(* ========================================================================== *)
(* FLYSPECK - BOOK FORMALIZATION *)
(* *)
(* Chapter: Local Fan *)
(* Author: Hoang Le Truong *)
(* Date: 2012-04-01 *)
(* ========================================================================= *)
(*
remaining conclusions from appendix to Local Fan chapter
*)
module Qknvmlb = struct
open Polyhedron;;
open Sphere;;
open Topology;;
open Fan_misc;;
open Planarity;;
open Conforming;;
open Hypermap;;
open Fan;;
open Topology;;
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;;
open Pcrttid;;
open Local_lemmas;;
open Pack_defs;;
open Hales_tactic;;
open Appendix;;
open Hypermap;;
open Fan;;
open Wrgcvdr_cizmrrh;;
open Local_lemmas;;
open Flyspeck_constants;;
open Pack_defs;;
open Hales_tactic;;
open Appendix;;
open Zithlqn;;
open Xwitccn;;
open Ayqjtmd;;
open Jkqewgv;;
let MOD_EQ_IMP_MOD_EQ_0=prove(
`~(k=0)/\ p MOD k = q MOD k/\ q<= p
==> (p-q) MOD k =0`,
REPEAT STRIP_TAC
THEN MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN MRESA_TAC
DIVISION[`q:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`p = p DIV k * k + q MOD k
/\ q = q DIV k * k + q MOD k
/\ q<=p
==> p-q = (p DIV k *k - q DIV k *k)`)
THEN ASM_REWRITE_TAC[]
THEN POP_ASSUM(fun th->
REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th1->
GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th1])
THEN REWRITE_TAC[]
THEN GEN_REWRITE_TAC( LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th])
THEN ASM_REWRITE_TAC[]
THEN MRESA_TAC
DIV_MONO[`q:num`;`p:num`;`k:num`]
THEN REWRITE_TAC[GSYM
RIGHT_SUB_DISTRIB]
THEN RESA_TAC
THEN MRESA_TAC MOD_MULT[`k:num`;`(p DIV k - q DIV k)`]
THEN ONCE_REWRITE_TAC[ARITH_RULE`(p DIV k - q DIV k) * k=k* (p DIV k - q DIV k) `]
THEN ASM_REWRITE_TAC[]);;
let SCS_K_PRIME_CASE_4=prove(`s' =
scs_half_slice_v39 s p q d' mkj
/\
is_scs_v39 s
/\
scs_k_v39 s=4
/\
scs_diag (
scs_k_v39 s) p q
==>
scs_k_v39 s' =3`,
let CASE_DIAGONAL_MOD=
fun (so:term) (so1:term) (so2:term)->
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`p:num`;so;`4`][ARITH_RULE`~(4=0)/\ 4 MOD 4=0/\ SUC 0 MOD 4=1/\ 1 MOD 4=1 /\ 2 MOD 4=2 /\ 3 MOD 4=3/\ 0 MOD 4=0 /\ SUC 1=2 /\ SUC 2=3/\ SUC 3=4`]
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN MRESAL_TAC
MOD_ADD_MOD[`q:num`;so1;`4`][ARITH_RULE`~(4=0) /\ 1 MOD 4=1/\ 1+4-0=5/\ 1+4-1=4 /\ 5 MOD 4=1/\ 4 MOD 4=0/\ 1+4-2=3 /\ 3 MOD 4=3/\ 2 MOD 4=2 /\ 1+4-3=2`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
DIVISION[`q:num`;`4:num`][ARITH_RULE`~(4=0)`]
THEN MP_TAC(ARITH_RULE`q MOD 4 < 4 ==> q MOD 4 =0 \/ q MOD 4 =1 \/ q MOD 4 =2 \/ q MOD 4 =3`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`(3) MOD 4=3 /\ 2 +1=3 /\ 3+0=3/\ 0+3=3 /\ 1+2=3`]
THEN MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`q:num`;so2;`4`][ARITH_RULE`~(4=0)/\ 4 MOD 4=0/\ SUC 0 MOD 4=1/\ 1 MOD 4=1 /\ 2 MOD 4=2 /\ 3 MOD 4=3/\ 0 MOD 4=0/\ SUC 3=4/\ SUC 1=2/\ SUC 2=3`]
in
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
scs_half_slice_v39;
LET_DEF;
LET_END_DEF;
scs_k_v39_explicit;
scs_diag]
THEN STRIP_TAC
THEN MRESAL_TAC
DIVISION[`p:num`;`4:num`][ARITH_RULE`~(4=0)`]
THEN MP_TAC(ARITH_RULE`p MOD 4 < 4 ==> p MOD 4 =0 \/ p MOD 4 =1 \/ p MOD 4 =2 \/ p MOD 4 =3`)
THEN RESA_TAC
THENL[
CASE_DIAGONAL_MOD `0` `5` `3`;
CASE_DIAGONAL_MOD `1` `4` `0`;
CASE_DIAGONAL_MOD `2` `3` `1`;
CASE_DIAGONAL_MOD `3` `2` `2`]);;
let SCS_K_PRIME_CASE_5=prove(
`s' =
scs_half_slice_v39 s p q d' mkj
/\ s'' =
scs_half_slice_v39 s q p d' mkj
/\
is_scs_v39 s
/\
scs_k_v39 s=5
/\
scs_diag (
scs_k_v39 s) p q
==> (
scs_k_v39 s' =3 /\
scs_k_v39 s'' =4) \/ (
scs_k_v39 s' =4 /\
scs_k_v39 s'' =3)`,
let CASE_DIAGONAL_MOD=
fun (so:term) (so1:term) (so2:term) ->
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`p:num`;so;`5`][ARITH_RULE`~(5=0)/\ 5 MOD 5=0/\ SUC 0 MOD 5=1/\ 1 MOD 5=1 /\ 2 MOD 5=2 /\ 3 MOD 5=3/\ 0 MOD 5=0 /\ SUC 1=2 /\ SUC 2=3/\ SUC 3=4/\ 4 MOD 5=4 /\ SUC 4=5`]
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC
THEN ASSUME_TAC
th)
THEN MRESAL_TAC
MOD_ADD_MOD[`q:num`;so1;`5`][ARITH_RULE`~(5=0) /\ 1 MOD 5=1/\ 1+5-0=6/\ 1+5-1=5 /\ 6 MOD 5=1/\ 5 MOD 5=0/\ 1+5-2=4 /\ 3 MOD 5=3/\ 2 MOD 5=2 /\ 1+5-3=3/\ 1+5-4=2`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
DIVISION[`q:num`;`5:num`][ARITH_RULE`~(5=0)`]THEN MP_TAC(ARITH_RULE`q MOD 5 < 5 ==> q MOD 5 =0 \/ q MOD 5 =1 \/ q MOD 5 =2 \/ q MOD 5 = 3\/ q MOD 5=4`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`(3) MOD 5=3 /\ 2 +1=3 /\ 3+0=3/\ 0+3=3 /\ 1+2=3`]
THEN MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`q:num`;so2;`5`][ARITH_RULE`~(5=0)/\ 5 MOD 5=0/\ SUC 0 MOD 5=1/\ 1 MOD 5=1 /\ 2 MOD 5=2 /\ 3 MOD 5=3/\ 0 MOD 5=0/\ SUC 3=4/\ SUC 1=2/\ SUC 2=3/\ SUC 4=5/\ 4 MOD 5=4`]
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`4`;`5`][ARITH_RULE`~(5=0) /\ 1 MOD 5=1/\ 1+5-0=6/\ 1+5-1=5 /\ 6 MOD 5=1/\ 5 MOD 5=0/\ 1+5-2=4 /\ 3 MOD 5=3/\ 2 MOD 5=2 /\ 1+5-3=3/\ 1+5-4=2/\ 4 MOD 5=4 /\ 0+4=4`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`3`;`5`][ARITH_RULE`~(5=0) /\ 1 MOD 5=1/\ 1+5-0=6/\ 1+5-1=5 /\ 6 MOD 5=1/\ 5 MOD 5=0/\ 1+5-2=4 /\ 3 MOD 5=3/\ 2 MOD 5=2 /\ 1+5-3=3/\ 1+5-4=2/\ 4 MOD 5=4 /\ 0+3=3 /\ 3+1=4/\ 1+3=4/\ 4+0=4`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`2`;`5`][ARITH_RULE`~(5=0) /\ 1 MOD 5=1/\ 1+5-0=6/\ 1+5-1=5 /\ 6 MOD 5=1/\ 5 MOD 5=0/\ 1+5-2=4 /\ 3 MOD 5=3/\ 2 MOD 5=2 /\ 1+5-3=3/\ 1+5-4=2/\ 4 MOD 5=4 /\ 0+3=3 /\ 3+1=4/\ 1+3=4/\ 4+0=4/\ 1+2=3`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`6`;`5`][ARITH_RULE`~(5=0) /\ 1 MOD 5=1/\ 1+5-0=6/\ 1+5-1=5 /\ 6 MOD 5=1/\ 5 MOD 5=0/\ 1+5-2=4 /\ 3 MOD 5=3/\ 2 MOD 5=2 /\ 1+5-3=3/\ 1+5-4=2/\ 4 MOD 5=4 /\ 0+3=3 /\ 3+1=4/\ 1+3=4/\ 4+0=4/\ 1+2=3/\ 2+1=3 /\ 4+4=8 /\ 8 MOD 5=3 /\ 2+2=4`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`5`;`5`][ARITH_RULE`~(5=0) /\ 1 MOD 5=1/\ 1+5-0=6/\ 1+5-1=5 /\ 6 MOD 5=1/\ 5 MOD 5=0/\ 1+5-2=4 /\ 3 MOD 5=3/\ 2 MOD 5=2 /\ 1+5-3=3/\ 1+5-4=2/\ 4 MOD 5=4 /\ 0+3=3 /\ 3+1=4/\ 1+3=4/\ 4+0=4/\ 1+2=3/\ 2+1=3 /\ 4+4=8 /\ 8 MOD 5=3 /\ 2+2=4/\ 3+0=3`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
in
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
scs_half_slice_v39;
LET_DEF;
LET_END_DEF;
scs_k_v39_explicit;
scs_diag]
THEN STRIP_TAC
THEN MRESAL_TAC
DIVISION[`p:num`;`5:num`][ARITH_RULE`~(5=0)`]
THEN MP_TAC(ARITH_RULE`p MOD 5 < 5 ==> p MOD 5 =0 \/ p MOD 5 =1 \/ p MOD 5 =2 \/ p MOD 5 =3\/ p MOD 5=4`)
THEN RESA_TAC
THENL[
CASE_DIAGONAL_MOD `0` `6` `4`;
CASE_DIAGONAL_MOD `1` `5` `0`;
CASE_DIAGONAL_MOD `2` `4` `1`;
CASE_DIAGONAL_MOD `3` `3` `2`;
CASE_DIAGONAL_MOD `4` `2` `3`]);;
let SCS_K_PRIME_CASE_6=prove(
`s' =
scs_half_slice_v39 s p q d' mkj
/\ s'' =
scs_half_slice_v39 s q p d' mkj
/\
is_scs_v39 s
/\
scs_k_v39 s=6
/\
scs_diag (
scs_k_v39 s) p q
==> (
scs_k_v39 s' =3 /\
scs_k_v39 s'' =5) \/ (
scs_k_v39 s' =5 /\
scs_k_v39 s'' =3) \/ (
scs_k_v39 s' =4 /\
scs_k_v39 s'' =4)`,
let CASE_DIAGONAL_MOD=
fun (so:term) (so1:term) (so2:term) ->
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`p:num`;so;`6`][ARITH_RULE`~(6=0)/\ 6 MOD 6=0/\ SUC 0 MOD 6=1/\ 1 MOD 6=1 /\ 2 MOD 6=2 /\ 3 MOD 6=3/\ 0 MOD 6=0 /\ SUC 1=2 /\ SUC 2=3/\ SUC 3=4/\ 4 MOD 6=4 /\ SUC 4=5/\ SUC 5=6 /\ 5 MOD 6=5`]
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC
THEN ASSUME_TAC
th)
THEN MRESAL_TAC
MOD_ADD_MOD[`q:num`;so1;`6`][ARITH_RULE`~(6=0) /\ 1 MOD 6=1/\ 1+6-0=7/\ 1+6-1=6 /\ 7 MOD 6=1/\ 6 MOD 6=0/\ 1+6-2=5 /\ 3 MOD 6=3/\ 2 MOD 6=2 /\ 1+6-3=4/\ 1+6-4=3/\ 1+6-5=2`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
DIVISION[`q:num`;`6:num`][ARITH_RULE`~(6=0)`]THEN MP_TAC(ARITH_RULE`q MOD 6 < 6 ==> q MOD 6 =0 \/ q MOD 6 =1 \/ q MOD 6 =2 \/ q MOD 6 = 3\/ q MOD 6=4\/ q MOD 6=5`)
THEN RESA_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`(3) MOD 6=3 /\ 2 +1=3 /\ 3+0=3/\ 0+3=3 /\ 1+2=3`]
THEN MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`q:num`;so2;`6`][ARITH_RULE`~(6=0)/\ 6 MOD 6=0/\ SUC 0 MOD 6=1/\ 1 MOD 6=1 /\ 2 MOD 6=2 /\ 3 MOD 6=3/\ 0 MOD 6=0/\ SUC 3=4/\ SUC 1=2/\ SUC 2=3/\ SUC 4=5/\ SUC 5=6/\ 4 MOD 6=4/\ 5 MOD 6=5`]
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`4`;`6`][ARITH_RULE`~(6=0) /\ 1 MOD 6=1/\ 1+6-0=7/\ 1+6-1=6 /\ 7 MOD 6=1/\ 6 MOD 6=0/\ 1+6-2=5 /\ 3 MOD 6=3/\ 2 MOD 6=2 /\ 1+6-3=4/\ 1+6-4=3/\ 4 MOD 6=4 /\ 0+3=3 /\ 3+1=4/\ 1+3=4/\ 4+0=4/\ 1+2=3/\ 2+1=3 /\ 4+4=8 /\ 8 MOD 6=2 /\ 2+2=4/\ 3+0=3/\ 1+6-5=2`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`3`;`6`][ARITH_RULE`~(6=0) /\ 1 MOD 6=1/\ 1+6-0=7/\ 1+6-1=6 /\ 7 MOD 6=1/\ 6 MOD 6=0/\ 1+6-2=5 /\ 3 MOD 6=3/\ 2 MOD 6=2 /\ 1+6-3=4/\ 1+6-4=3/\ 4 MOD 6=4 /\ 0+3=3 /\ 3+1=4/\ 1+3=4/\ 4+0=4/\ 1+2=3/\ 2+1=3 /\ 4+4=8 /\ 8 MOD 6=2 /\ 2+2=4/\ 3+0=3`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`2`;`6`][ARITH_RULE`~(6=0) /\ 1 MOD 6=1/\ 1+6-0=7/\ 1+6-1=6 /\ 7 MOD 6=1/\ 6 MOD 6=0/\ 1+6-2=5 /\ 3 MOD 6=3/\ 2 MOD 6=2 /\ 1+6-3=4/\ 1+6-4=3/\ 4 MOD 6=4 /\ 0+3=3 /\ 3+1=4/\ 1+3=4/\ 4+0=4/\ 1+2=3/\ 2+1=3 /\ 4+4=8 /\ 8 MOD 6=2 /\ 2+2=4/\ 3+0=3`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`6`;`6`][ARITH_RULE`~(6=0) /\ 1 MOD 6=1/\ 1+6-0=7/\ 1+6-1=6 /\ 7 MOD 6=1/\ 6 MOD 6=0/\ 1+6-2=5 /\ 3 MOD 6=3/\ 2 MOD 6=2 /\ 1+6-3=4/\ 1+6-4=3/\ 4 MOD 6=4 /\ 0+3=3 /\ 3+1=4/\ 1+3=4/\ 4+0=4/\ 1+2=3/\ 2+1=3 /\ 4+4=8 /\ 8 MOD 6=2 /\ 2+2=4/\ 3+0=3`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`5`;`6`][ARITH_RULE`~(6=0) /\ 1 MOD 6=1/\ 1+6-0=7/\ 1+6-1=6 /\ 7 MOD 6=1/\ 6 MOD 6=0/\ 1+6-2=5 /\ 3 MOD 6=3/\ 2 MOD 6=2 /\ 1+6-3=4/\ 1+6-4=3/\ 4 MOD 6=4 /\ 0+3=3 /\ 3+1=4/\ 1+3=4/\ 4+0=4/\ 1+2=3/\ 2+1=3 /\ 4+4=8 /\ 8 MOD 6=2 /\ 2+2=4/\ 3+0=3/\ 5 MOD 6=5/\ 0+5=5/\0+4= 4/\ 4+1=5/\ 1+4=5 /\ 5+0=5`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`7`;`6`][ARITH_RULE`~(6=0) /\ 1 MOD 6=1/\ 1+6-0=7/\ 1+6-1=6 /\ 7 MOD 6=1/\ 6 MOD 6=0/\ 1+6-2=5 /\ 3 MOD 6=3/\ 2 MOD 6=2 /\ 1+6-3=4/\ 1+6-4=3/\ 4 MOD 6=4 /\ 0+3=3 /\ 3+1=4/\ 1+3=4/\ 4+0=4/\ 1+2=3/\ 2+1=3 /\ 4+4=8 /\ 8 MOD 6=2 /\ 2+2=4/\ 3+0=3/\ 5 MOD 6=5/\ 0+5=5/\0+4= 4/\ 4+1=5/\ 1+4=5 /\ 5+0=5/\ 4+5=9/\ 9 MOD 6=3/\ 2+3=5/\ 5+5=10/\ 10 MOD 6=4/\ 5+4=9/\ 3+2=5`;
ADD1]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
in
REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
scs_half_slice_v39;
LET_DEF;
LET_END_DEF;
scs_k_v39_explicit;
scs_diag]
THEN STRIP_TAC
THEN MRESAL_TAC
DIVISION[`p:num`;`6:num`][ARITH_RULE`~(6=0)`]
THEN MP_TAC(ARITH_RULE`p MOD 6 < 6 ==> p MOD 6 =0 \/ p MOD 6 =1 \/ p MOD 6 =2 \/ p MOD 6 =3\/ p MOD 6=4\/ p MOD 6=5`)
THEN RESA_TAC
THENL[
CASE_DIAGONAL_MOD `0` `7` `5`;
CASE_DIAGONAL_MOD `1` `6` `0`;
CASE_DIAGONAL_MOD `2` `5` `1`;
CASE_DIAGONAL_MOD `3` `4` `2`;
CASE_DIAGONAL_MOD `4` `3` `3`;
CASE_DIAGONAL_MOD `5` `2` `4`]);;
let SCS_K_PRIME_CASE_3=prove(`
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q
==> 3<
scs_k_v39 s`,
let LEMMA_TAC=
fun (so:term) (so1:term)->
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`p:num`;so;`3`][ARITH_RULE`~(3=0)/\ 3 MOD 3=0/\ SUC 0 MOD 3=1/\ 1 MOD 3=1 /\ 2 MOD 3=2 /\ 0 MOD 3=0 /\ SUC 1=2 /\ SUC 2=3/\ SUC 3=4`]
THEN MRESAL_TAC
DIVISION[`q:num`;`3:num`][ARITH_RULE`~(3=0)`]
THEN MP_TAC(ARITH_RULE`q MOD 3< 3 ==> q MOD 3 =0\/ q MOD 3 =1 \/ q MOD 3 =2`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`q:num`;so1;`3`][ARITH_RULE`~(3=0)/\ 3 MOD 3=0/\ SUC 0 MOD 3=1/\ 1 MOD 3=1 /\ 2 MOD 3=2 /\ 0 MOD 3=0 /\ SUC 1=2 /\ SUC 2=3/\ SUC 3=4`]
in
REWRITE_TAC[
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit]
THEN STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==>
scs_k_v39 s =3\/ 3<
scs_k_v39 s`)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN REPLICATE_TAC 3 (POP_ASSUM MP_TAC)
THEN MRESAL_TAC
DIVISION[`p:num`;`3:num`][ARITH_RULE`~(3=0)`]
THEN MP_TAC(ARITH_RULE`p MOD 3< 3 ==> p MOD 3 =0\/ p MOD 3 =1 \/ p MOD 3 =2`)
THEN RESA_TAC
THENL[LEMMA_TAC `0` `2`;
LEMMA_TAC `1` `0`;
LEMMA_TAC `2` `1`]);;
let SCS_HALF_SLICE_IS_SCS=prove_by_refinement(
` s' =
scs_half_slice_v39 s p q d' mkj /\
s'' =
scs_half_slice_v39 s q p d'' mkj /\
(
scs_J_v39 s' 0 (
scs_k_v39 s' -1))= mkj
/\
(~(
scs_J_v39 s' 0 (
scs_k_v39 s' -1)))
/\ vv' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s')+p MOD (
scs_k_v39 s)))
/\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q
/\
d'< #0.9 /\
scs_bm_v39 s p q < &4 /\
(4<
scs_k_v39 s ==>
scs_bm_v39 s p q <= cstab)
==>
is_scs_v39 s'`,
(* {{{ proof *)
[
STRIP_TAC
THEN MRESA_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`p:num`;`q:num`;`d':real`;`
scs_J_v39 s' p q`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;]
THEN REPLICATE_TAC 5 (POP_ASSUM MP_TAC)
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit]
THEN ABBREV_TAC`k'=(q + 1 +
scs_k_v39 s - p MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN REPEAT RESA_TAC;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
REAL_ARITH_TAC;
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (17)(REMOVE_ASSUM_TAC)
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`i MOD k' + p MOD
scs_k_v39 s`;`j MOD k' + p MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 3 (POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC
;
REAL_ARITH_TAC;
MP_TAC(ARITH_RULE`3<=k' ==> ~(0=k'-1)`)
THEN RESA_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', i MOD k'} = {0, k'-1} \/ ~({i MOD k', i MOD k'} = {0, k'-1})`);
MP_TAC(SET_RULE`{i MOD k', i MOD k'} = {0, k'-1}==> i MOD k'= 0 /\ i MOD k'= k'-1`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC;
ASM_REWRITE_TAC[];
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `q:num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s (q MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `p:num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
DIVISION[`p:num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`q:num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN REPLICATE_TAC 23 (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 25(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 4(POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(j MOD k' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (((j MOD k' + p MOD
scs_k_v39 s) MOD
scs_k_v39 s))`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `((i MOD k' + p MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN REPLICATE_TAC (34-15)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN MRESAL_TAC
DIVISION[`(i MOD k' + p MOD
scs_k_v39 s):num`;`
scs_k_v39 s:num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j MOD k' + p MOD
scs_k_v39 s):num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN STRIP_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i MOD k':num`;`j MOD k':num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`j:num`;`k':num`]
THEN MP_TAC(ARITH_RULE`i< k' /\ k'<
scs_k_v39 s /\ j<k' ==>
i<
scs_k_v39 s /\ j<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`j:num`;`
scs_k_v39 s`]
;
DISJ_CASES_TAC(SET_RULE`{i MOD 3, SUC i MOD 3} = {0, 3-1} \/ ~({i MOD 3, SUC i MOD 3} = {0, 3-1})`);
ASM_REWRITE_TAC[]
;
ASM_REWRITE_TAC[]
THEN MRESAL_TAC
DIVISION[`i:num`;`k':num`][ARITH_RULE`~(3=0)`]
THEN MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s`)
THEN RESA_TAC
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`(i ) MOD 3 < 3 ==> (i ) MOD 3 =0 \/ (i ) MOD 3 =1 \/ (i ) MOD 3 =2`)
THEN RESA_TAC
;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`0`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 0+A=A /\ 1+A= SUC A`]
THEN REPLICATE_TAC (37-17)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL1_TAC
th `p MOD
scs_k_v39 s` [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3`;cstab])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`1`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
THEN REPLICATE_TAC (37-17)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL1_TAC
th `1+p MOD
scs_k_v39 s` [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3`;cstab])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`2`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th;ARITH_RULE`3-1=2`]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th;ARITH_RULE`3-1=2`]
THEN REPEAT STRIP_TAC)
THEN REPLICATE_TAC (4) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
DISJ_CASES_TAC(SET_RULE`{i MOD k', SUC i MOD k'} = {0, k'-1} \/ ~({i MOD k', SUC i MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`3<k' /\ k'<
scs_k_v39 s ==> 4<
scs_k_v39 s`)
THEN RESA_TAC
THEN REPLICATE_TAC (6) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN RESA_TAC
;
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'`)
THEN RESA_TAC
THEN MRESAL_TAC
DIVISION[`i:num`;`k':num`][ARITH_RULE`~(3=0)`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_ADD_MOD[`i:num`;`1:num`;`k':num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LT[`1`;`k':num`]
THEN MP_TAC(ARITH_RULE`3<k' /\ i MOD k' < k' ==> i MOD k' =k'-1 \/ i MOD k' +1 < k' `)
THEN RESA_TAC
;
MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESAL_TAC MOD_MULT[`k':num`;`1`][ARITH_RULE`A* 1=A`]
THEN
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`k'-1`;`k':num`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN REPLICATE_TAC (2) REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN REPLICATE_TAC (8) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
MRESAL_TAC
MOD_LT[`i MOD k' +1`;`k':num`][ARITH_RULE`(i MOD k' + 1) + p MOD
scs_k_v39 s= SUC(i MOD k' + p MOD
scs_k_v39 s)`]
THEN REPLICATE_TAC (42-17)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[];
MP_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} ==> (i MOD k'= 0 /\ j MOD k' = k'-1)\/ (i MOD k'= k'-1 /\ j MOD k' = 0)`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESAL_TAC MOD_MULT[`k':num`;`1`][ARITH_RULE`A* 1=A`]
THEN
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`j:num`;`k'-1`;`k':num`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESAL_TAC MOD_MULT[`k':num`;`1`][ARITH_RULE`A* 1=A`]
THEN
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`k'-1`;`k':num`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (31-18)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`][ARITH_RULE`SUC(A+B)= SUC A+B`])
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`j MOD k':num`;`SUC(i MOD k'):num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC
DIVISION[`(i):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MP_TAC(ARITH_RULE`j MOD k' < k' /\ k'<
scs_k_v39 s
/\ i MOD k' < k' ==> j MOD k' <
scs_k_v39 s /\ SUC (i MOD k') <
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`j MOD k'`;`
scs_k_v39 s:num`]
THEN MRESA_TAC
MOD_LT[`SUC(i MOD k')`;`
scs_k_v39 s:num`]
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k' /\ i MOD k' < k' ==> i MOD k' =k'-1 \/ i MOD k' +1 < k' `)
THEN RESA_TAC
;
POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (4) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_ADD_MOD[`i:num`;`1:num`;`k':num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LT[`1`;`k':num`]
THEN MRESA_TAC
MOD_LT[`i MOD k' + 1`;`k':num`]
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i MOD k':num`;`SUC(j MOD k'):num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC
DIVISION[`(i):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MP_TAC(ARITH_RULE`j MOD k' < k' /\ k'<
scs_k_v39 s
/\ i MOD k' < k' ==> SUC(j MOD k') <
scs_k_v39 s /\ (i MOD k') <
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i MOD k'`;`
scs_k_v39 s:num`]
THEN MRESA_TAC
MOD_LT[`SUC(j MOD k')`;`
scs_k_v39 s:num`]
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k' /\ j MOD k' < k' ==> j MOD k' =k'-1 \/ j MOD k' +1 < k' `)
THEN RESA_TAC
;
POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (5) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_ADD_MOD[`j:num`;`1:num`;`k':num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LT[`1`;`k':num`]
THEN MRESA_TAC
MOD_LT[`j MOD k' + 1`;`k':num`]
;
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (33)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`])
;
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (33)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`])
;
SUBGOAL_THEN`
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}
SUBSET {i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION {(k'-1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}` ASSUME_TAC;
REWRITE_TAC[
SUBSET;
UNION;
IN_ELIM_THM;
IN_SING;
IMAGE]
THEN GEN_TAC
THEN RESA_TAC;
REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`x'<k'==> x'= k'-1\/ SUC x' < k' `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x':num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`k':num`]
THEN DISJ_CASES_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} \/ ~({x', SUC x'} = {0, k' - 1})`)
;
MP_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} ==> (x' =0 /\ SUC x' = k'-1)\/ (x' =k'-1/\ SUC x' =0)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`SUC 0= k'-1<=> k'=2`]
THEN STRIP_TAC
THEN POP_ASSUM (fun th->
REPLICATE_TAC (40-32)(REMOVE_ASSUM_TAC)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th])
THEN ARITH_TAC
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MP_TAC(ARITH_RULE`k' <
scs_k_v39 s /\ 3<= k' /\ x'< k' /\ SUC x' < k' ==> ~(
scs_k_v39 s=0) /\ x'<
scs_k_v39 s/\ SUC x' <
scs_k_v39 s/\ 1<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC
DIVISION[`x' + p MOD
scs_k_v39 s`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MRESA_TAC
MOD_LT[`x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`x':num`;`p:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`SUC x':num`;`p:num`;`
scs_k_v39 s`]
THEN REPLICATE_TAC (10)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s (x' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s ((SUC x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s ((x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC ((x' + p) MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th;
ADD1])
THEN MRESA_TAC
MOD_LT[`1:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`(x' + p):num`;`1:num`;`
scs_k_v39 s`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`(x' + p) + 1=(x' + 1) + p`]
;
REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`x'<k'==> x'= k'-1\/ SUC x' < k' `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x':num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`k':num`]
THEN DISJ_CASES_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} \/ ~({x', SUC x'} = {0, k' - 1})`)
;
MP_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} ==> (x' =0 /\ SUC x' = k'-1)\/ (x' =k'-1/\ SUC x' =0)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`SUC 0= k'-1<=> k'=2`]
THEN STRIP_TAC
THEN POP_ASSUM (fun th->
REPLICATE_TAC (40-32)(REMOVE_ASSUM_TAC)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th])
THEN ARITH_TAC
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MP_TAC(ARITH_RULE`k' <
scs_k_v39 s /\ 3<= k' /\ x'< k' /\ SUC x' < k' ==> ~(
scs_k_v39 s=0) /\ x'<
scs_k_v39 s/\ SUC x' <
scs_k_v39 s/\ 1<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC
DIVISION[`x' + p MOD
scs_k_v39 s`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MATCH_MP_TAC(SET_RULE`B==>A \/ B`)
THEN MRESA_TAC
MOD_LT[`x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`x':num`;`p:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`SUC x':num`;`p:num`;`
scs_k_v39 s`]
THEN REPLICATE_TAC (10)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (x' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s ((SUC x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s ((x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC ((x' + p) MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th;
ADD1])
THEN MRESA_TAC
MOD_LT[`1:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`(x' + p):num`;`1:num`;`
scs_k_v39 s`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`(x' + p) + 1=(x' + 1) + p`]
;
SUBGOAL_THEN`
CARD(
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))})
=
CARD {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}`ASSUME_TAC
;
MATCH_MP_TAC
CARD_IMAGE_INJ
THEN STRIP_TAC;
REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s /\ x<k' /\y< k' ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)
/\ x<
scs_k_v39 s/\ y<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`y:num`;`
scs_k_v39 s`]
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`x:num`;`y:num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k'`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`FINITE
({i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION
{(k' - 1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s})`ASSUME_TAC
;
REWRITE_TAC[
FINITE_UNION;
FINITE_SING]
THEN MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
MRESA_TAC
CARD_SUBSET[`
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}`;`{i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION {(k'-1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}`]
THEN SUBGOAL_THEN`FINITE
({i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))})`ASSUME_TAC
;
REWRITE_TAC[
FINITE_UNION;
FINITE_SING]
THEN MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
MRESAL_TAC
CARD_UNION_LE[`{i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}`;
`{(k' - 1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}`][
FINITE_SING;Geomdetail.CARD_SING]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REPLICATE_TAC (8) REMOVE_ASSUM_TAC
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`a=
CARD
({i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION
{(k' - 1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s})`
THEN ABBREV_TAC`b=
CARD
{i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}`
THEN ABBREV_TAC`c=
CARD
(
IMAGE (\x. (x + p MOD
scs_k_v39 s) MOD
scs_k_v39 s)
{i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))})`
THEN MP_TAC(ARITH_RULE`c<=a /\ a<= b+1 /\ k'<
scs_k_v39 s/\ b +
scs_k_v39 s <= 6 ==> c+k' <=6`)
THEN RESA_TAC;]);;
let SCS_HALF_SLICE_IS_SCS_4=prove_by_refinement(` s' =
scs_half_slice_v39 s p q d' mkj /\
s'' =
scs_half_slice_v39 s q p d'' mkj /\
(
scs_J_v39 s' 0 (
scs_k_v39 s' -1))= mkj
/\
(~(
scs_J_v39 s' 0 (
scs_k_v39 s' -1)))
/\ vv' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s')+p MOD (
scs_k_v39 s)))
/\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q
/\
d'< #0.9 /\
scs_bm_v39 s p q < &4 /\
scs_k_v39 s =4
==>
is_scs_v39 s'`,
[STRIP_TAC
THEN MRESA_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`p:num`;`q:num`;`d':real`;`
scs_J_v39 s' p q`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;]
THEN REPLICATE_TAC 5 (POP_ASSUM MP_TAC)
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit]
THEN ABBREV_TAC`k'=(q + 1 +
scs_k_v39 s - p MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN REPEAT RESA_TAC;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
REAL_ARITH_TAC;
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (17)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MRESA_TAC
th[`i MOD k' + p MOD
scs_k_v39 s`;`j MOD k' + p MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 3 (POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC;
REAL_ARITH_TAC;
MP_TAC(ARITH_RULE`3<=k' ==> ~(0=k'-1)`)
THEN RESA_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', i MOD k'} = {0, k'-1} \/ ~({i MOD k', i MOD k'} = {0, k'-1})`);
MP_TAC(SET_RULE`{i MOD k', i MOD k'} = {0, k'-1}==> i MOD k'= 0 /\ i MOD k'= k'-1`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC;
ASM_REWRITE_TAC[];
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `q:num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s (q MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `p:num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
DIVISION[`p:num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`q:num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN REPLICATE_TAC 23 (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 25(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 4(POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(j MOD k' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (((j MOD k' + p MOD
scs_k_v39 s) MOD
scs_k_v39 s))`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `((i MOD k' + p MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN REPLICATE_TAC (34-15)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN MRESAL_TAC
DIVISION[`(i MOD k' + p MOD
scs_k_v39 s):num`;`
scs_k_v39 s:num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j MOD k' + p MOD
scs_k_v39 s):num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN STRIP_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i MOD k':num`;`j MOD k':num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`j:num`;`k':num`]
THEN MP_TAC(ARITH_RULE`i< k' /\ k'<
scs_k_v39 s /\ j<k' ==>
i<
scs_k_v39 s /\ j<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`j:num`;`
scs_k_v39 s`]
;
DISJ_CASES_TAC(SET_RULE`{i MOD 3, SUC i MOD 3} = {0, 3-1} \/ ~({i MOD 3, SUC i MOD 3} = {0, 3-1})`);
ASM_REWRITE_TAC[]
;
ASM_REWRITE_TAC[]
THEN MRESAL_TAC
