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products/Sources/formale Sprachen/Isabelle/Archive-of-Formal-Proofs/thys/LOFT/document/ Datei vom 31.4.2026 mit Größe 789 B

Quelle Valuation2.thy

Sprache: Isabelle

(** Valuation2
 author HidetsuneKobayashi
 Group You Santo
 Department of Mathematics
 Nihon University
 h_coba@math.cst.-..jp
 2005
 July20 2007(** Valuation2

 Mathematics
 section 8. approximation(continued)

 **)

theory
imports
begin Valuation1

lemma (OstrowskiTr8:<>valuation <in> carrier K;
 0 < v (1_r ±a x)]
 0 < 0 <v1<subr <lusminus -_a x)] ==>
apply (cut_tac field_is_ring, frule Ring.ring_is_agchapter. a
apply (frule aGroup.ag_mOp_closed[ofsection()
 frule Ring*java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 frule aGroup.ag_pOp_closed[of0 < < (v 1<subr ±r ±&hyphen
 frule OstrowskiTr32cut_tacRingfjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
apply case_tac ^>_{, simpRing.ring_one[ "

 add:aGroup.ag_r_inv1, simp add:Ring.ring_t,

 simp add:aGroup.ag_r_zero, cut_tac val_field_1_neq_0 (case_tac " \ strowskiTr32assumption

 cut_tac, simpsimp:Ring,

 simp add add.ag_r_zero val_field_1_neq_0,

apply (frule .ag_pOp_closed K ^" " <><subr (1java.lang.NullPointerException

 rule

apply (cut_tac invf_closed ptionmOp_closedion

apply ( ield_one_plus_frac3suc

 subst val_t2plfsimp

apply (rule aGroupthin_tacr (-_^r",

ule.nCle

 assumption+,

 eadd__of"v<r<>-<sub,

 (1assumption

 sminusa x^^{= (1_\<^sub>r (1java.lang.NullPointerException

 sucdde_of_one

 \^padd assumption

apply substb2ng.ring_l_one,

 rule aGroupring_distrib1istrib1,tionng_r_one

 oup:_rulessumption

apply rulemption

 frule Ring.,

 simp addnv1_2ption

 frule aGroup



apply applyontrapos_pp

 frule eq_elems_eq_val " \>\<^ubsubstpaopa py(smadaro.ag_nv_i[of "\r

 thin_tac "x ⋅1\^>r = \sub 1\<ubr

 addv_iu_qalu__n)

 plyi d:_t2,

java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25



 subst

 assumption+,

 rulel igrn_tO_lsd assumppv (x ⋅_lera_O_losf "" u+

 simp add:val, subst val_t2p[of v], assumption+,

 rule aadd_pos_poss[of "v xv1subr ±

 simp add:value_of_one,

 cut_tac aadd_pos_poss[of "v (1₎ 1<^s>r ±

 simp add:aadd_0_r,rule val_axi, assumption+)

 apply (subst Ring.ring_distrib2, assumption+, spadRn.ig(leRin.rin_t a+

 subst a ( co, +,

 subst aGroup.pOp_assocTr4R frue in.ro K "\sub <>-java.lang.NullPointerException

 rule Ring.ring_tOp_closed, assumption+,

 simp addsimpGroupmmute1\^subr"],

 subst Ring.ring_inv1_2, assumption+ m,asmto+

java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 11

apply (rule contrapos_pp, simp+,

 frule Ring.ring_tOp_closed[of K x "(1_r ± -_\<^sub>r (-<^> x ± 1_a apply(rulering_tOp_closed

 paGroup<"],

 frule aGroup.ag_eq_diffzero[THEN sym, of K "x ⋅

 assumption..,osed

apply (simpaddval_minus_eq_eqN, v]

 e_ofr(-^> <plusminussub<^a1ub,

 thin_tac java.lang.NullPointerException

 simp addapply (simp add:aless_imp_le, assum

 simp add:val_t2p,

 frule aadd_pos_possof " x"v (-\sub>a x \<>

 simp

done

lemma (in Corps) OstrowskiTr9:"[

r" "x ⋅>r ±

apply "1\^>r ±_r \<plusminus a x) ≠

apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],

 frule aGroup.ag_mOp_closed[of "

 frulering_one(.ag_pOp_commute

 frule aGroup.ag_pOp_closed,

 contrapos_pp

 cut_tac[of^r ±value_of_inv 1subcdotr (1_a x)"], assumption+,

apply simp

asmp+,

 ruleru Ring.ri, assumption+)

(*

apply

 ._Oof K "\^>r\plusminus<^sub x]assumptionptionn ply

 dGroupmuteone:minus_eq

 frulestEN of> " 🚫_r ±

 apply (rulsimdd:lminus_eqvaueof_on,

 e ru.gmOp_lse,as+)

 apply (si add:val_t2

 apply (fule eq[ fru Rin.in[fK x "1>rplusminusa x)"], assumption+,

 thin_tac "x ⋅

 simp value_less_eqK] )

 simp add

 apply

 apply (cut_tac aadd_pos_poss[of

 applyimpddqEN]java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54

 apply (subst aGroupommute+

 apply (rule val_axiom4[of v java.lang.NullPointerException

 apply (simp add:aless_imp_le, assumption) *)

apply (subst value_of_inv[of v "1"v"x]

 rule aGroup.ag_pOp_closed,assumption

 rule (cut_tac, Ringring_is_ag[of,

 frule value_less_eq[THEN symof "\^br "- (a x ±a 1java.lang.NullPointerException

 subst

 simp rulefx \dot<<^ 1_a 1java.lang.NullPointerException

 subst aadd_commute,

 rule aadd_pos_poss[of java.lang.NullPointerException

 simp, assumptionfr aag_pOp_closed[ofK "\ ering_tOp_closed,

 simp"x\<\ub

apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of K],

 frule .ing_o,

 rule contrapos_pp simp+,

 frule Ring.ring_tOp_closed[of K x "(1_r ±, v "1,

 rule .ag_p, assumption+

 rule aGroup.ag_mOp_c

 frule aGroup.ag_mOp_closed[of K x], assumption+)frule value_l[THEN sym, of v"java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

apply (simp add:aGroup.ag_pOp_commute[of K "fruleval_nonzero_z "v" "x",assumptionly trowskiTr6x,sumption

 leiffzero[THEN \dot<> (<^ x\lusminus \^subr) -sub 1_r"],

 simp add:aGroup.ag_pOp_commute,

 rule aGroup.ag_mOp_closed (inule con, s add:aeg_le, v[of "",

 simp add:aGroup.ag_inv_inv,

 frule eq_elems_eq_val[of "x apply ( add,

 thin_tac<^>r -<subx ±a 1java.lang.NullPointerException

 simpup ",sumtio

 frule_tac a ng.rg_of",

 simpfrule rotate_tacac

 apply [ ubstnv <r>x<><subr < -java.lang.NullPointerException

 cut_tac aadd_pos_poss a= gg_tOp_closedon

roupsumption

 alue_less_eq>^ub1^r ±r"], assumption+)

 simp d:value_of_one val_minus_eq, simp add:value_of_one)



apply a

done

lemma (in orps) O:"<lbrakkvaluation

 ¬

apply (fruleassumption

 invf_closed1 plusminusr(_a x)"], simp,

 erule conjE, simp add:aneg_le, fru_ta "< v_equiv K(( 0)) vv",

 (erule conjE)+, assumption+, erule conjE)

applycut_tac f field_is_ring, frul Ring.ring_is_ag[[ "",

 frule aGroup.ag_mOp_closed[of " "" of"""x", assumption erule

 frulering_one""

 )

 subst

apply (

 assumption )Ostrowski_firstSucjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60

 rule.ring_tOp_closed+ule, umption

 subst value_less_eqrule_tac nd="vv ( 0)" in Nset_Suc0 conjE

 "x ⋅ a = 0 in forall si only:Ostrowski_elem_de

applydrule_tac a = " ( field_is_ringfrule.ring_is_ag",

 one_plus_x_nonze[of " -^> xssumption

 frule_tac

 subst aleption

apply (simp add:val_t2pt_tac= 0 forall_spec,

 frule[THEN, of java.lang.NullPointerException

 simp add:value_of_one, simp add:val_minus_eq,

 simp add

 frule [of "v

 :_zpz :ant_0 sym

 assumption)

done

lemma (in Corps) Ostrowski_first:"vals_nonequiv Sc0)v

 <Lt\ ))"

applyrule,

 cut_tac Nset_Suc0, apply rulejava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

 simp add)

apply (rotate_tac =composeSuc (skipinjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77

 frule_tacin_pec

 rotate_tac+(rule

 rule_tac 0" foralspesi)

"ucssumptionion

 rotate_tac

 drule_tac a (ule_tac dt ._of" a+,

,rl ale_nqesssssu+)

 drule_tac a = "Suc 0" in a fl_a s == s an t =t in OrowkkiT,sspton

 frule_tac v = "vv

 nonequiv_ex_Ostrowski_elem, assumption,

 erule

apply (erule conjE n "Suc"m Suc" vv = vvi

 frule_tac v = "vv (Suc

 nonequiv_ex_Ostrowski_elem+java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

 erulebexE

 thin_tac "¬, as,

 equiv K (vv 0 (vi, ig"assumptionmption

apply (rename_tac s t) (* we show s and t are non-zero in the following 4

 lines *)

apply (erule conjE,

 nd"vv "val_neg_nonzeroion

applyapplyumptionabove,pouldn

 ule_tac ="

 assumption+, un less_

applythin_tac "vals_nonequivSuc

apply (frule_tac s = s and

 assumption+, rule ( orpsOstrowskihin_tac K(Succompose{.<> Suc (skip",

(<>∈

 assumption+, rule ale_neq_less, assumption+)

apply (subgoal_tac "t ⋅SucSuc vvvm=0

 simp onlyx

 simp only: nset_m_m, assumptionnandvals_nonequiv2

 (* Here simp add:nset_m_m[of "Suc 0"] wouldn't work *)

applydrule_tac h <> Sucn}vv 2"in fo,

 rule, (erbexE)+)

 frule_tac = sa :compose_defs,,

 assumption+, rule ale_neq_less, assucutc "0 \le( ucjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 41

 frule_tac 0 0 ≤

 applycut_tacxs\plusminus"n iv_closed bl

 apply assumption (* in the above two lines, simp wouapp (case_tac " =uc simp

done

(** subsection on inequality **)

lemma (in Corps) Ostrowski (mp

 ( v= (0) t

apply (induct_tac n,

rule, rule impIjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

(** case (Suc n) **)

apply (rule v =vvand=tin,

 frule_tac ,

 frule_tac and =vv

 frule_tac a = "compose {h. h ≤-_" in

 assumption,

 assumption+ring_tOp +, si)

 assumption+ (e bxE))

apply (ru =" andx t

 uleRingring_is_ag K]

(** case * * * **)

apply blast

 case_tac "vals_nonequiv (suOstr,

 frule_tac vv = vv an s = s aa t = t OstrowskiTr5, ssumption+)

apply blas

(** case * * * **)

apply(simp

 case_tac "0 ≤ Succompose 2java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64

 rule_tac Sucand m " (Suc0))"and = in

 vals_nonequiv_valuation2

 simpfrule_tac Suc "Suc Suc )" vv and= in

 frule_tac v =vals_nonequiv_va,simp

 assumption+,

\sub 1<subplusminus s <>\ -java.lang.NullPointerException

 frule_tac(le

 Ring(rule_tac<>nset (Suc 0) ( n).

applysubgoal_tac " K (Suc (S n)) v

 (t ⋅thin_tac " SuclI

 blast)( {h ≤

apply (subst Ostrowski_elem_def,frule_tac (Sucn"and

 rule conjI,

 thin_tac "Kc

 (compose {h. h ≤

 thin_tacvals_nonequivc

 (composeh h < Sucsimpef

 thin_tac "vals_nonequiv K (Suc n) (compose {h. h≤"

 thin_tac " l> (vv (Suc 0) s)",

 Suc nd min

 ,

 rule_tac " 0" x = in

 OstrowskiTr8+)

apply (simp addOstrowski_elem_def (erule )+,

 thin_tac

 0 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 simp add:ompose_def_efkip_def

 rule ballI,

 thin_tac "0 ≤x \noteq 0"

 thin_tac "Ostrowski_elem K (Suc n)

 (compose {h. h \<ledd, rul ym ipolyan1[TE sm,

 frule_tac n = "Suc (Suc n)" and vv = vv and m =

 vals_nonequiv_valuationk>va_oeqi Scn v;

 simp add:nset_def, simp add:Ostrowski_elem_def, (erule conjE)+)

(** case * * * **)(** case * * * **)

apply (case_tac "j = Suc 0", simp,

 drule_tac x = "Suc 0" in bspec,

 simp add:nset_def,

 simp add:compose_def skip_def,

 rule_tac v = "vv (Suc 0apply_acfield_is_ring[ ",

 OstrowskiTr9, assumption+,

 frule_tac j = j n = n in nset_T, ass,

 drule_tac x = "j - Suc 0" in bspec, assumption+,

 simp add:co frule vals_nonequiv_valu[of "Suc vv",

(** case * * * **)

apply (case_tac "j = Suc (Suc 0)", simp) imp,

 rule_tac v =" (Suc ( 0)) and = inwskiTr100

 tion

 subgoal_tac "¬"0", sim,

 rule_tac v = " j"and n

 OstrowskiTr9) apply (simp add:nset_def, assumption+)

pplyaddn_e,(rl onE+ uene_r2, asupin,

 thin_tac "vals_nonequiv K "<su>" "-_a x"], assumption+)

 (compose <java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

 thin_tac vals_nonequiv

 (compose {h. h ≤

 thin_tac "Ostrowski_elem K (Suc n)

 (compose {h. h ≤x" +,

apply(subgoal_tac_r ±_r ±a s))^{∈

apply(case_tacx=1<sub<bsub^", simp add:aGroup.ag_r_inv1,

 (s ⋅of",

prefer(cut_tac inf_ge_any[of "1"], simp add: less_le)

apply (frule_tac n = "Suc (Suc n)" and m = "Suc 0" and vv = vv in

 vals_nonequiv_valuation, simp,

 frule_tac v = "vv (Suc 0)" and x = s in OstrowskiTr6, assumption+,

 rule Ring.ring_tOp_closed, assumptionfrule aGroupag_neq_diffnonzeroof java.lang.NullPointerException

 simp :leprigTE sm,

 simp dOtosiee_e)

apply(uecnI

apply (rule_tac v = "vv 0" and x = s in OstrowskiTr8,

 simp add:vals_nonequiv_valuation, assum (thin (0 <

 apply , r sy,sm

 thin_tac "thin_tacvv<^> x^^{x^\^K m}K m}

 erule)+,

 "frule value_zero_nonzero[of " ""ssumption

 <ompose ) vv (Sucjsjava.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70

 simp add:compose_def " K (Suc n)

(** case *** *** **)

proximation1_5Tr3:"<lbrakkvals_nonequiv<<>r\^>r 1sub<-<a s))^{∈

 rule_tacincarrier K; Ostrowski_elem K ((<cdotr ((1_\<^sub>r (1_a s))^{)", blast)

 simp add:vals_nop

 rule_tac v = "" an x = s in Ostro,

 simp add:vals_nonequiv_valuation nset_def, assumptivals_nonequiv_valuation, simp,

 (frule v ="c" x = s in Ostro, as+, u Ring.in_ne[[o ""

 j = j in nset_Tr51, assumption+,

 drule_tac x = "j - Sucxjava.lang.NullPointerException

apply (sim ad:Ostrowski_eem_def)

done

lemma(in Corps) val_1_:"<valuation m = jin[ "Suc "vv

 x ≠+)

apply ((( ,

 simp(conjE

 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 simp del 0 <composeh >Suc n)} vv0 s

done

lemma (simp addvalue_of_onese_tacdet_def

n_val K (vv 0) = vv vdasprod_n_0



 orall \le \longrightarrow> vv((1_K\esup \plusminusa<><^subr (xjava.lang.NullPointerException

apply (cut_tac K xjSuc" bspec, assumption+)

 frule Ring.ring_one[of "K]

 \><sub(x^{(vv x"

 frule vals_nonequiv_valuation[of "Suc n" "vv" "0"],java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 simp,

 (frul_a j in als_nonei_p(

 frule val_1_nonzero[of "vvrotate_tac1HEN ing sumption

apply (frule vals_nonequiv_valuation[of

 frule val_nonzero_noninf[of"0 a +va[THEN sym], aasumpo+, simp)

 frule val_unit_cond[of "vv 0" "x"], assumption+,

ingnpClose[f "]assumption,

 frule aGroupforalla<in 0; x <in field_is_ring Ring[of (n)vvinSuc)<> .[ ""java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

 frule aGroup.ag_pOp_closed "K "<sub-java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

apply (subgoal_tac val_1_nonzero <n j (a \cdot><> \^up\esup) = vvja + n <sub j")

 frule_tac x = "a ⋅

 value_less_eq[of "vv 0"],

 rule Ring.ring_tOp_closed, assumption+,

 rule dval_t2p2java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

 frulesimpadd,

 conjunct2 Ostrowski_elem_def

apply (ase_tac" \^\^sbK<a ueRg.ri_Ocoed, ssmtn

 frule_tace_a = m ini.pZero_sub[o "]imp

 simp:value_of_zeroalue_zero_nonzero"0"x+,

apply (cut_tacsimpa andxinjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56

applyrotate_tac1, drule not_sym,

 frule

 simp

 simpdl_exp_ring

 cut_tac n1 =a<> carrier K; a ≠

apply cut_tac"nd b = ""00 < m" _ imp

 assumption)

apply"0 m)= (0< nt m)",

 frule val_nonzero_z[of "vv 0" "1_,siadd:nset_m_m[o[of "Suc (simp.ag_mOp_closed simpaddjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56

 erule

 simp:ant_1[( vals_nonequiv_valuation " 0]

apply (subst aGroup.ag_pOp_commute[of "K", aa+,

 rule Ring.npC, assu[of "K,

 assumption+ n=n nrestrict_vals_nonequiv"],

 rotate_tac -1, drule sym, rule_tac n n = n in rest[of "x" _ ""],

 thin_tac "vv 0 (a ⋅

apply add,

 frule value_zero_nonzerofrule_tacsimp:asprod_n_0

 simp

 simp add:aadd_0_r,

 cut_tac =m in[of ( Corps:"\<lbrakkvals_nonequiv

done

lemma (in Corps) Approximation1_5Tr3:"[

 x\in carrierK;Ostrowski_elem K (Suc) x; j ∈

 ==>j (1<sub\plusminus₎ = 0"

apply (frule Ostrowski_elem_not_one[of "n" "vv" "x"], assumption+,

 cut_tafield_is_vv j ((1\\ \<x^^bsupK m\m<^esup>±r (x^^{) = 0)"},

 frule Ring.ring_one[of "K"]

 frule aGroup.ag_pOp_closed[of "K" "1\simp add:nse)

apply (simp add:aGroup.ag_mOp_closed, simp add:nset_def,

 j in vals_nonequiv_valuat[[of "nvv

 simp,

 frule_tacv1j"and 1= "1^ub>rplusminus<subx" and n1 = m in

 val_exp_ring[THEN sym], assumption+)

apply " and=1\>and=-<^>x n

 value_less_eq, assumption+, simp add:aGroup.ag_mOp_closed)rule Suc" 0"mp

applyimpcut_tac_om

 simpaa

apply (simp add:value_of_one=nstrowskily

 addprod_n_0_0

done

lemma (in Corps) Approximation1_5Tr4

 aa ∈) Approximation1_5Tr5frule_tacx proximation1_5Tr1"n"v"

