(* ========================================================================== *)
(* FLYSPECK - BOOK FORMALIZATION                                              *)
(*                                                                            *)
(* Chapter: Local Fan                                              *)
(* Author: Hoang Le Truong                                        *)
(* Date: 2012-04-01                                                           *)
(* ========================================================================= *)




module   Mtuwlun = struct



open Polyhedron;;
open Sphere;;
open Fan_defs;;
open Hypermap;;
open Vol1;;
open Fan;;
open Topology;;		
open Fan_misc;;
open Planarity;; 
open Conforming;;
open Sphere;;
open Hypermap;;
open Fan;;
open Topology;;
open Prove_by_refinement;;
open Pack_defs;;
open Wrgcvdr_cizmrrh;;
open Local_lemmas;;
open Collect_geom;;
open Dih2k_hypermap;;
open Wjscpro;;
open Tecoxbm;;
open Hdplygy;;
open Nkezbfc_local;;
open Flyspeck_constants;;
open Gbycpxs;;


let IS_SY = new_definition`IS_SY (s:stable_sy) p q={x| ?n m. n<m /\ m< k_sy s /\ ((f_sy s) POWER n) x=q /\ ((f_sy s) POWER m) x=p }`;;

let kl_sy = new_definition`kl_sy (s:stable_sy) p q=CARD(IS_SY (s:stable_sy) p q)`;;

let fl_sy = new_definition`fl_sy (s:stable_sy) p q i= (if ~(i=q) then f_sy s i else p)`;;



let COVER1_SY=new_definition`COVER1_SY p q (s:stable_sy) (s1:stable_sy) (s2:stable_sy) <=> I_SY s1= IS_SY s p q /\ I_SY s2= IS_SY s q p
/\ f_sy s1= fl_sy s p q /\ f_sy s2= fl_sy s q p`;;

let COVER2_SY=new_definition`COVER2_SY (s:stable_sy) (s1:stable_sy) (s2:stable_sy) <=> d_sy s <= d_sy s1 + d_sy s2 `;;


let COVER3_SY=new_definition`COVER3_SY p q (s:stable_sy) (s1:stable_sy) (s2:stable_sy) <=> J_SY s1 SUBSET J_SY s UNION {{p,q}}/\ 
J_SY s2 SUBSET J_SY s UNION {{p,q}}`;;

let COVER4_SY=new_definition`COVER4_SY p q (s:stable_sy) (s1:stable_sy) (s2:stable_sy) <=>
(!i j. ~({i,j }={p,q}) /\ i IN I_SY s1 /\ j IN I_SY s1
==> a_sy s1 (i,j)=a_sy s (i,j)/\ b_sy s1 (i,j)=b_sy s (i,j))
/\ 
(!i j. ~({i,j }={p,q}) /\ i IN I_SY s2 /\ j IN I_SY s2
==> a_sy s2 (i,j)=a_sy s (i,j)/\ b_sy s2 (i,j)=b_sy s (i,j))`;;


let COVER5_SY=new_definition`COVER5_SY p q (s:stable_sy) (s1:stable_sy) (s2:stable_sy) <=>
a_sy s1 (p,q)<= a_sy s (p,q)/\ a_sy s2 (p,q)<= a_sy s (p,q)/\ a_sy s (p,q)<=b_sy s1 (p,q)
/\ a_sy s (p,q)<=b_sy s2 (p,q)`;;

let COVER6_SY=new_definition`COVER6_SY p q (s:stable_sy) (s1:stable_sy) (s2:stable_sy) <=>
(({p,q} IN J_SY s1 <=> {p,q} IN J_SY s2 )
/\ ({p,q} IN J_SY s1 <=> ear_sy s1 \/ ear_sy s2))`;;

let COVER_SY=new_definition`COVER_SY p q (s:stable_sy) (s1:stable_sy) (s2:stable_sy) <=> COVER1_SY p q (s:stable_sy) (s1:stable_sy) (s2:stable_sy)
/\ COVER2_SY (s:stable_sy) (s1:stable_sy) (s2:stable_sy)
/\ COVER3_SY p q (s:stable_sy) (s1:stable_sy) (s2:stable_sy)
/\ COVER4_SY p q (s:stable_sy) (s1:stable_sy) (s2:stable_sy)
/\ COVER5_SY p q (s:stable_sy) (s1:stable_sy) (s2:stable_sy)
/\ COVER6_SY p q (s:stable_sy) (s1:stable_sy) (s2:stable_sy)`;;



let pmat1=new_definition`pmat1 (A:real^N^M)= (lambda i j. if i< dimindex(:P) then A$i$j else A$ dimindex(:M)$j):real^N^P`;;


let pmat2=new_definition`pmat2 (A:real^N^M)= (lambda i j.   A$(dimindex(:M)- dimindex(:P)+i)$j):real^N^P`;;


let DIA_SY=new_definition`DIA_SY p q (s:stable_sy) <=> ~(p=q) /\ p IN I_SY(s) /\ q IN I_SY s /\ ~(?i. i IN I_SY s /\ {p,q}= {i, f_sy s i})`;;






let CARD_I_SY_LT_3=prove(`
DIA_SY p q s
==> 3< CARD (I_SY s)`,
MRESA1_TAC stable_sy_lemma`s:stable_sy`
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[stable_system;constraint_system;torsor;
HAS_SIZE;]
THEN REWRITE_TAC[DIA_SY]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC(ARITH_RULE`3<= A /\ ~(A=3) ==> 3< A`)
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN MATCH_MP_TAC MONO_NOT
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[th;ARITH_RULE`~(3<3)`])
THEN STRIP_TAC
THEN STRIP_TAC
THEN DISCH_THEN(LABEL_TAC"THY1")
THEN DISCH_THEN(LABEL_TAC"THY2")
THEN DISCH_THEN(LABEL_TAC"THY3")
THEN DISCH_THEN(LABEL_TAC"THY4")
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_THEN"THY1" (fun th-> MRESA1_TAC th`p:num` THEN MRESA1_TAC th`f_sy (s:stable_sy) (p:num)`)
THEN REMOVE_THEN"THY3" (fun th-> MRESAL_TAC th[`SUC 0`;`p:num`][ARITH_RULE`0<SUC 0 /\ SUC 0<3`;POWER;o_DEF;I_DEF] THEN MRESAL_TAC th[`SUC (SUC 0)`;`p:num`][ARITH_RULE`0<SUC (SUC 0) /\ SUC (SUC 0)<3`;POWER;o_DEF;I_DEF]
THEN MRESAL_TAC th[`SUC 0`;`f_sy (s:stable_sy) (p:num):num`][ARITH_RULE`0<SUC 0 /\ SUC 0<3`;POWER;o_DEF;I_DEF])
THEN MRESA_TAC Geomdetail.CARD3[`p:num`;`f_sy (s:stable_sy) (p:num)`;`f_sy (s:stable_sy) (f_sy (s:stable_sy) (p:num))`]
THEN MP_TAC(SET_RULE`p IN I_SY s
/\ f_sy s p IN I_SY s
/\ f_sy s (f_sy s p) IN I_SY (s:stable_sy)
==> {p, f_sy s p, f_sy s (f_sy s p)} SUBSET I_SY s `)
THEN RESA_TAC
THEN MRESA_TAC CARD_SUBSET_EQ[`{p, f_sy (s:stable_sy) p, f_sy s (f_sy s p)}`;`I_SY (s:stable_sy)`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN ASM_REWRITE_TAC[SET_RULE`A IN{B,C,D}<=> A= B \/ A=C\/ A=D`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]
THENL[EXISTS_TAC`p:num`
THEN ASM_REWRITE_TAC[];
EXISTS_TAC`f_sy s (f_sy (s:stable_sy)p)`
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN"THY4" (fun th-> MRESA1_TAC th`p:num`)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`3=SUC(SUC(SUC 0))`;POWER;o_DEF;I_DEF]
THEN RESA_TAC
THEN SET_TAC[]]);;

let COVER_NOT_EAR_SY=prove(`!(s:stable_sy).
DIA_SY p q s
==> ~(ear_sy s)`,
REWRITE_TAC[ear_sy;]
THEN GEN_TAC THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL CARD_I_SY_LT_3)[`p:num`;`q:num`;`s:stable_sy`]
THEN REWRITE_TAC[DE_MORGAN_THM]
THEN MP_TAC(ARITH_RULE`3 < CARD (I_SY s) ==> ~(CARD (I_SY (s:stable_sy))= 3)`)
THEN RESA_TAC);;

let DIAGONAL_SY=prove(`DIA_SY p q s
==> ~({p,q} IN J_SY s) `,
MRESA1_TAC stable_sy_lemma`s:stable_sy`
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[stable_system;constraint_system;DIA_SY]
THEN REPEAT STRIP_TAC
THEN MP_TAC(SET_RULE`J_SY s SUBSET {{i, f_sy s i} | i | i IN I_SY (s:stable_sy)}
/\ {p, q} IN J_SY s
==> {p, q} IN {{i, f_sy s i} | i | i IN I_SY (s:stable_sy)}`)
THEN ASM_REWRITE_TAC[IN_ELIM_THM]
THEN STRIP_TAC);;

let SCHANGE= new_definition`SCHANGE (f:num->num) (s:stable_sy) (s1:stable_sy)<=>
k_sy s= k_sy s1
/\ d_sy s= d_sy s1
/\ (I_SY s)=IMAGE f (I_SY s1)
/\ (!p q. {f p, f q} IN J_SY s<=> {p,q} IN J_SY s1)
/\ (!p q. a_sy s (f p, f q)=a_sy s1 (p,q))
/\ (!p q. b_sy s (f p, f q)=b_sy s1 (p,q))
/\ f_sy s1 =(f_sy s) o f
`;;



let IK_SY=prove(`1<= p/\ I_SY s= 0..(p-1) ==> k_sy (s:stable_sy)= p`,
STRIP_TAC
THEN MP_TAC(ARITH_RULE`1<= p ==> (p - 1 + 1)-0 = p`)
THEN RESA_TAC
THEN MRESA1_TAC stable_sy_lemma`s:stable_sy`
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[stable_system;constraint_system;torsor;HAS_SIZE;CARD_NUMSEG;]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]);;


let K_SY_LE2=prove(`2< k_sy (s:stable_sy)`,
MRESA1_TAC stable_sy_lemma`s:stable_sy`
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[stable_system;constraint_system;torsor;HAS_SIZE;CARD_NUMSEG;]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN STRIP_TAC
THEN REPEAT(STRIP_TAC THEN REMOVE_ASSUM_TAC)
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC);;

let IN_J_IMP_IN_J1_SY=prove(` k_sy s1=p
/\ I_SY s1= 0..p-1
/\ f_sy s1=(\i. ((1+i):num MOD p))
/\ {0, p - 1} IN J_SY s1
==> p-1,p IN J1_SY (s1:stable_sy)`,
REPEAT STRIP_TAC
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM]
THEN MP_TAC(ARITH_RULE`2<p ==> SUC(p-1)= p/\1<= p-1 /\ p-1<=p /\ p-1< p /\ ~(p=0) /\ 1+p-1=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`]
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p`]
THEN EXISTS_TAC`p-1:num`
THEN ASM_REWRITE_TAC[IN_NUMSEG]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN ASM_REWRITE_TAC[]);;



