TAC;
      CONJ2_TAC;
        FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
        FIRST_X_ASSUM MATCH_MP_TAC;
        BY(ASM_REWRITE_TAC[] THEN NUM_REDUCE_TAC);
      ONCE_REWRITE_TAC[DIST_SYM];
      TYPIFY `v p = v (SUC (p+3))` ENOUGH_TO_SHOW_TAC;
        DISCH_THEN SUBST1_TAC;
        FIRST_X_ASSUM_ST `d <= scs_b_v39 s i j` MP_TAC;
        BY(MESON_TAC[]);
      BY(ASM_REWRITE_TAC[arith `SUC (p+3) = p+4`;Oxl_2012.MOD_ADD_CANCEL]);
    MATCH_MP_TAC (arith `c < d ==> c <= d`);
    FIRST_X_ASSUM_ST `cstab < d` (C INTRO_TAC [`p+1`;`p+3`]);
    ANTS_TAC;
      REWRITE_TAC[arith `p+3 = SUC(SUC (p+1))`];
      MATCH_MP_TAC Tfitskc.SCS_DIAG_2;
      BY(NUM_REDUCE_TAC);
    BY(DISCH_THEN (unlist REWRITE_TAC));
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `?e4. &0 < e4 /\ (!t. abs t < e4 ==> dist(v p,v (p+1)) + t IN real_interval(a,b))` (C SUBGOAL_THEN MP_TAC);
    INTRO_TAC Zlzthic.real_interval_contains_0_ball [`a-dist(v p,v (p+1))`;`b - dist(v p,v (p+1))`;`&1`];
    ASM_REWRITE_TAC[arith `&0 < &1`;arith `a - d < &0 <=> a < d`;arith `&0 < b - d <=> d < b`];
    ANTS_TAC;
      BY(FIRST_X_ASSUM_ST `dist s IN real_interval(a,b)` MP_TAC THEN REWRITE_TAC[IN_REAL_INTERVAL]);
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `e''''` EXISTS_TAC;
    ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM (C INTRO_TAC [`t`]);
    ASM_REWRITE_TAC[IN_REAL_INTERVAL];
    BY(REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC Cuxvzoz.epsilon_pair [`e4`;`e7`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  TYPIFY `?t. abs t < e'''' /\ t < &0` (C SUBGOAL_THEN MP_TAC);
    TYPIFY `-- e'''' / &2` EXISTS_TAC;
    BY(REPLICATE_TAC 3 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  REPLICATE_TAC 2 (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  REPLICATE_TAC 7 (FIRST_X_ASSUM_ST `abs t < e ==> a` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM (C INTRO_TAC [`dist(v p,v(p+1)) + t`;`dist(v p,v (p+1))`]);
  ANTS_TAC;
    ASM_REWRITE_TAC[];
    BY(FIRST_X_ASSUM_ST `t < &0` MP_TAC THEN REAL_ARITH_TAC);
  REWRITE_TAC[arith `~(x < y) <=> y <= x`];
  FIRST_X_ASSUM_ST `norm (f (v p) t) = norm (v p)` (C INTRO_TAC [`t`]);
  ANTS_TAC;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!d. t < &0 ==> d + -- abs t = d + t` (C SUBGOAL_THEN MP_TAC);
    BY(REAL_ARITH_TAC);
  ASM_REWRITE_TAC[];
  DISCH_THEN (unlist REWRITE_TAC);
  (REWRITE_TAC[DIST_SYM;arith `d + &0 = d`] THEN REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[]);
  (COMMENT "back to root, next tau");
  TYPIFY `sum {i | i < 4} (\i. rho_fun (norm (v i)) * azim (vec 0) (v i) (v (i + 1)) (v (i + 4 - 1))) <=    sum {i | i < 4} (\i. rho_fun (norm (f (v i) t)) * azim (vec 0) (f (v i) t) (f(v(i+1)) t) (f (v (i + 4 - 1)) t))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    MATCH_MP_TAC Cuxvzoz.MMs_minimize_tau_fun;
    TYPIFY `s` EXISTS_TAC;
    ASM_REWRITE_TAC[arith `3 < 4`];
    FIRST_X_ASSUM MATCH_MP_TAC;
    BY(ASM_REWRITE_TAC[]);
  POP_ASSUM MP_TAC;
  TYPIFY `{i | i < 4} = 0..(4-1)` (C SUBGOAL_THEN SUBST1_TAC);
    BY(REWRITE_TAC[EXTENSION;IN_ELIM_THM;IN_NUMSEG] THEN ARITH_TAC);
  REPEAT ( GMATCH_SIMP_TAC (GSYM Oxl_def.periodic_sum));
  TYPIFY `p` EXISTS_TAC;
  CONJ_TAC;
    NUM_REDUCE_TAC;
    FIRST_ASSUM_ST `periodic` MP_TAC THEN REWRITE_TAC[Oxl_def.periodic];
    REWRITE_TAC[arith `(i+4)+k = (i+k)+4`];
    BY(DISCH_THEN (unlist REWRITE_TAC));
  TYPIFY `p` EXISTS_TAC;
  CONJ_TAC;
    NUM_REDUCE_TAC;
    FIRST_ASSUM_ST `periodic` MP_TAC THEN REWRITE_TAC[Oxl_def.periodic];
    REWRITE_TAC[arith `(i+4)+k = (i+k)+4`];
    BY(DISCH_THEN (unlist REWRITE_TAC));
  NUM_REDUCE_TAC;
  REWRITE_TAC[arith `3+p = SUC(SUC(SUC p))`;SUM_CLAUSES_NUMSEG;SUM_SING_NUMSEG;arith `p <= SUC p /\ p <= SUC (SUC p) /\ p <= SUC(SUC(SUC p))`];
  REWRITE_TAC[arith `SUC i = i+1`;arith `(i + j) + (k:num) = i+(j+k)`];
  NUM_REDUCE_TAC;
  ASM_REWRITE_TAC[];
  REPEAT (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM_ST `rho_fun` MP_TAC THEN ASM_REWRITE_TAC[Appendix.rho_rho_fun];
  TYPIFY `f (v(p+1)) t = v(p+1) /\ f (v (p+2)) t = v(p+2) /\ f(v (p+3)) t = v(p+3)` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `~(w = v p) ==> a` (REPEAT o GMATCH_SIMP_TAC);
    BY(ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  POP_ASSUM (fun t -> ASM_TAC THEN unlist REWRITE_TAC t THEN REPEAT DISCH_TAC THEN ASSUME_TAC t);
  FIRST_X_ASSUM_ST `azim` MP_TAC;
  FIRST_X_ASSUM_ST `azim` MP_TAC THEN BURY_MP_TAC;
  FIRST_ASSUM_ST `azim` ((unlist REWRITE_TAC) o GSYM);
  FIRST_ASSUM_ST `azim (vec 0) (v (p + 3)) (f (v p) t) (v (p + 2))` ((unlist REWRITE_TAC) o GSYM);
  REWRITE_TAC[arith `(a+b)+c = a + b + c`];
  REWRITE_TAC[arith `r0 * (a013) + r1 * (a123 + a130) + r2 * a231 + r3 * (a301+a312)  <=  r0 * (f013) + r1 * (a123 + f130) + r2 * a231 + r3 *(f301 +a312) <=> r0 * a013 + r1*a130 + r3* a301 <=  r0 * f013 + r1 * f130 + r3 * f301`];
  MATCH_MP_TAC (arith `a - (pi+sol0) = a' /\ b - (pi+sol0) = b' ==> (a <= b ==> a' <= b')`);
  REWRITE_TAC[arith `(a + b+c) - (pi + sol0) = a + b + c - (pi + sol0)`];
  CONJ_TAC;
    GMATCH_SIMP_TAC (GSYM Cuxvzoz.tau3_azim);
    GMATCH_SIMP_TAC Cuxvzoz.tau3_taum_nonplanar;
    ASM_REWRITE_TAC[];
    REWRITE_TAC[DIST_SYM];
    BY(MESON_TAC[Terminal.taum_sym]);
  FIRST_X_ASSUM_ST `norm a  = norm b` (SUBST1_TAC o GSYM);
  FIRST_X_ASSUM_ST `dist s + t` (SUBST1_TAC o GSYM);
  GMATCH_SIMP_TAC (GSYM Cuxvzoz.tau3_azim);
  GMATCH_SIMP_TAC Cuxvzoz.tau3_taum_nonplanar;
  REWRITE_TAC[DIST_SYM];
  REWRITE_TAC[CONJ_ASSOC] THEN CONJ2_TAC;
    ASM_REWRITE_TAC[DIST_SYM];
    BY(MESON_TAC[Terminal.taum_sym]);
  ASM_REWRITE_TAC[];
  CONJ_TAC;
    FIRST_X_ASSUM MATCH_MP_TAC;
    BY(ASM_REWRITE_TAC[]);
  BY(FIRST_X_ASSUM_ST `azim (vec 0) (f (v p) t) (v (p + 1)) (v (p + 3)) < pi` MP_TAC THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let LEMMA_PWE1 = prove_by_refinement(
(* main_work4 ==> *) ` main_nonlinear_terminal_v11 ==> (!s v p.
