g KIZHLTL1_concl;;  
e (GEN_TAC);;
e (ASM_CASES_TAC `saturated V /\ packing (V:real^3->bool)`);;
e (UP_ASM_TAC THEN STRIP_TAC);;


e (NEW_GOAL `!r. &1 <= r
         ==> sum {X | X SUBSET ball (vec 0,r) /\ mcell_set V X} vol <= 
             vol (ball (vec 0, r))`);;
e (REPEAT STRIP_TAC);;
e (ABBREV_TAC `S = {X | X SUBSET ball (vec 0,r) /\ mcell_set V X}`);;
e (REWRITE_WITH `sum S vol = vol (UNIONS S)`);;
e (ONCE_REWRITE_TAC[EQ_SYM_EQ]);;
e (MATCH_MP_TAC MEASURE_NEGLIGIBLE_UNIONS);;
e (REPEAT STRIP_TAC);;
e (EXPAND_TAC "S");;
e (ASM_SIMP_TAC[FINITE_MCELL_SET_LEMMA]);;
e (REWRITE_TAC[GSYM HAS_MEASURE_MEASURE]);;
e (UP_ASM_TAC THEN EXPAND_TAC "S" THEN REWRITE_TAC[IN;IN_ELIM_THM]);;
e (REWRITE_TAC[mcell_set; IN_ELIM_THM] THEN REPEAT STRIP_TAC);;
e (ASM_REWRITE_TAC[]);;
e (MATCH_MP_TAC MEASURABLE_MCELL);;
e (ASM_REWRITE_TAC[]);;
e (UP_ASM_TAC THEN REWRITE_TAC[IN]);;

e (ASM_CASES_TAC `~NULLSET (s INTER t)`);;
e (NEW_GOAL `F`);;
e (UNDISCH_TAC `s:real^3->bool IN S` THEN UNDISCH_TAC `t:real^3->bool IN S`);;
e (EXPAND_TAC "S" THEN REWRITE_TAC[IN;IN_ELIM_THM]);;
e (REWRITE_TAC[mcell_set; IN_ELIM_THM] THEN REPEAT STRIP_TAC);;
e (NEW_GOAL `s = t:real^3->bool`);;
e (REWRITE_TAC[ASSUME `t = mcell i V ul`; ASSUME `s = mcell i' V ul'`]);;
e (ABBREV_TAC `j = if i <= 4 then i else 4`);;
e (ABBREV_TAC `j' = if i' <= 4 then i' else 4`);;
e (REWRITE_WITH `mcell i V ul = mcell j V ul`);;
e (EXPAND_TAC "j" THEN COND_CASES_TAC);;
e (REFL_TAC);;
e (ASM_SIMP_TAC[ARITH_RULE `~(i <= 4) ==> i >= 4`; 
   ARITH_RULE `4 >= 4`; MCELL_EXPLICIT]);;
e (REWRITE_WITH `mcell i' V ul' = mcell j' V ul'`);;
e (EXPAND_TAC "j'" THEN COND_CASES_TAC);;
e (REFL_TAC);;
e (ASM_SIMP_TAC[ARITH_RULE `~(i <= 4) ==> i >= 4`; 
   ARITH_RULE `4 >= 4`; MCELL_EXPLICIT]);;
e (REWRITE_WITH `j' = j /\ mcell j' V ul' = mcell j V ul`);;
e (MATCH_MP_TAC Ajripqn.AJRIPQN);;

e (REPEAT STRIP_TAC);;
e (ASM_REWRITE_TAC[]);;
e (ASM_REWRITE_TAC[]);;
e (ASM_MESON_TAC[IN]);;
e (ASM_MESON_TAC[IN]);;
e (EXPAND_TAC "j'" THEN COND_CASES_TAC);;
e (UP_ASM_TAC THEN REWRITE_TAC[ARITH_RULE `i <= 4 <=> i=0\/i=1\/i=2\/i=3\/i=4`]
   THEN SET_TAC[]);;
e (SET_TAC[]);;
e (EXPAND_TAC "j" THEN COND_CASES_TAC);;
e (UP_ASM_TAC THEN REWRITE_TAC[ARITH_RULE `i <= 4 <=> i=0\/i=1\/i=2\/i=3\/i=4`]
   THEN SET_TAC[]);;
e (SET_TAC[]);;
e (UP_ASM_TAC);;
e (REWRITE_WITH `mcell j V ul = mcell i V ul`);;
e (EXPAND_TAC "j" THEN COND_CASES_TAC);;
e (REFL_TAC);;
e (ASM_SIMP_TAC[ARITH_RULE `~(i <= 4) ==> i >= 4`; 
   ARITH_RULE `4 >= 4`; MCELL_EXPLICIT]);;
e (REWRITE_WITH `mcell j' V ul' = mcell i' V ul'`);;
e (EXPAND_TAC "j'" THEN COND_CASES_TAC);;
e (REFL_TAC);;
e (ASM_SIMP_TAC[ARITH_RULE `~(i <= 4) ==> i >= 4`; 
   ARITH_RULE `4 >= 4`; MCELL_EXPLICIT]);;
e (ASM_MESON_TAC[]);;
e (ASM_MESON_TAC[]);;
e (ASM_MESON_TAC[]);;
e (ASM_MESON_TAC[]);;

e (MATCH_MP_TAC MEASURE_SUBSET);;
e (REWRITE_TAC[MEASURABLE_BALL]);;
e (REPEAT STRIP_TAC);;
e (EXPAND_TAC "S" THEN MATCH_MP_TAC MEASURABLE_UNIONS);;
e (REPEAT STRIP_TAC);;
e (ASM_SIMP_TAC[FINITE_MCELL_SET_LEMMA]);;
e (UP_ASM_TAC THEN REWRITE_TAC[IN;IN_ELIM_THM; mcell_set]);;
e (REPEAT STRIP_TAC);;
e (ASM_SIMP_TAC[MEASURABLE_MCELL]);;
e (EXPAND_TAC "S" THEN SET_TAC[]);;

(* ----------------------------------------------------------------------- *)

e (NEW_GOAL `?c. !r. &1 <= r
         ==> vol (ball (vec 0, r)) + c * r pow 2 <= 
            sum (V INTER ball (vec 0,r)) (\u:real^3. vol (voronoi_open V u))`);;
e (EXISTS_TAC `-- (&24 / &3) * pi`);;
e (REPEAT STRIP_TAC);;

e (ASM_CASES_TAC `r < &6`);;
e (NEW_GOAL `&0 <= sum (V INTER ball (vec 0,r)) (\u:real^3. vol (voronoi_open V u))`);;
e (MATCH_MP_TAC SUM_POS_LE);;
e (ASM_SIMP_TAC[Packing3.KIUMVTC]);;
e (REPEAT STRIP_TAC);;
e (MATCH_MP_TAC MEASURE_POS_LE);;
e (ASM_SIMP_TAC[Pack1.measurable_voronoi]);;

e (NEW_GOAL `vol (ball ((vec 0):real^3,r)) + (--(&24 / &3) * pi) * r pow 2 <= &0`);;
e (REWRITE_TAC[REAL_ARITH `a + (--b * c) * d <= &0 <=> a <= b * c * d`]);;
e (ASM_SIMP_TAC [VOLUME_BALL; REAL_ARITH `&1 <= r ==> &0 <= r`]);;
e (REWRITE_TAC[REAL_ARITH `&4 / &3 * pi * r pow 3 <= &24 / &3 * pi * r pow 2
   <=> &0 <= &4 / &3 * pi * r pow 2 * (&6 - r)`]);;
e (MATCH_MP_TAC REAL_LE_MUL);;
e (STRIP_TAC);;
e (MATCH_MP_TAC REAL_LE_DIV THEN REAL_ARITH_TAC);;
e (MATCH_MP_TAC REAL_LE_MUL);;
e (REWRITE_TAC[PI_POS_LE]);;
e (MATCH_MP_TAC REAL_LE_MUL);;
e (REWRITE_TAC[REAL_LE_POW_2]);;
e (ASM_REAL_ARITH_TAC);;
e (ASM_REAL_ARITH_TAC);;


e (NEW_GOAL `vol (ball (vec 0,r - &2)) <=  
   sum (V INTER ball (vec 0,r)) (\u:real^3. vol (voronoi_open V u))`);;
e (REWRITE_WITH 
   `sum (V INTER ball (vec 0,r)) (\u:real^3. vol (voronoi_open V u)) = 
    sum (V INTER ball (vec 0,r)) (\u:real^3. vol (voronoi_closed V u))`);;
e (MATCH_MP_TAC SUM_EQ);;
e (REPEAT STRIP_TAC);;
e (REWRITE_TAC[BETA_THM]);;
e (REWRITE_TAC[GSYM Pack2.MEASURE_VORONOI_CLOSED_OPEN]);;