let D_FUN_COVER_SY=prove_by_refinement(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
==> d_fun(s,l) <= d_fun(s1,l1) + d_fun(s3,l2)`,
[REPEAT STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p `)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN MRESA_TAC (GEN_ALL COVER_NOT_EAR_SY)[`0`;`p-1`;`s:stable_sy`] 
THEN MRESA1_TAC sigma_sy`s:stable_sy`
THEN MRESA_TAC (GEN_ALL DIAGONAL_SY)[`0:num`;`p-1`;`s:stable_sy`]
THEN DISJ_CASES_TAC(SET_RULE`{0,p-1} IN J_SY (s1:stable_sy)\/ ~({0,p-1} IN J_SY (s1:stable_sy))`);
FIND_ASSUM MP_TAC `COVER_SY 0 (p - 1) s s1 (s2:stable_sy)`
THEN REWRITE_TAC[COVER_SY;COVER3_SY;COVER6_SY]
THEN RESA_TAC;
ASM_REWRITE_TAC[d_fun]
THEN MRESA1_TAC sigma_sy`s1:stable_sy`
THEN FIND_ASSUM MP_TAC`SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 (s3:stable_sy)`
THEN REWRITE_TAC[SCHANGE]
THEN RESA_TAC
THEN SUBGOAL_THEN`{0,1} IN J_SY (s3:stable_sy)`ASSUME_TAC;
REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`0`;`1`])
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th;ARITH_RULE`A+0=A/\ ~(1=0)`])
THEN MP_TAC(ARITH_RULE`2<p ==> p-2+1 =p-1`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`];
MRESA_TAC (GEN_ALL IN_J_IMP_IN_J1_SY)[`p:num`;`s1:stable_sy`]
THEN SUBGOAL_THEN`(k-p+2,1) IN J1_SY (s3:stable_sy)`ASSUME_TAC;
REWRITE_TAC[J1_SY;IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN ASM_REWRITE_TAC[IN_NUMSEG]
THEN STRIP_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`k-p+2`
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\1+0=1/\ SUC 0=1 /\ k - p + 2 <= k - p + 2`]
THEN MRESA_TAC MOD_LT[`1:num`;`k-p+2:num`];
MRESAL_TAC SUM_DELETE[`(\x. cstab - norm (row (FST x) (vecmats l1) - row (SND x) (vecmats (l1:real^(P,3)finite_product))))`;`J1_SY (s1:stable_sy)`;`p-1,p`][FINITE_J1_SY;REAL_ARITH`A=B-C<=> B=A+C`]
THEN MRESAL_TAC SUM_DELETE[`(\x. cstab - norm (row (FST x) (vecmats l2) - row (SND x) (vecmats (l2:real^(N,3)finite_product))))`;`J1_SY (s3:stable_sy)`;`k-p+2,1`][FINITE_J1_SY;REAL_ARITH`A=B-C<=> B=A+C`]
THEN SUBGOAL_THEN`J1_SY s1  DELETE (p-1,p) DELETE (p,1) SUBSET J1_SY (s:stable_sy)` ASSUME_TAC;
ASM_REWRITE_TAC[SUBSET;J1_SY;IN_ELIM_THM;DELETE;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ;]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i< p:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`];
MP_TAC(ARITH_RULE`i<p /\ 2<p ==> i=p-1 \/ i< p-1:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p ==> SUC(p-1)=p /\ p-1<p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
STRIP_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<p-1 /\ p < k /\ 2<p ==> i< k /\ i<=k:num /\ 1+i<k/\ 1+i< p/\ ~(i=1+i)/\ ~(i=p-1)/\ ~(p=2)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[th] THEN REPEAT STRIP_TAC)
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`1+i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`1+i:num`;`p:num`]
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[SYM th] )
THEN FIND_ASSUM MP_TAC`J_SY s1 SUBSET J_SY (s:stable_sy) UNION {{0, p - 1}}`
THEN REWRITE_TAC[SUBSET;UNION;IN_ELIM_THM;IN_SING]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{i,(1+i) MOD p}`)
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC MOD_LT[`1+i:num`;`p:num`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`{i, 1 + i} = {0, p - 1} /\ ~(i=1+i) /\ ~(i=p-1) ==> i=0 /\ 1+i= p-1`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`1+0=p-1<=> p=2`];
SUBGOAL_THEN`IMAGE (\(x,y). (p - 2 + x), (p - 2 + y)) (J1_SY s3  DELETE (k-p+2,1)) SUBSET J1_SY (s:stable_sy)` ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;IMAGE;DELETE;SUBSET;IN_ELIM_THM;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ;]
THEN MP_TAC(ARITH_RULE`i<=k-p+2 ==> i=k-p+2 \/ i< k-p+2:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`];
MP_TAC(ARITH_RULE`i<k-p+2 /\ 2<k-p+2 ==> i=k-p+1 \/ i< k-p+1:num`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN RESA_TAC
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`2<p /\ 2< k-p+2 /\ 2< k ==> p-2+k-p+1=k-1/\ SUC(k-p+1)=k-p+2 /\ k-p+1<k-p+2/\ p - 2 + k - p + 2=k/\ 1<=k-1 /\ k-1<=k/\ k-1<k/\ 1+k-1=k /\ SUC(k-1)=k/\ 1 + k - p + 1= k - p + 2 /\ ~(k-p+1=0)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`k-p+1:num`;`k-p+2:num`]
THEN MRESA_TAC MOD_LT[`k-1:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`]
THEN EXISTS_TAC`k-1`
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`{(k - p + 1) MOD (k - p + 2), (f_sy s2 o
                                     (\x. if x = 0
                                          then 0
                                          else (p - 2 + x) MOD k))
                                    ((k - p + 1) MOD (k - p + 2))} IN
      J_SY (s3:stable_sy)`
THEN FIND_ASSUM(fun th-> REWRITE_TAC[SYM th])`(\i. (1 + i) MOD (k - p + 2)) =
      f_sy (s2:stable_sy) o (\x. if x = 0 then 0 else (p - 2 + x) MOD k)`
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`]
THEN STRIP_TAC
THEN FIND_ASSUM (fun th-> MP_TAC( ISPECL [`k-p+1`;`0`]th))`!p' q.
          {(if p' = 0 then 0 else (p - 2 + p') MOD k), (if q = 0
                                                        then 0
                                                        else (p - 2 + q) MOD
                                                             k)} IN
          J_SY s2 <=>
          {p', q} IN J_SY (s3:stable_sy)`
THEN POP_ASSUM (fun th-> REWRITE_TAC[th;ARITH_RULE`0=0`])
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`J_SY s2 SUBSET J_SY (s:stable_sy) UNION {{0, p - 1}}`
THEN REWRITE_TAC[SUBSET;UNION;IN_ELIM_THM;IN_SING]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{k-1,0}`);
POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<=k-p /\2<p==> ~(k-1=p-1) /\ ~(0= p-1)`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(k-1=p-1) /\ ~(0= p-1)==> ~(p-1 IN {k-1,0})`)
THEN ASM_REWRITE_TAC[]
THEN SET_TAC[];
STRIP_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`p-2+i:num`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<k - p + 1 /\ 2<p /\ 1<=i==> p-2+i< k/\ p-2+i<=k /\ 1<=p-2+i/\ 1+p-2+i<k /\ 1+i< k-p+2/\ p-2+1+i<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+1+i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`i<k - p + 1 /\ 2<p /\ 1<=i==>  ~(i=0)/\ ~(1+i=0)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`1+i:num`;`k-p+2:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN REWRITE_TAC[ARITH_RULE`SUC (p - 2 + i)= p - 2 + SUC i`]
THEN MRESA_TAC MOD_LT[`1+p-2+i:num`;`k:num`]
THEN FIND_ASSUM (fun th-> MP_TAC( ISPECL [`i MOD (k - p + 2)`;`(f_sy (s2:stable_sy) o (\x. if x = 0 then 0 else (p - 2 + x) MOD k))
                          (i MOD (k - p + 2))`] th))`!p' q.
          {(if p' = 0 then 0 else (p - 2 + p') MOD k), (if q = 0
                                                        then 0
                                                        else (p - 2 + q) MOD
                                                             k)} IN
          J_SY s2 <=>
          {p', q} IN J_SY  (s3:stable_sy)`
THEN FIND_ASSUM(fun th-> REWRITE_TAC[th])`{i MOD (k - p + 2), (f_sy s2 o (\x. if x = 0 then 0 else (p - 2 + x) MOD k))
                          (i MOD (k - p + 2))} IN
      J_SY (s3:stable_sy)` 
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN FIND_ASSUM(fun th-> REWRITE_TAC[SYM th])`(\i. (1 + i) MOD (k - p + 2)) =
      f_sy (s2:stable_sy) o (\x. if x = 0 then 0 else (p - 2 + x) MOD k)`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`J_SY s2 SUBSET J_SY (s:stable_sy) UNION {{0, p - 1}}`
THEN REWRITE_TAC[SUBSET;UNION;IN_ELIM_THM;IN_SING]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{p-2+i,p-2+1+i}`);
ASM_REWRITE_TAC[ARITH_RULE`1+p-2+i= p-2+1+i`];
MP_TAC(SET_RULE`{p - 2 + i, p - 2 + 1 + i} = {0, p - 1} ==> (p-2+i=0) \/ (p-2+1+i=0)`)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`1<=i /\ 2< p==> ~(p-2+i=0) /\ ~(p-2+1+i=0)`)
THEN ASM_REWRITE_TAC[];
MRESAL_TAC FINITE_DELETE_IMP[`J1_SY (s1:stable_sy)`;`p-1,p`][FINITE_J1_SY]
THEN MRESAL_TAC FINITE_DELETE_IMP[`J1_SY (s3:stable_sy)`;`k-p+2,1`][FINITE_J1_SY]
THEN DISJ_CASES_TAC(SET_RULE`ear_sy s3 \/ ~(ear_sy (s3:stable_sy))`);
ASM_REWRITE_TAC[sigma_sy]
THEN MATCH_MP_TAC(REAL_ARITH`
d_sy s <=
 d_sy s1 +
   d_sy s3
/\ 
&0<= sum (J1_SY s)
 (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats l)))
/\
&0<=  sum (J1_SY s1 DELETE (p - 1,p))
   (\x. cstab - norm (row (FST x) (vecmats l1) - row (SND x) (vecmats l1)))
/\ &0<=   cstab - norm (row (p - 1) (vecmats l1) - row p (vecmats l1))
/\
&0<=
 sum (J1_SY s3 DELETE (k - p + 2,1))
  (\x. cstab - norm (row (FST x) (vecmats l2) - row (SND x) (vecmats l2))) 
/\ &0<=  cstab - norm (row (k - p + 2) (vecmats l2) - row 1 (vecmats l2))
==>
d_sy s +
 #0.1 *
 -- &1 *
 sum (J1_SY s)
 (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats l))) <=
 (d_sy s1 +
  #0.1 *
  &1 *
  (sum (J1_SY s1 DELETE (p - 1,p))
   (\x. cstab - norm (row (FST x) (vecmats l1) - row (SND x) (vecmats l1))) +
   cstab - norm (row (p - 1) (vecmats l1) - row p (vecmats l1)))) +
 d_sy s3 +
 #0.1 *
 &1 *
 (sum (J1_SY (s3:stable_sy) DELETE (k - p + 2,1))
  (\x. cstab - norm (row (FST x) (vecmats l2) - row (SND x) (vecmats l2))) +
  cstab - norm (row (k - p + 2) (vecmats l2) - row 1 (vecmats l2)))`)
THEN RESA_TAC;
FIND_ASSUM MP_TAC `COVER2_SY s s1 (s2:stable_sy)`
THEN REWRITE_TAC[COVER2_SY]
THEN RESA_TAC;
STRIP_TAC;
MATCH_MP_TAC SUM_POS_LE
THEN REWRITE_TAC[FINITE_J1_SY]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;IN_NUMSEG]
THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL B_SY_LE_CSTAB)[`i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC;
MATCH_MP_TAC SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(SET_RULE`x=p:num,1 \/ ~(x=p:num,1 )`);
ASM_REWRITE_TAC[]
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN RESA_TAC
THEN SUBGOAL_THEN`row 1 ((pmat1 (vecmats l)):real^3^P) = row 1 (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2<p==> 1<=1 /\ 1<p `)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p`]
THEN MRESAL_TAC (GEN_ALL B_SY_LE_CSTAB)[`k:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;][ARITH_RULE`SUC 0=1`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
SUBGOAL_THEN`x IN J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1)` ASSUME_TAC;
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[IN_ELIM_THM;DELETE]
THEN SET_TAC[];
MP_TAC(SET_RULE`x IN J1_SY s1 DELETE (p - 1,p) DELETE (p,1)
/\ J1_SY s1 DELETE (p - 1,p) DELETE (p,1) SUBSET J1_SY s
==> x IN J1_SY (s:stable_sy)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN DISCH_THEN(LABEL_TAC"THY1")
THEN REPEAT STRIP_TAC
THEN REMOVE_THEN"THY" MP_TAC
THEN REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM;IN_NUMSEG]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_THEN"THY1" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=p==> i=p \/ i<p:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< p /\ 2<p ==> i=p -1 \/ SUC i<p:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p ==> p-1< p /\ SUC (p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
RESA_TAC
THEN RESA_TAC
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN SUBGOAL_THEN`row i ((pmat1 (vecmats l)):real^3^P) = row i (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat1 (vecmats l)):real^3^P) = row (SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`SUC i < p==> 1<= SUC i /\ SUC i <= p`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;IN_NUMSEG]
THEN STRIP_TAC
THEN MP_TAC(ARITH_RULE`SUC i < p /\ 1<=k-p ==> i <= k/\ i<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC (GEN_ALL B_SY_LE_CSTAB)[`i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC;
SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`p <= p`]
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN RESA_TAC
THEN SUBGOAL_THEN`row (p-1) ((pmat1 (vecmats l)):real^3^P) = row (p-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2< p ==> p-1 <= p /\ 1<= p-1/\ p-1<p`)
THEN RESA_TAC
THEN  ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN FIND_ASSUM (fun th-> MP_TAC(ISPECL [`p-1`;`k:num`] th) )`!i j.
          {i MOD k, j MOD k} IN J_SY s UNION {{0, p - 1}}
          ==> norm (row i (vecmats l) - row j (vecmats (l:real^(M,3)finite_product))) <= cstab`
THEN REWRITE_TAC[IN_ELIM_THM;UNION;IN_SING]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`;SET_RULE`{p-1,0}={0,p-1}`]
THEN REAL_ARITH_TAC;
STRIP_TAC;
MATCH_MP_TAC SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(SET_RULE`x=k-p+1,k-p+2 \/ ~(x=k-p+1,k-p+2 )`);
ASM_REWRITE_TAC[]
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN SUBGOAL_THEN`row (k-p+1) ((pmat2 (vecmats l)):real^3^N) = row (k-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+1<=k-p+2 /\ 1<= k-p+1 /\ k - (k - p + 2) + k - p + 1= k-1`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (k-p+2) ((pmat2 (vecmats l)):real^3^N) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+2<=k-p+2 /\ 1<= k-p+2 /\ k - (k - p + 2) + k - p + 2= k`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE` 2<k ==> 1<=k-1 /\ k-1<= k /\ SUC (k - 1)= k /\ k-1<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`k-1:num`;`k:num`]
THEN MRESAL_TAC (GEN_ALL B_SY_LE_CSTAB)[`k-1:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;][ARITH_RULE`SUC 0=1`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN DISCH_THEN(LABEL_TAC"THY1")
THEN REMOVE_THEN"THY" MP_TAC
THEN REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM;IN_NUMSEG]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_THEN"THY1" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=k-p+2==> i=k-p+2 \/ i<k-p+2:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< k-p+2 /\ 2<p ==> i=k-p +1 \/ i< k-p+1:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p /\ 1<= k-p ==> k-p+1< k-p+2 /\ SUC (k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`k-p+1:num`;`k-p+2:num`];
RESA_TAC
THEN RESA_TAC
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN SUBGOAL_THEN`row i ((pmat2 (vecmats l)):real^3^N) = row (p-2+i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + i= p-2+i`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat2 (vecmats l)):real^3^N) = row (p-2+ SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p/\ i< k-p+1 ==> k - (k - p + 2) + SUC i= p-2+SUC i/\ 1<= SUC i /\ SUC i<= k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE` i< k-p+1 /\ 2< p /\ 1<= k-p ==> 1<=p-2+i /\ p-2+i <= k /\ SUC (p-2+i)= p-2+ SUC i /\ p-2+i<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MRESAL_TAC (GEN_ALL B_SY_LE_CSTAB)[`p-2+i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;][ARITH_RULE`SUC 0=1`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
SUBGOAL_THEN`row (k-p+2) ((pmat2 (vecmats l)):real^3^N) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+2<=k-p+2 /\ 1<= k-p+2 /\ k - (k - p + 2) + k - p + 2= k`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row 1 ((pmat2 (vecmats l)):real^3^N) = row (p-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + 1= p-1/\ 1<=1 /\ 1<=k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM (fun th-> MP_TAC(ISPECL [`k:num`;`p-1:num`] th) )`!i j.
          {i MOD k, j MOD k} IN J_SY s UNION {{0, p - 1}}
          ==> norm (row i (vecmats l) - row j (vecmats (l:real^(M,3)finite_product))) <= cstab`
THEN REWRITE_TAC[IN_ELIM_THM;UNION;IN_SING]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`;SET_RULE`{p-1,0}={0,p-1}`]
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[sigma_sy]
THEN SUBGOAL_THEN `cstab - norm (row (p - 1) (vecmats (l1:real^(P,3)finite_product)) - row p (vecmats l1)) =cstab - norm (row (k-p + 2) (vecmats (l2:real^(N,3)finite_product)) - row 1 (vecmats l2))` ASSUME_TAC;
REWRITE_TAC[REAL_ARITH`A-B=A-C<=> B=C`]
THEN EXPAND_TAC"l1"
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN SUBGOAL_THEN`row 1 ((pmat2 (vecmats l)):real^3^N) = row (p-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + 1= p-1/\ 1<=1 /\ 1<=k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (k-p+2) ((pmat2 (vecmats l)):real^3^N) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+2<=k-p+2 /\ 1<= k-p+2 /\ k - (k - p + 2) + k - p + 2= k`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (p-1) ((pmat1 (vecmats l)):real^3^P) = row (p-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2< p ==> p-1 <= p /\ 1<= p-1/\ p-1<p`)
THEN RESA_TAC
THEN  ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`p <= p`]
THEN RESA_TAC
THEN SIMP_TAC[NORM_SUB];
ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN`(!x y.
           x IN J1_SY s3 DELETE (k - p + 2,1) /\
           y IN J1_SY (s3:stable_sy) DELETE (k - p + 2,1) /\
           (\(x,y). p - 2 + x,p - 2 + y) x = (\(x,y). p - 2 + x,p - 2 + y) y
           ==> x = y)`ASSUME_TAC;
REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ;ARITH_RULE`p-2+B=p-2+C<=> B=C`];
SUBGOAL_THEN`!x. x IN (J1_SY (s3:stable_sy) DELETE (k - p + 2,1))
==> 
      ((\x. cstab - norm (row (FST x) (vecmats (l:real^(M,3)finite_product)) - row (SND x) (vecmats l))) o
       (\(x,y). p - 2 + x,p - 2 + y)) x
= (\x. cstab - norm (row (FST x) (vecmats (l2:real^(N,3)finite_product)) - row (SND x) (vecmats l2))) x`
ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM;o_DEF;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A-B=A-C<=> B=C`]
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`i<=k-p+2==> i=k-p+2 \/ i<k-p+2:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN SUBGOAL_THEN`row i ((pmat2 (vecmats l)):real^3^N) = row (p-2+i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + i= p-2+i`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat2 (vecmats l)):real^3^N) = row (p-2+ SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p/\ i< k-p+2 ==> k - (k - p + 2) + SUC i= p-2+SUC i/\ 1<= SUC i /\ SUC i<= k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[];
MRESA_TAC SUM_EQ[`      ((\x. cstab - norm (row (FST x) (vecmats (l:real^(M,3)finite_product)) - row (SND x) (vecmats l))) o
       (\(x,y). p - 2 + x,p - 2 + y))`;`(\x. cstab - norm (row (FST x) (vecmats (l2:real^(N,3)finite_product)) - row (SND x) (vecmats l2)))`;`(J1_SY (s3:stable_sy) DELETE (k - p + 2,1))`]
THEN
MRESA_TAC SUM_IMAGE[`(\(x,y). p - 2 + x,p - 2 + y)`;`(\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product))))`;`J1_SY (s3:stable_sy) DELETE (k - p + 2,1)`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN MRESAL_TAC SUM_DIFF[`(\x. cstab - norm (row (FST x) (vecmats (l:real^(M,3)finite_product)) - row (SND x) (vecmats l)))`;`J1_SY(s:stable_sy)`;`(IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY (s3:stable_sy) DELETE (k - p + 2,1)))`][FINITE_J1_SY]
THEN MATCH_MP_TAC(REAL_ARITH`
d_sy s <= d_sy s1+  d_sy s3
/\ &0<= sum (J1_SY s1 DELETE (p - 1,p))
   (\x. cstab - norm (row (FST x) (vecmats l1) - row (SND x) (vecmats l1)))
/\ &0<=
sum (J1_SY s)
 (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats l)))
- sum (IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY s3 DELETE (k - p + 2,1)))
  (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats l)))
==> 
d_sy s +
 #0.1 *
 -- &1 *
 sum (J1_SY (s:stable_sy))
 (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats l))) <=
 (d_sy s1 +
  #0.1 *
  &1 *
  (sum (J1_SY s1 DELETE (p - 1,p))
   (\x. cstab - norm (row (FST x) (vecmats (l1:real^(P,3)finite_product)) - row (SND x) (vecmats l1))) +
   cstab - norm (row (k - p + 2) (vecmats l2) - row 1 (vecmats l2)))) +
 d_sy s3 +
 #0.1 *
 -- &1 *
 (sum (IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY s3 DELETE (k - p + 2,1)))
  (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product)))) +
  cstab - norm (row (k - p + 2) (vecmats (l2:real^(N,3)finite_product)) - row 1 (vecmats l2)))`)
THEN RESA_TAC;
FIND_ASSUM MP_TAC `COVER2_SY s s1 (s2:stable_sy)`
THEN REWRITE_TAC[COVER2_SY]
THEN RESA_TAC;
STRIP_TAC;
MATCH_MP_TAC SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(SET_RULE`x=p:num,1 \/ ~(x=p:num,1 )`);
ASM_REWRITE_TAC[]
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN RESA_TAC
THEN SUBGOAL_THEN`row 1 ((pmat1 (vecmats l)):real^3^P) = row 1 (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2<p==> 1<=1 /\ 1<p `)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p`]
THEN MRESAL_TAC (GEN_ALL B_SY_LE_CSTAB)[`k:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;][ARITH_RULE`SUC 0=1`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
SUBGOAL_THEN`x IN J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1)` ASSUME_TAC;
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[IN_ELIM_THM;DELETE]
THEN SET_TAC[];
MP_TAC(SET_RULE`x IN J1_SY s1 DELETE (p - 1,p) DELETE (p,1)
/\ J1_SY s1 DELETE (p - 1,p) DELETE (p,1) SUBSET J1_SY s
==> x IN J1_SY (s:stable_sy)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN DISCH_THEN(LABEL_TAC"THY1")
THEN REPEAT STRIP_TAC
THEN REMOVE_THEN"THY" MP_TAC
THEN REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM;IN_NUMSEG]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_THEN"THY1" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=p==> i=p \/ i<p:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< p /\ 2<p ==> i=p -1 \/ SUC i<p:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p ==> p-1< p /\ SUC (p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
RESA_TAC
THEN RESA_TAC
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN SUBGOAL_THEN`row i ((pmat1 (vecmats l)):real^3^P) = row i (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat1 (vecmats l)):real^3^P) = row (SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`SUC i < p==> 1<= SUC i /\ SUC i <= p`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;IN_NUMSEG]
THEN STRIP_TAC
THEN MP_TAC(ARITH_RULE`SUC i < p /\ 1<=k-p ==> i <= k/\ i<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC (GEN_ALL B_SY_LE_CSTAB)[`i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN MATCH_MP_TAC SUM_POS_LE
THEN STRIP_TAC;
MATCH_MP_TAC FINITE_DIFF
THEN REWRITE_TAC[FINITE_J1_SY];
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;IN_NUMSEG;DIFF]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC (GEN_ALL B_SY_LE_CSTAB)[`i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[d_fun]
THEN MRESA1_TAC sigma_sy`s3:stable_sy`
THEN FIND_ASSUM MP_TAC`SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 (s3:stable_sy)`
THEN REWRITE_TAC[SCHANGE]
THEN RESA_TAC
THEN SUBGOAL_THEN`{0,1} IN J_SY (s3:stable_sy)`ASSUME_TAC;
REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`0`;`1`])
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th;ARITH_RULE`A+0=A/\ ~(1=0)`])
THEN MP_TAC(ARITH_RULE`2<p ==> p-2+1 =p-1`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`];
MRESA_TAC (GEN_ALL IN_J_IMP_IN_J1_SY)[`p:num`;`s1:stable_sy`]
THEN SUBGOAL_THEN`(k-p+2,1) IN J1_SY (s3:stable_sy)`ASSUME_TAC;
REWRITE_TAC[J1_SY;IN_ELIM_THM]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN ASM_REWRITE_TAC[IN_NUMSEG]
THEN STRIP_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`k-p+2`
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\1+0=1/\ SUC 0=1 /\ k - p + 2 <= k - p + 2`]
THEN MRESA_TAC MOD_LT[`1:num`;`k-p+2:num`];
MRESAL_TAC SUM_DELETE[`(\x. cstab - norm (row (FST x) (vecmats l1) - row (SND x) (vecmats (l1:real^(P,3)finite_product))))`;`J1_SY (s1:stable_sy)`;`p-1,p`][FINITE_J1_SY;REAL_ARITH`A=B-C<=> B=A+C`]
THEN MRESAL_TAC SUM_DELETE[`(\x. cstab - norm (row (FST x) (vecmats l2) - row (SND x) (vecmats (l2:real^(N,3)finite_product))))`;`J1_SY (s3:stable_sy)`;`k-p+2,1`][FINITE_J1_SY;REAL_ARITH`A=B-C<=> B=A+C`]
THEN SUBGOAL_THEN`J1_SY s1  DELETE (p-1,p) DELETE (p,1) SUBSET J1_SY (s:stable_sy)` ASSUME_TAC;
ASM_REWRITE_TAC[SUBSET;J1_SY;IN_ELIM_THM;DELETE;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ;]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i< p:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`];
MP_TAC(ARITH_RULE`i<p /\ 2<p ==> i=p-1 \/ i< p-1:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p ==> SUC(p-1)=p /\ p-1<p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
STRIP_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<p-1 /\ p < k /\ 2<p ==> i< k /\ i<=k:num /\ 1+i<k/\ 1+i< p/\ ~(i=1+i)/\ ~(i=p-1)/\ ~(p=2)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[th] THEN REPEAT STRIP_TAC)
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`1+i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`1+i:num`;`p:num`]
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[SYM th] )
THEN FIND_ASSUM MP_TAC`J_SY s1 SUBSET J_SY (s:stable_sy) UNION {{0, p - 1}}`
THEN REWRITE_TAC[SUBSET;UNION;IN_ELIM_THM;IN_SING]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{i,(1+i) MOD p}`)
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC MOD_LT[`1+i:num`;`p:num`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`{i, 1 + i} = {0, p - 1} /\ ~(i=1+i) /\ ~(i=p-1) ==> i=0 /\ 1+i= p-1`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`1+0=p-1<=> p=2`];
SUBGOAL_THEN`IMAGE (\(x,y). (p - 2 + x), (p - 2 + y)) (J1_SY s3  DELETE (k-p+2,1)) SUBSET J1_SY (s:stable_sy)` ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;IMAGE;DELETE;SUBSET;IN_ELIM_THM;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ;]
THEN MP_TAC(ARITH_RULE`i<=k-p+2 ==> i=k-p+2 \/ i< k-p+2:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`];
MP_TAC(ARITH_RULE`i<k-p+2 /\ 2<k-p+2 ==> i=k-p+1 \/ i< k-p+1:num`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN RESA_TAC
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`2<p /\ 2< k-p+2 /\ 2< k ==> p-2+k-p+1=k-1/\ SUC(k-p+1)=k-p+2 /\ k-p+1<k-p+2/\ p - 2 + k - p + 2=k/\ 1<=k-1 /\ k-1<=k/\ k-1<k/\ 1+k-1=k /\ SUC(k-1)=k/\ 1 + k - p + 1= k - p + 2 /\ ~(k-p+1=0)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`k-p+1:num`;`k-p+2:num`]
THEN MRESA_TAC MOD_LT[`k-1:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`]
THEN EXISTS_TAC`k-1`
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`{(k - p + 1) MOD (k - p + 2), (f_sy s2 o
                                     (\x. if x = 0
                                          then 0
                                          else (p - 2 + x) MOD k))
                                    ((k - p + 1) MOD (k - p + 2))} IN
      J_SY (s3:stable_sy)`
THEN FIND_ASSUM(fun th-> REWRITE_TAC[SYM th])`(\i. (1 + i) MOD (k - p + 2)) =
      f_sy (s2:stable_sy) o (\x. if x = 0 then 0 else (p - 2 + x) MOD k)`
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`]
THEN STRIP_TAC
THEN FIND_ASSUM (fun th-> MP_TAC( ISPECL [`k-p+1`;`0`]th))`!p' q.
          {(if p' = 0 then 0 else (p - 2 + p') MOD k), (if q = 0
                                                        then 0
                                                        else (p - 2 + q) MOD
                                                             k)} IN
          J_SY s2 <=>
          {p', q} IN J_SY (s3:stable_sy)`
THEN POP_ASSUM (fun th-> REWRITE_TAC[th;ARITH_RULE`0=0`])
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`J_SY s2 SUBSET J_SY (s:stable_sy) UNION {{0, p - 1}}`
THEN REWRITE_TAC[SUBSET;UNION;IN_ELIM_THM;IN_SING]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{k-1,0}`);
POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<=k-p /\2<p==> ~(k-1=p-1) /\ ~(0= p-1)`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(k-1=p-1) /\ ~(0= p-1)==> ~(p-1 IN {k-1,0})`)
THEN ASM_REWRITE_TAC[]
THEN SET_TAC[];
STRIP_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`p-2+i:num`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<k - p + 1 /\ 2<p /\ 1<=i==> p-2+i< k/\ p-2+i<=k /\ 1<=p-2+i/\ 1+p-2+i<k /\ 1+i< k-p+2/\ p-2+1+i<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+1+i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`i<k - p + 1 /\ 2<p /\ 1<=i==>  ~(i=0)/\ ~(1+i=0)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`1+i:num`;`k-p+2:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN REWRITE_TAC[ARITH_RULE`SUC (p - 2 + i)= p - 2 + SUC i`]
THEN MRESA_TAC MOD_LT[`1+p-2+i:num`;`k:num`]
THEN FIND_ASSUM (fun th-> MP_TAC( ISPECL [`i MOD (k - p + 2)`;`(f_sy (s2:stable_sy) o (\x. if x = 0 then 0 else (p - 2 + x) MOD k))
                          (i MOD (k - p + 2))`] th))`!p' q.
          {(if p' = 0 then 0 else (p - 2 + p') MOD k), (if q = 0
                                                        then 0
                                                        else (p - 2 + q) MOD
                                                             k)} IN
          J_SY s2 <=>
          {p', q} IN J_SY  (s3:stable_sy)`
THEN FIND_ASSUM(fun th-> REWRITE_TAC[th])`{i MOD (k - p + 2), (f_sy s2 o (\x. if x = 0 then 0 else (p - 2 + x) MOD k))
                          (i MOD (k - p + 2))} IN
      J_SY (s3:stable_sy)` 
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN FIND_ASSUM(fun th-> REWRITE_TAC[SYM th])`(\i. (1 + i) MOD (k - p + 2)) =
      f_sy (s2:stable_sy) o (\x. if x = 0 then 0 else (p - 2 + x) MOD k)`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`J_SY s2 SUBSET J_SY (s:stable_sy) UNION {{0, p - 1}}`
THEN REWRITE_TAC[SUBSET;UNION;IN_ELIM_THM;IN_SING]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{p-2+i,p-2+1+i}`);
ASM_REWRITE_TAC[ARITH_RULE`1+p-2+i= p-2+1+i`];
MP_TAC(SET_RULE`{p - 2 + i, p - 2 + 1 + i} = {0, p - 1} ==> (p-2+i=0) \/ (p-2+1+i=0)`)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`1<=i /\ 2< p==> ~(p-2+i=0) /\ ~(p-2+1+i=0)`)
THEN ASM_REWRITE_TAC[];
MRESAL_TAC FINITE_DELETE_IMP[`J1_SY (s1:stable_sy)`;`p-1,p`][FINITE_J1_SY]
THEN MRESAL_TAC FINITE_DELETE_IMP[`J1_SY (s3:stable_sy)`;`k-p+2,1`][FINITE_J1_SY]
THEN DISJ_CASES_TAC(SET_RULE`ear_sy s1 \/ ~(ear_sy (s1:stable_sy))`);
ASM_REWRITE_TAC[sigma_sy]
THEN MATCH_MP_TAC(REAL_ARITH`
d_sy s <=
 d_sy s1 +
   d_sy s3
/\ 
&0<= sum (J1_SY s)
 (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats l)))
/\
&0<=  sum (J1_SY s1 DELETE (p - 1,p))
   (\x. cstab - norm (row (FST x) (vecmats l1) - row (SND x) (vecmats l1)))
/\ &0<=   cstab - norm (row (p - 1) (vecmats l1) - row p (vecmats l1))
/\
&0<=
 sum (J1_SY s3 DELETE (k - p + 2,1))
  (\x. cstab - norm (row (FST x) (vecmats l2) - row (SND x) (vecmats l2))) 
/\ &0<=  cstab - norm (row (k - p + 2) (vecmats l2) - row 1 (vecmats l2))
==>
d_sy s +
 #0.1 *
 -- &1 *
 sum (J1_SY s)
 (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats l))) <=
 (d_sy s1 +
  #0.1 *
  &1 *
  (sum (J1_SY s1 DELETE (p - 1,p))
   (\x. cstab - norm (row (FST x) (vecmats l1) - row (SND x) (vecmats l1))) +
   cstab - norm (row (p - 1) (vecmats l1) - row p (vecmats l1)))) +
 d_sy s3 +
 #0.1 *
 &1 *
 (sum (J1_SY (s3:stable_sy) DELETE (k - p + 2,1))
  (\x. cstab - norm (row (FST x) (vecmats l2) - row (SND x) (vecmats l2))) +
  cstab - norm (row (k - p + 2) (vecmats l2) - row 1 (vecmats l2)))`)
THEN RESA_TAC;
FIND_ASSUM MP_TAC `COVER2_SY s s1 (s2:stable_sy)`
THEN REWRITE_TAC[COVER2_SY]
THEN RESA_TAC;
STRIP_TAC;
MATCH_MP_TAC SUM_POS_LE
THEN REWRITE_TAC[FINITE_J1_SY]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;IN_NUMSEG]
THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL B_SY_LE_CSTAB)[`i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC;
MATCH_MP_TAC SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(SET_RULE`x=p:num,1 \/ ~(x=p:num,1 )`);
ASM_REWRITE_TAC[]
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN RESA_TAC
THEN SUBGOAL_THEN`row 1 ((pmat1 (vecmats l)):real^3^P) = row 1 (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2<p==> 1<=1 /\ 1<p `)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p`]
THEN MRESAL_TAC (GEN_ALL B_SY_LE_CSTAB)[`k:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;][ARITH_RULE`SUC 0=1`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
SUBGOAL_THEN`x IN J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1)` ASSUME_TAC;
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[IN_ELIM_THM;DELETE]
THEN SET_TAC[];
MP_TAC(SET_RULE`x IN J1_SY s1 DELETE (p - 1,p) DELETE (p,1)
/\ J1_SY s1 DELETE (p - 1,p) DELETE (p,1) SUBSET J1_SY s
==> x IN J1_SY (s:stable_sy)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN DISCH_THEN(LABEL_TAC"THY1")
THEN REPEAT STRIP_TAC
THEN REMOVE_THEN"THY" MP_TAC
THEN REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM;IN_NUMSEG]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_THEN"THY1" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=p==> i=p \/ i<p:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< p /\ 2<p ==> i=p -1 \/ SUC i<p:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p ==> p-1< p /\ SUC (p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
RESA_TAC
THEN RESA_TAC
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN SUBGOAL_THEN`row i ((pmat1 (vecmats l)):real^3^P) = row i (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat1 (vecmats l)):real^3^P) = row (SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`SUC i < p==> 1<= SUC i /\ SUC i <= p`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;IN_NUMSEG]
THEN STRIP_TAC
THEN MP_TAC(ARITH_RULE`SUC i < p /\ 1<=k-p ==> i <= k/\ i<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC (GEN_ALL B_SY_LE_CSTAB)[`i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
STRIP_TAC;
SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`p <= p`]
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN RESA_TAC
THEN SUBGOAL_THEN`row (p-1) ((pmat1 (vecmats l)):real^3^P) = row (p-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2< p ==> p-1 <= p /\ 1<= p-1/\ p-1<p`)
THEN RESA_TAC
THEN  ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN FIND_ASSUM (fun th-> MP_TAC(ISPECL [`p-1`;`k:num`] th) )`!i j.
          {i MOD k, j MOD k} IN J_SY s UNION {{0, p - 1}}
          ==> norm (row i (vecmats l) - row j (vecmats (l:real^(M,3)finite_product))) <= cstab`
THEN REWRITE_TAC[IN_ELIM_THM;UNION;IN_SING]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`;SET_RULE`{p-1,0}={0,p-1}`]
THEN REAL_ARITH_TAC;
STRIP_TAC;
MATCH_MP_TAC SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(SET_RULE`x=k-p+1,k-p+2 \/ ~(x=k-p+1,k-p+2 )`);
ASM_REWRITE_TAC[]
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN SUBGOAL_THEN`row (k-p+1) ((pmat2 (vecmats l)):real^3^N) = row (k-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+1<=k-p+2 /\ 1<= k-p+1 /\ k - (k - p + 2) + k - p + 1= k-1`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (k-p+2) ((pmat2 (vecmats l)):real^3^N) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+2<=k-p+2 /\ 1<= k-p+2 /\ k - (k - p + 2) + k - p + 2= k`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE` 2<k ==> 1<=k-1 /\ k-1<= k /\ SUC (k - 1)= k /\ k-1<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`k-1:num`;`k:num`]
THEN MRESAL_TAC (GEN_ALL B_SY_LE_CSTAB)[`k-1:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;][ARITH_RULE`SUC 0=1`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN DISCH_THEN(LABEL_TAC"THY1")
THEN REMOVE_THEN"THY" MP_TAC
THEN REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM;IN_NUMSEG]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_THEN"THY1" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=k-p+2==> i=k-p+2 \/ i<k-p+2:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< k-p+2 /\ 2<p ==> i=k-p +1 \/ i< k-p+1:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p /\ 1<= k-p ==> k-p+1< k-p+2 /\ SUC (k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`k-p+1:num`;`k-p+2:num`];
RESA_TAC
THEN RESA_TAC
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN SUBGOAL_THEN`row i ((pmat2 (vecmats l)):real^3^N) = row (p-2+i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + i= p-2+i`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat2 (vecmats l)):real^3^N) = row (p-2+ SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p/\ i< k-p+1 ==> k - (k - p + 2) + SUC i= p-2+SUC i/\ 1<= SUC i /\ SUC i<= k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE` i< k-p+1 /\ 2< p /\ 1<= k-p ==> 1<=p-2+i /\ p-2+i <= k /\ SUC (p-2+i)= p-2+ SUC i /\ p-2+i<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MRESAL_TAC (GEN_ALL B_SY_LE_CSTAB)[`p-2+i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;][ARITH_RULE`SUC 0=1`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
SUBGOAL_THEN`row (k-p+2) ((pmat2 (vecmats l)):real^3^N) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+2<=k-p+2 /\ 1<= k-p+2 /\ k - (k - p + 2) + k - p + 2= k`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row 1 ((pmat2 (vecmats l)):real^3^N) = row (p-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + 1= p-1/\ 1<=1 /\ 1<=k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM (fun th-> MP_TAC(ISPECL [`k:num`;`p-1:num`] th) )`!i j.
          {i MOD k, j MOD k} IN J_SY s UNION {{0, p - 1}}
          ==> norm (row i (vecmats l) - row j (vecmats (l:real^(M,3)finite_product))) <= cstab`
THEN REWRITE_TAC[IN_ELIM_THM;UNION;IN_SING]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`;SET_RULE`{p-1,0}={0,p-1}`]
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[sigma_sy]
THEN SUBGOAL_THEN `cstab - norm (row (p - 1) (vecmats (l1:real^(P,3)finite_product)) - row p (vecmats l1)) =cstab - norm (row (k-p + 2) (vecmats (l2:real^(N,3)finite_product)) - row 1 (vecmats l2))` ASSUME_TAC;
REWRITE_TAC[REAL_ARITH`A-B=A-C<=> B=C`]
THEN EXPAND_TAC"l1"
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN SUBGOAL_THEN`row 1 ((pmat2 (vecmats l)):real^3^N) = row (p-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + 1= p-1/\ 1<=1 /\ 1<=k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (k-p+2) ((pmat2 (vecmats l)):real^3^N) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+2<=k-p+2 /\ 1<= k-p+2 /\ k - (k - p + 2) + k - p + 2= k`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (p-1) ((pmat1 (vecmats l)):real^3^P) = row (p-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2< p ==> p-1 <= p /\ 1<= p-1/\ p-1<p`)
THEN RESA_TAC
THEN  ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`p <= p`]
THEN RESA_TAC
THEN SIMP_TAC[NORM_SUB];
DISJ_CASES_TAC(SET_RULE`(p,1) IN  J1_SY s1 DELETE (p - 1,p) \/ ~((p,1) IN  J1_SY (s1:stable_sy) DELETE (p - 1,p))`);
SUBGOAL_THEN`k,1 IN J1_SY (s:stable_sy)`ASSUME_TAC;
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;DELETE;PAIR_EQ;IN_NUMSEG]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p /\ 1+0=1`]
THEN MP_TAC(ARITH_RULE`2<p /\ 2< k==> 1<p /\ 1<k /\ k<=k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`1:num`;`p:num`]
THEN STRIP_TAC
THEN EXISTS_TAC`k:num`
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1/\ 1+0=1`]
THEN MRESA_TAC MOD_LT[`1:num`;`k:num`]
THEN FIND_ASSUM MP_TAC`J_SY s1 SUBSET J_SY (s:stable_sy) UNION {{0, p - 1}}`
THEN REWRITE_TAC[SUBSET;UNION;IN_ELIM_THM;IN_SING]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{0,1}`);
MP_TAC(ARITH_RULE`2<p ==> ~(p-1=0) /\ ~(p-1=1)`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(p-1=0) /\ ~(p-1=1) ==> ~(p-1 IN {0,1})`)
THEN ASM_REWRITE_TAC[]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
MRESAL_TAC SUM_DELETE[`(\x. cstab - norm (row (FST x) (vecmats l1) - row (SND x) (vecmats (l1:real^(P,3)finite_product))))`;`J1_SY (s1:stable_sy) DELETE (p-1,p)`;`p:num,1`][FINITE_J1_SY;REAL_ARITH`A=B-C<=> B=A+C`]
THEN MRESAL_TAC SUM_DELETE[`(\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product))))`;`J1_SY (s:stable_sy)`;`k:num,1`][FINITE_J1_SY;REAL_ARITH`A=B-C<=> B=A+C`]
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN RESA_TAC
THEN SUBGOAL_THEN`row 1 ((pmat1 (vecmats l)):real^3^P) = row 1 (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2<p==> 1<=1 /\ 1<p `)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`J1_SY s1 DELETE (p - 1,p) DELETE (p,1) SUBSET J1_SY (s:stable_sy) DELETE (k,1)`ASSUME_TAC;
MATCH_MP_TAC(SET_RULE`~(x IN A) /\ A SUBSET B==> A SUBSET B DELETE x`)
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC(SET_RULE`~(x IN A) ==> ~(x IN A DELETE y)`)
THEN MATCH_MP_TAC(SET_RULE`~(x IN A) ==> ~(x IN A DELETE y)`)
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;PAIR_EQ;IN_NUMSEG]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<= k-p ==> ~(k<=p)`)
THEN RESA_TAC;
ASM_REWRITE_TAC[REAL_ARITH`(A+ #0.1 * -- &1 *(C+B)) + D+ #0.1 * &1* (E+B)= (A+ #0.1 * -- &1 * C) + D+ #0.1 * &1 * E`]
THEN MRESAL_TAC FINITE_DELETE_IMP[`J1_SY (s:stable_sy)`;`k:num,1`][FINITE_J1_SY]
THEN MRESAL_TAC SUM_DIFF[`(\x. cstab - norm (row (FST x) (vecmats (l:real^(M,3)finite_product)) - row (SND x) (vecmats l)))`;`J1_SY(s:stable_sy) DELETE (k,1)`;`J1_SY (s1:stable_sy) DELETE (p-1,p:num) DELETE (p:num,1)`][FINITE_J1_SY;]
THEN MATCH_MP_TAC(REAL_ARITH`
A1<= A+D /\ &0<= C1 -C /\ &0<= E
==> A1+ #0.1 * -- &1 * (C1+B) <=(A+ #0.1 * -- &1 *(C+B)) + D+ #0.1 * &1* E`)
THEN STRIP_TAC;
FIND_ASSUM MP_TAC `COVER2_SY s s1 (s2:stable_sy)`
THEN REWRITE_TAC[COVER2_SY]
THEN RESA_TAC;
STRIP_TAC;
SUBGOAL_THEN`!x. x IN (J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1))
==> (\x. cstab -
      norm
      (row (FST x) ((pmat1 (vecmats (l:real^(M,3)finite_product)):real^3^P)) - row (SND x) ((pmat1 (vecmats l)):real^3^P))) x
= (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product)))) x` ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;DELETE;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A-B=A-C<=> B=C`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i< p:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< p /\ 2<p ==> i=p -1 \/ SUC i<p:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p ==> p-1< p /\ SUC (p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
RESA_TAC
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN SUBGOAL_THEN`row i ((pmat1 (vecmats l)):real^3^P) = row i (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat1 (vecmats l)):real^3^P) = row (SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`SUC i < p==> 1<= SUC i /\ SUC i <= p`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[];
MRESA_TAC SUM_EQ[`(\x. cstab -
      norm
      (row (FST x) ((pmat1 (vecmats (l:real^(M,3)finite_product)):real^3^P)) - row (SND x) ((pmat1 (vecmats l)):real^3^P)))`;`(\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product))))`;`(J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1))`]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN MATCH_MP_TAC SUM_POS_LE
THEN STRIP_TAC;
MATCH_MP_TAC FINITE_DIFF
THEN REWRITE_TAC[FINITE_J1_SY; FINITE_DELETE];
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;IN_NUMSEG;DIFF;DELETE]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC (GEN_ALL B_SY_LE_CSTAB)[`i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MATCH_MP_TAC SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(SET_RULE`x=k-p+1,k-p+2 \/ ~(x=k-p+1,k-p+2 )`);
ASM_REWRITE_TAC[]
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN SUBGOAL_THEN`row (k-p+1) ((pmat2 (vecmats l)):real^3^N) = row (k-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+1<=k-p+2 /\ 1<= k-p+1 /\ k - (k - p + 2) + k - p + 1= k-1`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (k-p+2) ((pmat2 (vecmats l)):real^3^N) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+2<=k-p+2 /\ 1<= k-p+2 /\ k - (k - p + 2) + k - p + 2= k`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE` 2<k ==> 1<=k-1 /\ k-1<= k /\ SUC (k - 1)= k /\ k-1<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`k-1:num`;`k:num`]
THEN MRESAL_TAC (GEN_ALL B_SY_LE_CSTAB)[`k-1:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;][ARITH_RULE`SUC 0=1`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN DISCH_THEN(LABEL_TAC"THY1")
THEN REMOVE_THEN"THY" MP_TAC
THEN REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM;IN_NUMSEG]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_THEN"THY1" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=k-p+2==> i=k-p+2 \/ i<k-p+2:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< k-p+2 /\ 2<p ==> i=k-p +1 \/ i< k-p+1:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p /\ 1<= k-p ==> k-p+1< k-p+2 /\ SUC (k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`k-p+1:num`;`k-p+2:num`];
RESA_TAC
THEN RESA_TAC
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN SUBGOAL_THEN`row i ((pmat2 (vecmats l)):real^3^N) = row (p-2+i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + i= p-2+i`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat2 (vecmats l)):real^3^N) = row (p-2+ SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p/\ i< k-p+1 ==> k - (k - p + 2) + SUC i= p-2+SUC i/\ 1<= SUC i /\ SUC i<= k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE` i< k-p+1 /\ 2< p /\ 1<= k-p ==> 1<=p-2+i /\ p-2+i <= k /\ SUC (p-2+i)= p-2+ SUC i /\ p-2+i<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MRESAL_TAC (GEN_ALL B_SY_LE_CSTAB)[`p-2+i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;][ARITH_RULE`SUC 0=1`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MP_TAC(SET_RULE`~(p,1 IN J1_SY s1 DELETE (p - 1,p))
==> J1_SY (s1:stable_sy) DELETE (p - 1,p) = J1_SY s1 DELETE (p - 1,p) DELETE (p,1)`)
THEN RESA_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[th])
THEN MATCH_MP_TAC(REAL_ARITH`
A1<= A+D /\ &0<= C1 -C /\ &0<= E
==> A1+ #0.1 * -- &1 * C1 <=(A+ #0.1 * -- &1 *(C+B)) + D+ #0.1 * &1* (E+B)`)
THEN STRIP_TAC;
FIND_ASSUM MP_TAC `COVER2_SY s s1 (s2:stable_sy)`
THEN REWRITE_TAC[COVER2_SY]
THEN RESA_TAC;
STRIP_TAC;
SUBGOAL_THEN`!x. x IN (J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1))
==> (\x. cstab -
      norm
      (row (FST x) ((pmat1 (vecmats (l:real^(M,3)finite_product)):real^3^P)) - row (SND x) ((pmat1 (vecmats l)):real^3^P))) x
= (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product)))) x` ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;DELETE;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A-B=A-C<=> B=C`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i< p:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< p /\ 2<p ==> i=p -1 \/ SUC i<p:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p ==> p-1< p /\ SUC (p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
RESA_TAC
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN SUBGOAL_THEN`row i ((pmat1 (vecmats l)):real^3^P) = row i (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat1 (vecmats l)):real^3^P) = row (SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`SUC i < p==> 1<= SUC i /\ SUC i <= p`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[];
MRESA_TAC SUM_EQ[`(\x. cstab -
      norm
      (row (FST x) ((pmat1 (vecmats (l:real^(M,3)finite_product)):real^3^P)) - row (SND x) ((pmat1 (vecmats l)):real^3^P)))`;`(\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product))))`;`(J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1))`]
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN ASM_REWRITE_TAC[]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN MRESAL_TAC SUM_DIFF[`(\x. cstab - norm (row (FST x) (vecmats (l:real^(M,3)finite_product)) - row (SND x) (vecmats l)))`;`J1_SY(s:stable_sy) `;`J1_SY (s1:stable_sy) DELETE (p-1,p:num) DELETE (p:num,1)`][FINITE_J1_SY;]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN MATCH_MP_TAC SUM_POS_LE
THEN STRIP_TAC;
MATCH_MP_TAC FINITE_DIFF
THEN REWRITE_TAC[FINITE_J1_SY; FINITE_DELETE];
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;IN_NUMSEG;DIFF;DELETE]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC (GEN_ALL B_SY_LE_CSTAB)[`i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MATCH_MP_TAC SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN DISJ_CASES_TAC(SET_RULE`x=k-p+1,k-p+2 \/ ~(x=k-p+1,k-p+2 )`);
ASM_REWRITE_TAC[]
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN SUBGOAL_THEN`row (k-p+1) ((pmat2 (vecmats l)):real^3^N) = row (k-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+1<=k-p+2 /\ 1<= k-p+1 /\ k - (k - p + 2) + k - p + 1= k-1`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (k-p+2) ((pmat2 (vecmats l)):real^3^N) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+2<=k-p+2 /\ 1<= k-p+2 /\ k - (k - p + 2) + k - p + 2= k`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE` 2<k ==> 1<=k-1 /\ k-1<= k /\ SUC (k - 1)= k /\ k-1<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`k-1:num`;`k:num`]
THEN MRESAL_TAC (GEN_ALL B_SY_LE_CSTAB)[`k-1:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;][ARITH_RULE`SUC 0=1`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN DISCH_THEN(LABEL_TAC"THY1")
THEN REMOVE_THEN"THY" MP_TAC
THEN REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM;IN_NUMSEG]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_THEN"THY1" MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=k-p+2==> i=k-p+2 \/ i<k-p+2:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< k-p+2 /\ 2<p ==> i=k-p +1 \/ i< k-p+1:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p /\ 1<= k-p ==> k-p+1< k-p+2 /\ SUC (k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`k-p+1:num`;`k-p+2:num`];
RESA_TAC
THEN RESA_TAC
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN SUBGOAL_THEN`row i ((pmat2 (vecmats l)):real^3^N) = row (p-2+i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + i= p-2+i`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat2 (vecmats l)):real^3^N) = row (p-2+ SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p/\ i< k-p+1 ==> k - (k - p + 2) + SUC i= p-2+SUC i/\ 1<= SUC i /\ SUC i<= k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE` i< k-p+1 /\ 2< p /\ 1<= k-p ==> 1<=p-2+i /\ p-2+i <= k /\ SUC (p-2+i)= p-2+ SUC i /\ p-2+i<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MRESAL_TAC (GEN_ALL B_SY_LE_CSTAB)[`p-2+i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;][ARITH_RULE`SUC 0=1`]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
FIND_ASSUM MP_TAC `COVER_SY 0 (p - 1) s s1 (s2:stable_sy)`
THEN REWRITE_TAC[COVER_SY;COVER3_SY;COVER6_SY;]
THEN ASM_REWRITE_TAC[DE_MORGAN_THM]
THEN RESA_TAC
THEN ASM_REWRITE_TAC[d_fun]
THEN MRESA1_TAC sigma_sy`s1:stable_sy`
THEN MRESA1_TAC sigma_sy`s3:stable_sy`
THEN FIND_ASSUM MP_TAC`SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 (s3:stable_sy)`
THEN REWRITE_TAC[SCHANGE]
THEN RESA_TAC
THEN SUBGOAL_THEN`~({0,1} IN J_SY (s3:stable_sy))`ASSUME_TAC;
FIND_ASSUM MP_TAC`~({0, p - 1} IN J_SY (s2:stable_sy))`
THEN MATCH_MP_TAC MONO_NOT
THEN STRIP_TAC
THEN FIND_ASSUM (fun th-> MP_TAC(ISPECL [`0`;`1`]th))`!p' q.
          {(if p' = 0 then 0 else (p - 2 + p') MOD k), (if q = 0
                                                        then 0
                                                        else (p - 2 + q) MOD
                                                             k)} IN
          J_SY s2 <=>
          {p', q} IN J_SY (s3:stable_sy)`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th;ARITH_RULE`(0=0) /\ ~(1=0)`])
THEN MP_TAC(ARITH_RULE`2<p/\ 1<= k-p ==> p-2+1=p-1 /\ p-1<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`];
SUBGOAL_THEN`~(p - 1,p IN J1_SY (s1:stable_sy))`ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;PAIR_EQ;IN_NUMSEG]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`2<p/\ 1<= k-p ==> p-2+1=p-1 /\ p-1<p/\ 1+p-1=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`]
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B}={B,A}`]
THEN RESA_TAC;
SUBGOAL_THEN`~((k-p+2,1) IN J1_SY (s3:stable_sy))`ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;PAIR_EQ;IN_NUMSEG]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN STRIP_TAC
THEN STRIP_TAC
THEN MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ 1+0=1`]
THEN MRESA_TAC MOD_LT[`1:num`;`k-p+2:num`];
SUBGOAL_THEN`J1_SY s1  DELETE (p-1,p) DELETE (p,1) SUBSET J1_SY (s:stable_sy)` ASSUME_TAC;
ASM_REWRITE_TAC[SUBSET;J1_SY;IN_ELIM_THM;DELETE;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ;]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i< p:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`];
MP_TAC(ARITH_RULE`i<p /\ 2<p ==> i=p-1 \/ i< p-1:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p ==> SUC(p-1)=p /\ p-1<p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
STRIP_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<p-1 /\ p < k /\ 2<p ==> i< k /\ i<=k:num /\ 1+i<k/\ 1+i< p/\ ~(i=1+i)/\ ~(i=p-1)/\ ~(p=2)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[th] THEN REPEAT STRIP_TAC)
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`1+i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`1+i:num`;`p:num`]
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[SYM th] )
THEN FIND_ASSUM MP_TAC`J_SY s1 SUBSET J_SY (s:stable_sy) UNION {{0, p - 1}}`
THEN REWRITE_TAC[SUBSET;UNION;IN_ELIM_THM;IN_SING]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{i,(1+i) MOD p}`)
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC MOD_LT[`1+i:num`;`p:num`]
THEN STRIP_TAC
THEN MP_TAC(SET_RULE`{i, 1 + i} = {0, p - 1} /\ ~(i=1+i) /\ ~(i=p-1) ==> i=0 /\ 1+i= p-1`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`1+0=p-1<=> p=2`];
SUBGOAL_THEN`IMAGE (\(x,y). (p - 2 + x), (p - 2 + y)) (J1_SY s3  DELETE (k-p+2,1)) SUBSET J1_SY (s:stable_sy)` ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;IMAGE;DELETE;SUBSET;IN_ELIM_THM;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ;]
THEN MP_TAC(ARITH_RULE`i<=k-p+2 ==> i=k-p+2 \/ i< k-p+2:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`];
MP_TAC(ARITH_RULE`i<k-p+2 /\ 2<k-p+2 ==> i=k-p+1 \/ i< k-p+1:num`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN RESA_TAC
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`2<p /\ 2< k-p+2 /\ 2< k ==> p-2+k-p+1=k-1/\ SUC(k-p+1)=k-p+2 /\ k-p+1<k-p+2/\ p - 2 + k - p + 2=k/\ 1<=k-1 /\ k-1<=k/\ k-1<k/\ 1+k-1=k /\ SUC(k-1)=k/\ 1 + k - p + 1= k - p + 2 /\ ~(k-p+1=0)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`k-p+1:num`;`k-p+2:num`]
THEN MRESA_TAC MOD_LT[`k-1:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`]
THEN EXISTS_TAC`k-1`
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`{(k - p + 1) MOD (k - p + 2), (f_sy s2 o
                                     (\x. if x = 0
                                          then 0
                                          else (p - 2 + x) MOD k))
                                    ((k - p + 1) MOD (k - p + 2))} IN
      J_SY (s3:stable_sy)`
THEN FIND_ASSUM(fun th-> REWRITE_TAC[SYM th])`(\i. (1 + i) MOD (k - p + 2)) =
      f_sy (s2:stable_sy) o (\x. if x = 0 then 0 else (p - 2 + x) MOD k)`
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1`]
THEN STRIP_TAC
THEN FIND_ASSUM (fun th-> MP_TAC( ISPECL [`k-p+1`;`0`]th))`!p' q.
          {(if p' = 0 then 0 else (p - 2 + p') MOD k), (if q = 0
                                                        then 0
                                                        else (p - 2 + q) MOD
                                                             k)} IN
          J_SY s2 <=>
          {p', q} IN J_SY (s3:stable_sy)`
THEN POP_ASSUM (fun th-> REWRITE_TAC[th;ARITH_RULE`0=0`])
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`J_SY s2 SUBSET J_SY (s:stable_sy) UNION {{0, p - 1}}`
THEN REWRITE_TAC[SUBSET;UNION;IN_ELIM_THM;IN_SING]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{k-1,0}`);
POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<=k-p /\2<p==> ~(k-1=p-1) /\ ~(0= p-1)`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(k-1=p-1) /\ ~(0= p-1)==> ~(p-1 IN {k-1,0})`)
THEN ASM_REWRITE_TAC[]
THEN SET_TAC[];
STRIP_TAC
THEN STRIP_TAC
THEN EXISTS_TAC`p-2+i:num`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<k - p + 1 /\ 2<p /\ 1<=i==> p-2+i< k/\ p-2+i<=k /\ 1<=p-2+i/\ 1+p-2+i<k /\ 1+i< k-p+2/\ p-2+1+i<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+1+i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`i<k - p + 1 /\ 2<p /\ 1<=i==>  ~(i=0)/\ ~(1+i=0)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`1+i:num`;`k-p+2:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN REWRITE_TAC[ARITH_RULE`SUC (p - 2 + i)= p - 2 + SUC i`]
THEN MRESA_TAC MOD_LT[`1+p-2+i:num`;`k:num`]
THEN FIND_ASSUM (fun th-> MP_TAC( ISPECL [`i MOD (k - p + 2)`;`(f_sy (s2:stable_sy) o (\x. if x = 0 then 0 else (p - 2 + x) MOD k))
                          (i MOD (k - p + 2))`] th))`!p' q.
          {(if p' = 0 then 0 else (p - 2 + p') MOD k), (if q = 0
                                                        then 0
                                                        else (p - 2 + q) MOD
                                                             k)} IN
          J_SY s2 <=>
          {p', q} IN J_SY  (s3:stable_sy)`
THEN FIND_ASSUM(fun th-> REWRITE_TAC[th])`{i MOD (k - p + 2), (f_sy s2 o (\x. if x = 0 then 0 else (p - 2 + x) MOD k))
                          (i MOD (k - p + 2))} IN
      J_SY (s3:stable_sy)` 
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN FIND_ASSUM(fun th-> REWRITE_TAC[SYM th])`(\i. (1 + i) MOD (k - p + 2)) =
      f_sy (s2:stable_sy) o (\x. if x = 0 then 0 else (p - 2 + x) MOD k)`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`J_SY s2 SUBSET J_SY (s:stable_sy) UNION {{0, p - 1}}`
THEN REWRITE_TAC[SUBSET;UNION;IN_ELIM_THM;IN_SING]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{p-2+i,p-2+1+i}`);
ASM_REWRITE_TAC[ARITH_RULE`1+p-2+i= p-2+1+i`];
MP_TAC(SET_RULE`{p - 2 + i, p - 2 + 1 + i} = {0, p - 1} ==> (p-2+i=0) \/ (p-2+1+i=0)`)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`1<=i /\ 2< p==> ~(p-2+i=0) /\ ~(p-2+1+i=0)`)
THEN ASM_REWRITE_TAC[];
MP_TAC(SET_RULE`~(k - p + 2,1 IN J1_SY s3) ==>  J1_SY (s3:stable_sy) =J1_SY s3 DELETE (k - p + 2,1)`)
THEN RESA_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[th])
THEN MP_TAC(SET_RULE`~(p -1,p IN J1_SY s1) ==>  J1_SY (s1:stable_sy) =J1_SY s1 DELETE (p -1,p)`)
THEN RESA_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[th]);
DISJ_CASES_TAC(SET_RULE`(p,1) IN  J1_SY s1 DELETE (p - 1,p) \/ ~((p,1) IN  J1_SY (s1:stable_sy) DELETE (p - 1,p))`);
SUBGOAL_THEN`k,1 IN J1_SY (s:stable_sy)`ASSUME_TAC;
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;DELETE;PAIR_EQ;IN_NUMSEG]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p /\ 1+0=1`]
THEN MP_TAC(ARITH_RULE`2<p /\ 2< k==> 1<p /\ 1<k /\ k<=k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`1:num`;`p:num`]
THEN STRIP_TAC
THEN EXISTS_TAC`k:num`
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p /\ SUC 0=1/\ 1+0=1`]
THEN MRESA_TAC MOD_LT[`1:num`;`k:num`]
THEN FIND_ASSUM MP_TAC`J_SY s1 SUBSET J_SY (s:stable_sy) UNION {{0, p - 1}}`
THEN REWRITE_TAC[SUBSET;UNION;IN_ELIM_THM;IN_SING]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`{0,1}`);
MP_TAC(ARITH_RULE`2<p ==> ~(p-1=0) /\ ~(p-1=1)`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(p-1=0) /\ ~(p-1=1) ==> ~(p-1 IN {0,1})`)
THEN ASM_REWRITE_TAC[]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
MRESAL_TAC FINITE_DELETE_IMP[`J1_SY (s1:stable_sy)`;`p-1,p`][FINITE_J1_SY]
THEN MRESAL_TAC FINITE_DELETE_IMP[`J1_SY (s3:stable_sy)`;`k-p+2,1`][FINITE_J1_SY]
THEN MRESAL_TAC SUM_DELETE[`(\x. cstab - norm (row (FST x) (vecmats l1) - row (SND x) (vecmats (l1:real^(P,3)finite_product))))`;`J1_SY (s1:stable_sy) DELETE (p-1,p)`;`p:num,1`][FINITE_J1_SY;REAL_ARITH`A=B-C<=> B=A+C`]
THEN MRESAL_TAC SUM_DELETE[`(\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product))))`;`J1_SY (s:stable_sy)`;`k:num,1`][FINITE_J1_SY;REAL_ARITH`A=B-C<=> B=A+C`]
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN RESA_TAC
THEN SUBGOAL_THEN`row 1 ((pmat1 (vecmats l)):real^3^P) = row 1 (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2<p==> 1<=1 /\ 1<p `)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`J1_SY s1 DELETE (p - 1,p) DELETE (p,1) SUBSET J1_SY (s:stable_sy) DELETE (k,1)`ASSUME_TAC;
MATCH_MP_TAC(SET_RULE`~(x IN A) /\ A SUBSET B==> A SUBSET B DELETE x`)
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC(SET_RULE`~(x IN A) ==> ~(x IN A DELETE y)`)
THEN MATCH_MP_TAC(SET_RULE`~(x IN A) ==> ~(x IN A DELETE y)`)
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;PAIR_EQ;IN_NUMSEG]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<= k-p ==> ~(k<=p)`)
THEN RESA_TAC;
ASM_REWRITE_TAC[REAL_ARITH`
A1+ #0.1 * -- &1 * (C1+B) <=(A+ #0.1 * -- &1 *(C+B)) + D+ #0.1 * -- &1* E
<=> A1+ #0.1 * -- &1 * (C1) <=(A+ #0.1 * -- &1 *(C)) + D+ #0.1 * -- &1* E
`]
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN MATCH_MP_TAC(REAL_ARITH`
A1<= A+D /\ &0<=C1-(C+E)
==>
A1+ #0.1 * -- &1 * (C1) <=(A+ #0.1 * -- &1 *(C)) + D+ #0.1 * -- &1* E`)
THEN STRIP_TAC;
FIND_ASSUM MP_TAC `COVER2_SY s s1 (s2:stable_sy)`
THEN REWRITE_TAC[COVER2_SY]
THEN RESA_TAC;
SUBGOAL_THEN`!x. x IN (J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1))
==> (\x. cstab -
      norm
      (row (FST x) ((pmat1 (vecmats (l:real^(M,3)finite_product)):real^3^P)) - row (SND x) ((pmat1 (vecmats l)):real^3^P))) x
= (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product)))) x` ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;DELETE;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A-B=A-C<=> B=C`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i< p:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< p /\ 2<p ==> i=p -1 \/ SUC i<p:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p ==> p-1< p /\ SUC (p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
RESA_TAC
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN SUBGOAL_THEN`row i ((pmat1 (vecmats l)):real^3^P) = row i (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat1 (vecmats l)):real^3^P) = row (SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`SUC i < p==> 1<= SUC i /\ SUC i <= p`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[];
MRESA_TAC SUM_EQ[`(\x. cstab -
      norm
      (row (FST x) ((pmat1 (vecmats (l:real^(M,3)finite_product)):real^3^P)) - row (SND x) ((pmat1 (vecmats l)):real^3^P)))`;`(\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product))))`;`(J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1))`]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN SUBGOAL_THEN`!x. x IN (J1_SY (s3:stable_sy) DELETE (k - p + 2,1))
==> 
      ((\x. cstab - norm (row (FST x) (vecmats (l:real^(M,3)finite_product)) - row (SND x) (vecmats l))) o
       (\(x,y). p - 2 + x,p - 2 + y)) x
= (\x. cstab - norm (row (FST x) (vecmats (l2:real^(N,3)finite_product)) - row (SND x) (vecmats l2))) x`
ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM;o_DEF;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A-B=A-C<=> B=C`]
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`i<=k-p+2==> i=k-p+2 \/ i<k-p+2:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN SUBGOAL_THEN`row i ((pmat2 (vecmats l)):real^3^N) = row (p-2+i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + i= p-2+i`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat2 (vecmats l)):real^3^N) = row (p-2+ SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p/\ i< k-p+2 ==> k - (k - p + 2) + SUC i= p-2+SUC i/\ 1<= SUC i /\ SUC i<= k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[];
MRESA_TAC SUM_EQ[`      ((\x. cstab - norm (row (FST x) (vecmats (l:real^(M,3)finite_product)) - row (SND x) (vecmats l))) o
       (\(x,y). p - 2 + x,p - 2 + y))`;`(\x. cstab - norm (row (FST x) (vecmats (l2:real^(N,3)finite_product)) - row (SND x) (vecmats l2)))`;`(J1_SY (s3:stable_sy) DELETE (k - p + 2,1))`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN SUBGOAL_THEN`(!x y.
           x IN J1_SY s3 DELETE (k - p + 2,1) /\
           y IN J1_SY (s3:stable_sy) DELETE (k - p + 2,1) /\
           (\(x,y). p - 2 + x,p - 2 + y) x = (\(x,y). p - 2 + x,p - 2 + y) y
           ==> x = y)`ASSUME_TAC;
REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ;ARITH_RULE`p-2+B=p-2+C<=> B=C`];
MRESA_TAC SUM_IMAGE[`(\(x,y). p - 2 + x,p - 2 + y)`;`(\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product))))`;`J1_SY (s3:stable_sy) DELETE (k - p + 2,1)`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN SUBGOAL_THEN`J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1) INTER (IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY s3 DELETE (k - p + 2,1)))={}` ASSUME_TAC;
ASM_REWRITE_TAC[SET_RULE`A ={} <=> ~(?a. a IN A)`; INTER;IN_ELIM_THM;J1_SY;DELETE;IMAGE;IN_NUMSEG]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i< p:num`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i<p /\ 2< p ==> i=p-1 \/ i< p-1:num`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE` 2< p ==> p-1< p:num/\ SUC (p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
MP_TAC(ARITH_RULE`1<= i /\ i< p-1 /\ 1<= i' ==> ~(i= p-2+i')`)
THEN RESA_TAC;
MRESA_TAC FINITE_IMAGE[`(\(x,y). p - 2 + x,p - 2 + y)`;`(J1_SY (s3:stable_sy) DELETE (k - p + 2,1))`]
THEN MRESAL_TAC (GEN_ALL SUM_UNION_EQ)[`(\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product))))`;`J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1)`;`(IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY (s3:stable_sy) DELETE (k - p + 2,1)))`;`J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1) UNION (IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY (s3:stable_sy) DELETE (k - p + 2,1)))`][FINITE_UNION;FINITE_DELETE]
THEN SUBGOAL_THEN`IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY s3 DELETE (k - p + 2,1)) SUBSET
      J1_SY s DELETE (k,1)` ASSUME_TAC;
MATCH_MP_TAC(SET_RULE`~(x IN A) /\ A SUBSET B ==> A SUBSET B DELETE x`)
THEN ASM_REWRITE_TAC[]
THEN ASM_REWRITE_TAC[IMAGE;IN_ELIM_THM;DELETE;J1_SY;IN_NUMSEG]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ]
THEN MP_TAC(ARITH_RULE`i<= k-p+2 ==> i=k-p+2 \/ i< k-p+2`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< k-p+2  /\ 1<= k-p /\ 2< p ==> ~(k=p-2 +i)`)
THEN RESA_TAC;
MP_TAC(SET_RULE`IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY s3 DELETE (k - p + 2,1)) SUBSET
      J1_SY s DELETE (k,1)
/\ J1_SY s1 DELETE (p - 1,p) DELETE (p,1) SUBSET J1_SY s DELETE (k,1)
==> J1_SY s1 DELETE (p - 1,p) DELETE (p,1) UNION (IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY s3 DELETE (k - p + 2,1))) SUBSET J1_SY (s:stable_sy) DELETE (k,1)`)
THEN RESA_TAC
THEN MRESAL_TAC SUM_DIFF[`(\x. cstab - norm (row (FST x) (vecmats (l:real^(M,3)finite_product)) - row (SND x) (vecmats l)))`;`J1_SY(s:stable_sy) DELETE (k,1)`;`J1_SY (s1:stable_sy) DELETE (p-1,p:num) DELETE (p:num,1) UNION (IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY s3 DELETE (k - p + 2,1)))`][FINITE_J1_SY;FINITE_DELETE]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN MATCH_MP_TAC SUM_POS_LE
THEN STRIP_TAC;
MATCH_MP_TAC FINITE_DIFF
THEN REWRITE_TAC[FINITE_J1_SY; FINITE_DELETE];
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;IN_NUMSEG;DIFF;DELETE]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC (GEN_ALL B_SY_LE_CSTAB)[`i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MP_TAC(SET_RULE`~(p,1 IN J1_SY s1 DELETE (p - 1,p))
==> J1_SY (s1:stable_sy) DELETE (p - 1,p) = J1_SY s1 DELETE (p - 1,p) DELETE (p,1)`)
THEN RESA_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[th])
THEN SUBGOAL_THEN`J1_SY s1 DELETE (p - 1,p) DELETE (p,1) SUBSET J1_SY (s:stable_sy) DELETE (k,1)`ASSUME_TAC;
MATCH_MP_TAC(SET_RULE`~(x IN A) /\ A SUBSET B==> A SUBSET B DELETE x`)
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC(SET_RULE`~(x IN A) ==> ~(x IN A DELETE y)`)
THEN MATCH_MP_TAC(SET_RULE`~(x IN A) ==> ~(x IN A DELETE y)`)
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;PAIR_EQ;IN_NUMSEG]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<= k-p ==> ~(k<=p)`)
THEN RESA_TAC;
ASM_REWRITE_TAC[REAL_ARITH`
A1+ #0.1 * -- &1 * (C1+B) <=(A+ #0.1 * -- &1 *(C+B)) + D+ #0.1 * -- &1* E
<=> A1+ #0.1 * -- &1 * (C1) <=(A+ #0.1 * -- &1 *(C)) + D+ #0.1 * -- &1* E
`]
THEN MATCH_MP_TAC(REAL_ARITH`
A1<= A+D /\ &0<=C1-(C+E)
==>
A1+ #0.1 * -- &1 * (C1) <=(A+ #0.1 * -- &1 *(C)) + D+ #0.1 * -- &1* E`)
THEN STRIP_TAC;
FIND_ASSUM MP_TAC `COVER2_SY s s1 (s2:stable_sy)`
THEN REWRITE_TAC[COVER2_SY]
THEN RESA_TAC;
SUBGOAL_THEN`!x. x IN (J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1))
==> (\x. cstab -
      norm
      (row (FST x) ((pmat1 (vecmats (l:real^(M,3)finite_product)):real^3^P)) - row (SND x) ((pmat1 (vecmats l)):real^3^P))) x
= (\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product)))) x` ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;DELETE;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A-B=A-C<=> B=C`]
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i< p:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i< p /\ 2<p ==> i=p -1 \/ SUC i<p:num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p ==> p-1< p /\ SUC (p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
RESA_TAC
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN SUBGOAL_THEN`row i ((pmat1 (vecmats l)):real^3^P) = row i (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat1 (vecmats l)):real^3^P) = row (SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`SUC i < p==> 1<= SUC i /\ SUC i <= p`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[];
MRESA_TAC SUM_EQ[`(\x. cstab -
      norm
      (row (FST x) ((pmat1 (vecmats (l:real^(M,3)finite_product)):real^3^P)) - row (SND x) ((pmat1 (vecmats l)):real^3^P)))`;`(\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product))))`;`(J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1))`]
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN ASM_REWRITE_TAC[]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN SUBGOAL_THEN`!x. x IN (J1_SY (s3:stable_sy) DELETE (k - p + 2,1))
==> 
      ((\x. cstab - norm (row (FST x) (vecmats (l:real^(M,3)finite_product)) - row (SND x) (vecmats l))) o
       (\(x,y). p - 2 + x,p - 2 + y)) x
= (\x. cstab - norm (row (FST x) (vecmats (l2:real^(N,3)finite_product)) - row (SND x) (vecmats l2))) x`
ASSUME_TAC;
ASM_REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM;o_DEF;IN_NUMSEG]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[REAL_ARITH`A-B=A-C<=> B=C`]
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID;]
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`i<=k-p+2==> i=k-p+2 \/ i<k-p+2:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN SUBGOAL_THEN`row i ((pmat2 (vecmats l)):real^3^N) = row (p-2+i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + i= p-2+i`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat2 (vecmats l)):real^3^N) = row (p-2+ SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p/\ i< k-p+2 ==> k - (k - p + 2) + SUC i= p-2+SUC i/\ 1<= SUC i /\ SUC i<= k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[];
MRESA_TAC SUM_EQ[`      ((\x. cstab - norm (row (FST x) (vecmats (l:real^(M,3)finite_product)) - row (SND x) (vecmats l))) o
       (\(x,y). p - 2 + x,p - 2 + y))`;`(\x. cstab - norm (row (FST x) (vecmats (l2:real^(N,3)finite_product)) - row (SND x) (vecmats l2)))`;`(J1_SY (s3:stable_sy) DELETE (k - p + 2,1))`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN SUBGOAL_THEN`(!x y.
           x IN J1_SY s3 DELETE (k - p + 2,1) /\
           y IN J1_SY (s3:stable_sy) DELETE (k - p + 2,1) /\
           (\(x,y). p - 2 + x,p - 2 + y) x = (\(x,y). p - 2 + x,p - 2 + y) y
           ==> x = y)`ASSUME_TAC;
REWRITE_TAC[J1_SY;DELETE;IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ;ARITH_RULE`p-2+B=p-2+C<=> B=C`];
MRESA_TAC SUM_IMAGE[`(\(x,y). p - 2 + x,p - 2 + y)`;`(\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product))))`;`J1_SY (s3:stable_sy) DELETE (k - p + 2,1)`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN SUBGOAL_THEN`J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1) INTER (IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY s3 DELETE (k - p + 2,1)))={}` ASSUME_TAC;
ASM_REWRITE_TAC[SET_RULE`A ={} <=> ~(?a. a IN A)`; INTER;IN_ELIM_THM;J1_SY;DELETE;IMAGE;IN_NUMSEG]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i< p:num`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
MP_TAC(ARITH_RULE`i<p /\ 2< p ==> i=p-1 \/ i< p-1:num`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE` 2< p ==> p-1< p:num/\ SUC (p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
MP_TAC(ARITH_RULE`1<= i /\ i< p-1 /\ 1<= i' ==> ~(i= p-2+i')`)
THEN RESA_TAC;
MRESAL_TAC FINITE_IMAGE[`(\(x,y). p - 2 + x,p - 2 + y)`;`(J1_SY (s3:stable_sy) DELETE (k - p + 2,1))`][FINITE_J1_SY;FINITE_DELETE]
THEN MRESAL_TAC (GEN_ALL SUM_UNION_EQ)[`(\x. cstab - norm (row (FST x) (vecmats l) - row (SND x) (vecmats (l:real^(M,3)finite_product))))`;`J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1)`;`(IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY (s3:stable_sy) DELETE (k - p + 2,1)))`;`J1_SY (s1:stable_sy) DELETE (p - 1,p) DELETE (p,1) UNION (IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY (s3:stable_sy) DELETE (k - p + 2,1)))`][FINITE_UNION;FINITE_DELETE;FINITE_J1_SY]
THEN MP_TAC(SET_RULE`IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY s3 DELETE (k - p + 2,1)) SUBSET
      J1_SY s 
/\ J1_SY s1 DELETE (p - 1,p) DELETE (p,1) SUBSET J1_SY s 
==> J1_SY s1 DELETE (p - 1,p) DELETE (p,1) UNION (IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY s3 DELETE (k - p + 2,1))) SUBSET J1_SY (s:stable_sy) `)
THEN RESA_TAC
THEN MRESAL_TAC SUM_DIFF[`(\x. cstab - norm (row (FST x) (vecmats (l:real^(M,3)finite_product)) - row (SND x) (vecmats l)))`;`J1_SY(s:stable_sy) `;`J1_SY (s1:stable_sy) DELETE (p-1,p:num) DELETE (p:num,1) UNION (IMAGE (\(x,y). p - 2 + x,p - 2 + y) (J1_SY s3 DELETE (k - p + 2,1)))`][FINITE_J1_SY;FINITE_DELETE]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN MATCH_MP_TAC SUM_POS_LE
THEN STRIP_TAC;
MATCH_MP_TAC FINITE_DIFF
THEN REWRITE_TAC[FINITE_J1_SY; FINITE_DELETE];
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[J1_SY;IN_ELIM_THM;IN_NUMSEG;DIFF;DELETE]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC (GEN_ALL B_SY_LE_CSTAB)[`i:num`;`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`;]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]);;



