(* ========================================================================== *)
(* FLYSPECK - BOOK FORMALIZATION                                              *)
(* Section: Conclusions                                                       *)
(* Chapter: Local Fan                                                         *)
(* Author: Thomas C. Hales                                                    *)
(* Date: 2013-04-17                                                           *)
(* ========================================================================== *)

(*
Terminal cases of main estimate.
*)

flyspeck_needs "local/appendix_main_estimate.hl";;

module Terminal = struct


open Hales_tactic;;

(*
let filter_flypaper tl = 
  List.flatten (map (function
	    | Flypaper s -> s
	    | _ -> []) tl);;

let has_flypaper_tag sl ind = 
  let tl = ind.tags in
    List.length (intersect (filter_flypaper tl) sl) > 0;;

let main_ineq_data = 
  filter (fun ind ->
	    has_flypaper_tag ["UPONLFY";"SAUZWSD";"EDZEPIH";"OMKYNLT";"FHOLLLW";"TNNOPSI"] ind) (!Ineq.ineqs);;

let main_nonlinear = 
  let ineql = map (fun idv -> idv.ineq) main_ineq_data in
  let main_ineq_conj = end_itlist (curry mk_conj) ineql in
  let _ = new_definition (mk_eq (`main_nonlinear_v39:bool`,main_ineq_conj)) in
    ();;

let is_main = function 
	    | Main_estimate -> true
	    | _ -> false;;

let has_main ind = 
    exists (is_main) ind.tags;;

let main_ineq_data = 
  filter has_main (!Ineq.ineqs);;

let get_main_nonlinear = 
  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_nonlinear_v39: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
    fun s ->
      let i = index s sl in
      let th2 = funpow i CONJUNCT2 th1 in
	co1 th2;;

*)


let get_main_nonlinear = 
  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
  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_nonlinear_terminal_v11: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;;

(* Start with OWZLKVY. *)
let LET_THM = CONJ LET_DEF LET_END_DEF;;

let sqrt8_flyspeck = 
prove_by_refinement(
  `#2.828427 < sqrt8 /\
     sqrt8 <  #2.828428`,

  (* {{{ proof *)
  [
    BY(REWRITE_TAC[Flyspeck_constants.bounds])
  ]);;
  (* }}} *)

let SOL_SOLID_TRIANGLE_ALT = 
prove_by_refinement(
  `!v0 v1 v2 v3.
         ~coplanar {v0, v1, v2, v3} 
         ==> sol v0 (convex hull {v0, v1, v2, v3}) = dihV v0 v1 v2 v3 + dihV v0 v2 v3 v1 + dihV v0 v3 v1 v2 - pi`,

  (* {{{ proof *)
  [
    BY(MESON_TAC[Tskajxy_lemmas.SOL_SOLID_TRIANGLE])
  ]);;
  (* }}} *)

let sol_x_nn = 
prove_by_refinement(
  `!x1 x2 x3 x4 x5 x6.
    &0 < x1 /\ &0 < x2 /\ &0 < x3 /\
            &0 < ups_x x1 x2 x6 /\
            &0 < ups_x x2 x3 x4 /\
            &0 < ups_x x1 x3 x5 /\
            &0 < eulerA_x x1 x2 x3 x4 x5 x6 /\
            &0 < delta_x x1 x2 x3 x4 x5 x6 ==>
    &0 < sol_x x1 x2 x3 x4 x5 x6`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  GMATCH_SIMP_TAC Merge_ineq.sol_x_sol_euler_x;
  ASM_REWRITE_TAC[Sphere.sol_euler_x;LET_DEF;LET_END_DEF];
  REPEAT (GMATCH_SIMP_TAC SQRT_MUL);
  REWRITE_TAC[GSYM Sphere.eulerA_x];
  GMATCH_SIMP_TAC REAL_LE_MUL;
  ASM_SIMP_TAC[arith `&0 < x ==> &0 <= x`];
  MATCH_MP_TAC (arith `&0 < x ==> &0 < &2 * x `);
  GMATCH_SIMP_TAC Merge_ineq.ATN2_POS;
  CONJ_TAC;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  GMATCH_SIMP_TAC REAL_LT_RSQRT;
  BY(FIRST_X_ASSUM MP_TAC THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)


let DIH_X_NN = 
prove_by_refinement(
  `!x1 x2 x3 x4 x5 x6. &0 < x1 /\ &0 <= delta_x x1 x2 x3 x4 x5 x6 
           ==> &0 <= dih_x x1 x2 x3 x4 x5 x6`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  ASM_CASES_TAC `&0 < delta_x x1 x2 x3 x4 x5 x6`;
    MATCH_MP_TAC Merge_ineq.dih_x_nn;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `delta_x x1 x2 x3 x4 x5 x6 = &0` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  ASM_REWRITE_TAC[Sphere.dih_x;LET_DEF;LET_END_DEF];
  REWRITE_TAC[arith `x * &0 = &0`;SQRT_0];
  TYPED_ABBREV_TAC `d4 = -- delta_x4 x1 x2 x3 x4 x5 x6`;
  INTRO_TAC Merge_ineq.atn2_0 [`d4`];
  REPEAT WEAKER_STRIP_TAC;
  ASSUME_TAC PI_POS;
  MP_TAC (arith `&0 < d4 \/ d4 < &0 \/ d4 = &0`);
  DISCH_THEN DISJ_CASES_TAC;
    ASM_SIMP_TAC[];
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM DISJ_CASES_TAC;
    ASM_SIMP_TAC[];
    BY(REAL_ARITH_TAC);
  ASM_SIMP_TAC[];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let DIH_Y_NN = 
prove_by_refinement(
  `!y1 y2 y3 y4 y5 y6.
    &0 < y1 /\ &0 <= delta_y y1 y2 y3 y4 y5 y6 ==>
    &0 <= dih_y y1 y2 y3 y4 y5 y6`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Sphere.dih_y;LET_DEF;LET_END_DEF];
  MATCH_MP_TAC DIH_X_NN;
  GMATCH_SIMP_TAC REAL_LT_MUL_EQ;
  ASM_REWRITE_TAC[];
  FIRST_X_ASSUM MP_TAC;
  BY(REWRITE_TAC[Sphere.delta_y])
  ]);;
  (* }}} *)

let RHO_LB = 
prove_by_refinement(
  `!y. &2 <= y  ==> &1 <= rho y`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Nonlinear_lemma.rho_alt];
  MATCH_MP_TAC (arith `&0 <= x ==> &1 <= &1 + x `);
  GMATCH_SIMP_TAC REAL_LE_MUL;
  REWRITE_TAC[GSYM Nonlinear_lemma.sol0_over_pi_EQ_const1];
  GMATCH_SIMP_TAC REAL_LE_RDIV_EQ;
  GMATCH_SIMP_TAC REAL_LE_RDIV_EQ;
  REWRITE_TAC[arith `&0 * x = &0`];
  REWRITE_TAC[PI_POS];
  INTRO_TAC Flyspeck_constants.bounds [];
  FIRST_X_ASSUM MP_TAC;
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let DIH_Y_LT_RHAZIM = 
prove_by_refinement(
  `!y1 y2 y3 y4 y5 y6.
    &2 <= y1 /\ &0 <= delta_y y1 y2 y3 y4 y5 y6 ==>
    dih_y y1 y2 y3 y4 y5 y6 <= rhazim y1 y2 y3 y4 y5 y6`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Sphere.rhazim];
  MATCH_MP_TAC (arith `&0 <= (r - &1) * d ==> d <= r * d`);
  GMATCH_SIMP_TAC REAL_LE_MUL;
  CONJ2_TAC;
    MATCH_MP_TAC DIH_Y_NN;
    ASM_REWRITE_TAC[];
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  MATCH_MP_TAC (arith `&1 <= r ==> &0 <= r - &1`);
  MATCH_MP_TAC RHO_LB;
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let taum_taum_x = 
prove_by_refinement(
  `!y1 y2 y3 y4 y5 y6.
    &0 <= y1 /\ &0 <= y2 /\ &0 <= y3 /\ &0 <= y4 /\ &0 <= y5 /\ &0 <= y6 ==>
    taum y1 y2 y3 y4 y5 y6 = y_of_x taum_x y1 y2 y3 y4 y5 y6`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Sphere.y_of_x;Sphere.taum_x;Sphere.rhazim_x;Sphere.rhazim2_x;Sphere.rhazim3_x];
  REWRITE_TAC[Nonlinear_lemma.taum_123];
  REPEAT WEAKER_STRIP_TAC;
  BY(ASM_SIMP_TAC[Nonlinear_lemma.sqrtxx])
  ]);;
  (* }}} *)

(* Now do OCBICBY *)

let BBs_terminal = `!s vv. MEM s scs_terminal_v116 /\  BBs_v39 s vv ==> &0 <= taustar_v39 s vv`;;

let scs_unadorned_explicit = 
prove_by_refinement(
  `(!k d a b.
             scs_k_v39 (mk_unadorned_v39 k d a b) = k) /\
        (!k d a b .
             scs_d_v39 (mk_unadorned_v39 k d a b) = d) /\
        (!k d a b .
             scs_a_v39 (mk_unadorned_v39 k d a b) = a) /\
        (!k d a b .
             scs_am_v39 (mk_unadorned_v39 k d a b) = a) /\
        (!k d a b .
             scs_b_v39 (mk_unadorned_v39 k d a b) = b) /\
        (!k d a b .
             scs_bm_v39 (mk_unadorned_v39 k d a b) = b) /\
        (!k d a b .
             scs_J_v39 (mk_unadorned_v39 k d a b) = (\ i j. F)) /\
        (!k d a b .
             scs_lo_v39 (mk_unadorned_v39 k d a b) = {}) /\
        (!k d a b .
             scs_hi_v39 (mk_unadorned_v39 k d a b) = {}) /\
        (!k d a b .
             scs_str_v39 (mk_unadorned_v39 k d a b) = {})`,

  (* {{{ proof *)
  [
  BY(REWRITE_TAC[Appendix.scs_v39_explicit;Appendix.mk_unadorned_v39])
  ]);;
  (* }}} *)

let UNADORNED_NOT_EAR = 
prove_by_refinement(
  `!k d a b.
    ~(is_ear_v39 (mk_unadorned_v39 k d a b))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.is_ear_v39;scs_unadorned_explicit;EMPTY_GSPEC];
  BY(MESON_TAC[NOT_IN_EMPTY;IN_SING])
  ]);;
  (* }}} *)

let dsv_unadorned = 
prove_by_refinement(
  `!k d a b vv.
    dsv_v39 (mk_unadorned_v39 k d a b) vv = d `,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.dsv_v39;UNADORNED_NOT_EAR;scs_unadorned_explicit;EMPTY_GSPEC;SUM_CLAUSES];
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let dsv_F = 
prove_by_refinement(
  `!k d a a' b' b  s s' s'' vv.
    dsv_v39 (scs_v39 (k, d, a, a', b', b, (\i j. F), s, s',s'')) vv = d `,

  (* {{{ proof *)
  [
  ASM_REWRITE_TAC[Appendix.dsv_v39;Appendix.scs_v39_explicit;EMPTY_GSPEC;SUM_CLAUSES];
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let SUM_INTER = 
prove_by_refinement(
  `!(A:A->bool) B f. sum (A INTER B) f = sum A (\i. if (i IN B) then f i else &0)`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC SUM_SUPERSET [`(\i. if (i IN B) then f i else &0)`;`A INTER B`;`A`];
  ANTS_TAC;
    CONJ_TAC;
      BY(SET_TAC[]);
    BY(SET_TAC[]);
  DISCH_THEN SUBST1_TAC;
  MATCH_MP_TAC SUM_EQ;
  REWRITE_TAC[IN_INTER];
  BY(MESON_TAC[])
  ]);;
  (* }}} *)

let dsv_fun3 = 
prove_by_refinement(
  `! d a a' b' b f  s s' s'' vv.
    dsv_v39 (scs_v39 (3, d, a, a', b', b, f  , s, s',s'')) vv = 
    d +
  #0.1 *
 (if is_ear_v39 (scs_v39 (3,d,a,a',b',b,f ,s,s',s''))
  then &1
  else -- &1) *
 ((if f 0 1 then cstab - dist (vv 0,vv 1) else &0) +
  (if f 1 2 then cstab - dist (vv 1,vv 2) else &0) +
  (if f 2 3 then cstab - dist (vv 2,vv 3) else &0) +
  &0) `,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  ASM_REWRITE_TAC[Appendix.dsv_v39;Appendix.scs_v39_explicit;EMPTY_GSPEC;SUM_CLAUSES];
  TYPIFY `{i | i < 3 /\ f i (SUC i) } = {0,1,2} INTER {i | f i (SUC i) }` (C SUBGOAL_THEN SUBST1_TAC);
    REWRITE_TAC[EXTENSION;IN_ELIM_THM;IN_INTER;IN_INSERT;NOT_IN_EMPTY];
    BY(REWRITE_TAC[arith `x < 3 <=> (x = 0 \/ x = 1 \/ x = 2)`]);
  REWRITE_TAC[SUM_INTER;IN_ELIM_THM];
  REPEAT (GMATCH_SIMP_TAC (CONJUNCT2 SUM_CLAUSES));
  REWRITE_TAC[NOT_IN_EMPTY;IN_INSERT;FINITE_EMPTY;FINITE_INSERT;arith `~(0 = 1) /\ ~(0=2) /\ ~(1 = 2)`;SUM_CLAUSES];
  BY(REWRITE_TAC[arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3`])
  ]);;
  (* }}} *)

let dsv_fun4 = 
prove_by_refinement(
  `! d a a' b' b f  s s' s'' vv.
    dsv_v39 (scs_v39 (4, d, a, a', b', b, f  , s, s',s'')) vv = 
    d -
  #0.1 *
 ((if f 0 1 then cstab - dist (vv 0,vv 1) else &0) +
  (if f 1 2 then cstab - dist (vv 1,vv 2) else &0) +
  (if f 2 3 then cstab - dist (vv 2,vv 3) else &0) +
  (if f 3 4 then cstab - dist (vv 3,vv 4) else &0) +
  &0) `,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  ASM_REWRITE_TAC[Appendix.dsv_v39;Appendix.scs_v39_explicit;EMPTY_GSPEC;SUM_CLAUSES];
  REWRITE_TAC[Appendix.is_ear_v39;Appendix.scs_v39_explicit;arith `~(4 = 3)`];
  MATCH_MP_TAC (arith `(s = s') ==> d + a * -- &1 * s = d - a * s'`);
  TYPIFY `{i | i < 4 /\ f i (SUC i) } = {0,1,2,3} INTER {i | f i (SUC i) }` (C SUBGOAL_THEN SUBST1_TAC);
    REWRITE_TAC[EXTENSION;IN_ELIM_THM;IN_INTER;IN_INSERT;NOT_IN_EMPTY];
    BY(REWRITE_TAC[arith `x < 4 <=> (x = 0 \/ x = 1 \/ x = 2 \/ x = 3)`]);
  REWRITE_TAC[SUM_INTER;IN_ELIM_THM];
  REPEAT (GMATCH_SIMP_TAC (CONJUNCT2 SUM_CLAUSES));
  REWRITE_TAC[NOT_IN_EMPTY;IN_INSERT;FINITE_EMPTY;FINITE_INSERT;arith `~(0 = 1) /\ ~(0=2) /\ ~(1 = 2) /\ ~(2 = 3) /\ ~(0=3) /\ ~(1 = 3)`;SUM_CLAUSES];
  BY(REWRITE_TAC[arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4`])
  ]);;
  (* }}} *)

let IMAGE_SUBSET_IN = 
prove_by_refinement(
  `!(f:A->B) A B. IMAGE f A SUBSET B <=> (!a. a IN A ==> f a IN B)`,

  (* {{{ proof *)
  [
  SET_TAC[]
  ]);;
  (* }}} *)

let taustar3  = 
prove_by_refinement(
  `!d a b. 
			     (let s = mk_unadorned_v39     3 d a b in
				((!v0 v1 v2. 
				    &2 <= norm v0 /\ norm v0 <= &2 * h0 /\
				    &2 <= norm v1 /\ norm v1 <= &2 * h0 /\
				    &2 <= norm v2 /\ norm v2 <= &2 * h0 /\
				    a 0 1 <= dist(v0,v1) /\ dist(v0,v1) <= b 0 1 /\
				    a 1 2 <= dist(v1,v2) /\ dist(v1,v2) <= b 1 2 /\
				    a 0 2 <= dist(v0,v2) /\ dist(v0,v2) <= b 0 2 ==>
				    d <= tau3 v0 v1 v2) ==>
    (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv)))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Appendix.taustar_v39;scs_unadorned_explicit;arith `3 <= 3`];
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REWRITE_TAC[dsv_unadorned];
  REWRITE_TAC[arith `&0 <= t - d <=> d <= t`];
  FIRST_X_ASSUM MATCH_MP_TAC;
  FIRST_X_ASSUM MP_TAC;
  REWRITE_TAC[Appendix.BBs_v39;scs_unadorned_explicit;arith `3 <= 3`];
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REWRITE_TAC[IMAGE_SUBSET_IN;IN_UNIV;Fnjlbxs.in_ball_annulus];
  BY(MESON_TAC[])
  ]);;
  (* }}} *)

let taustar3_fun = 
prove_by_refinement(
  `!d a b f.
  (let s = scs_v39 (3,d,a,a,b,b,f,{},{},{}) in
    ((!v0 v1 v2.
   &2 <= norm v0 /\ norm v0 <= &2 * h0 /\
				    &2 <= norm v1 /\ norm v1 <= &2 * h0 /\
				    &2 <= norm v2 /\ norm v2 <= &2 * h0 /\
				    a 0 1 <= dist(v0,v1) /\ dist(v0,v1) <= b 0 1 /\
				    a 1 2 <= dist(v1,v2) /\ dist(v1,v2) <= b 1 2 /\
				    a 0 2 <= dist(v0,v2) /\ dist(v0,v2) <= b 0 2 ==>
				     d +
  #0.1 *
 (if is_ear_v39 s
  then &1
  else -- &1) *
 ((if f 0 1 then cstab - dist (v0,v1) else &0) +
  (if f 1 2 then cstab - dist (v1,v2) else &0) +
  (if f 2 3 then cstab - dist (v2,v0) else &0) +
  &0) <= tau3 v0 v1 v2) ==>
    (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv)))	`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Appendix.taustar_v39;Appendix.scs_v39_explicit;arith `3 <= 3`];
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REWRITE_TAC[dsv_fun3];
  REWRITE_TAC[arith `&0 <= t - d <=> d <= t`];
  TYPIFY `vv 3 = vv 0` (C SUBGOAL_THEN SUBST1_TAC);
    FIRST_X_ASSUM MP_TAC;
    REWRITE_TAC[Appendix.BBs_v39];
    REWRITE_TAC[Appendix.scs_v39_explicit];
    REWRITE_TAC[LET_DEF;LET_END_DEF];
    REWRITE_TAC[Oxl_def.periodic];
    BY(MESON_TAC[arith `0 + 3 = 3`]);
  FIRST_X_ASSUM MATCH_MP_TAC;
  FIRST_X_ASSUM MP_TAC;
  REWRITE_TAC[Appendix.BBs_v39;Appendix.scs_v39_explicit;arith `3 <= 3`];
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REWRITE_TAC[IMAGE_SUBSET_IN;IN_UNIV;Fnjlbxs.in_ball_annulus];
  BY(MESON_TAC[])
  ]);;
  (* }}} *)

let NONPARALLEL_BALL_ANNULUS40 =
prove_by_refinement(`!v w. &2<= norm(v-w) /\ norm(v-w)< &4 /\ v IN ball_annulus /\ w IN ball_annulus
==> ~(collinear ({vec 0} UNION {v,w}))`,

[
  REPEAT GEN_TAC;
  REWRITE_TAC[SET_RULE`{A} UNION {B,C}={A,B,C}`;GSYM NORM_CAUCHY_SCHWARZ_EQUAL;NORM_LE_SQUARE;NORM_LT_SQUARE;];
  ONCE_REWRITE_TAC[REAL_ARITH`A<=B <=> B >= A`];
  REWRITE_TAC[NORM_GE_SQUARE;REAL_ARITH`~(&2<= &0) /\ &0 < &2`];
  ONCE_REWRITE_TAC[REAL_ARITH`A>=B <=> B <= A`];
  REWRITE_TAC[DOT_RSUB;DOT_LSUB;DOT_SQUARE_NORM];
  DISJ_CASES_TAC(REAL_ARITH`&0<= v dot w \/ &0<= --(v dot (w:real^3))`);
    POP_ASSUM MP_TAC;
    REWRITE_TAC[GSYM REAL_ABS_REFL];
    RESA_TAC;
    REPEAT STRIP_TAC;
    POP_ASSUM(fun t-> ASM_TAC THEN REWRITE_TAC[t] THEN REPEAT STRIP_TAC THEN MP_TAC t);
    REWRITE_TAC[];
    ONCE_REWRITE_TAC[DOT_SYM];
    STRIP_TAC;
    POP_ASSUM(fun t-> ASM_TAC THEN REWRITE_TAC[t] );
    REWRITE_TAC[REAL_ARITH`A pow 2 -A*B-(A*B-B pow 2)=(A-B) pow 2`;GSYM REAL_LE_SQUARE_ABS;Pack_defs.ball_annulus;IN_ELIM_THM;DIFF;ball;cball;dist;VECTOR_ARITH`vec 0 -A= --A`;NORM_NEG;REAL_ARITH`~(a<b)<=> b<=a`];
    DISJ_CASES_TAC(REAL_ARITH`&0<= norm (v:real^3) -norm w \/ &0<= --(norm v -norm (w:real^3))`);
      POP_ASSUM MP_TAC;
      REWRITE_TAC[GSYM REAL_ABS_REFL];
      RESA_TAC;
      REPEAT STRIP_TAC;
      MP_TAC(REAL_ARITH`norm (v:real^3) <= &2 * h0 /\ &2 <= norm (w:real^3) ==> norm v - norm w <= &2*(h0 - &1)`);
      ASM_REWRITE_TAC[Sphere.h0];
      ASM_TAC;
      BY(REAL_ARITH_TAC);
    POP_ASSUM MP_TAC;
    REWRITE_TAC[GSYM REAL_ABS_REFL;REAL_ABS_NEG];
    RESA_TAC;
    ONCE_REWRITE_TAC[REAL_ARITH`--(A-B)=B-A`];
    REPEAT STRIP_TAC;
    MP_TAC(REAL_ARITH`norm (w:real^3) <= &2 * h0 /\ &2 <= norm (v:real^3) ==> norm w - norm v <= &2*(h0 - &1)`);
    ASM_REWRITE_TAC[Sphere.h0];
    ASM_TAC;
    BY(REAL_ARITH_TAC);
  POP_ASSUM MP_TAC;
  REWRITE_TAC[GSYM REAL_ABS_REFL;REAL_ABS_NEG];
  RESA_TAC;
  ONCE_REWRITE_TAC[REAL_ARITH`--A=B<=> A= --B`];
  REPEAT STRIP_TAC;
  POP_ASSUM(fun t-> ASM_TAC THEN REWRITE_TAC[t] THEN REPEAT STRIP_TAC THEN MP_TAC t);
  REWRITE_TAC[];
  ONCE_REWRITE_TAC[DOT_SYM];
  STRIP_TAC;
  POP_ASSUM(fun t-> ASM_TAC THEN REWRITE_TAC[t] );
  REWRITE_TAC[REAL_ARITH`A pow 2 -(--(A*B))-(--(A*B)-B pow 2)=(A+B) pow 2`;GSYM REAL_LE_SQUARE_ABS;Pack_defs.ball_annulus;IN_ELIM_THM;DIFF;ball;cball;dist;VECTOR_ARITH`vec 0 -A= --A`;NORM_NEG;REAL_ARITH`~(a<b)<=> b<=a`];
  STRIP_TAC;
  REMOVE_ASSUM_TAC;
  MP_TAC(REAL_ARITH`&0<= norm (v:real^3) /\ &0<= norm (w:real^3)==> &0<= norm v + norm w`);
  SIMP_TAC[NORM_POS_LE];
  REWRITE_TAC[GSYM REAL_ABS_REFL];
  RESA_TAC;
  REPEAT STRIP_TAC;
  MP_TAC(REAL_ARITH`&2<=norm (w:real^3) /\ &2 <= norm (v:real^3) ==> &4<= norm v + norm w `);
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  FIRST_X_ASSUM_ST `pow` MP_TAC;
  REWRITE_TAC[];
  REWRITE_TAC[arith `~(x2 < y2) <=> y2 <= x2`];
  MATCH_MP_TAC Collect_geom2.POS_IMP_POW2;
  ASM_REWRITE_TAC[];
  BY(REAL_ARITH_TAC)
]
);;
let NONPARALLEL_BALL_ANNULUS_ALT = 
prove_by_refinement(
  `!v w. &2 <= dist(v,w) /\
     dist(v,w) <= #3.62 /\ // was cstab.
     v IN ball_annulus /\
     w IN ball_annulus
     ==> ~collinear ({vec 0,v, w})`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM MP_TAC;
  TYPIFY `{vec 0,v,w} = {vec 0} UNION {v,w}` (C SUBGOAL_THEN SUBST1_TAC);
    BY(SET_TAC[]);
  REWRITE_TAC[];
  MATCH_MP_TAC Dih2k_hypermap.NONPARALLEL_BALL_ANNULUS362;
  REWRITE_TAC[GSYM dist];
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let NONPARALLEL_BALL_ANNULUS40_ALT = 
prove_by_refinement(
  `!v w. &2 <= dist(v,w) /\
     dist(v,w) < &4 /\ 
     v IN ball_annulus /\
     w IN ball_annulus
     ==> ~collinear ({vec 0,v, w})`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM MP_TAC;
  TYPIFY `{vec 0,v,w} = {vec 0} UNION {v,w}` (C SUBGOAL_THEN SUBST1_TAC);
    BY(SET_TAC[]);
  REWRITE_TAC[];
  MATCH_MP_TAC NONPARALLEL_BALL_ANNULUS40;
  REWRITE_TAC[GSYM dist];
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let tau3_taum = 
prove_by_refinement(
  `!(v0:real^3) v1 v2.
    v0 IN ball_annulus /\ v1 IN ball_annulus /\ v2 IN ball_annulus /\
    &2 <= dist(v0,v1) /\ &2 <= dist(v0,v2) /\ &2 <= dist(v1,v2) /\
    dist(v0,v1) <= #3.62 /\ dist(v0,v2) <= #3.62 /\ dist(v1,v2) <= #3.62 ==>
    tau3 v0 v1 v2 = taum (norm v0) (norm v1) (norm v2) (dist(v1,v2)) (dist(v0,v2)) (dist(v0,v1))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.tau3;Nonlinear_lemma.taum_123;Sphere.rhazim2;Sphere.rhazim3;Sphere.node2_y;Sphere.node3_y];
  REWRITE_TAC[Sphere.rhazim;Nonlinear_lemma.sol0_const1];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `dihV (vec 0) v0 v1 v2 = dih_y (norm v0) (norm v1) (norm v2) (dist (v1,v2)) (dist (v0,v2)) (dist (v0,v1)) /\ dihV (vec 0) v1 v2 v0  = dih_y (norm v1) (norm v2) (norm v0) (dist (v0,v2)) (dist (v0,v1)) (dist (v1,v2)) /\ dihV (vec 0) v2 v0 v1 = dih_y (norm v2) (norm v0) (norm v1) (dist (v0,v1)) (dist (v1,v2))  (dist (v0,v2))` ENOUGH_TO_SHOW_TAC;
    DISCH_THEN (unlist REWRITE_TAC);
    BY(REAL_ARITH_TAC);
  REPEAT (GMATCH_SIMP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] Merge_ineq.DIHV_EQ_DIH_Y));
  REWRITE_TAC[DIST_0];
  TYPIFY `dist(v2,v0) = dist(v0,v2) /\ dist(v2,v1) = dist(v1,v2) /\ dist(v1,v0) = dist(v0,v1)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[DIST_SYM]);
  ASM_REWRITE_TAC[];
  REPEAT (GMATCH_SIMP_TAC NONPARALLEL_BALL_ANNULUS_ALT);
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let tau3_taum_40 = 
prove_by_refinement(
  `!(v0:real^3) v1 v2.
    v0 IN ball_annulus /\ v1 IN ball_annulus /\ v2 IN ball_annulus /\
    &2 <= dist(v0,v1) /\ &2 <= dist(v0,v2) /\ &2 <= dist(v1,v2) /\
    dist(v0,v1) < &4 /\ dist(v0,v2) < &4 /\ dist(v1,v2) < &4 ==>
    tau3 v0 v1 v2 = taum (norm v0) (norm v1) (norm v2) (dist(v1,v2)) (dist(v0,v2)) (dist(v0,v1))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.tau3;Nonlinear_lemma.taum_123;Sphere.rhazim2;Sphere.rhazim3;Sphere.node2_y;Sphere.node3_y];
  REWRITE_TAC[Sphere.rhazim;Nonlinear_lemma.sol0_const1];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `dihV (vec 0) v0 v1 v2 = dih_y (norm v0) (norm v1) (norm v2) (dist (v1,v2)) (dist (v0,v2)) (dist (v0,v1)) /\ dihV (vec 0) v1 v2 v0  = dih_y (norm v1) (norm v2) (norm v0) (dist (v0,v2)) (dist (v0,v1)) (dist (v1,v2)) /\ dihV (vec 0) v2 v0 v1 = dih_y (norm v2) (norm v0) (norm v1) (dist (v0,v1)) (dist (v1,v2))  (dist (v0,v2))` ENOUGH_TO_SHOW_TAC;
    DISCH_THEN (unlist REWRITE_TAC);
    BY(REAL_ARITH_TAC);
  REPEAT (GMATCH_SIMP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] Merge_ineq.DIHV_EQ_DIH_Y));
  REWRITE_TAC[DIST_0];
  TYPIFY `dist(v2,v0) = dist(v0,v2) /\ dist(v2,v1) = dist(v1,v2) /\ dist(v1,v0) = dist(v0,v1)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[DIST_SYM]);
  ASM_REWRITE_TAC[];
  REPEAT (GMATCH_SIMP_TAC NONPARALLEL_BALL_ANNULUS40_ALT);
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let DELTA_Y_POS_4POINTS = 
prove_by_refinement(
  `!v0 v1 v2 (v3:real^3).
    &0 <= delta_y (dist(v0,v1)) (dist(v0,v2)) (dist(v0,v3)) (dist(v2,v3)) (dist(v1,v3)) (dist(v1,v2))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Sphere.delta_y;arith `x * x = x pow 2`];
  REWRITE_TAC[GSYM Merge_ineq.delta_delta_x];
  BY(REWRITE_TAC[Collect_geom.DELTA_POS_4POINTS])
  ]);;
  (* }}} *)

