(** Algebra5
author Hidetsune Kobayashi
Group You Santo
Department of Mathematics
Nihon University
h_coba@math.cst.nihon-u.ac.jp
May 3, 2004.
April 6, 2007 (revised).
chapter 4. Ring theory (continued)
section 6. operation of ideals
section 7. direct product1, general case
section 8. Chinese remainder theorem
section 9. addition of finite elements of a ring and ideal_multiplication
section 10. extension and contraction
section 11. complete system of representatives
section 12. Polynomial ring
section 13. addition and multiplication of polyn_exprs
section 14. the degree of a polynomial
**)
theory Algebra5 imports Algebra4 begin
section "6. Operation of ideals"
lemma (in Ring) ideal_sumTr1:"[|ideal R A; ideal R B|] ==>
A \<minusplus> B = \<Inter> {J. ideal R J ∧ (A ∪ B) ⊆ J}"
apply (frule sum_ideals[of "A" "B"], assumption,
frule sum_ideals_la1[of "A" "B"], assumption,
frule sum_ideals_la2[of "A" "B"], assumption)
apply (rule equalityI)
(* A \<minusplus>R B ⊆ \<Inter>{J. ideal R J ∧ A ∪ B ⊆ J} *)
apply (rule_tac A = "{J. ideal R J ∧ (A ∪ B) ⊆ J}" and C = "A \<minusplus> B" in
Inter_greatest)
apply (simp, (erule conjE)+)
apply (rule_tac A = A and B = B and I = X in sum_ideals_cont,
simp add:ideal_subset1, simp add:ideal_subset1, assumption+)
apply (rule_tac B = "A \<minusplus> B" and A = "{J. ideal R J ∧ (A ∪ B) ⊆ J}" in
Inter_lower)
apply simp
done
lemma (in Ring) sum_ideals_commute:"[|ideal R A; ideal R B|] ==>
A \<minusplus> B = B \<minusplus> A"
apply (frule ideal_sumTr1 [of "B"], assumption+,
frule ideal_sumTr1 [of "A" "B"], assumption+)
apply (simp add:Un_commute[of "B" "A"])
done
lemma (in Ring) ideal_prod_mono1:"[|ideal R A; ideal R B; ideal R C;
A ⊆ B |] ==> A ♦r C ⊆ B ♦r C"
apply (frule ideal_prod_ideal[of "B" "C"], assumption+)
apply (rule ideal_prod_subTr[of "A" "C" "B ♦r C"], assumption+)
apply (rule ballI)+
apply (frule_tac c = i in subsetD[of "A" "B"], assumption+)
apply (rule_tac i = i and j = j in prod_mem_prod_ideals[of "B" "C"],
assumption+)
done
lemma (in Ring) ideal_prod_mono2:"[|ideal R A; ideal R B; ideal R C;
A ⊆ B |] ==> C ♦r A ⊆ C ♦r B"
apply (frule ideal_prod_mono1[of "A" "B" "C"], assumption+)
apply (simp add:ideal_prod_commute)
done
lemma (in Ring) cont_ideal_prod:"[|ideal R A; ideal R B; ideal R C;
A ⊆ C; B ⊆ C |] ==> A ♦r B ⊆ C"
apply (simp add:ideal_prod_def)
apply (rule subsetI, simp)
apply (frule ideal_prod_ideal[of "A" "B"], assumption,
frule_tac a = "A ♦r B" in forall_spec,
thin_tac "∀xa. ideal R xa ∧ {x. ∃i∈A. ∃j∈B. x = i ·r j} ⊆ xa --> x ∈ xa",
simp)
apply (rule subsetI, simp, (erule bexE)+, simp add:prod_mem_prod_ideals)
apply (frule ideal_prod_la1[of "A" "B"], assumption,
frule_tac c = x in subsetD[of "A ♦r B" "A"], assumption+,
simp add:subsetD)
done
lemma (in Ring) ideal_distrib:"[|ideal R A; ideal R B; ideal R C|] ==>
A ♦r (B \<minusplus> C) = A ♦r B \<minusplus> A ♦r C"
apply (frule sum_ideals[of "B" "C"], assumption,
frule ideal_prod_ideal[of "A" "B \<minusplus> C"], assumption+,
frule ideal_prod_ideal[of "A" "B"], assumption+,
frule ideal_prod_ideal[of "A" "C"], assumption+,
frule sum_ideals[of "A ♦r B" "A ♦r C"], assumption)
apply (rule equalityI)
apply (rule ideal_prod_subTr[of "A" "B \<minusplus> C" "A ♦r B \<minusplus> A ♦r C"], assumption+)
apply (rule ballI)+
apply (frule_tac x = j in ideals_set_sum[of B C], assumption+,
(erule bexE)+, simp) apply (
thin_tac "j = h ± k",
frule_tac h = i in ideal_subset[of "A"], assumption+,
frule_tac h = h in ideal_subset[of "B"], assumption+,
frule_tac h = k in ideal_subset[of "C"], assumption+)
apply (simp add:ring_distrib1)
apply (frule_tac i = i and j = h in prod_mem_prod_ideals[of "A" "B"],
assumption+,
frule_tac i = i and j = k in prod_mem_prod_ideals[of "A" "C"],
assumption+)
apply (frule sum_ideals_la1[of "A ♦r B" "A ♦r C"], assumption+,
frule sum_ideals_la2[of "A ♦r B" "A ♦r C"], assumption+)
apply (frule_tac c = "i ·r h" in subsetD[of "A ♦r B" "A ♦r B \<minusplus> A ♦r C"],
assumption+,
frule_tac c = "i ·r k" in subsetD[of "A ♦r C" "A ♦r B \<minusplus> A ♦r C"],
assumption+)
apply (simp add:ideal_pOp_closed)
apply (rule sum_ideals_cont[of "A ♦r (B \<minusplus> C)" "A ♦r B" "A ♦r C" ],
assumption+)
apply (rule ideal_prod_subTr[of "A" "B" "A ♦r (B \<minusplus> C)"], assumption+)
apply (rule ballI)+
apply (frule sum_ideals_la1[of "B" "C"], assumption+,
frule_tac c = j in subsetD[of "B" "B \<minusplus> C"], assumption+,
rule_tac i = i and j = j in prod_mem_prod_ideals[of "A" "B \<minusplus> C"],
assumption+)
apply (rule ideal_prod_subTr[of "A" "C" "A ♦r (B \<minusplus> C)"], assumption+)
apply (rule ballI)+
apply (frule sum_ideals_la2[of "B" "C"], assumption+,
frule_tac c = j in subsetD[of "C" "B \<minusplus> C"], assumption+,
rule_tac i = i and j = j in prod_mem_prod_ideals[of "A" "B \<minusplus> C"],
assumption+)
done
constdefs (structure R)
coprime_ideals::"[_, 'a set, 'a set] => bool"
"coprime_ideals R A B == A \<minusplus> B = carrier R"
lemma (in Ring) coprimeTr:"[|ideal R A; ideal R B|] ==>
coprime_ideals R A B = (∃a ∈ A. ∃b ∈ B. a ± b = 1r)"
apply (rule iffI)
apply (simp add:coprime_ideals_def)
apply (cut_tac ring_one, frule sym, thin_tac "A \<minusplus> B = carrier R", simp,
thin_tac "carrier R = A \<minusplus> B", frule ideals_set_sum[of A B],
assumption+, (erule bexE)+, frule sym, blast)
apply (erule bexE)+
apply (frule ideal_subset1[of A], frule ideal_subset1[of B])
apply (frule_tac a = a and b = b in set_sum_mem[of _ A _ B], assumption+)
apply (simp add:coprime_ideals_def)
apply (frule sum_ideals[of "A" "B"], assumption+,
frule ideal_inc_one[of "A \<minusplus> B"], assumption)
apply (simp add:coprime_ideals_def)
done
lemma (in Ring) coprime_int_prod:"[|ideal R A; ideal R B; coprime_ideals R A B|]
==> A ∩ B = A ♦r B"
apply (frule ideal_prod_la1[of "A" "B"], assumption+,
frule ideal_prod_la2[of "A" "B"], assumption+)
apply (rule equalityI)
defer
(** A ♦R B ⊆ A ∩ B **)
apply simp
apply (simp add:coprime_ideals_def)
apply (frule int_ideal[of "A" "B"], assumption)
apply (frule idealprod_whole_r[of "A ∩ B"])
apply (frule sym, thin_tac "A \<minusplus> B = carrier R", simp)
apply (simp add:ideal_distrib)
apply (simp add:ideal_prod_commute[of "A ∩ B" "A"])
apply (cut_tac Int_lower1[of "A" "B"], cut_tac Int_lower2[of "A" "B"])
apply (frule ideal_prod_mono2[of "A ∩ B" "B" "A"], assumption+,
frule ideal_prod_mono1[of "A ∩ B" "A" "B"], assumption+)
apply (frule ideal_prod_ideal[of "A ∩ B" "B"], assumption+,
frule ideal_prod_ideal[of "A" "A ∩ B"], assumption+,
frule ideal_subset1[of "A ♦r (A ∩ B)"],
frule ideal_subset1[of "(A ∩ B) ♦r B"])
apply (frule ideal_prod_ideal[of A B], assumption+,
frule sum_ideals_cont[of "A ♦r B" "A ♦r (A ∩ B)" "(A ∩ B) ♦r B"],
assumption+)
apply simp
done
lemma (in Ring) coprime_elems:"[|ideal R A; ideal R B; coprime_ideals R A B|] ==>
∃a∈A. ∃b∈B. a ± b = 1r"
by (simp add:coprimeTr)
lemma (in Ring) coprime_elemsTr:"[|ideal R A; ideal R B; a∈A; b∈B; a ± b = 1r|]
==> pj R A b = 1r(qring R A) ∧ pj R B a = 1r(qring R B)"
apply (frule pj_Hom [OF Ring, of "A"],
frule pj_Hom [OF Ring, of "B"])
apply (frule qring_ring[of "A"], frule qring_ring[of "B"])
apply (cut_tac ring_is_ag,
frule Ring.ring_is_ag[of "R /r A"],
frule Ring.ring_is_ag[of "R /r B"])
apply (frule_tac a = a and b = b in aHom_add[of "R" "R /r A" "pj R A"],
assumption+,
simp add:rHom_def, simp add:ideal_subset,
simp add:ideal_subset, simp)
apply (frule ideal_subset[of "A" "a"], assumption,
frule ideal_subset[of "B" "b"], assumption+)
apply (simp only:pj_zero[OF Ring, THEN sym, of "A" "a"],
frule rHom_mem[of "pj R A" "R" "qring R A" "b"], assumption+,
simp add:aGroup.l_zero[of "R /r A"])
apply (simp add:rHom_one[OF Ring, of "qring R A"])
apply (frule_tac a = a and b = b in aHom_add[of "R" "R /r B" "pj R B"],
assumption+,
simp add:rHom_def, simp add:ideal_subset,
simp add:ideal_subset, simp)
apply (simp only:pj_zero[OF Ring, THEN sym, of "B" "b"],
frule rHom_mem[of "pj R B" "R" "qring R B" "a"], assumption+,
simp add:aGroup.ag_r_zero[of "R /r B"])
apply (simp add:rHom_one[OF Ring, of "qring R B"])
done
lemma (in Ring) partition_of_unity:"[|ideal R A; a ∈ A; b ∈ carrier R;
a ± b = 1r; u ∈ carrier R; v ∈ carrier R|] ==>
pj R A (a ·r v ± b ·r u ) = pj R A u"
apply (frule pj_Hom [OF Ring, of "A"],
frule ideal_subset [of "A" "a"], assumption+,
frule ring_tOp_closed[of "a" "v"], assumption+,
frule ring_tOp_closed[of "b" "u"], assumption+,
frule qring_ring[of "A"])
apply (simp add:ringhom1[OF Ring, of "qring R A" "a ·r v" "b ·r u" "pj R A"])
apply (frule ideal_ring_multiple1[of "A" "a" "v"], assumption+,
simp add:pj_zero[OF Ring, THEN sym, of "A" "a ·r v"],
frule rHom_mem[of "pj R A" "R" "qring R A" "b ·r u"], assumption+,
simp add:Ring.l_zero, simp add:rHom_tOp[OF Ring])
apply (frule ringhom1[OF Ring, of "qring R A" "a" "b" "pj R A"], assumption+,
simp,
simp add:pj_zero[OF Ring, THEN sym, of "A" "a"],
frule rHom_mem[of "pj R A" "R" "qring R A" "b"], assumption+,
simp add:Ring.l_zero)
apply (simp add:rHom_one[OF Ring, of "qring R A" "pj R A"],
rotate_tac -2, frule sym, thin_tac "1rR /r A = pj R A b",
simp,
rule Ring.ring_l_one[of "qring R A" "pj R A u"], assumption)
apply (simp add:rHom_mem)
done
lemma (in Ring) coprimes_commute:"[|ideal R A; ideal R B; coprime_ideals R A B |]
==> coprime_ideals R B A"
apply (simp add:coprime_ideals_def)
apply (simp add:sum_ideals_commute)
done
lemma (in Ring) coprime_surjTr:"[|ideal R A; ideal R B; coprime_ideals R A B;
X ∈ carrier (qring R A); Y ∈ carrier (qring R B) |] ==>
∃r∈carrier R. pj R A r = X ∧ pj R B r = Y"
apply (frule qring_ring [of "A"],
frule qring_ring [of "B"],
frule coprime_elems [of "A" "B"], assumption+,
frule pj_Hom [OF Ring, of "A"],
frule pj_Hom [OF Ring, of "B"])
apply (erule bexE)+
apply (simp add:qring_carrier[of "A"],
simp add:qring_carrier[of "B"], (erule bexE)+,
rename_tac a b u v)
apply (rotate_tac -1, frule sym, thin_tac "v \<uplus>R B = Y",
rotate_tac -3, frule sym, thin_tac "u \<uplus>R A = X", simp)
apply (frule_tac h = a in ideal_subset[of "A"], assumption+,
frule_tac h = b in ideal_subset[of "B"], assumption+,
frule_tac x = a and y = v in ring_tOp_closed, assumption+,
frule_tac x = b and y = u in ring_tOp_closed, assumption+,
cut_tac ring_is_ag,
frule_tac x = "a ·r v" and y = "b ·r u" in aGroup.ag_pOp_closed[of "R"],
assumption+)
apply (frule_tac a = a and b = b and u = u and v = v in
partition_of_unity[of "A"], assumption+,
simp add:pj_mem[OF Ring, of "A"])
apply (frule_tac a = b and b = a and u = v and v = u in
partition_of_unity[of "B"], assumption+,
subst aGroup.ag_pOp_commute[of "R"], assumption+,
simp add:pj_mem[OF Ring, of "B"])
apply (frule_tac x = "b ·r u" and y = "a ·r v" in
aGroup.ag_pOp_commute[of "R"], assumption+, simp)
apply (simp add:pj_mem[OF Ring], blast)
done
lemma (in Ring) coprime_n_idealsTr0:"[|ideal R A; ideal R B; ideal R C;
coprime_ideals R A C; coprime_ideals R B C |] ==>
coprime_ideals R (A ♦r B) C"
apply (simp add:coprimeTr[of "A" "C"],
simp add:coprimeTr[of "B" "C"], (erule bexE)+)
apply (cut_tac ring_is_ag,
frule_tac h = a in ideal_subset[of "A"], assumption+,
frule_tac h = aa in ideal_subset[of "B"], assumption+,
frule_tac h = b in ideal_subset[of "C"], assumption+,
frule_tac h = ba in ideal_subset[of "C"], assumption+,
frule_tac x = a and y = b in aGroup.ag_pOp_closed, assumption+)
apply (frule_tac x = "a ± b" and y = aa and z = ba in ring_distrib1,
assumption+) apply (
rotate_tac 6, frule sym, thin_tac "a ± b = 1r",
frule sym, thin_tac "aa ± ba = 1r")
apply (simp only:ring_distrib2)
apply (rotate_tac -1, frule sym, thin_tac "1r = a ± b", simp,
thin_tac "aa ± ba = 1r")
apply (simp add:ring_r_one,
thin_tac "a ·r aa ± b ·r aa ± (a ·r ba ± b ·r ba) ∈ carrier R",
thin_tac "a ± b = a ·r aa ± b ·r aa ± (a ·r ba ± b ·r ba)")
apply (frule_tac x = a and y = aa in ring_tOp_closed, assumption+,
frule_tac x = b and y = aa in ring_tOp_closed, assumption+,
frule_tac x = a and y = ba in ring_tOp_closed, assumption+,
frule_tac x = b and y = ba in ring_tOp_closed, assumption+,
frule_tac x = "a ·r ba" and y = "b ·r ba" in aGroup.ag_pOp_closed,
assumption+)
apply (simp add:aGroup.ag_pOp_assoc,
frule sym, thin_tac "1r = a ·r aa ± (b ·r aa ± (a ·r ba ± b ·r ba))")
apply (frule_tac i = a and j = aa in prod_mem_prod_ideals[of "A" "B"],
assumption+)
apply (frule_tac x = ba and r = a in ideal_ring_multiple[of "C"],
assumption+,
frule_tac x = ba and r = b in ideal_ring_multiple[of "C"],
assumption+,
frule_tac x = b and r = aa in ideal_ring_multiple1[of "C"],
assumption+)
apply (frule_tac I = C and x = "a ·r ba" and y = "b ·r ba" in
ideal_pOp_closed, assumption+,
frule_tac I = C and x = "b ·r aa" and y = "a ·r ba ± b ·r ba" in
ideal_pOp_closed, assumption+)
apply (frule ideal_prod_ideal[of "A" "B"], assumption)
apply (subst coprimeTr, assumption+, blast)
done
lemma (in Ring) coprime_n_idealsTr1:"ideal R C ==>
(∀k ≤ n. ideal R (J k)) ∧ (∀i ≤ n. coprime_ideals R (J i) C) -->
coprime_ideals R (iΠR,n J) C"
apply (induct_tac n)
apply (rule impI)
apply (erule conjE)
apply simp
apply (rule impI)
apply (erule conjE)
apply (cut_tac n = "Suc n" in n_in_Nsetn)
apply (cut_tac n = n in Nset_Suc) apply simp
apply (cut_tac n = n and J = J in n_prod_ideal,
rule allI, simp)
apply (rule_tac A = "iΠR,n J" and B = "J (Suc n)" in
coprime_n_idealsTr0[of _ _ "C"], simp+)
done
lemma (in Ring) coprime_n_idealsTr2:"[|ideal R C; (∀k ≤ n. ideal R (J k));
(∀i ≤ n. coprime_ideals R (J i) C) |] ==>
coprime_ideals R (iΠR,n J) C"
by (simp add:coprime_n_idealsTr1)
lemma (in Ring) coprime_n_idealsTr3:"(∀k ≤ (Suc n). ideal R (J k)) ∧
(∀i ≤ (Suc n). ∀j ≤ (Suc n). i ≠ j -->
coprime_ideals R (J i) (J j)) --> coprime_ideals R (iΠR,n J) (J (Suc n))"
apply (rule impI, erule conjE)
apply (case_tac "n = 0", simp)
apply (simp add:Nset_1)
apply (cut_tac nat_eq_le[of "Suc n" "Suc n"],
frule_tac a = "Suc n" in forall_spec, assumption)
apply (rotate_tac 1, frule_tac a = "Suc n" in forall_spec, assumption,
thin_tac "∀j≤Suc n. Suc n ≠ j --> coprime_ideals R (J (Suc n)) (J j)")
apply (rule_tac C = "J (Suc n)" and n = n and J = J in coprime_n_idealsTr2,
assumption, rule allI, simp)
apply (rule allI, rule impI)
apply (frule_tac a = i in forall_spec, simp,
thin_tac "∀i≤Suc n. ∀j≤Suc n. i ≠ j --> coprime_ideals R (J i) (J j)",
rotate_tac -1,
frule_tac a = "Suc n" in forall_spec, assumption+)
apply simp+
done
lemma (in Ring) coprime_n_idealsTr4:"[|(∀k ≤ (Suc n). ideal R (J k)) ∧
(∀i ≤ (Suc n). ∀j ≤ (Suc n). i ≠ j -->
coprime_ideals R (J i) (J j))|] ==> coprime_ideals R (iΠR,n J) (J (Suc n))"
apply (simp add:coprime_n_idealsTr3)
done
section "7. direct product1, general case"
constdefs
prod_tOp::"['i set, 'i => ('a, 'm) Ring_scheme] =>
('i => 'a) => ('i => 'a) => ('i => 'a)"
"prod_tOp I A == λf∈carr_prodag I A. λg∈carr_prodag I A.
λx∈I. (f x) ·r(A x) (g x)"
(** Let x ∈ I, then A x is a ring, {A x | x ∈ I} is a set of rings. **)
prod_one::"['i set, 'i => ('a, 'm) Ring_scheme] => ('i => 'a)"
"prod_one I A == λx∈I. 1r(A x)"
(** 1(A x) is the unit of a ring A x. **)
constdefs
prodrg :: "['i set, 'i => ('a, 'more) Ring_scheme] => ('i => 'a) Ring"
"prodrg I A == (|carrier = carr_prodag I A, pop = prod_pOp I A, mop =
prod_mOp I A, zero = prod_zero I A, tp = prod_tOp I A,
un = prod_one I A |)),"
(** I is the index set **)
syntax
"@PRODRING" :: "['i set, 'i => ('a, 'more) Ring_scheme] =>
('i => 'a ) Ring" ("(rΠ_/ _)" [72,73]72)
translations
"rΠI A" == "prodrg I A"
constdefs
augm_func::"[nat, nat => 'a,'a set, nat, nat => 'a, 'a set] => nat => 'a"
"augm_func n f A m g B == λi∈{j. j ≤ (n + m)}. if i ≤ n then f i else
if (Suc n) ≤ i ∧ i ≤ n + m then g ((sliden (Suc n)) i) else arbitrary"
(* Remark. g is a function of Nset (m - 1) -> B *)
ag_setfunc::"[nat, nat => ('a, 'more) Ring_scheme, nat,
nat => ('a, 'more) Ring_scheme] => (nat => 'a) set => (nat => 'a) set
=> (nat => 'a) set"
"ag_setfunc n B1 m B2 X Y
== {f. ∃g. ∃h. (g∈X) ∧(h∈Y) ∧(f = (augm_func n g (Un_carrier {j. j ≤ n} B1)
m h (Un_carrier {j. j ≤ (m - 1)} B2)))}"
(* Later we consider X = ac_Prod_Rg (Nset n) B1 and Y = ac_Prod_Rg (Nset (m - 1)) B2 *)
consts
ac_fProd_Rg::"[nat, nat => ('a, 'more) Ring_scheme] =>
(nat => 'a) set"
primrec
fprod_0: "ac_fProd_Rg 0 B = carr_prodag {0::nat} B"
frpod_n: "ac_fProd_Rg (Suc n) B = ag_setfunc n B (Suc 0) (compose {0::nat}
B (slide (Suc n))) (carr_prodag {j. j ≤ n} B) (carr_prodag {0} (compose {0} B (slide (Suc n))))"
constdefs
prodB1 :: "[('a, 'm) Ring_scheme, ('a, 'm) Ring_scheme] =>
(nat => ('a, 'm) Ring_scheme)"
"prodB1 R S == λk. if k=0 then R else if k=Suc 0 then S else
arbitrary"
Prod2Rg :: "[('a, 'm) Ring_scheme, ('a, 'm) Ring_scheme]
=> (nat => 'a) Ring" (infixl "\<Oplus>r" 80)
"A1 \<Oplus>r A2 == prodrg {0, Suc 0} (prodB1 A1 A2)"
text {* Don't try (Prod_ring (Nset n) B) \<Oplus>r (B (Suc n)) *}
lemma carr_prodrg_mem_eq:"[|f ∈ carrier (rΠI A); g ∈ carrier (rΠI A);
∀i∈I. f i = g i |] ==> f = g"
by (simp add:prodrg_def carr_prodag_def, (erule conjE)+,
rule funcset_eq[of _ I], assumption+)
lemma prod_tOp_mem:"[|∀k∈I. Ring (A k); X ∈ carr_prodag I A;
Y ∈ carr_prodag I A|] ==> prod_tOp I A X Y ∈ carr_prodag I A"
apply (subst carr_prodag_def, simp)
apply (rule conjI)
apply (simp add:prod_tOp_def restrict_def extensional_def)
apply (rule conjI)
apply (rule univar_func_test, rule ballI)
apply (simp add:Un_carrier_def prod_tOp_def)
apply (simp add:carr_prodag_def, (erule conjE)+)
apply (frule_tac b = x in forball_spec1, assumption,
thin_tac "∀k∈I. Ring (A k)",
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀i∈I. X i ∈ carrier (A i)",
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀i∈I. Y i ∈ carrier (A i)")
apply (frule_tac R = "A x" and x = "X x" and y = "Y x" in
Ring.ring_tOp_closed, assumption+, blast)
apply (rule ballI)
apply (simp add:prod_tOp_def)
apply (simp add:carr_prodag_def, (erule conjE)+)
apply (simp add:Ring.ring_tOp_closed)
done
lemma prod_tOp_func:"∀k∈I. Ring (A k) ==>
prod_tOp I A ∈ carr_prodag I A -> carr_prodag I A -> carr_prodag I A"
apply (rule bivar_func_test)
apply (rule ballI)+
apply (simp add:prod_tOp_mem)
done
lemma prod_one_func:"∀k∈I. Ring (A k) ==>
prod_one I A ∈ carr_prodag I A"
apply (simp add:prod_one_def carr_prodag_def)
apply (rule conjI)
apply (rule univar_func_test) apply (rule ballI)
apply (simp add:Un_carrier_def)
apply (frule_tac b = x in forball_spec1, assumption,
thin_tac "∀k∈I. Ring (A k)",
frule_tac R = "A x" in Ring.ring_one)
apply blast
apply (rule ballI)
apply (rule Ring.ring_one, simp)
done
lemma prodrg_carrier:"∀k∈I. Ring (A k) ==>
carrier (prodrg I A) = carrier (prodag I A)"
by (simp add:prodrg_def prodag_def)
lemma prodrg_ring:"∀k∈I. Ring (A k) ==> Ring (prodrg I A)"
apply (rule Ring.intro)
apply (simp add:prodrg_def)
apply (rule prod_pOp_func,
rule ballI, simp add:Ring.ring_is_ag)
apply (simp add:prodrg_def, rule prod_pOp_assoc,
simp add:Ring.ring_is_ag, assumption+)
apply (simp add:prodrg_def, rule prod_pOp_commute,
simp add:Ring.ring_is_ag, assumption+)
apply (simp add:prodrg_def, rule prod_mOp_func,
simp add:Ring.ring_is_ag)
apply (simp add:prodrg_def)
apply (cut_tac X = a in prod_mOp_mem[of "I" "A"])
apply (simp add:Ring.ring_is_ag, assumption)
apply (cut_tac X = "prod_mOp I A a" and Y = a in prod_pOp_mem[of "I" "A"],
simp add:Ring.ring_is_ag, assumption+,
cut_tac prod_zero_func[of "I" "A"])
apply (rule carr_prodag_mem_eq[of "I" "A"],
simp add:Ring.ring_is_ag, assumption+,
rule ballI, simp add:prod_pOp_def,
subst prod_mOp_mem_i, simp add:Ring.ring_is_ag, assumption+,
subst prod_zero_i, simp add:Ring.ring_is_ag, assumption+,
rule aGroup.l_m, simp add:Ring.ring_is_ag,
simp add:prodag_comp_i, simp add:Ring.ring_is_ag)
apply (simp add:prodrg_def,
rule prod_zero_func, simp add:Ring.ring_is_ag)
apply (simp add:prodrg_def,
cut_tac prod_zero_func[of "I" "A"],
cut_tac X = "prod_zero I A" and Y = a in prod_pOp_mem[of "I" "A"],
simp add:Ring.ring_is_ag, assumption+,
rule carr_prodag_mem_eq[of "I" "A"],
simp add:Ring.ring_is_ag, assumption+)
apply (rule ballI)
apply (simp add:prod_pOp_def prod_zero_def)
apply (rule aGroup.ag_l_zero, simp add:Ring.ring_is_ag)
apply (simp add:prodag_comp_i, simp add:Ring.ring_is_ag)
apply (simp add:prodrg_def,
rule prod_tOp_func, simp add:Ring.ring_is_ag)
apply (simp add:prodrg_def)
apply (frule_tac X = a and Y = b in prod_tOp_mem[of "I" "A"], assumption+,
frule_tac X = "prod_tOp I A a b" and Y = c in
prod_tOp_mem[of "I" "A"], assumption+,
frule_tac X = b and Y = c in prod_tOp_mem[of "I" "A"], assumption+,
frule_tac X = a and Y = "prod_tOp I A b c" in
prod_tOp_mem[of "I" "A"], assumption+)
apply (rule carr_prodag_mem_eq[of "I" "A"],
simp add:Ring.ring_is_ag, assumption+, rule ballI)
apply (simp add:prod_tOp_def)
apply (rule Ring.ring_tOp_assoc, simp, (simp add:prodag_comp_i)+)
apply (simp add:prodrg_def)
apply (frule_tac X = a and Y = b in prod_tOp_mem[of "I" "A"], assumption+,
frule_tac X = b and Y = a in prod_tOp_mem[of "I" "A"], assumption+,
rule carr_prodag_mem_eq[of "I" "A"],
simp add:Ring.ring_is_ag, assumption+)
apply (rule ballI, simp add:prod_tOp_def)
apply (rule Ring.ring_tOp_commute, (simp add:prodag_comp_i)+)
apply (simp add:prodrg_def, rule prod_one_func, assumption)
apply (simp add:prodrg_def)
apply (cut_tac X = b and Y = c in prod_pOp_mem[of "I" "A"],
simp add:Ring.ring_is_ag, assumption+,
cut_tac X = a and Y = b in prod_tOp_mem[of "I" "A"], assumption+,
cut_tac X = a and Y = c in prod_tOp_mem[of "I" "A"], assumption+,
cut_tac X = "prod_tOp I A a b" and Y = "prod_tOp I A a c" in
prod_pOp_mem[of "I" "A"], simp add:Ring.ring_is_ag, assumption+)
