(* Main estimate file*) (* Author: PHAN HOANG CHON *) (*========================================================*) (* needs "database_more.ml";; needs "definitions_kepler.ml";; (* removed convex.ml by thales, not compatible with other loads *) (* needs "Multivariate/convex.ml";; *) needs "geomdetail.ml";; (* Writen by Nguyen Quang TRuong *) *) (*========================================================*) (* These definitions are defined by other authors*) (*========================================================*) (*Some constant*) (*arctan2 function is defined in definition_kepler.ml*) let delta_tet = new_definition (`delta_tet = sqrt8 * atn2(&5, sqrt2)`);; let delta_oct = new_definition (`delta_oct = (( &3* pi_rt18 - delta_tet)*( &1 / &2 ))`);; (*========================================================*) let lambda_v = new_definition (`lambda_v = (-- &4)* delta_oct `);; let lambda_s = new_definition (`lambda_s = (&1)/(&3) `);; let lambda_oct = new_definition (`lambda_oct = (lambda_v , lambda_s) `);; (*========================================================*) (* benign redefinition *) let open_ball = new_definition `open_ball (x:real^3) (r:real)= { y | norm(y-x)< r }`;; (*========================================================*) (* There is a mistake in this definition. For example: let d={x,y,z,t} be a quarter with 2*t0= &2 * t0 ))`;; *) (*========================================================*) (* benign redef *) let diag = new_definition ` diag d q s = ( quarter q s /\ ( ? x y. ! v1 v2. x IN q /\ y IN q /\ v1 IN q /\ v2 IN q /\ { x, y } = d /\ d3 x y >= &2 * t0 /\ d3 x y >= d3 v1 v2 ))`;; (* benign *) let find_diagonal = new_definition ` find_diagonal q s:real^3->bool = @{x, y}. {x , y} SUBSET q /\ diag {x , y} q s `;; let find_dia = new_definition ` find_dia q (s:real^3->bool) = let d = find_diagonal q s in @(u,v). u IN d /\ v IN d /\ ~( u = v ) `;; (* thales, nov 11. anchor is already defined in geomdetail.ml *) let anchor_alt = new_definition ` anchor_alt (v:real^3) v1 v2 = ( d3 v1 v2 <= sqrt ( &8 ) /\ d3 v1 v2 >= &2 * t0 /\ d3 v v1 <= &2 * t0 /\ d3 v v2 <= &2 * t0 )`;; (* Definition 7.20 =============================================*) let context = new_definition ` context (v , w) s (p:num, r) = ( CARD { d | diag { v , w } d s } = p /\ CARD { t | &2 * t0 <= d3 v w /\ anchor_alt t v w } = (p + r ))`;; let cotext = new_definition ` cotext (v , w) (s:real^3->bool) = @( (p:num) , (r:num) ). CARD { d | diag { v , w } d s } = p /\ CARD { t | &2 * t0 <= d3 v w /\ anchor_alt t v w } = (p + r )`;; (* Definition 7.21 =============================================*) let VC1 = new_definition ` VC1 v s d = VC v d INTER conv s `;; let VCt = new_definition ` VCt x t s d = VC1 x s d INTER open_ball x t `;; (* Definition 7.22 =============================================*) let sv = new_definition ` sv x p d = lambda_v * vol(VC1 x p d) + lambda_s * solid x (conv0 p) `;; let gamma = new_definition ` gamma (v1 , v2 , v3 , v4) d = ( &1 / &4 ) * ( sv v1 { v1 , v2 , v3 , v4 } d + sv v2 { v1 , v2 , v3 , v4 } d + sv v3 { v1 , v2 , v3 , v4 } d + sv v4 { v1 , v2 , v3 , v4 } d )`;; let volan = new_definition ` volan v0 (v0 , v1, v2, v3 ) = let x01 = dist(v0,v1) pow 2 in let x02 = dist(v0,v2) pow 2 in let x03 = dist(v0,v3) pow 2 in let x12 = dist(v1,v2) pow 2 in let x13 = dist(v1,v3) pow 2 in let x23 = dist(v2,v3) pow 2 in ( x01 * ( x02 + x12 - x01 ) * ( chi_x x23 x13 x03 x01 x02 x12 ) ) / ( &48 * (ups_x x01 x02 x12) * sqrt( delta_x x01 x02 x03 x23 x13 x12 )) + ( x01 * ( x03 + x13 - x01 ) * ( chi_x x23 x12 x02 x01 x03 x13 ) ) / ( &48 * (ups_x x01 x03 x13) * sqrt( delta_x x01 x02 x03 x23 x13 x12 )) + ( x02 * ( x01 + x12 - x02 ) * ( chi_x x13 x23 x03 x02 x01 x12 ) ) / ( &48 * (ups_x x02 x01 x12) * sqrt( delta_x x01 x02 x03 x23 x13 x12 )) + ( x01 * ( x03 + x23 - x02 ) * ( chi_x x13 x12 x01 x02 x03 x23 ) ) / ( &48 * (ups_x x02 x03 x23) * sqrt( delta_x x01 x02 x03 x23 x13 x12 )) + ( x02 * ( x01 + x13 - x03 ) * ( chi_x x12 x23 x02 x03 x01 x13 ) ) / ( &48 * (ups_x x03 x01 x13) * sqrt( delta_x x01 x02 x03 x23 x13 x12 )) + ( x02 * ( x02 + x23 - x03 ) * ( chi_x x12 x13 x01 x03 x02 x23 ) ) / ( &48 * (ups_x x03 x02 x23) * sqrt( delta_x x01 x02 x03 x23 x13 x12 )) `;; let svan =new_definition ` svan v0 ( v0 , v1, v2, v3) = lambda_v * volan v0 ( v0 , v1, v2, v3) + lambda_s * solid v0 ( conv0 { v0 , v1, v2, v3}) `;; (* Definition 7.23 =================================================*) let eta_pos = new_definition ` eta_pos (q:real^3->bool) (s:real^3->bool) = let d = find_diagonal q s in let f1 = d UNION ( @{u}. ( q DIFF d ) u ) in let f2 = d UNION ( @{v}. ( q DIFF f1) v ) in ( max_real ( radV f1 ) ( radV f2) )`;; (* The definition 7.24 =============================================*) let sv0 = new_definition ` sv0 v s d = sovo v (VCt v t0 s d) lambda_oct `;; let v_hat = new_definition `v_hat v q s = if ( quarter q s ) /\ ( v IN find_diagonal q s ) then ( @u. u IN (find_diagonal q s DIFF {v} )) else v `;; let sigma = new_definition ` sigma v0 ( v0 , v1 , v2, v3 ) ( s:real^3->bool ) = let q = { v0 , v1 , v2 , v3 } in if ( quasi_reg_tet q s ) then if radV q < sqrt2 then gamma ( v0 , v1 , v2, v3 ) s else svan v0 ( v0 , v1 , v2, v3 ) else if ( eta_pos q s ) < sqrt2 then if ( cotext ( find_dia q s ) s = (1,1) ) \/( cotext ( find_dia q s ) s = (4,0) ) then gamma (v0 , v1 , v2, v3 ) s else gamma ( v0 , v1 , v2, v3 ) s + ( &1 / &2 )* ( ( sv0 v0 q s ) - ( sv0 ( v_hat v0 q s ) q s )) else if ( cotext ( find_dia q s ) s = (1,1) ) then svan v0 ( v0 , v1, v2, v3 ) else ( if ( cotext ( find_dia q s ) s = (4,0) ) then ( &1 / &2 )* ( ( svan v0 ( v0 , v1, v2, v3 )) + ( svan ( v_hat v0 q s ) ( v0 , v1, v2, v3 ))) else ( &1 / &2 )* ( ( svan v0 ( v0 , v1, v2, v3 )) + ( svan ( v_hat v0 q s ) ( v0 , v1, v2, v3 ))) + ( &1 / &2 )* ( ( sv0 v0 q s - sv0 ( v_hat v0 q s) q s ))) `;; let A_1 = new_definition ` A_1 v0 ( v0 , v1 , v2 , v3 ) (s:real^3->bool) = let q = { v0 , v1 , v2 , v3 } in -- vol( VC1 v0 q s ) + (( solid v0 q )/ ( &3 * delta_oct )) + (( sigma v0 ( v0 , v1 , v2, v3 ) s ) / (&4 * delta_oct )) `;; (*=====================================================================*) (* this definition is writen by anhtamct ==============================*) let ball3_lambda = new_definition ` ball3_lambda (x:real^3) (r:real) (Lambda:real^3 -> bool) = ((open_ball x r ) INTER ( UNIONS ( IMAGE (\v. open_ball v (&1) ) Lambda ) ))`;; let truncated_packing = new_definition ` truncated_packing x r Lambda = Lambda INTER (ball3_lambda x r