let F_SY_COVER_EQ=prove_by_refinement(`!(l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ 2<p
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ (!i j. 1<= i /\ i<= dimindex (:M) /\ 1<= j /\ j<= dimindex (:M)
/\ row i (vecmats l)= row j (vecmats l)
==> i=j)
==> (F_SY(vecmats l1) DELETE (row (p-1) (vecmats l1), row p (vecmats l1))) UNION (F_SY(vecmats l2) DELETE (row (k-p+2) (vecmats l2), row 1 (vecmats l2))) = F_SY(vecmats l)`,
[REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN REWRITE_TAC[F_SY;UNION;EXTENSION;IN_ELIM_THM;DELETE]
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p `)
THEN RESA_TAC
THEN GEN_TAC
THEN EQ_TAC
THEN RESA_TAC;
EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID]
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i<p:num`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN "THY" MP_TAC
THEN SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN SUBGOAL_THEN`row 1 ((pmat1 (vecmats l)):real^3^P) = row 1 (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2<p==> 1<=1 /\ 1<p `)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC
THEN EXISTS_TAC`k:num`
THEN ASM_REWRITE_TAC[ARITH_RULE`k<=k`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN ARITH_TAC;
MP_TAC(ARITH_RULE` i< p/\ 2<p ==> i= p-1 \/ SUC i< p`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE` 2< p ==> p-1< p:num /\ SUC(p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
MP_TAC(ARITH_RULE` i< p /\ 1<= k-p ==> i< k:num /\ i<=k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN STRIP_TAC
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
THEN REMOVE_THEN "THY" MP_TAC
THEN SUBGOAL_THEN`row i ((pmat1 (vecmats l)):real^3^P) = row i (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat1 (vecmats l)):real^3^P) = row (SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`SUC i < p==> 1<= SUC i /\ SUC i <= p`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[];
POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`i <= k-p+2 ==> i=k-p+2 \/ i <k-p+2`)
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID]
THEN REMOVE_THEN "THY" MP_TAC
THEN SUBGOAL_THEN`row i ((pmat2 (vecmats l)):real^3^N) = row (p-2+i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + i= p-2+i`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat2 (vecmats l)):real^3^N) = row (p-2+ SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p/\ i< k-p+2 ==> k - (k - p + 2) + SUC i= p-2+SUC i/\ 1<= SUC i /\ SUC i<= k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
MP_TAC(ARITH_RULE` i< k-p+2 /\ 1<= k-p /\ 2<p==> p-2+ i< k:num /\ p-2+i <=k/\ 1<= p-2+i/\ p-2+ SUC i= SUC(p-2+i)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC
THEN EXISTS_TAC`p-2+i`
THEN ASM_REWRITE_TAC[];
DISJ_CASES_TAC(ARITH_RULE`i< p-1\/ p-1<= i`);
MATCH_MP_TAC(SET_RULE`A==> A\/B`)
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<p-1 /\ 1<= k-p ==> i< k /\ i<= p /\ i<p /\ SUC i< p`)
THEN RESA_TAC
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID]
THEN REMOVE_THEN "THY" MP_TAC
THEN SUBGOAL_THEN`row i ((pmat1 (vecmats l)):real^3^P) = row i (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i) ((pmat1 (vecmats l)):real^3^P) = row (SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`SUC i < p==> 1<= SUC i /\ SUC i <= p`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC;
EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[];
REMOVE_THEN "THY" MP_TAC
THEN SUBGOAL_THEN`row (p-1) ((pmat1 (vecmats l)):real^3^P) = row (p-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2< p ==> p-1 <= p /\ 1<= p-1/\ p-1<p`)
THEN RESA_TAC
THEN  ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN ASM_REWRITE_TAC[PAIR_EQ]
THEN STRIP_TAC
THEN MP_TAC(ARITH_RULE`2<p /\ 1<= k-p /\ i< p-1==> 1<= p-1 /\ p-1<= k /\ ~(i=p-1)`)
THEN RESA_TAC
THEN REMOVE_THEN"THY"(fun th-> MRESA_TAC th[`i:num`;`p-1`]);
MP_TAC(ARITH_RULE`i<= k ==> i=k \/ i< k:num`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN MATCH_MP_TAC(SET_RULE`A==> A\/B`)
THEN STRIP_TAC;
EXISTS_TAC`p:num`
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID]
THEN REMOVE_THEN "THY" MP_TAC
THEN SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN SUBGOAL_THEN`row 1 ((pmat1 (vecmats l)):real^3^P) = row 1 (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2<p==> 1<=1 /\ 1<p `)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`];
REWRITE_TAC[PAIR_EQ]
THEN REMOVE_THEN "THY" MP_TAC
THEN SUBGOAL_THEN`row (p-1) ((pmat1 (vecmats l)):real^3^P) = row (p-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2< p ==> p-1 <= p /\ 1<= p-1/\ p-1<p`)
THEN RESA_TAC
THEN  ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
DISCH_THEN(LABEL_TAC"THY")
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID]
THEN STRIP_TAC
THEN REMOVE_THEN"THY"(fun th-> MRESA_TAC th[`k:num`;`p-1`])
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`2< p /\ 1<= k-p ==> 1<= p-1 /\ ~(k=p-1)`)
THEN RESA_TAC;
MATCH_MP_TAC(SET_RULE`A==> B\/A`)
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`p-1<= i /\ 2< p/\ i< k /\ 1<= k-p ==> 1<= i- (p-2) /\ i-(p-2)<= k-p+2/\ k - (k - p + 2) = p-2 /\ i-(p-2)< k- p+2/\  SUC (i - (p - 2))= SUC i - (p - 2)/\ k<=k`)
THEN RESA_TAC
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID]
THEN REMOVE_THEN "THY" MP_TAC
THEN SUBGOAL_THEN`row (i-(p-2)) ((pmat2 (vecmats l)):real^3^N) = row i (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE` 2<p /\ p-1<= i ==> 
p-2 + i-(p-2) =i`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row (SUC i -(p-2)) ((pmat2 (vecmats l)):real^3^N) = row (SUC i) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p/\ i< k /\ p-1<=i ==> p-2 + SUC i -(p-2)= SUC i/\ 1<= SUC i -(p-2) /\ SUC i - (p-2)<= k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC;
EXISTS_TAC`i-(p-2)`
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`i-(p-2):num`;`k-p+2:num`];
POP_ASSUM MP_TAC
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID]
THEN SUBGOAL_THEN`row (k-p+2) ((pmat2 (vecmats l)):real^3^N) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p ==> k-p+2<=k-p+2 /\ 1<= k-p+2 /\ k - (k - p + 2) + k - p + 2= k`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
DISCH_THEN(LABEL_TAC"THY")
THEN ASM_REWRITE_TAC[PAIR_EQ]
THEN STRIP_TAC
THEN REMOVE_THEN"THY"(fun th->MRESA_TAC th[`i:num`;`k:num`])
THEN MP_TAC(ARITH_RULE`i<k==> ~(i=k:num)`)
THEN RESA_TAC]);;