apply (rule carr_prodag_mem_eq[of "I" "A"],
simp add:Ring.ring_is_ag, rule prod_tOp_mem[of "I" "A"], assumption+)
apply (rule ballI, simp add:prod_tOp_def prod_pOp_def)
apply (rule Ring.ring_distrib1, (simp add:prodag_comp_i)+)
apply (simp add:prodrg_def,
cut_tac prod_one_func[of "I" "A"],
cut_tac X = "prod_one I A" and Y = a in prod_tOp_mem[of "I" "A"],
assumption+)
apply (rule carr_prodag_mem_eq[of "I" "A"],
simp add:Ring.ring_is_ag, assumption+,
rule ballI, simp add:prod_tOp_def prod_one_def,
rule Ring.ring_l_one, simp, simp add:prodag_comp_i)
apply simp
done
lemma prodrg_elem_extensional:"[|∀k∈I. Ring (A k); f ∈ carrier (prodrg I A)|]
==> f ∈ extensional I"
apply (simp add:prodrg_carrier)
apply (simp add:prodag_def carr_prodag_def)
done
lemma prodrg_pOp:"∀k∈I. Ring (A k) ==>
pop (prodrg I A) = prod_pOp I A"
apply (simp add:prodrg_def)
done
lemma prodrg_mOp:"∀k∈I. Ring (A k) ==>
mop (prodrg I A) = prod_mOp I A"
apply (simp add:prodrg_def)
done
lemma prodrg_zero:"∀k∈I. Ring (A k) ==>
zero (prodrg I A) = prod_zero I A"
apply (simp add:prodrg_def)
done
lemma prodrg_tOp:"∀k∈I. Ring (A k) ==>
tp (prodrg I A) = prod_tOp I A"
apply (simp add:prodrg_def)
done
lemma prodrg_one:"∀k∈I. Ring (A k) ==>
un (prodrg I A) = prod_one I A"
apply (simp add:prodrg_def)
done
lemma prodrg_sameTr5:"[|∀k∈I. Ring (A k); ∀k∈I. A k = B k|]
==> prod_tOp I A = prod_tOp I B"
apply (frule prod_tOp_func)
apply (subgoal_tac "carr_prodag I A = carr_prodag I B", simp)
apply (frule prod_tOp_func [of "I" "B"])
apply (rule funcset_eq [of _ "carr_prodag I B" _])
apply (simp add:prod_tOp_def extensional_def)
apply (simp add:prod_tOp_def extensional_def)
apply (rule ballI)
apply (frule_tac x = x in funcset_mem [of "prod_tOp I A" "carr_prodag I B"
"carr_prodag I B -> carr_prodag I B"], assumption+)
apply (frule_tac x = x in funcset_mem [of "prod_tOp I B" "carr_prodag I B"
"carr_prodag I B -> carr_prodag I B"], assumption+)
apply (thin_tac " prod_tOp I A
∈ carr_prodag I B -> carr_prodag I B -> carr_prodag I B")
apply (thin_tac "prod_tOp I B
∈ carr_prodag I B -> carr_prodag I B -> carr_prodag I B")
apply (rule funcset_eq [of _ "carr_prodag I B"])
apply (simp add:prod_tOp_def extensional_def)
apply (simp add:prod_tOp_def extensional_def)
apply (rule ballI)
apply (frule_tac f = "prod_tOp I A x" and A = "carr_prodag I B" and
x = xa in funcset_mem, assumption+)
apply (frule_tac f = "prod_tOp I B x" and A = "carr_prodag I B" and
x = xa in funcset_mem, assumption+)
apply (subgoal_tac "∀k∈I. aGroup (B k)")
apply (rule carr_prodag_mem_eq, assumption+)
apply (rule ballI, simp add:prod_tOp_def)
apply (rule ballI, rule Ring.ring_is_ag, simp)
apply (subgoal_tac "∀k∈I. aGroup (A k)")
apply (simp add:prodag_sameTr1)
apply (rule ballI, rule Ring.ring_is_ag, simp)
done
lemma prodrg_sameTr6:"[|∀k∈I. Ring (A k); ∀k∈I. A k = B k|]
==> prod_one I A = prod_one I B"
apply (frule prod_one_func [of "I" "A"])
apply (cut_tac prodag_sameTr1[of "I" "A" "B"])
apply (rule carr_prodag_mem_eq[of I A "prod_one I A" "prod_one I B"])
apply (simp add:Ring.ring_is_ag, assumption)
apply (cut_tac prod_one_func [of "I" "B"], simp)
apply simp
apply (rule ballI, simp add:prod_one_def)
apply (simp add:Ring.ring_is_ag, simp)
done
lemma prodrg_same:"[|∀k∈I. Ring (A k); ∀k∈I. A k = B k|]
==> prodrg I A = prodrg I B"
apply (subgoal_tac "∀k∈I. aGroup (A k)")
apply (frule prodag_sameTr1, assumption+)
apply (frule prodag_sameTr2, assumption+)
apply (frule prodag_sameTr3, assumption+)
apply (frule prodag_sameTr4, assumption+)
apply (frule prodrg_sameTr5, assumption+)
apply (frule prodrg_sameTr6, assumption+)
apply (simp add:prodrg_def)
apply (rule ballI, rule Ring.ring_is_ag, simp)
done
lemma prodrg_component:"[|f ∈ carrier (prodrg I A); i ∈ I|] ==>
f i ∈ carrier (A i)"
by (simp add:prodrg_def carr_prodag_def)
lemma project_rhom:"[|∀k∈I. Ring (A k); j ∈ I|] ==>
PRoject I A j ∈ rHom ( prodrg I A) (A j)"
apply (simp add:rHom_def)
apply (rule conjI)
apply (simp add:aHom_def)
apply (rule conjI)
apply (rule univar_func_test, rule ballI)
apply (simp add:prodrg_carrier)
apply (cut_tac prodag_carrier[of I A], simp)
apply (simp add:PRoject_def)
apply (cut_tac prodag_carrier[of I A], simp,
thin_tac "carrier (aΠI A) = carr_prodag I A")
apply (simp add:prodag_comp_i)
apply (rule ballI, simp add:Ring.ring_is_ag)
apply (rule ballI, simp add:Ring.ring_is_ag)
apply (subgoal_tac "∀k∈I. aGroup (A k)")
apply (frule project_aHom [of "I" "A" "j"], assumption+)
apply (rule conjI)
apply (simp add:prodrg_carrier aHom_def)
apply (simp add:prodrg_carrier)
apply (simp add:prodrg_pOp)
apply (simp add:prodag_pOp[THEN sym])
apply (simp add:aHom_def)
apply (rule ballI, simp add:Ring.ring_is_ag)
apply (rule conjI)
apply (rule ballI)+
apply (simp add:prodrg_carrier)
apply (cut_tac prodag_carrier[of I A], simp)
apply (frule_tac X = x and Y = y in prod_tOp_mem[of I A], assumption+)
apply (simp add:prodrg_tOp)
apply (simp add:PRoject_def)
apply (simp add:prod_tOp_def)
apply (rule ballI)
apply (simp add:Ring.ring_is_ag)
apply (frule prodrg_ring [of "I" "A"])
apply (frule Ring.ring_one[of "rΠI A"])
apply (simp add:prodrg_carrier)
apply (cut_tac prodag_carrier[of I A], simp)
apply (simp add:PRoject_def) apply (simp add:prodrg_def)
apply (fold prodrg_def) apply (simp add:prod_one_def)
apply (rule ballI)
apply (simp add:Ring.ring_is_ag)
done
lemma augm_funcTr:"[|∀k ≤(Suc n). Ring (B k);
f ∈ carr_prodag {i. i ≤ (Suc n)} B|] ==>
f = augm_func n (restrict f {i. i ≤ n}) (Un_carrier {i. i ≤ n} B) (Suc 0)
(λx∈{0::nat}. f (x + Suc n))
(Un_carrier {0} (compose {0} B (slide (Suc n))))"
apply (rule carr_prodag_mem_eq [of "{i. i ≤ (Suc n)}" "B" "f"
"augm_func n (restrict f {i. i ≤ n}) (Un_carrier {i. i ≤ n} B) (Suc 0)
(λx∈{0}. f (x + Suc n)) (Un_carrier {0} (compose {0} B (slide (Suc n))))"])
apply (rule ballI, simp add:Ring.ring_is_ag)
apply (simp add:augm_func_def)
apply (simp add:carr_prodag_def)
apply (rule conjI)
apply (simp add:augm_func_def)
apply (rule conjI)
apply (rule univar_func_test)
apply (rule ballI)
apply (simp add:augm_func_def sliden_def)
apply (erule conjE)+
apply (frule_tac x = x in funcset_mem[of f "{i. i ≤ Suc n}"
"Un_carrier {i. i ≤ Suc n} B"]) apply simp
apply simp
apply (rule impI)
apply (rule_tac x = "Suc n" in funcset_mem[of f "{i. i ≤ Suc n}"
"Un_carrier {i. i ≤ Suc n} B"], assumption) apply simp
apply (rule allI, (erule conjE)+)
apply (simp add:augm_func_def)
apply (case_tac "i ≤ n", simp add:sliden_def)
apply (simp add:sliden_def, rule impI)
apply (simp add:nat_not_le_less,
frule_tac m = n and n = i in Suc_leI,
frule_tac m = i and n = "Suc n" in Nat.le_anti_sym, assumption+,
simp)
apply (rule ballI, simp)
apply (simp add:augm_func_def sliden_def)
apply (rule impI, simp add:nat_not_le_less)
apply (frule_tac n = l in Suc_leI[of n _])
apply (frule_tac m = l and n = "Suc n" in Nat.le_anti_sym, assumption+,
simp)
done
lemma A_to_prodag_mem:"[|Ring A; ∀k∈I. Ring (B k); ∀k∈I. (S k) ∈
rHom A (B k); x ∈ carrier A |] ==> A_to_prodag A I S B x ∈ carr_prodag I B"
apply (simp add:carr_prodag_def)
apply (rule conjI)
apply (simp add:A_to_prodag_def extensional_def)
apply (rule conjI)
apply (rule univar_func_test) apply (rule ballI)
apply (simp add: A_to_prodag_def)
apply (subgoal_tac "(S xa) ∈ rHom A (B xa)") prefer 2 apply simp
apply (thin_tac "∀k∈I. S k ∈ rHom A (B k)")
apply (frule_tac f = "S xa" and A = A and R = "B xa" in rHom_mem, assumption+)
apply (simp add:Un_carrier_def) apply blast
apply (rule ballI)
apply (simp add:A_to_prodag_def)
apply (rule_tac f = "S i" and A = A and R = "B i" and a = x in rHom_mem)
apply simp
apply assumption
done
lemma A_to_prodag_rHom:"[|Ring A; ∀k∈I. Ring (B k); ∀k∈I. (S k) ∈
rHom A (B k) |] ==> A_to_prodag A I S B ∈ rHom A (rΠI B)"
apply (simp add:rHom_def [of "A" "rΠI B"])
apply (rule conjI)
apply (cut_tac A_to_prodag_aHom[of A I B S])
apply (subst aHom_def, simp)
apply (simp add:prodrg_carrier)
apply (simp add:aHom_def)
apply (simp add:prodrg_def)
apply (cut_tac prodag_pOp[of I B], simp)
apply (rule ballI, simp add:Ring.ring_is_ag,
simp add:Ring.ring_is_ag,
rule ballI, simp add:Ring.ring_is_ag)
apply (rule ballI)
apply (simp add:rHom_def)
apply (rule conjI)
apply (rule ballI)+
apply (frule_tac x = x and y = y in Ring.ring_tOp_closed[of A], assumption+)
apply (frule_tac x = "x ·rA y" in A_to_prodag_mem[of A I B S], assumption+,
frule_tac x = x in A_to_prodag_mem[of A I B S], assumption+,
frule_tac x = y in A_to_prodag_mem[of A I B S], assumption+)
apply (simp add:prodrg_tOp[of I B])
apply (frule_tac X = "A_to_prodag A I S B x " and Y = "A_to_prodag A I S B y" in prod_tOp_mem[of I B], assumption+)
apply (cut_tac X = "A_to_prodag A I S B (x ·rA y)" and Y = "prod_tOp I B (A_to_prodag A I S B x) (A_to_prodag A I S B y)" in carr_prodag_mem_eq[of I B])
apply (rule ballI, simp add:Ring.ring_is_ag) apply assumption+
apply (rule ballI, simp add:prod_tOp_def A_to_prodag_def)
apply (frule_tac b = l in forball_spec1, assumption,
thin_tac "∀k∈I. Ring (B k)",
frule_tac b = l in forball_spec1, assumption,
thin_tac "∀k∈I. S k ∈ rHom A (B k)")
apply (simp add:rHom_tOp) apply simp
apply (simp add:prodrg_one[of I B])
apply (frule prod_one_func[of I B])
apply (frule Ring.ring_one[of A],
frule_tac x = "1rA" in A_to_prodag_mem[of A I B S], assumption+)
apply (cut_tac X = "A_to_prodag A I S B 1rA" and Y = "prod_one I B" in
carr_prodag_mem_eq[of I B])
apply (rule ballI, simp add:Ring.ring_is_ag)
apply assumption+
apply (rule ballI)
apply (subst A_to_prodag_def, simp add:prod_one_def)
apply (frule_tac b = l in forball_spec1, assumption,
thin_tac "∀k∈I. Ring (B k)",
frule_tac b = l in forball_spec1, assumption,
thin_tac "∀k∈I. S k ∈ rHom A (B k)")
apply (simp add:rHom_one)
apply assumption
done
lemma ac_fProd_ProdTr1:"∀k ≤ (Suc n). Ring (B k) ==>
ag_setfunc n B (Suc 0) (compose {0::nat} B (slide (Suc n)))
(carr_prodag {i. i ≤ n} B) (carr_prodag {0}
(compose {0} B (slide (Suc n)))) ⊆ carr_prodag {i. i ≤ (Suc n)} B"
apply (rule subsetI)
apply (simp add:ag_setfunc_def)
apply (erule exE, erule conjE, erule exE, erule conjE)
apply (simp,
thin_tac "x =
augm_func n g (Un_carrier {j. j ≤ n} B) (Suc 0) h
(Un_carrier {0} (compose {0} B (slide (Suc n))))")
apply (simp add:carr_prodag_def [of "{j. j ≤ (Suc n)}" "B"])
apply (rule conjI)
apply (simp add:augm_func_def)
apply (rule conjI)
apply (simp add:Pi_def) apply (rule allI) apply (rule impI)
apply (simp add:augm_func_def)
apply (case_tac "x ≤ n")
apply simp apply (simp add:carr_prodag_def)
apply (erule conjE)+ apply (frule_tac x = x in mem_of_Nset [of _ "n"])
apply (frule_tac f = g and x = x in funcset_mem[of _ "{j. j ≤ n}"
"Un_carrier {j. j ≤ n} B"], assumption+)
apply (thin_tac "g ∈ {j. j ≤ n} -> Un_carrier {j. j ≤ n} B",
thin_tac "h ∈ {0} -> Un_carrier {0} (compose {0} B (slide (Suc n)))")
apply (simp add:Un_carrier_def,
erule exE, erule conjE, erule exE, simp, erule conjE,
frule_tac x = i and y = n and z = "Suc n" in le_less_trans,
simp,
frule_tac x = i and y = "Suc n" in less_imp_le, blast)
apply (simp add:sliden_def)
apply (simp add:carr_prodag_def Un_carrier_def, (erule conjE)+)
apply (simp add:compose_def slide_def)
apply (cut_tac n_in_Nsetn[of "Suc n"], blast)
apply (rule allI, rule impI)
apply (simp add:augm_func_def)
apply (case_tac "i ≤ n", simp)
apply (simp add:carr_prodag_def [of "{i. i ≤ n}" _])
apply simp apply (thin_tac "g ∈ carr_prodag {i. i ≤ n} B")
apply (simp add: not_less [symmetric, of _ n],
frule_tac n = i in Suc_leI[of n],
frule_tac m = i and n = "Suc n" in le_anti_sym, assumption+, simp)
apply (simp add:carr_prodag_def compose_def slide_def sliden_def)
done
lemma ac_fProd_Prod:"∀k ≤ n. Ring (B k) ==>
ac_fProd_Rg n B = carr_prodag {j. j ≤ n} B"
apply (case_tac "n = 0")
apply simp
apply (subgoal_tac "∃m. n = Suc m")
apply (subgoal_tac "∀m. n = Suc m -->
ac_fProd_Rg n B = carr_prodag {j. j ≤ n} B")
apply blast apply (thin_tac "∃m. n = Suc m")
apply (rule allI, rule impI) apply (simp, thin_tac "n = Suc m")
apply (rule equalityI)
apply (simp add:ac_fProd_ProdTr1)
apply (rule subsetI)
apply (rename_tac m f)
apply (frule augm_funcTr, assumption+)
apply (simp add:ag_setfunc_def)
apply (subgoal_tac "(restrict f {j. j ≤ m}) ∈ carr_prodag {j. j ≤ m} B")
apply (subgoal_tac "(λx∈{0::nat}. f (Suc (x + m))) ∈ carr_prodag {0}
(compose {0} B (slide (Suc m)))")
apply blast
apply (thin_tac "f =
augm_func m (restrict f {i. i ≤ m}) (Un_carrier {i. i ≤ m} B)
(Suc 0) (λx∈{0}. f (Suc (x + m)))
(Un_carrier {0} (compose {0} B (slide (Suc m))))")
apply (simp add:carr_prodag_def)
apply (rule conjI)
apply (simp add:Pi_def restrict_def)
apply (simp add:Un_carrier_def compose_def slide_def)
apply (simp add:compose_def slide_def)
apply (thin_tac "f =
augm_func m (restrict f {i. i ≤ m}) (Un_carrier {i. i ≤ m} B)
(Suc 0) (λx∈{0}. f (Suc (x + m)))
(Un_carrier {0} (compose {0} B (slide (Suc m))))")
apply (simp add:carr_prodag_def)
apply (simp add:Un_carrier_def)
apply (simp add:Pi_def)
apply (rule allI) apply (rule impI)
apply (erule conjE)+
apply (rotate_tac 1)
apply (frule_tac a = x in forall_spec, simp)
apply (erule exE,
thin_tac "∀x≤Suc m. ∃xa. (∃i≤Suc m. xa = carrier (B i)) ∧ f x ∈ xa")
apply (frule_tac a = x in forall_spec, simp)
apply blast
apply (cut_tac t = n in Suc_pred[THEN sym], simp)
apply blast
done
text{* A direct product of a finite number of rings defined with
ac_fProd_Rg is equal to that defined by using carr_prodag. *}
constdefs
fprodrg::"[nat, nat => ('a, 'more) Ring_scheme] =>
(|carrier:: (nat => 'a) set, pop::[(nat => 'a), (nat => 'a)]
=> (nat => 'a), mop:: (nat => 'a) => (nat => 'a), zero::(nat => 'a),
tp :: [(nat => 'a), (nat => 'a)] => (nat => 'a), un :: (nat => 'a) |)),"
"fprodrg n B == (| carrier = ac_fProd_Rg n B,
pop = λf. λg. prod_pOp {i. i ≤ n} B f g, mop = λf. prod_mOp {i. i ≤ n} B f,
zero = prod_zero {i. i ≤ n} B, tp = λf. λg. prod_tOp {i. i ≤ n} B f g,
un = prod_one {i. i ≤ n} B |)),"
fPRoject::"[nat, nat => ('a, 'more) Ring_scheme, nat]
=> (nat => 'a) => 'a"
"fPRoject n B x == λf∈ac_fProd_Rg n B. f x"
syntax
"@FPRODRING" :: "[nat, nat => ('a, 'more) Ring_scheme] =>
(nat => 'a) set" ("(frΠ_/ _)" [72,73]72)
translations
"frΠn B" == "fprodrg n B"
lemma fprodrg_ring:"∀k ≤ n. Ring (B k) ==> Ring (fprodrg n B)"
apply (simp add:fprodrg_def)
apply (frule ac_fProd_Prod)
apply simp
apply (fold prodrg_def)
apply (simp add:prodrg_ring)
done
section "8. Chinese remainder theorem"
lemma Chinese_remTr1:"[|Ring A; ∀k ≤ (n::nat). ideal A (J k);
∀k ≤ n. B k = qring A (J k); ∀k ≤ n. S k = pj A (J k) |] ==>
kerA,(rΠ{j. j ≤ n} B) (A_to_prodag A {j. j ≤ n} S B) =
\<Inter> {I. ∃k∈{j. j ≤ n}. I = (J k)}"
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:ker_def)
apply (rule allI, rule impI)
apply (rename_tac a K, erule conjE)
apply (simp add:prodrg_def, simp add:A_to_prodag_def prod_one_def)
apply (erule exE, erule conjE)
apply (subgoal_tac "(λk∈{j. j ≤ n}. S k a) k = (λx∈{j. j ≤ n}. \<zero>B x) k")
apply (thin_tac "(λk∈{j. j ≤ n}. S k a) = prod_zero {j. j ≤ n} B")
apply simp apply (frule_tac I = "J k" in Ring.qring_zero [of "A"])
apply simp
apply (frule_tac I = "J k" and x = a in pj_mem [of "A"]) apply simp
apply assumption apply simp
apply (frule_tac I = "J k" and a = a in Ring.a_in_ar_coset [of "A"])
apply simp apply assumption apply simp
apply (simp add:prod_zero_def)
apply (rule subsetI)
apply (simp add:CollectI ker_def)
apply (cut_tac Nset_inc_0[of n])
apply (frule_tac a = "J 0" in forall_spec, blast)
apply (frule_tac a = 0 in forall_spec1, simp)
apply (frule_tac h = x in Ring.ideal_subset [of "A" "J 0"], simp+)
apply (thin_tac "x ∈ J 0")
apply (simp add:A_to_prodag_def prodrg_def)
apply (simp add:prod_zero_def)
apply (rule funcset_eq [of _ "{j. j ≤ n}"])
apply (simp add:extensional_def restrict_def)+
apply (rule allI, rule impI)
apply (simp add:Ring.qring_zero)
apply (frule_tac a = xa in forall_spec, assumption,
thin_tac "∀k ≤ n. ideal A (J k)")
apply (subst pj_mem [of "A"], assumption+)
apply (frule_tac I = "J xa" and a = x in Ring.a_in_ar_coset [of "A"],
assumption+)
apply (rule_tac a = x and I = "J xa" in Ring.Qring_fix1 [of "A"], assumption+)
apply blast
done
lemma (in Ring) coprime_prod_int2Tr:
"((∀k ≤ (Suc n). ideal R (J k)) ∧
(∀i ≤ (Suc n). ∀j ≤ (Suc n). (i ≠j --> coprime_ideals R (J i) (J j))))
--> (\<Inter> {I. ∃k ≤ (Suc n). I = (J k)} = ideal_n_prod R (Suc n) J)"
apply (induct_tac n)
apply (rule impI)
apply (erule conjE)
apply (simp add:Nset_1)
apply (subst coprime_int_prod [THEN sym, of "J 0" "J (Suc 0)"], simp+)
apply (rule equalityI, rule subsetI)
apply (simp, blast)
apply (rule subsetI, simp, rule allI, rule impI, erule exE, (erule conjE)+)
apply simp
apply (simp add:Nset_1_1, erule disjE, (simp del:ideal_n_prodSn)+)
(** n **)
apply (rule impI)
apply (subgoal_tac "\<Inter>{I. ∃k ≤ (Suc (Suc n)). I = J k} =
(\<Inter>{I. ∃k ≤ (Suc n). I = J k}) ∩ (J (Suc (Suc n)))")
apply (subgoal_tac "\<Inter>{I. ∃k ≤ (Suc n). I = J k} = (iΠR,(Suc n) J)")
(* apply (simp del:ideal_n_prodSn)*)
apply (frule_tac n = "Suc n" and J = J in coprime_n_idealsTr4)
apply (simp del:ideal_n_prodSn)
apply (subst coprime_int_prod)
apply (rule n_prod_ideal)
apply (rule allI, simp, simp, assumption)
apply simp apply (cut_tac n = "Suc n" in Nsetn_sub_mem1)
apply simp
apply (thin_tac "(∀k≤Suc n. ideal R (J k)) ∧
(∀i≤Suc n. ∀j≤Suc n. i ≠ j --> coprime_ideals R (J i) (J j)) -->
\<Inter>{I. ∃k≤Suc n. I = J k} = iΠR,Suc n J",
thin_tac "(∀k≤Suc (Suc n). ideal R (J k)) ∧
(∀i≤Suc (Suc n).
∀j≤Suc (Suc n). i ≠ j --> coprime_ideals R (J i) (J j))")
apply (rule equalityI, rule subsetI, simp)
apply (rule conjI,
rule allI, rule impI, erule exE, erule conjE, simp,
frule_tac a = xa in forall_spec,
frule_tac x = k and y = "Suc n" and z = "Suc (Suc n)" in
le_less_trans, simp,
frule_tac x = k and y = "Suc (Suc n)" in less_imp_le, blast)
apply simp
apply (frule_tac a = "J (Suc (Suc n))" in forall_spec,
cut_tac n = "Suc (Suc n)" in le_refl, blast, simp)
apply (rule subsetI, simp, rule allI, rule impI)
apply (erule exE, erule conjE)
apply (erule conjE,
case_tac "k = Suc (Suc n)", simp)
apply (frule_tac m = k and n = "Suc (Suc n)" in noteq_le_less, assumption,
thin_tac "k ≤ Suc (Suc n)")
apply (frule_tac x = k and n = "Suc n" in Suc_less_le)
apply (frule_tac a = xa in forall_spec,
blast,
thin_tac "∀xa. (∃k≤Suc n. xa = J k) --> x ∈ xa",
simp)
done
lemma (in Ring) coprime_prod_int2:"[| ∀k ≤ (Suc n). ideal R (J k);
∀i ≤ (Suc n). ∀j ≤ (Suc n). (i ≠j --> coprime_ideals R (J i) (J j))|]
==> (\<Inter> {I. ∃k ≤ (Suc n). I = (J k)} = ideal_n_prod R (Suc n) J)"
apply (simp add:coprime_prod_int2Tr)
done
lemma (in Ring) coprime_2_n:"[|ideal R A; ideal R B|] ==>
(qring R A) \<Oplus>r (qring R B) = rΠ{j. j ≤ (Suc 0)} (prodB1 (qring R A) (qring R B))"
apply (simp add:Prod2Rg_def Nset_1)
done
text{* In this and following lemmata, ideals A and B are of type ('a, 'more) RingType_scheme. Don't try (rΠ(Nset n) B) \<Oplus>r B (Suc n) *}
lemma (in Ring) A_to_prodag2_hom:"[|ideal R A; ideal R B; S 0 = pj R A;
S (Suc 0) = pj R B|] ==>
A_to_prodag R {j. j ≤ (Suc 0)} S (prodB1 (qring R A) (qring R B)) ∈
rHom R (qring R A \<Oplus>r qring R B)"
apply (subst coprime_2_n [of "A" "B"], assumption+)
apply (rule A_to_prodag_rHom, rule Ring_axioms)
apply (rule ballI)
apply (case_tac "k = 0")
apply (simp add:prodB1_def)
apply (simp add:qring_ring)
apply (simp)
apply (frule_tac n = k in Suc_leI[of 0], thin_tac "0 < k")
apply (frule_tac m = k and n = "Suc 0" in le_anti_sym, assumption)
apply (simp, simp add:prodB1_def, simp add:qring_ring)
apply (rule ballI)
apply (simp add:Nset_1)
apply (erule disjE)
apply (simp add:prodB1_def, rule pj_Hom, rule Ring_axioms, assumption)
apply (simp, simp add:prodB1_def)
apply (rule pj_Hom, assumption+)
done
lemma (in Ring) A2coprime_rsurjecTr:"[|ideal R A; ideal R B; S 0 = pj R A;
S (Suc 0) = pj R B|] ==>
(carrier (qring R A \<Oplus>r qring R B)) =
carr_prodag {j. j ≤ (Suc 0)} (prodB1 (qring R A) (qring R B))"
apply (simp add:Prod2Rg_def prodrg_def Nset_1)
done
lemma (in Ring) A2coprime_rsurjec:"[|ideal R A; ideal R B; S 0 = pj R A;
S (Suc 0) = pj R B; coprime_ideals R A B|] ==>
surjecR,((qring R A) \<Oplus>r (qring R B))
(A_to_prodag R {j. j≤(Suc 0)} S (prodB1 (qring R A) (qring R B)))"
apply (frule A_to_prodag2_hom [of "A" "B" "S"], assumption+)
apply (simp add:surjec_def)
apply (rule conjI,
simp add:rHom_def)
apply (rule surj_to_test[of "A_to_prodag R {j. j ≤ (Suc 0)} S
(prodB1 (qring R A) (qring R B))" "carrier R"
"carrier (qring R A \<Oplus>r qring R B)"])
apply (simp add:rHom_def aHom_def)
apply (rule ballI)
apply (frule rHom_func[of "A_to_prodag R {j. j ≤ (Suc 0)} S
(prodB1 (R /r A) (R /r B))" R],
thin_tac "A_to_prodag R {j. j ≤ (Suc 0)} S (prodB1 (R /r A) (R /r B))
∈ rHom R (R /r A \<Oplus>r R /r B)")
apply (simp add:A2coprime_rsurjecTr[of A B S])
apply (simp add:Nset_1)
apply (frule_tac X = "b 0" and Y = "b (Suc 0)" in
coprime_surjTr[of A B], assumption+)
apply (simp add:carr_prodag_def prodB1_def,
simp add:carr_prodag_def prodB1_def)
apply (erule bexE)
apply (frule_tac x = r in funcset_mem[of "A_to_prodag R {0, Suc 0} S
(prodB1 (R /r A) (R /r B))"
"carrier R" "carr_prodag {0, Suc 0} (prodB1 (R /r A) (R /r B))"],
assumption+)
apply (cut_tac X = "A_to_prodag R {0, Suc 0} S (prodB1 (R /r A) (R /r B)) r"
and Y = b in
carr_prodag_mem_eq[of "{0, Suc 0}" "prodB1 (R /r A) (R /r B)"])
apply (rule ballI)
apply (simp, erule disjE)
apply (simp add:prodB1_def, fold prodB1_def,
simp add:qring_ring Ring.ring_is_ag)
apply (simp add:prodB1_def, fold prodB1_def,
simp add:qring_ring Ring.ring_is_ag)
apply assumption+
apply (rule ballI, simp, erule disjE, simp)