Lambda) `;; (*=====================================================================*) (* these definition are writen by Hoang Le Truong =====================*) let graph = new_definition `graph E <=> (!e. E e ==> (e HAS_SIZE 2))`;; let fan1 = new_definition`fan1(x,V,E):bool <=> FINITE V /\ ~(V SUBSET {}) `;; let fan2 = new_definition`fan2(x,V,E):bool <=> ~(x IN V)`;; let fan3=new_definition`fan3(x,V,E):bool <=> (!v. (v IN V) ==> cyclic_set {w | {v,w} IN E } x v)`;; let fan4 = new_definition`fan4(x,V,E):bool<=> (!e. (e IN E) ==> (aff_gt {x} e INTER V={}))`;; let fan5 = new_definition`fan5(x,V,E):bool<=> (!e f. (e IN E)/\ (f IN E) /\ ~(e=f) ==> (aff_gt {x} e INTER aff_gt {x} f ={}))`;; let fan = new_definition`fan(x,V,E)<=> ((UNIONS E) SUBSET V) /\ graph(E)/\ fan1(x,V,E)/\ fan2(x,V,E)/\ fan3(x,V,E)/\ fan4 (x,V,E) /\ fan5(x,V,E)`;; let X= new_definition`X fan(x,V,E)={v | ?e. (E e)/\(v IN aff_ge {x} e)}`;; let Y= new_definition`Y fan(x,V,E)={v:real^3 | ?e. (E e)/\(~(v IN aff_ge {x} e))}`;; (*=====================================================================*) (* Definition 8.9 =====================================================*) let tru_pack =new_definition ` tru_pack (v:real^3) (r:real) (s:real^3->bool) = { x | center_pac s v /\ x IN s /\ x IN open_ball v r }`;; let v_std = new_definition ` v_std (s:real^3->bool) (v0:real^3) = ( tru_pack v0 ( &2 * t0 ) s ) DIFF {v0}`;; let e_std = new_definition ` e_std (s:real^3->bool) (v0:real^3) = { { u , v } | u IN (tru_pack v0 ( &2 * t0 ) s ) DIFF {v0} /\ v IN (tru_pack v0 ( &2 * t0 ) s ) DIFF {v0} /\ ~(u = v) /\ d3 u v <= ( &2 * t0 ) }`;; (*=====================================================================*) (* Definition 26 in collection_geom file ==============================*) let condC = new_definition ` condC (M13:real) (m12:real) (m14:real) (M24:real) (m34:real) (m23:real) = (( m12 + m23 >= M13 ) /\ ( m14 + m34 >= M13 ) /\ ( m12 + m14 > M24 ) /\ ( m23 + m34 > M24 ) /\ ( delta_x (M13 pow 2) (m12 pow 2) (m14 pow 2) (M24 pow 2) (m34 pow 2) (m23 pow 2) >= &0 ))` ;; (*=====================================================================*) (* Begin to prove lemma 8.30 *) (*=====================================================================*) (* lemma_grap*) let in_lemma2 = REWRITE_CONV[IN_ELIM_THM] `e IN {{u, v} | u IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u = v) /\ d3 u v <= &2 * t0}`;; let in_inv = prove (`{{u, v} | u IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u = v) /\ d3 u v <= &2 * t0} e <=> e IN {{u, v} | u IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u = v) /\ d3 u v <= &2 * t0}`,SET_TAC[]);; let lemma_graph = prove(`center_pac (s:real^3->bool) (v0:real^3) ==> graph (e_std s v0)`, REWRITE_TAC[e_std;tru_pack;graph] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ x IN open_ball v0 (&2 * t0)}= s INTER open_ball v0 (&2 * t0)`] THEN GEN_TAC THEN REWRITE_TAC[in_inv] THEN REWRITE_TAC[in_lemma2] THEN REWRITE_TAC[ARITH_RULE `2 = SUC (SUC 0)`;HAS_SIZE_CLAUSES]THEN REPEAT STRIP_TAC THEN EXISTS_TAC`u:real^3` THEN EXISTS_TAC`{v:real^3}` THEN ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY] THEN EXISTS_TAC `v:real^3` THEN EXISTS_TAC `{}:real^3->bool` THEN ASM_REWRITE_TAC[IN_INSERT;NOT_IN_EMPTY]);; (*==============================================================================*) (* end lemma_graph*) (*==============================================================================*) (*unions_lemma*) let uni_lemma1 = prove( ` UNIONS {{u, v} | u IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u = v) /\ d3 u v <= &2 * t0} =UNIONS {{u1, v1} | u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0}`,SET_TAC[]);; let in_lemma1 = REWRITE_CONV[IN_ELIM_THM] `u IN {{u1, v1} | u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0}`;; let in_lemma = ONCE_REWRITE_CONV[IN_ELIM_THM] `x IN {y | ?u. u IN {{u1, v1} | u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0} /\ y IN u}`;; let exist_lemma = prove(` (?u. (?u1 v1. (u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0) /\ u = {u1, v1}) /\ x IN u) <=> (?u1 v1. (u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0 /\ x IN { u1, v1 }))`, MESON_TAC[]);; let cha_lemma1 = prove (`{x | ?u. u IN {{u1, v1} | u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0} /\ x IN u} = {y | ?u. u IN {{u1, v1} | u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0} /\ y IN u}`,SET_TAC[]);; let exist_lemma1 = prove(` (?u1 v1. u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0 /\ x = u1 \/ u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0 /\ x = v1) <=> ( ?v1. x IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(x = v1) /\ d3 x v1 <= &2 * t0 ) \/ ( ?u1. u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ x IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = x) /\ d3 u1 x <= &2 * t0 )`,MESON_TAC[]);; let unions_lemma = prove (`!(v0:real^3) (s:real^3->bool). center_pac (s:real^3->bool) (v0:real^3) ==> UNIONS (e_std s v0) SUBSET v_std s v0`, REPEAT GEN_TAC THEN REWRITE_TAC[e_std;v_std;tru_pack] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SET_RULE` {x | x IN s /\ x IN open_ball v0 (&2 * t0)}= s INTER open_ball v0 (&2 * t0)`] THEN ONCE_REWRITE_TAC[uni_lemma1] THEN REWRITE_TAC[SUBSET;UNIONS] THEN ONCE_REWRITE_TAC[cha_lemma1] THEN ONCE_REWRITE_TAC[in_lemma] THEN GEN_TAC THEN ONCE_REWRITE_TAC[in_lemma1] THEN REWRITE_TAC[exist_lemma] THEN REWRITE_TAC[SET_RULE ` x IN {u1 , v1} <=> x = u1 \/ x = v1`] THEN REWRITE_TAC[TAUT ` u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0 /\ (x = u1 \/ x = v1) <=> ( u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0 /\ x = u1) \/ ( u1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ v1 IN s INTER open_ball v0 (&2 * t0) DIFF {v0} /\ ~(u1 = v1) /\ d3 u1 v1 <= &2 * t0 /\ x = v1 )`] THEN REWRITE_TAC[exist_lemma1] THEN STRIP_TAC);; (*==================================================================================*) (* end of unions_lemma*) (*==================================================================================*) (*fan1_lemma*) let lemma7_1 = new_axiom `!(v0:real^3) (s:real^3->bool) r:real. center_pac (s:real^3->bool) (v0:real^3) ==> FINITE (tru_pack v0 r s )`;; let fini_lemma = prove(` FINITE (tru_pack v0 (&2 * t0) s) ==> FINITE (tru_pack v0 (&2 * t0) s DIFF {v0})`, REWRITE_TAC[FINITE_DIFF]);; let infi_lemma2 = prove(`center_pac s v0 ==> FINITE (tru_pack v0 (&2 * t0) s) `, REWRITE_TAC[lemma7_1]);; let fini_lemma1 = prove(`!