(* ========================================================================== *)
(* FLYSPECK - BOOK FORMALIZATION                                              *)
(*                                                                            *)
(* Chapter: Packing                                                           *)
(* Lemma: OXLZLEZ                                                             *)
(* Author: Nguyen Quang Truong                                                *)
(* Date: 2012-12-21                                                           *)
(* ========================================================================== *)



module Oxl_2012 = struct


open Oxl_def;;


(* 

needs "/home/user1/flyspeck/working/boot.hl";;



===================== dont worry about this ================
needs "/home/user1/flyspeck/working/update_database_310.ml";;
needs "/home/user1/flyspeck/working/asm_search.ml";;
needs "/home/user1/flyspeck/working/Oxl_deff.hl";;



needs "/home/user1/flyspeck/working/oxl_2012.hl";;



*)





let asms_search0 sths =
  let rec immediatesublist l1 l2 =
    match (l1,l2) with
      [],_ -> true
    | _,[] -> false
    | (h1::t1,h2::t2) -> h1 = h2 & immediatesublist t1 t2 in
  let rec sublist l1 l2 =
    match (l1,l2) with
      [],_ -> true
    | _,[] -> false
    | (h1::t1,h2::t2) -> immediatesublist l1 l2 or sublist l1 t2 in
  let exists_subterm_satisfying p (n,th) = can (find_term p) (concl th)
  and name_contains s (n,th) = sublist (explode s) (explode n) in
  let rec filterpred tm =
    match tm with
      Comb(Var("<omit this pattern>",_),t) -> not o filterpred t
    | Comb(Var("<match theorem name>",_),Var(pat,_)) -> name_contains pat
    | Comb(Var("<match aconv>",_),pat) -> exists_subterm_satisfying (aconv pat)
    | pat -> exists_subterm_satisfying (can (term_match [] pat)) in
  fun pats ->
    let triv,nontriv = partition is_var pats in
    (if triv <> [] then
      warn true
         ("Ignoring plain variables in search: "^
          end_itlist (fun s t -> s^", "^t) (map (fst o dest_var) triv))
     else ());
    (if nontriv = [] & triv <> [] then []
     else itlist (filter o filterpred) pats sths);;

let asms_search tms =
  let gstk = !current_goalstack in
  match gstk with
    [] -> []
  | (meta,gl::_,just)::_
       -> let (sths,_) = gl in
          map snd (asms_search0 sths tms)
  | _  -> failwith "asm_searchs: Invalid goal state";;

let ASMS_SEARCH_TCL (tms:(term)list) (thstac:(thm)list->tactic) : tactic =
  fun (gl:goal) ->
  let (sths,tm) = gl in
  let ths1 = map snd (asms_search0 sths tms) in
  thstac ths1 gl;;

let ASM_SEARCH_TCL (tms:(term)list) (thstac:thm->tactic) : tactic =
  let foo ths = match ths with
                  [] -> failwith "ASM_SEARCH_TCL: No matching asms found"
                | _  -> thstac (hd ths) in
  ASMS_SEARCH_TCL tms foo;;




let ATTACH thm = MATCH_MP (MESON[]` ! a b. ( a ==> b ) ==> ( a <=> a /\ b )`) thm;;

let NHANH tm = ONCE_REWRITE_TAC[ ATTACH (SPEC_ALL ( tm ))];;

let PHA = REWRITE_TAC[ MESON[] ` (a/\b)/\c <=> a/\ b /\ c `; MESON[]`
a ==> b ==> c <=> a /\ b ==> c `];;

let DOWN_TAC = REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[IMP_IMP] THEN PHA;;


let DOWN = FIRST_X_ASSUM MP_TAC;;

let DOWNS n = REPLICATE_TAC n DOWN THEN PHA;;

let REMOVE_TAC = FIRST_X_ASSUM MP_TAC THEN MATCH_MP_TAC (TAUT` a ==> b ==> a`);;


let types_thm th = let cl = concl th in
List.map dest_var (frees cl );;

let seans_fn () =
let (tms,tm) = top_goal () in
let vss = map frees (tm::tms) in
let vs = setify (flat vss) in
map dest_var vs;;





let PAT_REWRITE_TAC tm thms =
(CONV_TAC (PAT_CONV tm (REWRITE_CONV thms )));;

let FOR_ASM th =
let th1 = REWRITE_RULE[MESON[]` a /\ b ==> c <=>
a ==> b ==> c `] th in
let th2 = SPEC_ALL th1 in UNDISCH_ALL th2;;

(* change a th having form |- A ==> t to the form A |- t
to get ready to some other commands


|- A ==> t
----------- FOR_ASM
A |- t
*)

let ASSUME_TAC2 = ASSUME_TAC o FOR_ASM;;


let PAT_ONCE_REWRITE_TAC tm thms =
(CONV_TAC (PAT_CONV tm (ONCE_REWRITE_CONV thms )));;

let ASM_PAT_RW_TAC tm thms = EVERY_ASSUM (fun th ->
(CONV_TAC (PAT_CONV tm (ONCE_REWRITE_CONV
( th ::[ thms ] )))));;

let PAT_TH_TAC tm th =
(CONV_TAC (PAT_CONV tm (REWRITE_CONV[th] )));;


let IMP_TO_EQ_RULE th = MATCH_MP (TAUT` (a ==> b ) ==>
( a <=> a /\ b )`) (SPEC_ALL th);;

let NHANH_PAT tm th = PAT_ONCE_REWRITE_TAC tm
[ IMP_TO_EQ_RULE th ];;


let MAKE_FIRST_TAC tm = UNDISCH_TAC tm THEN DISCH_TAC;;


(* ---------- BG TEST ------------- *)



let rec els L = match L with
[] -> p ()
| (x::l) -> e x; els l;;

let rec bls L = match L with
[] -> p ()
| (x::l) -> b (); bls l;;





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

let QU_OR_QXY = 
prove (` ! i. cc_qu_v11 cc i \/ cc_qx_v11 cc i \/ cc_qy_v11 cc i `,

REWRITE_TAC[cc_qu_v11; cc_qx_v11; cc_qy_v11] THEN MESON_TAC[]);;

let QX_NN0 = REWRITE_RULE[REAL_ARITH` #0.0 = &0`] QX_NN;;
let QY_NN0 = REWRITE_RULE[REAL_ARITH` #0.0 = &0`] QY_NN;;


let MOD_REFL = REWRITE_RULE[MULT_CLAUSES] (SPECL [`m:num `;`1`] MOD_MULT);;


let SUM_POS_LT_NUMSEG = 
prove_by_refinement (
`!m n f. m <= n /\ (!p. m <= p /\ p <= n ==> &0 < f p) ==> &0 < sum (m..n) f`,

[GEN_TAC;
INDUCT_TAC;
REWRITE_TAC[LE; SUM_CLAUSES_NUMSEG];
SIMP_TAC[];
GEN_TAC;
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC` 0`));
ASM_REWRITE_TAC[LE];


GEN_TAC;
ASM_CASES_TAC` m = SUC n `;
ASM_REWRITE_TAC[SUM_SING_NUMSEG];
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC` SUC n `));
FIRST_X_ASSUM MP_TAC;
CONV_TAC TAUT;

STRIP_TAC;
SUBGOAL_THEN`  &0 < sum (m..n) f ` MP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
CONJ_TAC;
DOWN_TAC;
ARITH_TAC;
FIRST_X_ASSUM MP_TAC;
MESON_TAC[ARITH_RULE` a <= p ==> a <= SUC p `];
REWRITE_TAC[SUM_CLAUSES_NUMSEG];
ASM_REWRITE_TAC[];
STRIP_TAC;
MATCH_MP_TAC REAL_LT_ADD;
ASM_REWRITE_TAC[];
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[LE_REFL]]);;




let periodic_mult = 
prove_by_refinement (
` periodic f n <=> !k p. f (k * n + p ) = f p `,

[REWRITE_TAC[periodic];
EQ_TAC;
STRIP_TAC;
INDUCT_TAC;
REWRITE_TAC[MULT; ADD];

REWRITE_TAC[ADD1; NUM_RING` (k + 1) * n = k * n + n `];
REWRITE_TAC[ARITH_RULE` (a + b) + c = a + b + (c:num) `];
DOWN_TAC;
MESON_TAC[ADD_SYM];

DISCH_THEN (MP_TAC o (SPEC`1 `));
REWRITE_TAC[MULT_CLAUSES];
MESON_TAC[ADD_SYM]]);;




(* =================================================== *)
CHQSQEY_concl;;




let glt = `cc_bool_model_v11 cc /\
 cc_real_model_v11 cc /\
 sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 /\
 indSet = 0..cc_card_v11 cc - 1
 ==> (?i j k.
          ~(i = j \/ j = k \/ k = i) /\
          i IN indSet /\
          j IN indSet /\
          k IN indSet /\
          cc_4cell_v11 cc i /\
          cc_4cell_v11 cc j /\
          cc_4cell_v11 cc k)`;;


let e1 = (STRIP_TAC THEN ASSUME_TAC2 CC_CARD2 THEN 
DOWN_TAC THEN NHANH periodic_fn THEN 
ASM_CASES_TAC` ~ (?i. i IN indSet /\ cc_4cell_v11 cc i)` THENL [


SUBGOAL_THEN` ! i. i IN indSet ==> cc_qy_v11 cc i ` MP_TAC THENL 
[MP_TAC QU_OR_QXY THEN 
REWRITE_TAC[cc_qu_v11; cc_qx_v11] THEN 
FIRST_X_ASSUM MP_TAC THEN 
MESON_TAC[];

REWRITE_TAC[cc_real_model_v11; cc_bool_model_v11]] THEN 

REPEAT STRIP_TAC THEN 
SUBGOAL_THEN` ! i. i IN indSet ==> &0 < cc_gg_v11 cc i ` MP_TAC THENL [


UNDISCH_TAC`!i. i IN indSet ==> cc_qy_v11 cc i` THEN
DISCH_THEN NHANH THEN
UNDISCH_TAC`!i. cc_qy_v11 cc i ==> #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i` THEN
STRIP_TAC THEN
FIRST_ASSUM NHANH THEN
GEN_TAC THEN

UNDISCH_TAC` !i. #0.606 <= cc_azim_v11 cc i ` THEN
DISCH_TAC THEN
FIRST_ASSUM (MP_TAC o (SPEC` i: num `)) THEN
REAL_ARITH_TAC;

STRIP_TAC] THEN 
SUBGOAL_THEN` &0 < sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) ` MP_TAC THENL


[MATCH_MP_TAC SUM_POS_LT_NUMSEG THEN CONJ_TAC THENL 
[UNDISCH_TAC` ~ (cc_card_v11 cc = 0) ` THEN 
ARITH_TAC;

REPEAT STRIP_TAC THEN 
FIRST_X_ASSUM MATCH_MP_TAC THEN 
ASM_REWRITE_TAC[IN_NUMSEG]];


UNDISCH_TAC` sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ` THEN 
REAL_ARITH_TAC];

FIRST_X_ASSUM MP_TAC] THEN 
REWRITE_TAC[] THEN STRIP_TAC THEN STRIP_TAC THEN 
ASM_CASES_TAC` ~ ( ?j. j IN indSet /\ ~(j = i) /\ cc_4cell_v11 cc j)`);;



let e2 = (FIRST_X_ASSUM MP_TAC THEN 
REWRITE_TAC [
MESON[]` ~(?j. j IN indSet /\ ~(j = i) /\ cc_4cell_v11 cc j) <=>
!j. j IN indSet /\ ~(j = i)  ==> ~ cc_4cell_v11 cc j `] THEN 

STRIP_TAC THEN 
SUBGOAL_THEN`!j. j IN indSet /\ ~(j = i) ==> cc_qy_v11 cc j ` MP_TAC THENL [



FIRST_ASSUM NHANH THEN 
REPEAT STRIP_TAC THEN 
MP_TAC (SPEC `j:num` QU_OR_QXY) THEN 
FIRST_ASSUM MP_TAC THEN 
SIMP_TAC[cc_qu_v11; cc_qx_v11];


SUBGOAL_THEN` cc_qu_v11 cc i \/ cc_qx_v11 cc i ` MP_TAC] THENL [

MP_TAC (SPEC_ALL QU_OR_QXY) THEN 
MATCH_MP_TAC (TAUT` ~ c ==> a \/ b \/ c ==> a \/ b `) THEN
ASM_REWRITE_TAC[cc_qy_v11];


STRIP_TAC THEN 
STRIP_TAC THEN 
SUBGOAL_THEN` cc_qy_v11 cc (i + 1)` MP_TAC]);;




let e3 = (ASM_CASES_TAC` i < cc_card_v11 cc - 1 ` THENL [
FIRST_ASSUM (MP_TAC o SPEC` i + 1 `) THEN ANTS_TAC THENL [
ASM_REWRITE_TAC[ARITH_RULE`~( i + 1 = i ) `; IN_NUMSEG] THEN 
FIRST_ASSUM MP_TAC THEN 
ARITH_TAC;

REWRITE_TAC[]];



UNDISCH_TAC` (i:num) IN indSet ` THEN 
ASM_REWRITE_TAC[IN_NUMSEG] THEN 
STRIP_TAC THEN 
ASSUME_TAC2 (ARITH_RULE` ~(i < cc_card_v11 cc - 1) /\ i <= cc_card_v11 cc - 1 ==> cc_card_v11 cc - 1 = i `) THEN 
UNDISCH_TAC` periodic (cc_qy_v11 cc) (cc_card_v11 cc) ` THEN 
EXPAND_TAC "i" THEN 
DOWN_TAC THEN 
REWRITE_TAC[cc_bool_model_v11] THEN 
NHANH (ARITH_RULE` ~( a = 0) ==> a - 1 + 1 = a `) THEN 
SIMP_TAC[periodic] THEN 


STRIP_TAC THEN 
ONCE_REWRITE_TAC[ARITH_RULE` a = 0 + a `] THEN 
ASM_REWRITE_TAC[] THEN 
FIRST_X_ASSUM MATCH_MP_TAC THEN 
ASM_REWRITE_TAC[IN_NUMSEG] THEN 

EXPAND_TAC "i" THEN 
REWRITE_TAC[LE] THEN 
MATCH_MP_TAC (ARITH_RULE` 2 <= a ==> ~( 0 = a - 1) `) THEN 
FIRST_ASSUM ACCEPT_TAC]);;


let e4 = (


(* ------------------------------------- *)
DOWN_TAC THEN 
REWRITE_TAC[cc_real_model_v11] THEN 
STRIP_TAC THEN 
UNDISCH_TAC` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1)
          ==> cc_eps <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) ` THEN 
STRIP_TAC THEN 
FIRST_ASSUM (MP_TAC o SPEC_ALL) THEN 
ANTS_TAC THENL [ASM_REWRITE_TAC[];
MAKE_FIRST_TAC` !i. cc_qy_v11 cc i
          ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i `] THEN 

SUBGOAL_THEN` cc_gg3a_v11 cc ( i + 1) <= cc_gg_v11 cc ( i + 1) ` MP_TAC THENL [
MATCH_MP_TAC REAL_LE_TRANS THEN 
EXISTS_TAC` cc_gg3a_v11 cc (i + 1) + cc_gg3b_v11 cc (i + 1)` THEN 

CONJ_TAC THENL [
REWRITE_TAC[REAL_LE_ADDR] THEN 
FIRST_X_ASSUM MATCH_MP_TAC THEN 
ASM_REWRITE_TAC[];

FIRST_X_ASSUM MATCH_MP_TAC THEN 
ASM_REWRITE_TAC[]];


PHA] THEN 
NHANH (REAL_ARITH` a <= b /\ c <= d + a ==> c <= d + b `));;



(* 
`(cc_gg3a_v11 cc (i + 1) <= cc_gg_v11 cc (i + 1) /\
  cc_eps <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1)) /\
 cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1)
 ==> (?i j k.
          ~(i = j \/ j = k \/ k = i) /\
          i IN indSet /\
          j IN indSet /\
          k IN indSet /\
          cc_4cell_v11 cc i /\
          cc_4cell_v11 cc j /\
          cc_4cell_v11 cc k)`

*)

(* =============================================
================================================
let WKR_COMPTED = prove_by_refinement (glt ,


g glt;;



els
[e1 ; e2 ; e3; e4;

STRIP_TAC THEN 
MP_TAC (SPECL [` cc_gg_v11 cc `;` cc_card_v11 cc`] periodic_sum) THEN 
ANTS_TAC THENL [
ASM_REWRITE_TAC[] THEN 
UNDISCH_TAC` cc_bool_model_v11 cc ` THEN 
SIMP_TAC[cc_bool_model_v11];
DISCH_THEN (ASSUME_TAC o (SPEC` i:num`))] THEN 
SUBGOAL_THEN` &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) ` MP_TAC;

ASM_CASES_TAC` i < cc_card_v11 cc - 1 `;
SUBGOAL_THEN` 0..cc_card_v11 cc - 1 = ((0..cc_card_v11 cc - 1) DIFF {i, i +1} ) UNION {i, i + 1}` MP_TAC THENL [
MATCH_MP_TAC (SET_RULE` s SUBSET S ==> S = (S DIFF s) UNION s `) THEN 
UNDISCH_TAC` (i:num) IN indSet` THEN 
ASM_SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN 

REWRITE_TAC[IN_NUMSEG] THEN 
FIRST_ASSUM MP_TAC THEN 
ARITH_TAC;


DISCH_THEN SUBST1_TAC THEN 
REWRITE_TAC[]];

SUBGOAL_THEN` ! f.  sum ((0..cc_card_v11 cc - 1) DIFF {i, i + 1} UNION {i, i + 1}) f = sum ((0..cc_card_v11 cc - 1) DIFF {i, i + 1}) f + sum ({i, i + 1}) f ` MP_TAC THENL [
GEN_TAC THEN 
MATCH_MP_TAC SUM_UNION THEN
CONJ_TAC THENL [
MESON_TAC[FINITE_NUMSEG; FINITE_DIFF];
CONJ_TAC] THENL [
MESON_TAC[FINITE_EMPTY; FINITE_INSERT];
SET_TAC[]];


SIMP_TAC[] THEN
STRIP_TAC THEN 
MATCH_MP_TAC REAL_LE_ADD THEN 
CONJ_TAC];


MATCH_MP_TAC SUM_POS_LE THEN CONJ_TAC THENL [
MESON_TAC[FINITE_NUMSEG; FINITE_DIFF];
REPEAT STRIP_TAC];



REWRITE_TAC[REAL_ARITH` &0 = #0.0 `] THEN 
MATCH_MP_TAC QY_NN THEN 
ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11] THEN
FIRST_ASSUM MATCH_MP_TAC
 THEN
ASM_REWRITE_TAC[] THEN 
FIRST_ASSUM MP_TAC THEN 
REWRITE_TAC[IN_DIFF; IN_INSERT] THEN 
MESON_TAC[];



MP_TAC (MATCH_MP Geomdetail.SUM_DIS2 (ARITH_RULE` ~( i = i + 1 ) `)) THEN
STRIP_TAC THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC` cc_eps:real` THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[cc_eps] THEN
REAL_ARITH_TAC;


ASM_CASES_TAC` i = cc_card_v11 cc - 1 ` THENL [
SUBGOAL_THEN` 0..cc_card_v11 cc - 1 = ((0..cc_card_v11 cc - 1) DIFF {i, 0}) UNION {i, 0}` MP_TAC THENL [
MATCH_MP_TAC (SET_RULE` s SUBSET S ==> S = (S DIFF s) UNION s`) THEN
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_NUMSEG] THEN
FIRST_X_ASSUM MP_TAC THEN
UNDISCH_TAC` (i:num) IN indSet` THEN
ASM_REWRITE_TAC[IN_NUMSEG] THEN
SIMP_TAC[LE; LE_0];

DISCH_THEN SUBST1_TAC] THEN 

SUBGOAL_THEN` FINITE ((0..cc_card_v11 cc - 1) DIFF {i, 0}) /\
     FINITE {i, 0} /\
     DISJOINT ((0..cc_card_v11 cc - 1) DIFF {i, 0}) {i, 0} ` MP_TAC THENL [
REWRITE_TAC[SET_RULE` DISJOINT (S DIFF s) s`] THEN 
MESON_TAC[FINITE_NUMSEG; FINITE_INSERT; FINITE_EMPTY; FINITE_DIFF];
NHANH (ISPECL[` cc_gg_v11 cc`;` (0..cc_card_v11 cc - 1) DIFF {i, 0}`;` {i, 0}`] SUM_UNION) THEN 
SIMP_TAC[]] THEN 

STRIP_TAC THEN 
MATCH_MP_TAC REAL_LE_ADD THEN 
CONJ_TAC THENL [
MATCH_MP_TAC SUM_POS_LE THEN
ASM_REWRITE_TAC[] THEN
GEN_TAC THEN
REWRITE_TAC[IN_DIFF; IN_INSERT] THEN
STRIP_TAC THEN
MATCH_MP_TAC QY_NN0 THEN
ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM MP_TAC THEN
CONV_TAC TAUT;


SUBGOAL_THEN` ~( i = 0) ` MP_TAC THENL [
ASM_REWRITE_TAC[] THEN 
UNDISCH_TAC` 2 <= cc_card_v11 cc ` THEN 
ARITH_TAC; STRIP_TAC]] THEN 




ASSUME_TAC2 (ISPECL [`i:num `;` 0`; ` cc_gg_v11 cc `] Geomdetail.SUM_DIS2) THEN
UNDISCH_TAC` cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) ` THEN
ASSUME_TAC2 (ARITH_RULE` 2 <= cc_card_v11 cc ==> cc_card_v11 cc - 1 + 1 = cc_card_v11 cc `) THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC` periodic (cc_gg_v11 cc) (cc_card_v11 cc) ` THEN
REWRITE_TAC[periodic] THEN
STRIP_TAC THEN
FIRST_ASSUM (MP_TAC o (SPEC` 0 `)) THEN
SIMP_TAC[ARITH_RULE` 0 + a = a `] THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC` cc_eps:real ` THEN
ASM_REWRITE_TAC[cc_eps] THEN
REAL_ARITH_TAC;


UNDISCH_TAC` (i:num) IN indSet` THEN
ASM_REWRITE_TAC[IN_NUMSEG] THEN
REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN
ARITH_TAC];

ASM_REWRITE_TAC[REAL_ARITH` a <= b <=> ~( b < a ) `];

ASSUME_TAC2 QX_NN0 THEN
SUBGOAL_THEN` &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc)` MP_TAC THENL [


MATCH_MP_TAC SUM_POS_LE_NUMSEG THEN
REPEAT STRIP_TAC THEN
ASM_CASES_TAC` p = (i:num) ` THENL [
ASM_REWRITE_TAC[];

MATCH_MP_TAC QY_NN0 THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[IN_NUMSEG]];
ASM_REWRITE_TAC[REAL_ARITH` a <= b <=> ~( b < a ) `]];

STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc)` MP_TAC;


MATCH_MP_TAC SUM_POS_LE_NUMSEG;
REPEAT STRIP_TAC;
ASM_CASES_TAC` p = (i:num) `;
ASM_REWRITE_TAC[];
MATCH_MP_TAC QX_NN0;

ASM_REWRITE_TAC[];
MATCH_MP_TAC QY_NN0;

ASM_REWRITE_TAC[];
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[IN_NUMSEG];

ASM_REWRITE_TAC[REAL_ARITH` a <= b <=> ~( b < a ) `];


(* *********** *)

FIRST_X_ASSUM MP_TAC THEN 
REWRITE_TAC[] THEN 
STRIP_TAC THEN 
ASM_CASES_TAC` ~(? k. (k IN indSet /\ ~(k = i) /\ ~(k = j)) /\ cc_4cell_v11 cc k)`;


FIRST_X_ASSUM MP_TAC THEN
REWRITE_TAC[MESON[]` ~(?x. p x /\ q x) <=> !x. p x ==> ~ q x `] THEN
STRIP_TAC THEN

SUBGOAL_THEN` !k. k IN indSet /\ ~(k = i) /\ ~(k = j) ==> cc_qy_v11 cc k ` MP_TAC THENL [
FIRST_X_ASSUM NHANH THEN
REPEAT STRIP_TAC THEN
MP_TAC (SPEC` k:num` QU_OR_QXY) THEN
ASM_REWRITE_TAC[cc_qu_v11; cc_qx_v11] ;

STRIP_TAC THEN
ASM_CASES_TAC`~( ?qy. qy IN indSet /\ cc_qy_v11 cc qy)` THEN
FIRST_X_ASSUM MP_TAC THEN
FIRST_X_ASSUM MP_TAC THEN
PHA THEN
NHANH (SET_RULE` (!k. k IN indSet /\ ~(k = i) /\ ~(k = j) ==> cc_qy_v11 cc k) /\
 ~(?qy. qy IN indSet /\ cc_qy_v11 cc qy) ==> indSet SUBSET {i, j} `)] THEN 
DOWN_TAC THEN
REWRITE_TAC[cc_real_model_v11] THEN
STRIP_TAC THEN
UNDISCH_TAC` sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) = &2 * pi ` THEN
UNDISCH_TAC` !i. #0.606 <= cc_azim_v11 cc i ` THEN
NHANH (REAL_ARITH` #0.606 <= f i ==> &0 <= f i `) THEN
STRIP_TAC THEN STRIP_TAC;



SUBGOAL_THEN` sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) <= sum {i, j} (cc_azim_v11 cc) ` MP_TAC;


MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN
UNDISCH_TAC` indSet SUBSET {i, j:num} ` THEN
ASM_SIMP_TAC[] THEN
MESON_TAC[FINITE_EMPTY; FINITE_INSERT];


ASSUME_TAC2 (ISPECL [`j:num `;` i:num `;` cc_azim_v11 cc `] Geomdetail.SUM_DIS2) THEN
ONCE_REWRITE_TAC[INSERT_COMM] THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC` cc_4cell_v11 cc j ` THEN
UNDISCH_TAC` cc_4cell_v11 cc i ` THEN
UNDISCH_TAC` !i. cc_4cell_v11 cc i ==> cc_azim_v11 cc i < #2.8 ` THEN
DISCH_TAC THEN
FIRST_ASSUM NHANH THEN
REPEAT STRIP_TAC;

MP_TAC (MESON[Flyspeck_constants.bounds]` #3.14159 < pi`) THEN
STRIP_TAC THEN
ASSUME_TAC2 (
REAL_ARITH` &2 * pi <= cc_azim_v11 cc j + cc_azim_v11 cc i /\
cc_azim_v11 cc j < #2.8 /\ cc_azim_v11 cc i < #2.8 ==>
pi < #2.8 `) THEN
REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN
REAL_ARITH_TAC;




(* 

 46 [`qy IN indSet`]
 47 [`cc_qy_v11 cc qy`]
 48 [`!i. #0.606 <= cc_azim_v11 cc i /\ &0 <= cc_azim_v11 cc i`]
 49 [`sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) = &2 * pi`]

`?i j k.
     ~(i = j \/ j = k \/ k = i) /\
     i IN indSet /\
     j IN indSet /\
     k IN indSet /\
     cc_4cell_v11 cc i /\
     cc_4cell_v11 cc j /\
     cc_4cell_v11 cc k`

*)



SUBGOAL_THEN` ?ii jj. {ii, jj} = {i:num, j} /\ cc_qy_v11 cc (ii + 1)` MP_TAC;

ASM_CASES_TAC` cc_qy_v11 cc ( (i:num) + 1) ` THENL [
EXISTS_TAC` i:num ` THEN
EXISTS_TAC` j:num ` THEN
FIRST_X_ASSUM MP_TAC THEN
SIMP_TAC[];

EXISTS_TAC` j:num ` ] THEN
EXISTS_TAC` i:num ` THEN
REWRITE_TAC[ INSERT_COMM] THEN
SUBGOAL_THEN` ! m. ~cc_qy_v11 cc m ==> (m) MOD (cc_card_v11 cc)  = (i:num)
\/ m MOD (cc_card_v11 cc) = j ` MP_TAC;

GEN_TAC THEN
ASSUME_TAC2 (ARITH_RULE` 2 <= cc_card_v11 cc ==> ~( cc_card_v11 cc = 0)`) THEN
ABBREV_TAC ` nn = cc_card_v11 cc ` THEN
ASSUME_TAC2 (SPECL [`m:num `;` nn:num `] MOD_IN_NUMSEG) THEN
ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn: num `;` m:num `] periodic_mod) THEN
ABBREV_TAC`m1 = m MOD nn ` THEN
ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
SUBGOAL_THEN` m1 = (i:num) \/ m1 = j ` MP_TAC;

UNDISCH_TAC` !k. k IN indSet /\ ~(k = i) /\ ~(k = j) ==> cc_qy_v11 cc k` THEN
DISCH_THEN (MP_TAC o (SPEC` m1 : num `)) THEN
UNDISCH_TAC` ~ cc_qy_v11 cc m1 ` THEN
UNDISCH_TAC` m1 IN 0..nn - 1 ` THEN
ASM_REWRITE_TAC[] THEN
CONV_TAC TAUT;

SIMP_TAC[];
(* *)
 
(* 
 50 [`~cc_qy_v11 cc (i + 1)`]

`(!m. ~cc_qy_v11 cc m ==> m MOD cc_card_v11 cc = i \/ m MOD cc_card_v11 cc = j)
 ==> cc_qy_v11 cc (j + 1)`

*)



ABBREV_TAC` nn = cc_card_v11 cc ` THEN 
STRIP_TAC THEN 
ASM_CASES_TAC` cc_qy_v11 cc (j + 1) ` THENL [
FIRST_X_ASSUM ACCEPT_TAC;
SUBGOAL_THEN` ! m. ~(m IN indSet /\ (m + 1) MOD nn = m)` MP_TAC];

GEN_TAC THEN 
ASM_REWRITE_TAC[IN_NUMSEG] THEN 
STRIP_TAC THEN 
ASM_CASES_TAC` m = nn - 1 `;

ASSUME_TAC2 (ARITH_RULE` 2 <= nn /\ m = nn - 1 ==> m + 1 = 1 * nn + 0  `) THEN
MP_TAC (SPECL [` m + 1`;` nn: num `;` 1`; ` 0`] MOD_UNIQ) THEN

ANTS_TAC THENL [
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC` 2 <= nn ` THEN
ARITH_TAC;

DISCH_THEN SUBST_ALL_TAC THEN
UNDISCH_TAC` 0 = m` THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC` 2 <= nn ` THEN
ARITH_TAC ];

ASSUME_TAC2 (ARITH_RULE` m <= nn - 1 /\ ~(m = nn - 1) ==> m + 1 < nn`) THEN 
MP_TAC (SPECL[` m + 1 `;` nn: num `;` 0 `;` m + 1 `] MOD_UNIQ) THEN 
ANTS_TAC THENL [
ASM_REWRITE_TAC[] THEN ARITH_TAC;
DISCH_THEN SUBST_ALL_TAC] THEN 
UNDISCH_TAC` m + 1 = m`  THEN ARITH_TAC;


FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i + 1 `));
FIRST_ASSUM (ASSUME_TAC2 o (SPEC` j + 1 `));
STRIP_TAC;
FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i:num`));
FIRST_ASSUM (ASSUME_TAC2 o (SPEC` j:num`));
REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC);
UNDISCH_TAC` (i:num) IN indSet `;
UNDISCH_TAC` (j:num) IN indSet`;
SIMP_TAC[];
REPEAT STRIP_TAC;
UNDISCH_TAC` (i + 1) MOD nn = i \/ (i + 1) MOD nn = j `;
UNDISCH_TAC` (j + 1) MOD nn = i \/ (j + 1) MOD nn = j`;
ASM_REWRITE_TAC[];

ASM_CASES_TAC` j + 1 = nn `;
ASM_REWRITE_TAC[];

(* 
`nn MOD nn = i ==> ~((i + 1) MOD nn = j)`

*)


ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> ~( nn = 0) `) THEN 
ASSUME_TAC2 (SPEC`nn:num` (GEN_ALL MOD_REFL)) THEN 
ASM_REWRITE_TAC[] THEN 
DISCH_THEN (SUBST_ALL_TAC o SYM) THEN 
REWRITE_TAC[ADD] THEN 
ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> 1 < nn `) THEN 
FIRST_ASSUM MP_TAC THEN 
NHANH MOD_LT THEN 
SIMP_TAC[] THEN 
REPEAT STRIP_TAC THEN 
FIRST_X_ASSUM (SUBST_ALL_TAC o SYM) THEN 
SUBST_ALL_TAC (ARITH_RULE` 1 + 1 = 2 `) THEN 

UNDISCH_TAC` qy IN (indSet: num -> bool) ` THEN 
ASM_REWRITE_TAC[IN_NUMSEG] THEN 
UNDISCH_TAC` 2 = nn ` THEN 
DISCH_THEN (SUBST_ALL_TAC o SYM) THEN 
REWRITE_TAC[ARITH_RULE` 0 <= x /\ x <= 2 - 1 <=> x = 0 \/ x = 1 `];


STRIP_TAC THENL [

FIRST_X_ASSUM SUBST_ALL_TAC THEN 
UNDISCH_TAC` cc_4cell_v11 cc 0 ` THEN 
UNDISCH_TAC` cc_qy_v11 cc 0 ` THEN 
ASM_REWRITE_TAC[cc_qy_v11];

FIRST_X_ASSUM SUBST_ALL_TAC THEN 
UNDISCH_TAC` cc_4cell_v11 cc 1 ` THEN 
UNDISCH_TAC` cc_qy_v11 cc 1 ` THEN 
ASM_REWRITE_TAC[cc_qy_v11]];

STRIP_TAC THEN 
SUBGOAL_THEN` j + 1 < (nn: num) ` MP_TAC THENL [
UNDISCH_TAC` j:num IN indSet` THEN 
ASM_REWRITE_TAC[IN_NUMSEG] THEN 
UNDISCH_TAC` ~( j + 1 = nn ) ` THEN 
UNDISCH_TAC` 2 <= nn ` THEN 
ARITH_TAC;

NHANH MOD_LT THEN 
ASM_REWRITE_TAC[] THEN 
SIMP_TAC[]];

REWRITE_TAC[ARITH_RULE` (j + 1) + 1 = j + 2 `] THEN
ASM_CASES_TAC` j + 2 < nn` THENL [
ASSUME_TAC2 (SPECL [`j + 2`;` nn:num `] MOD_LT) THEN
FIRST_X_ASSUM SUBST1_TAC THEN
ARITH_TAC;
STRIP_TAC];

ASSUME_TAC2  (ARITH_RULE` ~(j + 2 < nn) /\ j + 1 < nn ==> j + 2 = nn `) THEN
ASM_REWRITE_TAC[] THEN
ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> ~(nn = 0) `) THEN
ASSUME_TAC2 (SPEC`nn:num ` (GEN_ALL MOD_REFL)) THEN
ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
FIRST_X_ASSUM (SUBST_ALL_TAC o SYM) THEN
UNDISCH_TAC` qy:num IN indSet ` THEN
ASM_REWRITE_TAC[IN_NUMSEG] THEN
EXPAND_TAC "nn" THEN
REWRITE_TAC[ARITH_RULE` 0 <= qy /\ qy <= (0 + 2) - 1 <=> qy = 0 \/ qy = 1`] THEN
STRIP_TAC THENL [

UNDISCH_TAC` cc_qy_v11 cc qy ` THEN
UNDISCH_TAC` cc_4cell_v11 cc 0 ` THEN
ASM_SIMP_TAC[cc_qy_v11];