apply (subst A_to_prodag_def, simp)
apply (subst A_to_prodag_def, simp)
apply blast
done
lemma (in Ring) prod2_n_Tr1:"[|∀k ≤ (Suc 0). ideal R (J k);
∀k ≤ (Suc 0). B k = qring R (J k);
∀k ≤ (Suc 0). S k = pj R (J k) |] ==>
A_to_prodag R {j. j ≤ (Suc 0)} S
(prodB1 (qring R (J 0)) (qring R (J (Suc 0)))) =
A_to_prodag R {j. j ≤ (Suc 0)} S B"
apply (subgoal_tac "∀k ≤ (Suc 0). (prodB1 (qring R (J 0)) (qring R (J (Suc 0)))) k = B k")
apply (simp add:A_to_prodag_def)
apply (rule allI, rule impI)
apply (case_tac "k = 0", simp add:Nset_1_1)
apply (simp add:prodB1_def)
apply (simp add:Nset_1_1)
apply (simp add:prodB1_def)
done
lemma (in aGroup) restrict_prod_Suc:"[|∀k ≤ (Suc (Suc n)). ideal R (J k);
∀k ≤ (Suc (Suc n)). B k = R /r J k;
∀k ≤ (Suc (Suc n)). S k = pj R (J k);
f ∈ carrier (rΠ{j. j ≤ (Suc (Suc n))} B)|] ==>
restrict f {j. j ≤ (Suc n)} ∈ carrier (rΠ{j. j ≤ (Suc n)} B)"
apply (cut_tac Nsetn_sub_mem1[of "Suc n"])
apply (simp add:prodrg_def)
apply (simp add:carr_prodag_def, (erule conjE)+)
apply (simp add:Un_carrier_def)
apply (rule univar_func_test, rule ballI)
apply simp
apply (frule_tac x = x in funcset_mem[of f "{j. j ≤ (Suc (Suc n))}"
"\<Union>{X. ∃i ≤ (Suc (Suc n)). X = carrier (R /r J i)}"],
simp)
apply simp
apply (erule exE, erule conjE, erule exE, erule conjE, simp)
apply (rotate_tac -5)
apply (frule_tac a = x in forall_spec) apply simp
apply blast
done
lemma (in Ring) Chinese_remTr2:"(∀k ≤ (Suc n). ideal R (J k)) ∧
(∀k≤(Suc n). B k = qring R (J k)) ∧
(∀k≤(Suc n). S k = pj R (J k)) ∧
(∀i≤(Suc n). ∀j≤ (Suc n). (i ≠j -->
coprime_ideals R (J i) (J j))) -->
surjecR,(rΠ{j. j≤ (Suc n)} B)
(A_to_prodag R {j. j≤(Suc n)} S B)"
apply (cut_tac Ring)
apply (induct_tac n)
(* case n = 0, i.e. two coprime_ideals *) (** use coprime_surjTr **)
apply (rule impI) apply (erule conjE)+
apply (frule A_to_prodag_rHom [of R "{j. j ≤ Suc 0}" "B" "S"])
apply (rule ballI, simp add:Ring.qring_ring)
apply (rule ballI, simp add:pj_Hom)
apply (frule rHom_func[of "A_to_prodag R {j. j ≤ (Suc 0)} S B" R
"rΠ{j. j ≤ (Suc 0)} B"])
apply (simp add:surjec_def)
apply (rule conjI)
apply (simp add:rHom_def)
apply (rule surj_to_test, assumption+)
apply (rule ballI) apply (simp add:Nset_1)
apply (cut_tac coprime_elems [of "J 0" "J (Suc 0)"])
apply (rename_tac f)
apply (erule bexE, erule bexE)
apply (simp add:prodrg_def) apply (fold prodrg_def)
apply (cut_tac X = "f 0" and Y = "f (Suc 0)" in
coprime_surjTr[of "J 0" "J (Suc 0)"], simp+)
apply (simp add:carr_prodag_def, simp add:carr_prodag_def)
apply (erule bexE, (erule conjE)+)
apply (frule_tac x = r in funcset_mem[of "A_to_prodag R {0, Suc 0} S B"
"carrier R" "carr_prodag {0, Suc 0} B"], assumption+)
apply (cut_tac X = "A_to_prodag R {0, Suc 0} S B r" and Y = f in
carr_prodag_mem_eq[of "{0, Suc 0}" B])
apply (rule ballI, simp, erule disjE, simp add:qring_ring
Ring.ring_is_ag,
simp add:Ring.qring_ring Ring.ring_is_ag)
apply assumption+
apply (rule ballI, simp, erule disjE, simp)
apply (simp add:A_to_prodag_def, simp add:A_to_prodag_def)
apply blast apply simp+
apply (rule impI, (erule conjE)+)
(**** n ****)
apply (cut_tac n = "Suc n" in Nsetn_sub_mem1)
apply (subgoal_tac "∀k∈{i. i ≤ Suc (Suc n)}. Ring (B k)")
apply (frule_tac I = "{i. i ≤ Suc (Suc n)}" in A_to_prodag_rHom [of "R" _ "B" "S"])
apply assumption
apply (rule ballI)
apply (simp add:pj_Hom)
apply simp
apply (subst surjec_def, rule conjI,
simp add:rHom_def)
apply (cut_tac n = "Suc n" in coprime_n_idealsTr4[of _ J])
apply blast
apply (frule_tac f = "A_to_prodag R {j. j ≤ (Suc (Suc n))} S B" and
A = R in rHom_func)
apply (rule_tac f = "A_to_prodag R {j. j ≤ (Suc (Suc n))} S B" and
A = "carrier R" and B = "carrier (rΠ{j. j ≤ (Suc (Suc n))} B)" in
surj_to_test, assumption+)
apply (rule ballI)
apply (cut_tac n = "Suc n" in n_prod_ideal[of _ J])
apply (rule allI, simp)
apply (frule_tac A = "iΠR,(Suc n) J" and B = "J (Suc (Suc n))" in
coprime_elems,
cut_tac n = "Suc (Suc n)" in n_in_Nsetn,
blast, assumption)
apply (erule bexE, erule bexE) apply (rename_tac n f a b)
apply (thin_tac " coprime_ideals R (iΠR,(Suc n) J) (J (Suc (Suc n)))")
apply (cut_tac n = "Suc n" and a = a and J = J in ele_n_prod,
rule allI, simp, assumption)
apply (cut_tac ring_is_ag)
apply (frule_tac n = n and f = f in aGroup.restrict_prod_Suc[of R _ R J B S],
assumption+)
apply (frule_tac S = "rΠ{j. j ≤ (Suc n)} B" and
f = "A_to_prodag R {j. j ≤ (Suc n)} S B" in surjec_surj_to[of R])
apply (frule_tac f = "A_to_prodag R {j. j ≤ (Suc n)} S B" and A = "carrier R"
and B = "carrier (rΠ{j. j ≤ (Suc n)} B)" and
b = "restrict f {j. j ≤ (Suc n)}" in surj_to_el2, assumption)
apply (erule bexE, rename_tac n f a b u)
apply (cut_tac n = "Suc (Suc n)" in n_in_Nsetn,
frule_tac f = f and I = "{j. j ≤ (Suc (Suc n))}" and A = B and
i = "Suc (Suc n)" in prodrg_component, assumption)
apply simp
apply (frule_tac J = "J (Suc (Suc n))" and X = "f (Suc (Suc n))" in
pj_surj_to[of R], simp, assumption)
apply (erule bexE, rename_tac n f a b u v)
apply (frule_tac a = "Suc (Suc n)" in forall_spec, simp,
frule_tac I = "J (Suc (Suc n))" and h = b in Ring.ideal_subset[of R],
assumption+,
cut_tac I = "iΠR,n J ♦rR J (Suc n)" and h = a in
Ring.ideal_subset[of R], assumption+)
apply (frule_tac x = b and y = u in Ring.ring_tOp_closed[of R], assumption+,
frule_tac x = a and y = v in Ring.ring_tOp_closed[of R], assumption+,
frule Ring.ring_is_ag[of R],
frule_tac x = "b ·rR u" and y = "a ·rR v" in aGroup.ag_pOp_closed[of R],
assumption+)
apply (frule_tac f = "A_to_prodag R {j. j ≤ (Suc (Suc n))} S B" and
A = "carrier R" and B = "carrier (rΠ{j. j ≤ (Suc (Suc n))} B)" and
x = "b ·rR u ±R a ·rR v" in funcset_mem, assumption+)
apply (frule_tac f = "A_to_prodag R {j. j ≤ (Suc (Suc n))} S B
(b ·rR u ±R a ·rR v)" and I = "{j. j ≤ (Suc (Suc n))}"
and g = f in carr_prodrg_mem_eq, simp)
apply (rule ballI)
apply (subst A_to_prodag_def, simp)
apply (frule_tac I = "J i" in pj_Hom[of R], simp)
apply (simp add: rHom_add)
apply (frule_tac a = i in forall_spec, assumption,
thin_tac "∀k ≤ (Suc (Suc n)). ideal R (J k)",
frule_tac I = "J i" in Ring.qring_ring[of R], assumption,
thin_tac "∀k ≤ (Suc (Suc n)). Ring (R /r J k)",
frule_tac R = "R /r (J i)" and x = b and y = u and f = "pj R (J i)" in
rHom_tOp[of R], assumption+, simp,
thin_tac "pj R (J i) (b ·rR u) = pj R (J i) b ·rR /r J i pj R (J i) u",
frule_tac R = "R /r (J i)" and x = a and y = v and f = "pj R (J i)" in
rHom_tOp[of R], simp add:Ring.qring_ring, assumption+)
apply (frule_tac f = "pj R (J i)" and R = "R /r J i" and a = v in
rHom_mem[of _ R], assumption+,
frule_tac f = "pj R (J i)" and R = "R /r J i" and a = u in
rHom_mem[of _ R], assumption+,
frule_tac f = "pj R (J i)" and R = "R /r J i" and a = b in
rHom_mem[of _ R], assumption+,
frule_tac f = "pj R (J i)" and R = "R /r J i" and a = a in
rHom_mem[of _ R], assumption+)
apply (frule_tac R = "R /r J i" in Ring.ring_is_ag)
apply (case_tac "i ≤ (Suc n)")
apply (frule_tac I1 = "J i" and x1 = a in pj_zero[THEN sym, of R ],
assumption+, simp,
thin_tac "pj R (J i) (a ·rR v) = \<zero>R /r J i ·rR /r J i pj R (J i) v")
apply (simp add:Ring.ring_times_0_x)
apply (frule_tac f = "pj R (J i)" and A = R and R = "R /r J i" and
x = a and y = b in rHom_add, assumption+, simp,
thin_tac "A_to_prodag R {j. j ≤ Suc (Suc n)} S B
(b ·r u) ±rΠ{i. i ≤ Suc (Suc n)} B
A_to_prodag R {j. j ≤ Suc (Suc n)} S B (a ·r v)
∈ carrier (rΠ{j. j ≤ Suc (Suc n)} B)")
apply (simp add:aGroup.ag_l_zero)
apply (rotate_tac -1, frule sym, thin_tac " pj R (J i) 1rR = pj R (J i) b",
simp add:rHom_one) apply (simp add:Ring.ring_l_one)
apply (simp add:aGroup.ag_r_zero)
apply (frule_tac f = "A_to_prodag R {j. j ≤ (Suc n)} S B u" and
g = "restrict f {j. j ≤ (Suc n)}" and x = i in eq_fun_eq_val,
thin_tac "A_to_prodag R {j. j≤(Suc n)} S B u = restrict f {j. j≤(Suc n)}")
apply (simp add:A_to_prodag_def)
apply simp
apply (frule_tac m = i and n = "Suc n" in nat_not_le,
frule_tac m = "Suc n" and n = i in Suc_leI,
frule_tac m = i and n = "Suc (Suc n)" in Nat.le_anti_sym, assumption+,
simp)
apply (frule_tac I1 = "J (Suc (Suc n))" and x1 = b in pj_zero[THEN sym, of
R ], assumption+, simp add:Ring.ring_times_0_x)
apply (frule_tac f = "pj R (J (Suc (Suc n)))" and A = R and
R = "R /r J (Suc (Suc n))" and
x = a and y = b in rHom_add, assumption+, simp)
apply (simp add:aGroup.ag_r_zero)
apply (rotate_tac -1, frule sym,
thin_tac "pj R (J (Suc (Suc n))) 1rR = pj R (J (Suc (Suc n))) a",
simp add:rHom_one,
simp add:Ring.ring_l_one)
apply (simp add:aGroup.ag_l_zero)
apply blast
apply (rule ballI, simp add:Ring.qring_ring)
done
lemma (in Ring) Chinese_remTr3:"[|∀k ≤ (Suc n). ideal R (J k);
∀k ≤ (Suc n). B k = qring R (J k); ∀k≤ (Suc n). S k = pj R (J k);
∀i ≤ (Suc n). ∀j ≤ (Suc n). (i ≠j --> coprime_ideals R (J i) (J j))|] ==>
surjecR,(rΠ{j. j ≤ (Suc n)} B)
(A_to_prodag R {j. j ≤ (Suc n)} S B)"
apply (simp add:Chinese_remTr2 [of "n" "J" "B" "S"])
done
lemma (in Ring) imset:"[|∀k≤ (Suc n). ideal R (J k)|]
==> {I. ∃k≤ (Suc n). I = J k} = {J k| k. k ∈ {j. j ≤ (Suc n)}}"
apply blast
done
theorem (in Ring) Chinese_remThm:"[|(∀k ≤ (Suc n). ideal R (J k));
∀k≤(Suc n). B k = qring R (J k); ∀k ≤ (Suc n). S k = pj R (J k);
∀i ≤ (Suc n). ∀j ≤ (Suc n). (i ≠j --> coprime_ideals R (J i) (J j))|]
==> bijec(qring R (\<Inter> {J k | k. k∈{j. j ≤ (Suc n)}})),(rΠ{j. j ≤ (Suc n)} B)
((A_to_prodag R {j. j ≤ (Suc n)} S B)°R,(prodrg {j. j ≤ (Suc n)} B))"
apply (frule Chinese_remTr3, assumption+)
apply (cut_tac I = "{j. j ≤ (Suc n)}" and A = B in prodrg_ring)
apply (rule ballI, simp add:qring_ring)
apply (cut_tac surjec_ind_bijec [of "R" "rΠ{j. j ≤ (Suc n)} B"
"A_to_prodag R {j. j ≤ (Suc n)} S B"])
apply (cut_tac Ring,
simp add:Chinese_remTr1 [of "R" "Suc n" "J" "B" "S"])
apply (simp add:imset, rule Ring_axioms, assumption+)
apply (rule A_to_prodag_rHom, rule Ring_axioms)
apply (rule ballI)
apply (simp add:qring_ring)
apply (rule ballI, simp, rule pj_Hom, assumption, simp)
apply assumption
done
lemma (in Ring) prod_prime:"[|ideal R A; ∀k≤(Suc n). prime_ideal R (P k);
∀l≤(Suc n). ¬ (A ⊆ P l);
∀k≤ (Suc n). ∀l≤ (Suc n). k = l ∨ ¬ (P k) ⊆ (P l)|] ==>
∀i ≤ (Suc n). (nprod R (ppa R P A i) n ∈ A ∧
(∀l ∈ {j. j≤(Suc n)} - {i}. nprod R (ppa R P A i) n ∈ P l) ∧
(nprod R (ppa R P A i) n ∉ P i))"
apply (rule allI, rule impI)
apply (rule conjI)
apply (frule_tac i = i in prod_primeTr1[of n P A], assumption+)
apply (frule_tac n = n and f = "ppa R P A i" in ideal_nprod_inc[of A])
apply (rule allI, rule impI)
apply (rotate_tac -2,
frule_tac a = ia in forall_spec, assumption,
thin_tac "∀l ≤ n.
ppa R P A i l ∈ A ∧
ppa R P A i l ∈ P (skip i l) ∧ ppa R P A i l ∉ P i",
simp add:ideal_subset)
apply (rotate_tac -1,
frule_tac a = n in forall_spec, simp,
thin_tac "∀l≤ n.
ppa R P A i l ∈ A ∧
ppa R P A i l ∈ P (skip i l) ∧ ppa R P A i l ∉ P i",
(erule conjE)+,
blast, assumption)
apply (frule_tac i = i in prod_primeTr1[of n P A], assumption+)
apply (rule conjI)
apply (rule ballI)
apply (frule_tac a = l in forall_spec, simp,
frule_tac I = "P l" in prime_ideal_ideal)
apply (frule_tac n = n and f = "ppa R P A i" and A = "P l" in ideal_nprod_inc)
apply (rule allI) apply (simp add:ppa_mem, simp)
apply (case_tac "l < i",
thin_tac "∀l≤ (Suc n). ¬ A ⊆ P l",
thin_tac "∀k≤ (Suc n). ∀l ≤ (Suc n). k = l ∨ ¬ P k ⊆ P l")
apply (erule conjE,
frule_tac x = l and y = i and z = "Suc n" in less_le_trans,
assumption,
frule_tac x = l and n = n in Suc_less_le)
apply (rotate_tac 2,
frule_tac a = l in forall_spec, assumption,
thin_tac "∀l≤n. ppa R P A i l ∈ A ∧
ppa R P A i l ∈ P (skip i l) ∧ ppa R P A i l ∉ P i",
thin_tac "l < Suc n")
apply (simp only:skip_im_Tr1_2, blast)
apply (frule_tac m = l and n = i in nat_not_less,
thin_tac "¬ l < i",
cut_tac x = l and A = "{j. j ≤ (Suc n)}" and a = i in in_diff1)
apply simp
apply (erule conjE,
frule not_sym, thin_tac "l ≠ i",
frule_tac x = i and y = l in le_imp_less_or_eq,
erule disjE, thin_tac "i ≤ l",
frule_tac x = i and n = l in less_le_diff)
apply (cut_tac i = i and n = n and x = "l - Suc 0" in skip_im_Tr2_1,
simp, assumption+, simp,
frule_tac x = l and n = n in le_Suc_diff_le)
apply (rotate_tac -7,
frule_tac a = "l - Suc 0" in forall_spec, assumption,
thin_tac "∀l≤n. ppa R P A i l ∈ A ∧
ppa R P A i l ∈ P (skip i l) ∧ ppa R P A i l ∉ P i",
simp, (erule conjE)+)
apply blast
apply simp
apply assumption
apply (frule_tac a = i in forall_spec, assumption,
thin_tac "∀k≤ (Suc n). prime_ideal R (P k)")
apply (rule_tac P = "P i" and n = n and f = "ppa R P A i" in
prime_nprod_exc, assumption+)
apply (rule allI, rule impI)
apply (rotate_tac -3,
frule_tac a = ia in forall_spec, assumption,
thin_tac "∀l ≤ n.
ppa R P A i l ∈ A ∧
ppa R P A i l ∈ P (skip i l) ∧ ppa R P A i l ∉ P i",
simp add:ideal_subset)
apply (rule allI, rule impI) apply (
rotate_tac 4,
frule_tac a = l in forall_spec, assumption,
thin_tac "∀l≤ n.
ppa R P A i l ∈ A ∧
ppa R P A i l ∈ P (skip i l) ∧ ppa R P A i l ∉ P i",
simp)
done
lemma skip_im1:"[|i ≤ (Suc n); P ∈ {j. j ≤ (Suc n)} -> Collect (prime_ideal R)|]
==>
compose {j. j ≤ n} P (skip i) ` {j. j ≤ n} = P ` ({j. j ≤ (Suc n)} - {i})"
apply (cut_tac skip_fun[of i n])
apply (subst setim_cmpfn[of _ _ _ _ "{X. prime_ideal R X}"], assumption+)
apply simp
apply (simp add:skip_fun_im)
done
lemma (in Ring) mutch_aux1:"[|ideal R A; i ≤ (Suc n);
P ∈ {j. j ≤ (Suc n)} -> Collect (prime_ideal R)|] ==>
compose {j. j ≤ n} P (skip i) ∈ {j. j ≤ n} -> Collect (prime_ideal R)"
apply (cut_tac skip_fun[of i n])
apply (simp add:composition[of "skip i" "{j. j ≤ n}" "{j. j ≤ (Suc n)}" P
"Collect (prime_ideal R)"])
done
lemma (in Ring) prime_ideal_cont1Tr:"ideal R A ==>
∀P. ((P ∈ {j. j ≤ (n::nat)} -> {X. prime_ideal R X}) ∧
(A ⊆ \<Union> P ` {j. j ≤ n})) --> (∃i≤ n. A ⊆ (P i))"
apply (induct_tac n)
apply (rule allI, rule impI)
apply (erule conjE)
apply simp
(** n **)
apply (rule allI, rule impI)
apply (erule conjE)+
apply (case_tac "∃i ≤ (Suc n). ∃j≤ (Suc n). (i ≠ j ∧ P i ⊆ P j)")
apply ((erule exE, erule conjE)+, erule conjE)
apply (frule_tac f = P and n = n and X = "{X. prime_ideal R X}" and
A = A and i = i and j = j in Un_less_Un, assumption+, simp+)
apply (frule mutch_aux1, assumption+)
apply (frule_tac a = "compose {j. j ≤ n} P (skip i)" in forall_spec,
simp, erule exE)
apply (cut_tac i = i and n = n and x = ia in skip_fun_im1,
simp+, erule conjE, simp add:compose_def,blast)
(** last_step induction assumption is of no use **)
apply (thin_tac "∀P. P ∈ {j. j ≤ n} -> {X. prime_ideal R X} ∧
A ⊆ \<Union>P ` {j. j ≤ n} -->
(∃i≤n. A ⊆ P i)",
rule contrapos_pp, simp+)
apply (cut_tac n = n and P = P in prod_prime [of A], assumption)
apply (rule allI, rule impI,
frule_tac f = P and A = "{j. j ≤ (Suc n)}" and B = "{X. prime_ideal R X}"
and x = k in funcset_mem, simp, simp, assumption+)
apply (frule_tac n = "Suc n" and
f = "λi∈{j. j ≤ (Suc n)}. (nprod R (ppa R P A i) n)" in
nsum_ideal_inc[of A], rule allI, rule impI, simp)
apply (subgoal_tac "(nsum R (λi∈{j. j ≤ (Suc n)}. nprod R (ppa R P A i) n)
(Suc n)) ∉ (\<Union>x∈{j. j ≤ (Suc n)}. P x)")
apply blast
apply (simp del:nsum_suc)
apply (rule allI, rule impI) apply (rename_tac n P l)
apply (frule_tac f = P and A = "{j. j ≤ (Suc n)}" and
B = "{X. prime_ideal R X}"
and x = l in funcset_mem, simp, simp del:nsum_suc,
frule_tac I = "P l" in prime_ideal_ideal)
apply (rule_tac A = "P l" and n = "Suc n" and
f = "λi∈{j. j ≤ (Suc n)}. (nprod R (ppa R P A i) n)" in
nsum_ideal_exc, simp+, rule allI, simp add:ideal_subset)
apply (rule contrapos_pp, simp+)
apply (rotate_tac -1,
frule_tac a = l in forall_spec, simp,
thin_tac "∀j≤Suc n.
(∃la∈{i. i ≤ Suc n} - {j}. eΠR,n ppa R P A la ∉ P l) ∨
eΠR,n ppa R P A j ∈ P l",
thin_tac "∀i≤Suc n. ∀j≤Suc n. i = j ∨ ¬ P i ⊆ P j",
thin_tac "∀i≤Suc n. ¬ A ⊆ P i")
apply (erule disjE, erule bexE)
apply (frule_tac a = la in forall_spec, simp,
thin_tac "∀i≤Suc n.
eΠR,n ppa R P A i ∈ A ∧
(∀l∈{j. j ≤ Suc n} - {i}. eΠR,n ppa R P A i ∈ P l) ∧
eΠR,n ppa R P A i ∉ P i",
(erule conjE)+)
apply blast
apply blast
done
lemma (in Ring) prime_ideal_cont1:"[|ideal R A; ∀i ≤ (n::nat).
prime_ideal R (P i); A ⊆ \<Union> {X. (∃i ≤ n. X = (P i))} |] ==>
∃i≤ n. A⊆(P i)"
apply (frule prime_ideal_cont1Tr[of A n])
apply (frule_tac a = P in forall_spec,
thin_tac "∀P. P ∈ {j. j ≤ n} -> {X. prime_ideal R X} ∧
A ⊆ \<Union>P ` {j. j ≤ n} --> (∃i≤n. A ⊆ P i)")
apply (rule conjI,
rule univar_func_test, rule ballI, simp,
rule subsetI, simp,
frule_tac c = x in subsetD[of A "\<Union>{X. ∃i≤n. X = P i}"], assumption+,
simp, blast)
apply assumption
done
lemma (in Ring) prod_n_ideal_contTr0:"(∀l≤ n. ideal R (J l)) -->
iΠR,n J ⊆ \<Inter>{X. (∃k≤n. X = (J k))}"
apply (induct_tac n)
apply simp
apply (rule impI)
apply (cut_tac n = n in Nsetn_sub_mem1,
simp)
apply (cut_tac n = n in n_prod_ideal[of _ J], simp)
apply (cut_tac I = "iΠR,n J" and J = "J (Suc n)" in
ideal_prod_sub_Int) apply assumption apply simp
apply (frule_tac A = "iΠR,n J ♦rR J (Suc n)" and
B = "iΠR,n J ∩ J (Suc n)" and
C = "\<Inter>{X. ∃k≤ n. X = J k} ∩ J (Suc n)" in subset_trans)
apply (rule_tac A = "iΠR,n J" and B = "\<Inter>{X. ∃k≤n. X = J k}" and
C = "J (Suc n)" in inter_mono, assumption)
apply (rule_tac A = "iΠR,n J ♦r J (Suc n)" and
B = "\<Inter>{X. ∃k≤ n. X = J k} ∩ J (Suc n)" and
C = "\<Inter>{X. ∃k≤ (Suc n). X = J k}" in subset_trans,
assumption)
apply (rule subsetI)
apply simp
apply (rule allI, rule impI)
apply (erule exE, (erule conjE)+)
apply (case_tac "k = Suc n", simp)
apply (frule_tac m = k and n = "Suc n" in noteq_le_less, assumption)
apply (thin_tac " k ≤ Suc n")
apply (frule_tac x = k and n = "Suc n" in less_le_diff,
thin_tac "k < Suc n", simp, thin_tac "∀l≤Suc n. ideal R (J l)")
apply (frule_tac a = xa in forall_spec, blast,
thin_tac "∀xa. (∃k≤n. xa = J k) --> x ∈ xa",
simp)
done
lemma (in Ring) prod_n_ideal_contTr:"[|∀l≤ n. ideal R (J l)|] ==>
iΠR,n J ⊆ \<Inter>{X. (∃k ≤ n. X = (J k))}"
apply (simp add:prod_n_ideal_contTr0)
done
lemma (in Ring) prod_n_ideal_cont2:"[|∀l≤ (n::nat). ideal R (J l);
prime_ideal R P; \<Inter>{X. (∃k≤ n. X = (J k))} ⊆ P|] ==>
∃l≤ n. (J l) ⊆ P"
apply (frule prod_n_ideal_contTr[of n J])
apply (frule_tac A = "iΠR,n J" and B = "\<Inter>{X. ∃k≤ n. X = J k}" and C = P
in subset_trans, assumption+)
apply (thin_tac "\<Inter>{X. ∃k≤ n. X = J k} ⊆ P",
thin_tac "iΠR,n J ⊆ \<Inter>{X. ∃k≤ n. X = J k}")
apply (simp add:ideal_n_prod_prime)
done
lemma (in Ring) prod_n_ideal_cont3:"[|∀l≤ (n::nat). ideal R (J l);
prime_ideal R P; \<Inter>{X. (∃k≤ n. X = (J k))} = P|] ==>
∃l≤ n. (J l) = P"
apply (frule prod_n_ideal_cont2[of n J P], assumption+)
apply simp
apply (erule exE)
apply (subgoal_tac "J l = P")
apply blast
apply (rule equalityI, simp)
apply (rule subsetI)
apply (rotate_tac -4, frule sym, thin_tac "\<Inter>{X. ∃k≤ n. X = J k} = P")
apply simp
apply blast
done
constdefs (structure R)
ideal_quotient::"[_ , 'a set, 'a set] => 'a set"
"ideal_quotient R A B == {x| x. x ∈ carrier R ∧ (∀b∈B. x ·r b ∈ A)}"
syntax
"@IDEALQT"::"['a set, _ , 'a set] => 'a set"
("(3_/ \<dagger>_/ _)" [82,82,83]82)
translations
"A \<dagger>R B" =="ideal_quotient R A B"
lemma (in Ring) ideal_quotient_is_ideal:
"[|ideal R A; ideal R B|] ==> ideal R (ideal_quotient R A B)"
apply (rule ideal_condition)
apply (rule subsetI)
apply (simp add:ideal_quotient_def CollectI)
apply (simp add:ideal_quotient_def)
apply (cut_tac ring_zero)
apply (subgoal_tac "∀b∈B. \<zero> ·r b ∈ A")
apply blast
apply (rule ballI)
apply (frule_tac h = b in ideal_subset[of B], assumption)
apply (frule_tac x = b in ring_times_0_x )
apply (simp add:ideal_zero)
apply (rule ballI)+
apply (simp add:ideal_quotient_def, (erule conjE)+,
rule conjI)
apply (rule ideal_pOp_closed)
apply (simp add:whole_ideal, assumption+)
apply (cut_tac ring_is_ag)
apply (simp add:aGroup.ag_mOp_closed)
apply (rule ballI)
apply (subst ring_distrib2)
apply (simp add:ideal_subset, assumption)
apply (cut_tac ring_is_ag, simp add: aGroup.ag_mOp_closed)
apply (frule_tac a1 = y and b1 = b in ring_inv1_1 [THEN sym])
apply (simp add:ideal_subset, simp)
apply (rule ideal_pOp_closed, assumption+, simp)
apply (rule ideal_inv1_closed, assumption+, simp)
apply (rule ballI)+
apply (simp add:ideal_quotient_def)
apply (rule conjI)
apply (erule conjE)
apply (simp add:ring_tOp_closed)
apply (erule conjE)
apply (rule ballI)
apply (subst ring_tOp_assoc, assumption+, simp add:ideal_subset)
apply (simp add:ideal_ring_multiple [of "A"])
done
section "9. Addition of finite elements of a ring and ideal_multiplication"
text{* We consider sum in an abelian group *}
lemma (in aGroup) nsum_mem1Tr:" A +> J ==>
(∀j ≤ n. f j ∈ J) --> nsum A f n ∈ J"
apply (induct_tac n)
apply (rule impI)
apply simp
apply (rule impI)
apply simp
apply (rule asubg_pOp_closed, assumption+)
apply simp
done
lemma (in aGroup) fSum_mem:"[|∀j ∈ nset (Suc n) m. f j ∈ carrier A; n < m|] ==>
fSum A f (Suc n) m ∈ carrier A"
apply (simp add:fSum_def)
apply (rule nsum_mem)
apply (rule allI, simp add:cmp_def slide_def)
apply (rule impI)
apply (frule_tac b = "Suc (n + j)" in forball_spec1)
apply (simp add:nset_def, arith)
done
lemma (in aGroup) nsum_mem1:"[|A +> J; ∀j ≤ n. f j ∈ J|] ==> nsum A f n ∈ J"
apply (simp add:nsum_mem1Tr)
done
lemma (in aGroup) nsum_eq_i:"[|∀j≤n. f j ∈ carrier A; ∀j≤n. g j ∈ carrier A;
i ≤ n; ∀l ≤ i. f l = g l|] ==> nsum A f i = nsum A g i"
apply (rule nsum_eq)
apply (rule allI, rule impI, simp)+
done
lemma (in aGroup) nsum_cmp_eq:"[|f ∈ {j. j≤(n::nat)} -> carrier A;
h1 ∈ {j. j ≤ n} -> {j. j ≤ n}; h2 ∈ {j. j ≤ n} -> {j. j ≤ n}; i ≤ n|] ==>
nsum A (cmp f (cmp h2 h1)) i = nsum A (cmp (cmp f h2) h1) i"
apply (rule nsum_eq_i [of n "cmp f (cmp h2 h1)" "cmp (cmp f h2) h1" i])