(v0:real^3) (s:real^3->bool). center_pac s v0 ==> FINITE ( v_std s v0 )`, REPEAT GEN_TAC THEN REWRITE_TAC[v_std] THEN DISCH_TAC THEN MATCH_MP_TAC ( fini_lemma) THEN UNDISCH_TAC `center_pac (s:real^3->bool) (v0:real^3)` THEN REWRITE_TAC[infi_lemma2]);; let fan1_lemma = prove(`!(v0:real^3) (s:real^3->bool). center_pac (s:real^3->bool) (v0:real^3) /\ ~( v_std s v0 = {}) ==> fan1 (v0 , v_std s v0 , e_std s v0 )` , REPEAT GEN_TAC THEN REWRITE_TAC[fan1] THEN REWRITE_TAC[TAUT `center_pac s v0 /\ ~(v_std s v0 = {} ) ==> FINITE ( v_std s v0 ) /\ ~(v_std s v0 SUBSET {}) <=> ( center_pac s v0 /\ ~( v_std s v0 = {}) ==> FINITE ( v_std s v0 ) ) /\ ( center_pac s v0 /\ ~( v_std s v0 = {}) ==> ~(v_std s v0 SUBSET {}))`] THEN REWRITE_TAC[SET_RULE ` ~( v_std s v0 SUBSET {}) <=> ~( v_std s v0 = {})`] THEN MESON_TAC[fini_lemma1]);; (*======================================================================================*) (* end of fan1_lemma*) (*======================================================================================*) (* fan2_lemma*) let fan2_lemma = prove(`!(v0:real^3) (s:real^3->bool). center_pac s v0 ==> fan2 (v0,v_std s v0,e_std s v0 )`, REPEAT GEN_TAC THEN REWRITE_TAC[fan2;v_std;DIFF] THEN SET_TAC[]);; (*======================================================================================*) (*end of fan2_lemma*) (*======================================================================================*) (* fan3_lemma*) let lemmaf3 = prove(`{v, w} IN {{x, y} | x IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ y IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ ~(x = y) /\ d3 x y <= &2 * t0} <=> (?x y. (x IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ y IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ ~(x = y) /\ d3 x y <= &2 * t0) /\ {v, w} = {x, y})`, REWRITE_TAC[IN_ELIM_THM]);; let lemma_subset = prove ( `{w | ?x y. (x IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ y IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ ~(x = y) /\ d3 x y <= &2 * t0) /\ {v, w} = {x, y}} SUBSET tru_pack v0 (&2 * t0) s DIFF {v0}`, SET_TAC[]);; let lemmaf32 = prove (` FINITE (tru_pack v0 (&2 * t0) s DIFF {v0}) ==> FINITE {w | ?x y. (x IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ y IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ ~(x = y) /\ d3 x y <= &2 * t0) /\ {v, w} = {x, y}} <=> ( FINITE (tru_pack v0 (&2 * t0) s DIFF {v0}) /\ {w | ?x y. (x IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ y IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ ~(x = y) /\ d3 x y <= &2 * t0) /\ {v, w} = {x, y}} SUBSET tru_pack v0 (&2 * t0) s DIFF {v0} ) ==> FINITE {w | ?x y. (x IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ y IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ ~(x = y) /\ d3 x y <= &2 * t0) /\ {v, w} = {x, y}}`, MESON_TAC[lemma_subset]);; let lemmaf33 = prove(` FINITE (tru_pack v0 (&2 * t0) s DIFF {v0}) ==> FINITE {w | ?x y. (x IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ y IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ ~(x = y) /\ d3 x y <= &2 * t0) /\ {v, w} = {x, y}}`, REWRITE_TAC[lemmaf32] THEN REWRITE_TAC[FINITE_SUBSET]);; let lemmaf34 = prove (` condC (#2.6) (&2) (&2) (#2.6) (&2) (&2)`, REWRITE_TAC[condC;delta_x] THEN REAL_ARITH_TAC);; (* prove lemma have_not_prove*) let lemma_c = REWRITE_CONV[IN_ELIM_THM]`&1 / (h + &1) % p + h / (h + &1) % v IN {w | ?a b. &0 <= a /\ &0 <= b /\ a + b = &1 /\ w = a % p + b % v}`;; let lemma_2 = REWRITE_CONV[IN_ELIM_THM]`&1 / (h + &1) % p + h / (h + &1) % v IN {w | ?a b. &0 <= a /\ &0 <= b /\ a + b = &1 /\ w = a % q + b % v0}`;; let lemma_c1 = prove (` &0 <= h ==> &0 <= &1 / (h + &1)`, DISCH_TAC THEN MATCH_MP_TAC (REAL_LE_DIV) THEN UNDISCH_TAC `&0 <= h` THEN REWRITE_TAC[MESON[]` &0 <= h ==> &0 <= &1 /\ &0 <= h + &1 <=> ( &0 <= h ==> &0 <= &1 ) /\ (&0 <= h ==> &0 <= h + &1 )`] THEN REAL_ARITH_TAC);; let lemma_c11 = prove (`&0 < &1 / ( h + &1) ==> &0 <= &1 / (h + &1) `, REAL_ARITH_TAC);; let lemma_c111 = prove (`h >= &0 ==> &0 <= &1 / (h + &1)`, DISCH_TAC THEN MATCH_MP_TAC (lemma_c1) THEN UNDISCH_TAC `h:real >= &0` THEN REAL_ARITH_TAC);; let lemma_ch = prove (` &0 <= h ==> &0 <= h / (h + &1)`, DISCH_TAC THEN MATCH_MP_TAC (REAL_LE_DIV) THEN UNDISCH_TAC `&0 <= h` THEN REWRITE_TAC[MESON[]` &0 <= h ==> &0 <= h /\ &0 <= h + &1 <=> ( &0 <= h ==> &0 <= h ) /\ (&0 <= h ==> &0 <= h + &1 )`] THEN REAL_ARITH_TAC);; let lemma_ch1 = prove (`&0 < h / ( h + &1) ==> &0 <= h / (h + &1) `, REAL_ARITH_TAC);; let lemma_ch11 = prove (`h >= &0 ==> &0 <= h / (h + &1)`, DISCH_TAC THEN MATCH_MP_TAC (lemma_ch) THEN UNDISCH_TAC `h:real >= &0` THEN REAL_ARITH_TAC);; let lemma_1 = prove (` h >= &0 ==> &1 / (h + &1) + h / (h + &1) = &1`, STRIP_TAC THEN REWRITE_TAC[REAL_ARITH `&1 / (h + &1) + h / (h + &1) = (h + &1)/ (h + &1)`] THEN MATCH_MP_TAC (REAL_DIV_REFL) THEN UNDISCH_TAC `h >= &0` THEN REAL_ARITH_TAC);; let le1_diag_trape = prove (` p:real^3 = q + (h:real) % (v0 - v) /\ h >= &0 ==> &1 / (h + &1) % p + h / (h + &1) % v IN {w | ?a b. &0 <= a /\ &0 <= b /\ a + b = &1 /\ w = a % p + b % v}`, REWRITE_TAC[lemma_c]THEN STRIP_TAC THEN EXISTS_TAC ` &1 / (h+ &1)` THEN EXISTS_TAC ` h:real / (h+ &1)` THEN UNDISCH_TAC `h:real >= &0` THEN REWRITE_TAC[MESON[] ` h >= &0 ==> &0 <= &1 / (h + &1) /\ &0 <= h / (h + &1) /\ &1 / (h + &1) + h / (h + &1) = &1 /\ &1 / (h + &1) % p + h / (h + &1) % v = &1 / (h + &1) % p + h / (h + &1) % v <=> ( h >= &0 ==> &0 <= &1 / (h + &1) ) /\ ( h >= &0 ==> &0 <= h / (h + &1) ) /\ ( h >= &0 ==> &1 / (h + &1) + h / (h + &1) = &1 ) /\ ( h >= &0 ==> &1 / (h + &1) % p + h / (h + &1) % v = &1 / (h + &1) % p + h / (h + &1) % v )`] THEN REWRITE_TAC[lemma_c111;lemma_ch11;lemma_1]);; let diag_trape = prove (` !