DIVISION[`i:num`;`k':num`][ARITH_RULE`~(3=0)`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`(i ) MOD 3 < 3 ==> (i ) MOD 3 =0 \/ (i ) MOD 3 =1 \/ (i ) MOD 3 =2`)
THEN RESA_TAC
;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`0`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 0+A=A /\ 1+A= SUC A`]
THEN REPLICATE_TAC (41-22)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL1_TAC
th `p MOD
scs_k_v39 s` [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3/\ 3<4`;cstab])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`1`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
THEN REPLICATE_TAC (41-22)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL1_TAC
th `1+p MOD
scs_k_v39 s` [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3/\3<4`;cstab])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`2`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th;ARITH_RULE`3-1=2`]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th;ARITH_RULE`3-1=2`]
THEN REPEAT STRIP_TAC)
THEN REPLICATE_TAC (3) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC
;
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[];
MP_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} ==> (i MOD k'= 0 /\ j MOD k' = k'-1)\/ (i MOD k'= k'-1 /\ j MOD k' = 0)`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESAL_TAC MOD_MULT[`k':num`;`1`][ARITH_RULE`A* 1=A`]
THEN
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`j:num`;`k'-1`;`k':num`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESAL_TAC MOD_MULT[`k':num`;`1`][ARITH_RULE`A* 1=A`]
THEN
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`k'-1`;`k':num`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-23)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`][ARITH_RULE`SUC(A+B)= SUC A+B`])
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`j MOD k':num`;`SUC(i MOD k'):num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC
DIVISION[`(i):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MP_TAC(ARITH_RULE`j MOD k' < k' /\ k'<
scs_k_v39 s
/\ i MOD k' < k' ==> j MOD k' <
scs_k_v39 s /\ SUC (i MOD k') <
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`j MOD k'`;`
scs_k_v39 s:num`]
THEN MRESA_TAC
MOD_LT[`SUC(i MOD k')`;`
scs_k_v39 s:num`]
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k' /\ i MOD k' < k' ==> i MOD k' =k'-1 \/ i MOD k' +1 < k' `)
THEN RESA_TAC
;
POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (4) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_ADD_MOD[`i:num`;`1:num`;`k':num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LT[`1`;`k':num`]
THEN MRESA_TAC
MOD_LT[`i MOD k' + 1`;`k':num`]
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i MOD k':num`;`SUC(j MOD k'):num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC
DIVISION[`(i):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MP_TAC(ARITH_RULE`j MOD k' < k' /\ k'<
scs_k_v39 s
/\ i MOD k' < k' ==> SUC(j MOD k') <
scs_k_v39 s /\ (i MOD k') <
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i MOD k'`;`
scs_k_v39 s:num`]
THEN MRESA_TAC
MOD_LT[`SUC(j MOD k')`;`
scs_k_v39 s:num`]
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k' /\ j MOD k' < k' ==> j MOD k' =k'-1 \/ j MOD k' +1 < k' `)
THEN RESA_TAC
;
POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (5) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_ADD_MOD[`j:num`;`1:num`;`k':num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LT[`1`;`k':num`]
THEN MRESA_TAC
MOD_LT[`j MOD k' + 1`;`k':num`]
;
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (33)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`])
;
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (33)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`])
;
SUBGOAL_THEN`
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}
SUBSET {i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION {(k'-1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}` ASSUME_TAC;
REWRITE_TAC[
SUBSET;
UNION;
IN_ELIM_THM;
IN_SING;
IMAGE]
THEN GEN_TAC
THEN RESA_TAC;
REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`x'<k'==> x'= k'-1\/ SUC x' < k' `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x':num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`k':num`]
THEN DISJ_CASES_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} \/ ~({x', SUC x'} = {0, k' - 1})`)
;
MP_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} ==> (x' =0 /\ SUC x' = k'-1)\/ (x' =k'-1/\ SUC x' =0)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`SUC 0= k'-1<=> k'=2`]
THEN STRIP_TAC
THEN POP_ASSUM (fun th->
REPLICATE_TAC (40-32)(REMOVE_ASSUM_TAC)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th])
THEN ARITH_TAC
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MP_TAC(ARITH_RULE`k' <
scs_k_v39 s /\ 3<= k' /\ x'< k' /\ SUC x' < k' ==> ~(
scs_k_v39 s=0) /\ x'<
scs_k_v39 s/\ SUC x' <
scs_k_v39 s/\ 1<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC
DIVISION[`x' + p MOD
scs_k_v39 s`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MRESA_TAC
MOD_LT[`x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`x':num`;`p:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`SUC x':num`;`p:num`;`
scs_k_v39 s`]
THEN REPLICATE_TAC (10)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s (x' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s ((SUC x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s ((x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC ((x' + p) MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th;
ADD1])
THEN MRESA_TAC
MOD_LT[`1:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`(x' + p):num`;`1:num`;`
scs_k_v39 s`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`(x' + p) + 1=(x' + 1) + p`]
;
REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`x'<k'==> x'= k'-1\/ SUC x' < k' `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x':num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`k':num`]
THEN DISJ_CASES_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} \/ ~({x', SUC x'} = {0, k' - 1})`)
;
MP_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} ==> (x' =0 /\ SUC x' = k'-1)\/ (x' =k'-1/\ SUC x' =0)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`SUC 0= k'-1<=> k'=2`]
THEN STRIP_TAC
THEN POP_ASSUM (fun th->
REPLICATE_TAC (40-32)(REMOVE_ASSUM_TAC)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th])
THEN ARITH_TAC
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MP_TAC(ARITH_RULE`k' <
scs_k_v39 s /\ 3<= k' /\ x'< k' /\ SUC x' < k' ==> ~(
scs_k_v39 s=0) /\ x'<
scs_k_v39 s/\ SUC x' <
scs_k_v39 s/\ 1<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC
DIVISION[`x' + p MOD
scs_k_v39 s`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MATCH_MP_TAC(SET_RULE`B==>A \/ B`)
THEN MRESA_TAC
MOD_LT[`x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`x':num`;`p:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`SUC x':num`;`p:num`;`
scs_k_v39 s`]
THEN REPLICATE_TAC (10)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (x' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s ((SUC x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s ((x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC ((x' + p) MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th;
ADD1])
THEN MRESA_TAC
MOD_LT[`1:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`(x' + p):num`;`1:num`;`
scs_k_v39 s`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`(x' + p) + 1=(x' + 1) + p`]
;
SUBGOAL_THEN`
CARD(
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))})
=
CARD {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}`ASSUME_TAC
;
MATCH_MP_TAC
CARD_IMAGE_INJ
THEN STRIP_TAC;
ASM_REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s /\ x<k' /\y< k' ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)
/\ x<
scs_k_v39 s/\ y<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`y:num`;`
scs_k_v39 s`]
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`x:num`;`y:num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k'`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`FINITE
({i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION
{(k' - 1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s})`ASSUME_TAC
;
REWRITE_TAC[
FINITE_UNION;
FINITE_SING]
THEN MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESA_TAC
CARD_SUBSET[`
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}`;`{i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION {(k'-1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}`]
THEN SUBGOAL_THEN`FINITE
({i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))})`ASSUME_TAC
;
REWRITE_TAC[
FINITE_UNION;
FINITE_SING]
THEN MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESAL_TAC
CARD_UNION_LE[`{i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}`;`{(k' - 1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}`][
FINITE_SING;Geomdetail.CARD_SING]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th->
REPLICATE_TAC (5) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th])
THEN ABBREV_TAC`a={i | i < 4 /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}`
THEN ABBREV_TAC`b=
CARD
(
IMAGE (\x. (x + p MOD 4) MOD 4)
{i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD 4) (SUC i MOD k' + p MOD 4)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD 4) (SUC i MOD k' + p MOD 4)))})`
THEN ABBREV_TAC`c=
CARD (a
UNION {(k' - 1 + p MOD 4) MOD 4})`
THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC;]);;
let SCS_HALF_SLICE_IS_SCS_4_PRIME=prove_by_refinement(
` s' =
scs_half_slice_v39 s p q d' mkj /\
s'' =
scs_half_slice_v39 s q p d'' mkj /\
(
scs_J_v39 s' 0 (
scs_k_v39 s' -1))= mkj
/\
(
is_ear_v39 s'')
/\ vv' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s')+p MOD (
scs_k_v39 s)))
/\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q
/\
d'< #0.9 /\
scs_bm_v39 s p q < &4 /\
scs_k_v39 s =4
==>
is_scs_v39 s'`,
[
STRIP_TAC
THEN MRESA_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`p:num`;`q:num`;`d':real`;`
scs_J_v39 s' p q`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;]
THEN REPLICATE_TAC 5 (POP_ASSUM MP_TAC)
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit]
THEN ABBREV_TAC`k'=(q + 1 +
scs_k_v39 s - p MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN REPEAT RESA_TAC;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
REAL_ARITH_TAC;
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (17)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MRESA_TAC
th[`i MOD k' + p MOD
scs_k_v39 s`;`j MOD k' + p MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 3 (POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC;
REAL_ARITH_TAC;
MP_TAC(ARITH_RULE`3<=k' ==> ~(0=k'-1)`)
THEN RESA_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', i MOD k'} = {0, k'-1} \/ ~({i MOD k', i MOD k'} = {0, k'-1})`);
MP_TAC(SET_RULE`{i MOD k', i MOD k'} = {0, k'-1}==> i MOD k'= 0 /\ i MOD k'= k'-1`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC;
ASM_REWRITE_TAC[];
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `q:num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s (q MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `p:num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
DIVISION[`p:num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`q:num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN REPLICATE_TAC 23 (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 25(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 4(POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(j MOD k' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (((j MOD k' + p MOD
scs_k_v39 s) MOD
scs_k_v39 s))`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `((i MOD k' + p MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN REPLICATE_TAC (34-15)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN MRESAL_TAC
DIVISION[`(i MOD k' + p MOD
scs_k_v39 s):num`;`
scs_k_v39 s:num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j MOD k' + p MOD
scs_k_v39 s):num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN STRIP_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i MOD k':num`;`j MOD k':num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`j:num`;`k':num`]
THEN MP_TAC(ARITH_RULE`i< k' /\ k'<
scs_k_v39 s /\ j<k' ==>
i<
scs_k_v39 s /\ j<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`j:num`;`
scs_k_v39 s`]
;
DISJ_CASES_TAC(SET_RULE`{i MOD 3, SUC i MOD 3} = {0, 3-1} \/ ~({i MOD 3, SUC i MOD 3} = {0, 3-1})`);
ASM_REWRITE_TAC[]
;
ASM_REWRITE_TAC[]
THEN MRESAL_TAC
DIVISION[`i:num`;`k':num`][ARITH_RULE`~(3=0)`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`(i ) MOD 3 < 3 ==> (i ) MOD 3 =0 \/ (i ) MOD 3 =1 \/ (i ) MOD 3 =2`)
THEN RESA_TAC
;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`0`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 0+A=A /\ 1+A= SUC A`]
THEN REPLICATE_TAC (41-22)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL1_TAC
th `p MOD
scs_k_v39 s` [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3/\ 3<4`;cstab])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`1`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
THEN REPLICATE_TAC (41-22)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL1_TAC
th `1+p MOD
scs_k_v39 s` [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3/\3<4`;cstab])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`2`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th;ARITH_RULE`3-1=2`]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th;ARITH_RULE`3-1=2`]
THEN REPEAT STRIP_TAC)
THEN REPLICATE_TAC (3) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC
;
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[];
MP_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} ==> (i MOD k'= 0 /\ j MOD k' = k'-1)\/ (i MOD k'= k'-1 /\ j MOD k' = 0)`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESAL_TAC MOD_MULT[`k':num`;`1`][ARITH_RULE`A* 1=A`]
THEN
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`j:num`;`k'-1`;`k':num`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESAL_TAC MOD_MULT[`k':num`;`1`][ARITH_RULE`A* 1=A`]
THEN
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`k'-1`;`k':num`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-23)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`][ARITH_RULE`SUC(A+B)= SUC A+B`])
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`j MOD k':num`;`SUC(i MOD k'):num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC
DIVISION[`(i):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MP_TAC(ARITH_RULE`j MOD k' < k' /\ k'<
scs_k_v39 s
/\ i MOD k' < k' ==> j MOD k' <
scs_k_v39 s /\ SUC (i MOD k') <
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`j MOD k'`;`
scs_k_v39 s:num`]
THEN MRESA_TAC
MOD_LT[`SUC(i MOD k')`;`
scs_k_v39 s:num`]
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k' /\ i MOD k' < k' ==> i MOD k' =k'-1 \/ i MOD k' +1 < k' `)
THEN RESA_TAC
;
POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (4) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_ADD_MOD[`i:num`;`1:num`;`k':num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LT[`1`;`k':num`]
THEN MRESA_TAC
MOD_LT[`i MOD k' + 1`;`k':num`]
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i MOD k':num`;`SUC(j MOD k'):num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC
DIVISION[`(i):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MP_TAC(ARITH_RULE`j MOD k' < k' /\ k'<
scs_k_v39 s
/\ i MOD k' < k' ==> SUC(j MOD k') <
scs_k_v39 s /\ (i MOD k') <
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i MOD k'`;`
scs_k_v39 s:num`]
THEN MRESA_TAC
MOD_LT[`SUC(j MOD k')`;`
scs_k_v39 s:num`]
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k' /\ j MOD k' < k' ==> j MOD k' =k'-1 \/ j MOD k' +1 < k' `)
THEN RESA_TAC
;
POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (5) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_ADD_MOD[`j:num`;`1:num`;`k':num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LT[`1`;`k':num`]
THEN MRESA_TAC
MOD_LT[`j MOD k' + 1`;`k':num`]
;
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-3)(POP_ASSUM MP_TAC)
THEN POP_ASSUM (fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
is_ear_v39;
scs_half_slice_v39;
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit])
THEN STRIP_TAC
THEN ABBREV_TAC`k''=(p + 1 + 4 - q MOD 4) MOD 4`
THEN SUBGOAL_THEN`k'' -1
IN {i | i < 3 /\
(if {i MOD k'', SUC i MOD k''} = {0, k'' - 1}
then T
else
scs_J_v39 s (i MOD k'' + q MOD 4) (SUC i MOD k'' + q MOD 4))}` ASSUME_TAC
;
REWRITE_TAC[
IN_ELIM_THM]
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2 /\ 2<3 /\ 2 MOD 3=2/\ SUC 2=3/\ 3 MOD 3=0 `];
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`;SET_RULE`( a = b <=> b=a)`;
IN_SING]
THEN REPLICATE_TAC (3)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2 /\ 2<3 /\ 2 MOD 3=2/\ SUC 2=3/\ 3 MOD 3=0 `];
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`])
;
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-3)(POP_ASSUM MP_TAC)
THEN POP_ASSUM (fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
is_ear_v39;
scs_half_slice_v39;
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit])
THEN STRIP_TAC
THEN ABBREV_TAC`k''=(p + 1 + 4 - q MOD 4) MOD 4`
THEN SUBGOAL_THEN`k'' -1
IN {i | i < 3 /\
(if {i MOD k'', SUC i MOD k''} = {0, k'' - 1}
then T
else
scs_J_v39 s (i MOD k'' + q MOD 4) (SUC i MOD k'' + q MOD 4))}` ASSUME_TAC
;
REWRITE_TAC[
IN_ELIM_THM]
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2 /\ 2<3 /\ 2 MOD 3=2/\ SUC 2=3/\ 3 MOD 3=0 `];
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`;SET_RULE`( a = b <=> b=a)`;
IN_SING]
THEN REPLICATE_TAC (2)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2 /\ 2<3 /\ 2 MOD 3=2/\ SUC 2=3/\ 3 MOD 3=0 `]
THEN RESA_TAC;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`])
;
SUBGOAL_THEN`
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}
SUBSET {i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION {(k'-1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}` ASSUME_TAC;
REWRITE_TAC[
SUBSET;
UNION;
IN_ELIM_THM;
IN_SING;
IMAGE]
THEN GEN_TAC
THEN RESA_TAC;
REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`x'<k'==> x'= k'-1\/ SUC x' < k' `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x':num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`k':num`]
THEN DISJ_CASES_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} \/ ~({x', SUC x'} = {0, k' - 1})`)
;
MP_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} ==> (x' =0 /\ SUC x' = k'-1)\/ (x' =k'-1/\ SUC x' =0)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`SUC 0= k'-1<=> k'=2`]
THEN STRIP_TAC
THEN POP_ASSUM (fun th->
REPLICATE_TAC (40-32)(REMOVE_ASSUM_TAC)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th])
THEN ARITH_TAC
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MP_TAC(ARITH_RULE`k' <
scs_k_v39 s /\ 3<= k' /\ x'< k' /\ SUC x' < k' ==> ~(
scs_k_v39 s=0) /\ x'<
scs_k_v39 s/\ SUC x' <
scs_k_v39 s/\ 1<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC
DIVISION[`x' + p MOD
scs_k_v39 s`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MRESA_TAC
MOD_LT[`x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`x':num`;`p:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`SUC x':num`;`p:num`;`
scs_k_v39 s`]
THEN REPLICATE_TAC (10)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s (x' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s ((SUC x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s ((x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC ((x' + p) MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th;
ADD1])
THEN MRESA_TAC
MOD_LT[`1:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`(x' + p):num`;`1:num`;`
scs_k_v39 s`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`(x' + p) + 1=(x' + 1) + p`]
;
REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`x'<k'==> x'= k'-1\/ SUC x' < k' `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x':num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`k':num`]
THEN DISJ_CASES_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} \/ ~({x', SUC x'} = {0, k' - 1})`)
;
MP_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} ==> (x' =0 /\ SUC x' = k'-1)\/ (x' =k'-1/\ SUC x' =0)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`SUC 0= k'-1<=> k'=2`]
THEN STRIP_TAC
THEN POP_ASSUM (fun th->
REPLICATE_TAC (40-32)(REMOVE_ASSUM_TAC)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th])
THEN ARITH_TAC
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MP_TAC(ARITH_RULE`k' <
scs_k_v39 s /\ 3<= k' /\ x'< k' /\ SUC x' < k' ==> ~(
scs_k_v39 s=0) /\ x'<
scs_k_v39 s/\ SUC x' <
scs_k_v39 s/\ 1<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC
DIVISION[`x' + p MOD
scs_k_v39 s`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MATCH_MP_TAC(SET_RULE`B==>A \/ B`)
THEN MRESA_TAC
MOD_LT[`x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`x':num`;`p:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`SUC x':num`;`p:num`;`
scs_k_v39 s`]
THEN REPLICATE_TAC (10)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (x' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s ((SUC x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s ((x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC ((x' + p) MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th;
ADD1])
THEN MRESA_TAC
MOD_LT[`1:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`(x' + p):num`;`1:num`;`
scs_k_v39 s`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`(x' + p) + 1=(x' + 1) + p`]
;
SUBGOAL_THEN`
CARD(
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))})
=
CARD {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}`ASSUME_TAC
;
MATCH_MP_TAC
CARD_IMAGE_INJ
THEN STRIP_TAC;
ASM_REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s /\ x<k' /\y< k' ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)
/\ x<
scs_k_v39 s/\ y<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`y:num`;`
scs_k_v39 s`]
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`x:num`;`y:num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k'`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`FINITE
({i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION
{(k' - 1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s})`ASSUME_TAC
;
REWRITE_TAC[
FINITE_UNION;
FINITE_SING]
THEN MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESA_TAC
CARD_SUBSET[`
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}`;`{i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION {(k'-1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}`]
THEN SUBGOAL_THEN`FINITE
({i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))})`ASSUME_TAC
;
REWRITE_TAC[
FINITE_UNION;
FINITE_SING]
THEN MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
POP_ASSUM MP_TAC
THEN RESA_TAC
THEN MRESAL_TAC
CARD_UNION_LE[`{i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}`;`{(k' - 1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}`][
FINITE_SING;Geomdetail.CARD_SING]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM (fun th->
REPLICATE_TAC (5) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th])
THEN ABBREV_TAC`a={i | i < 4 /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}`
THEN ABBREV_TAC`b=
CARD
(
IMAGE (\x. (x + p MOD 4) MOD 4)
{i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD 4) (SUC i MOD k' + p MOD 4)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD 4) (SUC i MOD k' + p MOD 4)))})`
THEN ABBREV_TAC`c=
CARD (a
UNION {(k' - 1 + p MOD 4) MOD 4})`
THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC]);;
let SCS_HALF_SLICE_IS_SCS_PRIME=prove_by_refinement(
` s' =
scs_half_slice_v39 s p q d' mkj /\
s'' =
scs_half_slice_v39 s q p d'' mkj /\
(
scs_J_v39 s' 0 (
scs_k_v39 s' -1))= mkj
/\
(
is_ear_v39 s'')
/\ vv' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s')+p MOD (
scs_k_v39 s)))
/\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q
/\
d'< #0.9 /\
scs_bm_v39 s p q < &4 /\
(4<
scs_k_v39 s ==>
scs_bm_v39 s p q <= cstab)
==>
is_scs_v39 s'`,
[
STRIP_TAC
THEN MRESA_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`p:num`;`q:num`;`d':real`;`
scs_J_v39 s' p q`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;]
THEN REPLICATE_TAC 5 (POP_ASSUM MP_TAC)
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit]
THEN ABBREV_TAC`k'=(q + 1 +
scs_k_v39 s - p MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN REPEAT RESA_TAC;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic]
THEN SET_TAC[]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`]
;
REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
;
MRESAL_TAC
MOD_EQ [`i+k':num`;`i:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`i+p+k':num`;`i+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
MRESAL_TAC
MOD_EQ [`j+k':num`;`j:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ (i + k') + p = i+p+k'`]
THEN MRESAL_TAC
MOD_EQ [`j+p+k':num`;`j+p:num`;`k':num`;`1`][ARITH_RULE`1* k'=k'/\ i + p + k' = (i + p) + k'`];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
GEN_REWRITE_TAC(LAND_CONV o ONCE_DEPTH_CONV)[SET_RULE`({A,B}={C,D})<=> ({B,A}={C,D}) `]
THEN REWRITE_TAC[];
REAL_ARITH_TAC;
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (17)(REMOVE_ASSUM_TAC)
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`i MOD k' + p MOD
scs_k_v39 s`;`j MOD k' + p MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 3 (POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC
;
REAL_ARITH_TAC;
MP_TAC(ARITH_RULE`3<=k' ==> ~(0=k'-1)`)
THEN RESA_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', i MOD k'} = {0, k'-1} \/ ~({i MOD k', i MOD k'} = {0, k'-1})`);
MP_TAC(SET_RULE`{i MOD k', i MOD k'} = {0, k'-1}==> i MOD k'= 0 /\ i MOD k'= k'-1`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN RESA_TAC;
ASM_REWRITE_TAC[];
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `q:num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s (q MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `p:num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
DIVISION[`p:num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`q:num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN REPLICATE_TAC 23 (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 25(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`])
THEN REPLICATE_TAC 4(POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(j MOD k' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (((j MOD k' + p MOD
scs_k_v39 s) MOD
scs_k_v39 s))`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `((i MOD k' + p MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN REPLICATE_TAC (34-15)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN MRESAL_TAC
DIVISION[`(i MOD k' + p MOD
scs_k_v39 s):num`;`
scs_k_v39 s:num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j MOD k' + p MOD
scs_k_v39 s):num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN STRIP_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i MOD k':num`;`j MOD k':num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`j:num`;`k':num`]
THEN MP_TAC(ARITH_RULE`i< k' /\ k'<
scs_k_v39 s /\ j<k' ==>
i<
scs_k_v39 s /\ j<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`j:num`;`
scs_k_v39 s`]
;
DISJ_CASES_TAC(SET_RULE`{i MOD 3, SUC i MOD 3} = {0, 3-1} \/ ~({i MOD 3, SUC i MOD 3} = {0, 3-1})`);
ASM_REWRITE_TAC[]
;
ASM_REWRITE_TAC[]
THEN MRESAL_TAC
DIVISION[`i:num`;`k':num`][ARITH_RULE`~(3=0)`]
THEN MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s`)
THEN RESA_TAC
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`(i ) MOD 3 < 3 ==> (i ) MOD 3 =0 \/ (i ) MOD 3 =1 \/ (i ) MOD 3 =2`)
THEN RESA_TAC
;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`0`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 0+A=A /\ 1+A= SUC A`]
THEN REPLICATE_TAC (37-17)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL1_TAC
th `p MOD
scs_k_v39 s` [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3`;cstab])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`1`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
THEN REPLICATE_TAC (37-17)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL1_TAC
th `1+p MOD
scs_k_v39 s` [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3`;cstab])
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`2`;`3`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th;ARITH_RULE`3-1=2`]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th;ARITH_RULE`3-1=2`]
THEN REPEAT STRIP_TAC)
THEN REPLICATE_TAC (4) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
DISJ_CASES_TAC(SET_RULE`{i MOD k', SUC i MOD k'} = {0, k'-1} \/ ~({i MOD k', SUC i MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`3<k' /\ k'<