apply (rule allI, rule impI, simp add:cmp_def)
apply ((rule funcset_mem, assumption)+, simp)
apply (rule allI, rule impI, simp add:cmp_def,
(rule funcset_mem, assumption)+, simp+)
apply (rule allI, rule impI, simp add:cmp_def)
done
lemma (in aGroup) nsum_cmp_eq_transpos:"[| ∀j≤(Suc n). f j ∈ carrier A;
i ≤ n;i ≠ n |] ==>
nsum A (cmp f (cmp (transpos i n) (cmp (transpos n (Suc n)) (transpos i n))))
(Suc n) = nsum A (cmp f (transpos i (Suc n))) (Suc n)"
apply (rule nsum_eq [of "Suc n" "cmp f (cmp (transpos i n)
(cmp (transpos n (Suc n)) (transpos i n)))"
"cmp f (transpos i (Suc n))"])
apply (rule allI, rule impI)
apply (simp add:cmp_def)
apply (cut_tac i = i and n = "Suc n" and j = n and l = j in transpos_mem,
simp+)
apply (cut_tac i = n and n = "Suc n" and j = "Suc n" and l = "transpos i n j"
in transpos_mem, simp+)
apply (cut_tac i = i and n = "Suc n" and j = n and
l = "transpos n (Suc n) (transpos i n j)" in transpos_mem,
simp+)
apply (rule allI, rule impI, simp add:cmp_def)
apply (cut_tac i = i and n = "Suc n" and j = "Suc n" and l = j in transpos_mem,
simp+)
apply (rule allI, rule impI)
apply (simp add:cmp_def)
apply (thin_tac "∀j≤Suc n. f j ∈ carrier A",
rule eq_elems_eq_val[of _ _ f])
apply (simp add:transpos_def)
done
lemma transpos_Tr_n1:"Suc (Suc 0) ≤ n ==>
transpos (n - Suc 0) n n = n - Suc 0"
apply (simp add:transpos_def)
done
lemma transpos_Tr_n2:"Suc (Suc 0) ≤ n ==>
transpos (n - (Suc 0)) n (n - (Suc 0)) = n"
apply (simp add:transpos_def)
done
lemma (in aGroup) additionTr0:"[|0 < n; ∀j ≤ n. f j ∈ carrier A|]
==> nsum A (cmp f (transpos (n - 1) n)) n = nsum A f n"
apply (case_tac "n ≤ 1")
apply simp
apply (frule Suc_leI [of "0" "n"])
apply (frule le_anti_sym [of "n" "Suc 0"], assumption+, simp)
apply (simp add:cmp_def)
apply (subst transpos_ij_1[of 0 "Suc 0"], simp+)
apply (subst transpos_ij_2[of 0 "Suc 0"], simp+)
apply (rule ag_pOp_commute, simp+)
apply (frule nat_not_le[of n "Suc 0"])
apply (frule_tac Suc_leI [of "Suc 0" "n"],
thin_tac "¬ n ≤ Suc 0")
apply (cut_tac nsum_suc[of A f "n - Suc 0"], simp)
apply (cut_tac nsum_suc[of A "cmp f (transpos (n - Suc 0) n)" "n - Suc 0"],
simp,
thin_tac "Σe A f n = Σe A f (n - Suc 0) ± f n",
thin_tac "Σe A (cmp f (transpos (n - Suc 0) n)) n =
Σe A (cmp f (transpos (n - Suc 0) n)) (n - Suc 0) ±
(cmp f (transpos (n - Suc 0) n) n)")
apply (case_tac "n = Suc (Suc 0)", simp)
apply (cut_tac transpos_id_1[of "Suc 0" "Suc (Suc 0)" "Suc (Suc 0)" 0],
cut_tac transpos_ij_1[of "Suc 0" "Suc (Suc 0)" "Suc (Suc 0)"],
cut_tac transpos_ij_2[of "Suc 0" "Suc (Suc 0)" "Suc (Suc 0)"],
simp add:cmp_def,
thin_tac "n = Suc (Suc 0)",
thin_tac "transpos (Suc 0) (Suc (Suc 0)) 0 = 0",
thin_tac "transpos (Suc 0) (Suc (Suc 0)) (Suc 0) = Suc (Suc 0)",
thin_tac "transpos (Suc 0) (Suc (Suc 0)) (Suc (Suc 0)) = Suc 0")
apply (subst ag_pOp_assoc, simp+)
apply (subst ag_pOp_commute[of "f (Suc (Suc 0))" "f (Suc 0)"], simp+)
apply (subst ag_pOp_assoc[THEN sym], simp+)
apply (frule not_sym)
apply (frule noteq_le_less[of "Suc (Suc 0)" n], assumption,
thin_tac "Suc (Suc 0) ≤ n")
apply (cut_tac nsum_suc[of A f "n - Suc 0 - Suc 0"])
apply (cut_tac Suc_pred[of "n - Suc 0"], simp del:nsum_suc)
apply (cut_tac nsum_suc[of A "cmp f (transpos (n - Suc 0) n)"
"n - Suc (Suc 0)"], simp del:nsum_suc,
thin_tac "Σe A f (n - Suc 0) = Σe A f (n - Suc (Suc 0)) ± f (n - Suc 0)",
thin_tac "Suc (n - Suc (Suc 0)) = n - Suc 0",
thin_tac "Σe A (cmp f (transpos (n - Suc 0) n)) (n - Suc 0) =
Σe A (cmp f (transpos (n - Suc 0) n)) (n - Suc (Suc 0)) ±
(cmp f (transpos (n - Suc 0) n)) (n - Suc 0)")
apply (cut_tac nsum_eq_i[of n "cmp f (transpos (n - Suc 0) n)" f
"n - Suc (Suc 0)"], simp,
thin_tac "Σe A (cmp f (transpos (n - Suc 0) n)) (n - Suc (Suc 0)) =
Σe A f (n - Suc (Suc 0))")
apply (simp add:cmp_def)
apply (cut_tac transpos_ij_1[of "n - Suc 0" n n], simp)
apply (cut_tac transpos_ij_2[of "n - Suc 0" n n], simp)
apply (subst ag_pOp_assoc,
rule nsum_mem, rule allI, rule impI)
apply (frule_tac x = j and y = "n - Suc (Suc 0)" and z = n in
le_less_trans, simp, frule_tac x = j and y = n in less_imp_le)
apply simp+
apply (subst ag_pOp_commute[of "f n"], simp, simp)
apply (subst ag_pOp_assoc[THEN sym],
rule nsum_mem, rule allI, rule impI,
frule_tac x = j and y = "n - Suc (Suc 0)" and z = n in
le_less_trans, simp, frule_tac x = j and y = n in less_imp_le)
apply simp+
apply (rule allI, rule impI, simp add:cmp_def)
apply (cut_tac i = "n - Suc 0" and n = n and j = n and l = j in transpos_mem,
simp+)
apply (rule allI, rule impI)
apply (simp add:cmp_def)
apply (cut_tac i = "n - Suc 0" and n = n and j = n and x = l in transpos_id,
simp+)
apply (cut_tac x = l and y = "n - Suc (Suc 0)" and z = n in le_less_trans,
assumption) apply simp
apply arith
apply simp
apply arith
done
lemma (in aGroup) additionTr1:"[| ∀f. ∀h. f ∈ {j. j≤(Suc n)} -> carrier A ∧
h ∈ {j. j≤(Suc n)} -> {j. j≤(Suc n)} ∧ inj_on h {j. j≤(Suc n)} -->
nsum A (cmp f h) (Suc n) = nsum A f (Suc n);
f ∈ {j. j≤(Suc (Suc n))} -> carrier A;
h ∈ {j. j≤(Suc (Suc n))} -> {j. j≤(Suc (Suc n))};
inj_on h {j. j≤(Suc (Suc n))}; h (Suc (Suc n)) = Suc (Suc n)|]
==> nsum A (cmp f h) (Suc (Suc n)) = nsum A f (Suc (Suc n))"
apply (subgoal_tac "f ∈ {j. j≤(Suc n)} -> carrier A")
apply (subgoal_tac "h ∈ {j. j≤(Suc n)} -> {j. j≤(Suc n)}")
apply (subgoal_tac "inj_on h {j. j≤(Suc n)}")
apply (subgoal_tac "nsum A (cmp f h) (Suc n) = nsum A f (Suc n)")
apply (thin_tac "∀f. ∀h. f ∈ {j. j≤(Suc n)} -> carrier A ∧
h ∈ {j. j≤(Suc n)} -> {j. j≤(Suc n)} ∧ inj_on h {j. j≤(Suc n)} -->
nsum A (cmp f h) (Suc n) = nsum A f (Suc n)")
prefer 2 apply simp
apply simp
apply (thin_tac "nsum A (cmp f h) n ± (cmp f h (Suc n)) = nsum A f n ± (f (Suc n))")
apply (simp add:cmp_def)
apply (thin_tac "∀f h. (f ∈ {j. j ≤ Suc n} -> carrier A) ∧
(h ∈ {j. j≤Suc n} -> {j. j≤Suc n}) ∧ (inj_on h {j. j≤Suc n}) -->
Σe A (cmp f h) (Suc n) = Σe A f (Suc n)")
apply (frule Nset_injTr0 [of "h" "Suc n"], assumption+, simp)
apply (frule Nset_injTr0 [of "h" "Suc n"], assumption+, simp)
apply (thin_tac "∀f h. f ∈ {j. j ≤ Suc n} -> carrier A ∧
h ∈ {j. j ≤ Suc n} -> {j. j ≤ Suc n} ∧ inj_on h {j. j ≤ Suc n} -->
Σe A (cmp f h) (Suc n) = Σe A f (Suc n)")
apply (thin_tac "h ∈ {j. j ≤ Suc (Suc n)} -> {j. j ≤ Suc (Suc n)}")
apply (simp add:Pi_def [of "{j. j ≤ (Suc n)}"])
apply (rule allI, rule impI)
apply (subgoal_tac "x ∈ {j. j ≤ (Suc (Suc n))}", simp add:funcset_mem)
apply simp
done
lemma (in aGroup) additionTr1_1:"[|∀f. ∀h. f ∈ {j. j≤Suc n} -> carrier A ∧
h ∈ {j. j≤Suc n} -> {j. j≤Suc n} ∧ inj_on h {j. j≤Suc n} -->
nsum A (cmp f h) (Suc n) = nsum A f (Suc n);
f ∈ {j. j≤Suc (Suc n)} -> carrier A; i ≤ n|] ==>
nsum A (cmp f (transpos i (Suc n))) (Suc (Suc n)) = nsum A f (Suc (Suc n))"
apply (rule additionTr1 [of "n" "f" "transpos i (Suc n)"], assumption+)
apply (rule transpos_hom [of "i" "Suc (Suc n)" "Suc n"])
apply simp+
apply (rule transpos_inj [of "i" "Suc (Suc n)" "Suc n"])
apply simp+
apply (subst transpos_id[of i "Suc (Suc n)" "Suc n" "Suc (Suc n)"])
apply simp+
done
lemma (in aGroup) additionTr1_2:"[|∀f. ∀h. f ∈ {j. j≤Suc n} -> carrier A ∧
h ∈ {j. j≤Suc n} -> {j. j≤Suc n} ∧
inj_on h {j. j≤Suc n} -->
nsum A (cmp f h) (Suc n) = nsum A f (Suc n);
f ∈ {j. j≤ Suc (Suc n)} -> carrier A; i ≤ (Suc n)|] ==>
nsum A (cmp f (transpos i (Suc (Suc n)))) (Suc (Suc n)) =
nsum A f (Suc (Suc n))"
apply (case_tac "i = Suc n")
apply (simp del:nsum_suc)
apply (cut_tac additionTr0 [of "Suc (Suc n)" "f"], simp, simp,
rule allI, rule impI, rule funcset_mem[of f "{j. j ≤ Suc (Suc n)}"
"carrier A"], (simp del:nsum_suc)+)
apply (subst nsum_cmp_eq_transpos [THEN sym, of "Suc n" f i],
rule allI, rule impI, rule funcset_mem[of f "{j. j ≤ Suc (Suc n)}"
"carrier A"], assumption+,
simp, assumption+)
apply (subst nsum_cmp_eq [of "f" "Suc (Suc n)"
"cmp (transpos (Suc n) (Suc(Suc n))) (transpos i (Suc n))"
"transpos i (Suc n)" "Suc (Suc n)"], assumption+,
rule univar_func_test, rule ballI, simp add:cmp_def,
rule transpos_mem, (simp del:nsum_suc)+,
rule transpos_mem, (simp del:nsum_suc)+,
rule univar_func_test, rule ballI, simp,
rule transpos_mem, (simp del:nsum_suc)+)
apply (subst nsum_cmp_eq [of "cmp f (transpos i (Suc n))" "Suc (Suc n)"
"(transpos i (Suc n))" "transpos (Suc n) (Suc (Suc n))" "Suc (Suc n)"],
rule univar_func_test, rule ballI, simp add:cmp_def,
rule funcset_mem[of f "{j. j ≤ Suc (Suc n)}" "carrier A"], assumption,
simp,
rule transpos_mem, (simp del:nsum_suc)+,
(rule univar_func_test, rule ballI, simp,
rule transpos_mem, (simp del:nsum_suc)+)+)
apply (subst additionTr1_1 [of "n" "cmp (cmp f (transpos i (Suc n)))
(transpos (Suc n) (Suc (Suc n)))" "i"], assumption+,
rule cmp_fun [of _ "{j. j ≤ (Suc (Suc n))}"
"{j. j ≤ (Suc (Suc n))}" _ "carrier A"],
rule transpos_hom, simp, simp, simp,
rule cmp_fun [of _ "{j. j ≤ (Suc (Suc n))}"
"{j. j ≤ (Suc (Suc n))}" "f" "carrier A"],
rule transpos_hom, simp, simp, assumption+, arith)
apply (cut_tac additionTr0 [of "Suc (Suc n)" "cmp f (transpos i (Suc n))"],
simp del:nsum_suc,
thin_tac "nsum A (cmp
(cmp f (transpos i (Suc n))) (transpos (Suc n) (Suc (Suc n)))) (Suc (Suc n))
= nsum A (cmp f (transpos i (Suc n))) (Suc (Suc n))")
apply (rule additionTr1_1, assumption+, arith, simp,
rule allI, rule impI, simp add:cmp_def,
rule funcset_mem[of f "{j. j ≤ Suc (Suc n)}" "carrier A"],
assumption)
apply (simp add:transpos_mem)
done
lemma (in aGroup) additionTr2:" ∀f. ∀h. f ∈ {j. j ≤ (Suc n)} -> carrier A ∧
h ∈ {j. j ≤ (Suc n)} -> {j. j ≤ (Suc n)} ∧
inj_on h {j. j ≤ (Suc n)} -->
nsum A (cmp f h) (Suc n) = nsum A f (Suc n)"
apply (induct_tac n)
apply (rule allI)+
apply (rule impI, (erule conjE)+)
apply (simp add:cmp_def)
apply (case_tac "h 0 = 0")
apply (simp add:Nset_1)
apply (frule_tac f = h in funcset_mem [of _ "{0, Suc 0}"
"{0, Suc 0}" "Suc 0"], simp)
apply (simp add:inj_on_def, simp add:Nset_1)
apply (frule_tac f = h in funcset_mem [of _ "{0, Suc 0}" "{0, Suc 0}" "0"],
simp, frule not_sym, simp)
apply (frule_tac f = h in funcset_mem [of _ "{0, Suc 0}" "{0, Suc 0}"
"Suc 0"], simp, simp)
apply (rule ag_pOp_commute)
apply (rule funcset_mem, assumption+, simp,
rule funcset_mem, assumption+, simp)
(************* n *****************)
apply (rule allI)+
apply (rule impI, (erule conjE)+)
apply (case_tac "h (Suc (Suc n)) = Suc (Suc n)")
apply (rule additionTr1, assumption+)
apply (frule_tac f = h and n = "Suc (Suc n)" in inj_surj, assumption+)
apply (frule sym, thin_tac "h ` {i. i ≤ Suc (Suc n)} = {i. i ≤ Suc (Suc n)}")
apply (cut_tac n = "Suc (Suc n)" in n_in_Nsetn)
apply (frule_tac a = "Suc (Suc n)" and A = "{i. i ≤ Suc (Suc n)}" and
B = "h ` {i. i ≤ Suc (Suc n)}" in eq_set_inc, assumption+)
apply (thin_tac "{i. i ≤ Suc (Suc n)} = h ` {i. i ≤ Suc (Suc n)}")
apply (simp del:nsum_suc add:image_def)
apply (erule exE, erule conjE)
apply (frule sym, thin_tac "Suc (Suc n) = h x")
apply (frule_tac i = x and n = "Suc (Suc n)" and j = "Suc (Suc n)" in
transpos_ij_2, simp del:nsum_suc add:n_in_Nsetn)
apply (rule contrapos_pp, (simp del:nsum_suc)+)
apply (frule_tac x = "transpos x (Suc (Suc n)) (Suc (Suc n))" and y = x and
f = h in eq_elems_eq_val,
thin_tac "transpos x (Suc (Suc n)) (Suc (Suc n)) = x",
simp del:nsum_suc)
apply (frule_tac f = h and A = "{i. i ≤ Suc (Suc n)}" and x = x and
y = "Suc (Suc n)" in inj_onTr2, simp, simp,
frule not_sym, simp)
apply (cut_tac f1 = "cmp f h" and n1 = n and i1 = x in
additionTr1_2[THEN sym], assumption)
apply (rule cmp_fun, simp, assumption, arith)
apply (simp del:nsum_suc,
thin_tac "Σe A (cmp f h) (Suc (Suc n)) =
Σe A (cmp (cmp f h) (transpos x (Suc (Suc n)))) (Suc (Suc n))")
apply (frule_tac f = f and n = "Suc n" and A = "carrier A" in func_pre)
apply (cut_tac f = "cmp h (transpos x (Suc (Suc n)))" and A = "{j. j ≤ (Suc ( Suc n))}" and ?A1.0 = "{j. j ≤ (Suc n)}" in restrict_inj)
apply (rule_tac f = "transpos x (Suc (Suc n))" and A = "{j. j ≤ Suc (Suc n)}"
and B = "{j. j ≤ Suc (Suc n)}" and g = h and C = "{j. j ≤ Suc (Suc n)}" in
cmp_inj, simp,
rule transpos_hom, simp, simp, assumption+,
rule transpos_inj, simp, simp, assumption+,
rule subsetI, simp)
apply (subst nsum_cmp_assoc,
rule allI, rule impI, simp add:funcset_mem,
rule transpos_hom, assumption, simp, assumption+)
apply (cut_tac f = "cmp f (cmp h (transpos x (Suc (Suc n))))" and n = "Suc n"
in nsum_suc[of A ], simp del:nsum_suc,
thin_tac "Σe A (cmp f (cmp h (transpos x (Suc (Suc n))))) (Suc (Suc n)) =
Σe A (cmp f (cmp h (transpos x (Suc (Suc n))))) (Suc n) ±
cmp f (cmp h (transpos x (Suc (Suc n)))) (Suc (Suc n))")
apply (frule_tac a = f in forall_spec1,
thin_tac "∀f h. f ∈ {j. j ≤ Suc n} -> carrier A ∧
h ∈ {j. j ≤ Suc n} -> {j. j ≤ Suc n} ∧
inj_on h {j. j ≤ Suc n} -->
Σe A (cmp f h) (Suc n) = Σe A f (Suc n)")
apply (frule_tac a = "cmp h (transpos x (Suc (Suc n)))" in forall_spec,
thin_tac "∀h. f ∈ {j. j ≤ Suc n} -> carrier A ∧
h ∈ {j. j ≤ Suc n} -> {j. j ≤ Suc n} ∧ inj_on h {j. j ≤ Suc n} -->
Σe A (cmp f h) (Suc n) = Σe A f (Suc n)")
apply simp
apply (rule univar_func_test, rule ballI)
apply (simp add:cmp_def)
apply (case_tac "xa = x", simp)
apply (cut_tac i = x and n = "Suc (Suc n)" and j = "Suc (Suc n)" in
transpos_ij_1, simp, simp, simp, simp,
frule_tac x = "Suc (Suc n)" and f = h and A = "{j. j ≤ Suc (Suc n)}"
and B = "{j. j ≤ Suc (Suc n)}" in funcset_mem, simp,
thin_tac "h ∈ {j. j ≤ Suc (Suc n)} -> {j. j ≤ Suc (Suc n)}")
apply (cut_tac m = "h (Suc (Suc n))" and n = "Suc (Suc n)" in noteq_le_less,
simp, simp,
rule_tac x = "h (Suc (Suc n))" and n = "Suc n" in Suc_less_le,
assumption)
apply (subst transpos_id, simp, simp, simp, simp,
frule_tac x = xa and f = h and A = "{j. j ≤ Suc (Suc n)}" and
B = "{j. j ≤ Suc (Suc n)}" in funcset_mem, simp)
apply (frule_tac f = h and A = "{j. j ≤ Suc (Suc n)}" and x = xa and y = x
in injective, simp, simp, assumption)
apply (cut_tac m = "h xa" and n = "Suc (Suc n)" in noteq_le_less, simp,
simp)
apply (rule Suc_less_le, assumption,
thin_tac "∀h. f ∈ {j. j ≤ Suc n} -> carrier A ∧
h ∈ {j. j ≤ Suc n} -> {j. j ≤ Suc n} ∧ inj_on h {j. j ≤ Suc n} -->
Σe A (cmp f h) (Suc n) = Σe A f (Suc n)")
apply (simp del:nsum_suc add:cmp_def)
apply simp
done
lemma (in aGroup) addition2:"[|f ∈ {j. j ≤ (Suc n)} -> carrier A;
h ∈ {j. j ≤ (Suc n)} -> {j. j ≤ (Suc n)}; inj_on h {j. j ≤ (Suc n)}|] ==>
nsum A (cmp f h) (Suc n) = nsum A f (Suc n)"
apply (simp del:nsum_suc add:additionTr2)
done
lemma (in aGroup) addition21:"[|f ∈ {j. j ≤ n} -> carrier A;
h ∈ {j. j ≤ n} -> {j. j ≤ n}; inj_on h {j. j ≤ n}|] ==>
nsum A (cmp f h) n = nsum A f n"
apply (case_tac "n = 0")
apply simp
apply (subgoal_tac "h 0 = 0", simp add:cmp_def)
apply (cut_tac Nset_inc_0[of 0])
apply (frule_tac f = h and A = "{0}" and B = "{0}" in funcset_mem, simp,
simp)
apply (cut_tac f = f and n = "n - Suc 0" and h = h in addition2)
apply simp+
done
lemma (in aGroup) addition3:"[|∀j ≤ (Suc n). f j ∈ carrier A; j ≤ (Suc n);
j ≠ Suc n|] ==> nsum A f (Suc n) = nsum A (cmp f (transpos j (Suc n))) (Suc n)"
apply (rule addition2 [THEN sym,of "f" "n" "transpos j (Suc n)"])
apply (rule univar_func_test, rule ballI, simp)
apply (rule transpos_hom, assumption+, simp, assumption)
apply (rule transpos_inj, simp+)
done
lemma (in aGroup) nsum_splitTr:"(∀j ≤ (Suc (n + m)). f j ∈ carrier A) -->
nsum A f (Suc (n + m)) = nsum A f n ± (nsum A (cmp f (slide (Suc n))) m)"
apply (induct_tac m)
apply (rule impI) apply (simp add:slide_def cmp_def)
apply (rule impI, simp del:nsum_suc)
apply (cut_tac n = "Suc (n + na)" in nsum_suc[of A f],
simp del:nsum_suc,
thin_tac "Σe A f (Suc (Suc (n + na))) =
Σe A f n ± Σe A (cmp f (slide (Suc n))) na ± f (Suc (Suc (n + na)))")
apply (cut_tac f = "cmp f (slide (Suc n))" and n = na in nsum_suc[of A],
simp del:nsum_suc)
apply (subst ag_pOp_assoc)
apply (rule nsum_mem, rule allI, simp)
apply (rule_tac n = na in nsum_mem,
thin_tac "Σe A (cmp f (slide (Suc n))) (Suc na) =
Σe A (cmp f (slide (Suc n))) na ± (cmp f (slide (Suc n)) (Suc na))")
apply (rule allI, rule impI,
simp add:cmp_def slide_def, simp)
apply (simp add:cmp_def slide_def)
done
lemma (in aGroup) nsum_split:"∀j ≤ (Suc (n + m)). f j ∈ carrier A ==>
nsum A f (Suc (n + m)) = nsum A f n ± (nsum A (cmp f (slide (Suc n))) m)"
by (simp del:nsum_suc add:nsum_splitTr)
lemma (in aGroup) nsum_split1:"[|∀j ≤ m. f j ∈ carrier A; n < m|] ==>
nsum A f m = nsum A f n ± (fSum A f (Suc n) m)"
apply (cut_tac nsum_split[of n "m - n - Suc 0" f])
apply simp
apply (simp add:fSum_def)
apply simp
done
lemma (in aGroup) nsum_minusTr:" (∀j ≤ n. f j ∈ carrier A) -->
-a (nsum A f n) = nsum A (λx∈{j. j ≤ n}. -a (f x)) n"
apply (induct_tac n)
apply (rule impI, simp)
apply (rule impI)
apply (subst nsum_suc, subst nsum_suc)
apply (subst ag_p_inv)
apply (rule_tac n = n in nsum_mem [of _ f],
rule allI, simp, simp)
apply (subgoal_tac "∀j≤n. f j ∈ carrier A", simp)
apply (rule_tac a = "Σe A (λu. if u ≤ n then -a (f u) else arbitrary) n"
and b = "Σe A (λx∈{j. j ≤ (Suc n)}. -a (f x)) n" and c = "-a (f (Suc n))"
in ag_pOp_add_r,
rule_tac n = n in nsum_mem,
rule allI, rule impI, simp,
rule ag_mOp_closed, simp)
apply (rule_tac n = n in nsum_mem,
rule allI, rule impI, simp,
rule ag_mOp_closed, simp,
rule ag_mOp_closed, simp)
apply (rule_tac f = "λu. if u ≤ n then -a (f u) else arbitrary" and
n = n and g = "λx∈{j. j ≤ (Suc n)}. -a (f x)" in nsum_eq,
rule allI, rule impI,
simp, rule ag_mOp_closed, simp,
rule allI, simp, rule impI, rule ag_mOp_closed, simp)
apply (rule allI, simp)
apply (rule allI, simp)
done
lemma (in aGroup) nsum_minus:"∀j ≤ n. f j ∈ carrier A ==>
-a (nsum A f n) = nsum A (λx∈{j. j ≤ n}. -a (f x)) n"
apply (simp add:nsum_minusTr)
done
lemma (in aGroup) ring_nsum_zeroTr:"(∀j ≤ (n::nat). f j ∈ carrier A) ∧
(∀j ≤ n. f j = \<zero>) --> nsum A f n = \<zero>"
apply (induct_tac n)
apply (rule impI) apply (erule conjE)+ apply simp
apply (rule impI, (erule conjE)+)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (simp add:ag_inc_zero)
apply (cut_tac ag_inc_zero,
simp add:ag_r_zero)
done
lemma (in aGroup) ring_nsum_zero:"∀j ≤ (n::nat). f j = \<zero> ==> Σe A f n = \<zero>"
apply (cut_tac ring_nsum_zeroTr[of n f])
apply (simp add:ag_inc_zero)
done
lemma (in aGroup) ag_nsum_1_nonzeroTr:
"∀f. (∀j ≤ n. f j ∈ carrier A) ∧
(l ≤ n ∧ (∀j ∈ {j. j ≤ n} - {l}. f j = \<zero>))
--> nsum A f n = f l"
apply (induct_tac n)
apply simp
apply (rule allI,
rule impI, (erule conjE)+)
apply (case_tac "l = Suc n")
apply simp
apply (subgoal_tac "{j. j ≤ Suc n} - {Suc n} = {j. j ≤ n}", simp,
frule ring_nsum_zero, simp)
apply (rule ag_l_zero, simp)
apply (rule equalityI, rule subsetI, simp,
rule subsetI, simp)
apply (frule_tac m = l and n = "Suc n" in noteq_le_less, assumption,
thin_tac "l ≤ Suc n",
frule_tac x = l and n = n in Suc_less_le)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (thin_tac "∀f. (∀j≤n. f j ∈ carrier A) ∧
(∀j∈{j. j ≤ n} - {l}. f j = \<zero>) -->
Σe A f n = f l",
frule_tac a = l in forall_spec, simp)
apply (simp add:ag_r_zero)
done
lemma (in aGroup) ag_nsum_1_nonzero:"[|∀j ≤ n. f j ∈ carrier A; l ≤ n;
∀j∈({j. j ≤ n} - {l}). f j = \<zero> |] ==> nsum A f n = f l"
apply (simp add:ag_nsum_1_nonzeroTr[of n l])
done
constdefs (structure R)
set_mult::"[_ , 'a set, 'a set] => 'a set"
"set_mult R A B == {z. ∃x∈A. ∃y∈B. x ·r y = z}"
sum_mult::"[_ , 'a set, 'a set] => 'a set"
"sum_mult R A B == {x. ∃n. ∃f ∈ {j. j ≤ (n::nat)}
-> set_mult R A B. nsum R f n = x}"
(*
zero_fini::"[_ , 'a set, 'a set] => (nat => 'a)"
"zero_fini R A B i == \<zero> ·r \<zero>" *)
lemma (in Ring) set_mult_sub:"[|A ⊆ carrier R; B ⊆ carrier R|] ==>
set_mult R A B ⊆ carrier R"
apply (rule subsetI, simp add:set_mult_def, (erule bexE)+,
frule sym, thin_tac "xa ·r y = x", simp)
apply (rule ring_tOp_closed, (simp add:subsetD)+)
done
lemma (in Ring) set_mult_mono:"[|A1 ⊆ carrier R; A2 ⊆ carrier R; A1 ⊆ A2;
B ⊆ carrier R|] ==> set_mult R A1 B ⊆ set_mult R A2 B"
apply (rule subsetI)
apply (simp add:set_mult_def, (erule bexE)+)
apply (frule_tac c = xa in subsetD[of A1 A2], assumption+)
apply blast
done
lemma (in Ring) sum_mult_Tr1:"[|A ⊆ carrier R; B ⊆ carrier R|] ==>
(∀j ≤ n. f j ∈ set_mult R A B) --> nsum R f n ∈ carrier R"
apply (cut_tac ring_is_ag)
apply (induct_tac n)
apply (rule impI, simp)
apply (frule set_mult_sub[of A B], assumption, simp add:subsetD)
apply (rule impI)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (frule set_mult_sub[of A B], assumption)
apply (frule_tac a = "Suc n" in forall_spec, simp,
frule_tac c = "f (Suc n)" in subsetD[of "set_mult R A B" "carrier R"],
assumption)
apply (rule aGroup.ag_pOp_closed, assumption+)
done
lemma (in Ring) sum_mult_mem:"[|A ⊆ carrier R; B ⊆ carrier R;
∀j ≤ n. f j ∈ set_mult R A B|] ==> nsum R f n ∈ carrier R"
apply (cut_tac ring_is_ag)
apply (simp add:sum_mult_Tr1)
done
lemma (in Ring) sum_mult_mem1:"[|A ⊆ carrier R; B ⊆ carrier R;
x ∈ sum_mult R A B|] ==>
∃n. ∃f∈{j. j ≤ (n::nat)} -> set_mult R A B. nsum R f n = x"
by (simp add:sum_mult_def)
lemma (in Ring) sum_mult_subR:"[|A ⊆ carrier R; B ⊆ carrier R|] ==>
sum_mult R A B ⊆ carrier R"
apply (rule subsetI)
apply (frule_tac x = x in sum_mult_mem1[of A B], assumption+)
apply (erule exE, erule bexE, frule sym, thin_tac "Σe R f n = x", simp)
apply (cut_tac ring_is_ag)
apply (rule aGroup.nsum_mem[of R], assumption)
apply (rule allI, rule impI)
apply (frule_tac f = f and A = "{j. j ≤ n}" and B = "set_mult R A B" and
x = j in funcset_mem, simp)
apply (frule set_mult_sub[of A B], assumption)
apply (simp add:subsetD)
done
lemma (in Ring) times_mem_sum_mult:"[|A ⊆ carrier R; B ⊆ carrier R;
a ∈ A; b ∈ B |] ==> a ·r b ∈ sum_mult R A B"
apply (simp add:sum_mult_def)
apply (subgoal_tac "(λi∈{j. j ≤ (0::nat)}. a ·r b) ∈ {j. j ≤ 0} -> set_mult R A B")