e (ABBREV_TAC `S:real^3->bool = V INTER ball (vec 0, r)`);;
e (ABBREV_TAC `g = (\t:real^3. voronoi_closed V t)`);;

e (REWRITE_WITH `sum S (\u:real^3. vol (voronoi_closed V u)) = sum S (\t. vol (g t))`);;
e (EXPAND_TAC "g" THEN REWRITE_TAC[]);;
e (REWRITE_WITH `sum S (\t:real^3. vol (g t)) = measure (UNIONS (IMAGE g S))`);;
e (ONCE_REWRITE_TAC[EQ_SYM_EQ]);;
e (MATCH_MP_TAC MEASURE_NEGLIGIBLE_UNIONS_IMAGE);;
e (ASM_REWRITE_TAC[] THEN EXPAND_TAC "g");;
e (REPEAT STRIP_TAC);;
e (EXPAND_TAC "S");;
e (ASM_SIMP_TAC[Packing3.KIUMVTC]);;
e (MATCH_MP_TAC Pack2.MEASURABLE_VORONOI_CLOSED);;
e (ASM_REWRITE_TAC[]);;
e (MATCH_MP_TAC Pack2.NEGLIGIBLE_INTER_VORONOI_CLOSED);;
e (ASM_SET_TAC[]);;
e (EXPAND_TAC "g" THEN REWRITE_TAC[IMAGE]);;
e (MATCH_MP_TAC MEASURE_SUBSET);;
e (REWRITE_TAC[MEASURABLE_BALL]);;
e (REPEAT STRIP_TAC);;
e (MATCH_MP_TAC MEASURABLE_UNIONS);;
e (STRIP_TAC);;
e (MATCH_MP_TAC FINITE_IMAGE_EXPAND);;
e (EXPAND_TAC "S");;
e (ASM_SIMP_TAC[Packing3.KIUMVTC]);;

e (REWRITE_TAC[IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC);;
e (ASM_REWRITE_TAC[]);;
e (MATCH_MP_TAC Pack2.MEASURABLE_VORONOI_CLOSED);;
e (ASM_REWRITE_TAC[]);;
e (REWRITE_TAC[SUBSET; IN_BALL; IN_UNIONS]);;
e (REPEAT STRIP_TAC);;
e (MP_TAC (ASSUME `saturated (V:real^3->bool)`));;
e (REWRITE_TAC[saturated] THEN STRIP_TAC);;
e (NEW_GOAL `?y. y IN V /\ dist (x:real^3,y) < &2`);;
e (ASM_MESON_TAC[]);;
e (UP_ASM_TAC THEN STRIP_TAC);;
e (NEW_GOAL `?v:real^3. v IN V /\ x IN voronoi_closed V v`);;
e (MATCH_MP_TAC Packing3.TIWWFYQ);;
e (ASM_REWRITE_TAC[]);;
e (UP_ASM_TAC THEN STRIP_TAC);;

e (EXISTS_TAC `voronoi_closed V (v:real^3)`);;
e (ASM_REWRITE_TAC[]);;

e (ONCE_REWRITE_TAC[IN] THEN REWRITE_TAC[IN_ELIM_THM]);;
e (EXISTS_TAC `v:real^3`);;
e (STRIP_TAC);;
e (EXPAND_TAC "S" THEN REWRITE_TAC[IN_INTER]);;
e (ASM_REWRITE_TAC[IN_BALL]);;

e (NEW_GOAL `dist (vec 0,v) <= dist (vec 0,x) + dist (x, v:real^3)`);;
e (REWRITE_TAC[DIST_TRIANGLE]);;
e (NEW_GOAL `dist (x, v:real^3) < &2`);;
e (NEW_GOAL `dist (x, v) <= dist (x, y:real^3)`);;
e (UNDISCH_TAC `x:real^3 IN voronoi_closed V v`);;
e (REWRITE_TAC[IN; voronoi_closed; IN_ELIM_THM]);;
e (STRIP_TAC);;
e (FIRST_ASSUM MATCH_MP_TAC);;
e (ASM_SET_TAC[]);;
e (ASM_REAL_ARITH_TAC);;
e (ASM_REAL_ARITH_TAC);;
e (REFL_TAC);;

e (NEW_GOAL `vol (ball (vec 0,r)) + (--(&24 / &3) * pi) * r pow 2 <= 
             vol (ball (vec 0,r - &2))`);;
e (ASM_SIMP_TAC[VOLUME_BALL; REAL_ARITH `~(r < &6) ==> &0 <= r`;
   REAL_ARITH `~(r < &6) ==> &0 <= (r - &2)` ]);;
e (REWRITE_TAC[REAL_ARITH 
 `&4 / &3 * pi * r pow 3 + (--(&24 / &3) * pi) * r pow 2 <=
  &4 / &3 * pi * (r - &2) pow 3 <=> 
  &0 <= &4 / &3 * pi * (&12 * r - &8)`]);;
e (MATCH_MP_TAC REAL_LE_MUL);;
e (STRIP_TAC);;
e (MATCH_MP_TAC REAL_LE_DIV THEN REAL_ARITH_TAC);;
e (MATCH_MP_TAC REAL_LE_MUL);;
e (STRIP_TAC);;
e (REWRITE_TAC[PI_POS_LE]);;
e (NEW_GOAL `&12 * r >= &72`);;
e (ASM_REAL_ARITH_TAC);;
e (ASM_REAL_ARITH_TAC);;

e (ASM_REAL_ARITH_TAC);;
e (UP_ASM_TAC THEN STRIP_TAC);;

e (EXISTS_TAC `c:real`);;
e (REPEAT STRIP_TAC);;

e (NEW_GOAL `sum {X | X SUBSET ball (vec 0,r) /\ mcell_set V X} vol <=
              vol (ball (vec 0,r))`);;
e (ASM_SIMP_TAC[]);;
e (NEW_GOAL `vol (ball (vec 0,r)) + c * r pow 2 <=
            sum (V INTER ball (vec 0,r)) (\u:real^3. vol (voronoi_open V u))`);;
e (ASM_SIMP_TAC[]);;
e (ABBREV_TAC `a1 = sum {X | X SUBSET ball (vec 0,r) /\ mcell_set V X} vol`);;
e (ABBREV_TAC `a2 = vol (ball ((vec 0):real^3,r))`);;
e (ABBREV_TAC `a3 = sum (V INTER ball (vec 0,r)) (\u:real^3. vol (voronoi_open V u))`);;
e (ASM_REAL_ARITH_TAC);;

e (EXISTS_TAC `&0`);;
e (REPEAT STRIP_TAC);;
e (NEW_GOAL `F`);;
e (ASM_MESON_TAC[]);;
e (ASM_MESON_TAC[]);;

let KIZHLTL1 = top_thm();;

(* ------------------------------------------------------------------------ *)
(* ======================================================================== *)

g KIZHLTL2_concl;;
e (REPEAT STRIP_TAC);;
e (ASM_CASES_TAC `saturated V /\ packing V`);;
e (NEW_GOAL 
  `?C. !r. &1 <= r ==> 
   &(CARD (V INTER ball ((vec 0):real^3,r) DIFF V INTER ball (vec 0,r - &8))) <=
   C * r pow 2`);;
e (REWRITE_WITH `!r p. V INTER ball (p:real^3,r) DIFF V INTER ball (p,r - &8)
    = V INTER ball (p:real^3,r + &0) DIFF V INTER ball (p,r - &8)`);;
e (ASM_REWRITE_TAC[REAL_ARITH `a + &0 = a`]);;
e (ASM_SIMP_TAC[PACKING_BALL_BOUNDARY]);;
e (TAKE_TAC);;
e (EXISTS_TAC `(&2 * mm1 / pi) * (&4 * pi) * (--C)`);;
e (REPEAT STRIP_TAC);;
e (NEW_GOAL `&(CARD (V INTER ball ((vec 0):real^3,r) DIFF 
                     V INTER ball (vec 0,r - &8))) <= C * r pow 2`);;
e (ASM_SIMP_TAC[]);;
e (NEW_GOAL `total_solid V = (\X. total_solid V X)`);;
e (REWRITE_TAC[GSYM ETA_AX]);;
e (ONCE_ASM_REWRITE_TAC[] THEN DEL_TAC);;
e (REWRITE_TAC[total_solid]);;
e (ABBREV_TAC `B = {X | X SUBSET ball (vec 0,r) /\ mcell_set V X}`);;
e (NEW_GOAL `FINITE (B:(real^3->bool) ->bool)`);;
e (EXPAND_TAC "B" THEN MATCH_MP_TAC FINITE_MCELL_SET_LEMMA);;
e (ASM_REWRITE_TAC[]);;
e (ABBREV_TAC `A1:real^3->bool = V INTER ball (vec 0,r)`);;
e (ABBREV_TAC `A2:real^3->bool = V INTER ball (vec 0,r - &8)`);;
e (NEW_GOAL `FINITE (A1:real^3->bool)`);;
e (EXPAND_TAC "A1" THEN MATCH_MP_TAC FINITE_PACK_LEMMA);;
e (ASM_REWRITE_TAC[]);;
e (NEW_GOAL `FINITE (A2:real^3->bool)`);;
e (EXPAND_TAC "A2" THEN MATCH_MP_TAC FINITE_PACK_LEMMA);;
e (ASM_REWRITE_TAC[]);;