let F_SY_COVER_INTER=prove_by_refinement(`!(l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ 2<p
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ (!i j. 1<= i /\ i<= dimindex (:M) /\ 1<= j /\ j<= dimindex (:M)
/\ row i (vecmats l)= row j (vecmats l)
==> i=j)
==> (F_SY(vecmats l1) DELETE (row (p-1) (vecmats l1), row p (vecmats l1))) INTER (F_SY(vecmats l2) DELETE (row (k-p+2) (vecmats l2), row 1 (vecmats l2))) = {} `,
[REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN REWRITE_TAC[F_SY;INTER;IN_ELIM_THM;DELETE;SET_RULE`A={}<=> ~(?a. a IN A)`]
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p `)
THEN RESA_TAC
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY1")
THEN STRIP_TAC
THEN REMOVE_THEN "THY1"MP_TAC
THEN ASM_REWRITE_TAC[PAIR_EQ]
THEN EXPAND_TAC"l1"
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID]
THEN REMOVE_THEN "THY" MP_TAC
THEN SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN SUBGOAL_THEN`row i' ((pmat2 (vecmats l)):real^3^N) = row (p-2+i') (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + i'= p-2+i'`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i<p:num`)
THEN RESA_TAC;
DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY1")
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=i' /\ 1<= k-p /\ i'<= k-p+2 ==> k<=k /\ 1<= p-2+i' /\ p-2+i'<=k/\ ~(i'=0)`)
THEN RESA_TAC
THEN REMOVE_THEN "THY"(fun th-> MRESA_TAC th[`k:num`;`p-2+i'`] 
THEN POP_ASSUM MP_TAC
THEN MP_TAC th
THEN DISCH_THEN(LABEL_TAC"THY"))
THEN STRIP_TAC
THEN MP_TAC(ARITH_RULE`k = p - 2 + i'/\ 2<p /\ 1<= k-p ==> i'= k-p+2`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESAL_TAC MOD_MULT[`i':num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN REMOVE_THEN"THY1"MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`SUC 0=1`]
THEN REMOVE_THEN"THY" MP_TAC
THEN SUBGOAL_THEN`row 1 ((pmat2 (vecmats l)):real^3^N) = row (p-1) (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + 1= p-1/\ 1<=1 /\ 1<=k-p+2`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ];
SUBGOAL_THEN`row 1 ((pmat1 (vecmats l)):real^3^P) = row 1 (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
MP_TAC(ARITH_RULE`2<p==> 1<=1 /\ 1<p `)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC
THEN REMOVE_THEN "THY"(fun th-> MRESA_TAC th[`1:num`;`p-1`] )
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`2< p /\ 1<= k-p ==> 1<=1 /\ 1<= p-1 /\ p-1<=k /\ ~(1=p-1)`)
THEN RESA_TAC;
SUBGOAL_THEN`row i ((pmat1 (vecmats l)):real^3^P) = row i (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC;
ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
ASM_REWRITE_TAC[]
THEN DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY1")
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC
THEN STRIP_TAC
THEN REMOVE_THEN "THY"(fun th-> MRESA_TAC th[`i:num`;`p-2+i'`] )
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`i<p /\ 1<= k-p /\ 2< p /\ 1<=i'/\ i'<= k-p+2
==> i <= k /\ 1 <= p - 2 + i' /\ p - 2 + i' <= k /\ p-1<= p-2+i'`)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`i<p /\ p-1<= p-2+i' /\ 2< p==> (i=p-1 /\ i'=1) \/ i< p-2+i'`)
THEN RESA_TAC;
REMOVE_THEN"THY1"MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<p /\ p-1<= p-2+i' /\ 2< p==> p-1<p /\ SUC (p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`];
POP_ASSUM MP_TAC
THEN ARITH_TAC]);;