let tau3_taum_d = 
prove_by_refinement(
  `!d  a01 a12 a02 b01 b12 b02. 
    (&2 <= a01 /\ &2 <= a12 /\ &2 <= a02 /\ b01 <= #3.62 /\ b12 <= #3.62 /\ b02 <= #3.62 /\
				(!y1 y2 y3 y4 y5 y6. 
				    &2 <= y1 /\ y1 <= &2 * h0 /\
				    &2 <= y2 /\ y2 <= &2 * h0 /\
				    &2 <= y3 /\ y3 <= &2 * h0 /\
				    a01 <= y6 /\ y6 <= b01 /\
				    a12 <= y4 /\ y4 <= b12 /\
				    a02 <= y5 /\ y5 <= b02 /\
				    &0 <= delta_y y1 y2 y3 y4 y5 y6 ==>
				    d <= taum y1 y2 y3 y4 y5 y6) ==>
    (!v0 v1 v2. 
				    &2 <= norm v0 /\ norm v0 <= &2 * h0 /\
				    &2 <= norm v1 /\ norm v1 <= &2 * h0 /\
				    &2 <= norm v2 /\ norm v2 <= &2 * h0 /\
				    a01 <= dist(v0,v1) /\ dist(v0,v1) <= b01 /\
				    a12 <= dist(v1,v2) /\ dist(v1,v2) <= b12 /\
				    a02 <= dist(v0,v2) /\ dist(v0,v2) <= b02 ==>
				    d <= tau3 v0 v1 v2))`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  GMATCH_SIMP_TAC tau3_taum;
  CONJ_TAC;
    REWRITE_TAC[Fnjlbxs.in_ball_annulus];
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM MATCH_MP_TAC;
  TYPIFY `&0 <= delta_y (norm v0) (norm v1) (norm v2) (dist (v1,v2)) (dist (v0,v2))  (dist (v0,v1))` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    REWRITE_TAC[GSYM Trigonometry1.DIST_L_ZERO];
    BY(REWRITE_TAC[DELTA_Y_POS_4POINTS]);
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let tau3_taum_dfun = 
prove_by_refinement(
  `!d  a01 a12 a02 b01 b12 b02 f. 
    (&2 <= a01 /\ &2 <= a12 /\ &2 <= a02 /\ b01 <= #3.62 /\ b12 <= #3.62 /\ b02 <= #3.62 /\
				(!y1 y2 y3 y4 y5 y6. 
				    &2 <= y1 /\ y1 <= &2 * h0 /\
				    &2 <= y2 /\ y2 <= &2 * h0 /\
				    &2 <= y3 /\ y3 <= &2 * h0 /\
				    a01 <= y6 /\ y6 <= b01 /\
				    a12 <= y4 /\ y4 <= b12 /\
				    a02 <= y5 /\ y5 <= b02 /\
				    &0 <= delta_y y1 y2 y3 y4 y5 y6 ==>
				    d + f y4 y5 y6 <= taum y1 y2 y3 y4 y5 y6) ==>
    (!v0 v1 v2. 
				    &2 <= norm v0 /\ norm v0 <= &2 * h0 /\
				    &2 <= norm v1 /\ norm v1 <= &2 * h0 /\
				    &2 <= norm v2 /\ norm v2 <= &2 * h0 /\
				    a01 <= dist(v0,v1) /\ dist(v0,v1) <= b01 /\
				    a12 <= dist(v1,v2) /\ dist(v1,v2) <= b12 /\
				    a02 <= dist(v0,v2) /\ dist(v0,v2) <= b02 ==>
				    d + f (dist(v1,v2)) (dist(v0,v2)) (dist(v0,v1)) <= tau3 v0 v1 v2))`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  GMATCH_SIMP_TAC tau3_taum;
  CONJ_TAC;
    REWRITE_TAC[Fnjlbxs.in_ball_annulus];
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM MATCH_MP_TAC;
  TYPIFY `&0 <= delta_y (norm v0) (norm v1) (norm v2) (dist (v1,v2)) (dist (v0,v2))  (dist (v0,v1))` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    REWRITE_TAC[GSYM Trigonometry1.DIST_L_ZERO];
    BY(REWRITE_TAC[DELTA_Y_POS_4POINTS]);
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let taustar_taum = 
prove_by_refinement(
    `!d a b.   
    (&2 <= a 0 1 /\ &2 <= a 1 2 /\ &2 <= a 0 2 /\ b 0 1 <= #3.62 /\ b 1 2 <= #3.62 /\ b 0 2 <= #3.62 /\
				(!y1 y2 y3 y4 y5 y6. 
				    &2 <= y1 /\ y1 <= &2 * h0 /\
				    &2 <= y2 /\ y2 <= &2 * h0 /\
				    &2 <= y3 /\ y3 <= &2 * h0 /\
				    a 0 1 <= y6 /\ y6 <= b 0 1 /\
				    a 1 2 <= y4 /\ y4 <= b 1 2 /\
				    a 0 2 <= y5 /\ y5 <= b 0 2 /\
				    &0 <= delta_y y1 y2 y3 y4 y5 y6 ==>
				    d <= taum y1 y2 y3 y4 y5 y6) ==>
    (let s = mk_unadorned_v39     3 d a b in  (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv)))
`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  MATCH_MP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] taustar3);
  MATCH_MP_TAC tau3_taum_d;
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let taustar_taum_dfun = 
prove_by_refinement(
    `!d a b f. (let s = scs_v39 (3,d,a,a,b,b,f,{},{},{}) in
    (&2 <= a 0 1 /\ &2 <= a 1 2 /\ &2 <= a 0 2 /\ b 0 1 <= #3.62 /\ b 1 2 <= #3.62 /\ b 0 2 <= #3.62 /\
				(!y1 y2 y3 y4 y5 y6. 
				    &2 <= y1 /\ y1 <= &2 * h0 /\
				    &2 <= y2 /\ y2 <= &2 * h0 /\
				    &2 <= y3 /\ y3 <= &2 * h0 /\
				    a 0 1 <= y6 /\ y6 <= b 0 1 /\
				    a 1 2 <= y4 /\ y4 <= b 1 2 /\
				    a 0 2 <= y5 /\ y5 <= b 0 2 /\
				    &0 <= delta_y y1 y2 y3 y4 y5 y6 ==>
				    d  + #0.1 *
 (if is_ear_v39 s
  then &1
  else -- &1) *
 ((if f 0 1 then cstab - y6 else &0) +
  (if f 1 2 then cstab - y4 else &0) +
  (if f 2 3 then cstab - y5 else &0) +
  &0) <= taum y1 y2 y3 y4 y5 y6) ==>
      (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv)))
`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REPEAT GEN_TAC;
  DISCH_TAC;
  MATCH_MP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] taustar3_fun);
  REPEAT GEN_TAC;
  TYPIFY `dist(v2,(v0:real^3)) = dist(v0,v2)` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[DIST_SYM]);
  REPEAT WEAKER_STRIP_TAC;
  REPLICATE_TAC 12 (FIRST_X_ASSUM MP_TAC);
  REWRITE_TAC[TAUT `(a ==> b ==> c) <=> (a /\ b) ==> c` ];
  REWRITE_TAC[TAUT `(a /\ b) /\ c <=> (a /\ b /\ c)`];
  SPEC_TAC (`v2:real^3`,`v2:real^3`);
  SPEC_TAC (`v1:real^3`,`v1:real^3`);
  SPEC_TAC (`v0:real^3`,`v0:real^3`);
  MATCH_MP_TAC tau3_taum_dfun;
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let taum_sym = 
prove_by_refinement(
  `!y1 y2 y3 y4 y5 y6. taum y1 y2 y3 y4 y5 y6 = taum y2 y1 y3 y5 y4 y6 /\
    taum y1 y2 y3 y4 y5 y6 = taum y1 y3 y2 y4 y6 y5`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Nonlinear_lemma.taum_123;Sphere.rhazim2;Sphere.rhazim3;Sphere.node2_y;Sphere.node3_y];
  REWRITE_TAC[Sphere.rhazim];
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC (REAL_RING `((e1:real) = e1' /\ e2 = e2' /\ e3 = e3' /\ e1 = e1'' /\ e2 = e2'' /\ e3 = e3'') ==> ((r1 * e1 + r2 * e2 + r3 * e3 - c = r2 * e2' + r1 * e1' + r3 * e3' - c) /\ (r1 * e1 + r2 * e2 + r3 * e3 - c = r1 * e1'' + r3 * e3'' + r2 * e2'' - c))`);
  BY(MESON_TAC[Nonlinear_lemma.dih_y_sym;Nonlinear_lemma.dih_y_sym2])
  ]);;
  (* }}} *)

let MOD_4_EXPLICIT = 
prove_by_refinement(
  `0 MOD 4 = 0 /\ 1 MOD 4 = 1 /\ 2 MOD 4 = 2 /\ 3 MOD 4 = 3 /\ 4 MOD 4 = 0 /\
     5 MOD 4 = 1 /\ 6 MOD 4 = 2`,

  (* {{{ proof *)
  [
  ASM_SIMP_TAC[MOD_LT;MOD_MULT_ADD;arith `0 < 4 /\ 1 < 4 /\ 2 < 4 /\ 3 < 4 /\ 5 = 1*4 + 1 /\ 6 = 1*4 + 2`];
  BY(ASM_SIMP_TAC[Oxlzlez.MOD_REFL_ALT;arith `~(4 =0)`])
  ]);;
  (* }}} *)

let FUNLIST_EXPLICIT = 
prove_by_refinement(
`(!data d k i j.
     funlist_v39 data d k i j =
     (if i MOD k = j MOD k
      then &0
      else ASSOCD_v39 (psort k (i,j)) (MAP (\ (u,d). psort k u,d) data) d)) /\
 (!data u u' k i j.    funlistA_v39 data (u:A) u' k i j =  (if i MOD k = j MOD k
       then u
       else ASSOCD_v39 (psort k (i,j)) (MAP (\ (u,d). psort k u,d) data) u')) /\
     0 MOD 3 = 0 /\
     1 MOD 3 = 1 /\
     2 MOD 3 = 2 /\
     3 MOD 3 = 0 /\
     (! (x:real) . x = x) /\
     ~(0 = 1) /\
     ~(0 = 2) /\
     ~(1 = 2) /\
     ~(0 = 3) /\
     ~(1 = 3) /\
     ~(2 = 3) /\
     psort 3 (0,1) = 0,1 /\
     psort 3 (0,2) = 0,2 /\
     psort 3 (1,2) = 1,2 /\
       psort 3 (2,3) = 0,2 /\
	 psort 4 (0,1) = 0,1 /\
	   psort 4 (0,2) = 0,2 /\
	     psort 4 (0,3) = 0,3 /\
	       psort 4 (1,2) = (1,2) /\
    psort 4 (1,3) = (1,3) /\
    psort 4 (2,3) = (2,3) /\
    psort 4 (1,0) = (0,1) /\
    psort 4 (2,0) = (0,2) /\
    psort 4 (3,0) = (0,3) /\
    psort 4 (2,1) = (1,2) /\
    psort 4 (3,1) = (1,3) /\
    psort 4 (3,2) = (2,3) /\
      (!(a:A) (b:B) c d. (a,b) = (c,d) <=> (a = c) /\ (b = d)) /\ 
      (!(a:A) (d:B).  ASSOCD_v39 a [] d = d) /\
      (!(a:A) (d:B) h t. ASSOCD_v39 a (CONS h t) d =
 	 (if a = FST h then SND h else ASSOCD_v39 a t d)) /\
     (!k. MAP (\ (u,(d:B)). psort k u,d) [] = [])  /\
     (!p d. ASSOCD_v39 (p:A) [] (d:B) = d)  /\
     (!k i j r t.
          MAP (\ (u,(d:B)). psort k u,d) (CONS ((i,j),r) t) =
          CONS (psort k (i,j),r) (MAP (\ (u,d). psort k u,d) t))
`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Appendix.funlist_v39;Appendix.funlistA_v39;Appendix.ASSOCD_v39;PAIR_EQ];
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REWRITE_TAC[arith `~(0  = 1) /\ ~(0=2) /\ ~(1=2) /\      ~(0 = 3) /\     ~(1 = 3) /\     ~(2 = 3)`];
  REWRITE_TAC[MAP];
  REWRITE_TAC[Appendix.psort;LET_DEF;LET_END_DEF];
  REWRITE_TAC[MOD_4_EXPLICIT;PAIR_EQ];
  TYPIFY `0 MOD 3 = 0 /\ 1 MOD 3 = 1 /\ 2 MOD 3 = 2` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT (GMATCH_SIMP_TAC MOD_LT);
    BY(ARITH_TAC);
  ASM_REWRITE_TAC[];
  TYPIFY `3 MOD 3 = 0` (C SUBGOAL_THEN SUBST1_TAC);
    BY(ASM_SIMP_TAC[Oxlzlez.MOD_REFL_ALT;arith `~(3 = 0)`]);
    ARITH_TAC
  ]);;
  (* }}} *)

let periodic2_funlist = 
prove_by_refinement(
  `!a a0 k. periodic2 (funlist_v39 a a0 k) k`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.periodic2;Appendix.funlist_v39];
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  TYPIFY `!i. (i + k) MOD k = i MOD k` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    ONCE_REWRITE_TAC[arith `i + (k:num) = 1 * k + i`];
    BY(REWRITE_TAC[MOD_MULT_ADD]);
  TYPIFY `psort k (i + k,j) = psort k (i,j) /\ psort k (i,j+k) = psort k (i,j)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    BY(ASM_REWRITE_TAC[Appendix.psort]);
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let periodic2_funlistA = 
prove_by_refinement(
  `!j1 j' j'' k. periodic2 (funlistA_v39 j1 j' (j'':A) k) k`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.periodic2;Appendix.funlistA_v39];
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  TYPIFY `!i. (i + k) MOD k = i MOD k` (C SUBGOAL_THEN ASSUME_TAC);
    GEN_TAC;
    ONCE_REWRITE_TAC[arith `i + (k:num) = 1 * k + i`];
    BY(REWRITE_TAC[MOD_MULT_ADD]);
  TYPIFY `psort k (i + k,j) = psort k (i,j) /\ psort k (i,j+k) = psort k (i,j)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    BY(ASM_REWRITE_TAC[Appendix.psort]);
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let psort_sym = 
prove_by_refinement(
  `!k i j. psort k (i,j) = psort k (j,i)`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Appendix.psort];
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  BY(REPEAT COND_CASES_TAC THEN REWRITE_TAC[PAIR_EQ] THEN (REPEAT (FIRST_X_ASSUM MP_TAC)) THEN ARITH_TAC)
  ]);;
  (* }}} *)

let funlist_sym = 
prove_by_refinement(
  `!k a a0 i j. funlist_v39 a a0 k i j = funlist_v39 a a0 k j i`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.funlist_v39];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `psort k (i,j) = psort k (j,i)` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[psort_sym]);
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  BY(COND_CASES_TAC THEN ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let funlistA_sym = 
prove_by_refinement(
  `!k a (a0:A) a1 i j. funlistA_v39 a a0 a1 k i j = funlistA_v39 a a0 a1 k j i`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.funlistA_v39];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `psort k (i,j) = psort k (j,i)` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[psort_sym]);
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  BY(COND_CASES_TAC THEN ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let funlist_diag = 
prove_by_refinement(
  `!a (a0) k i. funlist_v39 a a0 k i i = &0`,

  (* {{{ proof *)
  [
  BY(REWRITE_TAC[Appendix.funlist_v39;LET_DEF;LET_END_DEF])
  ]);;
  (* }}} *)

let funlistA_diag = 
prove_by_refinement(
  `!j1 j2 j3 k i. funlistA_v39 j1 j2 j3 k i i = (j2:A)`,

  (* {{{ proof *)
  [
  BY(REWRITE_TAC[Appendix.funlistA_v39;LET_DEF;LET_END_DEF])
  ]);;
  (* }}} *)

let funlistA_empty = 
prove_by_refinement(
  `!k. funlistA_v39 [] F F k = ( \ i j. F) `,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.funlistA_v39;FUN_EQ_THM;MAP;LET_DEF;LET_END_DEF;Appendix.ASSOCD_v39];
  BY(MESON_TAC[])
  ]);;
  (* }}} *)

let funlist_kik = 
prove_by_refinement(
  `!b b0 k i. funlist_v39 b b0 k i k = funlist_v39 b b0 k 0 i`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `k MOD k = 0 MOD k` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[arith `k = 1*k + 0`;MOD_MULT_ADD]);
  REWRITE_TAC[Appendix.funlist_v39;LET_DEF;LET_END_DEF];
  TYPIFY `psort k (i,k) = psort k (0,i)` (C SUBGOAL_THEN SUBST1_TAC);
    ASM_REWRITE_TAC[Appendix.psort;LET_DEF;LET_END_DEF];
    BY(COND_CASES_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[PAIR_EQ] THEN (REPEAT (FIRST_X_ASSUM MP_TAC)) THEN ARITH_TAC);
  ASM_REWRITE_TAC[];
  BY(COND_CASES_TAC THEN ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let periodic_mod_reduce = 
prove_by_refinement(
  `!P k.  ~(k = 0) /\ periodic P k /\ (!i. i < k ==> P i) ==> (!i. P i)`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  GMATCH_SIMP_TAC Oxl_def.periodic_mod;
  TYPIFY `k` EXISTS_TAC;
  ASM_REWRITE_TAC[];
  FIRST_X_ASSUM MATCH_MP_TAC;
  BY(ASM_MESON_TAC[DIVISION])
  ]);;
  (* }}} *)

let periodic2_mod_reduce = 
prove_by_refinement(
  `!P k. ~(k = 0) /\ periodic2 P k /\ (!i j. i < k /\ j < k ==> P i j) ==> (!i j. P i j)`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.periodic2];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `!i. periodic (P i) k` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[Oxl_def.periodic];
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `!j. periodic (\i. P i j) k` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[Oxl_def.periodic];
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `(\ i. P i j) i  = (\ i. P i j) (i MOD k)` ((C SUBGOAL_THEN MP_TAC));
    MATCH_MP_TAC Oxl_def.periodic_mod;
    BY(ASM_REWRITE_TAC[]);
  REWRITE_TAC[];
  DISCH_THEN SUBST1_TAC;
  TYPIFY `P (i MOD k) j = P (i MOD k) (j MOD k)` (C SUBGOAL_THEN MP_TAC);
    MATCH_MP_TAC Oxl_def.periodic_mod;
    BY(ASM_REWRITE_TAC[]);
  DISCH_THEN SUBST1_TAC;
  FIRST_X_ASSUM MATCH_MP_TAC;
  BY(ASM_MESON_TAC[DIVISION])
  ]);;
  (* }}} *)

let periodic2_mod_sym_reduce = 
prove_by_refinement(
  `!P k. ~(k = 0) /\ periodic2 P k /\ (!i. P i i) /\
    (!i j. P i j = P j i) /\ (!i j. i < j /\ j < k ==> P i j) ==> (!i j. P i j)`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  GMATCH_SIMP_TAC periodic2_mod_reduce;
  TYPIFY `k` EXISTS_TAC;
  ASM_REWRITE_TAC[];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `(i:num) < j` ASM_CASES_TAC;
    FIRST_X_ASSUM MATCH_MP_TAC;
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `(j:num) < i \/ (j = i)` (C SUBGOAL_THEN MP_TAC);
    BY(FIRST_X_ASSUM MP_TAC THEN ARITH_TAC);
  DISCH_THEN DISJ_CASES_TAC;
    BY(ASM_MESON_TAC[]);
  BY(ASM_MESON_TAC[])
  ]);;
  (* }}} *)

let periodic2_SUC_periodic = 
prove_by_refinement(
  `!f i. periodic2 (f:num->num->A) k ==> periodic (\i. f i (SUC i)) k`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.periodic2;Oxl_def.periodic];
  REPEAT WEAKER_STRIP_TAC;
  ASM_REWRITE_TAC[];
  TYPIFY `SUC (i + k) = SUC i + k` (C SUBGOAL_THEN SUBST1_TAC);
    BY(ARITH_TAC);
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

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

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Geomdetail.EQ_EXPAND];
  CONJ2_TAC;
    BY(ASM_MESON_TAC[Oxl_def.periodic_mod]);
  INTRO_TAC Oxl_def.periodic_mod [`vv`;`k`];
  ASM_REWRITE_TAC[];
  DISCH_THEN (unlist ONCE_REWRITE_TAC);
  BY(ASM_MESON_TAC[DIVISION])
  ]);;
  (* }}} *)

let I_LT_J_LT_3_EXPLICIT = 
prove_by_refinement(
  `!i j. (i < j /\ j < 3) <=> ((i = 0 /\ j = 1) \/ (i = 0 /\ j = 2) \/ (i = 1 /\ j = 2))`,

  (* {{{ proof *)
  [
  BY(ARITH_TAC)
  ]);;
  (* }}} *)

let I_LT_J_LT_4_EXPLICIT = 
prove_by_refinement(
  `!i j. (i < j /\ j < 4) <=> ((i = 0 /\ j = 1) \/ (i = 0 /\ j = 2) \/ (i = 0 /\ j = 3) \/ (i = 1 /\ j = 2) \/
			      (i = 1 /\ j = 3) \/ (i = 2 /\ j = 3))`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Geomdetail.EQ_EXPAND];
  CONJ2_TAC;
    BY(ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `j = 1 \/ j = 2 \/ j = 3` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC);
  REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[] THEN REPEAT (FIRST_X_ASSUM MP_TAC);
      BY(ARITH_TAC);
    BY(ARITH_TAC);
  BY(ARITH_TAC)
  ]);;
  (* }}} *)

let is_scs_funlist = 
prove_by_refinement(
  `!k d a0 b0 j0 a b j1 u u' u''. d < #0.9 /\ 3 <= k /\ k <= 6 /\
  periodic u k /\ periodic u' k /\ periodic u'' k /\
  (!i j. i < j /\ j < k ==> funlist_v39 a a0 k i j <= funlist_v39 b b0 k i j) /\
 (!i j. i < j /\ j < k  ==> &2 <= funlist_v39 a a0 k i j) /\
 (!i. i < 3 /\ k = 3 ==> funlist_v39 b b0 k i (SUC i) < &4) /\
 (!i. i < k /\ 3 < k ==> funlist_v39 b b0 k i (SUC i) <= cstab) /\
 (!i j. i < j /\ j < k /\
      funlistA_v39 j1 F j0 k i j
      ==> funlist_v39 a a0 k i j = sqrt8 /\ funlist_v39 b b0 k i j = cstab) /\
 (!i j. i < j /\ j < k /\
     funlistA_v39 j1 F j0 k i j
     ==> j  = SUC i \/ (i = 0 /\ SUC j = k)) /\ 
 CARD
 {i | i < k /\
      (&2 * h0 < funlist_v39 b b0 k i (SUC i) \/
       &2 < funlist_v39 a a0 k i (SUC i))} +
 k <=  6 ==>
    is_scs_v39 (scs_v39 (k,d,
			 funlist_v39 a a0 k,funlist_v39 a a0 k,
			 funlist_v39 b b0 k,funlist_v39 b b0 k,
			 funlistA_v39 j1 F j0 k,u,u',u''))`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  ASM_REWRITE_TAC[Appendix.is_scs_v39;Appendix.scs_v39_explicit;arith `x <= x`;Appendix.periodic_empty];
  REWRITE_TAC[periodic2_funlist;periodic2_funlistA];
  REWRITE_TAC[funlist_sym;funlistA_sym;funlist_diag];
  CONJ_TAC;
    MATCH_MP_TAC periodic2_mod_sym_reduce;
    TYPIFY `k` EXISTS_TAC;
    ASM_REWRITE_TAC[TAUT `(a /\ a' ==> b ==> c) <=> (a /\ a' /\ b ==> c)` ];
    ASM_SIMP_TAC[arith `3 <= k ==> ~(k=0)`];
    REWRITE_TAC[funlist_diag;arith `&0 <= &0`];
    CONJ_TAC;
      REWRITE_TAC[Appendix.periodic2];
      BY(MESON_TAC[periodic2_funlist;Appendix.periodic2]);
    BY(MESON_TAC[funlist_sym]);
  COMMENT "next goal";
  CONJ_TAC;
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `i < (j:num)` ASM_CASES_TAC;
      BY(ASM_MESON_TAC[]);
    ONCE_REWRITE_TAC[funlist_sym];
    FIRST_X_ASSUM MATCH_MP_TAC;
    BY(ASM_SIMP_TAC[arith `~(i = j) /\ ~(i < j) ==> (j < (i:num))`]);
  COMMENT "next goal again";
  CONJ_TAC;
    REWRITE_TAC[RIGHT_FORALL_IMP_THM];
    DISCH_TAC;
    MATCH_MP_TAC periodic_mod_reduce;
    TYPIFY `k` EXISTS_TAC;
    ASM_SIMP_TAC [arith `k=3 ==> ~(k=0)`];
    REWRITE_TAC[Oxl_def.periodic];
    BY(ASM_MESON_TAC[periodic2_funlist;periodic2_SUC_periodic;Oxl_def.periodic]);
  COMMENT "next";
  CONJ_TAC;
    REWRITE_TAC[RIGHT_FORALL_IMP_THM];
    DISCH_TAC;
    MATCH_MP_TAC periodic_mod_reduce;
    TYPIFY `k` EXISTS_TAC;
    ASM_SIMP_TAC [arith `3 < k ==> ~(k=0)`];
    REWRITE_TAC[Oxl_def.periodic];
    BY(ASM_MESON_TAC[periodic2_funlist;periodic2_SUC_periodic;Oxl_def.periodic]);
  COMMENT "pentultimate";
  CONJ2_TAC;
    MATCH_MP_TAC periodic2_mod_sym_reduce;
    TYPIFY `k` EXISTS_TAC;
    ASM_SIMP_TAC [arith `3 <= k ==> ~(k=0)`];
    CONJ_TAC;
      REWRITE_TAC[Appendix.periodic2];
      BY(REWRITE_TAC[ (REWRITE_RULE[Appendix.periodic2] periodic2_funlist); (REWRITE_RULE[Appendix.periodic2] periodic2_funlistA)]);
    REWRITE_TAC[funlistA_diag];
    BY(MESON_TAC[funlistA_sym;funlist_sym]);
  COMMENT "final";
  MATCH_MP_TAC periodic2_mod_sym_reduce;
  TYPIFY `k` EXISTS_TAC;
  ASM_SIMP_TAC [arith `3 <= k ==> ~(k=0)`];
  CONJ_TAC;
    REWRITE_TAC[Appendix.periodic2];
    (REWRITE_TAC[ (REWRITE_RULE[Appendix.periodic2] periodic2_funlist); (REWRITE_RULE[Appendix.periodic2] periodic2_funlistA)]);
    REWRITE_TAC[arith `SUC (i + k) = SUC i + k`];
    ONCE_REWRITE_TAC[arith `(i + k) = (1 * k + i)`];
    BY(REWRITE_TAC[MOD_MULT_ADD]);
  REWRITE_TAC[funlistA_diag];
  CONJ_TAC;
    BY(MESON_TAC[funlistA_sym;funlist_sym]);
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM (C INTRO_TAC [`i`;`j`]);
  ASM_REWRITE_TAC[];
  DISCH_THEN DISJ_CASES_TAC;
    BY(ASM_REWRITE_TAC[]);
  ASM_REWRITE_TAC[];
  TYPIFY `k MOD k = (1 * k + 0) MOD k` (C SUBGOAL_THEN SUBST1_TAC);
    AP_THM_TAC;
    AP_TERM_TAC;
    BY(ARITH_TAC);
  BY(REWRITE_TAC[MOD_MULT_ADD])
  ]);;
  (* }}} *)

let is_ear_scs3 = 
prove_by_refinement(
 `!a b jf. is_ear_v39 (scs_v39 (3, #0.11, a, a, b, b, jf, {}, {},{})) <=>  
    is_scs_v39 (scs_v39 (3, #0.11, a,a,b,b,jf, {},{},{})) /\
    (!i. b i i = &0) /\
  (?i. {i | i < 3 /\ jf i (SUC i)} = {i} /\
      a i (SUC i) = sqrt8 /\ b i (SUC i) = cstab /\
      (!j. j < 3 /\ ~(j = i) ==> a j (SUC j) = &2 /\ b j (SUC j) = &2 * h0))`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Appendix.is_ear_v39;Appendix.scs_v39_explicit];
  ASM_CASES_TAC `~is_scs_v39 (scs_v39 (3, #0.11,a,a,b,b,jf,{},{},{}))`;
    BY(ASM_REWRITE_TAC[]);
  RULE_ASSUM_TAC (REWRITE_RULE[ (TAUT `~ ~ x = x`)]);
  ASM_REWRITE_TAC[];
  ASM_CASES_TAC `~(?i. {i | i < 3 /\ jf i (SUC i)} = {i} /\       a i (SUC i) = sqrt8 /\      b i (SUC i) = cstab /\      (!j. j < 3 /\ ~(j = i) ==> a j (SUC j) = &2 /\ b j (SUC j) = &2 * h0))`;
    BY(ASM_REWRITE_TAC[]);
  RULE_ASSUM_TAC (REWRITE_RULE[ (TAUT `~ ~ x = x`)]);
  ASM_REWRITE_TAC[];
  BY(REWRITE_TAC[Appendix.unadorned_v39;Appendix.scs_v39_explicit])
  ]);;
  (* }}} *)

let is_scs_scs3 = 
prove_by_refinement(
  `! d a a0 b b0 jf j0 .  d < #0.9 /\
    funlist_v39 a a0 3 0 1 <= funlist_v39 b b0 3 0 1 /\
    funlist_v39 a a0 3 0 2 <= funlist_v39 b b0 3 0 2 /\
    funlist_v39 a a0 3 1 2 <= funlist_v39 b b0 3 1 2 /\
   &2 <= funlist_v39 a a0 3 0 1 /\ 
   &2 <= funlist_v39 a a0 3 1 2 /\ 
   &2 <= funlist_v39 a a0 3 0 2 /\ 
   funlist_v39 b b0 3 0 1  < &4 /\
   funlist_v39 b b0 3 0 2  < &4 /\
   funlist_v39 b b0 3 1 2  < &4 /\
   (!i j.
     i < j /\ j < 3 /\ funlistA_v39 jf F j0 3 i j 
     ==> funlist_v39 a a0 3 i j = sqrt8 /\ funlist_v39 b b0 3 i j = cstab)
   ==>
   is_scs_v39
             (scs_v39
             (3,
              d,
              funlist_v39 a a0 3,
              funlist_v39 a a0 3,
              funlist_v39 b b0 3,
              funlist_v39 b b0 3,
              funlistA_v39 jf F j0 3,
              {},{},
              {}))`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC is_scs_funlist;
  ASM_REWRITE_TAC[Appendix.periodic_empty;arith `3 <= 3 /\ 3 <= 6 /\ ~(3 < 3)`];
  REWRITE_TAC[arith `x + 3 <= 6 <=> x <= 3`];
  TYPIFY `CARD  {i | i < 3 /\      (&2 * h0 < funlist_v39 b b0 3 i (SUC i) \/       &2 < funlist_v39 a a0 3 i (SUC i))} <= 3` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    ENOUGH_TO_SHOW_TAC `CARD  {i | i < 3 /\      (&2 * h0 < funlist_v39 b b0 3 i (SUC i) \/       &2 < funlist_v39 a a0 3 i (SUC i))} <= CARD {i | i < 3}`;
      TYPIFY `{i | i < 3} = (0..2)` (C SUBGOAL_THEN SUBST1_TAC);
        REWRITE_TAC[IN_NUMSEG;EXTENSION;IN_ELIM_THM];
        BY(ARITH_TAC);
      BY(REWRITE_TAC[CARD_NUMSEG;arith `2 + 1 = 3 /\ 3 - 0 = 3`]);
    MATCH_MP_TAC CARD_SUBSET;
    CONJ_TAC;
      BY(SET_TAC[]);
    BY(REWRITE_TAC[FINITE_NUMSEG_LT]);
  REWRITE_TAC[I_LT_J_LT_3_EXPLICIT];
  CONJ_TAC;
    REPEAT WEAKER_STRIP_TAC;
    BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[]);
  CONJ_TAC;
    REPEAT WEAKER_STRIP_TAC;
    BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[]);
  CONJ_TAC;
    REWRITE_TAC[arith `i < 3 <=> (i = 0 \/ i = 1 \/ i = 2)`];
    REPEAT WEAKER_STRIP_TAC;
    BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3`;funlist_kik]);
  REWRITE_TAC[TAUT `(a /\ b /\ c ==> d) <=> (a /\ b ==> (c ==> d))`];
  REWRITE_TAC[I_LT_J_LT_3_EXPLICIT];
  REPEAT WEAKER_STRIP_TAC;
  BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3`])
  ]);;
  (* }}} *)

let is_scs_ear_3603097872 = 
prove_by_refinement(
  `is_scs_v39 (scs_v39
   (3,
     #0.11,
    funlist_v39 [(0,1),sqrt8] (&2) 3,
    funlist_v39 [(0,1),sqrt8] (&2) 3,
    funlist_v39 [(0,1),cstab] (&2 * h0) 3,
    funlist_v39 [(0,1),cstab] (&2 * h0) 3,
    funlistA_v39 [(0,1),T] F F 3,
    {},{},
    {}))`,