apply (subgoal_tac "nsum R (λi∈{j. j ≤ (0::nat)}. a ·r b) 0 = a ·r b")
apply blast
apply simp
apply (rule univar_func_test, rule ballI,
simp add:set_mult_def, blast)
done
lemma (in Ring) mem_minus_sum_multTr2:"[|A ⊆ carrier R; B ⊆ carrier R;
∀j ≤ n. f j ∈ set_mult R A B; i ≤ n|] ==> f i ∈ carrier R"
apply (frule_tac a = i in forall_spec, simp)
apply (frule set_mult_sub[of A B], assumption, simp add:subsetD)
done
lemma (in aGroup) nsum_jointfun:"[|∀j ≤ n. f j ∈ carrier A;
∀j ≤ m. g j ∈ carrier A|] ==>
Σe A (jointfun n f m g) (Suc (n + m)) = Σe A f n ± (Σe A g m)"
apply (subst nsum_split)
apply (rule allI, rule impI)
apply (frule_tac f = f and n = n and A = "carrier A" and g = g and m = m
and B = "carrier A" in jointfun_mem, assumption+, simp)
apply (subgoal_tac "nsum A (jointfun n f m g) n = nsum A f n")
apply (subgoal_tac "nsum A (cmp (jointfun n f m g) (slide (Suc n))) m =
nsum A g m")
apply simp
apply (thin_tac "nsum A (jointfun n f m g) n = nsum A f n")
apply (rule nsum_eq)
apply (rule allI, rule impI,
simp add:cmp_def jointfun_def slide_def sliden_def,
assumption)
apply (rule allI, simp add:cmp_def jointfun_def slide_def sliden_def)
apply (rule nsum_eq)
apply (rule allI, rule impI,
simp add:jointfun_def, assumption)
apply (rule allI, rule impI)
apply (simp add:jointfun_def)
done
lemma (in Ring) sum_mult_pOp_closed:"[|A ⊆ carrier R; B ⊆ carrier R;
a ∈ sum_mult R A B; b ∈ sum_mult R A B |] ==> a ±R b ∈ sum_mult R A B"
apply (cut_tac ring_is_ag)
apply (simp add:sum_mult_def)
apply ((erule exE)+, (erule bexE)+)
apply (rename_tac n m f g)
apply (frule sym, thin_tac "Σe R f n = a", frule sym,
thin_tac "Σe R g m = b", simp)
apply (frule set_mult_sub[of A B], assumption)
apply (subst aGroup.nsum_jointfun[THEN sym, of R], assumption)
apply (rule allI, rule impI,
frule_tac f = f and A = "{j. j ≤ n}" and B = "set_mult R A B" and
x = j in funcset_mem, simp, simp add:subsetD)
apply (rule allI, rule impI,
frule_tac f = g and A = "{j. j ≤ m}" and B = "set_mult R A B" and
x = j in funcset_mem, simp, simp add:subsetD)
apply (frule_tac f = f and n = n and A = "set_mult R A B" and g = g and m = m
and B = "set_mult R A B" in jointfun_hom, assumption+)
apply (simp del:nsum_suc)
apply blast
done
lemma (in Ring) set_mult_mOp_closed:"[|A ⊆ carrier R; ideal R B;
x ∈ set_mult R A B|] ==> -a x ∈ set_mult R A B"
apply (cut_tac ring_is_ag,
simp add:set_mult_def,
(erule bexE)+, frule sym, thin_tac "xa ·r y = x", simp,
frule_tac c = xa in subsetD[of A "carrier R"], assumption+,
frule ideal_subset1[of B],
frule_tac c = y in subsetD[of B "carrier R"], assumption+,
simp add:ring_inv1_2,
frule_tac I = B and x = y in ideal_inv1_closed,
assumption+)
apply blast
done
lemma (in Ring) set_mult_ring_times_closed:"[|A ⊆ carrier R; ideal R B;
x ∈ set_mult R A B; r ∈ carrier R|] ==> r ·r x ∈ set_mult R A B"
apply (cut_tac ring_is_ag,
simp add:set_mult_def,
(erule bexE)+, frule sym, thin_tac "xa ·r y = x", simp,
frule_tac c = xa in subsetD[of A "carrier R"], assumption+,
frule ideal_subset1[of B],
frule_tac c = y in subsetD[of B "carrier R"], assumption,
frule_tac x = r and y = "xa ·r y" in ring_tOp_commute,
simp add:ring_tOp_closed, simp,
subst ring_tOp_assoc, assumption+)
apply (frule_tac x = y and y = r in ring_tOp_commute, assumption+,
simp,
frule_tac x = y and r = r in ideal_ring_multiple [of B], assumption+)
apply blast
done
lemma (in Ring) set_mult_sub_sum_mult:"[|A ⊆ carrier R; ideal R B|] ==>
set_mult R A B ⊆ sum_mult R A B"
apply (rule subsetI)
apply (simp add:sum_mult_def)
apply (cut_tac f = "(λi∈{j. j ≤ (0::nat)}. x)" in nsum_0[of R])
apply (cut_tac n_in_Nsetn[of 0],
simp del:nsum_0)
apply (cut_tac f = "λi∈{j. j ≤ (0::nat)}. x" and B = "set_mult R A B" in
univar_func_test[of "{j. j ≤ 0}"],
rule ballI, simp)
apply (subgoal_tac "Σe R (λi∈{j. j ≤ 0}. x) 0 = x")
apply blast
apply simp
done
lemma (in Ring) sum_mult_pOp_closedn:"[|A ⊆ carrier R; ideal R B|] ==>
(∀j ≤ n. f j ∈ set_mult R A B) --> Σe R f n ∈ sum_mult R A B"
apply (induct_tac n)
apply (rule impI, simp)
apply (frule set_mult_sub_sum_mult[of A B], assumption+, simp add:subsetD)
apply (rule impI)
apply simp
apply (frule_tac a = "Suc n" in forall_spec, simp)
apply (frule set_mult_sub_sum_mult[of A B], assumption+,
frule_tac c= "f (Suc n)" in
subsetD[of "set_mult R A B" "sum_mult R A B"], assumption+)
apply (rule sum_mult_pOp_closed, assumption,
simp add:ideal_subset1, assumption+)
done
lemma (in Ring) mem_minus_sum_multTr4:"[|A ⊆ carrier R; ideal R B|] ==>
(∀j ≤ n. f j ∈ set_mult R A B) --> -a (nsum R f n) ∈ sum_mult R A B"
apply (cut_tac ring_is_ag)
apply (induct_tac n)
apply (rule impI)
apply (cut_tac n_in_Nsetn[of 0])
apply (frule_tac x = "f 0" in set_mult_mOp_closed[of A B], assumption+)
apply simp
apply (frule set_mult_sub_sum_mult[of A B], assumption+,
simp add:subsetD)
apply (rule impI)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (frule sum_mult_subR[of A B], simp add:ideal_subset1)
apply (frule_tac n = n and f = f in sum_mult_pOp_closedn[of A B], assumption,
cut_tac n = n in Nsetn_sub_mem1, simp)
apply (frule_tac c = "Σe R f n" in subsetD[of "sum_mult R A B" "carrier R"],
assumption+,
frule_tac a = "Suc n" in forall_spec, simp,
thin_tac "∀j≤Suc n. f j ∈ set_mult R A B",
frule set_mult_sub[of A B], simp add:ideal_subset1,
frule_tac c = "f (Suc n)" in subsetD[of "set_mult R A B" "carrier R"],
assumption+ )
apply (frule_tac x = "Σe R f n" and y = "f (Suc n)" in aGroup.ag_p_inv[of R],
assumption+, simp)
apply (rule_tac a = "-a (Σe R f n)" and b = "-a (f (Suc n))" in
sum_mult_pOp_closed[of A B], assumption+,
simp add:ideal_subset1, assumption)
apply (frule_tac x = "f (Suc n)" in set_mult_mOp_closed[of A B], assumption+,
frule set_mult_sub_sum_mult[of A B], assumption+)
apply (simp add:subsetD)
done
lemma (in Ring) sum_mult_iOp_closed:"[|A ⊆ carrier R; ideal R B;
x ∈ sum_mult R A B |] ==> -a x ∈ sum_mult R A B"
apply (frule sum_mult_mem1 [of A B x],
simp add:ideal_subset1, assumption)
apply (erule exE, erule bexE, frule sym, thin_tac "Σe R f n = x")
apply simp
apply (frule_tac n = n and f = f in mem_minus_sum_multTr4[of A B],
assumption+)
apply (subgoal_tac "∀j≤n. f j ∈ set_mult R A B", simp)
apply (rule allI, simp add:funcset_mem)
done
lemma (in Ring) sum_mult_ring_multiplicationTr:
"[|A ⊆ carrier R; ideal R B; r ∈ carrier R|] ==>
(∀j ≤ n. f j ∈ set_mult R A B) --> r ·r (nsum R f n) ∈ sum_mult R A B"
apply (cut_tac ring_is_ag)
apply (induct_tac n)
apply (rule impI, simp)
apply (simp add:set_mult_def)
apply ((erule bexE)+, frule sym, thin_tac "x ·r y = f 0", simp)
apply (frule_tac c = x in subsetD[of A "carrier R"], assumption+)
apply (frule ideal_subset1[of B],
frule_tac c = y in subsetD[of B "carrier R"], assumption,
frule_tac x = r and y = "x ·r y" in ring_tOp_commute,
simp add:ring_tOp_closed, simp,
subst ring_tOp_assoc, assumption+)
apply (frule_tac x = y and y = r in ring_tOp_commute, assumption+,
simp,
frule_tac x = y and r = r in ideal_ring_multiple [of B], assumption+)
apply (rule times_mem_sum_mult, assumption+)
apply (rule impI)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (frule_tac f = f and n = n in aGroup.nsum_mem,
frule set_mult_sub [of "A" "B"], simp add:ideal_subset1,
rule allI, rule impI, cut_tac n = n in Nsetn_sub_mem1,
simp add: subsetD,
frule_tac a = "Suc n" in forall_spec, simp)
apply (frule set_mult_sub[of A B], simp add:ideal_subset1,
frule_tac c = "f (Suc n)" in subsetD[of "set_mult R A B" "carrier R"],
assumption)
apply (simp add: ring_distrib1)
apply (rule sum_mult_pOp_closed[of A B], assumption+,
simp add:ideal_subset1, assumption)
apply (frule_tac x = "f (Suc n)" in set_mult_ring_times_closed [of A B _ r],
assumption+, simp, assumption,
frule set_mult_sub_sum_mult[of A B], assumption+,
simp add:subsetD)
done
lemma (in Ring) sum_mult_ring_multiplication:"[|A ⊆ carrier R; ideal R B;
r ∈ carrier R; a ∈ sum_mult R A B|] ==> r ·r a ∈ sum_mult R A B"
apply (cut_tac ring_is_ag)
apply (frule sum_mult_mem1[of A B a],
simp add:ideal_subset1, assumption)
apply (erule exE, erule bexE, frule sym, thin_tac "Σe R f n = a", simp)
apply (subgoal_tac "∀j ≤ n. f j ∈ set_mult R A B")
apply (simp add:sum_mult_ring_multiplicationTr)
apply (rule allI, rule impI, simp add:funcset_mem)
done
lemma (in Ring) ideal_sum_mult:"[|A ⊆ carrier R; A ≠ {}; ideal R B|] ==>
ideal R (sum_mult R A B)"
apply (simp add:ideal_def [of _ "sum_mult R A B"])
apply (cut_tac ring_is_ag)
apply (rule conjI)
apply (rule aGroup.asubg_test, assumption+)
apply (rule subsetI)
apply (frule_tac x = x in sum_mult_mem1[of A B],
simp add:ideal_subset1, assumption,
erule exE, erule bexE, frule sym, thin_tac "Σe R f n = x", simp)
apply (rule_tac f = f and n = n in sum_mult_mem[of A B _ _], assumption+)
apply (simp add:ideal_subset1)
apply (rule allI, simp add:funcset_mem)
apply (frule nonempty_ex[of A], erule exE)
apply (frule ideal_zero[of B])
apply (frule_tac a = x and b = \<zero> in times_mem_sum_mult[of A B],
simp add:ideal_subset1, assumption+) apply blast
apply (rule ballI)+
apply (rule_tac a = a and b = "-a b" in sum_mult_pOp_closed[of A B],
assumption, simp add:ideal_subset1, assumption+,
rule_tac x = b in sum_mult_iOp_closed[of A B], assumption+)
apply (rule ballI)+
apply (rule sum_mult_ring_multiplication, assumption+)
done
lemma (in Ring) ideal_inc_set_multTr:"[|A ⊆ carrier R; ideal R B; ideal R C;
set_mult R A B ⊆ C |] ==>
∀f ∈ {j. j ≤ (n::nat)} -> set_mult R A B. Σe R f n ∈ C"
apply (induct_tac n)
apply (rule ballI, simp,
frule_tac x = 0 and f = f in funcset_mem[of _ "{0}" "set_mult R A B"],
simp, simp add:subsetD)
apply (rule ballI)
apply (
frule_tac f = f and A = "{j. j ≤ Suc n}" and x = "Suc n" and
B = "set_mult R A B"in funcset_mem, simp,
frule_tac c = "f (Suc n)" in subsetD[of "set_mult R A B" "C"],
assumption+, simp)
apply (rule ideal_pOp_closed[of C], assumption+,
cut_tac n = n in Nsetn_sub_mem1,
frule_tac b = f in forball_spec1, simp)
apply (rule univar_func_test, rule ballI, simp add:funcset_mem, assumption+)
done
lemma (in Ring) ideal_inc_set_mult:"[|A ⊆ carrier R; ideal R B; ideal R C;
set_mult R A B ⊆ C |] ==> sum_mult R A B ⊆ C"
apply (rule subsetI)
apply (frule_tac x = x in sum_mult_mem1[of A B],
simp add:ideal_subset1, assumption+)
apply (erule exE, erule bexE, frule sym, thin_tac "Σe R f n = x", simp,
thin_tac "x = Σe R f n", simp add:subsetD)
apply (simp add:ideal_inc_set_multTr)
done
lemma (in Ring) AB_inc_sum_mult:"[|ideal R A; ideal R B|] ==>
sum_mult R A B ⊆ A ∩ B"
apply (frule ideal_subset1[of A], frule ideal_subset1[of B])
apply (frule ideal_inc_set_mult [of "A" "B" "A"], assumption+)
apply (rule subsetI,
simp add:set_mult_def, (erule bexE)+, frule sym, thin_tac "xa ·r y = x",
simp,
frule_tac c = xa in subsetD[of A "carrier R"], assumption+,
frule_tac c = y in subsetD[of B "carrier R"], assumption+,
subst ring_tOp_commute, assumption+,
simp add:ideal_ring_multiple)
apply (frule ideal_inc_set_mult [of "A" "B" "B"], assumption+)
apply (rule subsetI,
simp add:set_mult_def, (erule bexE)+, frule sym, thin_tac "xa ·r y = x",
simp,
frule_tac c = xa in subsetD[of A "carrier R"], assumption+,
simp add:ideal_ring_multiple)
apply simp
done
lemma (in Ring) sum_mult_is_ideal_prod:"[|ideal R A; ideal R B|] ==>
sum_mult R A B = A ♦r B"
apply (rule equalityI)
apply (frule ideal_prod_ideal [of "A" "B"], assumption+)
apply (rule ideal_inc_set_mult)
apply (simp add:ideal_subset1)+
apply (rule subsetI)
apply (simp add:set_mult_def ideal_prod_def)
apply (auto del:subsetI)
apply (rule subsetI)
apply (simp add:ideal_prod_def)
apply (frule ideal_subset1[of A],
frule ideal_sum_mult[of A B],
frule ideal_zero[of A], blast, assumption)
apply (frule_tac a = "sum_mult R A B" in forall_spec, simp)
apply (rule subsetI, simp,
thin_tac "∀xa. ideal R xa ∧ {x. ∃i∈A. ∃j∈B. x = i ·r j} ⊆ xa --> x ∈ xa",
(erule bexE)+, simp)
apply (rule times_mem_sum_mult, assumption,
simp add:ideal_subset1, assumption+)
done
lemma (in Ring) ideal_prod_assocTr0:"[|ideal R A; ideal R B; ideal R C; y ∈ C;
z ∈ set_mult R A B|] ==> z ·r y ∈ sum_mult R A (B ♦r C)"
apply (simp add:set_mult_def, (erule bexE)+,
frule sym, thin_tac "x ·r ya = z", simp)
apply (frule_tac h = x in ideal_subset[of A], assumption,
frule_tac h = ya in ideal_subset[of B], assumption,
frule_tac h = y in ideal_subset[of C], assumption,
subst ring_tOp_assoc, assumption+)
apply (frule ideal_subset1[of A],
frule ideal_subset1[of B],
frule ideal_subset1[of C],
frule ideal_prod_ideal[of B C], assumption,
frule ideal_subset1[of "B ♦r C"])
apply (rule times_mem_sum_mult[of A "B ♦r C"], assumption+,
subst sum_mult_is_ideal_prod[of B C, THEN sym], assumption+,
rule times_mem_sum_mult[of B C], assumption+)
done
lemma (in Ring) ideal_prod_assocTr1:"[|ideal R A; ideal R B; ideal R C; y ∈ C|]
==> ∀f ∈ {j. j≤(n::nat)} -> set_mult R A B. (Σe R f n) ·r y ∈ A ♦r (B ♦r C)"
apply (cut_tac ring_is_ag)
apply (frule ideal_prod_ideal[of "B" "C"], assumption+,
subst sum_mult_is_ideal_prod[of A "B ♦r C", THEN sym], assumption+)
apply (induct_tac n)
apply (rule ballI, simp,
frule_tac x = 0 and f = f in funcset_mem[of _ "{0}" "set_mult R A B"],
simp)
apply (simp add:ideal_prod_assocTr0)
apply (rule ballI,
frule_tac b = f in forball_spec1,
thin_tac "∀f∈{j. j ≤ n} -> set_mult R A B.
Σe R f n ·r y ∈ sum_mult R A (B ♦r C)",
rule univar_func_test, rule ballI, simp,
frule_tac f = f and A = "{j. j ≤ Suc n}" and B = "set_mult R A B" and
x = x in funcset_mem, simp, assumption)
apply simp
apply (frule ideal_subset1[of A], frule ideal_subset1[of B],
frule set_mult_sub[of A B], assumption,
frule_tac f = f and A = "{j. j ≤ (Suc n)}" in extend_fun[of _ _
"set_mult R A B"
"carrier R"], assumption,
subst ring_distrib2,
simp add:ideal_subset)
apply (rule aGroup.nsum_mem, assumption) apply (rule allI,
simp add:funcset_mem) apply (simp add:funcset_mem,
frule_tac f = f and A = "{j. j ≤ Suc n}" and B = "set_mult R A B" and
x = "Suc n" in funcset_mem, simp)
apply (frule ideal_subset1[of A],
frule ideal_zero[of A],
frule ideal_sum_mult[of A "B ♦r C"], blast, assumption)
apply (rule ideal_pOp_closed, assumption+)
apply (simp add:ideal_prod_assocTr0)
done
lemma (in Ring) ideal_quotient_idealTr:"[|ideal R A; ideal R B; ideal R C;
x ∈ carrier R;∀c∈C. x ·r c ∈ ideal_quotient R A B|] ==>
f∈{j. j ≤ n} -> set_mult R B C --> x ·r (nsum R f n) ∈ A"
apply (frule ideal_subset1 [of "A"],
frule ideal_subset1 [of "B"])
apply (induct_tac n)
apply (rule impI)
apply (cut_tac n_in_Nsetn[of 0])
apply (frule funcset_mem, assumption+)
apply (thin_tac "f ∈ {j. j ≤ 0} -> set_mult R B C")
apply (simp add:set_mult_def)
apply (erule bexE)+
apply (frule sym, thin_tac "xa ·r y = f 0", simp)
apply (frule_tac h = xa in ideal_subset[of B], assumption,
frule_tac h = y in ideal_subset[of C], assumption)
apply (frule_tac x = xa and y = y in ring_tOp_commute, assumption+,
simp)
apply (subst ring_tOp_assoc[THEN sym], assumption+)
apply (frule_tac b = y in forball_spec1, assumption,
thin_tac "∀c∈C. x ·r c ∈ A \<dagger>R B")
apply (simp add:ideal_quotient_def)
(****** n ********)
apply (rule impI)
apply (frule func_pre) apply simp
apply (cut_tac ring_is_ag)
apply (frule ideal_subset1[of B], frule ideal_subset1[of C],
frule set_mult_sub[of B C], assumption+)
apply (cut_tac n = n in nsum_memr [of _ "f"],
rule allI, rule impI,
frule_tac x = i in funcset_mem, simp, simp add:subsetD)
apply (frule_tac a = n in forall_spec, simp)
apply (thin_tac "∀l≤n. Σe R f l ∈ carrier R",
frule_tac f = f and A = "{j. j ≤ Suc n}" and B = "set_mult R B C"
and x = "Suc n" in funcset_mem, simp,
frule_tac c = "f (Suc n)" in subsetD[of "set_mult R B C" " carrier R"],
assumption+)
apply (subst ring_distrib1, assumption+)
apply (rule ideal_pOp_closed[of A], assumption+)
apply (simp add: set_mult_def, (erule bexE)+,
fold set_mult_def,
frule sym, thin_tac "xa ·r y = f (Suc n)", simp)
apply (frule_tac c = xa in subsetD[of B "carrier R"], assumption+,
frule_tac c = y in subsetD[of C "carrier R"], assumption+,
frule_tac x = xa and y = y in ring_tOp_commute, assumption, simp,
subst ring_tOp_assoc[THEN sym], assumption+)
apply (simp add:ideal_quotient_def)
done
lemma (in Ring) ideal_quotient_ideal:"[|ideal R A; ideal R B; ideal R C|] ==>
A \<dagger>R B \<dagger>R C = A \<dagger>R B ♦r C"
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:ideal_quotient_def [of _ _ "C"])
apply (erule conjE)
apply (simp add:ideal_quotient_def [of _ _ "B ♦r C"])
apply (rule ballI)
apply (simp add:sum_mult_is_ideal_prod [THEN sym])
apply (simp add:sum_mult_def)
apply (erule exE, erule bexE)
apply (rename_tac x c n f)
apply (frule sym) apply simp
apply (simp add:ideal_quotient_idealTr)
apply (rule subsetI)
apply (simp add:sum_mult_is_ideal_prod [THEN sym])
apply (simp add:ideal_quotient_def)
apply (erule conjE)
apply (rule ballI)
apply (rename_tac x c)
apply (frule ideal_subset [of "C"], assumption+)
apply (simp add:ring_tOp_closed)
apply (rule ballI)
apply (rename_tac x v u)
apply (frule ideal_subset [of "B"], assumption+)
apply (subst ring_tOp_assoc, assumption+)
apply (frule ideal_subset1[of B],
frule ideal_subset1[of C],
frule_tac a = u and b = v in times_mem_sum_mult[of B C], assumption+)
apply (frule_tac x = u and y = v in ring_tOp_commute, assumption,
simp)
done
lemma (in Ring) ideal_prod_assocTr:"[|ideal R A; ideal R B; ideal R C|] ==>
∀f. (f ∈ {j. j ≤ (n::nat)} -> set_mult R (A ♦r B) C -->
(Σe R f n) ∈ A ♦r (B ♦r C))"
apply (subgoal_tac "∀x∈(A ♦r B). ∀y∈C. x ·r y ∈ A ♦r (B ♦r C)")
apply (induct_tac n)
apply (rule allI) apply (rule impI)
apply (frule_tac f = f and A = "{j. j ≤ 0}" and B = "set_mult R (A ♦r B) C"
and x = 0 in funcset_mem, simp, simp)
apply (thin_tac "f ∈ {0} -> set_mult R (A ♦r B) C")
apply (simp add:set_mult_def)
apply ((erule bexE)+, frule sym, thin_tac "x ·r y = f 0", simp)
apply (rule allI, rule impI)
apply (frule func_pre)
apply (frule_tac a = f in forall_spec, simp,
thin_tac "∀f. f ∈ {j. j ≤ n} -> set_mult R (A ♦r B) C -->
Σe R f n ∈ A ♦r (B ♦r C)",
frule ideal_prod_ideal[of "B" "C"], assumption+,
frule ideal_prod_ideal[of "A" "B ♦r C"], assumption+, simp)
apply (rule ideal_pOp_closed[of "A ♦r (B ♦r C)"], assumption+)
apply (cut_tac n = "Suc n" in n_in_Nsetn,
frule_tac f = f and A = "{j. j ≤ Suc n}" and
B = "set_mult R (A ♦r B) C" and x = "Suc n" in funcset_mem, assumption)
apply (thin_tac "f ∈ {j. j ≤ n} -> set_mult R (A ♦r B) C",
thin_tac "f ∈ {j. j ≤ Suc n} -> set_mult R (A ♦r B) C")
apply (simp add:set_mult_def)
apply ((erule bexE)+,
frule sym, thin_tac "x ·r y = f (Suc n)", simp)
apply (rule ballI)+
apply (simp add:sum_mult_is_ideal_prod[of A B, THEN sym])
apply (frule ideal_subset1[of A], frule ideal_subset1[of B],
frule_tac x = x in sum_mult_mem1[of A B], assumption+)
apply (erule exE, erule bexE, frule sym, thin_tac "Σe R f n = x",
simp)
apply (simp add:ideal_prod_assocTr1)
done
lemma (in Ring) ideal_prod_assoc:"[|ideal R A; ideal R B; ideal R C|] ==>
(A ♦r B) ♦r C = A ♦r (B ♦r C)"
apply (rule equalityI)
apply (rule subsetI)
apply (frule ideal_prod_ideal[of "A" "B"], assumption+)
apply (frule sum_mult_is_ideal_prod[of "A ♦r B" "C"], assumption+)
apply (frule sym) apply (thin_tac "sum_mult R (A ♦r B) C = (A ♦r B) ♦r C")
apply simp apply (thin_tac "(A ♦r B) ♦r C = sum_mult R (A ♦r B) C")
apply (thin_tac "ideal R (A ♦r B)")
apply (frule ideal_prod_ideal[of "B" "C"], assumption+)
apply (simp add:sum_mult_def)
apply (erule exE, erule bexE)
apply (frule sym, thin_tac "Σe R f n = x", simp)
apply (simp add:ideal_prod_assocTr)
apply (rule subsetI)
apply (frule ideal_prod_ideal[of "B" "C"], assumption+)
apply (simp add:ideal_prod_commute [of "A" "B ♦r C"])
apply (frule ideal_prod_ideal[of "A" "B"], assumption+)
apply (simp add:ideal_prod_commute[of "A ♦r B" "C"])
apply (simp add:ideal_prod_commute[of "A" "B"])
apply (simp add:ideal_prod_commute[of "B" "C"])
apply (frule ideal_prod_ideal[of "C" "B"], assumption+)
apply (frule sum_mult_is_ideal_prod[of "C ♦r B" "A"], assumption+)
apply (frule sym) apply (thin_tac "sum_mult R (C ♦r B) A = (C ♦r B) ♦r A")
apply simp apply (thin_tac "(C ♦r B) ♦r A = sum_mult R (C ♦r B) A")
apply (thin_tac "ideal R (C ♦r B)")
apply (frule ideal_prod_ideal[of "B" "A"], assumption+)
apply (simp add:sum_mult_def)
apply (erule exE, erule bexE)
apply (frule sym, thin_tac "Σe R f n = x", simp)
apply (simp add:ideal_prod_assocTr)
done
lemma (in Ring) prod_principal_idealTr0:" [|a ∈ carrier R; b ∈ carrier R;
z ∈ set_mult R (R ♦p a) (R ♦p b)|] ==> z ∈ R ♦p (a ·r b)"
apply (simp add:set_mult_def, (erule bexE)+,
simp add:Rxa_def, (erule bexE)+, simp)
apply (frule_tac x = r and y = a and z = "ra ·r b" in ring_tOp_assoc,
assumption+, simp add:ring_tOp_closed, simp)
apply (simp add:ring_tOp_assoc[THEN sym, of a _ b])
apply (frule_tac x = a and y = ra in ring_tOp_commute, assumption+, simp)
apply (simp add:ring_tOp_assoc[of _ a b],
frule_tac x = a and y = b in ring_tOp_closed, assumption)
apply (simp add:ring_tOp_assoc[THEN sym, of _ _ "a ·r b"],
frule sym, thin_tac "r ·r ra ·r (a ·r b) = z", simp,
frule_tac x = r and y = ra in ring_tOp_closed, assumption+)
apply blast
done
lemma (in Ring) prod_principal_idealTr1:" [|a ∈ carrier R; b ∈ carrier R|] ==>
∀f ∈ {j. j ≤ (n::nat)} -> set_mult R (R ♦p a) (R ♦p b).
Σe R f n ∈ R ♦p (a ·r b)"
apply (induct_tac n)
apply (rule ballI,
frule_tac f = f in funcset_mem[of _ "{j. j ≤ 0}"
"set_mult R (R ♦p a) (R ♦p b)"], simp)
apply (simp add:prod_principal_idealTr0)
apply (rule ballI,
frule func_pre,
frule_tac b = f in forball_spec1, assumption,
thin_tac "∀f∈{j. j ≤ n} -> set_mult R (R ♦p a) (R ♦p b).