(p:real^3) (q:real^3) (v0:real^3) (v:real^3) (h:real). p = q + h % (v0 - v) /\ h >= &0 ==> ~(conv {p, v} INTER conv {q, v0} = {})`, REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[SET_RULE ` ~(conv {p, v} INTER conv {q, v0} = {}) <=> ? x. x IN conv {p, v} /\ x IN conv {q, v0}`] THEN EXISTS_TAC ` (&1/(h + &1)) % (p:real^3) + ((h:real)/(h + &1)) % (v:real^3)` THEN UNDISCH_TAC `p:real^3 = q + h % (v0 - v) /\ h >= &0 ` THEN REWRITE_TAC[CONV_SET2] THEN REWRITE_TAC[MESON[] ` p = q + h % (v0 - v) /\ h >= &0 ==> &1 / (h + &1) % p + h / (h + &1) % v IN {w | ?a b. &0 <= a /\ &0 <= b /\ a + b = &1 /\ w = a % p + b % v} /\ &1 / (h + &1) % p + h / (h + &1) % v IN {w | ?a b. &0 <= a /\ &0 <= b /\ a + b = &1 /\ w = a % q + b % v0} <=> ( p = q + h % (v0 - v) /\ h >= &0 ==> &1 / (h + &1) % p + h / (h + &1) % v IN {w | ?a b. &0 <= a /\ &0 <= b /\ a + b = &1 /\ w = a % p + b % v} ) /\ ( p = q + h % (v0 - v) /\ h >= &0 ==> &1 / (h + &1) % p + h / (h + &1) % v IN {w | ?a b. &0 <= a /\ &0 <= b /\ a + b = &1 /\ w = a % q + b % v0} )`] THEN REWRITE_TAC[le1_diag_trape] THEN REWRITE_TAC[lemma_2] THEN STRIP_TAC THEN EXISTS_TAC ` &1 / (h+ &1)` THEN EXISTS_TAC ` h:real / (h+ &1)` THEN UNDISCH_TAC `h >= &0` THEN REWRITE_TAC[MESON[] ` h >= &0 ==> &0 <= &1 / (h + &1) /\ &0 <= h / (h + &1) /\ &1 / (h + &1) + h / (h + &1) = &1 /\ &1 / (h + &1) % p + h / (h + &1) % v = &1 / (h + &1) % q + h / (h + &1) % v0 <=> ( h >= &0 ==> &0 <= &1 / (h + &1) ) /\ ( h >= &0 ==> &0 <= h / (h + &1) ) /\ ( h >= &0 ==> &1 / (h + &1) + h / (h + &1) = &1 ) /\ ( h >= &0 ==> &1 / (h + &1) % p + h / (h + &1) % v = &1 / (h + &1) % q + h / (h + &1) % v0 )`] THEN REWRITE_TAC[lemma_c111; lemma_ch11; lemma_1] THEN DISCH_TAC THEN REWRITE_TAC[VECTOR_ARITH `&1 / (h + &1) % p + h / (h + &1) % v = &1 / (h + &1) % q + h / (h + &1) % v0 <=> &1 / (h + &1) % p = &1 / (h + &1) % q + h / (h + &1) % v0 - ( h / (h + &1) % v )`] THEN REWRITE_TAC[VECTOR_ARITH `&1 / (h + &1) % p = &1 / (h + &1) % q + h / (h + &1) % v0 - h / (h + &1) % v <=> &1 / (h + &1) % p = &1 / (h + &1) % q + h / (h + &1) % (v0 - v)`] THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC ) ;; let inv_diag = prove(`!(p:real^3) (q:real^3) (v0:real^3) (v:real^3) (h:real). p = q + h % (v0 - v) /\ conv {p, v} INTER conv {q, v0} = {} ==> ~(h >= &0)`, REWRITE_TAC[TAUT `p = q + h % (v0 - v) /\ conv {p, v} INTER conv {q, v0} = {} ==> ~(h >= &0) <=> p = q + h % (v0 - v) ==> ( conv {p, v} INTER conv {q, v0} = {} ==> ~(h >= &0) )`] THEN REWRITE_TAC[TAUT ` conv {p, v} INTER conv {q, v0} = {} ==> ~(h >= &0) <=> ( h >= &0 ==> ~(conv {p, v} INTER conv {q, v0} = {}))`] THEN REWRITE_TAC[TAUT `p = q + h % (v0 - v)==> h >= &0 ==> ~(conv {p, v} INTER conv {q, v0} = {}) <=> ( p = q + h % (v0 - v) /\ h >= &0 ) ==> ~(conv {p, v} INTER conv {q, v0} = {})`] THEN REWRITE_TAC[diag_trape]);; let inv1_diag = prove (`!(p:real^3) (q:real^3) (v0:real^3) (v:real^3) (h:real). conv {p, v} INTER conv {q, v0} = {} /\ h >= &0 ==> ~(p = q + h % (v0 - v))`, MESON_TAC[diag_trape]);; let vec_le = prove (` (h:real) % ( v0:real^3 - v ) = -- h % (v - v0)`, VECTOR_ARITH_TAC);; let diag_trape1 = prove ( ` !(p:real^3) (q:real^3) (v0:real^3) (v:real^3) (h:real). p = q + h % (v0 - v) /\ h <= &0 ==> ~(conv {p, v0} INTER conv {q, v} = {})`, REWRITE_TAC[REAL_ARITH ` h <= &0 <=> -- h >= &0`] THEN ONCE_REWRITE_TAC[vec_le] THEN REWRITE_TAC[diag_trape]);; let inv_diag1 = prove(`!(p:real^3) (q:real^3) (v0:real^3) (v:real^3) (h:real). p = q + h % (v0 - v) /\ conv {p, v0} INTER conv {q, v} = {} ==> ~(h <= &0)`, REWRITE_TAC[TAUT `p = q + h % (v0 - v) /\ conv {p, v0} INTER conv {q, v} = {} ==> ~(h <= &0) <=> p = q + h % (v0 - v) ==> ( conv {p, v0} INTER conv {q, v} = {} ==> ~(h <= &0) )`] THEN REWRITE_TAC[TAUT ` conv {p, v0} INTER conv {q, v} = {} ==> ~(h <= &0) <=> ( h <= &0 ==> ~(conv {p, v0} INTER conv {q, v} = {}))`] THEN REWRITE_TAC[TAUT `p = q + h % (v0 - v)==> h <= &0 ==> ~(conv {p, v0} INTER conv {q, v} = {}) <=> ( p = q + h % (v0 - v) /\ h <= &0 ) ==> ~(conv {p, v0} INTER conv {q, v} = {})`] THEN REWRITE_TAC[diag_trape1]);; let inv1_diag1 = prove (`!(p:real^3) (q:real^3) (v0:real^3) (v:real^3) (h:real). conv {p, v0} INTER conv {q, v} = {} /\ h <= &0 ==> ~(p = q + h % (v0 - v))`, MESON_TAC[diag_trape1]);; let lemma_3_4 = new_axiom (` !(v1:real^3) (v2:real^3) (v3:real^3) (v4:real^3). !(m12:real) (m23:real) (m34:real) (m14:real) (M13:real) (M24:real). d3 v1 v2 >= m12 /\ d3 v2 v3 >= m23 /\ d3 v3 v4 >= m34 /\ d3 v4 v1 >= m14 /\ d3 v1 v3 < M13 /\ d3 v2 v4 <= M24 /\ condC M13 m12 m14 M24 m34 m23 ==> conv {v1 , v3} INTER conv {v2 , v4} = {}`);; let lemma_3_4_c = new_axiom (` !(p:real^3) (q:real^3) (v:real^3) (v0:real^3). d3 p q >= m12 /\ d3 q v >= m23 /\ d3 v v0 >= m34 /\ d3 v0 p >= m14 /\ d3 p v < M13 /\ d3 q v0 <= M24 /\ condC M13 m12 m14 M24 m34 m23 ==> conv {p , v} INTER conv {q , v0} = {}`);; let affine_two_points = prove (`!(x:real^3) (y:real^3). affine {z | ?u v. u + v = &1 /\ z = u % x + v % y}`, REPEAT GEN_TAC THEN REWRITE_TAC[affine] THEN REPEAT GEN_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `( u:real) * u' + v * u''` THEN EXISTS_TAC ` u:real * v' + v * v''` THEN REWRITE_TAC[REAL_RING `(u * u' + v * u'') + u * v' + v * v'' = u * (u' + v') + v * (u'' + v'')`] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[REAL_RING `u * &1 + v * &1 = u + v`] THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);; let ch_le = prove (`(!x y u v. x IN t /\ y IN t /\ u + v = &1 ==> u % x + v % y IN t) <=> (!v1 v2 r s. v1 IN t /\ v2 IN t /\ r + s = &1 ==> r % v1 + s % v2 IN t)`, MESON_TAC[]);; let sub_aff = prove ( ` ! (t:real^3->bool) v1:real^3 v2:real^3 . affine t /\ {v1 , v2} SUBSET t ==> {w | ?r s. r + s = &1 /\ w = r % v1 + s % v2} SUBSET t`, REPEAT GEN_TAC THEN REWRITE_TAC[affine] THEN REWRITE_TAC[SET_RULE `{v1, v2} SUBSET t <=> v1 IN t /\ v2 IN t`] THEN STRIP_TAC THEN REWRITE_TAC[SUBSET] THEN GEN_TAC THEN ONCE_REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC `!x:real^3 y:real^3 u:real v. x IN (t:real^3->bool) /\ y IN t /\ u + v = &1 ==> u % x + v % y IN t` THEN ONCE_REWRITE_TAC[ch_le] THEN UNDISCH_TAC `v1:real^3 IN t:real^3->bool` THEN UNDISCH_TAC `v2:real^3 IN t:real^3->bool` THEN UNDISCH_TAC `r + s = &1 ` THEN MESON_TAC[]);; let ans = prove (` INTERS {t:real^3->bool | affine t /\ {v1:real^3, v2} SUBSET t} = {w:real^3 | ?r:real s. r + s = &1 /\ w = r % v1 + s % v2} <=> INTERS {t | affine t /\ {v1, v2} SUBSET t} SUBSET {w | ?r s. r + s = &1 /\ w = r % v1 + s % v2} /\ {w | ?r s. r + s = &1 /\ w = r % v1 + s % v2} SUBSET INTERS {t | affine t /\ {v1, v2} SUBSET t} `, SET_TAC[]);; let anss = SET_RULE `(!t:real^3->bool. affine t /\ {v1, v2} SUBSET t ==> {w | ?r s. r + s = &1 /\ w = r % v1 + s % v2} SUBSET t ) ==> {w | ?r s. r + s = &1 /\ w = r % v1 + s % v2} SUBSET INTERS {t | affine t /\ {v1, v2} SUBSET t}`;; let chon = SET_RULE `!s:A->bool. s IN { t | P t} ==> INTERS { t| P t} SUBSET s`;; let subset_two_points = SET_RULE `{a , b } SUBSET s <=> a IN s /\ b IN s`;; let sub_1 = prove (`{w | ?r s. r + s = &1 /\ w = r % (v1:real^3) + s % (v2:real^3)} SUBSET INTERS {(t:real^3->bool) | affine t /\ {v1, v2} SUBSET t}`, MATCH_MP_TAC(anss) THEN REWRITE_TAC[sub_aff]);; let AFF_HULL_TWO_POINTS = prove (` ! v1:real^3 v2:real^3 . affine hull {v1 , v2} = { w | ? r s . r + s = &1 /\ w = r % v1 + s % v2 }`, REPEAT GEN_TAC THEN REWRITE_TAC[hull] THEN REWRITE_TAC[ans] THEN REWRITE_TAC[sub_1] THEN MATCH_MP_TAC(chon) THEN REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[affine_two_points] THEN