e (NEW_GOAL `sum B (\X. sum {u | u IN A2 /\ VX V X u} (\u. sol u X)) 
          <= sum B (\X. sum (VX V X) (\x. sol x X))`);;
e (MATCH_MP_TAC SUM_LE);;
e (ASM_REWRITE_TAC[BETA_THM]);;
e (REPEAT STRIP_TAC);;
e (MATCH_MP_TAC SUM_SUBSET_SIMPLE);;
e (REPEAT STRIP_TAC);;
e (REWRITE_TAC[VX] THEN COND_CASES_TAC);;
e (REWRITE_TAC[FINITE_EMPTY]);;
e (LET_TAC);;
e (COND_CASES_TAC);;
e (REWRITE_TAC[FINITE_EMPTY]);;
e (REWRITE_TAC[FINITE_SET_OF_LIST]);;
e (SET_TAC[]);;
e (REWRITE_TAC[BETA_THM]);;

e (UNDISCH_TAC `x:real^3->bool IN B`);;
e (EXPAND_TAC "B" THEN REWRITE_TAC[IN; IN_ELIM_THM; mcell_set_2]);;
e (REPEAT STRIP_TAC);;
e (ASM_REWRITE_TAC[]);;

e (NEW_GOAL `eventually_radial x' (mcell i V ul)`);;
e (MATCH_MP_TAC Urrphbz2.URRPHBZ2);;
e (ASM_REWRITE_TAC[]);;
e (SUBGOAL_THEN `x' IN (VX V x)` MP_TAC);;
e (ASM_SET_TAC[]);;
e (REWRITE_TAC[VX]);;
e (COND_CASES_TAC);;
e (SET_TAC[]);;
e (LET_TAC);;
e (COND_CASES_TAC);;
e (SET_TAC[]);;
e (STRIP_TAC);;

e (UNDISCH_TAC `cell_params V x = k,ul'`);;
e (ONCE_REWRITE_TAC[EQ_SYM_EQ]);;
e (REWRITE_TAC[cell_params]);;
e (ABBREV_TAC `P = (\(k,ul). k <= 4 /\ ul IN barV V 3 /\ x = mcell k V ul)`);;
e (DISCH_TAC);;
e (NEW_GOAL `(P:(num#(real^3)list->bool)) (k,ul')`);;
e (ASM_REWRITE_TAC[]);;
e (MATCH_MP_TAC SELECT_AX);;
e (EXISTS_TAC `(i:num,ul:(real^3)list)`);;
e (EXPAND_TAC "P");;
e (REWRITE_TAC[BETA_THM]);;
e (REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[]);;
e (UP_ASM_TAC THEN EXPAND_TAC "P" THEN REWRITE_TAC[BETA_THM] THEN STRIP_TAC);;
e (NEW_GOAL `set_of_list (truncate_simplex (k - 1) ul') SUBSET V:real^3->bool`);;
e (MATCH_MP_TAC Packing3.BARV_SUBSET);;
e (EXISTS_TAC `k - 1`);;
e (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV);;
e (EXISTS_TAC `3`);;
e (STRIP_TAC);;
e (ASM_SET_TAC[]);;
e (ASM_ARITH_TAC);;
e (ASM_SET_TAC[]);;


e (UP_ASM_TAC THEN REWRITE_TAC[eventually_radial]);;
e (REPEAT STRIP_TAC);;
e (REWRITE_WITH `sol x' (mcell i V ul) = 
  &3 * vol ((mcell i V ul) INTER normball x' r') / r' pow 3`);;
e (MATCH_MP_TAC sol);;
e (ASM_REWRITE_TAC[GSYM Marchal_cells_2_new.RADIAL_VS_RADIAL_NORM; NORMBALL_BALL]);;
e (MATCH_MP_TAC MEASURABLE_INTER);;
e (ASM_SIMP_TAC[MEASURABLE_BALL; MEASURABLE_MCELL]);;
e (MATCH_MP_TAC REAL_LE_MUL);;
e (REWRITE_TAC[REAL_ARITH `&0 <= &3`] THEN MATCH_MP_TAC REAL_LE_DIV);;
e (STRIP_TAC);;
e (MATCH_MP_TAC MEASURE_POS_LE);;
e (MATCH_MP_TAC MEASURABLE_INTER);;
e (ASM_SIMP_TAC[MEASURABLE_BALL; NORMBALL_BALL; MEASURABLE_MCELL]);;
e (MATCH_MP_TAC REAL_POW_LE THEN ASM_REAL_ARITH_TAC);;

(* ------------------------------------------------------------------------- *)

e (NEW_GOAL `sum B  (\X. sum {u | u IN A2 /\ VX V X u} (\u. sol u X)) = 
             sum A2 (\u. sum  {X | X IN B /\ VX V X u} (\X. sol u X))`);;
e (MATCH_MP_TAC SUM_SUM_RESTRICT);;
e (ASM_REWRITE_TAC[]);;

e (NEW_GOAL `sum A2 (\u. sum {X | X IN B /\ VX V X u} (\X. sol u X)) = 
             sum A2 (\u. sum {X | mcell_set V X /\ u IN VX V X} 
                    (\X. sol u X))`);;
e (MATCH_MP_TAC SUM_EQ);;
e (EXPAND_TAC "A2" THEN REWRITE_TAC[IN_INTER; IN_DIFF] THEN 
   REWRITE_TAC[IN_BALL; IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC);;
e (REWRITE_WITH `{X | B X /\ VX V X x} = 
                 {X | mcell_set V X /\ VX V X x}`);;
e (REWRITE_TAC[SET_RULE `a = b <=> a SUBSET b /\ b SUBSET a`]);;
e (STRIP_TAC);;
e (EXPAND_TAC "B");;
e (SET_TAC[]);;
e (REWRITE_WITH `!x:real^3->bool. B x <=> x IN B`);;
e (REWRITE_TAC[IN]);;
e (EXPAND_TAC "B" THEN REWRITE_TAC[SUBSET; IN_INTER; IN_DIFF] THEN  
   REWRITE_TAC[IN_BALL; IN; IN_ELIM_THM]);;
e (REWRITE_TAC[MESON[] `A /\ X /\ Y ==> (B /\ A) /\ X /\ Y <=> A /\ X /\ Y ==> B`]);;
e (REPEAT STRIP_TAC);;
e (NEW_GOAL `x IN VX V x'`);;
e (ASM_REWRITE_TAC[IN]);;
e (NEW_GOAL `~NULLSET x'`);;
e (UNDISCH_TAC `x IN VX V x'` THEN REWRITE_TAC[VX] THEN COND_CASES_TAC);;
e (SET_TAC[]);;
e (MESON_TAC[]);;

e (NEW_GOAL `dist (vec 0, x'':real^3) <= dist (vec 0, x) + dist (x, x'')`);;
e (REWRITE_TAC[DIST_TRIANGLE]);;
e (NEW_GOAL `?p. x' SUBSET ball (p:real^3,&4)`);;
e (MATCH_MP_TAC MCELL_SUBSET_BALL_4);;
e (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]);;
e (UP_ASM_TAC THEN REWRITE_TAC[SUBSET; IN_BALL] THEN STRIP_TAC);;
e (NEW_GOAL `dist (x, x'':real^3) <= dist (x, p) + dist (p, x'')`);;
e (REWRITE_TAC[DIST_TRIANGLE]);;
e (NEW_GOAL `dist (x, p:real^3) < &4`);;
e (ONCE_REWRITE_TAC[DIST_SYM]);;
e (FIRST_ASSUM MATCH_MP_TAC);;
e (NEW_GOAL `VX V x' = V INTER x'`);;
e (MATCH_MP_TAC Hdtfnfz.HDTFNFZ);;
e (UNDISCH_TAC `mcell_set V x'` THEN REWRITE_TAC[mcell_set; IN_ELIM_THM] THEN STRIP_TAC);;
e (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num`);;
e (ASM_REWRITE_TAC[]);;
e (ASM_SET_TAC[]);;
e (ASM_SET_TAC[]);;
e (NEW_GOAL `dist (p:real^3,x'') < &4`);;
e (FIRST_ASSUM MATCH_MP_TAC);;
e (ASM_REWRITE_TAC[IN]);;
e (ASM_REAL_ARITH_TAC);;
e (ASM_REWRITE_TAC[]);;
e (ASM_REWRITE_TAC[]);;