    is_scs_v39 s /\ scs_basic_v39 s /\ scs_k_v39 s = 4 /\ v IN MMs_v39 s /\ 
    (!i j. scs_diag 4 i j ==> scs_a_v39 s i j <= cstab /\ cstab < dist(v i,v j) /\ &4*h0 < scs_b_v39 s i j) /\
    (!i. (scs_a_v39 s i (i+1) = &2 /\ scs_b_v39 s i (i+1)= &2*h0) \/
       (scs_a_v39 s i (i+1) = &2 * h0 /\ scs_b_v39 s i (i+1)= cstab)) /\
      scs_generic v /\
    ~scs_is_str s v (p+1) /\ ~scs_is_str s v (p+3) /\
    scs_is_str s v p /\ norm (v p) = &2 /\
    scs_a_v39 s p (p+1) = &2 /\
    scs_a_v39 s (p+3) (p) = &2 /\
    dist(v (p+1),v(p+2)) <= &2 * h0 ==>
    pi / &2 < dihV (vec 0) (v (p+1)) (v (p+2)) (v p))`,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[arith `x < y <=> ~(y <= x)`] THEN DISCH_TAC;
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `BBprime2_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.scs_basic;Ayqjtmd.unadorned_MMs;IN]);
  TYPIFY `BBprime_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime2_v39]);
  TYPIFY `BBs_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime_v39]);
  INTRO_TAC Appendix.BBs_v39 [`s`;`v`];
  ASM_REWRITE_TAC[LET_DEF;LET_END_DEF;Terminal.IMAGE_SUBSET_IN;IN_UNIV;arith `~(4 <= 3)`];
  REPEAT WEAKER_STRIP_TAC;
  TYPED_ABBREV_TAC `V = IMAGE v (:num)`;
  TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`;
  TYPED_ABBREV_TAC `FF = IMAGE (\i. (v i,v (SUC i))) (:num)`;
  TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    EXPAND_TAC "V";
    REWRITE_TAC[IN_IMAGE;IN_UNIV];
    BY(MESON_TAC[]);
  INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`4`];
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  TYPIFY `!i. interior_angle1 (vec 0) FF (v i) = azim (vec 0) (v i) (v (i+1)) (v (i+3))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    EXPAND_TAC "FF";
    GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] (GSYM Ocbicby.INTERIOR_ANGLE1_AZIM));
    TYPIFY `s` EXISTS_TAC;
    BY(ASM_REWRITE_TAC[IN;arith `4 - 1 = 3`;arith `3 < 4`;arith `SUC i = i+1`]);
  INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`];
  ASM_REWRITE_TAC[LET_THM;IN;arith `4-1 = 3`;arith `SUC i = i+1`];
  DISCH_TAC;
  TYPIFY `!i. v (i MOD 4) = v i` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC (GSYM Oxl_def.periodic_mod);
    BY(ASM_REWRITE_TAC[arith `~(0=4)`]);
  TYPIFY `v 4 = v 0 /\ v 5 = v 1 /\ v 6 = v 2 /\ v 7 = v 3 /\ v 8  = v 4` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[Terminal.MOD_4_EXPLICIT;arith `7 = 1*4 + 3`;arith `8 = 2*4 + 0`;MOD_MULT_ADD]);
  TYPIFY `generic V E` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM_ST `scs_generic` MP_TAC THEN ASM_REWRITE_TAC[Appendix.scs_generic]);
  TYPIFY `v (p+4) = v p /\ v (p+5) = v (p+1) /\ v (p+6) = v (p+2) /\ v (p+7) = v (p+3) /\ v (p+8)  = v (p)` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `MOD` ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[arith `p+4 = 1*4 + p /\ p+5 = 1*4+(p+1) /\ p+6=1*4+(p+2) /\ p+7 = 1*4 + (p+3) /\ p+8 = 2*4 + p`;MOD_MULT_ADD]);
  TYPIFY `!i. &2 <= norm (v i) /\ norm (v i) <= &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    REWRITE_TAC[GSYM Fnjlbxs.in_ball_annulus];
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> &2 <= dist(v (p + i),v (p+j))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    MATCH_MP_TAC Ocbicby.scs_lb_2;
    TYPIFY `s` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    (NUM_REDUCE_TAC);
    REPEAT (GMATCH_SIMP_TAC MOD_LT);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `scs_b_v39 s (p+2) ((p+2)+1) <= cstab /\ scs_b_v39 s p (p+1) = &2 * h0 /\ scs_b_v39 s (p+3) ((p+3)+1) = &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    CONJ2_TAC;
      TYPIFY `~(&2 = &2 * h0)` (C SUBGOAL_THEN ASSUME_TAC);
        BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
      TYPIFY `scs_a_v39 s (p+3) p = scs_a_v39 s (p+3) ((p+3)+1)` ENOUGH_TO_SHOW_TAC;
        BY(ASM_MESON_TAC[]);
      REWRITE_TAC[arith `((p+3)+1) = p+4`];
      FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
      BY(ASM_MESON_TAC[Appendix.periodic2;Oxl_def.periodic]);
    FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM_ST `x <= cstab` (C INTRO_TAC [`p+2`]) THEN ASM_REWRITE_TAC[arith `3 < 4`];
    BY(REWRITE_TAC[arith `SUC (p+2) = (p+2)+1`]);
  FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[arith `((p:num) + i ) + j = p + (i+j)`] THEN NUM_REDUCE_TAC;
  DISCH_TAC;
  TYPIFY `dist(v (p+2),v(p+3)) <= cstab /\ dist(v p,v(p+1)) <= &2 * h0 /\ dist(v(p+3),v(p+4)) <= &2 * h0` (C SUBGOAL_THEN MP_TAC);
    FIRST_X_ASSUM MP_TAC;
    FIRST_X_ASSUM_ST `scs_b_v39` MP_TAC;
    BY(MESON_TAC[REAL_LE_TRANS]);
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "collinear";
  TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0,v i, v j}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT GEN_TAC THEN DISCH_TAC;
    MATCH_MP_TAC Zlzthic.PROPERTIES_GENERIC_LOCAL_FAN_ALT;
    GEXISTL_TAC [`V`;`E`;`FF`];
    BY(ASM_REWRITE_TAC[IN_IMAGE;IN_UNIV]);
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 20 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
  INTRO_TAC PERIODIC_INJ_MOD [`v`;`4`];
  ASM_REWRITE_TAC[arith `~(4 = 0)`];
  DISCH_TAC;
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i=j) ==> ~(v (p +i) = v (p+j))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM MP_TAC THEN ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    (NUM_REDUCE_TAC);
    REPEAT (GMATCH_SIMP_TAC MOD_LT);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. 0 < i /\ i < 4 ==> ~(v (p+i) = v p)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[arith `p+0 = p`;arith `0< 4`;arith `0 < i ==> ~(i=0)`]);
  TYPIFY `#3.3 <= dist(v (p+2),v p)` ASM_CASES_TAC;
    COMMENT "long diag";
    INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "6184614449") [`norm (v (p+1))`;`norm(v (p+2))`;`norm(v p)`;`dist(v(p+2),v p)`;`dist(v(p+1),v p)`;`dist(v(p+1),v(p+2))`];
    FIRST_X_ASSUM_ST `norm x = &2` kill;
    REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
    ASM_REWRITE_TAC[];
    DISCH_THEN MP_TAC THEN ANTS_TAC;
      CONJ_TAC;
        INTRO_TAC DIST_TRIANGLE [`v (p+2)`;`(vec 0):real^3`;`v (p)`];
        REWRITE_TAC[DIST_0];
        MATCH_MP_TAC (arith `x <= &2 * h0 /\ y <= &2 * h0 ==> (d <= x + y ==> d <= &4 * h0)`);
        BY(ASM_REWRITE_TAC[]);
      nCONJ_TAC 1;
        BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
      ASM_REWRITE_TAC[];
      TYPIFY `v p = v (p+0)` (C SUBGOAL_THEN SUBST1_TAC);
        ONCE_REWRITE_TAC[arith `p+0=p`];
        BY(REWRITE_TAC[]);
      BY(CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    DISCH_TAC;
    FIRST_X_ASSUM_ST `dihV` MP_TAC;
    REWRITE_TAC[arith `~(x <= y) <=> y < x`];
    GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] Merge_ineq.DIHV_EQ_DIH_Y);
    ASM_REWRITE_TAC[DIST_0];
    CONJ_TAC;
      BY(CONJ_TAC THEN (REPEAT (FIRST_X_ASSUM MATCH_MP_TAC)) THEN NUM_REDUCE_TAC);
    MATCH_MP_TAC Ocbicby.delta4_y_imp_obtuse;
    CONJ_TAC;
      BY(ASM_MESON_TAC[arith `&2 <= x==> &0 < x`]);
    CONJ_TAC;
      BY(REWRITE_TAC[REWRITE_RULE[LET_THM] Tame_lemmas.delta_y_pos]);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. ~scs_is_str s v i ==> ~coplanar {vec 0, v i, v(i+1), v(i+3)}` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC THEN ASM_REWRITE_TAC[Appendix.scs_is_str;arith `SUC i = i+1`;GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR];
    NUM_REDUCE_TAC;
    DISCH_THEN (unlist REWRITE_TAC);
    MATCH_MP_TAC (arith `&0 < a ==> ~(a = &0)`);
    INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`;`i`];
    BY(ASM_REWRITE_TAC[LET_THM;IN;arith `SUC i = i+1`;arith `4 - 1 = 3`]);
  FIRST_X_ASSUM MP_TAC THEN BURY_MP_TAC;
  COMMENT "short diag";
  INTRO_TAC Terminal.vv_split_azim_generic [`v`;`4`;`p`;`2`;`3`];
  ASM_REWRITE_TAC[arith `SUC p = p+1`;LET_THM];
  NUM_REDUCE_TAC;
  DISCH_TAC;
  TYPIFY `azim (vec 0) (v p) (v (p+1)) (v (p+3)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
    MATCH_MP_TAC Local_lemmas.CONVEX_LOFA_IMP_INANGLE_LE_PI;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `azim (vec 0) (v p) (v (p + 1)) (v (p + 2)) <= pi /\  azim (vec 0) (v p) (v (p + 2)) (v (p + 3)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC (arith `&0 < pi /\ &0 <= a /\ &0 <= b /\ a + b <= pi ==> a <= pi /\ b <= pi`);
    REWRITE_TAC[PI_POS];
    REWRITE_TAC[Counting_spheres.AZIM_NN];
    BY(REPLICATE_TAC 2 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM_ST `scs_is_str` MP_TAC;
  REWRITE_TAC[Appendix.scs_is_str];
  ASM_REWRITE_TAC[];
  NUM_REDUCE_TAC;
  REWRITE_TAC[arith `SUC p = p+1`];
  DISCH_THEN (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
  INTRO_TAC AZIM_DIHV_SAME_STRONG [`(vec 0):real^3`;`v p`;`v (p+2)`;`v (p+3)`];
  INTRO_TAC AZIM_DIHV_SAME_STRONG [`(vec 0):real^3`;`v p`;`v (p+1)`;`v (p+2)`];
  FIRST_X_ASSUM_ST `pi = a` (MP_TAC o SYM) THEN DISCH_TAC;
  ASM_REWRITE_TAC[];
  TYPIFY `~collinear {vec 0, v p, v (p + 1)} /\ ~collinear {vec 0, v p, v (p + 2)} /\  ~collinear {vec 0, v p, v (p + 3)}` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  REPEAT (FIRST_X_ASSUM_ST `dihV` MP_TAC) THEN REWRITE_TAC[] THEN REPEAT (GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] Merge_ineq.DIHV_EQ_DIH_Y));
  ASM_REWRITE_TAC[];
  CONJ_TAC;
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM_ST `+` MP_TAC THEN ASM_REWRITE_TAC[];
  (DISCH_TAC);
  TYPIFY `cstab <= dist(v p,v(p+2))` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `cstab < d` (C INTRO_TAC [`p`;`p+2`]);
    ANTS_TAC;
      REWRITE_TAC[arith `p+2 = SUC(SUC (p))`];
      MATCH_MP_TAC Tfitskc.SCS_DIAG_2;
      BY(NUM_REDUCE_TAC);
    BY(REAL_ARITH_TAC);
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "1348932091 delta") [`norm (v (p))`;`norm(v (p+1))`;`norm(v (p+2))`;`&2 * h0`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+1))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  TYPIFY `(&2 <= norm (v p) /\ norm (v p) <= &2) /\   (&2 <= norm (v (p + 1)) /\ norm (v (p + 1)) <=  #2.52) /\   (&2 <= norm (v (p + 2)) /\ norm (v (p + 2)) <=  #2.52) /\   ( #2.52 <= &2 * h0 /\ &2 * h0 <=  #2.52) /\   ( #3.01 <= dist (v p,v (p + 2)) /\ dist (v p,v (p + 2)) <=  #3.3) /\   &2 <= dist (v p,v (p + 1)) /\   dist (v p,v (p + 1)) <=  #2.52` (C SUBGOAL_THEN ASSUME_TAC);
    TYPIFY `#2.52 = &2 * h0` (C SUBGOAL_THEN SUBST1_TAC);
      BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
    ASM_REWRITE_TAC[arith `(x <= y /\ y <= x) <=> y = x`];
    CONJ2_TAC;
      FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`0`;`1`]);
      BY(REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    CONJ2_TAC;