  (* {{{ proof *)
  [
  MATCH_MP_TAC is_scs_scs3;
  TYPIFY `(!i j.      i < j /\ j < 3 /\ funlistA_v39 [(0,1),T] F F 3 i j      ==> funlist_v39 [(0,1),sqrt8] (&2) 3 i j = sqrt8 /\          funlist_v39 [(0,1),cstab] (&2 * h0) 3 i j = cstab)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    REWRITE_TAC[TAUT `(a /\ b /\ c ==> d) <=> (a /\ b ==> (c ==> d))`];
    REWRITE_TAC[I_LT_J_LT_3_EXPLICIT];
    REPEAT WEAKER_STRIP_TAC;
    FIRST_X_ASSUM MP_TAC;
    BY((REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[FUNLIST_EXPLICIT]));
  REWRITE_TAC[FUNLIST_EXPLICIT;arith `#0.11 < #0.9`];
  REWRITE_TAC[Appendix.cstab;Sphere.h0];
  INTRO_TAC Flyspeck_constants.bounds [];
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let is_ear_3603097872 = 
prove_by_refinement(
  `is_ear_v39 (scs_v39
   (3,
     #0.11,
    funlist_v39 [(0,1),sqrt8] (&2) 3,
    funlist_v39 [(0,1),sqrt8] (&2) 3,
    funlist_v39 [(0,1),cstab] (&2 * h0) 3,
    funlist_v39 [(0,1),cstab] (&2 * h0) 3,
    funlistA_v39 [(0,1),T] F F 3,
    {},{},
    {})) `,

  (* {{{ proof *)
  [
  REWRITE_TAC[ is_ear_scs3;is_scs_ear_3603097872;funlist_diag];
  EXISTS_TAC `0`;
  REWRITE_TAC[arith `SUC 0 = 1`;arith `!j. (j < 3 /\ ~(j = 0) <=> (j = 1 \/ j = 2))`];
  CONJ_TAC;
    REWRITE_TAC[EXTENSION;IN_SING;IN_ELIM_THM];
    GEN_TAC;
    MP_TAC (arith `x = 0 \/ x = 1 \/ x = 2 \/ ~(x < 3)`);
    DISCH_TAC;
    BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_SIMP_TAC[arith `0 < 3 /\ 1 < 3 /\ 2 < 3 /\ (~(x < 3 ) ==> ~(x = 0))`;FUNLIST_EXPLICIT;arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3`]);
  TYPIFY `(!j. j = 1 \/ j = 2       ==> funlist_v39 [(0,1),sqrt8] (&2) 3 j (SUC j) = &2 /\          funlist_v39 [(0,1),cstab] (&2 * h0) 3 j (SUC j) = &2 * h0)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    REPEAT WEAKER_STRIP_TAC;
    BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_SIMP_TAC[arith `SUC 1 = 2 /\ SUC 2 = 3`;FUNLIST_EXPLICIT;funlist_kik]);
  BY(REWRITE_TAC[FUNLIST_EXPLICIT])
  ]);;
  (* }}} *)

let REAL_FINITE_MIN_EXISTS =
prove(`!S:real->bool. FINITE S /\ ~(S = {}) ==> ?m. m IN S /\ (!x. x IN S ==> m <= x)`,

		MESON_TAC[INF_FINITE]);;
let REAL_WLOG_SQUARE_LEMMA = 
prove_by_refinement(
  `!P. (!y1 y2 y3 y4 y5 y6. P y1 y2 y3 y4 y5 y6 = P y2 y3 y4 y1 y6 y5) /\
    (!y1 y2 y3 y4 y5 y6. (y2 <= y1) /\ (y3 <= y1) /\ (y4 <= y1) 
         ==>
       P y1 y2 y3 y4 y5 y6) ==>
    (!y1 y2 y3 y4 y5 y6. P y1 y2 y3 y4 y5 y6)`,

  (* {{{ proof *)
  [
  REPEAT WEAK_STRIP_TAC;
  SUBGOAL_THEN `?a.  a IN {y1,y2,y3,y4} /\ (!x. x IN {y1,y2,y3,y4} ==> x <= a)` MP_TAC;
    MATCH_MP_TAC Merge_ineq.REAL_FINITE_MAX_EXISTS;
    BY(REWRITE_TAC[ FINITE_INSERT ; FINITE_EMPTY;NOT_INSERT_EMPTY]);
  REPEAT WEAK_STRIP_TAC;
  REPEAT (FIRST_X_ASSUM_ST `IN` MP_TAC);
  REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY];
  REWRITE_TAC[MESON[] `(!x. x = y1 \/ x = y2 \/ x = y3 \/ x = y4 ==> x <= a) = (y1 <= a /\ y2 <= a /\ y3 <= a /\ y4 <= a)`];
  BY(DISCH_THEN STRIP_ASSUME_TAC THEN ASM_MESON_TAC[])
  ]);;
  (* }}} *)

let REAL_WLOG_SQUARE2_LEMMA = 
prove_by_refinement(
`!P. (!y1 y2 y3 y4 y5 y6. P y1 y2 y3 y4 y5 y6 = P y2 y3 y4 y1 y6 y5 
    /\ P y1 y2 y3 y4 y5 y6 = P y1 y4 y3 y2 y6 y5) /\
    (!y1 y2 y3 y4 y5 y6. (y2 <= y1) /\ (y3 <= y1) /\ (y4 <= y1) /\ (y4 <= y2)
         ==>
       P y1 y2 y3 y4 y5 y6) ==>
    (!y1 y2 y3 y4 y5 y6. P y1 y2 y3 y4 y5 y6)`,

  (* {{{ proof *)
  [
  GEN_TAC;
  DISCH_TAC;
  MATCH_MP_TAC REAL_WLOG_SQUARE_LEMMA;
  ASM_REWRITE_TAC[];
  BY(ASM_MESON_TAC[arith `y2 <= y4 \/ y4 <= y2`])
  ]);;
  (* }}} *)

let EE_vv = 
prove_by_refinement(
  `!vv k i.
  periodic (vv:num->A) k /\ 3 <= k /\
  (!i (j:num). i < k /\ j < k /\ vv i = vv j ==> i = j) ==> 
    (EE  (vv i) (IMAGE (\i. {vv i, vv (SUC i)}) (:num))) = { vv (SUC i) , vv (i + (k -1)) }`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `~(k=0)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC);
  REWRITE_TAC[Lvducxu.IN_EE_IFF_IN_E;EXTENSION;IN_INSERT;NOT_IN_EMPTY];
  X_GEN_TAC `ww:A`;
  REWRITE_TAC[IN_IMAGE;IN_UNIV];
  REWRITE_TAC[Geomdetail.EQ_EXPAND];
  CONJ2_TAC;
    DISCH_THEN DISJ_CASES_TAC;
      BY(ASM_MESON_TAC[]);
    TYPIFY `i + k - 1` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    REWRITE_TAC[Geomdetail.PAIR_EQ_EXPAND];
    DISJ2_TAC;
    ASM_SIMP_TAC[arith `3 <= k ==> SUC(i + k - 1) = i +k`];
    BY(ASM_MESON_TAC[Oxl_def.periodic]);
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM MP_TAC;
  REWRITE_TAC[Geomdetail.PAIR_EQ_EXPAND];
  DISCH_THEN DISJ_CASES_TAC;
    ASM_REWRITE_TAC[];
    DISJ1_TAC;
    FIRST_X_ASSUM MP_TAC;
    INTRO_TAC Oxl_def.periodic_mod [`vv`;`k`];
    REWRITE_TAC[GSYM RIGHT_IMP_FORALL_THM];
    ANTS_TAC;
      BY(ASM_REWRITE_TAC[]);
    DISCH_THEN (unlist ONCE_REWRITE_TAC);
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `i MOD k = x MOD k` (C SUBGOAL_THEN ASSUME_TAC);
      BY(ASM_MESON_TAC[DIVISION]);
    AP_TERM_TAC;
    REWRITE_TAC[arith `SUC x = x + 1`];
    GMATCH_SIMP_TAC (GSYM MOD_ADD_MOD);
    FIRST_X_ASSUM (SUBST1_TAC o GSYM);
    GMATCH_SIMP_TAC MOD_ADD_MOD;
    BY(ASM_REWRITE_TAC[]);
  COMMENT "last case";
  ASM_REWRITE_TAC[];
  DISJ2_TAC;
  FIRST_X_ASSUM MP_TAC;
  INTRO_TAC Oxl_def.periodic_mod [`vv`;`k`];
  ASM_REWRITE_TAC[];
  DISCH_THEN (unlist ONCE_REWRITE_TAC);
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `i MOD k = SUC x MOD k` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[DIVISION]);
  AP_TERM_TAC;
  GMATCH_SIMP_TAC (GSYM MOD_ADD_MOD);
  ASM_REWRITE_TAC[];
  GMATCH_SIMP_TAC MOD_ADD_MOD;
  ASM_SIMP_TAC[arith `3 <= k ==> (SUC x + k - 1 = 1 * k + x)`];
  BY(REWRITE_TAC[MOD_MULT_ADD])
  ]);;
  (* }}} *)

let tau_fun_azim = 
prove_by_refinement(
  `!vv k .
  (let V = IMAGE vv (:num) in
   let E = IMAGE (\i. {vv i, vv (SUC i)}) (:num) in
   let f = IMAGE (\i. vv i,vv (SUC i)) (:num) in
    (periodic vv k /\ 3 <= k /\
     (!i j. i < k /\ j < k /\ vv i = vv j ==> (i = j)) ==>
       tau_fun V E f = sum {i | i < k} (\i. rho_fun (norm (vv i)) *
	 azim (vec 0) (vv i) (vv (i+1)) (vv (i + (k-1))))  - (pi + sol0) * (&k - &2)))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_DEF;LET_END_DEF;Appendix.tau_fun];
  REPEAT WEAKER_STRIP_TAC;
  GMATCH_SIMP_TAC (GSYM Oxlzlez.PERIODIC_IMAGE);
  TYPIFY `k` EXISTS_TAC;
  CONJ_TAC;
    ASM_SIMP_TAC[arith `3 <= k ==> ~(0 = k)`];
    FIRST_X_ASSUM_ST `periodic` MP_TAC;
    REWRITE_TAC[Oxl_def.periodic;PAIR_EQ];
    BY(MESON_TAC[arith `SUC (i + k) = SUC i + k`]);
  MATCH_MP_TAC (arith `s = s' /\ b = b' ==> s - (pi+sol0) * b = s' - (pi+sol0) * b'`);
  GMATCH_SIMP_TAC CARD_IMAGE_INJ;
  ASM_REWRITE_TAC[IN_ELIM_THM;PAIR_EQ;FINITE_NUMSEG_LT;CARD_NUMSEG_LT];
  CONJ_TAC;
    BY(ASM_MESON_TAC[]);
  CONJ2_TAC;
    GMATCH_SIMP_TAC REAL_OF_NUM_SUB;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC);
  GMATCH_SIMP_TAC SUM_IMAGE;
  ASM_REWRITE_TAC[IN_ELIM_THM;PAIR_EQ;FINITE_NUMSEG_LT;CARD_NUMSEG_LT];
  CONJ_TAC;
    BY(ASM_MESON_TAC[]);
  MATCH_MP_TAC SUM_EQ;
  REWRITE_TAC[IN_ELIM_THM;o_THM];
  REPEAT WEAKER_STRIP_TAC;
  AP_TERM_TAC;
  REWRITE_TAC[Localization.azim_in_fan;LET_DEF;LET_END_DEF];
  GMATCH_SIMP_TAC EE_vv;
  TYPIFY `k` EXISTS_TAC;
  ASM_REWRITE_TAC[];
  COND_CASES_TAC;
    REWRITE_TAC[Local_lemmas.AZIM_CYCLE_TWO_POINT_SET];
    BY(REWRITE_TAC[arith `SUC x = x + 1`]);
  PROOF_BY_CONTR_TAC;
  FIRST_X_ASSUM_ST `CARD` MP_TAC;
  TYPIFY `~(k = 0)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC);
  REWRITE_TAC[];
  MATCH_MP_TAC (arith `c = 2 ==> c > 1`);
  MATCH_MP_TAC Hypermap.CARD_TWO_ELEMENTS;
  INTRO_TAC Oxl_def.periodic_mod [`vv`;`k`];
  ASM_REWRITE_TAC[];
  DISCH_THEN (unlist ONCE_REWRITE_TAC);
  DISCH_TAC;
  TYPIFY `SUC x MOD k = (x + k - 1) MOD k` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[DIVISION]);
  INTRO_TAC Oxlzlez.MOD_INJ1_ALT [`k-2`;`k`];
  ANTS_TAC;
    BY(REPEAT (FIRST_X_ASSUM_ST `3 <= k` MP_TAC) THEN ARITH_TAC);
  DISCH_TAC;
  FIRST_X_ASSUM (C INTRO_TAC [`SUC x`]);
  ASM_REWRITE_TAC[];
  BY(ASM_SIMP_TAC[arith `3 <= k ==> SUC x + k - 2 = x + k - 1`])
  ]);;
  (* }}} *)

let vv_rho_node1 = 
prove_by_refinement(
  `!vv k .
  (let V = IMAGE vv (:num) in
   let f = IMAGE (\i. vv i,vv (SUC i)) (:num) in
    (periodic vv k /\ 3 <= k /\
     (!i j. i < k /\ j < k /\ vv i = vv j ==> (i = j)) ==>
       (!i. rho_node1 f (vv i) = vv (SUC i))))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Local_lemmas.rho_node1];
  TYPIFY `~( k = 0)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC);
  SELECT_TAC;
    RULE_ASSUM_TAC(REWRITE_RULE[IN_IMAGE;IN_UNIV;PAIR_EQ]);
    REPEAT (FIRST_X_ASSUM MP_TAC) THEN REPEAT WEAKER_STRIP_TAC;
    ASM_REWRITE_TAC[];
    REPLICATE_TAC 4 (FIRST_X_ASSUM MP_TAC);
    INTRO_TAC periodic_vv_inj [`vv`;`k`];
    ASM_REWRITE_TAC[];
    DISCH_THEN (unlist ONCE_REWRITE_TAC);
    REWRITE_TAC[arith `SUC u = u + 1`];
    REPEAT WEAKER_STRIP_TAC;
    ONCE_REWRITE_TAC [GSYM MOD_ADD_MOD];
    GMATCH_SIMP_TAC (GSYM MOD_ADD_MOD);
    TYPIFY `x' MOD k = i MOD k` (C SUBGOAL_THEN SUBST1_TAC);
      BY(ASM_MESON_TAC[]);
    GMATCH_SIMP_TAC MOD_ADD_MOD;
    BY(ASM_REWRITE_TAC[]);
  PROOF_BY_CONTR_TAC;
  FIRST_X_ASSUM_ST `IMAGE` MP_TAC;
  REWRITE_TAC[];
  TYPIFY `vv(SUC i)` EXISTS_TAC;
  BY(SET_TAC[])
  ]);;
  (* }}} *)

let ITER_vv_rho_node1 = 
prove_by_refinement(
  `!vv k j. (let V = IMAGE vv (:num) in
   let f = IMAGE (\i. vv i,vv (SUC i)) (:num) in
    (periodic vv k /\ 3 <= k /\
     (!i j. i < k /\ j < k /\ vv i = vv j ==> (i = j)) ==>
       (!i. ITER j (rho_node1 f) (vv i) = vv (i+j))))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REPEAT WEAKER_STRIP_TAC;
  SPEC_TAC (`j:num`,`j:num`);
  INDUCT_TAC;
    BY(REWRITE_TAC[ITER;arith `i + 0 = i`]);
  ASM_SIMP_TAC[ITER];
  INTRO_TAC vv_rho_node1 [`vv`;`k`];
  ASM_REWRITE_TAC[LET_DEF;LET_END_DEF];
  DISCH_THEN (unlist REWRITE_TAC);
  AP_TERM_TAC;
  BY(ARITH_TAC)
  ]);;
  (* }}} *)

let PRIOR_TO_LESS_THAN_PI_LEMMA_ALT = 
prove_by_refinement(
  `!V E FF v. convex_local_fan (V,E,FF) /\ v IN V
     ==> (!w. w IN V
              ==> azim (vec 0) v (rho_node1 FF v) w <=
                  azim (vec 0) v (rho_node1 FF v)
                  (azim_cycle (EE v E) (vec 0) v (rho_node1 FF v)))`,

  (* {{{ proof *)
  [
  MESON_TAC[Local_lemmas.PRIOR_TO_LESS_THAN_PI_LEMMA]
  ]);;
  (* }}} *)

let vv_azim_le = 
prove_by_refinement(
  `!vv k i j. 
    (let V = IMAGE vv (:num) in
   let E = IMAGE (\i. {vv i, vv (SUC i)}) (:num) in
   let f = IMAGE (\i. vv i,vv (SUC i)) (:num) in
    (periodic vv k /\ 3 <= k /\
   V SUBSET ball_annulus /\
   dist(vv i, vv (SUC i)) < &4 /\
   dist(vv i, vv (i+k - 1)) < &4 /\
   dist(vv i, vv j) < &4 /\
   ~(i MOD k = j MOD k) /\
   (!i j. ~(vv i = vv j) ==> &2 <= dist(vv i,vv j) ) /\
   convex_local_fan (V,E,f) /\
        (!i j. i < k /\ j < k /\ vv i = vv j ==> (i = j))
    ==> azim (vec 0) (vv i) (vv (SUC i)) (vv j) <= azim (vec 0) (vv i) (vv (SUC i)) (vv (i + k - 1))))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `~(k=0)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC);
  INTRO_TAC PRIOR_TO_LESS_THAN_PI_LEMMA_ALT [`IMAGE vv (:num)`; `IMAGE (\i. {vv i, vv (SUC i)}) (:num)`; `IMAGE (\i. vv i,vv (SUC i)) (:num)`;`vv i`];
  ASM_REWRITE_TAC[];
  ANTS_TAC;
    BY(SET_TAC[]);
  DISCH_THEN (C INTRO_TAC [`vv j`]);
  ANTS_TAC;
    BY(SET_TAC[]);
  INTRO_TAC vv_rho_node1 [`vv`;`k`];
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  ANTS_TAC;
    BY(ASM_REWRITE_TAC[]);
  DISCH_THEN (unlist REWRITE_TAC);
  GMATCH_SIMP_TAC EE_vv;
  TYPIFY `k` EXISTS_TAC;
  ASM_REWRITE_TAC[];
  REWRITE_TAC[Local_lemmas.AZIM_CYCLE_TWO_POINT_SET];
  ]);;
  (* }}} *)

let vv_split_azim = 
prove_by_refinement(
  `!vv k i j. 
    (let V = IMAGE vv (:num) in
   let E = IMAGE (\i. {vv i, vv (SUC i)}) (:num) in
   let f = IMAGE (\i. vv i,vv (SUC i)) (:num) in
    (periodic vv k /\ 3 <= k /\
   V SUBSET ball_annulus /\
   dist(vv i, vv (SUC i)) < &4 /\
   dist(vv i, vv (i+k - 1)) < &4 /\
   dist(vv i, vv j) < &4 /\
   ~(i MOD k = j MOD k) /\
   (!i j. ~(vv i = vv j) ==> &2 <= dist(vv i,vv j) ) /\
   convex_local_fan (V,E,f) /\
        (!i j. i < k /\ j < k /\ vv i = vv j ==> (i = j))
    ==> azim (vec 0) (vv i) (vv (SUC i)) (vv (i + k - 1)) = 
	azim (vec 0) (vv i) (vv (SUC i)) (vv j) +
	  azim (vec 0) (vv i) (vv j) (vv (i + k - 1))))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `~(k=0)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC);
  INTRO_TAC vv_azim_le [`vv`;`k`;`i`;`j`];
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  ANTS_TAC;
    BY(ASM_REWRITE_TAC[]);
  DISCH_TAC;
  MATCH_MP_TAC Fan.sum4_azim_fan;
  ASM_REWRITE_TAC[];
  REPEAT (GMATCH_SIMP_TAC NONPARALLEL_BALL_ANNULUS40_ALT);
  ASM_REWRITE_TAC[];
  RULE_ASSUM_TAC (REWRITE_RULE[IMAGE_SUBSET_IN;IN_UNIV]);
  ASM_REWRITE_TAC[];
  REPEAT (FIRST_ASSUM_ST `dist` GMATCH_SIMP_TAC);
  INTRO_TAC periodic_vv_inj [`vv`;`k`];
  ANTS_TAC;
    BY(ASM_REWRITE_TAC[]);
  DISCH_THEN (unlist ASM_REWRITE_TAC);
  REWRITE_TAC[arith `SUC i = (i +1)`];
  BY(ASM_MESON_TAC[Oxlzlez.MOD_INJ1_ALT;arith `3 <= k ==> (k - 1 < k)`;arith `3 <=k ==> 1 < k`;arith `3 <=k ==> ~(k- 1=0)`;arith `~(1 = 0)`])
  ]);;
  (* }}} *)

let EGHNAVX1_ALT = 
prove_by_refinement(
  `!V E FF bta v0 ww k. convex_local_fan (V,E,FF) /\
     v0 IN V /\
     CARD V = k /\
     (!v. v IN V /\ ~(v = v0) ==> ~collinear {vec 0, v0, v}) /\
     (!i. ITER i (rho_node1 FF) v0 = ww i) /\
     (!i. azim (vec 0) v0 (ww 1) (ww i) = bta i)
     ==>          (!i j. i < j /\ j < k ==> bta i <= bta j)`,

  (* {{{ proof *)
  [
  REPEAT GEN_TAC;
  DISCH_THEN (MP_TAC o (MATCH_MP Local_lemmas.EGHNAVX));
  BY(DISCH_THEN (unlist REWRITE_TAC))
  ]);;
  (* }}} *)

let vv_split_azim_generic = 
prove_by_refinement(
  `!vv k i j j'. 
    (let V = IMAGE vv (:num) in
   let E = IMAGE (\i. {vv i, vv (SUC i)}) (:num) in
   let f = IMAGE (\i. vv i,vv (SUC i)) (:num) in
    (periodic vv k /\ 3 <= k /\
   (0< j) /\ (j < j') /\ (j' < k) /\
   generic V E /\
   convex_local_fan (V,E,f) /\
        (!i j. i < k /\ j < k /\ vv i = vv j ==> (i = j))
    ==> azim (vec 0) (vv i) (vv (SUC i)) (vv (i+j')) = 
	azim (vec 0) (vv i) (vv (SUC i)) (vv (i+j)) +
	  azim (vec 0) (vv i) (vv (i+j)) (vv (i+j'))))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `~(k=0)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC);
  TYPIFY `!i. vv i IN IMAGE vv (:num)` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[IN_IMAGE;IN_UNIV];
    BY(ASM_MESON_TAC[]);
  INTRO_TAC EGHNAVX1_ALT [`IMAGE vv (:num)`; `IMAGE (\i. {vv i, vv (SUC i)}) (:num)`; `IMAGE (\i. vv i,vv (SUC i)) (:num)`;`\j. azim (vec 0) (vv i) (vv (SUC i)) (vv (i + j))`;`vv i`;`\j. ITER j (rho_node1 (IMAGE (\i. vv i,vv (SUC i)) (:num))) (vv i)`;`k`];
  ASM_REWRITE_TAC[];
  INTRO_TAC ITER_vv_rho_node1 [`vv`;`k`];
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  ASM_REWRITE_TAC[];
  DISCH_THEN (unlist REWRITE_TAC);
  REWRITE_TAC[arith `i + 1 = SUC i`];
  INTRO_TAC Nkezbfc_local.PROPERTIES_GENERIC_LOCAL_FAN [`IMAGE vv (:num)`;`IMAGE (\i. {vv i, vv (SUC i)}) (:num)`;`IMAGE (\i. vv i,vv (SUC i)) (:num)`;`vv i`];
  ASM_REWRITE_TAC[];
  ANTS_TAC;
    MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO;
    BY(ASM_REWRITE_TAC[]);
  DISCH_TAC;
  ASM_REWRITE_TAC[];
  ANTS_TAC;
    GMATCH_SIMP_TAC (GSYM Oxlzlez.PERIODIC_IMAGE);
    TYPIFY `k` EXISTS_TAC;
    ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC CARD_IMAGE_INJ;
    BY(ASM_REWRITE_TAC[IN_ELIM_THM;CARD_NUMSEG_LT;FINITE_NUMSEG_LT ]);
  DISCH_THEN (C INTRO_TAC [`j`;`j'`]);
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  MATCH_MP_TAC Fan.sum4_azim_fan;
  ASM_REWRITE_TAC[];
  FIRST_X_ASSUM_ST `IMAGE` (REPEAT o GMATCH_SIMP_TAC);
  ASM_REWRITE_TAC[];
  INTRO_TAC periodic_vv_inj [`vv`;`k`];
  ANTS_TAC;
    BY(ASM_REWRITE_TAC[]);
  DISCH_THEN (unlist REWRITE_TAC);
  REWRITE_TAC[arith `SUC i = i + 1`];
  BY(REPEAT CONJ_TAC THEN MATCH_MP_TAC (REWRITE_RULE[GSYM RIGHT_FORALL_IMP_THM] (GSYM Oxlzlez.MOD_INJ1_ALT)) THEN ASM_REWRITE_TAC[] THEN REPEAT (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC)
  ]);;
  (* }}} *)

let muR_ALT = 
prove_by_refinement(
  `!y1 y2 y3 y4 y5 y6 y7 y8 y9. muR y1 y2 y3 y4 y5 y6 y7 y8 y9  =
         cayleyR (y6 * y6) (y5 * y5) (y1 * y1) (y7 * y7) (y4 * y4) (y2 * y2)
         (y8 * y8)
         (y3 * y3)
         (y9 * y9)
          `,

  (* {{{ proof *)
  [
    BY(REWRITE_TAC[FUN_EQ_THM;Mur.muR])
  ]);;
  (* }}} *)

let enclosed4_lemma = 
prove_by_refinement(
  `!(v0:real^3) v1 v2 v3. 
   (let y1 = norm (v1) in
   let y2 = norm (v2) in
   let y3 = norm (v0) in
   let y4 = dist(v0, v2) in
   let y5 = dist(v0, v1) in
   let y6 = dist(v1, v2) in
   let y7 = norm (v3) in
   let y8 = dist(v0,v3) in
   let y9 = dist(v2,v3) in
    (
    &0 < ups_x (norm (v0) * norm (v0)) (norm (v2) * norm (v2)) (dist(v0,v2) * dist(v0,v2)) /\
      chi_msb [(vec 0);v0;v2] v1 * chi_msb [(vec 0);v0;v2] v3 <= &0
    ==> dist(v1 ,v3) = 
	enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Enclosed.enclosed];
  INTRO_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] Collect_geom2.CAYLEYR_5POINTS) [`(vec 0):real^3`;`v0`;`v2`;`v1`;`v3`];
  REWRITE_TAC[DIST_0];
  TYPIFY `dist(v2,v1) = dist(v1,v2)` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[DIST_SYM]);
  DISCH_TAC;
  TYPED_ABBREV_TAC `p = muR (norm (v1:real^3)) (dist (v0,v1)) (dist (v1,v2)) (dist (v0,v2)) (norm v2)       (norm v0)      (norm v3)      (dist (v0,v3))     (dist (v2,v3))`;
  TYPIFY `quadratic_root_plus (abc_of_quadratic p) = dist (v1,v3) pow 2` ENOUGH_TO_SHOW_TAC;
    DISCH_THEN SUBST1_TAC;
    GMATCH_SIMP_TAC POW_2_SQRT;
    BY(REWRITE_TAC[DIST_POS_LE]);
  FIRST_X_ASSUM_ST `muR` MP_TAC;
  REWRITE_TAC[muR_ALT];
  DISCH_THEN (SUBST1_TAC o GSYM);
  TYPED_ABBREV_TAC  `a = ups_x (norm (v0) * norm (v0)) (norm (v2) * norm (v2)) (dist(v0,v2) * dist(v0,v2))` ;
  TYPED_ABBREV_TAC  `b = cayleytr (norm v0 * norm v0) (norm v2 * norm v2) (norm v1 * norm v1)   (norm v3 * norm v3)  (dist (v0,v2) * dist (v0,v2))  (dist (v0,v1) * dist (v0,v1))  (dist (v0,v3) * dist (v0,v3))  (dist (v1,v2) * dist (v1,v2))  (dist (v2,v3) * dist (v2,v3)) (&0)` ;
  TYPED_ABBREV_TAC  `c = cayleyR (norm v0 * norm v0) (norm v2 * norm v2) (norm v1 * norm v1)   (norm v3 * norm v3)  (dist (v0,v2) * dist (v0,v2))  (dist (v0,v1) * dist (v0,v1))  (dist (v0,v3) * dist (v0,v3))  (dist (v1,v2) * dist (v1,v2))  (dist (v2,v3) * dist (v2,v3)) (&0)` ;
  TYPIFY ` cayleyR (norm v0 * norm v0) (norm v2 * norm v2) (norm v1 * norm v1)   (norm v3 * norm v3)  (dist (v0,v2) * dist (v0,v2))  (dist (v0,v1) * dist (v0,v1))  (dist (v0,v3) * dist (v0,v3))  (dist (v1,v2) * dist (v1,v2))  (dist (v2,v3) * dist (v2,v3))  = (\x. a * x pow 2 + b * x + c)` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[FUN_EQ_THM];
    ONCE_REWRITE_TAC[Collect_geom.LTCTBAN];
    BY(ASM_REWRITE_TAC[]);
  ASM_REWRITE_TAC[Nonlinear_lemma.abc_quadratic];
  TYPIFY `b pow 2 - &4 * a * c = &16 *  delta_y (norm v0) (norm v2) (norm v1) (dist (v2,v1)) (dist (v0,v1)) (dist (v0,v2)) * delta_y (norm v0) (norm v2) (norm v3) (dist (v2,v3)) (dist (v0,v3)) (dist (v0,v2))` (C SUBGOAL_THEN ASSUME_TAC);
    EXPAND_TAC "a";
    EXPAND_TAC "b";
    EXPAND_TAC "c";
    REWRITE_TAC[Collect_geom.DISCRIMINANT_OF_CAY];
    REWRITE_TAC[Merge_ineq.delta_delta_x;GSYM Sphere.delta_y];
    BY(MESON_TAC[DIST_SYM]);
  TYPIFY `&0 <= delta_y (norm v0) (norm v2) (norm v1) (dist (v2,v1)) (dist (v0,v1))       (dist (v0,v2))` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[REWRITE_RULE[LET_DEF;LET_END_DEF] Tame_lemmas.delta_y_pos]);
  TYPIFY `&0 <= delta_y (norm v0) (norm v2) (norm v3) (dist (v2,v3)) (dist (v0,v3))       (dist (v0,v2))` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[REWRITE_RULE[LET_DEF;LET_END_DEF] Tame_lemmas.delta_y_pos]);
  TYPIFY `&0 <= b pow 2 - &4 * a * c` (C SUBGOAL_THEN ASSUME_TAC);
    ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC REAL_LE_MUL;
    CONJ_TAC;
      BY(REAL_ARITH_TAC);
    GMATCH_SIMP_TAC REAL_LE_MUL;
    BY(ASM_REWRITE_TAC[]);
  TYPED_ABBREV_TAC  `(x:real) = dist(v1,v3) pow 2` ;
  TYPIFY `a * x pow 2 + b * x + c = &0` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `cayleyR` MP_TAC;
    REWRITE_TAC[FUN_EQ_THM];
    DISCH_THEN (C INTRO_TAC [`x`]);
    DISCH_THEN (SUBST1_TAC o GSYM);
    FIRST_X_ASSUM kill;
    BY(ASM_REWRITE_TAC[arith ` x * x = x pow 2`]);
  ASM_CASES_TAC `b pow 2 - &4 * a * c = &0`;
    MATCH_MP_TAC Tame_lemmas.quadratic_root_plus_disc0_eq;
    FIRST_X_ASSUM_ST `&0 < a` MP_TAC;
    REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC);
    BY(REAL_ARITH_TAC);
  COMMENT "pos discr. case";
  MATCH_MP_TAC Tame_lemmas.quadratic_root_plus_gt_eq;
  TYPED_ABBREV_TAC `(n:real^3) =  v0 cross v2`;
  TYPED_ABBREV_TAC  `(v1':real^3) = reflection n v1` ;
  TYPED_ABBREV_TAC  `(y:real) = dist(v1',v3) pow 2` ;
  TYPIFY `y` EXISTS_TAC;
  SUBCONJ_TAC;
    BY(ASM_MESON_TAC[]);
  DISCH_TAC;
  CONJ_TAC;
    FIRST_X_ASSUM_ST `&0 <= x` MP_TAC;
    BY(REWRITE_TAC[]);
  FIRST_X_ASSUM_ST `a * x pow 2` MP_TAC;
  DISCH_THEN (SUBST1_TAC);
  REWRITE_TAC[];
  COMMENT "cayleyR again";
  INTRO_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] Collect_geom2.CAYLEYR_5POINTS) [`(vec 0):real^3`;`v0`;`v2`;`v1'`;`v3`];
  ASM_REWRITE_TAC[];
  EXPAND_TAC "v1'";
  TYPIFY `!v. dist(v,reflection n v1) =dist(reflection n v1,v)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    BY(MESON_TAC[DIST_SYM]);
  REPEAT (GMATCH_SIMP_TAC Tame_lemmas.dist_reflection_special);
  EXPAND_TAC "n";
  REWRITE_TAC[DOT_LZERO];
  REWRITE_TAC[Collect_geom2.ORTHOGONAL_CROSS_PRODUCT];
  REWRITE_TAC[DIST_0];
  TYPIFY `dist(v1,v0) = dist(v0,v1)` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[DIST_SYM]);
  ASM_REWRITE_TAC[arith `x pow 2 = x * x`];
  DISCH_THEN SUBST1_TAC;
  REWRITE_TAC[];
  COMMENT "y < x";
  EXPAND_TAC "x";
  EXPAND_TAC "y";
  EXPAND_TAC "v1'";
  REWRITE_TAC[Tame_lemmas.dist_reflection_lemma];
  MATCH_MP_TAC (arith `&0 < ( --r )/ nn   ==> d + &4 * r / nn < d`);
  GMATCH_SIMP_TAC REAL_LT_DIV;
  TYPIFY `~(n = vec 0)` (C SUBGOAL_THEN ASSUME_TAC);
    EXPAND_TAC "n";
    FIRST_X_ASSUM_ST `&0 < a` MP_TAC;
    EXPAND_TAC "a";
    REWRITE_TAC[GSYM DIST_0;arith `x * x = x pow 2`];
    REWRITE_TAC[GSYM Collect_geom2.NORM_CROSS_PRODUCT_UPS_X];
    REWRITE_TAC[arith `&0 < &4 * x <=> &0 < x`;arith `(v0:real^3) - vec 0 = v0`];
    REWRITE_TAC[GSYM NORM_POS_LT];
    REWRITE_TAC[GSYM Trigonometry2.NOT_ZERO_EQ_POW2_LT];
    BY(MESON_TAC[NORM_POS_LE;arith `&0 <= x /\ ~(x = &0) ==> &0 < x`]);
  CONJ2_TAC;
    BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[DOT_POS_LT]);
  REWRITE_TAC[arith `&0 < -- r <=> r < &0`];
  EXPAND_TAC "n";
  TYPIFY `!v. v dot (v0 cross v2) = chi_msb [vec 0;v0;v2] v` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    REWRITE_TAC[Leaf_cell.chi_msb;Basics.EL_EXPLICIT];
    REWRITE_TAC[arith `(v:real^3) - vec 0 = v`];
    BY(MESON_TAC[DOT_SYM]);
  FIRST_X_ASSUM_ST `chi_msb` MP_TAC;
  TYPIFY `~(chi_msb [vec 0;v0;v2] v1 * chi_msb [vec 0;v0;v2] v3 = &0)` ENOUGH_TO_SHOW_TAC;
    BY(REAL_ARITH_TAC);
  REWRITE_TAC[REAL_ENTIRE;DE_MORGAN_THM];
  REWRITE_TAC[GSYM Leaf_cell.CHI_MSB_COPLANAR];
  REWRITE_TAC[Oxlzlez.coplanar_delta_y;DIST_0];
  REWRITE_TAC[arith `&0 < x <=> (&0 <= x /\ ~(x = &0))`];
  ASM_REWRITE_TAC[];
  REWRITE_TAC[GSYM DE_MORGAN_THM;GSYM REAL_ENTIRE];
  DISCH_TAC;
  REPEAT (FIRST_X_ASSUM_ST `b pow 2 - &4 * a *c` MP_TAC);
  ASM_REWRITE_TAC[];
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let IN_V_IMP_AZIM_LESS_PI_ALT = 
prove_by_refinement(
  `!V E FF v. (convex_local_fan (V,E,FF) /\ v IN V
  ==> (!w. w IN V ==> azim (vec 0) v (rho_node1 FF v) w <= pi))`,