Σe R f n ∈ R ♦p (a ·r b)")
apply (frule ring_tOp_closed[of a b], assumption)
apply (frule principal_ideal[of "a ·r b"], simp,
rule ideal_pOp_closed, assumption+)
apply (cut_tac n = "Suc n" in n_in_Nsetn,
frule_tac f = f and A = "{j. j ≤ Suc n}" and
B = "set_mult R (R ♦p a) (R ♦p b)" in funcset_mem, assumption)
apply (simp add:prod_principal_idealTr0)
done
lemma (in Ring) prod_principal_ideal:"[|a ∈ carrier R; b ∈ carrier R|] ==>
(Rxa R a) ♦r (Rxa R b) = Rxa R (a ·r b)"
apply (frule principal_ideal[of "a"],
frule principal_ideal[of "b"])
apply (subst sum_mult_is_ideal_prod[THEN sym, of "Rxa R a" "Rxa R b"],
assumption+)
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:sum_mult_def)
apply (erule exE, erule bexE)
apply (frule sym, thin_tac "Σe R f n = x", simp, thin_tac "x = Σe R f n")
apply (simp add:prod_principal_idealTr1)
apply (rule subsetI)
apply (simp add:Rxa_def, fold Rxa_def)
apply (erule bexE)
apply (simp add:ring_tOp_assoc[THEN sym])
apply (frule ideal_subset1[of "R ♦p a"],
frule ideal_subset1[of "R ♦p b"])
apply (rule_tac a = "r ·r a" and b = b in times_mem_sum_mult[of "R ♦p a"
"R ♦p b"], assumption+)
apply (simp add:Rxa_def, blast)
apply (simp add:a_in_principal)
done
lemma (in Ring) principal_ideal_n_pow1:"a ∈ carrier R ==>
(Rxa R a)♦R n = Rxa R (a^R n )"
apply (cut_tac ring_one)
apply (induct_tac n)
apply simp
apply (cut_tac a_in_principal[of "1r"])
apply (frule principal_ideal[of "1r"])
apply (frule ideal_inc_one, assumption, simp)
apply (simp add:ring_one)
apply simp
apply (frule_tac n = n in npClose[of a],
subst prod_principal_ideal, assumption+)
apply (simp add:ring_tOp_commute)
done
lemma (in Ring) principal_ideal_n_pow:"[|a ∈ carrier R; I = Rxa R a|] ==>
I ♦R n = Rxa R (a^R n)"
apply simp
apply (rule principal_ideal_n_pow1[of "a" "n"], assumption+)
done
text{* more about ideal_n_prod *}
lemma (in Ring) nprod_eqTr:" f ∈ {j. j ≤ (n::nat)} -> carrier R ∧
g ∈ {j. j ≤ n} -> carrier R ∧ (∀j ≤ n. f j = g j) -->
nprod R f n = nprod R g n"
apply (induct_tac n)
apply simp
apply (rule impI, (erule conjE)+)
apply (frule func_pre[of f], frule func_pre[of g],
cut_tac n = n in Nsetn_sub_mem1, simp)
done
lemma (in Ring) nprod_eq:"[|∀j ≤ n. f j ∈ carrier R; ∀j ≤ n. g j ∈ carrier R;
(∀j ≤ (n::nat). f j = g j)|] ==> nprod R f n = nprod R g n"
apply (cut_tac nprod_eqTr[of f n g])
apply simp
apply (cut_tac univar_func_test[of "{j. j ≤ n}" f "carrier R"],
cut_tac univar_func_test[of "{j. j ≤ n}" g "carrier R"])
apply simp
apply (rule ballI, simp)
apply (rule ballI) apply simp
done
constdefs
mprod_expR::"[('b, 'm) Ring_scheme, nat => nat, nat => 'b, nat]
=> 'b"
"mprod_expR R e f n == nprod R (λj. ((f j)^R (e j))) n"
(** Note that e j is a natural number for all j in Nset n **)
lemma (in Ring) mprodR_Suc:"[|e ∈ {j. j ≤ (Suc n)} -> {j. (0::nat) ≤ j};
f ∈ {j. j ≤ (Suc n)} -> carrier R|] ==>
mprod_expR R e f (Suc n) =
(mprod_expR R e f n) ·r ((f (Suc n))^R (e (Suc n)))"
apply (simp add:mprod_expR_def)
done
lemma (in Ring) mprod_expR_memTr:"e ∈ {j. j ≤ n} -> {j. (0::nat) ≤ j} ∧
f ∈ {j. j ≤ n} -> carrier R --> mprod_expR R e f n ∈ carrier R"
apply (induct_tac n)
apply (rule impI, (erule conjE)+)
apply (cut_tac n_in_Nsetn[of 0],
simp add: mprod_expR_def)
apply (rule npClose,
simp add:funcset_mem)
apply (rule impI, (erule conjE)+)
apply (frule func_pre[of "e"], frule func_pre[of "f"])
apply simp
apply (simp add:mprodR_Suc)
apply (rule ring_tOp_closed, assumption+)
apply (rule npClose, cut_tac n = "Suc n" in n_in_Nsetn)
apply (simp add:funcset_mem)
done
lemma (in Ring) mprod_expR_mem:"[| e ∈ {j. j ≤ n} -> {j. (0::nat) ≤ j};
f ∈ {j. j ≤ n} -> carrier R|] ==> mprod_expR R e f n ∈ carrier R"
apply (simp add:mprod_expR_memTr)
done
lemma (in Ring) prod_n_principal_idealTr:"e ∈ {j. j≤n} -> {j. (0::nat)≤j} ∧
f ∈ {j. j≤n} -> carrier R ∧ (∀k ≤ n. J k = (Rxa R (f k))♦R (e k)) -->
ideal_n_prod R n J = Rxa R (mprod_expR R e f n)"
apply (induct_tac n)
apply (rule impI) apply (erule conjE)+
apply (simp add:mprod_expR_def)
apply (subgoal_tac "J 0 = R ♦p (f 0) ♦R (e 0)")
apply simp
apply (rule principal_ideal_n_pow[of "f 0" "R ♦p (f 0)"])
apply (cut_tac n_in_Nsetn[of 0], simp add:funcset_mem) apply simp
apply (cut_tac n_in_Nsetn[of 0], simp)
apply (rule impI, (erule conjE)+)
apply (frule func_pre[of "e"], frule func_pre[of "f"])
apply (cut_tac n = n in Nsetn_sub_mem1,
simp add:mprodR_Suc)
apply (cut_tac n = "Suc n" in n_in_Nsetn, simp)
apply (frule_tac A = "{j. j ≤ Suc n}" and x = "Suc n" in funcset_mem[of "f" _ "carrier R"], simp)
apply (frule_tac a = "f (Suc n)" and I = "R ♦p (f (Suc n))" and n = "e (Suc n)" in principal_ideal_n_pow) apply simp
apply (subst prod_principal_ideal[THEN sym])
apply (simp add:mprod_expR_mem)
apply (rule npClose, assumption+) apply simp
done
(************* used in Valuation2.thy *****************)
lemma (in Ring) prod_n_principal_ideal:"[|e ∈ {j. j≤n} -> {j. (0::nat)≤j};
f ∈ {j. j≤n} -> carrier R; ∀k≤ n. J k = (Rxa R (f k))♦R (e k)|] ==>
ideal_n_prod R n J = Rxa R (mprod_expR R e f n)"
apply (simp add:prod_n_principal_idealTr[of e n f J])
done
(*******************************************************)
lemma (in Idomain) a_notin_n_pow1:"[|a ∈ carrier R; ¬ Unit R a; a ≠ \<zero>; 0 < n|]
==> a ∉ (Rxa R a) ♦R (Suc n)"
apply (rule contrapos_pp)
apply (simp del:ipSuc) apply (simp del:ipSuc)
apply (frule principal_ideal[of "a"])
apply (frule principal_ideal_n_pow[of "a" "R ♦p a" "Suc n"])
apply simp apply (simp del:ipSuc)
apply (thin_tac "R ♦p a ♦R (Suc n) = R ♦p (a^R n ·r a)")
apply (thin_tac "ideal R (R ♦p a)")
apply (simp add:Rxa_def)
apply (erule bexE)
apply (frule npClose[of "a" "n"])
apply (simp add:ring_tOp_assoc[THEN sym])
apply (frule ring_l_one[THEN sym, of "a"])
apply (subgoal_tac "1r ·r a = r ·r a^R n ·r a")
apply (cut_tac b = "r ·r (a^R n)" in idom_mult_cancel_r[of "1r" _ "a"])
apply (simp add:ring_one) apply (simp add:ring_tOp_closed)
apply assumption+
apply (thin_tac "1r ·r a = r ·r a^R n ·r a",
thin_tac "a = 1r ·r a",
thin_tac "a = r ·r a^R n ·r a")
apply (subgoal_tac "1r = r ·r (a^R (Suc (n - Suc 0)))") prefer 2
apply (simp del:ipSuc)
apply (thin_tac "1r = r ·r a^R n")
apply (simp del:Suc_pred)
apply (frule npClose[of "a" "n - Suc 0"])
apply (simp add:ring_tOp_assoc[THEN sym])
apply (frule_tac x = r and y = "a^R (n - Suc 0)" in ring_tOp_closed, assumption)
apply (simp add:ring_tOp_commute[of _ a])
apply (simp add:Unit_def) apply blast
apply simp
done
lemma (in Idomain) a_notin_n_pow2:"[|a ∈ carrier R; ¬ Unit R a; a ≠ \<zero>;
0 < n|] ==> a^R n ∉ (Rxa R a) ♦R (Suc n)"
apply (rule contrapos_pp)
apply (simp del:ipSuc, simp del:ipSuc)
apply (frule principal_ideal[of "a"])
apply (frule principal_ideal_n_pow[of "a" "R ♦p a" "Suc n"])
apply (simp, simp del:ipSuc)
apply (thin_tac "R ♦p a ♦R (Suc n) = R ♦p (a^R n ·r a)")
apply (thin_tac "ideal R (R ♦p a)")
apply (simp add:Rxa_def)
apply (erule bexE)
apply (frule idom_potent_nonzero[of "a" "n"], assumption+)
apply (frule npClose[of "a" "n"])
apply (frule ring_l_one[THEN sym, of "a^R n "])
apply (subgoal_tac "1r ·r (a^R n) = r ·r ((a^R n) ·r a)")
prefer 2 apply simp
apply (thin_tac "a^R n = 1r ·r a^R n",
thin_tac "a^R n = r ·r (a^R n ·r a)")
apply (simp add:ring_tOp_commute[of "a^R n" a])
apply (simp add:ring_tOp_assoc[THEN sym])
apply (cut_tac ring_one,
frule_tac b = "r ·r a" in idom_mult_cancel_r[of "1r" _ "a^R n"],
simp add:ring_tOp_closed,
assumption+)
apply (simp add:ring_tOp_commute[of _ a])
apply (simp add:Unit_def, blast)
done
lemma (in Idomain) n_pow_not_prime:"[|a ∈ carrier R; a ≠ \<zero>; 0 < n|]
==> ¬ prime_ideal R ((Rxa R a) ♦R (Suc n))"
apply (case_tac "n = 0")
apply simp
apply (case_tac "Unit R a")
apply (simp del:ipSuc add:prime_ideal_def, rule impI)
apply (frule principal_ideal[of "a"])
apply (frule principal_ideal_n_pow[of "a" "R ♦p a" "Suc n"])
apply simp apply (simp del:npow_suc)
apply (simp del:npow_suc add:idom_potent_unit [of "a" "Suc n"])
apply (thin_tac "R ♦p a ♦r R ♦p a ♦R n = R ♦p (a^R (Suc n))")
apply (frule npClose[of "a" "Suc n"])
apply (frule a_in_principal[of "a^R (Suc n)"])
apply (simp add: ideal_inc_unit)
apply (frule a_notin_n_pow1[of "a" "n"], assumption+)
apply (frule a_notin_n_pow2[of "a" "n"], assumption+)
apply (frule npClose[of "a" "n"])
apply (frule principal_ideal[of "a"])
apply (frule principal_ideal_n_pow[of "a" "R ♦p a" "Suc n"])
apply simp apply (simp del:ipSuc npow_suc)
apply (thin_tac "R ♦p a ♦R (Suc n) = R ♦p (a^R (Suc n))")
apply (subst prime_ideal_def)
apply (simp del:npow_suc) apply (rule impI)
apply (subgoal_tac "(a^R n) ·r a ∈ R ♦p (a^R (Suc n))")
apply blast
apply (simp add:Rxa_def)
apply (frule ring_tOp_closed[of "a" "a^R n"], assumption+)
apply (frule ring_l_one[THEN sym, of "a ·r (a^R n)"])
apply (cut_tac ring_one)
apply (simp add:ring_tOp_commute[of _ a], blast)
done
lemma (in Idomain) principal_pow_prime_condTr:
"[|a ∈ carrier R; a ≠ \<zero>; prime_ideal R ((Rxa R a) ♦R (Suc n))|] ==> n = 0"
apply (rule contrapos_pp, (simp del:ipSuc)+)
apply (frule n_pow_not_prime[of "a" "n"], assumption+)
apply (simp del:ipSuc)
done
lemma (in Idomain) principal_pow_prime_cond:
"[|a ∈ carrier R; a ≠ \<zero>; prime_ideal R ((Rxa R a) ♦R n)|] ==> n = Suc 0"
apply (case_tac "n = 0")
apply simp
apply (simp add:prime_ideal_def) apply (erule conjE)
apply (cut_tac ring_one, simp)
apply (subgoal_tac "prime_ideal R (R ♦p a ♦R (Suc (n - Suc 0)))")
apply (frule principal_pow_prime_condTr[of "a" "n - Suc 0"], assumption+)
apply simp apply simp
done
section "10. extension and contraction"
locale TwoRings = Ring +
fixes R' (structure)
assumes secondR: "Ring R'"
constdefs
i_contract::"['a => 'b, ('a, 'm1) Ring_scheme, ('b, 'm2) Ring_scheme,
'b set] => 'a set"
"i_contract f R R' J == invim f (carrier R) J"
i_extension::"['a => 'b, ('a, 'm1) Ring_scheme, ('b, 'm2) Ring_scheme,
'a set] => 'b set"
"i_extension f R R' I == sum_mult R' (f ` I) (carrier R')"
lemma (in TwoRings) i_contract_sub:"[|f ∈ rHom R R'; ideal R' J |] ==>
(i_contract f R R' J) ⊆ carrier R"
apply (simp add:i_contract_def invim_def)
apply blast
done
lemma (in TwoRings) i_contract_ideal:"[|f ∈ rHom R R'; ideal R' J |] ==>
ideal R (i_contract f R R' J)"
apply (cut_tac Ring,
cut_tac secondR)
apply (rule ideal_condition)
apply (simp add:i_contract_sub)
apply (simp add:i_contract_def invim_def)
apply (cut_tac ring_zero)
apply (cut_tac Ring)
apply (frule rHom_0_0[of R R' f], assumption+,
cut_tac Ring.ideal_zero[of R' J])
apply (frule sym, thin_tac "f \<zero> = \<zero>R'", simp, blast,
assumption+)
apply (rule ballI)+
apply (simp add:i_contract_def invim_def, (erule conjE)+)
apply (cut_tac ring_is_ag,
frule_tac x = y in aGroup.ag_mOp_closed[of R], assumption)
apply (simp add:aGroup.ag_pOp_closed)
apply (simp add:rHom_add)
apply (frule_tac x = y in rHom_inv_inv[of R R' _ f], assumption+, simp,
thin_tac "f (-a y) = -aR' (f y)",
frule_tac x = "f y" in Ring.ideal_inv1_closed[of R' J], assumption+,
rule Ring.ideal_pOp_closed[of R'], assumption+)
apply ((rule ballI)+,
simp add:i_contract_def invim_def, erule conjE,
simp add:ring_tOp_closed,
simp add:rHom_tOp)
apply (frule_tac a = r in rHom_mem[of f R R'], assumption,
simp add:Ring.ideal_ring_multiple[of R' J])
done
lemma (in TwoRings) i_contract_mono:"[|f ∈ rHom R R'; ideal R' J1; ideal R' J2;
J1 ⊆ J2 |] ==> i_contract f R R' J1 ⊆ i_contract f R R' J2"
apply (rule subsetI)
apply (simp add:i_contract_def invim_def) apply (erule conjE)
apply (rule subsetD, assumption+)
done
lemma (in TwoRings) i_contract_prime:"[|f ∈ rHom R R'; prime_ideal R' P|] ==>
prime_ideal R (i_contract f R R' P)"
apply (cut_tac Ring,
cut_tac secondR)
apply (simp add:prime_ideal_def, (erule conjE)+)
apply (simp add:i_contract_ideal)
apply (rule conjI)
apply (rule contrapos_pp, simp+)
apply (simp add:i_contract_def invim_def, erule conjE)
apply (simp add:rHom_one)
apply (rule ballI)+
apply (frule_tac a = x in rHom_mem[of "f" "R" "R'"], assumption+,
frule_tac a = y in rHom_mem[of "f" "R" "R'"], assumption+)
apply (rule impI)
apply (simp add:i_contract_def invim_def, erule conjE)
apply (simp add:rHom_tOp)
done
lemma (in TwoRings) i_extension_ideal:"[|f ∈ rHom R R'; ideal R I |] ==>
ideal R' (i_extension f R R' I)"
apply (cut_tac Ring, cut_tac secondR)
apply (simp add:i_extension_def)
apply (rule Ring.ideal_sum_mult [of "R'" "f ` I" "carrier R'"], assumption+)
apply (rule subsetI)
apply (simp add:image_def)
apply (erule bexE, frule_tac a = xa in rHom_mem[of f R R'],
rule ideal_subset, assumption+, simp)
apply (frule ideal_zero, simp, blast)
apply (simp add:Ring.whole_ideal[of R'])
done
lemma (in TwoRings) i_extension_mono:"[|f ∈ rHom R R'; ideal R I1; ideal R I2;
I1 ⊆ I2 |] ==> (i_extension f R R' I1) ⊆ (i_extension f R R' I2)"
apply (rule subsetI)
apply (simp add:i_extension_def)
apply (simp add:sum_mult_def)
apply (erule exE, erule bexE)
apply (cut_tac Ring.set_mult_mono[of R' "f ` I1" "f ` I2" "carrier R'"])
apply (frule_tac f = fa and A = "{j. j ≤ n}" in extend_fun[of _ _
"set_mult R' (f ` I1) (carrier R')" "set_mult R' (f ` I2) (carrier R')"],
assumption+) apply blast
apply (simp add:secondR)
apply (simp add:image_def, rule subsetI, simp, erule bexE,
frule_tac h = xb in ideal_subset[of I1], assumption, simp add:rHom_mem)
apply (simp add:image_def, rule subsetI, simp, erule bexE,
frule_tac h = xb in ideal_subset[of I2], assumption, simp add:rHom_mem)
apply (rule subsetI,
simp add:image_def, erule bexE,
frule_tac c = xb in subsetD[of I1 I2], assumption+, blast)
apply simp
done
lemma (in TwoRings) e_c_inc_self:"[|f ∈ rHom R R'; ideal R I|] ==>
I ⊆ i_contract f R R' (i_extension f R R' I)"
apply (rule subsetI)
apply (simp add:i_contract_def i_extension_def invim_def)
apply (simp add:ideal_subset)
apply (cut_tac secondR,
frule Ring.ring_one [of "R'"])
apply (frule_tac h = x in ideal_subset[of I], assumption,
frule_tac f = f and A = R and R = R' and a = x in rHom_mem, assumption)
apply (frule_tac t = "f x" in Ring.ring_r_one[THEN sym, of R'], assumption)
apply (frule_tac a = "f x" and b = "1rR'" in Ring.times_mem_sum_mult[of R'
"f ` I" "carrier R'"],
rule subsetI,
simp add:image_def, erule bexE,
frule_tac h = xb in ideal_subset[of I], assumption,
simp add:rHom_mem, simp,
simp add:image_def, blast, assumption+)
apply simp
done
lemma (in TwoRings) c_e_incd_self:"[|f ∈ rHom R R'; ideal R' J |] ==>
i_extension f R R' (i_contract f R R' J) ⊆ J"
apply (rule subsetI)
apply (simp add:i_extension_def)
apply (simp add:sum_mult_def)
apply (erule exE, erule bexE)
apply (cut_tac secondR,
frule_tac n = n and f = fa in Ring.ideal_nsum_closed[of R' J ],
assumption)
apply (rule allI, rule impI) apply (
frule_tac f = fa and A = "{j. j ≤ n}" and
B = "set_mult R' (f ` i_contract f R R' J) (carrier R')" and x = j in
funcset_mem, simp) apply (
thin_tac "fa ∈ {j. j ≤ n} -> set_mult R' (f ` i_contract f R R' J) (carrier R')")
apply (simp add:set_mult_def, (erule bexE)+,
simp add:i_contract_def invim_def, erule conjE)
apply (frule_tac x = "f xa" and r = y in Ring.ideal_ring_multiple1[of R' J],
assumption+, simp)
apply simp
done
lemma (in TwoRings) c_e_c_eq_c:"[|f ∈ rHom R R'; ideal R' J |] ==>
i_contract f R R' (i_extension f R R' (i_contract f R R' J))
= i_contract f R R' J"
apply (frule i_contract_ideal [of "f" "J"], assumption)
apply (frule e_c_inc_self [of "f" "i_contract f R R' J"], assumption+)
apply (frule c_e_incd_self [of "f" "J"], assumption+)
apply (frule i_contract_mono [of "f"
"i_extension f R R' (i_contract f R R' J)" "J"])
apply (rule i_extension_ideal, assumption+)
apply (rule equalityI, assumption+)
done
lemma (in TwoRings) e_c_e_eq_e:"[|f ∈ rHom R R'; ideal R I |] ==>
i_extension f R R' (i_contract f R R' (i_extension f R R' I))
= i_extension f R R' I"
apply (frule i_extension_ideal [of "f" "I"], assumption+)
apply (frule c_e_incd_self [of "f" "i_extension f R R' I"], assumption+)
apply (rule equalityI, assumption+)
apply (thin_tac "i_extension f R R' (i_contract f R R' (i_extension f R R' I))
⊆ i_extension f R R' I")
apply (frule e_c_inc_self [of "f" "I"], assumption+)
apply (rule i_extension_mono [of "f" "I"
"i_contract f R R' (i_extension f R R' I)"], assumption+)
apply (rule i_contract_ideal, assumption+)
done
section "11. complete system of representatives"
constdefs (structure R)
csrp_fn :: "[_, 'a set] => 'a set => 'a"
"csrp_fn R I == λx∈carrier (R /r I). (if x = I then \<zero> else SOME y. y ∈ x)"
csrp :: "[_ , 'a set] => 'a set"
"csrp R I == (csrp_fn R I) ` (carrier (R /r I))"
(** complete system of representatives having 1-1 correspondence with
carrier (R /r I) **)
lemma (in Ring) csrp_mem:"[|ideal R I; a ∈ carrier R|] ==>
csrp_fn R I (a \<uplus>R I) ∈ a \<uplus>R I"
apply (simp add:csrp_fn_def qring_carrier)
apply (case_tac "a \<uplus>R I = I") apply simp
apply (rule conjI, rule impI)
apply (simp add:ideal_zero)
apply (rule impI)
apply (cut_tac ring_zero)
apply (frule_tac b = \<zero> in forball_spec1, assumption+)
apply (thin_tac "∀a∈carrier R. a \<uplus>R I ≠ I")
apply (frule ideal_zero[of "I"])
apply (frule ar_coset_same4[of "I" "\<zero>"], assumption+, simp)
apply simp
apply (rule conjI)
apply (rule impI, rule someI2_ex)
apply (frule a_in_ar_coset[of "I" "a"], assumption+, blast, assumption+)
apply (rule impI)
apply (frule_tac b = a in forball_spec1, assumption+,
thin_tac "∀aa∈carrier R. aa \<uplus>R I ≠ a \<uplus>R I", simp)
done
lemma (in Ring) csrp_same:"[|ideal R I; a ∈ carrier R|] ==>
csrp_fn R I (a \<uplus>R I) \<uplus>R I = a \<uplus>R I"
apply (frule csrp_mem[of "I" "a"], assumption+)
apply (rule ar_cos_same[of "a" "I" "csrp_fn R I (a \<uplus>R I)"], assumption+)
done
lemma (in Ring) csrp_mem1:"[|ideal R I; x ∈ carrier (R /r I)|] ==>
csrp_fn R I x ∈ x"
apply (simp add:qring_carrier, erule bexE, frule sym,
thin_tac "a \<uplus>R I = x", simp)
apply (simp add:csrp_mem)
done
lemma (in Ring) csrp_fn_mem:"[|ideal R I; x ∈ carrier (R /r I)|] ==>
(csrp_fn R I x) ∈ carrier R"
apply (simp add:qring_carrier, erule bexE, frule sym,
thin_tac "a \<uplus>R I = x", simp,
frule_tac a = a in csrp_mem[of "I"], assumption+)
apply (rule_tac a = a and x = "csrp_fn R I (a \<uplus>R I)" in
ar_coset_subsetD[of "I"], assumption+)
done
lemma (in Ring) csrp_eq_coset:"[|ideal R I; x ∈ carrier (R /r I)|] ==>
(csrp_fn R I x) \<uplus>R I = x"
apply (simp add:qring_carrier, erule bexE)
apply (frule sym, thin_tac "a \<uplus>R I = x", simp)
apply (frule_tac a = a in csrp_mem[of "I"], assumption+)
apply (rule ar_cos_same, assumption+)
done
lemma (in Ring) csrp_nz_nz:"[|ideal R I; x ∈ carrier (R /r I);
x ≠ \<zero>(R /r I)|] ==> (csrp_fn R I x) ≠ \<zero>"
apply (rule contrapos_pp, simp+)
apply (frule csrp_eq_coset[of "I" "x"], assumption+, simp)
apply (simp add:qring_zero[of "I"])
apply (frule ideal_zero[of "I"]) apply (
cut_tac ring_zero)
apply (simp add:Qring_fix1 [of "\<zero>" "I"])
done
lemma (in Ring) csrp_diff_in_vpr:"[|ideal R I; x ∈ carrier R|] ==>
x ± (-a (csrp_fn R I (pj R I x))) ∈ I"
apply (frule csrp_mem[of "I" "x"],
frule csrp_same[of "I" "x"],
simp add:pj_mem, assumption,
frule ar_coset_subsetD[of I x "csrp_fn R I (x \<uplus>R I)"],
assumption+)
apply (frule belong_ar_coset2[of I x "csrp_fn R I (x \<uplus>R I)"], assumption+,
frule ideal_inv1_closed[of I "csrp_fn R I (x \<uplus>R I) ± -a x"], assumption+,
cut_tac ring_is_ag,
frule aGroup.ag_mOp_closed[of R x], assumption,
simp add:aGroup.ag_pOp_commute[of R "csrp_fn R I (x \<uplus>R I)" "-a x"])
apply (simp add:aGroup.ag_p_inv[of R "-a x" "csrp_fn R I (x \<uplus>R I)"],
simp add:aGroup.ag_inv_inv,
cut_tac Ring, simp add:pj_mem[of R I x])
done
lemma (in Ring) csrp_pj:"[|ideal R I; x ∈ carrier (R /r I)|] ==>
(pj R I) (csrp_fn R I x) = x"
apply(cut_tac Ring,
frule csrp_fn_mem[of "I" "x"], assumption+,
simp add:pj_mem[of "R" "I" "csrp_fn R I x"],
simp add:csrp_eq_coset)
done
section "12. polynomial ring"
text{* In this section, we treat a ring of polynomials over a ring S.
Numbers are of type ant *}
constdefs
pol_coeff::"[('a, 'more) Ring_scheme, (nat × (nat => 'a))] => bool"
"pol_coeff S c == (∀j ≤ (fst c). (snd c) j ∈ carrier S)"
c_max::"[('a, 'more) Ring_scheme, nat × (nat => 'a)] => nat"
"c_max S c == if {j. j ≤ (fst c) ∧ (snd c) j ≠ \<zero>S} = {} then 0 else
n_max {j. j ≤ (fst c) ∧ (snd c) j ≠ \<zero>S}"
polyn_expr::"[('a, 'more) Ring_scheme, 'a, nat, nat × (nat => 'a)] => 'a"
"polyn_expr R X k c == nsum R (λj. ((snd c) j) ·rR (X^R j)) k"
algfree_cond::"[('a, 'm) Ring_scheme, ('a, 'm1) Ring_scheme,
'a] => bool"
"algfree_cond R S X == ∀c. pol_coeff S c ∧ (∀k ≤ (fst c).
(nsum R (λj. ((snd c) j) ·rR (X^R j)) k = \<zero>R -->
(∀j ≤ k. (snd c) j = \<zero>S)))"
locale PolynRg = Ring +
fixes S (structure)
fixes X (structure)
assumes X_mem_R:"X ∈ carrier R"
and not_zeroring:"¬ Zero_ring S"
and subring: "Subring R S"
and algfree: "algfree_cond R S X"
and S_X_generate:"x ∈ carrier R ==>
∃f. pol_coeff S f ∧ x = polyn_expr R X (fst f) f"
(** a polynomial is an element of a polynomial ring **)
section "13. addition and multiplication of polyn_exprs"
subsection "simple properties of a polyn_ring"
lemma Subring_subset:"Subring R S ==> carrier S ⊆ carrier R"
by (simp add:Subring_def)
lemma (in Ring) subring_Ring:"Subring R S ==> Ring S"
by (simp add:Subring_def)
lemma (in Ring) mem_subring_mem_ring:"[|Subring R S; x ∈ carrier S|] ==>
x ∈ carrier R"
by (simp add:Subring_def, (erule conjE)+, simp add: subsetD)
lemma (in Ring) Subring_pOp_ring_pOp:"[|Subring R S; a ∈ carrier S;
b ∈ carrier S |] ==> a ±S b = a ± b"
apply (simp add:Subring_def, (erule conjE)+)
apply (frule rHom_add[of "ridmap S" S R a b], assumption+)
apply (cut_tac Ring.ring_is_ag[of S],
frule aGroup.ag_pOp_closed[of S a b], assumption+,
simp add:ridmap_def, assumption)
done
lemma (in Ring) Subring_tOp_ring_tOp:"[|Subring R S; a ∈ carrier S;
b ∈ carrier S |] ==> a ·rS b = a ·r b"
apply (simp add:Subring_def, (erule conjE)+)
apply (frule rHom_tOp[of "S" "R" "a" "b" "ridmap S"], rule Ring_axioms, assumption+)
apply (frule Ring.ring_tOp_closed[of "S" "a" "b"], assumption+,
simp add:ridmap_def)
done
lemma (in Ring) Subring_one_ring_one:"Subring R S ==> 1rS = 1r"
apply (simp add:Subring_def, (erule conjE)+)
apply (frule rHom_one[of "S" "R" "ridmap S"], rule Ring_axioms, assumption+)
apply (simp add:ridmap_def, simp add:Ring.ring_one[of S])
done
lemma (in Ring) Subring_zero_ring_zero:"Subring R S ==> \<zero>S = \<zero>"
apply (simp add:Subring_def, (erule conjE)+,
frule rHom_0_0[of "S" "R" "ridmap S"], rule Ring_axioms, assumption+,
simp add:ridmap_def, simp add:Ring.ring_zero[of "S"])
done
lemma (in Ring) Subring_minus_ring_minus:"[|Subring R S; x ∈ carrier S|]
==> -aS x = -a x"
apply (simp add:Subring_def, (erule conjE)+, simp add:rHom_def, (erule conjE)+)
apply (cut_tac ring_is_ag, frule Ring.ring_is_ag[of "S"])
apply (frule aHom_inv_inv[of "S" "R" "ridmap S" "x"], assumption+,
frule aGroup.ag_mOp_closed[of "S" "x"], assumption+)
apply (simp add:ridmap_def)
done
lemma (in PolynRg) Subring_pow_ring_pow:"x ∈ carrier S ==>
x^S n = x^R n"
apply (cut_tac subring, frule subring_Ring)
apply (induct_tac n)
apply (simp, simp add:Subring_one_ring_one)
apply (frule_tac n = n in Ring.npClose[of S x], assumption+)
apply (simp add:Subring_tOp_ring_tOp)
done
lemma (in PolynRg) is_Ring:"Ring R"
by unfold_locales
lemma (in PolynRg) polyn_ring_nonzero:"1r ≠ \<zero>"
apply (cut_tac Ring, cut_tac subring)
apply (simp add:Subring_zero_ring_zero[THEN sym])
apply (simp add:Subring_one_ring_one[THEN sym])
apply (simp add:not_zeroring)
done
lemma (in PolynRg) polyn_ring_S_nonzero:"1rS ≠ \<zero>S"
apply (cut_tac subring)
apply (simp add:Subring_zero_ring_zero)
apply (simp add:Subring_one_ring_one)
apply (simp add:polyn_ring_nonzero)
done
lemma (in PolynRg) polyn_ring_X_nonzero:"X ≠ \<zero>"
apply (cut_tac algfree,
cut_tac subring)
apply (simp add:algfree_cond_def)
apply (rule contrapos_pp, simp+)
apply (drule_tac m = "Suc 0" in nat_forall_spec)
apply (subgoal_tac "pol_coeff S ((Suc 0),
(λj∈{l. l ≤ (Suc 0)}. if j = 0 then \<zero>S else 1rS))")
apply (drule_tac a = "λj∈{l. l ≤ (Suc 0)}. if j = 0 then \<zero>S else 1rS" in
forall_spec1)
apply (erule conjE, simp)
apply (simp only:Nset_1)
apply (drule_tac a = "Suc 0" in forall_spec, simp)
apply simp
apply (cut_tac subring, simp add:Subring_zero_ring_zero,
simp add:Subring_one_ring_one, cut_tac ring_zero, cut_tac ring_one,
simp add:ring_r_one, simp add:ring_times_x_0, cut_tac ring_is_ag,
simp add:aGroup.ag_r_zero,
drule_tac a = "Suc 0" in forall_spec, simp, simp)
apply (cut_tac polyn_ring_S_nonzero, simp add:Subring_zero_ring_zero)
apply (thin_tac "∀b. pol_coeff S (Suc 0, b) ∧
(∀k≤Suc 0. Σe R (λj. b j ·r \<zero>^R j) k = \<zero> --> (∀j≤k. b j = \<zero>S))",
simp add:pol_coeff_def,
rule allI,
simp add:Subring_def, simp add:Ring.ring_zero,
(rule impI)+,
simp add:Ring.ring_one)
done
subsection "coefficients of a polynomial"
lemma (in PolynRg) pol_coeff_split:"pol_coeff S f = pol_coeff S (fst f, snd f)"
by simp
lemma (in PolynRg) pol_coeff_cartesian:"pol_coeff S c ==>
(fst c, snd c) = c"
by simp
lemma (in PolynRg) split_pol_coeff:"[|pol_coeff S c; k ≤ (fst c)|] ==>
pol_coeff S (k, snd c)"
by (simp add:pol_coeff_def)
lemma (in PolynRg) pol_coeff_pre:"pol_coeff S ((Suc n), f) ==>
pol_coeff S (n, f)"
apply (simp add:pol_coeff_def)
done
lemma (in PolynRg) pol_coeff_le:"[|pol_coeff S c; n ≤ (fst c)|] ==>
pol_coeff S (n, (snd c))"
apply (simp add:pol_coeff_def)
done
lemma (in PolynRg) pol_coeff_mem:"[|pol_coeff S c; j ≤ (fst c)|] ==>
((snd c) j) ∈ carrier S"
by (simp add:pol_coeff_def)