(* ------------------------------------------------------------------------ *)

e (UP_ASM_TAC);;
e (REWRITE_WITH `sum A2 (\u. sum {X | mcell_set V X /\ u IN VX V X} (\X. sol u X)) = sum A2 (\u. &4 * pi)`);;
e (MATCH_MP_TAC SUM_EQ);;
e (REWRITE_TAC[BETA_THM] THEN REPEAT STRIP_TAC);;
e (MATCH_MP_TAC Qzyzmjc.QZYZMJC);;
e (ASM_REWRITE_TAC[]);;
e (ASM_SET_TAC[]);;
e (ASM_SIMP_TAC[SUM_CONST]);;
e (STRIP_TAC);;

e (ABBREV_TAC `s1 = sum B (\X. sum (VX V X) (\x. sol x X))`);;
e (NEW_GOAL `&(CARD (A2:real^3->bool)) * &4 * pi <= s1`);;
e (ABBREV_TAC `s2 = sum B (\X. sum {u | u IN A2 /\ VX V X u} (\u. sol u X))`);;
e (ABBREV_TAC `s3 = sum A2 (\u. sum {X | X IN B /\ VX V X u} (\X. sol u X))`);;
e (ASM_REAL_ARITH_TAC);;
e (NEW_GOAL `(&2 * mm1 / pi) * &(CARD (A2:real^3->bool)) * &4 * pi <= (&2 * mm1 / pi) * s1`);;
e (REWRITE_TAC[REAL_ARITH `a * b <= a * c <=> &0 <= (c - b) * a`]);;
e (MATCH_MP_TAC REAL_LE_MUL);;
e (STRIP_TAC);;
e (ASM_REAL_ARITH_TAC);;
e (MATCH_MP_TAC REAL_LE_MUL);;
e (STRIP_TAC);;
e (REAL_ARITH_TAC);;
e (MATCH_MP_TAC REAL_LE_DIV);;
e (REWRITE_TAC[PI_POS_LE]);;
e (NEW_GOAL `#1.012080 < mm1`);;
e (REWRITE_TAC[Flyspeck_constants.bounds]);;
e (UP_ASM_TAC THEN REAL_ARITH_TAC);;
 
e (NEW_GOAL `&(CARD (A1:real^3->bool)) * &8 * mm1 + 
             ((&2 * mm1 / pi) * (&4 * pi) * --C) * r pow 2 <= 
             (&2 * mm1 / pi) * &(CARD (A2:real^3->bool)) * &4 * pi`);;
e (REWRITE_TAC[REAL_ARITH `((&2 * mm1 / pi) * (&4 * pi) * --C) * r pow 2 = 
  (--C * r pow 2) * (&8 * mm1) * (pi / pi)`]);;
e (REWRITE_TAC[REAL_ARITH `(&2 * mm1 / pi) * &(CARD A2) * &4 * pi = 
  &(CARD A2) * (&8 * mm1) * (pi / pi)`]);;
e (REWRITE_WITH `pi / pi = &1`);;
e (MATCH_MP_TAC REAL_DIV_REFL);;
e (MP_TAC PI_POS THEN REAL_ARITH_TAC);;
e (REWRITE_TAC[REAL_MUL_RID; REAL_ARITH `a * b + c * b <= d * b <=>
  &0 <= (d - a - c) * b `]);;
e (MATCH_MP_TAC REAL_LE_MUL);;
e (STRIP_TAC);;
e (REWRITE_TAC[REAL_ARITH `&0 <= a - b - (--c * x) <=> b - a <= c * x`]);;

e (NEW_GOAL `A2 SUBSET A1:real^3->bool`);;
e (EXPAND_TAC "A1" THEN EXPAND_TAC "A2");;
e (MATCH_MP_TAC (SET_RULE `A SUBSET B ==> V INTER A SUBSET V INTER B`));;
e (MATCH_MP_TAC SUBSET_BALL);;
e (REAL_ARITH_TAC);;
e (REWRITE_WITH `&(CARD (A1:real^3->bool)) - &(CARD (A2:real^3->bool)) = 
   &(CARD A1 - CARD A2)`);;
e (MATCH_MP_TAC REAL_OF_NUM_SUB);;
e (MATCH_MP_TAC CARD_SUBSET);;
e (ASM_REWRITE_TAC[]);;

e (REWRITE_WITH `CARD (A1:real^3->bool) - CARD (A2:real^3->bool) = 
   CARD (A1 DIFF A2)`);;
e (ONCE_REWRITE_TAC[EQ_SYM_EQ]);;
e (MATCH_MP_TAC CARD_DIFF);;
e (ASM_REWRITE_TAC[]);;
e (ASM_REWRITE_TAC[]);;

e (MATCH_MP_TAC REAL_LE_MUL);;
e (NEW_GOAL `#1.012080 < mm1`);;
e (REWRITE_TAC[Flyspeck_constants.bounds]);;
e (UP_ASM_TAC THEN REAL_ARITH_TAC);;


e (ASM_REAL_ARITH_TAC);;
e (EXISTS_TAC `&0`);;
e (REPEAT STRIP_TAC);;
e (NEW_GOAL `F`);;
e (ASM_MESON_TAC[]);;
e (ASM_MESON_TAC[]);;

let KIZHLTL2 = top_thm();;

(* ========================================================================== *)
open Flyspeck_constants;;

let SUM_PAIR_2_SET = 
prove
(`!f s:real^N->bool d.
 FINITE s
==> sum {(m,n) | m IN s /\ n IN s /\ ~(m = n) /\ dist(m,n) <= d}
 (\(m,n). f {m, n}) = &2 *
 sum {{m, n} | m IN s /\ n IN s /\ ~(m = n) /\ dist(m,n) <= d} f`,

REPEAT STRIP_TAC THEN MP_TAC(ISPECL
 [`\(m:real^N,n). {m,n}`;`(\(m,n). f {m,n}):real^N#real^N->real`;
  `{(m,n) | m IN s /\ n IN s /\ ~(m:real^N = n) /\ dist(m,n) <= d}`;
  `{{m,n} | m IN s /\ n IN s /\ ~(m:real^N = n) /\ dist(m,n) <= d}`]
  SUM_GROUP) THEN
  ANTS_TAC THENL
 [CONJ_TAC THENL
 [ONCE_REWRITE_TAC[SET_RULE
 `m IN s /\ n IN s /\ P m n <=>
  m IN s /\ n IN {n | n IN s /\ P m n}`] THEN
  MATCH_MP_TAC FINITE_PRODUCT_DEPENDENT THEN
  ASM_SIMP_TAC[FINITE_RESTRICT];
  REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN SET_TAC[]];
  DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[GSYM SUM_LMUL] THEN
  MATCH_MP_TAC SUM_EQ THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN
  MAP_EVERY X_GEN_TAC [`m:real^N`; `n:real^N`] THEN STRIP_TAC THEN
  MATCH_MP_TAC EQ_TRANS THEN
  EXISTS_TAC `sum {(m:real^N,n), (n,m)} (\(m,n). f {m,n})` THEN
  CONJ_TAC THENL
  [AP_THM_TAC THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN
  REWRITE_TAC[FORALL_PAIR_THM] THEN
  REWRITE_TAC[IN_ELIM_THM; PAIR_EQ; IN_INSERT; NOT_IN_EMPTY;
  SET_RULE `{a,b} = {c,d} <=> a = c /\ b = d \/ a = d /\ b = c`] THEN
  ASM_MESON_TAC[DIST_SYM];
  SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
  ASM_REWRITE_TAC[IN_SING; NOT_IN_EMPTY; PAIR_EQ; REAL_ADD_RID] THEN
  REWRITE_TAC[INSERT_AC] THEN REAL_ARITH_TAC]]);;
(* ========================================================================== *)

let H0_LT_SQRT2 = 
prove_by_refinement (`h0 < sqrt (&2)`,

[REWRITE_TAC[h0; GSYM sqrt2]; 
 NEW_GOAL `#1.414213 < sqrt2`;
 REWRITE_TAC[bounds];
 ASM_REAL_ARITH_TAC]);;
(* ========================================================================== *)

let gamma_y_pos_le = 
prove_by_refinement (
 `!V X e. packing V /\ saturated V /\ mcell_set V X /\ edgeX V X e
                        ==> &0 <= gammaY V X lmfun e`,