scs_k_v39 s ==> 4<
scs_k_v39 s`)
THEN RESA_TAC
THEN REPLICATE_TAC (6) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN RESA_TAC
;
ASM_REWRITE_TAC[]
THEN MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'`)
THEN RESA_TAC
THEN MRESAL_TAC
DIVISION[`i:num`;`k':num`][ARITH_RULE`~(3=0)`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0)`)
THEN RESA_TAC
THEN REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_ADD_MOD[`i:num`;`1:num`;`k':num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LT[`1`;`k':num`]
THEN MP_TAC(ARITH_RULE`3<k' /\ i MOD k' < k' ==> i MOD k' =k'-1 \/ i MOD k' +1 < k' `)
THEN RESA_TAC
;
MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESAL_TAC MOD_MULT[`k':num`;`1`][ARITH_RULE`A* 1=A`]
THEN
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`k'-1`;`k':num`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN REPLICATE_TAC (2) REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN REPLICATE_TAC (8) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN SET_TAC[];
MRESAL_TAC
MOD_LT[`i MOD k' +1`;`k':num`][ARITH_RULE`(i MOD k' + 1) + p MOD
scs_k_v39 s= SUC(i MOD k' + p MOD
scs_k_v39 s)`]
THEN REPLICATE_TAC (42-17)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[];
MP_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} ==> (i MOD k'= 0 /\ j MOD k' = k'-1)\/ (i MOD k'= k'-1 /\ j MOD k' = 0)`)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESAL_TAC MOD_MULT[`k':num`;`1`][ARITH_RULE`A* 1=A`]
THEN
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`j:num`;`k'-1`;`k':num`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESAL_TAC MOD_MULT[`k':num`;`1`][ARITH_RULE`A* 1=A`]
THEN
MRESAL_TAC(GEN_ALL
IMP_SUC_MOD_EQ)[`i:num`;`k'-1`;`k':num`][ARITH_RULE`0 MOD 3=0 /\ 1 MOD 3=1/\ 2 MOD 3=2/\ SUC 0=1/\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ ~(3=0)/\ 2+A=SUC(1+A) `]
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (31-18)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`][ARITH_RULE`SUC(A+B)= SUC A+B`])
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`j MOD k':num`;`SUC(i MOD k'):num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC
DIVISION[`(i):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MP_TAC(ARITH_RULE`j MOD k' < k' /\ k'<
scs_k_v39 s
/\ i MOD k' < k' ==> j MOD k' <
scs_k_v39 s /\ SUC (i MOD k') <
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`j MOD k'`;`
scs_k_v39 s:num`]
THEN MRESA_TAC
MOD_LT[`SUC(i MOD k')`;`
scs_k_v39 s:num`]
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k' /\ i MOD k' < k' ==> i MOD k' =k'-1 \/ i MOD k' +1 < k' `)
THEN RESA_TAC
;
POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (4) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_ADD_MOD[`i:num`;`1:num`;`k':num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LT[`1`;`k':num`]
THEN MRESA_TAC
MOD_LT[`i MOD k' + 1`;`k':num`]
;
MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)`)
THEN RESA_TAC
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i MOD k':num`;`SUC(j MOD k'):num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC
DIVISION[`(i):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`(j):num`;`k':num`][ARITH_RULE`~(6=0)`]
THEN MP_TAC(ARITH_RULE`j MOD k' < k' /\ k'<
scs_k_v39 s
/\ i MOD k' < k' ==> SUC(j MOD k') <
scs_k_v39 s /\ (i MOD k') <
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i MOD k'`;`
scs_k_v39 s:num`]
THEN MRESA_TAC
MOD_LT[`SUC(j MOD k')`;`
scs_k_v39 s:num`]
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k' /\ j MOD k' < k' ==> j MOD k' =k'-1 \/ j MOD k' +1 < k' `)
THEN RESA_TAC
;
POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (5) REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_ADD_MOD[`j:num`;`1:num`;`k':num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LT[`1`;`k':num`]
THEN MRESA_TAC
MOD_LT[`j MOD k' + 1`;`k':num`]
;
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-3)(POP_ASSUM MP_TAC)
THEN POP_ASSUM (fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
is_ear_v39;
scs_half_slice_v39;
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit])
THEN STRIP_TAC
THEN ABBREV_TAC`k''=(p + 1 +
scs_k_v39 s - q MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN SUBGOAL_THEN`k'' -1
IN {i | i < 3 /\
(if {i MOD k'', SUC i MOD k''} = {0, k'' - 1}
then T
else
scs_J_v39 s (i MOD k'' + q MOD
scs_k_v39 s)
(SUC i MOD k'' + q MOD
scs_k_v39 s))}` ASSUME_TAC
;
REWRITE_TAC[
IN_ELIM_THM]
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2 /\ 2<3 /\ 2 MOD 3=2/\ SUC 2=3/\ 3 MOD 3=0 `;SET_RULE`{A,B}={B,A}`];
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`;SET_RULE`( a = b <=> b=a)`;
IN_SING]
THEN REPLICATE_TAC (3)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2 /\ 2<3 /\ 2 MOD 3=2/\ SUC 2=3/\ 3 MOD 3=0 `]
THEN RESA_TAC;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`])
;
POP_ASSUM MP_TAC
THEN DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1} \/ ~({i MOD k', j MOD k'} = {0, k' - 1})`)
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-3)(POP_ASSUM MP_TAC)
THEN POP_ASSUM (fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
is_ear_v39;
scs_half_slice_v39;
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit])
THEN STRIP_TAC
THEN ABBREV_TAC`k''=(p + 1 +
scs_k_v39 s - q MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN SUBGOAL_THEN`k'' -1
IN {i | i < 3 /\
(if {i MOD k'', SUC i MOD k''} = {0, k'' - 1}
then T
else
scs_J_v39 s (i MOD k'' + q MOD
scs_k_v39 s)
(SUC i MOD k'' + q MOD
scs_k_v39 s))}` ASSUME_TAC
;
REWRITE_TAC[
IN_ELIM_THM]
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2 /\ 2<3 /\ 2 MOD 3=2/\ SUC 2=3/\ 3 MOD 3=0 `;SET_RULE`{A,B}={B,A}`];
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`;SET_RULE`( a = b <=> b=a)`;
IN_SING]
THEN REPLICATE_TAC (2)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2 /\ 2<3 /\ 2 MOD 3=2/\ SUC 2=3/\ 3 MOD 3=0 `]
THEN RESA_TAC;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN REPLICATE_TAC (36-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i MOD k' + p MOD
scs_k_v39 s)`;`(j MOD k' + p MOD
scs_k_v39 s)`])
;
SUBGOAL_THEN`
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}
SUBSET {i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION {(k'-1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}` ASSUME_TAC;
REWRITE_TAC[
SUBSET;
UNION;
IN_ELIM_THM;
IN_SING;
IMAGE]
THEN GEN_TAC
THEN RESA_TAC;
REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`x'<k'==> x'= k'-1\/ SUC x' < k' `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x':num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`k':num`]
THEN DISJ_CASES_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} \/ ~({x', SUC x'} = {0, k' - 1})`)
;
MP_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} ==> (x' =0 /\ SUC x' = k'-1)\/ (x' =k'-1/\ SUC x' =0)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`SUC 0= k'-1<=> k'=2`]
THEN STRIP_TAC
THEN POP_ASSUM (fun th->
REPLICATE_TAC (40-32)(REMOVE_ASSUM_TAC)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th])
THEN ARITH_TAC
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MP_TAC(ARITH_RULE`k' <
scs_k_v39 s /\ 3<= k' /\ x'< k' /\ SUC x' < k' ==> ~(
scs_k_v39 s=0) /\ x'<
scs_k_v39 s/\ SUC x' <
scs_k_v39 s/\ 1<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC
DIVISION[`x' + p MOD
scs_k_v39 s`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MRESA_TAC
MOD_LT[`x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`x':num`;`p:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`SUC x':num`;`p:num`;`
scs_k_v39 s`]
THEN REPLICATE_TAC (10)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s (x' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s ((SUC x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s ((x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC ((x' + p) MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th;
ADD1])
THEN MRESA_TAC
MOD_LT[`1:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`(x' + p):num`;`1:num`;`
scs_k_v39 s`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`(x' + p) + 1=(x' + 1) + p`]
;
REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`x'<k'==> x'= k'-1\/ SUC x' < k' `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x':num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`k':num`]
THEN DISJ_CASES_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} \/ ~({x', SUC x'} = {0, k' - 1})`)
;
MP_TAC(SET_RULE`{x', SUC x'} = {0, k' - 1} ==> (x' =0 /\ SUC x' = k'-1)\/ (x' =k'-1/\ SUC x' =0)`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`SUC 0= k'-1<=> k'=2`]
THEN STRIP_TAC
THEN POP_ASSUM (fun th->
REPLICATE_TAC (40-32)(REMOVE_ASSUM_TAC)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th])
THEN ARITH_TAC
;
ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN MATCH_MP_TAC(SET_RULE`A==>A \/ B`)
THEN MP_TAC(ARITH_RULE`k' <
scs_k_v39 s /\ 3<= k' /\ x'< k' /\ SUC x' < k' ==> ~(
scs_k_v39 s=0) /\ x'<
scs_k_v39 s/\ SUC x' <
scs_k_v39 s/\ 1<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC
DIVISION[`x' + p MOD
scs_k_v39 s`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MATCH_MP_TAC(SET_RULE`B==>A \/ B`)
THEN MRESA_TAC
MOD_LT[`x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`SUC x':num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`x':num`;`p:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`SUC x':num`;`p:num`;`
scs_k_v39 s`]
THEN REPLICATE_TAC (10)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s (x' + p MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> POP_ASSUM MP_TAC THEN REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s ((SUC x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x' + p MOD
scs_k_v39 s):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_a_v39 s ((x' + p) MOD
scs_k_v39 s)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(SUC ((x' + p) MOD
scs_k_v39 s)):num`)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th;
ADD1])
THEN MRESA_TAC
MOD_LT[`1:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_ADD_MOD[`(x' + p):num`;`1:num`;`
scs_k_v39 s`]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`(x' + p) + 1=(x' + 1) + p`]
;
SUBGOAL_THEN`
CARD(
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))})
=
CARD {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}`ASSUME_TAC
;
MATCH_MP_TAC
CARD_IMAGE_INJ
THEN STRIP_TAC;
REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<= k' /\ k' <
scs_k_v39 s /\ x<k' /\y< k' ==> 3<
scs_k_v39 s/\ ~(k'=0) /\ 1< k'/\ k'-1< k'/\ SUC(k'-1)= k'/\ ~(
scs_k_v39 s =0)
/\ x<
scs_k_v39 s/\ y<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`
scs_k_v39 s`]
THEN MRESA_TAC
MOD_LT[`y:num`;`
scs_k_v39 s`]
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`x:num`;`y:num`;`p MOD
scs_k_v39 s`;`
scs_k_v39 s`][
ADD_SYM]
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k'`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`FINITE
({i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION
{(k' - 1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s})`ASSUME_TAC
;
REWRITE_TAC[
FINITE_UNION;
FINITE_SING]
THEN MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
MRESA_TAC
CARD_SUBSET[`
IMAGE (\x. (x+p MOD
scs_k_v39 s )MOD
scs_k_v39 s ) {i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))}`;`{i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION {(k'-1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}`]
THEN SUBGOAL_THEN`FINITE
({i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))})`ASSUME_TAC
;
REWRITE_TAC[
FINITE_UNION;
FINITE_SING]
THEN MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
MRESAL_TAC
CARD_UNION_LE[`{i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}`;`{(k' - 1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s}`][
FINITE_SING;Geomdetail.CARD_SING]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN REPLICATE_TAC (8) REMOVE_ASSUM_TAC
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`a=
CARD
({i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}
UNION
{(k' - 1 + p MOD
scs_k_v39 s) MOD
scs_k_v39 s})`
THEN ABBREV_TAC`b=
CARD
{i | i <
scs_k_v39 s /\
(&2 * h0 <
scs_b_v39 s i (SUC i) \/ &2 <
scs_a_v39 s i (SUC i))}`
THEN ABBREV_TAC`c=
CARD
(
IMAGE (\x. (x + p MOD
scs_k_v39 s) MOD
scs_k_v39 s)
{i | i < k' /\
(&2 * h0 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_bm_v39 s p q
else
scs_b_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)) \/
&2 <
(if {i MOD k', SUC i MOD k'} = {0, k' - 1}
then
scs_am_v39 s p q
else
scs_a_v39 s (i MOD k' + p MOD
scs_k_v39 s)
(SUC i MOD k' + p MOD
scs_k_v39 s)))})`
THEN MP_TAC(ARITH_RULE`c<=a /\ a<= b+1 /\ k'<
scs_k_v39 s/\ b +
scs_k_v39 s <= 6 ==> c+k' <=6`)
THEN RESA_TAC]);;
let SUC_MOD_NOT_EQ=prove(` 1<k ==> !i. ~(i MOD k= SUC i MOD k)`,
STRIP_TAC
THEN INDUCT_TAC
THENL[ MP_TAC(ARITH_RULE`1<k==> 0<k/\ SUC 0=1`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`0`;`k:num`]
THEN MRESAL_TAC
MOD_LT[`1`;`k:num`][
ADD1]
THEN ARITH_TAC;
POP_ASSUM MP_TAC
THEN REWRITE_TAC[
ADD1]
THEN MP_TAC(ARITH_RULE`1<k ==> ~(k=0)/\ SUC 0=1/\ i<= i+1 `)
THEN RESA_TAC
THEN STRIP_TAC
THEN MRESAL_TAC (GEN_ALL Hdplygy.MOD_EQ_MOD1)[`i+1:num`;`i:num`;`1`;`k:num`][
ADD_SYM]]);;
let IS_SCS_NOT_COLLINEAR_BBs_CASE_LE_PRIME_3=prove_by_refinement(
`3<
scs_k_v39 s/\
is_scs_v39 s
/\ vv
IN BBs_v39 s
==> ~collinear{
vec 0, vv (i MOD (
scs_k_v39 s)) ,vv ((SUC i) MOD (
scs_k_v39 s))}`,
[
REWRITE_TAC[Local_lemmas.collinear_fan22;aff;
AFFINE_HULL_2;
IN_ELIM_THM;VECTOR_ARITH`A %
vec 0+B=B`;cstab]
THEN STRIP_TAC
THEN MRESA_TAC( GEN_ALL
IS_SCS_POINT_IN_BBS_IS_NOT_0_LE_3)[`s:scs_v39`;`vv:num->real^3`;`i MOD
scs_k_v39 s`]
THEN RESA_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
IN]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[
scs_k_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
LET_DEF;
LET_END_DEF;
BBs_v39;ARITH_RULE`3<=3`;
mk_unadorned_v39;
CS_ADJ;
is_scs_v39]
THEN POP_ASSUM(fun th->
STRIP_TAC
THEN ASSUME_TAC
th)
THEN REPEAT STRIP_TAC;
MP_TAC(ARITH_RULE`3 <
scs_k_v39 s==> ~(
scs_k_v39 s <= 3)`)
THEN RESA_TAC;
ABBREV_TAC`k=
scs_k_v39 s`
THEN POP_ASSUM MP_TAC
THEN MP_TAC(ARITH_RULE`3<=k ==> ~(k=0)/\ 1<k`)
THEN RESA_TAC
THEN STRIP_TAC
THEN MRESA_TAC
DIVISION[`i:num`;`k:num`]
THEN MRESA_TAC
DIVISION[`SUC i:num`;`k:num`]
THEN POP_ASSUM MP_TAC
THEN MP_TAC
SUC_MOD_NOT_EQ
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`i:num`)
THEN STRIP_TAC
THEN REPLICATE_TAC (35-14) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`i MOD k`;`(SUC i) MOD k`][
scs_k_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
LET_DEF;
LET_END_DEF;
BBs_v39;ARITH_RULE`3<=3`;
mk_unadorned_v39;
CS_ADJ;
ARITH_RULE`1 MOD 3 =1 /\ 2 MOD 3=2/\ ~(1=2) /\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0/\ 1<3/\ 2<3`;
dist;VECTOR_ARITH`A- B%A=(&1-B)%A`;
NORM_MUL])
THEN REPLICATE_TAC (35-25)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN REPLICATE_TAC (34-22) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESAL_TAC
th[`i MOD k`;`(SUC i) MOD k`][
scs_k_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
LET_DEF;
LET_END_DEF;
BBs_v39;ARITH_RULE`3<=3`;
mk_unadorned_v39;
CS_ADJ;
ARITH_RULE`1 MOD 3 =1 /\ 2 MOD 3=2/\ ~(1=2) /\ SUC 1=2 /\ SUC 2=3/\ 3 MOD 3=0`;
dist;VECTOR_ARITH`A- B%A=(&1-B)%A`;
NORM_MUL])
THEN REPLICATE_TAC (35-20)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN SUBGOAL_THEN`(?x. x
IN (:num) /\ vv (i MOD k) = (vv:num->real^3) x)`ASSUME_TAC;
EXISTS_TAC`i MOD k`
THEN ASM_REWRITE_TAC[SET_RULE`(a:num)
IN (:num)`];
SUBGOAL_THEN`(?x. x
IN (:num) /\ vv ((SUC i) MOD k) = (vv:num->real^3) x)`ASSUME_TAC;
EXISTS_TAC`SUC(i) MOD k`
THEN ASM_REWRITE_TAC[SET_RULE`(a:num)
IN (:num)`];
REWRITE_TAC[
IMAGE;
SUBSET;
IN_ELIM_THM]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESAL1_TAC
th`(vv:num->real^3) (i MOD k)`[
ball_annulus;
IN_ELIM_THM;
DIFF;
cball;
ball;
dist;VECTOR_ARITH`
vec 0- B= --B`;
NORM_NEG;
NORM_MUL;REAL_ARITH`~(a<b) <=> b<=a`]
THEN MRESAL1_TAC
th`(vv:num->real^3) ((SUC i) MOD k)`[
ball_annulus;
IN_ELIM_THM;
DIFF;
cball;
ball;
dist;VECTOR_ARITH`
vec 0- B= --B`;
NORM_NEG;
NORM_MUL;REAL_ARITH`~(a<b) <=> b<=a`])
THEN MP_TAC(REAL_ARITH`&0<= &1-v\/ &0<= --( &1-v)`)
THEN STRIP_TAC;
MP_TAC(REAL_ARITH`&0<= v\/ &0<= --( v)`)
THEN STRIP_TAC;
MRESA1_TAC Trigonometry2.ABS_REFL `v:real`
THEN MRESA1_TAC Trigonometry2.ABS_REFL `&1- v:real`
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN MP_TAC(REAL_ARITH`&2 <=
scs_a_v39 s (i MOD k) ((SUC i) MOD k)/\
scs_a_v39 s (i MOD k) ((SUC i) MOD k)<= (&1 - v) *
norm (vv (i MOD k))
/\ &2 <= v *
norm (vv (i MOD k)) ==> &4<=
norm ((vv:num->real^3) (i MOD k))`)
THEN REPLICATE_TAC 6(POP_ASSUM MP_TAC)
THEN ASM_REWRITE_TAC[h0]
THEN REAL_ARITH_TAC;
MRESAL1_TAC Trigonometry2.ABS_REFL `-- v:real`[
REAL_ABS_NEG]
THEN MRESA1_TAC Trigonometry2.ABS_REFL `&1- v:real`
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th; ]
THEN REPEAT STRIP_TAC)
THEN MP_TAC(REAL_ARITH`&2 <= --v *
norm (vv (i MOD k))
/\ &2 <=
norm (vv (i MOD k)) ==> &4<=(&1- v) *norm ((vv:num->real^3) (i MOD k) )`)
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (45-15)(POP_ASSUM MP_TAC)
THEN POP_ASSUM (fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA1_TAC
th`i:num`
THEN ASM_REWRITE_TAC[])
THEN ASM_TAC
THEN REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
THEN REPLICATE_TAC (2)(POP_ASSUM MP_TAC)
THEN MRESAL_TAC (GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s i`][periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`SUC i:num`)
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC (GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s ((SUC i MOD k))`][periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`i:num`)
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN REPLICATE_TAC (44-33)(POP_ASSUM MP_TAC)
THEN POP_ASSUM (fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[cstab]
THEN REAL_ARITH_TAC;
MP_TAC(REAL_ARITH`&0<= v\/ &0<= --( v)`)
THEN STRIP_TAC;
MRESAL1_TAC Trigonometry2.ABS_REFL ` v:real`[
REAL_ABS_NEG]
THEN MRESAL1_TAC Trigonometry2.ABS_REFL `--(&1- v):real` [
REAL_ABS_NEG]
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th; ]
THEN REPEAT STRIP_TAC)
THEN MP_TAC(REAL_ARITH`&2 <=
norm (vv (i MOD k))
/\ &2 <=
scs_a_v39 s (i MOD k) ((SUC i) MOD k)/\
scs_a_v39 s (i MOD k) ((SUC i) MOD k)<= --(&1 - v) *
norm (vv (i MOD k)) ==> &4<= v *norm ((vv:num->real^3) (i MOD k))`)
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC 3(POP_ASSUM MP_TAC)
THEN POP_ASSUM (fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[h0]
THEN REAL_ARITH_TAC;
MRESAL1_TAC Trigonometry2.ABS_REFL ` --v:real`[
REAL_ABS_NEG]
THEN MRESAL1_TAC Trigonometry2.ABS_REFL `--(&1- v):real` [
REAL_ABS_NEG]
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th; ]
THEN REPEAT STRIP_TAC)
THEN MP_TAC(REAL_ARITH`&2 <=
norm (vv (i MOD k))
/\ &2 <= --v *
norm (vv (i MOD k)) ==> &4<= (&1-v) *norm ((vv:num->real^3) (i MOD k))`)
THEN ASM_REWRITE_TAC[REAL_ARITH`~(a<= b) <=> b<a`]
THEN REPLICATE_TAC (45-32)(POP_ASSUM MP_TAC)
THEN POP_ASSUM (fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN REPLICATE_TAC (44-32)(POP_ASSUM MP_TAC)
THEN POP_ASSUM (fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[]
THEN REAL_ARITH_TAC;]);;
let TECOXBMv2=prove_by_refinement(
`
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q
/\
dist(u ,w) <= cstab
/\ (vv:num->real^3) (p MOD (
scs_k_v39 s))=u
/\ (vv:num->real^3) (q MOD (
scs_k_v39 s))=w
/\
IMAGE (vv:num->real^3) (:num)=V/\
IMAGE (\i. {vv i,(vv:num->real^3) (SUC i)}) (:num) =E/\
IMAGE (\i. (vv i,(vv:num->real^3) (SUC i))) (:num)=
FF /\
BBs_v39 s vv
==>
(!x. x
IN FF ==> aff_gt {
vec 0} {u, w}
SUBSET wedge_in_fan_gt x E)`,
[
REWRITE_TAC[
dist]
THEN REPEAT STRIP_TAC
THEN MP_TAC
IS_SCS_STABLE_SYSTEM
THEN ASM_REWRITE_TAC[ARITH_RULE`3<4`]
THEN MP_TAC
SCS_K_PRIME_CASE_3
THEN RESA_TAC
THEN MP_TAC
SCS_K_LE_6
THEN RESA_TAC
THEN STRIP_TAC
THEN ABBREV_TAC`s1 =stable_sy((
scs_k_v39 s),(
scs_d_v39 s),(0..scs_k_v39 s - 1),
(
change_type_v3 (
scs_a_v39 s)),
(
change_type_v3 (
scs_b_v39 s)),
(
change_type_v2 (
scs_J_v39 s) (
scs_k_v39 s)),
(\i. (1 + i) MOD
scs_k_v39 s))`
THEN MRESA_TAC
stable_sy_explicit[`(
scs_k_v39 s)`;`(
scs_d_v39 s)`;`(0..scs_k_v39 s - 1)`;`
(
change_type_v3 (
scs_a_v39 s))`;`
(
change_type_v3 (
scs_b_v39 s))`;`
(
change_type_v2 (
scs_J_v39 s) (
scs_k_v39 s))`;`
(\i. (1 + i) MOD
scs_k_v39 s)`]
THEN MP_TAC(ARITH_RULE`3 <
scs_k_v39 s /\
scs_k_v39 s<=6 ==>
scs_k_v39 s=4 \/
scs_k_v39 s=5 \/
scs_k_v39 s=6`)
THEN RESA_TAC;
POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th)
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`v=
vector[(vv:num->real^3) 1;(vv:num->real^3) 2;(vv:num->real^3) 3;(vv:num->real^3) 0]:real^3^(2+2)`
THEN ABBREV_TAC`a=matvec (v:real^3^(2+2))`
THEN MP_TAC(INST_TYPE [`:2+2`,`:M`]
V_E_FF_IS_SCS_CASES_4)
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_4;
IN]
THEN STRIP_TAC
THEN MP_TAC(INST_TYPE [`:2+2`,`:M`]
IN_IS_SCS_CASE_4)
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_4;
IN]
THEN STRIP_TAC
THEN SUBGOAL_THEN`{u, w}
SUBSET V_SY (v:real^3^(2+2))` ASSUME_TAC;
REWRITE_TAC[SET_RULE`{a,b}
SUBSET A<=> a
IN A /\ b
IN A`]
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`p MOD
scs_k_v39 s`;`vv:num->real^3`]
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`q MOD
scs_k_v39 s`;`vv:num->real^3`];
SUBGOAL_THEN`&2 <=
norm (u - w:real^3)` ASSUME_TAC;
ASM_TAC
THEN REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
IN;
mk_unadorned_v39;
scs_diag]
THEN REPEAT RESA_TAC;
ASM_TAC
THEN ARITH_TAC;
REPLICATE_TAC (51-32)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th [`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`] [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3`;cstab;
dist])
THEN MRESAL_TAC
DIVISION[`p:num`;`
scs_k_v39 s`][ARITH_RULE`~(4=0)`]
THEN MRESAL_TAC
DIVISION[`q:num`;`
scs_k_v39 s`][ARITH_RULE`~(4=0)`]
THEN REPLICATE_TAC (56-15)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th [`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`] [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3`;cstab;
dist])
THEN REPLICATE_TAC (58-48)(POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC;
MP_TAC
DIAG_NOT_IN_EDGES
THEN ASM_REWRITE_TAC[
dist;
IN]
THEN STRIP_TAC
THEN MRESA_TAC(GEN_ALL (INST_TYPE [`:2+2`,`:M`]Tecoxbm.TECOXBM))[`
scs_d_v39 s`;
`(
change_type_v2 (
scs_J_v39 s) (
scs_k_v39 s))`;`(
scs_k_v39 s)`;`(
change_type_v3 (
scs_a_v39 s))`;`(
change_type_v3 (
scs_b_v39 s))`;`u:real^3`;`w:real^3`;`matvec (v:real^3^(2+2))`]
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"a"
THEN REWRITE_TAC[Dih2k_hypermap.VECMATS_MATVEC_ID]
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_4;
IN;
B_SY1;ARITH_RULE`2<4`]
THEN STRIP_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
IN]
THEN REPEAT RESA_TAC;
POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th)
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`v=
vector[(vv:num->real^3) 1;(vv:num->real^3) 2;(vv:num->real^3) 3;(vv:num->real^3) 4;(vv:num->real^3) 0]:real^3^(2+3)`
THEN ABBREV_TAC`a=matvec (v:real^3^(2+3))`
THEN MP_TAC(INST_TYPE [`:2+3`,`:M`]
V_E_FF_IS_SCS_CASES_5)
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_5;
IN]
THEN STRIP_TAC
THEN MP_TAC(INST_TYPE [`:2+3`,`:M`]
IN_IS_SCS_CASE_5)
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_5;
IN]
THEN STRIP_TAC
THEN SUBGOAL_THEN`{u, w}
SUBSET V_SY (v:real^3^(2+3))` ASSUME_TAC;
REWRITE_TAC[SET_RULE`{a,b}
SUBSET A<=> a
IN A /\ b
IN A`]
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`p MOD
scs_k_v39 s`;`vv:num->real^3`]
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`q MOD
scs_k_v39 s`;`vv:num->real^3`];
SUBGOAL_THEN`&2 <=
norm (u - w:real^3)` ASSUME_TAC;
ASM_TAC
THEN REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
IN;
mk_unadorned_v39;
scs_diag]
THEN REPEAT RESA_TAC;
ASM_TAC
THEN ARITH_TAC;
REPLICATE_TAC (50-31)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th [`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`] [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3`;cstab;
dist])
THEN MRESAL_TAC
DIVISION[`p:num`;`
scs_k_v39 s`][ARITH_RULE`~(5=0)`]
THEN MRESAL_TAC
DIVISION[`q:num`;`
scs_k_v39 s`][ARITH_RULE`~(5=0)`]
THEN REPLICATE_TAC (56-15)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th [`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`] [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3`;cstab;
dist])
THEN REPLICATE_TAC (57-48)(POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC;
MP_TAC
DIAG_NOT_IN_EDGES
THEN ASM_REWRITE_TAC[
dist;
IN]
THEN STRIP_TAC
THEN MRESA_TAC(GEN_ALL (INST_TYPE [`:2+3`,`:M`]Tecoxbm.TECOXBM))[`
scs_d_v39 s`;
`(
change_type_v2 (
scs_J_v39 s) (
scs_k_v39 s))`;`(
scs_k_v39 s)`;`(
change_type_v3 (
scs_a_v39 s))`;`(
change_type_v3 (
scs_b_v39 s))`;`u:real^3`;`w:real^3`;`matvec (v:real^3^(2+3))`]
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"a"
THEN REWRITE_TAC[Dih2k_hypermap.VECMATS_MATVEC_ID]
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_5;
IN;
B_SY1;ARITH_RULE`2<5`]
THEN STRIP_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
IN]
THEN REPEAT RESA_TAC;
POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th)
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`v=
vector[(vv:num->real^3) 1;(vv:num->real^3) 2;(vv:num->real^3) 3;(vv:num->real^3) 4;(vv:num->real^3) 5;(vv:num->real^3) 0]:real^3^(3+3)`
THEN ABBREV_TAC`a=matvec (v:real^3^(3+3))`
THEN MP_TAC(INST_TYPE [`:3+3`,`:M`]
V_E_FF_IS_SCS_CASES_6)
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_6;
IN]
THEN STRIP_TAC
THEN MP_TAC(INST_TYPE [`:3+3`,`:M`]
IN_IS_SCS_CASE_6)
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_6;
IN]
THEN STRIP_TAC
THEN SUBGOAL_THEN`{u, w}
SUBSET V_SY (v:real^3^(3+3))` ASSUME_TAC;
REWRITE_TAC[SET_RULE`{a,b}
SUBSET A<=> a
IN A /\ b
IN A`]
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`p MOD
scs_k_v39 s`;`vv:num->real^3`]
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`q MOD
scs_k_v39 s`;`vv:num->real^3`];
SUBGOAL_THEN`&2 <=
norm (u - w:real^3)` ASSUME_TAC;
ASM_TAC
THEN REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
IN;
mk_unadorned_v39;
scs_diag]
THEN REPEAT RESA_TAC;
ASM_TAC
THEN ARITH_TAC;
REPLICATE_TAC (50-31)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th [`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`] [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3`;cstab;
dist])
THEN MRESAL_TAC
DIVISION[`p:num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN MRESAL_TAC
DIVISION[`q:num`;`
scs_k_v39 s`][ARITH_RULE`~(6=0)`]
THEN REPLICATE_TAC (56-15)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th [`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`] [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3`;cstab;
dist])
THEN REPLICATE_TAC (57-48)(POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC;
MP_TAC
DIAG_NOT_IN_EDGES
THEN ASM_REWRITE_TAC[
dist;
IN]
THEN STRIP_TAC
THEN MRESA_TAC(GEN_ALL (INST_TYPE [`:3+3`,`:M`]Tecoxbm.TECOXBM))[`
scs_d_v39 s`;
`(
change_type_v2 (
scs_J_v39 s) (
scs_k_v39 s))`;`(
scs_k_v39 s)`;`(
change_type_v3 (
scs_a_v39 s))`;`(
change_type_v3 (
scs_b_v39 s))`;`u:real^3`;`w:real^3`;`matvec (v:real^3^(3+3))`]
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC"a"
THEN REWRITE_TAC[Dih2k_hypermap.VECMATS_MATVEC_ID]
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_6;
IN;
B_SY1;ARITH_RULE`2<6`]
THEN STRIP_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
IN]
THEN REPEAT RESA_TAC;
]);;
let W_EW_K_SCS_ADD_P=prove_by_refinement(
`
scs_half_slice_v39 s p q d' mkj= s' /\
scs_k_v39 s'=k' /\
scs_k_v39 s =k /\
(vv:num->real^3) (p MOD k)=u /\
(vv:num->real^3) (q MOD k)=w /\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q /\
BBs_v39 s vv
==> vv (k' - 1 + p MOD k) = w`,
[
ABBREV_TAC`V=
IMAGE (vv:num->real^3) (:num)`
THEN ABBREV_TAC`E=
IMAGE (\i. {vv i,(vv:num->real^3) (SUC i)}) (:num)`
THEN ABBREV_TAC`
FF=
IMAGE (\i. (vv i,(vv:num->real^3) (SUC i))) (:num)`
THEN REPEAT STRIP_TAC
THEN MP_TAC
SCS_K_PRIME_CASE_3
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<
scs_k_v39 s ==> ~(
scs_k_v39 s<=3)`)
THEN RESA_TAC
THEN POP_ASSUM (fun th->
REPLICATE_TAC 7 (POP_ASSUM MP_TAC)
THEN ASSUME_TAC
th)
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit]
THEN REPEAT RESA_TAC
THEN REPLICATE_TAC (1) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN ASSUME_TAC
th
THEN MP_TAC
th
THEN REWRITE_TAC[
convex_local_fan]
THEN STRIP_TAC)
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`p MOD
scs_k_v39 s`;`vv:num->real^3`]
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`q MOD
scs_k_v39 s`;`vv:num->real^3`]
THEN MRESA_TAC (GEN_ALL Local_lemmas1.POINT_PRESENTED_IN_RHOND1)[`E:(real^3->bool)->bool`;`V:real^3->bool`;`FF:real^3#real^3->bool`;`u:real^3`;`w:real^3`]
THEN SUBGOAL_THEN`(!i.