  (* {{{ proof *)
  [
    BY(MESON_TAC[Local_lemmas.IN_V_IMP_AZIM_LESS_PI])
  ]);;
  (* }}} *)

let vv_enclosed4 = 
prove_by_refinement(
  `!vv i.  
    (let V = IMAGE vv (:num) in
     let E = IMAGE (\i. {vv i, vv (SUC i)}) (:num) in
     let f = IMAGE (\i. vv i,vv (SUC i)) (:num) in
     let y1 = norm (vv (i+1)) in
   let y2 = norm (vv (i+2)) in
   let y3 = norm (vv (i)) in
   let y4 = dist(vv (i), vv (i+2)) in
   let y5 = dist(vv (i), vv (i+1)) in
   let y6 = dist(vv (i+1), vv (i+2)) in
   let y7 = norm (vv (i+3)) in
   let y8 = dist(vv (i),vv (i+3)) in
   let y9 = dist(vv (i+2),vv (i+3)) in
       (periodic vv 4 /\
	  dist (vv i,vv ( i+1)) < &4 /\
	  dist (vv i,vv (i + 3)) < &4 /\
	  (!i j. ~(vv i = vv j) ==> &2 <= dist (vv i,vv j)) /\
          V SUBSET ball_annulus /\
         dist (vv i,vv (i+2)) < &4 /\
         convex_local_fan (V,E,f) /\
         (!i j. i < 4 /\ j < 4 /\ vv i = vv j ==> i = j)
         ==> dist (vv (i+1), vv(i+3)) = enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_THM];
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] enclosed4_lemma);
  TYPIFY `!i. vv i IN ball_annulus` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `ball_annulus` MP_TAC;
    BY(SET_TAC[IN_UNIV]);
  TYPIFY `!i. vv i IN IMAGE vv (:num)` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[IN_IMAGE;IN_UNIV];
    BY(MESON_TAC[]);
  SUBCONJ_TAC;
    REWRITE_TAC[GSYM DIST_0];
    REWRITE_TAC[GSYM Collect_geom2.NOT_COL_EQ_UPS_X_POS;arith `x * x = x pow 2`];
    MATCH_MP_TAC NONPARALLEL_BALL_ANNULUS40_ALT;
    ASM_REWRITE_TAC[];
    FIRST_X_ASSUM MATCH_MP_TAC;
    GMATCH_SIMP_TAC periodic_vv_inj;
    TYPIFY `4` EXISTS_TAC;
    ASM_REWRITE_TAC[arith `~(4 = 0)`];
    BY(ASM_MESON_TAC[ Oxlzlez.MOD_INJ1_ALT;arith `~(4 = 0) /\ ~(2 = 0) /\ (2 < 4)`]);
  DISCH_TAC;
  COMMENT "case delta=0";
  TYPIFY `chi_msb [vec 0; vv i; vv (i + 2)] (vv (i + 1)) = &0` ASM_CASES_TAC;
    BY(ASM_REWRITE_TAC[arith `&0 * x <= &0`]);
  TYPIFY `chi_msb [vec 0; vv i; vv (i + 2)] (vv (i + 3)) = &0` ASM_CASES_TAC;
    BY(ASM_REWRITE_TAC[arith ` x * &0 <= &0`]);
  TYPIFY `~coplanar {vec 0, vv i, vv (i + 2), vv (i + 1)} /\ ~coplanar {vec 0, vv i, vv (i + 2), vv (i + 3)}` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_REWRITE_TAC[ Leaf_cell.CHI_MSB_COPLANAR]);
  TYPIFY `~collinear {vec 0, vv  i, vv (i+1)} /\ ~collinear {vec 0,vv i, vv (i+3)} /\ ~collinear {vec 0,vv i, vv(i+2)}` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Planarity.notcoplanar_imp_notcollinear_fan]);
  COMMENT "angle 0";
  INTRO_TAC IN_V_IMP_AZIM_LESS_PI_ALT [ `IMAGE vv (:num)`;`IMAGE (\i. {vv i, vv (SUC i)}) (:num)`;`IMAGE (\i. vv i,vv (SUC i)) (:num)`;`vv i`];
  ASM_REWRITE_TAC[];
  DISCH_THEN (C INTRO_TAC [`vv (i +3)`]);
  ASM_REWRITE_TAC[];
  GMATCH_SIMP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] vv_rho_node1);
  CONJ_TAC;
    TYPIFY `4` EXISTS_TAC;
    BY(ASM_REWRITE_TAC[arith `3 <= 4`]);
  DISCH_TAC;
  COMMENT "v2 in wedge";
  TYPIFY `vv (i +2) IN wedge_ge (vec 0) (vv i) (vv ( i+1)) (vv (i + 3))` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[Local_lemmas.WEDGE_GE_AZIM_LE];
    REWRITE_TAC[arith `i + 1 = SUC i /\ i + 3 = i + 4 -1`];
    INTRO_TAC (REWRITE_RULE[LET_THM] vv_azim_le) [`vv`;`4`;`i`;`i + 2`];
    ANTS_TAC;
      ASM_REWRITE_TAC[arith `3 <= 4`];
      REWRITE_TAC[arith `SUC i = i + 1 /\ i + 4 -1 = i + 3`];
      ASM_REWRITE_TAC[];
      BY(MESON_TAC[ Oxlzlez.MOD_INJ1_ALT;arith `~(4 = 0) /\ ~(2 = 0) /\ (2 < 4)`]);
    BY(REWRITE_TAC[]);
  COMMENT " <  pi ";
  TYPIFY `azim (vec 0) (vv i) (vv (SUC i)) (vv (i + 3)) < pi` ASM_CASES_TAC;
    FIRST_X_ASSUM_ST `wedge_ge` MP_TAC;
    GMATCH_SIMP_TAC Local_lemmas.WEDGE_GE_EQ_AFF_GE;
    ASM_REWRITE_TAC[arith `i + 1 = SUC i`];
    GMATCH_SIMP_TAC Marchal_cells_2_new.AFF_GE_2_2;
    CONJ_TAC;
      TYPIFY `DISJOINT {vec 0, vv i} {vv (SUC i)} /\ DISJOINT {vec 0, vv i} {vv (i + 3)}` ENOUGH_TO_SHOW_TAC;
        REWRITE_TAC[DISJOINT];
        BY(SET_TAC[]);
      BY(ASM_MESON_TAC[Fan.th3a;arith `SUC i = i + 1`]);
    REWRITE_TAC[IN_ELIM_THM];
    REPEAT WEAKER_STRIP_TAC;
    ASM_REWRITE_TAC[];
    ONCE_REWRITE_TAC[Leaf_cell.chi_msb_swap_23];
    GMATCH_SIMP_TAC Leaf_cell.chi_msb_additive_d;
    TYPIFY `t3 % vv (SUC i) + t4 % vv (i + 3) =  t4 % vv (i + 3) + t3 % vv (SUC i)` (C SUBGOAL_THEN SUBST1_TAC);
      BY(VECTOR_ARITH_TAC);
    GMATCH_SIMP_TAC Leaf_cell.chi_msb_additive_d;
    CONJ_TAC;
      BY(FIRST_X_ASSUM_ST `&1` MP_TAC THEN REAL_ARITH_TAC);
    CONJ_TAC;
      BY(FIRST_X_ASSUM_ST `&1` MP_TAC THEN REAL_ARITH_TAC);
    TYPIFY `chi_msb [vec 0; vv i; vv (SUC i)] (vv (i + 3)) = -- chi_msb [vec 0; vv i; vv (i + 3)] (vv (SUC i))` (C SUBGOAL_THEN SUBST1_TAC);
      BY(MESON_TAC[Leaf_cell.chi_msb_swap_23]);
    REWRITE_TAC[arith `-- (t4 * -- c) * -- (t3 * c) <= &0 <=> &0 <= t4 * t3 * c pow 2`];
    GMATCH_SIMP_TAC REAL_LE_MUL;
    ASM_REWRITE_TAC[];
    GMATCH_SIMP_TAC REAL_LE_MUL;
    ASM_REWRITE_TAC[];
    BY(REWRITE_TAC[ REAL_LE_POW_2]);
  COMMENT "azim = pi";
  TYPIFY `azim (vec 0) (vv i) (vv (SUC i)) (vv (i + 3)) = pi` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPLICATE_TAC 2 (FIRST_X_ASSUM_ST `azim` MP_TAC) THEN REAL_ARITH_TAC);
  FIRST_X_ASSUM MP_TAC;
  GMATCH_SIMP_TAC Ldurdpn.LDURDPN;
  ASM_REWRITE_TAC[arith `SUC i = i + 1`];
  REWRITE_TAC[EXTENSION;NOT_IN_EMPTY;NOT_FORALL_THM];
  REWRITE_TAC[IN_INTER;Geomdetail.CONV0_SET2;IN_ELIM_THM];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC Leaf_cell.AFFINE_IMP_CHI_MSB_0 [`[vec 0;vv i; vv (i+2)]`;`x`];
  REWRITE_TAC[LENGTH;arith `SUC(SUC(SUC 0)) = 3`];
  REWRITE_TAC[Bump.set_of_list3_explicit];
  ANTS_TAC;
    TYPIFY `aff {vec 0, vv i} SUBSET affine hull {vec 0, vv i, vv (i + 2)}` ENOUGH_TO_SHOW_TAC;
      BY(FIRST_X_ASSUM_ST `aff` MP_TAC THEN SET_TAC[]);
    MATCH_MP_TAC Collect_geom.AFFINE_CONTAIN_LINE;
    REWRITE_TAC[AFFINE_AFFINE_HULL];
    MATCH_MP_TAC SUBSET_TRANS;
    TYPIFY `{vec 0, vv i, vv (i + 2)}` EXISTS_TAC;
    CONJ_TAC;
      BY(SET_TAC[]);
    BY(REWRITE_TAC[HULL_SUBSET]);
  ASM_REWRITE_TAC[];
  GMATCH_SIMP_TAC Leaf_cell.CHI_MSB_ADDITIVE;
  ASM_REWRITE_TAC[];
  DISCH_TAC;
  TYPIFY `!c c'. ((c':real) = (-- a/b) * c)  ==> c * c' <= &0` (C SUBGOAL_THEN MATCH_MP_TAC);
    REPEAT WEAKER_STRIP_TAC;
    ASM_REWRITE_TAC[];
    REWRITE_TAC[ (arith `c * -- a /b * c <= &0 <=> &0 <= (a / b) * c pow 2`) ];
    GMATCH_SIMP_TAC REAL_LE_MUL;
    GMATCH_SIMP_TAC REAL_LE_RDIV_EQ;
    ASM_REWRITE_TAC[];
    REWRITE_TAC[ REAL_LE_POW_2];
    BY(REPEAT (FIRST_X_ASSUM_ST `&0 < x` MP_TAC) THEN REAL_ARITH_TAC);
  MATCH_MP_TAC REAL_EQ_LCANCEL_IMP;
  TYPIFY `b` EXISTS_TAC;
  FIRST_X_ASSUM MP_TAC;
  REWRITE_TAC[arith `a + b = &0 <=> b = --a`];
  DISCH_THEN SUBST1_TAC;
  CONJ_TAC;
    BY(FIRST_X_ASSUM_ST `&0 < b` MP_TAC THEN REAL_ARITH_TAC);
  Calc_derivative.CALC_ID_TAC;
  BY(FIRST_X_ASSUM_ST `&0 < b` MP_TAC THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let enclosed_sym = 
prove_by_refinement(
  `!y1 y2 y3 y4 y5 y6 y7 y8 y0. enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 =
    enclosed y1 y6 y5 y4 y3 y2 y7 y9 y8`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Enclosed.enclosed];
  REPLICATE_TAC 3 AP_TERM_TAC;
  REWRITE_TAC[muR_ALT];
  REWRITE_TAC[FUN_EQ_THM];
  GEN_TAC;
  REWRITE_TAC[Collect_geom2.cayleyR];
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let enclosed_sym2 = 
prove_by_refinement(
  `!y1 y2 y3 y4 y5 y6 y7 y8 y0. enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 =
    enclosed y7 y8 y9 y4 y2 y3 y1 y5 y6`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Enclosed.enclosed];
  REPLICATE_TAC 3 AP_TERM_TAC;
  REWRITE_TAC[muR_ALT];
  REWRITE_TAC[FUN_EQ_THM];
  GEN_TAC;
  REWRITE_TAC[Collect_geom2.cayleyR];
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let convex_local_fan_azim_le_pi = 
prove_by_refinement(
  `!vv k i. (let V = IMAGE vv (:num) in
   let E = IMAGE (\i. {vv i, vv (SUC i)}) (:num) in
   let f = IMAGE (\i. vv i,vv (SUC i)) (:num) in
    (periodic vv k /\ 
   3 <= k /\ (!i j. i < k /\ j < k /\ vv i = vv j ==> i = j) /\
       convex_local_fan (V,E,f) /\
   V SUBSET ball_annulus 
    ==>
   (azim (vec 0) (vv i) (vv (SUC i)) (vv (i + k - 1)) <= pi)))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_THM];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC Localization.convex_local_fan [`IMAGE (\i. vv i,vv (SUC i)) (:num)`;`IMAGE vv (:num)`;`IMAGE (\i. {vv i, vv (SUC i)}) (:num)`;];
  ASM_REWRITE_TAC[IN_IMAGE;IN_UNIV];
  REPEAT WEAKER_STRIP_TAC;
  FIRST_X_ASSUM (C INTRO_TAC [ `vv i, vv (SUC i)`]);
  ANTS_TAC;
    BY(MESON_TAC[]);
  GMATCH_SIMP_TAC Local_lemmas.LOFA_DETERMINE_AZIM_IN_FA;
  CONJ_TAC;
    GEXISTL_TAC [`IMAGE vv (:num)`;`IMAGE (\i. vv i,vv (SUC i)) (:num)`];
    ASM_REWRITE_TAC[];
    BY(MESON_TAC[IN_IMAGE;IN_UNIV]);
  GMATCH_SIMP_TAC EE_vv;
  TYPIFY `k` EXISTS_TAC;
  CONJ_TAC;
    BY(ASM_REWRITE_TAC[]);
  REWRITE_TAC[Local_lemmas.AZIM_CYCLE_TWO_POINT_SET];
  BY(MESON_TAC[])
  ]);;
  (* }}} *)

let vv_quad_split012 = 
prove_by_refinement(
  `!vv. 
    (let V = IMAGE vv (:num) in
   let E = IMAGE (\i. {vv i, vv (SUC i)}) (:num) in
   let f = IMAGE (\i. vv i,vv (SUC i)) (:num) in
    (periodic vv 4 /\
   V SUBSET ball_annulus /\
   dist(vv 0, vv 1) < &4 /\
   dist(vv 0, vv 2) < &4 /\
   dist(vv 0, vv 3) < &4 /\
   dist(vv 1,vv 2) < &4 /\
   dist(vv 2,vv 3) < &4 /\
   (!i j. ~(vv i = vv j) ==> &2 <= dist(vv i,vv j) ) /\
   convex_local_fan (V,E,f) /\
        (!i j. i < 4 /\ j < 4 /\ vv i = vv j ==> (i = j))
     ==> (rho_fun (norm (vv 0)) * azim (vec 0) (vv 0) (vv 1) (vv 3) +
          rho_fun (norm (vv 1)) * azim (vec 0) (vv 1) (vv 2) (vv 0) +
          rho_fun (norm (vv 2)) * azim (vec 0) (vv 2) (vv 3) (vv 1) +
          rho_fun (norm (vv 3)) * azim (vec 0) (vv 3) (vv 0) (vv 2)) -
         (pi + sol0) * &2 =
         tau3 (vv 0) (vv 1) (vv 2) + tau3 (vv 2) (vv 3) (vv 0)))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_THM];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY ` azim (vec 0) (vv 0) (vv (SUC 0)) (vv (0 + 4 - 1)) <= pi /\ azim (vec 0) (vv 1) (vv (SUC 1)) (vv (1 + 4 - 1)) <= pi /\ azim (vec 0) (vv 2) (vv (SUC 2)) (vv (2 + 4 - 1)) <= pi /\ azim (vec 0) (vv 3) (vv (SUC 3)) (vv (3 + 4 -1)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT CONJ_TAC THEN MATCH_MP_TAC (REWRITE_RULE[LET_THM] convex_local_fan_azim_le_pi) THEN ASM_REWRITE_TAC[arith `3 <= 4`]);
  REWRITE_TAC[Appendix.tau3];
  TYPIFY `vv 6 = vv 2 /\ vv 5 = vv 1 /\ vv 4 = vv 0` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (CONJ_TAC) THEN GMATCH_SIMP_TAC periodic_vv_inj THEN TYPIFY `4` EXISTS_TAC THEN ASM_REWRITE_TAC[MOD_4_EXPLICIT;arith `~(4 = 0)`]);
  TYPIFY `azim (vec 0) ((vv 0)) ((vv (SUC 0))) ((vv (0 + 4 - 1))) = azim (vec 0) ((vv 0)) ((vv (SUC 0))) ((vv 2)) + azim(vec 0) ((vv 0)) ((vv 2)) ((vv (0 + 4 - 1))) /\ (azim (vec 0) ((vv 2)) ((vv (SUC 2))) ((vv (2 + 4 - 1))) = azim (vec 0) ((vv 2)) ((vv (SUC 2))) ((vv 0)) + azim(vec 0) ((vv 2)) ((vv 0)) ((vv (2 + 4 -1))))` (C SUBGOAL_THEN ASSUME_TAC);
    CONJ_TAC THEN MATCH_MP_TAC (REWRITE_RULE[LET_THM] vv_split_azim) THEN ASM_REWRITE_TAC[arith `3 <= 4 /\ ~( 0 = 2)`;MOD_4_EXPLICIT;arith `SUC  0 = 1 /\ 0 + 4 - 1 = 3`;arith `SUC 2 = 3 /\ 2 + 4 - 1 = 5`];
    BY(ASM_MESON_TAC[DIST_SYM]);
  RULE_ASSUM_TAC (REWRITE_RULE[arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4 /\ 0+4-1 = 3 /\ 1+4-1=4 /\ 2+4-1=5 /\ 3+4-1=6`]);
  FIRST_X_ASSUM_ST `6` (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
  ASM_REWRITE_TAC[];
  TYPIFY `azim (vec 0) (vv 0) (vv 1) (vv 2) <= pi /\ azim (vec 0) (vv 0) (vv 2) (vv 3) <= pi /\ azim (vec 0) (vv 2 ) (vv 3) (vv 0) <= pi /\ azim (vec 0) (vv 2) (vv 0) (vv 1) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Counting_spheres.AZIM_NN;arith `&0 <= x /\ &0 <= y /\ x + y <= pi ==> (x <= pi /\ y <= pi)`]);
  COMMENT "azim to dih";
  TYPIFY `~collinear {vec 0,vv 0,vv 1} /\ ~collinear {vec 0,vv 0, vv 2} /\ ~collinear {vec 0,vv 0,vv 3} /\ ~collinear {vec 0,vv 1,vv 2} /\ ~collinear {vec 0,vv 2, vv 3}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT (GMATCH_SIMP_TAC NONPARALLEL_BALL_ANNULUS40_ALT);
    RULE_ASSUM_TAC (REWRITE_RULE[IMAGE_SUBSET_IN;IN_UNIV]);
    ASM_REWRITE_TAC[];
    BY(REPEAT CONJ_TAC THEN (FIRST_X_ASSUM_ST `dist` MATCH_MP_TAC) THEN GMATCH_SIMP_TAC periodic_vv_inj THEN (TYPIFY `4` EXISTS_TAC) THEN ASM_REWRITE_TAC[arith `~(4 = 0)`;MOD_4_EXPLICIT;arith `~(2 = 3) /\ ~(1 = 2) /\ ~(0=3) /\ ~(0=1) /\ ~(0=2)`]);
  COMMENT "azim to dih";
  TYPIFY `azim (vec 0) (vv 0) (vv 1) (vv 2) = dihV (vec 0) (vv 0) (vv 1) (vv 2) /\ azim (vec 0) (vv 0) (vv 2) (vv 3) = dihV (vec 0) (vv 0) (vv 2) (vv 3) /\ azim (vec 0) (vv 1) (vv 2) (vv 0) = dihV (vec 0) (vv 1) (vv 2) (vv 0) /\ azim (vec 0) (vv 2) (vv 3) (vv 0) = dihV (vec 0) (vv 2) (vv 3) (vv 0) /\ azim (vec 0) (vv 2) (vv 0) (vv 1) = dihV (vec 0) (vv 2) (vv 0) (vv 1) /\ azim (vec 0) (vv 3) (vv 0) (vv 2) = dihV (vec 0) (vv 3) (vv 0) (vv 2)` ENOUGH_TO_SHOW_TAC;
    DISCH_THEN (unlist REWRITE_TAC);
    REWRITE_TAC[Appendix.rho_rho_fun];
    BY(REAL_ARITH_TAC);
  BY(REPEAT CONJ_TAC THEN MATCH_MP_TAC (REWRITE_RULE[TAUT `(a ==> b ==> c) <=> (a /\ b ==> c)`] Local_lemmas.AZIM_LE_PI_EQ_DIHV) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SET_RULE `{a,b,c}={a,c,b}`] THEN ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let vv_quad_split123 = 
prove_by_refinement(
  `!vv. 
    (let V = IMAGE vv (:num) in
   let E = IMAGE (\i. {vv i, vv (SUC i)}) (:num) in
   let f = IMAGE (\i. vv i,vv (SUC i)) (:num) in
    (periodic vv 4 /\
   V SUBSET ball_annulus /\
   dist(vv 0, vv 1) < &4 /\
   dist(vv 0, vv 3) < &4 /\
   dist(vv 1,vv 2) < &4 /\
   dist(vv 1,vv 3) < &4 /\
   dist(vv 2,vv 3) < &4 /\
   (!i j. ~(vv i = vv j) ==> &2 <= dist(vv i,vv j) ) /\
   convex_local_fan (V,E,f) /\
        (!i j. i < 4 /\ j < 4 /\ vv i = vv j ==> (i = j))
     ==> (rho_fun (norm (vv 0)) * azim (vec 0) (vv 0) (vv 1) (vv 3) +
          rho_fun (norm (vv 1)) * azim (vec 0) (vv 1) (vv 2) (vv 0) +
          rho_fun (norm (vv 2)) * azim (vec 0) (vv 2) (vv 3) (vv 1) +
          rho_fun (norm (vv 3)) * azim (vec 0) (vv 3) (vv 0) (vv 2)) -
         (pi + sol0) * &2 =
         tau3 (vv 1) (vv 2) (vv 3) + tau3 (vv 3) (vv 0) (vv 1)))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_THM];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY ` azim (vec 0) (vv 0) (vv (SUC 0)) (vv (0 + 4 - 1)) <= pi /\ azim (vec 0) (vv 1) (vv (SUC 1)) (vv (1 + 4 - 1)) <= pi /\ azim (vec 0) (vv 2) (vv (SUC 2)) (vv (2 + 4 - 1)) <= pi /\ azim (vec 0) (vv 3) (vv (SUC 3)) (vv (3 + 4 -1)) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT CONJ_TAC THEN MATCH_MP_TAC (REWRITE_RULE[LET_THM] convex_local_fan_azim_le_pi) THEN ASM_REWRITE_TAC[arith `3 <= 4`]);
  REWRITE_TAC[Appendix.tau3];
  TYPIFY `vv 6 = vv 2 /\ vv 5 = vv 1 /\ vv 4 = vv 0` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (CONJ_TAC) THEN GMATCH_SIMP_TAC periodic_vv_inj THEN TYPIFY `4` EXISTS_TAC THEN ASM_REWRITE_TAC[MOD_4_EXPLICIT;arith `~(4 = 0)`]);
  TYPIFY `azim (vec 0) ((vv 1)) ((vv (SUC 1))) ((vv (1 + 4 - 1))) = azim (vec 0) ((vv 1)) ((vv (SUC 1))) ((vv 3)) + azim(vec 0) ((vv 1)) ((vv 3)) ((vv (1 + 4 - 1))) /\ (azim (vec 0) ((vv 3)) ((vv (SUC 3))) ((vv (3 + 4 - 1))) = azim (vec 0) ((vv 3)) ((vv (SUC 3))) ((vv 1)) + azim(vec 0) ((vv 3)) ((vv 1)) ((vv (3 + 4 -1))))` (C SUBGOAL_THEN ASSUME_TAC);
    CONJ_TAC THEN MATCH_MP_TAC (REWRITE_RULE[LET_THM] vv_split_azim) THEN ASM_REWRITE_TAC[arith `3 <= 4 /\ ~( 1 = 3)`;MOD_4_EXPLICIT;arith `SUC  1 = 2 /\ 1 + 4 - 1 = 4`;arith `SUC 3 = 4 /\ 3 + 4 - 1 = 6`];
      BY(ASM_MESON_TAC[DIST_SYM]);
    BY(ASM_MESON_TAC[DIST_SYM]);
  RULE_ASSUM_TAC (REWRITE_RULE[arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4 /\ 0+4-1 = 3 /\ 1+4-1=4 /\ 2+4-1=5 /\ 3+4-1=6`]);
  FIRST_X_ASSUM_ST `vv 6 = vv 2` (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
  ASM_REWRITE_TAC[];
  TYPIFY `azim (vec 0) (vv 1) (vv 2) (vv 3) <= pi /\ azim (vec 0) (vv 1) (vv 3) (vv 0) <= pi /\ azim (vec 0) (vv 3 ) (vv 0) (vv 1) <= pi /\ azim (vec 0) (vv 3) (vv 1) (vv 2) <= pi` (C SUBGOAL_THEN ASSUME_TAC);
    BY(ASM_MESON_TAC[Counting_spheres.AZIM_NN;arith `&0 <= x /\ &0 <= y /\ x + y <= pi ==> (x <= pi /\ y <= pi)`]);
  COMMENT "collinearity";
  TYPIFY `~collinear {vec 0,vv 0,vv 1} /\ ~collinear {vec 0,vv 0, vv 3} /\ ~collinear {vec 0,vv 1,vv 3} /\ ~collinear {vec 0,vv 1,vv 2} /\ ~collinear {vec 0,vv 2, vv 3}` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT (GMATCH_SIMP_TAC NONPARALLEL_BALL_ANNULUS40_ALT);
    RULE_ASSUM_TAC (REWRITE_RULE[IMAGE_SUBSET_IN;IN_UNIV]);
    ASM_REWRITE_TAC[];
    BY(REPEAT CONJ_TAC THEN (FIRST_X_ASSUM_ST `dist` MATCH_MP_TAC) THEN GMATCH_SIMP_TAC periodic_vv_inj THEN (TYPIFY `4` EXISTS_TAC) THEN ASM_REWRITE_TAC[arith `~(4 = 0)`;MOD_4_EXPLICIT;arith `~(2 = 3) /\ ~(1 = 2) /\ ~(0=3) /\ ~(0=1) /\ ~(1=3)`]);
  COMMENT "azim to dih";
  TYPIFY `azim (vec 0) (vv 0) (vv 1) (vv 3) = dihV (vec 0) (vv 0) (vv 1) (vv 3) /\ azim (vec 0) (vv 1) (vv 2) (vv 3) = dihV (vec 0) (vv 1) (vv 2) (vv 3) /\ azim (vec 0) (vv 1) (vv 3) (vv 0) = dihV (vec 0) (vv 1) (vv 3) (vv 0) /\ azim (vec 0) (vv 2) (vv 3) (vv 1) = dihV (vec 0) (vv 2) (vv 3) (vv 1) /\ azim (vec 0) (vv 3) (vv 0) (vv 1) = dihV (vec 0) (vv 3) (vv 0) (vv 1) /\ azim (vec 0) (vv 3) (vv 1) (vv 2) = dihV (vec 0) (vv 3) (vv 1) (vv 2)` ENOUGH_TO_SHOW_TAC;
    DISCH_THEN (unlist REWRITE_TAC);
    REWRITE_TAC[Appendix.rho_rho_fun];
    BY(REAL_ARITH_TAC);
  BY(REPEAT CONJ_TAC THEN MATCH_MP_TAC (REWRITE_RULE[TAUT `(a ==> b ==> c) <=> (a /\ b ==> c)`] Local_lemmas.AZIM_LE_PI_EQ_DIHV) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SET_RULE `{a,b,c}={a,c,b}`] THEN ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let vv_quad_split_short = 
prove_by_refinement(
  `!vv. (let V = IMAGE vv (:num) in
          let E = IMAGE (\i. {vv i, vv (SUC i)}) (:num) in
          let f = IMAGE (\i. vv i,vv (SUC i)) (:num) in
          (periodic vv 4 /\
          V SUBSET ball_annulus /\
          dist (vv 0,vv 1) < &4 /\
          dist (vv 0,vv 3) < &4 /\
          dist (vv 0,vv 2) < &4 /\
          dist (vv 1,vv 2) < &4 /\
          dist (vv 1,vv 3) < &4 /\
          dist (vv 2,vv 3) < &4 /\
          (!i j. ~(vv i = vv j) ==> &2 <= dist (vv i,vv j)) /\
          convex_local_fan (V,E,f) /\
          (!i j. i < 4 /\ j < 4 /\ vv i = vv j ==> i = j)
          ==> (?i. i < 2 /\ (dist(vv i,vv (i+2)) <= dist(vv (i+1),vv(i+3))) /\
	       (rho_fun (norm (vv 0)) * azim (vec 0) (vv 0) (vv 1) (vv 3) +
               rho_fun (norm (vv 1)) * azim (vec 0) (vv 1) (vv 2) (vv 0) +
               rho_fun (norm (vv 2)) * azim (vec 0) (vv 2) (vv 3) (vv 1) +
               rho_fun (norm (vv 3)) * azim (vec 0) (vv 3) (vv 0) (vv 2)) -
              (pi + sol0) * &2 =
              tau3 (vv i) (vv (i+1)) (vv (i+2)) + tau3 (vv (i+2)) (vv (i+3)) (vv i))))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[LET_THM];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `dist(vv 0,vv 2) <= dist(vv 1,vv 3)` ASM_CASES_TAC;
    TYPIFY `0` EXISTS_TAC;
    ASM_REWRITE_TAC[arith `0+x = x /\ 0 < 2`];
    MATCH_MP_TAC (REWRITE_RULE[LET_THM] vv_quad_split012);
    BY(ASM_REWRITE_TAC[]);
  TYPIFY `1` EXISTS_TAC;
  REWRITE_TAC[arith `1+2 = 3 /\ 1+1 = 2 /\ 1 + 3 = 4 /\ 1 < 2`];
  TYPIFY `vv 4 = vv 0` (C SUBGOAL_THEN SUBST1_TAC);
    BY( GMATCH_SIMP_TAC periodic_vv_inj THEN TYPIFY `4` EXISTS_TAC THEN ASM_REWRITE_TAC[MOD_4_EXPLICIT;arith `~(4 = 0)`]);
  CONJ_TAC;
    BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[DIST_SYM;arith `~(x <= y) ==> y <= x`]);
  MATCH_MP_TAC (REWRITE_RULE[LET_THM] vv_quad_split123);
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

let SUC_EXPLICIT = 
prove_by_refinement(
  `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4 /\ SUC 4 = 5 /\ SUC 5 = 6 /\ SUC 6 = 7 /\ SUC 7 = 8 /\ SUC 8 = 9`,