      BY(FIRST_X_ASSUM_ST `#3.3` MP_TAC THEN REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC);
    BY(ASM_REWRITE_TAC[GSYM Sphere.cstab]);
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "1348932091") [`norm (v (p))`;`norm(v (p+1))`;`norm(v (p+2))`;`&2 * h0`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+1))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  FIRST_X_ASSUM_ST `#3.01` (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC Ocbicby.dih_y_mono [`norm (v (p))`;`norm(v (p+1))`;`norm(v (p+2))`;`dist(v (p+1),v(p+2))`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+1))`;`&2 * h0`];
  ANTS_TAC;
    ASM_REWRITE_TAC[arith `&0 < &2`];
    CONJ_TAC;
      FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`1`;`2`]);
      BY(NUM_REDUCE_TAC THEN REAL_ARITH_TAC);
    CONJ2_TAC;
      BY(FIRST_X_ASSUM_ST `delta_y` MP_TAC THEN REAL_ARITH_TAC);
    FIRST_X_ASSUM_ST `norm (v p) = &2` (SUBST1_TAC o GSYM);
    FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`p+1`]);
    ANTS_TAC;
      BY(ASM_REWRITE_TAC[]);
    REWRITE_TAC[arith `(p+1) +i = p+(1+i)`] THEN NUM_REDUCE_TAC;
    REWRITE_TAC[Oxlzlez.coplanar_delta_y];
    ASM_REWRITE_TAC[DIST_SYM;DIST_0];
    MATCH_MP_TAC (arith `a = b ==> &0 < a ==> &0 < b`);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  (DISCH_TAC);
  COMMENT "5557";
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "5557288534 delta") [`norm (v (p))`;`norm(v (p+3))`;`norm(v (p+2))`;`cstab`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+3))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  TYPIFY `(&2 <= norm (v p) /\ norm (v p) <= &2) /\   (&2 <= norm (v (p + 3)) /\ norm (v (p + 3)) <=  #2.52) /\   (&2 <= norm (v (p + 2)) /\ norm (v (p + 2)) <=  #2.52) /\   ( #3.01 <= cstab /\ cstab <=  #3.01) /\   ( #3.01 <= dist (v p,v (p + 2)) /\ dist (v p,v (p + 2)) <=  #3.3) /\   &2 <= dist (v p,v (p + 3)) /\   dist (v p,v (p + 3)) <=  #2.52` (C SUBGOAL_THEN ASSUME_TAC);
    TYPIFY `#2.52 = &2 * h0` (C SUBGOAL_THEN SUBST1_TAC);
      BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
    ASM_REWRITE_TAC[arith `(x <= y /\ y <= x) <=> y = x`;Sphere.cstab];
    CONJ2_TAC;
      CONJ_TAC;
        FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`0`;`3`]);
        BY(REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN MATCH_MP_TAC THEN NUM_REDUCE_TAC);
      BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
    CONJ2_TAC;
      BY(FIRST_X_ASSUM_ST `#3.3` MP_TAC THEN REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC);
    BY(ASM_REWRITE_TAC[GSYM Sphere.cstab]);
  FIRST_ASSUM (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "5557288534") [`norm (v (p))`;`norm(v (p+3))`;`norm(v (p+2))`;`cstab`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+3))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  FIRST_X_ASSUM_ST `#3.01` (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC Ocbicby.dih_y_mono [`norm (v (p))`;`norm(v (p+3))`;`norm(v (p+2))`;`dist(v (p+3),v(p+2))`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+3))`;`cstab`];
  ANTS_TAC;
    TYPIFY_GOAL_THEN `&0 < norm (v p)` (unlist REWRITE_TAC);
      BY(ASM_REWRITE_TAC[arith `&0 < &2`]);
    FIRST_X_ASSUM_ST `norm (v p) =  &2` kill;
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
    CONJ_TAC;
      FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`3`;`2`]);
      BY(NUM_REDUCE_TAC THEN REAL_ARITH_TAC);
    CONJ2_TAC;
      BY(FIRST_X_ASSUM_ST `delta_y` MP_TAC THEN REAL_ARITH_TAC);
    FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`p+3`]);
    ANTS_TAC;
      BY(ASM_REWRITE_TAC[]);
    REWRITE_TAC[arith `(p+3) +i = p+(3+i)`] THEN NUM_REDUCE_TAC;
    REWRITE_TAC[Oxlzlez.coplanar_delta_y];
    ASM_REWRITE_TAC[DIST_SYM;DIST_0];
    MATCH_MP_TAC (arith `a = b ==> &0 < a ==> &0 < b`);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  (DISCH_TAC);
  TYPIFY `dih_y (norm (v p)) (norm (v (p + 3))) (norm (v (p + 2)))      (dist (v (p + 3),v (p + 2)))      (dist (v p,v (p + 2)))      (dist (v p,v (p + 3))) + dih_y (norm (v p)) (norm (v (p + 1))) (norm (v (p + 2)))      (dist (v (p + 1),v (p + 2)))      (dist (v p,v (p + 2)))      (dist (v p,v (p + 1))) <= #3.1` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 5 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  TYPIFY `#3.1 < pi` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MP_TAC Flyspeck_constants.bounds THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM_ST `a = pi` MP_TAC THEN REPLICATE_TAC 2 (POP_ASSUM MP_TAC);
  REWRITE_TAC[DIST_0;DIST_SYM];
  TYPIFY `dih_y (norm (v p)) (norm (v (p + 2))) (norm (v (p + 3)))       (dist (v (p + 2),v (p + 3)))       (dist (v p,v (p + 3)))       (dist (v p,v (p + 2))) = dih_y (norm (v p)) (norm (v (p + 3))) (norm (v (p + 2))) (dist (v (p + 2),v (p + 3))) (dist (v p,v (p + 2))) (dist (v p,v (p + 3)))` (C SUBGOAL_THEN SUBST1_TAC);
    BY(REWRITE_TAC[Nonlinear_lemma.dih_y_sym]);
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let LEMMA_PWE2 = prove_by_refinement(
(* main_work4 ==> *) ` main_nonlinear_terminal_v11 ==> (!s v p.
    is_scs_v39 s /\ scs_basic_v39 s /\ scs_k_v39 s = 4 /\ v IN MMs_v39 s /\ 
    (!i j. scs_diag 4 i j ==> scs_a_v39 s i j <= cstab /\ cstab < dist(v i,v j) /\ &4*h0 < scs_b_v39 s i j) /\
    (!i. (scs_a_v39 s i (i+1) = &2 /\ scs_b_v39 s i (i+1)= &2*h0) \/
       (scs_a_v39 s i (i+1) = &2 * h0 /\ scs_b_v39 s i (i+1)= cstab)) /\
      scs_generic v /\
    ~scs_is_str s v (p+1) /\ ~scs_is_str s v (p+3) /\
    scs_is_str s v p /\ norm (v p) = &2 /\
    scs_a_v39 s p (p+1) = &2 /\
    scs_a_v39 s (p+3) (p) = &2 /\
    dist(v (p+2),v(p+3)) <= &2 * h0 ==>
    pi / &2 < dihV (vec 0) (v (p+3)) (v (p)) (v (p+2)))`,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[arith `x < y <=> ~(y <= x)`] THEN DISCH_TAC;
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `BBprime2_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.scs_basic;Ayqjtmd.unadorned_MMs;IN]);
  TYPIFY `BBprime_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime2_v39]);
  TYPIFY `BBs_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime_v39]);
  INTRO_TAC Appendix.BBs_v39 [`s`;`v`];
  ASM_REWRITE_TAC[LET_DEF;LET_END_DEF;Terminal.IMAGE_SUBSET_IN;IN_UNIV;arith `~(4 <= 3)`];
  REPEAT WEAKER_STRIP_TAC;
  TYPED_ABBREV_TAC `V = IMAGE v (:num)`;
  TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`;
  TYPED_ABBREV_TAC `FF = IMAGE (\i. (v i,v (SUC i))) (:num)`;
  TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    EXPAND_TAC "V";
    REWRITE_TAC[IN_IMAGE;IN_UNIV];
    BY(MESON_TAC[]);
  INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`4`];
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  TYPIFY `!i. interior_angle1 (vec 0) FF (v i) = azim (vec 0) (v i) (v (i+1)) (v (i+3))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    EXPAND_TAC "FF";
    GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] (GSYM Ocbicby.INTERIOR_ANGLE1_AZIM));
    TYPIFY `s` EXISTS_TAC;
    BY(ASM_REWRITE_TAC[IN;arith `4 - 1 = 3`;arith `3 < 4`;arith `SUC i = i+1`]);
  INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`];
  ASM_REWRITE_TAC[LET_THM;IN;arith `4-1 = 3`;arith `SUC i = i+1`];
  DISCH_TAC;
  TYPIFY `!i. v (i MOD 4) = v i` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC (GSYM Oxl_def.periodic_mod);
    BY(ASM_REWRITE_TAC[arith `~(0=4)`]);
  TYPIFY `v 4 = v 0 /\ v 5 = v 1 /\ v 6 = v 2 /\ v 7 = v 3 /\ v 8  = v 4` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[Terminal.MOD_4_EXPLICIT;arith `7 = 1*4 + 3`;arith `8 = 2*4 + 0`;MOD_MULT_ADD]);
  TYPIFY `generic V E` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM_ST `scs_generic` MP_TAC THEN ASM_REWRITE_TAC[Appendix.scs_generic]);
  TYPIFY `v (p+4) = v p /\ v (p+5) = v (p+1) /\ v (p+6) = v (p+2) /\ v (p+7) = v (p+3) /\ v (p+8)  = v (p)` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `MOD` ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[arith `p+4 = 1*4 + p /\ p+5 = 1*4+(p+1) /\ p+6=1*4+(p+2) /\ p+7 = 1*4 + (p+3) /\ p+8 = 2*4 + p`;MOD_MULT_ADD]);
  TYPIFY `!i. &2 <= norm (v i) /\ norm (v i) <= &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    REWRITE_TAC[GSYM Fnjlbxs.in_ball_annulus];
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> &2 <= dist(v (p + i),v (p+j))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    MATCH_MP_TAC Ocbicby.scs_lb_2;
    TYPIFY `s` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    (NUM_REDUCE_TAC);
    REPEAT (GMATCH_SIMP_TAC MOD_LT);
    BY(ASM_REWRITE_TAC[]);
  COMMENT "edge lengths";
  TYPIFY `scs_b_v39 s (p+1) ((p+1)+1) <= cstab /\ scs_b_v39 s p (p+1) = &2 * h0 /\ scs_b_v39 s (p+3) ((p+3)+1) = &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    CONJ2_TAC;
      TYPIFY `~(&2 = &2 * h0)` (C SUBGOAL_THEN ASSUME_TAC);
        BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
      TYPIFY `scs_a_v39 s (p+3) p = scs_a_v39 s (p+3) ((p+3)+1)` ENOUGH_TO_SHOW_TAC;
        BY(ASM_MESON_TAC[]);
      REWRITE_TAC[arith `((p+3)+1) = p+4`];
      FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
      BY(ASM_MESON_TAC[Appendix.periodic2;Oxl_def.periodic]);
    FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM_ST `x <= cstab` (C INTRO_TAC [`p+1`]) THEN ASM_REWRITE_TAC[arith `3 < 4`];
    BY(REWRITE_TAC[arith `SUC (p+1) = (p+1)+1`]);
  FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[arith `((p:num) + i ) + j = p + (i+j)`] THEN NUM_REDUCE_TAC;
  DISCH_TAC;
  TYPIFY `dist(v (p+1),v(p+2)) <= cstab /\ dist(v p,v(p+1)) <= &2 * h0 /\ dist(v(p+3),v(p+4)) <= &2 * h0` (C SUBGOAL_THEN MP_TAC);
    FIRST_X_ASSUM MP_TAC;
    FIRST_X_ASSUM_ST `scs_b_v39` MP_TAC;
    BY(MESON_TAC[REAL_LE_TRANS]);
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "collinear";
  TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0,v i, v j}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT GEN_TAC THEN DISCH_TAC;
    MATCH_MP_TAC Zlzthic.PROPERTIES_GENERIC_LOCAL_FAN_ALT;
    GEXISTL_TAC [`V`;`E`;`FF`];
    BY(ASM_REWRITE_TAC[IN_IMAGE;IN_UNIV]);
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 20 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
  INTRO_TAC PERIODIC_INJ_MOD [`v`;`4`];
  ASM_REWRITE_TAC[arith `~(4 = 0)`];
  DISCH_TAC;
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i=j) ==> ~(v (p +i) = v (p+j))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM MP_TAC THEN ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    (NUM_REDUCE_TAC);