scs_b_v39 s i (SUC i) <= cstab)`ASSUME_TAC;
REPLICATE_TAC (45-27)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
REPLICATE_TAC (43-3)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN EXPAND_TAC"s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
THEN ASM_SIMP_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`p:num`;`q:num`;`d':real`;`
scs_J_v39 s' p q`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;
scs_half_slice_v39]
THEN ASM_SIMP_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`p MOD k< k==> 0< q + 1 + k - p MOD k`)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k/\ 3<= (q + 1 + k - p MOD k) MOD k /\ p MOD k<k ==>
(((q + 1 + k - p MOD k) MOD k - 1 + p MOD k) + k)
=(q + 1 + k - p MOD k) MOD k +(k- 1 + p MOD k)
`)
THEN RESA_TAC
THEN REPLICATE_TAC (52-32)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN REWRITE_TAC[periodic]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`(q + 1 + k - p MOD k) MOD k - 1 + p MOD k`
THEN POP_ASSUM(fun th1-> REWRITE_TAC[SYM
th1])
THEN ASSUME_TAC
th)
THEN MRESA_TAC
MOD_MOD_REFL[`(q + 1 + k - p MOD k)`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`(q + 1 + k - p MOD k) :num`;`k - 1 + p MOD k:num`;`
scs_k_v39 s`]
THEN MP_TAC(ARITH_RULE`p MOD k< k /\ 3<=k==> (q + 1 + k - p MOD k) + k - 1 + p MOD k =q+ 2*k`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`q :num` THEN MRESA1_TAC
th`q +2*k:num` THEN MRESA1_TAC
th `((q + 1 + k - p MOD k) MOD k + k - 1 + p MOD k):num`)
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_ADD_MOD[`(q + 1 + k - p MOD k) MOD k:num`;`k - 1 + p MOD k:num`;`
scs_k_v39 s`]
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN REPLICATE_TAC (5)(REMOVE_ASSUM_TAC)
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`q:num` THEN MRESAL1_TAC
th`q+k:num`[ARITH_RULE`(A+B)+B=A+2*B`])]);;
let CARD_V_EQ_SCS_K=prove_by_refinement(
`
scs_k_v39 s =k /\
IMAGE (vv:num->real^3) (:num)=V/\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q /\
BBs_v39 s vv
==>
CARD V=k`,
[
REPEAT STRIP_TAC
THEN MP_TAC
IS_SCS_STABLE_SYSTEM
THEN ASM_REWRITE_TAC[ARITH_RULE`3<4`]
THEN MP_TAC
SCS_K_PRIME_CASE_3
THEN RESA_TAC
THEN MP_TAC
SCS_K_LE_6
THEN RESA_TAC
THEN STRIP_TAC
THEN ABBREV_TAC`s1 =stable_sy((
scs_k_v39 s),(
scs_d_v39 s),(0..scs_k_v39 s - 1),
(
change_type_v3 (
scs_a_v39 s)),
(
change_type_v3 (
scs_b_v39 s)),
(
change_type_v2 (
scs_J_v39 s) (
scs_k_v39 s)),
(\i. (1 + i) MOD
scs_k_v39 s))`
THEN MRESA_TAC
stable_sy_explicit[`(
scs_k_v39 s)`;`(
scs_d_v39 s)`;`(0..scs_k_v39 s - 1)`;`
(
change_type_v3 (
scs_a_v39 s))`;`
(
change_type_v3 (
scs_b_v39 s))`;`
(
change_type_v2 (
scs_J_v39 s) (
scs_k_v39 s))`;`
(\i. (1 + i) MOD
scs_k_v39 s)`]
THEN MP_TAC(ARITH_RULE`3 <
scs_k_v39 s /\
scs_k_v39 s<=6 ==>
scs_k_v39 s=4 \/
scs_k_v39 s=5 \/
scs_k_v39 s=6`)
THEN RESA_TAC;
POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th)
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`v=
vector[(vv:num->real^3) 1;(vv:num->real^3) 2;(vv:num->real^3) 3;(vv:num->real^3) 0]:real^3^(2+2)`
THEN ABBREV_TAC`a=matvec (v:real^3^(2+2))`
THEN MP_TAC(INST_TYPE [`:2+2`,`:M`]
V_E_FF_IS_SCS_CASES_4)
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_4;
IN]
THEN STRIP_TAC
THEN MRESAL_TAC (INST_TYPE [`:2+2`,`:M`]
VECTOR_3_4)[`(vv:num->real^3) 1`;`(vv:num->real^3) 2`;`(vv:num->real^3) 3`;`(vv:num->real^3) 0`][ARITH_RULE`1<=4/\ 2<=4/\ 3<=4/\ 4<=4/\ 4 MOD 4= 0 /\ 1 MOD 4= 1 /\ 2 MOD 4= 2/\ 3 MOD 4= 3/\ ~(2=1)
/\ SUC 4 MOD 4= 1 /\ SUC 1 MOD 4= 2 /\ SUC 2 MOD 4= 3/\ SUC 3 MOD 4= 0/\ ~(3=1)/\ ~(1=0)`;Basics.DIMINDEX_4]
THEN EXPAND_TAC "V"
THEN REWRITE_TAC[
V_SY;
rows;Basics.DIMINDEX_4;ARITH_RULE`1<=i /\ i<=4 <=> i=1\/ i=2\/ i=3 \/ i=4`;SET_RULE`{
row i v | i = 1 \/ i = 2 \/ i = 3 \/ i = 4}={
row 1 v,
row 2 v,
row 3 v,
row 4 v}`]
THEN ASM_REWRITE_TAC[]
THEN MP_TAC
VV_INJ
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MRESAL_TAC
th[`1`;`2`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<4/\ 1<4/\ 2<4/\ 3<4`]
THEN MRESAL_TAC
th[`1`;`3`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<4/\ 1<4/\ 2<4/\ 3<4`]
THEN MRESAL_TAC
th[`1`;`0`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<4/\ 1<4/\ 2<4/\ 3<4`]
THEN MRESAL_TAC
th[`2`;`0`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<4/\ 1<4/\ 2<4/\ 3<4`]
THEN MRESAL_TAC
th[`3`;`0`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<4/\ 1<4/\ 2<4/\ 3<4`]
THEN MRESAL_TAC
th[`2`;`3`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<4/\ 1<4/\ 2<4/\ 3<4`])
THEN SIMP_TAC[
HAS_SIZE;
CARD_CLAUSES;
FINITE_INSERT;
FINITE_EMPTY;
IN_INSERT;
NOT_IN_EMPTY]
THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC;
POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th)
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`v=
vector[(vv:num->real^3) 1;(vv:num->real^3) 2;(vv:num->real^3) 3;(vv:num->real^3) 4;(vv:num->real^3) 0]:real^3^(2+3)`
THEN ABBREV_TAC`a=matvec (v:real^3^(2+3))`
THEN MP_TAC(INST_TYPE [`:2+3`,`:M`]
V_E_FF_IS_SCS_CASES_5)
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_5;
IN]
THEN STRIP_TAC
THEN MRESAL_TAC (INST_TYPE [`:2+3`,`:M`]
VECTOR_3_5)[`(vv:num->real^3) 1`;`(vv:num->real^3) 2`;`(vv:num->real^3) 3`;`(vv:num->real^3) 4`;`(vv:num->real^3) 0`;][ARITH_RULE`1<=5/\ 2<=5/\ 3<=5/\ 4<=5/\ 5<=5/\ 0 MOD 5= 0 /\ 1 MOD 5= 1 /\ 2 MOD 5= 2/\ 3 MOD 5= 3 /\ 4 MOD 5= 4 /\ 5 MOD 5=0 /\ ~(2=1)
/\ SUC 0 MOD 5= 1 /\ SUC 1 MOD 5= 2 /\ SUC 2 MOD 5= 3/\ SUC 3 MOD 5= 4
/\ SUC 4 MOD 5= 0 /\ SUC 5 MOD 5 = 1 /\ ~(3=1)/\ ~(1=0)`;SET_RULE`(a:num)
IN (:num)`;Basics.DIMINDEX_5]
THEN EXPAND_TAC "V"
THEN REWRITE_TAC[
V_SY;
rows;Basics.DIMINDEX_5;ARITH_RULE`1<=i /\ i<=5 <=> i=1\/ i=2\/ i=3 \/ i=4 \/ i=5`;SET_RULE`{
row i v | i = 1 \/ i = 2 \/ i = 3 \/ i = 4\/ i=5}={
row 1 v,
row 2 v,
row 3 v,
row 4 v,
row 5 v}`]
THEN ASM_REWRITE_TAC[]
THEN MP_TAC
VV_INJ
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MRESAL_TAC
th[`1`;`2`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<5/\ 1<5/\ 2<5/\ 3<5/\ 4<5 /\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)`]
THEN MRESAL_TAC
th[`1`;`3`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<5/\ 1<5/\ 2<5/\ 3<5/\ 4<5 /\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)`]
THEN MRESAL_TAC
th[`1`;`0`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<5/\ 1<5/\ 2<5/\ 3<5/\ 4<5 /\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)`]
THEN MRESAL_TAC
th[`2`;`0`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<5/\ 1<5/\ 2<5/\ 3<5/\ 4<5 /\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)`]
THEN MRESAL_TAC
th[`3`;`0`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<5/\ 1<5/\ 2<5/\ 3<5/\ 4<5 /\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)`]
THEN MRESAL_TAC
th[`2`;`3`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<5/\ 1<5/\ 2<5/\ 3<5/\ 4<5 /\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)`]
THEN MRESAL_TAC
th[`1`;`4`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<5/\ 1<5/\ 2<5/\ 3<5/\ 4<5 /\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)`]
THEN MRESAL_TAC
th[`2`;`4`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<5/\ 1<5/\ 2<5/\ 3<5/\ 4<5 /\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)`]
THEN MRESAL_TAC
th[`3`;`4`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<5/\ 1<5/\ 2<5/\ 3<5/\ 4<5 /\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)`]
THEN MRESAL_TAC
th[`0`;`4`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ 0<5/\ 1<5/\ 2<5/\ 3<5/\ 4<5 /\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)`])
THEN SIMP_TAC[
HAS_SIZE;
CARD_CLAUSES;
FINITE_INSERT;
FINITE_EMPTY;
IN_INSERT;
NOT_IN_EMPTY]
THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC;
POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th)
THEN REPEAT STRIP_TAC
THEN ABBREV_TAC`v=
vector[(vv:num->real^3) 1;(vv:num->real^3) 2;(vv:num->real^3) 3;(vv:num->real^3) 4;(vv:num->real^3) 5;(vv:num->real^3) 0]:real^3^(3+3)`
THEN ABBREV_TAC`a=matvec (v:real^3^(3+3))`
THEN MP_TAC(INST_TYPE [`:3+3`,`:M`]
V_E_FF_IS_SCS_CASES_6)
THEN ASM_REWRITE_TAC[Basics.DIMINDEX_6;
IN]
THEN STRIP_TAC
THEN MRESAL_TAC (INST_TYPE [`:3+3`,`:M`]
VECTOR_3_6)[`(vv:num->real^3) 1`;`(vv:num->real^3) 2`;`(vv:num->real^3) 3`;`(vv:num->real^3) 4`;`(vv:num->real^3) 5`;`(vv:num->real^3) 0`][ARITH_RULE`1<=6/\ 2<=6/\ 3<=6/\ 4<=6/\ 5<=6 /\ 6<=6 /\ 0 MOD 6= 0 /\ 1 MOD 6= 1 /\ 2 MOD 6= 2/\ 3 MOD 6= 3 /\ 4 MOD 6= 4 /\ 5 MOD 6=5 /\ 6 MOD 6=0 /\ ~(2=1)
/\ SUC 0 MOD 6= 1 /\ SUC 1 MOD 6= 2 /\ SUC 2 MOD 6= 3/\ SUC 3 MOD 6= 4
/\ SUC 4 MOD 6= 5 /\ SUC 5 MOD 6 = 0 /\ SUC 6 MOD 6 =1 /\ ~(3=1)/\ ~(1=0)`;Basics.DIMINDEX_6]
THEN EXPAND_TAC "V"
THEN REWRITE_TAC[
V_SY;
rows;Basics.DIMINDEX_6;ARITH_RULE`1<=i /\ i<=6 <=> i=1\/ i=2\/ i=3 \/ i=4 \/ i=5\/ i=6`;SET_RULE`{
row i v | i = 1 \/ i = 2 \/ i = 3 \/ i = 4\/ i=5 \/ i=6}={
row 1 v,
row 2 v,
row 3 v,
row 4 v,
row 5 v,
row 6 v}`]
THEN ASM_REWRITE_TAC[]
THEN MP_TAC
VV_INJ
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MRESAL_TAC
th[`1`;`2`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0) /\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`1`;`3`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`1`;`0`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`2`;`0`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`3`;`0`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`2`;`3`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`1`;`4`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`2`;`4`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`3`;`4`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`0`;`4`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`1`;`5`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`2`;`5`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`3`;`5`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`0`;`5`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`]
THEN MRESAL_TAC
th[`4`;`5`][ARITH_RULE`~(1=2)/\ ~(1=0) /\ ~(1=3)/\ ~(2=3)/\ ~(2=0)/\ ~(3=0)/\ ~(3=4)/\ ~(1=4)/\ ~(2=4)/\ ~(0=4)/\ 0<6/\ 1<6/\ 2<6/\ 3<6/\ 4<6/\ 5<6 /\ ~(0=5) /\ ~(1=5) /\ ~(2=5)/\ ~(3=5)/\ ~(4=5)`])
THEN SIMP_TAC[
HAS_SIZE;
CARD_CLAUSES;
FINITE_INSERT;
FINITE_EMPTY;
IN_INSERT;
NOT_IN_EMPTY]
THEN ASM_REWRITE_TAC[]
THEN ARITH_TAC]);;
let QKNVMLB1_LE4F= prove_by_refinement(
` s' =
scs_half_slice_v39 s p q d' mkj /\
s'' =
scs_half_slice_v39 s q p d'' mkj /\
vv' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s')+p MOD (
scs_k_v39 s)))
/\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q
/\
d'< #0.9 /\
(4<
scs_k_v39 s /\
scs_bm_v39 s p q <= cstab)
/\
MMs_v39 s vv
==>
BBs_v39 s' vv'`,
[
STRIP_TAC
THEN POP_ASSUM MP_TAC
(*THEN MP_TAC SCS_HALF_SLICE_IS_SCS
THEN RESA_TAC*)
THEN MRESA_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`p:num`;`q:num`;`d':real`;`
scs_J_v39 s' p q`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;]
THEN MP_TAC
SCS_K_PRIME_CASE_3
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<
scs_k_v39 s ==> ~(
scs_k_v39 s<=3)`)
THEN RESA_TAC
THEN REPLICATE_TAC 7 (POP_ASSUM MP_TAC)
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit]
THEN ABBREV_TAC`k'=(q + 1 +
scs_k_v39 s - p MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN ABBREV_TAC`k=
scs_k_v39 s`
THEN REPEAT RESA_TAC;
ASM_TAC
THEN REWRITE_TAC[
IMAGE;
SUBSET;
IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (51-38) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN EXISTS_TAC`((x') MOD k'+p MOD k)`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN (:num)`];
REWRITE_TAC[periodic;]
THEN GEN_TAC
THEN MRESAL_TAC
MOD_MULT_ADD[`1:num`;`k':num`;`i:num`][ARITH_RULE`1 * k' + i = (i+k')`]
;
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC 2 (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`((i) MOD k'+p MOD k)`;`((j) MOD k'+p MOD k)`])
THEN STRIP_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`
scs_am_v39 s ((i) MOD k'+p MOD k) ((j) MOD k'+p MOD k)
=
scs_am_v39 s p q`ASSUME_TAC;
MP_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1}
==> (i MOD k' =0 /\ j MOD k'=k'-1)\/ (i MOD k' =k'-1 /\ j MOD k'=0)`)
THEN RESA_TAC
THEN REWRITE_TAC[ARITH_RULE`0+p=p`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0) /\ 0<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `k'+(k-1)+p MOD k:num`)
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC
MOD_LT[`k':num`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`k':num`;`k-1+p MOD k`;`k:num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC
MOD_ADD_MOD[`q + 1 + k - p MOD k`;`k-1+p MOD k`;`k:num`]
THEN EXPAND_TAC"k'"
THEN MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`p MOD k< k==>(((q + 1 + k - p MOD k) + k - 1 + p MOD k))
=(q+ 2*k) `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_EQ[`q+2*k`;`q:num`;`k:num`;`2`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `q:num`)
THEN REPLICATE_TAC (60-14) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`0<k/\ 3<= k'==> (k' - 1 + p MOD k) + k =k' + k - 1 + p MOD k`)
THEN RESA_TAC
THEN MRESA_TAC
th[`k'-1+p MOD k`;`p:num`]
THEN REMOVE_ASSUM_TAC
THEN ASSUME_TAC
th)
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s (k' - 1 + p MOD k)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `p:num`)
;
POP_ASSUM(fun th1->ASM_REWRITE_TAC[SYM
th1])
;
ASM_REWRITE_TAC[];
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC 1 (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`((i) MOD k'+p MOD k)`;`((j) MOD k'+p MOD k)`])
THEN STRIP_TAC
THEN SUBGOAL_THEN`
scs_bm_v39 s ((i) MOD k'+p MOD k) ((j) MOD k'+p MOD k)
=
scs_bm_v39 s p q`ASSUME_TAC;
MP_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1}
==> (i MOD k' =0 /\ j MOD k'=k'-1)\/ (i MOD k' =k'-1 /\ j MOD k'=0)`)
THEN RESA_TAC
THEN REWRITE_TAC[ARITH_RULE`0+p=p`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0) /\ 0<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_bm_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `k'+(k-1)+p MOD k:num`)
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC
MOD_LT[`k':num`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`k':num`;`k-1+p MOD k`;`k:num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC
MOD_ADD_MOD[`q + 1 + k - p MOD k`;`k-1+p MOD k`;`k:num`]
THEN EXPAND_TAC"k'"
THEN MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`p MOD k< k==>(((q + 1 + k - p MOD k) + k - 1 + p MOD k))
=(q+ 2*k) `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_EQ[`q+2*k`;`q:num`;`k:num`;`2`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_bm_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `q:num`)
THEN REPLICATE_TAC (60-15) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`0<k/\ 3<= k'==> (k' - 1 + p MOD k) + k =k' + k - 1 + p MOD k`)
THEN RESA_TAC
THEN MRESA_TAC
th[`k'-1+p MOD k`;`p:num`]
THEN REMOVE_ASSUM_TAC
THEN ASSUME_TAC
th)
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_bm_v39 s (k' - 1 + p MOD k)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `p:num`)
;
POP_ASSUM(fun th1->ASM_REWRITE_TAC[SYM
th1])
;
ASM_REWRITE_TAC[];
ABBREV_TAC`u= (vv:num->real^3) (p MOD k)`
THEN ABBREV_TAC `w= (vv:num->real^3) (q MOD k)`
THEN ABBREV_TAC`V=
IMAGE (vv:num->real^3) (:num)`
THEN ABBREV_TAC`E=
IMAGE (\i. {vv i,(vv:num->real^3) (SUC i)}) (:num)`
THEN ABBREV_TAC`
FF=
IMAGE (\i. (vv i,(vv:num->real^3) (SUC i))) (:num)`
THEN MP_TAC(ARITH_RULE`3<=k'==> k'<=3 \/ ~(k'<=3)`)
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`p MOD
scs_k_v39 s`;`vv:num->real^3`]
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`q MOD
scs_k_v39 s`;`vv:num->real^3`]
THEN SUBGOAL_THEN`(!i.
scs_b_v39 s i (SUC i) <= cstab)`ASSUME_TAC
;
REPLICATE_TAC (57-23)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
SUBGOAL_THEN`
norm (u - w:real^3) <= cstab`ASSUME_TAC
;
MP_TAC(ARITH_RULE`3<=k==> ~(k=0)`)
THEN RESA_TAC
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_bm_v39 s (p)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `q:num`)
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_bm_v39 s (q MOD k)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `p:num`)
THEN REPLICATE_TAC (61-49)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th [`p MOD
scs_k_v39 s`;`q MOD
scs_k_v39 s`] [ARITH_RULE`SUC 0=1/\ SUC 1=2/\ SUC 2=3`;
dist])
THEN POP_ASSUM MP_TAC
THEN REPLICATE_TAC (60-32)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC
;
MP_TAC
NOT_EQ_DIAG
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN STRIP_TAC
THEN MP_TAC
TECOXBMv2
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN STRIP_TAC
THEN MRESA_TAC(GEN_ALL Local_lemmas1.EJRCFJD)[`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))`;`(E:(real^3->bool)->bool)`;`w:real^3`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`V:real^3->bool`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (w,
rho_node1 FF w)`;`FF:real^3#real^3->bool`;`u:real^3`]
THEN MP_TAC
F_PRIME_EQ_F_vv
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN STRIP_TAC
THEN MP_TAC
E_PRIME_EQ_E_vv
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN STRIP_TAC
THEN MP_TAC
V_PRIME_EQ_V_vv
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN STRIP_TAC
THEN ASM_REWRITE_TAC[]]);;
let QKNVMLB1_EQ4F= prove_by_refinement(
` s' =
scs_half_slice_v39 s p q d' mkj /\
s'' =
scs_half_slice_v39 s q p d'' mkj /\
vv' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s')+p MOD (
scs_k_v39 s)))
/\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q
/\
d'< #0.9 /\
(4=
scs_k_v39 s /\
scs_bm_v39 s p q <= &4)
/\
MMs_v39 s vv
==>
BBs_v39 s' vv'`,
[
STRIP_TAC
THEN POP_ASSUM MP_TAC
(*THEN MP_TAC SCS_HALF_SLICE_IS_SCS
THEN RESA_TAC*)
THEN MRESA_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`p:num`;`q:num`;`d':real`;`
scs_J_v39 s' p q`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;]
THEN MP_TAC
SCS_K_PRIME_CASE_3
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<
scs_k_v39 s ==> ~(
scs_k_v39 s<=3)`)
THEN RESA_TAC
THEN REPLICATE_TAC 7 (POP_ASSUM MP_TAC)
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
scs_half_slice_v39;
is_scs_v39;
scs_diag;scs_v39_explicit]
THEN ABBREV_TAC`k'=(q + 1 +
scs_k_v39 s - p MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN ABBREV_TAC`k=
scs_k_v39 s`
THEN REPEAT RESA_TAC;
ASM_TAC
THEN REWRITE_TAC[
IMAGE;
SUBSET;
IN_ELIM_THM]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (51-38) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN EXISTS_TAC`((x') MOD k'+p MOD k)`
THEN ASM_REWRITE_TAC[SET_RULE`(a:num)
IN (:num)`];
REWRITE_TAC[periodic;]
THEN GEN_TAC
THEN MRESAL_TAC
MOD_MULT_ADD[`1:num`;`k':num`;`i:num`][ARITH_RULE`1 * k' + i = (i+k')`];
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC 2 (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`((i) MOD k'+p MOD k)`;`((j) MOD k'+p MOD k)`])
THEN STRIP_TAC
THEN STRIP_TAC
THEN SUBGOAL_THEN`
scs_am_v39 s ((i) MOD k'+p MOD k) ((j) MOD k'+p MOD k)
=
scs_am_v39 s p q`ASSUME_TAC;
MP_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1}
==> (i MOD k' =0 /\ j MOD k'=k'-1)\/ (i MOD k' =k'-1 /\ j MOD k'=0)`)
THEN RESA_TAC
THEN REWRITE_TAC[ARITH_RULE`0+p=p`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0) /\ 0<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `k'+(k-1)+p MOD k:num`)
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC
MOD_LT[`k':num`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`k':num`;`k-1+p MOD k`;`k:num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC
MOD_ADD_MOD[`q + 1 + k - p MOD k`;`k-1+p MOD k`;`k:num`]
THEN EXPAND_TAC"k'"
THEN MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`p MOD k< k==>(((q + 1 + k - p MOD k) + k - 1 + p MOD k))
=(q+ 2*k) `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_EQ[`q+2*k`;`q:num`;`k:num`;`2`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `q:num`)
THEN REPLICATE_TAC (60-14) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`0<k/\ 3<= k'==> (k' - 1 + p MOD k) + k =k' + k - 1 + p MOD k`)
THEN RESA_TAC
THEN MRESA_TAC
th[`k'-1+p MOD k`;`p:num`]
THEN REMOVE_ASSUM_TAC
THEN ASSUME_TAC
th)
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_am_v39 s (k' - 1 + p MOD k)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `p:num`);
POP_ASSUM(fun th1->ASM_REWRITE_TAC[SYM
th1]);
ASM_REWRITE_TAC[];
DISJ_CASES_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k'-1} \/ ~({i MOD k', j MOD k'} = {0, k'-1})`);
ASM_REWRITE_TAC[]
THEN REPLICATE_TAC 1 (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`((i) MOD k'+p MOD k)`;`((j) MOD k'+p MOD k)`])
THEN STRIP_TAC
THEN SUBGOAL_THEN`
scs_bm_v39 s ((i) MOD k'+p MOD k) ((j) MOD k'+p MOD k)
=
scs_bm_v39 s p q`ASSUME_TAC;
MP_TAC(SET_RULE`{i MOD k', j MOD k'} = {0, k' - 1}
==> (i MOD k' =0 /\ j MOD k'=k'-1)\/ (i MOD k' =k'-1 /\ j MOD k'=0)`)
THEN RESA_TAC
THEN REWRITE_TAC[ARITH_RULE`0+p=p`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<=
scs_k_v39 s ==> ~(
scs_k_v39 s = 0) /\ 0<
scs_k_v39 s`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_bm_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `k'+(k-1)+p MOD k:num`)
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC
MOD_LT[`k':num`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`k':num`;`k-1+p MOD k`;`k:num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN REMOVE_ASSUM_TAC
THEN MRESA_TAC
MOD_ADD_MOD[`q + 1 + k - p MOD k`;`k-1+p MOD k`;`k:num`]
THEN EXPAND_TAC"k'"
THEN MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`p MOD k< k==>(((q + 1 + k - p MOD k) + k - 1 + p MOD k))
=(q+ 2*k) `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_EQ[`q+2*k`;`q:num`;`k:num`;`2`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_bm_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `q:num`)
THEN REPLICATE_TAC (60-15) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`0<k/\ 3<= k'==> (k' - 1 + p MOD k) + k =k' + k - 1 + p MOD k`)
THEN RESA_TAC
THEN MRESA_TAC
th[`k'-1+p MOD k`;`p:num`]
THEN REMOVE_ASSUM_TAC
THEN ASSUME_TAC
th)
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_bm_v39 s (k' - 1 + p MOD k)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `p:num`);
POP_ASSUM(fun th1->ASM_REWRITE_TAC[SYM
th1]);
ASM_REWRITE_TAC[];
MP_TAC
SCS_K_PRIME_CASE_4
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN RESA_TAC
THEN REPLICATE_TAC (50-31)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC THEN REWRITE_TAC[SYM
th]
THEN REPEAT RESA_TAC)
THEN POP_ASSUM MP_TAC
THEN SUBGOAL_THEN`(!i.
scs_b_v39 s i (SUC i) <= cstab)`ASSUME_TAC;
REPLICATE_TAC (48-23)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
THEN ARITH_TAC]);;
let INTER_SLICE_SCS_EMPTY1=prove_by_refinement(
`
scs_half_slice_v39 s p q d' mkj =s'/\
scs_half_slice_v39 s q p d'' mkj =s'' /\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q /\
p MOD
scs_k_v39 s <q MOD
scs_k_v39 s
==>
{(i + p MOD
scs_k_v39 s) MOD
scs_k_v39 s | i <
scs_k_v39 s'-1 }
INTER {(i + q MOD
scs_k_v39 s)MOD
scs_k_v39 s | i <
scs_k_v39 s''-1 }={}
`,
[
REWRITE_TAC[SET_RULE`A
INTER B={}<=> ~(?a. a
IN A
INTER B)`;
INTER]
THEN STRIP_TAC
THEN EXPAND_TAC"s'"
THEN EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39;
IN_ELIM_THM]
THEN ABBREV_TAC`k'=(q + 1 +
scs_k_v39 s - p MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN ABBREV_TAC`k''=(p + 1 +
scs_k_v39 s - q MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN ABBREV_TAC`k=
scs_k_v39 s`
THEN REPEAT STRIP_TAC
THEN POP_ASSUM (fun th->ASM_TAC
THEN REWRITE_TAC[
th]
THEN REPEAT STRIP_TAC)
THEN MRESAL_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`p:num`;`q:num`;`d':real`;`
scs_J_v39 s' p q`][
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39;
IN_ELIM_THM]
THEN SUBGOAL_THEN`(k'-1 +p MOD k) <= q MOD k `ASSUME_TAC;
MP_TAC(ARITH_RULE`3<= k' /\ k'<k /\ p MOD k< q MOD k ==> ~(k=0)/\ (q +k -p MOD k)+1=q + 1 + k - p MOD k/\ 1<k `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LE[`q:num`;`k:num`]
THEN MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`q MOD k <= q /\ p MOD k< q MOD k/\ p MOD k< k ==> q+k- p MOD k= 1*k + (q- p MOD k)`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_MULT_ADD[`1`;`k:num`;`q-p MOD k`]
THEN MRESA_TAC
MOD_LT[`1`;`k:num`]
THEN MRESA_TAC
DIVISION[`q:num`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`q+k-p MOD k`;`1`;`k:num`]
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[
th])
THEN MP_TAC(ARITH_RULE`q MOD k <= q /\ p MOD k< q MOD k/\ p MOD k< k ==> (q DIV k * k + q MOD k)- p MOD k= (q DIV k)*k + (q MOD k- p MOD k)`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_MULT_ADD[`q DIV k `;`k:num`;`q MOD k-p MOD k`]
THEN MRESA_TAC
DIVISION[`q:num`;`k:num`]
THEN MP_TAC(ARITH_RULE` p MOD k< q MOD k/\ q MOD k< k ==> q MOD k- p MOD k<k`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`q MOD k - p MOD k`;`k:num`]
THEN STRIP_TAC
THEN MRESA_TAC
MOD_LE[`q MOD k - p MOD k +1:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`k' <= q MOD k - p MOD k + 1
/\ 3<= k' ==> k' -1 + p MOD k <= q MOD k`)
THEN RESA_TAC
THEN REPLICATE_TAC (32-26) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th->
REPEAT STRIP_TAC
THEN ONCE_REWRITE_TAC[SYM
th] THEN ASSUME_TAC(SYM
th))
THEN ASM_REWRITE_TAC[];
MP_TAC(ARITH_RULE`3<= k' /\ k'<k /\ p MOD k< q MOD k ==> ~(k=0)/\ (q +k -p MOD k)+1=q + 1 + k - p MOD k/\ 1<k `)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`q:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`i< k'-1 /\ k'-1 + p MOD k <= q MOD k/\ 3<=k' /\ q MOD k < k
==> i + p MOD k < q MOD k/\ i + p MOD k<k`)
THEN RESA_TAC
THEN SUBGOAL_THEN`(k'' +q MOD k) <= k+ p MOD k+1 `ASSUME_TAC;
MP_TAC(ARITH_RULE`(p +k -q MOD k)+1=(p + 1 + k - q MOD k) `)
THEN RESA_TAC
THEN EXPAND_TAC"k''"
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LT[`1`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`p+k-q MOD k`;`1`;`k:num`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[
th]
THEN MP_TAC(ARITH_RULE`(p DIV k * k + p MOD k) + k - q MOD k= (p DIV k)*k + (p MOD k + k - q MOD k)`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_MULT_ADD[`p DIV k `;`k:num`;`p MOD k + k - q MOD k`]
THEN ONCE_REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_LE[`p MOD k + k - q MOD k:num`;`k:num`]
THEN MRESA_TAC
MOD_LE[`(p MOD k + k - q MOD k) MOD k +1:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`(p MOD k + k - q MOD k) MOD k <= p MOD k + k - q MOD k
/\ ((p MOD k + k - q MOD k) MOD k + 1) MOD k <=
(p MOD k + k - q MOD k) MOD k + 1 /\ q MOD k <k
==> ((p MOD k + k - q MOD k) MOD k + 1) MOD k + q MOD k <= k+ p MOD k +1`)
THEN RESA_TAC;
ASM_TAC
THEN REWRITE_TAC[
scs_diag]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`q:num`;`p:num`;`d'':real`;`
scs_J_v39 s' p q`][
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
scs_diag;
is_scs_v39;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39;
IN_ELIM_THM]
THEN MP_TAC(ARITH_RULE`i'<k''-1 /\ 3<= k'' /\ k'' + q MOD k <= k + p MOD k + 1
==> i'+ q MOD k < k + p MOD k `)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`i'+ q MOD k< k \/ k <= i' + q MOD k `)
THEN RESA_TAC;
MRESA_TAC
MOD_LT[`i' + q MOD k`;`k:num`]
THEN MRESA_TAC
MOD_LT[`i + p MOD k`;`k:num`]
THEN MP_TAC(ARITH_RULE`q MOD k<= i'+ q MOD k`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[]
THEN ASM_REWRITE_TAC[ARITH_RULE`~(q MOD k <= i + p MOD k)<=> i + p MOD k< q MOD k`];
MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN
MP_TAC(ARITH_RULE`i' + q MOD k < k + p MOD k
/\ k <= i' + q MOD k /\ p MOD k<k
==> 1 * k + (i' + q MOD k) - k = i' + q MOD k /\ (i' + q MOD k) - k <k /\ (i' + q MOD k) - k < p MOD k`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`(i' + q MOD k)-k`;`k:num`]
THEN MRESA_TAC
MOD_LT[`i + p MOD k`;`k:num`]
THEN MRESA_TAC
MOD_MULT_ADD[`1 `;`k:num`;`(i' + q MOD k) -k`]
THEN MP_TAC(ARITH_RULE`i< k'-1 /\ k'-1 + p MOD k <= q MOD k/\ 3<=k' /\ q MOD k < k
==> p MOD k <= i + p MOD k/\ i + p MOD k < q MOD k/\ i + p MOD k<k`)
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC]);;
let QKNVMLB2 = prove_by_refinement(
`
scs_half_slice_v39 s p q d' mkj =s'/\
scs_half_slice_v39 s q p d'' mkj =s'' /\
vv' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s')+p MOD (
scs_k_v39 s)))
/\
vv'' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s'')+q MOD (
scs_k_v39 s)))
/\
MMs_v39 s vv /\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q /\
is_scs_slice_v39 s s' s'' p q /\
scs_d_v39 s <= d' + d'' ==>
dsv_v39 s vv <=
dsv_v39 s' vv' +
dsv_v39 s'' vv''`,
[
STRIP_TAC
THEN POP_ASSUM MP_TAC
(*THEN MP_TAC SCS_HALF_SLICE_IS_SCS
THEN RESA_TAC*)
THEN MRESA_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`p:num`;`q:num`;`d':real`;`
scs_J_v39 s' p q`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;]
THEN MRESA_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`q:num`;`p:num`;`d'':real`;`
scs_J_v39 s' p q`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
LET_DEF;
LET_END_DEF;]
THEN MP_TAC
SCS_K_PRIME_CASE_3
THEN RESA_TAC
THEN MP_TAC
DIAG_IS_NOT_EAR
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<
scs_k_v39 s ==> ~(
scs_k_v39 s<=3)`)
THEN RESA_TAC
THEN MP_TAC
DIAG_NOT_IN_SCS_J
THEN RESA_TAC
THEN REPLICATE_TAC 7 (POP_ASSUM MP_TAC)
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;]
THEN ABBREV_TAC`k'=(q + 1 +
scs_k_v39 s - p MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN ABBREV_TAC`k''=(p + 1 +
scs_k_v39 s - q MOD
scs_k_v39 s) MOD
scs_k_v39 s`
THEN ABBREV_TAC`k=
scs_k_v39 s`
THEN REPEAT RESA_TAC
;
REPLICATE_TAC (49-37) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th] THEN ASSUME_TAC(SYM
th))
THEN REPEAT STRIP_TAC
THEN REPLICATE_TAC (7) (POP_ASSUM MP_TAC)
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN REPEAT RESA_TAC
THEN SUBGOAL_THEN`FINITE {i | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
` ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s' -1)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
SUBGOAL_THEN`{i | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
INTER
{i| i=
scs_k_v39 s' - 1/\
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC (
scs_k_v39 s' - 1)) }={}` ASSUME_TAC
;
REWRITE_TAC[
INTER;SET_RULE`A={}<=> ~(?a. a
IN A)`;
IN_ELIM_THM;
IN_SING]
THEN ARITH_TAC;
SUBGOAL_THEN`{i | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
UNION
{i| i=
scs_k_v39 s' - 1/\
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC(
scs_k_v39 s'-1)) }={i | i <
scs_k_v39 s' /\
scs_J_v39 s' i (SUC i)}`
ASSUME_TAC;
REWRITE_TAC[
UNION;
IN_SING;
IN_ELIM_THM;
EXTENSION]
THEN GEN_TAC
THEN EXPAND_TAC "s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (47-43) REMOVE_ASSUM_TAC
THEN EQ_TAC
;
RESA_TAC
;
REPLICATE_TAC (47-40) (POP_ASSUM MP_TAC)
THEN ARITH_TAC
;
MP_TAC(ARITH_RULE`3<=k' ==>SUC (k' - 1) =k'/\ k'-1<k'`)
THEN RESA_TAC;
RESA_TAC
THEN MP_TAC(ARITH_RULE`x<k' /\ 3<= k' ==> x< k'-1 \/ (x=k'-1)`)
THEN RESA_TAC
THEN POP_ASSUM (fun th->
POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th]
THEN RESA_TAC)
;
SUBGOAL_THEN`FINITE
{i | i =
scs_k_v39 s' - 1 /\
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC (
scs_k_v39 s' - 1))}`ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`{(
scs_k_v39 s' -1)}`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_SING;
FINITE_SING]
THEN SET_TAC[];
MRESAL_TAC
SUM_INCL_EXCL[`{i | i <
scs_k_v39 s'-1 /\
scs_J_v39 s' i (SUC i)}`;`{i| i=
scs_k_v39 s' - 1/\
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC(
scs_k_v39 s'-1)) }`;`(\i. cstab -
dist
((vv:num->real^3) (i MOD
scs_k_v39 s' + p MOD k),
vv (SUC i MOD
scs_k_v39 s' + p MOD k)))`][
FINITE_SING;
SUM_SING;
SUM_CLAUSES;REAL_ARITH`a+ &0=a`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`FINITE {i | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
` ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s'' -1)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
SUBGOAL_THEN`{i | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
INTER
{i| i=
scs_k_v39 s'' - 1/\
scs_J_v39 s'' (
scs_k_v39 s'' - 1) (SUC (
scs_k_v39 s'' - 1)) }={}` ASSUME_TAC
;
REWRITE_TAC[
INTER;SET_RULE`A={}<=> ~(?a. a
IN A)`;
IN_ELIM_THM;
IN_SING]
THEN ARITH_TAC;
SUBGOAL_THEN`{i | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
UNION
{i| i=
scs_k_v39 s'' - 1/\
scs_J_v39 s'' (
scs_k_v39 s'' - 1) (SUC(
scs_k_v39 s''-1)) }={i | i <
scs_k_v39 s'' /\
scs_J_v39 s'' i (SUC i)}`
ASSUME_TAC;
REWRITE_TAC[
UNION;
IN_SING;
IN_ELIM_THM;
EXTENSION]
THEN GEN_TAC
THEN EXPAND_TAC "s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (47-43) REMOVE_ASSUM_TAC
THEN EQ_TAC
;
RESA_TAC
;
REPLICATE_TAC (47-40) (POP_ASSUM MP_TAC)
THEN ARITH_TAC
;
MP_TAC(ARITH_RULE`3<=k'' ==>SUC (k'' - 1) =k''/\ k''-1<k''`)
THEN RESA_TAC;
RESA_TAC
THEN MP_TAC(ARITH_RULE`x<k'' /\ 3<= k'' ==> x< k''-1 \/ (x=k''-1)`)
THEN RESA_TAC
THEN POP_ASSUM (fun th->
POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th]
THEN RESA_TAC)
;
SUBGOAL_THEN`FINITE
{i | i =
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' (
scs_k_v39 s'' - 1) (SUC (
scs_k_v39 s'' - 1))}`ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`{(
scs_k_v39 s'' -1)}`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_SING;
FINITE_SING]
THEN SET_TAC[];
MRESAL_TAC
SUM_INCL_EXCL[`{i | i <
scs_k_v39 s''-1 /\
scs_J_v39 s'' i (SUC i)}`;`{i| i=
scs_k_v39 s'' - 1/\
scs_J_v39 s'' (
scs_k_v39 s'' - 1) (SUC(
scs_k_v39 s''-1)) }`;`(\i. cstab -
dist
((vv:num->real^3) (i MOD
scs_k_v39 s'' + q MOD k),
vv (SUC i MOD
scs_k_v39 s'' + q MOD k)))`][
FINITE_SING;
SUM_SING;
SUM_CLAUSES;REAL_ARITH`a+ &0=a`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`
sum
{i | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
(\i. cstab -
dist
(vv (i MOD
scs_k_v39 s' + p MOD k),
(vv:num->real^3) (SUC i MOD
scs_k_v39 s' + p MOD k)))
=
sum
{(i + p MOD
scs_k_v39 s) MOD
scs_k_v39 s | i <
scs_k_v39 s'-1
/\
scs_J_v39 s' i (SUC i)}
(\i. cstab -
dist
(vv (i ),
vv (SUC i )))`ASSUME_TAC
;
MATCH_MP_TAC
SUM_EQ_GENERAL
THEN REWRITE_TAC[
IN_ELIM_THM]
THEN EXISTS_TAC`(\i. (i + p MOD
scs_k_v39 s) MOD
scs_k_v39 s)`
THEN ASM_REWRITE_TAC[
EXISTS_UNIQUE]
THEN REPEAT RESA_TAC;
EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
;
REPEAT STRIP_TAC
THEN REPLICATE_TAC (47-40) (POP_ASSUM MP_TAC)
THEN EXPAND_TAC "s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`i<k'-1/\ y'<k'-1 /\ k'< k/\ 3<=k' ==> i<k /\ y'<k/\ ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC
MOD_LT[`y':num`;`k:num`]
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i:num`;`y':num`;`p MOD k`;`k:num`][
ADD_SYM];
EXISTS_TAC`x:num`
THEN ASM_REWRITE_TAC[]
;
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC "s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`x<k'-1/\ 3<=k'/\k'<k ==> x<k'/\ ~(k'=0)/\ SUC x< k'/\ ~(k=0)/\ 1<k`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`SUC x:num`;`k':num`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num-> real^3`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x + p MOD k):num`
THEN MRESA1_TAC
th ` ((x + p MOD k) +1):num`
THEN MRESA1_TAC
th ` SUC ((x + p MOD k) MOD k):num`)
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_LT[`1`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`x+p MOD k`;`1`;`k:num`]
THEN REWRITE_TAC[ARITH_RULE`((x + p MOD k) + 1)= (x+ 1) + p MOD k`]
;
SUBGOAL_THEN`
sum
{i | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
(\i. cstab -
dist
(vv (i MOD
scs_k_v39 s'' + q MOD k),
(vv:num->real^3) (SUC i MOD
scs_k_v39 s'' + q MOD k)))
=
sum
{(i + q MOD
scs_k_v39 s) MOD
scs_k_v39 s | i <
scs_k_v39 s''-1
/\
scs_J_v39 s'' i (SUC i)}
(\i. cstab -
dist
(vv (i ),
vv (SUC i )))`ASSUME_TAC
;
MATCH_MP_TAC
SUM_EQ_GENERAL
THEN REWRITE_TAC[
IN_ELIM_THM]
THEN EXISTS_TAC`(\i. (i + q MOD
scs_k_v39 s) MOD
scs_k_v39 s)`
THEN ASM_REWRITE_TAC[
EXISTS_UNIQUE]
THEN REPEAT RESA_TAC;
EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
;
REPEAT STRIP_TAC
THEN REPLICATE_TAC (47-40) (POP_ASSUM MP_TAC)
THEN EXPAND_TAC "s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`i<k''-1/\ y'<k''-1 /\ k''< k/\ 3<=k'' ==> i<k /\ y'<k/\ ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC
MOD_LT[`y':num`;`k:num`]
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i:num`;`y':num`;`q MOD k`;`k:num`][
ADD_SYM];
EXISTS_TAC`x:num`
THEN ASM_REWRITE_TAC[]
;
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC "s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`x<k''-1/\ 3<=k''/\k''<k ==> x<k''/\ ~(k''=0)/\ SUC x< k''/\ ~(k=0)/\ 1<k`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`k'':num`]
THEN MRESA_TAC
MOD_LT[`SUC x:num`;`k'':num`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num-> real^3`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x + q MOD k):num`
THEN MRESA1_TAC
th ` ((x + q MOD k) +1):num`
THEN MRESA1_TAC
th ` SUC ((x + q MOD k) MOD k):num`)
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_LT[`1`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`x+q MOD k`;`1`;`k:num`]
THEN REWRITE_TAC[ARITH_RULE`((x + q MOD k) + 1)= (x+ 1) + q MOD k`]
;
ASM_REWRITE_TAC[
REAL_ARITH`a + b * (-- &1) *c <= (a1 + b * d1 *(c1+c2)) + a2 + b * d2 *(c3+c4)
<=> &0<= (a1+ a2-a) + b*(c+ d1*
c1 +d2*c3)+ b*(d1* c2+ d2* c4)`]
THEN MATCH_MP_TAC(REAL_ARITH`&0<=a /\ &0<=b /\ &0<=c==> &0<= a+b+c`)
THEN STRIP_TAC;
EXPAND_TAC "s'"
THEN EXPAND_TAC "s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
;
STRIP_TAC;
MATCH_MP_TAC(REAL_ARITH`&0<= a==> &0<= #0.1 *a`)
THEN SUBGOAL_THEN`-- &1<= if
is_ear_v39 s' then &1 else -- &1` ASSUME_TAC;
MP_TAC(SET_RULE`
is_ear_v39 s' \/ ~(
is_ear_v39 s')`)
THEN RESA_TAC
THEN REAL_ARITH_TAC;
SUBGOAL_THEN`-- &1<= if
is_ear_v39 s'' then &1 else -- &1` ASSUME_TAC;
MP_TAC(SET_RULE`
is_ear_v39 s'' \/ ~(
is_ear_v39 s'')`)
THEN RESA_TAC
THEN REAL_ARITH_TAC;
SUBGOAL_THEN`&0<=
sum {(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
(\i. cstab -
dist ((vv:num->real^3) i,vv (SUC i)))`ASSUME_TAC
;
MATCH_MP_TAC
SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG;ARITH_RULE`0<= a`]
THEN GEN_TAC
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k ==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + p MOD k`;`k:num`]
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT RESA_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPEAT RESA_TAC
THEN REPLICATE_TAC (75-6)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i + p MOD k) MOD k`;`SUC ((i + p MOD k) MOD k)`])
THEN REPLICATE_TAC (76-34)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA1_TAC
th`(i + p MOD k) MOD k`)
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC
;
SUBGOAL_THEN`&0<=
sum {(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
(\i. cstab -
dist ((vv:num->real^3) i,vv (SUC i)))`ASSUME_TAC
;
MATCH_MP_TAC
SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG;ARITH_RULE`0<= a`]
THEN GEN_TAC
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k ==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + q MOD k`;`k:num`]
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT RESA_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPEAT RESA_TAC
THEN REPLICATE_TAC (76-6)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i + q MOD k) MOD k`;`SUC ((i + q MOD k) MOD k)`])
THEN REPLICATE_TAC (77-34)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA1_TAC
th`(i + q MOD k) MOD k`)
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC
;
MRESA_TAC
REAL_LE_RMUL[`-- &1`;`(if
is_ear_v39 s' then &1 else -- &1)`;`
sum {(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
(\i. cstab -
dist (vv i,(vv:num->real^3) (SUC i)))`]
THEN MRESA_TAC
REAL_LE_RMUL[`-- &1`;`(if
is_ear_v39 s'' then &1 else -- &1)`;`
sum {(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
(\i. cstab -
dist (vv i,(vv:num->real^3) (SUC i)))`]
THEN SUBGOAL_THEN`FINITE
{(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}`ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG;ARITH_RULE`0<= a`]
THEN GEN_TAC
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k ==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + q MOD k`;`k:num`]
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
SUBGOAL_THEN`FINITE
{(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}`ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG;ARITH_RULE`0<= a`]
THEN GEN_TAC
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k ==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + q MOD k`;`k:num`]
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
SUBGOAL_THEN`&0 <=
sum {i | i < k /\
scs_J_v39 s i (SUC i)}
(\i. cstab -
dist (vv i,vv (SUC i))) -
(
sum {(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
(\i. cstab -
dist (vv i,vv (SUC i))) +
sum {(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
(\i. cstab -
dist (vv i,(vv:num->real^3) (SUC i))))` ASSUME_TAC
;
SUBGOAL_THEN`(!i.