 Ostrowski_elem K (Suc n) vv x; j \+,

 (acdot>_K m}

apply (frule Ostrowski_elem_nonzero[of "n" "vvrulenonzerojava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53

 +,

 cut_tac, frule[of)

apply (frule_tac m = j in vals_nonequiv_valuation vv j a \cdot>_a vv j x")

apply (subst val_t2p[of "vv j"], assumption+,

 rule Ring.npClose, assumption+,

 cut_tac field_is_idom,

 frule_tac v1 = vand x1 = x and 1 = m i

 val_exp_ring[THEN sym], assumption+, simp)

done

lemma ((n Corps)A:"lbrakk;

 a ∈ addnset_def

 Ostrowski_elem K (Suc n) vv x; j ∈+ rule.npClose+java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78

 ∃ Ring

apply ( K_gamma_hom<lewski_elem_nonzeron" ""x"umption

 ( :Zset_def

 simp,

 java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null

refer 2

apply (rule allI, rule Approximation1_5Tr4[of _ vv a x j],

 assumption+, simp add:nset_def)

apply cut_tac transpos_id[of " Suc" "j])

 simp add:nset_def,

 frule val_nonzero_z[of "vv j" "a

 imp ad:srws_emdef

 frule conjunct2, fold Os \forall<>(u \rbrakkLo>

 drule_tac x = j in bspec, assumption)

apply (frule Ostrowski_elem_nonzero[of "n" "vv" "x"], assumption+,

 frule val_nonzero_z[of "vv j" "x"], assumption+, erule exE, simp,

 frule_tac a = za and x = z in zmult_pos_bignumTr,

 simp add:asprod_amult a_z_z ap)

done

lemma (in Corps) Approximation1_5Tr6:"\rename_tac

 a ∈

 Ostrowski_elem K (Suc n) vv x; j ∈

 ∃l. ∀x, blast,p

apply(frule[of""]

 simp add:nset_def,

 frulesimpset_def

 assumption+, erule exE,

 cut_tacfield_is_ring ng K"]

 subgoal_tac "∀add:mp_defsubst transpos_ij_1 "0" " n"], simp

 vv j (1<sub ± -_K m}

apply (simp add:Approximation1_5Tr3, blastapplyeeI2_ex

apply (rule allI

 drule_tac<vals_nonequiv K n) vv;

 frule_tac x = "(1_r ±l∈ Suc n}. n_val K (vv l) = vv l; i ≤ Suc n]

 value_less_eq[of "vv]

 rule Ring.npClose, assumption

 case_tacp add

 simp add:aGroup.ag_mOp_closed)

apply (rule Ring.ring_tOp_closed, assumptionn1_5P"], simp , blast,

 rule Ring.npClose, sim add:Kronecker_delta_df,rulimpI, (erule conjE)+,

 simp add:Approximation1_5Tr3,

 frule sym, assumption)

done

lemma (in Corps) Approximation1_5Tr7:"[a ∈ j = j in[ "" ""]simp

 x ∈ ==>

 vals_nonequiv K (Suc n) simp add:cmp_def addtranspos_ij_1of

 (∃l. <forallapply

 ( j ((1^> ± -<^sub> )\bsupK m}< a ⋅_r (x^^m)) = 0)))"

apply (induct_tac n,

 rule impI, erule conjE, simp add:nset_m_m[of "Suc 0"],

 frule vals_nonequiv_valuation[of "Suc add:transpos_ij_1 simpKronecker_delta_def

 frule Approximation1_5Tr6[ofrule, blast

apply (frulevals_nonequiv_valuationof" "" "0"], simp,

 val_1_nonzero[of "vv assumption+ simp add,

 assumption)

(** case n **)

apply (rule impI x = j in bspec add,

 addtranspos_ij_2

 frule_tac n = n in

 assumption,

 erule exE,

 frule_tac"Suc" Suc n)" in Approx

 [of _ "vv" "a" "x"], assumption+,

 frule_tac n = "Suc (Suc n)" in vals_nonequiv_valuation[of _ "vv"

 "0",s

 rule val_1_nonzero[of "vv 0" "a"], assump hin_ "∃carrierK ((τ0 j}} \>
 simp add:nset_def)
apply (erule exE,
 subgoal_tac "\ (<for>ja∈.vv ((🚫
 vv j ((1n" )
 ,
 simp add:nset_Suc
done

lemma (in Corps) Approximation1_5P"<>vals K (Suc n) vv;
 n_val K (vv 0) = vv 0]j ≤ j]
 ∃carrier K. ((vv 0 x = 1) \<j\>nset (Suc 0) (Su n). (vv j x) = 0))"
apply (frule vals_nonequiv_valuation[of "Suc n" "vv" "0"], simp) apply (
 frule n_val_surj[of "vv 0", erule bexE) apply (
 rename_tac aa) apply (
 cut_tac n = n in(in) Approximation1_5lbrakkvals_nonequiv K (Sucv
 \forallj ≤ vj<> <Longrightarrow
 apply (
 erule bexE,
 frule_tac a = aa andpproximation1_5Tr1" ""],
 assumption+,
 simp, assumption+)
apply (frule_tac a = aa and x = x in Approximation1_5Tr7[of _ "vv""]
 simp, assumption,
 simp, erule exE,
 cut_tac(vv)p_base_{cut_tac b = "Suc l" in max.cobounded2)

 cut_tac n = inlessI,

 frule_tac x = l andapply(ruleApprozimation1_5P2,assumption+simp)

 less_le_trans, assumption+,

 thin_tac "Suc l ≤) Ostr"[ (Suc n) ]

 drule_tac a ="max 2 (Suc l)"in, simp

 drule_tac a = "max 2 (Suc l)" in \forallm∈ (Suc n)} - {l} <(vv )Ostrowski_baseKvv Suc n )"

apply (subgoal_tac "(1_r ± -_:Ostrowski_base_def,

carrierK

 blast,

 cut_tac field_is_ring, frule Ring.ring_is_ag[of "K" erule

 oup gnpClose

 rulelem_def

 simprulerowski_elem_def

 rule Ring.npClose, assumption+)

done

lemma K_gamma_hom:"ka (case_ " =0" sip ia:rnpsq

apply (simp add:Zset_def)

done

lemma transpos_eq:"(τ

by (simp add:transpos_def)

lemma

 j ≤ef

apply (simp addvals_nonequiv_def

apply (frule conjunct1, fold vals_nonequiv_def)

apply ( addvaluations_def conjI

apply (rule allI, rule "0 <v 0 (1_a x) ∧

apply (case_tac "ja =0,simp

 case_tac "j = 0", simp add:transpos_eq)

apply (subst transpos_ij_1[of "0" "Suc n" "j"], simp, assumption+

 rule not_sym, assumption, simp)

apply (case_tac "ja = j", simp)

apply (subst transpos_ij_2[of "0" "Suc n" "j"],(erule conjE)+

 simp add:vals_nonequiv_valuation)

apply (case_tac,simp:transpos_eq

apply (cut_tac x = ja in transpos_id_1[of 0 "Suc n" j], simp, assumption+,

 rule not_sym, assumption+)

apply simp:transpos_ij_2,

 (rule x = m bspecsimp :nset_def

apply (case_tac "j = 0", d)

 simp:vals_nonequiv_def

 cut_tac

apply (frule_tac x = ja (in) Ostrowski_baseTr1vals_nonequiv K (Suc n) vv; l ≤

 le (Suc n)}"], simp, simp, assumption+)

apply (cut_tac l = ja in transpos_mem[of "0" "Suc n" "j"], simp, assumption+,

 simp, assumption,

 cut_tac l = l in transpos_mem[of "0" " n" "j"], sim, assumption+,

 simp, assumpt)

apply (simp add:vals_nonequiv_def,

 simp, assumption, r not_sym, assumption)

done

definition

 Ostrowski_base :: "[_, nat ==>

 (‹ (Suc n)} \<ongrightarrowgrightarrow

Klam>j∈{h. h ≤ n}. (SOME x. x∈carrier K ∧

java.lang.NullPointerException

App_base :: "[_, nat ==> K (Su n) vv]

"App_base K vv n = (λj∈{h. h ≤ n}. -_bsub>K vvv (Suc n)}^^{≠

= 1) ∧ield_is_ring, frule Rin.ngi_[of""]

 (* Approximation base. *)

lemma Ostrowski_base_mem_1 "v""] aumpin,

 Ostrowski_base K vv (Suc n) ∈_j" "N"], assumption+,

apply rule,rename_tacl,

 simp add:Ostrowski_base_def,

 frule_tac j = l in transpos_vals_nonequiv[of n vv], simp,



apply (drule_tac a = "vv ∘r" "-_\<^bsubK vvS \esubK N}],
 rule someI2_ex, b simp)
done

lemma (in Corps) Ostrowski_base_mem:"vals_nonequiv K (Suc n) vv<Omega_{j^^{sumption

 ∀g_inc_zero

by (rule allI, rule impIthin_tacr ±-_K vv (Suc) j^^{)

 frule Ostrowski_base_hom[of "n" "vv"],

 simp add:funcset_mem deli_I

lemmaorps_[

 j ≤forall ≤ (Suc.<>m\longrightarrow

by (simp add:Ostrowski_base_mem)

lemma (in Corps) Ostrowski_base_nonzero:"[vals_nonequiv K (Suc n) vv;

 j ≤

apply(ipadOrkibed

 frule_tac j = j in transpos_vals_nonequiv[of "n" "vv"],

 +,

 cut_tac Ostrowski[of "n"],

 drule_tac a = "vv ∘0 j}}}
 rule
apply (thin_tac "∃
 erule conjE)
apply (rule_tac vv = "vv (<>\^K vv (Suc n)
 assumption+)
done

lemma (in Corps) Ostrowski_base_pos:"[v m (ΩK vv (Suc n)
 j ≤ Suc n; ja ≤assum simp add:less_ant_def, sip, simp,
apply (simp add:Ostrowski_base_def,
 frule_tac j = ja in transpos_vals_nonequiv[of "n" "vv"],
 assumption+,
 cut_tac Ostrowski[of "n"],
 drule_tac a = "vv ∘
apply (rule someI2_ex, blast,
 thin_tac "∃x∈
 simp add:Ostrowski_el, (erule conjE)+)
apply (case_tac " \^ub^>_{== n choose i"

 simp add:nset_def, frule Suc_leI[of "0" "j"],

 frule_tac a = j in forall_spec, sim (i in)xai__1x <> car R ==>

apply (case_tac "j = 0", simp,

 frule_tac x = ja in bspec, simp add:nset_def,

 cut_tac ttra[of "0" "Suc

apply (frule_tacne\forall>j:). (x^^{in> carrier R")

 cut_tac transpos_id[of "0" "Suc n" "ja" "j"], simp+)

done

lemma (in Corps) App_base_hom:"<done carrier R ==>r ±<bsup =

 ∀j ≤ (Suc n). n_val K (vv j ( (< i. (_bsub(Suc i)_{x^^{nplusminus 1java.lang.NullPointerException

 ∀j ≤ del binomial_Suc_Suc,

apply (rule allI, rule impI,

 rename_tac k,

 subst App_base_def[ 🚫i}}bsub>R<^esub> x^^{], assumption

applycase_tac, simp:transpos_eq

 frule Approximation1_5P[of " (c ring_one,

 someI2_ex, blast, s)

apply (frule_tac j = k idon

 simp add:nset_def,

 frule_tac vv = "vv ∘ carrier R ==>

apply (simp add:cmp_def, subst transpos_ij_1 n"], simp+,

 subst transpos_ij_1[of 0 "Suc n" k], simapp(frule ttail_[of "x" "n"],

apply (rule someI2_ex, blast, siimp del:nsum_subinomial_Suc_Suc npow_suc,

done

lemma (in Corps) Approzimation1_5P2:"[vals_nonequiv K (Suc n) vvlemmacut_tactac

 ∀l∈2pply

 ==>_{"

apply (simp add:App_base_ ule a Grnme smpo ue l,remI

applyase_tacj = 0",simp:transpos_eq,

 rule someI2_ex,

 frule Approximation1_5P[of "n" "vv"], simp , blast,

 simp add:Kronecker_delta_def, rule impI, (erulerule aGroup, assumption, simp:ring_one

 frule_tac bspec add, assumption

apply ( j = j intranspos_vals_nonequivof] ,

 frule Approximation1_5P[of "n" "vv ∘ impI,

 add:cmp_def, simp add:transpos_ij_1of 00" n" j])



apply (simp add:cmp_def,

 case_tac "i = 0", simp add:transpos_eq,

 simp add:transapply (s(subst nsMulDist assumption, imp add::npCl,

 rule someI2_ex, blast,

 thin_tac "\exists∈carrier K.

 vv j x = 1 ∧

 (erule conjE)+,

 drule_tac x=jin, simp:nset_def

 simp add:transpos_ij_2)

apply (simp add:Kronecker_delta_def,

 case_tac "i = j", addtranspos_ij_1, rule, blast)

apply (simp, rule someI2_ex, blast,

 thin_tacexists∈carrier K. vv(<><>0 j}

 (∀jajava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

 (erule conjE)+,

 drule_tac x = i in bspec, simp add:nset_def,

 cut_tac transpos_id[of 0 "Suc n" j i], simp+)

done

(*

lemma (in Corps) Approximation1_5:"\<lbrakk>vals_nonequiv K (Suc n) vv;

 \<forall>j \<le> (Suc n)}. n_val K (vv j) = vv j\<rbrakk> \<Longrightarrow>

 \<xists>x\<in>{.h <>(Suc n)} \<rightarrow> carrier K. \<foralli <e (Suc n). \<forall>j \<le> (Suc n).java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115

 ((vv i) (x j) = \<delta>\<^bsub>i j\<^esub>)" *)

lemma (in Corps) Approximation1_5:"[

 ip lncagelr[ carrier K; 0 ≤

 ∃x. ( <> v x \lev (n ×_x)"

 (vv)( )=\delta_{)"

apply (frule App_base_hom[of n vv], rule a (si add:value_of_zero,

apply(s "(∀ (Suc n). ∀ (Suc n).

 (vv i) ((App_base y = " \><bsub>K} amin_le_plus[o """,assumption

 blast

apply (rule (simpaddamin_def

apply (rule Approzimation1_5P2, assumption+, simp Ring[ofK]

done

lemma in) Ostrowski_baseTr0\lbrakk>nonequiv (Suc n) ]

 ==>

 (∀

apply (simp add:Ostrowski_base_def

 frule_tac j = l in transpos_vals_nonequiv[of< ( -_r ±K (Sucesup>)))"

 cut_tac Ostrowski[of "n"],

 drule_tac a = "vv τ_{in forall_spec, assumption

apply (erule bexEcase_tac<>java.lang.NullPointerException

 unfold Ostrowski_elem_def, frule Ringring_one "K" simp_ro

 fold Ostrowski_elem_def,

 conjIdef

apply Ring

 rule someI2_exoal_tac K (λSuc n}Suc i}}}>K icarrier (Vr K v",
 add:t
 rule someI2_ex, blast, simp)

apply (simp add:Ostrowski_elem_def,
 case_tac "l = 0", simp, simp add:transpos_eq,
 rule someI2_ex, blast,
 thin_tac "0 < vv 0 (1_a x) ∧
 (subst aGroup[of" "^>"], assumption+,
 rule ballI, simp add:nset_def)

apply (rule ball erule conjE,
 rule someI2_ex, blast,
 thin_tac "∀nset 0)Suc ) < ((τ0 l
 rule

apply (case_tac "m = 0", simp,
 drule_tac x = l in bspec, simp add:nset_def,
 simpt2p
 drule_tac x = m in bspec, simp addaddv
 simp add:transpos_id"nsumK(\ambda<^sub>n\^u>C\^sbi<e>+ _{_×KK i}
done

lemma (in Corps) Ostrowski_baseTr1:"[vals_nonequiv K (Suc n) vv; l ≤ (Suc n)]
 ==> 0 < ((vv l) (1_Suc nSuc i\^>K iesup"], assumption,
by (simp add:Ostrowsk)

lemma (in Corps) Ostrowski_baseTr2:"[ , rule)
 applycut_tac 0andj="v x<^b>K j\^ andk = "^Suc nSuc jKK j
 0 < ((vvin)
apply Ostrowski_baseTr0 +)
apply simp
done

lemma Nset_have_two:"j ∈ v xx] ass+,
apply (ul contrapos_pp, simp+,
 case_tac "j = Suc n", simp,
 drule_tac a = 0 in forall_spec, simp, arith)
apply (drule_tac a = "Suc n" in forall_spec, simp, simp)
done

lemma (in orps) Ostrow:"[ Suc n;
 vals_nonequiv K (Suc n) vv] \apply( delof_nat_0_less_iff)
 1= "j an x1 = 0 an y1 = "ant"in
apply (cut_tac field_is_ring, frule Ri.ring_is_ag[of ""]
 rule contrapos_pp, simp+,
 frule Ostrowski_base_mem_1[of "n" "vvjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 frule.npCloseof"\Omega<K vv (Suc n)
 frule Ring.ring_one[of "K"],
 frule aGroup.ag_ al_nscal_of ]supo+
 frule aGroup.ag_pOp_closed[of "K" "1\ simpng
 assumption+)
apply (frule aGroup.ag_pOp_add_rse_tac,p
 "(Ωadm,sp,ipd:zz
 simp add:aGroup.ag_inc_zero, assumption+, (
 thin_tac "1java.lang.NullPointerException
applyadd""\^>r" "-_K vv (Suc n)<^>"])
apply (simp add:aGroup.ag_l_inv1, simp add:aGroup.ag_r_zero aGroup.ag_l_zero)
 apply (subgoal_tacforallm ≤ m ⟶
 0 < (vv m assumption <j" th "(0 < j)=int 0<int
apply (cut_tac Nset_have_two[of "j" "n"],
 erule bexE, drule_tac a = m in forall_spec, simp,
 thin_tac "(Ω
 frule_tac f = "vv m" in eq_elems_eq_val[of "1java.lang.NullPointerException
 _c "\^r=<Omega\^b v(cn<esu>^<^bsu>K N
apply (frule_tac m = m in vals_nonequiv (1_a (aa^^{)^^{<esup> \ineK}}
 assumption+,
 frule_tac v1 = "vv " andn NnvleprgHNs,
 of _ "(Ω_{assumption

 simp add:Ostrowski_base_nonzeroupimpRing

apply (subgoal_tac "int N ≠

 frule_tac x = "vv m ((Ωjava.lang.NullPointerException

 assumption, simp:less_ant_def, simp

 rule allI, rule impI, rule impI,

 rule Ostrowski_baseTr2, assumption

done

abbreviation

 CHOOSE :: "[nat, nat] ==> nat" (‹

java.lang.NullPointerException

(i Corps) ApproximationTr2:"[ carrier K; aa ≠0

java.lang.NullPointerException

(cut_tac ring_one, frule npeSum2[of "1_r" "x" "n"], assumption+,

simp add:npOne, subgoal_tac "∀(j::nat). (x^^{"],

(simp add:ring_l_one, rule allI, simp add:npClose)