  (* {{{ proof *)
  [
    ARITH_TAC
  ]);;
  (* }}} *)

let cs_adj4_EXPLICIT = 
prove_by_refinement(
  `!a b. 
    cs_adj 4 a b 0 0 = &0 /\
    cs_adj 4 a b 0 1 = a /\
    cs_adj 4 a b 0 2 = b /\
    cs_adj 4 a b 0 3 = a /\
    cs_adj 4 a b 1 0 = a /\
    cs_adj 4 a b 1 1 = &0 /\
    cs_adj 4 a b 1 2 = a /\
    cs_adj 4 a b 1 3 = b /\
    cs_adj 4 a b 2 0 = b /\
    cs_adj 4 a b 2 1 = a /\
    cs_adj 4 a b 2 2 = &0 /\
    cs_adj 4 a b 2 3 = a /\
    cs_adj 4 a b 3 0 = a /\
    cs_adj 4 a b 3 1 = b /\
    cs_adj 4 a b 3 2 = a /\
    cs_adj 4 a b 3 3 = &0 
`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.cs_adj;MOD_4_EXPLICIT;SUC_EXPLICIT];
  BY(ARITH_TAC)
  ]);;
  (* }}} *)

let delta_4680581274 = 
prove_by_refinement(
  `!y1 y4.  cstab <= y1 /\ &4 <= y4 ==>
    delta_y y1 (&2) (&2) y4 (&2) cstab < &0`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Sphere.delta_y;Sphere.delta_x;arith `&2 * &2 = &4`];
  REPEAT WEAKER_STRIP_TAC;
  TYPED_ABBREV_TAC  `y = y1 * y1` ;
  TYPED_ABBREV_TAC  `z = y4 * y4` ;
  TYPED_ABBREV_TAC  `c = cstab * cstab` ;
  TYPIFY `y * z * (--y + &4 + &4 - z + &4 + c) + &4 * &4 * (y - &4 + &4 + z - &4 + c) + &4 * c * (y + &4 - &4 + z + &4 - c) - &4 * &4 * z - y * &4 * &4 - y * &4 * c - z * &4 * c = (-- &64 + &32 * c  - &4 * c pow 2) - y * z * (-- &12 - c  + y + z)` (C SUBGOAL_THEN SUBST1_TAC);
    BY(REAL_ARITH_TAC);
  REWRITE_TAC[arith `x - y < &0 <=> x < y`];
  MATCH_MP_TAC REAL_LT_TRANS;
  TYPIFY ` &0` EXISTS_TAC;
  CONJ_TAC;
    EXPAND_TAC "c";
    REWRITE_TAC[Appendix.cstab];
    BY(REAL_ARITH_TAC);
  EXPAND_TAC "y";
  EXPAND_TAC "z";
  GMATCH_SIMP_TAC REAL_LT_MUL_EQ;
  REPEAT (GMATCH_SIMP_TAC REAL_LT_MUL_EQ);
  TYPIFY `&0 < y1 /\ &0 < y4` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
  TYPIFY `&9 <= y1 * y1` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `&9 = &3 * &3`];
    GMATCH_SIMP_TAC Misc_defs_and_lemmas.ABS_SQUARE_LE;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
  TYPIFY `&16 <= y4 * y4` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `&16 = &4 * &4`];
    GMATCH_SIMP_TAC Misc_defs_and_lemmas.ABS_SQUARE_LE;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
  EXPAND_TAC "c";
  REWRITE_TAC[Appendix.cstab];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let delta_7697147739 = 
prove_by_refinement(
  `!y1 y4.  cstab <= y1 /\ &4 <= y4 ==>
    delta_y y1 (&2) (&2) y4 (&2) sqrt8 < &0`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Sphere.delta_y;Sphere.delta_x;arith `&2 * &2 = &4`;Nonlinear_lemma.sqrt8_2;arith `#8.0 = &8`];
  REPEAT WEAKER_STRIP_TAC;
  TYPED_ABBREV_TAC  `y = y1 * y1` ;
  TYPED_ABBREV_TAC  `z = y4 * y4` ;
  TYPIFY `y * z * (--y + &4 + &4 - z + &4 + &8) + &4 * &4 * (y - &4 + &4 + z - &4 + &8) + &4 * &8 * (y + &4 - &4 + z + &4 - &8) - &4 * &4 * z - y * &4 * &4 - y * &4 * &8 - z * &4 * &8 = -- &64 - y * z * (-- &20 + y + z)`  (C SUBGOAL_THEN SUBST1_TAC);
    BY(REAL_ARITH_TAC);
  REWRITE_TAC[arith `x - y < &0 <=> x < y`];
  MATCH_MP_TAC REAL_LT_TRANS;
  TYPIFY ` &0` EXISTS_TAC;
  CONJ_TAC;
    BY(REAL_ARITH_TAC);
  EXPAND_TAC "y";
  EXPAND_TAC "z";
  GMATCH_SIMP_TAC REAL_LT_MUL_EQ;
  REPEAT (GMATCH_SIMP_TAC REAL_LT_MUL_EQ);
  TYPIFY `&0 < y1 /\ &0 < y4` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
  TYPIFY `&9 <= y1 * y1` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `&9 = &3 * &3`];
    GMATCH_SIMP_TAC Misc_defs_and_lemmas.ABS_SQUARE_LE;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
  TYPIFY `&16 <= y4 * y4` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `&16 = &4 * &4`];
    GMATCH_SIMP_TAC Misc_defs_and_lemmas.ABS_SQUARE_LE;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let tau3_sym = 
prove_by_refinement(
  `!v0 v1 v2. tau3 v0 v1 v2 = tau3 v0 v2 v1 /\ tau3 v0 v1 v2 = tau3 v1 v0 v2`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.tau3];
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[DIHV_SYM];
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let INSERT_SUBSET = 
prove_by_refinement(
  `!(a:A) A S. a INSERT A SUBSET S <=> (a IN S) /\ A SUBSET S`,

  (* {{{ proof *)
  [
  BY(SET_TAC[])
  ]);;
  (* }}} *)

(* INEQUALITY MATERIAL STARTS HERE *)

let OWZLKVY0 = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==>
    (!y1 y2 y3 y4 y5 y6.
       &2 <= y1 /\ y1 <= &2 * h0 /\
    &2 <= y1  /\ y2 <= &2 * h0 /\
    &2 <= y2 /\ y2 <= &2 * h0 /\
    &2 <= y3 /\ y3 <= &2 * h0 /\
    cstab <= y4 /\ y4 <= #3.915 /\
    y5 = &2 /\
    y6 = &2 /\ &200 <= delta_y y1 y2 y3 y4 y5 y6 ==>
	&0 <= taum y1 y2 y3 y4 y5 y6)`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.cstab];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "6762190381") [];
  DISCH_THEN (C INTRO_TAC [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`]);
  REWRITE_TAC[Sphere.ineq];
  REPEAT (FIRST_X_ASSUM MP_TAC);
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

(* was EAR_ANGLE_ACUTE *)

let EAR_DELTA_X4 = 
prove_by_refinement(
  `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 /\
    cstab <= y4 /\ y4 <= #3.915 /\
    y5 = &2 /\
    y6 = &2 /\
	delta_y y1 y2 y3 y4 y5 y6 <= &200 ==>
	(delta_x4 (y1 * y1) (y2 * y2) (y3 * y3) (y4 * y4) (y5 * y5) (y6 * y6) <
      &0 /\
  &0 < delta_x4 (y2 * y2) (y3 * y3) (y1 * y1) (y5 * y5) (y6 * y6) (y4 * y4)
       /\
 &0 < delta_x4 (y3 * y3) (y1 * y1) (y2 * y2) (y6 * y6) (y4 * y4) (y5 * y5)
      ) )`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  CONJ_TAC;
    INTRO_TAC ((* Appendix. *) get_main_nonlinear "8346775862") [];
    DISCH_THEN (C INTRO_TAC [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`]);
    REWRITE_TAC[Sphere.ineq;Sphere.delta_y;Sphere.y_of_x];
    REWRITE_TAC[TAUT `a ==> b ==> c <=> a /\ b ==> c`];
    ANTS_TAC;
      BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
    BY(FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[Sphere.delta_y] THEN REAL_ARITH_TAC);
  CONJ_TAC;
    TYPIFY `delta_x4 (y2 * y2) (y3 * y3) (y1 * y1) (y5 * y5) (y6 * y6) (y4 * y4) = delta_x4 (y2 * y2) (y1 * y1) (y3 * y3) (y5 * y5) (y4 * y4) (y6 * y6)` (C SUBGOAL_THEN SUBST1_TAC);
      REWRITE_TAC[Sphere.delta_x4];
      BY(REAL_ARITH_TAC);
    INTRO_TAC ((* Appendix. *) get_main_nonlinear "8631418063") [];
    DISCH_THEN (C INTRO_TAC [`y2`;`y1`;`y3`;`y5`;`y4`;`y6`]);
    REWRITE_TAC[Sphere.ineq;Sphere.delta_y;Sphere.y_of_x];
    REWRITE_TAC[arith `x > &0 <=> &0 < x`];
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "8631418063") [];
  DISCH_THEN (C INTRO_TAC [`y3`;`y1`;`y2`;`y6`;`y4`;`y5`]);
  REWRITE_TAC[Sphere.ineq;Sphere.delta_y;Sphere.y_of_x];
  REWRITE_TAC[arith `x > &0 <=> &0 < x`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let EAR_DIH1_DELTA_0 = 
prove_by_refinement(
  `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 /\
    cstab <= y4 /\ y4 <= #3.915 /\
    y5 = &2 /\
    y6 = &2 /\
	delta_y y1 y2 y3 y4 y5 y6 = &0 ==>
	dih_y y1 y2 y3 y4 y5 y6 = pi)`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC (UNDISCH EAR_DELTA_X4) [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  ANTS_TAC;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
  DISCH_TAC;
  FIRST_X_ASSUM_ST `delta_y` MP_TAC;
  REWRITE_TAC[Sphere.dih_y;Sphere.dih_x;LET_DEF;LET_END_DEF;Sphere.delta_y];
  DISCH_THEN (unlist REWRITE_TAC);
  REWRITE_TAC[arith `x * &0 = &0`;SQRT_0];
  TYPED_ABBREV_TAC `d = delta_x4 (y1 * y1) (y2 * y2) (y3 * y3) (y4 * y4) (y5 * y5) (y6 * y6)`;
  INTRO_TAC Merge_ineq.atn2_0 [`-- d`];
  REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC);
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let OWZLKVY3 = 
prove_by_refinement(
  `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 /\
    cstab <= y4 /\ y4 <= #3.915 /\
    y5 = &2 /\
    y6 = &2 /\
	&0 <= delta_y y1 y2 y3 y4 y5 y6 /\
	dih_y y1 y2 y3 y4 y5 y6 = pi ==>
	sol0 * (y1 - &2 * h0) / (&2 * h0 - &2) <= taum y1 y2 y3 y4 y5 y6)`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  ASM_CASES_TAC `&200 <= delta_y y1 y2 y3 y4 y5 y6`;
    INTRO_TAC OWZLKVY0 [];
    ASM_REWRITE_TAC[];
    DISCH_THEN (C INTRO_TAC [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`]);
    ASM_REWRITE_TAC[];
    MATCH_MP_TAC (arith `&0 <= s * (-- u) /v ==> (&0 <= t ==>  s * u / v <= t)`);
    GMATCH_SIMP_TAC REAL_LE_MUL;
    CONJ_TAC;
      INTRO_TAC Flyspeck_constants.bounds [];
      BY(REAL_ARITH_TAC);
    GMATCH_SIMP_TAC REAL_LE_RDIV_EQ;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
  RULE_ASSUM_TAC (REWRITE_RULE[arith `~(x <= y) <=> y < x`]);
  REWRITE_TAC[Nonlinear_lemma.taum_123;Sphere.rhazim2;Sphere.rhazim3;Sphere.node2_y;Sphere.node3_y];
  REWRITE_TAC[Sphere.rhazim];
  TYPIFY `delta_y y1 y2 y3 y4 y5 y6 = &0` (C SUBGOAL_THEN ASSUME_TAC);
    PROOF_BY_CONTR_TAC;
    INTRO_TAC Oxlzlez.DIH_X_LT_PI [`y1 * y1`;`y2*y2`;`y3*y3`;`y4*y4`;`y5*y5`;`y6*y6`];
    ANTS_TAC;
      GMATCH_SIMP_TAC REAL_LT_MUL_EQ;
      REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.delta_y];
      MP_TAC (arith `&2 <= y1 ==> &0 < y1`);
      BY(REAL_ARITH_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.dih_y;LET_DEF;LET_END_DEF] THEN REAL_ARITH_TAC);
  COMMENT "...";
  INTRO_TAC (UNDISCH EAR_DELTA_X4) [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  ANTS_TAC;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `dih_y y2 y3 y1 y5 y6 y4 = &0` (C SUBGOAL_THEN SUBST1_TAC);
    REWRITE_TAC[Sphere.dih_y;Sphere.dih_x;LET_DEF;LET_END_DEF];
    REPEAT (FIRST_X_ASSUM_ST `x = &2` kill);
    RULE_ASSUM_TAC (REWRITE_RULE[Sphere.delta_y]);
    TYPIFY `delta_x (y2 * y2) (y3 * y3) (y1 * y1) (y5 * y5) (y6 * y6) (y4 * y4) = &0` (C SUBGOAL_THEN SUBST1_TAC);
      FIRST_X_ASSUM_ST `x = &0` MP_TAC;
      BY(MESON_TAC[Merge_ineq.delta_x_sym]);
    REWRITE_TAC[arith `x * &0 = &0`;SQRT_0];
    TYPED_ABBREV_TAC `d = delta_x4 (y2 * y2) (y3 * y3) (y1 * y1) (y5 * y5) (y6 * y6) (y4 * y4)`;
    INTRO_TAC Merge_ineq.atn2_0 [`-- d`];
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  COMMENT "..";
  TYPIFY `dih_y y3 y1 y2 y6 y4 y5 = &0` (C SUBGOAL_THEN SUBST1_TAC);
    REWRITE_TAC[Sphere.dih_y;Sphere.dih_x;LET_DEF;LET_END_DEF];
    REPEAT (FIRST_X_ASSUM_ST `x = &2` kill);
    RULE_ASSUM_TAC (REWRITE_RULE[Sphere.delta_y]);
    TYPIFY `delta_x (y3 * y3) (y1 * y1) (y2 * y2) (y6 * y6) (y4 * y4) (y5 * y5) = &0` (C SUBGOAL_THEN SUBST1_TAC);
      FIRST_X_ASSUM_ST `x = &0` MP_TAC;
      BY(MESON_TAC[Merge_ineq.delta_x_sym]);
    REWRITE_TAC[arith `x * &0 = &0`;SQRT_0];
    TYPED_ABBREV_TAC `d = delta_x4 (y3 * y3) (y1 * y1) (y2 * y2) (y6 * y6) (y4 * y4) (y5 * y5)`;
    INTRO_TAC Merge_ineq.atn2_0 [`-- d`];
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  COMMENT "A";
  TYPIFY `dih_y y1 y2 y3 y4 y5 y6 = pi` (C SUBGOAL_THEN SUBST1_TAC);
    REWRITE_TAC[Sphere.dih_y;Sphere.dih_x;LET_DEF;LET_END_DEF];
    REPEAT (FIRST_X_ASSUM_ST `x = &2` kill);
    RULE_ASSUM_TAC (REWRITE_RULE[Sphere.delta_y]);
    TYPIFY `delta_x (y1 * y1) (y2 * y2) (y3 * y3) (y4 * y4) (y5 * y5) (y6 * y6) = &0` (C SUBGOAL_THEN SUBST1_TAC);
      FIRST_X_ASSUM_ST `x = &0` MP_TAC;
      BY(MESON_TAC[Merge_ineq.delta_x_sym]);
    REWRITE_TAC[arith `x * &0 = &0`;SQRT_0];
    TYPED_ABBREV_TAC `d = delta_x4 (y1 * y1) (y2 * y2) (y3 * y3) (y4 * y4) (y5 * y5) (y6 * y6)`;
    INTRO_TAC Merge_ineq.atn2_0 [`-- d`];
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  REWRITE_TAC[arith `x * &0 = &0 /\ x + &0 = x`];
  REWRITE_TAC[Nonlinear_lemma.rho_alt;Nonlinear_lemma.sol0_const1];
  MATCH_MP_TAC (arith `x = y ==> x <= y`);
  Calc_derivative.CALC_ID_TAC;
  REWRITE_TAC[Sphere.h0];
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let OWZLKVY1 = 
prove_by_refinement(
  `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 /\
    cstab <= y4 /\ y4 <= #3.915 /\
    y5 = &2 /\
    y6 = &2 /\
	&0 <= delta_y y1 y2 y3 y4 y5 y6 ==>
	-- sol0 <= taum y1 y2 y3 y4 y5 y6)`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  ASM_CASES_TAC `&200 <= delta_y y1 y2 y3 y4 y5 y6`;
    INTRO_TAC OWZLKVY0 [];
    ASM_REWRITE_TAC[];
    DISCH_THEN (C INTRO_TAC [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`]);
    ASM_REWRITE_TAC[];
    MATCH_MP_TAC (arith `&0 <= s ==> (&0 <= t ==> -- s <= t)`);
    INTRO_TAC Flyspeck_constants.bounds [];
    BY(REAL_ARITH_TAC);
  COMMENT "delta=0";
  ASM_CASES_TAC `delta_y y1 y2 y3 y4 y5 y6 = &0`;
    INTRO_TAC (UNDISCH EAR_DIH1_DELTA_0) [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
    ANTS_TAC;
      BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
    DISCH_TAC;
    MATCH_MP_TAC REAL_LE_TRANS;
    EXISTS_TAC `sol0 * (y1 - &2 * h0) / (&2 * h0 - &2)`;
    CONJ_TAC;
      MATCH_MP_TAC (arith `&0 <= s * (&1 + y /z) ==> --s <= s * y /z`);
      GMATCH_SIMP_TAC REAL_LE_MUL;
      CONJ_TAC;
        INTRO_TAC Flyspeck_constants.bounds [];
        BY(REAL_ARITH_TAC);
      TYPIFY `&1 + (y1 - &2 * h0) / (&2 * h0 - &2) = (y1 - &2) / (&2 * h0 - &2)` (C SUBGOAL_THEN SUBST1_TAC);
        Calc_derivative.CALC_ID_TAC;
        BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
      GMATCH_SIMP_TAC REAL_LE_RDIV_EQ;
      BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
    MATCH_MP_TAC (UNDISCH OWZLKVY3);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  COMMENT "replace taum with solid angle";
  TYPIFY `&0 < delta_y y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN MP_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  DISCH_TAC;
  REWRITE_TAC[Nonlinear_lemma.taum_123;Sphere.rhazim2;Sphere.rhazim3;Sphere.node2_y;Sphere.node3_y];
  REWRITE_TAC[arith `-- sol0 <= r1 + r2 + r3 - (&1 + c)*pi <=> (-- sol0 + c * pi) + pi <= (r1 + r2 + r3)`];
  MATCH_MP_TAC REAL_LE_TRANS;
  EXISTS_TAC `dih_y y1 y2 y3 y4 y5 y6 + dih_y y2 y3 y1 y5 y6 y4 + dih_y y3 y1 y2 y6 y4 y5`;
  CONJ2_TAC;
    MATCH_MP_TAC (arith `d1 <= r1 /\ d2 <= r2 /\ d3x <= r3 ==> d1 + d2 + d3x <= r1 + r2 + r3`);
    REPEAT (GMATCH_SIMP_TAC DIH_Y_LT_RHAZIM);
    ASM_REWRITE_TAC[];
    BY(ASM_MESON_TAC[Merge_ineq.delta_y_sym]);
  REWRITE_TAC[Nonlinear_lemma.sol0_const1;arith `(-- (pi * c) + c * pi) + pi = pi`];
  RULE_ASSUM_TAC(REWRITE_RULE[arith `~(x <= d) <=> (d < x)`]);
  ENOUGH_TO_SHOW_TAC `&0 < sol_y y1 y2 y3 y4 y5 y6`;
    REWRITE_TAC[Sphere.sol_y];
    BY(REAL_ARITH_TAC);
  REWRITE_TAC[Merge_ineq.sol_y_sol_x];
  MATCH_MP_TAC sol_x_nn;
  REPEAT (GMATCH_SIMP_TAC REAL_LT_MUL_EQ);
  REWRITE_TAC[Trigonometry1.UPS_X_SQUARES];
  REPEAT (GMATCH_SIMP_TAC REAL_LT_MUL_EQ);
  TYPIFY `&0 < eulerA_x (y1 * y1) (y2 * y2) (y3 * y3) (y4 * y4) (y5 * y5) (y6 * y6)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    INTRO_TAC ((* Appendix. *) get_main_nonlinear "4821120729") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
    REWRITE_TAC[Sphere.ineq;Sphere.y_of_x;arith `x > &0 <=> &0 < x`];
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
  TYPIFY `&0 < delta_x (y1 * y1) (y2 * y2) (y3 * y3) (y4 * y4) (y5 * y5) (y6 * y6)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    FIRST_X_ASSUM MP_TAC;
    BY(REWRITE_TAC[Sphere.delta_y]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN (REWRITE_TAC[Appendix.cstab;Sphere.h0]) THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let OWZLKVY2 = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> ( !y1 y2 y3 y4 y5 y6.
    y1 = &2 * h0 /\
    &2 <= y2 /\ y2 <= &2 * h0 /\
    &2 <= y3 /\ y3 <= &2 * h0 /\
    cstab <= y4 /\ y4 <= #3.915 /\
    y5 = &2 /\
    y6 = &2 /\
	&0 <= delta_y y1 y2 y3 y4 y5 y6 ==>
	&0 <= taum y1 y2 y3 y4 y5 y6)`,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  ASM_CASES_TAC `&200 <= delta_y y1 y2 y3 y4 y5 y6`;
    INTRO_TAC OWZLKVY0 [];
    ASM_REWRITE_TAC[];
    DISCH_THEN (C INTRO_TAC [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`]);
    ASM_REWRITE_TAC[];
    DISCH_THEN MATCH_MP_TAC;
    BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
  TYPIFY `&0 <= delta_x (y1 * y1) (y2 * y2) (y3 * y3) (y4 * y4) (y5 * y5) (y6 * y6)` (C SUBGOAL_THEN ASSUME_TAC);
    FIRST_X_ASSUM_ST `&0 <= d` MP_TAC;
    BY(REWRITE_TAC[Sphere.delta_y] THEN REAL_ARITH_TAC);
  GMATCH_SIMP_TAC taum_taum_x;
  CONJ_TAC;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
  REWRITE_TAC[Sphere.y_of_x];
  GMATCH_SIMP_TAC Merge_ineq.tau_x_tau_residual_x;
  COMMENT "residue hypotheses";
  CONJ_TAC;
    GMATCH_SIMP_TAC Nonlinear_lemma.sqrtxx;
    GMATCH_SIMP_TAC REAL_LT_MUL_EQ;
    REPEAT (GMATCH_SIMP_TAC REAL_LT_MUL_EQ);
    INTRO_TAC (UNDISCH EAR_DELTA_X4) [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
    ANTS_TAC;
      BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
    DISCH_THEN (unlist REWRITE_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.delta_y;Sphere.h0;Appendix.cstab] THEN REAL_ARITH_TAC);
  GMATCH_SIMP_TAC REAL_LE_MUL;
  CONJ_TAC;
    GMATCH_SIMP_TAC SQRT_POS_LE;
    BY(FIRST_X_ASSUM (unlist REWRITE_TAC));
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "5202826650 a") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  REWRITE_TAC[Sphere.ineq;Sphere.y_of_x];
  REWRITE_TAC[TAUT `a ==> b ==> c <=> (a /\ b ==> c)`];
  ANTS_TAC;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab;Sphere.h0] THEN REAL_ARITH_TAC);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

(* was sqrt2_bounds *)

let sqrt8_bounds = 
prove_by_refinement(
  `sqrt8 <= #3.01 /\ &2 <= sqrt8 /\ sqrt8 <= #3.62`,

  (* {{{ proof *)
  [
  INTRO_TAC Flyspeck_constants.bounds [];
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

(* renamed from  terminal_std_tri_OMKYNLT_3336871894_empty *)

(*
let empty_3T2 = prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3 (&0) (funlist_v39 [] (&2) 3) (funlist_v39 [] (&2) 3) in
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;arith `&2 <= &2 /\ &2 <= #3.01 /\ &2 <= #3.62`];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "OMKYNLT 3336871894") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  REWRITE_TAC[Sphere.ineq;arith `x >= #0.0 <=> &0 <= x`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)
*)

let empty_3T2 = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
  	(let s = scs_3T2 in (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_3T2];
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;arith `&2 <= &2 /\ &2 <= #3.01 /\ &2 <= #3.62`];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ( get_main_nonlinear "OMKYNLT 3336871894") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  REWRITE_TAC[Sphere.ineq;arith `x >= #0.0 <=> &0 <= x`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)


(* renamed from  terminal_std_tri_4010906068_empty *)
let empty_3T3   = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3
  (#0.476)
  (funlist_v39 [] (&2*h0) 3)
  (funlist_v39 [] (cstab) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "4010906068") [`y1`;`y3`;`y2`;`y4`;`y6`;`y5`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  TYPIFY `taum y1 y3 y2 y4 y6 y5 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let empty_3T3   = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_3T3  in (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_3T3];
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "4010906068") [`y1`;`y3`;`y2`;`y4`;`y6`;`y5`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  TYPIFY `taum y1 y3 y2 y4 y6 y5 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

(* renamed from  terminal_std_tri_6833979866_empty *)
let empty_3T4   = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_3T4 in
  //mk_unadorned_v39
  //3
  //(#0.2759)
  //(funlist_v39 [(0,1),(&2)] (&2*h0) 3)
  //(funlist_v39 [(0,1),(&2 * h0)] (cstab) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_3T4];
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "6833979866") [`y3`;`y1`;`y2`;`y6`;`y4`;`y5`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  TYPIFY `taum y3 y1 y2 y6 y4 y5 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

(*   renamed from terminal_std_tri_5541487347_empty *)
let empty_3T5  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_3T5 in
  //mk_unadorned_v39
  //3
  //(#0.103)
  //(funlist_v39 [(0,1),(&2 * h0)] (&2) 3)
  //(funlist_v39 [(0,1),(sqrt8)] (&2 * h0) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_3T5];
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "5541487347") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