scs_b_v39 s i (SUC i) <= cstab)`ASSUME_TAC;
REPLICATE_TAC (67-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
MP_TAC
INTER_SLICE_SCS_EMPTY
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPLICATE_TAC (68-37)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th->
ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th THEN REPEAT RESA_TAC)
THEN SUBGOAL_THEN`{(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
INTER
{(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}={}`ASSUME_TAC
;
POP_ASSUM MP_TAC
THEN REWRITE_TAC[
INTER;
IN_ELIM_THM]
THEN SET_TAC[];
MRESAL_TAC
SUM_INCL_EXCL[`{(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}`;`{(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}`;`(\i. cstab -
dist ((vv:num->real^3) i,vv (SUC i)))`][
SUM_CLAUSES;REAL_ARITH`a+ &0=a`;
dist]
THEN SUBGOAL_THEN`{(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
UNION
{(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
SUBSET {i | i < k /\
scs_J_v39 s i (SUC i)}`ASSUME_TAC
;
REWRITE_TAC[
UNION;
IN_ELIM_THM;
SUBSET]
THEN REPEAT RESA_TAC
;
MP_TAC(ARITH_RULE`3<k==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + p MOD k`;`k:num`]
;
MP_TAC
SCS_J_PRIME_SUBSET_SCS_J
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
;
MP_TAC(ARITH_RULE`3<k==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + q MOD k`;`k:num`]
;
MRESA_TAC(GEN_ALL
SCS_J_PRIME_SUBSET_SCS_J)[`d'':real`;`mkj:bool`;`p:num`;`s'':scs_v39`;`i:num`;`q:num`;`s:scs_v39`]
THEN POP_ASSUM MATCH_MP_TAC
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`];
SUBGOAL_THEN`FINITE {i | i < k /\
scs_J_v39 s i (SUC i)}` ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0..k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
MRESA_TAC
SUM_DIFF[`(\i. cstab -
norm (vv i - (vv:num->real^3) (SUC i)))`;`{i | i < k /\
scs_J_v39 s i (SUC i)}`;`{(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
UNION
{(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}`;]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MATCH_MP_TAC
SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
DIFF;
IN_ELIM_THM;
IN_NUMSEG;ARITH_RULE`0<= a`]
THEN ARITH_TAC;
REWRITE_TAC[
IN_ELIM_THM;
DIFF]
THEN REPEAT RESA_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPEAT RESA_TAC
THEN REPLICATE_TAC (87-7)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`x:num`;`SUC (x)`])
THEN REPLICATE_TAC (88-78)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA1_TAC
th`x:num`)
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MATCH_MP_TAC(REAL_ARITH`-- &1 *a <=c* a /\ -- &1 *b<= d*b /\ &0<= a1-(a+b) ==> &0<= a1+ c*a+d*b`)
THEN ASM_REWRITE_TAC[]
;
MATCH_MP_TAC(REAL_ARITH`&0<=a ==> &0<= #0.1 *a`)
THEN MP_TAC
SCS_J_DIAG_EQ
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;]
THEN ASM_SIMP_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPLICATE_TAC (59-31)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN REWRITE_TAC[]
THEN REPLICATE_TAC (59-31)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN RESA_TAC
THEN SUBGOAL_THEN`
scs_k_v39 s'=k'`ASSUME_TAC
;
EXPAND_TAC"s'"
THEN EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
;
SUBGOAL_THEN`
scs_k_v39 s''=k''`ASSUME_TAC
;
EXPAND_TAC"s'"
THEN EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
;
MP_TAC(ARITH_RULE`3<=k'/\ 3<=k'' ==> SUC(k'-1)=k' /\ SUC (k''-1)=k''`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC (
scs_k_v39 s' - 1)) \/ ~(
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC (
scs_k_v39 s' - 1)))`)
THEN RESA_TAC;
ASM_TAC
THEN REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
SCS_HALF_SLICE_IS_A_SCS
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2]
THEN ASM_SIMP_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPLICATE_TAC (64-58)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (2)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th
THEN ASM_REWRITE_TAC[])
THEN REPLICATE_TAC (64-58)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (2)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC( LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN RESA_TAC
THEN REPLICATE_TAC (81-72)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`0`;`k'-1`;][ARITH_RULE`0+A=A`])
THEN SUBGOAL_THEN`(!i.
scs_b_v39 s i (SUC i) <= cstab)`ASSUME_TAC;
REPLICATE_TAC (83-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
MRESAL_TAC(GEN_ALL
SCS_HALF_SLICE_IS_A_SCS)[`s:scs_v39`;`s':scs_v39`;`q:num`;`p:num`;`s'':scs_v39`]
[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2]
THEN REPLICATE_TAC (85-35)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th->
ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th THEN REPEAT RESA_TAC)
THEN REPLICATE_TAC (85-36)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN ASM_REWRITE_TAC[SET_RULE`A\/B<=> B\/A`]
THEN REPLICATE_TAC (84-63)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th
THEN ASM_REWRITE_TAC[])
THEN REPLICATE_TAC (84-62)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC( LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN ASM_REWRITE_TAC[REAL_ARITH`a<=b+c<=> a<= c+b`]
THEN STRIP_TAC
THEN REPLICATE_TAC (101-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`0`;`k''-1`;][ARITH_RULE`0+A=A`])
THEN REPLICATE_TAC (102-57)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN RESA_TAC
THEN REWRITE_TAC[SET_RULE`{a|a=b}={b}`;
SUM_SING]
THEN MP_TAC(ARITH_RULE`3<=k'/\ 3<=k'' ==> (k'-1)<k' /\ (k''-1)<k''`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESA_TAC
MOD_LT[`k''-1`;`k'':num`]
THEN MRESA_TAC (GEN_ALL
W_EW_K_SCS_ADD_P)[
`d':real`;`mkj:bool`;`s':scs_v39`;`(vv:num->real^3)(p MOD k)`;`q:num`;`s:scs_v39`;`(vv:num->real^3)`;`k':num`;`p:num`;`k:num`;`(vv:num->real^3) (q MOD k)`]
THEN MRESA_TAC (GEN_ALL
W_EW_K_SCS_ADD_P)[
`d'':real`;`mkj:bool`;`s'':scs_v39`;`(vv:num->real^3)(q MOD k)`;`p:num`;`s:scs_v39`;`(vv:num->real^3)`;`k'':num`;`q:num`;`k:num`;`(vv:num->real^3) (p MOD k)`]
THEN ASM_TAC
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2]
THEN REPEAT RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k'/\ 3<=k'' ==> ~(k'=0) /\ ~(k''=0)`)
THEN RESA_TAC
THEN MRESAL_TAC MOD_MULT[`k':num`;`1:num`][ARITH_RULE`k'*1 =k'/\ 0+A=A`]
THEN MRESAL_TAC MOD_MULT[`k'':num`;`1:num`][ARITH_RULE`k'*1 =k'/\ 0+A=A`]
THEN SUBGOAL_THEN`
norm ((vv:num->real^3) (q MOD k) - vv (p MOD k)) =
norm (vv (p MOD k) - vv (q MOD k))`ASSUME_TAC
;
MP_TAC(VECTOR_ARITH`vv (q MOD k) - vv (p MOD k) = --(vv (p MOD k) - (vv:num->real^3) (q MOD k))`)
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[
th])
THEN REWRITE_TAC[
NORM_NEG];
ASM_REWRITE_TAC[ARITH_RULE`a*b+c*b=(a+c)*b`]
THEN MATCH_MP_TAC
REAL_LE_MUL
THEN STRIP_TAC;
REPLICATE_TAC (124-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN REAL_ARITH_TAC
;
REPLICATE_TAC (124-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
;
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
is_ear_v39]
THEN STRIP_TAC
THEN SUBGOAL_THEN`k'-1
IN {i | i < 3 /\
scs_J_v39 s' i (SUC i)}`ASSUME_TAC;
ASM_REWRITE_TAC[
IN_ELIM_THM;ARITH_RULE`3-1<3`]
THEN REPLICATE_TAC (132-88)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]);
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
IN_SING;ARITH_RULE`3-1=2`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[SYM
th])
THEN REPEAT STRIP_TAC
THEN SUBGOAL_THEN`d' =
scs_d_v39 s' `ASSUME_TAC
;
EXPAND_TAC"s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
;
MP_TAC
QKNVMLB1
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;]
THEN REPEAT RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`2`;`SUC 2`])
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`SUC 2 MOD 3=0/\ 0+A=A/\ 2 MOD 3=2`]
THEN REPLICATE_TAC (136-117)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`])
THEN RESA_TAC
THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
is_ear_v39]
THEN STRIP_TAC
THEN SUBGOAL_THEN`k''-1
IN {i | i < 3 /\
scs_J_v39 s'' i (SUC i)}`ASSUME_TAC;
ASM_REWRITE_TAC[
IN_ELIM_THM;ARITH_RULE`3-1<3`]
THEN REPLICATE_TAC (132-110)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]);
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
IN_SING;ARITH_RULE`3-1=2`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[SYM
th])
THEN REPEAT STRIP_TAC
THEN SUBGOAL_THEN`d'' =
scs_d_v39 s'' `ASSUME_TAC
;
EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
;
MRESAL_TAC (GEN_ALL
QKNVMLB1)[`vv:num->real^3`;`s:scs_v39`;`q:num`;`p:num`;`d'':real`;`mkj:bool`]
[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;]
THEN REPLICATE_TAC (134-9)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN REWRITE_TAC[REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`2`;`SUC 2`])
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`SUC 2 MOD 3=0/\ 0+A=A/\ 2 MOD 3=2`]
THEN REPLICATE_TAC (136-117)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`])
THEN RESA_TAC
THEN REAL_ARITH_TAC;
MP_TAC(VECTOR_ARITH`vv (q MOD k) - vv (p MOD k) = --(vv (p MOD k) - (vv:num->real^3) (q MOD k))`)
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[
th])
THEN REWRITE_TAC[
NORM_NEG];
ASM_REWRITE_TAC[ARITH_RULE`a*b+c*b=(a+c)*b`]
THEN MATCH_MP_TAC
REAL_LE_MUL
THEN STRIP_TAC;
REPLICATE_TAC (124-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN REAL_ARITH_TAC
;
REPLICATE_TAC (124-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
;
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
is_ear_v39]
THEN STRIP_TAC
THEN SUBGOAL_THEN`k'-1
IN {i | i < 3 /\
scs_J_v39 s' i (SUC i)}`ASSUME_TAC;
ASM_REWRITE_TAC[
IN_ELIM_THM;ARITH_RULE`3-1<3`]
THEN REPLICATE_TAC (132-88)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]);
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
IN_SING;ARITH_RULE`3-1=2`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[SYM
th])
THEN REPEAT STRIP_TAC
THEN SUBGOAL_THEN`d' =
scs_d_v39 s' `ASSUME_TAC
;
EXPAND_TAC"s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
;
MP_TAC
QKNVMLB1
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;]
THEN REPEAT RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`2`;`SUC 2`])
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`SUC 2 MOD 3=0/\ 0+A=A/\ 2 MOD 3=2`]
THEN REPLICATE_TAC (136-117)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`])
THEN RESA_TAC
THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
is_ear_v39]
THEN STRIP_TAC
THEN SUBGOAL_THEN`k''-1
IN {i | i < 3 /\
scs_J_v39 s'' i (SUC i)}`ASSUME_TAC;
ASM_REWRITE_TAC[
IN_ELIM_THM;ARITH_RULE`3-1<3`]
THEN REPLICATE_TAC (132-110)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]);
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
IN_SING;ARITH_RULE`3-1=2`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[SYM
th])
THEN REPEAT STRIP_TAC
THEN SUBGOAL_THEN`d'' =
scs_d_v39 s'' `ASSUME_TAC
;
EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
;
MRESAL_TAC (GEN_ALL
QKNVMLB1)[`vv:num->real^3`;`s:scs_v39`;`q:num`;`p:num`;`d'':real`;`mkj:bool`]
[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;]
THEN REPLICATE_TAC (134-9)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN REWRITE_TAC[REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`2`;`SUC 2`])
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`SUC 2 MOD 3=0/\ 0+A=A/\ 2 MOD 3=2`]
THEN REPLICATE_TAC (136-117)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`])
THEN RESA_TAC
THEN REAL_ARITH_TAC;
ASM_TAC
THEN REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
SCS_HALF_SLICE_IS_A_SCS
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2]
THEN ASM_SIMP_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPLICATE_TAC (64-58)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (2)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th
THEN ASM_REWRITE_TAC[])
THEN REPLICATE_TAC (64-58)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (2)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC( LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN RESA_TAC
THEN REPLICATE_TAC (81-72)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`0`;`k'-1`;][ARITH_RULE`0+A=A`])
THEN SUBGOAL_THEN`(!i.
scs_b_v39 s i (SUC i) <= cstab)`ASSUME_TAC;
REPLICATE_TAC (83-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
MRESAL_TAC(GEN_ALL
SCS_HALF_SLICE_IS_A_SCS)[`s:scs_v39`;`s':scs_v39`;`q:num`;`p:num`;`s'':scs_v39`]
[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2]
THEN REPLICATE_TAC (85-35)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th->
ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th THEN REPEAT RESA_TAC)
THEN REPLICATE_TAC (85-36)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN ASM_REWRITE_TAC[SET_RULE`A\/B<=> B\/A`]
THEN REPLICATE_TAC (84-63)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th
THEN ASM_REWRITE_TAC[])
THEN REPLICATE_TAC (84-62)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC( LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN ASM_REWRITE_TAC[REAL_ARITH`a<=b+c<=> a<= c+b`]
THEN STRIP_TAC
THEN REPLICATE_TAC (101-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`0`;`k''-1`;][ARITH_RULE`0+A=A`])
THEN REPLICATE_TAC (102-57)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN RESA_TAC
THEN REWRITE_TAC[SET_RULE`{a|F}={}`;
SUM_SING;
SUM_CLAUSES]
THEN REAL_ARITH_TAC;
(********************************)
REPLICATE_TAC (7) (POP_ASSUM MP_TAC)
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN REPEAT RESA_TAC
THEN SUBGOAL_THEN`FINITE {i | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
` ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s' -1)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
SUBGOAL_THEN`{i | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
INTER
{i| i=
scs_k_v39 s' - 1/\
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC (
scs_k_v39 s' - 1)) }={}` ASSUME_TAC
;
REWRITE_TAC[
INTER;SET_RULE`A={}<=> ~(?a. a
IN A)`;
IN_ELIM_THM;
IN_SING]
THEN ARITH_TAC;
SUBGOAL_THEN`{i | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
UNION
{i| i=
scs_k_v39 s' - 1/\
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC(
scs_k_v39 s'-1)) }={i | i <
scs_k_v39 s' /\
scs_J_v39 s' i (SUC i)}`
ASSUME_TAC;
REWRITE_TAC[
UNION;
IN_SING;
IN_ELIM_THM;
EXTENSION]
THEN GEN_TAC
THEN EXPAND_TAC "s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (47-43) REMOVE_ASSUM_TAC
THEN EQ_TAC
;
RESA_TAC
;
REPLICATE_TAC (47-40) (POP_ASSUM MP_TAC)
THEN ARITH_TAC
;
MP_TAC(ARITH_RULE`3<=k' ==>SUC (k' - 1) =k'/\ k'-1<k'`)
THEN RESA_TAC;
RESA_TAC
THEN MP_TAC(ARITH_RULE`x<k' /\ 3<= k' ==> x< k'-1 \/ (x=k'-1)`)
THEN RESA_TAC
THEN POP_ASSUM (fun th->
POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th]
THEN RESA_TAC)
;
SUBGOAL_THEN`FINITE
{i | i =
scs_k_v39 s' - 1 /\
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC (
scs_k_v39 s' - 1))}`ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`{(
scs_k_v39 s' -1)}`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_SING;
FINITE_SING]
THEN SET_TAC[];
MRESAL_TAC
SUM_INCL_EXCL[`{i | i <
scs_k_v39 s'-1 /\
scs_J_v39 s' i (SUC i)}`;`{i| i=
scs_k_v39 s' - 1/\
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC(
scs_k_v39 s'-1)) }`;`(\i. cstab -
dist
((vv:num->real^3) (i MOD
scs_k_v39 s' + p MOD k),
vv (SUC i MOD
scs_k_v39 s' + p MOD k)))`][
FINITE_SING;
SUM_SING;
SUM_CLAUSES;REAL_ARITH`a+ &0=a`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`FINITE {i | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
` ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. (
scs_k_v39 s'' -1)`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
SUBGOAL_THEN`{i | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
INTER
{i| i=
scs_k_v39 s'' - 1/\
scs_J_v39 s'' (
scs_k_v39 s'' - 1) (SUC (
scs_k_v39 s'' - 1)) }={}` ASSUME_TAC
;
REWRITE_TAC[
INTER;SET_RULE`A={}<=> ~(?a. a
IN A)`;
IN_ELIM_THM;
IN_SING]
THEN ARITH_TAC;
SUBGOAL_THEN`{i | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
UNION
{i| i=
scs_k_v39 s'' - 1/\
scs_J_v39 s'' (
scs_k_v39 s'' - 1) (SUC(
scs_k_v39 s''-1)) }={i | i <
scs_k_v39 s'' /\
scs_J_v39 s'' i (SUC i)}`
ASSUME_TAC;
REWRITE_TAC[
UNION;
IN_SING;
IN_ELIM_THM;
EXTENSION]
THEN GEN_TAC
THEN EXPAND_TAC "s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (47-43) REMOVE_ASSUM_TAC
THEN EQ_TAC
;
RESA_TAC
;
REPLICATE_TAC (47-40) (POP_ASSUM MP_TAC)
THEN ARITH_TAC
;
MP_TAC(ARITH_RULE`3<=k'' ==>SUC (k'' - 1) =k''/\ k''-1<k''`)
THEN RESA_TAC;
RESA_TAC
THEN MP_TAC(ARITH_RULE`x<k'' /\ 3<= k'' ==> x< k''-1 \/ (x=k''-1)`)
THEN RESA_TAC
THEN POP_ASSUM (fun th->
POP_ASSUM MP_TAC
THEN REWRITE_TAC[
th]
THEN RESA_TAC)
;
SUBGOAL_THEN`FINITE
{i | i =
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' (
scs_k_v39 s'' - 1) (SUC (
scs_k_v39 s'' - 1))}`ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`{(
scs_k_v39 s'' -1)}`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_SING;
FINITE_SING]
THEN SET_TAC[];
MRESAL_TAC
SUM_INCL_EXCL[`{i | i <
scs_k_v39 s''-1 /\
scs_J_v39 s'' i (SUC i)}`;`{i| i=
scs_k_v39 s'' - 1/\
scs_J_v39 s'' (
scs_k_v39 s'' - 1) (SUC(
scs_k_v39 s''-1)) }`;`(\i. cstab -
dist
((vv:num->real^3) (i MOD
scs_k_v39 s'' + q MOD k),
vv (SUC i MOD
scs_k_v39 s'' + q MOD k)))`][
FINITE_SING;
SUM_SING;
SUM_CLAUSES;REAL_ARITH`a+ &0=a`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`
sum
{i | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
(\i. cstab -
dist
(vv (i MOD
scs_k_v39 s' + p MOD k),
(vv:num->real^3) (SUC i MOD
scs_k_v39 s' + p MOD k)))
=
sum
{(i + p MOD
scs_k_v39 s) MOD
scs_k_v39 s | i <
scs_k_v39 s'-1
/\
scs_J_v39 s' i (SUC i)}
(\i. cstab -
dist
(vv (i ),
vv (SUC i )))`ASSUME_TAC
;
MATCH_MP_TAC
SUM_EQ_GENERAL
THEN REWRITE_TAC[
IN_ELIM_THM]
THEN EXISTS_TAC`(\i. (i + p MOD
scs_k_v39 s) MOD
scs_k_v39 s)`
THEN ASM_REWRITE_TAC[
EXISTS_UNIQUE]
THEN REPEAT RESA_TAC;
EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
;
REPEAT STRIP_TAC
THEN REPLICATE_TAC (47-40) (POP_ASSUM MP_TAC)
THEN EXPAND_TAC "s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`i<k'-1/\ y'<k'-1 /\ k'< k/\ 3<=k' ==> i<k /\ y'<k/\ ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC
MOD_LT[`y':num`;`k:num`]
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i:num`;`y':num`;`p MOD k`;`k:num`][
ADD_SYM];
EXISTS_TAC`x:num`
THEN ASM_REWRITE_TAC[]
;
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC "s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`x<k'-1/\ 3<=k'/\k'<k ==> x<k'/\ ~(k'=0)/\ SUC x< k'/\ ~(k=0)/\ 1<k`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`k':num`]
THEN MRESA_TAC
MOD_LT[`SUC x:num`;`k':num`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num-> real^3`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x + p MOD k):num`
THEN MRESA1_TAC
th ` ((x + p MOD k) +1):num`
THEN MRESA1_TAC
th ` SUC ((x + p MOD k) MOD k):num`)
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_LT[`1`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`x+p MOD k`;`1`;`k:num`]
THEN REWRITE_TAC[ARITH_RULE`((x + p MOD k) + 1)= (x+ 1) + p MOD k`]
;
SUBGOAL_THEN`
sum
{i | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
(\i. cstab -
dist
(vv (i MOD
scs_k_v39 s'' + q MOD k),
(vv:num->real^3) (SUC i MOD
scs_k_v39 s'' + q MOD k)))
=
sum
{(i + q MOD
scs_k_v39 s) MOD
scs_k_v39 s | i <
scs_k_v39 s''-1
/\
scs_J_v39 s'' i (SUC i)}
(\i. cstab -
dist
(vv (i ),
vv (SUC i )))`ASSUME_TAC
;
MATCH_MP_TAC
SUM_EQ_GENERAL
THEN REWRITE_TAC[
IN_ELIM_THM]
THEN EXISTS_TAC`(\i. (i + q MOD
scs_k_v39 s) MOD
scs_k_v39 s)`
THEN ASM_REWRITE_TAC[
EXISTS_UNIQUE]
THEN REPEAT RESA_TAC;
EXISTS_TAC`i:num`
THEN ASM_REWRITE_TAC[]
;
REPEAT STRIP_TAC
THEN REPLICATE_TAC (47-40) (POP_ASSUM MP_TAC)
THEN EXPAND_TAC "s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`i<k''-1/\ y'<k''-1 /\ k''< k/\ 3<=k'' ==> i<k /\ y'<k/\ ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`i:num`;`k:num`]
THEN MRESA_TAC
MOD_LT[`y':num`;`k:num`]
THEN MRESAL_TAC Hdplygy.MOD_EQ_MOD[`i:num`;`y':num`;`q MOD k`;`k:num`][
ADD_SYM];
EXISTS_TAC`x:num`
THEN ASM_REWRITE_TAC[]
;
POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN EXPAND_TAC "s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN MP_TAC(ARITH_RULE`x<k''-1/\ 3<=k''/\k''<k ==> x<k''/\ ~(k''=0)/\ SUC x< k''/\ ~(k=0)/\ 1<k`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`k'':num`]
THEN MRESA_TAC
MOD_LT[`SUC x:num`;`k'':num`]
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN REPEAT STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num-> real^3`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x + q MOD k):num`
THEN MRESA1_TAC
th ` ((x + q MOD k) +1):num`
THEN MRESA1_TAC
th ` SUC ((x + q MOD k) MOD k):num`)
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN REWRITE_TAC[
ADD1]
THEN MRESA_TAC
MOD_LT[`1`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`x+q MOD k`;`1`;`k:num`]
THEN REWRITE_TAC[ARITH_RULE`((x + q MOD k) + 1)= (x+ 1) + q MOD k`]
;
ASM_REWRITE_TAC[
REAL_ARITH`a + b * (-- &1) *c <= (a1 + b * d1 *(c1+c2)) + a2 + b * d2 *(c3+c4)
<=> &0<= (a1+ a2-a) + b*(c+ d1*
c1 +d2*c3)+ b*(d1* c2+ d2* c4)`]
THEN MATCH_MP_TAC(REAL_ARITH`&0<=a /\ &0<=b /\ &0<=c==> &0<= a+b+c`)
THEN STRIP_TAC;
EXPAND_TAC "s'"
THEN EXPAND_TAC "s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
;
STRIP_TAC;
MATCH_MP_TAC(REAL_ARITH`&0<= a==> &0<= #0.1 *a`)
THEN SUBGOAL_THEN`-- &1<= if
is_ear_v39 s' then &1 else -- &1` ASSUME_TAC;
MP_TAC(SET_RULE`
is_ear_v39 s' \/ ~(
is_ear_v39 s')`)
THEN RESA_TAC
THEN REAL_ARITH_TAC;
SUBGOAL_THEN`-- &1<= if
is_ear_v39 s'' then &1 else -- &1` ASSUME_TAC;
MP_TAC(SET_RULE`
is_ear_v39 s'' \/ ~(
is_ear_v39 s'')`)
THEN RESA_TAC
THEN REAL_ARITH_TAC;
SUBGOAL_THEN`&0<=
sum {(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
(\i. cstab -
dist ((vv:num->real^3) i,vv (SUC i)))`ASSUME_TAC
;
MATCH_MP_TAC
SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG;ARITH_RULE`0<= a`]
THEN GEN_TAC
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k ==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + p MOD k`;`k:num`]
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT RESA_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPEAT RESA_TAC
THEN REPLICATE_TAC (75-6)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i + p MOD k) MOD k`;`SUC ((i + p MOD k) MOD k)`])
THEN REPLICATE_TAC (76-34)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA1_TAC
th`(i + p MOD k) MOD k`)
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC
;
SUBGOAL_THEN`&0<=
sum {(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
(\i. cstab -
dist ((vv:num->real^3) i,vv (SUC i)))`ASSUME_TAC
;
MATCH_MP_TAC
SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG;ARITH_RULE`0<= a`]
THEN GEN_TAC
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k ==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + q MOD k`;`k:num`]
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT RESA_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPEAT RESA_TAC
THEN REPLICATE_TAC (76-6)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`(i + q MOD k) MOD k`;`SUC ((i + q MOD k) MOD k)`])
THEN REPLICATE_TAC (77-34)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA1_TAC
th`(i + q MOD k) MOD k`)
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC
;
MRESA_TAC
REAL_LE_RMUL[`-- &1`;`(if
is_ear_v39 s' then &1 else -- &1)`;`
sum {(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
(\i. cstab -
dist (vv i,(vv:num->real^3) (SUC i)))`]
THEN MRESA_TAC
REAL_LE_RMUL[`-- &1`;`(if
is_ear_v39 s'' then &1 else -- &1)`;`
sum {(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
(\i. cstab -
dist (vv i,(vv:num->real^3) (SUC i)))`]
THEN SUBGOAL_THEN`FINITE
{(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}`ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG;ARITH_RULE`0<= a`]
THEN GEN_TAC
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k ==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + q MOD k`;`k:num`]
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
SUBGOAL_THEN`FINITE
{(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}`ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG;ARITH_RULE`0<= a`]
THEN GEN_TAC
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k ==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + q MOD k`;`k:num`]
THEN POP_ASSUM MP_TAC
THEN ARITH_TAC;
SUBGOAL_THEN`&0 <=
sum {i | i < k /\
scs_J_v39 s i (SUC i)}
(\i. cstab -
dist (vv i,vv (SUC i))) -
(
sum {(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
(\i. cstab -
dist (vv i,vv (SUC i))) +
sum {(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
(\i. cstab -
dist (vv i,(vv:num->real^3) (SUC i))))` ASSUME_TAC
;
SUBGOAL_THEN`(!i.