(in Ring) tail_of_expansion:"x ∈ carrier R ==> val_ v "aa"], assu+, erule exE, simp)

java.lang.NullPointerException

(cut_tac ring_is_ag)

frule asinfsu[ofx" "Suc n"],

simp del:nsum_suc binomial_Suc_Suc npow_suc,

thin_tac "(1Ring.npClose[of K" "aa"], assu+,

(subst aGroup.nsumTail[of R n "λi. \<^ frule

rule allI, rule impI, rule nsClose, rule npClose, assumption)

(cut_tac ring_one,

simp del:nsum_suc binomial_Suc_Suc npow_suc add:aGroup.ag_l_zero)

(in Ring) tail_of_expansion1lone

(1_r ± x)^^{(Suc n)} = x ⋅

(frule tail_of_[of "x" "n"],

simp del:nsum_suc binomial_Suc_Suc npow_suc,

subgoal_tac "∀Suc n}}Suc i_{x^^{∈

cut_tac ring_one, cut_tac ring_is_ag)

2 apply(simp add: nsClose npClose)

(rule aGroup.ag_pOp_add_r[of "R" _ _ "1_j∈j <>\r ±a (b j)))) ∧

java.lang.NullPointerException

rule nsClose, rule npClose, assumption)

(rule ring_tOp_closed, assumption+,

rule aGroup.nsum_mem, assumption+, blast, simp add:ring_one)

(subst nsumMulEleL[of "λi. _{ut_tac field_i, frule Ring.ring_is_ag[of "K"])

(rule aGroup.nsum_ assumption, rule allI, rule impI, r nsClose,

rule npClose, assumption, rule allI, rule impI,

rule ring_tOp_closed, assumption+, rule nsClose, rule npClose,

assumption)

(rule aallI, rule impI)

(subst nsMulDistrL, assumption, simp add:npClose,

frule_tac n = j in n npClose[of x], simp add:ring_tOp_ommuof x]])

(in Corps) nsum_in_VrTr:"valuation K v ==>[of K "b 0"], ssu,

(∀ "K" "x 0" "-\^>a (b 0)"], assum+,

0 \<le _], a+,

(induct_tac n)

apply (rule impI, erule conjE, simp add:val_pos_mem_Vr)

(afrule aGroup.ag_mOp_clof "K"],

+)

frule_tac x = "f (Suc n)" and y = "nsum K f n" in

aGroup.ag_pOp_closed[of"VK v"],

subst val_pos_mem_Vr[THEN sym, of "v"], assumption+,

simp, simp, simp)

(simp, subst Vr_pOp_f_pOp[of "v", THEN sym], assumption+,

subst val_pos_mem_Vr[THEN sym, of v], assumption+,

simp+)

(subst aGroup.ag_pOp_commute, assumption+, simp add:val_pos_mem_Vr,

assumption)

java.lang.NullPointerException

∀j ≤ n. 0 ≤subgoal_tac "{h. h e n}", simp,

(simp add:nsum_in_VrTr)

(in Corps) nsum_mem_in_Vr:"[valuation K v;

∀ n. (f j) ∈j ≤ (v (f j)) ==>

(nsum K f n) ∈ carrier (Vr K v)"

(ru nsum_in_Vr)

(in Co val_nscal:"[ carrier K; 0 ≤

==> v x ≤, rule all, rule impI, simp,si)

(cut_tac field_is_ring, induct_tac n, simp)

(simp ad:value_of_zero,

simp,

frule_tac y = "n ×K}+,

rule Ring.nsClose, assumption+)

(simp add:amin_def,

frule Ring.ring_is_ag[of K],

frule_tac n = n in Ring.nsClose[of K x], assumption+,

simp add:aGroup.ag_pOp_commute)

(in Corps) ApproximationTr:"[_csd ssupo+, i,

v x ≤

(cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],

case_tac "x = 0_{psp

simp, frule Ring.ring_one[of "K"], simp add:aGroup.a ruRinrgtlsd smtn ip,

simp add:Ring.npOne, simp add:Ring.ring_l_one,simp add:aGroup.ag_r_inv1,

subst Ring.tail_of_expansion1[of "K" "x"], assumption+,

frule Ring.ring_one[of "K"])

(subgoal_tac "(nsum K (λi. _n

java.lang.NullPointerException

assumption+,

tacx=xandy "nsum KK (\lambdai. _{_{\<times\K i}}}}

Ring.ring_tOp_closed[of "K"], assumption+,

java.lang.NullPointerException

subst aGroup.ag_p_inv[of "K" "1_r"], assumption+,

subst aGroup.ag_pOp_assoc[THEN sym], assumption+,

simp add:aGroup.ag_mOp_closed, rule aGroup.ag_mOp_closed, assumption+,

delnilS_S ad:arupa__iv,sbsaGou.gleo

assumption+,

rule aGroup.ag_mOp_closed, (rule allI, rule impI)+

(subst val_t2p[of v], assumption+) apply (

simp add:val_pos_mem_Vr[THEN sym, of v

"nsum K (λsimp ad:transp,

java.lang.NullPointerException

"v x"], simp (in Corps) Ostrowski_ba:"[ (Suc n)]

(rule nsum_mem_in_Vr[of v n "λi._n

rule allI, rule impI) apply (rule Ring.nsClose, assumption+) apply (sip add:Ring.npC)

(rule allI, rule impI)

(cut_tac i = 0 and j = "v (x^⁾" and k = "v (_{_×KK j

in ae_rans

apply (case_tac simp

apply

frule val_nonzero_z[of v x], assumption+,

erule exE,

cut_tac m1 = 0 and n1 = j in of_nat_less_iff[THEN sym],

frule_tac a = "0 < j

assumption, thin_tac "0 < j = "Suc n" i fora, simp, simp)

(sm del: of_na)

(frule_tac w1 = "int j" an x1 = 0 a y1 = "ant z" i

asprod_pos_mono[THEN sym],

simp only:asprod_n_0)

apply(rule_tac x = "x^^{and n = "_{_{in

val_nscal_ge_selfTr[of v], assumption+,

, simp add:val_exp_ring[THEN sym],

frule val_nonzero_z[of "v" "x"], assumption+, contrapos_pp, simp,

apply (case_tac j= 0",, simp)

apply (subst asprod_amult, simp, simp add:a_z_z)

(

simp only:ant_0[THEN sym], simp only:ale_zle,

cut_tac m1 = 0 and n1 = j in of_nat_less_iff[THEN sym])

( frule_tac a = "0<j ,

assumption+, thin_tac "0 < jr" "-_\_{j^^{"],

frule_tac z = "int 0" and z' = "int j" in zless_imp_zle,

frule_tac i = "int 0" and j = "int j" and k = z in int_mult_le,

assumption+, simp add:mult.commute )

apply assumption

(in Corps) ApproximationTr0:"aa ∈r ± -_<O>_{s N}} = 0")

(1-_K N}}}}K N carrier K"

(cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],

ruleln.nCoe supin+,

rule aGroup.ag_pOp_closed, assumption+, simp add:Ring.ring_one,

rule aGroup.ag_mOp_clos, assump+, rule Ring.npClose, assump+)

(in Corps) ApproximationTr1:"aa ∈ carrier K ==> "\<Omega\ n)\^>) j^\^bsup>K N} -_K vv (Suc n)}K N carrier K",
1-_r ±a (aa^^{)^^{∈

cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],

frule ApproximationTr0[of aa N],

frule Ring.ring_one[of "K"], rule aGroup.ag_pOp_closed, assumption+,

rule aGroup.ag_mOp_closed, assumption+) assumption

(in Corps) ApproximationTr2:"[ K v; aa ∈ 0;

0 ≤

cut_tac l_is_rn,fue Rigrn_sa[f"",

case_tac "N = 0",

frule val_nonzero_z[of v "aa"], assumption+, erule exE, simp)

apply(frule Ring.ring_one[of "K"], simp add:aGroup.ag_r_inv1,

simp add:value_of_zero)

(frule_tac n = N in Ring.npClose[of "K" "aa"], assumption+,

java.lang.NullPointerException

rule aGroup.ag_mOp_closed, assumption+, simp add:val_minus_eq,

subst val_exp_ring[THEN sym, of v], assumption+,

simp add:asprod_pos_pos)

(simp add:val_minus_eq, simp add:val_exp_ring[THEN sym])

(in Corps) eSum_tr:"

∀ (inRing) exansso_of_sum1:"\inrier R \<ongrightarrow C_{_{x^^{n"

( ∀j ≤ (bj) \in c K) ∧ n ∧

( ∀j∈ n} -{l}). (g j = (x j) ⋅r (1_r \<lusminus

g l = (xl) ⋅^sub>r (-_a (b l))

⟶ (nsum K (λj ∈

nsum K g n"

(cut_tac field_is_, frule Ring.ring_i[of "K"])

(induct_tac n)

apply (simp, rule impI, (erule conjE)+,

simp, frule Ring.ring_one[of "K"], subst Ring.ring_distrib1,

assumption+,

mpad:aroup.agmOp_clos, ip add:Rinringin_rone,

frule aGroup.ag_mOp_closed[of K "b 0"], assumption+,

frule Ring.ring_tOp_closed[of "K" "x 0" "-_r ±R (Suc n)}e R (λi. _{_×R^{(Suc n)")

subst aGr.ag_pOp_commute[of "K" "x 0" _], assumption+,

subst a aGroup.ag_pOp_as, assumption+,

(cut_tac ring_one,

+)

apply (simp add:aGroup.a

(rule impI, (erule conjE)+)

apply (subgoal_tac "∀j ≤ (Suc n). ((x j) \ \<cdot\(Suc n)}}C_{×_{x^ 1_"

(case_tac "l = Suc n", simp)

apply (subgoal_tac "Σcarrier K",

subgoal_tac "{h. h ≤ring_is_a)

subgoal_tac "∀ add nsClose npClose)

frule_tac f = "λu. if u ≤.ag_pOp_add_r[of"R" _" _ _"1_], assumption+,

undefined" and n = n in aGroup.nsum_eq[of "K" _ _ "g"])

apply (rule allI, rule impI, simp,

rule allI, simp, rule allI, rule impI, simp, simp)

apply (ct_tac a ==" Sc n ⋅r (1--_and

b = "x (Suc n) ⋅_, rule allI, rule impI, rule nsClose,

c = "Σrule npCnpClose, assumption, rule allI, rule impI,

apply (rule aGroup.ag_pOp_closed, assumption+,

rule Ring.ring_tOp_closed, assumption+, simp,

rule aGroup.ag_pOp_closed, assumption+, simp add:Ring.ring_one,

rule aGroup.ag_mOp_closed, assumption, simp,

rule aGroup.ag_mOp_closed, assumption, simp,

rule Ring.ring_tOp_closed, assumption+, simp,

rule aGroup.ag_mOp_closed, assumption+, simp, assumption)

apply (subst Ring.ring_distrib1, assumption+, simp, simp add:Ring.rin,

simp add:aGroup.ag_mOp_closed,

simp add:Ring.ring_r_one) apply (

frule_tac x = "x (Suc n)" and y = "x (Suc n) ⋅ v ==>

aGroup.ag_pOp_commute [of "K"], simp,

simp add:Ring.ring_tOp_closed aGroup.ag_mOp_closed,

simp) apply (

subst aGroup.ag_pOp_assoc[of "K"], assumption+,

rule Ring.ring_tOp_closed, assumption+, simp,

(simp add:aGroup.ag_mOp_closed)+,

subst aGroup.ag_r_inv1, assumption+, simp,

simp add:Ring.ring_tOp_closed aGroup.ag_mOp_closed, simp,

rotate_tac -1, drule sym, simp) apply (

thin_tac "Σag_pOp_close[f ""Vr K v"],

Σ_e K g n ± (x (Suc n) ⋅_r (1_r ± -_a (b (Suc n))) ± -_a (x (Suc n)))")

apply (subst aGroup.ag_pOp_assoc[THEN sym], assumption+,

rule Ring.ring_tOp_closed, assumption+, simp,

subst val val_pos_mem_Vr[THEN sym, of "v"], assumption+,

rule a aGroup.ag_mOp_closed, assumption+, simp,

rule aGroup.ag_mOp_closed, assumption+, simp, simp,

simp, rule equalityI, rule subsetI, simp, rule subsetI, simp)

apply (rule aGroup.nsum_mem, assumption+,

rule allI, rule impI, simp)

apply (rule allI, rule impI)

apply (case_tac "j = l", simp,

rule Ring.ring_tOp_closed, assumption, simp,

aGroup.ag_pOpclo, assump+, simp add:Ring.ring_,

rule aGroup.ag_mOp_closed, assumption, simp, simp,

rule Ring.ring_tOp_closed, assumption, simp,

rule aGroup.ag_pOp_closed, assumption+, simp add:Ring.ring_one,

 rule aGroup.ag_mOp_closed, assumption, simp, simp) (* end defer *)

apply (subst aGroupsubst aGroupag_pOp_commute, assumption simp addval_pos_mem_Vr,

 rule aGroup.nsum_mem)

 rule allI, simp, rule

 rule aGroup.ag_pOp_closed, assumption+, simp add:Ring.ring_one,

 rule aGroup.ag_mOp_closed, assumption, simp,

 rule aGroup.ag_mOp_closed, assumption, simp,

 subst aGroup.ag_pOp_commute[of K _ java.lang.NullPointerException

 rule Ring.ring_tOp_closed, assumption, simp,

 rule aGroup.ag_pOp_closed, assumption+, simp add:Ring.ring_one)

apply (rule aGrou in C) nsummem"[

 rule aGroup, assumption,

 subst aGroup.ag_pOp_assoc[THEN sym], assumption K f nn

 rule aGroup.nsum_mem, assumption+,

 rule allI, rule impI,,

 rule aGroup.ag_mOp_closed, assumption, simp,

 rule Ring.ring_tOp_closed, simp

 rule aGroup.ag_pOp_closed, assumption+, simp add:Ring.ring_one,

 rule aGroup.ag_mOp_closed, assumption, simp)

 apply (subgoal_tac "Σ

 else undefined) n ±a (x l) =

 Σ_nsCloseut+)

 -\<^a ,

 rule aGroup.ag_pOp_addr[of [of K _ _ "\^suba ( )],assumption,

 rule n=n in Ring[of]assumption

 rule allI add.ag_pOp_commute

 rule aGroup.nsum_mem, assumption+,

 rule, rule, simp

 rule aGroupx<le r ±a ((1x)^\^bsup )esup>)))"

 (ut_tac field_is_ring, frule Rin.ring_[of "K"]

 rule allI, rule impI, simp, rule allI, rule impI)

 apply simp

 apply (rule allI, rule impI, simp)

done

lemma (in Corps) eSum_minus_x:""lbrakk>\<forall

 ∀j ≤

 ∀({h. h ≤_r ±a (b j)));

java.lang.NullPointerException

 (nK (\lambdaj\inh. h ≤ n}. (x j) ⋅_r <> -_a (x l)) =

 nsum K g n"

by (cut_tac[of"""" ""p

emma> carrier R ==>

(1_,

apply (cut_tac ring_one, cut_tac ring_is_ag,

 frulep_closedption

 frule aGroup.ag_pOp_closed[of "R" java.lang.NullPointerException

apply (induct_tac n)

apply (simp add:ring_r_one ring_l_one)

apply (simp del:npow_suc,

 frule_tac n =" in[of]java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49

 subst ring_distrib1, assumption+)

apply (rule aGroup.nsum_meme,eimpI

 simp add:npClose, rule npClose, assumption+,

 simp del:npow_suc,

 thin_tac "(1_r

apply (subst ring_distrib2, assumption+,

 simp del:npow_suc add:ring_l_one,

 substri, rule aGroup.ag_mOp_clo, assumption+)

 rule_tac x = "x^^{:npow_suc

 rule ring_tOp_closed, rule aGroup.ag_mOp_closed, assumption+)

apply (subst aGroup.ag_l_inv1, assumption x ="\sub>a x"and^<bsup" in ring_tOp_closed,

 add:aGroup.ag_r_zero,

 frule_tac x = "-y "^\^>R (Suc n)}

 assumption+)

apply (rule aGroup.ag_pOp_add_l[of R _ _ "1_r"], assumption+,

 rule aGroup.ag_mOp_closed, assumption+,

 rule npClose, assumption+,

 subst ring_inv1_1[THEN sym, of x], assumption,

 rule npClose, assumption,

 simp,

 subst ring_tOp_commute[of x], assumption+, simp)

done

lemma

 (nsum K (npow K x) n) ∈

apply (frule Vr[of v])

apply (induct_tac n)

simp

apply (frule Ring.ring_one[of "Vr K v"])

apply (simp add:Vr_1_f_1)

apply (simp del:npow_suc)

apply (frule Ring.ring_is_ag[of "Vr K v"])

apply (subst Vr_pOp_f_pOp[THEN sym, of v], assumption+)

apply (subst Vr_exp_f_exp[THEN sym, of v], assumption+)

apply (rule Ring.npClose[of "Vr K v"], assumption+)

apply (ru aGroup.ag_pOp_closed[of " ",ass+)

apply (subst Vr_exp_f_exp[THEN sym, of v], assumption+)

apply (rule Ring.npClose[of "Vrsubst[THEN, of assumption

done

lemma (in Corps) val_1mx_pos Ring[ofassumption)