[(REPEAT STRIP_TAC);
 (REWRITE_TAC[gammaY]);
 (NEW_GOAL `?u v. (VX V X u /\ VX V X v /\ ~(u = v)) /\ e = {u, v}`);
 (UP_ASM_TAC THEN REWRITE_TAC[edgeX; IN_ELIM_THM]);
 (UP_ASM_TAC THEN STRIP_TAC);
 (ASM_REWRITE_TAC[]);
 (ABBREV_TAC `f = (\u v. if {u, v} IN edgeX V X
             then dihX V X (u,v) * lmfun (hl [u; v])
             else &0)`);
 (REWRITE_WITH `(\({u, v}). if {u, v} IN edgeX V X
             then dihX V X (u,v) * lmfun (hl [u; v])
             else &0) = (\({u, v}). f u v)`);
 (EXPAND_TAC "f" THEN REWRITE_TAC[]);
 (REWRITE_WITH `(\({u:real^3, v}). f u v) {u, v} = (f:real^3->real^3->real) u v`);
 (MATCH_MP_TAC BETA_PAIR_THM);
 (EXPAND_TAC "f" THEN REPEAT STRIP_TAC);
 (REPEAT COND_CASES_TAC);
 (REWRITE_TAC[HL; set_of_list; SET_RULE `{a,b} = {b,a}`]);
 (REWRITE_WITH `dihX V X (u',v') = dihX V X (v',u')`);
 (MATCH_MP_TAC DIHX_SYM);
 (ASM_REWRITE_TAC[]);
 (NEW_GOAL `F`);
 (UP_ASM_TAC THEN REWRITE_TAC[]);
 (ONCE_REWRITE_TAC[SET_RULE `{a,b} = {b,a}`] THEN ASM_REWRITE_TAC[]);
 (ASM_MESON_TAC[]);
 (NEW_GOAL `F`);
 (UP_ASM_TAC THEN REWRITE_TAC[]);
 (ONCE_REWRITE_TAC[SET_RULE `{a,b} = {b,a}`] THEN ASM_REWRITE_TAC[]);
 (ASM_MESON_TAC[]);
 (REFL_TAC);
 (EXPAND_TAC "f");
 (COND_CASES_TAC);
 (MATCH_MP_TAC REAL_LE_MUL);
 (REWRITE_TAC[DIHX_POS;lmfun_pos_le]);
 (REAL_ARITH_TAC)]);;
(* ========================================================================== *)

let BETA_SET_2_THM = 
prove
 (`!g:(A->bool)->B u0 v0. (\({u, v}). g {u, v}) {u0, v0} = g {u0, v0}`,

 GEN_TAC THEN REWRITE_TAC[GABS_DEF;GEQ_DEF] THEN
 CONV_TAC SELECT_CONV THEN EXISTS_TAC `g:(A->bool)->B` THEN
 REWRITE_TAC[]);;
(* ========================================================================== *)
(* Begin to prove KIZHLTL4 *)
(* ========================================================================== *)
(*                                                               *)                                                                       
(*    #use "/home/ky/flyspeck/working/marchal_cells_3.hl";;                   *)
(*    #use "/home/ky/flyspeck/working/empty.hl";;                             *)
(*                                                                            *)                                                                       
(* ========================================================================== *)


(* ========================================================================== *)

let KIZHLTL4_concl = 
  `!V. ?c. !r. saturated V /\ packing V /\ &1 <= r
               ==> (&8 * mm2 / pi) *
                   sum {X | X SUBSET ball (vec 0,r) /\ mcell_set V X}
                   (\X. sum (edgeX V X)
                        (\({u, v}). if {u, v} IN edgeX V X
                                    then dihX V X (u,v) * lmfun (hl [u; v])
                                    else &0)) +
                   c * r pow 2 <=
                   &8 *
                   mm2 *
                   sum (V INTER ball (vec 0,r))
                   (\u. sum {v | v IN V /\ ~(u = v) /\ dist (u,v) <= &2 * h0}
                        (\v. lmfun (hl [u; v])))`;;

(* ========================================================================== *)

g KIZHLTL4_concl;;
e (REPEAT STRIP_TAC);;
e (ASM_CASES_TAC `saturated V /\ packing V`);;
e (ABBREV_TAC `c = &8 * mm2 * (&0)`);;
e (EXISTS_TAC `c:real`);;  (* choose d later *)

(* ------------------------------------------------------------------------- *)

e (REPEAT STRIP_TAC);;
e (ABBREV_TAC `S1 = {X | X SUBSET ball (vec 0,r) /\ mcell_set V X}`);;
e (ABBREV_TAC `V1:real^3->bool = V INTER ball (vec 0, r)`);;
e (ABBREV_TAC `T1 = {{u,v:real^3} | u IN V1 /\ v IN V1}`);;

e (NEW_GOAL `FINITE (S1:(real^3->bool)->bool)`);;
e (EXPAND_TAC "S1");;
e (ASM_SIMP_TAC[FINITE_MCELL_SET_LEMMA]);;

e (NEW_GOAL `FINITE (T1:(real^3->bool)->bool)`);;
e (EXPAND_TAC "T1");;
e (REWRITE_TAC[GSYM set_of_list]);;
e (MATCH_MP_TAC FINITE_SUBSET);;
e (EXISTS_TAC `IMAGE (set_of_list) {[u; v:real^3] | u IN V1 /\ v IN V1}`);;
e (STRIP_TAC);;
e (MATCH_MP_TAC FINITE_IMAGE);;
e (REWRITE_TAC[SET_RULE `{[u;v] | u IN s /\ v IN s} = 
                         {y | ?u0 u1. u0 IN s /\ u1 IN s /\ y = [u0; u1]}`]);;
e (MATCH_MP_TAC FINITE_LIST_KY_LEMMA_2);;
e (EXPAND_TAC "V1" THEN MATCH_MP_TAC Packing3.KIUMVTC);;
e (ASM_REWRITE_TAC[]);;
e (SET_TAC[]);;

e (ABBREV_TAC 
   `S2 = {X | X SUBSET ball (vec 0,r) /\ mcell_set V X /\ ~NULLSET X}`);;
e (NEW_GOAL `FINITE (S2:(real^3->bool)->bool)`);;
e (MATCH_MP_TAC FINITE_SUBSET);;
e (EXISTS_TAC `S1:(real^3->bool)->bool`);;
e (ASM_REWRITE_TAC[]);;
e (EXPAND_TAC "S1" THEN EXPAND_TAC "S2" THEN SET_TAC[]);;

e (ABBREV_TAC `g = (\X. (\({u, v}). 
                            if {u, v} IN edgeX V X
                            then dihX V X (u,v) * lmfun (hl [u; v])
                            else &0))`);;
e (REWRITE_WITH 
`sum S1  (\X. sum (edgeX V X)
                 (\({u, v}). if {u, v} IN edgeX V X
                             then dihX V X (u,v) * lmfun (hl [u; v])
                             else &0)) = 
 sum S1 (\X. sum (edgeX V X) (\({u, v}). g X {u,v}))`);;
e (MATCH_MP_TAC SUM_EQ);;

e (EXPAND_TAC "S1" THEN REWRITE_TAC[IN_ELIM_THM; IN; mcell_set_2]);;
e (REPEAT STRIP_TAC);;
e (MATCH_MP_TAC SUM_EQ);;
e (REWRITE_WITH `!x'. x' IN edgeX V x <=> 
                      ?u v. VX V x u /\ VX V x v /\ ~(u = v) /\ x' = {u, v}`);;
e (REWRITE_TAC[IN; IN_ELIM_THM; edgeX]);;
e (MESON_TAC[]);;
e (REPEAT STRIP_TAC);;

e (ABBREV_TAC `f_temp = (\u v. if edgeX V x {u, v}
             then dihX V x (u,v) * lmfun (hl [u; v])
             else &0)`);;
e (NEW_GOAL `!u v. (f_temp:real^3->real^3->real) u v = f_temp v u`);;
e (EXPAND_TAC "f_temp" THEN REWRITE_TAC[BETA_THM]);;
e (REWRITE_TAC[HL; set_of_list; SET_RULE `{a,b} = {b,a}`]);;
e (REPEAT GEN_TAC THEN COND_CASES_TAC);;
e (COND_CASES_TAC);;
e (REWRITE_WITH `dihX V x (u',v') = dihX V x (v',u')`);;
e (MATCH_MP_TAC DIHX_SYM);;
e (ASM_REWRITE_TAC[IN; mcell_set; IN_ELIM_THM]);;
e (EXISTS_TAC `i:num` THEN EXISTS_TAC `ul:(real^3)list` THEN ASM_REWRITE_TAC[]);;
e (UP_ASM_TAC THEN MESON_TAC[]);;
e (COND_CASES_TAC);;
e (UP_ASM_TAC THEN MESON_TAC[]);;
e (REFL_TAC);;

e (REWRITE_WITH `(\({u, v}). if edgeX V x {u, v}
             then dihX V x (u,v) * lmfun (hl [u; v])
             else &0) = (\({u, v}). f_temp u v)`);;
e (EXPAND_TAC "f_temp");;
e (REWRITE_TAC[]);;

e (REWRITE_TAC[BETA_SET_2_THM; ASSUME `x' = {u,v:real^3}`]);;
e (REWRITE_WITH `(\({u, v}). f_temp u v) {u, v} = 
                             (f_temp:real^3->real^3->real) u v`);;
e (MATCH_MP_TAC BETA_PAIR_THM);;
e (ASM_REWRITE_TAC[]);;

e (REWRITE_WITH `(g:(real^3->bool)->(real^3->bool)->real) x = 
                 (\({u, v}). f_temp u v)`);;
e (EXPAND_TAC "f_temp" THEN EXPAND_TAC "g" THEN REWRITE_TAC[IN]);;
e (ONCE_REWRITE_TAC[EQ_SYM_EQ]);;
e (MATCH_MP_TAC BETA_PAIR_THM);;
e (ASM_REWRITE_TAC[]);;