(*
let  terminal_std_tri_4528012043_empty  = prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3 (tame_table_d 0 3 +  #3.0 * (tame_table_d 2 1 -  #0.11))
   (funlist_v39 [] cstab 3)
   (funlist_v39 [] cstab 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;arith `x <= x /\ &2 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62`];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "4528012043") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_7459553847_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3
   (tame_table_d 0 3 +  #3.0 * (tame_table_d 2 1 -  #0.11))
   (funlist_v39 [(0,1),&2 * h0] cstab 3)
   (funlist_v39 [(0,1),&2 * h0] cstab 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "7459553847") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  TYPIFY `taum y3 y2 y1 y6 y5 y4 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_4143829594_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3
   (tame_table_d 0 3 +  #3.0 * (tame_table_d 2 1 -  #0.11))
   (funlist_v39 [(0,1),cstab] (&2 * h0) 3)
   (funlist_v39 [(0,1),cstab] (&2 * h0) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "4143829594") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_1080462150_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3
   (tame_table_d 0 3 +  #3.0 * (tame_table_d 2 1 -  #0.11))
   (funlist_v39 [] (&2 * h0) 3)
   (funlist_v39 [] (&2 * h0) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "1080462150") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_9816718044_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3
   (tame_table_d 1 2 +  #2.0 * (tame_table_d 2 1 -  #0.11))
   (funlist_v39 [(0,1),&2] cstab 3)
   (funlist_v39 [(0,1),&2] cstab 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "9816718044") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  TYPIFY `taum y3 y2 y1 y6 y5 y4 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_3106201101_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3
   (tame_table_d 1 2 +  #2.0 * (tame_table_d 2 1 -  #0.11))
   (funlist_v39 [(0,1),&2; (0,2),cstab] sqrt8 3)
   (funlist_v39 [(0,1),&2; (0,2),cstab] sqrt8 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "3106201101") [`y3`;`y1`;`y2`;`y6`;`y4`;`y5`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  TYPIFY `taum y3 y1 y2 y6 y4 y5 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_2200527225_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3
   (tame_table_d 1 2 +  #2.0 * (tame_table_d 2 1 -  #0.11))
   (funlist_v39 [(0,1),&2] sqrt8 3)
   (funlist_v39 [(0,1),&2] sqrt8 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "2200527225") [`y3`;`y1`;`y2`;`y6`;`y4`;`y5`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  TYPIFY `taum y3 y1 y2 y6 y4 y5 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_2900061606_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3 (tame_table_d 1 2 + tame_table_d 2 1 -  #0.11)
   (funlist_v39 [(0,1),&2 * h0; (0,2),cstab] (&2) 3)
   (funlist_v39 [(0,1),&2 * h0; (0,2),cstab] (&2) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "2900061606") [`y1`;`y3`;`y2`;`y4`;`y6`;`y5`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  TYPIFY `taum y1 y3 y2 y4 y6 y5 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_7097350062a_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3 (tame_table_d 1 2)
   (funlist_v39 [(0,1),&2 * h0; (0,2),sqrt8] (&2) 3)
   (funlist_v39 [(0,1),&2 * h0; (0,2),sqrt8] (&2) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "7097350062a") [`y1`;`y3`;`y2`;`y4`;`y6`;`y5`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  TYPIFY `taum y1 y3 y2 y4 y6 y5 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_OMKYNLT_1_2_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3 (tame_table_d 1 2) (funlist_v39 [(0,1),&2] (&2 * h0) 3)
   (funlist_v39 [(0,1),&2] (&2 * h0) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "OMKYNLT 1 2") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  TYPIFY `taum y3 y2 y1 y6 y5 y4 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_7645170609_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3 (tame_table_d 2 1) (funlist_v39 [(0,1),sqrt8] (&2) 3)
   (funlist_v39 [(0,1),sqrt8] (&2) 3) in
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "7645170609") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_OMKYNLT_2_1_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3 (tame_table_d 2 1) (funlist_v39 [(0,1),&2 * h0] (&2) 3)
   (funlist_v39 [(0,1),&2 * h0] (&2) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "OMKYNLT 2 1") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`];
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)
*)

let  terminal_std_tri_7881254908_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3 ( #0.696 -  #2.0 *  #0.11)
   (funlist_v39 [(0,1),sqrt8; (1,2),sqrt8] (&2 * h0) 3)
   (funlist_v39 [(0,1),cstab; (1,2),cstab] (&2 * h0) 3) in
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "7881254908") [`y2`;`y1`;`y3`;`y5`;`y4`;`y6`];
  TYPIFY `taum y2 y1 y3 y5 y4 y6 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

(*  renamed from terminal_std_tri_5026777310a_empty  *)
let empty_3T6  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_3T6' in
   //mk_unadorned_v39 3 ( #0.6548 -  #2.0 *  #0.11)
   //(funlist_v39 [(0,1),sqrt8; (1,2),sqrt8] (&2) 3)
   //(funlist_v39 [(0,1),cstab; (1,2),cstab] (&2 * h0) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_3T6];
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "5026777310a") [`y3`;`y1`;`y2`;`y6`;`y4`;`y5`];
  TYPIFY `taum y3 y1 y2 y6 y4 y5 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`;Sphere.tame_table_d];
  REWRITE_TAC[arith `4 + 2 * 1 > 3`];
  TYPIFY `delta_y y3 y1 y2 y6 y4 y5 = delta_y y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

(* deprecated 2013-06-17
let  terminal_std_tri_7720405539_empty  = prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3 ( #0.5518 /  #2.0 -  #0.2)
   (funlist_v39 [(0,1),cstab; (0,2),&2 * h0; (1,2),&2] (&2) 3)
   (funlist_v39 [(0,1), #3.41; (0,2),&2 * h0; (1,2),&2] (&2) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62 /\ #3.41 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "7720405539") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`];
  TYPIFY `taum y3 y2 y1 y6 y5 y4 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`;Sphere.tame_table_d];
  REWRITE_TAC[arith `2 + 2 * 2 > 3`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_2739661360_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3 ( #0.5518 /  #2.0 +  #0.2)
   (funlist_v39 [(0,1),cstab; (0,2),cstab; (1,2),&2] (&2) 3)
   (funlist_v39 [(0,1), #3.41; (0,2),cstab; (1,2),&2] (&2) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62 /\ #3.41 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "2739661360") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`];
  TYPIFY `taum y3 y2 y1 y6 y5 y4 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`;Sphere.tame_table_d];
  REWRITE_TAC[arith `2 + 2 * 2 > 3`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_std_tri_4922521904_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 3 ( #0.5518 /  #2.0)
   (funlist_v39 [(0,1),cstab; (0,2),&2 * h0; (1,2),&2] (&2) 3)
   (funlist_v39 [(0,1), #3.339; (0,2),&2 * h0; (1,2),&2] (&2) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62 /\ #3.41 <= #3.62 /\ #3.339 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "4922521904") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`];
  TYPIFY `taum y3 y2 y1 y6 y5 y4 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`;Sphere.tame_table_d];
  REWRITE_TAC[arith `2 + 2 * 2 > 3`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)
*)

(* renamed from  terminal_std_tri_2468307358_empty *)
let empty_3T7   = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_3T7 in
//   mk_unadorned_v39 3 ( #0.2565)
//   (funlist_v39 [(0,1),cstab; (0,2),cstab; (1,2),&2] (&2) 3)
//   (funlist_v39 [(0,1), #3.62; (0,2),cstab; (1,2),&2] (&2) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_3T7];
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62 /\ #3.41 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "2468307358") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`];
  (*  INTRO_TAC ((* Appendix. *) get_main_nonlinear "2739661360") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`]; edited 2013-06-17. *)
  TYPIFY `taum y3 y2 y1 y6 y5 y4 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`;Sphere.tame_table_d];
  REWRITE_TAC[arith `2 + 2 * 2 > 3`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let terminal_std_tri_ear_stab_empty = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = 
 mk_unadorned_v39 3  #0.11 (funlist_v39 [(0,2),cstab] (&2) 3)
   (funlist_v39 [(0,2),cstab] (&2 * h0) 3) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  MATCH_MP_TAC taustar_taum;
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62 /\ #3.41 <= #3.62 /\ #3.339 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "3603097872") [`y2`;`y1`;`y3`;`y5`;`y4`;`y6`];
  TYPIFY `taum y2 y1 y3 y5 y4 y6 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  TYPIFY `delta_y y2 y1 y3 y5 y4 y6 = delta_y y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`;Sphere.tame_table_d];
  MP_TAC sqrt8_bounds;
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)


(*
let  terminal_std_ear_3603097872_empty  = prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_v39
   (3,
     #0.11,
    funlist_v39 [(0,1),sqrt8] (&2) 3,
    funlist_v39 [(0,1),sqrt8] (&2) 3,
    funlist_v39 [(0,1),cstab] (&2 * h0) 3,
    funlist_v39 [(0,1),cstab] (&2 * h0) 3,
    funlistA_v39 [(0,1),T] F F 3,
    {},{},
    {}) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,
  (* {{{ proof *)
  [
  DISCH_TAC;
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  MATCH_MP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] taustar_taum_dfun);
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[is_ear_3603097872];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62 /\ #3.41 <= #3.62 /\ #3.339 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "3603097872") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`];
  TYPIFY `taum y3 y2 y1 y6 y5 y4 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  TYPIFY `delta_y y3 y2 y1 y6 y5 y4 = delta_y y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`;Sphere.tame_table_d];
  MP_TAC sqrt8_bounds;
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)


let  terminal_std_ear_jnull_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_v39 
   (3,
     #0.11,
    funlist_v39 [(0,1),sqrt8] (&2) 3,
    funlist_v39 [(0,1),sqrt8] (&2) 3,
    funlist_v39 [(0,1),cstab] (&2 * h0) 3,
    funlist_v39 [(0,1),cstab] (&2 * h0) 3,
    (\i j. F),
    {},{},
    {}) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  DISCH_TAC;
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  MATCH_MP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] taustar_taum_dfun);
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.is_ear_v39;Appendix.scs_v39_explicit;EMPTY_GSPEC;NOT_INSERT_EMPTY];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62 /\ #3.41 <= #3.62 /\ #3.339 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "3603097872") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`];
  TYPIFY `taum y3 y2 y1 y6 y5 y4 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  TYPIFY `delta_y y3 y2 y1 y6 y5 y4 = delta_y y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`;Sphere.tame_table_d];
  MP_TAC sqrt8_bounds;
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

*)

let  empty_3T1  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_3T1 in
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_3T1;Appendix.mk_unadorned_v39];
  DISCH_TAC;
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  MATCH_MP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] taustar_taum_dfun);
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.is_ear_v39;Appendix.scs_v39_explicit;EMPTY_GSPEC;NOT_INSERT_EMPTY];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62 /\ #3.41 <= #3.62 /\ #3.339 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "3603097872") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`];
  TYPIFY `taum y3 y2 y1 y6 y5 y4 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  TYPIFY `delta_y y3 y2 y1 y6 y5 y4 = delta_y y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`;Sphere.tame_table_d];
  MP_TAC sqrt8_bounds;
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let  terminal_tri_5405130650_empty  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_v39  (3,
     #0.477 -  #0.11,
    funlist_v39 [(0,1),sqrt8; (0,2),&2 * h0; (1,2),&2] (&2) 3,
    funlist_v39 [(0,1),sqrt8; (0,2),&2 * h0; (1,2),&2] (&2) 3,
    funlist_v39 [(0,1),cstab; (0,2),sqrt8; (1,2),&2 * h0] (&2 * h0) 3,
    funlist_v39 [(0,1),cstab; (0,2),sqrt8; (1,2),&2 * h0] (&2 * h0) 3,
    funlistA_v39 [(0,1),T] F F 3,
    {},{},
    {}) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  DISCH_TAC;
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  MATCH_MP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] taustar_taum_dfun);
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.is_ear_v39;Appendix.scs_v39_explicit;arith `~(#0.477 - #0.11 = #0.11)`];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62 /\ #3.41 <= #3.62 /\ #3.339 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "5405130650") [`y3`;`y2`;`y1`;`y6`;`y5`;`y4`];
  TYPIFY `taum y3 y2 y1 y6 y5 y4 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  TYPIFY `delta_y y3 y2 y1 y6 y5 y4 = delta_y y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`;Sphere.tame_table_d];
  MP_TAC sqrt8_bounds;
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

(*
let  terminal_tri_5766053833_empty  = prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_v39 (3,
     #0.696 -  #2.0 *  #0.11,
    funlist_v39 [(0,1),sqrt8; (1,2),sqrt8] (&2) 3,
    funlist_v39 [(0,1),sqrt8; (1,2),sqrt8] (&2) 3,
    funlist_v39 [(0,1),cstab; (1,2),cstab] (&2) 3,
    funlist_v39 [(0,1),cstab; (1,2),cstab] (&2) 3,
    funlistA_v39 [(0,1),T; (1,2),T] F F 3,
    {},{},
    {}) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,
  (* {{{ proof *)
  [
  DISCH_TAC;
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  MATCH_MP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] taustar_taum_dfun);
  REWRITE_TAC[FUNLIST_EXPLICIT];
  REWRITE_TAC[Appendix.is_ear_v39;Appendix.scs_v39_explicit;arith `~(#0.696 - #2.0* #0.11 = #0.11)`];
  REWRITE_TAC[Appendix.cstab;Sphere.h0;arith `x <= x /\ &2 <= #3.01 /\ &2 <= &2 * #1.26 /\ &2 * #1.26 <= #3.01 /\ &2 <= #3.62 /\ #3.01 <= #3.62 /\ &2 * #1.26 <= #3.62 /\ #3.41 <= #3.62 /\ #3.339 <= #3.62`];
  REWRITE_TAC[sqrt8_bounds];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "5766053833") [`y2`;`y1`;`y3`;`y5`;`y4`;`y6`];
  TYPIFY `taum y2 y1 y3 y5 y4 y6 = taum y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  TYPIFY `delta_y y2 y1 y3 y5 y4 y6 = delta_y y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[Merge_ineq.delta_y_sym]);
  REWRITE_TAC[Sphere.ineq;arith `x >= y <=> y <= x`;arith `x > y <=> y < x`;Sphere.tame_table_d];
  MP_TAC sqrt8_bounds;
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)
*)

(* ************************************************************************** *)
(* START OF QUAD CASES *)
(* ************************************************************************** *)

(* renamed from  terminal__adhoc_quad_5691615370_empty *)
let empty_4T1  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_4T1 in
//mk_unadorned_v39 4  #0.467 (cs_adj 4 (&2) (&3)) (cs_adj 4 (&2) (&6)) in
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))` ,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_4T1];
  DISCH_TAC;
  REWRITE_TAC[LET_DEF;LET_END_DEF];
  REWRITE_TAC[Appendix.BBs_v39;Appendix.mk_unadorned_v39;Appendix.cs_adj;Appendix.scs_v39_explicit];
  REWRITE_TAC[LET_DEF;LET_END_DEF;arith `~(4 <= 3)`];
  REPEAT WEAKER_STRIP_TAC;
  PROOF_BY_CONTR_TAC;
  REPLICATE_TAC 2 (FIRST_X_ASSUM kill);
  ASSUME_TAC MOD_4_EXPLICIT;
  ASSUME_TAC (arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4`);
  FIRST_ASSUM (C INTRO_TAC [`0`;`1`]);
  FIRST_ASSUM (C INTRO_TAC [`0`;`2`]);
  FIRST_ASSUM (C INTRO_TAC [`0`;`3`]);
  FIRST_ASSUM (C INTRO_TAC [`1`;`2`]);
  FIRST_ASSUM (C INTRO_TAC [`1`;`3`]);
  FIRST_X_ASSUM (C INTRO_TAC [`2`;`3`]);
  ASM_REWRITE_TAC[arith `~(1 = 3) /\ ~(2 = 3) /\ ~(1 = 2) /\ ~(0 = 1) /\ ~(0=2) /\ ~(0=3)`];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `dist(vv 2,vv 3) = &2 /\ dist(vv 1,vv 2) = &2 /\ dist(vv 0, vv 3) = &2 /\ dist(vv 0,vv 1) = &2` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  INTRO_TAC DELTA_Y_POS_4POINTS [`vv 0 `;`vv 1`;`vv 2`;`vv 3`];
  ASM_REWRITE_TAC[Sphere.delta_y;Sphere.delta_x];
  TYPIFY `&9 <= dist(vv 1, vv 3) * dist(vv 1, vv 3) /\ &9 <= dist(vv 0 , vv 2) * dist(vv 0,vv 2)` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `&9 = &3 * &3`];
    GMATCH_SIMP_TAC Misc_defs_and_lemmas.ABS_SQUARE_LE;
    GMATCH_SIMP_TAC Misc_defs_and_lemmas.ABS_SQUARE_LE;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  TYPED_ABBREV_TAC  `y2 = dist(vv 1,vv 3) * dist(vv 1,vv 3)` ;
  TYPED_ABBREV_TAC  `z2 = dist(vv 0,vv 2) * dist(vv 0,vv 2)` ;
  REWRITE_TAC[arith `&2 * &2 = &4`];
  REWRITE_TAC[arith `&4 * &4 * (-- &4 + z2 + &4 - &4 + y2 + &4) +   z2 * y2 * (&4 - z2 + &4 + &4 - y2 + &4) +   &4 * &4 * (&4 + z2 - &4 + &4 + y2 - &4) -   z2 * &4 * &4 -   &4 * &4 * y2 -   &4 * z2 * &4 -   &4 * y2 * &4  = -- ( y2 *z2 *(  y2 + z2 - &16))`];
  REWRITE_TAC[arith `~(&0 <= -- x ) <=> (&0 <  x)`];
  GMATCH_SIMP_TAC REAL_LT_MUL_EQ;
  GMATCH_SIMP_TAC REAL_LT_MUL_EQ;
  BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let ineq_5691615370_asym = 
prove_by_refinement(
   `main_nonlinear_terminal_v11 ==> (!y1 y2 y3 y4 y5 y6. ineq
  [
  (#3.0,y1,#3.0);
  (&2,y2,#2.52);
  (&2,y3,#2.52);
  (#3.0,y4,#3.0);
  (&2,y5,#2.52);
  (&2,y6,#2.52)
  ]
  ((delta_y y1 y2 y3 y4 y5 y6 < &0) \/ (y2 + y3 + y5 + y6 > #8.472)))`,

  (* {{{ proof *)
  [
  DISCH_TAC;
  ONCE_REWRITE_TAC[MESON[] `(!y1 y2 y3 y4 y5 y6. P y1 y2 y3 y4 y5 y6) <=> (! y2 y3 y5 y6 y1 y4. P y1 y2 y3 y4 y5 y6)`];
  MATCH_MP_TAC REAL_WLOG_SQUARE2_LEMMA;
  CONJ_TAC;
    REWRITE_TAC[Sphere.ineq];
    REPEAT WEAKER_STRIP_TAC;
    TYPIFY `y2' + y5 + y6 + y2 = y2 + y2' + y5 + y6` (C SUBGOAL_THEN SUBST1_TAC);
      BY(REAL_ARITH_TAC);
    TYPIFY `y2 + y6 + y5 + y2' = y2 + y2' + y5 + y6` (C SUBGOAL_THEN SUBST1_TAC);
      BY(REAL_ARITH_TAC);
    TYPIFY `delta_y y4 y2' y5 y1 y6 y2 = delta_y y1 y2 y2' y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
      BY(REWRITE_TAC[Sphere.delta_y;Sphere.delta_x] THEN REAL_ARITH_TAC);
    CONJ_TAC;
      BY(REWRITE_TAC[Sphere.delta_y;Sphere.delta_x] THEN REAL_ARITH_TAC);
    BY(REWRITE_TAC[Sphere.delta_y;Sphere.delta_x] THEN REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[Sphere.ineq];
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "5691615370") [`y1`;`y2`;`y2'`;`y4`;`y5`;`y6`];
  REWRITE_TAC[Sphere.ineq];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let terminal_quad_lemma = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
   (!d a b . 
	(let s = mk_unadorned_v39 4  d a b in
	   (!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> &0 < a i j) /\
   (!vv. IMAGE vv (:num) SUBSET ball_annulus /\
    periodic vv 4 /\
    (!i j. a i j <= dist (vv i,vv j) /\ dist (vv i,vv j) <= b i j) /\
    (convex_local_fan
      (IMAGE vv (:num),
       IMAGE (\i. {vv i, vv (SUC i)}) (:num),
       IMAGE (\i. vv i,vv (SUC i)) (:num))) ==>
 (d <=
 (rho_fun (norm (vv 0)) * azim (vec 0) (vv 0) (vv 1) (vv 3) +
  rho_fun (norm (vv 1)) * azim (vec 0) (vv 1) (vv 2) (vv 4) +
  rho_fun (norm (vv 2)) * azim (vec 0) (vv 2) (vv 3) (vv 5) +
  rho_fun (norm (vv 3)) * azim (vec 0) (vv 3) (vv 4) (vv 6) +
  &0) -
 (pi + sol0) * (&4 - &2) 
 )) ==>
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv)))` ,

  (* {{{ proof *)
  [
  DISCH_TAC;
  REWRITE_TAC[LET_THM];
  REWRITE_TAC[Appendix.BBs_v39;Appendix.mk_unadorned_v39;Appendix.scs_v39_explicit;Appendix.taustar_v39];
  REWRITE_TAC[LET_THM;arith `~(4 <= 3)`;dsv_fun4];
  REWRITE_TAC[arith `d -  #0.1 * (&0 + &0 + &0 + &0 + &0) = d`];
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "expand tau_fun";
  GMATCH_SIMP_TAC (REWRITE_RULE[LET_DEF;LET_END_DEF] tau_fun_azim);
  TYPIFY `4` EXISTS_TAC;
  ASM_SIMP_TAC[arith `3 <= 4`];
  CONJ_TAC;
    REPEAT WEAKER_STRIP_TAC;
    PROOF_BY_CONTR_TAC;
    FIRST_X_ASSUM (C INTRO_TAC [`i`;`j`]);
    ASM_REWRITE_TAC[DIST_REFL];
    REWRITE_TAC[DE_MORGAN_THM;arith `~(a <= &0) <=> &0 < a`];
    BY(ASM_MESON_TAC[]);
  TYPIFY `{i | i < 4 } = {0,1,2,3}` (C SUBGOAL_THEN SUBST1_TAC);
    REWRITE_TAC[EXTENSION;IN_ELIM_THM;IN_INSERT;NOT_IN_EMPTY];
    BY(ARITH_TAC);
  REPEAT (GMATCH_SIMP_TAC Marchal_cells_2_new.SUM_CLAUSES_alt);
  REWRITE_TAC[SUM_CLAUSES;IN_INSERT;NOT_IN_EMPTY;arith `~(2 = 3) /\ ~(0 = 1) /\ ~(0 = 2) /\ ~(0 = 3) /\ ~(1 = 2) /\ ~(1 = 3) /\ ~(1  = 2) /\ ~(1 = 3)`;FINITE_INSERT;FINITE_EMPTY];
  REWRITE_TAC[arith `0 + 1 = 1 /\ 1 + 1 = 2 /\ 2 + 1  = 3 /\ 3 + 1 = 4 /\ 0 + 4 -1 = 3 /\ 1 + 4 -1 = 4 /\ 2 + 4 - 1 = 5 /\ 3 + 4 - 1 = 6`];
  COMMENT "next";
  REWRITE_TAC[arith `&0 <= r - d <=> d <= r`];
  FIRST_X_ASSUM MATCH_MP_TAC;
  BY(ASM_REWRITE_TAC[])
  ]);;
  (* }}} *)

(* renamed from terminal_adhoc_quad_9563139965B_empty *)
let empty_4T2  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_4T2 in // mk_unadorned_v39 4  #0.467 (cs_adj 4 (&2) (&3)) (cs_adj 4 (&2 * h0) (&3)) in
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_4T2];
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[LET_THM];
  MATCH_MP_TAC (REWRITE_RULE[LET_THM] (UNDISCH terminal_quad_lemma));
  COMMENT "digression";
  TYPIFY `(!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> &2 <= cs_adj 4 (&2) (&3) i j) ` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `i < 4 <=> (i = 0 \/ i = 1 \/ i = 2 \/ i = 3)`];
    REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`;TAUT `a /\ (b \/ c) <=> (a /\ b) \/ a /\ c`];
    REWRITE_TAC[Appendix.cs_adj];
    REPEAT WEAKER_STRIP_TAC;
    BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4`;MOD_4_EXPLICIT;arith `&2 <= &2 /\ &2 <= &3`;arith `~(0=1) /\ ~(0=2) /\ ~(0=3) /\ ~(1 =2) /\ ~(1 = 3) /\ ~(2 = 3)`] THEN ASM_MESON_TAC[]);
  COMMENT "digress2";
  TYPIFY `!i j. i < 4 /\ j < 4 /\ ~(i = j) ==>  cs_adj 4 (&2 * h0) (&3) i j < &4` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `i < 4 <=> (i = 0 \/ i = 1 \/ i = 2 \/ i = 3)`];
    REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`;TAUT `a /\ (b \/ c) <=> (a /\ b) \/ a /\ c`];
    REWRITE_TAC[Appendix.cs_adj;Sphere.h0];
    REPEAT WEAKER_STRIP_TAC;
    BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4`;MOD_4_EXPLICIT;arith `&2 * #1.26 < &4 /\ &3 < &4`;arith `~(0=1) /\ ~(0=2) /\ ~(0=3) /\ ~(1 =2) /\ ~(1 = 3) /\ ~(2 = 3)`] THEN ASM_MESON_TAC[]);
  CONJ_TAC;
    BY(ASM_MESON_TAC[arith `&2 <= x ==> &0 < x`]);
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[arith `x + &0 = x /\ &4 - &2 = &2`];
  TYPIFY `(!i j. i < 4 /\ j < 4 /\ vv i = vv j ==> i = j)` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    PROOF_BY_CONTR_TAC;
    BY(ASM_MESON_TAC[DIST_REFL;REAL_LE_TRANS;arith `~(&2 <= &0)`]);
  COMMENT "vv";
  TYPIFY `vv 4 = vv 0 /\ vv 5 = vv 1 /\ vv 6 = vv 2` (C SUBGOAL_THEN ASSUME_TAC);
    BY((REPEAT CONJ_TAC) THEN GMATCH_SIMP_TAC periodic_vv_inj THEN TYPIFY `4` EXISTS_TAC THEN ASM_REWRITE_TAC[MOD_4_EXPLICIT;arith `~(4 = 0)`]);
  ASM_REWRITE_TAC[];
  GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] vv_quad_split012);
  ASM_REWRITE_TAC[];
  COMMENT "dist";
  SUBCONJ_TAC;
    TYPIFY `(!i j. ~(vv i = vv j) ==> &2 <= dist (vv i,vv j))` (C SUBGOAL_THEN ASSUME_TAC);
      MATCH_MP_TAC periodic2_mod_reduce;
      TYPIFY `4` EXISTS_TAC;
      REWRITE_TAC[Appendix.periodic2;arith `~(4 = 0)`];
      TYPIFY `!i. vv (i + 4) = vv i` (C SUBGOAL_THEN (unlist REWRITE_TAC));
        BY(ASM_MESON_TAC[Oxl_def.periodic;arith `~(4 = 0)`]);
      REPEAT WEAKER_STRIP_TAC;
      BY(ASM_MESON_TAC[REAL_LE_TRANS]);
    ASM_REWRITE_TAC[];
    BY(ASM_MESON_TAC[REAL_LET_TRANS;arith `0 < 4 /\ 1 < 4 /\ 2 < 4 /\ 3 < 4 /\ ~(0=1) /\ ~(0=2) /\ ~(0=3) /\ ~(1=2) /\ ~(2 =3)`]);
  COMMENT "tau3";
  REPEAT WEAKER_STRIP_TAC;
  REPEAT (GMATCH_SIMP_TAC tau3_taum_40);
  ASM_REWRITE_TAC[];
  RULE_ASSUM_TAC(REWRITE_RULE[IMAGE_SUBSET_IN;IN_UNIV]);
  ASM_REWRITE_TAC[];
  TYPIFY `dist (vv 2,vv 0) < &4 /\   dist (vv 3,vv 0) < &4 ` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    ONCE_REWRITE_TAC[DIST_SYM];
    BY(ASM_REWRITE_TAC[]);
  REWRITE_TAC[TAUT `a /\ b /\ c <=> (a /\ b) /\ c`];
  CONJ_TAC;
    BY(ASM_MESON_TAC[REAL_LE_TRANS;arith `1 < 4 /\ 2 < 4 /\ 3 < 4 /\ 0 < 4 /\ ~(2=3) /\ ~(2  =0) /\ ~(3 =0) /\ ~(0 = 1) /\ ~(0=2) /\ ~(1 = 2)`]);
  TYPIFY `dist(vv 3,vv 0) = dist(vv 0,vv 3) /\ dist(vv 2, vv 0) = dist(vv 0,vv 2)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    BY(MESON_TAC[DIST_SYM]);
  COMMENT "prep dist";
  TYPIFY `(!i j d. d <= cs_adj 4 (&2) (&3) i j  ==> d <= dist (vv i,vv j)) /\ (!i j d. cs_adj 4 (&2 * h0) (&3) i j        <= d ==>   dist (vv i,vv j) <= d)` (C SUBGOAL_THEN MP_TAC);
    BY(ASM_MESON_TAC[REAL_LE_TRANS]);
  REPEAT WEAKER_STRIP_TAC;
  INTRO_TAC cs_adj4_EXPLICIT [`&2`;`&3`];
  INTRO_TAC cs_adj4_EXPLICIT [`&2 * h0`;`&3`];
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "fist get";
  INTRO_TAC (UNDISCH ineq_5691615370_asym) [`dist (vv 0,vv 2)`;`dist(vv 0, vv 1)`;`dist(vv 0, vv 3)`;`dist(vv 1,vv 3)`;`dist(vv 2,vv 3)`;`dist(vv 2,vv 1)`];
  TYPIFY `~(delta_y (dist (vv 0,vv 2)) (dist (vv 0,vv 1)) (dist (vv 0,vv 3))       (dist (vv 1,vv 3))      (dist (vv 2,vv 3))      (dist (vv 2,vv 1)) <      &0)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    MATCH_MP_TAC (arith `&0 <= d ==> ~(d < &0)`);
    BY(REWRITE_TAC[DELTA_Y_POS_4POINTS]);
  REWRITE_TAC[];
  TYPIFY `dist(vv 2,vv 1) = dist(vv 1, vv 2)` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[DIST_SYM]);
  REWRITE_TAC[Sphere.ineq];
  REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ANTS_TAC;
    BY(REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0] THEN TRY REAL_ARITH_TAC);
  DISCH_TAC;
  COMMENT "norms";
  RULE_ASSUM_TAC(REWRITE_RULE[Fnjlbxs.in_ball_annulus]);
  COMMENT "second get";
  TYPIFY `dist (vv 0,vv 1) <= #2.472 /\ dist(vv 1,vv 2) <= #2.472 /\ dist(vv 2,vv 3) <= #2.472 /\ dist(vv 0,vv 3) <= #2.472` ASM_CASES_TAC;
    INTRO_TAC ((* Appendix. *) get_main_nonlinear "9563139965 d") [`norm(vv 1)`;`norm (vv 2)`;`norm (vv 0)`;`dist(vv 0,vv 2)`;`dist(vv 0,vv 1)`;`dist(vv 1,vv 2)`];
    REWRITE_TAC[Sphere.ineq];
    REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
    ANTS_TAC;
      ASM_REWRITE_TAC[];
      BY(REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0] THEN TRY REAL_ARITH_TAC);
    DISCH_TAC;
    INTRO_TAC ((* Appendix. *) get_main_nonlinear "9563139965 d") [`norm(vv 3)`;`norm (vv 2)`;`norm (vv 0)`;`dist(vv 0,vv 2)`;`dist(vv 0,vv 3)`;`dist(vv 2,vv 3)`];
    REWRITE_TAC[Sphere.ineq];
    REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
    ANTS_TAC;
      ASM_REWRITE_TAC[];
      BY(REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0] THEN TRY REAL_ARITH_TAC);
    DISCH_TAC;
    TYPIFY `taum (norm (vv 0)) (norm (vv 1)) (norm (vv 2)) (dist (vv 1,vv 2))  (dist (vv 0,vv 2)) (dist (vv 0,vv 1)) = taum (norm (vv 1)) (norm (vv 2)) (norm (vv 0)) (dist (vv 0,vv 2))      (dist (vv 0,vv 1))      (dist (vv 1,vv 2)) /\ taum (norm (vv 2)) (norm (vv 3)) (norm (vv 0)) (dist (vv 0,vv 3)) (dist (vv 0,vv 2)) (dist (vv 2,vv 3)) = taum (norm (vv 3)) (norm (vv 2)) (norm (vv 0)) (dist (vv 0,vv 2))      (dist (vv 0,vv 3))      (dist (vv 2,vv 3))` ENOUGH_TO_SHOW_TAC;
      DISCH_THEN (unlist REWRITE_TAC);
      BY(REPLICATE_TAC 4 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
    BY(MESON_TAC[taum_sym]);
  COMMENT "third get";
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "9563139965 f") [`norm(vv 1)`;`norm (vv 2)`;`norm (vv 0)`;`dist(vv 0,vv 2)`;`dist(vv 0,vv 1)`;`dist(vv 1,vv 2)`];
  REWRITE_TAC[Sphere.ineq];
  REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ANTS_TAC;
    ASM_REWRITE_TAC[];
    BY(REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0] THEN TRY REAL_ARITH_TAC);
  TYPIFY `taum (norm (vv 1)) (norm (vv 2)) (norm (vv 0)) (dist (vv 0,vv 2)) (dist (vv 0,vv 1)) (dist (vv 1,vv 2)) = taum (norm (vv 0)) (norm (vv 1)) (norm (vv 2)) (dist (vv 1,vv 2))     (dist (vv 0,vv 2))     (dist (vv 0,vv 1))` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  DISCH_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "9563139965 f") [`norm(vv 3)`;`norm (vv 2)`;`norm (vv 0)`;`dist(vv 0,vv 2)`;`dist(vv 0,vv 3)`;`dist(vv 2,vv 3)`];
  REWRITE_TAC[Sphere.ineq];
  REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ANTS_TAC;
    ASM_REWRITE_TAC[];
    BY(REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0] THEN TRY REAL_ARITH_TAC);
  TYPIFY `taum (norm (vv 3)) (norm (vv 2)) (norm (vv 0)) (dist (vv 0,vv 2)) (dist (vv 0,vv 3)) (dist (vv 2,vv 3)) = taum (norm (vv 2)) (norm (vv 3)) (norm (vv 0)) (dist (vv 0,vv 3))     (dist (vv 0,vv 2))     (dist (vv 2,vv 3))` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  DISCH_TAC;
  FIRST_X_ASSUM_ST `~` MP_TAC;
  REWRITE_TAC[DE_MORGAN_THM];
  COMMENT "four cases on top edge";
  DISCH_TAC THEN REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC);
        INTRO_TAC ((* Appendix. *) get_main_nonlinear "9563139965 e") [`norm(vv 1)`;`norm (vv 2)`;`norm (vv 0)`;`dist(vv 0,vv 2)`;`dist(vv 0,vv 1)`;`dist(vv 1,vv 2)`];
        REWRITE_TAC[Sphere.ineq];
        REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
        ANTS_TAC;
          ASM_REWRITE_TAC[];
          TYPIFY `#2.467 <= dist(vv 0,vv 1)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
            BY(FIRST_X_ASSUM MP_TAC THEN REAL_ARITH_TAC);
          BY(REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0] THEN FIRST_X_ASSUM MP_TAC THEN TRY REAL_ARITH_TAC);
        TYPIFY `taum (norm (vv 1)) (norm (vv 2)) (norm (vv 0)) (dist (vv 0,vv 2)) (dist (vv 0,vv 1)) (dist (vv 1,vv 2)) = taum (norm (vv 0)) (norm (vv 1)) (norm (vv 2)) (dist (vv 1,vv 2))     (dist (vv 0,vv 2))     (dist (vv 0,vv 1))` (C SUBGOAL_THEN SUBST1_TAC);
          BY(MESON_TAC[taum_sym]);
        BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
      COMMENT "3 left";
      INTRO_TAC ((* Appendix. *) get_main_nonlinear "9563139965 e") [`norm(vv 1)`;`norm (vv 0)`;`norm (vv 2)`;`dist(vv 0,vv 2)`;`dist(vv 1,vv 2)`;`dist(vv 0,vv 1)`];
      REWRITE_TAC[Sphere.ineq];
      REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
      ANTS_TAC;
        ASM_REWRITE_TAC[];
        TYPIFY `#2.467 <= dist(vv 1,vv 2)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
          BY(FIRST_X_ASSUM MP_TAC THEN REAL_ARITH_TAC);
        BY(REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0] THEN FIRST_X_ASSUM MP_TAC THEN TRY REAL_ARITH_TAC);
      TYPIFY `taum (norm (vv 1)) (norm (vv 0)) (norm (vv 2)) (dist (vv 0,vv 2)) (dist (vv 1,vv 2)) (dist (vv 0,vv 1)) = taum (norm (vv 0)) (norm (vv 1)) (norm (vv 2)) (dist (vv 1,vv 2))     (dist (vv 0,vv 2))     (dist (vv 0,vv 1))` (C SUBGOAL_THEN SUBST1_TAC);
        BY(MESON_TAC[taum_sym]);
      BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
    COMMENT "2 left";
    INTRO_TAC ((* Appendix. *) get_main_nonlinear "9563139965 e") [`norm(vv 3)`;`norm (vv 0)`;`norm (vv 2)`;`dist(vv 0,vv 2)`;`dist(vv 2,vv 3)`;`dist(vv 0,vv 3)`];
    REWRITE_TAC[Sphere.ineq];
    REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
    ANTS_TAC;
      ASM_REWRITE_TAC[];
      TYPIFY `#2.467 <= dist(vv 2,vv 3)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
        BY(FIRST_X_ASSUM MP_TAC THEN REAL_ARITH_TAC);
      BY(REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0] THEN FIRST_X_ASSUM MP_TAC THEN TRY REAL_ARITH_TAC);
    TYPIFY `taum (norm (vv 3)) (norm (vv 0)) (norm (vv 2)) (dist (vv 0,vv 2)) (dist (vv 2,vv 3)) (dist (vv 0,vv 3)) = taum (norm (vv 2)) (norm (vv 3)) (norm (vv 0)) (dist (vv 0,vv 3))     (dist (vv 0,vv 2))     (dist (vv 2,vv 3))` (C SUBGOAL_THEN SUBST1_TAC);
      BY(MESON_TAC[taum_sym]);
    BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  COMMENT "1 left";
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "9563139965 e") [`norm(vv 3)`;`norm (vv 2)`;`norm (vv 0)`;`dist(vv 0,vv 2)`;`dist(vv 0,vv 3)`;`dist(vv 2,vv 3)`];
  REWRITE_TAC[Sphere.ineq];
  REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ANTS_TAC;
    ASM_REWRITE_TAC[];
    TYPIFY `#2.467 <= dist(vv 0,vv 3)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
      BY(FIRST_X_ASSUM MP_TAC THEN REAL_ARITH_TAC);
    BY(REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0] THEN FIRST_X_ASSUM MP_TAC THEN TRY REAL_ARITH_TAC);
  TYPIFY `taum (norm (vv 3)) (norm (vv 2)) (norm (vv 0)) (dist (vv 0,vv 2)) (dist (vv 0,vv 3)) (dist (vv 2,vv 3)) =  taum (norm (vv 2)) (norm (vv 3)) (norm (vv 0)) (dist (vv 0,vv 3))     (dist (vv 0,vv 2))     (dist (vv 2,vv 3))` (C SUBGOAL_THEN SUBST1_TAC);
    BY(MESON_TAC[taum_sym]);
  BY(REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