scs_b_v39 s i (SUC i) <= cstab)`ASSUME_TAC;
REPLICATE_TAC (67-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
MP_TAC
INTER_SLICE_SCS_EMPTY
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPLICATE_TAC (68-37)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th->
ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th THEN REPEAT RESA_TAC)
THEN SUBGOAL_THEN`{(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
INTER
{(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}={}`ASSUME_TAC
;
POP_ASSUM MP_TAC
THEN REWRITE_TAC[
INTER;
IN_ELIM_THM]
THEN SET_TAC[];
MRESAL_TAC
SUM_INCL_EXCL[`{(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}`;`{(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}`;`(\i. cstab -
dist ((vv:num->real^3) i,vv (SUC i)))`][
SUM_CLAUSES;REAL_ARITH`a+ &0=a`;
dist]
THEN SUBGOAL_THEN`{(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
UNION
{(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}
SUBSET {i | i < k /\
scs_J_v39 s i (SUC i)}`ASSUME_TAC
;
REWRITE_TAC[
UNION;
IN_ELIM_THM;
SUBSET]
THEN REPEAT RESA_TAC
;
MP_TAC(ARITH_RULE`3<k==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + p MOD k`;`k:num`]
;
MP_TAC
SCS_J_PRIME_SUBSET_SCS_J
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
;
MP_TAC(ARITH_RULE`3<k==> ~(k=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`i + q MOD k`;`k:num`]
;
MRESA_TAC(GEN_ALL
SCS_J_PRIME_SUBSET_SCS_J)[`d'':real`;`mkj:bool`;`p:num`;`s'':scs_v39`;`i:num`;`q:num`;`s:scs_v39`]
THEN POP_ASSUM MATCH_MP_TAC
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`];
SUBGOAL_THEN`FINITE {i | i < k /\
scs_J_v39 s i (SUC i)}` ASSUME_TAC
;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0..k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
IN_ELIM_THM;
IN_NUMSEG]
THEN ARITH_TAC;
MRESA_TAC
SUM_DIFF[`(\i. cstab -
norm (vv i - (vv:num->real^3) (SUC i)))`;`{i | i < k /\
scs_J_v39 s i (SUC i)}`;`{(i + p MOD k) MOD k | i <
scs_k_v39 s' - 1 /\
scs_J_v39 s' i (SUC i)}
UNION
{(i + q MOD k) MOD k | i <
scs_k_v39 s'' - 1 /\
scs_J_v39 s'' i (SUC i)}`;]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MATCH_MP_TAC
SUM_POS_LE
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC;
MATCH_MP_TAC
FINITE_SUBSET
THEN EXISTS_TAC`0.. k`
THEN REWRITE_TAC[
FINITE_NUMSEG;
SUBSET;
DIFF;
IN_ELIM_THM;
IN_NUMSEG;ARITH_RULE`0<= a`]
THEN ARITH_TAC;
REWRITE_TAC[
IN_ELIM_THM;
DIFF]
THEN REPEAT RESA_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPEAT RESA_TAC
THEN REPLICATE_TAC (87-7)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA_TAC
th[`x:num`;`SUC (x)`])
THEN REPLICATE_TAC (88-78)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESA1_TAC
th`x:num`)
THEN POP_ASSUM MP_TAC
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MATCH_MP_TAC(REAL_ARITH`-- &1 *a <=c* a /\ -- &1 *b<= d*b /\ &0<= a1-(a+b) ==> &0<= a1+ c*a+d*b`)
THEN ASM_REWRITE_TAC[]
;
MATCH_MP_TAC(REAL_ARITH`&0<=a ==> &0<= #0.1 *a`)
THEN MP_TAC
SCS_J_DIAG_EQ
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;]
THEN ASM_SIMP_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPLICATE_TAC (59-31)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN REWRITE_TAC[]
THEN REPLICATE_TAC (59-31)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN RESA_TAC
THEN SUBGOAL_THEN`
scs_k_v39 s'=k'`ASSUME_TAC
;
EXPAND_TAC"s'"
THEN EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
;
SUBGOAL_THEN`
scs_k_v39 s''=k''`ASSUME_TAC
;
EXPAND_TAC"s'"
THEN EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
;
MP_TAC(ARITH_RULE`3<=k'/\ 3<=k'' ==> SUC(k'-1)=k' /\ SUC (k''-1)=k''`)
THEN RESA_TAC
THEN MP_TAC(SET_RULE`
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC (
scs_k_v39 s' - 1)) \/ ~(
scs_J_v39 s' (
scs_k_v39 s' - 1) (SUC (
scs_k_v39 s' - 1)))`)
THEN RESA_TAC;
ASM_TAC
THEN REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
SCS_HALF_SLICE_IS_A_SCS
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2]
THEN ASM_SIMP_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPLICATE_TAC (64-58)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (2)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th
THEN ASM_REWRITE_TAC[])
THEN REPLICATE_TAC (64-58)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (2)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC( LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN RESA_TAC
THEN REPLICATE_TAC (81-72)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`0`;`k'-1`;][ARITH_RULE`0+A=A`])
THEN SUBGOAL_THEN`(!i.
scs_b_v39 s i (SUC i) <= cstab)`ASSUME_TAC;
REPLICATE_TAC (83-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
MRESAL_TAC(GEN_ALL
SCS_HALF_SLICE_IS_A_SCS)[`s:scs_v39`;`s':scs_v39`;`q:num`;`p:num`;`s'':scs_v39`]
[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2]
THEN REPLICATE_TAC (85-35)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th->
ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th THEN REPEAT RESA_TAC)
THEN REPLICATE_TAC (85-36)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN ASM_REWRITE_TAC[SET_RULE`A\/B<=> B\/A`]
THEN REPLICATE_TAC (84-63)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th
THEN ASM_REWRITE_TAC[])
THEN REPLICATE_TAC (84-62)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC( LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN ASM_REWRITE_TAC[REAL_ARITH`a<=b+c<=> a<= c+b`]
THEN STRIP_TAC
THEN REPLICATE_TAC (101-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`0`;`k''-1`;][ARITH_RULE`0+A=A`])
THEN REPLICATE_TAC (102-57)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN RESA_TAC
THEN REWRITE_TAC[SET_RULE`{a|a=b}={b}`;
SUM_SING]
THEN MP_TAC(ARITH_RULE`3<=k'/\ 3<=k'' ==> (k'-1)<k' /\ (k''-1)<k''`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`k'-1`;`k':num`]
THEN MRESA_TAC
MOD_LT[`k''-1`;`k'':num`]
THEN MRESA_TAC (GEN_ALL
W_EW_K_SCS_ADD_P)[
`d':real`;`mkj:bool`;`s':scs_v39`;`(vv:num->real^3)(p MOD k)`;`q:num`;`s:scs_v39`;`(vv:num->real^3)`;`k':num`;`p:num`;`k:num`;`(vv:num->real^3) (q MOD k)`]
THEN MRESA_TAC (GEN_ALL
W_EW_K_SCS_ADD_P)[
`d'':real`;`mkj:bool`;`s'':scs_v39`;`(vv:num->real^3)(q MOD k)`;`p:num`;`s:scs_v39`;`(vv:num->real^3)`;`k'':num`;`q:num`;`k:num`;`(vv:num->real^3) (p MOD k)`]
THEN ASM_TAC
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2]
THEN REPEAT RESA_TAC
THEN MP_TAC(ARITH_RULE`3<=k'/\ 3<=k'' ==> ~(k'=0) /\ ~(k''=0)`)
THEN RESA_TAC
THEN MRESAL_TAC MOD_MULT[`k':num`;`1:num`][ARITH_RULE`k'*1 =k'/\ 0+A=A`]
THEN MRESAL_TAC MOD_MULT[`k'':num`;`1:num`][ARITH_RULE`k'*1 =k'/\ 0+A=A`]
THEN SUBGOAL_THEN`
norm ((vv:num->real^3) (q MOD k) - vv (p MOD k)) =
norm (vv (p MOD k) - vv (q MOD k))`ASSUME_TAC
;
MP_TAC(VECTOR_ARITH`vv (q MOD k) - vv (p MOD k) = --(vv (p MOD k) - (vv:num->real^3) (q MOD k))`)
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[
th])
THEN REWRITE_TAC[
NORM_NEG];
ASM_REWRITE_TAC[ARITH_RULE`a*b+c*b=(a+c)*b`]
THEN MATCH_MP_TAC
REAL_LE_MUL
THEN STRIP_TAC;
REPLICATE_TAC (124-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN REAL_ARITH_TAC
;
REPLICATE_TAC (124-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
;
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
is_ear_v39]
THEN STRIP_TAC
THEN SUBGOAL_THEN`k'-1
IN {i | i < 3 /\
scs_J_v39 s' i (SUC i)}`ASSUME_TAC;
ASM_REWRITE_TAC[
IN_ELIM_THM;ARITH_RULE`3-1<3`]
THEN REPLICATE_TAC (132-88)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]);
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
IN_SING;ARITH_RULE`3-1=2`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[SYM
th])
THEN REPEAT STRIP_TAC
THEN SUBGOAL_THEN`d' =
scs_d_v39 s' `ASSUME_TAC
;
EXPAND_TAC"s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
;
MP_TAC
QKNVMLB1
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;]
THEN REPEAT RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`2`;`SUC 2`])
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`SUC 2 MOD 3=0/\ 0+A=A/\ 2 MOD 3=2`]
THEN REPLICATE_TAC (136-117)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`])
THEN RESA_TAC
THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
is_ear_v39]
THEN STRIP_TAC
THEN SUBGOAL_THEN`k''-1
IN {i | i < 3 /\
scs_J_v39 s'' i (SUC i)}`ASSUME_TAC;
ASM_REWRITE_TAC[
IN_ELIM_THM;ARITH_RULE`3-1<3`]
THEN REPLICATE_TAC (132-110)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]);
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
IN_SING;ARITH_RULE`3-1=2`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[SYM
th])
THEN REPEAT STRIP_TAC
THEN SUBGOAL_THEN`d'' =
scs_d_v39 s'' `ASSUME_TAC
;
EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
;
MRESAL_TAC (GEN_ALL
QKNVMLB1)[`vv:num->real^3`;`s:scs_v39`;`q:num`;`p:num`;`d'':real`;`mkj:bool`]
[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;]
THEN REPLICATE_TAC (134-9)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN REWRITE_TAC[REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`2`;`SUC 2`])
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`SUC 2 MOD 3=0/\ 0+A=A/\ 2 MOD 3=2`]
THEN REPLICATE_TAC (136-117)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`])
THEN RESA_TAC
THEN REAL_ARITH_TAC;
MP_TAC(VECTOR_ARITH`vv (q MOD k) - vv (p MOD k) = --(vv (p MOD k) - (vv:num->real^3) (q MOD k))`)
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ONCE_REWRITE_TAC[
th])
THEN REWRITE_TAC[
NORM_NEG];
ASM_REWRITE_TAC[ARITH_RULE`a*b+c*b=(a+c)*b`]
THEN MATCH_MP_TAC
REAL_LE_MUL
THEN STRIP_TAC;
REPLICATE_TAC (124-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN REAL_ARITH_TAC
;
REPLICATE_TAC (124-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
;
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
is_ear_v39]
THEN STRIP_TAC
THEN SUBGOAL_THEN`k'-1
IN {i | i < 3 /\
scs_J_v39 s' i (SUC i)}`ASSUME_TAC;
ASM_REWRITE_TAC[
IN_ELIM_THM;ARITH_RULE`3-1<3`]
THEN REPLICATE_TAC (132-88)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]);
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
IN_SING;ARITH_RULE`3-1=2`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[SYM
th])
THEN REPEAT STRIP_TAC
THEN SUBGOAL_THEN`d' =
scs_d_v39 s' `ASSUME_TAC
;
EXPAND_TAC"s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
;
MP_TAC
QKNVMLB1
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;]
THEN REPEAT RESA_TAC
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`2`;`SUC 2`])
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`SUC 2 MOD 3=0/\ 0+A=A/\ 2 MOD 3=2`]
THEN REPLICATE_TAC (136-117)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`])
THEN RESA_TAC
THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
is_ear_v39]
THEN STRIP_TAC
THEN SUBGOAL_THEN`k''-1
IN {i | i < 3 /\
scs_J_v39 s'' i (SUC i)}`ASSUME_TAC;
ASM_REWRITE_TAC[
IN_ELIM_THM;ARITH_RULE`3-1<3`]
THEN REPLICATE_TAC (132-110)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]);
POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
IN_SING;ARITH_RULE`3-1=2`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[SYM
th])
THEN REPEAT STRIP_TAC
THEN SUBGOAL_THEN`d'' =
scs_d_v39 s'' `ASSUME_TAC
;
EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
;
MRESAL_TAC (GEN_ALL
QKNVMLB1)[`vv:num->real^3`;`s:scs_v39`;`q:num`;`p:num`;`d'':real`;`mkj:bool`]
[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;]
THEN REPLICATE_TAC (134-9)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN REWRITE_TAC[REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`2`;`SUC 2`])
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`SUC 2 MOD 3=0/\ 0+A=A/\ 2 MOD 3=2`]
THEN REPLICATE_TAC (136-117)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[ARITH_RULE`3-1=2`])
THEN RESA_TAC
THEN REAL_ARITH_TAC;
ASM_TAC
THEN REWRITE_TAC[periodic2]
THEN REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
SCS_HALF_SLICE_IS_A_SCS
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2]
THEN ASM_SIMP_TAC[REAL_ARITH`&0<= A+B -C<=> C<= A+B`]
THEN REPLICATE_TAC (64-58)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (2)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th
THEN ASM_REWRITE_TAC[])
THEN REPLICATE_TAC (64-58)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (2)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC( LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN RESA_TAC
THEN REPLICATE_TAC (81-72)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`0`;`k'-1`;][ARITH_RULE`0+A=A`])
THEN SUBGOAL_THEN`(!i.
scs_b_v39 s i (SUC i) <= cstab)`ASSUME_TAC;
REPLICATE_TAC (83-24)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
MRESAL_TAC(GEN_ALL
SCS_HALF_SLICE_IS_A_SCS)[`s:scs_v39`;`s':scs_v39`;`q:num`;`p:num`;`s'':scs_v39`]
[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2]
THEN REPLICATE_TAC (85-35)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th->
ASM_TAC
THEN REWRITE_TAC[
th]
THEN ASSUME_TAC
th THEN REPEAT RESA_TAC)
THEN REPLICATE_TAC (85-36)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN POP_ASSUM MP_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN ASM_REWRITE_TAC[SET_RULE`A\/B<=> B\/A`]
THEN REPLICATE_TAC (84-63)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC(LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th
THEN ASM_REWRITE_TAC[])
THEN REPLICATE_TAC (84-62)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REPLICATE_TAC (1)(POP_ASSUM MP_TAC)
THEN GEN_REWRITE_TAC( LAND_CONV o LAND_CONV o LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)[
th]
THEN ASSUME_TAC
th)
THEN ASM_REWRITE_TAC[REAL_ARITH`a<=b+c<=> a<= c+b`]
THEN STRIP_TAC
THEN REPLICATE_TAC (101-92)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MRESAL_TAC
th[`0`;`k''-1`;][ARITH_RULE`0+A=A`])
THEN REPLICATE_TAC (102-57)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN RESA_TAC
THEN REWRITE_TAC[SET_RULE`{a|F}={}`;
SUM_SING;
SUM_CLAUSES]
THEN REAL_ARITH_TAC;
]);;
let SUM_AZIM_EQ_ANGLE_EQ4=prove_by_refinement(
`(k'=3)/\
scs_half_slice_v39 s p q d' mkj =s'/\
scs_half_slice_v39 s q p d'' mkj =s'' /\
vv' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s')+p MOD (
scs_k_v39 s)))
/\
vv'' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s'')+q MOD (
scs_k_v39 s)))
/\
MMs_v39 s vv /\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q /\
is_scs_slice_v39 s s' s'' p q /\
scs_k_v39 s' =k' /\
scs_k_v39 s =k
/\(vv:num->real^3) (p MOD k)=u /\
(vv:num->real^3) (q MOD k)=w /\
IMAGE (vv:num->real^3) (:num)=V/\
IMAGE (\i. {vv i,(vv:num->real^3) (SUC i)}) (:num) =E/\
IMAGE (\i. (vv i,(vv:num->real^3) (SUC i))) (:num)=
FF /\
norm (u - w) <= cstab /\
convex_local_fan
(
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)),
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)),
face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
==>
tau3 (vv (0 MOD k' + p MOD k)) (vv (1 MOD k' + p MOD k))
(vv (2 MOD k' + p MOD k))
=
sum {i | i < k'}
(\i.
rho_fun (
norm (vv (i + p MOD k))) *
interior_angle1 (
vec 0)
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
(
ITER i (
rho_node1 FF) u))
- (
pi + sol0)`,
[
REWRITE_TAC[
LET_DEF;
LET_END_DEF;tau3;
MMs_v39;
BBprime2_v39;
BBprime_v39;]
THEN STRIP_TAC
THEN MATCH_MP_TAC(REAL_ARITH`a1+a2+a3=b==> a1+a2+a3-c= b-c`)
THEN ASM_REWRITE_TAC[
NUMSEG_2;
SUM_NUMSEG2;ARITH_RULE`0 MOD 3=0/\1 MOD 3=1/\ 2 MOD 3=2`;Appendix.rho_rho_fun]
THEN MP_TAC
V_PRIME_EQ_V_vv
THEN RESA_TAC
THEN MP_TAC
E_PRIME_EQ_E_vv
THEN RESA_TAC
THEN MP_TAC
F_PRIME_EQ_F_vv
THEN RESA_TAC
THEN MP_TAC
SCS_HALF_SLICE_IS_A_SCS
THEN RESA_TAC
THEN SUBGOAL_THEN`
BBs_v39 s' vv'`ASSUME_TAC;
MP_TAC
QKNVMLB1
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
IN;
mk_unadorned_v39;]
THEN RESA_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
LET_DEF;
LET_END_DEF;tau3;
MMs_v39;
BBprime2_v39;
BBprime_v39;
is_scs_slice_v39]
THEN REPEAT RESA_TAC
THEN SUBGOAL_THEN`d' =
scs_d_v39 s'`ASSUME_TAC
;
EXPAND_TAC"s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
;
EXPAND_TAC"s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
;
MRESAL_TAC (GEN_ALL(INST_TYPE [`:3`,`:M`]
V_E_FF_IS_SCS_CASES_3))[`s':scs_v39`;`vv':num->real^3`]
[DIMINDEX_3]
THEN MRESA_TAC(GEN_ALL
VV_SUC_EQ_RHO_NODE)[`
scs_k_v39 s`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`p:num`;`q:num`;`s:scs_v39`;`FF:real^3#real^3->bool`;`u:real^3`;`vv:num->real^3`;`p:num MOD k`]
THEN MRESA_TAC (GEN_ALL Nkezbfc_local.CONVEX_LOFA_IMP_INANGLE_EQ_AZIM_IVS)
[`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`]
THEN MATCH_MP_TAC(REAL_ARITH`a1= b1 /\ a2= b2 /\a3= b3 ==> a1+a2+a3= b1+b2+b3`)
THEN REPEAT STRIP_TAC
THEN REWRITE_TAC[
REAL_EQ_MUL_LCANCEL]
THEN MATCH_MP_TAC(SET_RULE`A==>B\/A`)
;
POP_ASSUM MP_TAC
THEN SUBGOAL_THEN`vv (0 + p MOD k)
IN
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
ASSUME_TAC;
REPLICATE_TAC (33-26)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REWRITE_TAC[SYM
th;
IN_ELIM_THM;
IMAGE])
THEN EXISTS_TAC`0`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`];
SUBGOAL_THEN`vv (1 + p MOD k)
IN
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
ASSUME_TAC;
REPLICATE_TAC (34-26)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REWRITE_TAC[SYM
th;
IN_ELIM_THM;
IMAGE])
THEN EXISTS_TAC`1`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`];
SUBGOAL_THEN`vv (2 + p MOD k)
IN
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
ASSUME_TAC;
REPLICATE_TAC (35-26)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REWRITE_TAC[SYM
th;
IN_ELIM_THM;
IMAGE])
THEN EXISTS_TAC`2`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`];
REPLICATE_TAC (3)(POP_ASSUM MP_TAC)
THEN REPLICATE_TAC (4) RESA_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`(vv:num->real^3) (0 + p MOD k)`)
THEN SUBGOAL_THEN`(
rho_node1
{(vv (0 MOD 3 + p MOD k),vv (1 MOD 3 + p MOD k)), (vv (1 MOD 3 + p MOD k),
vv (2 MOD 3 + p MOD k)),
(vv (2 MOD 3 + p MOD k),
vv (0 MOD 3 + p MOD k))}
(vv (0 + p MOD k))) = (vv:num->real^3) (1 MOD 3 + p MOD k)`ASSUME_TAC
;
MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN EXISTS_TAC`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (37-25)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
convex_local_fan]
THEN RESA_TAC)
THEN REWRITE_TAC[SET_RULE`a
IN {a1,b,c} <=> a=a1\/ a=b\/ a=c`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
;
ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN`(
ivs_rho_node1
{(vv (0 MOD 3 + p MOD k),vv (1 MOD 3 + p MOD k)), (vv (1 MOD 3 + p MOD k),
vv (2 MOD 3 + p MOD k)),
(vv (2 MOD 3 + p MOD k),
vv (0 MOD 3 + p MOD k))}
(vv (0 + p MOD k))) = (vv:num->real^3) (2 MOD 3 + p MOD k)`ASSUME_TAC
;
MATCH_MP_TAC (GEN_ALL Local_lemmas.IVS_RHO_NODE1_DETE)
THEN EXISTS_TAC`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN EXISTS_TAC`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (38-25)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
convex_local_fan]
THEN RESA_TAC)
THEN REWRITE_TAC[SET_RULE`a
IN {a1,b,c} <=> a=a1\/ a=b\/ a=c`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
;
SUBGOAL_THEN`(
rho_node1
{(vv (0 MOD 3 + p MOD k),vv (1 MOD 3 + p MOD k)), (vv (1 MOD 3 + p MOD k),
vv (2 MOD 3 + p MOD k)),
(vv (2 MOD 3 + p MOD k),
vv (0 MOD 3 + p MOD k))}
(vv (2 + p MOD k))) = (vv:num->real^3) (0 MOD 3 + p MOD k)`ASSUME_TAC
;
MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN EXISTS_TAC`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (39-25)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
convex_local_fan]
THEN RESA_TAC)
THEN REWRITE_TAC[SET_RULE`a
IN {a1,b,c} <=> a=a1\/ a=b\/ a=c`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
;
ASM_REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN MRESA_TAC (GEN_ALL Local_lemmas.IN_V_IMP_AZIM_LESS_PI)[`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`(vv:num->real^3) (0 + p MOD k)`]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`(vv:num->real^3) (2 + p MOD k)`)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN STRIP_TAC
THEN SUBGOAL_THEN`
local_fan
(
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)),
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)),
face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
`ASSUME_TAC
;
ASM_TAC
THEN REWRITE_TAC[
convex_local_fan]
THEN REPEAT RESA_TAC
;
MRESA_TAC (GEN_ALL Local_lemmas.LOCAL_FAN_CHARACTER_OF_RHO_NODE2)
[`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`(vv:num->real^3) (0 + p MOD k)`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL Local_lemmas.LOCAL_FAN_CHARACTER_OF_RHO_NODE2)
[`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`(vv:num->real^3) (2 + p MOD k)`]THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL Local_lemmas.AZIM_LE_PI_EQ_DIHV)[`
vec 0:real^3`;`(vv:num->real^3) (0 + p MOD k)`;`(vv:num->real^3) (1 + p MOD k)`;`(vv:num->real^3) (2 + p MOD k)`]
;
(******************)
POP_ASSUM MP_TAC
THEN SUBGOAL_THEN`vv (0 + p MOD k)
IN
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
ASSUME_TAC;
REPLICATE_TAC (33-26)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REWRITE_TAC[SYM
th;
IN_ELIM_THM;
IMAGE])
THEN EXISTS_TAC`0`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`];
SUBGOAL_THEN`vv (1 + p MOD k)
IN
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
ASSUME_TAC;
REPLICATE_TAC (34-26)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REWRITE_TAC[SYM
th;
IN_ELIM_THM;
IMAGE])
THEN EXISTS_TAC`1`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`];
SUBGOAL_THEN`vv (2 + p MOD k)
IN
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
ASSUME_TAC;
REPLICATE_TAC (35-26)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REWRITE_TAC[SYM
th;
IN_ELIM_THM;
IMAGE])
THEN EXISTS_TAC`2`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`];
REPLICATE_TAC (3)(POP_ASSUM MP_TAC)
THEN REPLICATE_TAC (4) RESA_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`(vv:num->real^3) (1 + p MOD k)`)
THEN SUBGOAL_THEN`(
rho_node1
{(vv (0 MOD 3 + p MOD k),vv (1 MOD 3 + p MOD k)), (vv (1 MOD 3 + p MOD k),
vv (2 MOD 3 + p MOD k)),
(vv (2 MOD 3 + p MOD k),
vv (0 MOD 3 + p MOD k))}
(vv (1 + p MOD k))) = (vv:num->real^3) (2 MOD 3 + p MOD k)`ASSUME_TAC
;
MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN EXISTS_TAC`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (37-25)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
convex_local_fan]
THEN RESA_TAC)
THEN REWRITE_TAC[SET_RULE`a
IN {a1,b,c} <=> a=a1\/ a=b\/ a=c`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
;
ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN`(
ivs_rho_node1
{(vv (0 MOD 3 + p MOD k),vv (1 MOD 3 + p MOD k)), (vv (1 MOD 3 + p MOD k),
vv (2 MOD 3 + p MOD k)),
(vv (2 MOD 3 + p MOD k),
vv (0 MOD 3 + p MOD k))}
(vv (1 + p MOD k))) = (vv:num->real^3) (0 MOD 3 + p MOD k)`ASSUME_TAC
;
MATCH_MP_TAC (GEN_ALL Local_lemmas.IVS_RHO_NODE1_DETE)
THEN EXISTS_TAC`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN EXISTS_TAC`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (38-25)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
convex_local_fan]
THEN RESA_TAC)
THEN REWRITE_TAC[SET_RULE`a
IN {a1,b,c} <=> a=a1\/ a=b\/ a=c`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
;
SUBGOAL_THEN`(
rho_node1
{(vv (0 MOD 3 + p MOD k),vv (1 MOD 3 + p MOD k)), (vv (1 MOD 3 + p MOD k),
vv (2 MOD 3 + p MOD k)),
(vv (2 MOD 3 + p MOD k),
vv (0 MOD 3 + p MOD k))}
(vv (0 + p MOD k))) = (vv:num->real^3) (1 MOD 3 + p MOD k)`ASSUME_TAC
;
MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN EXISTS_TAC`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (39-25)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
convex_local_fan]
THEN RESA_TAC)
THEN REWRITE_TAC[SET_RULE`a
IN {a1,b,c} <=> a=a1\/ a=b\/ a=c`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
;
ASM_REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN MRESA_TAC (GEN_ALL Local_lemmas.IN_V_IMP_AZIM_LESS_PI)[`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`(vv:num->real^3) (1 + p MOD k)`]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`(vv:num->real^3) (0 + p MOD k)`)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN STRIP_TAC
THEN SUBGOAL_THEN`
local_fan
(
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)),
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)),
face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
`ASSUME_TAC
;
ASM_TAC
THEN REWRITE_TAC[
convex_local_fan]
THEN REPEAT RESA_TAC
;
MRESA_TAC (GEN_ALL Local_lemmas.LOCAL_FAN_CHARACTER_OF_RHO_NODE2)
[`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`(vv:num->real^3) (1 + p MOD k)`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL Local_lemmas.LOCAL_FAN_CHARACTER_OF_RHO_NODE2)
[`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`(vv:num->real^3) (0 + p MOD k)`]THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL Local_lemmas.AZIM_LE_PI_EQ_DIHV)[`
vec 0:real^3`;`(vv:num->real^3) (1 + p MOD k)`;`(vv:num->real^3) (2 + p MOD k)`;`(vv:num->real^3) (0 + p MOD k)`]
;
(************)
POP_ASSUM MP_TAC
THEN SUBGOAL_THEN`vv (0 + p MOD k)
IN
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
ASSUME_TAC;
REPLICATE_TAC (33-26)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REWRITE_TAC[SYM
th;
IN_ELIM_THM;
IMAGE])
THEN EXISTS_TAC`0`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`];
SUBGOAL_THEN`vv (1 + p MOD k)
IN
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
ASSUME_TAC;
REPLICATE_TAC (34-26)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REWRITE_TAC[SYM
th;
IN_ELIM_THM;
IMAGE])
THEN EXISTS_TAC`1`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`];
SUBGOAL_THEN`vv (2 + p MOD k)
IN
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
ASSUME_TAC;
REPLICATE_TAC (35-26)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN REWRITE_TAC[SYM
th;
IN_ELIM_THM;
IMAGE])
THEN EXISTS_TAC`2`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`];
REPLICATE_TAC (3)(POP_ASSUM MP_TAC)
THEN REPLICATE_TAC (4) RESA_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`(vv:num->real^3) (2 + p MOD k)`)
THEN SUBGOAL_THEN`(
rho_node1
{(vv (0 MOD 3 + p MOD k),vv (1 MOD 3 + p MOD k)), (vv (1 MOD 3 + p MOD k),
vv (2 MOD 3 + p MOD k)),
(vv (2 MOD 3 + p MOD k),
vv (0 MOD 3 + p MOD k))}
(vv (2 + p MOD k))) = (vv:num->real^3) (0 MOD 3 + p MOD k)`ASSUME_TAC
;
MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN EXISTS_TAC`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (37-25)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
convex_local_fan]
THEN RESA_TAC)
THEN REWRITE_TAC[SET_RULE`a
IN {a1,b,c} <=> a=a1\/ a=b\/ a=c`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
;
ASM_REWRITE_TAC[]
THEN SUBGOAL_THEN`(
ivs_rho_node1
{(vv (0 MOD 3 + p MOD k),vv (1 MOD 3 + p MOD k)), (vv (1 MOD 3 + p MOD k),
vv (2 MOD 3 + p MOD k)),
(vv (2 MOD 3 + p MOD k),
vv (0 MOD 3 + p MOD k))}
(vv (2 + p MOD k))) = (vv:num->real^3) (1 MOD 3 + p MOD k)`ASSUME_TAC
;
MATCH_MP_TAC (GEN_ALL Local_lemmas.IVS_RHO_NODE1_DETE)
THEN EXISTS_TAC`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN EXISTS_TAC`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (38-25)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
convex_local_fan]
THEN RESA_TAC)
THEN REWRITE_TAC[SET_RULE`a
IN {a1,b,c} <=> a=a1\/ a=b\/ a=c`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
;
SUBGOAL_THEN`(
rho_node1
{(vv (0 MOD 3 + p MOD k),vv (1 MOD 3 + p MOD k)), (vv (1 MOD 3 + p MOD k),
vv (2 MOD 3 + p MOD k)),
(vv (2 MOD 3 + p MOD k),
vv (0 MOD 3 + p MOD k))}
(vv (1 + p MOD k))) = (vv:num->real^3) (2 MOD 3 + p MOD k)`ASSUME_TAC
;
MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN EXISTS_TAC`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`
THEN ASM_REWRITE_TAC[]
THEN REPLICATE_TAC (39-25)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[
convex_local_fan]
THEN RESA_TAC)
THEN REWRITE_TAC[SET_RULE`a
IN {a1,b,c} <=> a=a1\/ a=b\/ a=c`;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
;
ASM_REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN MRESA_TAC (GEN_ALL Local_lemmas.IN_V_IMP_AZIM_LESS_PI)[`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`(vv:num->real^3) (2 + p MOD k)`]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`(vv:num->real^3) (1 + p MOD k)`)
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN STRIP_TAC
THEN SUBGOAL_THEN`
local_fan
(
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)),
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)),
face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
`ASSUME_TAC
;
ASM_TAC
THEN REWRITE_TAC[
convex_local_fan]
THEN REPEAT RESA_TAC
;
MRESA_TAC (GEN_ALL Local_lemmas.LOCAL_FAN_CHARACTER_OF_RHO_NODE2)
[`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`(vv:num->real^3) (2 + p MOD k)`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL Local_lemmas.LOCAL_FAN_CHARACTER_OF_RHO_NODE2)
[`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`(vv:num->real^3) (1 + p MOD k)`]THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1/\ 2 MOD 3=2`]
THEN ONCE_REWRITE_TAC[SET_RULE`{A,B,C}={A,C,B}`]
THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL Local_lemmas.AZIM_LE_PI_EQ_DIHV)[`
vec 0:real^3`;`(vv:num->real^3) (2 + p MOD k)`;`(vv:num->real^3) (0 + p MOD k)`;`(vv:num->real^3) (1 + p MOD k)`]
;]);;
let CARD_FF_EQ_CARD_SLICE_FF= prove_by_refinement(
`
convex_local_fan (V,E,
FF) /\
v
IN V /\
w
IN V /\
~(v = w) /\
(!x. x
IN FF ==> aff_gt {
vec 0} {v, w}
SUBSET wedge_in_fan_gt x E) /\
HS = hypermap (
HYP (
vec 0,V,E
UNION {{v, w}})) /\
fv = face HS (v,
rho_node1 FF v) /\
fw = face HS (w,
rho_node1 FF w)
==>
CARD FF =
CARD fv +
CARD fw -2`,
[
STRIP_TAC
THEN MP_TAC Local_lemmas1.PROVE_THE_SLICE_ASSUMPTION
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`v:real^3`;`w:real^3`])
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
DIFF;
IN_ELIM_THM;
IN_SING]
THEN ONCE_REWRITE_TAC[SET_RULE`~(a=b)<=> ~(b=a)`]
THEN STRIP_TAC
THEN MP_TAC Nkezbfc_local.COMPATIBLE_BW_TWO_LEMMAS
THEN RESA_TAC
THEN SUBGOAL_THEN`{{w, v}} = {{v,w:real^3}}` ASSUME_TAC;
REWRITE_TAC[
EXTENSION;
IN_SING]
THEN SET_TAC[];
MRESA_TAC (GEN_ALL Nkezbfc_local.COMPATIBLE_BW_TWO_LEMMAS)
[`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{v, w}}))`;`V:real^3->bool`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{v, w}}))) (w,
rho_node1 FF w)`;`(E:(real^3->bool)->bool)`;`FF:real^3#real^3->bool`;`w:real^3`;`v:real^3`;]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`~(a=b)<=> ~(b=a)`]
THEN ONCE_REWRITE_TAC[SET_RULE`{a,b}={b,a}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{a,b}={b,a}`]
THEN RESA_TAC
THEN MRESA_TAC
CARD_SLICE_EQ[`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`FF:real^3#real^3->bool`;`v:real^3`;`w:real^3`;]
THEN MP_TAC Local_lemmas1.EJRCFJD
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ)
[`E:(real^3->bool)->bool`;`FF:real^3#real^3->bool`;`V:real^3->bool`]
THEN MRESA_TAC (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ)
[`
e_prime (E
UNION {{v, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (w,
rho_node1 FF w))`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (w,
rho_node1 FF w)`;`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (w,
rho_node1 FF w))`]
THEN MRESA_TAC (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ)
[`
e_prime (E
UNION {{v, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (v,
rho_node1 FF v))`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (v,
rho_node1 FF v)`;`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (v,
rho_node1 FF v))`]
THEN ASM_TAC
THEN REWRITE_TAC[
convex_local_fan]
THEN REPEAT RESA_TAC
THEN REPLICATE_TAC (25-16)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN MP_TAC
th
THEN REPEAT RESA_TAC)
THEN REPLICATE_TAC (25-13)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN MP_TAC
th
THEN REPEAT RESA_TAC)]);;
let CARD_SLICE_FF_LE_3= prove_by_refinement(
`
convex_local_fan (V,E,
FF) /\
v
IN V /\
w
IN V /\
~(v = w) /\
(!x. x
IN FF ==> aff_gt {
vec 0} {v, w}
SUBSET wedge_in_fan_gt x E) /\
HS = hypermap (
HYP (
vec 0,V,E
UNION {{v, w}})) /\
fv = face HS (v,
rho_node1 FF v) /\
fw = face HS (w,
rho_node1 FF w)
==> 3<=
CARD fv /\ 3<=
CARD fw`,
[
STRIP_TAC
THEN MP_TAC Local_lemmas1.PROVE_THE_SLICE_ASSUMPTION
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`v:real^3`;`w:real^3`])
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
DIFF;
IN_ELIM_THM;
IN_SING]
THEN ONCE_REWRITE_TAC[SET_RULE`~(a=b)<=> ~(b=a)`]
THEN STRIP_TAC
THEN MP_TAC Nkezbfc_local.COMPATIBLE_BW_TWO_LEMMAS
THEN RESA_TAC
THEN SUBGOAL_THEN`{{w, v}} = {{v,w:real^3}}` ASSUME_TAC;
REWRITE_TAC[
EXTENSION;
IN_SING]
THEN SET_TAC[];
MRESA_TAC (GEN_ALL Nkezbfc_local.COMPATIBLE_BW_TWO_LEMMAS)
[`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{v, w}}))`;`V:real^3->bool`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{v, w}}))) (w,
rho_node1 FF w)`;`(E:(real^3->bool)->bool)`;`FF:real^3#real^3->bool`;`w:real^3`;`v:real^3`;]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[SET_RULE`~(a=b)<=> ~(b=a)`]
THEN ONCE_REWRITE_TAC[SET_RULE`{a,b}={b,a}`]
THEN ASM_REWRITE_TAC[]
THEN ONCE_REWRITE_TAC[SET_RULE`{a,b}={b,a}`]
THEN RESA_TAC
THEN MRESA_TAC
CARD_SLICE_EQ[`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`FF:real^3#real^3->bool`;`v:real^3`;`w:real^3`;]
THEN MP_TAC Local_lemmas1.EJRCFJD
THEN RESA_TAC
THEN MRESA_TAC (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ)
[`E:(real^3->bool)->bool`;`FF:real^3#real^3->bool`;`V:real^3->bool`]
THEN MRESA_TAC (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ)
[`
e_prime (E
UNION {{v, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (w,
rho_node1 FF w))`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (w,
rho_node1 FF w)`;`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (w,
rho_node1 FF w))`]
THEN MRESA_TAC (GEN_ALL Local_lemmas.LOFA_IMP_CARD_FF_V_EQ)
[`
e_prime (E
UNION {{v, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (v,
rho_node1 FF v))`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (v,
rho_node1 FF v)`;`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{v, w}}))) (v,
rho_node1 FF v))`]
THEN ASM_TAC
THEN REWRITE_TAC[
convex_local_fan]
THEN REPEAT RESA_TAC
THEN REPLICATE_TAC (25-16)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN MP_TAC
th
THEN REPEAT RESA_TAC)
THEN REPLICATE_TAC (25-13)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN REWRITE_TAC[
th]
THEN MP_TAC
th
THEN REPEAT RESA_TAC)
THEN MRESA_TAC (GEN_ALL Nkezbfc_local.CARD_VERTEX_GE_3_LOCAL_FAN)
[`slicev (E:(real^3->bool)->bool)
FF w v`;`slicee (E:(real^3->bool)->bool)
FF w v`;`slicef (E:(real^3->bool)->bool)
FF w v`]
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC (GEN_ALL Nkezbfc_local.CARD_VERTEX_GE_3_LOCAL_FAN)
[`slicev (E:(real^3->bool)->bool)
FF v w`;`slicee (E:(real^3->bool)->bool)
FF v w`;`slicef (E:(real^3->bool)->bool)
FF v w`]
THEN POP_ASSUM MP_TAC
THEN REWRITE_TAC[
convex_local_fan]
THEN RESA_TAC;]);;
let QKNVMLB3_LE4=prove_by_refinement(
`~(
scs_k_v39 s =4 /\ ~(
scs_J_v39 s' 0 (
scs_k_v39 s' - 1)))/\
scs_half_slice_v39 s p q d' mkj =s'/\
scs_half_slice_v39 s q p d'' mkj =s'' /\
vv' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s')+p MOD (
scs_k_v39 s)))
/\
vv'' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s'')+q MOD (
scs_k_v39 s)))
/\
MMs_v39 s vv /\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q /\
is_scs_slice_v39 s s' s'' p q /\
scs_d_v39 s <= d' + d'' ==>
taustar_v39 s' vv' +
taustar_v39 s'' vv'' <=
taustar_v39 s vv`,
[REWRITE_TAC[
taustar_v39;
LET_DEF;
LET_END_DEF;
scs_diag]
THEN STRIP_TAC
THEN MP_TAC
SCS_K_PRIME_CASE_3
THEN REWRITE_TAC[
taustar_v39;
LET_DEF;
LET_END_DEF;
scs_diag]
THEN RESA_TAC
THEN ABBREV_TAC`k=
scs_k_v39 s`
THEN ABBREV_TAC`k'=
scs_k_v39 s'`
THEN ABBREV_TAC`k''=
scs_k_v39 s''`
THEN MP_TAC(ARITH_RULE`3<k==> ~(k<=3)`)
THEN RESA_TAC
THEN MP_TAC
SCS_K_LE_6
THEN RESA_TAC
THEN ABBREV_TAC`u= (vv:num->real^3) (p MOD k)`
THEN ABBREV_TAC `w= (vv:num->real^3) (q MOD k)`
THEN ABBREV_TAC`V=
IMAGE (vv:num->real^3) (:num)`
THEN ABBREV_TAC`E=
IMAGE (\i. {vv i,(vv:num->real^3) (SUC i)}) (:num)`
THEN ABBREV_TAC`
FF=
IMAGE (\i. (vv i,(vv:num->real^3) (SUC i))) (:num)`
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`p MOD
scs_k_v39 s`;`vv:num->real^3`]
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`q MOD
scs_k_v39 s`;`vv:num->real^3`]
THEN ASM_TAC
THEN REPLICATE_TAC 2 STRIP_TAC
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime_v39;
BBprime2_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN REPEAT RESA_TAC
;
REPLICATE_TAC (54-8)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[];
REPLICATE_TAC (54-8)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC
THEN SUBGOAL_THEN`(!i.