 0 < (v (1java.lang.NullPointerException

apply (cut_tac field_is_ring, frule Ring.ring_one[of "K"],

 frule Ring.[of K])

apply (frule aGroup.ag_mOp_closed[of "K" "x"], assumption+)

apply (frule aGroup.ag_pOp_closed[of "K" "1_r ±a x))]=0

apply (frule aGroup.ag_mOp_closed[of "K" "1java.lang.NullPointerException

apply (cut_tac x = x and y = "1_ng_gof"")

 eq_elems_eq_val)

apply (subst aGroup.ag_p_inv, assumption+,

 aGroup.ag_pOp_assoc[THEN s], assumption+,

 rule aGroup.ag_mOp_closed, assumption+,

 subst aGroup.ag_inv_inv, assumption+,

 subst aGroup.ag_r_inv1, assumption+,

 subst aGroup.ag_l_zero, assumption+,

 (simp add:aGroup.ag_inv_inv)+,

 frule value_less_eq[of v "1_r" "-\apply( aGroup, assumption

 assumptionjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20

apply (simp add:val_minus_eq value_of_one,

 simp add:value_of_one)

done

lemma (in Corps) val_1mx_pow:"[^>r "-_r ±a x)"],

 0 < (v (1_r ±

apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])

apply (subst Ring.one_m_x_times[THEN sym, of K x n], assumption+)

apply (frule Ring.ring_one[of "K"],

 frule

 subst val_pos_mem_Vr[THENsm, sutn,

 frule val_1mx_pos[ofl -\^> )\rbrakk><> 0<( \^ubr\plusminus^> x^^)"

 simp)

y,on

 rule aGroup.ag_pOp_closed, assumptionrule

 pdroupaddmem_f_mem

 frule val_pos_mem_Vr

 simp add: l_1mx_pos",a+,

apply(sim)

 simp add:aadd_0_l, simp add:aadd_commute)

done

lemma (in Corps) ApproximationTr3:"[vals_nonequiv K (Suc n) vv;

 ∀l ≤:.ag_mOp_closed add,

 ∃,of nsum,

 (x k) ⋅ :Vr_mem_f_mem)

 (Suc -java.lang.NullPointerException

apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])

apply (frule_tac vals_nonequiv_valuation[of

apply_c<orallN

-_a ((Ω_{≤ carrier K; j ≤ ==>

Σe K (λl∈( n)}. (fl noteq j then (x l) ⋅ 1r ±>a (1_^>r ±a

((\OmegabsubK vv (Suc n)}K N^)^^{else (x j) ⋅r (1_a (1java.lang.NullPointerException

 -_K vv (Suc n\^>) l)^\bsupK N<)^^{1java.lang.NullPointerException

apply (simp del:nsum_suc)

apply (thin_tac "∀(_c🚫r (1_a (1

-_a (Ω_{((Ω>K vv (Su n)})^^{) (Suc n) ±a (x j) =}

prefer 2 apply (rule allI)

apply (rule eSum_minus_x, assumption+)

apply (rule allI, rule impI) apply (rule ApproximationTr0)

:Ostrowski_base_mem)app assumption

apply (rule ballI, simp)

simp

apply (frule Ring.ring_one[of "K"])

apply (cut_tac aa = "(Ω_{<^sub(λ{h. h ≤-_r ±



apply (simp add:Ostrowski_base_mem)

apply (subst aGroup.ag_pOp_assoc, assumption+)

apply (rule aGroup.ag_mOp_closed, assumption+)+

apply (subst aGroup.ag_pOp_commute[of "K" _ java.lang.NullPointerException

apply (rule aGroup.ag_mOp_closed, assumption+)+

apply (subs aGroup.ag_pOp_assoc[THEN s], assumption+)

apply (rule aGroup.ag_mOp, assumption+)+

apply (simp add:aGroup.ag_r_inv1)

apply (subst aGroup.ag_l_zero, assumption+) apply (simp add:aGroup.ag_mOp_closed)

s (* subgoal 2 done**)

apply (subgoal_tac "∃L. ∀ "K" _-<subr"], assumption+)

java.lang.NullPointerException

(*

apply (subgoal_tac "∃L. ∀N. L < N ⟶ (an

 (x j)<>K (1_K -_K +(Omega_{l)^N)^java.lang.NullPointerException

+\ rule, assumption

apply (erule exE)

apply (subgoal_tac "∀N. L < N ⟶( aGro.ag_l_zero, assu+) apply (simp add:aGroup.ag_Op_

 ((an m) ≤ ((vv j) (eΣ K (λ

(1_K -<sub-<sK ((Ω_{l)^N)^<spN le( j) \cdot<^>K (1_K +_K -_K}

(1_K +_K -(Suc n). (an m) ≤ (λ{h. h ≤ j then (x l) ⋅r (1_<s> (1-_K vv (Suc n)})^^{elsej)\cdott<sb>r1^s>r\plusminus -_K vv (Suc n)}}) l)^^N)^^Na 1_r))) ja))")

apply blast

*)

apply (erule exE)

apply (rename_tac M)

apply (subgoal_tac "∀N. M < (N::nat) ⟶

 (an m) ≤ (vv j (Σ_e K (λl∈{h. h ≤ (Suc n)}. (if l ≠ j then

 (x l) ⋅_r (1_r ± -_a (1_r ± -_a ((Ω_{vv (Suc n)}) l)^^N)^^N)

 else (x j) ⋅_r (1_r ± -_a (1_r ± -_a ((Ω_{vv (Suc n)}) l)^^N)^^N

 ± -_a 1_r))) (Suc n)))")

apply blast

apply (rule allI, rule impI)

apply (drule_tac a = N in forall_spec, assumption)

apply (rule value_ge_add[of "vv j" "Suc n" _ "an m"], assumption+)

apply (rule allI, rule impI)

apply (frule Ring.ring_one[of "K"])

apply (case_tac "ja = j", simp)

apply (rule Ring.ring_tOp_closed, assumption+, simp)

apply (rule\ N^a (1_r ± -_a ((Ω_vv (Suc n)) j)^^N)^^N)) (Suc n)) ± -_a (x j)))))")

apply (rule aGroup.ag_mOp_closed, assumption+)

apply (rule Ring.npClose, assumption)

apply (rule aGroup.ag_pOp_closed, assumption+)

apply (rule aGroup.ag_mOp_closed, assumption)

apply (rule Ring.npClose, assumption)

apply (simp add:Ostrowski_base_mem)

apply (rule aGroup.ag_mOp_closed, assumption+)

apply simp

apply (rule Ring.ring_tOp_closed, assumption+, simp)

apply (rule aGroup.ag_pOp_closed, assumption+)+

apply (rule aGroup.ag_mOp_closed, assumption+)

apply (rule Ring.npClose, assumption)

apply (rule aGroup.ag_pOp_closed, assumption+)

apply (rule aGroup.ag_mOp_closed, assumption)

apply (rule Ring.npClose, assumption)

apply (simp add:Ostrowski_base_mem)

apply assumption

apply (subgoal_tac "∀N. ∀ja ≤ (Suc n). (1_r ± -_a (1_r ± -_a

((Ω_vv (Suc n)) ja)^^N)^^N) ∈ carrier K")

apply (subgoal_tac "∀N. (1_r ± -_a (1_r ± -_a ((Ω_vv (Suc n)) j)^^N)^^N

 ± -_a 1_r) ∈ carrier K")

apply (simp add:val_t2p)

apply (cut_tac multi_inequalityTr0[of "Suc n" "(vv j) ∘ x" "m"])

apply (subgoal_tac "∀ja ≤ (Suc n). (vv j ∘ x) ja ≠ - ∞", simp)

apply (erule exE)

apply (subgoal_tac "∀N. L < N ⟶ (∀ja ≤ (Suc n). (ja ≠ j ⟶

an m ≤ vv j (x ja) + (vv j (1_r ± -_a (1_r ± -_a ((Ω_vv (Suc n)) ja)^^N)^^N)))

∧ (ja = j ⟶ (an m) ≤ vv j (x j) + (vv j (1_r ± -_a (1_r ±

 -_a ((Ω_vv (Suc n)) j)^^N)^^N ± -_a (1_r)))))")

apply blast

apply (rule allI, rule impI)+

apply (case_tac "ja = j", simp)

apply (thin_tac "∀N. 1_r ± -_a (1_r ± -_a (Ω_vv (Suc n)) j^^N)^^N ± -_a 1_r ∈

 carrier K")

apply (thin_tac "∀l≤Suc n. x l ∈ carrier K")

apply (drule_tac x = N in spec)

apply (drule_tac a = j in forall_spec, assumption,

 thin_tac "∀ja≤Suc n. 1_r ± -_a (1_r ± -_a (Ω_vv (Suc n)) ja^^N)^^N

 ∈ carrier K")

apply (cut_tac N = N in ApproximationTr0 [of "(Ω_vv (Suc n)) j"])

apply (simp add:Ostrowski_base_mem)

apply (frule Ring.ring_one[of "K"], frule aGroup.ag_mOp_closed[of "K" "1_r"],

 assumption) apply (

 frule_tac x = "(1_r ± -_a ((Ω_vv (Suc n)) j)^^N)^^N" in

 aGroup.ag_mOp_closed[of "K"], assumption+)

apply (simp only:aGroup.ag_pOp_assoc)

apply (simp only:aGroup.ag_pOp_commute[of "K" _ "-_a 1_r"])

apply (simp only:aGroup.ag_pOp_assoc[THEN sym])

apply (simp add:aGroup.ag_r_inv1)

apply (simp add:aGroup.ag_l_zero) apply (simp only:val_minus_eq)

 apply (thin_tac "(1_r ±-<^sub(Ω<bsubK vv (Suc_K N}K N} ∈

 thin_tac "-_a (1_\<^sub>e K g n ±_<^ub>r ±a (b (Suc n))) ±a (x (Suc n)))")

apply (subst val_exp_ring[THEN sym, of "vv j"], assumption+)

 apply (rule aGroup.ag_pOp_closed[ofng ssumption

 apply (rule aGrouppg_pOp_closedpadde,

 apply (rule Ringrulelosed

ly_ow_not_onesimp

apply (drule_tac a = N in forall_spec, assumption)

apply (drule_tac a = j in forall_spec, assumption)

apply (frule Ostrowski_baseTr1[of "n"

apply (frule_tac n = "N - pply (rule alI,rule ip)

pddsrowsi_ase_mm) apppl ssumpion

rule RiRing.ingtp_cosed,asupto, smp,

apply hinta "0 <vv1\^ub>r <> K vv (Suc n)}

pply j (1_^a ((Ω_vv (Suc n)) j)^^N)" and N = N in

 asprod_ge) apply assumption apply simp

apply (cut_tac x = "an N" and y = "int N *_a vv j 1\^sub\plusminus -<suba ((Ω_{n)<esub^^{" in aadd_le_mono[of _ _ "vv j (x j)"], assumption)

apply (simp add:aadd_commute)

apply simp

apply (frule_tac aa = "( aGroup, assumption simp add.ring_one

 apply (simp add:Ostrowski_base_mem)

 apply (rule (subst.ag_pOp_assocassumption+java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

applyfrule_tacl=ja Ostrowski_baseTr0 """],asumpio+,

 erule rule aGro.ag_mp_cloe, asumpio simp

e_tac-1ful_tca= in forall_spec)appy ssumpton

ply ful_ta x= nbsp, imp

lyaless_imp_lepbat

rl Gpamp_oe supin+,ip

 drule_tac a = N in forall_spec, assumption)

apply (rotate_tac -2,

 drule_tac a = ja in forall_spec, assumption) apply (

 drule_tac a = ja in forall_spec, assumption)

apply (frule_tac l = ja in Ostrowski_baseTr0[of "n" "vv"], assumption+)

apply (erule conjE, rotate_tac -1,

 frule_tac a = j in forall_spec, assumption+)

 plytin_c "vv) ≠")

apply (cut_tac b = "vv j ((Ω<^ applyr 1<ubr ± -_a (b a))

apply simp apply simp

apply (frule_tac x = "an N" and y = "int N *_plusminus -)=

 "vv ja)"in aadd_l)

apply (frule_tac x = "int-<subxl" s,

 1<s>r ± -_r ±a ((Ω>K vv (Su n)}}K N}}}}K N}
 in aadd_le_mono)
apply (frule_tac i = "an N + vv j (x ja)" and
 j = "int N *_a vv llI ,imp
 k = java.lang.NullPointerException
 vv j (x ja)" in ale_trans, assumption+)
apply (subst aadd_commute)
apply (frule_tac x = "an m" and y = "vv j (x ja) + an N" in aless_imp_le)
apply (rule_tac j = "vv j (x ja) + an N" in ale_transimp
 assumption)
apply (simp add:aadd_commute)
apply (rule allI, rule impI, subst comp_def)
apply (frule_tac a = jain forall_spec, assumption)
apply (frule_tacj ≤ carrier K; l ≤n;
apply (simp add:aug_inf_def)

apply (rule allI)
 apply (rule aGroup.ag_pOp_closed, assumption+) l (xl) 🚫_r (-_a (b l)) ] ==>
apply ( aGroupag_mOp_closed, assumption Ring, assumptionjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78

apply ((rule allI)+, rule impI)
apply (rule_tac aa = "(ΩK vv (Suc n)" in ApproximationTr1
 simp add:Ostrowski_base_mem)
done

definition
 :: "[_ , nat, nat ==> ant, nat ==>
 (nat ==> nat)" (‹r" "-], assumption+)
"Ψ n)
(an m) ≤ (vv j (Σ_e K (λ:npow_suc,
(1_r, ru allI, rule impI,,
(** Approximation lower bound **)

lemma (in Corpsp_LB;
 ∀l≤ (Suc (subst, assumption+,
 ∀N. (ΨpOp_assocTr43 R] +,
 (vv j (Σe K (<lambda{ (Suc_r ±a (1java.lang.NullPointerException
 -java.lang.NullPointerException
applyuleofn""" x" """m",
 assumption+)
applypelum_suclb_defrulejava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
apply
apply (rule impI) apply blast
done

lemma (in Corps) ApplicationTr4Vr_pOp_f_pOpmption
∀
∃java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
 (vv_{. \le (Suc n)}. (x j) ⋅r (1\^>r\plusminus -<>a (1
 -_a ((Ω_{± -<> x))] java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90

apply (subgoal_tac java.lang.NullPointerException

 \j≤ (Suc n). (an m) ≤

 (vv j (Σe K (λj∈h\le> (Suc n)}.(x j) ⋅r (1_-a (1🚫

 -_K vv (Suc n)}K N⁾) (Su )\plusminussa (x j))))")
apply blast
apply (rule allI, rule)
apply (frule_tacsubst., assumption
 simpassumption
 subgoal_tac "(Ψ_{a.ag_mOp_closed assumption+,}
apply (frule_tac l = j and n = "Suc n" and f = "Ψ_{(Suc n) vv x m}
 frule_tac x = "(Ψsim ad:aGrou.ag_inv_inv)+,
 y = "m_max (Suc n) (Ψ_{(Suc n) vv x m} z = Ninle_less_trans
 assumption+)
done

theorem (in
∀ (Sucn)xj<> er ==>
<exists0r ± -_v<^>ra x^^{)"

apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])

apply (subgoal_tac "∃substne_m_x_timesssumption

apply (erulem_in_Vrof

apply (rename_tac M)

apply (subgoal_tac "∀j≤

java.lang.NullPointerException

java.lang.NullPointerException

apply (subgoal_tac "Σ_"v1_a x)"],

-_, add:aadd_com)

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

apply (rule roup.sum_mem[f" "n",+)

apply (rule allI, rule impI, simpL.(∀ ( m) ≤vv j ((Σ_{h. h \le n}

apply (rule Ring.ring_tOp_closed, assumption+, simp,

 xk \cdot_r ±a ((1_a (((Ω<>Kvv n\^sub) k^<>K N}K N

apply (subgoal_tac "M < Suc M") apply blast
apply simp
 rule [of x],+)
apply simp
done

definition
 distinct_pds \Rightarrow ('b ==>
 "distinct_pdsKnP\longleftrightarrow (∀ \bsub <eu> \>
 (∀ n. ∀ n. l ≠Pl ≠

(** pds --- prime divisors **)
lemma (in Corps) distinct_pds_restriction:"[inus_xsumption
 distinct_pds K n P"
apply (simp add:distinct_pds_def)
done

lemma (in Corps) ring_n_distinct_prime_divisors:"distinct_pds <Longrightarrow
 (Sr. x∈ (<>j<>n. 0 ≤\^> K(P )esub x))})"
apply add:distinct_) apply (erule conjE)
apply (cut_tac field_is_ring)
apply (rule Ring.Sr_ring, assumption+)
apply (subst sr_def)
apply (rule conjI)
apply (rule subsetI) apply simp
apply (rule conjI)
apply (simp add:Ring.ring_one)
apply (rule allI, rule impI)
apply (cut_tac P = "P j" in representative_of_pd_valuation, simp,
 simpoal_t "\exists\all <<longrightarrow
apply (rule ballI)+
apply simp
apply (frule Ring.ring_is_ag(al_tacL. ∀ ( m) ≤ (λl∈j then (x l) ⋅K (1_K-_K+<sub-\^>K(<>\K^K^N)
apply (frule_tac x y inaGroupag_mOp_closed[ "K"]ssumption
apply (frule_tac x = x and y = java.lang.NullPointerException
 assumption+)
apply simp
apply (rule conjI)
apply (rule allI, rule impI)
apply (rotate_tac -4, frule_tac a = j in foralmptin
 -3,
 drule_tac a = j in forall_spec, assumption)
apply (cut_tac P = "P j" in representative_of_pd_valuation, simp)
apply (frule_tac v = "ν_{. M < (N::nat \ongrightarrow

 assumption+)

apply (simp add:val_minus_eq)

apply (frule_tac x (<\bsub\^sub and y "<nu\^bsu>K (P j)}")
 apply simp
apply (rule_tac j = java.lang.NullPointerException
apply (simp add:Ring.ring_tOp_closed)

apply (rule all, rule impI,
 cut_tac P = "blast
 subst val_t2p [where v="ν
 rule aadd_two_pos, simp+)
done

lemma (i Co) distinct_pds_valuation:"lbrakk ≤ (Suc n);
 distinct_pds K (Suc n) P] ==> valuation aGroup, assumption+
apply (rule_tac =" "nepresentative_of_pd_valuation
pds_def
done

lemma (in Corps) distinct_pds_valuation1:"[
java.lang.NullPointerException
apply (rule distinct_pds_valuation[of "j" "n - Suc 0" "P"])
apply simp+
doneapplyleR.loe ssui)

lemma( or) ditntpsauto2🚫
 tion (<nu>\^b (( )<^esub)
apply (case_tac"n=0",
 simp add:distinct_pds_def,
 subgoal_tac "0 ∈
 simp n"(vv jj)∘
 )

apply (simp add:distinct_pds_valuation1[of "n"])
done

definition
 ring_n_pd :: "[(ing_scheme ('b <ghtarrowtarrow)^^{>-_r)))))")

 nat ] ==> ('b, 'm) R

java.lang.NullPointerException

 O^> P n carrier K ∧

 (∀N. 1\<>r(_a (Ω_{j^^{^^{±a 1\^>r ∈}}}

 applya\foralll<> n. x l ∈

lemma (in Corps) rthin_ta" Suc n. 1_a (1_a (Ω₎ ja\<^>KK N}
imp add:ring_p_def,sip add:ring_n_isnctt_pri_divisor