let quad_4680581274_derived = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> (!y0 y1 y2 y3 y4 y5 y6 y7 y8 y9.  ineq [
      (#3.01,y0,&4);
      (#2.0,y1,&2 * h0);
      (#2.0,y2,&2 * h0);
      (#2.0,y3,&2 * h0);
      (#3.01,y4,&4);
      (#2.0,y5,&2);
      (#2.0,y6,&2);
      (#2.0,y7,&2 * h0);
      (#2.0,y8,&2);
      (#3.01,y9,#3.01)]
( 
   (enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 = y0) /\ 
  ( y4 <= y0 ) /\ &0 <=  delta_y y0 y9 y8 y4 y5 y6 ==>
  // #0.616 - #0.11 
  #0.513
   < tauq y1 y2 y3 y4 y5 y6 y7 y8 y9 ))
`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `y4 <= #3.166` (C SUBGOAL_THEN ASSUME_TAC);
    PROOF_BY_CONTR_TAC;
    INTRO_TAC ((* Appendix. *) get_main_nonlinear "4680581274 delta top issue") [`y0`;`y9`;`y8`;`y4`;`y5`;`y6`];
    REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "4559601669") [`y1`;`y5`;`y6`;`y4`;`y2`;`y3`];
  REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ANTS_TAC;
    BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN TRY REAL_ARITH_TAC);
  TYPIFY `delta4_y y1 y5 y6 y4 y2 y3 = delta4_y y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    REWRITE_TAC[Sphere.delta4_y;Sphere.y_of_x;Sphere.delta_x4];
    BY(REAL_ARITH_TAC);
  DISCH_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "4680581274 delta issue") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`;`y7`;`y8`;`y9`];
  REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ANTS_TAC;
    BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN TRY REAL_ARITH_TAC);
  ASM_SIMP_TAC[arith `d4 < &0 ==> ~(d4 > &0)`;arith `c <= y0 ==> ~(y0 < c)`];
  REWRITE_TAC[arith `x > y <=> y < x`];
  DISCH_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "4680581274 a") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`;`y7`;`y8`;`y9`];
  REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ANTS_TAC;
    BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN TRY REAL_ARITH_TAC);
  ASM_SIMP_TAC[arith `d4 < &0 ==> ~(d4 > &0)`;arith `c <= y0 ==> ~(y0 < c)`;arith `c < d ==> ~(d < c)`];
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)

(* deprecated 2013-06-17
let quad_7697147739_derived = prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> (!y0 y1 y2 y3 y4 y5 y6 y7 y8 y9.  ineq [
      (#3.01,y0,&4);
      (#2.0,y1,&2 * h0);
      (#2.0,y2,&2 * h0);
      (#2.0,y3,&2 * h0);
      (#3.01,y4,&4);
      (#2.0,y5,&2);
      (#2.0,y6,&2);
      (#2.0,y7,&2 * h0);
      (#2.0,y8,&2);
      (sqrt8,y9,sqrt8)]
( 
   (enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 = y0) /\ 
  ( y4 <= y0 ) /\ &0 <=  delta_y y0 y9 y8 y4 y5 y6 ==>
  #0.616 - #0.11 < tauq y1 y2 y3 y4 y5 y6 y7 y8 y9 ))
`,
  (* {{{ proof *)
  [
  REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `y4 <= #3.108` (C SUBGOAL_THEN ASSUME_TAC);
    PROOF_BY_CONTR_TAC;
    INTRO_TAC ((* Appendix. *) get_main_nonlinear "7697147739 delta top issue") [`y0`;`y9`;`y8`;`y4`;`y5`;`y6`];
    REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "4559601669") [`y1`;`y5`;`y6`;`y4`;`y2`;`y3`];
  REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ANTS_TAC;
    BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN TRY REAL_ARITH_TAC);
  TYPIFY `delta4_y y1 y5 y6 y4 y2 y3 = delta4_y y1 y2 y3 y4 y5 y6` (C SUBGOAL_THEN SUBST1_TAC);
    REWRITE_TAC[Sphere.delta4_y;Sphere.y_of_x;Sphere.delta_x4];
    BY(REAL_ARITH_TAC);
  DISCH_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "7697147739 delta issue") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`;`y7`;`y8`;`y9`];
  REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ANTS_TAC;
    BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN TRY REAL_ARITH_TAC);
  ASM_SIMP_TAC[arith `d4 < &0 ==> ~(d4 > &0)`;arith `c <= y0 ==> ~(y0 < c)`];
  REWRITE_TAC[arith `x > y <=> y < x`];
  DISCH_TAC;
  INTRO_TAC ((* Appendix. *) get_main_nonlinear "7697147739 a") [`y1`;`y2`;`y3`;`y4`;`y5`;`y6`;`y7`;`y8`;`y9`];
  REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ANTS_TAC;
    BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Sphere.h0] THEN TRY REAL_ARITH_TAC);
  ASM_SIMP_TAC[arith `d4 < &0 ==> ~(d4 > &0)`;arith `c <= y0 ==> ~(y0 < c)`;arith `c < d ==> ~(d < c)`];
  BY(REAL_ARITH_TAC)
  ]);;
  (* }}} *)
*)


(* renamed from terminal_adhoc_quad_4680581274_empty, 2013-06-26, constant #0.616- #0.11 -> #0.513 *)
let empty_4T3  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_4T3 in
  //mk_unadorned_v39 4 ( #0.616 - #0.11)
  // (funlist_v39 [(0,1),cstab; (0,2),cstab; (1,3),cstab] (&2) 4)
  // (funlist_v39 [(0,1),cstab; (0,2),&6; (1,3),&6] (&2) 4) in 
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_4T3];
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[LET_THM];
  MATCH_MP_TAC (REWRITE_RULE[LET_THM] (UNDISCH terminal_quad_lemma));
  COMMENT "digression";
  TYPIFY `(!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> &2 <= funlist_v39 [(0,1),cstab; (0,2),cstab; (1,3),cstab] (&2) 4 i j) ` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `i < 4 <=> (i = 0 \/ i = 1 \/ i = 2 \/ i = 3)`];
    REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`;TAUT `a /\ (b \/ c) <=> (a /\ b) \/ a /\ c`];
    REPEAT WEAKER_STRIP_TAC;
    BY((REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4`;MOD_4_EXPLICIT;FUNLIST_EXPLICIT;arith `&2 <= &2 /\ &2 <= &3`;arith `~(0=1) /\ ~(0=2) /\ ~(0=3) /\ ~(1 =2) /\ ~(1 = 3) /\ ~(2 = 3)`;Appendix.cstab;arith `&2 <= #3.01`]) THEN ASM_MESON_TAC[]);
  COMMENT "conj1";
  CONJ_TAC;
    BY(ASM_MESON_TAC[arith `&2 <= x ==> &0 < x`]);
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "prep dist";
  TYPIFY `(!i j d. d <= funlist_v39 [(0,1),cstab; (0,2),cstab; (1,3),cstab] (&2) 4 i j  ==> d <= dist (vv i,vv j)) /\ (!i j d. funlist_v39 [(0,1),cstab; (0,2),&6; (1,3),&6] (&2) 4 i j   <= d ==>   dist (vv i,vv j) <= d)` ((C SUBGOAL_THEN MP_TAC));
    BY(ASM_MESON_TAC[REAL_LE_TRANS]);
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[arith `x + &0 = x /\ &4 - &2 = &2`];
  TYPIFY `(!i j. i < 4 /\ j < 4 /\ vv i = vv j ==> i = j)` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    PROOF_BY_CONTR_TAC;
    BY(ASM_MESON_TAC[DIST_REFL;REAL_LE_TRANS;arith `~(&2 <= &0)`]);
  COMMENT "vv";
  TYPIFY `vv 4 = vv 0 /\ vv 5 = vv 1 /\ vv 6 = vv 2` (C SUBGOAL_THEN ASSUME_TAC);
    BY((REPEAT CONJ_TAC) THEN GMATCH_SIMP_TAC periodic_vv_inj THEN TYPIFY `4` EXISTS_TAC THEN ASM_REWRITE_TAC[MOD_4_EXPLICIT;arith `~(4 = 0)`]);
  ASM_REWRITE_TAC[];
  INTRO_TAC vv_quad_split_short [`vv`];
  ASM_REWRITE_TAC[LET_THM];
  COMMENT "dist";
  TYPIFY `(!i j. ~(vv i = vv j) ==> &2 <= dist (vv i,vv j))` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC periodic2_mod_reduce;
    TYPIFY `4` EXISTS_TAC;
    REWRITE_TAC[Appendix.periodic2;arith `~(4 = 0)`];
    TYPIFY `!i. vv (i + 4) = vv i` (C SUBGOAL_THEN (unlist REWRITE_TAC));
      BY(ASM_MESON_TAC[Oxl_def.periodic;arith `~(4 = 0)`]);
    REPEAT WEAKER_STRIP_TAC;
    BY(ASM_MESON_TAC[REAL_LE_TRANS]);
  ASM_REWRITE_TAC[];
  COMMENT "edge lengths";
  TYPIFY `I(dist(vv 0,vv 1) = cstab /\ dist(vv 1,vv 2) = &2 /\ dist(vv 2, vv 3) = &2 /\ dist(vv 0,vv 3) = &2)` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[I_DEF];
    REWRITE_TAC[arith `d = c <=> d <= c /\ c <= d`];
    FIRST_X_ASSUM kill;
    BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[FUNLIST_EXPLICIT;MOD_4_EXPLICIT;Appendix.cstab] THEN TRY REAL_ARITH_TAC);
  TYPIFY `dist(vv 1,vv 0) = dist(vv 0,vv 1) /\ dist (vv 2, vv 0) = dist(vv 0,vv 2) /\ dist(vv 3,vv 0) = dist (vv 0, vv 3) /\ dist (vv 2,vv 1) = dist (vv 1, vv 2) /\ dist (vv 3,vv 2) = dist (vv 2, vv 3) /\ dist(vv 3,vv 1)=dist(vv 1,vv 3)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[DIST_SYM]);
  DISCH_TAC;
  COMMENT "diags";
  TYPIFY `dist(vv 0,vv 2) < &4` (C SUBGOAL_THEN ASSUME_TAC);
    PROOF_BY_CONTR_TAC;
    INTRO_TAC delta_4680581274 [`dist(vv 1,vv 3)`;`dist(vv 0,vv 2)`];
    ANTS_TAC;
      CONJ_TAC;
        (FIRST_X_ASSUM MATCH_MP_TAC) THEN REWRITE_TAC[FUNLIST_EXPLICIT;Appendix.cstab;MOD_4_EXPLICIT;arith `~(1 = 3)`];
        BY(REAL_ARITH_TAC);
      BY(FIRST_X_ASSUM MP_TAC THEN REAL_ARITH_TAC);
    REWRITE_TAC[arith `~(d < &0) <=> &0 <= d`];
    TYPIFY `delta_y (dist (vv 1,vv 3)) (&2) (&2) (dist (vv 0,vv 2)) (&2) cstab = delta_y (dist(vv 3,vv 1)) (dist(vv 3,vv 0)) (dist(vv 3,vv 2)) (dist(vv 0,vv 2)) (dist (vv 1, vv 2)) (dist (vv 1, vv 0))` ENOUGH_TO_SHOW_TAC;
      DISCH_THEN SUBST1_TAC;
      BY(REWRITE_TAC[ DELTA_Y_POS_4POINTS]);
    RULE_ASSUM_TAC (REWRITE_RULE[I_THM]);
    BY(ASM_REWRITE_TAC[]);
  COMMENT "second diag";
  TYPIFY `dist(vv 1,vv 3) < &4` (C SUBGOAL_THEN ASSUME_TAC);
    PROOF_BY_CONTR_TAC;
    INTRO_TAC delta_4680581274 [`dist(vv 2,vv 0)`;`dist(vv 1,vv 3)`];
    ANTS_TAC;
      CONJ_TAC;
        (FIRST_X_ASSUM MATCH_MP_TAC) THEN REWRITE_TAC[FUNLIST_EXPLICIT;Appendix.cstab;MOD_4_EXPLICIT;arith `~(0=2)`];
        BY(REAL_ARITH_TAC);
      BY(FIRST_X_ASSUM MP_TAC THEN REAL_ARITH_TAC);
    REWRITE_TAC[arith `~(d < &0) <=> &0 <= d`];
    TYPIFY `delta_y (dist (vv 2,vv 0)) (&2) (&2) (dist (vv 1,vv 3)) (&2) cstab = delta_y (dist(vv 2,vv 0)) (dist(vv 2,vv 1)) (dist(vv 2,vv 3)) (dist(vv 1,vv 3)) (dist (vv 0, vv 3)) (dist (vv 0, vv 1))` ENOUGH_TO_SHOW_TAC;
      DISCH_THEN SUBST1_TAC;
      BY(REWRITE_TAC[ DELTA_Y_POS_4POINTS]);
    RULE_ASSUM_TAC (REWRITE_RULE[I_THM]);
    BY(ASM_REWRITE_TAC[]);
  FIRST_X_ASSUM_ST `rho_fun` MP_TAC;
  ANTS_TAC;
    RULE_ASSUM_TAC (REWRITE_RULE[I_THM]) THEN ASM_REWRITE_TAC[];
    BY(REWRITE_TAC[Appendix.cstab] THEN REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  ASM_REWRITE_TAC[];
  COMMENT "enclosed";
  INTRO_TAC vv_enclosed4 [`vv`;`i`];
  REWRITE_TAC[LET_THM];
  ASM_REWRITE_TAC[];
  RULE_ASSUM_TAC(REWRITE_RULE[arith `i < 2 <=> i = 0 \/ i = 1`]);
  ANTS_TAC;
    FIRST_X_ASSUM DISJ_CASES_TAC;
      RULE_ASSUM_TAC(REWRITE_RULE[I_THM]);
      ASM_REWRITE_TAC[arith `0+x = x`;Appendix.cstab];
      BY(REAL_ARITH_TAC);
    RULE_ASSUM_TAC(REWRITE_RULE[I_THM]);
    ASM_REWRITE_TAC[arith `1+x = SUC x`;Appendix.cstab;SUC_EXPLICIT];
    BY(REAL_ARITH_TAC);
  DISCH_THEN (ASSUME_TAC o GSYM);
  COMMENT "relabel vars";
  TYPIFY `?v1 v2 v3 v4.  (    let y1 = norm v1 in    let y2 = norm v2 in    let y3 = norm v3 in    let y4 = dist(v2,v3) in    let y5 = dist(v1,v3) in    let y6 = dist(v1,v2) in    let y7 = norm v4 in    let y8 = dist(v3,v4) in    let y9 = dist(v2,v4) in    let y0 = enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 in      ({v1,v2,v3,v4} SUBSET ball_annulus /\ 	 tau3 (vv i) (vv (i + 1)) (vv (i + 2)) +	 tau3 (vv (i + 2)) (vv (i + 3)) (vv i) = tau3 v1 v2 v3 + tau3 v4 v2 v3 /\	  y4 <= y0 /\	  cstab <= y4 /\ y0 < &4 /\	  y9 = cstab /\ y8 = &2 /\ y5 = &2 /\ y6 = &2 /\ y0 = dist(v1,v4)))` (C SUBGOAL_THEN MP_TAC);
    FIRST_X_ASSUM DISJ_CASES_TAC;
      ASM_REWRITE_TAC[LET_THM;arith `0+x = x`];
      FIRST_X_ASSUM (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
      RULE_ASSUM_TAC (REWRITE_RULE[arith `0 + x = x`]);
      GEXISTL_TAC [`vv 3`;`vv 0`;`vv 2`;`vv 1`];
      ASM_REWRITE_TAC[];
      ONCE_REWRITE_TAC[enclosed_sym2];
      ONCE_REWRITE_TAC[enclosed_sym];
      ASM_REWRITE_TAC[];
      CONJ_TAC;
        RULE_ASSUM_TAC(REWRITE_RULE[IMAGE_SUBSET_IN;IN_UNIV]);
        ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY];
        BY(FIRST_X_ASSUM_ST `ball_annulus` MP_TAC THEN MESON_TAC[]);
      CONJ_TAC;
        BY(ASM_MESON_TAC[tau3_sym;arith `a + b = b + a`]);
      RULE_ASSUM_TAC(REWRITE_RULE[I_THM]);
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM MATCH_MP_TAC;
      REWRITE_TAC[FUNLIST_EXPLICIT;MOD_4_EXPLICIT];
      BY(REAL_ARITH_TAC);
    COMMENT "second case relabel";
    ASM_REWRITE_TAC[LET_THM;arith `1+x = SUC x`;SUC_EXPLICIT];
    FIRST_X_ASSUM (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
    RULE_ASSUM_TAC (REWRITE_RULE[arith `1 + x = SUC x`;SUC_EXPLICIT]);
    FIRST_X_ASSUM_ST `vv 4 = vv 0` (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
    GEXISTL_TAC [`vv 2`;`vv 1`;`vv 3`;`vv 0`];
    ASM_REWRITE_TAC[];
    ONCE_REWRITE_TAC[enclosed_sym];
    FIRST_X_ASSUM_ST `enclosed` MP_TAC;
    ASM_REWRITE_TAC[];
    DISCH_THEN SUBST1_TAC;
    CONJ_TAC;
      RULE_ASSUM_TAC(REWRITE_RULE[IMAGE_SUBSET_IN;IN_UNIV]);
      ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY];
      BY(FIRST_X_ASSUM_ST `ball_annulus` MP_TAC THEN MESON_TAC[]);
    CONJ_TAC;
      BY(MESON_TAC[tau3_sym]);
    FIRST_X_ASSUM_ST `<=` MP_TAC;
    RULE_ASSUM_TAC(REWRITE_RULE[I_THM]);
    ASM_REWRITE_TAC[];
    DISCH_THEN (unlist REWRITE_TAC);
    FIRST_X_ASSUM MATCH_MP_TAC;
    REWRITE_TAC[FUNLIST_EXPLICIT;MOD_4_EXPLICIT];
    BY(REAL_ARITH_TAC);
  COMMENT "final kill";
  FIRST_X_ASSUM_ST `main_nonlinear_terminal_v11` MP_TAC;
  REPEAT (FIRST_X_ASSUM kill);
  REWRITE_TAC[LET_THM];
  REPEAT WEAKER_STRIP_TAC;
  ASM_REWRITE_TAC[];
  RULE_ASSUM_TAC(ONCE_REWRITE_RULE[ISPEC `(a:real) = b` (GSYM I_THM)]);
  REPEAT (GMATCH_SIMP_TAC tau3_taum_40);
  RULE_ASSUM_TAC(REWRITE_RULE[INSERT_SUBSET]) THEN ASM_REWRITE_TAC[DIST_SYM];
  CONJ_TAC;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab;I_THM] THEN REAL_ARITH_TAC);
  CONJ_TAC;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab;I_THM] THEN REAL_ARITH_TAC);
  MATCH_MP_TAC (arith `x < y ==> x <= y`);
  COMMENT "get";
  MP_TAC (UNDISCH quad_4680581274_derived);
  REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  REWRITE_TAC[Sphere.tauq];
  DISCH_THEN MATCH_MP_TAC;
  TYPIFY `dist(v1,v4)` EXISTS_TAC;
  FIRST_X_ASSUM MP_TAC;
  REWRITE_TAC[I_THM] THEN DISCH_THEN (fun t -> RULE_ASSUM_TAC(REWRITE_RULE[t]) THEN SUBST1_TAC t);
  TYPIFY `&0 <= delta_y (dist (v1,v4)) (dist (v2,v4)) (dist (v3,v4)) (dist (v2,v3)) (dist (v1,v3)) (dist (v1,v2))` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    TYPIFY `dist (v1,v4) = dist(v4,v1) /\ dist(v2,v4) = dist(v4,v2) /\ dist(v3,v4) = dist(v4,v3)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
      BY(MESON_TAC[DIST_SYM]);
    BY(REWRITE_TAC[DELTA_Y_POS_4POINTS]);
  RULE_ASSUM_TAC(REWRITE_RULE[I_THM;Appendix.cstab;Fnjlbxs.in_ball_annulus]) THEN ASM_REWRITE_TAC[Appendix.cstab;arith `#2.0= &2`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)