REWRITE_TAC[subset_two_points] THEN REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL [EXISTS_TAC `&1`; EXISTS_TAC `&0`] THENL [EXISTS_TAC `&0`; EXISTS_TAC `&1`] THENL [REWRITE_TAC[REAL_ARITH ` &1 + &0 = &1`]; REWRITE_TAC[REAL_ARITH ` &0 + &1 = &1`]] THEN VECTOR_ARITH_TAC);; let fa4 = prove(`(p:real^3) = (q:real^3) + (h:real) % ((v0:real^3) - (v:real^3)) /\ h = &0 ==> p = q`, REWRITE_TAC[TAUT` p = q + h % (v0 - v) /\ h = &0 ==> p = q <=> h = &0 /\ p = q + h % (v0 - v) ==> p = q`]THEN REWRITE_TAC[TAUT `h = &0 /\ p = q + h % (v0 - v) ==> p = q <=> h = &0 ==> p = q + h % (v0 - v) ==> p = q`] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);; let tam = prove (` (p:real^3) = (q:real^3) + (h:real) % ((v0:real^3) - (v:real^3)) /\ ~(h = &0) ==> ~(conv {p, v} INTER conv {q, v0} = {}) \/ ~(conv {p, v0} INTER conv {q, v} = {})`, REWRITE_TAC[REAL_ARITH `~(h = &0) <=> (h > &0 \/ h < &0)`] THEN REWRITE_TAC[TAUT ` p = q + h % (v0 - v) /\ (h > &0 \/ h < &0) <=> p = q + h % (v0 - v) /\ h > &0 \/ p = q + h % (v0 - v) /\ h < &0 `] THEN STRIP_TAC THENL[ MATCH_MP_TAC(TAUT ` ~(conv {(p:real^3) , (v:real^3)} INTER conv {(q:real^3), (v0:real^3)} = {}) ==> ~(conv {p, v} INTER conv {q, v0} = {}) \/ ~(conv {p, v0} INTER conv {q, v} = {})`); MATCH_MP_TAC(TAUT ` ~(conv {(p:real^3) , (v0:real^3)} INTER conv {(q:real^3), (v:real^3)} = {}) ==> ~(conv {p, v} INTER conv {q, v0} = {}) \/ ~(conv {p, v0} INTER conv {q, v} = {})`)] THENL[MATCH_MP_TAC(diag_trape);MATCH_MP_TAC(diag_trape1)] THEN EXISTS_TAC ` h:real` THENL[STRIP_TAC THENL[ASM_REWRITE_TAC[]; UNDISCH_TAC ` h > &0 ` ] THEN REAL_ARITH_TAC ; STRIP_TAC THENL[ASM_REWRITE_TAC[]; UNDISCH_TAC ` h < &0 ` ] THEN REAL_ARITH_TAC]);; let tam1 = prove (` (p:real^3) = (q:real^3) + (h:real) % ( (v0:real^3) - (v:real^3)) /\ ~(p = q ) ==> ~(conv {p, v} INTER conv {q, v0} = {}) \/ ~(conv {p, v0} INTER conv {q, v} = {})`, STRIP_TAC THEN MATCH_MP_TAC(tam) THEN STRIP_TAC THENL[ASM_REWRITE_TAC[];UNDISCH_TAC `~(p:real^3 = q)` THEN UNDISCH_TAC `p:real^3 = q + h % (v0 - v)`] THEN REWRITE_TAC[TAUT` ~(p = q) ==> ~(h = &0) <=> h = &0 ==> p = q`] THEN MESON_TAC[fa4]);; let tam2 = prove (` ~(p:real^3 = q) /\ conv {p, v:real^3} INTER conv {q, v0:real^3} = {} /\ conv {p, v0} INTER conv {q, v} = {} ==> ~(p = q + (h:real) % (v0 - v) )`, MESON_TAC[tam1]);; let c_nov16 = prove (` ~(p:real^3 = q) ==> (!(x:real^3) (y:real^3). x IN (s:real^3->bool) /\ y IN s /\ ~(x = y) ==> norm (x - y) >= &2) /\ (v0:real^3) IN s ==> (v:real^3) IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0) ==> (((v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (p IN s /\ norm (p - v0) < &2 * t0 /\ ~(p = v0)) /\ ~(v = p) /\ norm (v - p) <= &2 * t0) \/ ((p IN s /\ norm (p - v0) < &2 * t0 /\ ~(p = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(p = v) /\ norm (p - v) <= &2 * t0)) ==> (((v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (q IN s /\ norm (q - v0) < &2 * t0 /\ ~(q = v0)) /\ ~(v = q) /\ norm (v - q) <= &2 * t0) \/ ((q IN s /\ norm (q - v0) < &2 * t0 /\ ~(q = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(q = v) /\ norm (q - v) <= &2 * t0)) ==> ~(p = q + h % (v0 - v))`, REWRITE_TAC[MESON[NORM_SUB]` (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (p IN s /\ norm (p - v0) < &2 * t0 /\ ~(p = v0)) /\ ~(v = p) /\ norm (v - p) <= &2 * t0 \/ (p IN s /\ norm (p - v0) < &2 * t0 /\ ~(p = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(p = v) /\ norm (p - v) <= &2 * t0 <=> (p IN s /\ norm (p - v0) < &2 * t0 /\ ~(p = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(p = v) /\ norm (p - v) <= &2 * t0 `] THEN REPEAT STRIP_TAC THEN UNDISCH_TAC `p:real^3 = q + (h:real) % (v0 - v)` THEN REWRITE_TAC[MESON[]`p = q + h % (v0 - v) ==> F <=> ~(p = q + h % (v0 - v))`] THEN MATCH_MP_TAC(tam2) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN MATCH_MP_TAC(lemma_3_4) THEN EXISTS_TAC ` &2` THEN EXISTS_TAC ` &2` THEN EXISTS_TAC ` &2` THEN EXISTS_TAC ` &2` THEN EXISTS_TAC ` #2.6` THEN EXISTS_TAC ` #2.6` THEN REWRITE_TAC[lemmaf34] THEN REWRITE_TAC[d3;dist] THENL[UNDISCH_TAC ` norm (q:real^3 - v0) < &2 * t0` ; UNDISCH_TAC` norm (q:real^3 - v) <= &2 * t0` ] THEN REWRITE_TAC[t0] THEN REWRITE_TAC[REAL_ARITH ` &2 * #1.255 = #2.51`] THENL[REWRITE_TAC[TAUT ` norm (q - v0) < #2.51 ==> norm (p - q) >= &2 /\ norm (q - v) >= &2 /\ norm (v - v0) >= &2 /\ norm (v0 - p) >= &2 /\ norm (p - v) < #2.6 /\ norm (q - v0) <= #2.6 <=> ( norm (q - v0) < #2.51 ==> norm (p - q) >= &2 /\ norm (q - v) >= &2 /\ norm (v - v0) >= &2 /\ norm (v0 - p) >= &2 /\ norm (p - v) < #2.6) /\ (norm (q - v0) < #2.51 ==> norm (q - v0) <= #2.6 )`] THEN CONJ_TAC THENL [DISCH_TAC ; REAL_ARITH_TAC]; REWRITE_TAC[TAUT `norm (q - v) <= #2.51 ==> norm (p - q) >= &2 /\ norm (q - v0) >= &2 /\ norm (v0 - v) >= &2 /\ norm (v - p) >= &2 /\ norm (p - v0) < #2.6 /\ norm (q - v) <= #2.6 <=> ( norm (q - v) <= #2.51 ==> norm (p - q) >= &2 /\ norm (q - v0) >= &2 /\ norm (v0 - v) >= &2 /\ norm (v - p) >= &2 /\ norm (p - v0) < #2.6 ) /\ ( norm (q - v) <= #2.51 ==> norm (q - v) <= #2.6 ) `] THEN CONJ_TAC THENL [DISCH_TAC ; REAL_ARITH_TAC]] THENL[UNDISCH_TAC `norm (p:real^3 - v) <= &2 * t0` THEN REWRITE_TAC[t0] THEN REWRITE_TAC[REAL_ARITH ` &2 * #1.255 = #2.51`] THEN REWRITE_TAC[TAUT ` norm (p - v) <= #2.51 ==> norm (p - q) >= &2 /\ norm (q - v) >= &2 /\ norm (v - v0) >= &2 /\ norm (v0 - p) >= &2 /\ norm (p - v) < #2.6 <=> ( norm (p - v) <= #2.51 ==> norm (p - q) >= &2 /\ norm (q - v) >= &2 /\ norm (v - v0) >= &2 /\ norm (v0 - p) >= &2 ) /\ (norm (p - v) <= #2.51 ==> norm (p - v) < #2.6 )`] THEN CONJ_TAC THENL [DISCH_TAC; REAL_ARITH_TAC] THEN UNDISCH_TAC ` ~(p:real^3 = v0)` THEN UNDISCH_TAC ` v0:real^3 IN s:real^3->bool` THEN UNDISCH_TAC` !x:real^3 y:real^3. x IN s:real^3->bool /\ y IN s /\ ~(x = y) ==> norm (x - y) >= &2` THEN UNDISCH_TAC ` p:real^3 IN s:real^3->bool` THEN UNDISCH_TAC `q:real^3 IN (s:real^3->bool)` THEN UNDISCH_TAC ` ~(p:real^3 = q)` THEN UNDISCH_TAC ` v:real^3 IN (s:real^3->bool)` THEN UNDISCH_TAC ` ~(q:real^3 = v)` THEN UNDISCH_TAC ` ~(v:real^3 = v0)` ; UNDISCH_TAC` norm (p:real^3 - v0) < &2 * t0` THEN REWRITE_TAC[t0] THEN REWRITE_TAC[REAL_ARITH ` &2 * #1.255 = #2.51`] THEN REWRITE_TAC[TAUT ` norm (p - v0) < #2.51 ==> norm (p - q) >= &2 /\ norm (q - v0) >= &2 /\ norm (v0 - v) >= &2 /\ norm (v - p) >= &2 /\ norm (p - v0) < #2.6 <=> (norm (p - v0) < #2.51 ==> norm (p - q) >= &2 /\ norm (q - v0) >= &2 /\ norm (v0 - v) >= &2 /\ norm (v - p) >= &2 ) /\ (norm (p - v0) < #2.51 ==> norm (p - v0) < #2.6)`] THEN CONJ_TAC THENL [DISCH_TAC; REAL_ARITH_TAC] THEN UNDISCH_TAC ` p:real^3 IN (s:real^3->bool) ` THEN UNDISCH_TAC ` q:real^3 IN (s:real^3->bool) ` THEN UNDISCH_TAC ` v:real^3 IN (s:real^3->bool) ` THEN UNDISCH_TAC ` v0:real^3 IN (s:real^3->bool) ` THEN UNDISCH_TAC ` ~(p:real^3 = q) ` THEN UNDISCH_TAC ` ~(p:real^3 = v) ` THEN UNDISCH_TAC ` ~(v:real^3 = v0) ` THEN UNDISCH_TAC ` ~(q:real^3 = v0) ` THEN UNDISCH_TAC ` !x:real^3 y:real^3. x IN (s:real^3->bool) /\ y IN s /\ ~(x = y) ==> norm (x - y) >= &2 `] THEN MESON_TAC[NORM_SUB]);; let c_nov17 = prove (` ( ~(p:real^3 = q) /\ (!