let ROW_IN_F_SY=prove(`! (l:real^(M,3)finite_product)
 (l1:real^(P,3)finite_product).
 dimindex(:P) =p 
/\ 2<p
/\ matvec(pmat1(vecmats l))= l1
==> (row (p - 1) (vecmats l1),row p (vecmats l1)) IN F_SY (vecmats l1)`,
REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[F_SY;IN_ELIM_THM]
THEN EXISTS_TAC`p-1`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`2<p==> p-1<p  /\ 1<= p-1 /\ p-1<=p/\ SUC(p-1)=p`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`]);;


let PMAT2_EQ_SY=prove(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ 1<= i /\ i<= k-p+2
==> row i (vecmats l2) = row (p-2+i ) (vecmats l)`,
REPEAT STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN EXPAND_TAC"l2"
THEN REWRITE_TAC[VECMATS_MATVEC_ID]
THEN MP_TAC(ARITH_RULE`1<=k-p/\ 2<p  ==> k - (k - p + 2) + i= p-2+i`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat2;LAMBDA_BETA;CART_EQ]);;

let PMAT1_EQ_SY=prove(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ 1<= i /\ i<= p-1
==> row i (vecmats l1) = row i (vecmats l)`,
REPEAT STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID]
THEN MP_TAC(ARITH_RULE`i<=p-1/\ 2<p  ==> i<= p /\ i< p`)
THEN RESA_TAC
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ]);;


let PMAT1_EQ_SY_P=prove(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
==> row p (vecmats l1) = row k (vecmats l)`,
REPEAT STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN EXPAND_TAC"l1"
THEN REWRITE_TAC[VECMATS_MATVEC_ID]
THEN SUBGOAL_THEN`1<= p/\ p<= dimindex(:P)==> row p ((pmat1 (vecmats l)):real^3^P) = row k (vecmats (l:real^(M,3)finite_product))` ASSUME_TAC
THENL[ASSUME_TAC(ARITH_RULE`p<=p /\ ~(p<p:num) `)
THEN ASM_SIMP_TAC[row;pmat1;LAMBDA_BETA;CART_EQ];
POP_ASSUM MATCH_MP_TAC 
THEN ASM_REWRITE_TAC[]]);;




let POWER_RHO_NODE_SY= prove_by_refinement(`!i u (l:real^(M,3)finite_product). local_fan (V_SY (vecmats (l:real^(M,3)finite_product)),E_SY (vecmats l),F_SY (vecmats l))
/\ 1<= i /\ i<= dimindex (:M) 
/\ row i (vecmats l)=u
/\ row 1 (vecmats l)=v
==>  ITER (i-1) (rho_node1 (F_SY (vecmats l))) v=u`,
[INDUCT_TAC;
ARITH_TAC;
POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN REPEAT GEN_TAC
THEN DISJ_CASES_TAC(ARITH_RULE`i=0 \/ 1<=i`);
ASM_REWRITE_TAC[ARITH_RULE`1-1=0/\ SUC 0=1`;ITER]
THEN RESA_TAC
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM th]);
STRIP_TAC
THEN MP_TAC(ARITH_RULE`1<=i /\ SUC i <= dimindex(:M) ==> SUC i- 1= SUC(i-1) /\ i<= dimindex(:M) /\ i< dimindex(:M)`)
THEN RESA_TAC
THEN REMOVE_THEN "THY"(fun th-> MRESA_TAC th[`row i (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`])
THEN ASM_REWRITE_TAC[ITER]
THEN MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`V_SY (vecmats (l:real^(M,3)finite_product))`
THEN EXISTS_TAC`E_SY (vecmats (l:real^(M,3)finite_product))`
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC EDGE_IN_F_SY
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC MOD_LT[`i:num`;`dimindex(:M)`]]);;

let RHO_NODE_K_SY=prove(`!(l:real^(M,3)finite_product). local_fan (V_SY (vecmats (l:real^(M,3)finite_product)),E_SY (vecmats l),F_SY (vecmats l))
/\ dimindex(:M)=k
/\ row k (vecmats l)=u
/\ row 1 (vecmats l)=v
==>   (rho_node1 (F_SY (vecmats l))) u=v`,
REPEAT STRIP_TAC
THEN MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`V_SY (vecmats (l:real^(M,3)finite_product))`
THEN EXISTS_TAC`E_SY (vecmats (l:real^(M,3)finite_product))`
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC (GEN_ALL EDGE_IN_F_SY)
THEN EXISTS_TAC`k:num`
THEN ASM_REWRITE_TAC[]
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MP_TAC(ARITH_RULE`1<=k==> k<=k /\ ~(k=0)/\ SUC 0=1`)
THEN RESA_TAC
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]);;


let RHO_NODE_SY=prove(`!i (l:real^(M,3)finite_product). local_fan (V_SY (vecmats (l:real^(M,3)finite_product)),E_SY (vecmats l),F_SY (vecmats l))
/\ 1<= i /\ i<= dimindex (:M) 
==>  (rho_node1 (F_SY (vecmats l))) (row i (vecmats l))=row (SUC (i MOD dimindex (:M))) (vecmats l)`,
REPEAT STRIP_TAC
THEN MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`V_SY (vecmats (l:real^(M,3)finite_product))`
THEN EXISTS_TAC`E_SY (vecmats (l:real^(M,3)finite_product))`
THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC (GEN_ALL EDGE_IN_F_SY)
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MP_TAC(ARITH_RULE`1<=k==> k<=k /\ ~(k=0)/\ SUC 0=1`)
THEN RESA_TAC
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]);;


let ORDER_COVER1_SY=prove(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ row k (vecmats l)= v
/\ row (p-1) (vecmats l)= w
==> order (rho_node1 (F_SY (vecmats l))) v w = p-1`,
REPEAT GEN_TAC THEN STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN MATCH_MP_TAC UNIQUE_ORDER
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p ==> 1<= p-1 /\ p-1<=k/\ p-1= SUC(p-1-1)`)
THEN RESA_TAC
THEN MRESA_TAC(GEN_ALL POWER_RHO_NODE_SY)[`row 1 (vecmats (l:real^(M,3)finite_product))`;`p-1`;`w:real^3`;`l:real^(M,3)finite_product`]
THEN STRIP_TAC
THENL[
POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC
THEN REMOVE_THEN"THY"(fun th-> ONCE_REWRITE_TAC[th;])
THEN REWRITE_TAC[POWER;o_DEF]
THEN MRESA_TAC (GEN_ALL RHO_NODE_K_SY)[`k:num`;`v:real^3`;`row 1 (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`];
STRIP_TAC
THENL[
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`0<i /\ i< p-1 /\ 1<= k-p ==> 1<=i /\ i<=k/\ i= SUC(i-1)`)
THEN RESA_TAC
THEN MRESA_TAC(GEN_ALL POWER_RHO_NODE_SY)[`row 1 (vecmats (l:real^(M,3)finite_product))`;`i:num`;`row i (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC
THEN REMOVE_THEN"THY"(fun th-> ONCE_REWRITE_TAC[th;])
THEN REWRITE_TAC[POWER;o_DEF]
THEN MRESA_TAC (GEN_ALL RHO_NODE_K_SY)[`k:num`;`v:real^3`;`row 1 (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN STRIP_TAC
THEN MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`i:num`;`p-1:num`])
THEN MP_TAC(ARITH_RULE`i< p-1 ==> ~(i=p-1)`)
THEN RESA_TAC;
STRIP_TAC
THEN MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`p-1:num`])
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<= k-p ==> k<=k/\ ~(k=p-1)`)
THEN RESA_TAC]]);;


let SLICEV_EQ_V_SY=prove_by_refinement(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ row k (vecmats l)= v
/\ row (p-1) (vecmats l)= w
==> slicev (E_SY (vecmats l)) (F_SY (vecmats l)) v w = V_SY(vecmats l1)`,
[REPEAT GEN_TAC THEN STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN REWRITE_TAC[slicev;V_SY;rows;IN_ELIM_THM;EXTENSION]
THEN MRESA_TAC (GEN_ALL ORDER_COVER1_SY)[`k:num`;`v:real^3`;`w:real^3`;`p:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN GEN_TAC
THEN EQ_TAC;
REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN DISJ_CASES_TAC(ARITH_RULE`n=0 \/ 0< n`);
ASM_REWRITE_TAC[ITER]
THEN EXISTS_TAC`p:num`
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`0<n /\ n<= p-1 /\ 1<= k-p ==> 1<=n /\ n<=k/\ n<=p/\ n= SUC(n-1)`)
THEN RESA_TAC
THEN MRESA_TAC(GEN_ALL POWER_RHO_NODE_SY)[`row 1 (vecmats (l:real^(M,3)finite_product))`;`n:num`;`row n (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC
THEN REMOVE_THEN"THY"(fun th-> ONCE_REWRITE_TAC[th;])
THEN REWRITE_TAC[POWER;o_DEF]
THEN MRESA_TAC (GEN_ALL RHO_NODE_K_SY)[`k:num`;`v:real^3`;`row 1 (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN EXISTS_TAC`n:num`
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`n:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<= p ==> i=p \/ i< p:num`)
THEN RESA_TAC;
ASM_REWRITE_TAC[ITER]
THEN EXISTS_TAC`0:num`
THEN ASM_REWRITE_TAC[ITER;ARITH_RULE`0<=0`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MP_TAC(ARITH_RULE`2<p==> 0<= p-1`)
THEN RESA_TAC;
FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`1<=i /\ i< p /\ 1<= k-p ==> 0<=i/\ 1<=i /\ i<=p-1/\ i<=k/\ i= SUC(i-1)`)
THEN RESA_TAC
THEN MRESA_TAC(GEN_ALL POWER_RHO_NODE_SY)[`row 1 (vecmats (l:real^(M,3)finite_product))`;`i:num`;`row i (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC
THEN REMOVE_THEN"THY"(fun th-> ONCE_REWRITE_TAC[th;])
THEN REWRITE_TAC[POWER;o_DEF]
THEN MRESA_TAC (GEN_ALL RHO_NODE_K_SY)[`k:num`;`v:real^3`;`row 1 (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN MP_TAC(ARITH_RULE`1<=i /\ i< p /\ 1<= k-p ==> SUC(i-1)=i`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]]);;