(in Corps) ring_n_pd_Suc:"distinct_pds K (Suc n) P ==>
carrier (O_{"K"]ruul Gopagmp_closd[f"K" "1\^r"],

(rule subsetI)

apply (simp add:ring_n_pd_def Sr_def)

(s only:aGroup.ag_pOp_commute[of "K" __ "_r])

y ∈K P n} x ±(O
(simp add:ring_n_pd_def Sr_def)

(in Corps) ring_n_pd_tOp_K_tOp:"[distinK n P; x \in>carr (O₎;
y ∈ carr thin_t "-_r ±a (Ω>K vv (Suc n)K NK N\<^esup
(simp add:ring_n_pd_def Sr_def)

(in Corps) ring_n_eSum_K_eSumTr:
(∀
(induct_tac m)
apply (rule impI, simp)

apply (rule impI, simp,
subst ring_n_pd_pOp_K_pOp, assumption+,
frule_tac n = n in rin[f"n""vv" j"],ass+)
frule_tac Ring.ring_is_ag, drule sym, simp)
apply (rule aGroup.nsum_mem, assumption+, simp+)

(in Corps) ring_n_eSum_K_eSum:"[distinct_pds K n P;
∀ m. f j ∈K P n ==>K P n= sm K f"
(simp add:ring_n_eSum_K_eSumTr)

(in Corps) ideal_eSum_closed:"[K P n
∀) apply assumption simp
(frule ring_n_pd[of "n" "P"]) thm Ring.ideal_nsum_closed
apply (frule_tac n = m in
Ring.ideal_nsum_closed[of "(O_{P n}
apply (subst ring_n_eSum_K_eSum [THEN sym, of n P m f], assumption+,
rule allI, simp add:Ring.ideal_subset)
apply assumption

prime_n_pd :: "[_, nat ==> ('b ==> ant) set,
trowski_baseTr0[of "n" "vv"][of "n" "vv"], assumption+,
(‹ a = jin forall_spec) appassum
"P_{P n} j = {x. x ∈ blas

(in Corps) zero_in_ring_drule_tac a = N in forall_spec, assumption)
0_{O_{P n}) apply (

(simp add:ring_n_pd_def Sr_def))

(in Corp one_in:"distinct_pds K n P ==>

1_r_{O_{P n} - ∞")

(simp add:ring_n_pd_def Sr_def)

(in Corps) mem_ring_n_pd_mem_K:"[distinct_pds K n P; x ∈ simp apply simp

==> x ∈ y = "int N *_K vv (Suc n)}

(simp add:ring_n_pd_def Sr_def)

(in Corps) ring_n_tOp_K_tOp:"[ carrier (O_{;

java.lang.NullPointerException

(simp add:ring_n_pd_def Sr_def)

(in Corps) ring_n_exp_K_exp:"[distinct_pds K n P; x ∈ carrier (O_{"vj (1^sub>r \<plusminus a (1-_\_{ja)^^{^^N

==> x^^m

(frule ring_n_pd[of "n" "P"])

(induct_tac m) apply simp

apply (simp add:one_in_assumption)

simp

apply (frule_ta n = na in Ring.npClose[of "O P n}} "x], assumpin)

apply (simp add:ring_n_tOp_K_tOp)

(in Corpa (sip add:aug_ininf_def)

java.lang.NullPointerException

(sust rime_ideal_def)

apply (rule conjI)

apply (simp add:ideal_def)

apply apply (le Gop.ag_mOpclosed, asuto, ue Rigrgoe,suptio)

apply (rule aGroup.asubg_test)

apply (frule ring_n_pd[of "n" "P"], simp add:Ring.ring_is_ag)

apply (rule subsetI, simp add:prime_n_pd_def)

apply (pply ((subgoal_tac "0_{O_{P n} ∈ P_{j")

apply blast

apply (simp add:zero_in_ring_n_pd_zero_K)

apply (simp add:prime_n_pd_def)

apply (simp add: ring_n_pd_def Sr_def)

t_tac field_is_ring, simp add:Ring.ring_zero)

apply (rule conjI) apply (rule allI, rule impI)

apply (cut_tac P = "P ja" in representav__pvlao,

simp add:distinct_pds_def, simp add:value_of_zero)

apply (cut_tac P = "P j" in representative_of_pd_valuation,

simp add:distinct_pds_def, simp add:value_of_zero)

apply (simp add:[THEN sym]

apply (rule ballI)+

apply (simp add:prime_n_ apply (rule onE+

apply (frule ring_n_pd [of "n" "P"], frule Ring(** Approx

java.lang.NullPointerException

apply (simp add:aGroup.ag_pOp_closed)

apply (thin_tac "Ring (O_{a:"[;

(simp ad add:ring_n_pddef Sr)

apply (erule conjE)+

apply (cut_tac v = "ν>K (P j)\j)\<^>"

amin_le_plus)

apply (ru (rule_tac P = "P j" in representative_of_pd_valuation,

simp add:distinct_pds_def)

apply assumption+

apply (cut_tac P = "P j" in representative_of_pd_valuation)

apply (simp add:distinct_pds_def)

apply (frule_tac x = "(ν_{(P j)}nsum_s add:app_lb_def) apply (ul al)

amin_gt[of "0"])

apply (simp add:val_minus_eq)

(frule_tac y = "amin ((ν_{(P j)}) a) ((ν

z = "(νK (P j)} a±a b)" in aless_le_trans[of "0"], assumption+)

(rule ballI)+

apply (frule ring_n_pd [of "n" "P"])

apply (frule_tac x = rand y x in Ring.r.ring_tOp_closed[of "O_{],

assumption+)

apply (simp add:prime_n_pd_def)

apply (cut_tac P = "P j" in representative_of_pd_valuation,

simp add::distinct_pds_def)

apply (thin_tac "Ring (O_{P n}subgoal_tac "\forall>N. (m_max (Suc n) (Ψ_{) < N

apply (simp add:prime_n_pd_def ring_n_pd_def Sr_def (erule c conjE)++,

simp add:val_t2p)

java.lang.NullPointerException

apply (simp add:aadd_pos_poss, simp)

apply (rule conjI,

rule contrapos_pp, simp+,

simp add:prime_n_pd_def,

(erule conjE)+, simp add: one_in_ring_n_pd_one_K,

simp add:distinct_pds_def, (erule conjE)+,

cut_tac representative_of_pd_valuation[of "P j"],

simp add:value_of_one, simp)

((rule ballI)+, rule impI)

apply (rule contrapos_pp, simp+, erule conjE,

simp add:prime_n_pd_def, (erule conjE)+,

simp add:ring_n_pd_def Sr_def, (erule conjE)+,

simp add:aneg_less,

frule_tac x = "(ν_{(P j)}) x" in ale_antisym[of _ "0"], simp,

frule_tac x = "(ν_{(P j)}) y" in ale_antisym[of _ "0"], simp)

apply (simp add:distinct_pds_def, (erule conjE)+,

cut_tac representative_of_pd_valuation[of "P j"],

simp add:val_t2p aadd_0_l,

simp)

(in Corps) n_eq_val_eq_idealTr:

[distinct_pds K n P; x ∈ carrier (O_{P n}); y ∈ carrier (O_{P n});

∀j ≤ n. ((ν_{(P j)}) x) ≤ ((ν_{(P j)}) y)] ==> Rxa (O_{P n}) y ⊆ Rxa (O_{P n} (erule exE)

"∀ n. valuation K (ν_{")

apply (case_tac "x = 0_{O_{j ( (\Sigma_{h. h ≤_r ± -_r ±

simp add:zero_in_)

apply (simp add:value_of_zero)

apply (subgoal_tac y = 0",simp

drule_tac a = n in forall_spec, simp,

drule_tac a=n in forall_spec, simp)

apply (cut_tac inf_ge_any[of "(ν_{(P n)} appblat

frule ale_antisym[of "(ν(rl aoupnmmm[f"" "Suc n"], ssum+)

apply (rule value_inf_zero, assumption+)

apply (simp add:mem_ring_n_pd_mem_K, assumption)

(frule ring_n_pd[of n P])

apply (subgoal_tac "∀j≤n. 0 ≤ ((ν "M < Suc

ubgoal_tac(y ⋅r (x^{) ∈ carrier (O_{")

apply (cut_tac field_frac_mul[of "y" "x"],

frule Ring.rxa_in_Rxa[of "O_{P n}

simp

frule Ring.principal_ideal[of "O_{P n}[_, nat, n ==> ant) set] ==>

apply (cut_tac Ring.ideal_cont_Rxa[of "O_{"(O_{♢p x" "y"],

assumption+,

simp add:mem_ring_n_pd_mem_K,

simp add:mem_ring_n_pd_mem_K,

simp add:zero_in_ring_n_pd_zero_K)

apply ((frule Ring.xan_RRxa[of "O\^subK P n\^sb""x" "y ⋅r (x^{"], assumption+,

simp add:ring_n_pd_def Sr_def,

(erule conjE)+,

cut_tac field_is_ring, rule Ring.ring_tOp_closed, assumption+,

cut_tac invf_closed1[of x], simp, simp,

simp add:ring_n_pd_def Sr_def)

apply (cut_tac Ring.ring_tOp_closed, assumption+,

cut_tac field_is_ring, assumption+, simp+,

cut_tac invf_closed1[of x], simp, simp)

apply (rule allI, rule impI, drule_tac a = j in forall_spec, assumption+,

cut_tac invf_closed1[of x], simp, erule conjE)

apply (subst val_t2p [where v="ν_{P j}"], simp,

rule mem_ring_n_pd_mem_K[of "n" "P" "y"], assumption+,

frule_tac x = j in spec, simp,

simp add:zero_in_ri

java.lang.NullPointerException

add:ring_n_pd_def Sr_def, assumption+)

apply (frule_tac x = "(ν allI, rule impI)

simp add:diff_ant_def,

simp add:mem_ring_n_pd_mem_K[of "n" "P" "x"] zero_in_ring_n_pd_zero_K)

(rule allI, rule impI,

simp add:distinct_pds_def, (erule conjE)+,

rule_tac P = "P j" in representative simp)

(in Corps) n_eq_val_eq_ideal:"[distinct_pds K n P; x ∈ calapply (frule(frule_tac x = y in a aGro.ag_mOp_closed[of "K"], assumption+)

y \<in K P n}} \forall>j ≤_{x) = ((νK (P j)} ==>

)

(rule equalityI)

apply (subgoal_tac "∀j≤ n. (νrule conjI)

apply (rule n_eq_val_eq_idealTr, assumption+)

apply (rule allI, rule impI, simp)

apply (subgoal_tac "∀j≤ n. (ν_{apply (rule n_eq_val_eq_idealTr, assumption+)

apply (rule allI, rule impI)

apply simp

mI_gen :: "[_ , nat ==> ('r ==> ant) set, nat, 'r set] ==> (simpad:vl_iu_e)

"mI_gen K P n I =frul_tacx "νK (P j)}) x" nd y = "(νK (P j)}

apply simp

mL :: "[_, nat ==> ('r ==> (simp add:Ring.ring_tOp_closed)

"mL K P I j = tna (LI K (ν_{(P j)}) I)"

(in Corps) mI_vals_nonempty:"[K P n}I; ≤

(ν) ` I ≠

(frule ring_n_pd[of "n" "P"])

java.lang.NullPointerException

(simp add:image_def)

blast

(in Corps) mI_vals_LB:"[distinct_pds K n P; ideal (O_{P n} add:distin

((ν_{(P j)}) `I) ⊆

rulesubsetI)

(simp add:image_def, erule bexE)

(frule ring_n_pd[of "n" "P"])

apply (frule_tac h = xa in Ring.ideal_subset[of "O_{"I"], assumption+)

apply (thin_tac "ideal (O_{P n\<^ (

apply (thin_tac "Ring (O_{P n}

apply (simp add: ring_n_pd_def Sr_def) apply (erule conjE)+

apply drule_ta a = j in forall_spec, simp)

(simp add:LBset_def ant_0)

(in Corps) mL_hom:"[K P n}

I "0 ∈}"

∀eo_pdvauatio[of " "]

(rule allI, rule impI)

apply (simp add:mL_def LI_def)

apply (simp add:Zset_def)

java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null

==> ∃

(frule_tac j = j in mI_vals_nonempty[of "n" "P" "I"], assumption+)

apply (frule_tac j = j in mI_vals_LB[of "n" "P" "I"], assumption+)

java.lang.NullPointerException

apply (simp add:LI_def)

(in Corps) distinct:"[ (Suc n);

apply (simp add:image_def, erule bexE)

apply (drule sym)

apply blast

(in Corps) val_LI_pos:"[

I ≠

apply (frule_tac j = j in mI_vals_nonempty[of n P I], assumption+)

apply (frule_tac j = j in mI_vals_LB[of n P I], assumption+)

apply (frule_tac A = "(\<nu>\<^bsub>K (P j)\<^esub>) ` I" and z = 0 in AMin_mem, assumption+)

apply (simp add:LI_def)

apply (frule subsetD[of "(\<nu>\<^bsub>K (P j)\<^esub>) ` I" "LBset (ant 0)" "AMin ((\<nu>\<^bsub>K (P j)\<^esub>) ` I)"], assumption+)

apply (simp add:LBset_def ant_0)

done

lemma (in Corps) val_LI_noninf:"\<lbrakk>distinct_pds K n P; ideal (O\<^bsub>K P n\<^esub>) I;

 I \<noteq> {\<zero>\<^bsub>(O\<^bsub>K P n\<^esub>)\<^esub>}; j \<le> n\<rbrakk> \<Longrightarrow> LI K (\<nu>\<^bsub>K (P j)\<^esub>) I \<noteq> \<infinity>"

apply (frule_tac j = j in mI_vals_nonempty[of "n" "P" "I"], assumption+)

apply (frule_tac j = j in mI_vals_LB[of "n" "P" "I"], assumption+)

apply (frule_tac A = "(\<nu>\<^bsub>K (P j)\<^esub>) ` I" and z = 0 in AMin, assumption+)

apply (thin_tac "(\<nu>\<^bsub>K (P j)\<^esub>) ` I \<subseteq> LBset (ant 0)",

 thin_tac "(\<nu>\<^bsub>K (P j)\<^esub> ) ` I \<noteq> {}")

apply (frule ring_n_pd[of "n" "P"])

apply (frule Ring.ideal_zero[of "O\<^bsub>K P n\<^esub>" "I"], assumption+)

apply (erule conjE, simp add:LI_def)

apply (frule singleton_sub[of "\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>" "I"])

apply (frule sets_not_eq[of "I" "{\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"],

 assumption+, erule bexE)

apply (simp add:zero_in_ring_n_pd_zero_K)

apply (subgoal_tac "\<exists>x\<in>I. AMin ((\<nu>\<^bsub>K (P j)\<^esub>) ` I) = (\<nu>\<^bsub>K (P j)\<^esub>) x",

 erule bexE) apply simp

apply (drule_tac x = a in bspec, assumption)

apply (thin_tac "AMin ((\<nu>\<^bsub>K (P j)\<^esub>) ` I) = (\<nu>\<^bsub>K (P j)\<^esub>) x")

apply (frule_tac h = a in Ring.ideal_subset[of "O\<^bsub>K P n\<^esub>" "I"], assumption+)

apply (frule_tac x = a in mem_ring_n_pd_mem_K[of n P], assumption+)

apply (simp add:distinct_pds_def, (erule conjE)+)

apply (cut_tac representative_of_pd_valuation[of "P j"])

defer apply simp apply blast

apply (frule_tac x = a in val_nonzero_z[of "\<nu>\<^bsub>K (P j)\<^esub>"], assumption+,

 erule exE, simp)

apply (thin_tac "\<forall>l \<le> n. \<forall>m \<le> n. l \<noteq> m \<longrightarrow> P l \<noteq> P m",

 thin_tac "(\<nu>\<^bsub>K (P j)\<^esub>) a = ant z")

apply (rule contrapos_pp, simp+)

apply (cut_tac x = "ant z" in inf_ge_any)

apply (frule_tac x = "ant z" in ale_antisym[of _ "\<infinity>"], assumption+)

apply simp

done

lemma (in Corps) Zleast_in_mI_pos:"\<lbrakk>distinct_pds K n P; ideal (O\<^bsub>K P n\<^esub>) I;

 I \<noteq> {\<zero>\<^bsub>(O\<^bsub>K P n\<^esub>)\<^esub>}; j \<le> n\<rbrakk> \<Longrightarrow> 0 \<le> mL K P I j"

apply (simp add:mL_def)

apply (frule ex_Zleast_in_mI[of "n" "P" "I" "j"], assumption+,

 erule bexE, frule sym, thin_tac "(\<nu>\<^bsub>K (P j)\<^esub>) x = LI K (\<nu>\<^bsub>K (P j)\<^esub>) I")

apply (subgoal_tac "LI K (\<nu>\<^bsub>K (P j)\<^esub>) I \<noteq> \<infinity>", simp)

apply (thin_tac "LI K (\<nu>\<^bsub>K (P j)\<^esub>) I = (\<nu>\<^bsub>K (P j)\<^esub>) x")

apply (frule ring_n_pd[of "n" "P"])

apply (frule_tac h = x in Ring.ideal_subset[of "O\<^bsub>K P n\<^esub>" "I"], assumption+)

apply (thin_tac "ideal (O\<^bsub>K P n\<^esub>) I")

apply (thin_tac "Ring (O\<^bsub>K P n\<^esub>)")

apply (simp add: ring_n_pd_def Sr_def) apply (erule conjE)

apply (drule_tac a = j in forall_spec, assumption)

apply (simp add:apos_tna_pos)

apply (rule val_LI_noninf, assumption+)

done

lemma (in Corps) Zleast_mL_I:"\<lbrakk>distinct_pds K n P; ideal (O\<^bsub>K P n\<^esub>) I; j \<le> n;