lemma (in PolynRg) pol_coeff_mem_R:"[|pol_coeff S c; j ≤ (fst c)|]
==> ((snd c) j) ∈ carrier R"
apply (cut_tac subring, frule subring_Ring)
apply (frule pol_coeff_mem[of c "j"], assumption+,
simp add:mem_subring_mem_ring)
done
lemma (in PolynRg) Slide_pol_coeff:"[|pol_coeff S c; n < (fst c)|] ==>
pol_coeff S (((fst c) - Suc n), (λx. (snd c) (Suc (n + x))))"
apply (simp add: pol_coeff_def)
done
subsection "addition of polyn_exprs"
lemma (in PolynRg) monomial_mem:"pol_coeff S c ==>
∀j ≤ (fst c). (snd c) j ·r X^R j ∈ carrier R"
apply (rule allI, rule impI)
apply (rule ring_tOp_closed)
apply (simp add:pol_coeff_mem_R[of c],
cut_tac X_mem_R, simp add:npClose)
done
lemma (in PolynRg) polyn_mem:"[|pol_coeff S c; k ≤ (fst c)|] ==>
polyn_expr R X k c ∈ carrier R"
apply (simp add:polyn_expr_def,
cut_tac ring_is_ag)
apply (rule aGroup.nsum_mem[of R k "λj. (snd c) j ·r X^R j"], assumption+)
apply (simp add:monomial_mem)
done
lemma (in PolynRg) polyn_exprs_eq:"[|pol_coeff S c; pol_coeff S d;
k ≤ (min (fst c) (fst d)); ∀j ≤ k. (snd c) j = (snd d) j|] ==>
polyn_expr R X k c = polyn_expr R X k d"
apply (cut_tac ring_is_ag,
simp add:polyn_expr_def,
cut_tac subring,
cut_tac X_mem_R)
apply (rule aGroup.nsum_eq[of R k "λj. (snd c) j ·r X^R j"
"λj. (snd d) j ·r X^R j"], assumption)
apply (simp add:monomial_mem)+
done
lemma (in PolynRg) polyn_expr_restrict:"pol_coeff S (Suc n, f) ==>
polyn_expr R X n (Suc n, f) = polyn_expr R X n (n, f)"
apply (cut_tac subring, frule subring_Ring,
cut_tac pol_coeff_le[of "(Suc n, f)" n])
apply (cut_tac polyn_exprs_eq[of "(Suc n, f)" "(n, f)" n],
(simp add:pol_coeff_split[THEN sym])+)
done
lemma (in PolynRg) polyn_expr_short:"[|pol_coeff S c; k ≤ (fst c)|] ==>
polyn_expr R X k c = polyn_expr R X k (k, snd c)"
apply (rule polyn_exprs_eq[of c "(k, snd c)" k], assumption+)
apply (simp add:pol_coeff_def)
apply (simp add:min_def)
apply simp
done
lemma (in PolynRg) polyn_expr0:"pol_coeff S c ==>
polyn_expr R X 0 c = (snd c) 0"
apply (simp add:polyn_expr_def)
apply (cut_tac subring,
cut_tac subring_Ring[of S])
apply (frule pol_coeff_mem[of c 0], simp)
apply (frule mem_subring_mem_ring [of S "(snd c) 0"], assumption)
apply (simp add:ring_r_one, assumption)
done
lemma (in PolynRg) polyn_expr_split:"
polyn_expr R X k f = polyn_expr R X k (fst f, snd f)"
by simp
lemma (in PolynRg) polyn_Suc:"Suc n ≤ (fst c) ==>
polyn_expr R X (Suc n) ((Suc n), (snd c)) =
polyn_expr R X n c ± ((snd c) (Suc n)) ·r (X^R (Suc n))"
by (simp add:polyn_expr_def)
lemma (in PolynRg) polyn_Suc_split:"pol_coeff S (Suc n, f) ==>
polyn_expr R X (Suc n) ((Suc n), f) =
polyn_expr R X n (n, f) ± (f (Suc n)) ·r (X^R (Suc n))"
apply (cut_tac polyn_Suc[of n "(Suc n, f)"])
apply (simp del:npow_suc)
apply (subst polyn_expr_short[of "(Suc n, f)" n], assumption+, simp)
apply (simp del:npow_suc)
apply simp
done
lemma (in PolynRg) polyn_n_m:"[|pol_coeff S c; n < m; m ≤ (fst c)|] ==>
polyn_expr R X m (m, (snd c)) = polyn_expr R X n (n, (snd c)) ±
(fSum R (λj. ((snd c) j) ·r (X^R j)) (Suc n) m)"
apply (simp add:polyn_expr_def, cut_tac ring_is_ag)
apply (rule aGroup.nsum_split1[of "R" m "λj. ((snd c) j) ·r (X^R j)" n],
assumption+)
apply (rule allI, rule impI)
apply (frule_tac monomial_mem[of c],
frule_tac i = j and j = m and k = "(fst c)" in le_trans, assumption+,
simp+)
done
lemma (in PolynRg) polyn_n_m1:"[|pol_coeff S c; n < m; m ≤ (fst c)|] ==>
polyn_expr R X m c = polyn_expr R X n c ±
(fSum R (λj. ((snd c) j) ·r (X^R j)) (Suc n) m)"
apply (subst polyn_expr_short[of c n], assumption)
apply (frule_tac x = n and y = m and z = "fst c" in less_le_trans, assumption,
simp add:less_imp_le)
apply (subst polyn_expr_short[of c m], assumption+)
apply (simp add:polyn_n_m)
done
lemma (in PolynRg) polyn_n_m_mem:"[|pol_coeff S c; n < m; m ≤ (fst c)|] ==>
(fSum R (λj. ((snd c) j) ·r (X^R j)) (Suc n) m) ∈ carrier R"
apply (simp add:fSum_def)
apply (cut_tac ring_is_ag,
rule_tac n = "m - Suc n" in aGroup.nsum_mem, assumption+)
apply (rule allI, rule impI,
simp del:npow_suc add:cmp_def slide_def)
apply (rule ring_tOp_closed)
apply (simp add:pol_coeff_def)
apply (frule_tac a = "Suc (n + j)" in forall_spec, arith)
apply (cut_tac subring)
apply (simp add:mem_subring_mem_ring)
apply (rule npClose)
apply (cut_tac X_mem_R,
simp del:npow_suc add:npClose)
done
lemma (in PolynRg) polyn_n_ms_eq:"[|pol_coeff S c; pol_coeff S d;
m ≤ min (fst c) (fst d); n < m;
∀j∈nset (Suc n) m. (snd c) j = (snd d) j|] ==>
(fSum R (λj. ((snd c) j) ·r (X^R j)) (Suc n) m) =
(fSum R (λj. ((snd d) j) ·r (X^R j)) (Suc n) m)"
apply (cut_tac ring_is_ag)
apply (cut_tac aGroup.fSum_eq1[of R "Suc n" m "λj. (snd c) j ·r X^R j"
"λj. (snd d) j ·r X^R j"],
assumption+)
apply (rule Suc_leI, assumption,
simp add:nset_def, simp add:monomial_mem)
apply (frule Suc_leI,
rule ballI, simp add:nset_def)
apply (simp add:monomial_mem)
apply simp
done
lemma (in PolynRg) polyn_addTr:
"(pol_coeff S (n, f)) ∧ (pol_coeff S (n, g)) -->
(polyn_expr R X n (n, f)) ± (polyn_expr R X n (n, g)) =
nsum R (λj. ((f j) ±S (g j)) ·r (X^R j)) n"
apply (cut_tac subring,
frule subring_Ring[of S])
apply (induct_tac n)
apply (rule impI, simp, erule conjE)
apply (simp add:polyn_expr0)
apply (cut_tac pol_coeff_mem[of "(0, f)" 0], simp,
cut_tac pol_coeff_mem[of "(0, g)" 0], simp,
frule mem_subring_mem_ring[of S "f 0"], assumption+,
frule mem_subring_mem_ring[of S "g 0"], assumption+,
frule Ring.ring_is_ag[of S],
frule aGroup.ag_pOp_closed[of S "f 0" "g 0"], assumption+,
frule mem_subring_mem_ring[of S "f 0 ±S g 0"], assumption+)
apply (simp add:ring_r_one)
apply (simp add:Subring_pOp_ring_pOp[of S "f 0" "g 0"])
apply (simp del:npow_suc)+
apply (rule impI, erule conjE)
apply (frule_tac n = n in pol_coeff_pre[of _ f],
frule_tac n = n in pol_coeff_pre[of _ g], simp del:npow_suc)
apply (cut_tac n = n and c = "(Suc n, f)" in polyn_Suc, simp del:npow_suc,
simp del:npow_suc,
thin_tac "polyn_expr R X (Suc n) (Suc n, f) =
polyn_expr R X n (Suc n, f) ± f (Suc n) ·r X^R (Suc n)")
apply (cut_tac n = n and c = "(Suc n, g)" in polyn_Suc, simp del:npow_suc,
simp del:npow_suc,
thin_tac "polyn_expr R X (Suc n) (Suc n, g) =
polyn_expr R X n (Suc n, g) ± g (Suc n) ·r X^R (Suc n)")
apply (cut_tac c = "(Suc n, f)" and k = n in polyn_mem, assumption,
simp del:npow_suc,
cut_tac k = n and c = "(Suc n, g)" in polyn_mem, assumption,
simp del:npow_suc)
apply (frule_tac j = "Suc n" and c = "(Suc n, f)" in pol_coeff_mem_R, simp,
frule_tac j = "Suc n" and c = "(Suc n, g)" in pol_coeff_mem_R, simp,
cut_tac X_mem_R,
frule_tac n = "Suc n" in npClose[of "X"], simp del:npow_suc)
apply (frule_tac x = "f (Suc n)" and y = "X^R (Suc n)" in ring_tOp_closed,
assumption+,
frule_tac x = "g (Suc n)" and y = "X^R (Suc n)" in ring_tOp_closed,
assumption+)
apply (cut_tac ring_is_ag,
subst aGroup.pOp_assocTr43, assumption+)
apply (frule_tac x = "f (Suc n) ·r X^R (Suc n)" and
y = "polyn_expr R X n (Suc n, g)" in aGroup.ag_pOp_commute[of R],
assumption+, simp del:npow_suc,
thin_tac "f (Suc n) ·r X^R (Suc n) ± polyn_expr R X n (Suc n, g) =
polyn_expr R X n (Suc n, g) ± f (Suc n) ·r X^R (Suc n)")
apply (subst aGroup.pOp_assocTr43[THEN sym], assumption+,
simp del:npow_suc add:polyn_expr_restrict)
apply (frule_tac c = "(Suc n, f)" and j = "Suc n" in pol_coeff_mem, simp,
frule_tac c = "(Suc n, g)" and j = "Suc n" in pol_coeff_mem, simp)
apply (subst ring_distrib2[THEN sym], assumption+)
apply (frule_tac c = "(Suc n, f)" and j = "Suc n" in pol_coeff_mem, simp,
frule_tac c = "(Suc n, g)" and j = "Suc n" in pol_coeff_mem, simp)
apply (frule_tac a = "f (Suc n)" and b = "g (Suc n)" in
Subring_pOp_ring_pOp[of S], simp, simp)
apply simp
done
lemma (in PolynRg) polyn_add_n:"[|pol_coeff S (n, f); pol_coeff S (n, g)|] ==>
(polyn_expr R X n (n, f)) ± (polyn_expr R X n (n, g)) =
nsum R (λj. ((f j) ±S (g j)) ·r (X^R j)) n"
by (simp add:polyn_addTr)
constdefs
add_cf::"[('a, 'm) Ring_scheme, nat × (nat => 'a), nat × (nat => 'a)] =>
nat × (nat => 'a)"
"add_cf S c d ==
if (fst c) < (fst d) then ((fst d), λj. (if j ≤ (fst c)
then (((snd c) j) ±S ((snd d) j)) else ((snd d) j)))
else if (fst c) = (fst d) then ((fst c), λj. ((snd c) j ±S (snd d) j))
else ((fst c), λj. (if j ≤ (fst d) then
((snd c) j ±S (snd d) j) else ((snd c) j)))"
lemma (in PolynRg) add_cf_pol_coeff:"[|pol_coeff S c; pol_coeff S d|]
==> pol_coeff S (add_cf S c d)"
apply (cut_tac subring,
frule subring_Ring[of S], frule Ring.ring_is_ag[of S])
apply (simp add:pol_coeff_def)
apply (rule allI, rule impI)
apply (case_tac "(fst c) < (fst d)", simp add:add_cf_def)
apply (rule impI, rule aGroup.ag_pOp_closed, assumption+, simp+)
apply (drule nat_not_less[of "fst c" "fst d"],
drule le_imp_less_or_eq[of "fst d" "fst c"])
apply (erule disjE)
apply (simp add:add_cf_def, rule impI)
apply (frule Ring.ring_is_ag[of S], rule aGroup.ag_pOp_closed, assumption,
simp+)
apply (simp add:add_cf_def)
apply (frule Ring.ring_is_ag[of S], rule aGroup.ag_pOp_closed, assumption,
simp+)
done
lemma (in PolynRg) add_cf_len:"[|pol_coeff S c; pol_coeff S d|]
==> fst (add_cf S c d) = (max (fst c) (fst d))"
by (simp add:max_def add_cf_def)
lemma (in PolynRg) polyn_expr_restrict1:"[|pol_coeff S (n, f);
pol_coeff S (Suc (m + n), g)|] ==>
polyn_expr R X (m + n) (add_cf S (n, f) (m + n, g)) =
polyn_expr R X (m + n) (m + n, snd (add_cf S (n, f) (Suc (m + n), g)))"
apply (frule pol_coeff_pre[of "m+n" g])
apply (frule add_cf_pol_coeff[of "(n, f)" "(Suc (m + n), g)"], assumption+,
frule add_cf_pol_coeff[of "(n, f)" "(m + n, g)"], assumption+)
apply (rule polyn_exprs_eq[of "add_cf S (n, f) (m + n, g)"
"(m + n, snd (add_cf S (n, f) (Suc (m + n), g)))" "m + n"], assumption+)
apply (rule split_pol_coeff[of "add_cf S (n, f) (Suc (m + n), g)" "m + n"],
assumption, simp add:add_cf_len max_def)
apply (simp add:add_cf_len max_def)
apply (rule allI, rule impI)
apply (simp add:add_cf_def)
done
lemma (in PolynRg) polyn_add_n1:"[|pol_coeff S (n, f); pol_coeff S (n, g)|] ==>
(polyn_expr R X n (n, f)) ± (polyn_expr R X n (n, g)) =
polyn_expr R X n (add_cf S (n, f) (n, g))"
apply (subst polyn_add_n, assumption+)
apply (simp add:polyn_expr_def add_cf_def)
done
lemma (in PolynRg) add_cf_val_hi:"(fst c) < (fst d) ==>
snd (add_cf S c d) (fst d) = (snd d) (fst d)"
by (simp add:add_cf_def)
lemma (in PolynRg) add_cf_commute:"[|pol_coeff S c; pol_coeff S d|]
==> ∀j ≤ (max (fst c) (fst d)). snd (add_cf S c d) j =
snd (add_cf S d c) j"
apply (cut_tac subring, frule subring_Ring,
frule Ring.ring_is_ag[of S])
apply (simp add:add_cf_def)
apply (case_tac "(fst c) = (fst d)", simp add:pol_coeff_def)
apply (rule allI, rule impI,
rule aGroup.ag_pOp_commute[of S], simp+)
apply (case_tac "(fst d) < (fst c)", simp,
rule allI, rule impI, rule impI,
rule aGroup.ag_pOp_commute, assumption+)
apply (frule_tac x = j and y = "fst d" and z = "fst c" in le_less_trans,
assumption+, frule_tac x = j and y = "fst c" in less_imp_le,
thin_tac "j < fst c", simp add:pol_coeff_mem, simp add:pol_coeff_mem)
apply simp
apply (frule nat_not_less[of "fst d" "fst c"],
frule noteq_le_less[of "fst c" "fst d"], assumption,
rule allI, rule impI, rule impI,
simp add:max_def)
apply (rule aGroup.ag_pOp_commute, assumption+,
simp add:pol_coeff_mem,
frule_tac x = j and y = "fst c" and z = "fst d" in le_less_trans,
assumption+, frule_tac x = j and y = "fst d" in less_imp_le,
thin_tac "j < fst d", simp add:pol_coeff_mem)
done
lemma (in PolynRg) polyn_addTr1:"pol_coeff S (n, f) ==>
∀g. pol_coeff S (n + m, g) -->
(polyn_expr R X n (n, f) ± (polyn_expr R X (n + m) ((n + m), g))
= polyn_expr R X (n + m) (add_cf S (n, f) ((n + m), g)))"
apply (cut_tac subring, frule subring_Ring)
apply (induct_tac m)
apply (rule allI, rule impI, simp)
apply (simp add:polyn_add_n1)
apply (simp add:add_commute[of n])
apply (rule allI, rule impI)
apply (frule_tac n = "na + n" and f = g in pol_coeff_pre)
apply (drule_tac a = g in forall_spec, assumption)
apply (cut_tac n = "na + n" and c = "(Suc (na + n), g)" in polyn_Suc,
simp, simp del:npow_suc,
thin_tac "polyn_expr R X (Suc (na + n)) (Suc (na + n), g) =
polyn_expr R X (na + n) (Suc (na + n), g) ±
g (Suc (na + n)) ·r X^R (Suc (na + n))")
apply (frule_tac c = "(n, f)" and k = n in polyn_mem, simp,
frule_tac c = "(Suc (na + n), g)" and k = "na + n" in polyn_mem, simp,
frule_tac c = "(Suc (na + n), g)" in monomial_mem)
apply (drule_tac a = "Suc (na + n)" in forall_spec, simp del:npow_suc,
cut_tac ring_is_ag,
subst aGroup.ag_pOp_assoc[THEN sym], assumption+, simp del:npow_suc)
apply (simp del:npow_suc add:polyn_expr_restrict)
apply (frule_tac c = "(n, f)" and d = "(Suc (na + n), g)" in
add_cf_pol_coeff, assumption+,
frule_tac c = "(n, f)" and d = "(na + n, g)" in
add_cf_pol_coeff, assumption+)
apply (frule_tac c = "add_cf S (n, f) (Suc (na + n), g)" and
n = "na + n" and m = "Suc (na + n)" in polyn_n_m, simp,
subst add_cf_len, assumption+, simp add:max_def)
apply (cut_tac k = "Suc (na + n)" and f = "add_cf S (n, f) (Suc (na + n), g)"
in polyn_expr_split)
apply (frule_tac c = "(n, f)" and d = "(Suc (na + n), g)" in
add_cf_len, assumption+, simp del:npow_suc add:max_def)
apply (thin_tac "polyn_expr R X (Suc (na + n))
(Suc (na + n), snd (add_cf S (n, f) (Suc (na + n), g))) =
polyn_expr R X (na + n)
(na + n, snd (add_cf S (n, f) (Suc (na + n), g))) ±
Σf R (λj. snd (add_cf S (n, f) (Suc (na + n), g)) j ·r
X^R j) (Suc (na + n)) (Suc (na + n))",
thin_tac "polyn_expr R X (Suc (na + n)) (add_cf S (n, f) (Suc (na + n),
g)) =
polyn_expr R X (na + n)
(na + n, snd (add_cf S (n, f) (Suc (na + n), g))) ±
Σf R (λj. snd (add_cf S (n, f) (Suc (na + n), g)) j ·r
X^R j) (Suc (na + n)) (Suc (na + n))")
apply (simp del:npow_suc add:fSum_def cmp_def slide_def)
apply (cut_tac d = "(Suc (na + n), g)" in add_cf_val_hi[of "(n, f)"],
simp, simp del:npow_suc,
thin_tac "snd (add_cf S (n, f) (Suc (na + n), g)) (Suc (na + n)) =
g (Suc (na + n))")
apply (frule_tac c = "add_cf S (n, f) (Suc (na + n), g)" and k = "na + n" in
polyn_mem, simp,
frule_tac c = "add_cf S (n, f) (na + n, g)" and k = "na + n" in
polyn_mem, simp )
apply (subst add_cf_len, assumption+, simp del:npow_suc
add:max_def)
apply (frule_tac a = "polyn_expr R X (na + n) (add_cf S (n, f) (na + n, g))"
and b = "polyn_expr R X (na + n) (add_cf S (n, f) (Suc (na + n), g))"
and c = "g (Suc (na + n)) ·r X^R (Suc (na + n))" in
aGroup.ag_pOp_add_r[of R], assumption+)
apply (rule_tac c = "add_cf S (n, f) (na + n, g)" and
d = "add_cf S (n, f) (Suc (na + n), g)" and k = "na + n" in
polyn_exprs_eq, assumption+, simp,
subst add_cf_len, assumption+)
apply (simp add:max_def)
apply (rule allI, rule impI,
(subst add_cf_def)+, simp,
frule_tac m = na and g = g in polyn_expr_restrict1[of n f], assumption,
simp del:npow_suc)
done
lemma (in PolynRg) polyn_add:"[|pol_coeff S (n, f); pol_coeff S (m, g)|]
==> polyn_expr R X n (n, f) ± (polyn_expr R X m (m, g))
= polyn_expr R X (max n m) (add_cf S (n, f) (m, g))"
apply (cut_tac less_linear[of n m])
apply (erule disjE,
frule polyn_addTr1[of n f "m - n"],
drule_tac a = g in forall_spec, simp, simp add:max_def)
apply (erule disjE,
simp add:polyn_add_n1)
apply (frule polyn_mem[of "(n, f)" n], simp,
frule polyn_mem[of "(m, g)" m], simp)
apply (cut_tac ring_is_ag, simp add:aGroup.ag_pOp_commute)
apply (frule polyn_addTr1[of m g "n - m"],
drule_tac a = f in forall_spec, simp, simp add:max_def,
frule add_cf_commute[of "(m, g)" "(n, f)"], assumption+,
simp add:max_def,
frule add_cf_pol_coeff[of "(n, f)" "(m, g)"], assumption+,
frule add_cf_pol_coeff[of "(m, g)" "(n, f)"], assumption+)
apply (rule polyn_exprs_eq[of "add_cf S (m, g) (n, f)"
"add_cf S (n, f) (m, g)" n], assumption+)
apply (simp add:add_cf_len, simp add:max_def)
done
lemma (in PolynRg) polyn_add1:"[|pol_coeff S c; pol_coeff S d|]
==> polyn_expr R X (fst c) c ± (polyn_expr R X (fst d) d)
= polyn_expr R X (max (fst c) (fst d)) (add_cf S c d)"
apply (cases c)
apply (cases d)
apply (simp add: polyn_add)
done
lemma (in PolynRg) polyn_minus_nsum:"[|pol_coeff S c; k ≤ (fst c)|] ==>
-a (polyn_expr R X k c) = nsum R (λj. ((-aS ((snd c) j)) ·r (X^R j))) k"
apply (cut_tac subring,
frule subring_Ring[of S],
frule Ring.ring_is_ag[of S],
cut_tac ring_is_ag,
cut_tac X_mem_R)
apply (simp add:polyn_expr_def,
subst aGroup.nsum_minus[of R], assumption)
apply (frule monomial_mem[of c], rule allI, rule impI,
frule_tac i = j and j = k and k = "fst c" in le_trans, assumption+,
simp)
apply (rule aGroup.nsum_eq, assumption,
rule allI, rule impI, simp,
rule aGroup.ag_mOp_closed, assumption)
apply (frule monomial_mem[of c],
frule_tac i = j and j = k and k = "fst c" in le_trans, assumption+,
simp)
apply (rule allI, rule impI,
rule ring_tOp_closed)
apply (frule_tac j = j in pol_coeff_mem[of c])
apply (frule_tac i = j and j = k and k = "fst c" in le_trans, assumption+,
simp add:Subring_minus_ring_minus,
frule_tac x = "(snd c) j" in mem_subring_mem_ring[of S], assumption,
simp add:aGroup.ag_mOp_closed,
simp add:npClose)
apply (rule allI, rule impI, simp,
cut_tac j = j in pol_coeff_mem[of c], assumption,
rule_tac i = j and j = k and k = "fst c" in le_trans, assumption+)
apply (simp add:Subring_minus_ring_minus,
frule_tac x = "(snd c) j" in mem_subring_mem_ring[of S], assumption)
apply (subst ring_inv1_1, assumption+)
apply (simp add:npClose, simp)
done
lemma (in PolynRg) minus_pol_coeff:"pol_coeff S c ==>
pol_coeff S ((fst c), (λj. (-aS ((snd c) j))))"
apply (simp add:pol_coeff_def)
apply (rule allI, rule impI)
apply (cut_tac subring, frule subring_Ring)
apply (frule Ring.ring_is_ag[of "S"])
apply (rule aGroup.ag_mOp_closed, assumption)
apply simp
done
lemma (in PolynRg) polyn_minus:"[|pol_coeff S c; k ≤ (fst c)|] ==>
-a (polyn_expr R X k c) =
polyn_expr R X k (fst c, (λj. (-aS ((snd c) j))))"
apply (cases c)
apply (subst polyn_minus_nsum)
apply (simp_all add: polyn_expr_def)
done
constdefs
m_cf ::"[('a, 'm) Ring_scheme, nat × (nat => 'a)] => nat × (nat => 'a)"
"m_cf S c == (fst c, (λj. (-aS ((snd c) j))))"
lemma (in PolynRg) m_cf_pol_coeff:"pol_coeff S c ==>
pol_coeff S (m_cf S c)"
by (simp add:m_cf_def, simp add:minus_pol_coeff)
lemma (in PolynRg) m_cf_len:"pol_coeff S c ==>
fst (m_cf S c) = fst c"
by (simp add:m_cf_def)
lemma (in PolynRg) polyn_minus_m_cf:"[|pol_coeff S c; k ≤ (fst c)|] ==>
-a (polyn_expr R X k c) =
polyn_expr R X k (m_cf S c)"
by (simp add:m_cf_def polyn_minus)
lemma (in PolynRg) polyn_zero_minus_zero:"[|pol_coeff S c; k ≤ (fst c)|] ==>
(polyn_expr R X k c = \<zero>) = (polyn_expr R X k (m_cf S c) = \<zero>)"
apply (cut_tac ring_is_ag)
apply (simp add:polyn_minus_m_cf[THEN sym])
apply (rule iffI, simp)
apply (simp add:aGroup.ag_inv_zero)
apply (frule polyn_mem[of c k], assumption)
apply (frule aGroup.ag_inv_inv[of "R" "polyn_expr R X k c"], assumption)
apply (simp add:aGroup.ag_inv_zero)
done
lemma (in PolynRg) coeff_0_pol_0:"[|pol_coeff S c; k ≤ fst c|] ==>
(∀j≤ k. (snd c) j = \<zero>S) = (polyn_expr R X k c = \<zero>)"
apply (rule iffI)
apply (cut_tac ring_is_ag, cut_tac subring,
frule subring_Ring)
apply (simp add:Subring_zero_ring_zero)
apply (simp add:polyn_expr_def,
rule aGroup.nsum_zeroA[of R], assumption)
apply (rule allI, rule impI,
cut_tac X_mem_R)
apply (drule_tac a = j in forall_spec, simp,
frule_tac n = j in npClose[of X], simp)
apply (simp add:ring_times_0_x)
apply (cases c)
using algfree [simplified algfree_cond_def] by (auto simp add: polyn_expr_def)
subsection "multiplication of pol_exprs"
subsection "multiplication"
constdefs
ext_cf ::"[('a, 'm) Ring_scheme, nat, nat × (nat => 'a)] =>
nat × (nat => 'a)"
"ext_cf S n c == (n + fst c, λi. if n ≤ i then (snd c) (sliden n i) else \<zero>S)"
(* 0 0 g(0) g(m)
0 n m+n , where (m, g) is a pol_coeff **)
sp_cf::"[('a, 'm) Ring_scheme, 'a, nat × (nat => 'a)] => nat × (nat => 'a)"
"sp_cf S a c == (fst c, λj. a ·rS ((snd c) j))" (* scalar times cf *)
special_cf ::"('a, 'm) Ring_scheme => nat × (nat => 'a)" ("C0")
"C0 S == (0, λj. 1rS)"
lemma (in PolynRg) special_cf_pol_coeff:"pol_coeff S (C0 S)"
apply (cut_tac subring, frule subring_Ring)
apply (simp add:pol_coeff_def special_cf_def)
apply (simp add:Ring.ring_one)
done
lemma (in PolynRg) special_cf_len:"fst (C0 S) = 0"
apply (simp add:special_cf_def)
done
lemma (in PolynRg) ext_cf_pol_coeff:"pol_coeff S c ==>
pol_coeff S (ext_cf S n c)"
apply (simp add: pol_coeff_def ext_cf_def sliden_def)
apply (rule impI)
apply (rule Ring.ring_zero)
apply (rule subring_Ring)
apply (rule subring)
done
lemma (in PolynRg) ext_cf_len:"pol_coeff S c ==>
fst (ext_cf S m c) = m + fst c"
by (simp add:ext_cf_def)
lemma (in PolynRg) ext_special_cf_len:"fst (ext_cf S m (C0 S)) = m"
apply (cut_tac special_cf_pol_coeff)
apply (simp add:ext_cf_len special_cf_def)
done
lemma (in PolynRg) ext_cf_self:"pol_coeff S c ==>
∀j ≤ (fst c). snd (ext_cf S 0 c) j = (snd c) j"
apply (rule allI, rule impI, simp add:ext_cf_def sliden_def)
done
lemma (in PolynRg) ext_cf_hi:"pol_coeff S c ==>
(snd c) (fst c) =
snd (ext_cf S n c) (n + (fst c))"
apply (subst ext_cf_def)
apply (simp add:sliden_def)
done
lemma (in PolynRg) ext_special_cf_hi:"snd (ext_cf S n (C0 S)) n = 1rS"
apply (cut_tac special_cf_pol_coeff)
apply (cut_tac ext_cf_hi[of "C0 S" n, THEN sym])
apply (simp add:special_cf_def, assumption)
done
lemma (in PolynRg) ext_cf_lo_zero:"[|pol_coeff S c; 0 < n; x ≤ (n - Suc 0)|]
==> snd (ext_cf S n c) x = \<zero>S"
apply (cut_tac Suc_le_mono[THEN sym, of x "n - Suc 0"], simp,
cut_tac x = x and y = "Suc x" and z = n in less_le_trans,
simp, assumption,
simp add:nat_not_le_less[THEN sym, of x n],
thin_tac "x ≤ n - Suc 0")
apply (simp add:ext_cf_def)
done
lemma (in PolynRg) ext_special_cf_lo_zero:"[|0 < n; x ≤ (n - Suc 0)|]
==> snd (ext_cf S n (C0 S)) x = \<zero>S"
by (cut_tac special_cf_pol_coeff,
frule ext_cf_lo_zero[of "C0 S" n], assumption+)
lemma (in PolynRg) sp_cf_pol_coeff:"[|pol_coeff S c; a ∈ carrier S|] ==>
pol_coeff S (sp_cf S a c)"
apply (cut_tac subring, frule subring_Ring)
apply (simp add:pol_coeff_def sp_cf_def,
rule allI, rule impI,
rule Ring.ring_tOp_closed, assumption+)
apply simp
done
lemma (in PolynRg) sp_cf_len:"[|pol_coeff S c; a ∈ carrier S|] ==>
fst (sp_cf S a c) = fst c"
by (simp add:sp_cf_def)
lemma (in PolynRg) sp_cf_val:"[|pol_coeff S c; j ≤ (fst c); a ∈ carrier S|] ==>
snd (sp_cf S a c) j = a ·rS ((snd c) j)"
by (simp add:sp_cf_def)
lemma (in PolynRg) polyn_ext_cf_lo_zero:"[|pol_coeff S c; 0 < j|] ==>
polyn_expr R X (j - Suc 0) (ext_cf S j c) = \<zero>"
apply (simp add:polyn_expr_def, cut_tac ring_is_ag,
rule aGroup.nsum_zeroA, assumption)
apply (rule allI, rule impI)
apply (frule_tac x = ja in ext_cf_lo_zero [of c j], assumption+)
apply (cut_tac X_mem_R, frule_tac n = ja in npClose[of X])
apply (cut_tac subring,
simp add:Subring_zero_ring_zero,
simp add:ring_times_0_x)
done
lemma (in PolynRg) monomial_d:"pol_coeff S c ==>
polyn_expr R X d (ext_cf S d c) = ((snd c) 0) ·r X^R d"
apply (cut_tac ring_is_ag,
cut_tac subring,
cut_tac X_mem_R,
frule subring_Ring[of S])
apply (frule pol_coeff_mem [of c 0], simp)
apply (case_tac "d = 0")
apply simp
apply (simp add:polyn_expr_def ext_cf_def sliden_def)
apply (frule ext_cf_pol_coeff[of c d])
apply (cut_tac polyn_Suc[of "d - Suc 0" "ext_cf S d c"])
apply (simp)
apply (cut_tac polyn_ext_cf_lo_zero[of c d], simp,
thin_tac "polyn_expr R X (d - Suc 0) (ext_cf S d c) = \<zero>")
apply (frule monomial_mem[of "ext_cf S d c"], simp add:ext_cf_len,
drule_tac a = d in forall_spec, simp, simp add:aGroup.ag_l_zero)
apply (subst polyn_expr_short[of "ext_cf S d c" d], assumption,
simp add:ext_cf_len)
apply (simp,
subst ext_cf_def, simp add:sliden_def , assumption+,
simp add:ext_cf_len)
done
lemma (in PolynRg) X_to_d:" X^R d = polyn_expr R X d (ext_cf S d (C0 S))"
apply (cut_tac special_cf_pol_coeff)
apply (subst monomial_d[of "C0 S" d], assumption+)
apply (subst special_cf_def, simp)
apply (cut_tac subring, frule subring_Ring)
apply (simp add:Subring_one_ring_one)
apply (cut_tac X_mem_R, frule_tac n = d in npClose[of X])
apply (simp add:ring_l_one)
done
lemma (in PolynRg) c_max_ext_special_cf:"c_max S (ext_cf S n (C0 S)) = n"
apply (cut_tac polyn_ring_S_nonzero,
cut_tac subring, frule subring_Ring)
apply (simp add:c_max_def special_cf_def ext_cf_def)
apply (cut_tac n_max[of "{j. (n ≤ j --> j = n) ∧ n ≤ j}" n])
apply (erule conjE)+ apply simp
apply (rule subsetI, simp, erule conjE, simp)
apply (cut_tac le_refl[of n], blast)
done
lemma (in PolynRg) scalar_times_polynTr:"a ∈ carrier S ==>
∀f. pol_coeff S (n, f) -->
a ·r (polyn_expr R X n (n, f)) = polyn_expr R X n (sp_cf S a (n, f))"
apply (cut_tac subring,
cut_tac X_mem_R,
frule_tac x = a in mem_subring_mem_ring, assumption)
apply (induct_tac n,
rule allI, rule impI, simp add:polyn_expr_def sp_cf_def,
cut_tac n_in_Nsetn[of "0"])
apply (cut_tac subring_Ring,
frule_tac c = "(0, f)" in pol_coeff_mem[of _ "0"], simp)
apply (simp,
frule_tac x = "f 0" in mem_subring_mem_ring, assumption)
apply ( simp add:Subring_tOp_ring_tOp,
frule_tac y = "f 0" in ring_tOp_closed[of a], assumption+,
cut_tac ring_one, simp add:ring_tOp_assoc, assumption)
apply (rule allI, rule impI,
frule subring_Ring,
frule_tac n = n and f = f in pol_coeff_pre,
drule_tac a = f in forall_spec1, simp)
apply (cut_tac n = n and c = "(Suc n, f)" in polyn_Suc, simp,
simp del:npow_suc,
thin_tac "polyn_expr R X (Suc n) (Suc n, f) =
polyn_expr R X n (Suc n, f) ± f (Suc n) ·r X^R (Suc n)")
apply (cut_tac n = n and c = "sp_cf S a (Suc n, f)" in polyn_Suc,
simp add:sp_cf_len)
apply (frule_tac c = "(Suc n, f)" and a = a in sp_cf_len, assumption+,
simp only:fst_conv)
apply (cut_tac k = "Suc n" and f = "sp_cf S a (Suc n, f)" in