UNDISCH_TAC` cc_qy_v11 cc qy ` THEN
UNDISCH_TAC` cc_4cell_v11 cc i ` THEN
ASM_SIMP_TAC[cc_qy_v11; ADD]];




STRIP_TAC THEN ABBREV_TAC` nn = cc_card_v11 cc ` THEN 
SUBGOAL_THEN` ? nj k. nj < nn /\ nj + 1 = k * nn + jj /\ cc_qy_v11 cc nj ` MP_TAC;

ASM_CASES_TAC` jj = 0 `;

EXISTS_TAC` nn - 1 ` THEN
EXISTS_TAC` 1 ` THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN
ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> nn - 1 < nn /\ nn - 1 + 1 = 1 * nn + 0 `) THEN
ASM_SIMP_TAC[] THEN
ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> ~( nn - 1 = 0) `) THEN
ASM_CASES_TAC` nn - 1 = ii' ` THENL [
UNDISCH_TAC` cc_qy_v11 cc (ii' + 1)` THEN
FIRST_X_ASSUM (SUBST_ALL_TAC o SYM) THEN
ASM_REWRITE_TAC[ARITH_RULE` 1 * n + 0 = 0 + n`] THEN
UNDISCH_TAC` periodic (cc_qy_v11 cc) nn ` THEN
SIMP_TAC[periodic] THEN
STRIP_TAC THEN
UNDISCH_TAC` {nn - 1, 0} = {i, j} ` THEN
UNDISCH_TAC` cc_4cell_v11 cc i ` THEN
UNDISCH_TAC` cc_4cell_v11 cc j ` THEN
REWRITE_TAC[cc_qy_v11] THEN
SET_TAC[];


ASSUME_TAC2 (SET_RULE` ~(nn - 1 = 0) /\ ~(nn - 1 = ii') /\ {ii', 0} = {i, j} ==> ~( nn - 1 = i) /\ ~( nn - 1 = j ) `) THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[IN_NUMSEG; LE_REFL] THEN
MATCH_MP_TAC (ARITH_RULE` 2 <= nn ==> 0 <= nn - 1 `) THEN
FIRST_X_ASSUM ACCEPT_TAC];


EXISTS_TAC` jj - 1 ` THEN
EXISTS_TAC` 0 ` THEN
REWRITE_TAC[MULT; ADD] THEN
CONJ_TAC THENL [
     MATCH_MP_TAC (ARITH_RULE` jj <= nn - 1 /\ ~( jj = 0) ==> jj - 1 < nn `) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN` jj:num IN indSet` MP_TAC THENL [


MP_TAC (SET_RULE` jj IN {ii', jj:num}`) THEN
ASM_REWRITE_TAC[] THEN
ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
UNDISCH_TAC` (i:num) IN indSet ` THEN
UNDISCH_TAC` (j:num) IN indSet ` THEN
ASM_REWRITE_TAC[] THEN
MESON_TAC[];

ASM_REWRITE_TAC[IN_NUMSEG] THEN
SIMP_TAC[]];

    CONJ_TAC THENL [FIRST_X_ASSUM MP_TAC THEN 
ARITH_TAC; REWRITE_TAC[]]];


ASM_CASES_TAC` jj - 1 = ii' ` THENL [

UNDISCH_TAC` cc_qy_v11 cc (ii' + 1) ` THEN
EXPAND_TAC "ii'" THEN
ASSUME_TAC2 (ARITH_RULE` ~( jj = 0) ==> jj - 1 + 1 = jj `) THEN
FIRST_X_ASSUM SUBST1_TAC THEN
MP_TAC (SET_RULE` jj IN {ii', jj:num} `) THEN
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; cc_qy_v11 ] THEN
UNDISCH_TAC` cc_4cell_v11 cc i ` THEN
UNDISCH_TAC` cc_4cell_v11 cc j ` THEN
MESON_TAC[];

FIRST_X_ASSUM MATCH_MP_TAC THEN
CONJ_TAC];



ASM_REWRITE_TAC[IN_NUMSEG] THEN
MATCH_MP_TAC (ARITH_RULE` ~(jj = 0) /\ 0 <= jj /\ jj <= nn - 1 ==> 0 <= jj - 1 /\ jj - 1 <= nn - 1 `) THEN
ASM_REWRITE_TAC[] THEN
MP_TAC (SET_RULE` jj IN {ii', jj:num} `) THEN
ASM_REWRITE_TAC[GSYM IN_NUMSEG] THEN
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
UNDISCH_TAC` (i:num) IN indSet ` THEN
UNDISCH_TAC` (j:num) IN indSet ` THEN
ASM_REWRITE_TAC[] THEN
MESON_TAC[];

(* 
 52 [`~(jj = 0)`]
 53 [`~(jj - 1 = ii')`]

`~(jj - 1 = i) /\ ~(jj - 1 = j)`

*)


SUBGOAL_THEN` ~( jj - 1 IN {ii', jj:num})` MP_TAC;
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY];
ASM_REWRITE_TAC[];
UNDISCH_TAC` ~( jj = 0) `;
ARITH_TAC;

ASM_REWRITE_TAC[];
SET_TAC[];

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

STRIP_TAC;
(* 

 53 [`nj < nn`]
 54 [`nj + 1 = k * nn + jj`]
 55 [`cc_qy_v11 cc nj`]

`?i j k.
     ~(i = j \/ j = k \/ k = i) /\
     i IN indSet /\
     j IN indSet /\
     k IN indSet /\
     cc_4cell_v11 cc i /\
     cc_4cell_v11 cc j /\
     cc_4cell_v11 cc k`


*)

ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> ~( nn = 0) `) THEN
ASSUME_TAC2 (SPECL [` ii' + 1 `;` nn:num `] DIVMOD_EXIST) THEN
FIRST_X_ASSUM MP_TAC THEN STRIP_TAC THEN
SUBGOAL_THEN` {nj, r, ii', jj:num} SUBSET indSet ` MP_TAC THENL [
ONCE_REWRITE_TAC[INSERT_SUBSET] THEN
ONCE_REWRITE_TAC[INSERT_SUBSET] THEN
ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL [
REWRITE_TAC[IN_NUMSEG] THEN 
UNDISCH_TAC` ~( nn = 0) ` THEN 
UNDISCH_TAC` nj < nn:num ` THEN 
ARITH_TAC;
CONJ_TAC] THENL [

  REWRITE_TAC[IN_NUMSEG] THEN 
UNDISCH_TAC` ~( nn = 0) ` THEN 
UNDISCH_TAC` r < nn:num ` THEN 
ARITH_TAC;

UNDISCH_TAC` i:num IN indSet`] THEN
UNDISCH_TAC` j:num IN indSet` THEN 
ASM_REWRITE_TAC[] THEN SET_TAC[];

STRIP_TAC];


SUBGOAL_THEN` &0 <= sum ({nj, r, ii', jj}) (cc_gg_v11 cc)` MP_TAC;


UNDISCH_TAC` cc_qy_v11 cc (ii' + 1) ` THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC` periodic (cc_qy_v11 cc) nn ` THEN
REWRITE_TAC[periodic_mult] THEN
SIMP_TAC[] THEN
STRIP_TAC THEN
UNDISCH_TAC` cc_qy_v11 cc nj ` THEN
REWRITE_TAC[cc_qy_v11] THEN
REPEAT STRIP_TAC THEN
ASSUME_TAC2 (SET_RULE` {ii', jj} = {i, j} /\ cc_4cell_v11 cc i /\ cc_4cell_v11 cc j /\
~ (cc_4cell_v11 cc nj) /\ ~( cc_4cell_v11 cc r) ==> DISJOINT {nj, r} {ii', jj}`) THEN
REWRITE_TAC[SET_RULE` {a,b,c,d} = {a,b} UNION {c,d} `] THEN
ASSUME_TAC (MESON[Geomdetail.FINITE6]` ! a (b:A). FINITE {a,b}`) THEN

SUBGOAL_THEN` ! f. sum ({nj, r:num} UNION {i, j}) f = sum {nj, r} f + sum {i, j} f ` MP_TAC THENL [
GEN_TAC THEN
MATCH_MP_TAC SUM_UNION THEN
UNDISCH_TAC` DISJOINT {nj, r} {ii', jj:num} ` THEN
ASM_SIMP_TAC[];


SIMP_TAC[] THEN
STRIP_TAC];


(*


61 [`!a b. FINITE {a, b}`]
 62 [`!f. sum ({nj, r} UNION {i, j}) f = sum {nj, r} f + sum {i, j} f`]

`&0 <= sum {nj, r} (cc_gg_v11 cc) + sum {i, j} (cc_gg_v11 cc)`


*)

SUBGOAL_THEN` (! i. cc_qy_v11 cc (i + 1)
          ==> &0 <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1)) /\
  (! i. cc_qy_v11 cc i
          ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)) ` MP_TAC;
CONJ_TAC;
(* AHTYRED ================= *)
GEN_TAC;
ASM_CASES_TAC` cc_qu_v11 cc i' `;
FIRST_X_ASSUM MP_TAC;
PHA;
UNDISCH_TAC` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1)
       ==> cc_eps <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `;
DISCH_THEN NHANH;
REWRITE_TAC[cc_eps];
REAL_ARITH_TAC;


MP_TAC (SPEC` i':num ` QU_OR_QXY);
ASM_REWRITE_TAC[];
ASM_SEARCH_TCL [`#0.0 <= cc_gg_v11 cc i `] NHANH;
STRIP_TAC;
ASM_SEARCH_TCL [`&0 <= cc_gg3a_v11 cc i `] NHANH;
FIRST_X_ASSUM MP_TAC;
REAL_ARITH_TAC;

ASM_SEARCH_TCL [` &0 <= cc_gg3a_v11 cc i`] NHANH;
STRIP_TAC;
ASM_SEARCH_TCL [` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i`] (ASSUME_TAC2 o (SPEC` i':num `));
MATCH_MP_TAC REAL_LE_ADD;
ASM_REWRITE_TAC[];
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` cc_gg3a_v11 cc i' + cc_gg3b_v11 cc i' `;
ASM_REWRITE_TAC[];
MATCH_MP_TAC REAL_LE_ADD;
CONJ_TAC;
ASM_SEARCH_TCL [` A ==> &0 <= cc_gg3a_v11 cc i' `] (MATCH_MP_TAC o (SPEC` i':num`));
FIRST_X_ASSUM ACCEPT_TAC;
ASM_SEARCH_TCL [` A ==> &0 <= cc_gg3b_v11 cc i' `] (MATCH_MP_TAC o (SPEC` i':num`));
FIRST_X_ASSUM ACCEPT_TAC;
(* AHTYRED ================= *)
(*

 60 [`DISJOINT {nj, nj} {ii', jj}`]
 61 [`!a b. FINITE {a, b}`]
 62 [`!f. sum ({nj, nj} UNION {i, j}) f = sum {nj, nj} f + sum {i, j} f`]

`!i. cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)`

*)

GEN_TAC;
ASM_CASES_TAC` cc_qu_v11 cc (i' + 1) `;
FIRST_X_ASSUM MP_TAC THEN PHA;
ASM_SEARCH_TCL [`cc_eps <= cc_gg3b_v11 cc i' + cc_gg_v11 cc i `] NHANH;
REWRITE_TAC[cc_eps];
REAL_ARITH_TAC;

SUBST_ALL_TAC (REAL_ARITH` #0.0 = &0 `);
ASM_SEARCH_TCL [` &0 <= cc_gg3b_v11 cc i`] NHANH;
MP_TAC (SPEC` i' + 1 ` QU_OR_QXY);
ASMS_SEARCH_TCL [` ~cc_qu_v11 cc (i' + 1) `] REWRITE_TAC;
ASM_SEARCH_TCL [` &0 <= cc_gg_v11 cc i `] NHANH;
ASM_SEARCH_TCL [` &0 <= cc_gg_v11 cc i `] NHANH;
ASM_SEARCH_TCL [` cc_gg3a_v11 cc i + ccy `] NHANH;
REPEAT STRIP_TAC;
(* 2 goals *)

MATCH_MP_TAC REAL_LE_ADD;
ASMS_SEARCH_TCL [` &0 <= x `] REWRITE_TAC;
MATCH_MP_TAC REAL_LE_ADD;
ASMS_SEARCH_TCL [` &0 <= cc_gg3b_v11 cc i  `] REWRITE_TAC;
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` cc_gg3a_v11 cc (i' + 1) + cc_gg3b_v11 cc (i' + 1)`;
ASM_REWRITE_TAC[];
MATCH_MP_TAC REAL_LE_ADD;
UNDISCH_TAC` cc_qy_v11 cc (i' + 1)`;
ASMS_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= x `] SIMP_TAC;




 (* end the 2nd goal
(!i. cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1))
*)

STRIP_TAC;
ASM_CASES_TAC` nj = (r:num) `;
FIRST_X_ASSUM (SUBST_ALL_TAC o SYM);
REWRITE_TAC[INSERT_INSERT; SUM_SING];

ASM_SEARCH_TCL [` {a,b} = {x,y} `] (SUBST1_TAC o SYM);
ASSUME_TAC2 (SET_RULE` ~(j = (i:num)) /\ {ii', jj} = {i, j} ==> ~( ii' = jj)`);
ASSUME_TAC2 (ISPECL [` ii':num `;` jj:num `;` cc_gg_v11 cc `] Geomdetail.SUM_DIS2);
FIRST_X_ASSUM SUBST1_TAC;
ASM_SEARCH_TCL [`  cc_qy_v11 cc (i + 1) ==> x `] (MP_TAC o (SPEC` ii':num `));
ANTS_TAC;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[cc_qy_v11];
ASM_SEARCH_TCL [` periodic `;` cc_gg3a_v11`] MP_TAC;
REWRITE_TAC[periodic_mult];
ASM_REWRITE_TAC[];
SIMP_TAC[];
STRIP_TAC;
ASM_SEARCH_TCL [`  cc_qy_v11 cc i ==> x `;` cc_gg3b_v11 `] (MP_TAC o (SPEC`nj: num`));
ANTS_TAC;
ASM_REWRITE_TAC[cc_qy_v11];
ASM_SEARCH_TCL [` periodic `;` cc_gg_v11`] MP_TAC;
REWRITE_TAC[periodic_mult];
ASM_REWRITE_TAC[];
SIMP_TAC[];
STRIP_TAC;
ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> a + b <= c`] (MP_TAC o (SPEC` nj: num `));
ANTS_TAC;
ASM_REWRITE_TAC[cc_qy_v11];
PHA;
REAL_ARITH_TAC;


ASM_SEARCH_TCL [` {a,b} = {c,d} `] (SUBST1_TAC o SYM);
ASSUME_TAC2 (ISPECL [`nj: num`;` r:num `;` cc_gg_v11 cc `] Geomdetail.SUM_DIS2);
ASSUME_TAC2 (SET_RULE` {ii', jj} = {i,j:num} /\ ~( j = i) ==> ~( ii' = jj) `);
ASSUME_TAC2 (ISPECL [`ii': num`;` jj:num `;` cc_gg_v11 cc `] Geomdetail.SUM_DIS2);
ASM_REWRITE_TAC[];
FIRST_X_ASSUM (MP_TAC o (SPEC ` nj: num`));
ANTS_TAC;
ASM_REWRITE_TAC[cc_qy_v11];
FIRST_X_ASSUM (MP_TAC o (SPEC ` ii': num`));
ANTS_TAC;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[cc_qy_v11];
ASM_REWRITE_TAC[];
ASM_SEARCH_TCL [` periodic `;` cc_gg3a_v11`] MP_TAC;
ASM_SEARCH_TCL [` periodic `;` cc_gg_v11`] MP_TAC;
SIMP_TAC[periodic_mult];
STRIP_TAC THEN STRIP_TAC;
ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> a + b <= c`] (MP_TAC o (SPEC` r:num `));
ANTS_TAC;
ASM_REWRITE_TAC[cc_qy_v11];
ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> a + b <= c`] (MP_TAC o (SPEC` nj:num `));
ANTS_TAC;
ASM_REWRITE_TAC[cc_qy_v11];
ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= x`;` cc_gg3b_v11 `] (MP_TAC o (SPEC` r:num `));
ANTS_TAC;
ASM_REWRITE_TAC[cc_qy_v11];
ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= x`;` cc_gg3a_v11 `] (MP_TAC o (SPEC` nj:num `));
ANTS_TAC;
ASM_REWRITE_TAC[cc_qy_v11];
REAL_ARITH_TAC;

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

FIRST_X_ASSUM MP_TAC;
ASM_REWRITE_TAC[];
ABBREV_TAC` sts = {nj, r, i, j:num} `;
NHANH (SET_RULE` s SUBSET S ==> (S DIFF s) UNION s = S `);
REPEAT STRIP_TAC;
SUBGOAL_THEN` &0 <= sum ((0..nn - 1) DIFF sts) (cc_gg_v11 cc) ` MP_TAC;
MATCH_MP_TAC SUM_POS_LE;
CONJ_TAC;
MATCH_MP_TAC FINITE_DIFF;
REWRITE_TAC[FINITE_NUMSEG];
GEN_TAC;
REWRITE_TAC[IN_DIFF];
STRIP_TAC;
MATCH_MP_TAC QY_NN0;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[cc_real_model_v11];
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` ~( (x:num) IN sts ) `;
EXPAND_TAC "sts";
REWRITE_TAC[IN_INSERT];
CONV_TAC TAUT;
STRIP_TAC;
SUBGOAL_THEN` &0 <= sum ((0..nn - 1)) (cc_gg_v11 cc)` MP_TAC;
SUBGOAL_THEN` sum (0..nn - 1) (cc_gg_v11 cc) = sum ((0..nn - 1) DIFF sts) (cc_gg_v11 cc) + sum sts (cc_gg_v11 cc)` MP_TAC;
ASM_SEARCH_TCL [` a UNION b = c `] (fun x -> PAT_ONCE_REWRITE_TAC`\x. x = y `[SYM x]);
MATCH_MP_TAC SUM_UNION;
CONJ_TAC;
MATCH_MP_TAC FINITE_DIFF;
REWRITE_TAC[FINITE_NUMSEG];
CONJ_TAC;
EXPAND_TAC "sts";
REWRITE_TAC[Geomdetail.FINITE6];
SET_TAC[];
DISCH_THEN SUBST1_TAC;
MATCH_MP_TAC REAL_LE_ADD;
ASM_REWRITE_TAC[];

NHANH (REAL_ARITH` &0 <= a ==> ~( a < &0) `);
ASM_REWRITE_TAC[];

FIRST_X_ASSUM MP_TAC;
REWRITE_TAC[];
STRIP_TAC;

EXISTS_TAC`i:num `;
EXISTS_TAC`j:num `;
EXISTS_TAC`k:num `;
ASM_REWRITE_TAC[]]);;
====================================== *)
(* xxxxx *)

(* ===========================
let oxl6142 = prove_by_refinement (CHQSQEY_concl,
[GEN_TAC;
ABBREV_TAC` indSet = 0..cc_card_v11 cc - 1 `;
STRIP_TAC;
MP_TAC WKR_COMPTED;
ANTS_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;

REWRITE_TAC[cc_size_v11];
SUBGOAL_THEN` {i,j,k} SUBSET {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i}` MP_TAC;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_ELIM_THM];
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN` FINITE {i | i IN indSet /\ cc_4cell_v11 cc i} ` MP_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` indSet:num -> bool `;
EXPAND_TAC "indSet";
REWRITE_TAC[FINITE_NUMSEG];
SET_TAC[];
FIRST_X_ASSUM MP_TAC;
PHA;
NHANH CARD_SUBSET;
STRIP_TAC;
ASM_SEARCH_TCL [` a \/ b`] MP_TAC;
REWRITE_TAC[GSYM Geomdetail.CARD3];
DISCH_THEN SUBST_ALL_TAC;
FIRST_X_ASSUM ACCEPT_TAC]);;


=============================== *)

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

(* let CHQSQEY = oxl6142;; *)

let SUM_BETA = 
prove(` sum S (\x. f x) = sum S f `,

MATCH_MP_TAC SUM_EQ THEN 
REWRITE_TAC[BETA_THM]);;


let LXDEYBO = prove_by_refinement (LXDEYBO_concl,
[ GEN_TAC;
REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11; cc_size_v11];
STRIP_TAC;
ABBREV_TAC` cd = CARD {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i}`;
ASM_CASES_TAC` 4 < cd `;
ABBREV_TAC` ss = {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i}`;
ABBREV_TAC` S = 0..cc_card_v11 cc - 1 `;
SUBGOAL_THEN` sum ss (cc_gg_v11 cc) <= sum S (cc_gg_v11 cc) ` MP_TAC;
MATCH_MP_TAC SUM_SUBSET_SIMPLE;
CONJ_TAC;
EXPAND_TAC "S";
REWRITE_TAC[FINITE_NUMSEG];
EXPAND_TAC "ss";
CONJ_TAC;
SET_TAC[];
REWRITE_TAC[IN_DIFF; IN_ELIM_THM; DE_MORGAN_THM];
REPEAT STRIP_TAC;
FIRST_X_ASSUM MP_TAC;
ASM_REWRITE_TAC[];
MATCH_MP_TAC QY_NN0;
ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11; cc_qy_v11];

STRIP_TAC;
SUBGOAL_THEN` sum ss (\i. a_spine5 + b_spine5 * cc_azim_v11 cc i ) <= sum ss (cc_gg_v11 cc ) ` MP_TAC;
MATCH_MP_TAC SUM_LE;
CONJ_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` S: num -> bool `;
CONJ_TAC;
EXPAND_TAC "S";
REWRITE_TAC[FINITE_NUMSEG];
EXPAND_TAC "ss";
SET_TAC[];
EXPAND_TAC "ss";
REWRITE_TAC[IN_ELIM_THM];
ASM_SEARCH_TCL [` cc_4cell_v11 cc i ==> a <= b `] NHANH;
SIMP_TAC[];

SUBGOAL_THEN` sum ss (cc_azim_v11 cc ) <= &2 * pi ` MP_TAC;
ASM_SEARCH_TCL [`sum S (cc_azim_v11 cc) = &2 * pi `] (SUBST1_TAC o SYM);
MATCH_MP_TAC SUM_SUBSET_SIMPLE;

CONJ_TAC;
EXPAND_TAC "S";
REWRITE_TAC[FINITE_NUMSEG];
CONJ_TAC;
EXPAND_TAC "ss";
SET_TAC[];

REPEAT STRIP_TAC;
ASM_SEARCH_TCL [` (!i. #0.606 <= cc_azim_v11 cc i) `] MP_TAC;
DISCH_THEN (MP_TAC o (SPEC`x:num `));
REAL_ARITH_TAC;


ONCE_REWRITE_TAC[MESON[ABS_SIMP]` (\i. a + f i ) = (\i. (\j. a) i + f i)`];
STRIP_TAC THEN STRIP_TAC;
SUBGOAL_THEN` sum ss (\i. (\j. a_spine5) i + b_spine5 * cc_azim_v11 cc i) =
sum ss (\j. a_spine5) + sum ss (\i. b_spine5 * cc_azim_v11 cc i)` MP_TAC;
MATCH_MP_TAC SUM_ADD;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` S:num -> bool`;
EXPAND_TAC "S";
REWRITE_TAC[FINITE_NUMSEG];
ASM_REWRITE_TAC[];
EXPAND_TAC "ss";
SET_TAC[];

REWRITE_TAC[SUM_LMUL];
REWRITE_TAC[Sphere.b_spine5; Sphere.a_spine5];
SUBGOAL_THEN` FINITE (S:num -> bool) ` MP_TAC;
EXPAND_TAC "S";
REWRITE_TAC[FINITE_NUMSEG];
STRIP_TAC;
SUBGOAL_THEN` ss SUBSET (S:num -> bool) ` ASSUME_TAC;
EXPAND_TAC "ss";
SET_TAC[];
SUBGOAL_THEN` FINITE (ss:num -> bool) ` MP_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC`S: num -> bool `;
ASM_REWRITE_TAC[FINITE_NUMSEG];
STRIP_TAC;
ASSUME_TAC2 (ISPECL [` #0.0560305 `;`ss:num -> bool`] SUM_CONST);
ASM_REWRITE_TAC[];
ASSUME_TAC2 (ARITH_RULE` 4 < cd ==> 5 <= cd `);
FIRST_X_ASSUM MP_TAC;
FIRST_X_ASSUM MP_TAC;
ASM_REWRITE_TAC[GSYM REAL_OF_NUM_LE];
REPEAT STRIP_TAC;

SUBGOAL_THEN` &0 <= sum ss (cc_gg_v11 cc)` ASSUME_TAC;
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` sum ss (\i. (\j. a_spine5) i + b_spine5 * cc_azim_v11 cc i) `;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[Sphere.a_spine5; Sphere.b_spine5];
REWRITE_TAC[REAL_ARITH` &0 <= a + -- b * x <=> b * x <= a `];
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` #0.0445813 * ( &2 * pi) `;
ASM_REWRITE_TAC[REAL_ARITH` #0.0445813 * a <= #0.0445813 * b <=> a <= b `];
ASM_REWRITE_TAC[ISPECL [`ss:num -> bool `;` cc_azim_v11 cc `] (GEN_ALL SUM_BETA)];
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` &5 * #0.0560305`;
ASM_REWRITE_TAC[REAL_ARITH` &5 * #0.0560305 <= &cd * #0.0560305 <=> &5 <= &cd`];
MP_TAC (prove(` pi < #3.1416 `, REWRITE_TAC[ Flyspeck_constants.bounds]));
REAL_ARITH_TAC;


SUBGOAL_THEN` &0 <= sum S (cc_gg_v11 cc)` MP_TAC;
UNDISCH_TAC` &0 <= sum ss (cc_gg_v11 cc) `;
UNDISCH_TAC` sum ss (cc_gg_v11 cc) <= sum S (cc_gg_v11 cc) `;
REAL_ARITH_TAC;
UNDISCH_TAC` sum S (cc_gg_v11 cc) < &0 `;
REAL_ARITH_TAC;

FIRST_X_ASSUM MP_TAC;
ARITH_TAC]);;


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

let IMP_TAC = ONCE_REWRITE_TAC[TAUT` a /\ b ==> c <=> a ==> b ==> c `];;



(* the concls *)

let MTMLSRF_concl = 
`!cc. cc_bool_model_v11 cc /\
        cc_bool_prep_v11 cc /\
        cc_real_model_v11 cc /\
        sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0
        ==> (?i. 0 < i /\
                 cc_gg_v11 cc i < &0 /\
                 cc_qu_v11 cc i /\
                 cc_4cell_v11 cc (i + 1) /\
                 cc_4cell_v11 cc (i - 1))`;;

let UNPNFVW_concl = 
`!cc. cc_bool_model_v11 cc /\
        cc_bool_prep_v11 cc /\
        cc_real_model_v11 cc /\
        sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0
        ==> cc_size_v11 cc (cc_qy_v11 cc) <= 1`;;

let IPVICGW_concl = 
 `!cc. cc_bool_model_v11 cc /\
        cc_bool_prep_v11 cc /\
        cc_real_model_v11 cc /\
        sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0
        ==> (!i. cc_small_v11 cc i)`;;
let RSIWAMP_concl = `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ 
   (sum (0..cc_card_v11 cc -1) (cc_gg_v11 cc) < &0) ==> (cc_card_v11 cc <= 4)`;;








let MTMLSRF = prove_by_refinement (MTMLSRF_concl,
[REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
REPEAT STRIP_TAC;
ASM_CASES_TAC` ~(? i. 0 < i /\ cc_gg_v11 cc i < &0) `;
FIRST_X_ASSUM MP_TAC;
REWRITE_TAC[MESON[]` ~(?i. p i /\ q i) <=> ! i. p i ==> ~(q i )`];
STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) ` MP_TAC;
MATCH_MP_TAC SUM_POS_LE_NUMSEG;
REPEAT STRIP_TAC;
ASM_CASES_TAC` p = 0 `;
ASM_SEARCH_TCL [` periodic `; `cc_gg_v11 `] MP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
REWRITE_TAC[periodic];
DISCH_THEN (SUBST1_TAC o SYM o (SPEC` 0`));
REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `];
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_SEARCH_TCL [` ~( x = 0) `] MP_TAC;
ARITH_TAC;
REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `];
FIRST_X_ASSUM MATCH_MP_TAC;
FIRST_X_ASSUM MP_TAC;
ARITH_TAC;
ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `];
FIRST_X_ASSUM MP_TAC;
REWRITE_TAC[];
STRIP_TAC;
ASM_CASES_TAC` ~( ?i. 0 < i /\
     cc_gg_v11 cc i < &0 /\
     cc_qu_v11 cc i /\
     cc_4cell_v11 cc (i + 1) /\
     cc_4cell_v11 cc (i - 1) ) `;
FIRST_X_ASSUM MP_TAC;
REWRITE_TAC[MESON[]` ~(?i.  p1 i /\ p2 i /\ p3 i /\ p4 i) <=> (! i. p1 i /\ p2 i /\
p3 i ==> ~( p4 i))`];
STRIP_TAC;
ABBREV_TAC` S = 1..cc_card_v11 cc `;
ABBREV_TAC` cb i <=> cc_gg_v11 cc i < &0 /\ cc_qu_v11 cc i `;
ABBREV_TAC` sa = { i | i IN S /\ cb i /\ cc_qy_v11 cc (i + 1) }`;
ABBREV_TAC` sb = { i | i IN S /\ cb i /\ cc_qy_v11 cc ( i - 1)}`;
ABBREV_TAC` ss = {i | (i:num) IN S /\ cb i} `;


ABBREV_TAC` saa = { i + 1 | i + 1 IN S /\ cc_qy_v11 cc (i + 1) /\ cb i} `;
ABBREV_TAC` sbb = { i | i IN S /\ cc_qy_v11 cc i /\ cb ( i + 1)}`;
ABBREV_TAC` sab = { i | i IN S /\ cc_qy_v11 cc i /\ (cb (i + 1) \/ (cb ( i - 1)))} `;

SUBGOAL_THEN` DISJOINT ss (sab: num -> bool) ` MP_TAC;
EXPAND_TAC "sab";
EXPAND_TAC "ss";
ASM_SEARCH_TCL [` a /\ b <=> f i  `] ( fun x -> REWRITE_TAC[GSYM x]);
REWRITE_TAC[cc_qu_v11; cc_qy_v11];
SET_TAC[];
SUBGOAL_THEN` FINITE (S:num -> bool) ` MP_TAC;
EXPAND_TAC "S";
REWRITE_TAC[FINITE_NUMSEG];
STRIP_TAC;
SUBGOAL_THEN` ss SUBSET (S:num -> bool) /\ sab SUBSET S ` ASSUME_TAC;
EXPAND_TAC "ss";
EXPAND_TAC "sab";
SET_TAC[];
SUBGOAL_THEN` FINITE (ss:num -> bool) ` MP_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC `S:num -> bool `;
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN` FINITE (sab:num -> bool) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC `S:num -> bool `;
ASM_REWRITE_TAC[];

STRIP_TAC;
SUBGOAL_THEN` sum saa (cc_gg3a_v11 cc) + sum sbb (cc_gg3b_v11 cc) + sum ss (cc_gg_v11 cc) <= sum (ss UNION sab) (cc_gg_v11 cc)`  MP_TAC;
ASM_SIMP_TAC[SUM_UNION];
REWRITE_TAC[REAL_ARITH` a + b + c <= c + d <=> a + b <= d `];
SUBGOAL_THEN` sbb SUBSET (sab:num -> bool) /\ saa SUBSET (sab:num -> bool) ` ASSUME_TAC;
EXPAND_TAC "saa";
EXPAND_TAC "sbb";
EXPAND_TAC "sab";
CONJ_TAC;
REWRITE_TAC[SUBSET; IN_ELIM_THM];
MESON_TAC[];
REWRITE_TAC[SUBSET; IN_ELIM_THM];
MESON_TAC[ARITH_RULE` (i + 1) - 1 = i `];
SUBGOAL_THEN` !x. x IN saa ==> cc_qy_v11 cc x ` ASSUME_TAC;
EXPAND_TAC "saa";
SET_TAC[];
SUBGOAL_THEN` !x. x IN sbb ==> cc_qy_v11 cc x ` ASSUME_TAC;
EXPAND_TAC "sbb";
SET_TAC[];
SUBGOAL_THEN` !x. x IN sab ==> cc_qy_v11 cc x ` ASSUME_TAC;
EXPAND_TAC "sab";
SET_TAC[];
SUBGOAL_THEN` sum saa (cc_gg3a_v11 cc) <= sum sab (cc_gg3a_v11 cc) ` MP_TAC;
MATCH_MP_TAC SUM_SUBSET_SIMPLE;
ASM_REWRITE_TAC[IN_DIFF];
FIRST_ASSUM NHANH;
ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> c <= cc_gg3a_v11 cc i `] NHANH;
SIMP_TAC[];

SUBGOAL_THEN` sum sbb (cc_gg3b_v11 cc) <=
     sum sab (cc_gg3b_v11 cc)` MP_TAC;
MATCH_MP_TAC SUM_SUBSET_SIMPLE;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[IN_DIFF];
FIRST_ASSUM NHANH;
ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> c <= cc_gg3b_v11 cc i `] NHANH THEN
SIMP_TAC[];
SUBGOAL_THEN` sum sab (cc_gg3a_v11 cc) + sum sab (cc_gg3b_v11 cc) <= sum sab (cc_gg_v11 cc) ` MP_TAC;
ASSUME_TAC2 (ISPECL [` cc_gg3a_v11 cc `;` cc_gg3b_v11 cc `;` sab:num -> bool `] SUM_ADD);
FIRST_X_ASSUM (SUBST1_TAC o SYM);
MATCH_MP_TAC SUM_LE;
ASM_REWRITE_TAC[];
FIRST_ASSUM NHANH;
ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> a + b <= c`] NHANH;
SIMP_TAC[];
REAL_ARITH_TAC;
STRIP_TAC;

SUBGOAL_THEN`&0 <= sum saa (cc_gg3a_v11 cc) +
      sum sbb (cc_gg3b_v11 cc) +
      sum ss (cc_gg_v11 cc) ` MP_TAC;
SUBGOAL_THEN` sa SUBSET (ss:num -> bool) ` MP_TAC;
EXPAND_TAC "sa";
EXPAND_TAC "ss";
SET_TAC[];
NHANH (SET_RULE` s SUBSET S ==> DISJOINT s (S DIFF s) /\ s UNION ( S DIFF s) = S`);
STRIP_TAC;
FIRST_ASSUM (SUBST1_TAC o SYM);
SUBGOAL_THEN` FINITE (sa:num -> bool) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` ss: num -> bool `;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` FINITE (ss DIFF (sa:num -> bool)) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_DIFF;
FIRST_X_ASSUM ACCEPT_TAC;
ASM_SIMP_TAC[SUM_UNION];


SUBGOAL_THEN` sum saa (cc_gg3a_v11 cc) = sum sa (\i. cc_gg3a_v11 cc (i + 1)) ` MP_TAC;
MATCH_MP_TAC (GSYM SUM_EQ_GENERAL);
ABBREV_TAC` nn = cc_card_v11 cc `;