 I \<noteq> {\<zero>\<^bsub>(O\<^bsub>K P n\<^esub>)\<^esub>}; x \<in> I\<rbrakk> \<Longrightarrow> ant (mL K P I j) \<le> ((\<nu>\<^bsub>K (P j)\<^esub>) x)"

apply (frule val_LI_pos[of "n" "P" "I" "j"], assumption+)

apply (frule apos_neq_minf[of "LI K (\<nu>\<^bsub>K (P j)\<^esub>) I"])

apply (frule val_LI_noninf[of "n" "P" "I" "j"], assumption+)

apply (simp add:mL_def LI_def)

apply (simp add:ant_tna)

apply (frule Zleast_in_mI_pos[of "n" "P" "I" "j"], assumption+)

apply (frule mI_vals_nonempty[of "n" "P" "I" "j"], assumption+)

apply (frule mI_vals_LB[of "n" "P" "I" "j"], assumption+)

apply (frule AMin[of "(\<nu>\<^bsub>K (P j)\<^esub>) `I" "0"], assumption+)

apply (erule conjE)

apply (frule Zleast_in_mI_pos[of "n" "P" "I" "j"], assumption+)

apply (simp add:mL_def LI_def)

done

lemma (in Corps) Zleast_LI:"\<lbrakk>distinct_pds K n P; ideal (O\<^bsub>K P n\<^esub>) I; j \<le> n;

 I \<noteq> {\<zero>\<^bsub>(O\<^bsub>K P n\<^esub>)\<^esub>}; x \<in> I\<rbrakk> \<Longrightarrow> (LI K (\<nu>\<^bsub>K (P j)\<^esub>) I) \<le> ((\<nu>\<^bsub>K (P j)\<^esub>) x)"

apply (frule mI_vals_nonempty[of "n" "P" "I" "j"], assumption+)

apply (frule mI_vals_LB[of "n" "P" "I" "j"], assumption+)

apply (frule AMin[of "(\<nu>\<^bsub>K (P j)\<^esub>) `I" "0"], assumption+)

apply (erule conjE)

apply (simp add:LI_def)

done

lemma (in Corps) mpdiv_vals_nonequiv:"distinct_pds K n P \<Longrightarrow>

 vals_nonequiv K n (\<lambda>j. \<nu>\<^bsub>K (P j)\<^esub>) "

apply (simp add:vals_nonequiv_def)

apply (rule conjI)

apply (simp add:valuations_def)

apply (rule allI, rule impI)

apply (rule representative_of_pd_valuation,

 simp add:distinct_pds_def)

apply ((rule allI, rule impI)+, rule impI)

apply (simp add:distinct_pds_def, erule conjE)

apply (rotate_tac 4) apply (

 drule_tac a = j in forall_spec, assumption)

apply (rotate_tac -1,

 drule_tac a = l in forall_spec, assumption, simp)

apply (simp add:distinct_p_divisors)

done

definition

 KbaseP :: "[_, nat \<Rightarrow> ('r \<Rightarrow> ant) set, nat] \<Rightarrow>

 (nat \<Rightarrow> 'r) \<Rightarrow> bool" where

 "KbaseP K P n f \<longleftrightarrow> (\<forall>j \<le> n. f j \<in> carrier K) \<and>

 (\<forall>j \<le> n. \<forall>l \<le> n. (\<nu>\<^bsub>K (P j)\<^esub>) (f l) = (\<delta>\<^bsub>j l\<^esub>))"

definition

 Kbase :: "[_, nat, nat \<Rightarrow> ('r \<Rightarrow> ant) set]

 \<Rightarrow> (nat \<Rightarrow> 'r)" (\<open>(3Kb\<^bsub>_ _ _\<^esub>)\<close> [95,95,96]95) where

 "Kb\<^bsub>K n P \<^esub> = (SOME f. KbaseP K P n f)"

lemma (in Corps) KbaseTr:"distinct_pds K n P \<Longrightarrow> \<exists>f. KbaseP K P n f"

apply (simp add: KbaseP_def)

apply (frule mpdiv_vals_nonequiv[of "n" "P"])

apply (case_tac "n = 0")

 apply (simp add:vals_nonequiv_def valuations_def)

 apply (simp add:distinct_pds_def)

 apply (frule n_val_n_val1[of "P 0"])

 apply (frule n_val_surj[of "\<nu>\<^bsub>K (P 0)\<^esub>"])

 apply (erule bexE)

 apply (subgoal_tac " ((\<lambda>j\<in>{0::nat}. x) (0::nat)) \<in> carrier K \<and>

 (\<nu>\<^bsub>K (P 0)\<^esub>) ((\<lambda>j\<in>{0::nat}. x) (0::nat)) = (\<delta>\<^bsub>0 0\<^esub>)")

 apply blast

 apply (rule conjI)

apply simp apply (simp add:Kronecker_delta_def)

apply (cut_tac Approximation1_5[of "n - Suc 0" "\<lambda>j. \<nu>\<^bsub>K (P j)\<^esub>"])

apply simp

apply simp+

apply (rule allI, rule impI)

apply (rule n_val_n_val1 )

apply (simp add:distinct_pds_def)

done

lemma (in Corps) KbaseTr1:"distinct_pds K n P \<Longrightarrow> KbaseP K P n (Kb\<^bsub>K n P \<^esub>)"

apply (subst Kbase_def)

apply (frule KbaseTr[of n P])

apply (erule exE)

apply (simp add:someI)

done

lemma (in Corps) Kbase_hom:"distinct_pds K n P \<Longrightarrow>

 \<forall>j \<le> n. (Kb\<^bsub>K n P\<^esub>) j \<in> carrier K"

apply (frule KbaseTr1[of "n" "P"])

apply (simp add:KbaseP_def)

done

lemma (in Corps) Kbase_Kronecker:"distinct_pds K n P \<Longrightarrow>

 \<forall>j \<le> n. \<forall>l \<le> n. (\<nu>\<^bsub>K (P j)\<^esub>) ((Kb\<^bsub>K n P\<^esub>) l) = \<delta>\<^bsub>j l\<^esub>"

apply (frule KbaseTr1[of n P])

apply (simp add:KbaseP_def)

done

lemma (in Corps) Kbase_nonzero:"distinct_pds K n P \<Longrightarrow>

 \<forall>j \<le> n. (Kb\<^bsub>K n P\<^esub>) j \<noteq> \<zero>"

apply (rule allI, rule impI)

apply (frule Kbase_Kronecker[of n P])

apply (subgoal_tac "(\<nu>\<^bsub>K (P j)\<^esub>) ((Kb\<^bsub>K n P\<^esub>) j) = \<delta>\<^bsub>j j\<^esub>")

apply (thin_tac "\<forall>j\<le>n. (\<forall>l\<le>n. ((\<nu>\<^bsub>K P j\<^esub>) ((Kb\<^bsub>K n P\<^esub>) l)) = \<delta>\<^bsub>j l\<^esub>)")

apply (simp add:Kronecker_delta_def)

apply (rule contrapos_pp, simp+)

apply (cut_tac P = "P j" in representative_of_pd_valuation)

apply (simp add:distinct_pds_def)

apply (simp only:value_of_zero, simp only:ant_1[THEN sym],

 frule sym, thin_tac " \<infinity> = ant 1", simp del:ant_1)

apply simp

done

lemma (in Corps) Kbase_hom1:"distinct_pds K n P \<Longrightarrow>

 \<forall>j \<le> n. (Kb\<^bsub>K n P\<^esub>) j \<in> carrier K - {\<zero>}"

by(simp add:Kbase_nonzero Kbase_hom)

definition

 Zl_mI :: "[_, nat \<Rightarrow> ('b \<Rightarrow> ant) set, 'b set]

 \<Rightarrow> nat \<Rightarrow> 'b" where

 "Zl_mI K P I j = (SOME x. (x \<in> I \<and> ( (\<nu>\<^bsub>K (P j)\<^esub>) x = LI K (\<nu>\<^bsub>K (P j)\<^esub>) I)))"

lemma (in Corps) value_Zl_mI:"\<lbrakk>distinct_pds K n P; ideal (O\<^bsub>K P n\<^esub>) I; j \<le> n\<rbrakk>

\<Longrightarrow> (Zl_mI K P I j \<in> I) \<and> (\<nu>\<^bsub>K (P j)\<^esub>) (Zl_mI K P I j) = LI K (\<nu>\<^bsub>K (P j)\<^esub>) I"

apply (subgoal_tac "\<exists>x. (x \<in> I \<and> ((\<nu>\<^bsub>K (P j)\<^esub>) x = LI K (\<nu>\<^bsub>K (P j)\<^esub>) I))")

apply (subst Zl_mI_def)+

apply (rule someI2_ex, assumption+)

apply (frule ex_Zleast_in_mI[of "n" "P" "I" "j"], assumption+)

apply (erule bexE, blast)

done

lemma (in Corps) Zl_mI_nonzero:"\<lbrakk>distinct_pds K n P; ideal (O\<^bsub>K P n\<^esub>) I;

 I \<noteq> {\<zero>\<^bsub>(O\<^bsub>K P n\<^esub>)\<^esub>}; j \<le> n\<rbrakk> \<Longrightarrow> Zl_mI K P I j \<noteq> \<zero>"

apply (case_tac "n = 0")

apply (simp add:distinct_pds_def)

apply (frule representative_of_pd_valuation[of "P 0"])

apply (subgoal_tac "O\<^bsub>K P 0\<^esub> = Vr K (\<nu>\<^bsub>K (P 0)\<^esub>)")

apply (subgoal_tac "Zl_mI K P I 0 = Ig K (\<nu>\<^bsub>K (P 0)\<^esub>) I")

apply simp apply (simp add:Ig_nonzero)

apply (simp add:Ig_def Zl_mI_def)

apply (simp add:ring_n_pd_def Vr_def)

apply (simp)

apply (frule value_Zl_mI[of n P I j], assumption+)

apply (erule conjE)

apply (rule contrapos_pp, simp+)

apply (frule distinct_pds_valuation1[of n j P], assumption+)

apply (simp add:value_of_zero)

apply (simp add:zero_in_ring_n_pd_zero_K)

apply (frule singleton_sub[of "\<zero>" "I"],

 frule sets_not_eq[of "I" "{\<zero>}"], assumption,

 erule bexE, simp)

apply (frule_tac x = a in Zleast_mL_I[of "n" "P" "I" "j"], assumption+)

apply (frule_tac x = a in val_nonzero_z[of "\<nu>\<^bsub>K (P j)\<^esub>"])

apply (frule ring_n_pd[of "n" "P"])

apply (frule_tac h = a in Ring.ideal_subset[of "O\<^bsub>K P n\<^esub>" "I"], assumption+)

apply (simp add:mem_ring_n_pd_mem_K) apply assumption

apply (simp add:zero_in_ring_n_pd_zero_K) apply assumption

apply (frule val_LI_noninf[THEN not_sym, of "n" "P" "I" "j"], assumption+)

apply (simp add:zero_in_ring_n_pd_zero_K) apply assumption

 java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

done

lemma (in Corps) Zl_mI_mem_K:"\<lbrakk>distinct_pds K n P; ideal (O\<^bsub>K P n\<^esub>) I; l \<le> n\<rbrakk>

 \<Longrightarrow> (Zl_mI K P I l) \<in> carrier K"

apply (frule value_Zl_mI[of "n" "P" "I" "l"], assumption+)

apply (erule conjE)

apply (frule ring_n_pd[of "n" "P"])

apply (frule Ring.ideal_subset[of "O\<^bsub>K P n\<^esub>" "I" "Zl_mI K P I l"], assumption+)

apply (simp add:mem_ring_n_pd_mem_K[of "n" "P" "Zl_mI K P I l"])

done

definition

 mprod_exp :: "[_, nat \<Rightarrow> int, nat \<Rightarrow> 'b, nat]

 \<Rightarrow> 'b" where

 "mprod_exp K e f n = nprod K (\<lambda>j. ((f j)\<^bsub>K\<^esub>\<^bsup>(e j)\<^esup>)) n"

lemma (in Corps) mprod_expR_memTr:"(\<forall>j\<le>n. f j \<in> carrier K) \<longrightarrow>

 mprod_expR K e f n \<in> carrier K"

apply (cut_tac field_is_ring)

apply (induct_tac n)

apply (rule impI, simp)

apply (simp add:mprod_expR_def)

apply (cut_tac Ring.npClose[of K "f 0" "e 0"], assumption+)

apply (rule impI)

apply simp

apply (subst Ring.mprodR_Suc, assumption+)

apply (simp)

apply (simp)

apply (rule Ring.ring_tOp_closed[of K], assumption+)

apply (rule Ring.npClose, assumption+)

apply simp

done

lemma (in Corps) mprod_expR_mem:"\<forall>j \<le> n. f j \<in> carrier K \<Longrightarrow>

 mprod_expR K e f n \<in> carrier K"

apply (cut_tac field_is_ring)

apply (cut_tac Ring.mprod_expR_memTr[of K e n f])

applysimp

apply (subgoal_tac "f \<in> {j. j \<le> n} \<rightarrow> carrier K", simp+)

done

lemma (in Corps) mprod_Suc:"\<lbrakk> \<forall>j\<le>(Suc n). e j \<in> Zset;

 <forall> \le>Sucn) fj\<> (carrierK-{\ero>}\rbrakk \<>

mprod_exp K e f (Suc n) = (mprod_exp K e f n) \<cdot>\<^sub>r ((f (Suc n))\<^bsub>K\<^esub (frulering_n_pd[of n"P")

apply (simp add:mprod_exp_def)

one

lemma (in Corps) mprod_memTr:"

(\<forall>j \<le> n. e j \<in> Zset) \<and> (\<forall>j \<le> n. f j \<in> ((carrier K) - {\<zero>})) \<longrightarrow>

 (mprod_exp K e f n) \<in> ((carrier K) - {\<zero>})"

apply (induct_tac n)

apply (simp, rule impI, (erule conjE)+,

 simp add:mprod_exp_def, simp add:npowf_mem,

 simp add:field_potent_nonzero1)

apply (rule impI, simp, erule conjE,

 cut_tac field_is_ring, cut_tac field_is_idom,

eruleconjE,simpadd:)

apply (rule conjI)

apply( Ring.ring_tOp_closed[of""] assumption+,

 simp add:npowf_mem)

apply (rule Idomain.idom_tOp_nonzeros, assumption+,

simp:, assumption

simp:)

done

thin_tac I\noteq>\zero\^bsub\bsubPn<

apply (cut_tac mprod_memTr[of n e f]) apply simp

done

lemma (in Corps) mprod_mprodR:"\<lbrakk>\<forall>j \<le> n. e j \<in> Zset; \<forall>j \<le> n. 0 \<le> (e j);

 mprod_exp K e f n = mprod_expR K (nat o e) f n"

 )

apply (simp add:mprod_exp_def mprod_expR_def)

apply (rule Ring.nprod_eq, assumption+)

apply (rule allI, rule impI, simp add:npowf_mem)

apply (rule allI, rule impI, rule Ring.npClose, assumption+, simp)

apply (rule allI, rule impI)

apply (simp add:npowf_def)

done

subsection "Representation of an ideal I as a product of prime ideals"

lemma (in Corps) ring_n_mprod_mprodRTr:"distinct_pds K n P \<Longrightarrow>

 (\<forall>j \<le> m. e j \<in> Zset) \<and> (\<forall>j \<le> m. 0 \<le> (e j)) \<and>

 (\<forall>j \<le> m. f j \<in> carrier (O\<^bsub>K P n\<^esub>)-{\<zero>\<^bsub>(O\<^bsub>K P n\<^esub>)\<^esub>}) \<longrightarrow>

 mprod_exp K e f m = mprod_expR (O\<^bsub>K P n\<^esub>) (nat o e) f m"

apply (frule ring_n_pd[of n P])

apply (induct_tac m)

apply (rule impI, (erule conjE)+,

 simp add:mprod_exp_def mprod_expR_def)

apply (erule conjE, simp add:npowf_def, simp add:ring_n_exp_K_exp)

apply (rule impI, (erule conjE)+, simp)

apply (subst mprod_Suc, assumption+,

 rule allI, rule impI,

 simp add:mem_ring_n_pd_mem_K,

 simp add:zero_in_ring_n_pd_zero_K)

 apply (subst Ring.mprodR_Suc, assumption+,

 simp add:cmp_def,

 simp)

 apply (simp add:ring_n_pd, simp add:npowf_def,

 simp add:ring_n_exp_K_exp)

apply (subst ring_n_tOp_K_tOp, assumption+,

 rule Ring.mprod_expR_mem, simp add:ring_n_pd,

 simp,

 simp)

apply (rule Ring.npClose, simp add:ring_n_pd, simp, simp)

done

lemma (in Corps) ring_n_mprod_mprodR:"\<lbrakk>distinct_pds K n P; \<forall>j \<le> m. e j \<in> Zset;

\<forall>j \<le> m. 0 \<le> (e j); \<forall>j \<le> m. f j \<in> carrier (O\<^bsub>K P n\<^esub>)-{\<zero>\<^bsub>(O\<^bsub>K P n\<^esub>)\<^esub>}\<rbrakk>

\<Longrightarrow> mprod_exp K e f m = mprod_expR (O\<^bsub>K P n\<^esub>) (nat o e) f m"

apply (simp add:ring_n_mprod_mprodRTr)

done

lemma (in Corps) value_mprod_expTr:"valuation K v \<Longrightarrow>

(\<forall>j \<le> n. e j \<in> Zset) \<and> (\<forall>j \<le> n. f j \<in> (carrier K - {\<zero>})) \<longrightarrow>

v (mprod_exp K e f n) = ASum (\<lambda>j. (e j) *\<^sub>a (v (f j))) n"

apply (induct_tac n)

apply simp

apply (rule impI, erule conjE)

apply(simp add:mprod_exp_def val_exp)

apply (rule impI, erule conjE)

apply simp

apply (subst mprod_Suc, assumption+)

apply (rule allI, rule impI, simp)

apply (subst val_t2p[of v], assumption+)

apply (cut_tac n = "n" in mprod_mem[of _ e f],

 (rule allI, rule impI, simp)+, simp)

apply (simp add:npowf_mem, simp add:field_potent_nonzero1)

apply (simp add:val_exp[THEN sym, of "v"])

done

lemma (in Corps) value_mprod_exp:"\<lbrakk>valuation K v; \<forall>j \<le> n. e j \<in> Zset;

 \<forall>j \<le> n. f j \<in> (carrier K - {\<zero>})\<rbrakk> \<Longrightarrow>

 v (mprod_exp K e f n) = ASum (\<lambda>j. (e j) *\<^sub>a (v (f j))) n"

apply (simp add:value_mprod_expTr)

done

lemma (in Corps) mgenerator0_1:"\<lbrakk>distinct_pds K (Suc n) P;

ideal (O\<^bsub>K P (Suc n)\<^esub>) I; I \<noteq> {\<zero>\<^bsub>(O\<^bsub>K P (Suc n)\<^esub>)\<^esub>};

I \<noteq> carrier (O\<^bsub>K P (Suc n)\<^esub>); j \<le> (Suc n)\<rbrakk> \<Longrightarrow>

((\<nu>\<^bsub>K (P j)\<^esub>) (mprod_exp K (mL K P I) (Kb\<^bsub>K (Suc n) P\<^esub>) (Suc n))) =