    REPEAT (GMATCH_SIMP_TAC MOD_LT);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. 0 < i /\ i < 4 ==> ~(v (p+i) = v p)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[arith `p+0 = p`;arith `0< 4`;arith `0 < i ==> ~(i=0)`]);
  COMMENT "long diag";
  TYPIFY `#3.3 <= dist(v (p+2),v p)` ASM_CASES_TAC;
    COMMENT "long diag";
    INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "6184614449") [`norm (v (p+3))`;`norm(v (p+2))`;`norm(v p)`;`dist(v(p+2),v p)`;`dist(v(p+3),v p)`;`dist(v(p+3),v(p+2))`];
    FIRST_X_ASSUM_ST `norm x = &2` kill;
    REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
    ASM_REWRITE_TAC[];
    DISCH_THEN MP_TAC THEN ANTS_TAC;
      CONJ_TAC;
        INTRO_TAC DIST_TRIANGLE [`v (p+2)`;`(vec 0):real^3`;`v (p)`];
        REWRITE_TAC[DIST_0];
        MATCH_MP_TAC (arith `x <= &2 * h0 /\ y <= &2 * h0 ==> (d <= x + y ==> d <= &4 * h0)`);
        BY(ASM_REWRITE_TAC[]);
      nCONJ_TAC 2;
        BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
      ASM_REWRITE_TAC[];
      TYPIFY `v p = v (p+0)` (C SUBGOAL_THEN SUBST1_TAC);
        ONCE_REWRITE_TAC[arith `p+0=p`];
        BY(REWRITE_TAC[]);
      BY(CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    DISCH_TAC;
    FIRST_X_ASSUM_ST `dihV` MP_TAC;
    REWRITE_TAC[arith `~(x <= y) <=> y < x`];
    GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] Merge_ineq.DIHV_EQ_DIH_Y);
    ASM_REWRITE_TAC[DIST_0];
    CONJ_TAC;
      BY(CONJ_TAC THEN (REPEAT (FIRST_X_ASSUM MATCH_MP_TAC)) THEN NUM_REDUCE_TAC);
    ONCE_REWRITE_TAC[Nonlinear_lemma.dih_y_sym];
    MATCH_MP_TAC Ocbicby.delta4_y_imp_obtuse;
    CONJ_TAC;
      BY(ASM_MESON_TAC[arith `&2 <= x==> &0 < x`]);
    INTRO_TAC DIST_SYM [`v p`;`v (p+2)`];
    DISCH_THEN SUBST1_TAC;
    CONJ_TAC;
      BY(REWRITE_TAC[REWRITE_RULE[LET_THM] Tame_lemmas.delta_y_pos]);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. ~scs_is_str s v i ==> ~coplanar {vec 0, v i, v(i+1), v(i+3)}` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC THEN ASM_REWRITE_TAC[Appendix.scs_is_str;arith `SUC i = i+1`;GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR];
    NUM_REDUCE_TAC;
    DISCH_THEN (unlist REWRITE_TAC);
    MATCH_MP_TAC (arith `&0 < a ==> ~(a = &0)`);
    INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`;`i`];
    BY(ASM_REWRITE_TAC[LET_THM;IN;arith `SUC i = i+1`;arith `4 - 1 = 3`]);
  FIRST_X_ASSUM MP_TAC THEN BURY_MP_TAC;
  COMMENT "short diag";
  INTRO_TAC Terminal.vv_split_azim_generic [`v`;`4`;`p`;`2`;`3`];
  ASM_REWRITE_TAC[arith `SUC p = p+1`;LET_THM];
  NUM_REDUCE_TAC;
  DISCH_TAC;
  TYPIFY `azim (vec 0) (v p) (v (p+1)) (v (p+3)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
    MATCH_MP_TAC Local_lemmas.CONVEX_LOFA_IMP_INANGLE_LE_PI;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `azim (vec 0) (v p) (v (p + 1)) (v (p + 2)) <= pi /\  azim (vec 0) (v p) (v (p + 2)) (v (p + 3)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC (arith `&0 < pi /\ &0 <= a /\ &0 <= b /\ a + b <= pi ==> a <= pi /\ b <= pi`);
    REWRITE_TAC[PI_POS];
    REWRITE_TAC[Counting_spheres.AZIM_NN];
    BY(REPLICATE_TAC 2 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM_ST `scs_is_str` MP_TAC;
  REWRITE_TAC[Appendix.scs_is_str];
  ASM_REWRITE_TAC[];
  NUM_REDUCE_TAC;
  REWRITE_TAC[arith `SUC p = p+1`];
  DISCH_THEN (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
  INTRO_TAC AZIM_DIHV_SAME_STRONG [`(vec 0):real^3`;`v p`;`v (p+2)`;`v (p+3)`];
  INTRO_TAC AZIM_DIHV_SAME_STRONG [`(vec 0):real^3`;`v p`;`v (p+1)`;`v (p+2)`];
  FIRST_X_ASSUM_ST `pi = a` (MP_TAC o SYM) THEN DISCH_TAC;
  ASM_REWRITE_TAC[];
  TYPIFY `~collinear {vec 0, v p, v (p + 1)} /\ ~collinear {vec 0, v p, v (p + 2)} /\  ~collinear {vec 0, v p, v (p + 3)}` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  REPEAT (FIRST_X_ASSUM_ST `dihV` MP_TAC) THEN REWRITE_TAC[] THEN REPEAT (GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] Merge_ineq.DIHV_EQ_DIH_Y));
  ASM_REWRITE_TAC[];
  CONJ_TAC;
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM_ST `+` MP_TAC THEN ASM_REWRITE_TAC[];
  (DISCH_TAC);
  TYPIFY `cstab <= dist(v p,v(p+2))` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `cstab < d` (C INTRO_TAC [`p`;`p+2`]);
    ANTS_TAC;
      REWRITE_TAC[arith `p+2 = SUC(SUC (p))`];
      MATCH_MP_TAC Tfitskc.SCS_DIAG_2;
      BY(NUM_REDUCE_TAC);
    BY(REAL_ARITH_TAC);
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "1348932091 delta") [`norm (v (p))`;`norm(v (p+3))`;`norm(v (p+2))`;`&2 * h0`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+3))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  TYPIFY `(&2 <= norm (v p) /\ norm (v p) <= &2) /\   (&2 <= norm (v (p + 3)) /\ norm (v (p + 3)) <=  #2.52) /\   (&2 <= norm (v (p + 2)) /\ norm (v (p + 2)) <=  #2.52) /\   ( #2.52 <= &2 * h0 /\ &2 * h0 <=  #2.52) /\   ( #3.01 <= dist (v p,v (p + 2)) /\ dist (v p,v (p + 2)) <=  #3.3) /\   &2 <= dist (v p,v (p + 3)) /\   dist (v p,v (p + 3)) <=  #2.52` (C SUBGOAL_THEN ASSUME_TAC);
    TYPIFY `#2.52 = &2 * h0` (C SUBGOAL_THEN SUBST1_TAC);
      BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
    ASM_REWRITE_TAC[arith `(x <= y /\ y <= x) <=> y = x`];
    nCONJ_TAC 3;
      BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
    ASM_REWRITE_TAC[];
    CONJ2_TAC;
      FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`0`;`3`]);
      BY(REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    CONJ2_TAC;
      BY(FIRST_X_ASSUM_ST `#3.3` MP_TAC THEN REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC);
    BY(ASM_REWRITE_TAC[GSYM Sphere.cstab]);
  FIRST_X_ASSUM_ST `norm x = &2` MP_TAC;
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "1348932091") [`norm (v (p))`;`norm(v (p+3))`;`norm(v (p+2))`;`&2 * h0`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+3))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  FIRST_X_ASSUM_ST `#3.01` (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC Ocbicby.dih_y_mono [`norm (v (p))`;`norm(v (p+3))`;`norm(v (p+2))`;`dist(v (p+3),v(p+2))`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+3))`;`&2 * h0`];
  ANTS_TAC;
    nCONJ_TAC 1;
      BY(FIRST_X_ASSUM_ST `norm x = &2` MP_TAC THEN REAL_ARITH_TAC);
    FIRST_X_ASSUM_ST `norm x = &2` kill;
    ASM_REWRITE_TAC[arith `&0 < &2`];
    CONJ_TAC;
      BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
    CONJ_TAC;
      FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`3`;`2`]);
      BY(NUM_REDUCE_TAC THEN REAL_ARITH_TAC);
    CONJ2_TAC;
      BY(FIRST_X_ASSUM_ST `delta_y` MP_TAC THEN REAL_ARITH_TAC);
    FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`p+3`]);
    ANTS_TAC;
      BY(ASM_REWRITE_TAC[]);
    REWRITE_TAC[arith `(p+3) +i = p+(3+i)`] THEN NUM_REDUCE_TAC;
    REWRITE_TAC[Oxlzlez.coplanar_delta_y];
    ASM_REWRITE_TAC[DIST_SYM;DIST_0];
    MATCH_MP_TAC (arith `a = b ==> &0 < a ==> &0 < b`);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  (DISCH_TAC);
  COMMENT "5557";
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "5557288534 delta") [`norm (v (p))`;`norm(v (p+1))`;`norm(v (p+2))`;`cstab`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+1))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  TYPIFY `(&2 <= norm (v p) /\ norm (v p) <= &2) /\   (&2 <= norm (v (p + 1)) /\ norm (v (p + 1)) <=  #2.52) /\   (&2 <= norm (v (p + 2)) /\ norm (v (p + 2)) <=  #2.52) /\   ( #3.01 <= cstab /\ cstab <=  #3.01) /\   ( #3.01 <= dist (v p,v (p + 2)) /\ dist (v p,v (p + 2)) <=  #3.3) /\   &2 <= dist (v p,v (p + 1)) /\   dist (v p,v (p + 1)) <=  #2.52` (C SUBGOAL_THEN ASSUME_TAC);
    TYPIFY `#2.52 = &2 * h0` (C SUBGOAL_THEN SUBST1_TAC);
      BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
    ASM_REWRITE_TAC[arith `(x <= y /\ y <= x) <=> y = x`;Sphere.cstab];
    CONJ2_TAC;
      FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`0`;`1`]);
      BY(REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    CONJ2_TAC;
      BY(FIRST_X_ASSUM_ST `#3.3` MP_TAC THEN REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC);
    BY(ASM_REWRITE_TAC[GSYM Sphere.cstab]);
  FIRST_ASSUM (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "5557288534") [`norm (v (p))`;`norm(v (p+1))`;`norm(v (p+2))`;`cstab`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+1))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  FIRST_X_ASSUM_ST `#3.01` (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC Ocbicby.dih_y_mono [`norm (v (p))`;`norm(v (p+1))`;`norm(v (p+2))`;`dist(v (p+1),v(p+2))`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+1))`;`cstab`];
  ANTS_TAC;
    TYPIFY_GOAL_THEN `&0 < norm (v p)` (unlist REWRITE_TAC);
      BY(ASM_REWRITE_TAC[arith `&0 < &2`]);
    FIRST_X_ASSUM_ST `norm (v p) =  &2` kill;
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`1`;`2`]);
      BY(NUM_REDUCE_TAC THEN REAL_ARITH_TAC);
    CONJ2_TAC;
      BY(FIRST_X_ASSUM_ST `delta_y` MP_TAC THEN REAL_ARITH_TAC);
    FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`p+1`]);
    ANTS_TAC;
      BY(ASM_REWRITE_TAC[]);
    REWRITE_TAC[arith `(p+1) +i = p+(1+i)`] THEN NUM_REDUCE_TAC;
    REWRITE_TAC[Oxlzlez.coplanar_delta_y];
    ASM_REWRITE_TAC[DIST_SYM;DIST_0];
    MATCH_MP_TAC (arith `a = b ==> &0 < a ==> &0 < b`);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  (DISCH_TAC);
  TYPIFY `dih_y (norm (v p)) (norm (v (p + 3))) (norm (v (p + 2)))      (dist (v (p + 3),v (p + 2)))      (dist (v p,v (p + 2)))      (dist (v p,v (p + 3))) + dih_y (norm (v p)) (norm (v (p + 1))) (norm (v (p + 2)))      (dist (v (p + 1),v (p + 2)))      (dist (v p,v (p + 2)))      (dist (v p,v (p + 1))) < #3.1` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 5 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  TYPIFY `#3.1 < pi` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MP_TAC Flyspeck_constants.bounds THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM_ST `a = pi` MP_TAC THEN REPLICATE_TAC 2 (POP_ASSUM MP_TAC);
  REWRITE_TAC[DIST_0;DIST_SYM];
  TYPIFY `dih_y (norm (v p)) (norm (v (p + 2))) (norm (v (p + 3)))       (dist (v (p + 2),v (p + 3)))       (dist (v p,v (p + 3)))       (dist (v p,v (p + 2))) = dih_y (norm (v p)) (norm (v (p + 3))) (norm (v (p + 2))) (dist (v (p + 2),v (p + 3))) (dist (v p,v (p + 2))) (dist (v p,v (p + 3)))` (C SUBGOAL_THEN SUBST1_TAC);
    BY(REWRITE_TAC[Nonlinear_lemma.dih_y_sym]);
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let LINDIH_11 = prove_by_refinement(
    (* main_work4 ==> *) `main_nonlinear_terminal_v11 ==>
    (!y1 y2 y3 y4 y5 y6.