(x:real^3) (y:real^3). x IN (s:real^3->bool) /\ y IN s /\ ~(x = y) ==> norm (x - y) >= &2) /\ (v0:real^3) IN s /\ (v:real^3) IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0) /\ (((v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (p IN s /\ norm (p - v0) < &2 * t0 /\ ~(p = v0)) /\ ~(v = p) /\ norm (v - p) <= &2 * t0) \/ ((p IN s /\ norm (p - v0) < &2 * t0 /\ ~(p = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(p = v) /\ norm (p - v) <= &2 * t0)) /\ (((v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (q IN s /\ norm (q - v0) < &2 * t0 /\ ~(q = v0)) /\ ~(v = q) /\ norm (v - q) <= &2 * t0) \/ ((q IN s /\ norm (q - v0) < &2 * t0 /\ ~(q = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(q = v) /\ norm (q - v) <= &2 * t0)) ) ==> ~(p = q + h % (v0 - v))`, REWRITE_TAC[TAUT ` ~(p = q) /\ (!x y. x IN s /\ y IN s /\ ~(x = y) ==> norm (x - y) >= &2) /\ v0 IN s /\ v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0) /\ ((v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (p IN s /\ norm (p - v0) < &2 * t0 /\ ~(p = v0)) /\ ~(v = p) /\ norm (v - p) <= &2 * t0 \/ (p IN s /\ norm (p - v0) < &2 * t0 /\ ~(p = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(p = v) /\ norm (p - v) <= &2 * t0) /\ ((v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (q IN s /\ norm (q - v0) < &2 * t0 /\ ~(q = v0)) /\ ~(v = q) /\ norm (v - q) <= &2 * t0 \/ (q IN s /\ norm (q - v0) < &2 * t0 /\ ~(q = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(q = v) /\ norm (q - v) <= &2 * t0) ==> ~(p = q + h % (v0 - v)) <=> ~(p:real^3 = q) ==> (!(x:real^3) (y:real^3). x IN (s:real^3->bool) /\ y IN s /\ ~(x = y) ==> norm (x - y) >= &2) /\ (v0:real^3) IN s ==> (v:real^3) IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0) ==> (((v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (p IN s /\ norm (p - v0) < &2 * t0 /\ ~(p = v0)) /\ ~(v = p) /\ norm (v - p) <= &2 * t0) \/ ((p IN s /\ norm (p - v0) < &2 * t0 /\ ~(p = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(p = v) /\ norm (p - v) <= &2 * t0)) ==> (((v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (q IN s /\ norm (q - v0) < &2 * t0 /\ ~(q = v0)) /\ ~(v = q) /\ norm (v - q) <= &2 * t0) \/ ((q IN s /\ norm (q - v0) < &2 * t0 /\ ~(q = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(q = v) /\ norm (q - v) <= &2 * t0)) ==> ~(p = q + h % (v0 - v))`] THEN REWRITE_TAC[c_nov16]);; let fa2 = REWRITE_CONV[IN_ELIM_THM] `{v, q} IN {{u, v'} | (u IN s /\ norm (u - v0) < &2 * t0 /\ ~(u = v0)) /\ (v' IN s /\ norm (v' - v0) < &2 * t0 /\ ~(v' = v0)) /\ ~(u = v') /\ norm (u - v') <= &2 * t0}`;; let fa1 = REWRITE_CONV[IN_ELIM_THM]`p IN {w | {v, w} IN e_std s v0}`;; let c_nov1_17 = prove( ` center_pac (s:real^3->bool) (v0:real^3) ==> (v IN v_std s v0 ==> (!p q h. {w | {v, w} IN e_std s v0} p /\ {w | {v, w} IN e_std s v0} q /\ p = q + h % (v0 - v) ==> p = q))`, REWRITE_TAC[SET_RULE `{w | {v, w} IN e_std s v0} p <=> p IN {w | {v, w} IN e_std s v0}`] THEN REWRITE_TAC[fa1] THEN REPEAT DISCH_TAC THEN REPEAT GEN_TAC THEN UNDISCH_TAC `(v:real^3) IN v_std (s:real^3->bool) v0` THEN UNDISCH_TAC `center_pac (s:real^3->bool) v0` THEN REWRITE_TAC[TAUT `center_pac s v0 ==> v IN v_std s v0 ==> {v, p} IN e_std s v0 /\ {v, q} IN e_std s v0 /\ p = q + h % (v0 - v) ==> p = q <=> ~(p = q ) /\ center_pac s v0 /\ v IN v_std s v0 /\ {v, p} IN e_std s v0 /\ {v, q} IN e_std s v0 ==> ~(p = q + h % (v0 - v))`] THEN REWRITE_TAC[TAUT ` ~(p = q) /\ center_pac s v0 /\ v IN v_std s v0 /\ {v, p} IN e_std s v0 /\ {v, q} IN e_std s v0 ==> ~(p = q + h % (v0 - v)) <=> ~(p = q) ==> ( center_pac s v0 ==> ( v IN v_std s v0 /\ {v, p} IN e_std s v0 /\ {v, q} IN e_std s v0 ==> ~(p = q + h % (v0 - v)))) `] THEN DISCH_TAC THEN DISCH_TAC THEN REWRITE_TAC[v_std;e_std;tru_pack] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SET_RULE ` v IN {x | x IN s /\ x IN open_ball v0 (&2 * t0)} DIFF {v0} <=> v IN s /\ v IN open_ball v0 (&2 * t0) /\ ~ ( v = v0)`] THEN ONCE_REWRITE_TAC[SET_RULE `{{u, v} | (u IN s /\ u IN open_ball v0 (&2 * t0) /\ ~(u = v0)) /\ (v IN s /\ v IN open_ball v0 (&2 * t0) /\ ~(v = v0)) /\ ~(u = v) /\ d3 u v <= &2 * t0} = {{u, v'} | (u IN s /\ u IN open_ball v0 (&2 * t0) /\ ~(u = v0)) /\ (v' IN s /\ v' IN open_ball v0 (&2 * t0) /\ ~(v' = v0)) /\ ~(u = v') /\ d3 u v' <= &2 * t0}`] THEN UNDISCH_TAC ` center_pac (s:real^3->bool) v0` THEN REWRITE_TAC[center_pac;packing_trg;d3;dist;open_ball] THEN REWRITE_TAC[SET_RULE `u IN {y | norm (y - v0) < &2 * t0} <=> norm (u - v0) < &2 * t0`] THEN REWRITE_TAC[SET_RULE ` (s:real^3->bool) x <=> x IN s`] THEN REWRITE_TAC[SET_RULE ` ~(v0 IN (=) u) <=> ~ (u = v0)`] THEN REWRITE_TAC[fa2] THEN REWRITE_TAC[SET_RULE ` {v, p} = {u, v'} <=> ( v = u /\ p = v') \/ ( v = v' /\ p = u )`] THEN REPEAT DISCH_TAC THEN UNDISCH_TAC `(p:real^3) = (q:real^3) + (h:real) % ((v0:real^3) - (v:real^3))` THEN REWRITE_TAC[TAUT ` p = q + h % (v0 - v) ==> F <=> ~(p = q + h % (v0 - v))`] THEN MATCH_MP_TAC(c_nov17) THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]);; let c_nov18 = prove(` ! x:real^3. x IN affine hull {v1:real^3 , v2:real^3} ==> norm ( x - v1) = norm (x - v2) + norm (v2 - v1) \/ norm ( x - v2) = norm (x - v1) + norm (v2 - v1) \/ norm ( v1 - v2) = norm ( x - v1) + norm (x - v2)`, GEN_TAC THEN REWRITE_TAC[AFF_HULL_TWO_POINTS] THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC ` (r:real) + (s:real) = &1 ` THEN REWRITE_TAC [REAL_ARITH ` r + s = &1 <=> r = &1 - s`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[VECTOR_ARITH ` ((&1 - s) % v1 + s % v2) - v1 = s % ( v2 - v1) `] THEN REWRITE_TAC[VECTOR_ARITH ` ((&1 - s) % v1 + s % v2) - v2 = (&1 - s) % (v1 - v2)`] THEN REWRITE_TAC[NORM_MUL] THEN REWRITE_TAC[TAUT ` abs s * norm (v2 - v1) = abs (&1 - s) * norm (v1 - v2) + norm (v2 - v1) \/ abs (&1 - s) * norm (v1 - v2) = abs s * norm (v2 - v1) + norm (v2 - v1) \/ norm (v1 - v2) = abs s * norm (v2 - v1) + abs (&1 - s) * norm (v1 - v2) <=> ~( abs s * norm (v2 - v1) = abs (&1 - s) * norm (v1 - v2) + norm (v2 - v1)) /\ ~( abs (&1 - s) * norm (v1 - v2) = abs s * norm (v2 - v1) + norm (v2 - v1)) ==> norm (v1 - v2) = abs s * norm (v2 - v1) + abs (&1 - s) * norm (v1 - v2)`] THEN DISCH_TAC THEN REWRITE_TAC[NORM_SUB] THEN REWRITE_TAC[REAL_ARITH ` abs s * norm (v1 - v2) + abs (&1 - s) * norm (v1 - v2) = (abs s + abs (&1 - s))*norm (v1 - v2)`] THEN MATCH_MP_TAC(REAL_ARITH` (abs s + abs (&1 - s)) = &1 ==> norm (v1 - v2) = (abs s + abs (&1 - s)) * norm (v1 - v2)`) THEN MATCH_MP_TAC(REAL_ARITH ` abs s = s /\ abs (&1 - s) = (&1 - s) ==> abs s + abs (&1 - s) = &1`) THEN REWRITE_TAC[REAL_ABS_REFL] THEN UNDISCH_TAC ` ~(abs (s:real) * norm (v2:real^3 - v1:real^3) = abs (&1 - s) * norm (v1 - v2) + norm (v2 - v1)) /\ ~(abs (&1 - s) * norm (v1 - v2) = abs s * norm (v2 - v1) + norm (v2 - v1))` THEN REWRITE_TAC[TAUT ` ~(abs s * norm (v2 - v1) = abs (&1 - s) * norm (v1 - v2) + norm (v2 - v1)) /\ ~(abs (&1 - s) * norm (v1 - v2) = abs s * norm (v2 - v1) + norm (v2 - v1)) ==> &0 <= s /\ &0 <= &1 - s <=> ~(&0 <= s /\ &0 <= &1 - s ) /\ ~(abs s * norm (v2 - v1) = abs (&1 - s) * norm (v1 - v2) + norm (v2 - v1)) ==> (abs (&1 - s) * norm (v1 - v2) = abs s * norm (v2 - v1) + norm (v2 - v1))`] THEN STRIP_TAC THEN REWRITE_TAC[NORM_SUB] THEN REWRITE_TAC[REAL_ARITH ` abs (&1 - s) * norm (v1 - v2) = abs s * norm (v1 - v2) + norm (v1 - v2) <=> abs (&1 - s) * norm (v1 - v2) - abs s * norm (v1 - v2) = norm (v1 - v2)`] THEN REWRITE_TAC[REAL_ARITH `abs (&1 - s) * norm (v1 - v2) - abs s * norm (v1 - v2) = ( abs (&1 - s) - abs s ) * norm (v1 - v2)`] THEN REWRITE_TAC[REAL_ARITH ` (abs (&1 - s) - abs s) * norm (v1 - v2) = norm (v1 - v2) <=> norm (v1 - v2) = (abs (&1 - s) - abs s) * norm (v1 - v2)`] THEN MATCH_MP_TAC( REAL_RING `(abs (&1 - s:real) - abs s) = &1 ==> norm (v1 - v2) = (abs (&1 - s) - abs s) * norm (v1 - v2)`) THEN MATCH_MP_TAC(REAL_ARITH ` abs (&1 - s) = (&1 - s) /\ abs s = -- s ==> abs (&1 - s) - abs s = &1`) THEN REWRITE_TAC[REAL_ARITH ` abs s = --s <=> abs (--s) = --s`] THEN REWRITE_TAC[REAL_ABS_REFL] THEN UNDISCH_TAC ` ~(abs s * norm ((v2:real^3) - (v1:real^3)) = abs (&1 - s) * norm (v1 - v2) + norm (v2 - v1))` THEN REWRITE_TAC[TAUT ` ~(abs s * norm (v2 - v1) = abs (&1 - s) * norm (v1 - v2) + norm (v2 - v1)) ==> &0 <= &1 - s /\ &0 <= --s <=> ~(&0 <= &1 - s /\ &0 <= --s) ==> abs s * norm (v2 - v1) = abs (&1 - s) * norm (v1 - v2) + norm (v2 - v1)`] THEN REWRITE_TAC[REAL_ARITH ` abs s * norm (v2 - v1) = abs (&1 - s) * norm (v1 - v2) + norm (v2 - v1) <=> norm (v2 - v1) = abs s * norm (v2 - v1) - abs (&1 - s) * norm (v1 - v2)`] THEN REWRITE_TAC[NORM_SUB] THEN REWRITE_TAC[REAL_ARITH ` abs s * norm (v1 - v2) - abs (&1 - s) * norm (v1 - v2) = ( abs s - abs (&1 - s)) * norm (v1 - v2)`] THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_RING ` (abs s - abs (&1 - s)) = &1 ==> norm (v1 - v2) = (abs s - abs (&1 - s)) * norm (v1 - v2)`) THEN REWRITE_TAC[REAL_ARITH ` abs (&1 - s) = abs (s - &1)`] THEN MATCH_MP_TAC(REAL_ARITH ` abs s = s /\ abs (s - &1) = (s - &1) ==> abs s - abs (s - &1) = &1`) THEN REWRITE_TAC[REAL_ABS_REFL] THEN UNDISCH_TAC `~(&0 <= &1 - s /\ &0 <= --s)` THEN UNDISCH_TAC ` ~(&0 <= s /\ &0 <= &1 - s)` THEN REAL_ARITH_TAC);; let c_inv_nov18 = prove (` ~(norm ( x:real^3 - v1:real^3) = norm (x - v2:real^3) + norm (v2 - v1)) /\ ~(norm ( x - v2) = norm (x - v1) + norm (v2 - v1)) /\ ~(norm ( v1 - v2) = norm ( x - v1) + norm (x - v2)) ==> ~(x IN affine hull {v1 , v2})`, MESON_TAC[c_nov18]);; let fa5 = prove( `!a:real b:real c:real. a >= &2 /\ a <= #2.51 /\ b >= &2 /\ b <= #2.51 /\ c >= &2 /\ c <= #2.51 ==> ~(a = b + c) /\ ~(b = a + c) /\ ~(c = a + b)`, REAL_ARITH_TAC);; let c_nov18e = prove( `(!x:real^3 y:real^3. x IN (s:real^3->bool) /\ y IN s /\ ~(x = y) ==> norm (x - y) >= &2) /\ v0:real^3 IN s ==> v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0) ==> (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (w IN s /\ norm (w - v0) < &2 * t0 /\ ~(w = v0)) /\ ~(v = w) /\ norm (v - w) <= &2 * t0 \/ (w IN s /\ norm (w - v0) < &2 * t0 /\ ~(w = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(w = v) /\ norm (w - v) <= &2 * t0 ==> ~(w IN affine hull {v0, v})`, REWRITE_TAC[MESON[NORM_SUB] ` (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (w IN s /\ norm (w - v0) < &2 * t0 /\ ~(w = v0)) /\ ~(v = w) /\ norm (v - w) <= &2 * t0 \/ (w IN s /\ norm (w - v0) < &2 * t0 /\ ~(w = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(w = v) /\ norm (w - v) <= &2 * t0 <=> (w IN s /\ norm (w - v0) < &2 * t0 /\ ~(w = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(w = v) /\ norm (w - v) <= &2 * t0 `] THEN REPEAT STRIP_TAC THEN UNDISCH_TAC ` w:real^3 IN affine hull {v0:real^3, v:real^3}` THEN REWRITE_TAC[TAUT ` w IN affine hull {v0, v} ==> F <=> ~(w IN affine hull {v0, v})`] THEN MATCH_MP_TAC(c_inv_nov18) THEN UNDISCH_TAC ` !x:real^3 y:real^3. x IN (s:real^3->bool) /\ y IN s /\ ~(x = y) ==> norm (x - y) >= &2` THEN UNDISCH_TAC ` v0:real^3 IN (s:real^3->bool)` THEN UNDISCH_TAC ` v:real^3 IN (s:real^3->bool)` THEN UNDISCH_TAC ` w:real^3 IN (s:real^3->bool)` THEN UNDISCH_TAC ` norm (v:real^3 - v0) < &2 * t0` THEN UNDISCH_TAC ` norm (w:real^3 - v0) < &2 * t0` THEN UNDISCH_TAC ` norm (w:real^3 - v) <= &2 * t0` THEN UNDISCH_TAC ` ~(v:real^3 = v0)` THEN UNDISCH_TAC ` ~(w:real^3 = v0)` THEN UNDISCH_TAC ` ~(w:real^3 = v)` THEN REWRITE_TAC[t0] THEN REWRITE_TAC[REAL_ARITH ` &2 * #1.255 = #2.51`] THEN REWRITE_TAC[NORM_SUB] THEN REPEAT DISCH_TAC THEN MATCH_MP_TAC(fa5) THEN UNDISCH_TAC `norm (v0:real^3 - w) < #2.51` THEN REWRITE_TAC[TAUT ` norm (v0 - w) < #2.51 ==> norm (v0 - w) >= &2 /\ norm (v0 - w) <= #2.51 /\ norm (v - w) >= &2 /\ norm (v - w) <= #2.51 /\ norm (v - v0) >= &2 /\ norm (v - v0) <= #2.51 <=> (norm (v0 - w) < #2.51 ==> norm (v0 - w) >= &2 /\ norm (v - w) >= &2 /\ norm (v - w) <= #2.51 /\ norm (v - v0) >= &2 /\ norm (v - v0) <= #2.51 ) /\ (norm (v0 - w) < #2.51 ==> norm (v0 - w) <= #2.51 )`] THEN CONJ_TAC THENL[DISCH_TAC; REAL_ARITH_TAC] THEN UNDISCH_TAC ` norm (v:real^3 - v0) < #2.51` THEN REWRITE_TAC[TAUT ` norm (v - v0) < #2.51 ==> norm (v0 - w) >= &2 /\ norm (v - w) >= &2 /\ norm (v - w) <= #2.51 /\ norm (v - v0) >= &2 /\ norm (v - v0) <= #2.51 <=> (norm (v - v0) < #2.51 ==> norm (v0 - w) >= &2 /\ norm (v - w) >= &2 /\ norm (v - w) <= #2.51 /\ norm (v - v0) >= &2 ) /\ (norm (v - v0) < #2.51 ==> norm (v - v0) <= #2.51 )`] THEN CONJ_TAC THENL[DISCH_TAC;REAL_ARITH_TAC] THEN ASM_MESON_TAC[NORM_SUB]);; let c_nov18e1 = prove ( `((!x:real^3 y:real^3. x IN (s:real^3->bool) /\ y IN s /\ ~(x = y) ==> norm (x - y) >= &2) /\ v0:real^3 IN s ) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ((v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (w IN s /\ norm (w - v0) < &2 * t0 /\ ~(w = v0)) /\ ~(v = w) /\ norm (v - w) <= &2 * t0 \/ (w IN s /\ norm (w - v0) < &2 * t0 /\ ~(w = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(w = v) /\ norm (w - v) <= &2 * t0 ) ==> ~(w IN affine hull {v0, v})`, REWRITE_TAC[TAUT ` ((!x y. x IN s /\ y IN s /\ ~(x = y) ==> norm (x - y) >= &2) /\ v0 IN s) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ((v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (w IN s /\ norm (w - v0) < &2 * t0 /\ ~(w = v0)) /\ ~(v = w) /\ norm (v - w) <= &2 * t0 \/ (w IN s /\ norm (w - v0) < &2 * t0 /\ ~(w = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(w = v) /\ norm (w - v) <= &2 * t0) ==> ~(w IN affine hull {v0, v}) <=> ((!x y. x IN s /\ y IN s /\ ~(x = y) ==> norm (x - y) >= &2) /\ v0 IN s) ==> (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) ==> ((v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ (w IN s /\ norm (w - v0) < &2 * t0 /\ ~(w = v0)) /\ ~(v = w) /\ norm (v - w) <= &2 * t0 \/ (w IN s /\ norm (w - v0) < &2 * t0 /\ ~(w = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(w = v) /\ norm (w - v) <= &2 * t0) ==> ~(w IN affine hull {v0, v})`] THEN REWRITE_TAC[c_nov18e]);; let c_nov18e2 = prove( `center_pac (s:real^3->bool) (v0:real^3) ==> (v:real^3 IN v_std s v0 ==> {w | {v, w} IN e_std s v0} INTER affine hull {v0, v} = {})`, DISCH_TAC THEN REWRITE_TAC[v_std;e_std;tru_pack] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[SET_RULE ` u IN {x | x IN s /\ x IN open_ball v0 (&2 * t0)} DIFF {v0} <=> u IN s /\ u IN open_ball v0 (&2 * t0) /\ ~(u = v0)`] THEN REWRITE_TAC[open_ball] THEN REWRITE_TAC[SET_RULE ` v IN {y | norm (y - v0) < &2 * t0} <=> norm (v - v0) < &2 * t0 `] THEN ONCE_REWRITE_TAC[SET_RULE ` {{u, v} | (u IN s /\ norm (u - v0) < &2 * t0 /\ ~(u = v0)) /\ (v IN s /\ norm (v - v0) < &2 * t0 /\ ~(v = v0)) /\ ~(u = v) /\ d3 u v <= &2 * t0} = {{u, v'} | (u IN s /\ norm (u - v0) < &2 * t0 /\ ~(u = v0)) /\ (v' IN s /\ norm (v' - v0) < &2 * t0 /\ ~(v' = v0)) /\ ~(u = v') /\ d3 u v' <= &2 * t0}`] THEN UNDISCH_TAC ` center_pac (s:real^3->bool) (v0:real^3)` THEN REWRITE_TAC[center_pac; packing_trg;d3;dist] THEN REWRITE_TAC[SET_RULE ` {w | {v, w} IN {{u, v'} | (u IN s /\ norm (u - v0) < &2 * t0 /\ ~(u = v0)) /\ (v' IN s /\ norm (v' - v0) < &2 * t0 /\ ~(v' = v0)) /\ ~(u = v') /\ norm (u - v') <= &2 * t0}} INTER affine hull {v0, v} = {} <=> !w. w IN {w | {v, w} IN {{u, v'} | (u IN s /\ norm (u - v0) < &2 * t0 /\ ~(u = v0)) /\ (v' IN s /\ norm (v' - v0) < &2 * t0 /\ ~(v' = v0)) /\ ~(u = v') /\ norm (u - v') <= &2 * t0}} ==> ~(w IN affine hull {v0, v})`] THEN DISCH_TAC THEN DISCH_TAC THEN GEN_TAC THEN DISCH_TAC THEN UNDISCH_TAC ` (!x:real^3 y:real^3. (s:real^3->bool) x /\ s y /\ ~(x = y) ==> norm (x - y) >= &2) /\ s (v0:real^3)` THEN REWRITE_TAC[SET_RULE ` (s:real^3->bool) x <=> x IN s `] THEN REWRITE_TAC[SET_RULE `~(y IN (=) x) <=> ~(x = y)`] THEN DISCH_TAC THEN UNDISCH_TAC `(w:real^3) IN {w | {v:real^3, w} IN {{u:real^3, v'} | (u IN (s:real^3->bool) /\ norm (u - v0:real^3) < &2 * t0 /\ ~(u = v0)) /\ (v' IN s /\ norm (v' - v0) < &2 * t0 /\ ~(v' = v0)) /\ ~(u = v') /\ norm (u - v') <= &2 * t0}}` THEN ONCE_REWRITE_TAC[IN_ELIM_THM] THEN REWRITE_TAC[fa2] THEN REWRITE_TAC[SET_RULE ` {v, w} = {u, v'} <=> ( v = u /\ w = v') \/ (v = v' /\ w = u)`] THEN REWRITE_TAC[TAUT ` ((u IN s /\ norm (u - v0) < &2 * t0 /\ ~(u = v0)) /\ (v' IN s /\ norm (v' - v0) < &2 * t0 /\ ~(v' = v0)) /\ ~(u = v') /\ norm (u - v') <= &2 * t0) /\ (v = u /\ w = v' \/ v = v' /\ w = u) <=> (((u IN s /\ norm (u - v0) < &2 * t0 /\ ~(u = v0)) /\ (v' IN s /\ norm (v' - v0) < &2 * t0 /\ ~(v' = v0)) /\ ~(u = v') /\ norm (u - v') <= &2 * t0) /\ v = u /\ w = v') \/ (((u IN s /\ norm (u - v0) < &2 * t0 /\ ~(u = v0)) /\ (v' IN s /\ norm (v' - v0) < &2 * t0 /\ ~(v' = v0)) /\ ~(u = v') /\ norm (u - v') <= &2 * t0) /\ v = v' /\ w = u)`] THEN DISCH_TAC THEN MATCH_MP_TAC(c_nov18e1) THEN ASM_MESON_TAC[]);; let c_nov18e3 = prove( `center_pac (s:real^3->bool) (v0:real^3) ==> (v:real^3 IN v_std s v0 ==> (!p q h. {w | {v, w} IN e_std s v0} p /\ {w | {v, w} IN e_std s v0} q /\ p = q + h % (v0 - v) ==> p = q)) /\ (v IN v_std s v0 ==> {w | {v, w} IN e_std s v0} INTER affine hull {v0, v} = {})`, REWRITE_TAC[TAUT ` center_pac s v0 ==> (v IN v_std s v0 ==> (!p q h. {w | {v, w} IN e_std s v0} p /\ {w | {v, w} IN e_std s v0} q /\ p = q + h % (v0 - v) ==> p = q)) /\ (v IN v_std s v0 ==> {w | {v, w} IN e_std s v0} INTER affine hull {v0, v} = {}) <=> ( center_pac s v0 ==> (v IN v_std s v0 ==> (!p q h. {w | {v, w} IN e_std s v0} p /\ {w | {v, w} IN e_std s v0} q /\ p = q + h % (v0 - v) ==> p = q))) /\ ( center_pac s v0 ==> (v IN v_std s v0 ==> {w | {v, w} IN e_std s v0} INTER affine hull {v0, v} = {}))`] THEN REWRITE_TAC[c_nov18e2;c_nov1_17]);; let fan3_lemma = prove(`!(v0:real^3) (s:real^3->bool). center_pac s v0 ==> fan3 (v0,v_std s v0,e_std s v0 )`, REPEAT GEN_TAC THEN REWRITE_TAC[fan3;cyclic_set] THEN DISCH_TAC THEN GEN_TAC THEN REWRITE_TAC[TAUT ` v IN v_std s v0 ==> ~(v0 = v) /\ FINITE {w | {v, w} IN e_std s v0} /\ (!p q h. {w | {v, w} IN e_std s v0} p /\ {w | {v, w} IN e_std s v0} q /\ p = q + h % (v0 - v) ==> p = q) /\ {w | {v, w} IN e_std s v0} INTER affine hull {v0, v} = {} <=> ( v IN v_std s v0 ==> ~(v0 = v)) /\ ( v IN v_std s v0 ==> FINITE {w | {v, w} IN e_std s v0}) /\ ( v IN v_std s v0 ==> (!p q h. {w | {v, w} IN e_std s v0} p /\ {w | {v, w} IN e_std s v0} q /\ p = q + h % (v0 - v) ==> p = q)) /\ ( v IN v_std s v0 ==> {w | {v, w} IN e_std s v0} INTER affine hull {v0, v} = {})`] THEN CONJ_TAC THENL[REWRITE_TAC[v_std;DIFF] THEN SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL[REWRITE_TAC[e_std] THEN ONCE_REWRITE_TAC[SET_RULE `{{u, v} | u IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ v IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ ~(u = v) /\ d3 u v <= &2 * t0} = {{x, y} | x IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ y IN tru_pack v0 (&2 * t0) s DIFF {v0} /\ ~(x = y) /\ d3 x y <= &2 * t0}`] THEN DISCH_TAC THEN ONCE_REWRITE_TAC[lemmaf3] THEN MATCH_MP_TAC (lemmaf33) THEN MATCH_MP_TAC (fini_lemma) THEN UNDISCH_TAC `center_pac (s:real^3->bool) (v0:real^3)` THEN REWRITE_TAC[infi_lemma2]; UNDISCH_TAC `center_pac (s:real^3->bool) (v0:real^3)` THEN REWRITE_TAC[c_nov18e3]]);; (*=====================================================================*) (* end of fan3_lemma*) (*=====================================================================*)