 ((\<nu>\<^bsub>K (P j)\<^esub>) (Zl_mI K P I j))"

apply (frule distinct_pds_valuation[of j n P], assumption+)

apply (frule mL_hom[of "Suc n" "P" "I"], assumption+)

apply (frule Kbase_hom1[of "Suc n" "P"])

apply (frule value_mprod_exp[of "\<nu>\<^bsub>K (P j)\<^esub>" "Suc n" "mL K P I"

 "Kb\<^bsub>K (Suc n) P\<^esub>"], assumption+)

apply (simp del:ASum_Suc)

apply (thin_tac "(\<nu>\<^bsub>K (P j)\<^esub>) (mprod_exp K (mL K P I) (Kb\<^bsub>K (Suc n) P\<^esub>) (Suc n)) =

 ASum (\<lambda>ja. (mL K P I ja) *\<^sub>a (\<nu>\<^bsub>K (P j)\<^esub>) ((Kb\<^bsub>K (Suc n) P\<^esub>) ja)) (Suc n)")

apply (subgoal_tac "ASum (\<lambda>ja. (mL K P I ja) *\<^sub>a

 ((\<nu>\<^bsub>K (P j)\<^esub>) ((Kb\<^bsub>K (Suc n) P\<^esub>) ja))) (Suc n) =

 ASum (\<lambda>ja. (mL K P I ja) *\<^sub>a (\<delta>\<^bsub>j ja\<^esub>)) (Suc n)")

apply (simp del:ASum_Suc)

apply (subgoal_tac "\<forall>h \<le> (Suc n). (\<lambda>ja. (mL K P I ja) *\<^sub>a (\<delta>\<^bsub>j ja\<^esub>)) h \<in> Z\<^sub>\<infinity>")

auto , , , . We demonstratethiswithabenchmarkcontaining 2267methodcalls

apply simp

apply (simp add:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

apply (rotate_tac -1, drule not_sym)

apply (simp add:mL_def[of "K" "P" "I" "j"])

apply (frule val_LI_noninf[of "Suc n" "P" "I" "j"], assumption+)

apply (rule not_sym, simp, simp)

apply (frule val_LI_pos[of "Suc n" "P" "I" "j"], assumption+,

 rotate_tac -2, frule not_sym, simp, simp)

apply (frule apos_neq_minf[of "LI K (\<nu>\<^bsub>K (P jjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 20

(:)

apply (simp add:value_Zl_mI[of "Suc n" "P" "I" "j"])

apply (rule allI, rule impI)

apply (simp add:Kdelta_in_Zinf, simp)

apply (rule ballI, simp)

apply(simpadd:Kronecker_delta_def, erule conjE)

apply (simp add:asprod_n_0)

apply (rule allI, rule impI)

apply (simp add:Kdelta_in_Zinf)

apply (frule Kbase_Kronecker[of "Suc n" "P"])

apply (rule ASum_eq,

 rule allI, rule impI,

 simp add:Kdelta_in_Zinf,

 rule allI, rule impI,

 simp add:Kdelta_in_Zinf)

ruleallI rule ) simp

done

lemma (in Corps) mgenerator0_2:"\<lbrakk> 0 < n; distinct_pds K n P; ideal (O\<^bsub>K P n\<^esub>) I (simp add:Zl_mI_mem_K)

 \<noteq>{\zero>\<^bsub>(\^bsub> Pn\<esub)\<^esub>} I\noteq>carrier (O\<bsubK \^esub>);j\<>n<> <Longrightarrow>

(<>\<^bsubK P )\^esub)(mprod_exp K(mL P I)(\^bsub>K P<^esub) n) =((<nu><bsubK( j)\^>) (l_mI K P j)"

apply (cut_tac mgenerator0_1[of "n - Suc 0" "P" "I" "j"])

apply simp+

done

lemma (in Corps) mgenerator1:"\<lbrakk>distinct_pds K n P; ideal (O\<^bsub>K P n\<^esub>) I;

I \<noteq> {\<zero>\<^bsub>(O\<^bsub>K P n\<^esub>)\<^esub>}; I \<noteq> carrier (O\<^bsub>K P n\<^esub>); j \<le> n\<rbrakk> \<Longrightarrow>

((\<nu>\<^bsub>K (P j)\<^esub>) (mprod_exp K (mL K P I) (Kb\<^bsub>K n P\<^esub>) n)) = ((\<nu>\<^bsub>K (P j)\<^esub>) (Zl_mI K P I j))"

 frule value_Zl_mI[of "n" "P" "I" "j"], assumption+,

 frule val_LI_noninf[of "n" "P" "I" "j"], assumption+,

 frule val_LI_pos[of "n" "P" "I" "j"], assumption+,

 frule apos_neq_minf[of "LI K (\<nu>\<^bsub>K (P j)\<apply (fruleval_LI_noninf[of "n" "P"I n],assumption, simp )java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73

 simp add:distinct_pds_def, erule conjE)

apply (cut_tac representative_of_pd_valuation[of "P j"], simp+,

 simp add:mprod_exp_def,

 subst val_exp[THEN sym, of "\<nu>\<^bsub>K (P 0)\<^esub>" "(Kb\<^bsub>K 0 P\<^esub>) 0"], assumption+,

 cut_tac Kbase_hom[of "0" "P"], simp,

 simp add:distinct_pds_def,

 cut_tac Kbase_nonzero[of "0" "P"], simp+,

 simp add:distinct_pds_def)

apply (cut_tac Kbase_nonzero[of "0" "P"], simp add:distinct_pds_def)

apply (cut_tac Kbase_Kronecker[of "0" "P"], simp add:distinct_pds_def)

apply (simp add:Kronecker_delta_def, simp add:mL_def, simp add:ant_tna)

apply (simp add:distinct_pds_def)+

apply (cut_tac mgenerator0_2[of "n" "P" "I" "j"], simp+)

apply (simp add:distinct_pds_def) apply simp+

done



lemma (in Corps) mgenerator2Tr1:"\<lbrakk>0 < n; j \<le> n; k \<le> n; distinct_pds K napply( "\>K(lambdak Zl_mI KPIk\cdot\^subr

 (\<nu>\<^bsub>K (P j)\<^esub>) (mprod_exp K (\<lambda>l. \<gamma>\<^bsub>k l\<^esub> ) (Kb\<^bsub>K n P\<^esub>) n) = (\<gamma>\<^bsub>k j\<^esub>) *\<^sub>a (\<delta>\<^bsub>j j\<^esub>)"

apply (frule distinct_pds_valuation1[of "n" "j" "P"], assumption+)

apply (frule K_gamma_hom[of k n])

apply (subgoal_tac "\<forall>j \<le> n. (Kb\<^bsub>K n P\<^esub>) j \<in> carrier K - {\<zero>}")

apply (simp add:value_mprod_exp[of "\<nu>\<^bsub>K (P j)\<^esub>" n "K_gamma k" "(Kb\<^bsub>K n P\<^esub>)"])

apply (subgoal_tac "ASum (\<lambda>ja. (\<gamma>\

 = ASum (\<lambda>ja. (((\<gamma>\<^bsub>k ja\<^esub>) *\<^sub>a (\<delta>\<^bsub>j ja\<^esub>)))) n")

apply simp

apply (subgoal_tac "\<forall>j \<le> n. (\<lambda>ja. (\<gamma>\<^bsub>k ja\<^esub>) *\<^sub>a (\<delta>\<^bsub>j ja\<^esub>)) j \<in> Z\<^sub>\<infinity>")

apply (cut_tac eSum_single[of n "\<lambda>ja. ((\<gamma>\<^bsub>k ja\<^esub>) *\<^sub>a (\<delta>\<^bsub>j ja\<^esub>))" "j"], simp)

apply (rule allI, rule impI, simp add:Kronecker_delta_def,

 rule impI, simp add:asprod_n_0 Zero_in_aug_inf, assumption+)

apply (rule ballI, simp)

onecker_delta_def

 apply (rule apply ( add:aadd_0_r)

 apply (simp add:Kronecker_delta_def, simp add:K_gamma_def)

apply (simp add:ant_0 Zero_in_aug_inf)

apply (cut_tac lemma( Corps) mI_gen_in_I"\lbrakk0 <n;distinct_pds K nP;ideal (\^bsub> P n\esub> I

apply (rule ASum_eq)

 apply(rule allI,rule )

 apply (simp add:K_gamma_def, simp add:Zero_in_aug_inf)

 apply (rule impI, rule value_in_aug_inf, assumption+, simp)

 apply (simp add:K_gamma_def Zero_in_aug_inf Kdelta_in_Zinf1)

 apply (rule allI, rule impI)

 apply (simp add:Kbase_Kronecker[of "n" "P"])

 apply (rule Kbase_hom1, assumption+)

done

lemma (in Corps) mgenerator2Tr2:"\<lbrakk>0 < n; j \<le> n; k \<le> n; distinct_pds K n P\<rbrakk> \<Longrightarrow>

 (\<nu>\<^bsub>K (P j)\<^esub>) ((mprod_exp K (\<lambda>l. \<gamma>\<^bsub>k

apply (frule K_gamma_hom[of k n])

apply (frule Kbase_hom1[of "n" "P"])

apply (frule mprod_mem[of n "K_gamma k" "Kb\<^bsub>K n P\<^esub>"], assumption+)

apply (frule distinct_pds_valuation1[of "n" "j" "P"], assumption+)

apply (simp, erule conjE)

apply (simp add:val_exp[THEN sym])

apply (simp add:mgenerator2Tr1)

apply (simp add:K_gamma_def Kronecker_delta_def)

apply (rule impI)

apply (simp add:asprod_def a_z_z)

done

lemma (in Corps) mgenerator2Tr3_1:"\<lbrakk>0 < n; j \<le> n; k \<le> n; j = k;

 distinct_pds K n P\<rbrakk> \<Longrightarrow>

 \<nu\^bsub> (Pj)\^esub>) (mprod_exp K (<>l \gamma\^> \^>))(\^bsub>KnP<esub) n)\bsubK\esub\^bsup>\^> 0"

(simp add:mgenerator2Tr2) apply (simp add:K_gamma_def)

(in Corps) mgenerator2Tr3_2:"[0 < n; j ≤ n; k ≤ n; j ≠ k;

distinct_pds K n P] ==>

(ν_{(P j)}) ((mprod_exp K (λl. (γ_l)) (Kb_{n P}) n)) = ant m"

(simp add:mgenerator2Tr2) apply (simp add:K_gamma_def)

(in Corps) mgeneratorTr4:"[0 < n; distinct_pds K n P; ideal (O_{P n}) I;

I ≠ {0_{_{P n}}}; I ≠ carrier (O_{P n})] ==>

mprod_exp K (mL K P I) (Kb_{n P}) n ∈ carrier (O_{P n})"

(subst ring_n_pd_def)

(simp add:Sr_def)

apply (frule mL_hom[of "n" "P" "I"], assumption+)

apply (frule mprod_mem[of n "mL K P I" "Kb_{n P}"])

apply (rule Kbase_hom1, assumption+)

apply (simp add:mprod_mem)

(rule allI, rule impI)

apply (simp add:mgenerator1)

apply (simp add:value_Zl_mI)

apply (simp add:val_LI_pos)

m_zmax_pdsI_hom :: "[_, nat ==> ('b ==> ant) set, 'b set] ==> nat ==> int" where

"m_zmax_pdsI_hom K P I = (λj. tna (AMin ((ν_{(P j)}) ` I)))"

m_zmax_pdsI :: "[_, nat, nat ==> ('b ==> ant) set, 'b set] ==> int" where

"m_zmax_pdsI K n P I = (m_zmax n (m_zmax_pdsI_hom K P I)) + 1"

(in Corps) value_Zl_mI_pos:"[0 < n; distinct_pds K n P; ideal (O_{P n}) I;

I ≠ {0_{O_{P n})}}; I ≠ carrier (O_{P n}); j ≤ n; l ≤ n] ==>

0 ≤ ((ν_{(P j)}) (Zl_mI K P I l))"

(frule value_Zl_mI[of "n" "P" "I" "l"], assumption+)

(erule conjE)

apply (frule ring_n_pd[of "n" "P"])

apply (frule Ring.ideal_subset[of "O_{P n}" "I" "Zl_mI K P I l"], assumption+)

apply (thin_tac "ideal (O_{P n}) I")

apply (thin_tac "I ≠ {0_{_{P n}}}")

apply (thin_tac "I ≠ carrier (O_{P n})")

apply (thin_tac "Ring (O_{P n})")

apply (simp add:ring_n_pd_def Sr_def)

(in Corps) value_mI_genTr1:"[0 < n; distinct_pds K n P; ideal (O_{P n}) I;

I ≠ {0_{_{P n}}}; I ≠ carrier (O_{P n}); j ≤ n] ==>

(mprod_exp K (K_gamma j) (Kb_{n P}) n)^{m_zmax_pdsI K n P I)} ∈ carrier K"

(frule K_gamma_hom[of "j" "n"])

(frule mprod_mem[of n "K_gamma j" "Kb_{n P}"])

apply (rule Kbase_hom1, assumption+)

(rule npowf_mem)

apply simp+

(in Corps) value_mI_genTr1_0:"[0 < n; distinct_pds K n P;

ideal (O_{P n}) I; I ≠ {0_{_{P n}}}; I ≠ carrier (O_{P n}); j ≤ n]

==> (mprod_exp K (K_gamma j) (Kb_{n P}) n) ∈ carrier K"

(frule K_gamma_hom[of "j" "n"])

(frule mprod_mem[of n "K_gamma j" "Kb_{n P}"])

apply (rule Kbase_hom1, assumption+)

apply simp

(in Corps) value_mI_genTr2:"[0 < n; distinct_pds K n P; ideal (O_{P n}) I;

I ≠ {0_{_{P n}}}; I ≠ carrier (O_{P n}); j ≤ n] ==>

(mprod_exp K (K_gamma j) (Kb_{n P}) n)^{m_zmax_pdsI K n P I)} ≠ 0"

apply (frule K_gamma_hom[of "j" "n"])

apply (frule mprod_mem[of n "K_gamma j" "Kb_{n P}"])

apply (rule Kbase_hom1, assumption+) apply simp apply (erule conjE)

apply (simp add: field_potent_nonzero1)

(in Corps) value_mI_genTr3:"[0 < n; distinct_pds K n P; ideal (O_{P n}) I;

I ≠ {0_{_{P n}}}; I ≠ carrier (O_{P n}); j ≤ n] ==>

(Zl_mI K P I j) ⋅_r ((mprod_exp K (K_gamma j) (Kb_{n P}) n)^{m_zmax_pdsI K n P I)})

∈ carrier K"

(cut_tac field_is_ring)

(rule Ring.ring_tOp_closed, assumption+)

(simp add:Zl_mI_mem_K)

(simp add:value_mI_genTr1)

(in Corps) value_mI_gen:"[0 < n; distinct_pds K n P; ideal (O_{P n}) I;

I ≠ {0_{O_{P n})}}; I ≠ carrier (O_{P n}); j ≤ n] ==>

ν_{(P j)}) (nsum K (λk. ((Zl_mI K P I k) ⋅_r ((mprod_exp K (λl. (γ_l)) (Kb_{n P}) n)^{m_zmax_pdsI K n P I)}))) n) = LI K (ν_{(P j)}) I"

(cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])

(case_tac "j = n", simp)

apply (cut_tac nsum_suc[of K "λk. Zl_mI K P I k ⋅_r

mprod_exp K (K_gamma k) (Kb_{n P}) n^{K n P I}" "n - Suc 0"],

simp,

thin_tac "Σ_e K (λk. Zl_mI K P I k ⋅_r

mprod_exp K (K_gamma k) (Kb_{n P}) n^{K n P I}) n =

Σ_e K (λk. Zl_mI K P I k ⋅_r

mprod_exp K (K_gamma k) (Kb_{n P})

n^{K n P I}) (n - Suc 0) ±

Zl_mI K P I n ⋅_r

mprod_exp K (K_gamma n) (Kb_{n P}) n^{K n P I}")

apply (cut_tac distinct_pds_valuation[of "n" "n - Suc 0" "P"])

2 apply simp

2 apply simp

apply (subst value_less_eq1[THEN sym, of "ν_{(P n)}"

"(Zl_mI K P I n)⋅_r (mprod_exp K (K_gamma n) (Kb_{n P}) n^{K n P I})"

"nsum K (λk.(Zl_mI K P I k)⋅_r (mprod_exp K (K_gamma k) (Kb_{n P}) n^{K n P I})) (n - Suc 0)"], assumption+)

apply (simp add:value_mI_genTr3)

apply (frule Ring.ring_is_ag[of K])

apply (rule aGroup.nsum_mem[of _ "n - Suc 0"], assumption+)

apply (rule allI, rule impI)

apply (simp add:value_mI_genTr3)

apply (subst val_t2p[of "ν_{(P n)}"], assumption+)

apply (simp add:Zl_mI_mem_K)

apply (simp add:value_mI_genTr1)

apply (simp add:mgenerator2Tr3_1[of "n" "n" "n" "P"])

apply (simp add:aadd_0_r)

(frule value_Zl_mI[of "n" "P" "I" "n"], assumption+, simp)

apply (erule conjE)

apply (frule_tac f = "λk. (Zl_mI K P I k) ⋅_r

(mprod_exp K (K_gamma k) (Kb_{n P}) n^{K n P I})" in

value_ge_add[of "ν_{(P n)}" "n - Suc 0" _

"ant (m_zmax_pdsI K n P I)"])

apply (rule allI, rule impI)

apply (rule Ring.ring_tOp_closed, assumption+)

apply (simp add:Zl_mI_mem_K)

apply (simp add:value_mI_genTr1)

apply (rule allI, rule impI) apply (simp add:cmp_def)

apply (subst val_t2p [where v="ν_{P n}"], assumption+)

apply (simp add:Zl_mI_mem_K)

apply (simp add:value_mI_genTr1)

apply (cut_tac e = "K_gamma ja" in mprod_mem[of n _ "Kb_{n P}"])

apply (simp add:Zset_def) apply (rule Kbase_hom1, assumption+)

apply (subst val_exp[of "ν_{(P n)}", THEN sym], assumption+)

apply simp+

apply (subst mgenerator2Tr1[of "n" "n" _ "P"], assumption+, simp, ‹

assumption+)

apply (simp add:K_gamma_def Kronecker_delta_def)

apply (frule_tac l = ja in value_Zl_mI_pos[of "n" "P" "I" "n"],

assumption+, simp, simp)

dNe_per)

apply (frule_tac y = "(ν_{(P n)}) (Zl_mI K P I ja

aadd_le_mono[of "0" _ "ant (m_zmax_pdsI K n P I)"]) apply (simp add:aadd_0_l)

apply (subgoal_tac "LI K (ν_{(P n)}

apply simp

apply (rule aless_le_trans[of "LI K (ν_{I"