let SLICEE_EQ_E_SY=prove_by_refinement(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ row k (vecmats l)= v
/\ row (p-1) (vecmats l)= w
==> slicee (E_SY (vecmats l)) (F_SY (vecmats l)) v w = E_SY(vecmats l1)`,
[REPEAT GEN_TAC THEN STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN REWRITE_TAC[EXTENSION;]
THEN REWRITE_TAC[slicee;rows;IN_ELIM_THM;UNION;IN_SING]
THEN MRESA_TAC (GEN_ALL SLICEV_EQ_V_SY)[`k:num`;`p:num`;`v:real^3`;`w:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN ASM_REWRITE_TAC[V_SY;rows;E_SY;IN_ELIM_THM;DELETE]
THEN GEN_TAC
THEN EQ_TAC;
RESA_TAC;
FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`2<p /\ i<=p /\ 1<= k-p ==> i<= k/\ 1<=1/\ 1<= p-1 `)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL RHO_NODE_SY)[`i:num`;`l:real^(M,3)finite_product`]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i<p:num`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN EXISTS_TAC`p:num`
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESA_TAC (GEN_ALL RHO_NODE_K_SY)[`k:num`;`v:real^3`;`row 1 (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
MP_TAC(ARITH_RULE`1<=i /\ i< p /\ 1<= k-p ==> 0<=i/\ 1<=i /\ i<=p-1/\ i<k/\ i<=k/\ i= SUC(i-1)/\ 1<= SUC i`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<p ==> i= p-1 \/ SUC i<= p-1`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN ASM_TAC
THEN REPEAT RESA_TAC;
MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`SUC(i):num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
EXISTS_TAC`p-1`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`2<p  /\ 1<= k-p ==> p-1<= p/\ 1<=p-1/\ p-1<p /\SUC(p-1)=p /\ p-1<=p-1`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`p-1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`2<p /\ i<=p /\ 1<= k-p ==> i<= k/\ 1<=1/\ 1<= p-1 `)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`i<=p ==> i= p \/ i< p:num`)
THEN RESA_TAC;
MATCH_MP_TAC (SET_RULE`A==> A\/B`)
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN EXISTS_TAC`v:real^3`
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESA_TAC (GEN_ALL RHO_NODE_K_SY)[`k:num`;`v:real^3`;`row 1 (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN STRIP_TAC;
EXISTS_TAC`p:num`
THEN ASM_REWRITE_TAC[];
STRIP_TAC
THEN MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-1:num`;`k:num`])
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 2< p==> p-1<=k /\ k<=k /\ ~(p-1=k)`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`i<p ==> i= p-1 \/ SUC i<= p-1`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN MATCH_MP_TAC(SET_RULE`A==> B\/A`)
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`2<p  /\ 1<= k-p ==> p-1<= p/\ 1<=p-1/\ p-1<p /\SUC(p-1)=p /\ p-1<=p-1`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`p-1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
MATCH_MP_TAC(SET_RULE`A==> A\/B`)
THEN MP_TAC(ARITH_RULE`1<=i /\ i< p /\ 1<= k-p ==> 0<=i/\ 1<=i /\ i<=p-1/\ i<k/\ i<=k/\ i= SUC(i-1)/\ 1<= SUC i`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`SUC(i):num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN EXISTS_TAC`row i (vecmats (l:real^(M,3)finite_product))`
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC (GEN_ALL RHO_NODE_SY)[`i:num`;`l:real^(M,3)finite_product`]
THEN STRIP_TAC;
EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[];
STRIP_TAC
THEN MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-1:num`;`i:num`])
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 2< p /\ SUC i<= p-1 ==> p-1<=k  /\ ~(p-1=i)`)
THEN RESA_TAC]);;

let SLICEF_EQ_F_SY=prove_by_refinement(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ row k (vecmats l)= v
/\ row (p-1) (vecmats l)= w
==> slicef (E_SY (vecmats l)) (F_SY (vecmats l)) v w = F_SY(vecmats l1)`,
[REPEAT GEN_TAC THEN STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN REWRITE_TAC[EXTENSION;]
THEN REWRITE_TAC[slicef;rows;IN_ELIM_THM;UNION;IN_SING]
THEN MRESA_TAC (GEN_ALL SLICEV_EQ_V_SY)[`k:num`;`p:num`;`v:real^3`;`w:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN ASM_REWRITE_TAC[V_SY;rows;E_SY;IN_ELIM_THM;DELETE]
THEN GEN_TAC
THEN EQ_TAC;
STRIP_TAC;
ONCE_REWRITE_TAC[F_SY;]
THEN REWRITE_TAC[IN_ELIM_THM]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`2<p /\ i<=p /\ 1<= k-p ==> i<= k/\ 1<=1/\ 1<= p-1 `)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL RHO_NODE_SY)[`i:num`;`l:real^(M,3)finite_product`]
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i<p:num`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN EXISTS_TAC`p:num`
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESA_TAC (GEN_ALL RHO_NODE_K_SY)[`k:num`;`v:real^3`;`row 1 (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
MP_TAC(ARITH_RULE`1<=i /\ i< p /\ 1<= k-p ==> 0<=i/\ 1<=i /\ i<=p-1/\ i<k/\ i<=k/\ i= SUC(i-1)/\ 1<= SUC i`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<p ==> i= p-1 \/ SUC i<= p-1`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN ASM_TAC
THEN REPEAT RESA_TAC;
MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`SUC(i):num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
REWRITE_TAC[F_SY;IN_ELIM_THM]
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`p-1`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`2<p  /\ 1<= k-p ==> p-1<= p/\ 1<=p-1/\ p-1<p /\SUC(p-1)=p /\ p-1<=p-1`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`p-1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
GEN_REWRITE_TAC(LAND_CONV o DEPTH_CONV)[F_SY;IN_ELIM_THM]
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`2<p /\ i<=p /\ 1<= k-p ==> i<= k/\ 1<=1/\ 1<= p-1 `)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`i<=p ==> i= p \/ i< p:num`)
THEN RESA_TAC;
MATCH_MP_TAC (SET_RULE`A==> A\/B`)
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN EXISTS_TAC`v:real^3`
THEN ASM_REWRITE_TAC[]
THEN MRESAL_TAC MOD_MULT[`p:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESA_TAC (GEN_ALL RHO_NODE_K_SY)[`k:num`;`v:real^3`;`row 1 (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN STRIP_TAC;
EXISTS_TAC`p:num`
THEN ASM_REWRITE_TAC[];
STRIP_TAC
THEN MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-1:num`;`k:num`])
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 2< p==> p-1<=k /\ k<=k /\ ~(p-1=k)`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`i<p ==> i= p-1 \/ SUC i<= p-1`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN MATCH_MP_TAC(SET_RULE`A==> B\/A`)
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`2<p  /\ 1<= k-p ==> p-1<= p/\ 1<=p-1/\ p-1<p /\SUC(p-1)=p /\ p-1<=p-1`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`p:num`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`p-1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
MATCH_MP_TAC(SET_RULE`A==> A\/B`)
THEN MP_TAC(ARITH_RULE`1<=i /\ i< p /\ 1<= k-p ==> 0<=i/\ 1<=i /\ i<=p-1/\ i<k/\ i<=k/\ i= SUC(i-1)/\ 1<= SUC i`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`p:num`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`SUC(i):num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN EXISTS_TAC`row i (vecmats (l:real^(M,3)finite_product))`
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC (GEN_ALL RHO_NODE_SY)[`i:num`;`l:real^(M,3)finite_product`]
THEN STRIP_TAC;
EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[];
STRIP_TAC
THEN MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-1:num`;`i:num`])
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 2< p /\ SUC i<= p-1 ==> p-1<=k  /\ ~(p-1=i)`)
THEN RESA_TAC]);;

let ORDER_COVER2_SY=prove(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ row k (vecmats l)= v
/\ row (p-1) (vecmats l)= w
==> order (rho_node1 (F_SY (vecmats l))) w v = k-p+1`,
REPEAT GEN_TAC THEN STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN MATCH_MP_TAC UNIQUE_ORDER
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p ==> 1<= p-1 /\ p-1<=k/\ p-1= SUC(p-1-1) /\ (k - p + 1) + p - 1 - 1 =k-1`)
THEN RESA_TAC
THEN MRESA_TAC(GEN_ALL POWER_RHO_NODE_SY)[`row 1 (vecmats (l:real^(M,3)finite_product))`;`p-1`;`w:real^3`;`l:real^(M,3)finite_product`]
THEN STRIP_TAC
THENL[POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER]
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN STRIP_TAC
THEN REWRITE_TAC[POWER;o_DEF]
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM th;GSYM Hypermap.lemma_add_exponent_function])
THEN MRESA_TAC(GEN_ALL POWER_RHO_NODE_SY)[`row 1 (vecmats (l:real^(M,3)finite_product))`;`k:num`;`v:real^3`;`l:real^(M,3)finite_product`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER]
THEN STRIP_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN ARITH_TAC;
STRIP_TAC
THENL[
POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THY")
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`0<i /\ i< k-p+1 /\ 1<= k-p/\ 2<p ==> 1<=i+p-1 /\ i+p-1<=k/\ i= SUC(i-1)/\ (i + p - 1) - 1 =i + p - 1 - 1`)
THEN RESA_TAC
THEN REMOVE_THEN"THY"(fun th-> ASM_REWRITE_TAC[SYM th;GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER;GSYM Hypermap.lemma_add_exponent_function])
THEN MRESA_TAC(GEN_ALL POWER_RHO_NODE_SY)[`row 1 (vecmats (l:real^(M,3)finite_product))`;`i+p-1:num`;`row (i+p-1) (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER]
THEN RESA_TAC
THEN STRIP_TAC
THEN MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`i+p-1:num`;`k:num`])
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`i< k-p+1 /\2<p /\ 1<=k-p ==> ~(i+p-1=k) /\ k<=k`)
THEN RESA_TAC;
STRIP_TAC
THEN MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`p-1:num`])
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<= k-p ==> k<=k/\ ~(k=p-1)`)
THEN RESA_TAC]]);;

let SLICEV_EQ_V2_SY=prove(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ row k (vecmats l)= v
/\ row (p-1) (vecmats l)= w
==> slicev (E_SY (vecmats l)) (F_SY (vecmats l)) w v = V_SY(vecmats l2)`,
REPEAT GEN_TAC THEN STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN REWRITE_TAC[slicev;V_SY;rows;IN_ELIM_THM;EXTENSION]
THEN MRESA_TAC (GEN_ALL ORDER_COVER2_SY)[`w:real^3`;`v:real^3`;`k:num`;`p:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN GEN_TAC
THEN EQ_TAC
THENL[
RESA_TAC
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p /\ 0<=n /\ n<= k-p+1 ==> 1<= p-1 /\ p-1<=k/\ p-1= SUC(p-1-1) /\ (k - p + 1) + p - 1 - 1 =k-1/\ 1<= n+p-1 /\ n+p-1<=k
/\ (n+p-1)-1=n+p-1-1 /\ 1<= n+1 /\ n+1<= k-p+2/\ p-2+n+1=n+p-1`)
THEN RESA_TAC
THEN MRESA_TAC(GEN_ALL POWER_RHO_NODE_SY)[`row 1 (vecmats (l:real^(M,3)finite_product))`;`p-1`;`w:real^3`;`l:real^(M,3)finite_product`]
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM th;GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER;GSYM Hypermap.lemma_add_exponent_function])
THEN MRESA_TAC(GEN_ALL POWER_RHO_NODE_SY)[`row 1 (vecmats (l:real^(M,3)finite_product))`;`n+p-1:num`;`row (n+p-1) (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER]
THEN RESA_TAC
THEN EXISTS_TAC`n+1`
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`n+1`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN EXISTS_TAC`i-1`
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p /\ 1<=i /\ i<=k-p+2 ==> 0<= i-1 /\ i-1<=k-p+1/\ 1<= p-1 /\ p-1<=k/\ 1<=p-2+i /\ p-2+i<=k /\ (p - 2 + i) - 1=i - 1 + p - 1 - 1`)
THEN RESA_TAC
THEN MRESA_TAC(GEN_ALL POWER_RHO_NODE_SY)[`row 1 (vecmats (l:real^(M,3)finite_product))`;`p-1`;`w:real^3`;`l:real^(M,3)finite_product`]
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM th;GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER;GSYM Hypermap.lemma_add_exponent_function])
THEN MRESA_TAC(GEN_ALL POWER_RHO_NODE_SY)[`row 1 (vecmats (l:real^(M,3)finite_product))`;`p-2+i:num`;`row (p-2+i) (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[GSYM Wrgcvdr_cizmrrh.POWER_TO_ITER]
THEN RESA_TAC]);;



let SLICEE_EQ_E2_SY=prove_by_refinement(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ row k (vecmats l)= v
/\ row (p-1) (vecmats l)= w
==> slicee (E_SY (vecmats l)) (F_SY (vecmats l)) w v = E_SY(vecmats l2)`,
[REPEAT GEN_TAC THEN STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN REWRITE_TAC[EXTENSION;]
THEN REWRITE_TAC[slicee;rows;IN_ELIM_THM;UNION;IN_SING]
THEN MRESA_TAC (GEN_ALL SLICEV_EQ_V2_SY)[`k:num`;`p:num`;`w:real^3`;`v:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN ASM_REWRITE_TAC[V_SY;rows;E_SY;IN_ELIM_THM;DELETE]
THEN GEN_TAC
THEN EQ_TAC;
STRIP_TAC;
ASM_REWRITE_TAC[]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p /\ 1<=i /\ i<=k-p+2 ==>  1<=p-2+i /\ p-2+i<=k  `)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL RHO_NODE_SY)[`p-2+i:num`;`l:real^(M,3)finite_product`]
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<= k-p+2==> i= k-p+2 \/ i< k-p+2`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p  ==> p-2+k-p+2=k`)
THEN RESA_TAC
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p /\ 1<=k-p /\ i<k-p+2 ==>  p-2+i<k  /\ 1<= SUC i /\ SUC i<= k-p+2 /\ SUC (p - 2 + i) =p-2+ SUC i`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`SUC i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
ASM_REWRITE_TAC[]
THEN EXISTS_TAC`k-p+2`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`1<=k-p /\ 2<p==> p-2+k-p+2=k/\ k-p+2<=k-p+2/\ 1<=1/\ p-2+1=p-1`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`k-p+2`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`1`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MP_TAC(ARITH_RULE`i<=k-p+2==> i= k-p+2 \/ i< k-p+2`)
THEN RESA_TAC;
MATCH_MP_TAC(SET_RULE`A==> B\/A`)
THEN MP_TAC(ARITH_RULE`1<=k-p /\ 2<p==> p-2+k-p+2=k/\ k-p+2<=k-p+2/\ 1<=1/\ p-2+1=p-1`)
THEN RESA_TAC
THEN MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`1`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
MATCH_MP_TAC(SET_RULE`A==> A\/B`)
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p /\ i<k-p+2 ==>  p-2+i<k  /\ 1<= SUC i /\ SUC i<= k-p+2 /\ SUC (p - 2 + i) =p-2+ SUC i`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`SUC i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN EXISTS_TAC`row i (vecmats (l2:real^(N,3)finite_product))`
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p /\ 1<=i /\ i<=k-p+2 ==>  1<=p-2+i /\ p-2+i<=k  `)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL RHO_NODE_SY)[`p-2+i:num`;`l:real^(M,3)finite_product`]
THEN STRIP_TAC;
EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[];
STRIP_TAC
THEN MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`p-2+i:num`])
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<= k-p/\ i< k-p+2 /\ 2<p ==> k<=k/\ ~(k=p-2+i)`)
THEN RESA_TAC]);;

let SLICEF_EQ_F2_SY=prove_by_refinement(`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ row k (vecmats l)= v
/\ row (p-1) (vecmats l)= w
==> slicef (E_SY (vecmats l)) (F_SY (vecmats l)) w v = F_SY(vecmats l2)`,
[REPEAT GEN_TAC THEN STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN REWRITE_TAC[EXTENSION;]
THEN REWRITE_TAC[slicef;rows;IN_ELIM_THM;UNION;IN_SING]
THEN MRESA_TAC (GEN_ALL SLICEV_EQ_V2_SY)[`k:num`;`p:num`;`w:real^3`;`v:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN ASM_REWRITE_TAC[V_SY;rows;IN_ELIM_THM;DELETE]
THEN GEN_TAC
THEN EQ_TAC;
STRIP_TAC;
REWRITE_TAC[F_SY;IN_ELIM_THM]
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p /\ 1<=i /\ i<=k-p+2 ==>  1<=p-2+i /\ p-2+i<=k  `)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL RHO_NODE_SY)[`p-2+i:num`;`l:real^(M,3)finite_product`]
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<= k-p+2==> i= k-p+2 \/ i< k-p+2`)
THEN RESA_TAC;
POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p  ==> p-2+k-p+2=k`)
THEN RESA_TAC
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(th))
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p /\ 1<=k-p /\ i<k-p+2 ==>  p-2+i<k  /\ 1<= SUC i /\ SUC i<= k-p+2 /\ SUC (p - 2 + i) =p-2+ SUC i`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`SUC i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
REWRITE_TAC[F_SY;IN_ELIM_THM]
THEN ASM_REWRITE_TAC[]
THEN EXISTS_TAC`k-p+2`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`1<=k-p /\ 2<p==> p-2+k-p+2=k/\ k-p+2<=k-p+2/\ 1<=1/\ p-2+1=p-1`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`k-p+2`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`1`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
GEN_REWRITE_TAC(LAND_CONV o DEPTH_CONV)[F_SY;IN_ELIM_THM]
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MP_TAC(ARITH_RULE`i<=k-p+2==> i= k-p+2 \/ i< k-p+2`)
THEN RESA_TAC;
MATCH_MP_TAC(SET_RULE`A==> B\/A`)
THEN MP_TAC(ARITH_RULE`1<=k-p /\ 2<p==> p-2+k-p+2=k/\ k-p+2<=k-p+2/\ 1<=1/\ p-2+1=p-1`)
THEN RESA_TAC
THEN MRESAL_TAC MOD_MULT[`k-p+2:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`1`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`];
MATCH_MP_TAC(SET_RULE`A==> A\/B`)
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p /\ i<k-p+2 ==>  p-2+i<k  /\ 1<= SUC i /\ SUC i<= k-p+2 /\ SUC (p - 2 + i) =p-2+ SUC i`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`i:num`;`k-p+2:num`]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`SUC i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN EXISTS_TAC`row i (vecmats (l2:real^(N,3)finite_product))`
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MP_TAC(ARITH_RULE`2<p /\ 1<=k-p /\ 1<=i /\ i<=k-p+2 ==>  1<=p-2+i /\ p-2+i<=k  `)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL RHO_NODE_SY)[`p-2+i:num`;`l:real^(M,3)finite_product`]
THEN STRIP_TAC;
EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[];
STRIP_TAC
THEN MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`p-2+i:num`])
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`1<= k-p/\ i< k-p+2 /\ 2<p ==> k<=k/\ ~(k=p-2+i)`)
THEN RESA_TAC]);;




let TAU_STAR_COVER_concl=`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ row k (vecmats l)= v
/\ row (p-1) (vecmats l)= w
/\ (!u u1.
               u IN {v, w} /\ u1 IN V_SY(vecmats l) /\ ~(u = u1)
               ==> ~collinear {vec 0, u, u1})
==> tau_star s1 l1 + tau_star s3 l2 <= tau_star s l`
;;

let TAU_STAR_COVER_concl2 = mk_imp (EJRCFJD_concl, TAU_STAR_COVER_concl);;