       ineq
      [
	 (&2,y1,&2);
	 (&2,y2,#2.52);
	 (&2,y3,#2.52);
	 (&2,y4,&2);
	 (#3.01,y5,#3.55);
	 (&2 * h0,y6,#3.01)]      
      ((dih_y y1 y2 y3 y4 y5 y6  < #1.1) \/ delta_y y1 y2 y3 y4 y5 y6 < &0))`,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `delta_y y1 y2 y3 y4 y5 y6 < &0` ASM_CASES_TAC;
    BY(ASM_REWRITE_TAC[]);
  ASM_REWRITE_TAC[];
  REWRITE_TAC[Sphere.dih_y;LET_THM];
  GMATCH_SIMP_TAC Merge_ineq.lindih_lt;
  REWRITE_TAC[GSYM CONJ_ASSOC];
  GMATCH_SIMP_TAC REAL_LT_MUL_EQ;
  CONJ_TAC;
    BY(ASM_TAC THEN REAL_ARITH_TAC);
  CONJ_TAC;
    BY(MP_TAC Flyspeck_constants.bounds THEN REAL_ARITH_TAC);
  CONJ_TAC;
    BY(ASM_TAC THEN REAL_ARITH_TAC);
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "9368433105") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  ASM_REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;LET_THM];
  REWRITE_TAC[Sphere.delta4_y;Sphere.y_of_x];
  DISCH_TAC;
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "8405387449") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  ASM_REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;LET_THM];
  ASM_SIMP_TAC [arith `d > &0 ==> &0 < d`];
  DISCH_TAC;
  TYPIFY `&0 <= (&4 *  (y1 * y1) *  delta_x (y1 * y1) (y2 * y2) (y3 * y3) (y4 * y4) (y5 * y5) (y6 * y6))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT (GMATCH_SIMP_TAC REAL_LE_MUL);
    FIRST_X_ASSUM_ST `delta_y` MP_TAC THEN REWRITE_TAC[Sphere.delta_y];
    BY(ASM_TAC THEN REAL_ARITH_TAC);
  GMATCH_SIMP_TAC Pack1.bp_bdt;
  GMATCH_SIMP_TAC SQRT_POS_LE;
  ASM_REWRITE_TAC[];
  CONJ_TAC;
    GMATCH_SIMP_TAC REAL_LE_MUL;
    GMATCH_SIMP_TAC TAN_POS_PI2_LE;
    ASM_SIMP_TAC[arith `x > &0 ==> &0 <= x`];
    BY(MP_TAC Flyspeck_constants.bounds THEN ASM_TAC THEN REAL_ARITH_TAC);
  GMATCH_SIMP_TAC SQRT_POW_2;
  ASM_REWRITE_TAC[];
  MATCH_MP_TAC REAL_LT_TRANS;
  TYPIFY `#3.86 * delta4_squared_y y1 y2 y3 y4 y5 y6` EXISTS_TAC;
  CONJ_TAC;
    FIRST_X_ASSUM_ST `x1_delta_y` MP_TAC;
    BY(REWRITE_TAC[Sphere.x1_delta_y;Sphere.delta4_squared_y;Sphere.y_of_x;Sphere.x1_delta_x]);
  REWRITE_TAC[Sphere.delta4_squared_y;Sphere.y_of_x;Sphere.delta4_squared_x];
  REWRITE_TAC[arith `(x * y) pow 2 = x pow 2 * y pow 2`];
  GMATCH_SIMP_TAC REAL_LT_RMUL_EQ;
  CONJ_TAC;
    REWRITE_TAC[GSYM Trigonometry2.NOT_ZERO_EQ_POW2_LT];
    BY(ASM_TAC THEN REAL_ARITH_TAC);
  TYPIFY `#3.86 = sqrt(#3.86) pow 2` (C SUBGOAL_THEN SUBST1_TAC);
    GMATCH_SIMP_TAC SQRT_POW_2;
    BY(REAL_ARITH_TAC);
  MATCH_MP_TAC Tame_inequalities.REAL_LT_SQUARE_POS;
  GMATCH_SIMP_TAC SQRT_POS_LT;
  CONJ_TAC;
    BY(REAL_ARITH_TAC);
  TYPIFY `atn(sqrt  #3.86) < atn(tan  #1.1)` ENOUGH_TO_SHOW_TAC;
    BY(REWRITE_TAC[ATN_MONO_LT_EQ]);
  GMATCH_SIMP_TAC TAN_ATN;
  CONJ_TAC;
    BY(MP_TAC Flyspeck_constants.bounds THEN REAL_ARITH_TAC);
  BY(REWRITE_TAC[Flyspeck_constants.calc `atn (sqrt  #3.86) <  #1.1`])
  ]);;
  (* }}} *)

let LEMMA_PWE3 = prove_by_refinement(
(* main_work4 ==> *) ` main_nonlinear_terminal_v11 ==> (!s v p.
    is_scs_v39 s /\ scs_basic_v39 s /\ scs_k_v39 s = 4 /\ v IN MMs_v39 s /\ 
    (!i j. scs_diag 4 i j ==> scs_a_v39 s i j <= cstab /\ cstab < dist(v i,v j) /\ &4*h0 < scs_b_v39 s i j) /\
    (!i. (scs_a_v39 s i (i+1) = &2 /\ scs_b_v39 s i (i+1)= &2*h0) \/
       (scs_a_v39 s i (i+1) = &2 * h0 /\ scs_b_v39 s i (i+1)= cstab)) /\
      scs_generic v /\
    ~scs_is_str s v (p+1) /\ ~scs_is_str s v (p+3) /\
    scs_is_str s v p /\ norm (v p) = &2 /\
    scs_a_v39 s p (p+1) = &2 /\
    scs_a_v39 s (p+3) (p) = &2 * h0 /\
    dist(v (p+2),v(p+3)) =  &2 /\
    &2 * h0 <= dist(v(p+1),v(p+2)) ==>
    pi / &2 < dihV (vec 0) (v (p+1)) (v (p)) (v (p+2)))`,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[arith `x < y <=> ~(y <= x)`] THEN DISCH_TAC;
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `BBprime2_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.scs_basic;Ayqjtmd.unadorned_MMs;IN]);
  TYPIFY `BBprime_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime2_v39]);
  TYPIFY `BBs_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime_v39]);
  INTRO_TAC Appendix.BBs_v39 [`s`;`v`];
  ASM_REWRITE_TAC[LET_DEF;LET_END_DEF;Terminal.IMAGE_SUBSET_IN;IN_UNIV;arith `~(4 <= 3)`];
  REPEAT WEAKER_STRIP_TAC;
  TYPED_ABBREV_TAC `V = IMAGE v (:num)`;
  TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`;
  TYPED_ABBREV_TAC `FF = IMAGE (\i. (v i,v (SUC i))) (:num)`;
  TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    EXPAND_TAC "V";
    REWRITE_TAC[IN_IMAGE;IN_UNIV];
    BY(MESON_TAC[]);
  INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`4`];
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  TYPIFY `!i. interior_angle1 (vec 0) FF (v i) = azim (vec 0) (v i) (v (i+1)) (v (i+3))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    EXPAND_TAC "FF";
    GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] (GSYM Ocbicby.INTERIOR_ANGLE1_AZIM));
    TYPIFY `s` EXISTS_TAC;
    BY(ASM_REWRITE_TAC[IN;arith `4 - 1 = 3`;arith `3 < 4`;arith `SUC i = i+1`]);
  INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`];
  ASM_REWRITE_TAC[LET_THM;IN;arith `4-1 = 3`;arith `SUC i = i+1`];
  DISCH_TAC;
  TYPIFY `!i. v (i MOD 4) = v i` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC (GSYM Oxl_def.periodic_mod);
    BY(ASM_REWRITE_TAC[arith `~(0=4)`]);
  TYPIFY `v 4 = v 0 /\ v 5 = v 1 /\ v 6 = v 2 /\ v 7 = v 3 /\ v 8  = v 4` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[Terminal.MOD_4_EXPLICIT;arith `7 = 1*4 + 3`;arith `8 = 2*4 + 0`;MOD_MULT_ADD]);
  TYPIFY `generic V E` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM_ST `scs_generic` MP_TAC THEN ASM_REWRITE_TAC[Appendix.scs_generic]);
  TYPIFY `v (p+4) = v p /\ v (p+5) = v (p+1) /\ v (p+6) = v (p+2) /\ v (p+7) = v (p+3) /\ v (p+8)  = v (p)` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `MOD` ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[arith `p+4 = 1*4 + p /\ p+5 = 1*4+(p+1) /\ p+6=1*4+(p+2) /\ p+7 = 1*4 + (p+3) /\ p+8 = 2*4 + p`;MOD_MULT_ADD]);
  TYPIFY `!i. &2 <= norm (v i) /\ norm (v i) <= &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    REWRITE_TAC[GSYM Fnjlbxs.in_ball_annulus];
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> &2 <= dist(v (p + i),v (p+j))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    MATCH_MP_TAC Ocbicby.scs_lb_2;
    TYPIFY `s` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    (NUM_REDUCE_TAC);
    REPEAT (GMATCH_SIMP_TAC MOD_LT);
    BY(ASM_REWRITE_TAC[]);
  COMMENT "edge lengths";
  TYPIFY `scs_b_v39 s (p+1) ((p+1)+1) <= cstab /\ scs_b_v39 s p (p+1) = &2 * h0 /\ scs_b_v39 s (p+3) ((p+3)+1) = cstab` (C SUBGOAL_THEN ASSUME_TAC);
    TYPIFY `~(&2 = &2 * h0)` (C SUBGOAL_THEN ASSUME_TAC);
      BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
    CONJ2_TAC;
      TYPIFY `scs_a_v39 s (p+3) p = scs_a_v39 s (p+3) ((p+3)+1)` ENOUGH_TO_SHOW_TAC;
        BY(ASM_MESON_TAC[]);
      REWRITE_TAC[arith `((p+3)+1) = p+4`];
      FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
      BY(ASM_MESON_TAC[Appendix.periodic2;Oxl_def.periodic]);
    FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM_ST `x <= cstab` (C INTRO_TAC [`p+1`]) THEN ASM_REWRITE_TAC[arith `3 < 4`];
    BY(REWRITE_TAC[arith `SUC (p+1) = (p+1)+1`]);
  FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[arith `((p:num) + i ) + j = p + (i+j)`] THEN NUM_REDUCE_TAC;
  DISCH_TAC;
  TYPIFY `dist(v (p+1),v(p+2)) <= cstab /\ dist(v p,v(p+1)) <= &2 * h0 /\ dist(v(p+3),v(p+4)) <= cstab` (C SUBGOAL_THEN MP_TAC);
    FIRST_X_ASSUM MP_TAC;
    FIRST_X_ASSUM_ST `scs_b_v39` MP_TAC;
    BY(MESON_TAC[REAL_LE_TRANS]);
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "collinear";
  TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0,v i, v j}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT GEN_TAC THEN DISCH_TAC;
    MATCH_MP_TAC Zlzthic.PROPERTIES_GENERIC_LOCAL_FAN_ALT;
    GEXISTL_TAC [`V`;`E`;`FF`];
    BY(ASM_REWRITE_TAC[IN_IMAGE;IN_UNIV]);
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 20 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
  INTRO_TAC PERIODIC_INJ_MOD [`v`;`4`];
  ASM_REWRITE_TAC[arith `~(4 = 0)`];
  DISCH_TAC;
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i=j) ==> ~(v (p +i) = v (p+j))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM MP_TAC THEN ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    (NUM_REDUCE_TAC);
    REPEAT (GMATCH_SIMP_TAC MOD_LT);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. 0 < i /\ i < 4 ==> ~(v (p+i) = v p)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[arith `p+0 = p`;arith `0< 4`;arith `0 < i ==> ~(i=0)`]);
  COMMENT "long diag";
  TYPIFY `#2.52 = &2 * h0 /\ #3.01 = cstab` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REWRITE_TAC[Sphere.h0;Sphere.cstab] THEN REAL_ARITH_TAC);
  TYPIFY `#3.55 <= dist(v (p+2),v p)` ASM_CASES_TAC;
    INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "2073661826") [`norm (v (p+1))`;`norm(v (p+2))`;`norm(v p)`;`dist(v(p+2),v p)`;`dist(v(p+1),v p)`;`dist(v(p+1),v(p+2))`];
    FIRST_X_ASSUM_ST `norm x = &2` kill;
    REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
    ASM_REWRITE_TAC[];
    DISCH_THEN MP_TAC THEN ANTS_TAC;
      CONJ_TAC;
        INTRO_TAC DIST_TRIANGLE [`v (p+2)`;`(vec 0):real^3`;`v (p)`];
        REWRITE_TAC[DIST_0];
        MATCH_MP_TAC (arith `x <= &2 * h0 /\ y <= &2 * h0 ==> (d <= x + y ==> d <= &4 * h0)`);
        BY(ASM_REWRITE_TAC[]);
      CONJ2_TAC;
        BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
      TYPIFY `v p = v (p+0)` (C SUBGOAL_THEN SUBST1_TAC);
        ONCE_REWRITE_TAC[arith `p+0=p`];
        BY(REWRITE_TAC[]);
      BY(FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    DISCH_TAC;
    FIRST_X_ASSUM_ST `dihV` MP_TAC;
    REWRITE_TAC[arith `~(x <= y) <=> y < x`];
    GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] Merge_ineq.DIHV_EQ_DIH_Y);
    ASM_REWRITE_TAC[DIST_0];
    CONJ_TAC;
      BY(CONJ_TAC THEN (REPEAT (FIRST_X_ASSUM MATCH_MP_TAC)) THEN NUM_REDUCE_TAC);
    ONCE_REWRITE_TAC[Nonlinear_lemma.dih_y_sym];
    MATCH_MP_TAC Ocbicby.delta4_y_imp_obtuse;
    CONJ_TAC;
      BY(ASM_MESON_TAC[arith `&2 <= x==> &0 < x`]);
    INTRO_TAC DIST_SYM [`v p`;`v (p+2)`];
    DISCH_THEN SUBST1_TAC;
    CONJ_TAC;
      BY(REWRITE_TAC[REWRITE_RULE[LET_THM] Tame_lemmas.delta_y_pos]);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. ~scs_is_str s v i ==> ~coplanar {vec 0, v i, v(i+1), v(i+3)}` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC THEN ASM_REWRITE_TAC[Appendix.scs_is_str;arith `SUC i = i+1`;GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR];
    NUM_REDUCE_TAC;
    DISCH_THEN (unlist REWRITE_TAC);
    MATCH_MP_TAC (arith `&0 < a ==> ~(a = &0)`);
    INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`;`i`];
    BY(ASM_REWRITE_TAC[LET_THM;IN;arith `SUC i = i+1`;arith `4 - 1 = 3`]);
  FIRST_X_ASSUM MP_TAC THEN BURY_MP_TAC;
  COMMENT "short diag";
  INTRO_TAC Terminal.vv_split_azim_generic [`v`;`4`;`p`;`2`;`3`];
  ASM_REWRITE_TAC[arith `SUC p = p+1`;LET_THM];
  NUM_REDUCE_TAC;
  DISCH_TAC;
  TYPIFY `azim (vec 0) (v p) (v (p+1)) (v (p+3)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
    MATCH_MP_TAC Local_lemmas.CONVEX_LOFA_IMP_INANGLE_LE_PI;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `azim (vec 0) (v p) (v (p + 1)) (v (p + 2)) <= pi /\  azim (vec 0) (v p) (v (p + 2)) (v (p + 3)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC (arith `&0 < pi /\ &0 <= a /\ &0 <= b /\ a + b <= pi ==> a <= pi /\ b <= pi`);
    REWRITE_TAC[PI_POS];
    REWRITE_TAC[Counting_spheres.AZIM_NN];
    BY(REPLICATE_TAC 2 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM_ST `scs_is_str` MP_TAC;
  REWRITE_TAC[Appendix.scs_is_str];
  ASM_REWRITE_TAC[];
  NUM_REDUCE_TAC;
  REWRITE_TAC[arith `SUC p = p+1`];
  DISCH_THEN (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
  INTRO_TAC AZIM_DIHV_SAME_STRONG [`(vec 0):real^3`;`v p`;`v (p+2)`;`v (p+3)`];
  INTRO_TAC AZIM_DIHV_SAME_STRONG [`(vec 0):real^3`;`v p`;`v (p+1)`;`v (p+2)`];
  FIRST_X_ASSUM_ST `pi = a` (MP_TAC o SYM) THEN DISCH_TAC;
  ASM_REWRITE_TAC[];
  TYPIFY `~collinear {vec 0, v p, v (p + 1)} /\ ~collinear {vec 0, v p, v (p + 2)} /\  ~collinear {vec 0, v p, v (p + 3)}` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  REPEAT (FIRST_X_ASSUM_ST `dihV` MP_TAC) THEN REWRITE_TAC[] THEN REPEAT (GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] Merge_ineq.DIHV_EQ_DIH_Y));
  ASM_REWRITE_TAC[];
  CONJ_TAC;
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM_ST `+` MP_TAC THEN ASM_REWRITE_TAC[];
  (DISCH_TAC);
  TYPIFY `cstab <= dist(v p,v(p+2))` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `cstab < d` (C INTRO_TAC [`p`;`p+2`]);
    ANTS_TAC;
      REWRITE_TAC[arith `p+2 = SUC(SUC (p))`];
      MATCH_MP_TAC Tfitskc.SCS_DIAG_2;
      BY(NUM_REDUCE_TAC);
    BY(REAL_ARITH_TAC);
  INTRO_TAC (UNDISCH2 LINDIH_11) [`norm (v (p))`;`norm(v (p+3))`;`norm(v (p+2))`;`dist(v(p+3),v(p+2))`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+3))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  TYPIFY `(&2 <= norm (v p) /\ norm (v p) <= &2) /\   (&2 <= norm (v (p + 3)) /\ norm (v (p + 3)) <=  #2.52) /\   (&2 <= norm (v (p + 2)) /\ norm (v (p + 2)) <=  #2.52) /\   (&2 <= dist (v (p + 3),v (p + 2)) /\ dist (v (p + 3),v (p + 2)) <= &2) /\   ( #3.01 <= dist (v p,v (p + 2)) /\ dist (v p,v (p + 2)) <=  #3.55) /\   &2 * h0 <= dist (v p,v (p + 3)) /\   dist (v p,v (p + 3)) <=  #3.01` (C SUBGOAL_THEN ASSUME_TAC);
    ASM_REWRITE_TAC[arith `(x <= y /\ y <= x) <=> y = x`];
    nCONJ_TAC 3;
      BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
    CONJ_TAC;
      BY(FIRST_X_ASSUM_ST `#3.55` MP_TAC THEN REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC);
    TYPIFY `scs_a_v39 s (p+3) p <= dist(v(p+3),v p) ` ENOUGH_TO_SHOW_TAC;
      REWRITE_TAC[DIST_SYM] THEN MATCH_MP_TAC (arith `a = b ==> (a <= d ==> b <= d)`);
      BY(ASM_REWRITE_TAC[]);
    BY(ASM_REWRITE_TAC[]);
  FIRST_ASSUM (unlist REWRITE_TAC);
  REWRITE_TAC[DE_MORGAN_THM];
  CONJ2_TAC;
    REWRITE_TAC[arith `~(d < &0) <=> &0 <= d`];
    FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`p+3`]);
    ANTS_TAC;
      BY(ASM_REWRITE_TAC[]);
    REWRITE_TAC[arith `(p+3) +i = p+(3+i)`] THEN NUM_REDUCE_TAC;
    REWRITE_TAC[Oxlzlez.coplanar_delta_y];
    ASM_REWRITE_TAC[DIST_SYM;DIST_0];
    MATCH_MP_TAC (arith `a = b ==> &0 < a ==> &0 <= b`);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  (DISCH_TAC);
  COMMENT "5550";
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "5550839403 delta") [`norm (v (p))`;`norm(v (p+1))`;`norm(v (p+2))`;`cstab`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+1))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  TYPIFY `(&2 <= norm (v p) /\ norm (v p) <= &2) /\   (&2 <= norm (v (p + 1)) /\ norm (v (p + 1)) <=  #2.52) /\   (&2 <= norm (v (p + 2)) /\ norm (v (p + 2)) <=  #2.52) /\   ( #3.01 <= cstab /\ cstab <=  #3.01) /\   ( #3.01 <= dist (v p,v (p + 2)) /\ dist (v p,v (p + 2)) <=  #3.55) /\   &2 <= dist (v p,v (p + 1)) /\   dist (v p,v (p + 1)) <= &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    ASM_REWRITE_TAC[arith `(x <= y /\ y <= x) <=> y = x`];
    FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`0`;`1`]);
    BY(REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  FIRST_ASSUM (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "5550839403") [`norm (v (p))`;`norm(v (p+1))`;`norm(v (p+2))`;`cstab`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+1))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  FIRST_X_ASSUM_ST `#3.01` (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC Ocbicby.dih_y_mono [`norm (v (p))`;`norm(v (p+1))`;`norm(v (p+2))`;`dist(v (p+1),v(p+2))`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+1))`;`cstab`];
  ANTS_TAC;
    TYPIFY_GOAL_THEN `&0 < norm (v p)` (unlist REWRITE_TAC);
      BY(ASM_REWRITE_TAC[arith `&0 < &2`]);
    FIRST_X_ASSUM_ST `norm (v p) =  &2` kill;
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`1`;`2`]);
      BY(NUM_REDUCE_TAC THEN REAL_ARITH_TAC);
    CONJ2_TAC;
      BY(FIRST_X_ASSUM_ST `delta_y` MP_TAC THEN REAL_ARITH_TAC);
    FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`p+1`]);
    ANTS_TAC;
      BY(ASM_REWRITE_TAC[]);
    REWRITE_TAC[arith `(p+1) +i = p+(1+i)`] THEN NUM_REDUCE_TAC;
    REWRITE_TAC[Oxlzlez.coplanar_delta_y];
    ASM_REWRITE_TAC[DIST_SYM;DIST_0];
    MATCH_MP_TAC (arith `a = b ==> &0 < a ==> &0 < b`);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  (DISCH_TAC);
  TYPIFY `dih_y (norm (v p)) (norm (v (p + 3))) (norm (v (p + 2)))      (dist (v (p + 3),v (p + 2)))      (dist (v p,v (p + 2)))      (dist (v p,v (p + 3))) + dih_y (norm (v p)) (norm (v (p + 1))) (norm (v (p + 2)))      (dist (v (p + 1),v (p + 2)))      (dist (v p,v (p + 2)))      (dist (v p,v (p + 1))) < #3.1` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 5 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  TYPIFY `#3.1 < pi` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MP_TAC Flyspeck_constants.bounds THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM_ST `a = pi` MP_TAC THEN REPLICATE_TAC 2 (POP_ASSUM MP_TAC);
  REWRITE_TAC[DIST_0;DIST_SYM];
  FIRST_X_ASSUM_ST `dist s = &2` MP_TAC;
  DISCH_THEN ((unlist REWRITE_TAC) o GSYM);
  TYPIFY `dih_y (norm (v p)) (norm (v (p + 2))) (norm (v (p + 3)))       (dist (v (p + 2),v (p + 3)))       (dist (v p,v (p + 3)))       (dist (v p,v (p + 2))) = dih_y (norm (v p)) (norm (v (p + 3))) (norm (v (p + 2))) (dist (v (p + 2),v (p + 3))) (dist (v p,v (p + 2))) (dist (v p,v (p + 3)))` (C SUBGOAL_THEN SUBST1_TAC);
    BY(REWRITE_TAC[Nonlinear_lemma.dih_y_sym]);
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let LEMMA_PWE4 = prove_by_refinement(
(* main_work4 ==> *) ` main_nonlinear_terminal_v11 ==> (!s v p.