EXISTS_TAC`\i. if i = nn then 1 else i + 1 `;
CONJ_TAC;
GEN_TAC;
EXPAND_TAC "sa";
EXPAND_TAC "saa";
REWRITE_TAC[IN_ELIM_THM; EXISTS_UNIQUE];
STRIP_TAC;
ASM_CASES_TAC` 0 < i' `;
EXISTS_TAC` i':num `;
ASM_REWRITE_TAC[];
CONJ_TAC;
CONJ_TAC;
UNDISCH_TAC` i' + 1 IN S `;
FIRST_X_ASSUM MP_TAC;
EXPAND_TAC "S";
REWRITE_TAC[IN_NUMSEG];
ARITH_TAC;
SUBGOAL_THEN` ~(i' = (nn:num)) ` MP_TAC;
UNDISCH_TAC` i' + 1 IN S `;
EXPAND_TAC "S";
REWRITE_TAC[IN_NUMSEG];
ARITH_TAC;
SIMP_TAC[];
GEN_TAC;

EXPAND_TAC "S";
REWRITE_TAC[IN_NUMSEG];
ASM_CASES_TAC` y' = (nn:num) `;
ASM_REWRITE_TAC[];
FIRST_X_ASSUM SUBST_ALL_TAC;
NHANH (ARITH_RULE` a + 1 = 1 ==> ~( 0 < a) `);
ASM_REWRITE_TAC[];

ASM_REWRITE_TAC[];
ARITH_TAC;
SUBST_ALL_TAC (ARITH_RULE` ~( 0 < i') <=> i' = 0 `);
EXISTS_TAC `nn:num `;
ASM_REWRITE_TAC[];
CONJ_TAC;
EXPAND_TAC "S";
REWRITE_TAC[IN_NUMSEG; LE_REFL; ADD];
MP_TAC (SPEC_ALL CC_CARD2);
ANTS_TAC;
ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
ASM_SIMP_TAC[ARITH_RULE` 2 <= nn ==> 1 <= nn `];
STRIP_TAC;
ASM_SEARCH_TCL [` periodic`; `cc_gg_v11 `] MP_TAC;
ASM_SEARCH_TCL [` cc_bool_prep_v11 `] MP_TAC;
REWRITE_TAC[cc_bool_prep_v11];
MP_TAC (SPEC_ALL periodic_fn);
ANTS_TAC;
ASM_REWRITE_TAC[cc_bool_model_v11];
ASM_REWRITE_TAC[];
STRIP_TAC;
SIMP_TAC[Oxl_def.periodic];
STRIP_TAC;
UNDISCH_TAC`(cb: num -> bool) i'`;
ASM_REWRITE_TAC[];
ASM_SEARCH_TCL [` a /\ b <=> c `] (fun x -> REWRITE_TAC[GSYM x]);
FIRST_X_ASSUM MP_TAC;
UNDISCH_TAC` cc_qy_v11 cc (i' + 1) `;
ASM_SEARCH_TCL [` periodic `;` cc_qu_v11`] MP_TAC;
ASM_SEARCH_TCL [` periodic `;` cc_qy_v11`] MP_TAC;
ASM_SEARCH_TCL [` periodic `;` cc_gg_v11`] MP_TAC;
SIMP_TAC[Oxl_def.periodic];
UNDISCH_TAC` i' = 0 `;
SIMP_TAC[ADD;ADD_SYM];
MESON_TAC[ADD; ADD_SYM];

GEN_TAC;
EXPAND_TAC "S";
REWRITE_TAC[IN_NUMSEG];
ASM_CASES_TAC` y' = (nn:num) `;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[];
ARITH_TAC;


GEN_TAC;
EXPAND_TAC "sa";
EXPAND_TAC "saa";
REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
CONJ_TAC;
ASM_CASES_TAC` x = (nn: num) `;
EXISTS_TAC` 0`;
ASM_REWRITE_TAC[ADD];
EXPAND_TAC "S";
REWRITE_TAC[IN_NUMSEG];
CONJ_TAC;
MP_TAC (SPEC_ALL CC_CARD2);
ANTS_TAC;
ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
ASM_REWRITE_TAC[];
ARITH_TAC;


FIRST_X_ASSUM SUBST_ALL_TAC;
REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC);
ASM_SEARCH_TCL [` a /\ b <=> c `] (fun x -> REWRITE_TAC[GSYM x]);
ASM_SEARCH_TCL [` periodic `;` cc_gg_v11 `] MP_TAC;
MP_TAC (SPEC_ALL periodic_fn);
ANTS_TAC;
ASM_REWRITE_TAC[cc_bool_model_v11];
STRIP_TAC;
ASM_SEARCH_TCL [` periodic `;` cc_qy_v11 `] MP_TAC;
ASM_SEARCH_TCL [` periodic `;` cc_qu_v11 `] MP_TAC;
SIMP_TAC[Oxl_def.periodic];
ASM_SEARCH_TCL [` cc_card_v11 cc = nn `] SUBST1_TAC;

MESON_TAC[ADD; ADD_SYM];
EXISTS_TAC `x:num `;
ASM_REWRITE_TAC[];
UNDISCH_TAC` (x:num) IN S `;
EXPAND_TAC "S";
REWRITE_TAC[IN_NUMSEG];
FIRST_X_ASSUM MP_TAC;
ARITH_TAC;

ASM_CASES_TAC` x = (nn:num) `;
ASM_REWRITE_TAC[];
ASM_SEARCH_TCL [` periodic (cc_gg3a_v11 cc) `] MP_TAC;
REWRITE_TAC[Oxl_def.periodic];
MESON_TAC[ADD_SYM];
ASM_REWRITE_TAC[];

MP_TAC (SPEC_ALL periodic_fn);
ANTS_TAC;
ASM_REWRITE_TAC[cc_bool_model_v11];
ABBREV_TAC` nn = cc_card_v11 cc `;
STRIP_TAC;
SUBGOAL_THEN ` ss DIFF sa SUBSET (sb: num -> bool) ` MP_TAC;
UNDISCH_TAC` {i | (i:num) IN S /\ cb i} = ss `;
STRIP_TAC;
EXPAND_TAC "ss";
EXPAND_TAC "sa";
EXPAND_TAC "sb";

REWRITE_TAC[SUBSET; IN_DIFF; IN_ELIM_THM];
SUBGOAL_THEN` !x. (x IN S /\ cb x) ==>  ~(cc_4cell_v11 cc (x + 1) /\ cc_4cell_v11 cc (x - 1))` MP_TAC;
GEN_TAC THEN STRIP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` (x:num) IN S `;
EXPAND_TAC "S";
REWRITE_TAC[IN_NUMSEG];
ARITH_TAC;
REWRITE_TAC[cc_qy_v11];
MESON_TAC[];


SUBGOAL_THEN` sum sbb (cc_gg3b_v11 cc) = sum sb (\i. cc_gg3b_v11 cc ( i - 1))` MP_TAC;
MATCH_MP_TAC SUM_EQ_GENERAL;
EXISTS_TAC`\i. if i = nn then 1 else i + 1 `;
CONJ_TAC;
REWRITE_TAC[EXISTS_UNIQUE];
GEN_TAC;
EXPAND_TAC "sb";
REWRITE_TAC[IN_ELIM_THM];
EXPAND_TAC "sbb";
EXPAND_TAC "S";
REWRITE_TAC[IN_NUMSEG];
STRIP_TAC;
ASM_CASES_TAC` y = 1 `;
EXISTS_TAC` nn:num`;
REWRITE_TAC[IN_ELIM_THM];
CONJ_TAC;
ASM_REWRITE_TAC[];
CONJ_TAC;
STRIP_TAC;
MP_TAC (SPEC_ALL CC_CARD2);
ANTS_TAC;
ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
ASM_REWRITE_TAC[];

ARITH_TAC;
REWRITE_TAC[LE_REFL];
FIRST_X_ASSUM SUBST_ALL_TAC;
FIRST_X_ASSUM MP_TAC;
FIRST_X_ASSUM MP_TAC;
ASM_SEARCH_TCL [` a /\ b <=> c `] MP_TAC;
DISCH_THEN (fun x -> REWRITE_TAC[GSYM x]);
ASM_SEARCH_TCL [` periodic (cc_gg_v11 cc) `] MP_TAC;
ASM_SEARCH_TCL [` periodic (cc_qy_v11 cc) `] MP_TAC;
ASM_SEARCH_TCL [` periodic (cc_qu_v11 cc) `] MP_TAC;
REWRITE_TAC[Oxl_def.periodic; SUB_REFL];
MESON_TAC[ADD; ADD_SYM];
GEN_TAC;
ASM_CASES_TAC` y' = (nn:num) `;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[];
ARITH_TAC;
EXISTS_TAC` y - 1 `;
CONJ_TAC;
ASSUME_TAC2 (ARITH_RULE` 1 <= y /\ y <= nn ==> ~( y - 1 = nn) `);
ASM_REWRITE_TAC[IN_ELIM_THM];
ASSUME_TAC2 (ARITH_RULE` 1 <= y /\ y <= nn ==> y - 1 + 1 = y /\ y - 1 <= nn`);
ASM_REWRITE_TAC[];
UNDISCH_TAC` 1 <= y `;
UNDISCH_TAC` ~( y = 1) `;
ARITH_TAC;
GEN_TAC;

ASM_CASES_TAC` y' = (nn:num) `;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[];
STRIP_TAC;
FIRST_X_ASSUM MP_TAC;
UNDISCH_TAC` 1 <= y `;
ARITH_TAC;
GEN_TAC;
EXPAND_TAC "sbb";
EXPAND_TAC "sb";
EXPAND_TAC "S";
REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG];
STRIP_TAC;
ASM_CASES_TAC` x = (nn: num) `;
ASM_REWRITE_TAC[];
FIRST_X_ASSUM SUBST_ALL_TAC;
CONJ_TAC;
CONJ_TAC;
UNDISCH_TAC` 1 <= nn `;
ARITH_TAC;

UNDISCH_TAC` (cb:num -> bool) (nn + 1) `;
UNDISCH_TAC` cc_qy_v11 cc nn `;
ASM_SEARCH_TCL [` a /\ b <=> c `] (fun x -> REWRITE_TAC[GSYM x]);
ASM_SEARCH_TCL [` periodic (cc_gg_v11 cc) `] MP_TAC;
ASM_SEARCH_TCL [` periodic (cc_qu_v11 cc) `] MP_TAC;
ASM_SEARCH_TCL [` periodic (cc_qy_v11 cc) `] MP_TAC;
REWRITE_TAC[Oxl_def.periodic; SUB_REFL];
MESON_TAC[ADD; ADD_SYM];

REWRITE_TAC[SUB_REFL];
ASM_SEARCH_TCL [` periodic (cc_gg3b_v11 cc) `] MP_TAC;
REWRITE_TAC[Oxl_def.periodic];
MESON_TAC[ADD; ADD_SYM];

ASM_REWRITE_TAC[ARITH_RULE` (a + 1) - 1 = a /\ 1 <= a + 1`];
UNDISCH_TAC` x <= (nn:num) `;
FIRST_X_ASSUM MP_TAC;
ARITH_TAC;


SIMP_TAC[];

REPEAT STRIP_TAC;
SUBGOAL_THEN` sum (ss DIFF sa) (\i. cc_gg3b_v11 cc (i - 1)) <= sum sb (\i. cc_gg3b_v11 cc (i - 1))` MP_TAC;
MATCH_MP_TAC SUM_SUBSET_SIMPLE;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` sb SUBSET (S:num -> bool) ` ASSUME_TAC;
EXPAND_TAC "sb";
SET_TAC[];
CONJ_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` S: num -> bool `;
ASM_REWRITE_TAC[];

EXPAND_TAC "sb";
REWRITE_TAC[IN_DIFF; IN_ELIM_THM];
ASM_SEARCH_TCL  [` cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i `] NHANH;
SIMP_TAC[];

STRIP_TAC;
SUBGOAL_THEN` &0 <= sum sa (\i. cc_gg3a_v11 cc (i + 1)) +
         sum sa (cc_gg_v11 cc) ` ASSUME_TAC;
ASM_SIMP_TAC[GSYM SUM_ADD];
MATCH_MP_TAC SUM_POS_LE;
ASM_REWRITE_TAC[];
EXPAND_TAC "sa";
REWRITE_TAC[IN_ELIM_THM];
ASM_SEARCH_TCL [` a /\ b <=> c `] (fun x -> REWRITE_TAC[ GSYM x]);
ASM_SEARCH_TCL [` x /\ cc_qy_v11 cc (i + 1) ==> a `] MP_TAC;
REWRITE_TAC[cc_eps];
MESON_TAC[REAL_ADD_SYM; REAL_ARITH` &0 <= #0.0057`; REAL_LE_TRANS];


SUBGOAL_THEN` &0 <= sum (ss DIFF sa) (\i. cc_gg3b_v11 cc (i - 1)) + sum (ss DIFF sa) (cc_gg_v11 cc) ` ASSUME_TAC;
SUBGOAL_THEN` FINITE (ss DIFF (sa:num -> bool)) ` MP_TAC;
MATCH_MP_TAC FINITE_DIFF;
FIRST_X_ASSUM ACCEPT_TAC;
SIMP_TAC[GSYM SUM_ADD];
STRIP_TAC;
MATCH_MP_TAC SUM_POS_LE;
ASM_REWRITE_TAC[];
UNDISCH_TAC` ss DIFF sa SUBSET (sb:num -> bool) `;
REWRITE_TAC[SUBSET];
DISCH_THEN NHANH;
EXPAND_TAC "sb";
EXPAND_TAC "S";
REWRITE_TAC[IN_NUMSEG; IN_ELIM_THM];
NHANH (ARITH_RULE` 1 <= x ==> x - 1 + 1 = x `);
ASM_SEARCH_TCL [` a /\ b <=> v`] (fun x -> REWRITE_TAC[GSYM x]);
ASM_SEARCH_TCL  [` cc_qu_v11 cc (i + 1) /\ cc_qy_v11 cc i ==> a `] MP_TAC;
REWRITE_TAC[cc_eps];
MESON_TAC[REAL_ADD_SYM; REAL_ARITH` &0 <= #0.0057`; REAL_LE_TRANS];
REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC);
REAL_ARITH_TAC;

SUBGOAL_THEN` ss UNION sab SUBSET (S:num -> bool) ` MP_TAC;
EXPAND_TAC "ss";
EXPAND_TAC "sab";
SET_TAC[];
NHANH (SET_RULE` s SUBSET S ==> (S DIFF s ) UNION s = S /\ DISJOINT (S DIFF s) s`);
ABBREV_TAC` sg = ss UNION (sab:num -> bool) `;
STRIP_TAC;
SUBGOAL_THEN` FINITE (sg: num -> bool) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` S:num -> bool `;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` FINITE (S DIFF (sg:num -> bool)) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_DIFF;
FIRST_X_ASSUM ACCEPT_TAC;
STRIP_TAC;
SUBGOAL_THEN` &0 <= sum S (cc_gg_v11 cc) ` MP_TAC;
ASM_SIMP_TAC[SUM_UNION];
EXPAND_TAC "S";
REPLICATE_TAC 4 (FIRST_X_ASSUM MP_TAC);
ASM_SIMP_TAC[SUM_UNION];
REPEAT STRIP_TAC;
MATCH_MP_TAC REAL_LE_ADD;
CONJ_TAC;
MATCH_MP_TAC SUM_POS_LE;
ASM_REWRITE_TAC[];

GEN_TAC;
EXPAND_TAC "sg";
EXPAND_TAC "ss";
REWRITE_TAC[IN_DIFF; IN_ELIM_THM; IN_UNION];
ASM_SEARCH_TCL [` a /\ b <=> c `] (fun x -> REWRITE_TAC[GSYM x]);
REWRITE_TAC[DE_MORGAN_THM];
IMP_TAC;
SIMP_TAC[];
REPEAT STRIP_TAC;
UNDISCH_TAC` ~( cc_gg_v11 cc x < &0 ) `;
REAL_ARITH_TAC;
SUBGOAL_THEN ` cc_bool_model_v11 cc ` MP_TAC;
ASM_REWRITE_TAC[cc_bool_model_v11];
NHANH (SPECL [`cc: cc_v11`;` x:num `] QY_QX_QU);
ASM_REWRITE_TAC[];
SUBGOAL_THEN` cc_real_model_v11 cc ` MP_TAC;
ASM_REWRITE_TAC[cc_real_model_v11];
MESON_TAC[QY_NN0; QX_NN0];
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` sum saa (cc_gg3a_v11 cc) +
      sum sbb (cc_gg3b_v11 cc) +
      sum ss (cc_gg_v11 cc) `;
ASM_REWRITE_TAC[];
ABBREV_TAC` nn = cc_card_v11 cc `;
STRIP_TAC;
UNDISCH_TAC` ~( nn = 0 ) `;
ASM_SEARCH_TCL [` periodic (cc_gg_v11 cc) `] MP_TAC;
PHA;
NHANH periodic_sum;
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC ` 1 `));
ASSUME_TAC2 (ARITH_RULE` ~( nn = 0) ==> nn - 1 + 1 = nn `);
ASM_REWRITE_TAC[];
STRIP_TAC;
UNDISCH_TAC` &0 <= sum S (cc_gg_v11 cc) `;
ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0)`];
FIRST_X_ASSUM MP_TAC;
SIMP_TAC[]]);;




let MOD_INJ1 = 
prove_by_refinement
(` ~( n = 0) /\ k < n /\ ~( k = 0) ==> (! x. ~( x MOD n = (x + k) MOD n)) `,

[REPEAT STRIP_TAC;
MP_TAC (SPECL [` n: num `;` x:num `;` x:num `;` x + k:num`] MOD_INJ);
ANTS_TAC;
ASM_REWRITE_TAC[IN_NUMSEG];
DOWN_TAC;
SIMP_TAC[LE_REFL; ARITH_RULE` ~( n = 0) ==> x <= n - 1 + x `; ARITH_RULE` a <= a + x:num `];
STRIP_TAC;
UNDISCH_TAC` k < (n:num) `;
ARITH_TAC;
UNDISCH_TAC` ~( k = 0) `;
ARITH_TAC]);;




(* TTTTTT *)



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



let WKR_COMPTED = 
prove_by_refinement (`cc_bool_model_v11 cc /\
cc_bool_prep_v11 cc /\
 cc_real_model_v11 cc /\
 sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 /\
 indSet = 0..cc_card_v11 cc - 1
 ==> (?i j k.
          ~(i = j \/ j = k \/ k = i) /\
          i IN indSet /\
          j IN indSet /\
          k IN indSet /\
          cc_4cell_v11 cc i /\
          cc_4cell_v11 cc j /\
          cc_4cell_v11 cc k)`,
 [STRIP_TAC;
ASSUME_TAC2 (SPEC_ALL MTMLSRF);
ASSUME_TAC2 Oxl_def.CC_CARD2;
DOWN_TAC;
REWRITE_TAC[cc_real_model_v11; cc_qu_v11; cc_bool_model_v11];
STRIP_TAC;
ASM_CASES_TAC` cc_card_v11 cc = 2 `;
ASM_SEARCH_TCL [` x = &2 * pi `] MP_TAC;
FIRST_ASSUM SUBST1_TAC;
REWRITE_TAC[ARITH_RULE` 2 - 1 = 1 `];
MP_TAC (ARITH_RULE` 0 < 1 /\ 0 <= 1 `);
SIMP_TAC[SUM_CLAUSES_RIGHT; SUB_REFL; SUM_SING_NUMSEG];
STRIP_TAC THEN STRIP_TAC;
ABBREV_TAC` nn = cc_card_v11 cc `;
SUBGOAL_THEN` {i MOD nn, (i + 1) MOD nn} SUBSET 0..nn - 1 ` MP_TAC;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];

UNDISCH_TAC` ~(nn = 0) `;
MESON_TAC[Oxl_def.MOD_IN_NUMSEG];
MP_TAC (SPECL [`1 `;` 2 `] (GEN_ALL MOD_INJ1));
ANTS_TAC;
ARITH_TAC;
MP_TAC (SPECL [` 0`;` 2 - 1 `] CARD_NUMSEG);
ASM_REWRITE_TAC[ARITH_RULE` (2 - 1 + 1) - 0 = 2 `];
REPEAT STRIP_TAC;
SUBGOAL_THEN` {i MOD 2, (i + 1) MOD 2} = 0..2 - 1` MP_TAC;
MATCH_MP_TAC CARD_SUBSET_EQ;
ASM_REWRITE_TAC[FINITE_NUMSEG; Geomdetail.CARD_SET2];


DISCH_THEN (ASSUME_TAC o SYM);

ASM_SEARCH_TCL [` sum s f = &2 * pi `] MP_TAC;
UNDISCH_TAC` nn = 2 `;
SIMP_TAC[];
ASM_SIMP_TAC[];
FIRST_X_ASSUM (MP_TAC o (SPEC` i: num `));
SIMP_TAC[Geomdetail.SUM_DIS2];
REPEAT STRIP_TAC;
SUBGOAL_THEN` cc_4cell_v11 cc (i MOD 2) /\ cc_4cell_v11 cc ((i + 1) MOD 2) ` MP_TAC;
UNDISCH_TAC ` ~( nn = 0 ) `;
UNDISCH_TAC` periodic (cc_4cell_v11 cc) nn `;
PHA;
UNDISCH_TAC` cc_4cell_v11 cc (i) `;
UNDISCH_TAC` cc_4cell_v11 cc (i + 1) `;
ASM_REWRITE_TAC[];
MESON_TAC[Oxl_def.periodic_mod];
ASM_SEARCH_TCL [` cc_azim_v11 `;` #2.8`] (NHANH_PAT` \x. x ==> y `);

DOWN;
MP_TAC (prove(` #3.14159 < pi`, REWRITE_TAC[ Flyspeck_constants.bounds]));
REAL_ARITH_TAC;


ABBREV_TAC` nn = cc_card_v11 cc `;
EXISTS_TAC` (i - 1) MOD nn`;
EXISTS_TAC` (i) MOD nn`;
EXISTS_TAC` (i + 1) MOD nn`;
MP_TAC (ISPECL [` 1 `;`nn: num `] (GEN_ALL MOD_INJ1));
MP_TAC (ISPECL [` 2 `;`nn: num `] (GEN_ALL MOD_INJ1));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` ~( 2 = 0) /\ (2 < nn <=> 2 <= nn /\ ~( nn = 2))`];
STRIP_TAC;
ANTS_TAC;
ASM_SIMP_TAC[ARITH_RULE` ~( 1 = 0) /\ ( 2 <= x ==> 1 < x)`];
STRIP_TAC;
FIRST_ASSUM (MP_TAC o (SPEC` i: num `));
FIRST_X_ASSUM (MP_TAC o (SPEC` i - 1 `));
FIRST_ASSUM (MP_TAC o (SPEC` i - 1 `));
UNDISCH_TAC` 0 < i `;
SIMP_TAC[ARITH_RULE` 0 < i ==> i - 1 + 1 = i /\ i - 1 + 2 = i + 1 `];
REPLICATE_TAC 4 STRIP_TAC;
ASM_SIMP_TAC[MOD_IN_NUMSEG];
ASM_SEARCH_TCL [` periodic `;` cc_4cell_v11 `] MP_TAC;
UNDISCH_TAC` cc_4cell_v11 cc (i - 1) `;
UNDISCH_TAC` cc_4cell_v11 cc (i) `;
UNDISCH_TAC` cc_4cell_v11 cc (i + 1) `;
UNDISCH_TAC` ~( nn = 0) `;
MESON_TAC[Oxl_def.periodic_mod]]);;




let oxl6142 = prove_by_refinement (CHQSQEY_concl,
[GEN_TAC;
ABBREV_TAC` indSet = 0..cc_card_v11 cc - 1 `;
STRIP_TAC;
MP_TAC WKR_COMPTED;
ANTS_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;

REWRITE_TAC[cc_size_v11];
SUBGOAL_THEN` {i,j,k} SUBSET {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i}` MP_TAC;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_ELIM_THM];
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN` FINITE {i | i IN indSet /\ cc_4cell_v11 cc i} ` MP_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` indSet:num -> bool `;
EXPAND_TAC "indSet";
REWRITE_TAC[FINITE_NUMSEG];
SET_TAC[];
FIRST_X_ASSUM MP_TAC;
PHA;
NHANH CARD_SUBSET;
STRIP_TAC;
ASM_SEARCH_TCL [` a \/ b`] MP_TAC;
REWRITE_TAC[GSYM Geomdetail.CARD3];
DISCH_THEN SUBST_ALL_TAC;
FIRST_X_ASSUM ACCEPT_TAC]);;


let CHQSQEY = oxl6142;;


let THREE_LE_CC_CARD = 
prove_by_refinement (` cc_bool_model_v11 cc /\
          cc_bool_prep_v11 cc /\
          cc_real_model_v11 cc /\
          sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0
          ==> 3 <= cc_card_v11 cc `,

[NHANH CHQSQEY;
REWRITE_TAC[cc_size_v11];
ABBREV_TAC` sn = 0..cc_card_v11 cc - 1 `;
STRIP_TAC;
ASSUME_TAC (SET_RULE` {i | i IN sn /\ cc_4cell_v11 cc i} SUBSET sn`);
SUBGOAL_THEN` FINITE (sn:num -> bool) ` MP_TAC;
EXPAND_TAC "sn";
REWRITE_TAC[FINITE_NUMSEG];
FIRST_X_ASSUM MP_TAC;
PHA;
NHANH CARD_SUBSET;
STRIP_TAC;
ABBREV_TAC` c4 = CARD {i | i IN sn /\ cc_4cell_v11 cc i} `;
ASSUME_TAC2 (ARITH_RULE` 3 <= c4 /\ c4 <= CARD (sn:num -> bool) ==> 3 <= CARD sn`);
FIRST_X_ASSUM MP_TAC;
EXPAND_TAC "sn";
REWRITE_TAC[CARD_NUMSEG];
ARITH_TAC]);;




let THREE_MOD_CONSECUTIVES = 
prove_by_refinement (
` 0 < i /\ 3 <= n ==> ~( (i - 1) MOD n = i MOD n \/ i MOD n = (i + 1) MOD n \/
(i + 1) MOD n = (i - 1) MOD n )`,

[NHANH (ARITH_RULE` 3 <= n ==> 2 < n /\ 1 < n /\ ~(n = 0)`);
STRIP_TAC;
MP_TAC (SPECL [` 2`;` n:num `] (GEN_ALL MOD_INJ1));
ANTS_TAC;
ASM_REWRITE_TAC[];
ARITH_TAC;
MP_TAC (SPECL [` 1`;` n:num `] (GEN_ALL MOD_INJ1));
ANTS_TAC;
ASM_REWRITE_TAC[];
ARITH_TAC;
STRIP_TAC;
FIRST_ASSUM (ASSUME_TAC o (SPEC` i - 1 `));
FIRST_X_ASSUM (ASSUME_TAC o (SPEC` i:num `));
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC` i - 1 `));
REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC);
PHA;
ASSUME_TAC2 (ARITH_RULE` 0 < i ==> i - 1 + 1 = i /\ i - 1 + 2 = i + 1 `);
ASM_REWRITE_TAC[];
MESON_TAC[]]);;



let MOD_PERIOD_BOUNDED = 
prove_by_refinement (
` ~( n = 0) /\ ~( k = 0) ==> (! x. (x + k) MOD n = x MOD n ==> n <= k ) `,

[REPEAT STRIP_TAC;
ASM_CASES_TAC` n <= (k:num) `;
ASM_REWRITE_TAC[];
MP_TAC MOD_INJ1;
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` a < (b:num) <=> ~(b <= a ) `];
ASM_MESON_TAC[]]);;



let MOD_PERIOD_BOUNDED2 = 
prove_by_refinement
(` ~(nn = 0) /\ m <= k /\
(i + k ) MOD nn IN { (i + x) MOD nn | x < m } ==> nn <= k `,

[REWRITE_TAC[IN_ELIM_THM] ;
STRIP_TAC ;
MP_TAC (SPECL [` nn:num `;` k - (x:num) `] (GEN_ALL MOD_PERIOD_BOUNDED)) ;
ANTS_TAC ;
ASM_REWRITE_TAC[] ;
ASM_ARITH_TAC ;
DISCH_THEN (MP_TAC o (SPEC` i + (x:num) `)) ;
ANTS_TAC ;
ASSUME_TAC2 (ARITH_RULE` x < m /\ m <= (k:num) ==> (i + x) + k - x = i + k `) ;
ASM_REWRITE_TAC[] ;
ARITH_TAC]);;



let CARD_EMPTY = el 0 (CONJUNCTS CARD_CLAUSES);;

let CARD_INSERT = el 1 (CONJUNCTS CARD_CLAUSES);;
let SWITCH_TAC tm = UNDISCH_TAC tm THEN DISCH_THEN (ASSUME_TAC o GSYM);;



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




let UNPNFVW = prove_by_refinement ( UNPNFVW_concl,

[ NHANH THREE_LE_CC_CARD;
NHANH MTMLSRF;
NHANH CHQSQEY;
REPEAT STRIP_TAC;
ABBREV_TAC` indSet = 0..cc_card_v11 cc - 1 `;
MP_TAC WKR_COMPTED;
ANTS_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;
ASSUME_TAC2 (SPEC_ALL LXDEYBO);
ABBREV_TAC` sz = cc_size_v11 cc (cc_4cell_v11 cc) ` ;
ASM_CASES_TAC` cc_size_v11 cc (cc_4cell_v11 cc) = 3 `;
ABBREV_TAC` nn = cc_card_v11 cc `;
UNDISCH_TAC` 3 <= nn `;
UNDISCH_TAC` 0 < i `;
PHA;
NHANH THREE_MOD_CONSECUTIVES;
REWRITE_TAC[GSYM Geomdetail.CARD3];
ABBREV_TAC` ia = (i - 1) MOD nn `;
ABBREV_TAC` ib = i MOD nn `;
ABBREV_TAC ` ic = (i + 1) MOD nn `;
SUBGOAL_THEN` cc_4cell_v11 cc ia /\ cc_4cell_v11 cc ib /\ cc_4cell_v11 cc ic ` MP_TAC;
UNDISCH_TAC` cc_qu_v11 cc i `;
REWRITE_TAC[cc_qu_v11];
ASM_SEARCH_TCL [` cc_bool_model_v11 `] MP_TAC;
REWRITE_TAC[cc_bool_model_v11];
ASM_REWRITE_TAC[];
ASM_MESON_TAC[periodic_mod];
REPEAT STRIP_TAC;
ASSUME_TAC2 (ARITH_RULE` 3 <= nn ==> ~( nn = 0) `);
ASM_SEARCH_TCL [` cc_qu_v11 `] MP_TAC;
REWRITE_TAC[cc_qu_v11] THEN STRIP_TAC;
SUBGOAL_THEN`! i. cc_4cell_v11 cc (i MOD nn) = cc_4cell_v11 cc i ` MP_TAC;
GEN_TAC;
MATCH_MP_TAC (GSYM periodic_mod);
ASM_REWRITE_TAC[];
ASM_SEARCH_TCL [` cc_bool_model_v11 `] MP_TAC;
SIMP_TAC[cc_bool_model_v11];
ASM_REWRITE_TAC[];
SIMP_TAC[];
STRIP_TAC;

SUBGOAL_THEN` {ia, ib, ic} SUBSET indSet:num -> bool ` MP_TAC;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
EXPAND_TAC "indSet";
EXPAND_TAC "ia";
EXPAND_TAC "ib";
EXPAND_TAC "ic";
UNDISCH_TAC` ~(nn = 0) `;
SIMP_TAC[MOD_IN_NUMSEG];
STRIP_TAC;
SUBGOAL_THEN` {ia, ib, ic} SUBSET {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} ` MP_TAC;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_ELIM_THM];
FIRST_X_ASSUM MP_TAC;
ASM_SIMP_TAC[INSERT_SUBSET];
STRIP_TAC;
EXPAND_TAC "ia";
EXPAND_TAC "ib";
EXPAND_TAC "ic";
ASMS_SEARCH_TCL [` X <=> cc_4cell_v11 cc i `] REWRITE_TAC;
ASM_REWRITE_TAC[];
FIRST_X_ASSUM MP_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;


SUBGOAL_THEN` FINITE (indSet:num -> bool) ` ASSUME_TAC;
EXPAND_TAC "indSet";
REWRITE_TAC[FINITE_NUMSEG];
SUBGOAL_THEN`{ia, ib, ic} = {i | i IN indSet /\ cc_4cell_v11 cc i} ` ASSUME_TAC;
MATCH_MP_TAC CARD_SUBSET_EQ;
ASM_REWRITE_TAC[];
CONJ_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` indSet:num -> bool `;
ASM_REWRITE_TAC[];
SET_TAC[];
UNDISCH_TAC` cc_size_v11 cc (cc_4cell_v11 cc) = 3 `;
REWRITE_TAC[cc_size_v11];
EXPAND_TAC "indSet";
EXPAND_TAC "nn";
SIMP_TAC[];

ASM_CASES_TAC` cc_4cell_v11 cc ( (i + 2) MOD nn) `;
SUBGOAL_THEN` (i + 2) MOD nn IN {ia, ib, ic} ` MP_TAC;
ASM_REWRITE_TAC[];
REWRITE_TAC[IN_ELIM_THM];
ASM_REWRITE_TAC[];
EXPAND_TAC "indSet";
MATCH_MP_TAC MOD_IN_NUMSEG;
FIRST_X_ASSUM ACCEPT_TAC;
STRIP_TAC;
SUBGOAL_THEN` nn = 3 ` ASSUME_TAC;
FIRST_X_ASSUM MP_TAC;
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY];
STRIP_TAC;
FIRST_X_ASSUM MP_TAC;
ASSUME_TAC2 (ARITH_RULE` 0 < i ==> i + 2 = (i - 1) + 3 `);
ASM_REWRITE_TAC[];
EXPAND_TAC "ia";
STRIP_TAC;
MP_TAC (ARITH_RULE` ~( 3 = 0) `);
UNDISCH_TAC` ~( nn = 0) `;
PHA;
NHANH MOD_PERIOD_BOUNDED;
STRIP_TAC;
REWRITE_TAC[ARITH_RULE` a = 3 <=> a <= 3 /\ 3 <= a `];
ASM_REWRITE_TAC[];
FIRST_X_ASSUM MATCH_MP_TAC;
EXISTS_TAC` i - 1 `;
ASM_REWRITE_TAC[];

DOWN;
EXPAND_TAC "ib";
ASSUME_TAC (ARITH_RULE` ~( 2 = 0) `);
ASSUME_TAC2 (SPECL [` nn: num `;` 2 `] (GEN_ALL MOD_PERIOD_BOUNDED));
FIRST_ASSUM NHANH;
NHANH (ARITH_RULE` n <= 2 ==> ~( 3 <= n ) `);
ASM_REWRITE_TAC[];
DOWN;
EXPAND_TAC "ic";
ASSUME_TAC (ARITH_RULE` ~( 1 = 0) `);
ASSUME_TAC2 (SPECL [` nn: num `;` 1 `] (GEN_ALL MOD_PERIOD_BOUNDED));
REWRITE_TAC[ARITH_RULE` a + 2 = (a + 1) + 1 `];
FIRST_ASSUM NHANH;
NHANH (ARITH_RULE` n <= 1 ==> ~( 3 <= n ) `);
ASM_REWRITE_TAC[];
SUBGOAL_THEN` CARD (indSet:num -> bool) = nn ` MP_TAC;
EXPAND_TAC "indSet";
REWRITE_TAC[CARD_NUMSEG];
UNDISCH_TAC` ~( nn = 0) `;
ARITH_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;
MP_TAC (ISPECL [` {ia, ib, ic:num} `;` indSet:num -> bool `] CARD_SUBSET_EQ);
ANTS_TAC;
ASM_REWRITE_TAC[];