(* deprecated 2013-06-17:
let terminal_adhoc_quad_7697147739_empty = prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = mk_unadorned_v39 4 ( #0.616 -  #0.11)
   (funlist_v39 [(0,1),sqrt8; (0,2),cstab; (1,3),cstab] (&2) 4)
   (funlist_v39 [(0,1),sqrt8; (0,2),&6; (1,3),&6] (&2) 4) in
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))`,
  (* {{{ proof *)
  [
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[LET_THM];
  MATCH_MP_TAC (REWRITE_RULE[LET_THM] (UNDISCH terminal_quad_lemma));
  COMMENT "sqrt8 bounds";
  TYPIFY `#2.828427 < sqrt8 /\ sqrt8 <  #2.828428` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REWRITE_TAC[Flyspeck_constants.bounds]);
  TYPIFY `&2 <= sqrt8` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC);
  COMMENT "digression";
  TYPIFY `(!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> &2 <= funlist_v39 [(0,1),sqrt8; (0,2),cstab; (1,3),cstab] (&2) 4 i j) ` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `i < 4 <=> (i = 0 \/ i = 1 \/ i = 2 \/ i = 3)`];
    REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`;TAUT `a /\ (b \/ c) <=> (a /\ b) \/ a /\ c`];
    REPEAT WEAKER_STRIP_TAC;
    BY(((REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4`;MOD_4_EXPLICIT;FUNLIST_EXPLICIT;arith `&2 <= &2 /\ &2 <= &3`;arith `~(0=1) /\ ~(0=2) /\ ~(0=3) /\ ~(1 =2) /\ ~(1 = 3) /\ ~(2 = 3)`;Appendix.cstab;arith `&2 <= #3.01`])) THEN TRY (ASM_MESON_TAC[]));
  COMMENT "conj1";
  CONJ_TAC;
    BY(ASM_MESON_TAC[arith `&2 <= x ==> &0 < x`]);
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "prep dist";
  TYPIFY `(!i j d. d <= funlist_v39 [(0,1),sqrt8; (0,2),cstab; (1,3),cstab] (&2) 4 i j  ==> d <= dist (vv i,vv j)) /\ (!i j d. funlist_v39 [(0,1),sqrt8; (0,2),&6; (1,3),&6] (&2) 4 i j   <= d ==>   dist (vv i,vv j) <= d)` ((C SUBGOAL_THEN MP_TAC));
    BY(ASM_MESON_TAC[REAL_LE_TRANS]);
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[arith `x + &0 = x /\ &4 - &2 = &2`];
  TYPIFY `(!i j. i < 4 /\ j < 4 /\ vv i = vv j ==> i = j)` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    PROOF_BY_CONTR_TAC;
    BY(ASM_MESON_TAC[DIST_REFL;REAL_LE_TRANS;arith `~(&2 <= &0)`]);
  COMMENT "vv";
  TYPIFY `vv 4 = vv 0 /\ vv 5 = vv 1 /\ vv 6 = vv 2` (C SUBGOAL_THEN ASSUME_TAC);
    BY((REPEAT CONJ_TAC) THEN GMATCH_SIMP_TAC periodic_vv_inj THEN TYPIFY `4` EXISTS_TAC THEN ASM_REWRITE_TAC[MOD_4_EXPLICIT;arith `~(4 = 0)`]);
  ASM_REWRITE_TAC[];
  INTRO_TAC vv_quad_split_short [`vv`];
  ASM_REWRITE_TAC[LET_THM];
  COMMENT "dist";
  TYPIFY `(!i j. ~(vv i = vv j) ==> &2 <= dist (vv i,vv j))` (C SUBGOAL_THEN ASSUME_TAC);
    MATCH_MP_TAC periodic2_mod_reduce;
    TYPIFY `4` EXISTS_TAC;
    REWRITE_TAC[Appendix.periodic2;arith `~(4 = 0)`];
    TYPIFY `!i. vv (i + 4) = vv i` (C SUBGOAL_THEN (unlist REWRITE_TAC));
      BY(ASM_MESON_TAC[Oxl_def.periodic;arith `~(4 = 0)`]);
    REPEAT WEAKER_STRIP_TAC;
    BY(ASM_MESON_TAC[REAL_LE_TRANS]);
  ASM_REWRITE_TAC[];
  COMMENT "edge lengths";
  TYPIFY `I(dist(vv 0,vv 1) = sqrt8 /\ dist(vv 1,vv 2) = &2 /\ dist(vv 2, vv 3) = &2 /\ dist(vv 0,vv 3) = &2)` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[I_DEF];
    REWRITE_TAC[arith `d = c <=> d <= c /\ c <= d`];
    FIRST_X_ASSUM kill;
    BY((REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[FUNLIST_EXPLICIT;MOD_4_EXPLICIT;Appendix.cstab] THEN TRY REAL_ARITH_TAC));
  TYPIFY `dist(vv 1,vv 0) = dist(vv 0,vv 1) /\ dist (vv 2, vv 0) = dist(vv 0,vv 2) /\ dist(vv 3,vv 0) = dist (vv 0, vv 3) /\ dist (vv 2,vv 1) = dist (vv 1, vv 2) /\ dist (vv 3,vv 2) = dist (vv 2, vv 3) /\ dist(vv 3,vv 1)=dist(vv 1,vv 3)` (C SUBGOAL_THEN ASSUME_TAC);
    BY(MESON_TAC[DIST_SYM]);
  DISCH_TAC;
  COMMENT "diags";
  TYPIFY `dist(vv 0,vv 2) < &4` (C SUBGOAL_THEN ASSUME_TAC);
    PROOF_BY_CONTR_TAC;
    INTRO_TAC delta_7697147739 [`dist(vv 1,vv 3)`;`dist(vv 0,vv 2)`];
    ANTS_TAC;
      CONJ_TAC;
        (FIRST_X_ASSUM MATCH_MP_TAC) THEN REWRITE_TAC[FUNLIST_EXPLICIT;Appendix.cstab;MOD_4_EXPLICIT;arith `~(1 = 3)`];
        BY(REAL_ARITH_TAC);
      BY(FIRST_X_ASSUM MP_TAC THEN REAL_ARITH_TAC);
    REWRITE_TAC[arith `~(d < &0) <=> &0 <= d`];
    TYPIFY `delta_y (dist (vv 1,vv 3)) (&2) (&2) (dist (vv 0,vv 2)) (&2) sqrt8 = delta_y (dist(vv 3,vv 1)) (dist(vv 3,vv 0)) (dist(vv 3,vv 2)) (dist(vv 0,vv 2)) (dist (vv 1, vv 2)) (dist (vv 1, vv 0))` ENOUGH_TO_SHOW_TAC;
      DISCH_THEN SUBST1_TAC;
      BY(REWRITE_TAC[ DELTA_Y_POS_4POINTS]);
    RULE_ASSUM_TAC (REWRITE_RULE[I_THM]);
    BY(ASM_REWRITE_TAC[]);
  COMMENT "second diag";
  TYPIFY `dist(vv 1,vv 3) < &4` (C SUBGOAL_THEN ASSUME_TAC);
    PROOF_BY_CONTR_TAC;
    INTRO_TAC delta_7697147739 [`dist(vv 2,vv 0)`;`dist(vv 1,vv 3)`];
    ANTS_TAC;
      CONJ_TAC;
        (FIRST_X_ASSUM MATCH_MP_TAC) THEN REWRITE_TAC[FUNLIST_EXPLICIT;Appendix.cstab;MOD_4_EXPLICIT;arith `~(0=2)`];
        BY(REAL_ARITH_TAC);
      BY(FIRST_X_ASSUM MP_TAC THEN REAL_ARITH_TAC);
    REWRITE_TAC[arith `~(d < &0) <=> &0 <= d`];
    TYPIFY `delta_y (dist (vv 2,vv 0)) (&2) (&2) (dist (vv 1,vv 3)) (&2) sqrt8 = delta_y (dist(vv 2,vv 0)) (dist(vv 2,vv 1)) (dist(vv 2,vv 3)) (dist(vv 1,vv 3)) (dist (vv 0, vv 3)) (dist (vv 0, vv 1))` ENOUGH_TO_SHOW_TAC;
      DISCH_THEN SUBST1_TAC;
      BY(REWRITE_TAC[ DELTA_Y_POS_4POINTS]);
    RULE_ASSUM_TAC (REWRITE_RULE[I_THM]);
    BY(ASM_REWRITE_TAC[]);
  FIRST_X_ASSUM_ST `rho_fun` MP_TAC;
  ANTS_TAC;
    RULE_ASSUM_TAC (REWRITE_RULE[I_THM]) THEN ASM_REWRITE_TAC[];
    BY(REPEAT (FIRST_X_ASSUM_ST `sqrt8 < c` MP_TAC) THEN REAL_ARITH_TAC);
  REPEAT WEAKER_STRIP_TAC;
  ASM_REWRITE_TAC[];
  COMMENT "enclosed";
  INTRO_TAC vv_enclosed4 [`vv`;`i`];
  REWRITE_TAC[LET_THM];
  ASM_REWRITE_TAC[];
  RULE_ASSUM_TAC(REWRITE_RULE[arith `i < 2 <=> i = 0 \/ i = 1`]);
  ANTS_TAC;
    FIRST_X_ASSUM DISJ_CASES_TAC;
      RULE_ASSUM_TAC(REWRITE_RULE[I_THM]);
      ASM_REWRITE_TAC[arith `0+x = x`;Appendix.cstab];
      BY(REPEAT (FIRST_X_ASSUM_ST `sqrt8 < c` MP_TAC) THEN REAL_ARITH_TAC);
    RULE_ASSUM_TAC(REWRITE_RULE[I_THM]);
    ASM_REWRITE_TAC[arith `1+x = SUC x`;Appendix.cstab;SUC_EXPLICIT];
    BY(REPEAT (FIRST_X_ASSUM_ST `sqrt8 < c` MP_TAC) THEN REAL_ARITH_TAC);
  DISCH_THEN (ASSUME_TAC o GSYM);
  COMMENT "relabel vars";
  TYPIFY `?v1 v2 v3 v4.  (    let y1 = norm v1 in    let y2 = norm v2 in    let y3 = norm v3 in    let y4 = dist(v2,v3) in    let y5 = dist(v1,v3) in    let y6 = dist(v1,v2) in    let y7 = norm v4 in    let y8 = dist(v3,v4) in    let y9 = dist(v2,v4) in    let y0 = enclosed y1 y5 y6 y4 y2 y3 y7 y8 y9 in      ({v1,v2,v3,v4} SUBSET ball_annulus /\ 	 tau3 (vv i) (vv (i + 1)) (vv (i + 2)) +	 tau3 (vv (i + 2)) (vv (i + 3)) (vv i) = tau3 v1 v2 v3 + tau3 v4 v2 v3 /\	  y4 <= y0 /\	  cstab <= y4 /\ y0 < &4 /\	  y9 = sqrt8 /\ y8 = &2 /\ y5 = &2 /\ y6 = &2 /\ y0 = dist(v1,v4)))` (C SUBGOAL_THEN MP_TAC);
    FIRST_X_ASSUM DISJ_CASES_TAC;
      ASM_REWRITE_TAC[LET_THM;arith `0+x = x`];
      FIRST_X_ASSUM (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
      RULE_ASSUM_TAC (REWRITE_RULE[arith `0 + x = x`]);
      GEXISTL_TAC [`vv 3`;`vv 0`;`vv 2`;`vv 1`];
      ASM_REWRITE_TAC[];
      ONCE_REWRITE_TAC[enclosed_sym2];
      ONCE_REWRITE_TAC[enclosed_sym];
      ASM_REWRITE_TAC[];
      CONJ_TAC;
        RULE_ASSUM_TAC(REWRITE_RULE[IMAGE_SUBSET_IN;IN_UNIV]);
        ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY];
        BY(FIRST_X_ASSUM_ST `ball_annulus` MP_TAC THEN MESON_TAC[]);
      CONJ_TAC;
        BY(ASM_MESON_TAC[tau3_sym;arith `a + b = b + a`]);
      RULE_ASSUM_TAC(REWRITE_RULE[I_THM]);
      ASM_REWRITE_TAC[];
      FIRST_X_ASSUM MATCH_MP_TAC;
      REWRITE_TAC[FUNLIST_EXPLICIT;MOD_4_EXPLICIT];
      BY(REAL_ARITH_TAC);
    COMMENT "second case relabel";
    ASM_REWRITE_TAC[LET_THM;arith `1+x = SUC x`;SUC_EXPLICIT];
    FIRST_X_ASSUM (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
    RULE_ASSUM_TAC (REWRITE_RULE[arith `1 + x = SUC x`;SUC_EXPLICIT]);
    FIRST_X_ASSUM_ST `vv 4 = vv 0` (RULE_ASSUM_TAC o (unlist REWRITE_RULE));
    GEXISTL_TAC [`vv 2`;`vv 1`;`vv 3`;`vv 0`];
    ASM_REWRITE_TAC[];
    ONCE_REWRITE_TAC[enclosed_sym];
    FIRST_X_ASSUM_ST `enclosed` MP_TAC;
    ASM_REWRITE_TAC[];
    DISCH_THEN SUBST1_TAC;
    CONJ_TAC;
      RULE_ASSUM_TAC(REWRITE_RULE[IMAGE_SUBSET_IN;IN_UNIV]);
      ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY];
      BY(FIRST_X_ASSUM_ST `ball_annulus` MP_TAC THEN MESON_TAC[]);
    CONJ_TAC;
      BY(MESON_TAC[tau3_sym]);
    FIRST_X_ASSUM_ST `<=` MP_TAC;
    RULE_ASSUM_TAC(REWRITE_RULE[I_THM]);
    ASM_REWRITE_TAC[];
    DISCH_THEN (unlist REWRITE_TAC);
    FIRST_X_ASSUM MATCH_MP_TAC;
    REWRITE_TAC[FUNLIST_EXPLICIT;MOD_4_EXPLICIT];
    BY(REAL_ARITH_TAC);
  COMMENT "final kill";
  FIRST_X_ASSUM_ST `main_nonlinear_terminal_v11` MP_TAC;
  FIRST_X_ASSUM_ST `sqrt8 < c` MP_TAC;
  REPEAT (FIRST_X_ASSUM kill);
  REWRITE_TAC[LET_THM];
  REPEAT WEAKER_STRIP_TAC;
  ASM_REWRITE_TAC[];
  RULE_ASSUM_TAC(ONCE_REWRITE_RULE[ISPEC `(a:real) = b` (GSYM I_THM)]);
  REPEAT (GMATCH_SIMP_TAC tau3_taum_40);
  RULE_ASSUM_TAC(REWRITE_RULE[INSERT_SUBSET]) THEN ASM_REWRITE_TAC[DIST_SYM];
  CONJ_TAC;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab;I_THM] THEN REAL_ARITH_TAC);
  CONJ_TAC;
    BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[Appendix.cstab;I_THM] THEN REAL_ARITH_TAC);
  MATCH_MP_TAC (arith `x < y ==> x <= y`);
  COMMENT "get";
  MP_TAC (UNDISCH quad_7697147739_derived);
  REWRITE_TAC[Sphere.ineq;TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  REWRITE_TAC[Sphere.tauq];
  DISCH_THEN MATCH_MP_TAC;
  TYPIFY `dist(v1,v4)` EXISTS_TAC;
  FIRST_X_ASSUM MP_TAC;
  REWRITE_TAC[I_THM] THEN DISCH_THEN (fun t -> RULE_ASSUM_TAC(REWRITE_RULE[t]) THEN SUBST1_TAC t);
  TYPIFY `&0 <= delta_y (dist (v1,v4)) (dist (v2,v4)) (dist (v3,v4)) (dist (v2,v3)) (dist (v1,v3)) (dist (v1,v2))` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    TYPIFY `dist (v1,v4) = dist(v4,v1) /\ dist(v2,v4) = dist(v4,v2) /\ dist(v3,v4) = dist(v4,v3)` (C SUBGOAL_THEN (unlist REWRITE_TAC));
      BY(MESON_TAC[DIST_SYM]);
    BY(REWRITE_TAC[DELTA_Y_POS_4POINTS]);
  RULE_ASSUM_TAC(REWRITE_RULE[I_THM;Appendix.cstab;Fnjlbxs.in_ball_annulus]) THEN ASM_REWRITE_TAC[Appendix.cstab;arith `#2.0= &2`];
  BY(REPEAT (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC)
  ]);;
  (* }}} *)
*)

let empty_4T4  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_4T4 in
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_4T4];
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[LET_THM];
  MATCH_MP_TAC (REWRITE_RULE[LET_THM] (UNDISCH terminal_quad_lemma));
  COMMENT "digression";
  TYPIFY `(!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> &2 <= funlist_v39 [(0,1),&2 * h0; (0,2),sqrt8; (1,3),sqrt8] (&2) 4 i j)` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `i < 4 <=> (i = 0 \/ i = 1 \/ i = 2 \/ i = 3)`];
    REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`;TAUT `a /\ (b \/ c) <=> (a /\ b) \/ a /\ c`];
    REWRITE_TAC[TAUT `(a \/ b) \/ c <=> a \/ b \/ c`;arith `~(i = k /\ j = k /\ ~(i = (j:num)))`];
    REPEAT WEAKER_STRIP_TAC;
    BY((REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[FUNLIST_EXPLICIT;arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4`;MOD_4_EXPLICIT;arith `&2 <= &2 /\ &2 <= &3`;arith `~(0=1) /\ ~(0=2) /\ ~(0=3) /\ ~(1 =2) /\ ~(1 = 3) /\ ~(2 = 3)`;Sphere.h0] THEN MP_TAC sqrt8_flyspeck  THEN TRY REAL_ARITH_TAC));
  COMMENT "digress2";
  TYPIFY `!i j. (i=0 /\ j=1) \/ (i=0 /\ j=3) \/ (i=1 /\ j=2) \/ (i=1 /\ j=3) \/ (i=2 /\ j=3) ==>  funlist_v39 [(0,1),sqrt8; (0,2),&6; (1,3),cstab] (&2 * h0) 4 i j < &4` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `i < 4 <=> (i = 0 \/ i = 1 \/ i = 2 \/ i = 3)`];
    REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`;TAUT `a /\ (b \/ c) <=> (a /\ b) \/ a /\ c`];
    REWRITE_TAC[TAUT `(a \/ b) \/ c <=> a \/ b \/ c`;arith `~(i = k /\ j = k /\ ~(i = (j:num)))`];
    REWRITE_TAC[Appendix.cs_adj;Sphere.h0];
    REPEAT WEAKER_STRIP_TAC;
    BY((REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[FUNLIST_EXPLICIT;arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4`;MOD_4_EXPLICIT;arith `&2 * #1.26 < &4 /\ &3 < &4`;arith `~(0=1) /\ ~(0=2) /\ ~(0=3) /\ ~(1 =2) /\ ~(1 = 3) /\ ~(2 = 3)`;Sphere.cstab] THEN MP_TAC sqrt8_flyspeck THEN TRY REAL_ARITH_TAC));
  CONJ_TAC;
    BY(ASM_MESON_TAC[arith `&2 <= x ==> &0 < x`]);
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[arith `x + &0 = x /\ &4 - &2 = &2`];
  TYPIFY `(!i j. i < 4 /\ j < 4 /\ vv i = vv j ==> i = j)` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    PROOF_BY_CONTR_TAC;
    BY(ASM_MESON_TAC[DIST_REFL;REAL_LE_TRANS;arith `~(&2 <= &0)`]);
  COMMENT "vv";
  TYPIFY `vv 4 = vv 0 /\ vv 5 = vv 1 /\ vv 6 = vv 2` (C SUBGOAL_THEN ASSUME_TAC);
    BY((REPEAT CONJ_TAC) THEN GMATCH_SIMP_TAC periodic_vv_inj THEN TYPIFY `4` EXISTS_TAC THEN ASM_REWRITE_TAC[MOD_4_EXPLICIT;arith `~(4 = 0)`]);
  ASM_REWRITE_TAC[];
  GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] vv_quad_split123);
  ASM_REWRITE_TAC[];
  COMMENT "dist";
  SUBCONJ_TAC;
    TYPIFY `(!i j. ~(vv i = vv j) ==> &2 <= dist (vv i,vv j))` (C SUBGOAL_THEN ASSUME_TAC);
      MATCH_MP_TAC periodic2_mod_reduce;
      TYPIFY `4` EXISTS_TAC;
      REWRITE_TAC[Appendix.periodic2;arith `~(4 = 0)`];
      TYPIFY `!i. vv (i + 4) = vv i` (C SUBGOAL_THEN (unlist REWRITE_TAC));
        BY(ASM_MESON_TAC[Oxl_def.periodic;arith `~(4 = 0)`]);
      REPEAT WEAKER_STRIP_TAC;
      BY(ASM_MESON_TAC[REAL_LE_TRANS]);
    ASM_REWRITE_TAC[];
    BY(ASM_MESON_TAC[REAL_LET_TRANS]);
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "tau3";
  REPEAT (GMATCH_SIMP_TAC tau3_taum_40);
  ASM_REWRITE_TAC[];
  RULE_ASSUM_TAC(REWRITE_RULE[IMAGE_SUBSET_IN;IN_UNIV]);
  ASM_REWRITE_TAC[DIST_SYM];
  TYPIFY `dist (vv 3,vv 0) < &4 ` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    ONCE_REWRITE_TAC[DIST_SYM];
    BY(ASM_REWRITE_TAC[]);
  REWRITE_TAC[TAUT `a /\ b /\ c <=> (a /\ b) /\ c`];
  CONJ_TAC;
    FIRST_X_ASSUM (REPEAT o GMATCH_SIMP_TAC);
    FIRST_X_ASSUM_ST `i < 4` MP_TAC;
    BY(MESON_TAC[arith `1 < 4 /\ 2 < 4 /\ 3 < 4 /\ 0 < 4 /\ ~(2=3) /\ ~(2  =1) /\ ~(1 =0) /\ ~(1 = 3) /\ ~(3=0 ) /\ ~(1 = 2)`]);
  COMMENT "prep dist";
  TYPED_ABBREV_TAC `a = funlist_v39 [(0,1),&2 * h0; (0,2),sqrt8; (1,3),sqrt8] (&2) 4`;
  TYPED_ABBREV_TAC `b = funlist_v39 [(0,1),sqrt8; (0,2),&6; (1,3),cstab] (&2 * h0) 4`;
  TYPIFY `(!i j d. d <= a i j  ==> d <= dist (vv i,vv j)) /\ (!i j d. b i j        <= d ==>   dist (vv i,vv j) <= d)` (C SUBGOAL_THEN MP_TAC);
    BY(ASM_MESON_TAC[REAL_LE_TRANS]);
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `a 1 0 = &2 * h0 /\ a 2 1 = &2 /\ a 2 3 = &2 /\ a 3 0 = &2 /\ a 1 3 = sqrt8 /\ b 1 0 = sqrt8 /\ b 2 1 = &2 * h0 /\ b 2 3 = &2 * h0 /\ b 3 0 = &2 * h0 /\ b 1 3 = cstab` (C SUBGOAL_THEN ASSUME_TAC);
    EXPAND_TAC "a";
    EXPAND_TAC "b";
    BY(REWRITE_TAC[FUNLIST_EXPLICIT;MOD_4_EXPLICIT]);
  RULE_ASSUM_TAC(REWRITE_RULE[Fnjlbxs.in_ball_annulus]);
  TYPIFY `#2.52 = &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
  COMMENT "fist get";
  INTRO_TAC (get_main_nonlinear "5405130650") [`norm (vv 0)`;`norm (vv 3)`;`norm (vv 1)`;`dist(vv 3,vv 1)`;`dist(vv 0,vv 1)`;`dist(vv 0,vv 3)`];
  REWRITE_TAC[Sphere.ineq];
  REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  REWRITE_TAC[arith `(d < &0) <=> ~(&0 <= d)`];
  REWRITE_TAC[REWRITE_RULE[LET_THM] Tame_lemmas.delta_y_pos];
  ASM_REWRITE_TAC[DIST_SYM;arith `#2.0 = &2`];
  ANTS_TAC;
    BY((REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0;Sphere.cstab] THEN TRY REAL_ARITH_TAC));
  DISCH_TAC;
  COMMENT "second get";
  INTRO_TAC (get_main_nonlinear "3603097872") [`norm (vv 2)`;`norm (vv 3)`;`norm (vv 1)`;`dist(vv 3,vv 1)`;`dist(vv 2,vv 1)`;`dist(vv 2,vv 3)`];
  REWRITE_TAC[Sphere.ineq];
  REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  REWRITE_TAC[arith `(d < &0) <=> ~(&0 <= d)`];
  REWRITE_TAC[REWRITE_RULE[LET_THM] Tame_lemmas.delta_y_pos];
  ASM_REWRITE_TAC[DIST_SYM;arith `#2.0 = &2`];
  ANTS_TAC;
    BY((REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0;Sphere.cstab] THEN TRY REAL_ARITH_TAC));
  COMMENT "final kill";
  FIRST_X_ASSUM MP_TAC;
  REWRITE_TAC[DIST_SYM];
  TYPIFY `taum (norm (vv 0)) (norm (vv 3)) (norm (vv 1)) (dist (vv 1,vv 3)) (dist (vv 1,vv 0)) (dist (vv 3,vv 0)) = taum (norm (vv 3)) (norm (vv 0)) (norm (vv 1)) (dist (vv 1,vv 0))     (dist (vv 1,vv 3))     (dist (vv 3,vv 0)) /\ taum (norm (vv 2)) (norm (vv 3)) (norm (vv 1)) (dist (vv 1,vv 3))     (dist (vv 2,vv 1))     (dist (vv 2,vv 3)) = taum (norm (vv 1)) (norm (vv 2)) (norm (vv 3)) (dist (vv 2,vv 3))     (dist (vv 1,vv 3))     (dist (vv 2,vv 1))` ENOUGH_TO_SHOW_TAC;
    DISCH_THEN (unlist REWRITE_TAC);
    BY(REAL_ARITH_TAC);
  BY(REWRITE_TAC[(* Terminal. *) taum_sym])
  ]);;
  (* }}} *)

let empty_4T5  = 
prove_by_refinement(
  `main_nonlinear_terminal_v11 ==> 
	(let s = scs_4T5 in
	   (!vv. BBs_v39 s vv ==> &0 <= taustar_v39 s vv))`,

  (* {{{ proof *)
  [
  REWRITE_TAC[Appendix.scs_4T5];
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[LET_THM];
  MATCH_MP_TAC (REWRITE_RULE[LET_THM] (UNDISCH terminal_quad_lemma));
  COMMENT "digression";
  TYPED_ABBREV_TAC `a = funlist_v39 [(0,1),&2 * h0; (0,2),cstab; (1,3),cstab] (&2) 4`;
  TYPED_ABBREV_TAC `b = funlist_v39 [(0,1),cstab; (0,2),&6; (1,3),cstab] (&2 * h0) 4`;
  TYPIFY `(!i j. i < 4 /\ j < 4 /\ ~(i = j) ==> &2 <= a i j)` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `i < 4 <=> (i = 0 \/ i = 1 \/ i = 2 \/ i = 3)`];
    REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`;TAUT `a /\ (b \/ c) <=> (a /\ b) \/ a /\ c`];
    REWRITE_TAC[TAUT `(a \/ b) \/ c <=> a \/ b \/ c`;arith `~(i = k /\ j = k /\ ~(i = (j:num)))`];
    REPEAT WEAKER_STRIP_TAC;
    EXPAND_TAC "a";
    BY((REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[FUNLIST_EXPLICIT;arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4`;MOD_4_EXPLICIT;arith `&2 <= &2 /\ &2 <= &3`;arith `~(0=1) /\ ~(0=2) /\ ~(0=3) /\ ~(1 =2) /\ ~(1 = 3) /\ ~(2 = 3)`;Sphere.h0;Sphere.cstab] THEN MP_TAC sqrt8_flyspeck THEN TRY REAL_ARITH_TAC));
  COMMENT "digress2";
  TYPIFY `!i j. (i=0 /\ j=1) \/ (i=0 /\ j=3) \/ (i=1 /\ j=2) \/ (i=1 /\ j=3) \/ (i=2 /\ j=3) ==>  b i j < &4` (C SUBGOAL_THEN ASSUME_TAC);
    REWRITE_TAC[arith `i < 4 <=> (i = 0 \/ i = 1 \/ i = 2 \/ i = 3)`];
    REWRITE_TAC[TAUT `(a \/ b) /\ c <=> a /\ c \/ b /\ c`;TAUT `a /\ (b \/ c) <=> (a /\ b) \/ a /\ c`];
    REWRITE_TAC[TAUT `(a \/ b) \/ c <=> a \/ b \/ c`;arith `~(i = k /\ j = k /\ ~(i = (j:num)))`];
    EXPAND_TAC "b";
    REPEAT WEAKER_STRIP_TAC;
    BY((REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN ASM_REWRITE_TAC[FUNLIST_EXPLICIT;arith `SUC 0 = 1 /\ SUC 1 = 2 /\ SUC 2 = 3 /\ SUC 3 = 4`;MOD_4_EXPLICIT;arith `&2 * #1.26 < &4 /\ &3 < &4`;arith `~(0=1) /\ ~(0=2) /\ ~(0=3) /\ ~(1 =2) /\ ~(1 = 3) /\ ~(2 = 3)`;Sphere.cstab;Sphere.h0] THEN MP_TAC sqrt8_flyspeck THEN TRY REAL_ARITH_TAC));
  CONJ_TAC;
    BY(ASM_MESON_TAC[arith `&2 <= x ==> &0 < x`]);
  REPEAT WEAKER_STRIP_TAC;
  REWRITE_TAC[arith `x + &0 = x /\ &4 - &2 = &2`];
  TYPIFY `(!i j. i < 4 /\ j < 4 /\ vv i = vv j ==> i = j)` (C SUBGOAL_THEN ASSUME_TAC);
    REPEAT WEAKER_STRIP_TAC;
    PROOF_BY_CONTR_TAC;
    BY(ASM_MESON_TAC[DIST_REFL;REAL_LE_TRANS;arith `~(&2 <= &0)`]);
  COMMENT "vv";
  TYPIFY `vv 4 = vv 0 /\ vv 5 = vv 1 /\ vv 6 = vv 2` (C SUBGOAL_THEN ASSUME_TAC);
    BY((REPEAT CONJ_TAC) THEN GMATCH_SIMP_TAC periodic_vv_inj THEN TYPIFY `4` EXISTS_TAC THEN ASM_REWRITE_TAC[MOD_4_EXPLICIT;arith `~(4 = 0)`]);
  ASM_REWRITE_TAC[];
  GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] vv_quad_split123);
  ASM_REWRITE_TAC[];
  COMMENT "dist";
  SUBCONJ_TAC;
    TYPIFY `(!i j. ~(vv i = vv j) ==> &2 <= dist (vv i,vv j))` (C SUBGOAL_THEN ASSUME_TAC);
      MATCH_MP_TAC periodic2_mod_reduce;
      TYPIFY `4` EXISTS_TAC;
      REWRITE_TAC[Appendix.periodic2;arith `~(4 = 0)`];
      TYPIFY `!i. vv (i + 4) = vv i` (C SUBGOAL_THEN (unlist REWRITE_TAC));
        BY(ASM_MESON_TAC[Oxl_def.periodic;arith `~(4 = 0)`]);
      REPEAT WEAKER_STRIP_TAC;
      BY(ASM_MESON_TAC[REAL_LE_TRANS]);
    ASM_REWRITE_TAC[];
    BY(ASM_MESON_TAC[REAL_LET_TRANS]);
  REPEAT WEAKER_STRIP_TAC;
  COMMENT "tau3";
  REPEAT (GMATCH_SIMP_TAC tau3_taum_40);
  ASM_REWRITE_TAC[];
  RULE_ASSUM_TAC(REWRITE_RULE[IMAGE_SUBSET_IN;IN_UNIV]);
  ASM_REWRITE_TAC[DIST_SYM];
  TYPIFY `dist (vv 3,vv 0) < &4 ` (C SUBGOAL_THEN (unlist REWRITE_TAC));
    ONCE_REWRITE_TAC[DIST_SYM];
    BY(ASM_REWRITE_TAC[]);
  REWRITE_TAC[TAUT `a /\ b /\ c <=> (a /\ b) /\ c`];
  CONJ_TAC;
    FIRST_X_ASSUM (REPEAT o GMATCH_SIMP_TAC);
    FIRST_X_ASSUM_ST `i < 4` MP_TAC;
    BY(MESON_TAC[arith `1 < 4 /\ 2 < 4 /\ 3 < 4 /\ 0 < 4 /\ ~(2=3) /\ ~(2  =1) /\ ~(1 =0) /\ ~(1 = 3) /\ ~(3=0 ) /\ ~(1 = 2)`]);
  COMMENT "prep dist";
  TYPIFY `(!i j d. d <= a i j  ==> d <= dist (vv i,vv j)) /\ (!i j d. b i j        <= d ==>   dist (vv i,vv j) <= d)` (C SUBGOAL_THEN MP_TAC);
    BY(ASM_MESON_TAC[REAL_LE_TRANS]);
  REPEAT WEAKER_STRIP_TAC;
  TYPIFY `a 1 0 = &2 * h0 /\ a 2 1 = &2 /\ a 2 3 = &2 /\ a 3 0 = &2 /\ a 1 3 = cstab /\ b 1 0 = cstab /\ b 2 1 = &2 * h0 /\ b 2 3 = &2 * h0 /\ b 3 0 = &2 * h0 /\ b 1 3 = cstab` (C SUBGOAL_THEN ASSUME_TAC);
    EXPAND_TAC "a";
    EXPAND_TAC "b";
    BY(REWRITE_TAC[FUNLIST_EXPLICIT;MOD_4_EXPLICIT]);
  RULE_ASSUM_TAC(REWRITE_RULE[Fnjlbxs.in_ball_annulus]);
  TYPIFY `#2.52 = &2 * h0` (C SUBGOAL_THEN ASSUME_TAC);
    BY(REWRITE_TAC[Sphere.h0] THEN REAL_ARITH_TAC);
  INTRO_TAC (get_main_nonlinear "9096461391") [`norm (vv 1)`;`norm (vv 3)`;`norm (vv 0)`;`dist(vv 3,vv 0)`;`dist(vv 1,vv 0)`;`dist(vv 1,vv 3)`];
  REWRITE_TAC[Sphere.ineq];
  REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ASM_REWRITE_TAC[DIST_SYM;arith `#2.0 = &2`];
  ANTS_TAC;
    BY((REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0;Sphere.cstab] THEN TRY REAL_ARITH_TAC));
  DISCH_TAC;
  COMMENT "second get";
  INTRO_TAC (get_main_nonlinear "2445657182") [`norm (vv 1)`;`norm (vv 3)`;`norm (vv 2)`;`dist(vv 3,vv 2)`;`dist(vv 1,vv 2)`;`dist(vv 1,vv 3)`];
  REWRITE_TAC[Sphere.ineq];
  REWRITE_TAC[TAUT `(a ==> b==> c) <=> (a /\ b) ==> c `;TAUT `(a /\ b) /\ c <=> a /\ b /\ c`];
  ASM_REWRITE_TAC[DIST_SYM;arith `#2.0 = &2`];
  ANTS_TAC;
    BY((REPEAT CONJ_TAC THEN (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[Sphere.h0;Sphere.cstab] THEN TRY REAL_ARITH_TAC));
  COMMENT "final kill";
  FIRST_X_ASSUM MP_TAC;
  REWRITE_TAC[DIST_SYM];
  TYPIFY `taum (norm (vv 1)) (norm (vv 3)) (norm (vv 0)) (dist (vv 3,vv 0)) (dist (vv 1,vv 0)) (dist (vv 1,vv 3)) = taum (norm (vv 3)) (norm (vv 0)) (norm (vv 1)) (dist (vv 1,vv 0))     (dist (vv 1,vv 3))     (dist (vv 3,vv 0)) /\ taum (norm (vv 1)) (norm (vv 3)) (norm (vv 2)) (dist (vv 2,vv 3))     (dist (vv 2,vv 1))     (dist (vv 1,vv 3))  = taum (norm (vv 1)) (norm (vv 2)) (norm (vv 3)) (dist (vv 2,vv 3))     (dist (vv 1,vv 3))     (dist (vv 2,vv 1))` ENOUGH_TO_SHOW_TAC;
    DISCH_THEN (unlist REWRITE_TAC);
    BY(REAL_ARITH_TAC);
  BY(REWRITE_TAC[(* Terminal. *)taum_sym])
  ]);;
  (* }}} *)



  end;;