(* ----------------------------------------------------------------------- *)

e (REWRITE_WITH `sum S1 (\X. sum (edgeX V X) (\({u, v}). g X {u, v})) = 
                 sum S1 (\X. sum (edgeX V X)  (g X))`);;
e (MATCH_MP_TAC SUM_EQ);;
e (REWRITE_TAC[] THEN REPEAT STRIP_TAC);;
e (MATCH_MP_TAC SUM_EQ);;
e (REWRITE_TAC[edgeX; IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC);;
e (ASM_REWRITE_TAC[BETA_SET_2_THM]);;

e (REWRITE_WITH `sum S1 (\X. sum (edgeX V X) (g X)) = 
                 sum S2 (\X. sum (edgeX V X) (g X))`);;
e (MATCH_MP_TAC SUM_SUPERSET);;
e (STRIP_TAC);;
e (EXPAND_TAC "S1" THEN EXPAND_TAC "S2" THEN SET_TAC[]);;
e (EXPAND_TAC "S1" THEN EXPAND_TAC "S2" THEN REWRITE_TAC[IN; IN_ELIM_THM]);;
e (REPEAT STRIP_TAC);;
e (MATCH_MP_TAC SUM_EQ_0);;
e (REPEAT STRIP_TAC);;

e (ABBREV_TAC `f_temp = (\u v. if edgeX V x {u, v}
             then dihX V x (u,v) * lmfun (hl [u; v])
             else &0)`);;
e (NEW_GOAL `!u v. (f_temp:real^3->real^3->real) u v = f_temp v u`);;
e (EXPAND_TAC "f_temp" THEN REWRITE_TAC[BETA_THM]);;
e (REWRITE_TAC[HL; set_of_list; SET_RULE `{a,b} = {b,a}`]);;
e (REPEAT GEN_TAC THEN COND_CASES_TAC);;
e (COND_CASES_TAC);;
e (REWRITE_WITH `dihX V x (u,v) = dihX V x (v,u)`);;

e (MATCH_MP_TAC DIHX_SYM);;
e (ASM_REWRITE_TAC[IN; mcell_set; IN_ELIM_THM]);;
e (UP_ASM_TAC THEN MESON_TAC[]);;
e (COND_CASES_TAC);;
e (UP_ASM_TAC THEN MESON_TAC[]);;
e (REFL_TAC);;

e (REWRITE_WITH `(g:(real^3->bool)->(real^3->bool)->real) x = 
                 (\({u, v}). f_temp u v)`);;
e (EXPAND_TAC "f_temp" THEN EXPAND_TAC "g" THEN REWRITE_TAC[IN]);;
e (UNDISCH_TAC `x' IN edgeX V x` THEN REWRITE_TAC[IN;IN_ELIM_THM; edgeX]);;
e (STRIP_TAC);;
e (ASM_SIMP_TAC[BETA_PAIR_THM]);;
e (EXPAND_TAC "f_temp");;
e (COND_CASES_TAC);;
e (REWRITE_TAC[dihX]);;
e (COND_CASES_TAC);;
e (REAL_ARITH_TAC);;
e (NEW_GOAL `F`);;
e (ASM_MESON_TAC[]);;
e (UP_ASM_TAC THEN MESON_TAC[]);;
e (REFL_TAC);;

e (REWRITE_WITH 
 `sum S2 (\X. sum (edgeX V X) (g X)) =
  sum S2 (\X. sum {e | e IN T1 /\ edgeX V X e} (g X))`);;
e (MATCH_MP_TAC SUM_EQ);;
e (EXPAND_TAC "S2" THEN REWRITE_TAC[IN; IN_ELIM_THM]);;
e (REPEAT STRIP_TAC);;
e (AP_THM_TAC THEN AP_TERM_TAC);;
e (REWRITE_TAC[SET_RULE `a = b <=> b SUBSET a /\ a SUBSET b`]);;
e (STRIP_TAC);;
e (SET_TAC[]);;
e (REWRITE_TAC[SET_RULE `A SUBSET {y | T2 y /\ A y} <=> A SUBSET T2`]);;
e (EXPAND_TAC "T1" THEN REWRITE_TAC[SUBSET; edgeX; IN; IN_ELIM_THM]);;
e (REPEAT STRIP_TAC);;
e (EXPAND_TAC "V1" THEN REWRITE_TAC[IN_ELIM_THM]);;
e (EXISTS_TAC `u:real^3` THEN EXISTS_TAC `v:real^3`);;
e (REWRITE_TAC[ASSUME `x' = {u, v:real^3}`; IN_INTER;
               MESON[IN] `V (x:real^3) <=> x IN V`]);;
e (NEW_GOAL `VX V x = V INTER x`);;
e (MATCH_MP_TAC Hdtfnfz.HDTFNFZ);;
e (UNDISCH_TAC `mcell_set V x` THEN REWRITE_TAC[mcell_set; IN; IN_ELIM_THM]);;
e (STRIP_TAC);;
e (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num`);;
e (ASM_REWRITE_TAC[]);;
e (ASM_SET_TAC[]);;

e (REWRITE_WITH 
  `sum S2 (\X. sum {e | e IN T1 /\ edgeX V X e} (g X)) = 
   sum T1 (\x. sum {X | X IN S2 /\ edgeX V X x} (\X. g X x))`);;
e (REWRITE_WITH 
  `sum S2 (\X. sum {e | e IN T1 /\ edgeX V X e} (g X)) = 
   sum S2 (\X. sum {e | e IN T1 /\ edgeX V X e} (\x.g X x))`);;
e (REWRITE_TAC[ETA_AX]);;
e (ASM_SIMP_TAC[SUM_SUM_RESTRICT]);;

(* May 09 - OK here *)

e (ABBREV_TAC 
  `T2 = {{u0:real^3, u1} | u0 IN V1 /\ u1 IN V1 /\ ~(u0 = u1) /\
                           hl[u0;u1] <= h0}`);;
e (NEW_GOAL `sum T1 (\x. sum {X | X IN S2 /\ edgeX V X x} (\X. g X x)) = 
             sum T2 (\x. sum {X | X IN S2 /\ edgeX V X x} (\X. g X x))`);;
e (MATCH_MP_TAC SUM_SUPERSET);;
e (EXPAND_TAC "T1" THEN EXPAND_TAC "T2" THEN REWRITE_TAC[IN; IN_ELIM_THM] THEN 
   REPEAT STRIP_TAC);;
e (SET_TAC[]);;
e (MATCH_MP_TAC SUM_EQ_0);;
e (EXPAND_TAC "g" THEN REPEAT STRIP_TAC);;
e (ASM_REWRITE_TAC[]);;

e (ABBREV_TAC `f_temp = (\u v. if {u, v} IN edgeX V x'
             then dihX V x' (u,v) * lmfun (hl [u; v])
             else &0)`);;
e (REWRITE_WITH `(\({u, v}). if {u, v} IN edgeX V x'
             then dihX V x' (u,v) * lmfun (hl [u; v])
             else &0) = (\({u, v}). f_temp u v)`);;
e (EXPAND_TAC "f_temp");;
e (REWRITE_TAC[]);;
e (ASM_REWRITE_TAC[]);;