    is_scs_v39 s /\ scs_basic_v39 s /\ scs_k_v39 s = 4 /\ v IN MMs_v39 s /\ 
    (!i j. scs_diag 4 i j ==> scs_a_v39 s i j <= cstab /\ cstab < dist(v i,v j) /\ &4*h0 < scs_b_v39 s i j) /\
    (!i. (scs_a_v39 s i (i+1) = &2 /\ scs_b_v39 s i (i+1)= &2*h0) \/
       (scs_a_v39 s i (i+1) = &2 * h0 /\ scs_b_v39 s i (i+1)= cstab)) /\
      scs_generic v /\
    ~scs_is_str s v (p+1) /\ ~scs_is_str s v (p+3) /\
    scs_is_str s v p /\ norm (v p) = &2 /\
    scs_a_v39 s (p+3) p = &2 /\
    scs_a_v39 s p (p+1) = &2 * h0 /\
    dist(v (p+2),v(p+1)) =  &2 /\
    &2 * h0 <= dist(v(p+3),v(p+2)) ==>
    pi / &2 < dihV (vec 0) (v (p+3)) (v (p)) (v (p+2)))`,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[arith `x < y <=> ~(y <= x)`] THEN DISCH_TAC;
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `BBprime2_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.scs_basic;Ayqjtmd.unadorned_MMs;IN]);
  TYPIFY `BBprime_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime2_v39]);
  TYPIFY `BBs_v39 s v` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Appendix.BBprime_v39]);
  INTRO_TAC Appendix.BBs_v39 [`s`;`v`];
  ASM_REWRITE_TAC[LET_DEF;LET_END_DEF;Terminal.IMAGE_SUBSET_IN;IN_UNIV;arith `~(4 <= 3)`];
  REPEAT WEAKER_STRIP_TAC;
  TYPED_ABBREV_TAC `V = IMAGE v (:num)`;
  TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`;
  TYPED_ABBREV_TAC `FF = IMAGE (\i. (v i,v (SUC i))) (:num)`;
  TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    EXPAND_TAC "V";
    REWRITE_TAC[IN_IMAGE;IN_UNIV];
    BY(MESON_TAC[]);
  INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`4`];
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  TYPIFY `!i. interior_angle1 (vec 0) FF (v i) = azim (vec 0) (v i) (v (i+1)) (v (i+3))` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    EXPAND_TAC "FF";
    GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] (GSYM Ocbicby.INTERIOR_ANGLE1_AZIM));
    TYPIFY `s` EXISTS_TAC;
    BY(ASM_REWRITE_TAC[IN;arith `4 - 1 = 3`;arith `3 < 4`;arith `SUC i = i+1`]);
  INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`];
  ASM_REWRITE_TAC[LET_THM;IN;arith `4-1 = 3`;arith `SUC i = i+1`];
  DISCH_TAC;
  TYPIFY `!i. v (i MOD 4) = v i` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    MATCH_MP_TAC (GSYM Oxl_def.periodic_mod);
    BY(ASM_REWRITE_TAC[arith `~(0=4)`]);
  TYPIFY `v 4 = v 0 /\ v 5 = v 1 /\ v 6 = v 2 /\ v 7 = v 3 /\ v 8  = v 4` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[Terminal.MOD_4_EXPLICIT;arith `7 = 1*4 + 3`;arith `8 = 2*4 + 0`;MOD_MULT_ADD]);
  TYPIFY `generic V E` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM_ST `scs_generic` MP_TAC THEN ASM_REWRITE_TAC[Appendix.scs_generic]);
  TYPIFY `v (p+4) = v p /\ v (p+5) = v (p+1) /\ v (p+6) = v (p+2) /\ v (p+7) = v (p+3) /\ v (p+8)  = v (p)` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `MOD` ((unlist ONCE_REWRITE_TAC) o GSYM);
    BY(REWRITE_TAC[arith `p+4 = 1*4 + p /\ p+5 = 1*4+(p+1) /\ p+6=1*4+(p+2) /\ p+7 = 1*4 + (p+3) /\ p+8 = 2*4 + p`;MOD_MULT_ADD]);
  TYPIFY `!i. &2 <= norm (v i) /\ norm (v i) <= &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    REWRITE_TAC[GSYM Fnjlbxs.in_ball_annulus];
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> &2 <= dist(v (p + i),v (p+j))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    MATCH_MP_TAC Ocbicby.scs_lb_2;
    TYPIFY `s` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    (NUM_REDUCE_TAC);
    REPEAT (GMATCH_SIMP_TAC MOD_LT);
    BY(ASM_REWRITE_TAC[]);
  COMMENT "edge lengths";
  TYPIFY `scs_b_v39 s (p+3) ((p+3)+1) = &2 * h0 /\ scs_b_v39 s p (p+1) = cstab /\ scs_b_v39 s (p+2) ((p+2)+1) <= cstab` (C SUBGOAL_THEN ASSUME_TAC);
    TYPIFY `~(&2 = &2 * h0)` (C SUBGOAL_THEN ASSUME_TAC);
      BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
    CONJ2_TAC;
      CONJ_TAC;
        BY(ASM_MESON_TAC[]);
      FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
      FIRST_X_ASSUM_ST `scs_b_v39 s i j <= cstab` (C INTRO_TAC [`p+2`]);
      ASM_REWRITE_TAC[arith `SUC (p+2) = (p+2)+1`];
      BY(DISCH_THEN MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    TYPIFY `scs_a_v39 s (p+3) p = scs_a_v39 s (p+3) ((p+3)+1)` ENOUGH_TO_SHOW_TAC;
      BY(ASM_MESON_TAC[]);
    REWRITE_TAC[arith `((p+3)+1) = p+4`];
    FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC;
    BY(ASM_MESON_TAC[Appendix.periodic2;Oxl_def.periodic]);
  FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[arith `((p:num) + i ) + j = p + (i+j)`] THEN NUM_REDUCE_TAC;
  DISCH_TAC;
  TYPIFY `dist(v (p+3),v(p+4)) <= &2 *h0 /\ dist(v p,v(p+1)) <= cstab /\ dist(v(p+2),v(p+3)) <= cstab` (C SUBGOAL_THEN MP_TAC);
    FIRST_X_ASSUM MP_TAC;
    FIRST_X_ASSUM_ST `scs_b_v39` MP_TAC;
    BY(MESON_TAC[REAL_LE_TRANS]);
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "collinear";
  TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0,v i, v j}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT GEN_TAC THEN DISCH_TAC;
    MATCH_MP_TAC Zlzthic.PROPERTIES_GENERIC_LOCAL_FAN_ALT;
    GEXISTL_TAC [`V`;`E`;`FF`];
    BY(ASM_REWRITE_TAC[IN_IMAGE;IN_UNIV]);
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 20 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]);
  INTRO_TAC PERIODIC_INJ_MOD [`v`;`4`];
  ASM_REWRITE_TAC[arith `~(4 = 0)`];
  DISCH_TAC;
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i=j) ==> ~(v (p +i) = v (p+j))` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM MP_TAC THEN ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT;
    (NUM_REDUCE_TAC);
    REPEAT (GMATCH_SIMP_TAC MOD_LT);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. 0 < i /\ i < 4 ==> ~(v (p+i) = v p)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[arith `p+0 = p`;arith `0< 4`;arith `0 < i ==> ~(i=0)`]);
  COMMENT "long diag";
  TYPIFY `#2.52 = &2 * h0 /\ #3.01 = cstab` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REWRITE_TAC[Sphere.h0;Sphere.cstab] THEN REAL_ARITH_TAC);
  TYPIFY `#3.55 <= dist(v (p+2),v p)` ASM_CASES_TAC;
    INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "2073661826") [`norm (v (p+3))`;`norm(v (p+2))`;`norm(v p)`;`dist(v(p+2),v p)`;`dist(v(p+3),v p)`;`dist(v(p+3),v(p+2))`];
    FIRST_X_ASSUM_ST `norm x = &2` kill;
    REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
    ASM_REWRITE_TAC[];
    DISCH_THEN MP_TAC THEN ANTS_TAC;
      CONJ_TAC;
        INTRO_TAC DIST_TRIANGLE [`v (p+2)`;`(vec 0):real^3`;`v (p)`];
        REWRITE_TAC[DIST_0];
        MATCH_MP_TAC (arith `x <= &2 * h0 /\ y <= &2 * h0 ==> (d <= x + y ==> d <= &4 * h0)`);
        BY(ASM_REWRITE_TAC[]);
      CONJ2_TAC;
        BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
      TYPIFY `v p = v (p+0)` (C SUBGOAL_THEN SUBST1_TAC);
        ONCE_REWRITE_TAC[arith `p+0=p`];
        BY(REWRITE_TAC[]);
      BY(FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    DISCH_TAC;
    FIRST_X_ASSUM_ST `dihV` MP_TAC;
    REWRITE_TAC[arith `~(x <= y) <=> y < x`];
    GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] Merge_ineq.DIHV_EQ_DIH_Y);
    ASM_REWRITE_TAC[DIST_0];
    CONJ_TAC;
      BY(CONJ_TAC THEN (REPEAT (FIRST_X_ASSUM MATCH_MP_TAC)) THEN NUM_REDUCE_TAC);
    ONCE_REWRITE_TAC[Nonlinear_lemma.dih_y_sym];
    MATCH_MP_TAC Ocbicby.delta4_y_imp_obtuse;
    CONJ_TAC;
      BY(ASM_MESON_TAC[arith `&2 <= x==> &0 < x`]);
    INTRO_TAC DIST_SYM [`v p`;`v (p+2)`];
    DISCH_THEN SUBST1_TAC;
    CONJ_TAC;