ASM_REWRITE_TAC[cc_size_v11];
EXPAND_TAC "indSet";
UNDISCH_TAC` nn = 3 `;
SIMP_TAC[cc_qy_v11];
NHANH (SET_RULE` { i | i IN S /\ p i } = S ==> {i | i IN S /\ ~( p i)} = {} `);
SIMP_TAC[CARD_CLAUSES];
REPEAT STRIP_TAC;
ARITH_TAC;
ASM_SEARCH_TCL [` cc_bool_prep_v11 `] MP_TAC;
REWRITE_TAC[cc_bool_prep_v11; cc_qy_v11];

ASM_SEARCH_TCL [` p ( x MOD n) <=> p x `] (fun x -> ONCE_REWRITE_TAC[GSYM x]) ;
DISCH_THEN (ASSUME_TAC2 o (SPEC` (i + 2)`)) ;
SUBGOAL_THEN` { ((i - 1) + x) MOD nn | x < 3}  = {ia, ib, ic} ` MP_TAC ;
REWRITE_TAC[EXTENSION; IN_ELIM_THM] ;
GEN_TAC ;
REWRITE_TAC[ARITH_RULE` x < 3 <=> x = 0 \/ x = 1 \/ x = 2 `; IN_ELIM_THM; NOT_IN_EMPTY; IN_INSERT] ;
EXPAND_TAC "ia" ;
EXPAND_TAC "ib" ;
EXPAND_TAC "ic" ;
UNDISCH_TAC` 0 < i ` ;
NHANH (ARITH_RULE` 0 < i ==> (i - 1 ) + 0 = i - 1 /\ ( i - 1) + 1 = i /\ ( i - 1) + 2 = i + 1 `) ;


MESON_TAC[];
STRIP_TAC;
SUBGOAL_THEN` (i + 3) MOD nn IN {(i - 1 + x) MOD nn | x < 3} ` MP_TAC;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[IN_ELIM_THM];
EXPAND_TAC "indSet";
CONJ_TAC;
MATCH_MP_TAC MOD_IN_NUMSEG;
FIRST_X_ASSUM ACCEPT_TAC;
DOWN THEN DOWN;
ASM_REWRITE_TAC[ARITH_RULE` a + 3 = (a + 2 ) + 1 `];
SIMP_TAC[];

ASSUME_TAC2 (ARITH_RULE` 0 < i ==> i + 3 = (i -1) + 4 `);
FIRST_ASSUM SUBST1_TAC;
MP_TAC (ARITH_RULE` 3 <= 4 `);
UNDISCH_TAC` ~( nn = 0) `;
PHA;
NHANH MOD_PERIOD_BOUNDED2;
STRIP_TAC;


ABBREV_TAC` id = ((i - 1) + 4) MOD nn `;
SUBGOAL_THEN` (! x. x IN {ia, ib, ic} ==> cc_4cell_v11 cc x)` ASSUME_TAC;
UNDISCH_TAC` cc_4cell_v11 cc ia `;
UNDISCH_TAC` cc_4cell_v11 cc ib `;
UNDISCH_TAC` cc_4cell_v11 cc ic `;
SET_TAC[];

SUBGOAL_THEN` ~( ((i + 2) MOD nn) IN {ia, ib, ic})` MP_TAC;
UNDISCH_TAC` ~cc_4cell_v11 cc (((i + 2)) MOD nn) `;
DOWN;
MESON_TAC[];



ABBREV_TAC` ie = (i + 2) MOD nn `;
STRIP_TAC;
MP_TAC (prove(` FINITE {ia, ib, ic:num}`, REWRITE_TAC[Geomdetail.FINITE6]));
NHANH (ISPEC`ie: num ` CARD_INSERT);
ABBREV_TAC` cells = {i | i IN indSet /\ cc_4cell_v11 cc i} `;
ASM_REWRITE_TAC[ADD1];
STRIP_TAC;
SUBGOAL_THEN` CARD (indSet:num -> bool) = nn ` ASSUME_TAC;
EXPAND_TAC "indSet";
REWRITE_TAC[CARD_NUMSEG];
MATCH_MP_TAC (ARITH_RULE` ~(nn = 0) ==> (nn - 1 + 1) - 0 = nn `);
FIRST_X_ASSUM ACCEPT_TAC;
SUBGOAL_THEN` {ie, ia, ib, ic:num } SUBSET indSet ` MP_TAC;
ONCE_REWRITE_TAC[INSERT_SUBSET];
ASM_REWRITE_TAC[];
EXPAND_TAC "ie";
EXPAND_TAC "indSet";
MATCH_MP_TAC MOD_IN_NUMSEG;
FIRST_ASSUM ACCEPT_TAC;
STRIP_TAC;
SUBGOAL_THEN` {ie, ia, ib, ic:num} = indSet` MP_TAC;
MATCH_MP_TAC CARD_SUBSET_EQ;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` CARD ((ie:num) INSERT cells) <= CARD (indSet:num -> bool) ` MP_TAC;
MATCH_MP_TAC CARD_SUBSET;
DOWN;
ASM_SIMP_TAC[];


ASM_REWRITE_TAC[];
UNDISCH_TAC` nn <= 4 `;
ARITH_TAC;

ASM_REWRITE_TAC[cc_size_v11; cc_qy_v11];
STRIP_TAC;
ASSUME_TAC2 (
SET_RULE` cc_4cell_v11 cc ia /\ cc_4cell_v11 cc ib /\ cc_4cell_v11 cc ic /\
{ia, ib, ic} = cells /\ ~cc_4cell_v11 cc ie /\ ie INSERT cells = indSet
==> {i | i IN indSet /\ ~cc_4cell_v11 cc i} = {ie} `);
ASM_REWRITE_TAC[];
REWRITE_TAC[LE_REFL; Geomdetail.CARD_SING];
ASM_CASES_TAC` cc_size_v11 cc (cc_qy_v11 cc) <= 1 `;
FIRST_X_ASSUM ACCEPT_TAC;

DOWN;
REWRITE_TAC[ARITH_RULE` ~( x <= 1) <=> 2 <= x `; cc_size_v11];
ASM_REWRITE_TAC[];
ABBREV_TAC` qyset = {i | i IN indSet /\ cc_qy_v11 cc i} `;
SUBGOAL_THEN` qyset SUBSET (indSet:num -> bool) ` MP_TAC;
EXPAND_TAC "qyset";
SET_TAC[];
STRIP_TAC;
SUBGOAL_THEN` FINITE (indSet:num -> bool) ` ASSUME_TAC;
EXPAND_TAC "indSet";
REWRITE_TAC[FINITE_NUMSEG];
SUBGOAL_THEN` FINITE (qyset:num -> bool) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` indSet:num -> bool `;
ASM_REWRITE_TAC[];
DOWN;
NHANH CHOOSE_SUBSET;
STRIP_TAC;
FIRST_X_ASSUM NHANH;
REWRITE_TAC[HAS_SIZE_2_EXISTS; 
SET_RULE` (!z. z IN t <=> z = x \/ z = y) <=> {x,y} = t  `];
STRIP_TAC;
UNDISCH_TAC` ~(cc_size_v11 cc (cc_4cell_v11 cc) = 3) `;
ASM_REWRITE_TAC[];
STRIP_TAC;
ASSUME_TAC2 (ARITH_RULE` 3 <= sz /\ ~( sz = 3) /\ sz <= 4 ==> sz = 4 `);
DOWN;
EXPAND_TAC "sz";
REWRITE_TAC[cc_size_v11];
ASM_REWRITE_TAC[];
ABBREV_TAC` cells = {i | i IN indSet /\ cc_4cell_v11 cc i}`;
STRIP_TAC;

SUBGOAL_THEN` indSet = qyset UNION cells /\ DISJOINT qyset (cells: num -> bool)` MP_TAC;
EXPAND_TAC "qyset";
EXPAND_TAC "cells";
REWRITE_TAC[cc_qy_v11];
SET_TAC[];
STRIP_TAC;
MP_TAC (SET_RULE` {i | i IN indSet /\ cc_4cell_v11 cc i} SUBSET indSet`);
SWITCH_TAC` indSet = qyset UNION (cells: num -> bool) `;
ASM_REWRITE_TAC[];
UNDISCH_TAC` FINITE (indSet:num -> bool)`;
PHA;
NHANH (FINITE_SUBSET);
STRIP_TAC;
REPLICATE_TAC 27 DOWN;
PHA;
ASM_SEARCH_TCL [` cc_real_model_v11 `] MP_TAC;
ASM_SEARCH_TCL [` cc_bool_model_v11 `] MP_TAC;
REWRITE_TAC[cc_real_model_v11; cc_bool_model_v11];
REPEAT STRIP_TAC;

SUBGOAL_THEN` &0 <= sum indSet (cc_gg_v11 cc) ` MP_TAC;
EXPAND_TAC "indSet";
SUBGOAL_THEN` sum (qyset UNION cells) (cc_gg_v11 cc) = sum qyset (cc_gg_v11 cc) + sum cells (cc_gg_v11 cc) ` SUBST1_TAC;
MATCH_MP_TAC SUM_UNION;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` ! j. cc_qy_v11 cc j ==> &0 <= cc_gg_v11 cc j ` ASSUME_TAC;
REPEAT STRIP_TAC;
MATCH_MP_TAC QY_NN0;
ASM_REWRITE_TAC[];

SUBGOAL_THEN` sum t (cc_gg_v11 cc) <= sum qyset (cc_gg_v11 cc)` ASSUME_TAC;
MATCH_MP_TAC SUM_SUBSET_SIMPLE;
ASM_REWRITE_TAC[];
EXPAND_TAC "qyset";
REWRITE_TAC[IN_DIFF; IN_ELIM_THM];
FIRST_ASSUM NHANH;


SIMP_TAC[];
SUBGOAL_THEN` &2 * #0.606 * #0.008 <= sum qyset (cc_gg_v11 cc) ` ASSUME_TAC;
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` sum t (cc_gg_v11 cc) `;
ASM_REWRITE_TAC[];
EXPAND_TAC "t";
SUBGOAL_THEN` sum {x, y} (cc_gg_v11 cc) = cc_gg_v11 cc x + cc_gg_v11 cc y ` SUBST1_TAC;
MATCH_MP_TAC Geomdetail.SUM_DIS2;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` cc_qy_v11 cc x /\ cc_qy_v11 cc y ` MP_TAC;
UNDISCH_TAC` t SUBSET (qyset: num -> bool) ` ;
EXPAND_TAC "t";
EXPAND_TAC "qyset";
SET_TAC[];
STRIP_TAC;
ASM_SEARCH_TCL [`#0.008 `] (ASSUME_TAC2 o (SPEC` x:num `));
DOWN;
ASM_SEARCH_TCL [`#0.008 `] (ASSUME_TAC2 o (SPEC` y:num `));
DOWN;
ASM_SEARCH_TCL [`#0.606`] (MP_TAC o (SPEC` x:num `));
ASM_SEARCH_TCL [`#0.606`] (MP_TAC o (SPEC` y:num `));
REAL_ARITH_TAC;


SUBGOAL_THEN` ! i. i IN cells ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i` ASSUME_TAC;
EXPAND_TAC "cells";
REWRITE_TAC[IN_ELIM_THM];
ASM_SEARCH_TCL [` cc_4cell_v11 cc i ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i`] NHANH;
SIMP_TAC[];
SUBGOAL_THEN` sum cells (\i. (\j. a_spine5) i + b_spine5 * cc_azim_v11 cc i) <= sum cells (cc_gg_v11 cc)` MP_TAC;
MATCH_MP_TAC SUM_LE;
ASM_REWRITE_TAC[];

ASM_SIMP_TAC[SUM_ADD; SUM_CONST; SUM_LMUL];
REWRITE_TAC[ISPECL [` cells:num -> bool `; `cc_azim_v11 cc `] (GEN_ALL SUM_BETA)];
STRIP_TAC;
SUBGOAL_THEN` ! f. sum (indSet:num -> bool) f = sum cells f + sum qyset f ` ASSUME_TAC;
GEN_TAC;
EXPAND_TAC "indSet";
ONCE_REWRITE_TAC[REAL_ADD_SYM];
MATCH_MP_TAC SUM_UNION;
ASM_REWRITE_TAC[];
ASM_SEARCH_TCL [` x = &2 * pi`] MP_TAC;
ASM_SEARCH_TCL [` 0..m = t `] MP_TAC;
SIMP_TAC[];
ASM_REWRITE_TAC[];

STRIP_TAC;
SUBGOAL_THEN` sum (qyset DIFF t) (cc_azim_v11 cc) = sum qyset (cc_azim_v11 cc) - sum t (cc_azim_v11 cc) ` ASSUME_TAC;
MATCH_MP_TAC SUM_DIFF;
ASM_REWRITE_TAC[];
DOWN;
EXPAND_TAC "t";
UNDISCH_TAC` ~( x = (y:num)) `;
SIMP_TAC[Geomdetail.SUM_DIS2];
STRIP_TAC;
SUBGOAL_THEN` !s. FINITE s ==> &0 <= sum s (cc_azim_v11 cc) ` ASSUME_TAC;
GEN_TAC;
STRIP_TAC;
MATCH_MP_TAC SUM_POS_LE;
ASM_REWRITE_TAC[];
REPEAT STRIP_TAC;
ASM_SEARCH_TCL [` #0.606 <= x `] (MP_TAC o (SPEC` x':num `));
REAL_ARITH_TAC;
FIRST_ASSUM (MP_TAC o (SPEC` qyset DIFF {x, y:num}`));
ANTS_TAC;
MATCH_MP_TAC FINITE_DIFF;
ASM_REWRITE_TAC[];
PHA;
NHANH (REAL_ARITH` &0 <= x /\ x = y - z ==> z <= y `);
STRIP_TAC;
SUBGOAL_THEN` &2 * #0.606 <= sum qyset (cc_azim_v11 cc) ` MP_TAC;
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` cc_azim_v11 cc x + cc_azim_v11 cc y `;
ASM_REWRITE_TAC[];
ASM_SEARCH_TCL [` #0.606 <= x `] (MP_TAC o (SPEC` x:num `));
ASM_SEARCH_TCL [` #0.606 <= x `] (MP_TAC o (SPEC` y:num `));
REAL_ARITH_TAC;
STRIP_TAC;
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` &2 * #0.606 * #0.008 + ( &4 * a_spine5 + b_spine5 * sum cells (cc_azim_v11 cc)) ` ;
ONCE_REWRITE_TAC[TAUT` a /\ b <=> b /\ a `];
CONJ_TAC;
MATCH_MP_TAC REAL_LE_ADD2;
ASM_REWRITE_TAC[];

ASM_REWRITE_TAC[Sphere.a_spine5; Sphere.b_spine5];
UNDISCH_TAC` sum cells (cc_azim_v11 cc) + sum qyset (cc_azim_v11 cc) = &2 * pi `;
SIMP_TAC[REAL_ARITH` a + b = c <=> a = c - b `];
STRIP_TAC;
UNDISCH_TAC` &2 * #0.606 <= sum qyset (cc_azim_v11 cc) `;
MP_TAC (prove(` pi < #3.1416 `, REWRITE_TAC[ Flyspeck_constants.bounds]));
REAL_ARITH_TAC;
UNDISCH_TAC` sum indSet (cc_gg_v11 cc) < &0 `;
REAL_ARITH_TAC]);;




let CC_QU_SMALL_SITUATION = 
prove_by_refinement(` cc_bool_model_v11 cc /\ cc_real_model_v11 cc
==> !i. cc_qu_v11 cc i /\ 
   (~cc_small_v11 cc (i + 1) \/ ~cc_small_v11 cc (i))
    ==> cc_eps <= cc_gg_v11 cc i `,

[REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
STRIP_TAC;
ASM_SEARCH_TCL [` x ==> cc_small_v11 cc i`] MP_TAC;
UNDISCH_TAC ` !i. cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc i
          ==> cc_eps <= cc_gg_v11 cc i`;
UNDISCH_TAC`!i. cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc (i + 1)
          ==> cc_eps <= cc_gg_v11 cc i`;
MESON_TAC[]]);;




let CC_4CELL_QUQX = 
prove(` cc_4cell_v11 cc i <=> cc_qu_v11 cc i \/ cc_qx_v11 cc i `,

REWRITE_TAC[cc_qx_v11; cc_qu_v11] THEN CONV_TAC TAUT);;



let QX_NN00 = 
prove(`  cc_bool_model_v11 cc /\ cc_real_model_v11 cc ==> (! i. cc_qx_v11 cc i
           ==> &0 <= cc_gg_v11 cc i) `,

REPEAT STRIP_TAC THEN ASM_SIMP_TAC[QX_NN0]);;


let QY_NN00 = 
prove(`  cc_bool_model_v11 cc /\ cc_real_model_v11 cc ==> (! i. cc_qy_v11 cc i
           ==> &0 <= cc_gg_v11 cc i) `,

REPEAT STRIP_TAC THEN ASM_SIMP_TAC[QY_NN0]);;




let NUMSEG_SET_VER = 
prove(` 0..m = { i | i <= m } `,

REWRITE_TAC[numseg] THEN 
MP_TAC (ARITH_RULE`! x. (0 <= x /\ x <= m  <=> x <= m)`) THEN 
SET_TAC[]);;


let FINITE_INITIAL_SEG = REWRITE_RULE[NUMSEG_SET_VER] (SPEC `0` FINITE_NUMSEG);;


let FINITE_INITIAL_SEG2 = 
prove(` FINITE {i | i < (n:num)} `,

MATCH_MP_TAC (ISPECL [` {i | i < n:num } `;` {i | i <= (n:num) }`] FINITE_SUBSET)
THEN REWRITE_TAC[FINITE_INITIAL_SEG; SUBSET; IN_ELIM_THM; LT_IMP_LE]);;




let CARD_MOD_NUMSEG = 
prove_by_refinement(
` !m. m <= n ==> CARD {(i + k) MOD n | k | k < m} = m `,


[INDUCT_TAC;
STRIP_TAC;
MP_TAC (ARITH_RULE`(!x. ~( x < 0))`);
SIMP_TAC[SET_RULE` (!x. ~(x < 0)) ==> {(i + k) MOD n | k < 0} = {} `; CARD_EMPTY];
NHANH (ARITH_RULE` SUC m <= n ==> m <= n `);
FIRST_X_ASSUM NHANH;
REWRITE_TAC[ARITH_RULE` k < SUC m <=> k < m \/ k = m `];
REWRITE_TAC[SET_RULE` {(i + k) MOD n | k < m \/ k = m} = (i + m) MOD n INSERT {(i + k) MOD n | k < m }`];
STRIP_TAC;
SUBGOAL_THEN` ~( (i + m) MOD n IN {(i + k) MOD n | k < m})` ASSUME_TAC;

REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
MP_TAC (let tt = GEN_ALL MOD_INJ1 in
SPECL [` m - (k: num) `;` n: num `] tt);
ANTS_TAC; ASM_ARITH_TAC;
DISCH_THEN (ASSUME_TAC o (SPEC` i + (k:num) `));
DOWN;
ASSUME_TAC2 (ARITH_RULE` k < (m:num) ==> (i + k) + m - k = i + m `);
ASM_REWRITE_TAC[];
MP_TAC (ISPECL [`\k. ( i + k) MOD n `; `{ k  | k < m:num} `] FINITE_IMAGE);
ANTS_TAC;
REWRITE_TAC[ FINITE_INITIAL_SEG2; IMAGE; IN_ELIM_THM];
REWRITE_TAC[ IMAGE; IN_ELIM_THM];

ASM_REWRITE_TAC[SET_RULE` {y | ?x. x < m /\ y = (i + x) MOD n} = {(i + k) MOD n | k < m} `];

NHANH (ISPEC ` (i + m) MOD n ` CARD_INSERT);
ASM_REWRITE_TAC[];
SIMP_TAC[]]);;




let MOD_NUMSEG = 
prove_by_refinement(` ~( n = 0 ) ==> (! i. 0..(n - 1) = {(i + k) MOD n | k | k < n })`,

[STRIP_TAC;
GEN_TAC;
SUBGOAL_THEN` {(i + k) MOD n | k | k < n} SUBSET 0..(n - 1) ` ASSUME_TAC;
REWRITE_TAC[SUBSET];
GEN_TAC;
REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
ASM_REWRITE_TAC[];
ASM_SIMP_TAC[MOD_IN_NUMSEG];
MP_TAC (REWRITE_RULE[LE_REFL] (SPEC`n:num ` CARD_MOD_NUMSEG));
STRIP_TAC;
MATCH_MP_TAC (GSYM CARD_SUBSET_EQ);
ASM_REWRITE_TAC[FINITE_NUMSEG; CARD_NUMSEG];
UNDISCH_TAC` ~( n = 0) `;
ARITH_TAC]);;





(*  IPVICGW_concl;;  *)



let IPVICGW_C33 = 
prove_by_refinement(`! cc. (cc_bool_model_v11 cc /\
       cc_bool_prep_v11 cc /\
       cc_real_model_v11 cc /\
       sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\
      cc_card_v11 cc = 3 
==> (!i. cc_small_v11 cc i) `,


[NHANH LXDEYBO;
NHANH CHQSQEY;
REWRITE_TAC[cc_size_v11];
GEN_TAC; STRIP_TAC THEN ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00; DOWN_TAC;
ABBREV_TAC` cells = {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} `;
ABBREV_TAC` nn = cc_card_v11 cc `;
REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11; cc_bool_prep_v11];
STRIP_TAC;
SUBGOAL_THEN` cells SUBSET 0..nn - 1 ` ASSUME_TAC;
EXPAND_TAC "cells";
SET_TAC[];
ASSUME_TAC (SPECL [` 0`;` nn - 1 `] FINITE_NUMSEG);
MP_TAC (SPECL [` 0`;` nn - 1 `] CARD_NUMSEG);
ASM_SIMP_TAC[ARITH_RULE`(3 - 1 + 1) - 0 = 3 `];
ASSUME_TAC2 (ISPECL [` cells:num -> bool `;` 0.. nn - 1 `] CARD_SUBSET);
REPLICATE_TAC 3 DOWN THEN PHA;
ASM_REWRITE_TAC[];
ASM_SIMP_TAC[ARITH_RULE` 3 <= x ==> ( x <= y /\ y = 3 <=> x = 3 /\ y = 3) `];
STRIP_TAC;
SUBGOAL_THEN` 0..3 - 1 = cells ` MP_TAC;
MATCH_MP_TAC (GSYM CARD_SUBSET_EQ);
ASM_REWRITE_TAC[];
STRIP_TAC;
GEN_TAC;
ASM_CASES_TAC` ! i. i IN cells ==> cc_qy_v11 cc i `;
SUBGOAL_THEN` ! i. i IN cells ==> &0 <= cc_gg_v11 cc i ` ASSUME_TAC;
REPEAT STRIP_TAC;
ASM_SEARCH_TCL [` !i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `] MATCH_MP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
FIRST_ASSUM ACCEPT_TAC;
SUBGOAL_THEN` &0 <= sum cells (cc_gg_v11 cc) ` ASSUME_TAC;
MATCH_MP_TAC SUM_POS_LE;
ASM_REWRITE_TAC[];
EXPAND_TAC "cells";
FIRST_X_ASSUM ACCEPT_TAC;
UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `;
DOWN;
ASM_REWRITE_TAC[];
REAL_ARITH_TAC;

ASM_CASES_TAC` (!i. i IN cells ==> cc_qx_v11 cc i) `;
SUBGOAL_THEN` (!i. i IN cells ==> &0 <= cc_gg_v11 cc i)` MP_TAC;
GEN_TAC THEN STRIP_TAC;
ASM_SEARCH_TCL [` (!i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i) `] MATCH_MP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
FIRST_X_ASSUM ACCEPT_TAC;
UNDISCH_TAC` FINITE (0..3 - 1 ) `;
PHA;
ASM_REWRITE_TAC[];
NHANH SUM_POS_LE;
ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `];
STRIP_TAC;
UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `;
ASM_REWRITE_TAC[];
DOWN;
REWRITE_TAC[ISPECL [`cells: num -> bool `;` cc_gg_v11 cc `] (GEN_ALL SUM_BETA)];
SIMP_TAC[];
DOWN;
REWRITE_TAC[MESON[]` ~(! x. P x ==> q x) <=> (? x. P x /\ ~( q x)) `];
STRIP_TAC;
SUBGOAL_THEN` cc_4cell_v11 cc i' ` MP_TAC;
REPLICATE_TAC 2 DOWN THEN PHA;

UNDISCH_TAC` {i | i IN 0..nn - 1 /\ cc_4cell_v11 cc i} = cells `;
DISCH_THEN (SUBST1_TAC o GSYM);

SIMP_TAC[IN_ELIM_THM];
ASM_REWRITE_TAC[CC_4CELL_QUQX];
ASM_SEARCH_TCL [` !i. cc_qu_v11 cc i ==> --cc_eps <= cc_gg_v11 cc i `] NHANH;
REWRITE_TAC[cc_qu_v11; cc_hassmall_v11];
STRIP_TAC;
ASM_CASES_TAC` cc_small_v11 cc (i' + 2) `;
UNDISCH_TAC` ~( cc_card_v11 cc = 0) `;
ASM_REWRITE_TAC[];
NHANH MOD_NUMSEG;
ASM_REWRITE_TAC[];
STRIP_TAC;
FIRST_X_ASSUM (ASSUME_TAC o (SPEC` i': num `));
SUBGOAL_THEN` ! j. j IN cells ==> cc_small_v11 cc j ` ASSUME_TAC;
ASM_REWRITE_TAC[IN_ELIM_THM];
REPEAT STRIP_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` ~( 3 = 0) `;
UNDISCH_TAC ` periodic (cc_small_v11 cc) (cc_card_v11 cc) `;
ASM_REWRITE_TAC[];
PHA;
SIMP_TAC[GSYM Oxl_def.periodic_mod];
REPLICATE_TAC 9 DOWN;
REWRITE_TAC[ARITH_RULE` k < 3 <=> k = 0 \/ k = 1 \/ k = 2 `];
MESON_TAC[ADD_0];

UNDISCH_TAC` ~( 3 = 0) `;
UNDISCH_TAC ` periodic (cc_small_v11 cc) (cc_card_v11 cc) `;
ASM_REWRITE_TAC[];
PHA;
NHANH ( GEN_ALL (
SPEC` i:num` (GEN`m:num ` (SPEC_ALL Oxl_def.periodic_mod))));
SIMP_TAC[];
STRIP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
EXPAND_TAC "cells";
MATCH_MP_TAC MOD_IN_NUMSEG;
ASM_REWRITE_TAC[];


SUBGOAL_THEN` ~(cc_qu_v11 cc (i' + 1)) /\ ~(cc_qu_v11 cc (i' + 2)) ` MP_TAC;
ASM_REWRITE_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (x + 1) + 1 = x + 2 `];
STRIP_TAC;
SUBGOAL_THEN`! i. cc_4cell_v11 cc i ` ASSUME_TAC;
GEN_TAC;
UNDISCH_TAC` ~( cc_card_v11 cc = 0) `;
UNDISCH_TAC ` periodic (cc_4cell_v11 cc) (cc_card_v11 cc) `;
ASM_REWRITE_TAC[];
PHA;
NHANH ( GEN_ALL (
SPEC` i:num` (GEN`m:num ` (SPEC_ALL Oxl_def.periodic_mod))));
SIMP_TAC[];
STRIP_TAC;
SUBGOAL_THEN` i MOD 3 IN cells ` MP_TAC;
EXPAND_TAC "cells";
MATCH_MP_TAC MOD_IN_NUMSEG;
ARITH_TAC;
ASM_SEARCH_TCL [` {i | i IN 0..nn - 1 /\ cc_4cell_v11 cc i} = cells `] (SUBST1_TAC o SYM);
SET_TAC[];
FIRST_ASSUM (MP_TAC o (SPEC` i' + 1 `));
FIRST_ASSUM (MP_TAC o (SPEC` i' + 2 `));
DOWN;
ASM_REWRITE_TAC[CC_4CELL_QUQX];

REWRITE_TAC[GSYM CC_4CELL_QUQX];
REPEAT STRIP_TAC;
SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc (i' + 1 ) ` ASSUME_TAC;
ASM_SEARCH_TCL [` !i. cc_qx_v11 cc i /\ cc_small_v11 cc i /\ ~cc_small_v11 cc (i + 1)
       ==> cc_eps <= cc_gg_v11 cc i`] MATCH_MP_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 1) + 1 = i + 2 `];
SUBGOAL_THEN` cc_qu_v11 cc i' ` MP_TAC;
ASM_REWRITE_TAC[cc_qu_v11; cc_hassmall_v11];
ASM_SEARCH_TCL [`!i. cc_qu_v11 cc i ==> --cc_eps <= cc_gg_v11 cc i`] NHANH;
ASM_SEARCH_TCL [` cc_qx_v11 cc i ==> &0 <= r`] (ASSUME_TAC2 o (SPEC` i' + 2 `));
STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (i'.. i' + 2) (cc_gg_v11 cc) ` ASSUME_TAC;
MP_TAC (ARITH_RULE` i' <= i' + 2 `);
SIMP_TAC[SUM_CLAUSES_LEFT];
STRIP_TAC;
MP_TAC (ARITH_RULE` i' + 1 <= i' + 2 `);
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` (i + 1) + 1 = i + 2`; SUM_SING_NUMSEG];
STRIP_TAC;

UNDISCH_TAC` --cc_eps <= cc_gg_v11 cc i' `;
UNDISCH_TAC` cc_eps <= cc_gg_v11 cc (i' + 1) `;
UNDISCH_TAC` &0 <= cc_gg_v11 cc (i' + 2) `;
PHA;
REAL_ARITH_TAC;

UNDISCH_TAC` ~( cc_card_v11 cc = 0) `;
ASM_SEARCH_TCL [` periodic (cc_gg_v11 cc) `] MP_TAC;
PHA;
NHANH Oxl_def.periodic_sum;
ASM_REWRITE_TAC[];
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC` i':num `));
REWRITE_TAC[ARITH_RULE` 3 - 1 + i = i + 2 `];
DISCH_THEN (ASSUME_TAC o SYM);
UNDISCH_TAC` sum (0.. nn - 1 ) (cc_gg_v11 cc) < &0 `;
ASM_REWRITE_TAC[REAL_ARITH` a < &0 <=> ~( &0 <= a) `]]);;







let UNIQUE_QY = 
prove_by_refinement
(` !cc. cc_bool_model_v11 cc /\
          cc_bool_prep_v11 cc /\
          cc_real_model_v11 cc /\
          sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0
          ==> (! j. cc_qy_v11 cc j /\ j IN 0..cc_card_v11 cc - 1  ==> 
          (! k. ~( k = j) /\ k IN 0..cc_card_v11 cc - 1 
             ==> ~(cc_qy_v11 cc k))) `,

[NHANH UNPNFVW;
REWRITE_TAC[cc_size_v11];
REPEAT STRIP_TAC;
ABBREV_TAC ` qys = {i | i IN 0..cc_card_v11 cc - 1 /\ cc_qy_v11 cc i} `;
SUBGOAL_THEN` {j,k} SUBSET (qys: num -> bool) ` ASSUME_TAC;
EXPAND_TAC "qys";
REWRITE_TAC[SUBSET; EMPTY_SUBSET; IN_ELIM_THM; IN_INSERT; NOT_IN_EMPTY];
REPLICATE_TAC 6 DOWN THEN PHA;
MESON_TAC[];
ASSUME_TAC (SPECL [` 0`;` cc_card_v11 cc - 1 `] FINITE_NUMSEG);
SUBGOAL_THEN` FINITE (qys: num -> bool) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` 0..cc_card_v11 cc - 1 `;
ASM_REWRITE_TAC[];
EXPAND_TAC "qys";
SET_TAC[];
SUBGOAL_THEN` CARD {j:num, k} <= CARD (qys:num -> bool) ` MP_TAC;
MATCH_MP_TAC CARD_SUBSET;
ASM_REWRITE_TAC[];
STRIP_TAC;
UNDISCH_TAC` ~( k = (j:num)) `;
REWRITE_TAC[];

ASM_CASES_TAC` k = (j:num) `;
FIRST_X_ASSUM ACCEPT_TAC;
DOWN;
REWRITE_TAC[GSYM Geomdetail.CARD2];
DOWN;
REWRITE_TAC[INSERT_COMM];
PHA;
NHANH (ARITH_RULE` a <= b /\ a = 2 ==> ~( b <= 1 )`);
ASM_REWRITE_TAC[]]);;



let UNIQUE_QY2 = 
prove(`!cc. cc_bool_model_v11 cc /\
          cc_bool_prep_v11 cc /\
          cc_real_model_v11 cc /\
          sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0
          ==> (! j. cc_qy_v11 cc j /\ j IN 0..cc_card_v11 cc - 1  ==> 
          (! k. cc_qy_v11 cc k/\ k IN 0..cc_card_v11 cc - 1 
             ==> k = j))  `,

NHANH UNIQUE_QY THEN GEN_TAC THEN STRIP_TAC THEN 
FIRST_X_ASSUM NHANH THEN MESON_TAC[]);;



let ASM_SEARCH_TC = asms_search;;



let IPVICGW_C43_C44 = 
prove_by_refinement
(`!cc. (cc_bool_model_v11 cc /\
           cc_bool_prep_v11 cc /\
           cc_real_model_v11 cc /\
           sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\
          cc_card_v11 cc = 4
          ==> (!i. cc_small_v11 cc i) `,

[
NHANH CHQSQEY;
NHANH LXDEYBO;
GEN_TAC THEN STRIP_TAC;
ASSUME_TAC2 QX_NN00;
ASSUME_TAC2 QY_NN00;
SUBGOAL_THEN` cc_bool_model_v11 cc /\
 cc_bool_prep_v11 cc /\
 cc_real_model_v11 cc /\
 sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ` MP_TAC;
ASM_REWRITE_TAC[];
NHANH Oxl_def.periodic_fn;
REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11; cc_bool_prep_v11];
STRIP_TAC;
ABBREV_TAC` nn = cc_card_v11 cc `;
ASM_CASES_TAC` ! i. cc_small_v11 cc i `;
FIRST_X_ASSUM ACCEPT_TAC;
FIRST_X_ASSUM MP_TAC;
REWRITE_TAC[NOT_FORALL_THM];
STRIP_TAC;
SUBGOAL_THEN` ~ cc_qu_v11 cc i /\ ~ cc_qu_v11 cc ( i + 3) ` MP_TAC;
ASM_REWRITE_TAC[cc_qu_v11; cc_hassmall_v11; DE_MORGAN_THM];
DISJ1_TAC;
DISJ2_TAC;

ASM_SEARCH_TCL [` periodic (cc_small_v11 cc) `] MP_TAC;
REWRITE_TAC[periodic];
ASM_SIMP_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];
STRIP_TAC;
MP_TAC (SPEC` i:num` QU_OR_QXY);
MP_TAC (SPEC` i + 3` QU_OR_QXY);
ASM_REWRITE_TAC[];
ONCE_REWRITE_TAC[DISJ_SYM];
STRIP_TAC;
STRIP_TAC;
DOWN THEN DOWN;