scs_b_v39 s i (SUC i) <= cstab)`ASSUME_TAC;
REPLICATE_TAC (54-32)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MATCH_MP_TAC
th)
THEN ASM_REWRITE_TAC[];
REPLICATE_TAC (55-8)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[ARITH_RULE`SUC(3-1)=3`])
THEN STRIP_TAC
THEN MP_TAC
NOT_EQ_DIAG
THEN MP_TAC
DIST_DIAG_LE_CSTAB
THEN MP_TAC
CARD_V_EQ_SCS_K
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime_v39;
BBprime2_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN STRIP_TAC
THEN REPEAT RESA_TAC
THEN MP_TAC
TECOXBMv2
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
tau_fun]
THEN STRIP_TAC
THEN MRESA_TAC(GEN_ALL Local_lemmas1.EJRCFJD)[`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))`;`(E:(real^3->bool)->bool)`;`w:real^3`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`V:real^3->bool`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (w,
rho_node1 FF w)`;`FF:real^3#real^3->bool`;`u:real^3`]
THEN POP_ASSUM MP_TAC
THEN MP_TAC
V_PRIME_EQ_V_vv
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASSUME_TAC(SYM
th))
THEN MRESA_TAC (GEN_ALL
V_PRIME_EQ_V_vv)
[`d'':real`;`d':real`;`mkj:bool`;`s':scs_v39`;`s'':scs_v39`;`p:num`;`s:scs_v39`; `vv:num->real^3`;`k'':num`;`q:num`;`k:num`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`u:real^3`;`FF:real^3#real^3->bool`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39;]
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`u-w= --(w-u):real^3`]
THEN ASM_REWRITE_TAC[
NORM_NEG]
THEN SUBGOAL_THEN`{{w, u}} = {{u,w:real^3}}` ASSUME_TAC
;
REWRITE_TAC[
EXTENSION;
IN_SING]
THEN SET_TAC[];
RESA_TAC
THEN POP_ASSUM(fun th-> ASSUME_TAC(SYM
th))
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`(\i.
rho_fun (
norm ((vv:num->real^3) (i+ p MOD k))))`)
THEN SUBGOAL_THEN`
sum {i | i < k}
(\i.
rho_fun (
norm ((vv:num->real^3) (i+p MOD k))) *
interior_angle1 (
vec 0)
FF (
ITER i (
rho_node1 FF) u))
=
sum FF (\e.
rho_fun (
norm (
FST e)) *
azim_in_fan e E)
`ASSUME_TAC
;
MATCH_MP_TAC
SUM_EQ_GENERAL
THEN EXISTS_TAC`(\i. (vv:num->real^3) (i+ p MOD k), vv (SUC i + p MOD k))`
THEN SUBGOAL_THEN` (q+1+k -p MOD k)MOD k =
scs_k_v39 s'` ASSUME_TAC;
EXPAND_TAC"s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
;
MRESA_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`p:num`;`q:num`;`d':real`;`
scs_J_v39 s' p q`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
scs_half_slice_v39;
PAIR_EQ;REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`]
THEN STRIP_TAC
THEN MP_TAC(ARITH_RULE`3<= k' /\ k'<k ==> 0<k' /\ ~(k=0) /\ k'-1<k'/\ 1<k`)
THEN RESA_TAC
THEN STRIP_TAC
;
EXPAND_TAC"FF"
THEN REWRITE_TAC[
IN_ELIM_THM;
IMAGE;
EXISTS_UNIQUE]
THEN ASM_REWRITE_TAC[
PAIR_EQ]
THEN REPEAT RESA_TAC
THEN EXISTS_TAC`(x MOD k + k -p MOD k) MOD k`
THEN ASM_REWRITE_TAC[ARITH_RULE`A-0=A`;
PAIR_EQ]
THEN MRESA_TAC
DIVISION[`x MOD k + k - p MOD k:num`;`k:num`]
THEN MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN MRESA_TAC
MOD_LT[`p MOD k`;`k:num`]
THEN MRESA_TAC
MOD_MOD_REFL[`x:num`;`k:num`]
THEN MRESAL_TAC
MOD_MULT_ADD[`1`;`k:num`;`x MOD k`][ARITH_RULE`1 * k + x MOD k = x MOD k +k`]
THEN MRESA_TAC
MOD_ADD_MOD[`x MOD k + k - p MOD k:num`;`p MOD k:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`1<k/\ p MOD k<k ==>(x MOD k+ k- p MOD k) + p MOD k = x MOD k+k /\ (x + p MOD k) + k- p MOD k= x+k`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x MOD k + k - p MOD k) MOD k + p MOD k:num`
THEN MRESA1_TAC
th ` x:num`)
THEN MRESA_TAC
MOD_ADD_MOD[`x:num`;`1`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`(x MOD k + k - p MOD k) MOD k + p MOD k`;`1`;`k:num`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;ARITH_RULE`SUC ((x MOD k + k - p MOD k) MOD k) + p MOD k = ((x MOD k + k - p MOD k) MOD k + p MOD k)+1`]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `((x MOD k + k - p MOD k) MOD k + p MOD k)+1:num`
THEN MRESA1_TAC
th ` x+1:num`)
THEN REWRITE_TAC[
ADD1]
THEN REPEAT STRIP_TAC
THEN MRESA_TAC
DIVISION[`x:num`;`k:num`]
THEN MRESA_TAC
DIVISION[`y' + p MOD k:num`;`k:num`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `y' + p MOD k:num`)
THEN MP_TAC
VV_INJ
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
scs_half_slice_v39;
PAIR_EQ;REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`;
dist]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`(y'+ p MOD k) MOD k:num`;`(x MOD k + k)MOD k`])
THEN MRESA_TAC
MOD_LT[`y':num`;`k:num`]
THEN MRESAL_TAC(GEN_ALL Hdplygy.MOD_EQ_MOD)[`y':num`;`x MOD k + k- p MOD k`;`p MOD k`;`k:num`][]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[
ADD_SYM]
THEN RESA_TAC
;
REWRITE_TAC[
IN_ELIM_THM;]
THEN REPEAT STRIP_TAC;
EXPAND_TAC"FF"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`x+ p MOD k`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN (:num)`;
ADD1;ARITH_RULE`(x + 1) + p MOD k= (x + p MOD k)+1`]
;
MRESAL_TAC(GEN_ALL
VV_SUC_EQ_RHO_NODE)[`
scs_k_v39 s`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`p:num`;`q:num`;`s:scs_v39`;`FF:real^3#real^3->bool`;`u:real^3`;`vv:num->real^3`;`p:num MOD k`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN SUBGOAL_THEN`(vv:num->real^3) (x + p MOD k)
IN V`ASSUME_TAC
;
EXPAND_TAC"V"
THEN REWRITE_TAC[
IN_ELIM_THM;
IMAGE]
THEN EXISTS_TAC`x+ p MOD k`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN (:num)`];
MP_TAC Nkezbfc_local.CONVEX_LOFA_IMP_INANGLE_EQ_AZIM
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`(vv:num->real^3) (x + p MOD k )`])
THEN REMOVE_ASSUM_TAC
THEN REMOVE_ASSUM_TAC
THEN POP_ASSUM(fun th-> ASSUME_TAC
th
THEN MRESAL1_TAC
th`SUC x`[
ITER])
;
POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MP_TAC
F_PRIME_EQ_F_vv
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[
th])
THEN MRESAL_TAC (GEN_ALL
F_PRIME_EQ_F_vv)[`d'':real`;`d':real`;`mkj:bool`;`s':scs_v39`;`s'':scs_v39`;`p:num`;`s:scs_v39`;`vv:num->real^3`;`k'':num`;`q:num`;`k:num`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`u:real^3`;`FF:real^3#real^3->bool`;`w:real^3`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`u-w= --(w-u):real^3`]
THEN ASM_REWRITE_TAC[
NORM_NEG]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[
th])
THEN MP_TAC
V_SLICE_EQ_NUMSEG
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN RESA_TAC
THEN MP_TAC
SCS_SLICE_SYM
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39;]
THEN STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
V_SLICE_EQ_NUMSEG)[
`d'':real`;`mkj:bool`;`s:scs_v39`;`s'':scs_v39`;`s':scs_v39`;`p:num`;`FF:real^3#real^3->bool`;`w:real^3`;`vv:num->real^3`;`q:num`;`k:num`;`k'':num`]
[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39;]
THEN SUBGOAL_THEN`
sum
{i | i < k /\
ITER i (
rho_node1 FF) u
IN IMAGE (\i. vv (i MOD k'' + q MOD k)) (:num)}
(\i.
rho_fun (
norm ((vv:num->real^3) (i + p MOD k))) *
interior_angle1 (
vec 0)
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))
(
ITER i (
rho_node1 FF) u))
=
sum
{i | i < k /\
ITER i (
rho_node1 FF) w
IN IMAGE (\i. vv (i MOD k'' + q MOD k)) (:num)}
(\i.
rho_fun (
norm (vv (i + q MOD k))) *
interior_angle1 (
vec 0)
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))
(
ITER i (
rho_node1 FF) w))` ASSUME_TAC
;
MATCH_MP_TAC
SUM_EQ_GENERAL
THEN MRESAL_TAC(GEN_ALL
VV_SUC_EQ_RHO_NODE)[`
scs_k_v39 s`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`p:num`;`q:num`;`s:scs_v39`;`FF:real^3#real^3->bool`;`u:real^3`;`vv:num->real^3`;`p:num MOD k`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MRESAL_TAC(GEN_ALL
VV_SUC_EQ_RHO_NODE)[`
scs_k_v39 s`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`p:num`;`q:num`;`s:scs_v39`;`FF:real^3#real^3->bool`;`w:real^3`;`vv:num->real^3`;`q:num MOD k`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MP_TAC
SCS_K_PRIME_LE_GE
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN STRIP_TAC
THEN SUBGOAL_THEN` (p+1+k -q MOD k)MOD k =
scs_k_v39 s''` ASSUME_TAC;
EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[]
;
MRESAL_TAC (GEN_ALL
SCS_K_PRIME_LE_GE)[`s:scs_v39`;`q:num`;`p:num`;`d'':real`;`
scs_J_v39 s' p q`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39;
IN_ELIM_THM;
IMAGE]
THEN MP_TAC(ARITH_RULE`3<=k'/\ k'<k ==> ~(k=0)`)
THEN RESA_TAC
THEN MP_TAC(ARITH_RULE`p MOD k<= q MOD k \/ q MOD k<= p MOD k`)
THEN RESA_TAC;
EXISTS_TAC`(\i. (i+ p MOD k + k - q MOD k) MOD k)`
THEN STRIP_TAC;
REWRITE_TAC[
IN_ELIM_THM;
IMAGE;
EXISTS_UNIQUE]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`(y + q MOD k - p MOD k) MOD k `
THEN MRESA_TAC
DIVISION[`q:num`;`k:num`]
THEN MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN MRESA_TAC
DIVISION[`y + q MOD k - p MOD k:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`y<k'' /\ k''<k/\ q MOD k<k /\ p MOD k<= q MOD k /\ ~(p MOD k = q MOD k) ==> y<k:num /\ p MOD k + k - q MOD k =k -(q MOD k - p MOD k)
/\ k- (q MOD k - p MOD k)<k /\ (y + q MOD k - p MOD k) + k - (q MOD k - p MOD k) = y+ k/\ (y + q MOD k - p MOD k) + p MOD k= y + q MOD k
`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`y:num`;`k'':num`]
THEN MRESA_TAC
MOD_LT[`y:num`;`k:num`]
THEN MRESA_TAC
MOD_LT[`p MOD k:num`;`k:num`]
THEN MRESA_TAC
MOD_LT[`k-(q MOD k- p MOD k):num`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`y + q MOD k- p MOD k`;`k -(q MOD k - p MOD k)`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`y + q MOD k- p MOD k`;`(p MOD k)`;`k:num`]
THEN MRESAL_TAC
MOD_MULT_ADD[`1:num`;`k:num`;`y:num`][ARITH_RULE`1 * k' + i = (i+k')`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(y + q MOD k - p MOD k) MOD k + p MOD k:num`
THEN MRESA1_TAC
th`y+ q MOD k`)
THEN STRIP_TAC
;
EXISTS_TAC`y:num`
THEN ASM_REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`]
;
REPEAT STRIP_TAC
THEN POP_ASSUM (fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MP_TAC(ARITH_RULE`y<k'' /\ k''<k/\ q MOD k<k /\ p MOD k<= q MOD k /\ ~(p MOD k = q MOD k) ==> (y' + k - (q MOD k - p MOD k)) + q MOD k - p MOD k = y'+k /\ q MOD k - p MOD k < k`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`q MOD k - p MOD k`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`y' + k - (q MOD k - p MOD k)`;`q MOD k - p MOD k`;`k:num`]
THEN MRESAL_TAC
MOD_MULT_ADD[`1:num`;`k:num`;`y':num`][ARITH_RULE`1 * k' + i = (i+k')`]
THEN MRESA_TAC
MOD_LT[`y':num`;`k:num`]
;
REPEAT RESA_TAC
;
MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x + p MOD k):num`
THEN MRESA1_TAC
th`x' MOD k''+ q MOD k`)
THEN MP_TAC(ARITH_RULE`3<= k'' ==> ~(k''=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`q:num`;`k:num`]
THEN MRESA_TAC
DIVISION[`x':num`;`k'':num`]
THEN MRESA_TAC
DIVISION[`x +p MOD k`;`k:num`]
THEN MRESA_TAC
DIVISION[`x' MOD k'' +q MOD k`;`k:num`]
THEN MP_TAC(ARITH_RULE`p MOD k<= q MOD k /\ q MOD k < k/\ x' MOD k''<k'' /\ k''<k ==> p MOD k + x' MOD k''+ q MOD k - p MOD k = q MOD k +x' MOD k'' /\ (p MOD k + k - q MOD k) + q MOD k - p MOD k + x' MOD k'' = k + x' MOD k'' /\ x' MOD k'' < k`)
THEN RESA_TAC
THEN MP_TAC
VV_INJ
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
scs_half_slice_v39;
PAIR_EQ;REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`;
dist]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`(x+ p MOD k) MOD k:num`;`(x' MOD k'' + q MOD k)MOD k`])
THEN MRESAL_TAC(GEN_ALL Hdplygy.MOD_EQ_MOD)[`x:num`;`x' MOD k'' + q MOD k - p MOD k`;`p MOD k`;`k:num`][]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[
ADD_SYM]
THEN RESA_TAC
THEN MRESA_TAC
MOD_ADD_MOD[`(p MOD k + k - q MOD k)`;`x:num`;`k:num`]
THEN POP_ASSUM (fun th-> REWRITE_TAC[SYM
th])
THEN MRESA_TAC
MOD_ADD_MOD[`(p MOD k + k - q MOD k)`;`q MOD k - p MOD k + x' MOD k'':num`;`k:num`]
THEN MRESAL_TAC
MOD_MULT_ADD[`1:num`;`k:num`;`x' MOD k'':num`][ARITH_RULE`1 * k' + i = (k' +i)`]
THEN MRESA_TAC
MOD_LT[`x' MOD k''`;`k:num`];
MRESA_TAC
DIVISION[`q:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`q MOD k< k ==>(x + p MOD k + k - q MOD k) + q MOD k =
((x + p MOD k) + k) `)
THEN RESA_TAC
THEN MRESAL_TAC
MOD_MULT_ADD[`1:num`;`k:num`;`x+ p MOD k:num`][ARITH_RULE`1 * k' + i = (i+ k')`]
THEN MRESA_TAC
MOD_LT[`q MOD k`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`(x + p MOD k + k - q MOD k)`;`q MOD k:num`;`k:num`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th->
MRESA1_TAC
th`x+ p MOD k`
THEN MRESA1_TAC
th `(x + p MOD k + k - q MOD k) MOD k + q MOD k:num`
)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th]);
EXISTS_TAC`(\i. (i+ p MOD k - q MOD k) MOD k)`
THEN STRIP_TAC;
REWRITE_TAC[
IN_ELIM_THM;
IMAGE;
EXISTS_UNIQUE]
THEN REPEAT STRIP_TAC
THEN EXISTS_TAC`(y + q MOD k +k - p MOD k) MOD k `
THEN MRESA_TAC
DIVISION[`p:num`;`k:num`]
THEN MRESA_TAC
DIVISION[`q:num`;`k:num`]
THEN MRESA_TAC
DIVISION[`y + q MOD k +k - p MOD k:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`y<k'' /\ k''<k/\ p MOD k<k /\ q MOD k<= p MOD k /\ ~(q MOD k = p MOD k) ==> y<k:num /\ (y + q MOD k + k - p MOD k) + p MOD k - q MOD k =y+k/\ p MOD k - q MOD k < k /\ (y + q MOD k + k - p MOD k) + p MOD k = (y + q MOD k) + k
`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`y:num`;`k'':num`]
THEN MRESA_TAC
MOD_LT[`y:num`;`k:num`]
THEN MRESA_TAC
MOD_LT[`p MOD k:num`;`k:num`]
THEN MRESA_TAC
MOD_LT[`(p MOD k- q MOD k):num`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`y+q MOD k +k - p MOD k`;`(p MOD k - q MOD k)`;`k:num`]
THEN MRESAL_TAC
MOD_MULT_ADD[`1:num`;`k:num`;`y:num`][ARITH_RULE`1 * k' + i = (i+k')`]
THEN MRESAL_TAC
MOD_MULT_ADD[`1:num`;`k:num`;`y+ q MOD k:num`][ARITH_RULE`1 * k' + i = (i+k')`]
THEN MRESA_TAC
MOD_ADD_MOD[`y+q MOD k +k - p MOD k`;`p MOD k `;`k:num`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`y+ q MOD k` THEN MRESA1_TAC
th `((y + q MOD k + k - p MOD k) MOD k + p MOD k):num`
)
THEN STRIP_TAC
;
EXISTS_TAC`y:num`
THEN ASM_REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`]
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th]);
REPEAT STRIP_TAC
THEN POP_ASSUM (fun th-> ASM_REWRITE_TAC[SYM
th])
THEN MP_TAC(ARITH_RULE`y<k'' /\ k''<k/\ p MOD k<k /\ q MOD k<= p MOD k /\ ~(q MOD k = p MOD k) ==> q MOD k + k - p MOD k< k/\ (y' + p MOD k - q MOD k) + q MOD k + k - p MOD k = y'+k`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`q MOD k + k - p MOD k`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`y' + p MOD k - q MOD k`;`q MOD k + k - p MOD k`;`k:num`]
THEN MRESAL_TAC
MOD_MULT_ADD[`1:num`;`k:num`;`y':num`][ARITH_RULE`1 * k' + i = (i+k')`]
THEN MRESA_TAC
MOD_LT[`y':num`;`k:num`]
;
REPEAT RESA_TAC
;
MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(x + p MOD k):num`
THEN MRESA1_TAC
th`x' MOD k''+ q MOD k`)
THEN MP_TAC(ARITH_RULE`3<= k'' ==> ~(k''=0)`)
THEN RESA_TAC
THEN MRESA_TAC
DIVISION[`x +p MOD k`;`k:num`]
THEN MRESA_TAC
DIVISION[`x' MOD k'' +q MOD k`;`k:num`]
THEN MRESA_TAC
DIVISION[`x':num`;`k'':num`]
THEN MP_TAC
VV_INJ
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
scs_half_slice_v39;
PAIR_EQ;REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`;
dist]
THEN STRIP_TAC
THEN POP_ASSUM(fun th-> MRESA_TAC
th[`(x+ p MOD k) MOD k:num`;`(x' MOD k'' + q MOD k)MOD k`])
THEN MP_TAC(ARITH_RULE`q MOD k<= p MOD k /\ x' MOD k''< k'' /\ k''<k ==> (x + p MOD k - q MOD k) + q MOD k = x + p MOD k /\ x' MOD k''< k`)
THEN RESA_TAC
THEN MRESAL_TAC(GEN_ALL Hdplygy.MOD_EQ_MOD)[`x + p MOD k - q MOD k`;`x' MOD k'' `;`q MOD k`;`k:num`][]
THEN POP_ASSUM MP_TAC
THEN ONCE_REWRITE_TAC[
ADD_SYM]
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`x' MOD k''`;`k:num`];
MRESA_TAC
DIVISION[`q:num`;`k:num`]
THEN MP_TAC(ARITH_RULE`q MOD k<= p MOD k ==> (x + p MOD k - q MOD k) + q MOD k =
(x + p MOD k) `)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`q MOD k`;`k:num`]
THEN MRESA_TAC
MOD_ADD_MOD[`(x + p MOD k - q MOD k)`;`q MOD k:num`;`k:num`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th->
MRESA1_TAC
th`x+ p MOD k`
THEN MRESA1_TAC
th `((x + p MOD k - q MOD k) MOD k + q MOD k):num`
)
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th])
THEN POP_ASSUM(fun th-> ASM_REWRITE_TAC[SYM
th]);
ASM_REWRITE_TAC[]
THEN MP_TAC
E_PRIME_EQ_E_vv
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL
E_PRIME_EQ_E_vv)
[`d'':real`;`d':real`;`mkj:bool`;`s':scs_v39`;`s'':scs_v39`;`p:num`;`s:scs_v39`; `vv:num->real^3`;`k'':num`;`q:num`;`k:num`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`u:real^3`;`FF:real^3#real^3->bool`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39;]
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`u-w= --(w-u):real^3`]
THEN ASM_REWRITE_TAC[
NORM_NEG]
THEN STRIP_TAC
THEN MP_TAC
F_PRIME_EQ_F_vv
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN STRIP_TAC
THEN MRESA_TAC (GEN_ALL
F_PRIME_EQ_F_vv)
[`d'':real`;`d':real`;`mkj:bool`;`s':scs_v39`;`s'':scs_v39`;`p:num`;`s:scs_v39`; `vv:num->real^3`;`k'':num`;`q:num`;`k:num`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`u:real^3`;`FF:real^3#real^3->bool`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39;]
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`u-w= --(w-u):real^3`]
THEN ASM_REWRITE_TAC[
NORM_NEG]
THEN STRIP_TAC
THEN SUBGOAL_THEN`
BBs_v39 s' vv'`ASSUME_TAC;
MP_TAC
QKNVMLB1
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;]
THEN RESA_TAC
THEN POP_ASSUM MATCH_MP_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
LET_DEF;
LET_END_DEF;tau3;
MMs_v39;
BBprime2_v39;
BBprime_v39;
is_scs_slice_v39]
THEN REPEAT RESA_TAC
THEN SUBGOAL_THEN`d' =
scs_d_v39 s'`ASSUME_TAC
;
EXPAND_TAC"s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
;
EXPAND_TAC"s'"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
;
SUBGOAL_THEN`
BBs_v39 s'' vv''`ASSUME_TAC;
MRESAL_TAC (GEN_ALL
QKNVMLB1)[`vv:num->real^3`;`s:scs_v39`;`q:num`;`p:num`;`d'':real`;`mkj:bool`]
[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;]
THEN POP_ASSUM MATCH_MP_TAC
THEN ASM_TAC
THEN REWRITE_TAC[
LET_DEF;
LET_END_DEF;tau3;
MMs_v39;
BBprime2_v39;
BBprime_v39;
is_scs_slice_v39]
THEN REPEAT RESA_TAC
THEN SUBGOAL_THEN`d'' =
scs_d_v39 s''`ASSUME_TAC
;
EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
;
EXPAND_TAC"s''"
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
PAIR_EQ;
dist;periodic2;
scs_half_slice_v39]
THEN ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[]
;
MP_TAC
SCS_HALF_SLICE_IS_A_SCS
THEN ASM_REWRITE_TAC[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN STRIP_TAC
THEN MRESAL_TAC (GEN_ALL
SCS_HALF_SLICE_IS_A_SCS)[`s:scs_v39`;`s':scs_v39`;`q:num`;`p:num`;`s'':scs_v39`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN MRESAL_TAC(GEN_ALL
VV_SUC_EQ_RHO_NODE_PRIME)[`V:real^3->bool`;`E:(real^3->bool)->bool`;`k:num`;`s:scs_v39`;`FF:real^3#real^3->bool`;`u:real^3`;`vv:num->real^3`;`p:num MOD k`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MRESAL_TAC(GEN_ALL
VV_SUC_EQ_RHO_NODE_PRIME)[`V:real^3->bool`;`E:(real^3->bool)->bool`;`k:num`;`s:scs_v39`;`FF:real^3#real^3->bool`;`w:real^3`;`vv:num->real^3`;`q:num MOD k`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MP_TAC(ARITH_RULE`~(k'<= 3)\/ k'<= 3 `)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`~(k''<= 3)\/ k''<= 3 `)
THEN RESA_TAC;
MP_TAC(ARITH_RULE`3<=k'==> ~(k'=0)/\ 0<k'`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`0`;`k':num`]
THEN MRESAL_TAC(GEN_ALL
VV_SUC_EQ_RHO_NODE_PRIME)[`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;`k':num`;`s':scs_v39`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`u:real^3`;`vv':num->real^3`;`0:num MOD k'`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MRESAL_TAC (GEN_ALL
SUM_AZIM_EQ_ANGLE_LE4)
[`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`vv':num->real^3`;`0`;`s':scs_v39`;`u:real^3`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)/\ 0+A=A/\ A+0=A`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`
sum {i | i < k'}
(\i.
rho_fun (
norm (vv (i MOD k' + p MOD k))) *
interior_angle1 (
vec 0)
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
(vv (i MOD k' + p MOD k)))
=
sum {i | i < k'}
(\i.
rho_fun (
norm (vv (i + p MOD k))) *
interior_angle1 (
vec 0)
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
(vv (i + p MOD k)))` ASSUME_TAC
;
MATCH_MP_TAC
SUM_EQ
THEN REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`k':num`];
MP_TAC(ARITH_RULE`3<=k''==> ~(k''=0)/\ 0<k''`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`0`;`k'':num`]
THEN MRESAL_TAC(GEN_ALL
VV_SUC_EQ_RHO_NODE_PRIME)[`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))`;`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))`;`k'':num`;`s'':scs_v39`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)`;`w:real^3`;`vv'':num->real^3`;`0:num MOD k'`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MRESAL_TAC (GEN_ALL
SUM_AZIM_EQ_ANGLE_LE4)
[`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))`;
`vv'':num->real^3`;`0`;`s'':scs_v39`;`w:real^3`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)`;`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))`;]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)/\ 0+A=A/\ A+0=A`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`
sum {i | i < k''}
(\i.
rho_fun (
norm (vv (i MOD k'' + q MOD k))) *
interior_angle1 (
vec 0)
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))
(vv (i MOD k'' + q MOD k)))
=
sum {i | i < k''}
(\i.
rho_fun (
norm (vv (i + q MOD k))) *
interior_angle1 (
vec 0)
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))
(vv (i + q MOD k)))` ASSUME_TAC
;
MATCH_MP_TAC
SUM_EQ
THEN REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`k'':num`];
ASM_REWRITE_TAC[]
;
MRESA_TAC (GEN_ALL
CARD_FF_EQ_CARD_SLICE_FF)
[`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`u:real^3`;`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))`;`w:real^3`;`FF:real^3#real^3->bool`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (w,
rho_node1 FF w)`]
THEN MRESA_TAC (GEN_ALL
CARD_SLICE_FF_LE_3)[`V:real^3->bool`;`E:(real^3->bool)->bool`;`u:real^3`;`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))`;`FF:real^3#real^3->bool`;`w:real^3`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (w,
rho_node1 FF w)`]
THEN MP_TAC(ARITH_RULE`
3<=
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
/\ 3<=
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))
==> (
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)) +
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)) -
2) -
2=
(
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)) -2)+
(
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)) -
2)`)
THEN RESA_TAC
THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD]
THEN MP_TAC
QKNVMLB2
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN RESA_TAC
THEN MATCH_MP_TAC(REAL_ARITH`d2<= d+d1 ==> a-c*b-d + a1-c*b1-d1<= (a+a1) - c*(b+b1)-d2`)
THEN ASM_REWRITE_TAC[]
;
(******3<k' /\ k''=3 ********)
MP_TAC(ARITH_RULE`3<=k'==> ~(k'=0)/\ 0<k'`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`0`;`k':num`]
THEN MRESAL_TAC(GEN_ALL
VV_SUC_EQ_RHO_NODE_PRIME)[`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;`k':num`;`s':scs_v39`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`u:real^3`;`vv':num->real^3`;`0:num MOD k'`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MRESAL_TAC (GEN_ALL
SUM_AZIM_EQ_ANGLE_LE4)
[`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;
`vv':num->real^3`;`0`;`s':scs_v39`;`u:real^3`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))`;]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)/\ 0+A=A/\ A+0=A`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`
sum {i | i < k'}
(\i.