let TAU_STAR_COVER=prove_by_refinement(TAU_STAR_COVER_concl2,
[DISCH_THEN(LABEL_TAC"THY")
THEN REPEAT STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN REWRITE_TAC[tau_star]
THEN MRESA1_TAC stable_sy_lemma`s:stable_sy`
THEN MP_TAC(ARITH_RULE` 2<p /\ 1<=k-p 
==> p - 2 + k - p + 2= k/\ p-2+1=p-1 /\ 1<=p-1 /\ p-1<=k /\ p-1<k /\ ~(k=p-1)/\ k<=k`)
THEN RESA_TAC
THEN SUBGOAL_THEN`{v, w} SUBSET V_SY (vecmats (l:real^(M,3)finite_product))`ASSUME_TAC;
MATCH_MP_TAC(SET_RULE`A IN C /\ B IN C==> {A,B}SUBSET C`)
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[IN_ELIM_THM;V_SY;rows];
EXISTS_TAC`k:num`
THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC;
EXISTS_TAC`p-1`
THEN ASM_REWRITE_TAC[];
SUBGOAL_THEN`norm (v - w:real^3) <= cstab`ASSUME_TAC;
MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN FIND_ASSUM (fun th-> MP_TAC(ISPECL [`k:num`;`p-1`]th))`!i j.
          {i MOD k, j MOD k} IN J_SY s UNION {{0, p - 1}}
          ==> norm (row i (vecmats l) - row j (vecmats (l:real^(M,3)finite_product))) <= cstab`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th; SET_RULE`{0,p-1} IN J_SY s UNION {{0,p-1}}`])
THEN ASM_REWRITE_TAC[];
SUBGOAL_THEN`~({v, w} IN E_SY (vecmats (l:real^(M,3)finite_product)))`ASSUME_TAC;
ASM_REWRITE_TAC[E_SY;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;local_fan]
THEN LET_TAC
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MRESA_TAC (GEN_ALL EDGE_IN_E_SY)[`i:num`;`row (i) (vecmats(l:real^(M,3)finite_product))`;`
      (row (SUC (i MOD dimindex (:M))) (vecmats (l:real^(M,3)finite_product)))`;`(l:real^(M,3)finite_product)`]
THEN MRESA_TAC remark1_fan[`vec 0:real^3`;`V_SY (vecmats(l:real^(M,3)finite_product))`;`E_SY (vecmats (l:real^(M,3)finite_product))`;`
      (row (SUC (i MOD dimindex (:M))) (vecmats (l:real^(M,3)finite_product)))`;`row i (vecmats(l:real^(M,3)finite_product))`]
THEN DISJ_CASES_TAC(SET_RULE`row k (vecmats l)= row (p - 1) (vecmats l)
\/ ~(row k (vecmats l)= row (p - 1) (vecmats (l:real^(M,3)finite_product)))`);
MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`p-1`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`1 <= SUC (i' MOD dimindex (:M)) /\
      SUC (i' MOD dimindex (:M)) <= dimindex (:M)` ASSUME_TAC;
ASM_REWRITE_TAC[ARITH_RULE`1<= SUC a`]
THEN MRESAL_TAC DIVISION[`i':num`;`dimindex (:M):num`][DIMINDEX_NONZERO]
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MP_TAC(SET_RULE`{row k (vecmats l), row (p - 1) (vecmats l)} =
   {row i (vecmats l), row (SUC (i MOD k)) (vecmats l)}
/\ ~(row k (vecmats l) = row (p - 1) (vecmats l))
/\ ~(row i (vecmats l) = row (SUC (i MOD k)) (vecmats l))
==> (row i (vecmats l) = row (k) (vecmats l)
/\ row (p-1) (vecmats l) = row (SUC (i MOD k)) (vecmats l))
\/ (row i (vecmats l) = row (p-1) (vecmats l)
/\ row (k) (vecmats l) = row (SUC (i MOD k)) (vecmats (l:real^(M,3)finite_product)))`)
THEN RESA_TAC;
MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`i:num`;`k:num`] THEN  MRESA_TAC th[`p-1:num`;`SUC(i MOD k):num`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MP_TAC(ARITH_RULE`2<p==> ~(p-1=1)/\ 1<=1`)
THEN RESA_TAC;
MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`i:num`;`p-1:num`] THEN  MRESA_TAC th[`k:num`;`SUC(i MOD k):num`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`2<p /\ 1<= k-p ==> ~(k= SUC (p-1))/\ 1<= SUC(p-1)/\ SUC(p-1)<=k`)
THEN RESA_TAC;
MRESA_TAC (GEN_ALL PROPERTIES_OF_FAN_IN_B_SY)[`d_sy (s:stable_sy)`;`J_SY (s:stable_sy)`;`k_sy (s:stable_sy)`;`k:num`;`p-1:num`;`a_sy (s:stable_sy)`;`b_sy (s:stable_sy)`;`row k (vecmats (l:real^(M,3)finite_product))`;`row (p-1) (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`] 
THEN MRESAL_TAC(GEN_ALL TECOXBM1)[`d_sy (s:stable_sy)`;`J_SY(s:stable_sy)`;`k:num`;`a_sy(s:stable_sy)`;`b_sy(s:stable_sy)`;`row k (vecmats (l:real^(M,3)finite_product))`;`row (p-1) (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`][SET_RULE`{A} UNION {B,C}={A,B,C}`]
THEN MP_TAC(SET_RULE`{v, w} SUBSET V_SY (vecmats l)==> v IN V_SY (vecmats l)
/\ w IN V_SY (vecmats (l:real^(M,3)finite_product))`)
THEN RESA_TAC
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MRESA_TAC (GEN_ALL SLICEV_EQ_V_SY)[`k:num`;`p:num`;`v:real^3`;`w:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEE_EQ_E_SY)[`k:num`;`p:num`;`v:real^3`;`w:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEF_EQ_F_SY)[`k:num`;`p:num`;`v:real^3`;`w:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEF_EQ_F2_SY)[`k:num`;`p:num`;`w:real^3`;`v:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEE_EQ_E2_SY)[`k:num`;`p:num`;`w:real^3`;`v:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEV_EQ_V2_SY)[`k:num`;`p:num`;`w:real^3`;`v:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN REMOVE_THEN"THY"(fun th-> MRESA_TAC th[`V_SY(vecmats (l:real^(M,3)finite_product))`;`E_SY(vecmats (l:real^(M,3)finite_product))`;`F_SY(vecmats (l:real^(M,3)finite_product))`;
`v:real^3`;`w:real^3`])
THEN MRESA_TAC(GEN_ALL D_FUN_COVER_SY)[`k:num`;`p:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC]);;



let IN_B_SY_COVER_concl=`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ row k (vecmats l)= v
/\ row (p-1) (vecmats l)= w
/\ (!u u1.
               u IN {v, w} /\ u1 IN V_SY(vecmats l) /\ ~(u = u1)
               ==> ~collinear {vec 0, u, u1})
/\ norm(v-w)<= min (b_sy s1 (p-1,0)) (b_sy s2 (p-1,0))
==> l1 IN (B_SY1 (a_sy s1) (b_sy s1))`;;


let IN_B_SY_COVER_concl2 = mk_imp (EJRCFJD_concl, IN_B_SY_COVER_concl);;





let IN_B_SY_COVER=prove_by_refinement(IN_B_SY_COVER_concl2,
[DISCH_THEN(LABEL_TAC"THY")
THEN REPEAT STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN REWRITE_TAC[tau_star]
THEN MRESA1_TAC stable_sy_lemma`s:stable_sy`
THEN MP_TAC(ARITH_RULE` 2<p /\ 1<=k-p 
==> p - 2 + k - p + 2= k/\ p-2+1=p-1 /\ 1<=p-1 /\ p-1<=k /\ p-1<k /\ ~(k=p-1)/\ k<=k`)
THEN RESA_TAC
THEN SUBGOAL_THEN`{v, w} SUBSET V_SY (vecmats (l:real^(M,3)finite_product))`ASSUME_TAC;
MATCH_MP_TAC(SET_RULE`A IN C /\ B IN C==> {A,B}SUBSET C`)
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[IN_ELIM_THM;V_SY;rows];
EXISTS_TAC`k:num`
THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC;
EXISTS_TAC`p-1`
THEN ASM_REWRITE_TAC[];
SUBGOAL_THEN`norm (v - w:real^3) <= cstab`ASSUME_TAC;
MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN FIND_ASSUM (fun th-> MP_TAC(ISPECL [`k:num`;`p-1`]th))`!i j.
          {i MOD k, j MOD k} IN J_SY s UNION {{0, p - 1}}
          ==> norm (row i (vecmats l) - row j (vecmats (l:real^(M,3)finite_product))) <= cstab`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th; SET_RULE`{0,p-1} IN J_SY s UNION {{0,p-1}}`])
THEN ASM_REWRITE_TAC[];
SUBGOAL_THEN`~({v, w} IN E_SY (vecmats (l:real^(M,3)finite_product)))`ASSUME_TAC;
ASM_REWRITE_TAC[E_SY;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;local_fan]
THEN LET_TAC
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MRESA_TAC (GEN_ALL EDGE_IN_E_SY)[`i:num`;`row (i) (vecmats(l:real^(M,3)finite_product))`;`
      (row (SUC (i MOD dimindex (:M))) (vecmats (l:real^(M,3)finite_product)))`;`(l:real^(M,3)finite_product)`]
THEN MRESA_TAC remark1_fan[`vec 0:real^3`;`V_SY (vecmats(l:real^(M,3)finite_product))`;`E_SY (vecmats (l:real^(M,3)finite_product))`;`
      (row (SUC (i MOD dimindex (:M))) (vecmats (l:real^(M,3)finite_product)))`;`row i (vecmats(l:real^(M,3)finite_product))`]
THEN DISJ_CASES_TAC(SET_RULE`row k (vecmats l)= row (p - 1) (vecmats l)
\/ ~(row k (vecmats l)= row (p - 1) (vecmats (l:real^(M,3)finite_product)))`);
MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`p-1`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`1 <= SUC (i' MOD dimindex (:M)) /\
      SUC (i' MOD dimindex (:M)) <= dimindex (:M)` ASSUME_TAC;
ASM_REWRITE_TAC[ARITH_RULE`1<= SUC a`]
THEN MRESAL_TAC DIVISION[`i':num`;`dimindex (:M):num`][DIMINDEX_NONZERO]
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MP_TAC(SET_RULE`{row k (vecmats l), row (p - 1) (vecmats l)} =
   {row i (vecmats l), row (SUC (i MOD k)) (vecmats l)}
/\ ~(row k (vecmats l) = row (p - 1) (vecmats l))
/\ ~(row i (vecmats l) = row (SUC (i MOD k)) (vecmats l))
==> (row i (vecmats l) = row (k) (vecmats l)
/\ row (p-1) (vecmats l) = row (SUC (i MOD k)) (vecmats l))
\/ (row i (vecmats l) = row (p-1) (vecmats l)
/\ row (k) (vecmats l) = row (SUC (i MOD k)) (vecmats (l:real^(M,3)finite_product)))`)
THEN RESA_TAC;
MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`i:num`;`k:num`] THEN  MRESA_TAC th[`p-1:num`;`SUC(i MOD k):num`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MP_TAC(ARITH_RULE`2<p==> ~(p-1=1)/\ 1<=1`)
THEN RESA_TAC;
MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`i:num`;`p-1:num`] THEN  MRESA_TAC th[`k:num`;`SUC(i MOD k):num`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`2<p /\ 1<= k-p ==> ~(k= SUC (p-1))/\ 1<= SUC(p-1)/\ SUC(p-1)<=k`)
THEN RESA_TAC;
MRESA_TAC (GEN_ALL PROPERTIES_OF_FAN_IN_B_SY)[`d_sy (s:stable_sy)`;`J_SY (s:stable_sy)`;`k_sy (s:stable_sy)`;`k:num`;`p-1:num`;`a_sy (s:stable_sy)`;`b_sy (s:stable_sy)`;`row k (vecmats (l:real^(M,3)finite_product))`;`row (p-1) (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`] 
THEN MRESAL_TAC(GEN_ALL TECOXBM1)[`d_sy (s:stable_sy)`;`J_SY(s:stable_sy)`;`k:num`;`a_sy(s:stable_sy)`;`b_sy(s:stable_sy)`;`row k (vecmats (l:real^(M,3)finite_product))`;`row (p-1) (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`][SET_RULE`{A} UNION {B,C}={A,B,C}`]
THEN MP_TAC(SET_RULE`{v, w} SUBSET V_SY (vecmats l)==> v IN V_SY (vecmats l)
/\ w IN V_SY (vecmats (l:real^(M,3)finite_product))`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL SLICEV_EQ_V_SY)[`k:num`;`p:num`;`v:real^3`;`w:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEE_EQ_E_SY)[`k:num`;`p:num`;`v:real^3`;`w:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEF_EQ_F_SY)[`k:num`;`p:num`;`v:real^3`;`w:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEF_EQ_F2_SY)[`k:num`;`p:num`;`w:real^3`;`v:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEE_EQ_E2_SY)[`k:num`;`p:num`;`w:real^3`;`v:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEV_EQ_V2_SY)[`k:num`;`p:num`;`w:real^3`;`v:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_THEN"THY"(fun th-> MRESA_TAC th[`V_SY(vecmats (l:real^(M,3)finite_product))`;`E_SY(vecmats (l:real^(M,3)finite_product))`;`F_SY(vecmats (l:real^(M,3)finite_product))`;
`v:real^3`;`w:real^3`])
THEN EXISTS_TAC`vecmats(l1:real^(P,3)finite_product)`
THEN ASM_REWRITE_TAC[MATVEC_VECMATS_ID]
THEN STRIP_TAC;
REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`i<=p ==> i=p \/ i<= p-1:num`)
THEN RESA_TAC;
MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA1_TAC th`k:num`);
MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA1_TAC th`i:num`)
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"MA")
THEN STRIP_TAC
THEN STRIP_TAC
THEN REMOVE_THEN"MA" MATCH_MP_TAC
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`i<=p /\1<=k-p ==> i<=k`)
THEN RESA_TAC;
ASM_REWRITE_TAC[CONDITION1_SY]
THEN FIND_ASSUM MP_TAC`COVER_SY 0 (p - 1) s s1 (s2:stable_sy)`
THEN REWRITE_TAC[COVER_SY;COVER5_SY;COVER4_SY;]
THEN RESA_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THYGIANG")
THEN DISCH_THEN(LABEL_TAC"THYGIANG1")
THEN DISCH_THEN(LABEL_TAC"THYGIANG2")
THEN DISCH_THEN(LABEL_TAC"THYGIANG3")
THEN DISCH_THEN(LABEL_TAC"THYGIANG4")
THEN DISCH_THEN(LABEL_TAC"THYGIANG5")
THEN REPEAT GEN_TAC
THEN STRIP_TAC;
MP_TAC(ARITH_RULE`i<=p==> i=p \/ i<p :num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`j<=p==> j=p \/ j<p :num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p/\ 2<k ==> ~(0= p-1)/\ 0<=0 /\ 0<= p-1/\ 1<p /\ 0<=k-1`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(0= p-1)==> ~({0,0}= {0,p-1})`)
THEN RESA_TAC
THEN REMOVE_THEN"THYGIANG"(fun th-> MRESAL_TAC th[`0:num`;`0:num`][IN_NUMSEG])
THEN MRESAL1_TAC stable_sy_lemma`s1:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`0:num`;`p:num`] THEN MRESA_TAC th[`p:num`;`p:num`])
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`p:num`;`p:num`;`p:num`]
THEN MRESAL_TAC MOD_MULT[`p:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESAL1_TAC stable_sy_lemma`s:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`0`)
THEN REWRITE_TAC[VECTOR_ARITH`A-A= vec 0`;NORM_0;REAL_ARITH`&0<= &0`]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`0`;`0`])
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[th]);
MP_TAC(ARITH_RULE`j<p /\ 2<p ==> j=p-1 \/ j<p-1 :num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p/\ 2<k ==> ~(0= p-1)/\ 0<=0 /\ 0<= p-1/\ 1<p /\ 0<=k-1/\ p-1<=p-1/\ 1<k`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(0= p-1)==> ~({0,0}= {0,p-1})`)
THEN RESA_TAC
THEN MRESAL1_TAC stable_sy_lemma`s1:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`p-1:num`;`p:num`] )
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`p:num`;`p:num`;`p:num`]
THEN MRESAL_TAC MOD_MULT[`p:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`p-1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`p-1`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL1_TAC stable_sy_lemma`s:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`p-1:num`;`k:num`] )
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`k:num`;`k:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN REMOVE_THEN"THYGIANG2"MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC;
REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
FIND_ASSUM MP_TAC`norm (v - w:real^3) <= min (b_sy s1 (p - 1,0)) (b_sy s2 (p - 1,0))`
THEN REAL_ARITH_TAC;
MP_TAC(ARITH_RULE`2<p/\ 2<k /\ j< p-1 /\ 1<= k-p==> ~(0= p-1)/\ 0<=0 /\ 0<= p-1/\ 1<p /\ 0<=k-1 /\ ~(j=p-1) /\ 0<=j /\ j<=p-1 /\ j<=k /\ 1<k`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(0= p-1)/\ ~(j=p-1)==> ~({0,j}= {0,p-1})`)
THEN RESA_TAC
THEN REMOVE_THEN"THYGIANG"(fun th-> MRESAL_TAC th[`0:num`;`j:num`][IN_NUMSEG])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL1_TAC stable_sy_lemma`s1:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`j:num`;`p:num`] THEN MRESA_TAC th[`p:num`;`p:num`])
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`p:num`;`p:num`;`p:num`]
THEN MRESAL_TAC MOD_MULT[`p:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN RESA_TAC
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`j:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`j:num`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL1_TAC stable_sy_lemma`s:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`j:num`;`k:num`] )
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`k:num`;`k:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN RESA_TAC
THEN RESA_TAC;
MP_TAC(ARITH_RULE`i<p /\ 2<p ==> i=p-1 \/ i<p-1 :num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`j<=p ==> j=p \/ j<p :num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p/\ 2<k ==> ~(0= p-1)/\ 0<=0 /\ 0<= p-1/\ 1<p /\ 0<=k-1/\ p-1<=p-1/\ 1<k`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(0= p-1)==> ~({0,0}= {0,p-1})`)
THEN RESA_TAC
THEN MRESAL1_TAC stable_sy_lemma`s1:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`p-1:num`;`p:num`] )
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`p:num`;`p:num`;`p:num`]
THEN MRESAL_TAC MOD_MULT[`p:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`p-1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN ONCE_REWRITE_TAC[NORM_SUB]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`p-1`])
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL1_TAC stable_sy_lemma`s:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`p-1:num`;`k:num`] )
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`k:num`;`k:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN REMOVE_THEN"THYGIANG2"MP_TAC
THEN ASM_REWRITE_TAC[]
THEN FIND_ASSUM MP_TAC`norm (v - w:real^3) <= min (b_sy s1 (p - 1,0)) (b_sy s2 (p - 1,0))`
THEN REAL_ARITH_TAC;
MP_TAC(ARITH_RULE`j<p /\ 2< p ==> j=p-1 \/ j<p-1 :num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE` 2< p ==> p-1<=p-1 `)
THEN RESA_TAC
THEN MRESAL1_TAC stable_sy_lemma`s1:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`p-1` )
THEN REWRITE_TAC[VECTOR_ARITH`A-A:real^3= vec 0`;NORM_0]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`p-1`;`p-1`])
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[th])
THEN REAL_ARITH_TAC;
MP_TAC(ARITH_RULE`2<p/\ 2<k /\ 1<=j /\ j< p-1 /\ 1<= k-p==> ~(0= p-1)/\ 0<=0 /\ 0<= p-1/\ 1<p /\ 0<=k-1 /\ ~(j=0) /\ 0<=j /\ j<=p-1 /\ j<=k /\ 1<k/\ p-1<=p-1`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(0= p-1)/\ ~(j=0)==> ~({p-1,j}= {0,p-1})`)
THEN RESA_TAC
THEN REMOVE_THEN"THYGIANG"(fun th-> MRESAL_TAC th[`p-1:num`;`j:num`][IN_NUMSEG])
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`p-1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`j:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-1:num`;`j:num`]);
MP_TAC(ARITH_RULE`j<=p ==> j=p \/ j<p :num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p/\ 2<k /\ i< p-1 /\ 1<= k-p==> ~(0= p-1)/\ 0<=0 /\ 0<= p-1/\ 1<p /\ 0<=k-1 /\ ~(i=p-1) /\ 0<=i /\ i<=p-1 /\ i<=k /\ 1<k`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(0= p-1)/\ ~(i=p-1)==> ~({0,i}= {0,p-1})`)
THEN RESA_TAC
THEN REMOVE_THEN"THYGIANG"(fun th-> MRESAL_TAC th[`0:num`;`i:num`][IN_NUMSEG])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL1_TAC stable_sy_lemma`s1:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`i:num`;`p:num`] THEN MRESA_TAC th[`p:num`;`p:num`])
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`p:num`;`p:num`;`p:num`]
THEN MRESAL_TAC MOD_MULT[`p:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN RESA_TAC
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY_P)[`p:num`;`k:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`i:num`;`k:num`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL1_TAC stable_sy_lemma`s:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`i:num`;`k:num`] )
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`k:num`;`k:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN RESA_TAC
THEN RESA_TAC;
MP_TAC(ARITH_RULE`j<p /\ 2< p ==> j=p-1 \/ j<p-1 :num`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`2<p/\ 2<k /\ 1<=i /\ i< p-1 /\ 1<= k-p==> ~(0= p-1)/\ 0<=0 /\ 0<= p-1/\ 1<p /\ 0<=k-1 /\ ~(i=0) /\ 0<=i /\ i<=p-1 /\ i<=k /\ 1<k/\ p-1<=p-1`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(0= p-1)/\ ~(i=0)==> ~({p-1,i}= {0,p-1})`)
THEN RESA_TAC
THEN REMOVE_THEN"THYGIANG"(fun th-> MRESAL_TAC th[`p-1:num`;`i:num`][IN_NUMSEG])
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`p-1:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`i:num`;`p-1:num`])
THEN MRESAL1_TAC stable_sy_lemma`s1:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[th])
THEN ASM_REWRITE_TAC[]
THEN MRESAL1_TAC stable_sy_lemma`s:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[th])
THEN ASM_REWRITE_TAC[];
MP_TAC(ARITH_RULE`2<p/\ 2<k /\ 1<=i /\ i< p-1 /\ 1<=j /\ j< p-1/\ 1<= k-p==> ~(0= p-1)/\ 0<=0 /\ 0<= p-1/\ 1<p /\ 0<=k-1 /\ ~(i=p-1)  /\ ~(j=p-1)/\ 0<=i /\ i<=p-1 /\ i<=k /\ 1<k/\ p-1<=p-1/\0<=j /\ j<=p-1 /\ j<=k`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`~(i= p-1)/\ ~(j=p-1)==> ~({i,j}= {0,p-1})`)
THEN RESA_TAC
THEN REMOVE_THEN"THYGIANG"(fun th-> MRESAL_TAC th[`i:num`;`j:num`][IN_NUMSEG])
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT1_EQ_SY)[`k:num`;`p:num`;`j:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`i:num`;`j:num`])]);;



















let IN_B_SY2_COVER_concl=`!(s:stable_sy) (s1:stable_sy) (s2:stable_sy) (s3:stable_sy) (l:real^(M,3)finite_product)
(l1:real^(P,3)finite_product) (l2:real^(N,3)finite_product).
dimindex(:M) =k /\ dimindex(:N) =k-p+2 /\ dimindex(:P) =p 
/\ 1<= k-p
/\ I_SY s= 0..k-1 /\ f_sy s=(\i. ((1+i):num MOD k)) 
/\ I_SY s1= 0..p-1 /\ f_sy s1=(\i. ((1+i):num MOD p))  
/\ I_SY s3= 0..(k-p+1) /\ f_sy s3=(\i. ((1+i):num MOD (k-p+2))) 
/\ (ear_sy s2 <=> ear_sy s3)
/\ DIA_SY 0 (p-1) s
/\ SCHANGE (\x. if x=0 then 0 else (p-2+x) MOD k) s2 s3
/\ COVER_SY 0 (p-1) s s1 s2 
/\ (l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))
/\ (!i j. {i MOD k ,j MOD k} IN J_SY s UNION {{0,p-1}}
==> norm (row i (vecmats l) -row j (vecmats l))<= cstab)
/\ matvec(pmat1(vecmats l))= l1
/\ matvec(pmat2(vecmats l))= l2
/\ row k (vecmats l)= v
/\ row (p-1) (vecmats l)= w
/\ (!u u1.
               u IN {v, w} /\ u1 IN V_SY(vecmats l) /\ ~(u = u1)
               ==> ~collinear {vec 0, u, u1})
/\ norm(v-w)<= min (b_sy s1 (p-1,0)) (b_sy s2 (p-1,0))
==> l2 IN (B_SY1 (a_sy s3) (b_sy s3))`;;