ASM_SEARCH_TCL [` periodic (cc_qy_v11 cc) `] MP_TAC;
ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `; `nn:num `;` i:num `] Oxl_def.periodic_mod);
ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `; `nn:num `;` i + 3`] Oxl_def.periodic_mod);
REPLICATE_TAC 2 (FIRST_X_ASSUM SUBST1_TAC);
ASM_REWRITE_TAC[];
STRIP_TAC THEN STRIP_TAC;
MP_TAC (SPEC_ALL UNIQUE_QY2);
ANTS_TAC;
UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `;
ASM_REWRITE_TAC[];

DISCH_THEN (MP_TAC o (SPEC` i MOD 4 `));
ANTS_TAC;
ASM_REWRITE_TAC[];
MATCH_MP_TAC MOD_IN_NUMSEG;
ARITH_TAC;
DISCH_THEN (MP_TAC o (SPEC` (i + 3) MOD 4 `));
ANTS_TAC;
ASM_REWRITE_TAC[];
MATCH_MP_TAC MOD_IN_NUMSEG;
ARITH_TAC;
MP_TAC (let th = GEN_ALL MOD_INJ1 in
SPECL [` 3 `;` 4 `] th);
ANTS_TAC;
ARITH_TAC;
SIMP_TAC[];

SUBGOAL_THEN` ~ cc_qy_v11 cc ((i + 1) MOD 4) /\ ~( cc_qy_v11 cc ((i + 2) MOD 4)) ` ASSUME_TAC;
MP_TAC (SPEC_ALL UNIQUE_QY);
ANTS_TAC;
UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `;
ASM_REWRITE_TAC[];
DISCH_THEN (MP_TAC o (SPEC`(i + 3) MOD 4 `));
ANTS_TAC;
ASM_REWRITE_TAC[];
SWITCH_TAC` nn = 4 `;
ASM_SIMP_TAC[GSYM periodic_mod; MOD_IN_NUMSEG];
STRIP_TAC;
FIRST_ASSUM (MP_TAC o (SPEC` (i + 1) MOD 4 `));
ANTS_TAC;
SWITCH_TAC` nn = 4 `;
ASM_SIMP_TAC[MOD_IN_NUMSEG];
MP_TAC (let t = GEN_ALL MOD_INJ1 in SPECL [`2`; ` 4 `] t);
ANTS_TAC;
ARITH_TAC;
DISCH_THEN (MP_TAC o (SPEC` i + 1 `));
ASM_SIMP_TAC[ARITH_RULE` (i + 1) + 2 = i + 3 `];


FIRST_X_ASSUM (MP_TAC o (SPEC` (i + 2) MOD 4 `));
ANTS_TAC;
SWITCH_TAC` nn = 4 `;
MP_TAC (let t = GEN_ALL MOD_INJ1 in SPECL [` 1 `;` 4 `] t);
ANTS_TAC;
ARITH_TAC;
ASM_SIMP_TAC[ARITH_RULE` i + 3 = (i + 2) + 1 `; MOD_IN_NUMSEG];
SIMP_TAC[];
MP_TAC (SPEC`(i + 1) MOD 4 ` QU_OR_QXY);
MP_TAC (SPEC`(i + 2) MOD 4 ` QU_OR_QXY);
ASM_SIMP_TAC[];
ONCE_REWRITE_TAC[DISJ_SYM];
STRIP_TAC THEN STRIP_TAC;
REPLICATE_TAC 3 DOWN THEN PHA;
SWITCH_TAC` nn = 4 `;
ASM_REWRITE_TAC[];
ASM_SIMP_TAC[GSYM periodic_mod];
STRIP_TAC;
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn: num `] periodic_sum);
SUBGOAL_THEN` &0 <= sum (i..nn - 1 + i) (cc_gg_v11 cc)` MP_TAC;
MATCH_MP_TAC SUM_POS_LE;
REWRITE_TAC[FINITE_NUMSEG];
GEN_TAC;
EXPAND_TAC "nn";
REWRITE_TAC[IN_NUMSEG; ARITH_RULE` i <= x /\ x <= 4 - 1 + i <=> x = i \/ x = i + 1 \/ x = i + 2 \/ x = i + 3 `];
REPLICATE_TAC 8 DOWN THEN PHA;
ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `] MP_TAC;
ASM_SEARCH_TCL [` cc_qx_v11 cc i ==> &0 <= x `] MP_TAC;
MESON_TAC[];
ASM_REWRITE_TAC[REAL_ARITH` &0 <= a <=> ~( a < &0) `];
SUBGOAL_THEN` cc_small_v11 cc (i + 1) /\ cc_small_v11 cc (i + 2) ` MP_TAC;
DOWN;
SWITCH_TAC` nn = 4 `;
ASM_SIMP_TAC[GSYM periodic_mod];
SIMP_TAC[ARITH_RULE` (i + 1) + 1 = i + 2 `; cc_qu_v11; cc_hassmall_v11];
STRIP_TAC;

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


SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc i ` MP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 1) ` MP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
UNDISCH_TAC` cc_qu_v11 cc ((i + 1) MOD 4 )`;
SWITCH_TAC` nn = 4 `;
ASM_SIMP_TAC[GSYM Oxl_def.periodic_mod];
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2)` MP_TAC;
ASM_SEARCH_TCL [` cc_qx_v11 cc i ==> &0 <= x `] MATCH_MP_TAC;
UNDISCH_TAC` cc_qx_v11 cc (( i + 2) MOD 4 ) `;
SWITCH_TAC` nn = 4 `;
ASM_SIMP_TAC[GSYM Oxl_def.periodic_mod];
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3) ` MP_TAC;
ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `] MATCH_MP_TAC;
FIRST_X_ASSUM ACCEPT_TAC;
REPEAT STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
ASM_REWRITE_TAC[];
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum);
FIRST_X_ASSUM (MP_TAC o SYM o (SPEC_ALL));
ASM_SIMP_TAC[];
STRIP_TAC;
SIMP_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 3 /\ i + 1 <= i + 3 /\ (i + 1) + 1 <= i + 3 /\ ((i +1) + 1) + 1 = i + 3 `];
REWRITE_TAC[SUM_SING_NUMSEG; ARITH_RULE` (i + 1) + 1 = i + 2 `];
UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 2) `;
UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 3) `;
UNDISCH_TAC` cc_eps <= cc_gg_v11 cc (i) `;
UNDISCH_TAC` --cc_eps <= cc_gg_v11 cc (i + 1) `;
PHA;
REAL_ARITH_TAC;



ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `];





REPLICATE_TAC 3 DOWN THEN PHA;
SWITCH_TAC` nn = 4 `;
ASM_SIMP_TAC[GSYM Oxl_def.periodic_mod];
STRIP_TAC;


ASM_SEARCH_TCL [` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1)
          ==> xac <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `] (MP_TAC o (SPEC` i + 2 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `];
ASM_REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `];
SUBGOAL_THEN` cc_gg3a_v11 cc (i + 3) + cc_gg3b_v11 cc (i + 3) <= cc_gg_v11 cc ( i + 3) ` MP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
FIRST_ASSUM ACCEPT_TAC;
STRIP_TAC;
SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc (i + 3) ` ASSUME_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
FIRST_X_ASSUM ACCEPT_TAC;
STRIP_TAC;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3)` MP_TAC;
REPLICATE_TAC 3 DOWN THEN PHA;
REAL_ARITH_TAC;



ASM_SEARCH_TCL [` cc_qx_v11 cc (i) ==> &0 <= x `] (ASSUME_TAC2 o (SPEC` i + 1 `));
ASM_SEARCH_TCL [` cc_qx_v11 cc (i) ==> &0 <= x `] (ASSUME_TAC2 o (SPEC` i + 1 `));
ASM_SEARCH_TCL [` cc_qx_v11 cc (i) ==> &0 <= x `] (ASSUME_TAC2 o (SPEC` i:num `));
STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..nn - 1 ) (cc_gg_v11 cc) ` MP_TAC;
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `; `nn:num `] Oxl_def.periodic_sum);
FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC` i:num `));
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `];
SIMP_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 3 /\ i + 1 <= i + 3 /\ (i + 1) + 1 <= i + 3 /\ ((i +1) + 1) + 1 = i + 3 `];
REWRITE_TAC[SUM_SING_NUMSEG; ARITH_RULE` (i + 1) + 1 = i + 2 `];
DOWN THEN DOWN THEN DOWN THEN PHA;
REAL_ARITH_TAC;
ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `];
REPLICATE_TAC 3 DOWN THEN PHA;
SWITCH_TAC` nn = 4 `;
ASM_SIMP_TAC[GSYM Oxl_def.periodic_mod];
STRIP_TAC;

(* =================== *)
ASM_SEARCH_TCL [` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1)
          ==> xax <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `] (MP_TAC o (SPEC` i + 2 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `];
ASM_REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `];
SUBGOAL_THEN` cc_gg3a_v11 cc (i + 3) + cc_gg3b_v11 cc (i + 3) <= cc_gg_v11 cc ( i + 3) ` MP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
FIRST_ASSUM ACCEPT_TAC;
STRIP_TAC;
SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc (i + 3) ` ASSUME_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
FIRST_X_ASSUM ACCEPT_TAC;
STRIP_TAC;

SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3)` MP_TAC;
REPLICATE_TAC 3 DOWN THEN PHA;
REAL_ARITH_TAC;
STRIP_TAC;
SUBGOAL_THEN` cc_small_v11 cc ( i + 1) ` MP_TAC;
UNDISCH_TAC` cc_qu_v11 cc (i + 1) `;
SIMP_TAC[cc_qu_v11; cc_hassmall_v11];
STRIP_TAC;
ASM_SEARCH_TCL [`!i. cc_qx_v11 cc i /\ cc_small_v11 cc (i + 1) /\ ~cc_small_v11 cc i
       ==> cc_eps <= cc_gg_v11 cc i `] (ASSUME_TAC2 o SPEC_ALL);
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc ( i + 1) ` MP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
FIRST_X_ASSUM ACCEPT_TAC;

REPEAT STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
ASM_REWRITE_TAC[];
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum);
FIRST_X_ASSUM (MP_TAC o SYM o (SPEC_ALL));


ASM_SIMP_TAC[];
STRIP_TAC;
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `];
SIMP_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 3 /\ i + 1 <= i + 3 /\ (i + 1) + 1 <= i + 3 /\ ((i +1) + 1) + 1 = i + 3 `];
REWRITE_TAC[SUM_SING_NUMSEG; ARITH_RULE` (i + 1) + 1 = i + 2 `];


REPLICATE_TAC 4 DOWN THEN PHA;
DOWN;
REAL_ARITH_TAC;



ASM_REWRITE_TAC[REAL_ARITH` &0 <= a <=> ~( a < &0) `];
STRIP_TAC;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) ` MP_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 1) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 1 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
STRIP_TAC;
ASM_SEARCH_TCL [` cc_qu_v11 cc ( i + 1) /\ cc_qy_v11 cc i `] (ASSUME_TAC2 o SPEC_ALL);
ASM_SEARCH_TCL [` cc_gg3a_v11 cc i + x <= y `] (ASSUME_TAC2 o SPEC_ALL);
ASM_SEARCH_TCL  [` x ==> &0 <= cc_gg3a_v11 cc i  `] (ASSUME_TAC2 o SPEC_ALL);
DOWNS 3;
REWRITE_TAC[];
REAL_ARITH_TAC;


STRIP_TAC;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3) ` MP_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 2) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 2) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];

MP_TAC (SPEC ` i + 2 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
SUBGOAL_THEN`~ cc_small_v11 cc (i + 4) ` MP_TAC;
UNDISCH_TAC` periodic (cc_small_v11 cc) nn `;
ASM_SIMP_TAC[Oxl_def.periodic];
ASM_SEARCH_TCL [` cc_qu_v11 cc i ==> -- x <= y `] NHANH;
STRIP_TAC;


REWRITE_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (i + 2) + 1 = i + 3 `];
STRIP_TAC;
ASM_SEARCH_TCL [` !i. cc_qx_v11 cc i /\ cc_small_v11 cc (i) /\ ~cc_small_v11 cc (i + 1)
       ==> cc_eps <= cc_gg_v11 cc i ` ] (MP_TAC o (SPEC` i + 3 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];
DOWN;
REAL_ARITH_TAC;
STRIP_TAC;

SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
SWITCH_TAC ` nn = 4 `;
ASM_REWRITE_TAC[];
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `; ` nn:num `] Oxl_def.periodic_sum);
FIRST_X_ASSUM (SUBST1_TAC o SYM o SPEC_ALL);
FIRST_X_ASSUM (SUBST1_TAC o SYM);
SIMP_TAC[ARITH_RULE`4 - 1 + i = i + 3 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + (j:num) /\ i + 1 <= i + 3 /\ (i + 1) + 1 <= i + 3  `; LE_TRANS];
REWRITE_TAC[ARITH_RULE` ((i + 1) + 1) + 1 = i + 3 `; SUM_SING_NUMSEG];

DOWNS 2;
REWRITE_TAC[ARITH_RULE` (i + 1) + 1 = i + 2 `];
REAL_ARITH_TAC;
ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `];


(* pkpk *)

SUBGOAL_THEN` &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 1) ` ;
SWITCH_TAC` nn = 4 `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `;
SWITCH_TAC` nn = 4 `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 1 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
ASM_SEARCH_TCL [` cc_qu_v11 cc i ==> -- x <= y `] NHANH;
STRIP_TAC;
SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc i ` MP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[];
DOWNS 2;
SIMP_TAC[cc_qu_v11; cc_hassmall_v11];
DOWN;
REAL_ARITH_TAC;

SUBGOAL_THEN` ~cc_small_v11 cc (i + 4) ` ASSUME_TAC;
ASM_SEARCH_TCL [` periodic (cc_small_v11 cc) `] MP_TAC;
SIMP_TAC[Oxl_def.periodic];
SWITCH_TAC` nn = 4 `;
ASM_SIMP_TAC[];
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3) ` MP_TAC;
ASM_CASES_TAC` cc_qy_v11 cc ( i + 2) \/ cc_qx_v11 cc (i + 2)`;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
DOWN THEN STRIP_TAC;
ASM_SIMP_TAC[];
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 2 ` QU_OR_QXY);
DOWN;
SIMP_TAC[DE_MORGAN_THM];
ASM_SEARCH_TCL [` cc_qu_v11 cc i ==> -- x <= y `] NHANH;
STRIP_TAC THEN STRIP_TAC;
SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc (i + 3) ` MP_TAC;
ASM_SEARCH_TCL [` !i. cc_qx_v11 cc i /\ cc_small_v11 cc i /\ ~cc_small_v11 cc (i + 1)
          ==> cc_eps <= cc_gg_v11 cc i `] MATCH_MP_TAC;
UNDISCH_TAC` ~cc_small_v11 cc (i + 4) `;
ASM_SIMP_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];
UNDISCH_TAC` cc_qu_v11 cc (i + 2) `;
SIMP_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (i + 2) + 1 = i + 3 `];
DOWN;
REAL_ARITH_TAC;


STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum);
FIRST_X_ASSUM (SUBST1_TAC o SYM o SPEC_ALL);
ASM_REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `];
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 3 /\ i + 1 <= i + 3 /\ (i + 1) + 1 = i + 2 /\ i + 2 <= i + 3 /\ (i + 2 ) + 1 = i + 3 /\ (i + 1) + 2 = i + 3 `];
REWRITE_TAC[SUM_SING_NUMSEG];
DOWNS 3;
REAL_ARITH_TAC;
ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]]);;





let EXISTS_QY_CARD5 = 
prove_by_refinement
(` (cc_bool_model_v11 cc /\
           cc_bool_prep_v11 cc /\
           cc_real_model_v11 cc /\
           sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\
          cc_card_v11 cc = 5 
==> (? i. i < 5 /\ cc_qy_v11 cc i)`,

[NHANH UNPNFVW;
NHANH LXDEYBO;
REWRITE_TAC[cc_size_v11; cc_qy_v11];
MP_TAC (SET_RULE` DISJOINT {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} {i | i IN 0..cc_card_v11 cc - 1 /\ ~cc_4cell_v11 cc i}`);
MP_TAC (SET_RULE` {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} UNION {i | i IN 0..cc_card_v11 cc - 1 /\ ~cc_4cell_v11 cc i} = 0..cc_card_v11 cc - 1`);
REPEAT STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
ABBREV_TAC ` s1 = {i | i IN 0..5 - 1 /\ cc_4cell_v11 cc i}`;
ABBREV_TAC ` s2 = {i | i IN 0..5 - 1 /\ ~cc_4cell_v11 cc i}`;
SUBGOAL_THEN` CARD (s1 UNION (s2:num -> bool)) = CARD s1 + CARD s2 ` MP_TAC;
MATCH_MP_TAC CARD_UNION;
ASM_REWRITE_TAC[GSYM DISJOINT];
ASM_REWRITE_TAC[GSYM FINITE_UNION; FINITE_NUMSEG];
ASM_SIMP_TAC[CARD_NUMSEG; ARITH_RULE` (5 - 1 + 1) - 0 = 5 `];
STRIP_TAC;
ASSUME_TAC2 (ARITH_RULE` 5 = CARD (s1:num -> bool) + CARD (s2:num -> bool) /\ CARD s1 <= 4 /\ CARD s2 <= 1 
==> CARD s2 = 1 `);
DOWN;
ASM_CASES_TAC` (s2:num -> bool) = {} `;
ASM_REWRITE_TAC[CARD_EMPTY; ARITH_RULE` ~( 0 = 1) `];
DOWN;
REWRITE_TAC[GSYM MEMBER_NOT_EMPTY];
FIRST_X_ASSUM (SUBST1_TAC o SYM);
EXPAND_TAC "s2";
SIMP_TAC[IN_ELIM_THM; IN_NUMSEG; ARITH_RULE`x <= 5 - 1 <=> x < 5 `; LE_0]]);;




let NUMSEG_INTER_22 = 
prove_by_refinement(
` ~(nn = 0) ==> ( (i <= x /\ x <= i + nn - 1) <=> (? k. k < nn /\ x = i + k )) `,

[STRIP_TAC;
EQ_TAC;
STRIP_TAC;
EXISTS_TAC` x - (i:num) `;
DOWN_TAC;
ARITH_TAC;
STRIP_TAC;
DOWN_TAC;
ARITH_TAC]);;


let PERIODIC_IMAGE_EQUAL = 
prove_by_refinement(
` ~( nn = 0) /\ periodic (f:num -> A) nn ==> IMAGE f (0..nn - 1) = IMAGE f (i..i + nn - 1) `,

[NHANH MOD_NUMSEG;
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC` i:num `));
SIMP_TAC[];
ASSUME_TAC2 (
REWRITE_RULE[RIGHT_FORALL_IMP_THM] (ISPECL [` f:num -> A `;` nn:num `] Oxl_def.periodic_mod));
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
ASM_SIMP_TAC[IMAGE; IN_ELIM_THM; IN_NUMSEG];
ASM_SIMP_TAC[NUMSEG_INTER_22];
DOWN;
SET_TAC[]]);;



let IMAGE_NUMSEG222 = 
prove_by_refinement
(`~(nn = 0) /\ periodic (f:num -> A) nn
     ==> ! i. { f k | k IN 0..nn - 1} = { f (i + k) | k IN 0..nn - 1 } `,

[STRIP_TAC THEN GEN_TAC THEN DOWN_TAC;
NHANH PERIODIC_IMAGE_EQUAL;
REWRITE_TAC[IMAGE; IN_NUMSEG; LE_0];
STRIP_TAC;
DOWN;
ASM_SIMP_TAC[NUMSEG_INTER_22; ARITH_RULE`~(n = 0) ==> (x < (n:num) <=> x <= n - 1) `];
DOWN_TAC;
SET_TAC[]]);;





let LITTLE_CC_GGB = 
prove(`(!i. cc_qy_v11 cc i
          ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ) /\
 (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3a_v11 cc i) /\
( !i. cc_qu_v11 cc (i + 1) /\ cc_qy_v11 cc i
          ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1))
==> cc_qu_v11 cc (i + 1) /\ cc_qy_v11 cc i
==> &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) `,

STRIP_TAC THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN 
FIRST_X_ASSUM NHANH THEN REAL_ARITH_TAC);;


let LITTLE_CC_GGA = 
prove(` (!i. cc_qy_v11 cc i
          ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ) /\
 (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i) /\
(!i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1)
          ==> &0 <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1)) 
==> cc_qu_v11 cc (i) /\ cc_qy_v11 cc (i + 1)
==> &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) `,

STRIP_TAC THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN 
FIRST_X_ASSUM NHANH THEN REAL_ARITH_TAC);;






let QU_IMP_SMALL =
prove(` cc_qu_v11 cc i ==> cc_small_v11 cc i /\ cc_small_v11 cc ( i + 1) `,

SIMP_TAC[cc_qu_v11; cc_hassmall_v11]);;





let IPVICGW_C5_QX_ALONG_NOT_SMALL = 
prove_by_refinement(
` !cc. ((cc_bool_model_v11 cc /\
           cc_bool_prep_v11 cc /\
           cc_real_model_v11 cc /\
           sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\
          cc_card_v11 cc = 5 ) /\
          ~cc_small_v11 cc i /\ cc_qx_v11 cc i /\ cc_qx_v11 cc ( i + 4) ==> F `,

[REPEAT STRIP_TAC;
ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00;
ASSUME_TAC2 EXISTS_QY_CARD5;
DOWN_TAC;
NHANH Oxl_def.periodic_fn;
REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `];
IMP_TAC;
IMP_TAC;
STRIP_TAC;
STRIP_TAC;
STRIP_TAC;

ABBREV_TAC` nn = cc_card_v11 cc `;
ASSUME_TAC2 (ISPECL [`nn:num `;` cc_qy_v11 cc `] (GEN_ALL IMAGE_NUMSEG222));
SUBGOAL_THEN` cc_qy_v11 cc i' IN {cc_qy_v11 cc k | k IN 0..nn - 1}` MP_TAC;
REWRITE_TAC[IN_ELIM_THM];
EXISTS_TAC` i': num `;
ASM_REWRITE_TAC[IN_NUMSEG_0; ARITH_RULE` a <= 5 - 1 <=> a < 5 `];
FIRST_ASSUM (SUBST1_TAC o (SPEC` i:num `));

ASM_REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG_0; ARITH_RULE` 5 - 1 = 4 `];
STRIP_TAC;
ASM_CASES_TAC` k = 0 `;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWN;
REWRITE_TAC[ADD_0];
UNDISCH_TAC` cc_qx_v11 cc i `;
SIMP_TAC[cc_qx_v11; cc_qy_v11];

ASM_CASES_TAC` k = 4`;
FIRST_X_ASSUM SUBST_ALL_TAC;
DOWNS 2;
UNDISCH_TAC` cc_qx_v11 cc (i + 4) `;
SIMP_TAC[cc_qx_v11; cc_qy_v11];
ASM_CASES_TAC` k = 1 `;

FIRST_X_ASSUM SUBST_ALL_TAC;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) ` MP_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 2) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];


ASM_CASES_TAC` cc_qx_v11 cc (i + 2 ) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 2 ` QU_OR_QXY);
ASM_SIMP_TAC[];

STRIP_TAC;
ASM_SEARCH_TCL [` cc_qu_v11 cc (i + 1) /\ x ==>  xpapxa <= cc_gg3b_v11 cc i + cc_gg_v11 cc j `] (MP_TAC o (SPEC ` i + 1 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 1) + 1 = i + 2 `];
STRIP_TAC;
MP_TAC (SPEC ` i + 1 ` (GEN`i:num ` LITTLE_CC_GGB));
ANTS_TAC;
ASM_REWRITE_TAC[];
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (x + 1) + 1 = x + 2 `];
ASM_SIMP_TAC[ARITH_RULE` (x + 1) + 1 = x + 2 `; Oxl_def.cc_eps];


STRIP_TAC;
SWITCH_TAC` nn = 5 `;
SUBGOAL_THEN`~ cc_small_v11 cc (i + 5) ` ASSUME_TAC;
ASM_SIMP_TAC[ Oxl_def.periodic];
ASM_SEARCH_TCL [` periodic (cc_small_v11 cc) `] MP_TAC;
SIMP_TAC[ Oxl_def.periodic];
ASM_REWRITE_TAC[];
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3) + cc_gg_v11 cc (i + 4) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc ( i + 3) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 3 ) ` ;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 3 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
NHANH QU_IMP_SMALL;
REWRITE_TAC[ARITH_RULE` (i + 3 ) + 1 = i + 4 `];
ASM_SEARCH_TCL [` cc_qu_v11 cc i ==> -- x <= y `] NHANH;
STRIP_TAC;
ASM_SEARCH_TCL [` a /\ b /\ ~ cc_small_v11 cc (i + 1) ==> cc_eps <= y `] (MP_TAC o (SPEC` i + 4 `));
ANTS_TAC;
ASM_SEARCH_TCL [` 5 = n `] (SUBST_ALL_TAC o SYM);
ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `];
UNDISCH_TAC` --cc_eps <= cc_gg_v11 cc ( i + 3) `;
REAL_ARITH_TAC;

SUBGOAL_THEN` &0 <= cc_gg_v11 cc i ` ASSUME_TAC;
ASM_SIMP_TAC[];
SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc ) ` MP_TAC;
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum);
FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC` i:num `));
EXPAND_TAC "nn";
SIMP_TAC[ARITH_RULE` 5 - 1 + i = i + 4 /\ i <= i + 4 /\ i + 1 <= i + 4 /\ (i +1) + 1 <= i + 4 /\ ((i + 1) + 1) + 1 <= i + 4`; SUM_CLAUSES_LEFT];
REWRITE_TAC[GSYM ADD_ASSOC];
CONV_TAC NUM_REDUCE_CONV;
REWRITE_TAC[SUM_SING_NUMSEG];
DOWNS 2;
UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) `;
REAL_ARITH_TAC;
ASM_REWRITE_TAC[ARITH_RULE` ~( &0 <= x) <=> x < &0 `];

(* ========== *)
ASM_CASES_TAC` k = 2 `;
FIRST_X_ASSUM SUBST_ALL_TAC;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 1) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 1 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
STRIP_TAC;
MP_TAC (SPEC `i + 1 ` (GEN `i: num ` LITTLE_CC_GGA));
ANTS_TAC;
ASM_REWRITE_TAC[];
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 1) + 1 = i + 2 `];
REWRITE_TAC[Oxl_def.cc_eps; ARITH_RULE` (a + 1) + 1 = a + 2 `];





SUBGOAL_THEN` ~ cc_small_v11 cc ( i + 5) ` ASSUME_TAC;
ASM_SEARCH_TCL [` periodic (cc_small_v11 cc) `] MP_TAC;
REWRITE_TAC[Oxl_def.periodic];
ASM_SIMP_TAC[];

SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 3) + cc_gg_v11 cc (i + 4) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 3 ) ` ;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 3) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 3 ` QU_OR_QXY);
ASM_REWRITE_TAC[];


(* ??? *)

ASM_SEARCH_TCL [` -- x <= y `] NHANH;
REWRITE_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (i + 3) + 1 = i + 4 `];
STRIP_TAC;
ASM_SEARCH_TCL [` a /\ b /\ ~ p (i + 1) ==> cc_eps <= x `] (MP_TAC o (SPEC` i + 4 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `];
DOWN;
REAL_ARITH_TAC;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc i ` ASSUME_TAC;
ASM_SIMP_TAC[];

SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum);


FIRST_X_ASSUM (MP_TAC o SYM o (SPEC` i :num `));
ASM_SIMP_TAC[ARITH_RULE` 5 - 1 + i = i + 4 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `];
STRIP_TAC;
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG];
DOWNS 4;
DOWN;
REAL_ARITH_TAC;

ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x) <=> x < &0 `];




ASM_CASES_TAC` k = 3 `;
FIRST_X_ASSUM SUBST_ALL_TAC;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 2) + cc_gg_v11 cc (i + 3) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc ( i + 2) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 2) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 2 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
STRIP_TAC;
MP_TAC (SPEC` i + 2 ` (GEN`i:num ` LITTLE_CC_GGA));
ANTS_TAC;
ASM_REWRITE_TAC[];
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `];

SIMP_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `; Oxl_def.cc_eps];
SUBGOAL_THEN` &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 1) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 1 ` QU_OR_QXY);
ASM_SIMP_TAC[];
ASM_SEARCH_TCL [` x ==> -- y <= z `] NHANH;
REWRITE_TAC[cc_qu_v11; cc_hassmall_v11];
STRIP_TAC;
ASM_SEARCH_TCL [` a /\  b (i + 1 ) /\ ~ c ==> cc_eps <= x `] (MP_TAC o SPEC_ALL);
ANTS_TAC;
ASM_REWRITE_TAC[];
DOWN;
REAL_ARITH_TAC;


SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 4 ) ` ASSUME_TAC THEN 
ASM_SIMP_TAC[];

SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum);
FIRST_X_ASSUM (MP_TAC o SYM o (SPEC` i :num `));
ASM_SIMP_TAC[ARITH_RULE` 5 - 1 + i = i + 4 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `];
STRIP_TAC;
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG];
REMOVE_TAC;
DOWNS 4;
REAL_ARITH_TAC;
ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x) <=> x < &0 `];

ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x ) <=> x < &0 `];
DOWNS 5;
UNDISCH_TAC` k <= 4 `;
ARITH_TAC]);;







let CARD5_1074_LE_QYI = 
prove_by_refinement(` (cc_bool_model_v11 cc /\
            cc_bool_prep_v11 cc /\
            cc_real_model_v11 cc /\
            sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ) /\
(!i. cc_4cell_v11 cc i
          ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i ) /\
( !i. cc_qy_v11 cc i ==> #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i ) /\
cc_qy_v11 cc i /\ cc_card_v11 cc = 5 /\
periodic (cc_gg_v11 cc) (cc_card_v11 cc ) /\
(!i. #0.606 <= cc_azim_v11 cc i ) /\
sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) = &2 * pi /\
 #1.074 <= cc_azim_v11 cc i 
==> &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) `,


[NHANH UNPNFVW;

REWRITE_TAC[TAUT` a /\ b /\ c <=> (a/\b) /\ c `];
IMP_TAC;
STRIP_TAC;
DISCH_TAC;
ABBREV_TAC` S = { i + k | k | k < 5 /\ cc_qy_v11 cc (i + k ) } ` ;
SUBGOAL_THEN` i IN (S:num -> bool) ` ASSUME_TAC;
EXPAND_TAC "S";
REWRITE_TAC[IN_ELIM_THM];
EXISTS_TAC` 0 `;
ASM_REWRITE_TAC[ADD_CLAUSES];
ARITH_TAC;
SUBGOAL_THEN` S SUBSET { i + k | k | k < 5 } ` ASSUME_TAC;
EXPAND_TAC "S";
CONV_TAC SET_RULE;
ASSUME_TAC (SPEC` 5 ` FINITE_NUMSEG_LT );
SUBGOAL_THEN` IMAGE (\k. i + k ) { m | m < 5} = {i + k | k < 5 }` ASSUME_TAC;
REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM];
GEN_TAC;
MESON_TAC[];
UNDISCH_TAC` FINITE {m | m < 5} `;
NHANH (ISPEC `(\x. (i:num) + x ) ` FINITE_IMAGE);
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN` sum {i:num} ( cc_azim_v11 cc ) <= sum S ( cc_azim_v11 cc ) ` ASSUME_TAC;
MATCH_MP_TAC SUM_SUBSET_SIMPLE;
CONJ_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` {i + k | k < 5 } ` ;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];
GEN_TAC THEN STRIP_TAC;
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` #0.606 `;
ASM_REWRITE_TAC[];
REAL_ARITH_TAC;

MP_TAC (ISPECL [` cc_gg_v11 cc `;` 5 `] Oxl_def.periodic_sum);
ANTS_TAC;
UNDISCH_TAC` periodic (cc_gg_v11 cc) (cc_card_v11 cc) `;
ASM_SIMP_TAC[];
STRIP_TAC;
ARITH_TAC;
DISCH_THEN (ASSUME_TAC o SYM o SPEC_ALL);
ABBREV_TAC` SS = {i + k | k | k < 5 /\ ~cc_qy_v11 cc (i + k)} `;
SUBGOAL_THEN` S INTER (SS: num -> bool) = {} ` ASSUME_TAC;
EXPAND_TAC "S";
EXPAND_TAC "SS";
CONV_TAC SET_RULE;
SUBGOAL_THEN` { i + k | k | k < 5 } = S UNION (SS: num -> bool)` ASSUME_TAC;
EXPAND_TAC "S";
EXPAND_TAC "SS";
CONV_TAC SET_RULE;
ASM_REWRITE_TAC[numseg];
SUBGOAL_THEN` {x | i <= x /\ x <= 5 - 1 + i} = {i + k | k < 5 } ` ASSUME_TAC;
REWRITE_TAC[EXTENSION; IN_ELIM_THM];
GEN_TAC;
EQ_TAC;
STRIP_TAC;
EXISTS_TAC` x - (i:num) `;
DOWN THEN DOWN;
ARITH_TAC;
STRIP_TAC;
DOWNS 2;
ARITH_TAC;
ASM_REWRITE_TAC[];
ASSUME_TAC2 UNIQUE_QY;
DOWN THEN PHA;
DISCH_THEN (MP_TAC o (SPEC` i MOD 5 ` ));
ASM_REWRITE_TAC[];
ANTS_TAC;
MP_TAC (ARITH_RULE` ~( 5 = 0 ) `);
SIMP_TAC[Oxl_def.MOD_IN_NUMSEG];
ASSUME_TAC2 Oxl_def.periodic_fn;
DOWN;
ASM_REWRITE_TAC[];
STRIP_TAC;
DOWN THEN PHA;
SIMP_TAC [ GSYM Oxl_def.periodic_mod];
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN` cc_qy_v11 cc (i MOD 5 ) ` ASSUME_TAC;
ASSUME_TAC2 Oxl_def.periodic_fn;
DOWN;
ASM_REWRITE_TAC[];
STRIP_TAC;
MP_TAC (ARITH_RULE` ~( 5 = 0 ) `);
DOWN THEN PHA;
SIMP_TAC [ GSYM Oxl_def.periodic_mod];
ASM_REWRITE_TAC[];

SUBGOAL_THEN` FINITE (S: num -> bool) ` MP_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` {i + k | k < 5} `;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` FINITE (SS: num -> bool) ` MP_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` {i + k | k < 5} `;
ASM_REWRITE_TAC[];
CONV_TAC SET_RULE;
PHA THEN STRIP_TAC;
SUBGOAL_THEN` DISJOINT S (SS: num -> bool) ` MP_TAC;
ASM_REWRITE_TAC[DISJOINT];
DOWN THEN DOWN THEN PHA;
SIMP_TAC[SUM_UNION];
STRIP_TAC;
SUBGOAL_THEN`!i.  cc_qy_v11 cc (i MOD 5 ) = cc_qy_v11 cc i ` ASSUME_TAC;
MATCH_MP_TAC (* Xwitccn. *) Oxl_def.PERIODIC_PROPERTY;
ASSUME_TAC2 Oxl_def.periodic_fn;
DOWN;
ASM_SIMP_TAC[];
STRIP_TAC;
ARITH_TAC;
ASSUME_TAC2 Oxl_def.periodic_fn;
DOWN THEN ASM_REWRITE_TAC[] THEN STRIP_TAC;