e (REWRITE_WITH `(\({u, v}). f_temp u v) {u, v} = 
                             (f_temp:real^3->real^3->real) u v`);;
e (MATCH_MP_TAC BETA_PAIR_THM);;
e (ASM_REWRITE_TAC[]);;
e (EXPAND_TAC "f_temp");;
e (REPEAT GEN_TAC THEN REPEAT COND_CASES_TAC);;
e (REWRITE_TAC[HL; set_of_list; SET_RULE `{a,b} ={b,a}`]);;
e (REWRITE_WITH `dihX V x' (u',v') = dihX V x' (v',u')`);;
e (MATCH_MP_TAC DIHX_SYM);;
e (ASM_REWRITE_TAC[]);;
e (UNDISCH_TAC `x' IN {X | S2 X /\ edgeX V X x}`);;
e (EXPAND_TAC "S2" THEN REWRITE_TAC[IN; IN_ELIM_THM]);;
e (MESON_TAC[]);;
e (NEW_GOAL `F`);;
e (UP_ASM_TAC THEN REWRITE_TAC[]);;
e (ONCE_REWRITE_TAC[SET_RULE `{a,b} = {b,a}`]);;
e (ASM_REWRITE_TAC[]);;
e (UP_ASM_TAC THEN MESON_TAC[]);;
e (NEW_GOAL `F`);;
e (UP_ASM_TAC THEN REWRITE_TAC[]);;
e (ONCE_REWRITE_TAC[SET_RULE `{a,b} = {b,a}`]);;
e (ASM_REWRITE_TAC[]);;
e (UP_ASM_TAC THEN MESON_TAC[]);;
e (REFL_TAC);;

e (EXPAND_TAC "f_temp");;
e (COND_CASES_TAC);;

e (ASM_CASES_TAC `hl [u; v:real^3] <= h0`);;
e (NEW_GOAL `F`);;
e (UNDISCH_TAC `~(?u0 u1.
            (V1 u0 /\ V1 u1 /\ ~(u0 = u1) /\ hl [u0; u1] <= h0) /\
            x = {u0, u1:real^3})` THEN REWRITE_TAC[]);;
e (EXISTS_TAC `u:real^3` THEN EXISTS_TAC `v:real^3`);;
e (ASM_REWRITE_TAC[]);;
e (UNDISCH_TAC `{u, v} IN edgeX V x'` THEN REWRITE_TAC[IN; IN_ELIM_THM; edgeX]);;
e (REPEAT STRIP_TAC);;
e (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]);;
e (UP_ASM_TAC THEN MESON_TAC[]);;
e (REWRITE_WITH `lmfun (hl [u; v:real^3]) = &0`);;
e (ASM_REWRITE_TAC[lmfun]);;
e (REAL_ARITH_TAC);;
e (REFL_TAC);;

e (ASM_REWRITE_TAC[]);;

(* ==================================================================== *)

e (MATCH_MP_TAC (REAL_ARITH `(?b. a <= b /\ b + d <= e) ==> a + d <= e`));;
e (EXISTS_TAC `(&8 * mm2 / pi) *
     sum T2 (\x. sum {X | mcell_set V X /\ x IN edgeX V X} (\X. g X x))`);;
e (STRIP_TAC);;
e (REWRITE_TAC[REAL_ARITH `a * b <= a * c <=> &0 <= a * (c - b)`]);;
e (MATCH_MP_TAC REAL_LE_MUL THEN STRIP_TAC);;
e (MATCH_MP_TAC REAL_LE_MUL THEN STRIP_TAC);;
e (REAL_ARITH_TAC);;
e (MATCH_MP_TAC REAL_LE_DIV THEN SIMP_TAC[ZERO_LE_MM2_LEMMA; PI_POS_LE]);;
e (REWRITE_TAC[REAL_ARITH `&0 <= a - b <=> b <= a`]);;
e (MATCH_MP_TAC SUM_LE THEN REPEAT STRIP_TAC);;
e (MATCH_MP_TAC FINITE_SUBSET);;
e (EXISTS_TAC `T1:(real^3->bool)->bool`);;
e (ASM_REWRITE_TAC[] THEN ASM_SET_TAC[]);;
e (REWRITE_TAC[BETA_THM]);;
e (MATCH_MP_TAC SUM_SUBSET_SIMPLE);;

e (STRIP_TAC);;
e (REWRITE_TAC[IN] THEN MATCH_MP_TAC Marchal_cells_3.FINITE_EDGE_X2);;
e (UP_ASM_TAC THEN EXPAND_TAC "T2" THEN REWRITE_TAC[IN; IN_ELIM_THM] 
   THEN STRIP_TAC);;
e (EXISTS_TAC `u0:real^3` THEN EXISTS_TAC `u1:real^3` THEN ASM_REWRITE_TAC[]);;
e (REPEAT STRIP_TAC);;

e (EXPAND_TAC "S2" THEN REWRITE_TAC[IN] THEN SET_TAC[]);;
e (REWRITE_TAC[BETA_THM]);;
e (REWRITE_WITH `g x' x = gammaY V x' lmfun x`);;
e (EXPAND_TAC "g" THEN REWRITE_TAC[gammaY]);;
e (MATCH_MP_TAC gamma_y_pos_le);;

e (UP_ASM_TAC THEN ASM_REWRITE_TAC[IN_DIFF; IN; IN_ELIM_THM]);;
e (MESON_TAC[]);;
e (REWRITE_WITH 
  `sum T2 (\x. sum {X | mcell_set V X /\ x IN edgeX V X} (\X. g X x)) = 
   sum T2 (\x. &2 * pi * lmfun (radV x))`);;
e (MATCH_MP_TAC SUM_EQ);;
e (REPEAT STRIP_TAC);;
e (REWRITE_TAC[BETA_THM]);;
e (EXPAND_TAC "g");;

e (REWRITE_TAC[HL; BETA_THM; set_of_list]);;
e (UP_ASM_TAC THEN EXPAND_TAC "T2" THEN REWRITE_TAC[IN; IN_ELIM_THM]);;
e (STRIP_TAC);;

e (REWRITE_WITH `sum {X | mcell_set V X /\ edgeX V X x}
 (\X. (\({u, v}). if edgeX V X {u, v}
                  then dihX V X (u,v) * lmfun (radV {u, v}) else &0) x) = 
                sum {X | mcell_set V X /\ edgeX V X x}
 (\X. (if edgeX V X {u0, u1}
                  then dihX V X (u0,u1) * lmfun (radV {u0, u1}) else &0))`);;
e (MATCH_MP_TAC SUM_EQ);;
e (ASM_REWRITE_TAC[IN; IN_ELIM_THM; BETA_THM] THEN REPEAT STRIP_TAC);;
e (ABBREV_TAC 
  `f_temp = (\u v. if edgeX V x' {u, v}
                   then dihX V x' (u,v) * lmfun (radV {u, v}) else &0)`);;
e (NEW_GOAL `!u:real^3 v:real^3. 
            (f_temp:real^3->real^3->real) u v = f_temp v u`);;
e (EXPAND_TAC "f_temp" THEN REWRITE_TAC[BETA_THM]);;
e (REWRITE_TAC[HL; set_of_list; SET_RULE `{a,b} = {b,a}`]);;
e (REPEAT GEN_TAC THEN COND_CASES_TAC);;
e (COND_CASES_TAC);;
e (REWRITE_WITH `dihX V x' (u,v) = dihX V x' (v,u)`);;

e (MATCH_MP_TAC DIHX_SYM);;
e (ASM_REWRITE_TAC[IN; mcell_set; IN_ELIM_THM]);;
e (UP_ASM_TAC THEN MESON_TAC[]);;
e (COND_CASES_TAC);;
e (UP_ASM_TAC THEN MESON_TAC[]);;
e (REFL_TAC);;

e (REWRITE_WITH 
 `(\({u, v}). if edgeX V x' {u, v}
              then dihX V x' (u,v) * lmfun (radV {u, v})
              else &0)
    = (\({u, v}). f_temp u v)`);;
e (EXPAND_TAC "f_temp" THEN REWRITE_TAC[IN]);;
e (REWRITE_WITH 
  `(if edgeX V x' {u0, u1}
   then dihX V x' (u0,u1) * lmfun (radV {u0, u1}) else &0) = 
   f_temp u0 u1`);;
e (EXPAND_TAC "f_temp" THEN REWRITE_TAC[IN]);;
e (MATCH_MP_TAC BETA_PAIR_THM);;
e (ASM_REWRITE_TAC[]);;

e (NEW_GOAL `FINITE {X | mcell_set V X /\ edgeX V X x}`);;
e (MATCH_MP_TAC Marchal_cells_3.FINITE_EDGE_X2);;
e (EXISTS_TAC `u0:real^3` THEN EXISTS_TAC `u1:real^3` THEN ASM_REWRITE_TAC[]);;
e (ASM_SIMP_TAC [SUM_CASES]);;