let IN_B_SY2_COVER_concl2 = mk_imp (EJRCFJD_concl, IN_B_SY2_COVER_concl);;







let IN_B_SY2_COVER=prove_by_refinement(IN_B_SY2_COVER_concl2,
[DISCH_THEN(LABEL_TAC"THY")
THEN REPEAT STRIP_TAC
THEN MRESA1_TAC DIMINDEX_GE_1`(:M)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:P)`
THEN MRESA1_TAC DIMINDEX_GE_1`(:N)`
THEN MP_TAC(ARITH_RULE`1<= k-p /\ 1<=p /\ 1<=k ==> 2< k-p+2 /\ 1<= k-p+2 /\ (k - p + 2) - 1=k - p + 1 /\ ~(k=0)/\ p-1< k/\ p< k /\ ~(p=0)
/\ ~(k-p+2=0)/\ 1< k-p+2/\ p<= k /\ p<=p /\ k-p+1<k-p+2/\ SUC(k-p+1)=k-p+2`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL IK_SY)[`s:stable_sy`;`k:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s1:stable_sy`;`p:num`]
THEN MRESA_TAC (GEN_ALL IK_SY)[`s3:stable_sy`;`k-p+2:num`]
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s:stable_sy`
THEN MRESA1_TAC (GEN_ALL K_SY_LE2)`s1:stable_sy`
THEN REWRITE_TAC[tau_star]
THEN MRESA1_TAC stable_sy_lemma`s:stable_sy`
THEN MP_TAC(ARITH_RULE` 2<p /\ 1<=k-p 
==> p - 2 + k - p + 2= k/\ p-2+1=p-1 /\ 1<=p-1 /\ p-1<=k /\ p-1<k /\ ~(k=p-1)/\ k<=k`)
THEN RESA_TAC
THEN SUBGOAL_THEN`{v, w} SUBSET V_SY (vecmats (l:real^(M,3)finite_product))`ASSUME_TAC;
MATCH_MP_TAC(SET_RULE`A IN C /\ B IN C==> {A,B}SUBSET C`)
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[IN_ELIM_THM;V_SY;rows];
EXISTS_TAC`k:num`
THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC;
EXISTS_TAC`p-1`
THEN ASM_REWRITE_TAC[];
SUBGOAL_THEN`norm (v - w:real^3) <= cstab`ASSUME_TAC;
MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN FIND_ASSUM (fun th-> MP_TAC(ISPECL [`k:num`;`p-1`]th))`!i j.
          {i MOD k, j MOD k} IN J_SY s UNION {{0, p - 1}}
          ==> norm (row i (vecmats l) - row j (vecmats (l:real^(M,3)finite_product))) <= cstab`
THEN POP_ASSUM(fun th-> REWRITE_TAC[th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[th; SET_RULE`{0,p-1} IN J_SY s UNION {{0,p-1}}`])
THEN ASM_REWRITE_TAC[];
SUBGOAL_THEN`~({v, w} IN E_SY (vecmats (l:real^(M,3)finite_product)))`ASSUME_TAC;
ASM_REWRITE_TAC[E_SY;IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;convex_local_fan;local_fan]
THEN LET_TAC
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN MRESA_TAC (GEN_ALL EDGE_IN_E_SY)[`i:num`;`row (i) (vecmats(l:real^(M,3)finite_product))`;`
      (row (SUC (i MOD dimindex (:M))) (vecmats (l:real^(M,3)finite_product)))`;`(l:real^(M,3)finite_product)`]
THEN MRESA_TAC remark1_fan[`vec 0:real^3`;`V_SY (vecmats(l:real^(M,3)finite_product))`;`E_SY (vecmats (l:real^(M,3)finite_product))`;`
      (row (SUC (i MOD dimindex (:M))) (vecmats (l:real^(M,3)finite_product)))`;`row i (vecmats(l:real^(M,3)finite_product))`]
THEN DISJ_CASES_TAC(SET_RULE`row k (vecmats l)= row (p - 1) (vecmats l)
\/ ~(row k (vecmats l)= row (p - 1) (vecmats (l:real^(M,3)finite_product)))`);
MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`p-1`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`1 <= SUC (i' MOD dimindex (:M)) /\
      SUC (i' MOD dimindex (:M)) <= dimindex (:M)` ASSUME_TAC;
ASM_REWRITE_TAC[ARITH_RULE`1<= SUC a`]
THEN MRESAL_TAC DIVISION[`i':num`;`dimindex (:M):num`][DIMINDEX_NONZERO]
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MP_TAC(SET_RULE`{row k (vecmats l), row (p - 1) (vecmats l)} =
   {row i (vecmats l), row (SUC (i MOD k)) (vecmats l)}
/\ ~(row k (vecmats l) = row (p - 1) (vecmats l))
/\ ~(row i (vecmats l) = row (SUC (i MOD k)) (vecmats l))
==> (row i (vecmats l) = row (k) (vecmats l)
/\ row (p-1) (vecmats l) = row (SUC (i MOD k)) (vecmats l))
\/ (row i (vecmats l) = row (p-1) (vecmats l)
/\ row (k) (vecmats l) = row (SUC (i MOD k)) (vecmats (l:real^(M,3)finite_product)))`)
THEN RESA_TAC;
MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`i:num`;`k:num`] THEN  MRESA_TAC th[`p-1:num`;`SUC(i MOD k):num`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESAL_TAC MOD_MULT[`k:num`;`1:num`][ARITH_RULE`p*1=p/\ SUC 0=1`]
THEN MP_TAC(ARITH_RULE`2<p==> ~(p-1=1)/\ 1<=1`)
THEN RESA_TAC;
MRESAL_TAC(GEN_ALL INJ_ROW_B_SY)[`k:num`;`s:stable_sy`;`l:real^(M,3)finite_product`][IN_NUMSEG;SET_RULE`(1 <= i /\ i <= k) /\
          (1 <= j /\ j <= k) /\
          row i (vecmats l) = row j (vecmats l) 
<=> 1 <= i /\ i <= k /\ 1 <= j /\ j <= k /\
          row i (vecmats l) = row j (vecmats l)`]
THEN POP_ASSUM(fun th-> MRESA_TAC th[`i:num`;`p-1:num`] THEN  MRESA_TAC th[`k:num`;`SUC(i MOD k):num`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`2<p /\ 1<= k-p ==> ~(k= SUC (p-1))/\ 1<= SUC(p-1)/\ SUC(p-1)<=k`)
THEN RESA_TAC;
MRESA_TAC (GEN_ALL PROPERTIES_OF_FAN_IN_B_SY)[`d_sy (s:stable_sy)`;`J_SY (s:stable_sy)`;`k_sy (s:stable_sy)`;`k:num`;`p-1:num`;`a_sy (s:stable_sy)`;`b_sy (s:stable_sy)`;`row k (vecmats (l:real^(M,3)finite_product))`;`row (p-1) (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`] 
THEN MRESAL_TAC(GEN_ALL TECOXBM1)[`d_sy (s:stable_sy)`;`J_SY(s:stable_sy)`;`k:num`;`a_sy(s:stable_sy)`;`b_sy(s:stable_sy)`;`row k (vecmats (l:real^(M,3)finite_product))`;`row (p-1) (vecmats (l:real^(M,3)finite_product))`;`l:real^(M,3)finite_product`][SET_RULE`{A} UNION {B,C}={A,B,C}`]
THEN MP_TAC(SET_RULE`{v, w} SUBSET V_SY (vecmats l)==> v IN V_SY (vecmats l)
/\ w IN V_SY (vecmats (l:real^(M,3)finite_product))`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL SLICEV_EQ_V_SY)[`k:num`;`p:num`;`v:real^3`;`w:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEE_EQ_E_SY)[`k:num`;`p:num`;`v:real^3`;`w:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEF_EQ_F_SY)[`k:num`;`p:num`;`v:real^3`;`w:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEF_EQ_F2_SY)[`k:num`;`p:num`;`w:real^3`;`v:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEE_EQ_E2_SY)[`k:num`;`p:num`;`w:real^3`;`v:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL SLICEV_EQ_V2_SY)[`k:num`;`p:num`;`w:real^3`;`v:real^3`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION2_SY;]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_THEN"THY"(fun th-> MRESA_TAC th[`V_SY(vecmats (l:real^(M,3)finite_product))`;`E_SY(vecmats (l:real^(M,3)finite_product))`;`F_SY(vecmats (l:real^(M,3)finite_product))`;
`v:real^3`;`w:real^3`])
THEN EXISTS_TAC`vecmats(l2:real^(N,3)finite_product)`
THEN ASM_REWRITE_TAC[MATVEC_VECMATS_ID]
THEN STRIP_TAC;
REPEAT STRIP_TAC
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN FIND_ASSUM (fun th-> MATCH_MP_TAC(ISPEC `p-2+i`th))`!i. 1 <= i /\ i <= dimindex (:M) ==> row i (vecmats (l:real^(M,3)finite_product)) IN ball_annulus`
THEN ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`1<=i /\ i<= k-p+2 /\ 2<p /\ 1<=k-p ==> 1<= p-2+i /\ p-2+i<=k`)
THEN RESA_TAC;
ASM_REWRITE_TAC[CONDITION1_SY]
THEN FIND_ASSUM MP_TAC`COVER_SY 0 (p - 1) s s1 (s2:stable_sy)`
THEN REWRITE_TAC[COVER_SY;COVER5_SY;COVER4_SY;]
THEN RESA_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN DISCH_THEN(LABEL_TAC"THYGIANG")
THEN DISCH_THEN(LABEL_TAC"THYGIANG1")
THEN DISCH_THEN(LABEL_TAC"THYGIANG2")
THEN DISCH_THEN(LABEL_TAC"THYGIANG3")
THEN DISCH_THEN(LABEL_TAC"THYGIANG4")
THEN DISCH_THEN(LABEL_TAC"THYGIANG5")
THEN REPEAT GEN_TAC
THEN STRIP_TAC;
MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`i:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN MRESA_TAC (GEN_ALL PMAT2_EQ_SY)[`k:num`;`p:num`;`j:num`;`(s:stable_sy)`;` (s1:stable_sy)`;` (s2:stable_sy)`;` (s3:stable_sy)`;` (l:real^(M,3)finite_product)`;`
(l1:real^(P,3)finite_product) `;`(l2:real^(N,3)finite_product)`]
THEN REMOVE_THEN "THYGIANG1"MP_TAC
THEN REMOVE_THEN "THYGIANG3"MP_TAC
THEN FIND_ASSUM MP_TAC`SCHANGE (\x. if x = 0 then 0 else (p - 2 + x) MOD k) s2 (s3:stable_sy)`
THEN REWRITE_TAC[SCHANGE]
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`i:num`;`j:num`])
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`i:num`;`j:num`])
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN MP_TAC(ARITH_RULE`i<=k-p+2 ==> i=k-p+2 \/ i< k-p+2`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`j<=k-p+2 ==> j=k-p+2 \/ j< k-p+2`)
THEN RESA_TAC;
STRIP_TAC
THEN STRIP_TAC
THEN MRESAL_TAC MOD_MULT[`k:num`;`1`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN SUBGOAL_THEN`0 IN IMAGE (\x. if x = 0 then 0 else (p - 2 + x) MOD k) (0..k - p + 1)` ASSUME_TAC;
REWRITE_TAC[IMAGE;IN_ELIM_THM;IN_NUMSEG]
THEN EXISTS_TAC`0`
THEN ASM_REWRITE_TAC[ARITH_RULE`0=0 /\ 0<= 0/\ 0<= k-p+1`];
MRESAL1_TAC stable_sy_lemma`s2:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`0`)
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`0:num`;`0:num`] )
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[th;VECTOR_ARITH`A-A= vec 0 `;NORM_0;])
THEN REAL_ARITH_TAC;
MRESAL_TAC MOD_MULT[`k:num`;`1`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN MP_TAC(ARITH_RULE`1<=j==> j=1 \/ 1<j`)
THEN RESA_TAC;
REWRITE_TAC[ARITH_RULE`~(1=0)`]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`2<k==> 1<k`)
THEN RESA_TAC
THEN STRIP_TAC
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`p-1:num`])
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL1_TAC stable_sy_lemma`s:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`p-1:num`;`k:num`] THEN MRESA_TAC th[`p:num`;`p:num`])
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`k:num`;`k:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN MRESAL1_TAC stable_sy_lemma`s2:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[th])
THEN FIND_ASSUM MP_TAC`norm (v - w:real^3) <= min (b_sy s1 (p - 1,0)) (b_sy (s2:stable_sy) (p - 1,0))`
THEN REAL_ARITH_TAC;
MP_TAC(ARITH_RULE`1<j /\ j<k-p+2 /\ 2<p /\ 1<=k-p ==> ~(j=0) /\ p-2+j<k /\ ~(p-2+j=p-1) /\ ~(0=p-1)/\ 0<=j /\ j<= k-p+1/\ 1<=p-2+j/\ p-2+j<=k /\1<k`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+j:num`;`k:num`]
THEN MP_TAC(SET_RULE`~(p-2+j=p-1) /\ ~(0=p-1)==> ~({0,p-2+j}={0,p-1})`)
THEN RESA_TAC
THEN SUBGOAL_THEN`0 IN IMAGE (\x. if x = 0 then 0 else (p - 2 + x) MOD k) (0..k - p + 1)` ASSUME_TAC;
REWRITE_TAC[IMAGE;IN_ELIM_THM;IN_NUMSEG]
THEN EXISTS_TAC`0`
THEN ASM_REWRITE_TAC[ARITH_RULE`0=0 /\ 0<= 0/\ 0<= k-p+1`];
SUBGOAL_THEN`p-2+j IN IMAGE (\x. if x = 0 then 0 else (p - 2 + x) MOD k) (0..k - p + 1)` ASSUME_TAC;
REWRITE_TAC[IMAGE;IN_ELIM_THM;IN_NUMSEG]
THEN EXISTS_TAC`j:num`
THEN ASM_REWRITE_TAC[ARITH_RULE`0=0 /\ 0<= 0/\ 0<= k-p+1`];
STRIP_TAC
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`0`;`p-2+j`])
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`k:num`;`p-2+j:num`])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL1_TAC stable_sy_lemma`s:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`p-2+j:num`;`k:num`] THEN MRESA_TAC th[`p:num`;`p:num`])
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`k:num`;`k:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN RESA_TAC
THEN RESA_TAC;
MRESAL_TAC MOD_MULT[`k:num`;`1`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN MP_TAC(ARITH_RULE`1<=i==> i=1 \/ 1<i`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`1<=j==> j=1 \/ 1<j`)
THEN RESA_TAC;
STRIP_TAC
THEN STRIP_TAC
THEN REWRITE_TAC[ARITH_RULE`~(1=0)`]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN SUBGOAL_THEN`p-1 IN IMAGE (\x. if x = 0 then 0 else (p - 2 + x) MOD k) (0..k - p + 1)` ASSUME_TAC;
REWRITE_TAC[IMAGE;IN_ELIM_THM;IN_NUMSEG]
THEN EXISTS_TAC`1:num`
THEN ASM_REWRITE_TAC[ARITH_RULE`~(1=0) /\ 0<= 1/\ 1<= k-p+1`];
MRESAL1_TAC stable_sy_lemma`s2:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC th`p-1`)
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`p-1:num`;`p-1:num`] )
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[th;VECTOR_ARITH`A-A= vec 0 `;NORM_0;])
THEN REAL_ARITH_TAC;
MP_TAC(ARITH_RULE`j<=k-p+2 ==> j=k-p+2 \/ j< k-p+2`)
THEN RESA_TAC;
REWRITE_TAC[ARITH_RULE`~(1=0)`]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`2<k==> 1<k`)
THEN RESA_TAC
THEN STRIP_TAC
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-1:num`;`k:num`])
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL1_TAC stable_sy_lemma`s:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`p-1:num`;`k:num`] THEN MRESA_TAC th[`p:num`;`p:num`])
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`k:num`;`k:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN REWRITE_TAC[NORM_SUB]
THEN DISCH_THEN(LABEL_TAC"THY21")
THEN DISCH_THEN(LABEL_TAC"THY22")
THEN MRESAL1_TAC stable_sy_lemma`s2:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN FIND_ASSUM MP_TAC`norm (v - w:real^3) <= min (b_sy s1 (p - 1,0)) (b_sy (s2:stable_sy) (p - 1,0))`
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[th])
THEN REMOVE_THEN"THY21"MP_TAC
THEN REMOVE_THEN"THY22"MP_TAC
THEN REAL_ARITH_TAC;
MP_TAC(ARITH_RULE`1<j /\ j<k-p+2 /\ 2<p /\ 1<=k-p ==> ~(j=0) /\ p-2+j<k /\ ~(p-2+j=0) /\ ~(0=p-1)/\ 0<=j /\ j<= k-p+1/\ 1<=p-2+j/\ p-2+j<=k /\1<k/\ ~(1=0)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+j:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN MP_TAC(SET_RULE`~(p-2+j=0) /\ ~(0=p-1)==> ~({p-1,p-2+j}={0,p-1})`)
THEN RESA_TAC
THEN SUBGOAL_THEN`p-1 IN IMAGE (\x. if x = 0 then 0 else (p - 2 + x) MOD k) (0..k - p + 1)` ASSUME_TAC;
REWRITE_TAC[IMAGE;IN_ELIM_THM;IN_NUMSEG]
THEN EXISTS_TAC`1`
THEN ASM_REWRITE_TAC[ARITH_RULE`0=0 /\ 0<= 1/\ 1<= k-p+1`];
SUBGOAL_THEN`p-2+j IN IMAGE (\x. if x = 0 then 0 else (p - 2 + x) MOD k) (0..k - p + 1)` ASSUME_TAC;
REWRITE_TAC[IMAGE;IN_ELIM_THM;IN_NUMSEG]
THEN EXISTS_TAC`j:num`
THEN ASM_REWRITE_TAC[ARITH_RULE`0=0 /\ 0<= 0/\ 0<= k-p+1`];
STRIP_TAC
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-1`;`p-2+j`])
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-1:num`;`p-2+j:num`]);
MP_TAC(ARITH_RULE`1<i /\ i<k-p+2 /\ 2<p /\ 1<=k-p/\ 1<=j ==> ~(i=0) /\ p-2+i<k /\ ~(p-2+i=0) /\ ~(0=p-1)/\ 0<=i /\ i<= k-p+1/\ 1<=p-2+i/\ p-2+i<=k /\1<k/\ ~(p-2+i=p-1)/\ ~(j=0)`)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+i:num`;`k:num`]
THEN MRESA_TAC MOD_LT[`p-1:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`j<=k-p+2 ==> j=k-p+2 \/ j< k-p+2`)
THEN RESA_TAC;
MP_TAC(SET_RULE`~(p-2+i=p-1) /\ ~(0=p-1)==> ~({p-2+i,0}={0,p-1})`)
THEN RESA_TAC
THEN SUBGOAL_THEN`0 IN IMAGE (\x. if x = 0 then 0 else (p - 2 + x) MOD k) (0..k - p + 1)` ASSUME_TAC;
REWRITE_TAC[IMAGE;IN_ELIM_THM;IN_NUMSEG]
THEN EXISTS_TAC`0`
THEN ASM_REWRITE_TAC[ARITH_RULE`0=0 /\ 0<= 0/\ 0<= k-p+1`];
SUBGOAL_THEN`p-2+i IN IMAGE (\x. if x = 0 then 0 else (p - 2 + x) MOD k) (0..k - p + 1)` ASSUME_TAC;
REWRITE_TAC[IMAGE;IN_ELIM_THM;IN_NUMSEG]
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[ARITH_RULE`0=0 /\ 0<= 0/\ 0<= k-p+1`];
STRIP_TAC
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-2+i`;`0`])
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-2+i:num`;`k:num`;])
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL1_TAC stable_sy_lemma`s:stable_sy`[stable_system;IN_NUMSEG;ARITH_RULE`0<= i:num`;]
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[constraint_system]
THEN STRIP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th-> MRESA_TAC th[`p-2+i:num`;`k:num`] THEN MRESA_TAC th[`p:num`;`p:num`])
THEN MRESA_TAC (GEN_ALL POWER_MOD_FUN)[`k:num`;`k:num`;`k:num`]
THEN MRESAL_TAC MOD_MULT[`k:num`;`2`][DIMINDEX_NONZERO;ARITH_RULE`A*1=A/\ p+p= p* 2`]
THEN RESA_TAC
THEN RESA_TAC;
MP_TAC(ARITH_RULE`j<k-p+2 /\ 2<p /\ 1<=k-p/\ 1<=j ==> ~(j=0) /\ p-2+j<k/\ 0<=j /\ j<=k-p+1/\ 1<= p-2+j /\ p-2+j<=k `)
THEN RESA_TAC
THEN MRESA_TAC MOD_LT[`p-2+j:num`;`k:num`]
THEN MP_TAC(SET_RULE`~(p-2+i=p-1) /\ ~(p-2+i=0)==> ~({p-2+i,p-2+j}={0,p-1})`)
THEN RESA_TAC
THEN SUBGOAL_THEN`p-2+j IN IMAGE (\x. if x = 0 then 0 else (p - 2 + x) MOD k) (0..k - p + 1)` ASSUME_TAC;
REWRITE_TAC[IMAGE;IN_ELIM_THM;IN_NUMSEG]
THEN EXISTS_TAC`j:num`
THEN ASM_REWRITE_TAC[ARITH_RULE`0=0 /\ 0<= 0/\ 0<= k-p+1`];
SUBGOAL_THEN`p-2+i IN IMAGE (\x. if x = 0 then 0 else (p - 2 + x) MOD k) (0..k - p + 1)` ASSUME_TAC;
REWRITE_TAC[IMAGE;IN_ELIM_THM;IN_NUMSEG]
THEN EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[ARITH_RULE`0=0 /\ 0<= 0/\ 0<= k-p+1`];
STRIP_TAC
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-2+i`;`p-2+j`])
THEN FIND_ASSUM MP_TAC`(l:real^(M,3)finite_product) IN (B_SY1 (a_sy s) (b_sy s))`
THEN REWRITE_TAC[B_SY1;IN_ELIM_THM;CONDITION1_SY]
THEN STRIP_TAC
THEN MRESA1_TAC (GEN_ALL VECMATS_MATVEC_ID)`v'':real^3^M`
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM th])
THEN STRIP_TAC
THEN POP_ASSUM (fun th-> SUBST_ALL_TAC(SYM th))
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC th[`p-2+i:num`;`p-2+j:num`;])]);;




let MTUWLUN1 = D_FUN_COVER_SY;;

let MTUWLUN2 = TAU_STAR_COVER;;

let MTUWLUN3 = IN_B_SY_COVER;;

let MTUWLUN4 = IN_B_SY2_COVER;;







end;;