      BY(REWRITE_TAC[REWRITE_RULE[LET_THM] Tame_lemmas.delta_y_pos]);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!i. ~scs_is_str s v i ==> ~coplanar {vec 0, v i, v(i+1), v(i+3)}` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC THEN ASM_REWRITE_TAC[Appendix.scs_is_str;arith `SUC i = i+1`;GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR];
    NUM_REDUCE_TAC;
    DISCH_THEN (unlist REWRITE_TAC);
    MATCH_MP_TAC (arith `&0 < a ==> ~(a = &0)`);
    INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`;`i`];
    BY(ASM_REWRITE_TAC[LET_THM;IN;arith `SUC i = i+1`;arith `4 - 1 = 3`]);
  FIRST_X_ASSUM MP_TAC THEN BURY_MP_TAC;
  COMMENT "short diag";
  INTRO_TAC Terminal.vv_split_azim_generic [`v`;`4`;`p`;`2`;`3`];
  ASM_REWRITE_TAC[arith `SUC p = p+1`;LET_THM];
  NUM_REDUCE_TAC;
  DISCH_TAC;
  TYPIFY `azim (vec 0) (v p) (v (p+1)) (v (p+3)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `interior_angle1` ((unlist REWRITE_TAC) o GSYM);
    MATCH_MP_TAC Local_lemmas.CONVEX_LOFA_IMP_INANGLE_LE_PI;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `azim (vec 0) (v p) (v (p + 1)) (v (p + 2)) <= pi /\  azim (vec 0) (v p) (v (p + 2)) (v (p + 3)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC (arith `&0 < pi /\ &0 <= a /\ &0 <= b /\ a + b <= pi ==> a <= pi /\ b <= pi`);
    REWRITE_TAC[PI_POS];
    REWRITE_TAC[Counting_spheres.AZIM_NN];
    BY(REPLICATE_TAC 2 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM_ST `scs_is_str` MP_TAC;
  REWRITE_TAC[Appendix.scs_is_str];
  ASM_REWRITE_TAC[];
  NUM_REDUCE_TAC;
  REWRITE_TAC[arith `SUC p = p+1`];
  DISCH_THEN (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
  INTRO_TAC AZIM_DIHV_SAME_STRONG [`(vec 0):real^3`;`v p`;`v (p+2)`;`v (p+3)`];
  INTRO_TAC AZIM_DIHV_SAME_STRONG [`(vec 0):real^3`;`v p`;`v (p+1)`;`v (p+2)`];
  FIRST_X_ASSUM_ST `pi = a` (MP_TAC o SYM) THEN DISCH_TAC;
  ASM_REWRITE_TAC[];
  TYPIFY `~collinear {vec 0, v p, v (p + 1)} /\ ~collinear {vec 0, v p, v (p + 2)} /\  ~collinear {vec 0, v p, v (p + 3)}` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  REPEAT (FIRST_X_ASSUM_ST `dihV` MP_TAC) THEN REWRITE_TAC[] THEN REPEAT (GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] Merge_ineq.DIHV_EQ_DIH_Y));
  ASM_REWRITE_TAC[];
  CONJ_TAC;
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC);
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM_ST `+` MP_TAC THEN ASM_REWRITE_TAC[];
  (DISCH_TAC);
  TYPIFY `cstab <= dist(v p,v(p+2))` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `cstab < d` (C INTRO_TAC [`p`;`p+2`]);
    ANTS_TAC;
      REWRITE_TAC[arith `p+2 = SUC(SUC (p))`];
      MATCH_MP_TAC Tfitskc.SCS_DIAG_2;
      BY(NUM_REDUCE_TAC);
    BY(REAL_ARITH_TAC);
  INTRO_TAC (UNDISCH2 LINDIH_11) [`norm (v (p))`;`norm(v (p+1))`;`norm(v (p+2))`;`dist(v(p+1),v(p+2))`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+1))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  TYPIFY `(&2 <= norm (v p) /\ norm (v p) <= &2) /\   (&2 <= norm (v (p + 1)) /\ norm (v (p + 1)) <=  #2.52) /\   (&2 <= norm (v (p + 2)) /\ norm (v (p + 2)) <=  #2.52) /\   (&2 <= dist (v (p + 1),v (p + 2)) /\ dist (v (p + 1),v (p + 2)) <= &2) /\   ( #3.01 <= dist (v p,v (p + 2)) /\ dist (v p,v (p + 2)) <=  #3.55) /\   &2 * h0 <= dist (v p,v (p + 1)) /\   dist (v p,v (p + 1)) <=  #3.01` (C SUBGOAL_THEN ASSUME_TAC);
    ASM_REWRITE_TAC[arith `(x <= y /\ y <= x) <=> y = x`];
    nCONJ_TAC 2;
      TYPIFY `scs_a_v39 s p (p+1) <= dist(v p,v(p+1)) ` ENOUGH_TO_SHOW_TAC;
        MATCH_MP_TAC (arith `a = b ==> (a <= d ==> b <= d)`);
        BY(ASM_REWRITE_TAC[]);
      BY(ASM_REWRITE_TAC[]);
    ASM_REWRITE_TAC[];
    ONCE_REWRITE_TAC[DIST_SYM];
    ASM_REWRITE_TAC[];
    BY(FIRST_X_ASSUM_ST `#3.55` MP_TAC THEN REAL_ARITH_TAC);
  FIRST_ASSUM (unlist REWRITE_TAC);
  REWRITE_TAC[DE_MORGAN_THM];
  CONJ2_TAC;
    REWRITE_TAC[arith `~(d < &0) <=> &0 <= d`];
    BY(REWRITE_TAC[REWRITE_RULE[LET_THM] Tame_lemmas.delta_y_pos]);
  (DISCH_TAC);
  COMMENT "5550";
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "5550839403 delta") [`norm (v (p))`;`norm(v (p+3))`;`norm(v (p+2))`;`cstab`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+3))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  TYPIFY `(&2 <= norm (v p) /\ norm (v p) <= &2) /\   (&2 <= norm (v (p + 3)) /\ norm (v (p + 3)) <=  #2.52) /\   (&2 <= norm (v (p + 2)) /\ norm (v (p + 2)) <=  #2.52) /\   ( #3.01 <= cstab /\ cstab <=  #3.01) /\   ( #3.01 <= dist (v p,v (p + 2)) /\ dist (v p,v (p + 2)) <=  #3.55) /\   &2 <= dist (v p,v (p + 3)) /\   dist (v p,v (p + 3)) <= &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    ASM_REWRITE_TAC[arith `(x <= y /\ y <= x) <=> y = x`];
    CONJ_TAC;
      FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`0`;`3`]);
      BY(REWRITE_TAC[arith `p+0=p`] THEN DISCH_THEN MATCH_MP_TAC THEN NUM_REDUCE_TAC);
    BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
  FIRST_ASSUM (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC ((* get_work *) Terminal.get_main_nonlinear "5550839403") [`norm (v (p))`;`norm(v (p+3))`;`norm(v (p+2))`;`cstab`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+3))`];
  REWRITE_TAC[Sphere.ineq;TAUT `a ==> b ==> c <=> a /\ b ==> c`;arith `x <= x`];
  FIRST_X_ASSUM_ST `#3.01` (unlist REWRITE_TAC);
  DISCH_TAC;
  INTRO_TAC Ocbicby.dih_y_mono [`norm (v (p))`;`norm(v (p+3))`;`norm(v (p+2))`;`dist(v (p+3),v(p+2))`;`dist(v(p),v (p+2))`;`dist(v(p),v(p+3))`;`cstab`];
  ANTS_TAC;
    TYPIFY_GOAL_THEN `&0 < norm (v p)` (unlist REWRITE_TAC);
      BY(ASM_REWRITE_TAC[arith `&0 < &2`]);
    FIRST_X_ASSUM_ST `norm (v p) =  &2` kill;
    ASM_REWRITE_TAC[];
    CONJ_TAC;
      BY(ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]);
    CONJ_TAC;
      FIRST_X_ASSUM_ST `&2 <= dist s` (C INTRO_TAC [`3`;`2`]);
      BY(NUM_REDUCE_TAC THEN REAL_ARITH_TAC);
    CONJ2_TAC;
      BY(FIRST_X_ASSUM_ST `delta_y` MP_TAC THEN REAL_ARITH_TAC);
    FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`p+3`]);
    ANTS_TAC;
      BY(ASM_REWRITE_TAC[]);
    REWRITE_TAC[arith `(p+3) +i = p+(3+i)`] THEN NUM_REDUCE_TAC;
    REWRITE_TAC[Oxlzlez.coplanar_delta_y];
    ASM_REWRITE_TAC[DIST_SYM;DIST_0];
    MATCH_MP_TAC (arith `a = b ==> &0 < a ==> &0 < b`);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  (DISCH_TAC);
  TYPIFY `dih_y (norm (v p)) (norm (v (p + 3))) (norm (v (p + 2)))      (dist (v (p + 3),v (p + 2)))      (dist (v p,v (p + 2)))      (dist (v p,v (p + 3))) + dih_y (norm (v p)) (norm (v (p + 1))) (norm (v (p + 2)))      (dist (v (p + 1),v (p + 2)))      (dist (v p,v (p + 2)))      (dist (v p,v (p + 1))) < #3.1` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 5 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  TYPIFY `#3.1 < pi` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MP_TAC Flyspeck_constants.bounds THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM_ST `a = pi` MP_TAC THEN REPLICATE_TAC 2 (POP_ASSUM MP_TAC);
  REWRITE_TAC[DIST_0;DIST_SYM];
  TYPIFY `dih_y (norm (v p)) (norm (v (p + 2))) (norm (v (p + 3)))       (dist (v (p + 2),v (p + 3)))       (dist (v p,v (p + 3)))       (dist (v p,v (p + 2))) = dih_y (norm (v p)) (norm (v (p + 3))) (norm (v (p + 2))) (dist (v (p + 2),v (p + 3))) (dist (v p,v (p + 2))) (dist (v p,v (p + 3)))` (C SUBGOAL_THEN SUBST1_TAC);
    BY(REWRITE_TAC[Nonlinear_lemma.dih_y_sym]);
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

end;;