e (REWRITE_TAC[SET_RULE 
 `{X | X IN {X | mcell_set V X /\ edgeX V X {u0, u1}} /\ edgeX V X {u0, u1}} =
  {X | mcell_set V X /\ edgeX V X {u0, u1}} /\ 
  {X | X IN {X | mcell_set V X /\ edgeX V X {u0, u1}} /\ 
      ~edgeX V X {u0, u1}} = {}`; SUM_CLAUSES; REAL_ARITH `a + &0 = a`]);;
e (REWRITE_TAC[SUM_RMUL]);;
e (REWRITE_WITH 
  `sum {X | mcell_set V X /\ edgeX V X {u0, u1}} (\X. dihX V X (u0,u1)) = 
   &2 * pi`);;
e (REWRITE_WITH 
  `{X | mcell_set V X /\ edgeX V X {u0, u1}} = 
   {X | mcell_set V X /\ {u0, u1} IN edgeX V X}`);;
e (REWRITE_TAC[IN]);;
e (MATCH_MP_TAC GRUTOTI);;
e (ASM_REWRITE_TAC[]);;
e (REPEAT STRIP_TAC);;
e (ASM_SET_TAC[]);;
e (ASM_SET_TAC[]);;
e (NEW_GOAL `h0 < sqrt (&2)`);;
e (REWRITE_TAC[H0_LT_SQRT2]);;
e (ASM_REAL_ARITH_TAC);;
e (REAL_ARITH_TAC);;
e (REWRITE_TAC[SUM_LMUL; REAL_ARITH `(&8 * mm2 / pi) * &2 * pi * a = 
                                     (&8 * mm2) * (pi / pi) * (&2 * a)`]);;
e (REWRITE_WITH `pi / pi = &1`);;
e (MATCH_MP_TAC REAL_DIV_REFL);;
e (REWRITE_TAC[PI_NZ]);;
e (REWRITE_TAC[REAL_ARITH `&1 * a = a`]);;
e (EXPAND_TAC "c");;
e (REWRITE_TAC[REAL_ARITH 
  `(&8 * mm2) * a + (&8 * mm2 * d) * b <= &8 * mm2 * c <=> 
   &0 <= (&8 * mm2) * (c - a - d * b)`]);;
e (MATCH_MP_TAC REAL_LE_MUL);;
e (SIMP_TAC[REAL_LE_MUL; REAL_ARITH `&0 <= &8`; ZERO_LE_MM2_LEMMA]);;
e (REWRITE_TAC[REAL_ARITH `&0 <= a - b - c <=> b + c <= a`]);;

e (EXPAND_TAC "T2");;
e (REWRITE_TAC[Marchal_cells_3.HL_2; 
   REAL_ARITH `inv (&2) * x <= y <=> x <= &2 * y`]);;
e (REWRITE_WITH `&2 *
 sum
 {{u0:real^3, u1} | u0 IN V1 /\ u1 IN V1 /\ 
                   ~(u0 = u1) /\ dist (u0,u1) <= &2 * h0}
 (\x. lmfun (radV x)) = 
 sum 
 {u0:real^3,u1 | u0 IN V1 /\ u1 IN V1 /\ 
                ~(u0 = u1) /\ dist (u0,u1) <= &2 * h0}
 (\(u0,u1). (\x. lmfun (radV x)) {u0, u1})`);;
e (ONCE_REWRITE_TAC[EQ_SYM_EQ]);;
e (MATCH_MP_TAC SUM_PAIR_2_SET);;
e (EXPAND_TAC "V1");;
e (ASM_SIMP_TAC [Packing3.KIUMVTC]);;
e (REWRITE_TAC[GSYM Marchal_cells_3.HL_2; HL;set_of_list]);;

e (ABBREV_TAC 
  `t = (\u:real^3. {v | v IN V /\ ~(u = v) /\ dist (u,v) <= &2 * h0})`);;
e (ABBREV_TAC `f_temp = (\u v. lmfun (radV {u:real^3, v}))`);;

e (REWRITE_WITH 
`sum V1
 (\u. sum {v | v IN V /\ ~(u = v) /\ dist (u,v:real^3) <= &2 * h0}
 (\v. lmfun (radV {u, v}))) = 
 sum V1 
 (\u:real^3. sum
 ((t:real^3->real^3->bool) u) ((f_temp:real^3->real^3->real) u))`);;

e (EXPAND_TAC "f_temp" THEN EXPAND_TAC "t");;
e (REWRITE_TAC[]);;
 
e (REWRITE_WITH 
 `sum V1 (\i. sum (t i) (f_temp i)) =
  sum {u0:real^3,u1:real^3 | u0 IN V1 /\ u1 IN t u0} (\(u0,u1). f_temp u0 u1)`);;
e (MATCH_MP_TAC SUM_SUM_PRODUCT);;
e (REPEAT STRIP_TAC);;
e (EXPAND_TAC "V1");;
e (ASM_SIMP_TAC [Packing3.KIUMVTC]);;
e (EXPAND_TAC "t");;
e (MATCH_MP_TAC FINITE_SUBSET);;
e (EXISTS_TAC `(V:real^3->bool) INTER ball (vec 0, r + &2 * h0)`);;
e (STRIP_TAC);;
e (ASM_SIMP_TAC [Packing3.KIUMVTC]);;
e (REWRITE_TAC[SUBSET; IN; IN_INTER; IN_BALL; IN_ELIM_THM]);;
e (REPEAT STRIP_TAC);;
e (ASM_REWRITE_TAC[]);;
e (NEW_GOAL `dist (vec 0, x) <= dist (vec 0, u0) + dist (u0, x:real^3)`);;
e (NORM_ARITH_TAC);;
e (NEW_GOAL `dist (vec 0, u0:real^3) < r`);;
e (REWRITE_TAC[GSYM IN_BALL]);;
e (UNDISCH_TAC `u0:real^3 IN V1` THEN EXPAND_TAC "V1" THEN SET_TAC[]);;
e (ASM_REAL_ARITH_TAC);;
e (EXPAND_TAC "t" THEN EXPAND_TAC "f_temp");;
e (REWRITE_TAC[IN; IN_ELIM_THM]);;
e (REWRITE_TAC[REAL_ARITH `a + &0 * r pow 2 = a`]);;
e (MATCH_MP_TAC SUM_SUBSET_SIMPLE);;
e (REPEAT STRIP_TAC);;

e (MATCH_MP_TAC FINITE_SUBSET);;
e (EXISTS_TAC 
  `{u0:real^3,u1:real^3 |u0 IN V1 /\u1 IN V INTER ball (vec 0, r + &2 * h0)}`);;
e (STRIP_TAC);;
e (MATCH_MP_TAC FINITE_PRODUCT);;
e (STRIP_TAC);;
e (EXPAND_TAC "V1");;
e (ASM_SIMP_TAC [Packing3.KIUMVTC]);;
e (ASM_SIMP_TAC [Packing3.KIUMVTC]);;
e (REWRITE_TAC[SUBSET; IN; IN_ELIM_THM]);;
e (REPEAT STRIP_TAC);;
e (EXISTS_TAC `u0:real^3` THEN EXISTS_TAC `u1:real^3`);;
e (ASM_REWRITE_TAC[]);;
e (REWRITE_TAC[MESON[IN] `(A:real^3->bool) s <=> s IN A`]);;
e (REWRITE_TAC[IN_BALL; IN_INTER; IN]);;
e (ASM_REWRITE_TAC[]);;

e (NEW_GOAL `dist (vec 0, u1) <= dist (vec 0, u0) + dist (u0, u1:real^3)`);;
e (NORM_ARITH_TAC);;
e (NEW_GOAL `dist (vec 0, u0:real^3) < r`);;
e (REWRITE_TAC[GSYM IN_BALL]);;
e (UNDISCH_TAC `(V1:real^3->bool) u0` THEN EXPAND_TAC "V1");;
e (REWRITE_TAC[MESON[IN] `(A:real^3->bool) s <=> s IN A`]);;
e (SET_TAC[]);;
e (ASM_REAL_ARITH_TAC);;
e (EXPAND_TAC "V1" THEN 
   REWRITE_TAC[MESON[IN] `(A:real^3->bool) s <=> s IN A`; IN_INTER] THEN    
   SET_TAC[]);;
e (UP_ASM_TAC THEN REWRITE_TAC[IN_DIFF; IN_ELIM_THM; IN]);;
e (REPEAT STRIP_TAC);;
e (ASM_REWRITE_TAC[]);;
e (REWRITE_TAC[lmfun_pos_le]);;

e (EXISTS_TAC `&0`);;
e (REPEAT STRIP_TAC);;
e (NEW_GOAL `F`);;
e (ASM_MESON_TAC[]);;
e (ASM_MESON_TAC[]);;

let KIZHLTL4 = top_thm();;