"ant (m_zmax_pdsI K n P I)"])

_maa_d_f

java.lang.NullPointerException

"m_zmax n (m_zmax_pdsI_hom K P I) + 1"])

(frule val_LI_noninf[of "n" "P" "I" "n"], assumption+, simp, simp)

frule val_LI_pos[of "n" "P" "I" "n"], assumption+, simp,

frule apos_neq_minf[of "LI K (ν_{(P n)}) I"], simp add:ant_tna)

apply (subst m_zmax_pdsI_hom_def)

applyut Ie

apply (cut_tac m_zmax_gt_each[of n "λu.(tna (AMin ((ν_{(\^) ` I)))"])

apply simp

apply (rule allI, rule impI)

apply (simp add:Zset_def, simp)

apply (subst val_t2p[of "νK (P n)}

apply (rule Zl_mI_mem_K, assumption+, simp)

apply (simp add:value_mI_genTr1)

apply (simp add:mgenerator2Tr3_1[of "n" "n" "n" "P" "m_zmax_pdsI K n P I"])

apply (simp add:aadd_0_r)

apply (simp add:value_Zl_mI[of "n" "P" "I" "n"])

(*** case j = n done ***)}

apply[k. (Zl_mIcdot>java.lang.NullPointerException

((mprod_exp K (K_gamma k) (Kb_n P) n)^{m_zmax_pdsI K n P I)})" "j"])}}}}}}}}}}}}

apply simp
apply (rule allI, rule impI)
apply (simp add:value_mI_genTr3) apply simp+

apply (thin_tac "Σ_e K (λk. Zl_mI K P I k java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
 mprod_exp K (K_gamma k) (Kb_n P) nring_n_pd_mem_K "n"""
 Σ_k. Zl_mI ⋅java.lang.NullPointerException
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply (cut_tac nsum_suc[of ,e conjE
 mprod_exp K (K_gamma k)(b<>K
apply (simp del:nsum_suc) apply (
 thin_tac java.lang.NullPointerException
 mprod_exp K (K_gamma k) (Kb_{n P}) n_{P n}_K P n> (Kb_{j"

 Σ_e Kga_in_principal

 mprod_exp K (K_gamma k KbK n P}Km_zmax_pdsI) (τ
 (n - Suc 0) ± (cmp (λk. Zl_mI K P I k ⋅
 mprod_exp K (K_gamma k) (Kb_n P) njava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
apply (cut_tac "
prefer 2 apply simp prefer 2 apply simp

apply (cut_tac n_in_Nsetn[of "n"])
apply (simp add:transpos_ij_2)
apply (subst value_less_eq1[THEN sym, of "ν_(P j)"
"(Zl_mI K P I j) ⋅_r (mprod_exp K (K_gamma j) (Kb_n P)
 n^K n P I)" "Σ_e K (λx.(Zl_mI K P I ((τ_n) x)) ⋅_r
(mprod_exp K (K_gamma ((τ_n) x)) (Kb_n P) n^K n P I)) (n - Suc 0)"], assumption+)
apply (simp add:value_mI_genTr3)
apply (rule aGroup.nsum_mem[of "K" "n - Suc 0"], assumption+)
apply (rule allI, rule impI)
apply (frule_tac l = ja in transpos_mem[of "j" "n" "n"], simp+)
apply (simp add:value_mI_genTr3)

apply (subst val_t2p[of "ν
apply (simp add:Zl_mI_mem_K)
apply (simp add:value_mI_genTr1)

apply (simp add:mgenerator2Tr3_1[of "n" "j" "j" "P"])

apply (frule value_Zl_mI[of "n" "P" "I" "j"], assumption+)
apply (erule conjE)
apply (simp add:aadd_0_r)
apply (cut_tac f = "λx. (Zl_mI K P I ((τ_n) x)) ⋅_r
 j nK)" in
 value_ge_add[of java.lang.NullPointerException
 u0 _ " (m_zmax_pdsInP )" smto+)
apply ( ultimaell o ma (f ++ g) x = (map_inv f +++ map_inv g) x"
apply (frule_tac l = ja in transpos_mem[of "j" "n" "n"], simp+)
applydd
applyrule, impIapply ( add)

apply(l = ja[ofj"n"+)

applyapply simp
 metis)
apply simp)
apply (cut_tac k = ja in transpos_noteqTr[of "n" _ "j"], simp+)
apply (subst:list_induct2
apply (cut_tac l = "(\\ggtrrow>v he \<>ome exE)
 +)
apply (frule_tac y = "
aadd_le_mono[of "0" "nt (m_zmax_pdsI K n P I"
apply (simp (casesh "
apply (subgoal_tac "LI K (java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
apply (rulemerge =
 "ant (m_zmax_pdsI K n P I)"], assumption+)

apply (simp_ ]map_of_list .empty
apply
 list_of_map_emptyp list_of_map.empty ["
apply
 frule val_LI_pos[of "n" "P" "I" "j"], assumption+,mfrom Fa have "map_of_list < Map.empty"
 frule apos_neq_minf[of "LI K (νK (P j)
apply (substjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
apply (
apply (subgoal_tac "\<forall ‹
apply (frule m_zmax_gt_each[of n " λ
apply simp
apply (rule allI, rule impI)

(subst val -
apply (rule Zl_mI_mem_K, assumption+)
apply (simp add:value_mI_genTr1)

apply (simp add:mgenerator2Tr3_1[of "n" "" "j"""
"m_zmax_pdsI K n P I"])
apply (simp add:aadd_0_r)
apply (simp add:value_Zl_mI[of "n" "P" "I" "j"])

(in Corps) mI_gen_in_I:"[0 < n; distinct_pds K n P; ideal (O_{P n}) I;
I ≠ {0_{O_{P n})}}; I ≠ carrier (O_{P n})] ==>
(nsum K (λk. ((Zl_mI K P I k) ⋅_r
((mprod_exp K (λl. (γ_l)) (Kb_{n P}) n)^{m_zmax_pdsI K n P I)}))) n) ∈ I"
(cut_tac field_is_ring, frule ring_n_pd[of n P])
(rule ideal_eSum_closed[of n P I n], assumption+)
(rule allI, rule impI)
apply (frule_tac j = j in value_Zl_mI[of "n" "P" "I"], assumption+)
apply (erule conjE)
apply (thin_tac "(ν_{(P j)}) (Zl_mI K P I j) = LI K (ν_{(P j)}) I")
apply (subgoal_tac "(mprod_exp K (K_gamma j) (Kb_{n P}) n)^{m_zmax_pdsI K n P I)}
∈ carrier (O_{P n})")
apply (frule_tac x = "Zl_mI K P I j" and
r = "(mprod_exp K (K_gamma j) (Kb_{n P}) n)^{m_zmax_pdsI K n P I)}"
in Ring.ideal_ring_multiple1[of "(O_{P n})" "I"], assumption+)
apply (frule_tac h = "Zl_mI K P I j" in
Ring.ideal_subset[of "O_{P n}" "I"], assumption+)
apply (simp add:ring_n_pd_tOp_K_tOp[of "n" "P"])

(subst ring_n_pd_def) apply (simp add:Sr_def)
apply (simp add:value_mI_genTr1)

apply (rule allI, rule impI)
apply (case_tac "j = ja")
apply (simp add:mgenerator2Tr3_1)

apply (simp add:mgenerator2Tr3_2)
apply (simp add:m_zmax_pdsI_def) apply (simp add:m_zmax_pdsI_hom_def)
apply (simp only:ant_0[THEN sym])
apply (simp add:aless_zless)
apply (subgoal_tac "∀l ≤ n. (λj. tna (AMin ((ν_{(P j)}) ` I))) l ∈ Zset")
apply (frule m_zmax_gt_each[of n "λj. tna (AMin ((ν_{(P j)}) ` I))"])
apply (rotate_tac -1, drule_tac a = ja in forall_spec, simp+)
apply (frule_tac j = ja in val_LI_pos[of "n" "P" "I"], assumption+)
apply (cut_tac j = "tna (LI K (ν_{(P ja)}) I)" in ale_zle[of "0"])
(frule_tac j = ja in val_LI_noninf[of "n" "P" "I"], assumption+,
frule_tac j = ja in val_LI_pos[of "n" "P" "I"], assumption+,
frule_tac a = "LI K (ν_{(P ja)}) I" in apos_neq_minf, simp add:ant_tna,
simp add:ant_0) apply (unfold LI_def)
apply (frule_tac y = "tna (AMin (ν_{(P ja)} ` I))" and z = "m_zmax n (λj. tna (AMin (ν_{(P j)} ` I)))" in order_trans[of "0"], assumption+)
apply (rule_tac y = "m_zmax n (λj. tna (AMin (ν_{(P j)} ` I)))" and
z = "m_zmax n (λj. tna (AMin (ν_{(P j)} ` I))) + 1" in order_trans[of "0"],
assumption+) apply simp

apply (rule allI, rule impI) apply (simp add:Zset_def)

‹We write the element
‹eΣ K (λk. (Zl_mI K P I k) ⋅_K ((mprod_exp K (K_gamma k) (Kb_{n P})
n)_K⁽m_zmax_pdsI K n P I))) n›
as ‹mIg_{G a i n P I}››

mIg :: "[_, nat, nat ==> ('b ==> ant) set,
'b set] ==> 'b" (‹(4mIg_{_ _ _ _})› [82,82,82,83]82) where
"mIg_{n P I} = Σ_e K (λk. (Zl_mI K P I k) ⋅_r
((mprod_exp K (K_gamma k) (Kb_{n P}) n)^{m_zmax_pdsI K n P I)})) n"

‹We can rewrite above two lemmas by using ‹mIg_{G a i n P I}››

(in Corps) value_mI_gen1:"[0 < n; distinct_pds K n P; ideal (O_{P n}) I;
I ≠ {0_{O_{P n})}}; I ≠ carrier (O_{P n})] ==>
∀j ≤ n.(ν_{(P j)}) (mIg_{n P I}) = LI K (ν_{(P j)}) I"
(rule allI, rule impI)
apply (simp add:mIg_def value_mI_gen)

(in Corps) mI_gen_in_I1:"[0 < n; distinct_pds K n P; ideal (O_{P n}) I;
I ≠ {0_{O_{P n})}}; I ≠ carrier (O_{P n})] ==> (mIg_{n P I}) ∈ I"
(simp add:mIg_def mI_gen_in_I)

(in Corps) mI_principalTr:"[0 < n; distinct_pds K n P; ideal (O_{P n}) I;
I ≠ {0_{O_{P n})}}; I ≠ carrier (O_{P n}); x ∈ I] ==>
∀j ≤ n. ((ν_{(P j)}) (mIg_{n P I})) ≤ ((ν_{(P j)}) x)"
(simp add:value_mI_gen1)
apply (rule allI, rule impI)
apply (rule Zleast_LI, assumption+)

(in Corps) mI_principal:"[0 < n; distinct_pds K n P; ideal (O_{P n}) I;
I ≠ {0_{O_{P n})}}; I ≠ carrier (O_{P n})] ==>
I = Rxa (O_{P n}) (mIg_{n P I})"
(frule ring_n_pd[of "n" "P"])
(rule equalityI)
apply (rule subsetI)
apply (frule_tac x = x in mI_principalTr[of "n" "P" "I"],
assumption+)
apply (frule_tac y = x in n_eq_val_eq_idealTr[of "n" "P" "mIg_{n P I}"])
apply (frule mI_gen_in_I1[of "n" "P" "I"], assumption+)
apply (simp add:Ring.ideal_subset)+
apply (thin_tac "∀j≤n. (ν_{(P j)}) (mIg_{K n P I}) ≤ (ν_{(P j)}) x")
apply (frule_tac h = x in Ring.ideal_subset[of "O_{P n}" "I"], assumption+)
apply (frule_tac a = x in Ring.a_in_principal[of "O_{P n}"], assumption+)
apply (simp add:subsetD)
apply (rule Ring.ideal_cont_Rxa[of "O_{P n}" "I" "mIg_{K n P I}"], assumption+)
apply (rule mI_gen_in_I1[of "n" "P" "I"], assumption+)

‹‹prime_n_pd››

(in Corps) prime_n_pd_principal:"[distinct_pds K n P; j ≤ n] ==>
(P_{P n} j) = Rxa (O_{P n}) (((Kb_{n P}) j))"
(frule ring_n_pd[of "n" "P"])
(frule prime_n_pd_prime[of "n" "P" "j"], assumption+)
(simp add:prime_ideal_def, frule conjunct1)
apply (fold prime_ideal_def)
apply (thin_tac "prime_ideal (O_{P n}) (P_{P n} j)")
(rule equalityI)
apply (rule subsetI)
apply (frule_tac y = x in n_eq_val_eq_idealTr[of n P "(Kb_{n P}) j"])
apply (thin_tac "Ring (O_{P n})", thin_tac "ideal (O_{P n}) (P_{P n} j)")
apply (simp add:ring_n_pd_def Sr_def)
apply (frule Kbase_hom[of "n" "P"], simp)
apply (rule allI, rule impI)
apply (frule Kbase_Kronecker[of "n" "P"])
apply (simp add:Kronecker_delta_def, rule impI)
apply (simp only:ant_0[THEN sym], simp only:ant_1[THEN sym])
apply (simp del:ant_1)
apply (simp add:prime_n_pd_def)

apply (rule allI, rule impI)
apply (frule Kbase_Kronecker[of "n" "P"])
apply simp
apply (thin_tac "∀j≤n. ∀l≤n. (ν_{(P j)}) ((Kb_{n P}) l) = δ_l")
apply (case_tac "ja = j", simp add:Kronecker_delta_def)
apply (thin_tac "ideal (O_{P n}) (P_{P n} j)")
apply (simp add:prime_n_pd_def, erule conjE)
apply (frule_tac x = x in mem_ring_n_pd_mem_K[of "n" "P"],
assumption+)
apply (case_tac "x = 0")
apply (frule distinct_pds_valuation2[of "j" "n" "P"], assumption+)
apply (rule gt_a0_ge_1, assumption)+

apply (simp add:Kronecker_delta_def)
apply (frule_tac j = ja in distinct_pds_valuation2[of _ "n" "P"],
assumption+)
apply (simp add:prime_n_pd_def, erule conjE)
apply (thin_tac "ideal (O_{P n}) {x. x ∈ carrier (O_{P n}) ∧ 0 < (ν_{(P j)}) x}")
apply (simp add:ring_n_pd_def Sr_def)
apply (cut_tac h = x in Ring.ideal_subset[of "O_{P n}" "P_{P n} j"])
apply (frule_tac a = x in Ring.a_in_principal[of "O_{P n}"])
apply (simp add:Ring.ideal_subset, assumption+)

(rule_tac c = x and A = "(O_{P n}) ♢_p x" and B = "(O_{P n}) ♢_p (Kb_{n P}) j"
in subsetD, assumption+)
(simp add:Ring.a_in_principal)
apply (rule Ring.ideal_cont_Rxa[of "O_{P n}" "P_{P n} j" "(Kb_{n P}) j"], assumption+)
apply (subst prime_n_pd_def, simp)
apply (frule Kbase_Kronecker[of "n" "P"])
apply (simp add:Kronecker_delta_def)
apply (simp only:ant_1[THEN sym], simp only:ant_0[THEN sym])
apply (simp del:ant_1 add:aless_zless)
(subst ring_n_pd_def, simp add:Sr_def)
apply (frule Kbase_hom[of "n" "P"])
apply simp
apply (rule allI)
apply (simp add:ant_0)
apply (rule impI)
apply (simp only:ant_1[THEN sym], simp only:ant_0[THEN sym])
apply (simp del:ant_1)

(in Corps) ring_n_prod_primesTr:"[0 < n; distinct_pds K n P;
ideal (O_{P n}) I; I ≠ {0_{_{P n}}}; I ≠ carrier (O_{P n})] ==>
∀j ≤ n.(ν_{(P j)}) (mprod_exp K (mL K P I) (Kb_{n P}) n) =
(ν_{(P j)}) (mIg_{n P I})"
(rule allI, rule impI)
apply (simp add:mgenerator1)
apply (simp add:value_mI_gen1)

apply (simp add:value_Zl_mI)

(in Corps) ring_n_prod_primesTr1:"[0 < n; distinct_pds K n P;
ideal (O_{P n}) I; I ≠ {0_{_{P n}}}; I ≠ carrier (O_{P n})] ==>
I = (O_{P n}) ♢_p (mprod_exp K (mL K P I) (Kb_{n P}) n)"
(frule ring_n_pd[of "n" "P"])
(subst n_eq_val_eq_ideal[of "n" "P" "mprod_exp K (mL K P I)
(Kb_{n P}) n" "mIg_{n P I}"], assumption+)
(simp add:mgeneratorTr4)
(frule mI_gen_in_I1[of "n" "P" "I"], assumption+)
(simp add:Ring.ideal_subset)
(simp add:ring_n_prod_primesTr)
(simp add:mI_principal)

(in Corps) ring_n_prod_primes:"[0 < n; distinct_pds K n P;
ideal (O_{P n}) I; I ≠ {0_{_{P n}}}; I ≠ carrier (O_{P n});
∀k ≤ n. J k = (P_{P n} k)^{♢(O_{P n}) (nat ((mL K P I) k))}] ==>
I = iΠ_{O_{P n}),n} J"
(simp add:prime_n_pd_principal[of "n" "P"])
(subst ring_n_prod_primesTr1[of "n" "P" "I"], assumption+)
(frule ring_n_pd[of "n" "P"])
(frule Ring.prod_n_principal_ideal[of "O_{P n}" "nat o (mL K P I)" "n"
"Kb_{n P}" "J"])
apply (frule Kbase_hom[of "n" "P"])
apply (simp add:nat_def)
apply (subst ring_n_pd_def) apply (simp add:Sr_def)
apply (rule Pi_I, simp)
apply (simp add:Kbase_Kronecker[of "n" "P"])
apply (simp add:Kronecker_delta_def)
apply (simp only:ant_1[THEN sym], simp only:ant_0[THEN sym])
apply (simp del:ant_1)
apply (simp add:Kbase_hom) apply simp

apply simp
apply (frule ring_n_mprod_mprodR[of "n" "P" n "mL K P I" "Kb_{n P}"])
apply (rule allI, rule impI, simp add:Zset_def)
apply (rule allI, rule impI)
apply (simp add: Zleast_in_mI_pos)

apply (rule allI, rule impI)
apply (subst ring_n_pd_def) apply (simp add:Sr_def)
apply (frule Kbase_hom1[of "n" "P"], simp)
apply (simp add:zero_in_ring_n_pd_zero_K)
apply (frule Kbase_Kronecker[of "n" "P"])
apply (simp add:Kronecker_delta_def)
apply (simp only:ant_1[THEN sym], simp only:ant_0[THEN sym])
apply (simp del:ant_1)
simp

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