rho_fun (
norm (vv (i MOD k' + p MOD k))) *
interior_angle1 (
vec 0)
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
(vv (i MOD k' + p MOD k)))
=
sum {i | i < k'}
(\i.
rho_fun (
norm (vv (i + p MOD k))) *
interior_angle1 (
vec 0)
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
(vv (i + p MOD k)))` ASSUME_TAC
;
MATCH_MP_TAC
SUM_EQ
THEN REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`k':num`];
MP_TAC(ARITH_RULE`(k''<=3) /\ 3<= k'' ==> k''=3`)
THEN RESA_TAC
THEN REPLICATE_TAC (125-61)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC)
THEN MRESA_TAC (GEN_ALL
SUM_AZIM_EQ_ANGLE_EQ4)
[`d'':real`;`d':real`;`mkj:bool`;`vv'':num->real^3`;`vv':num->real^3`;`s':scs_v39`;`s'':scs_v39`;`s:scs_v39`;`p:num`;`k'':num`;`vv:num->real^3`;`q:num`;`k:num`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`u:real^3`;`FF:real^3#real^3->bool`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39;]
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`u-w= --(w-u):real^3`]
THEN ASM_REWRITE_TAC[
NORM_NEG]
THEN RESA_TAC
;
MRESA_TAC (GEN_ALL
CARD_FF_EQ_CARD_SLICE_FF)
[`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`u:real^3`;`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))`;`w:real^3`;`FF:real^3#real^3->bool`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (w,
rho_node1 FF w)`]
THEN MRESA_TAC (GEN_ALL
CARD_SLICE_FF_LE_3)[`V:real^3->bool`;`E:(real^3->bool)->bool`;`u:real^3`;`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))`;`FF:real^3#real^3->bool`;`w:real^3`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (w,
rho_node1 FF w)`]
THEN MP_TAC(ARITH_RULE`
3<=
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
/\ 3<=
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))
==> (
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)) +
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)) -
2) -
2=
(
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)) -2)+
(
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)) -
2)`)
THEN RESA_TAC
THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD]
THEN MP_TAC
QKNVMLB2
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN RESA_TAC
THEN REPLICATE_TAC (131-124)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN ASM_REWRITE_TAC[
th]
THEN MP_TAC
th
THEN REPEAT STRIP_TAC)
THEN MRESAL_TAC(GEN_ALL
CARD_FF_EQ_SCS_K_3)[`k'':num`;`s'':scs_v39`;`vv'':num->real^3`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)`][
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3) /\ 3-2=1`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MATCH_MP_TAC(REAL_ARITH`d2<= d+d1 ==> a-c*b-d + a1-c-d1<= (a+a1) - c*(b+ &1)-d2`)
THEN ASM_REWRITE_TAC[]
;
MP_TAC(ARITH_RULE`~(k''<= 3)\/ k''<= 3 `)
THEN RESA_TAC;
(********k'=3 /\ 3< k''***********)
MP_TAC(ARITH_RULE`3<=k''==> ~(k''=0)/\ 0<k''`)
THEN RESA_TAC
THEN MRESA_TAC
MOD_LT[`0`;`k'':num`]
THEN MRESAL_TAC(GEN_ALL
VV_SUC_EQ_RHO_NODE_PRIME)[`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))`;`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))`;`k'':num`;`s'':scs_v39`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)`;`w:real^3`;`vv'':num->real^3`;`0:num MOD k''`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MRESAL_TAC (GEN_ALL
SUM_AZIM_EQ_ANGLE_LE4)
[`
v_prime V
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))`;
`vv'':num->real^3`;`0`;`s'':scs_v39`;`w:real^3`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)`;`
e_prime (E
UNION {{u, w}})
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))`;]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)/\ 0+A=A/\ A+0=A`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN SUBGOAL_THEN`
sum {i | i < k''}
(\i.
rho_fun (
norm (vv (i MOD k'' + q MOD k))) *
interior_angle1 (
vec 0)
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))
(vv (i MOD k'' + q MOD k)))
=
sum {i | i < k''}
(\i.
rho_fun (
norm (vv (i + q MOD k))) *
interior_angle1 (
vec 0)
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))
(vv (i + q MOD k)))` ASSUME_TAC
;
MATCH_MP_TAC
SUM_EQ
THEN REWRITE_TAC[
IN_ELIM_THM]
THEN REPEAT RESA_TAC
THEN MRESA_TAC
MOD_LT[`x:num`;`k'':num`];
ASM_REWRITE_TAC[]
;
MP_TAC(ARITH_RULE`(k'<=3) /\ 3<= k' ==> k'=3`)
THEN RESA_TAC
THEN REPLICATE_TAC (125-60)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC)
THEN MRESA_TAC (GEN_ALL
SUM_AZIM_EQ_ANGLE_EQ4)
[`d':real`;`d'':real`;`mkj:bool`;`vv':num->real^3`;`vv'':num->real^3`;`s'':scs_v39`;`s':scs_v39`;`s:scs_v39`;`q:num`;`k':num`;`vv:num->real^3`;`p:num`;`k:num`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`w:real^3`;`FF:real^3#real^3->bool`;`u:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39;]
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`u-w= --(w-u):real^3`]
THEN ASM_REWRITE_TAC[
NORM_NEG]
THEN RESA_TAC
;
MRESA_TAC (GEN_ALL
CARD_FF_EQ_CARD_SLICE_FF)
[`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`u:real^3`;`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))`;`w:real^3`;`FF:real^3#real^3->bool`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (w,
rho_node1 FF w)`]
THEN MRESA_TAC (GEN_ALL
CARD_SLICE_FF_LE_3)[`V:real^3->bool`;`E:(real^3->bool)->bool`;`u:real^3`;`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))`;`FF:real^3#real^3->bool`;`w:real^3`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (w,
rho_node1 FF w)`]
THEN MP_TAC(ARITH_RULE`
3<=
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
/\ 3<=
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))
==> (
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)) +
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)) -
2) -
2=
(
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)) -2)+
(
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)) -
2)`)
THEN RESA_TAC
THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD]
THEN MP_TAC
QKNVMLB2
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN RESA_TAC
THEN REPLICATE_TAC (131-124)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN ASM_REWRITE_TAC[
th]
THEN MP_TAC
th
THEN REPEAT STRIP_TAC)
THEN MRESAL_TAC(GEN_ALL
CARD_FF_EQ_SCS_K_3)[`k':num`;`s':scs_v39`;`vv':num->real^3`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`][
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3) /\ 3-2=1`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MATCH_MP_TAC(REAL_ARITH`d2<= d+d1 ==> a-c-d + a1-c*b-d1<= (a+a1) - c*(&1+b)-d2`)
THEN ASM_REWRITE_TAC[]
;
(*********k'=3 /\ k''=3********)
MP_TAC(ARITH_RULE`(k'<=3) /\ 3<= k' ==> k'=3`)
THEN RESA_TAC
THEN REPLICATE_TAC (120-60)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC)
THEN MRESA_TAC (GEN_ALL
SUM_AZIM_EQ_ANGLE_EQ4)
[`d':real`;`d'':real`;`mkj:bool`;`vv':num->real^3`;`vv'':num->real^3`;`s'':scs_v39`;`s':scs_v39`;`s:scs_v39`;`q:num`;`k':num`;`vv:num->real^3`;`p:num`;`k:num`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`w:real^3`;`FF:real^3#real^3->bool`;`u:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39;]
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`u-w= --(w-u):real^3`]
THEN ASM_REWRITE_TAC[
NORM_NEG]
THEN RESA_TAC
;
MP_TAC(ARITH_RULE`(k''<=3) /\ 3<= k'' ==> k''=3`)
THEN RESA_TAC
THEN REPLICATE_TAC (122-60)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th
THEN ASM_REWRITE_TAC[]
THEN STRIP_TAC)
THEN MRESA_TAC (GEN_ALL
SUM_AZIM_EQ_ANGLE_EQ4)
[`d'':real`;`d':real`;`mkj:bool`;`vv'':num->real^3`;`vv':num->real^3`;`s':scs_v39`;`s'':scs_v39`;`s:scs_v39`;`p:num`;`k'':num`;`vv:num->real^3`;`q:num`;`k:num`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`u:real^3`;`FF:real^3#real^3->bool`;`w:real^3`]
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;
scs_half_slice_v39;]
THEN ONCE_REWRITE_TAC[VECTOR_ARITH`u-w= --(w-u):real^3`]
THEN ASM_REWRITE_TAC[
NORM_NEG]
THEN RESA_TAC
;
MRESA_TAC (GEN_ALL
CARD_FF_EQ_CARD_SLICE_FF)
[`V:real^3->bool`;`(E:(real^3->bool)->bool)`;`u:real^3`;`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))`;`w:real^3`;`FF:real^3#real^3->bool`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (w,
rho_node1 FF w)`]
THEN MRESA_TAC (GEN_ALL
CARD_SLICE_FF_LE_3)[`V:real^3->bool`;`E:(real^3->bool)->bool`;`u:real^3`;`hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))`;`FF:real^3#real^3->bool`;`w:real^3`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (u,
rho_node1 FF u)`;`face (hypermap (
HYP (
vec 0,V,(E:(real^3->bool)->bool)
UNION {{u, w}}))) (w,
rho_node1 FF w)`]
THEN MP_TAC(ARITH_RULE`
3<=
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u))
/\ 3<=
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w))
==> (
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)) +
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)) -
2) -
2=
(
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)) -2)+
(
CARD
(face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)) -
2)`)
THEN RESA_TAC
THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD]
THEN MP_TAC
QKNVMLB2
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN RESA_TAC
THEN REPLICATE_TAC (128-118)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN ASM_REWRITE_TAC[
th]
THEN MP_TAC
th
THEN REPEAT STRIP_TAC)
THEN MRESAL_TAC(GEN_ALL
CARD_FF_EQ_SCS_K_3)[`k':num`;`s':scs_v39`;`vv':num->real^3`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (u,
rho_node1 FF u)`][
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3) /\ 3-2=1`;
mk_unadorned_v39;
scs_diag;
dist]
THEN REPLICATE_TAC (128-120)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> ASM_TAC
THEN ASM_REWRITE_TAC[
th]
THEN MP_TAC
th
THEN REPEAT STRIP_TAC)
THEN MRESAL_TAC(GEN_ALL
CARD_FF_EQ_SCS_K_3)[`k'':num`;`s'':scs_v39`;`vv'':num->real^3`;`face (hypermap (
HYP (
vec 0,V,E
UNION {{u, w}}))) (w,
rho_node1 FF w)`][
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3) /\ 3-2=1`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MATCH_MP_TAC(REAL_ARITH`d2<= d+d1 ==> a-c-d + a1-c-d1<= (a+a1) - c*(&1+ &1)-d2`)
THEN ASM_REWRITE_TAC[]
;
]);;
let DIAGE_VAL_P_Q=prove(`~(p MOD 4 = q MOD 4)/\
~(SUC p MOD 4 = q MOD 4)/\
~(p MOD 4 = SUC q MOD 4)
==> q MOD 4=(p MOD 4 +2) MOD 4`,
MRESAL_TAC
DIVISION[`p:num`;`4`][ARITH_RULE`~(4=0)`;
ADD1]
THEN MP_TAC(ARITH_RULE`p MOD 4<4==> p MOD 4=0 \/ p MOD 4=1 \/ p MOD 4=2 \/ p MOD 4=3`)
THEN RESA_TAC
THEN MRESAL_TAC
DIVISION[`q:num`;`4`][ARITH_RULE`~(4=0)`;
ADD1]
THEN MP_TAC(ARITH_RULE`q MOD 4<4==> q MOD 4=0 \/ q MOD 4=1 \/ q MOD 4=2 \/ q MOD 4=3`)
THEN RESA_TAC
THEN MRESAL_TAC
MOD_ADD_MOD[`p:num`;`1`;`4`][ARITH_RULE`~(4=0)`]
THEN MRESAL_TAC
MOD_ADD_MOD[`q:num`;`1`;`4`][ARITH_RULE`~(4=0)`]
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN ARITH_TAC);;
let QKNVMLB3_Eq4=prove_by_refinement(
`(
scs_k_v39 s =4 /\ ~(
scs_J_v39 s' 0 (
scs_k_v39 s' - 1)))/\
scs_half_slice_v39 s p q d' mkj =s'/\
scs_half_slice_v39 s q p d'' mkj =s'' /\
vv' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s')+p MOD (
scs_k_v39 s)))
/\
vv'' = (\i. (vv:num->real^3) (i MOD (
scs_k_v39 s'')+q MOD (
scs_k_v39 s)))
/\
MMs_v39 s vv /\
is_scs_v39 s /\
scs_diag (
scs_k_v39 s) p q /\
is_scs_slice_v39 s s' s'' p q /\
scs_d_v39 s <= d' + d'' ==>
taustar_v39 s' vv' +
taustar_v39 s'' vv'' <=
taustar_v39 s vv`,
[
REWRITE_TAC[
taustar_v39;
LET_DEF;
LET_END_DEF;
scs_diag]
THEN STRIP_TAC
THEN MP_TAC
SCS_K_PRIME_CASE_3
THEN REWRITE_TAC[
taustar_v39;
LET_DEF;
LET_END_DEF;
scs_diag]
THEN RESA_TAC
THEN ABBREV_TAC`k=
scs_k_v39 s`
THEN ABBREV_TAC`k'=
scs_k_v39 s'`
THEN ABBREV_TAC`k''=
scs_k_v39 s''`
THEN MP_TAC(ARITH_RULE`3<k==> ~(k<=3)`)
THEN RESA_TAC
THEN MP_TAC
SCS_K_LE_6
THEN RESA_TAC
THEN ABBREV_TAC`u= (vv:num->real^3) (p MOD k)`
THEN ABBREV_TAC `w= (vv:num->real^3) (q MOD k)`
THEN ABBREV_TAC`V=
IMAGE (vv:num->real^3) (:num)`
THEN ABBREV_TAC`E=
IMAGE (\i. {vv i,(vv:num->real^3) (SUC i)}) (:num)`
THEN ABBREV_TAC`
FF=
IMAGE (\i. (vv i,(vv:num->real^3) (SUC i))) (:num)`
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`p MOD
scs_k_v39 s`;`vv:num->real^3`]
THEN MRESA_TAC (GEN_ALL
IN_IMAGE_VV)[`q MOD
scs_k_v39 s`;`vv:num->real^3`]
THEN ASM_TAC
THEN REPLICATE_TAC 2 STRIP_TAC
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime_v39;
BBprime2_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;]
THEN REPEAT RESA_TAC
;
REPLICATE_TAC (54-9)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[ARITH_RULE`~(4<=3)`]
;
MP_TAC
SCS_K_PRIME_CASE_4
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime_v39;
BBprime2_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN STRIP_TAC
THEN MRESAL_TAC(GEN_ALL
SCS_K_PRIME_CASE_4)
[`d'':real`;`mkj:bool`;`s:scs_v39`;`q:num`;`p:num`;`s'':scs_v39`]
[
MMs_v39;
BBprime_v39;
BBprime2_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN MATCH_MP_TAC(REAL_ARITH`a2=a+a1 /\ b2<= b+b1==>a-b+a1-b1<=a2-b2`)
THEN MP_TAC
QKNVMLB2
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 0+A=A`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN RESA_TAC
THEN REWRITE_TAC[
tau_fun]
THEN SUBGOAL_THEN`
local_fan(V,E,
FF)`ASSUME_TAC
;
REPLICATE_TAC (57-9)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN REWRITE_TAC[
convex_local_fan]
THEN ASM_REWRITE_TAC[ARITH_RULE`~(4<=3)`]
THEN REPEAT RESA_TAC
;
MP_TAC Local_lemmas.LOFA_IMP_CARD_FF_V_EQ
THEN RESA_TAC
THEN MP_TAC
CARD_V_EQ_SCS_K
THEN REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ 4-2=2`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;Basics.DIMINDEX_4;
IN]
THEN RESA_TAC
THEN REWRITE_TAC[ARITH_RULE`4-2=2`]
THEN MP_TAC
SUM_AZIM_EQ_ANGLE_LE4
THEN
ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3/\ ~(4<=3)`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist;Basics.DIMINDEX_4;
IN]
THEN RESA_TAC
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th;
NUMSEG_3;
SUM_NUMSEG3])
THEN MRESAL_TAC(GEN_ALL
VV_SUC_EQ_RHO_NODE)[`
scs_k_v39 s`;`V:real^3->bool`;`E:(real^3->bool)->bool`;`p:num`;`q:num`;`s:scs_v39`;`FF:real^3#real^3->bool`;`u:real^3`;`vv:num->real^3`;`p:num MOD k`]
[
is_scs_v39;
scs_k_v39_explicit;
scs_d_v39_explicit;
scs_a_v39_explicit;
scs_b_v39_explicit;
BBs_v39;ARITH_RULE`3<=3`;
LET_DEF;
LET_END_DEF;ARITH_RULE`~(4<=3)`;
mk_unadorned_v39;
scs_diag;
dist]
THEN REPLICATE_TAC (61-9)(POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT STRIP_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[ARITH_RULE`~(4<=3)`]
THEN REPEAT RESA_TAC
THEN SUBGOAL_THEN`(vv:num->real^3) (0+p MOD 4)
IN V`ASSUME_TAC
;
EXPAND_TAC"V"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`0+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`];
SUBGOAL_THEN`(vv:num->real^3) (1+p MOD 4)
IN V`ASSUME_TAC
;
EXPAND_TAC"V"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`1+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`];
SUBGOAL_THEN`(vv:num->real^3) (2+p MOD 4)
IN V`ASSUME_TAC
;
EXPAND_TAC"V"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`2+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`];
SUBGOAL_THEN`(vv:num->real^3) (3+p MOD 4)
IN V`ASSUME_TAC
;
EXPAND_TAC"V"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`3+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`];
MP_TAC Nkezbfc_local.CONVEX_LOFA_IMP_INANGLE_EQ_AZIM_IVS
THEN RESA_TAC
THEN POP_ASSUM(fun th-> MRESA1_TAC
th`(vv:num->real^3) (0+p MOD 4)`
THEN MRESA1_TAC
th`(vv:num->real^3) (1+p MOD 4)`
THEN MRESA1_TAC
th`(vv:num->real^3) (2+p MOD 4)`
THEN MRESA1_TAC
th`(vv:num->real^3) (3+p MOD 4)`)
THEN SUBGOAL_THEN`(
rho_node1 FF (vv (0 + p MOD 4))) =vv (1 + p MOD 4)`ASSUME_TAC;
MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`V:real^3->bool`
THEN EXISTS_TAC`E:(real^3->bool)->bool`
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"FF"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`0+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`SUC(0+p MOD 4)= 1+ p MOD 4`];
SUBGOAL_THEN`(
rho_node1 FF (vv (1 + p MOD 4))) =vv (2 + p MOD 4)`ASSUME_TAC;
MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`V:real^3->bool`
THEN EXISTS_TAC`E:(real^3->bool)->bool`
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"FF"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`1+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`SUC(1+p MOD 4)= 2+ p MOD 4`];
SUBGOAL_THEN`(
rho_node1 FF (vv (2 + p MOD 4))) =vv (3 + p MOD 4)`ASSUME_TAC;
MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`V:real^3->bool`
THEN EXISTS_TAC`E:(real^3->bool)->bool`
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"FF"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`2+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`SUC(2+p MOD 4)= 3+ p MOD 4`];
SUBGOAL_THEN`(
rho_node1 FF (vv (3 + p MOD 4))) =vv (0 + p MOD 4)`ASSUME_TAC;
MATCH_MP_TAC (GEN_ALL Local_lemmas.DETER_RHO_NODE)
THEN EXISTS_TAC`V:real^3->bool`
THEN EXISTS_TAC`E:(real^3->bool)->bool`
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"FF"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`3+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`SUC(3+p MOD 4)= 4+ p MOD 4/\ 0+A=A`]
THEN MRESAL_TAC
MOD_MOD_REFL[`p:num`;`k:num`][ARITH_RULE`~(4=0)`;]
THEN MRESAL_TAC
MOD_MULT_ADD[`1`;`k:num`;`p MOD k`][ARITH_RULE`1 * k + x MOD k = k+ x MOD k`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(4+ p MOD 4)`) ;
SUBGOAL_THEN`(
ivs_rho_node1 FF (vv (0 + p MOD 4))) =vv (3 + p MOD 4)`ASSUME_TAC
;
MATCH_MP_TAC (GEN_ALL Local_lemmas.IVS_RHO_NODE1_DETE)
THEN EXISTS_TAC`V:real^3->bool`
THEN EXISTS_TAC`E:(real^3->bool)->bool`
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"FF"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`3+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`SUC(3+p MOD 4)= 4+ p MOD 4/\ 0+A=A`]
THEN MRESAL_TAC
MOD_MOD_REFL[`p:num`;`k:num`][ARITH_RULE`~(4=0)`;]
THEN MRESAL_TAC
MOD_MULT_ADD[`1`;`k:num`;`p MOD k`][ARITH_RULE`1 * k + x MOD k = k+ x MOD k`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th-> MRESA1_TAC
th `(4+ p MOD 4)`) ;
SUBGOAL_THEN`(
ivs_rho_node1 FF (vv (3 + p MOD 4))) =vv (2 + p MOD 4)`ASSUME_TAC;
MATCH_MP_TAC (GEN_ALL Local_lemmas.IVS_RHO_NODE1_DETE)
THEN EXISTS_TAC`V:real^3->bool`
THEN EXISTS_TAC`E:(real^3->bool)->bool`
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"FF"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`2+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`SUC(2+p MOD 4)= 3+ p MOD 4`];
SUBGOAL_THEN`(
ivs_rho_node1 FF (vv (1 + p MOD 4))) =vv (0 + p MOD 4)`ASSUME_TAC;
MATCH_MP_TAC (GEN_ALL Local_lemmas.IVS_RHO_NODE1_DETE)
THEN EXISTS_TAC`V:real^3->bool`
THEN EXISTS_TAC`E:(real^3->bool)->bool`
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"FF"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`0+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`SUC(0+p MOD 4)= 1+ p MOD 4`];
SUBGOAL_THEN`(
ivs_rho_node1 FF (vv (2 + p MOD 4))) =vv (1 + p MOD 4)`ASSUME_TAC;
MATCH_MP_TAC (GEN_ALL Local_lemmas.IVS_RHO_NODE1_DETE)
THEN EXISTS_TAC`V:real^3->bool`
THEN EXISTS_TAC`E:(real^3->bool)->bool`
THEN ASM_REWRITE_TAC[]
THEN EXPAND_TAC"FF"
THEN REWRITE_TAC[
IMAGE;
IN_ELIM_THM]
THEN EXISTS_TAC`1+p MOD 4`
THEN REWRITE_TAC[SET_RULE`(a:num)
IN(:num)`;ARITH_RULE`SUC(1+p MOD 4)= 2+ p MOD 4`];
ASM_REWRITE_TAC[]
THEN MP_TAC
VV_INJ
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
scs_half_slice_v39;
PAIR_EQ;REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`3<=3`;
dist]
THEN STRIP_TAC THEN SUBGOAL_THEN`(!i j. i < 4 /\ j < 4 /\ vv i = (vv:num->real^3) j ==> (i = j))`ASSUME_TAC
;
REPEAT GEN_TAC
THEN POP_ASSUM(fun th->
REPEAT STRIP_TAC
THEN MRESA_TAC
th[`i:num`;`j:num`])
;
SUBGOAL_THEN`
dist(vv 0, vv 1) < &4 /\
dist(vv 0, vv 3) < &4 /\
dist(vv 1,vv 2) < &4 /\
dist(vv 2,(vv:num->real^3) 3) < &4 `ASSUME_TAC
;
REWRITE_TAC[
dist]
THEN REPLICATE_TAC (79-16) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MRESA_TAC
th[`0`;`1`]
THEN MRESA_TAC
th[`0`;`3`]
THEN MRESA_TAC
th[`1`;`2`]
THEN MRESA_TAC
th[`2`;`3`])
THEN REPLICATE_TAC (82-32) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MRESAL1_TAC
th`0`[ARITH_RULE`SUC 0=1`]
THEN MRESAL1_TAC
th`1`[ARITH_RULE`SUC 1=2`]
THEN MRESAL1_TAC
th`2`[ARITH_RULE`SUC 2=3`]
THEN MRESAL1_TAC
th`3`[ARITH_RULE`SUC 3=4`])
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN REPLICATE_TAC (85-28) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MRESA_TAC
th[`0`;`1`]
THEN MRESA_TAC
th[`0`;`3`]
THEN MRESA_TAC
th[`1`;`2`]
THEN MRESA_TAC
th[`2`;`3`])
THEN REPLICATE_TAC (96-76) (POP_ASSUM MP_TAC)
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_b_v39 s 3`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESAL1_TAC
th `4:num`[ARITH_RULE`4 MOD 4=0`])
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th;cstab])
THEN REAL_ARITH_TAC;
REPLICATE_TAC (80-7) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MP_TAC
th
THEN RESA_TAC)
THEN SUBGOAL_THEN`!i j. ~(vv i = vv j) ==> &2 <=
dist(vv i,(vv:num->real^3) j) ` ASSUME_TAC
;
REWRITE_TAC[
dist]
THEN REPEAT GEN_TAC
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th-> MRESAL1_TAC
th `i:num`[ARITH_RULE`4 MOD 4=0`]
THEN MRESAL1_TAC
th `j:num`[ARITH_RULE`4 MOD 4=0`])
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN MRESAL_TAC
DIVISION[`i:num`;`4`][ARITH_RULE`~(4=0)`]
THEN MRESAL_TAC
DIVISION[`j:num`;`4`][ARITH_RULE`~(4=0)`]
THEN MP_TAC(SET_RULE`i MOD 4= j MOD 4\/ ~(i MOD 4= j MOD 4)`)
THEN RESA_TAC;
REPLICATE_TAC (85-78) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN POP_ASSUM MP_TAC
THEN MRESA_TAC
th[`i MOD 4`;`j MOD 4`])
;
REPLICATE_TAC (85-30) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MRESA_TAC
th[`i MOD 4`;`j MOD 4`])
THEN REPLICATE_TAC (85-7) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MRESA_TAC
th[`i MOD 4`;`j MOD 4`])
THEN REPLICATE_TAC (3) (POP_ASSUM MP_TAC)
THEN REAL_ARITH_TAC;
SUBGOAL_THEN`
dist((vv:num->real^3) (p MOD 4), vv (q MOD 4))< &4`ASSUME_TAC
;
REWRITE_TAC[
dist]
THEN REPLICATE_TAC (81-39) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MP_TAC
th)
THEN ASM_REWRITE_TAC[
MMs_v39;
BBprime2_v39;
BBprime_v39;
BBs_v39;
LET_DEF;
LET_END_DEF;
is_scs_v39;
scs_diag;scs_v39_explicit;
is_scs_slice_v39;
dsv_v39;
scs_slice_v39;
scs_half_slice_v39;
PAIR_EQ;REAL_ARITH`#0.11 < #0.9`;ARITH_RULE`~(4<=3)`;
dist]
THEN STRIP_TAC
THEN REPLICATE_TAC (87-15) (POP_ASSUM MP_TAC)
THEN POP_ASSUM(fun th-> REPEAT DISCH_TAC
THEN MRESA_TAC
th[`p MOD 4`;`q MOD 4`])
THEN ASM_TAC
THEN REWRITE_TAC[periodic2;]
THEN REPEAT STRIP_TAC
THEN REPLICATE_TAC (3) (POP_ASSUM MP_TAC)
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_bm_v39 s p`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESAL1_TAC
th `q:num`[ARITH_RULE`4 MOD 4=0`])
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`
scs_bm_v39 s (q MOD 4)`][ARITH_RULE`~(4=0)`;periodic]
THEN POP_ASSUM(fun th-> MRESAL1_TAC
th `p:num`[ARITH_RULE`4 MOD 4=0`])
THEN POP_ASSUM(fun th-> REWRITE_TAC[SYM
th])
THEN REAL_ARITH_TAC;
POP_ASSUM MP_TAC
THEN MRESAL_TAC
DIVISION[`p:num`;`4`][ARITH_RULE`~(4=0)`]
THEN MRESAL_TAC(GEN_ALL
PERIODIC_PROPERTY)[`
scs_k_v39 s`;`vv:num->real^3`][ARITH_RULE`~(4=0)`;]
THEN POP_ASSUM(fun th-> MRESAL1_TAC
th `4:num`[ARITH_RULE`4 MOD 4=0`]
THEN MRESAL1_TAC
th`5`[ARITH_RULE`5 MOD 4=1`]
THEN MRESAL1_TAC
th`6`[ARITH_RULE`6 MOD 4=2`])
THEN POP_ASSUM(fun th->
POP_ASSUM (fun th1->
POP_ASSUM(fun th2-> ASSUME_TAC(SYM
th2))
THEN ASSUME_TAC(SYM
th1))
THEN ASSUME_TAC (SYM
th))
THEN MP_TAC(ARITH_RULE`p MOD 4<4==> p MOD 4=0 \/ p MOD 4=1 \/ p MOD 4=2 \/ p MOD 4=3`)
THEN RESA_TAC;
MP_TAC
DIAGE_VAL_P_Q
THEN RESA_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1 /\ 2 MOD 3=2/\ A+0=A /\ 0+A=A/\ 1+2=3/\ 2 MOD 4=2 /\ 3 MOD 4=3/\ 2+2=4`;SET_RULE`(A /\ B) ==> B`]
THEN POP_ASSUM(fun th->
POP_ASSUM(fun th1->
ASM_TAC
THEN REWRITE_TAC[
th;
th1;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1 /\ 2 MOD 3=2/\ A+0=A /\ 0+A=A/\ 1+2=3/\ 2 MOD 4=2 /\ 3 MOD 4=3/\ 2+2=4`;]))
THEN REPEAT STRIP_TAC
THEN MRESAL1_TAC (GEN_ALL Terminal.vv_quad_split012)`vv:num->real^3`[
LET_DEF;
LET_END_DEF;]
;
MP_TAC
DIAGE_VAL_P_Q
THEN RESA_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1 /\ 2 MOD 3=2/\ A+0=A /\ 0+A=A/\ 1+2=3/\ 2 MOD 4=2 /\ 3 MOD 4=3/\ 2+2=4/\ 1+1=2 /\ 2+1=3/\ 1+3=4/\ 2+3=5/\3+1=4`;SET_RULE`(B /\ A) ==> B`]
THEN POP_ASSUM(fun th->
POP_ASSUM(fun th1->
ASM_TAC
THEN REWRITE_TAC[
th;
th1;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1 /\ 2 MOD 3=2/\ A+0=A /\ 0+A=A/\ 1+2=3/\ 2 MOD 4=2 /\ 3 MOD 4=3/\ 2+2=4`;]))
THEN REPEAT STRIP_TAC
THEN MRESAL1_TAC (GEN_ALL Terminal.vv_quad_split123)`vv:num->real^3`[
LET_DEF;
LET_END_DEF;]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC
;
MP_TAC
DIAGE_VAL_P_Q
THEN RESA_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1 /\ 2 MOD 3=2/\ A+0=A /\ 0+A=A/\ 1+2=3/\ 2 MOD 4=2 /\ 3 MOD 4=3/\ 2+2=4/\ 1+1=2 /\ 2+1=3/\ 1+3=4/\ 2+3=5/\3+1=4/\ 3+2=5/\ 4 MOD 4=0`;SET_RULE`(B /\ A) ==> B`]
THEN POP_ASSUM(fun th->
POP_ASSUM(fun th1->
ASM_TAC
THEN REWRITE_TAC[
th;
th1;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1 /\ 2 MOD 3=2/\ A+0=A /\ 0+A=A/\ 1+2=3/\ 2 MOD 4=2 /\ 3 MOD 4=3/\ 2+2=4/\ 1+1=2 /\ 2+1=3/\ 1+3=4/\ 2+3=5/\3+1=4/\ 3+2=5/\ 4 MOD 4=0`;]))
THEN REPEAT STRIP_TAC
THEN MRESAL1_TAC (GEN_ALL Terminal.vv_quad_split012)`vv:num->real^3`[
LET_DEF;
LET_END_DEF;
DIST_SYM]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
MP_TAC
DIAGE_VAL_P_Q
THEN RESA_TAC
THEN POP_ASSUM MP_TAC
THEN ASM_REWRITE_TAC[ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1 /\ 2 MOD 3=2/\ A+0=A /\ 0+A=A/\ 1+2=3/\ 2 MOD 4=2 /\ 3 MOD 4=3/\ 2+2=4/\ 1+1=2 /\ 2+1=3/\ 1+3=4/\ 2+3=5/\3+1=4/\ 3+2=5/\ 4 MOD 4=0/\ 5 MOD 4=1/\ 3+3=6`;]
THEN STRIP_TAC
THEN POP_ASSUM(fun th->
POP_ASSUM(fun th1->
ASM_TAC
THEN REWRITE_TAC[
th;
th1;ARITH_RULE`0 MOD 3=0/\ 1 MOD 3=1 /\ 2 MOD 3=2/\ A+0=A /\ 0+A=A/\ 1+2=3/\ 2 MOD 4=2 /\ 3 MOD 4=3/\ 2+2=4/\ 1+1=2 /\ 2+1=3/\ 1+3=4/\ 2+3=5/\3+1=4/\ 3+2=5/\ 4 MOD 4=0`;]))
THEN REPEAT STRIP_TAC
THEN MRESAL1_TAC (GEN_ALL Terminal.vv_quad_split123)`vv:num->real^3`[
LET_DEF;
LET_END_DEF;
DIST_SYM]
THEN POP_ASSUM MP_TAC
THEN REAL_ARITH_TAC;
]);;