SUBGOAL_THEN` S = {i:num } ` ASSUME_TAC;
REWRITE_TAC[EXTENSION; IN_INSERT; NOT_IN_EMPTY];
EXPAND_TAC "S";
REWRITE_TAC[IN_ELIM_THM];
GEN_TAC;
EQ_TAC;
STRIP_TAC;
ASM_CASES_TAC` k = 0 `;
DOWN THEN DOWN THEN ARITH_TAC;

UNDISCH_TAC` !k. ~(k = i MOD 5) /\ k IN 0..5 - 1 ==> ~cc_qy_v11 cc k `;
DISCH_THEN (MP_TAC o (SPEC` ( i + k ) MOD 5 `));
ANTS_TAC;
CONJ_TAC;


MP_TAC (SPECL [` k:num `;` 5 `] Oxl_def.MOD_INJ1_ALT);
ANTS_TAC;
ASM_REWRITE_TAC[];
ARITH_TAC;
SIMP_TAC[];
MATCH_MP_TAC Oxl_def.MOD_IN_NUMSEG;
ARITH_TAC;
ASM_REWRITE_TAC[];

STRIP_TAC;
EXISTS_TAC` 0 `;
ASM_REWRITE_TAC[ADD_CLAUSES];
ARITH_TAC;

SUBGOAL_THEN` SS = { i + k | k < 5 } DIFF S ` ASSUME_TAC;
ASM_REWRITE_TAC[];
MATCH_MP_TAC (SET_RULE` SS INTER S = {} ==> SS = ( S UNION SS ) DIFF S  `);
UNDISCH_TAC` (S:num -> bool) INTER SS = {} `;
ASM_REWRITE_TAC[INTER_COMM];
SUBGOAL_THEN` SS = { i + k | 0 < k /\ k < 5 } ` ASSUME_TAC;
DOWN;
FIRST_X_ASSUM SUBST1_TAC;
SIMP_TAC[];
DISCH_TAC;
REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_DIFF; IN_INSERT; NOT_IN_EMPTY];
GEN_TAC;
MESON_TAC[ARITH_RULE` ~( 0 < k ) <=> x + k = x `];

SUBGOAL_THEN` cc_bool_model_v11 cc /\ cc_real_model_v11 cc ` MP_TAC;
ASM_REWRITE_TAC[];
REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
STRIP_TAC;
SUBGOAL_THEN` ! i. i IN SS ==> 
       a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC;
EXPAND_TAC "SS";
REWRITE_TAC[IN_ELIM_THM];
GEN_TAC THEN STRIP_TAC;
ASM_SEARCH_TCL [` !i. cc_4cell_v11 cc i ==>
       a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i `] MATCH_MP_TAC;
ASM_REWRITE_TAC[];
DOWN THEN DOWN;
SIMP_TAC[cc_qy_v11];

DOWN;
UNDISCH_TAC` FINITE (SS: num -> bool) `;
PHA;
NHANH SUM_LE;
UNDISCH_TAC` {i + k | k < 5} = S UNION SS `;
DISCH_THEN (ASSUME_TAC o SYM);
ASM_REWRITE_TAC[];
UNDISCH_TAC` SS = {i + k | k < 5} DIFF S `;
UNDISCH_TAC` SS = {i + k | 0 < k /\ k < 5} `;
ASM_REWRITE_TAC[];
DISCH_THEN (ASSUME_TAC o SYM);
DISCH_THEN (ASSUME_TAC o SYM);
ASM_REWRITE_TAC[SUM_SING; ETA_AX];
SUBGOAL_THEN`! a f. (\(i:num). (a:real) + f i ) = (\i. (\i. a ) i + f i )` ASSUME_TAC; 
REWRITE_TAC[BETA_THM];
FIRST_X_ASSUM (fun x -> ONCE_REWRITE_TAC[x]);
IMP_TAC;
ASM_SIMP_TAC[SUM_ADD; Vol1.SUM_CONST2; SUM_LMUL];
SUBGOAL_THEN` CARD {i + k | k < 5 } = 5 ` ASSUME_TAC;
MATCH_MP_TAC Topology.CARD_FINITE_SERIES_EQ;
GEN_TAC THEN GEN_TAC;
ARITH_TAC;
SUBGOAL_THEN` CARD ({i + k | k < 5} DIFF {i}) = CARD {i + k | k < 5} - CARD {i} ` ASSUME_TAC;
MATCH_MP_TAC CARD_DIFF;
ASM_REWRITE_TAC[INSERT_SUBSET];

REWRITE_TAC[EMPTY_SUBSET; IN_ELIM_THM];
EXISTS_TAC` 0 `;
CONJ_TAC;
ARITH_TAC;
ARITH_TAC;
DOWN;
ASM_REWRITE_TAC[CARD_SING];
STRIP_TAC THEN STRIP_TAC;
MP_TAC (GSYM (SPECL [` cc_azim_v11 cc `;` 5`] Oxl_def.periodic_sum));
ANTS_TAC;
UNDISCH_TAC` periodic (cc_azim_v11 cc) (cc_card_v11 cc)`;
ASM_REWRITE_TAC[];
SIMP_TAC[];
STRIP_TAC;
ARITH_TAC;
ASM_REWRITE_TAC[ARITH_RULE` 5 - 1 = 4 `];
STRIP_TAC;
STRIP_TAC;
UNDISCH_TAC` sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) = &2 * pi `;
ASM_REWRITE_TAC[ARITH_RULE` 5 - 1 = 4 `];
SUBGOAL_THEN` i.. 4 + i = S UNION SS ` ASSUME_TAC;
ASM_REWRITE_TAC[];
REWRITE_TAC[numseg; EXTENSION; IN_ELIM_THM];
GEN_TAC;
EQ_TAC;
STRIP_TAC;
EXISTS_TAC` x - (i:num) `;
DOWN THEN DOWN;
ARITH_TAC;
STRIP_TAC;
DOWN THEN DOWN;
ARITH_TAC;


FIRST_ASSUM SUBST1_TAC;
ASSUME_TAC2 (ISPECL [` cc_azim_v11 cc `;` S: num -> bool `;` SS: num -> bool `] SUM_UNION);
FIRST_ASSUM SUBST1_TAC;
ASM_REWRITE_TAC[SUM_SING; REAL_ARITH` a + b = x <=> b = x - a` ];
STRIP_TAC;
UNDISCH_TAC` &4 * a_spine5 + b_spine5 * sum SS (\i. cc_azim_v11 cc i) <=
      sum SS (cc_gg_v11 cc) `;
ASM_REWRITE_TAC[ETA_AX];
UNDISCH_TAC` #1.074 <= cc_azim_v11 cc i `;
REWRITE_TAC[Sphere.a_spine5; Sphere.b_spine5];
MP_TAC (
REWRITE_RULE[] (
REWRITE_CONV[Flyspeck_constants.bounds]` #3.14159 < pi /\ pi < #3.1416 `));
STRIP_TAC THEN STRIP_TAC;
MATCH_MP_TAC (REAL_ARITH` &0 <= x + b ==> b <= c ==> &0 <= x + c `);
SUBGOAL_THEN` #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC;
ASM_SIMP_TAC[];
DOWNS 4;
REAL_ARITH_TAC]);;




let MOD_INJ11 = REWRITE_RULE[GSYM RIGHT_FORALL_IMP_THM] MOD_INJ1;;


let PI_BOUNDS = REWRITE_RULE[] (REWRITE_CONV[Flyspeck_constants.bounds]` #3.14159 < pi /\ pi < #3.1416 `);;




let CARD5_LT_1074_QYI =
prove_by_refinement (` (cc_bool_model_v11 cc /\
            cc_bool_prep_v11 cc /\
            cc_real_model_v11 cc /\
            sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ) /\

cc_qy_v11 cc i /\ cc_card_v11 cc = 5 /\
 cc_azim_v11 cc i <  #1.074
==> &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) `,

[REPEAT STRIP_TAC;
ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00;
ASSUME_TAC2 EXISTS_QY_CARD5;
DOWN_TAC;
NHANH Oxl_def.periodic_fn;
STRIP_TAC;
SUBGOAL_THEN` cc_bool_model_v11 cc /\
          cc_bool_prep_v11 cc /\
          cc_real_model_v11 cc ` MP_TAC;
ASM_REWRITE_TAC[];
REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
STRIP_TAC THEN ASSUME_TAC2 UNIQUE_QY;







ASM_CASES_TAC` cc_small_eta_v11 cc i \/ cc_small_eta_v11 cc ( i + 1) `;
SUBGOAL_THEN` a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC;
UNDISCH_TAC` !i. cc_qy_v11 cc i /\
          cc_small_eta_v11 cc i /\
          cc_small_eta_v11 cc (i + 1)
          ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i `;
UNDISCH_TAC` !i. cc_qy_v11 cc i /\
          ~cc_small_eta_v11 cc i /\
          cc_small_eta_v11 cc (i + 1) /\
          cc_azim_v11 cc i < #1.074
          ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i `;
UNDISCH_TAC` !i. cc_qy_v11 cc i /\
          cc_small_eta_v11 cc i /\
          ~cc_small_eta_v11 cc (i + 1) /\
          cc_azim_v11 cc i < #1.074
          ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i `;
UNDISCH_TAC` cc_azim_v11 cc i < #1.074 `;
UNDISCH_TAC` cc_qy_v11 cc i `;
DOWN;
MESON_TAC[];


MP_TAC (REWRITE_RULE[RIGHT_FORALL_IMP_THM ] (ISPECL [` cc_qy_v11 cc `;` 5 `] periodic_mod));
ANTS_TAC;
UNDISCH_TAC` periodic (cc_qy_v11 cc) (cc_card_v11 cc ) `;
ASM_REWRITE_TAC[];
SIMP_TAC[];
STRIP_TAC;
ARITH_TAC;
DISCH_THEN (ASSUME_TAC o GSYM);
UNDISCH_TAC` !j. cc_qy_v11 cc j
          ==> j IN 0..cc_card_v11 cc - 1
          ==> (!k. ~(k = j)
                   ==> k IN 0..cc_card_v11 cc - 1
                   ==> ~cc_qy_v11 cc k) `;
PHA;
STRIP_TAC;
FIRST_ASSUM (MP_TAC o (SPEC` i MOD 5 ` ));
ANTS_TAC;
ASM_REWRITE_TAC[];
ASSUME_TAC (ARITH_RULE` ~( 5 = 0) `);
ASM_SIMP_TAC[Oxl_def.MOD_IN_NUMSEG];
STRIP_TAC;
SUBGOAL_THEN` ! k. 0 < k /\ k < 5 ==> ~cc_qy_v11 cc ((i + k) MOD 5 ) ` ASSUME_TAC;
GEN_TAC THEN STRIP_TAC;


FIRST_X_ASSUM MATCH_MP_TAC;
ASSUME_TAC (ARITH_RULE` ~( 5 = 0) `);
ASM_SIMP_TAC[Oxl_def.MOD_IN_NUMSEG];
MATCH_MP_TAC (GSYM MOD_INJ11);
ASM_REWRITE_TAC[ARITH_RULE` ~( 0 = x) <=> 0 < x `];
ARITH_TAC;

SUBGOAL_THEN` !k. k < 5 ==> a_spine5 + b_spine5 * cc_azim_v11 cc (i + k ) <= cc_gg_v11 cc (i + k ) ` ASSUME_TAC;
GEN_TAC;
ASM_CASES_TAC` k = 0 `;
ASM_REWRITE_TAC[ADD_CLAUSES];
STRIP_TAC;
UNDISCH_TAC` !i. cc_4cell_v11 cc i
          ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i `;
DISCH_THEN MATCH_MP_TAC;
UNDISCH_TAC` !k. 0 < k /\ k < 5 ==> ~cc_qy_v11 cc ((i + k) MOD 5) `;
ASM_REWRITE_TAC[];
REWRITE_TAC[cc_qy_v11];
DISCH_THEN MATCH_MP_TAC;
DOWN THEN DOWN THEN SIMP_TAC[];
ARITH_TAC;
MP_TAC (ISPECL [` cc_gg_v11 cc `;` 5 `] Oxl_def.periodic_sum);
ANTS_TAC;
ASM_REWRITE_TAC[];

UNDISCH_TAC` periodic (cc_gg_v11 cc) (cc_card_v11 cc) `;
ASM_SIMP_TAC[];
ARITH_TAC;

DISCH_THEN (ASSUME_TAC o GSYM);
ASM_REWRITE_TAC[ARITH_RULE` 5 - 1 = 4 `];
SUBGOAL_THEN` ! j. j IN (i..4 + i) ==>  a_spine5 + b_spine5 * cc_azim_v11 cc (j) <=
              cc_gg_v11 cc (j) ` ASSUME_TAC;
GEN_TAC;
REWRITE_TAC[IN_NUMSEG];
ABBREV_TAC` ij = j - (i:num) `;
NHANH (ARITH_RULE` (i:num) <= j ==> j = i + ( j - i) ` );
ASM_REWRITE_TAC[];
STRIP_TAC;
UNDISCH_TAC` j = i + ij:num `;
DISCH_THEN SUBST1_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
EXPAND_TAC "ij";
UNDISCH_TAC` j <= 4 + i `;
ARITH_TAC;
UNDISCH_TAC` !j. j IN i..4 + i
          ==> a_spine5 + b_spine5 * cc_azim_v11 cc j <= cc_gg_v11 cc j ` ;
NHANH (REWRITE_RULE[GSYM IN_NUMSEG] SUM_LE_NUMSEG);


REWRITE_TAC[SUM_ADD_NUMSEG; ETA_AX;SUM_CONST_NUMSEG; SUM_LMUL];
REWRITE_TAC[ARITH_RULE` ((4 + i) + 1) - i = 5 `];
IMP_TAC;
STRIP_TAC;
SUBST_ALL_TAC (ARITH_RULE` 5 - 1 = 4 `);
ASM_REWRITE_TAC[];
MATCH_MP_TAC (REAL_ARITH` c <= b ==> b <= a ==> c <= a `);
UNDISCH_TAC` sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) = &2 * pi `;
MP_TAC (ISPECL [` cc_azim_v11 cc `;` 5 `] Oxl_def.periodic_sum);
ANTS_TAC;
UNDISCH_TAC` periodic (cc_azim_v11 cc) (cc_card_v11 cc) ` ;
ASM_REWRITE_TAC[];
SIMP_TAC[];
ARITH_TAC;


ASM_SIMP_TAC[ARITH_RULE` 5 - 1 + i = 4 + i `];
STRIP_TAC THEN STRIP_TAC;
REWRITE_TAC[Sphere.a_spine5; Sphere.b_spine5];
MP_TAC PI_BOUNDS;
REAL_ARITH_TAC;

UNDISCH_TAC` ~(cc_small_eta_v11 cc i \/ cc_small_eta_v11 cc (i + 1)) `;
REWRITE_TAC[DE_MORGAN_THM];
STRIP_TAC;


FIRST_ASSUM (MP_TAC o (SPEC` i MOD 5 `));
PHA;
MP_TAC (
REWRITE_RULE[RIGHT_FORALL_IMP_THM ] (ISPECL [` cc_qy_v11 cc `;` 5 `] Oxl_def.periodic_mod));
ANTS_TAC;
ABBREV_TAC` nn = cc_card_v11 cc `;
SUBST_ALL_TAC (MESON[]` nn = 5 <=> 5 = nn `);
ASM_REWRITE_TAC[];
DISCH_THEN (ASSUME_TAC o GSYM);


ASM_REWRITE_TAC[];
ANTS_TAC;
ASSUME_TAC (ARITH_RULE` ~( 5 = 0) `);
ASM_SIMP_TAC[MOD_IN_NUMSEG];
STRIP_TAC;
SUBGOAL_THEN` ! k. 0 < k /\ k < 5 ==> ~ cc_qy_v11 cc ( i + k ) ` ASSUME_TAC;
GEN_TAC THEN STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC` (i + k) MOD 5 `));
ANTS_TAC;
ASSUME_TAC (ARITH_RULE` ~( 5 = 0) `);
ASM_SIMP_TAC[MOD_IN_NUMSEG];
MATCH_MP_TAC (GSYM MOD_INJ11);
ASM_REWRITE_TAC[ARITH_RULE` ~( 0 = k ) <=> 0 < k `];
ASM_SIMP_TAC[];



ARITH_TAC;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` ! i. -- cc_eps <= cc_gg_v11 cc i ` ASSUME_TAC;
GEN_TAC;
ASM_CASES_TAC` cc_qy_v11 cc i'' ` ;
DOWN;
ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `] NHANH;
REWRITE_TAC[Oxl_def.cc_eps];
REAL_ARITH_TAC;
ASM_CASES_TAC` cc_qx_v11 cc i'' ` ;
DOWN;
ASM_SEARCH_TCL [` cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i `] NHANH;
REWRITE_TAC[Oxl_def.cc_eps];
REAL_ARITH_TAC;
MP_TAC (SPEC` i'':num ` QU_OR_QXY);
ASM_REWRITE_TAC[];












SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qu_v11 cc (i + 1 ) ` ;
ASM_SEARCH_TCL [`cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc i`;` cc_eps <= x`] (MP_TAC o (SPEC ` i + 1 `));
ANTS_TAC;
ASM_REWRITE_TAC[];

FIRST_X_ASSUM (MP_TAC o (SPEC ` i + 2 `));
PHA;
MATCH_MP_TAC (REAL_ARITH` &0 <= x ==> -- a <= z /\ a <= y ==> &0 <= x + y + z `);
UNDISCH_TAC` cc_qy_v11 cc i `;
ASM_REWRITE_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 1 ) `;



ASM_CASES_TAC` ~ cc_small_v11 cc ( i + 2 ) ` ;
SUBGOAL_THEN` ~ cc_qu_v11 cc ( i + 1) /\ ~ cc_qu_v11 cc ( i + 2) ` ASSUME_TAC;
DOWN;
SIMP_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (i + 1 ) + 1 = i + 2 `];
MP_TAC (SPEC` i + 1 ` QU_OR_QXY);
MP_TAC (SPEC` i + 2 ` QU_OR_QXY);
ASM_REWRITE_TAC[];

DISCH_TAC;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) ` ASSUME_TAC;
UNDISCH_TAC` !i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i `;
UNDISCH_TAC` !i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `;
DOWN;
MESON_TAC[];
SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 1) ` MP_TAC;
UNDISCH_TAC` cc_qx_v11 cc (i + 1) `;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i ` MP_TAC;
UNDISCH_TAC` cc_qy_v11 cc i `;
ASM_REWRITE_TAC[];
DOWN;
REAL_ARITH_TAC;

ASM_CASES_TAC` ~ cc_small_v11 cc (i + 1) `;
ASM_SEARCH_TCL [` cc_qx_v11 cc i /\ cc_small_v11 cc (i + 1) /\ ~cc_small_v11 cc i
       ==> cc_eps <= cc_gg_v11 cc i`] (MP_TAC o (SPEC` i + 1 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` ( i + 1) + 1 = i + 2 `];
UNDISCH_TAC` ~ ~cc_small_v11 cc (i + 2) `;
SIMP_TAC[];


FIRST_X_ASSUM (MP_TAC o (SPEC ` i + 2 `));
SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i ` MP_TAC;
ASM_SIMP_TAC[];
REAL_ARITH_TAC;

ASM_SEARCH_TCL [` !i. cc_qx_v11 cc (i + 1) /\ cc_hassmall_v11 cc (i + 1) /\ cc_qy_v11 cc i
       ==> cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `] (MP_TAC o (SPEC ` i :num ` ));
ANTS_TAC;
DOWN THEN DOWN;
ASM_SIMP_TAC[cc_hassmall_v11; ARITH_RULE` (i + 1) + 1 = i + 2 `];
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2) ` MP_TAC;
ASM_REWRITE_TAC[];
REAL_ARITH_TAC;


MP_TAC (SPEC` i + 1 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
SUBGOAL_THEN` ~ cc_qy_v11 cc (i + 1) ` MP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
ARITH_TAC;
SIMP_TAC[];

SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i + cc_gg_v11 cc ( i + 3) + cc_gg_v11 cc ( i + 4 ) ` ASSUME_TAC;
ASM_CASES_TAC` ~ cc_small_v11 cc ( i + 4 ) ` ;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3) /\ &0 <= cc_gg_v11 cc (i + 4 ) ` ASSUME_TAC;
SUBGOAL_THEN` ~cc_qu_v11 cc (i + 3) /\ ~cc_qu_v11 cc ( i + 4) ` MP_TAC;
DOWN;
SIMP_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (i + 3 ) + 1 = i + 4 `];
STRIP_TAC;
MP_TAC (SPEC` i + 3 ` QU_OR_QXY);
MP_TAC (SPEC` i + 4 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
UNDISCH_TAC` !i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i `;
UNDISCH_TAC` !i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `;
MESON_TAC[];
SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i ` MP_TAC;
ASM_SIMP_TAC[];
DOWN;
REAL_ARITH_TAC;
DOWN;
REWRITE_TAC[];
SUBGOAL_THEN` ~cc_small_eta_v11 cc ((i + 5 ) MOD 5 ) ` ASSUME_TAC;
MP_TAC (
REWRITE_RULE[RIGHT_FORALL_IMP_THM] (
ISPECL [` cc_small_eta_v11 cc `; ` 5 `] Oxl_def.periodic_mod));
ANTS_TAC;
ASM_REWRITE_TAC[];
ABBREV_TAC` nn = cc_card_v11 cc `;
SUBST_ALL_TAC (MESON[]` nn = 5 <=> 5 = nn `);
ASM_REWRITE_TAC[];

DISCH_THEN (ASSUME_TAC o GSYM);
ONCE_REWRITE_TAC[ARITH_RULE` i + 5 = 1 * 5 + i `];
REWRITE_TAC[MOD_MULT_ADD];
ASM_REWRITE_TAC[];
STRIP_TAC;
MP_TAC (
REWRITE_RULE[RIGHT_FORALL_IMP_THM] (
ISPECL [` cc_small_eta_v11 cc `; ` 5 `] Oxl_def.periodic_mod));
ANTS_TAC;
ASM_REWRITE_TAC[];
ABBREV_TAC` nn = cc_card_v11 cc `;
SUBST_ALL_TAC (MESON[]` nn = 5 <=> 5 = nn `);
ASM_REWRITE_TAC[];
DISCH_THEN (ASSUME_TAC o GSYM);

UNDISCH_TAC` ~cc_small_eta_v11 cc ((i + 5) MOD 5) `;
ASM_REWRITE_TAC[];
STRIP_TAC;
ASM_CASES_TAC` cc_qu_v11 cc (i + 4 ) ` ;
UNDISCH_TAC` !i. cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc (i + 1)
       ==> cc_eps <= cc_gg_v11 cc i `;
DISCH_THEN (MP_TAC o (SPEC` i + 4 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 4 ) + 1 = i + 5 `];
SUBGOAL_THEN` --cc_eps <= cc_gg_v11 cc (i + 3) ` MP_TAC;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i ` MP_TAC;
ASM_SIMP_TAC[];
REAL_ARITH_TAC;

ASM_CASES_TAC` cc_qx_v11 cc ( i + 4) `;
ASM_CASES_TAC`~ cc_small_v11 cc ( i + 5 ) `;
UNDISCH_TAC` !i. cc_qx_v11 cc i /\ cc_small_v11 cc i /\ ~cc_small_v11 cc (i + 1)
       ==> cc_eps <= cc_gg_v11 cc i `;
DISCH_THEN (MP_TAC o (SPEC` i + 4 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 4 ) + 1 = i + 5 ` ];
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc ( i + 3 ) ` MP_TAC;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i ` MP_TAC;
ASM_SIMP_TAC[];
REAL_ARITH_TAC;

UNDISCH_TAC` !i. cc_qx_v11 cc i /\ cc_hassmall_v11 cc i /\ cc_qy_v11 cc (i + 1)
       ==> cc_eps <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `;
DISCH_THEN (MP_TAC o (SPEC` i + 4 `));
ANTS_TAC;
DOWN;
ASM_SIMP_TAC[cc_hassmall_v11; ARITH_RULE` (i + 4) + 1 = i + 5 `];
STRIP_TAC;
SUBGOAL_THEN` cc_qy_v11 cc ((i + 5) MOD 5 ) ` MP_TAC;
ONCE_REWRITE_TAC[ARITH_RULE` i + 5 = 1 * 5 + i `];
REWRITE_TAC[MOD_MULT_ADD];
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[];

ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `];
UNDISCH_TAC` periodic (cc_gg3a_v11 cc) (cc_card_v11 cc) `;
ASM_REWRITE_TAC[];
STRIP_TAC;
MP_TAC (let tt = REWRITE_RULE[RIGHT_FORALL_IMP_THM] Oxl_def.periodic_mod in
ISPECL [` cc_gg3a_v11 cc `; ` 5 `] tt);
ANTS_TAC;
ASM_REWRITE_TAC[];
ARITH_TAC;
DISCH_THEN (ASSUME_TAC o GSYM);

SUBGOAL_THEN` cc_gg3a_v11 cc ((i + 5) MOD 5) = cc_gg3a_v11 cc (i) ` MP_TAC;
ONCE_REWRITE_TAC[ARITH_RULE` i + 5 = 1 * 5 + i `];
REWRITE_TAC[MOD_MULT_ADD];
ASM_REWRITE_TAC[];
ASM_SIMP_TAC[];

STRIP_TAC;
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 3 ) ` MP_TAC;
ASM_REWRITE_TAC[];
REAL_ARITH_TAC;
SUBGOAL_THEN` ~ cc_qy_v11 cc (( i + 4 ) MOD 5) ` ASSUME_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
ASSUME_TAC (ARITH_RULE` ~( 5 = 0) `);
ASM_SIMP_TAC[MOD_IN_NUMSEG];
MATCH_MP_TAC (GSYM MOD_INJ11);
ARITH_TAC;
MP_TAC (SPEC` i + 4 ` QU_OR_QXY);
DOWN;
ASM_REWRITE_TAC[];
SIMP_TAC[];
ASM_SEARCH_TCL [` cc_gg3a_v11 cc i + x <= y`] (ASSUME_TAC2 o SPEC_ALL);
MP_TAC (ARITH_RULE` ~( 5 = 0) `);
UNDISCH_TAC` periodic (cc_gg_v11 cc ) (cc_card_v11 cc) `;
ASM_REWRITE_TAC[];
PHA;
NHANH Oxl_def.periodic_sum;
STRIP_TAC;
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
ASM_SIMP_TAC[];
ONCE_REWRITE_TAC[ADD_SYM];
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` 5 - 1 = 4 /\ i <= i + 4 /\ i + 1 <= i + 4 /\
i + 2 <= i + 4 /\ i + 3 <= i + 4 /\ i + 4 <= i + 4 /\ (i + 1 ) + 1 = i + 2 `];
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` (i + 1) + 2 <= i + 4 /\ ((i + 1) + 2) + 1 = i + 4`];
REWRITE_TAC[SUM_SING_NUMSEG; ARITH_RULE` (i + 1) + 2 = i + 3 `];
UNDISCH_TAC` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i `;
UNDISCH_TAC` &0 <= cc_gg3a_v11 cc i + cc_gg_v11 cc (i + 3) + cc_gg_v11 cc (i + 4) `;
UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) `;
REAL_ARITH_TAC]);;







let PRE_IPVICGW = 
prove_by_refinement (
` (cc_bool_model_v11 cc /\
      cc_bool_prep_v11 cc /\
      cc_real_model_v11 cc /\
      sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\
     cc_card_v11 cc = 5 
==> &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) `,

[NHANH EXISTS_QY_CARD5;
STRIP_TAC;
ASM_CASES_TAC` cc_azim_v11 cc i < #1.074 `;
ASSUME_TAC2 CARD5_LT_1074_QYI;
ASM_REWRITE_TAC[];

MATCH_MP_TAC CARD5_1074_LE_QYI;
ASM_REWRITE_TAC[REAL_ARITH` x <= y <=> ~( y < x ) `];
ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00;
ASSUME_TAC2 EXISTS_QY_CARD5;
DOWN_TAC;
NHANH Oxl_def.periodic_fn;
REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `];
IMP_TAC;
IMP_TAC;
STRIP_TAC;
STRIP_TAC;
STRIP_TAC;

ABBREV_TAC` nn = cc_card_v11 cc `;
ASM_REWRITE_TAC[REAL_ARITH` ~( a < b) <=> b <= a `];
DOWN;
SUBST_ALL_TAC (MESON[]` nn = 5 <=> 5 = nn `);
ASM_REWRITE_TAC[]]);;



let CC_CARD_LESS_THAN_5 = 
prove_by_refinement(` cc_bool_model_v11 cc /\
            cc_bool_prep_v11 cc /\
            cc_real_model_v11 cc /\
            sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0
            ==> cc_card_v11 cc <= 5 `,



[NHANH LXDEYBO;
NHANH UNPNFVW;
REWRITE_TAC[cc_size_v11; cc_qy_v11];
MP_TAC (SET_RULE` DISJOINT {i | i IN 0..cc_card_v11 cc - 1 /\ ~cc_4cell_v11 cc i} {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} `);
MP_TAC (SET_RULE` {i | i IN 0..cc_card_v11 cc - 1 /\ ~cc_4cell_v11 cc i} UNION {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} = 0..cc_card_v11 cc - 1 `);
REPEAT STRIP_TAC;
ABBREV_TAC` s1 = {i | i IN 0..cc_card_v11 cc - 1 /\ ~cc_4cell_v11 cc i} `;
ABBREV_TAC` s2 = {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i}`;
SUBGOAL_THEN` FINITE (s1:num -> bool) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` 0.. cc_card_v11 cc - 1 `;
REWRITE_TAC[FINITE_NUMSEG];
EXPAND_TAC "s1";
SET_TAC[];
SUBGOAL_THEN` FINITE (s2:num -> bool) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` 0.. cc_card_v11 cc - 1 `;
REWRITE_TAC[FINITE_NUMSEG];
EXPAND_TAC "s2";
SET_TAC[];

MP_TAC (ISPECL [` s1:num -> bool `; ` s2: num -> bool `] CARD_UNION);
ANTS_TAC;
ASM_REWRITE_TAC[GSYM DISJOINT];



ASM_REWRITE_TAC[CARD_NUMSEG];
ASSUME_TAC2 (prove(` cc_bool_model_v11 cc ==> ~( cc_card_v11 cc = 0) `, SIMP_TAC[cc_bool_model_v11]));
ASM_SEARCH_TCL [` x <= 1 `] MP_TAC;
ASM_SEARCH_TCL [` x <= 4 `] MP_TAC;
ARITH_TAC]);;





let IPVICGW = 
prove_by_refinement(`!cc. cc_bool_model_v11 cc /\
        cc_bool_prep_v11 cc /\
        cc_real_model_v11 cc /\
        sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0
        ==> (!i. cc_small_v11 cc i)`,

[NHANH CC_CARD_LESS_THAN_5 ; NHANH THREE_LE_CC_CARD;
PHA; REWRITE_TAC[ARITH_RULE` 3 <= x /\ x <= 5 <=> x = 3 \/ x = 4 \/ x = 5 `];
MESON_TAC[IPVICGW_C43_C44; IPVICGW_C33; PRE_IPVICGW; REAL_ARITH` a < b ==> ~( b <= a )`]]);;




let RSIWAMP = 
prove_by_refinement (
`!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ 
   (sum (0..cc_card_v11 cc -1) (cc_gg_v11 cc) < &0) ==> (cc_card_v11 cc <= 4)`,

[NHANH CC_CARD_LESS_THAN_5;
REWRITE_TAC[ARITH_RULE` a <= 5 <=> a <= 4 \/ a = 5 `];
GEN_TAC THEN STRIP_TAC;
ASSUME_TAC2 PRE_IPVICGW;
DOWN;
ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0 ) `]]);;







let C5_QY_ALONG_NOT_SMALL1 = 
prove_by_refinement
(` (!i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i) /\
(!i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i) /\
(!i. cc_qy_v11 cc i
          ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i) /\
     (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3a_v11 cc i) /\
     (!i. cqu cc (i + 1) /\ cc_qy_v11 cc i
          ==> cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)) /\
   (!i. cc_qx_v11 cc i /\ cc_small_v11 cc i /\ ~cc_small_v11 cc (i + 1)
          ==> cc_eps <= cc_gg_v11 cc i) /\
( !i. cqu cc i ==> --cc_eps <= cc_gg_v11 cc i ) /\
~ cc_small_v11 cc i /\ ~cc_small_v11 cc (i + 5) /\
          ~cc_small_v11 cc i /\
          cc_qy_v11 cc i /\
          cqu cc (i + 1) /\ cc_qx_v11 cc (i + 4) /\
( !cc i.
         cqu cc i <=>
         cc_hassmall_v11 cc i /\ cc_4cell_v11 cc i /\ cc_subcrit_v11 cc i ) /\
( !cc i.
         cc_hassmall_v11 cc i <=>
         cc_small_v11 cc i /\ cc_small_v11 cc (i + 1) ) /\

bb = (cqu cc (i + 2) \/ cc_qx_v11 cc (i + 2) \/ cc_qy_v11 cc (i + 2)) /\
bb /\
(! i.  cqu cc i \/ cc_qx_v11 cc i \/ cc_qy_v11 cc i ) /\
(((!i. cc_qy_v11 cc i
       ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i) /\
  (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3a_v11 cc i) /\
  (!i. cqu cc (i + 1) /\ cc_qy_v11 cc i
       ==> cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1))
  ==> cqu cc (i + 1) /\ cc_qy_v11 cc i
  ==> cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1)))
==> &0 <= sum (i..i + 4) (cc_gg_v11 cc) `,

[ASM_SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `];
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG];
STRIP_TAC;
FIRST_X_ASSUM MP_TAC;
ANTS_TAC;
ASM_REWRITE_TAC[];
ANTS_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 2) ` ;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 2) ` MP_TAC;
ASM_SIMP_TAC[];
REWRITE_TAC[Oxl_def.cc_eps];
REAL_ARITH_TAC;
ASM_CASES_TAC` cc_qx_v11 cc (i + 2) ` ;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 2) ` MP_TAC;
ASM_SIMP_TAC[];
REWRITE_TAC[Oxl_def.cc_eps];
REAL_ARITH_TAC;
UNDISCH_TAC` bb: bool `;

FIRST_X_ASSUM SUBST1_TAC;
REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN SIMP_TAC[];
STRIP_TAC THEN STRIP_TAC;
ASM_REWRITE_TAC[];

SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3) + cc_gg_v11 cc (i + 4) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 3) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 3) ` ;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
FIRST_ASSUM (MP_TAC o (SPEC` i + 3 `));
REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN SIMP_TAC[];
FIRST_X_ASSUM NHANH;
ASM_REWRITE_TAC[cc_hassmall_v11; ARITH_RULE` (i + 3) + 1 = i + 4 `];
REPEAT STRIP_TAC;
SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc (i + 4) ` MP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `];
DOWN;
REAL_ARITH_TAC;
DOWNS 3;
REAL_ARITH_TAC]);;






let C5_QY_ALONG_NOT_SMALL2 = 
prove_by_refinement
(` (!i. (cqx: cc_v11 -> num -> bool) cc i ==> &0 <= (ggg: cc_v11 -> num -> real) cc i) /\
     (!i. cqy cc i ==> &0 <= (ggg: cc_v11 -> num -> real) cc i) /\
     (!i. cqy cc i ==> gga cc i + ggb cc i <= (ggg: cc_v11 -> num -> real) cc i) /\
     (!i. cqy cc i ==> &0 <= gga cc i) /\
     (!i. cqu cc (i + 1) /\ cqy cc i
          ==> cc_eps <= ggb cc i + (ggg: cc_v11 -> num -> real) cc (i + 1)) /\
     (!i. (cqx: cc_v11 -> num -> bool) cc i /\ csmall cc i /\ ~csmall cc (i + 1)
          ==> cc_eps <= (ggg: cc_v11 -> num -> real) cc i) /\
     (!i. cqu cc i ==> --cc_eps <= (ggg: cc_v11 -> num -> real) cc i) /\
     ~csmall cc i /\
     ~csmall cc (i + 5) /\
     ~csmall cc i /\
     cqy cc i /\
     cqu cc (i + 1) /\
     (cqx: cc_v11 -> num -> bool) cc (i + 4) /\
     (!cc i.
          cqu cc i <=>
          hasmal cc i /\ fcell cc i /\ subcrt cc i) /\
     (!cc i.
          hasmal cc i <=>
          csmall cc i /\ csmall cc (i + 1)) /\
     (bb <=> cqu cc (i + 2) \/ (cqx: cc_v11 -> num -> bool) cc (i + 2) \/ cqy cc (i + 2)) /\
     bb /\
     (!i. cqu cc i \/ (cqx: cc_v11 -> num -> bool) cc i \/ cqy cc i) /\
     ((!i. cqy cc i ==> gga cc i + ggb cc i <= (ggg: cc_v11 -> num -> real) cc i) /\
      (!i. cqy cc i ==> &0 <= gga cc i) /\
      (!i. cqu cc (i + 1) /\ cqy cc i
           ==> cc_eps <= ggb cc i + (ggg: cc_v11 -> num -> real) cc (i + 1))
      ==> cqu cc (i + 1) /\ cqy cc i
      ==> cc_eps <= (ggg: cc_v11 -> num -> real) cc i + (ggg: cc_v11 -> num -> real) cc (i + 1))
     ==> &0 <= sum (i..i + 4) ((ggg: cc_v11 -> num -> real) cc) `,

[ASM_SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `];
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG];
STRIP_TAC;
FIRST_X_ASSUM MP_TAC;
ANTS_TAC;
ASM_REWRITE_TAC[];
ANTS_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN` -- cc_eps <= (ggg: cc_v11 -> num -> real) cc (i + 2) ` ASSUME_TAC;
ASM_CASES_TAC` (cqy: cc_v11 -> num -> bool) cc (i + 2) ` ;
SUBGOAL_THEN` &0 <= (ggg: cc_v11 -> num -> real) cc ( i + 2) ` MP_TAC;
ASM_SIMP_TAC[];
REWRITE_TAC[Oxl_def.cc_eps];
REAL_ARITH_TAC;
ASM_CASES_TAC` (cqx: cc_v11 -> num -> bool) cc (i + 2) ` ;
SUBGOAL_THEN` &0 <= (ggg: cc_v11 -> num -> real) cc ( i + 2) ` MP_TAC;
ASM_SIMP_TAC[];
REWRITE_TAC[Oxl_def.cc_eps];
REAL_ARITH_TAC;
UNDISCH_TAC` bb: bool `;

FIRST_X_ASSUM SUBST1_TAC;
REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN SIMP_TAC[];
STRIP_TAC THEN STRIP_TAC;
ASM_REWRITE_TAC[];

SUBGOAL_THEN` &0 <= (ggg: cc_v11 -> num -> real) cc (i + 3) + (ggg: cc_v11 -> num -> real) cc (i + 4) ` ASSUME_TAC;
ASM_CASES_TAC` (cqy: cc_v11 -> num -> bool) cc (i + 3) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` (cqx: cc_v11 -> num -> bool) cc (i + 3) ` ;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
FIRST_ASSUM (MP_TAC o (SPEC` i + 3 `));
REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN SIMP_TAC[];
FIRST_X_ASSUM NHANH;
ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];
REPEAT STRIP_TAC;
SUBGOAL_THEN` cc_eps <= (ggg: cc_v11 -> num -> real) cc (i + 4) ` MP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `];
DOWN;
REAL_ARITH_TAC;
DOWNS 3;
REAL_ARITH_TAC]);;







let LITTLE_CC_GGAA = 
prove(` (!i. cc_qy_v11 cc i
          ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ) /\
 (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i) /\
(!i. cc_qx_v11 cc i /\ cc_hassmall_v11 cc i /\ cc_qy_v11 cc (i + 1)
          ==> cc_eps <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1)) 
==> cc_qx_v11 cc (i) /\ cc_hassmall_v11 cc i /\ cc_qy_v11 cc (i + 1)
==> cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) `,

STRIP_TAC THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN 
FIRST_X_ASSUM NHANH THEN REAL_ARITH_TAC);;


let LITTLE_CC_GGBB = 
prove(` (!i. cc_qy_v11 cc i
          ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ) /\
 (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3a_v11 cc i) /\
(!i. cc_qx_v11 cc (i + 1) /\ cc_hassmall_v11 cc (i + 1) /\ cc_qy_v11 cc i
          ==> cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)) 
==> cc_qx_v11 cc (i + 1) /\ cc_hassmall_v11 cc (i + 1) /\ cc_qy_v11 cc i
==> cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) `,

STRIP_TAC THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN 
FIRST_X_ASSUM NHANH THEN REAL_ARITH_TAC);;









let C5_QX_NEXT_TO_QY = 
prove_by_refinement(`((cc_bool_model_v11 cc /\
       cc_bool_prep_v11 cc /\
       cc_real_model_v11 cc /\
       sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\
      cc_card_v11 cc = 5) /\
     ~cc_small_v11 cc i /\
     cc_qy_v11 cc (i) /\
     cc_qx_v11 cc (i + 1)
     ==> F `,


[
REPEAT STRIP_TAC;
ASSUME_TAC2 UNIQUE_QY;
ASSUME_TAC2 QY_NN00; ASSUME_TAC2 QX_NN00;
ASSUME_TAC2 EXISTS_QY_CARD5;
DOWN_TAC;
NHANH Oxl_def.periodic_fn;

REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `];
IMP_TAC;
IMP_TAC;
STRIP_TAC;
STRIP_TAC;
STRIP_TAC;

ABBREV_TAC` nn = cc_card_v11 cc ` ;
SUBGOAL_THEN` ~ cc_small_v11 cc (i + nn) ` ASSUME_TAC;
ASM_SEARCH_TCL [` periodic ( cc_small_v11 cc) `] MP_TAC;
SIMP_TAC[Oxl_def.periodic];
ASM_REWRITE_TAC[];
SUBGOAL_THEN` ~ cc_qu_v11 cc (i + 4 ) ` ASSUME_TAC;
SWITCH_TAC` nn = 5 `;
ASM_REWRITE_TAC[cc_qu_v11; cc_hassmall_v11 ; ARITH_RULE` (i + 4) + 1 = i + 5 `];
ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn:num` ;` i:num `] Oxl_def.periodic_mod);
ASSUME_TAC2 (ISPECL [` cc_qx_v11 cc `;` nn:num` ;` i + 4 `] Oxl_def.periodic_mod);
ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn:num` ;` i + 4 `] Oxl_def.periodic_mod);
SUBGOAL_THEN` ~ cc_qy_v11 cc (i + 4) ` ASSUME_TAC;
DOWN THEN DOWN THEN DOWN;
REPLICATE_TAC 3 (DISCH_THEN (ASSUME_TAC o SYM));
FIRST_ASSUM (SUBST1_TAC o SYM);
SUBGOAL_THEN` cc_qy_v11 cc (i MOD nn) /\ (i MOD nn) IN 0..nn - 1` MP_TAC;
SWITCH_TAC` nn = 5 `;
ASM_SIMP_TAC[MOD_IN_NUMSEG];
ASM_SEARCH_TCL [` cc_qy_v11 cc x /\ x IN 0..nn - 1 `] NHANH;
STRIP_TAC;

FIRST_X_ASSUM MATCH_MP_TAC;
SWITCH_TAC` nn = 5 `;
ASM_SIMP_TAC[MOD_IN_NUMSEG];
MP_TAC (SPECL [` 4 `;` nn:num `] (GEN_ALL MOD_INJ1));
ANTS_TAC;
ASM_REWRITE_TAC[];
EXPAND_TAC "nn";
ARITH_TAC;
SIMP_TAC[];
MP_TAC (SPEC` i + 4 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
STRIP_TAC;
REPLICATE_TAC 5 DOWN;
REPLICATE_TAC 3 (DISCH_THEN (ASSUME_TAC o SYM));
STRIP_TAC;
SWITCH_TAC` nn = 5 `;
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2 ) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 2) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 2) ` ;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 2 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
ASM_SEARCH_TCL [` -- cc_eps <= x `] NHANH;
NHANH QU_IMP_SMALL;
STRIP_TAC;
ASM_CASES_TAC` cc_small_v11 cc (i + 1) `;
SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) ` ASSUME_TAC;
MP_TAC LITTLE_CC_GGBB;
PHA;
ANTS_TAC;
ASM_REWRITE_TAC[cc_hassmall_v11; ARITH_RULE` (i + 1) + 1 = i + 2 `];
SIMP_TAC[];
DOWNS 3;
REAL_ARITH_TAC;
SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc (i + 1) ` ASSUME_TAC;
ASM_SEARCH_TCL [` a /\ b (i + 1) /\ ~ c ==> cc_eps <= x `;` cc_small_v11 `] MATCH_MP_TAC;
ASM_SIMP_TAC[ARITH_RULE` (i + 1 ) + 1 = i + 2 `];
SUBGOAL_THEN` &0 <= cc_gg_v11 cc i ` MP_TAC;
ASM_SIMP_TAC[];
DOWNS 3;
REAL_ARITH_TAC;





SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3) + cc_gg_v11 cc (i + 4 ) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 3) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 3) ` ;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 3 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
NHANH QU_IMP_SMALL;
REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];
STRIP_TAC;
ASM_SEARCH_TCL [` a /\ b /\ ~ c (i + 1) ==> cc_eps <= x `] (MP_TAC o (SPEC` i + 4 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `];
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 3) ` MP_TAC;
ASM_SIMP_TAC[];
REAL_ARITH_TAC;


SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num ` ] Oxl_def.periodic_sum);
FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC_ALL));
EXPAND_TAC "nn";
SIMP_TAC[ARITH_RULE` 5 - 1 + i = i + 4 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `];
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG]; DOWNS 2;
REAL_ARITH_TAC;
ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x) <=> x < &0 `]]);;






let LEAST_BOUND_CC_GG = 
prove(` (!i. cc_qu_v11 cc i ==> --cc_eps <= cc_gg_v11 cc i) /\
  (!i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i) /\
 (!i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i) 
==> (! i. -- cc_eps <= cc_gg_v11 cc i) `,

SUBGOAL_THEN` -- cc_eps <= &0 ` MP_TAC THENL [
REWRITE_TAC[Oxl_def.cc_eps] THEN 
REAL_ARITH_TAC; MESON_TAC[REAL_LE_TRANS; QU_OR_QXY]]);;







let C5_QX_NEXT_TO_QY_LEFT = 
prove_by_refinement(` ((cc_bool_model_v11 cc /\
       cc_bool_prep_v11 cc /\
       cc_real_model_v11 cc /\
       sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\
      cc_card_v11 cc = 5) /\
     ~cc_small_v11 cc i /\
     cc_qx_v11 cc i /\
     cc_qy_v11 cc (i + 4) /\
cc_qx_v11 cc ( i + 3 )
     ==> F `,

[
REPEAT STRIP_TAC;
ASSUME_TAC2 UNIQUE_QY;
ASSUME_TAC2 QY_NN00; ASSUME_TAC2 QX_NN00;
ASSUME_TAC2 EXISTS_QY_CARD5;
DOWN_TAC;
NHANH Oxl_def.periodic_fn;

REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `];
IMP_TAC;
IMP_TAC;
STRIP_TAC;
STRIP_TAC;
STRIP_TAC;

ABBREV_TAC` nn = cc_card_v11 cc ` ;
SUBGOAL_THEN` ~ cc_small_v11 cc (i + nn) ` ASSUME_TAC;
ASM_SEARCH_TCL [` periodic ( cc_small_v11 cc) `] MP_TAC;
SIMP_TAC[Oxl_def.periodic];
ASM_REWRITE_TAC[];

SWITCH_TAC` nn = 5 `;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3) + cc_gg_v11 cc (i + 4) ` ASSUME_TAC;


ASM_CASES_TAC` cc_qy_v11 cc ( i + 2 ) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc ( i + 2 ) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 2 ` QU_OR_QXY);
ASM_SIMP_TAC[];
ASM_SEARCH_TCL [` cc_qu_v11 cc i ==> -- x <= y `] NHANH;
NHANH QU_IMP_SMALL;
REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `];
STRIP_TAC;
ASM_CASES_TAC` cc_small_v11 cc (i + 4) `;
MP_TAC (SPEC` i + 3 ` (GEN` i:num ` LITTLE_CC_GGAA));
PHA;
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `; cc_hassmall_v11];

ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];
ASSUME_TAC2 LEAST_BOUND_CC_GG;
FIRST_X_ASSUM (MP_TAC o (SPEC` i + 2 `));
REAL_ARITH_TAC;
ASM_SEARCH_TCL [` a /\ b /\ ~ c (i + 1)  ==> cc_eps <= y `] (MP_TAC o (SPEC` i + 3 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];
SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 4) ` MP_TAC;
ASM_SIMP_TAC[];
UNDISCH_TAC` --cc_eps <= cc_gg_v11 cc (i + 2) ` ;
REAL_ARITH_TAC;

SUBGOAL_THEN` &0 <= cc_gg_v11 cc i + cc_gg_v11 cc ( i + 1) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 1) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `;
MATCH_MP_TAC REAL_LE_ADD;
ASM_SIMP_TAC[];
MP_TAC (SPEC` i + 1 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
NHANH QU_IMP_SMALL;
STRIP_TAC;
ASM_SEARCH_TCL [` a /\ b (i + 1) /\ ~ c (i)  ==> cc_eps <= y `] (ASSUME_TAC2 o (SPEC` i:num `));
ASSUME_TAC2 LEAST_BOUND_CC_GG;
FIRST_X_ASSUM (MP_TAC o (SPEC` i + 1 `));
DOWN;
REAL_ARITH_TAC;


SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num ` ] Oxl_def.periodic_sum);
FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC_ALL));
EXPAND_TAC "nn";
SIMP_TAC[ARITH_RULE` 5 - 1 + i = i + 4 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `];
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG];



 DOWNS 2;
REAL_ARITH_TAC;
ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x) <=> x < &0 `]]);;






let BOOL_MODEL_PERIODIC_SMALL = 
prove(` cc_bool_model_v11 cc ==> 
periodic (cc_small_v11 cc ) (cc_card_v11 cc) `,

SIMP_TAC[cc_bool_model_v11]);;





let QY_UNIQUE_ANY_SEGMENGT = 
prove_by_refinement
(` cc_bool_model_v11 cc /\
            cc_bool_prep_v11 cc /\
            cc_real_model_v11 cc /\
            sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0
            ==> (!j. cc_qy_v11 cc j 
                     ==> (!k. ~(k = j) /\ k IN j..cc_card_v11 cc - 1 + j
                              ==> ~cc_qy_v11 cc k)) `,

[NHANH UNIQUE_QY;
NHANH Oxl_def.periodic_fn;
REWRITE_TAC[cc_bool_model_v11];
REPEAT STRIP_TAC;
ABBREV_TAC` nn = cc_card_v11 cc `;
FIRST_X_ASSUM (MP_TAC o (SPEC` j MOD nn`));
ANTS_TAC;
ASM_SIMP_TAC[MOD_IN_NUMSEG];
ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn:num `;`j:num `] Oxl_def.periodic_mod);
FIRST_X_ASSUM (SUBST1_TAC o SYM);
FIRST_ASSUM ACCEPT_TAC;
MP_TAC (SPECL [` k - (j:num) `;` nn:num `] (GEN_ALL MOD_INJ1));
ANTS_TAC;
UNDISCH_TAC` k IN j..nn - 1 + j `;
ASM_REWRITE_TAC[IN_NUMSEG; ARITH_RULE` j <= (k:num) <=> j = k \/ j < k `];
UNDISCH_TAC` ~( nn = 0) `;
ARITH_TAC;
UNDISCH_TAC` k IN j..nn - 1 + j `;
ASM_REWRITE_TAC[IN_NUMSEG];
STRIP_TAC;
DISCH_THEN (MP_TAC o (SPEC` j:num `));
ASM_SIMP_TAC[ARITH_RULE` j <= k ==> j + k - j = (k:num) `];
STRIP_TAC;
STRIP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC` k MOD nn `));
ANTS_TAC;
ASM_SIMP_TAC[GSYM IN_NUMSEG; MOD_IN_NUMSEG];
ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn:num `;`k:num `] Oxl_def.periodic_mod);
FIRST_X_ASSUM (SUBST1_TAC o SYM);
ASM_REWRITE_TAC[]]);;






let periodic_mod1 = 
REWRITE_RULE[RIGHT_FORALL_IMP_THM] periodic_mod;;





let LITTLE_CC_GG3BB = 
prove(`    (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i) /\
       (!i. cc_qu_v11 cc (i + 1) /\ cc_qy_v11 cc i
            ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)) /\
(! i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i ) /\
(! i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i ) /\
       cc_qy_v11 cc i
       ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `,

MP_TAC (SPEC` i + 1 ` QU_OR_QXY) THEN 
MESON_TAC[REAL_ARITH` &0 <= a /\ &0 <= b ==> &0 <= a + b `]);;



let MOD_ADD_CANCEL = 
REWRITE_RULE[ARITH_RULE` 1 * n + p = p + n ` ] (
SPEC` 1 ` MOD_MULT_ADD);;



let LITTLE_CC_GG3AA = 
prove_by_refinement(
`    (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3a_v11 cc i) /\
      (!i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1)
                 ==> &0 <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1)) /\
(! i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i ) /\
(! i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i ) /\
periodic (cc_qy_v11 cc ) nn /\
periodic (cc_gg3a_v11 cc ) nn /\
~( nn = 0)
       /\ cc_qy_v11 cc i
       ==> &0 <= cc_gg_v11 cc ( i + nn - 1 ) + cc_gg3a_v11 cc (i) `,

[ASM_CASES_TAC` cc_qx_v11 cc (i + nn - 1) \/ cc_qy_v11 cc ( i + nn - 1 ) `;
DOWN;
MESON_TAC[REAL_ARITH` &0 <= a /\ &0 <= b ==> &0 <= a + b `];
MP_TAC (SPEC` i + nn - 1 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
ABBREV_TAC` nii = i + nn - 1 `;
STRIP_TAC;
STRIP_TAC;
UNDISCH_TAC` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1)
          ==> &0 <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `;
DISCH_THEN (MP_TAC o (SPEC` nii: num `));
ANTS_TAC;
ASM_REWRITE_TAC[];
ASSUME_TAC2 (ARITH_RULE ` ~( nn = 0) ==> (i + nn - 1 ) + 1 = i + nn `);
DOWN;
ASM_REWRITE_TAC[];
SIMP_TAC[];
ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn:num `] periodic_mod1);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
SUBGOAL_THEN` cc_qy_v11 cc ((i + nn) MOD nn) ` ASSUME_TAC;
REWRITE_TAC[MOD_ADD_CANCEL];
ASM_REWRITE_TAC[];
DOWN;
ASM_SIMP_TAC[];

SUBGOAL_THEN` nii + 1 = i + nn ` ASSUME_TAC;
EXPAND_TAC "nii";
UNDISCH_TAC` ~( nn = 0) `;
ARITH_TAC;
ASM_REWRITE_TAC[];

ASSUME_TAC2 (ISPECL [` cc_gg3a_v11 cc `; `nn:num `] periodic_mod1);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
SUBGOAL_THEN` cc_gg3a_v11 cc ((i + nn) MOD nn) = cc_gg3a_v11 cc i ` MP_TAC;
REWRITE_TAC[MOD_ADD_CANCEL];
ASM_REWRITE_TAC[];

ASM_SIMP_TAC[]]);;



let SUM_NUMSEG2 = 
SIMP_RULE[ARITH_RULE`m <= m + 1`; SUM_SING_NUMSEG] (SPEC` m + 1` (GEN` n:num ` (SPEC_ALL SUM_CLAUSES_LEFT)));;

let SUM_NUMSEG33 = 
SIMP_RULE[ARITH_RULE`m <= m + 2 /\ m + 2 = (m + 1 ) + 1`; SUM_NUMSEG2] (SPEC` m + 2` (GEN` n:num ` (SPEC_ALL SUM_CLAUSES_LEFT)));;


let SUM_NUMSEG3 =
REWRITE_RULE[ARITH_RULE` (a + 1) + 1 = a + 2 `]  SUM_NUMSEG33;;


let SUM_NUMSEG44 = 
SIMP_RULE[ARITH_RULE`m <= m + 3 /\ m + 3 = (m + 1 ) + 2 `; SUM_NUMSEG3] (SPEC` m + 3` (GEN` n:num ` (SPEC_ALL SUM_CLAUSES_LEFT)));;

let SUM_NUMSEG4 = 
REWRITE_RULE[ARITH_RULE` (m + 1) + 2 = m + 3 /\ (m + 1) + 1 = m + 2 `] SUM_NUMSEG44;;





let UTEOITF_concl = `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ 
   (sum (0..cc_card_v11 cc -1) (cc_gg_v11 cc) < &0) ==> (!i. cc_4cell_v11 cc i)`;;



let UTEOITF = prove_by_refinement(UTEOITF_concl,
[GEN_TAC;
NHANH RSIWAMP;
STRIP_TAC THEN ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00;
ASM_CASES_TAC` !i. cc_4cell_v11 cc i `;
ASM_REWRITE_TAC[];
DOWN;
REWRITE_TAC[NOT_FORALL_THM; GSYM cc_qy_v11];
STRIP_TAC;
ASSUME_TAC2 RSIWAMP;
ASSUME_TAC2 THREE_LE_CC_CARD;
ASSUME_TAC2 IPVICGW;
ASSUME_TAC2 UNIQUE_QY;
DOWN_TAC;
NHANH Oxl_def.periodic_fn;
REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
ABBREV_TAC` nn = cc_card_v11 cc `;
STRIP_TAC;



ASM_CASES_TAC` nn = 3 `;
SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i /\ &0 <= cc_gg3b_v11 cc i ` MP_TAC;
ASM_SIMP_TAC[];
STRIP_TAC;
SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qy_v11 cc (i + 1) `;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 1) ` MP_TAC;
ASM_SIMP_TAC[];
UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i `;
REAL_ARITH_TAC;
ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 1) ` MP_TAC;
ASM_SIMP_TAC[];
UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i `;
REAL_ARITH_TAC;
MP_TAC (SPEC` i + 1 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
STRIP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[];
MP_TAC (GSYM (
ISPECL [` cc_qy_v11 cc `;` nn:num `] periodic_mod1));
ANTS_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` nn = 3 `;
DISCH_THEN (ASSUME_TAC o SYM);
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN` cc_qy_v11 cc ((1 * 3 + i ) MOD 3) ` MP_TAC;
REWRITE_TAC[MOD_MULT_ADD];
ASM_REWRITE_TAC[];


MP_TAC LITTLE_CC_GG3AA;
ANTS_TAC;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` cc_gg3a_v11 cc i + cc_gg3b_v11 cc (i) <= cc_gg_v11 cc i ` ASSUME_TAC;
ASM_SIMP_TAC[];
STRIP_TAC THEN STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
ASSUME_TAC2 (
ISPECL [` cc_gg_v11 cc `;` nn:num `] periodic_sum);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);

ASM_REWRITE_TAC[];
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 3 - 1 = 2 /\ 2 + i = i + 2`; SUM_NUMSEG3];

UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + nn - 1) + cc_gg3a_v11 cc i `;
UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `;
UNDISCH_TAC` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i `;
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 3 - 1 = 2 `];
PHA;
REAL_ARITH_TAC;

ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0 ) `];
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] periodic_sum);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
DOWN;
FIRST_X_ASSUM (MP_TAC o (SPEC` i MOD nn`));
ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `; `nn:num `] periodic_mod1);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
ANTS_TAC;
ASM_SIMP_TAC[MOD_IN_NUMSEG];
STRIP_TAC;
FIRST_ASSUM (MP_TAC o (SPEC` (i + 1) MOD nn `));
ANTS_TAC;
ASM_SIMP_TAC[MOD_IN_NUMSEG];
MATCH_MP_TAC (GSYM MOD_INJ11);
ASM_REWRITE_TAC[];
UNDISCH_TAC` 3 <= nn `;
ARITH_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;
ASM_CASES_TAC` cc_qx_v11 cc (i + 1 ) `;
ASM_SEARCH_TCL [` !i. cc_qx_v11 cc (i + 1) /\ cc_hassmall_v11 cc (i + 1) /\ cc_qy_v11 cc i
       ==> cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)`] (MP_TAC o (SPEC` i:num `));
ANTS_TAC;
ASM_REWRITE_TAC[cc_hassmall_v11];
STRIP_TAC;


SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3 ) + cc_gg3a_v11 cc ((i + 3) + 1 ) ` ASSUME_TAC;
ASM_CASES_TAC` cc_qu_v11 cc ( i + 3) `;
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[];
ASSUME_TAC2 (ARITH_RULE` ~( nn = 3) /\ 3 <= nn /\ nn <= 4 ==> 4 = nn`);
SUBGOAL_THEN` cc_qy_v11 cc ((i + 4 ) MOD 4 ) ` MP_TAC;
REWRITE_TAC[MOD_ADD_CANCEL];
ASM_REWRITE_TAC[];

ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];

REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];
ASSUME_TAC2 (ARITH_RULE` ~( nn = 3) /\ 3 <= nn /\ nn <= 4 ==> 4 = nn`);
SUBGOAL_THEN` cc_qy_v11 cc ((i + 4) MOD 4 ) ` ASSUME_TAC;
REWRITE_TAC[MOD_ADD_CANCEL];
ASM_REWRITE_TAC[];
DOWN;
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc (i + nn) ` MP_TAC;
ASM_SIMP_TAC[];
SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 3) ` MP_TAC;
MP_TAC (SPEC` i + 3 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
STRIP_TAC;
ASM_SIMP_TAC[];
ASM_SIMP_TAC[];
REAL_ARITH_TAC;

ASSUME_TAC2 LEAST_BOUND_CC_GG;
STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
ASM_REWRITE_TAC[];


ASSUME_TAC2 (ARITH_RULE` ~( nn = 3) /\ 3 <= nn /\ nn <= 4 ==> 4 = nn`);
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `; SUM_NUMSEG4];
ASSUME_TAC2 (
ISPECL [` cc_gg3a_v11 cc `;` nn:num `] periodic_mod1);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
SUBGOAL_THEN` cc_gg3a_v11 cc (((i + 3) + 1) MOD 4) = cc_gg3a_v11 cc i ` ASSUME_TAC;
REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `; MOD_ADD_CANCEL];
ASM_REWRITE_TAC[];
DOWN;
ASM_REWRITE_TAC[];
DISCH_THEN SUBST_ALL_TAC;
UNDISCH_TAC` cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `;
UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 3) + cc_gg3a_v11 cc i`;
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc ( i + 2) ` MP_TAC;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC;
ASM_SIMP_TAC[];
DOWN;
PHA;
REAL_ARITH_TAC;
ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0 ) `];
MP_TAC (SPEC` i + 1 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
STRIP_TAC;




(* ========================== *)
ASSUME_TAC2 LEAST_BOUND_CC_GG;
DOWN;

FIRST_ASSUM (MP_TAC o (SPEC ` (i + 2) MOD nn `));
ANTS_TAC;
ASM_SIMP_TAC[MOD_IN_NUMSEG];
MATCH_MP_TAC (GSYM MOD_INJ11);
ASM_REWRITE_TAC[ARITH_RULE` ~( 0 = 2) `];
UNDISCH_TAC` 3 <= nn `;
ARITH_TAC;
STRIP_TAC;
FIRST_ASSUM (MP_TAC o (SPEC ` (i + 3) MOD nn `));
ANTS_TAC;
ASM_SIMP_TAC[MOD_IN_NUMSEG];
MATCH_MP_TAC (GSYM MOD_INJ11);
ASM_REWRITE_TAC[ARITH_RULE` ~( 0 = 3) `];
UNDISCH_TAC` 3 <= nn `;
UNDISCH_TAC` ~( nn = 3)`;
ARITH_TAC;
STRIP_TAC;
DOWN THEN DOWN;
ASM_REWRITE_TAC[];
STRIP_TAC THEN STRIP_TAC;

ASM_CASES_TAC` cc_qx_v11 cc ( i + 2 ) `;
SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 2) ` ASSUME_TAC;
ASM_SIMP_TAC[];
MP_TAC LITTLE_CC_GG3BB;
ANTS_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;
MP_TAC LITTLE_CC_GG3AA;
ANTS_TAC;
ASM_REWRITE_TAC[];
ASSUME_TAC2 (ARITH_RULE` 3 <= nn /\ nn <= 4 /\ ~( nn = 3) ==> 4 = nn `);
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `];
SUBGOAL_THEN` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ` MP_TAC;
ASM_SIMP_TAC[];
REPEAT STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..nn - 1 ) (cc_gg_v11 cc) ` MP_TAC;
EXPAND_TAC "nn";
ASM_REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 /\ 3 + i = i + 3`];
REWRITE_TAC[SUM_NUMSEG4];
UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 3) + cc_gg3a_v11 cc i `;
UNDISCH_TAC` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i `;
UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `;
UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 2) `;
PHA;
REAL_ARITH_TAC;


ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0 ) `];
ASM_CASES_TAC` cc_qx_v11 cc ( i + 3) `;



ASSUME_TAC2 (ARITH_RULE` 3 <= nn /\ nn <= 4 /\ ~( nn = 3) ==> 4 = nn `);
SUBGOAL_THEN` cc_qy_v11 cc ((i + 4) MOD 4) ` ASSUME_TAC;
REWRITE_TAC[MOD_ADD_CANCEL];
ASM_REWRITE_TAC[];
DOWN;
ASM_REWRITE_TAC[];
STRIP_TAC;
ASM_SEARCH_TCL [` cc_eps <= x + cc_gg3a_v11 cc i `] (MP_TAC o (SPEC` i + 3 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `; cc_hassmall_v11];
STRIP_TAC;
MP_TAC LITTLE_CC_GG3BB;
ANTS_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;

STRIP_TAC;
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2) ` MP_TAC;
ASM_SIMP_TAC[];
STRIP_TAC;

SUBGOAL_THEN` cc_gg3a_v11 cc ((i + 4) MOD 4) = cc_gg3a_v11 cc i ` MP_TAC;
REWRITE_TAC[MOD_ADD_CANCEL];
ASM_REWRITE_TAC[];
ASSUME_TAC2 (ISPECL [` cc_gg3a_v11 cc `;` nn:num `] periodic_mod1);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
ASM_REWRITE_TAC[];
ASSUME_TAC2 (ISPECL [` cc_gg3a_v11 cc `;` nn:num `] periodic_mod1);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
ASM_REWRITE_TAC[];
STRIP_TAC;

STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc ) ` MP_TAC;
ASM_REWRITE_TAC[];
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `; SUM_NUMSEG4];
SUBGOAL_THEN` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ` MP_TAC;
ASM_SIMP_TAC[];

UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `;
UNDISCH_TAC` cc_eps <= cc_gg_v11 cc (i + 3) + cc_gg3a_v11 cc ((i + 3) + 1) `;
PHA;
ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];
SUBGOAL_THEN` --cc_eps <= cc_gg_v11 cc (i + 2) ` MP_TAC;
ASM_REWRITE_TAC[];
REAL_ARITH_TAC;
DOWN;
ASM_REWRITE_TAC[ARITH_RULE` &0 <= x <=> ~( x < &0 ) `];









MP_TAC (SPEC` i + 2 ` QU_OR_QXY);
MP_TAC (SPEC` i + 3 ` QU_OR_QXY);
ASM_REWRITE_TAC[];
STRIP_TAC;
STRIP_TAC;
SUBGOAL_THEN` #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC;
ASM_SIMP_TAC[];
ASSUME_TAC2 (ISPECL [` cc_azim_v11 cc `;` nn:num `] periodic_sum);
ASM_SEARCH_TCL [` &2 * pi `] MP_TAC;
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
DOWN;
SIMP_TAC[];
ASSUME_TAC2 (ARITH_RULE` 3 <= nn  /\ nn <= 4 /\ ~( nn = 3) ==> 4 = nn `);
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 /\ 4 - 1 = 3 `];
REPEAT STRIP_TAC;
UNDISCH_TAC` sum (i..i + 3) (cc_azim_v11 cc) = &2 * pi `;
REWRITE_TAC[SUM_NUMSEG4];
STRIP_TAC;

ASM_CASES_TAC` #2.089 <= cc_azim_v11 cc i ` ;
ASM_SEARCH_TCL [`#0.161517`] (ASSUME_TAC2 o (SPEC` i + 1 `));
ASM_SEARCH_TCL [`!i. cc_qu_v11 cc i
       ==> #0.161517 - #0.119482 * cc_azim_v11 cc i <= cc_gg_v11 cc i`] (ASSUME_TAC2 o (SPEC` i + 2 `));
ASM_SEARCH_TCL [`!i. cc_qu_v11 cc i
       ==> #0.161517 - #0.119482 * cc_azim_v11 cc i <= cc_gg_v11 cc i`] (ASSUME_TAC2 o (SPEC` i + 3 `));
SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc ) ` MP_TAC;
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `];
ASM_REWRITE_TAC[SUM_NUMSEG4];
DOWNS 4;
DOWN;
UNDISCH_TAC` #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i `;
MP_TAC (PI_BOUNDS);
REAL_ARITH_TAC;

ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `];


ASM_CASES_TAC` cc_azim_v11 cc i <= #1.946 `;
UNDISCH_TAC` !i. cc_qu_v11 cc i
       ==> -- #0.0659 + #0.042 * cc_azim_v11 cc i <= cc_gg_v11 cc i `;
STRIP_TAC;
FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i + 1 `));
FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i + 2 `));
FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i + 3 `));
SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `];
ASM_REWRITE_TAC[SUM_NUMSEG4];
DOWNS 3;
UNDISCH_TAC` cc_azim_v11 cc i <= #1.946 `;
UNDISCH_TAC` cc_azim_v11 cc i +
      cc_azim_v11 cc (i + 1) +
      cc_azim_v11 cc (i + 2) +
      cc_azim_v11 cc (i + 3) =
      &2 * pi `;
MP_TAC PI_BOUNDS;
UNDISCH_TAC` #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i `;
REAL_ARITH_TAC;

ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `];
ASM_CASES_TAC` cc_small_eta_v11 cc i /\
       cc_small_eta_v11 cc (i + 1) `;
UNDISCH_TAC` !i. cc_qy_v11 cc i /\
          cc_small_eta_v11 cc i /\
          cc_small_eta_v11 cc (i + 1) /\
          #1.946 <= cc_azim_v11 cc i /\
          cc_azim_v11 cc i <= #2.089
          ==> #3.0 * cc_eps <= cc_gg_v11 cc i `;
DISCH_THEN (MP_TAC o (SPEC` i:num `));
ANTS_TAC;
ASM_REWRITE_TAC[];
DOWNS 3;
REAL_ARITH_TAC;
STRIP_TAC;
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 1) /\ 
-- cc_eps <= cc_gg_v11 cc (i + 2) /\ 
-- cc_eps <= cc_gg_v11 cc (i + 3) ` MP_TAC;
ASM_REWRITE_TAC[];
STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc ) ` MP_TAC;
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `];
ASM_REWRITE_TAC[SUM_NUMSEG4];
DOWNS 4;
REAL_ARITH_TAC;
ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `];


UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `;
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `];
ASM_REWRITE_TAC[];
REWRITE_TAC[REAL_ARITH` a < &0 <=> ~( &0 <= a ) `; SUM_NUMSEG4];
STRIP_TAC;



UNDISCH_TAC`~(cc_small_eta_v11 cc i /\ cc_small_eta_v11 cc (i + 1)) `;
REWRITE_TAC[DE_MORGAN_THM];
STRIP_TAC;
ASSUME_TAC2 (ISPECL [` cc_small_eta_v11 cc `; `nn:num `] periodic_mod1);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
SUBGOAL_THEN` ~cc_small_eta_v11 cc ((i + 4) MOD 4)` ASSUME_TAC;
REWRITE_TAC[MOD_ADD_CANCEL];
ASM_REWRITE_TAC[];
DOWN;
ASM_REWRITE_TAC[];
STRIP_TAC;
UNDISCH_TAC` !i. cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc (i + 1)
          ==> cc_eps <= cc_gg_v11 cc i `;
DISCH_THEN (MP_TAC o (SPEC` i + 3 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];
STRIP_TAC;
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2 ) ` ASSUME_TAC;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `ASSUME_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i /\ cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC;
ASM_SIMP_TAC[];

SUBGOAL_THEN` &0 <=
        cc_gg_v11 cc i +
        cc_gg_v11 cc (i + 1) +
        cc_gg_v11 cc (i + 2) +
        cc_gg_v11 cc (i + 3) ` MP_TAC;
DOWNS 4;
REAL_ARITH_TAC;
ASM_REWRITE_TAC[];

UNDISCH_TAC` !i. cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc i ==> cc_eps <= cc_gg_v11 cc i `;
DISCH_THEN (MP_TAC o (SPEC` i + 1 `));
ANTS_TAC;
ASM_REWRITE_TAC[];

SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2 ) ` ASSUME_TAC;

ASM_REWRITE_TAC[];
UNDISCH_TAC` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1)
       ==> &0 <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `;
DISCH_THEN (MP_TAC o (SPEC` i + 3 `));
ANTS_TAC;
ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `];
SUBGOAL_THEN` cc_qy_v11 cc ((i + nn) MOD nn) ` MP_TAC;
REWRITE_TAC[MOD_ADD_CANCEL];
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[];
DOWN;
SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i /\ cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ` MP_TAC;
ASM_SIMP_TAC[];

SUBGOAL_THEN` cc_gg3a_v11 cc (((i + 3) + 1) MOD 4) = cc_gg3a_v11 cc i ` MP_TAC;
REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `; MOD_ADD_CANCEL];
ASM_SIMP_TAC[];
MATCH_MP_TAC (GSYM periodic_mod);
ASM_REWRITE_TAC[];


ASSUME_TAC2 (ISPECL [` cc_gg3a_v11 cc `;` nn:num `] periodic_mod1);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
ASM_REWRITE_TAC[];
SIMP_TAC[];
STRIP_TAC;
UNDISCH_TAC` ~(&0 <=
        cc_gg_v11 cc i +
        cc_gg_v11 cc (i + 1) +
        cc_gg_v11 cc (i + 2) +
        cc_gg_v11 cc (i + 3)) `;
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2) ` MP_TAC;
ASM_REWRITE_TAC[];
REAL_ARITH_TAC]);;



let GRHIDFA_ALT = 
prove_by_refinement(`!cc. cc_bool_model_v11 cc /\
      cc_bool_prep_v11 cc /\
      cc_real_model_v11 cc /\
      sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0
      ==> ~(cc_card_v11 cc = 4) `,

[NHANH UTEOITF;
NHANH IPVICGW;
REPEAT STRIP_TAC;
ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00;
DOWN_TAC;
NHANH Oxl_def.periodic_fn;
REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `];
ABBREV_TAC` nn = cc_card_v11 cc `;
STRIP_TAC;
ABBREV_TAC ` squ = { i | i <= 3 /\ cc_qu_v11 cc i } `;
SWITCH_TAC` nn = 4 `;
ASSUME_TAC (SPECL [` 0 `;` 3 `] FINITE_NUMSEG);
SUBGOAL_THEN` squ SUBSET 0..3 ` ASSUME_TAC;
EXPAND_TAC "squ";
REWRITE_TAC[numseg; LE_0];
CONV_TAC SET_RULE;
SUBGOAL_THEN` FINITE (squ: num -> bool) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` 0..3 `;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` ! i. cc_qu_v11 cc i \/ cc_qx_v11 cc i ` ASSUME_TAC;
ASM_REWRITE_TAC[cc_qx_v11];
MESON_TAC[];
ABBREV_TAC` sqx = { i | i <= 3 /\ cc_qx_v11 cc i } ` ;
SUBGOAL_THEN` sqx UNION squ = 0..3` ASSUME_TAC;
EXPAND_TAC "squ";
EXPAND_TAC "sqx";
REWRITE_TAC[numseg; LE_0];
UNDISCH_TAC` !i. cc_qu_v11 cc i \/ cc_qx_v11 cc i `;
REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_UNION];
MESON_TAC[];


SUBGOAL_THEN` DISJOINT sqx (squ:num -> bool) ` ASSUME_TAC;
EXPAND_TAC "squ" THEN EXPAND_TAC "sqx";
REWRITE_TAC[cc_qx_v11];
CONV_TAC SET_RULE;

SUBGOAL_THEN` sqx SUBSET 0..3 ` ASSUME_TAC;
EXPAND_TAC "sqx";
REWRITE_TAC[numseg; LE_0];
CONV_TAC SET_RULE;
SUBGOAL_THEN` FINITE (sqx:num -> bool) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_SUBSET;
EXISTS_TAC` 0..3 `;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` DISJOINT squ (sqx: num -> bool) ` ASSUME_TAC;
ONCE_REWRITE_TAC[DISJOINT_SYM];
ASM_REWRITE_TAC[];


SUBGOAL_THEN`! f. sum (squ UNION sqx) f = sum squ f + sum (sqx:num -> bool) f ` ASSUME_TAC;
GEN_TAC;
MATCH_MP_TAC SUM_UNION;
ASM_REWRITE_TAC[];

MP_TAC (ISPECL [` squ: num -> bool `;` sqx: num -> bool `] CARD_UNION);
ANTS_TAC;
ASM_REWRITE_TAC[GSYM DISJOINT];
UNDISCH_TAC` sqx UNION squ = 0..3 `;
REWRITE_TAC[UNION_COMM];
STRIP_TAC;
ASM_REWRITE_TAC[CARD_NUMSEG; ARITH_RULE` (3 + 1) - 0 = 4 `];
STRIP_TAC; SUBGOAL_THEN` !i. i IN squ
       ==> #0.161517 - #0.119482 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC;
EXPAND_TAC "squ";
REWRITE_TAC[IN_ELIM_THM];
GEN_TAC THEN STRIP_TAC;
ASM_SIMP_TAC[];

SUBGOAL_THEN` ! i. i IN sqx 
==> #0.213849 - #0.119482 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC;
EXPAND_TAC "sqx";
REWRITE_TAC[IN_ELIM_THM];
GEN_TAC THEN STRIP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[cc_hassmall_v11];

ASM_CASES_TAC` CARD (squ: num -> bool) <= 2 ` ;
SWITCH_TAC` squ UNION sqx = 0..3 `;
SUBGOAL_THEN` sum squ (\i. #0.161517 - #0.119482 * cc_azim_v11 cc i ) <=
 sum squ (cc_gg_v11 cc) ` ASSUME_TAC;
MATCH_MP_TAC SUM_LE;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` sum sqx (\i. #0.213849 - #0.119482 * cc_azim_v11 cc i ) <=
 sum sqx (cc_gg_v11 cc) ` ASSUME_TAC;
MATCH_MP_TAC SUM_LE;
ASM_REWRITE_TAC[];

UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `;
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `];
ASM_REWRITE_TAC[ARITH_RULE` ~( x < &0 ) <=> &0 <= x `];
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` sum squ (\i. #0.161517 - #0.119482 * cc_azim_v11 cc i)  + 
sum sqx (\i. #0.213849 - #0.119482 * cc_azim_v11 cc i ) `;
CONJ_TAC;
ASM_SIMP_TAC[SUM_SUB; SUM_LMUL; SUM_CONST];
UNDISCH_TAC` sum (0..nn - 1) (cc_azim_v11 cc) = &2 * pi `;
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `];
ASM_REWRITE_TAC[ETA_AX];
UNDISCH_TAC` nn = CARD (squ: num -> bool) + CARD (sqx: num -> bool) `;
UNDISCH_TAC` CARD (squ: num -> bool) <= 2 `;
REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_EQ; GSYM REAL_OF_NUM_ADD];
EXPAND_TAC "nn";
MP_TAC PI_BOUNDS;
REAL_ARITH_TAC;

DOWNS 2;
REAL_ARITH_TAC;


ASM_CASES_TAC` ! i. i < 4 ==> cc_qu_v11 cc i ` ;
SUBGOAL_THEN` ! i. 0 <= i /\ i <= 3 ==> -- #0.0659 + #0.042 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC;
REWRITE_TAC[ARITH_RULE` 0 <= x /\ x <= 3 <=> x < 4 `];
FIRST_X_ASSUM NHANH;
ASM_SIMP_TAC[];

DOWN;
PHA;
NHANH SUM_LE_NUMSEG;
REWRITE_TAC[SUM_ADD_NUMSEG; SUM_LMUL; ETA_AX; SUM_CONST_NUMSEG];
REWRITE_TAC[ARITH_RULE` ( 3 + 1 ) - 0 = 4`];
UNDISCH_TAC` sum (0..nn - 1) (cc_azim_v11 cc) = &2 * pi `;
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `];
SIMP_TAC[];
REPEAT STRIP_TAC;
SUBGOAL_THEN` &0 <= sum (0..3) (cc_gg_v11 cc) ` MP_TAC;
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` &4 * --  #0.0659 + #0.042 * &2 * pi `;
DOWN THEN SIMP_TAC[];
MP_TAC PI_BOUNDS;
REAL_ARITH_TAC;
ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x ) <=> x < &0 `];
UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `;
EXPAND_TAC "nn";
SIMP_TAC[ARITH_RULE` 4 - 1 = 3 `];
SUBGOAL_THEN` ? i. i < 4 /\ ~ cc_qu_v11 cc i ` ASSUME_TAC;
DOWN;
MESON_TAC[];
DOWN THEN STRIP_TAC;
SUBGOAL_THEN` cc_qx_v11 cc i ` ASSUME_TAC;
ASM_REWRITE_TAC[cc_qx_v11];

SUBGOAL_THEN` CARD (sqx:num -> bool) <= 1 ` ASSUME_TAC;
UNDISCH_TAC` nn = CARD (squ:num -> bool) + CARD (sqx:num -> bool) `;
UNDISCH_TAC` ~(CARD (squ:num -> bool) <= 2) `;
EXPAND_TAC "nn";
ARITH_TAC;
SUBGOAL_THEN` ! j. j <= 3 /\ ~(j = i) ==> cc_qu_v11 cc j ` ASSUME_TAC;
GEN_TAC THEN STRIP_TAC;
ASM_CASES_TAC` cc_qx_v11 cc j `;
SUBGOAL_THEN` {i, j:num} SUBSET sqx ` ASSUME_TAC;
EXPAND_TAC "sqx";
REWRITE_TAC[IN_ELIM_THM; INSERT_SUBSET; EMPTY_SUBSET];
ASM_REWRITE_TAC[];
UNDISCH_TAC` i < 4 `;
ARITH_TAC;

UNDISCH_TAC` ~( j = (i:num)) ` ;
REWRITE_TAC[GSYM Geomdetail.CARD2; INSERT_COMM];
STRIP_TAC;
SUBGOAL_THEN` CARD {i, j:num} <= CARD (sqx:num -> bool)` ASSUME_TAC;
MATCH_MP_TAC CARD_SUBSET;
ASM_REWRITE_TAC[];
DOWN;
ASM_REWRITE_TAC[];
NHANH (ARITH_RULE` 2 <= a ==> ~( a <= 1 ) `);
ASM_REWRITE_TAC[];
UNDISCH_TAC` !i. cc_qu_v11 cc i \/ cc_qx_v11 cc i `;
DISCH_THEN (MP_TAC o (SPEC` j:num `));
ASM_REWRITE_TAC[];
ASSUME_TAC2 (ISPECL [` cc_qu_v11 cc `;` nn:num `] periodic_mod1);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);

MP_TAC (ISPECL [` 3`] Hypermap.LE_MOD_SUC);
REWRITE_TAC[ARITH_RULE` SUC 3 = 4 `];
STRIP_TAC;

SUBGOAL_THEN`  i MOD nn = i ` ASSUME_TAC;
MATCH_MP_TAC MOD_LT;
UNDISCH_TAC` i < 4 `;
ASM_REWRITE_TAC[];
UNDISCH_TAC` !j. j <= 3 /\ ~(j = i) ==> cc_qu_v11 cc j `;
EXPAND_TAC "i";
STRIP_TAC;

SUBGOAL_THEN` cc_qu_v11 cc (( i + 1) MOD 4 ) ` ASSUME_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
UNDISCH_TAC` nn = CARD (squ:num -> bool) + CARD (sqx: num -> bool) `;
DISCH_THEN (ASSUME_TAC o SYM);
CONJ_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` 4 = nn `;
SIMP_TAC[];
STRIP_TAC;
MATCH_MP_TAC (GSYM MOD_INJ11);
EXPAND_TAC "nn";
ARITH_TAC;

SUBGOAL_THEN` cc_qu_v11 cc (( i + 2) MOD 4 ) ` ASSUME_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
UNDISCH_TAC` nn = CARD (squ:num -> bool) + CARD (sqx: num -> bool) `;
DISCH_THEN (ASSUME_TAC o SYM);
CONJ_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` 4 = nn `;
SIMP_TAC[];
STRIP_TAC;
MATCH_MP_TAC (GSYM MOD_INJ11);
EXPAND_TAC "nn";
ARITH_TAC;


SUBGOAL_THEN` cc_qu_v11 cc (( i + 3) MOD 4 ) ` ASSUME_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
UNDISCH_TAC` nn = CARD (squ:num -> bool) + CARD (sqx: num -> bool) `;
DISCH_THEN (ASSUME_TAC o SYM);
CONJ_TAC;
ASM_REWRITE_TAC[];
UNDISCH_TAC` 4 = nn `;
SIMP_TAC[];
STRIP_TAC;
MATCH_MP_TAC (GSYM MOD_INJ11);
EXPAND_TAC "nn";
ARITH_TAC;

DOWNS 3;
UNDISCH_TAC` nn = CARD (squ:num -> bool) + CARD (sqx: num -> bool) `;
DISCH_THEN (ASSUME_TAC o SYM);
ASM_REWRITE_TAC[];
ASM_CASES_TAC` cc_crit_v11 cc i `;
STRIP_TAC;
UNDISCH_TAC` nn = 4 /\
   (?i. cc_4cell_v11 cc i /\
        cc_crit_v11 cc i /\
        cc_qu_v11 cc (i + 1) /\
        cc_qu_v11 cc (i + 2) /\
        cc_qu_v11 cc (i + 3))
   ==> &0 <= sum (0..nn - 1) (cc_gg_v11 cc) `;
ANTS_TAC;
ASM_REWRITE_TAC[];
EXISTS_TAC` i:num `;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x ) <=> x < &0 `];
ASM_CASES_TAC` cc_subcrit_v11 cc i `;
UNDISCH_TAC` cc_qx_v11 cc i `;
REWRITE_TAC[cc_qx_v11; cc_qu_v11];
ASM_REWRITE_TAC[cc_hassmall_v11];


SUBGOAL_THEN` cc_crit_v11 cc i \/ cc_subcrit_v11 cc i \/ cc_supercrit_v11 cc i ` ASSUME_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[];

DOWN;
ASM_REWRITE_TAC[];

STRIP_TAC;
SUBGOAL_THEN` #0.00457511 + #0.00609451 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` MP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[cc_hassmall_v11];
STRIP_TAC;

SUBGOAL_THEN` ! i. cc_hassmall_v11 cc i ` ASSUME_TAC;
ASM_REWRITE_TAC[cc_hassmall_v11];
UNDISCH_TAC` !i. cc_qu_v11 cc i
       ==> -- #0.0142852 + #0.00609451 * cc_azim_v11 cc i <= cc_gg_v11 cc i `;
DISCH_THEN NHANH;
STRIP_TAC;





UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `;
UNDISCH_TAC` sum (0..nn - 1) (cc_azim_v11 cc) = &2 * pi `;
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;`nn:num `] periodic_sum);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
ASSUME_TAC2 (ISPECL [` cc_azim_v11 cc `;`nn:num `] periodic_sum);
FIRST_X_ASSUM (ASSUME_TAC o GSYM);
ASM_REWRITE_TAC[];

UNDISCH_TAC` 4 = nn `;
STRIP_TAC;
EXPAND_TAC "nn";
REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `];
REWRITE_TAC[SUM_NUMSEG4];

UNDISCH_TAC ` -- #0.0142852 + #0.00609451 * cc_azim_v11 cc (i + 3) <=
      cc_gg_v11 cc (i + 3) `;
UNDISCH_TAC ` -- #0.0142852 + #0.00609451 * cc_azim_v11 cc (i + 2) <=
      cc_gg_v11 cc (i + 2) `;
UNDISCH_TAC ` -- #0.0142852 + #0.00609451 * cc_azim_v11 cc (i + 1) <=
      cc_gg_v11 cc (i + 1) `;
UNDISCH_TAC` #0.00457511 + #0.00609451 * cc_azim_v11 cc i <= cc_gg_v11 cc i `;
PHA;
MP_TAC PI_BOUNDS;
REAL_ARITH_TAC]);;





let LUIKGMH_concl = `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ 
   (sum (0..cc_card_v11 cc -1) (cc_gg_v11 cc) < &0) ==> (4 <= cc_card_v11 cc)`;;



let GRHIDFA_concl = `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ 
   (sum (0..cc_card_v11 cc -1) (cc_gg_v11 cc) < &0) ==> (F)`;; 



let NOT_QY_LEM = 
prove(` !cc. cc_bool_model_v11 cc /\
          cc_bool_prep_v11 cc /\
          cc_real_model_v11 cc /\
          sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0
          ==> (!i. ~ cc_qy_v11 cc i) `,
 NHANH UTEOITF THEN SIMP_TAC[cc_qy_v11]);;



let LUIKGMH = prove_by_refinement(LUIKGMH_concl,
[NHANH NOT_QY_LEM;
REPEAT STRIP_TAC;
ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00;
ASSUME_TAC2 THREE_LE_CC_CARD;
DOWN_TAC;
NHANH Oxl_def.periodic_fn;
REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
STRIP_TAC;
ABBREV_TAC` nn = cc_card_v11 cc `;
ASM_CASES_TAC` nn = 3 `;
SUBGOAL_THEN` ! i. cc_qu_v11 cc i /\ cc_gg_v11 cc i < &0 ==>  cc_azim_v11 cc i < #1.65 ` ASSUME_TAC;
GEN_TAC THEN STRIP_TAC;
SUBGOAL_THEN` -- #0.0659 + #0.042 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` MP_TAC;
ASM_SIMP_TAC[];
DOWN THEN PHA;
NHANH (REAL_ARITH` x < &0 /\ a <= x ==> a < &0 `);
STRIP_TAC;
DOWN;
CONV_TAC REAL_FIELD;


ABBREV_TAC` sxx = { i | i <= 2 /\ cc_qu_v11 cc i /\ cc_gg_v11 cc i < &0 }`;
SUBGOAL_THEN` ! i. cc_qu_v11 cc i \/ cc_qx_v11 cc i ` ASSUME_TAC;
GEN_TAC;
MP_TAC (SPEC` i:num ` QU_OR_QXY);
ASM_REWRITE_TAC[];
ASM_CASES_TAC` (sxx:num -> bool) = {} `;
SUBGOAL_THEN` ! i. i <= 2 ==> &0 <= cc_gg_v11 cc i ` ASSUME_TAC;
GEN_TAC THEN STRIP_TAC;
ASM_CASES_TAC` cc_qx_v11 cc i `;
ASM_SIMP_TAC[];
FIRST_X_ASSUM (MP_TAC o (SPEC`i:num `));
ASM_REWRITE_TAC[];
STRIP_TAC;
ASM_CASES_TAC` cc_gg_v11 cc i < &0 `;
SUBGOAL_THEN` i IN (sxx:num -> bool) ` MP_TAC;
EXPAND_TAC "sxx";
REWRITE_TAC[IN_ELIM_THM];
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[NOT_IN_EMPTY];
DOWN;
REAL_ARITH_TAC;

UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc ) < &0 `;
ASM_REWRITE_TAC[ARITH_RULE` 3 - 1 = 0 + 2 `; SUM_NUMSEG3; ADD];
REWRITE_TAC[REAL_ARITH` x < &0 <=> ~( &0 <= x) `];
DOWN;

MESON_TAC[LE_0; SUM_POS_LE_NUMSEG];

DOWN;
REWRITE_TAC[Local_lemmas.EMPTY_NOT_EXISTS_IN];
STRIP_TAC;
ASM_CASES_TAC` ? y. y IN sxx /\ ~( y = x:num ) `;
DOWN THEN STRIP_TAC;
ASSUME_TAC (SET_RULE` DISJOINT {x,y:num} ( (0..2) DIFF {x,y}) `);
ABBREV_TAC` sc = (0..2) DIFF {x,y} `;
SUBGOAL_THEN` x <= 2 /\ y <= 2 /\ 0 <= x /\ 0 <= y` MP_TAC;
UNDISCH_TAC` x IN (sxx: num -> bool) `;
UNDISCH_TAC` y IN (sxx: num -> bool) `;
EXPAND_TAC "sxx";
SIMP_TAC[IN_ELIM_THM; LE_0];
STRIP_TAC;
SUBGOAL_THEN` 0..2 = {x,y} UNION sc ` ASSUME_TAC;
EXPAND_TAC "sc";

SUBGOAL_THEN` x IN 0..2 /\ y IN 0..2 ` MP_TAC;
ASM_REWRITE_TAC[IN_NUMSEG];
CONV_TAC SET_RULE;
SUBGOAL_THEN` CARD {x,y:num} = 2 ` ASSUME_TAC;
ASM_REWRITE_TAC[Geomdetail.CARD2];
UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc ) < &0 `;
ASM_REWRITE_TAC[ARITH_RULE` 3 - 1 = 2 `];
ASSUME_TAC (SPECL [` 0 `;` 2 `] FINITE_NUMSEG);
ASSUME_TAC (ISPECL [` x:num `;` y:num `] Hypermap.FINITE_TWO_ELEMENTS);
SUBGOAL_THEN` FINITE (sc:num -> bool) ` ASSUME_TAC;
EXPAND_TAC "sc";
MATCH_MP_TAC FINITE_DIFF;
FIRST_ASSUM ACCEPT_TAC;


STRIP_TAC;
UNDISCH_TAC` sum (0..nn - 1) (cc_azim_v11 cc) = &2 * pi `;
ASM_REWRITE_TAC[ARITH_RULE` 3 - 1 = 2 `];
ASM_SIMP_TAC[SUM_UNION];
ASSUME_TAC2 (
ISPECL [` {x:num, y} `;` sc:num -> bool `] (
REWRITE_RULE[GSYM DISJOINT] CARD_UNION));
DOWN;
ASM_REWRITE_TAC[];
MP_TAC (SPECL [` 0 `;` 2 `] CARD_NUMSEG);
ASM_SIMP_TAC[ARITH_RULE` ( 2 + 1) - 0 = 3 `];
PHA THEN STRIP_TAC;
SUBGOAL_THEN` sum {x,y} (cc_azim_v11 cc) <= &(CARD {x,y}) * ( #1.65) ` ASSUME_TAC;
MATCH_MP_TAC SUM_BOUND;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` {x,y} SUBSET (sxx: num -> bool) ` MP_TAC;
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET];

REWRITE_TAC[SUBSET];
DISCH_THEN NHANH;
GEN_TAC THEN STRIP_TAC;
MATCH_MP_TAC REAL_LT_IMP_LE;
FIRST_X_ASSUM MATCH_MP_TAC;
DOWN;
EXPAND_TAC "sxx";
REWRITE_TAC[IN_ELIM_THM];
SIMP_TAC[];


SUBGOAL_THEN` sum sc (cc_azim_v11 cc) < &(CARD sc) * ( #2.8) ` ASSUME_TAC;
MATCH_MP_TAC SUM_BOUND_LT_ALL;
ASM_REWRITE_TAC[];
CONJ_TAC;
STRIP_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC;
UNDISCH_TAC` 3 = 2 + (CARD ({}:num -> bool))` ;
REWRITE_TAC[CARD_EMPTY];
ARITH_TAC;
GEN_TAC THEN STRIP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC` x':num `));
SIMP_TAC[cc_qu_v11; cc_qx_v11];
MESON_TAC[];


DOWNS 2;
NHANH (REAL_ARITH` a <= aa /\ b < bb ==> a + b < aa + bb `);
ASM_REWRITE_TAC[];
STRIP_TAC;
UNDISCH_TAC` 3 = 2 + CARD (sc:num -> bool) `;
REWRITE_TAC[ARITH_RULE` 3 = 2 + x <=> x = 1 `];
DISCH_THEN (SUBST_ALL_TAC);
DOWN;
MP_TAC PI_BOUNDS;
REAL_ARITH_TAC;

SUBGOAL_THEN` ! i. #2.3 < cc_azim_v11 cc i ==> cc_eps <= cc_gg_v11 cc i ` ASSUME_TAC;
GEN_TAC;
FIRST_X_ASSUM (MP_TAC o (SPEC` i:num `));
STRIP_TAC;
STRIP_TAC;
MATCH_MP_TAC REAL_LE_TRANS;
EXISTS_TAC` -- #0.0659 + #0.042 * cc_azim_v11 cc i `;
ASM_SIMP_TAC[];
DOWN;
REWRITE_TAC[cc_eps];
REAL_ARITH_TAC;
ASM_SIMP_TAC[];

ASSUME_TAC2 (
SET_RULE` x IN sxx /\ ~(?y. y IN sxx /\ ~(y = x)) ==> sxx = {x:num} `);
ABBREV_TAC` sc = (0..2) DIFF sxx `;
ASSUME_TAC (SPECL [` 0 `;` 2 `] FINITE_NUMSEG);
SUBGOAL_THEN` FINITE ((0..2) DIFF sxx ) ` ASSUME_TAC;
DOWN;
REWRITE_TAC[FINITE_DIFF];
DOWNS 3;
ASM_REWRITE_TAC[];
STRIP_TAC;
UNDISCH_TAC` x IN (sxx:num -> bool) `;
EXPAND_TAC "sxx";
REWRITE_TAC[IN_ELIM_THM];
STRIP_TAC;
SUBGOAL_THEN` x IN (0..2) ` ASSUME_TAC;
ASM_REWRITE_TAC[IN_NUMSEG; LE_0];

DOWN;
NHANH (SET_RULE` x IN S ==> DISJOINT {x} (S DIFF {x}) `);
ASM_REWRITE_TAC[];
STRIP_TAC;
ASSUME_TAC2 (SET_RULE` x IN (0..2) ==> {x} UNION ((0..2) DIFF {x}) = (0..2) `);
MP_TAC (ISPECL [` {x:num } `;`sc: num -> bool `] CARD_UNION);
ANTS_TAC;
ASM_REWRITE_TAC[GSYM DISJOINT];
UNDISCH_TAC` FINITE ((0..2) DIFF {x})`;
ASM_REWRITE_TAC[FINITE_SING];
DOWN;
ASM_SIMP_TAC[CARD_SING; CARD_NUMSEG; ARITH_RULE` (2 + 1) - 0 = 3 `];
REWRITE_TAC[ARITH_RULE` 3 = 1 + x <=> x = 2 `];
PHA THEN STRIP_TAC;
SUBGOAL_THEN` FINITE (sc:num -> bool) ` ASSUME_TAC;
EXPAND_TAC "sc";
FIRST_X_ASSUM ACCEPT_TAC;
ASM_CASES_TAC` ! i. i IN sc ==> 
cc_azim_v11 cc i <= #2.3 ` ;
DOWNS 2;
NHANH SUM_LE;
REWRITE_TAC[ETA_AX];
IMP_TAC;
ASM_SIMP_TAC[SUM_CONST];
PHA THEN STRIP_TAC;

UNDISCH_TAC` sum (0..nn - 1) (cc_azim_v11 cc) = &2 * pi `;
ASM_REWRITE_TAC[ARITH_RULE` 3 - 1 = 2 `];
UNDISCH_TAC` {x} UNION sc = 0..2 `;
DISCH_THEN (SUBST1_TAC o SYM);
ASM_SIMP_TAC[SUM_UNION; FINITE_SING; SUM_SING];
SUBGOAL_THEN` cc_azim_v11 cc x < #1.65 ` MP_TAC;
FIRST_X_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[];
DOWN;
MP_TAC PI_BOUNDS;
REAL_ARITH_TAC;
DOWN;
REWRITE_TAC[MESON[]` (~(! i. p i ==> q i)) <=> (? i. p i /\ ~ q i ) `];
REWRITE_TAC[REAL_ARITH` ~( x <= y ) <=> y < x `];
STRIP_TAC;


FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i:num `));
ASSUME_TAC2 (SET_RULE` (i:num) IN sc ==> sc = {i} UNION (sc DIFF {i})`);
ABBREV_TAC` scc = sc DIFF {i:num} `;
SUBGOAL_THEN` FINITE ( sc DIFF {i:num}) ` ASSUME_TAC;
MATCH_MP_TAC FINITE_DIFF;
ASM_REWRITE_TAC[];
SUBGOAL_THEN` ! j. j IN sc ==> &0 <= cc_gg_v11 cc j ` ASSUME_TAC;
EXPAND_TAC "sc";
UNDISCH_TAC` sxx = {x:num} `;
DISCH_THEN (SUBST1_TAC o SYM);
EXPAND_TAC "sxx";
REWRITE_TAC[IN_DIFF; IN_NUMSEG; IN_ELIM_THM; LE_0; DE_MORGAN_THM];
GEN_TAC THEN IMP_TAC;
SIMP_TAC[];
PHA THEN STRIP_TAC;
MP_TAC (SPEC` j:num` QU_OR_QXY);
ASM_REWRITE_TAC[];
DOWN;
REAL_ARITH_TAC;

UNDISCH_TAC` sum (0.. nn - 1) (cc_gg_v11 cc ) < &0 `;
ASM_SIMP_TAC[ARITH_RULE` 3 - 1 = 2 `];
UNDISCH_TAC` {x} UNION sc = 0..2 `;
DISCH_THEN (SUBST1_TAC o SYM);
UNDISCH_TAC ` sc = {i:num} UNION scc `;
DISCH_THEN (ASSUME_TAC o SYM);
ASM_SIMP_TAC[SUM_UNION; FINITE_SING; SUM_SING];
FIRST_X_ASSUM (ASSUME_TAC o SYM);
UNDISCH_TAC` FINITE (sc DIFF {i:num}) `;

MP_TAC (SET_RULE` DISJOINT {i} (sc DIFF {i:num} ) `);
ASM_SIMP_TAC[SUM_UNION; FINITE_SING; SUM_SING];
PHA THEN STRIP_TAC;
SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc x ` ASSUME_TAC;
ASSUME_TAC2 (LEAST_BOUND_CC_GG);
ASM_REWRITE_TAC[];
SUBGOAL_THEN` &0 <= sum scc (cc_gg_v11 cc) ` ASSUME_TAC;
MATCH_MP_TAC SUM_POS_LE;
ASM_REWRITE_TAC[];
EXPAND_TAC "scc";
REWRITE_TAC[IN_DIFF];
FIRST_X_ASSUM NHANH;
SIMP_TAC[];
DOWNS 3;
UNDISCH_TAC` cc_eps <= cc_gg_v11 cc i `;
REAL_ARITH_TAC;
DOWN;
UNDISCH_TAC` 3 <= nn `;
ARITH_TAC]);;



let GRHIDFA = prove(GRHIDFA_concl, 
PHA THEN NHANH LUIKGMH THEN NHANH GRHIDFA_ALT THEN NHANH RSIWAMP
THEN GEN_TAC THEN STRIP_TAC THEN DOWNS 3 THEN ARITH_TAC);;



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

(* =================


search[name "RSI"];;



g ` !cc. ((cc_bool_model_v11 cc /\
           cc_bool_prep_v11 cc /\
           cc_real_model_v11 cc /\
           sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\
          cc_card_v11 cc = 5 ) /\
          ~cc_small_v11 cc i /\ cc_qx_v11 cc i /\ cc_qx_v11 cc ( i + 4) ==> F `;;


els [REPEAT STRIP_TAC;
ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00;
ASSUME_TAC2 EXISTS_QY_CARD5;
DOWN_TAC;
NHANH Oxl_def.periodic_fn;
REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11];
ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `];
IMP_TAC;
IMP_TAC;
STRIP_TAC;
STRIP_TAC;
STRIP_TAC;

ABBREV_TAC` nn = cc_card_v11 cc `];;







ASM_REWRITE_TAC[];
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `; ` nn:num `] Oxl_def.periodic_sum);
SUBGOAL_THEN` cc_gg_v11 cc i' + cc_gg_v11 cc (i' + 1) + cc_gg_v11 cc (i' + 3 - 1) = sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC;
FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC` i':num `));
ASM_SIMP_TAC[ARITH_RULE` 3 - 1 + i = i + 2 `];
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 2 /\ i + 1 <= i + 2 /\ (i + 1) + 1 = i + 2 `; SUM_SING_NUMSEG];
SIMP_TAC[ARITH_RULE` 3 - 1 = 2 `];
SIMP_TAC[Oxl_def.cc_eps];
NHANH (REAL_ARITH` &2 * #0.0057 <= x ==> ~( x < &0 ) `);
ASM_REWRITE_TAC[];

ASSUME_TAC2 (ARITH_RULE` nn <= 4 /\ 3 <= nn /\ ~( nn = 3) ==> nn = 4 `);
ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `; ` nn:num `] Oxl_def.periodic_sum);
UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0`;
FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC` i':num `));
ASM_SIMP_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `];
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 3 /\ i + 1 <= i + 3 /\ (i + 1) + 1 = i + 2 `; SUM_SING_NUMSEG];
SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 3 /\ (i + 2) + 1 = i + 3`; SUM_SING_NUMSEG];
ASSUME_TAC2 LEAST_BOUND_CC_GG;
FIRST_X_ASSUM (ASSUME_TAC o (SPEC `i' + 2 `));
DOWN;
UNDISCH_TAC` &2 * cc_eps <=
      cc_gg_v11 cc i' + cc_gg_v11 cc (i' + 1) + cc_gg_v11 cc (i' + nn - 1) `;
ASM_SIMP_TAC[ARITH_RULE` 4 - 1 = 3 `; Oxl_def.cc_eps]];;







*)


end;;