polyn_expr_split, simp del:npow_suc,
thin_tac "polyn_expr R X (Suc n) (Suc n, snd (sp_cf S a (Suc n, f))) =
polyn_expr R X n (sp_cf S a (Suc n, f)) ±
snd (sp_cf S a (Suc n, f)) (Suc n) ·r X^R (Suc n)",
thin_tac "polyn_expr R X (Suc n) (sp_cf S a (Suc n, f)) =
polyn_expr R X n (sp_cf S a (Suc n, f)) ±
snd (sp_cf S a (Suc n, f)) (Suc n) ·r X^R (Suc n)")
apply (frule_tac c = "(Suc n, f)" and a = a in sp_cf_pol_coeff, assumption)
apply (frule_tac c = "(Suc n, f)" and k = n in polyn_mem,
simp,
frule_tac c = "(Suc n, f)" in monomial_mem,
drule_tac a = "Suc n" in forall_spec, simp,
simp only:snd_conv)
apply (subst ring_distrib1, assumption+,
subst polyn_expr_restrict, assumption+, simp del:npow_suc,
subst sp_cf_val, assumption, simp, assumption,
simp only:snd_conv,
frule_tac c = "(Suc n, f)" and j = "Suc n" in pol_coeff_mem,
simp, simp only:snd_conv,
simp del:npow_suc add:Subring_tOp_ring_tOp,
subst ring_tOp_assoc[THEN sym, of a], assumption+,
simp add:mem_subring_mem_ring, rule npClose, assumption)
apply (cut_tac ring_is_ag,
rule aGroup.ag_pOp_add_r, assumption+,
rule polyn_mem, rule sp_cf_pol_coeff, assumption+,
simp add:sp_cf_len,
rule polyn_mem, assumption, simp add:sp_cf_len,
frule_tac c = "(Suc n, f)" and j = "Suc n" in pol_coeff_mem_R,
simp, simp only:snd_conv,
(rule ring_tOp_closed)+, assumption+, rule npClose, assumption)
apply (rule_tac c = "sp_cf S a (n, f)" and d = "sp_cf S a (Suc n, f)" and
k = n in polyn_exprs_eq, rule sp_cf_pol_coeff, assumption+,
simp add:sp_cf_len)
apply (rule allI, rule impI,
(subst sp_cf_def)+, simp)
done
lemma (in PolynRg) scalar_times_pol_expr:"[|a ∈ carrier S; pol_coeff S c;
n ≤ fst c|] ==>
a ·r (polyn_expr R X n c) = polyn_expr R X n (sp_cf S a c)"
apply (cases c) apply (simp only:)
apply (rename_tac m g)
apply (thin_tac "c = (m, g)")
apply (frule_tac c = "(m, g)" and k = n in polyn_expr_short, simp,
simp)
apply (frule scalar_times_polynTr[of a n],
drule_tac a = g in forall_spec1)
apply (frule_tac c = "(m, g)" and n = n in pol_coeff_le, simp, simp,
thin_tac "polyn_expr R X n (m, g) = polyn_expr R X n (n, g)",
thin_tac "a ·r polyn_expr R X n (n, g) =
polyn_expr R X n (sp_cf S a (n, g))")
apply (frule_tac c = "(m, g)" and n = n in pol_coeff_le, simp, simp,
frule_tac c = "(n, g)" and a = a in sp_cf_pol_coeff, assumption,
frule_tac c = "(m, g)" and a = a in sp_cf_pol_coeff, assumption)
apply (rule_tac c = "sp_cf S a (n, g)" and d = "sp_cf S a (m, g)" and
k = n in polyn_exprs_eq, assumption+)
apply (simp add:sp_cf_len)
apply (rule allI, (subst sp_cf_def)+, simp)
done
lemma (in PolynRg) sp_coeff_nonzero:"[|Idomain S; a ∈ carrier S; a ≠ \<zero>S;
pol_coeff S c; (snd c) j ≠ \<zero>S; j ≤ (fst c)|] ==>
snd (sp_cf S a c) j ≠ \<zero>S"
apply (simp add:sp_cf_def)
apply (frule_tac y = "(snd c) j" in Idomain.idom_tOp_nonzeros[of S a],
assumption+,
simp add:pol_coeff_def, simp add:funcset_mem, assumption+)
done
lemma (in PolynRg) ext_cf_inductTl:"pol_coeff S (Suc n, f) ==>
polyn_expr R X (n + j) (ext_cf S j (Suc n, f)) =
polyn_expr R X (n + j) (ext_cf S j (n, f))"
apply (frule pol_coeff_pre[of n f],
frule ext_cf_pol_coeff[of "(Suc n, f)" j],
frule ext_cf_pol_coeff[of "(n, f)" j],
rule polyn_exprs_eq[of "ext_cf S j (Suc n, f)" "ext_cf S j (n, f)"
"n + j"], assumption+)
apply (simp add:ext_cf_len)
apply (rule allI, (subst ext_cf_def)+, simp add:sliden_def)
done
lemma (in PolynRg) low_deg_terms_zeroTr:"
pol_coeff S (n, f) -->
polyn_expr R X (n + j) (ext_cf S j (n, f)) =
(X^R j) ·r (polyn_expr R X n (n, f))"
apply (cut_tac ring_is_ag,
cut_tac X_mem_R, frule npClose[of "X" "j"])
apply (induct_tac n)
apply (rule impI, simp)
apply (case_tac "j = 0", simp add:ext_cf_def sliden_def polyn_expr_def)
apply (frule_tac c = "(0, f)" and j = 0 in pol_coeff_mem_R, simp, simp)
apply (simp add:ring_r_one ring_l_one)
apply (cut_tac polyn_Suc[of "j - Suc 0" "ext_cf S j (0, f)"],
simp del:npow_suc)
apply (frule ext_cf_len[of "(0, f)" j],
cut_tac polyn_expr_split[of j "ext_cf S j (0, f)"], simp,
thin_tac "polyn_expr R X j (ext_cf S j (0, f)) =
polyn_expr R X (j - Suc 0) (ext_cf S j (0, f)) ±
snd (ext_cf S j (0, f)) j ·r X^R j")
apply (simp add:polyn_ext_cf_lo_zero[of "(0, f)" j],
thin_tac "polyn_expr R X j (j, snd (ext_cf S j (0, f))) =
\<zero> ± snd (ext_cf S j (0, f)) j ·r X^R j",
frule ext_cf_hi[THEN sym, of "(0, f)" j], simp add:polyn_expr_def)
apply (frule_tac c = "(0, f)" and j = 0 in pol_coeff_mem_R, simp, simp)
apply (subst aGroup.ag_l_zero, assumption, simp add:ring_tOp_closed,
simp add:ring_r_one, subst ring_tOp_commute, assumption+, simp)
apply (simp add:ext_cf_len)
apply (rule impI,
cut_tac subring,
cut_tac subring_Ring[of S],
frule_tac n = n in pol_coeff_pre[of _ "f"])
apply simp
apply (subst polyn_expr_split)
apply (cut_tac n = "n + j" and c = "ext_cf S j (Suc n, f)" in polyn_Suc,
simp add:ext_cf_len)
apply (subst ext_cf_len, assumption+, simp del:npow_suc add:add_commute[of j],
thin_tac "polyn_expr R X (Suc (n + j))
(Suc (n + j), snd (ext_cf S j (Suc n, f))) =
polyn_expr R X (n + j) (ext_cf S j (Suc n, f)) ±
snd (ext_cf S j (Suc n, f)) (Suc (n + j)) ·r X^R (Suc (n + j))",
subst ext_cf_inductTl, assumption+, simp del:npow_suc,
thin_tac "polyn_expr R X (n + j) (ext_cf S j (n, f)) =
X^R j ·r polyn_expr R X n (n, f)")
apply (cut_tac c1 = "(Suc n, f)" and n1 = j in ext_cf_hi[THEN sym],
assumption+,
simp del:npow_suc add:add_commute[of j])
apply (cut_tac n = n and c = "(Suc n, f)" in polyn_Suc, simp,
simp del:npow_suc)
apply (frule_tac c = "(Suc n, f)" and k = n in polyn_mem, simp,
frule_tac c = "(Suc n, f)" and j = "Suc n" in pol_coeff_mem_R, simp,
simp del:npow_suc,
frule_tac x = "f (Suc n)" and y = "X^R (Suc n)" in ring_tOp_closed,
rule npClose, assumption,
subst ring_distrib1, assumption+)
apply (subst polyn_expr_restrict, assumption+)
apply (rule_tac a = "f (Suc n) ·r X^R (Suc (n + j)) " and
b = "X^R j ·r (f (Suc n) ·r X^R (Suc n))" and
c = "X^R j ·r polyn_expr R X n (n, f)" in aGroup.ag_pOp_add_l,
assumption+,
rule ring_tOp_closed, assumption+, rule npClose, assumption,
(rule ring_tOp_closed, assumption+)+,
simp add:polyn_mem,
frule_tac n = "Suc n" in npClose[of X],
subst ring_tOp_assoc[THEN sym], assumption+,
subst ring_tOp_commute[of "X^R j"], assumption,
simp add:pol_coeff_mem,
subst ring_tOp_assoc, assumption+,
subst npMulDistr[of X], assumption, simp add:add_commute[of j])
apply simp
done
lemma (in PolynRg) low_deg_terms_zero:"pol_coeff S (n, f) ==>
polyn_expr R X (n + j) (ext_cf S j (n, f)) =
(X^R j) ·r (polyn_expr R X n (n, f))"
by (simp add:low_deg_terms_zeroTr)
lemma (in PolynRg) low_deg_terms_zero1:"pol_coeff S c ==>
polyn_expr R X ((fst c) + j) (ext_cf S j c) =
(X^R j) ·r (polyn_expr R X (fst c) c)"
by (cases c) (simp add: low_deg_terms_zeroTr)
lemma (in PolynRg) polyn_expr_tOpTr:"pol_coeff S (n, f) ==>
∀g. (pol_coeff S (m, g) --> (∃h. pol_coeff S ((n + m), h) ∧
h (n + m) = (f n) ·rS (g m) ∧
(polyn_expr R X (n + m) (n + m, h) =
(polyn_expr R X n (n, f)) ·r (polyn_expr R X m (m, g)))))"
apply (cut_tac subring,
cut_tac X_mem_R,
frule subring_Ring[of S])
apply (induct_tac m)
apply (rule allI, rule impI, simp)
apply (simp add:polyn_expr_def [of R X 0])
apply (frule_tac c = "(0,g)" in pol_coeff_mem[of _ 0], simp, simp,
frule_tac c = "(0,g)" in pol_coeff_mem_R[of _ 0], simp, simp)
apply (simp add:ring_r_one,
frule_tac c = "(n, f)" and k = n in polyn_mem, simp,
simp only:ring_tOp_commute[of "polyn_expr R X n (n, f)"],
subst scalar_times_pol_expr, assumption+, simp)
apply (cut_tac f = "sp_cf S (g 0) (n, f)" in pol_coeff_split)
apply (simp add:sp_cf_len)
apply (cut_tac f = "sp_cf S (g 0) (n, f)" in polyn_expr_split[of n],
simp only:sp_cf_len, simp only:fst_conv,
frule_tac a = "g 0" in sp_cf_pol_coeff[of "(n, f)"], assumption+,
simp,
subgoal_tac "snd (sp_cf S (g 0) (n, f)) n = (f n) ·rS (g 0)", blast)
apply (thin_tac "pol_coeff S (n, snd (sp_cf S (g 0) (n, f)))",
thin_tac "polyn_expr R X n (sp_cf S (g 0) (n, f)) =
polyn_expr R X n (n, snd (sp_cf S (g 0) (n, f)))",
thin_tac "pol_coeff S (sp_cf S (g 0) (n, f))")
apply (subst sp_cf_val[of "(n, f)" n], assumption+, simp, assumption, simp,
frule_tac c = "(n,f)" in pol_coeff_mem[of _ n], simp, simp,
simp add:Ring.ring_tOp_commute)
apply (rule allI, rule impI)
apply (frule_tac n = na and f = g in pol_coeff_pre,
drule_tac a = g in forall_spec, assumption+)
apply (erule exE, (erule conjE)+)
apply (cut_tac n = na and c = "(Suc na, g)" in polyn_Suc, (simp del:npow_suc)+,
thin_tac "polyn_expr R X (Suc na) (Suc na, g) =
polyn_expr R X na (Suc na, g) ± g (Suc na) ·r X^R (Suc na)",
subst polyn_expr_restrict, assumption)
apply (frule_tac c = "(n, f)" and k = n in polyn_mem,simp del:npow_suc,
frule_tac c = "(na, g)" and k = na in polyn_mem, simp del:npow_suc,
frule_tac c = "(Suc na, g)" in monomial_mem, simp del:npow_suc,
drule_tac a = "Suc na" in forall_spec, simp del:npow_suc)
apply (subst ring_distrib1, assumption+)
apply (rotate_tac 8, drule sym,
simp del:npow_suc)
apply (thin_tac "polyn_expr R X n (n, f) ·r polyn_expr R X na (na, g) =
polyn_expr R X (n + na) (n + na, h)")
apply (frule_tac c = "(Suc na, g)" and j ="Suc na" in pol_coeff_mem_R, simp,
simp del:npow_suc,
frule_tac c = "(Suc na, g)" and j ="Suc na" in pol_coeff_mem, simp,
simp del:npow_suc)
apply (subst ring_tOp_commute, assumption+,
subst ring_tOp_assoc, assumption+, rule npClose, assumption+,
subst low_deg_terms_zero[THEN sym], assumption+)
apply (frule_tac c = "(n, f)" and n = "Suc na" in ext_cf_pol_coeff)
apply (frule_tac c = "ext_cf S (Suc na) (n, f)" and a = "g (Suc na)" in
sp_cf_pol_coeff, assumption)
apply (subst scalar_times_pol_expr, assumption+,
simp add:ext_cf_len,
cut_tac k = "n + Suc na" and
f = "sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f))" in
polyn_expr_split,
simp only:sp_cf_len,
thin_tac "polyn_expr R X (n + Suc na)
(sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f))) =
polyn_expr R X (n + Suc na)
(fst (ext_cf S (Suc na) (n, f)),
snd (sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f))))",
simp only:ext_cf_len, simp only:fst_conv,
simp add:add_commute[of _ n])
apply (subst polyn_add, assumption+,
cut_tac f = "sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f))" in
pol_coeff_split, simp only:sp_cf_len, simp only:ext_cf_len,
simp add:add_commute[of _ n], simp add: max_def,
frule_tac c = "sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f))"
in pol_coeff_cartesian,
simp only:sp_cf_len, simp only:ext_cf_len,
simp add:add_commute[of _ n],
thin_tac "(Suc (n + na),
snd (sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f)))) =
sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f))",
frule_tac c = "(n + na, h)" and
d = "sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f))" in
add_cf_pol_coeff, assumption)
apply (cut_tac k = "Suc (n + na)" and f = "add_cf S (n + na, h)
(sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f)))" in polyn_expr_split,
simp only:mp,
thin_tac "polyn_expr R X (Suc (n + na))
(add_cf S (n + na, h)
(sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f)))) =
polyn_expr R X (Suc (n + na))
(fst (add_cf S (n + na, h)
(sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f)))),
snd (add_cf S (n + na, h)
(sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f)))))",
subst add_cf_len, assumption+,
simp add:sp_cf_len, simp add:ext_cf_len max_def,
cut_tac f = "add_cf S (n + na, h)
(sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f)))" in
pol_coeff_split,
simp only:add_cf_len,
simp only:sp_cf_len, simp add:ext_cf_len, simp add:max_def,
thin_tac "pol_coeff S
(add_cf S (n + na, h)
(sp_cf S (g (Suc na)) (ext_cf S (Suc na) (n, f))))",
subgoal_tac "snd (add_cf S (n + na, h) (sp_cf S (g (Suc na))
(ext_cf S (Suc na) (n, f)))) (Suc (n + na)) = f n ·rS g (Suc na)",
simp add:add_commute[of _ n], blast)
apply (subst add_cf_def, simp add:sp_cf_len ext_cf_len,
subst sp_cf_def, simp add:ext_cf_len,
subst ext_cf_def, simp add:sliden_def,
frule pol_coeff_mem[of "(n, f)" n], simp,
simp add:Ring.ring_tOp_commute)
done
lemma (in PolynRg) polyn_expr_tOp:"[|
pol_coeff S (n, f); pol_coeff S (m, g)|] ==> ∃e. pol_coeff S ((n + m), e) ∧
e (n + m) = (f n) ·rS (g m) ∧
polyn_expr R X (n + m)(n + m, e) =
(polyn_expr R X n (n, f)) ·r (polyn_expr R X m (m, g))"
by (simp add:polyn_expr_tOpTr)
lemma (in PolynRg) polyn_expr_tOp_c:"[|pol_coeff S c; pol_coeff S d|] ==>
∃e. pol_coeff S e ∧ (fst e = fst c + fst d) ∧
(snd e) (fst e) = (snd c (fst c)) ·rS (snd d) (fst d) ∧
polyn_expr R X (fst e) e =
(polyn_expr R X (fst c) c) ·r (polyn_expr R X (fst d) d)"
by (cases c, cases d) (simp add: polyn_expr_tOpTr)
section "14. the degree of a polynomial"
lemma (in PolynRg) polyn_degreeTr:"[|pol_coeff S c; k ≤ (fst c)|] ==>
(polyn_expr R X k c = \<zero> ) = ({j. j ≤ k ∧ (snd c) j ≠ \<zero>S} = {})"
apply (subst coeff_0_pol_0[THEN sym, of c k], assumption+)
apply blast
done
lemma (in PolynRg) higher_part_zero:"[|pol_coeff S c; k < fst c;
∀j∈nset (Suc k) (fst c). snd c j = \<zero>S|] ==>
Σf R (λj. snd c j ·r X^R j) (Suc k) (fst c) = \<zero>"
apply (cut_tac ring_is_ag,
rule aGroup.fSum_zero1[of R k "fst c" "λj. snd c j ·r X^R j"],
assumption+)
apply (rule ballI,
drule_tac b = j in forball_spec1, assumption, simp)
apply (cut_tac subring,
simp add:Subring_zero_ring_zero,
cut_tac X_mem_R,
frule_tac n = j in npClose[of X],
simp add:ring_times_0_x)
done
lemma (in PolynRg) coeff_nonzero_polyn_nonzero:"[|pol_coeff S c; k ≤ (fst c)|]
==> (polyn_expr R X k c ≠ \<zero>) = (∃j≤k. (snd c) j ≠ \<zero>S )"
by (simp add:coeff_0_pol_0[THEN sym, of c k])
lemma (in PolynRg) pol_expr_unique:"[|p ∈ carrier R; p ≠ \<zero>;
pol_coeff S c; p = polyn_expr R X (fst c) c; (snd c) (fst c) ≠ \<zero>S;
pol_coeff S d; p = polyn_expr R X (fst d) d; (snd d) (fst d) ≠ \<zero>S|] ==>
(fst c) = (fst d) ∧ (∀j ≤ (fst c). (snd c) j = (snd d) j)"
apply (cut_tac ring_is_ag,
cut_tac subring, frule subring_Ring,
frule Ring.ring_is_ag[of S])
apply (frule m_cf_pol_coeff[of d])
apply (frule polyn_minus_m_cf[of d "fst d"], simp)
apply (drule sym, drule sym, simp)
apply (rotate_tac -2, drule sym, drule sym)
apply (frule_tac x = p in aGroup.ag_r_inv1[of R], assumption, simp,
thin_tac "p = polyn_expr R X (fst c) c",
thin_tac "polyn_expr R X (fst d) d = polyn_expr R X (fst c) c",
thin_tac "-a (polyn_expr R X (fst c) c) =
polyn_expr R X (fst d) (m_cf S d)")
apply (frule polyn_add1[of c "m_cf S d"], assumption+, simp add:m_cf_len,
thin_tac "polyn_expr R X (fst c) c ± polyn_expr R X (fst d) (m_cf S d)
= polyn_expr R X (max (fst c) (fst d)) (add_cf S c (m_cf S d))",
thin_tac "polyn_expr R X (fst c) c ≠
polyn_expr R X (max (fst c) (fst d)) (add_cf S c (m_cf S d))")
apply (frule_tac add_cf_pol_coeff[of c "m_cf S d"], simp add:m_cf_len)
apply (cut_tac coeff_0_pol_0[THEN sym, of "add_cf S c (m_cf S d)"
"max (fst c) (fst d)"],
drule sym, simp,
thin_tac "polyn_expr R X (max (fst c) (fst d))
(add_cf S c (m_cf S d)) = \<zero>",
thin_tac "pol_coeff S (add_cf S c (m_cf S d))",
thin_tac "pol_coeff S (m_cf S d)")
apply (case_tac "fst c = fst d", simp)
apply (rule allI, rule impI,
drule_tac a = j in forall_spec, assumption)
apply (simp add:add_cf_def m_cf_def m_cf_len)
apply (frule_tac j = j in pol_coeff_mem[of c], simp,
frule_tac j = j in pol_coeff_mem[of d], simp)
apply (subst aGroup.ag_eq_diffzero[of S], assumption+)
apply (simp add:add_cf_def max_def)
apply (case_tac "¬ (fst c) ≤ (fst d)", simp)
apply (simp add:m_cf_len)
apply (drule_tac a = "fst c" in forall_spec, simp, simp)
apply simp
apply (drule_tac a = "fst d" in forall_spec, simp, simp add:m_cf_len)
apply (case_tac "fst c ≠ fst d",
frule noteq_le_less[of "fst c" "fst d"], assumption, simp)
apply (simp add:m_cf_def)
apply (frule pol_coeff_mem[of d "fst d"], simp)
apply (frule Ring.ring_is_ag[of S],
frule aGroup.ag_inv_inv[of S "snd d (fst d)"], assumption)
apply (simp add:aGroup.ag_inv_zero)
apply simp
apply simp
apply (simp add:add_cf_len m_cf_len)
done
lemma (in PolynRg) pol_expr_unique2:"[|pol_coeff S c; pol_coeff S d;
fst c = fst d|] ==>
(polyn_expr R X (fst c) c = polyn_expr R X (fst d) d ) =
(∀j ≤ (fst c). (snd c) j = (snd d) j)"
apply (cut_tac ring_is_ag,
cut_tac subring, frule subring_Ring,
frule Ring.ring_is_ag[of S])
apply (rule iffI)
apply (frule m_cf_pol_coeff[of d])
apply (frule polyn_mem[of c "fst c"], simp,
frule polyn_mem[of d "fst d"], simp)
apply (frule aGroup.ag_eq_diffzero[of R "polyn_expr R X (fst c) c"
"polyn_expr R X (fst d) d"], assumption+,
simp,
simp only:polyn_minus_m_cf[of d "fst d"],
drule sym, simp)
apply (frule polyn_add1[of c "m_cf S d"], assumption+, simp add:m_cf_len)
apply (thin_tac "polyn_expr R X (fst c) d ± polyn_expr R X (fst c)
(m_cf S d) =
polyn_expr R X (fst c) (add_cf S c (m_cf S d))",
thin_tac "polyn_expr R X (fst c) c = polyn_expr R X (fst c) d",
thin_tac "polyn_expr R X (fst c) d ∈ carrier R",
drule sym)
apply (frule_tac add_cf_pol_coeff[of c "m_cf S d"], simp add:m_cf_len)
apply (frule coeff_0_pol_0[THEN sym, of "add_cf S c (m_cf S d)"
"fst c"],
simp add:add_cf_len, simp add:m_cf_len,
thin_tac "\<zero> = polyn_expr R X (fst d) (add_cf S c (m_cf S d))",
thin_tac "pol_coeff S (add_cf S c (m_cf S d))")
apply (simp add:add_cf_def m_cf_def)
apply (rule allI, rule impI)
apply (drule_tac a = j in forall_spec, assumption)
apply (frule_tac j = j in pol_coeff_mem[of c], simp,
frule_tac j = j in pol_coeff_mem[of d], simp)
apply (simp add:aGroup.ag_eq_diffzero[THEN sym])
apply simp
apply (rule polyn_exprs_eq[of c d "fst d"], assumption+)
apply (simp add:min_def, assumption+)
done
lemma (in PolynRg) pol_expr_unique3:"[|pol_coeff S c; pol_coeff S d;
fst c < fst d|] ==>
(polyn_expr R X (fst c) c = polyn_expr R X (fst d) d ) =
((∀j ≤ (fst c). (snd c) j = (snd d) j) ∧
(∀j∈nset (Suc (fst c)) (fst d). (snd d) j = \<zero>S))"
apply (rule iffI)
apply (cut_tac ring_is_ag,
cut_tac subring, frule subring_Ring,
frule Ring.ring_is_ag[of S])
apply (frule m_cf_pol_coeff[of d])
apply (frule polyn_mem[of c "fst c"], simp,
frule polyn_mem[of d "fst d"], simp)
apply (frule aGroup.ag_eq_diffzero[of R "polyn_expr R X (fst c) c"
"polyn_expr R X (fst d) d"], assumption+,
simp,
simp only:polyn_minus_m_cf[of d "fst d"],
drule sym, simp)
apply (frule polyn_add1[of c "m_cf S d"], assumption+, simp add:m_cf_len,
thin_tac "polyn_expr R X (fst c) c ± polyn_expr R X (fst d)
(m_cf S d) =
polyn_expr R X (max (fst c) (fst d)) (add_cf S c (m_cf S d))",
thin_tac "polyn_expr R X (fst d) d = polyn_expr R X (fst c) c",
thin_tac "polyn_expr R X (fst c) c ∈ carrier R", drule sym)
apply (frule_tac add_cf_pol_coeff[of c "m_cf S d"], simp add:m_cf_len)
apply (frule coeff_0_pol_0[THEN sym, of "add_cf S c (m_cf S d)"
"max (fst c) (fst d)"],
simp add:add_cf_len m_cf_len, simp,
thin_tac "polyn_expr R X (max (fst c) (fst d))
(add_cf S c (m_cf S d)) = \<zero>",
thin_tac "pol_coeff S (add_cf S c (m_cf S d))")
apply (simp add:add_cf_def m_cf_def max_def)
apply (rule conjI)
apply (rule allI, rule impI,
frule_tac x = j and y = "fst c" and z = "fst d" in le_less_trans,
assumption+,
frule_tac x = j and y = "fst d" in less_imp_le)
apply (drule_tac a = j in forall_spec, simp, simp)
apply (frule_tac j = j in pol_coeff_mem[of c], simp,
frule_tac j = j in pol_coeff_mem[of d], simp)
apply (simp add:aGroup.ag_eq_diffzero[THEN sym])
apply (rule ballI, simp add:nset_def, erule conjE)
apply (cut_tac x = "fst c" and y = "Suc (fst c)" and z = j in
less_le_trans, simp, assumption)
apply (cut_tac m1 = "fst c" and n1 = j in nat_not_le_less[THEN sym], simp)
apply (drule_tac a = j in forall_spec, assumption, simp,
frule_tac j = j in pol_coeff_mem[of d], simp)
apply (frule_tac x = "snd d j" in aGroup.ag_inv_inv[of S], assumption,
simp add:aGroup.ag_inv_inv aGroup.ag_inv_zero)
apply (cut_tac polyn_n_m[of d "fst c" "fst d"])
apply (subst polyn_expr_split[of "fst d" d], simp,
thin_tac "polyn_expr R X (fst d) d =
polyn_expr R X (fst c) (fst c, snd d) ±
Σf R (λj. snd d j ·r X^R j) (Suc (fst c)) (fst d)", erule conjE)
apply (subst higher_part_zero[of d "fst c"], assumption+)
apply (frule pol_coeff_le[of d "fst c"], simp add:less_imp_le,
frule polyn_mem[of "(fst c, snd d)" "fst c"], simp,
cut_tac ring_is_ag,
simp add:aGroup.ag_r_zero,
subst polyn_expr_short[THEN sym, of d "fst c"], assumption+,
simp add:less_imp_le)
apply (rule polyn_exprs_eq[of c d "fst c"], assumption+)
apply (simp add:min_def, assumption+)
apply (simp add:less_imp_le)
done
lemma (in PolynRg) polyn_degree_unique:"[|pol_coeff S c; pol_coeff S d;
polyn_expr R X (fst c) c = polyn_expr R X (fst d) d|] ==>
c_max S c = c_max S d"
apply (cut_tac ring_is_ag,
cut_tac subring,
frule subring_Ring,
frule Ring.ring_is_ag[of S])
apply (case_tac "polyn_expr R X (fst d) d = \<zero>R")
apply (cut_tac coeff_0_pol_0[THEN sym, of d "fst d"], simp,
cut_tac coeff_0_pol_0[THEN sym, of c "fst c"], simp)
apply (simp add:c_max_def, assumption, simp, assumption, simp)
apply (frule polyn_mem[of c "fst c"], simp, frule polyn_mem[of d "fst d"],
simp)
apply (frule aGroup.ag_eq_diffzero[of "R" "polyn_expr R X (fst c) c"
"polyn_expr R X (fst d) d"], assumption+)
apply (simp only:polyn_minus_m_cf[of d "fst d"],
frule m_cf_pol_coeff [of d])
apply (frule polyn_add1[of c "m_cf S d"], assumption+,
simp only:m_cf_len)
apply (rotate_tac -1, drule sym, simp,
thin_tac "polyn_expr R X (fst d) d ±
polyn_expr R X (fst d) (m_cf S d) = \<zero>",
frule add_cf_pol_coeff[of c "m_cf S d"], assumption+)
apply (cut_tac coeff_0_pol_0[THEN sym, of "add_cf S c (m_cf S d)"
"fst (add_cf S c (m_cf S d))"],
simp add:add_cf_len m_cf_len,
thin_tac "polyn_expr R X (max (fst c) (fst d))
(add_cf S c (m_cf S d)) = \<zero>",
thin_tac "pol_coeff S (add_cf S c (m_cf S d))",
thin_tac "pol_coeff S (m_cf S d)")
apply (frule coeff_nonzero_polyn_nonzero[of d "fst d"], simp, simp)
apply (drule sym, simp)
apply (frule coeff_nonzero_polyn_nonzero[of c "fst c"], simp, simp)
apply (simp add:c_max_def, rule conjI, rule impI, blast,
rule conjI, rule impI, blast)
apply (rule n_max_eq_sets)
apply (rule equalityI)
apply (rule subsetI, simp)
apply (erule conjE)
apply (case_tac "fst c ≤ fst d")
apply (frule_tac i = x in le_trans[of _ "fst c" "fst d"], assumption+, simp,
rule contrapos_pp, simp+, simp add:max_def,
frule_tac i = x in le_trans[of _ "fst c" "fst d"], assumption+,
drule_tac a = x in forall_spec, assumption,
drule le_imp_less_or_eq[of "fst c" "fst d"],
erule disjE, simp add:add_cf_def m_cf_len m_cf_def,
frule_tac j = x in pol_coeff_mem[of c], assumption+,
simp add:aGroup.ag_inv_zero aGroup.ag_r_zero[of S])
apply (simp add:add_cf_def m_cf_len m_cf_def,
rotate_tac -1, drule sym, simp,
frule_tac j = x in pol_coeff_mem[of c], simp,
simp add:aGroup.ag_inv_zero aGroup.ag_r_zero[of S])
apply (simp add:nat_not_le_less) (* ¬ fst c ≤ fst d *)
apply (case_tac "¬ x ≤ (fst d)", simp,
simp add:nat_not_le_less,
frule_tac x = "fst d" and y = x and z = "fst c" in
less_le_trans, assumption+,
drule_tac a = x in forall_spec1, simp add:max_def,
simp add:add_cf_def m_cf_len m_cf_def)
apply (simp,
drule_tac a = x in forall_spec1, simp add:max_def,
rule contrapos_pp, simp+,
simp add:add_cf_def m_cf_len m_cf_def,
frule_tac j = x in pol_coeff_mem[of c],
frule_tac x = x and y = "fst d" and z = "fst c" in
le_less_trans, assumption+, simp add:less_imp_le,
simp add:aGroup.ag_inv_zero aGroup.ag_r_zero[of S])
apply (rule subsetI, simp, erule conjE,
case_tac "fst d ≤ fst c",
frule_tac i = x and j = "fst d" and k = "fst c" in le_trans,
assumption+, simp,
drule_tac a = x in forall_spec1, simp add:max_def,
rule contrapos_pp, simp+,
simp add:add_cf_def m_cf_len m_cf_def)
apply (case_tac "fst d = fst c", simp, rotate_tac -1, drule sym, simp,
frule_tac j = x in pol_coeff_mem[of d], assumption,
frule_tac x = "snd d x" in aGroup.ag_mOp_closed, assumption+,
simp add:aGroup.ag_l_zero,
frule_tac x = "snd d x" in aGroup.ag_inv_inv[of S],
assumption, simp add:aGroup.ag_inv_zero)
apply (drule noteq_le_less[of "fst d" "fst c"], assumption,
simp,
frule_tac j = x in pol_coeff_mem[of d], assumption,
frule_tac x = "snd d x" in aGroup.ag_mOp_closed, assumption+,
simp add:aGroup.ag_l_zero,
frule_tac x = "snd d x" in aGroup.ag_inv_inv[of S],
assumption, simp add:aGroup.ag_inv_zero)
apply (simp add:nat_not_le_less,
case_tac "¬ x ≤ fst c", simp,
simp add:nat_not_le_less,
drule_tac a = x in forall_spec1, simp add:max_def,
simp add:add_cf_def m_cf_len m_cf_def,
frule_tac j = x in pol_coeff_mem[of d], assumption,
frule_tac x = "snd d x" in aGroup.ag_mOp_closed, assumption+,
simp add:aGroup.ag_l_zero,
frule_tac x = "snd d x" in aGroup.ag_inv_inv[of S],
assumption, simp add:aGroup.ag_inv_zero)
apply (simp,
drule_tac a = x in forall_spec1, simp add:max_def,
rule contrapos_pp, simp+,
simp add:add_cf_def m_cf_len m_cf_def,
frule_tac x = x and y = "fst c" and z = "fst d" in le_less_trans,
assumption+, frule_tac x = x and y = "fst d" in less_imp_le,
frule_tac j = x in pol_coeff_mem[of d], assumption,
frule_tac x = "snd d x" in aGroup.ag_mOp_closed, assumption+,
simp add:aGroup.ag_l_zero,
frule_tac x = "snd d x" in aGroup.ag_inv_inv[of S],
assumption, simp add:aGroup.ag_inv_zero)
apply (thin_tac "∀j≤max (fst c) (fst d). snd (add_cf S c (m_cf S d)) j = \<zero>S")
apply (rotate_tac -1, drule sym, simp)
apply (simp add:coeff_0_pol_0[THEN sym, of c "fst c"])
apply blast
apply simp+
done
lemma (in PolynRg) ex_polyn_expr:"p ∈ carrier R ==>
∃c. pol_coeff S c ∧ p = polyn_expr R X (fst c) c"
apply (cut_tac S_X_generate[of p], blast)
apply assumption
done
lemma (in PolynRg) c_max_eqTr0:"[|pol_coeff S c; k ≤ (fst c);
polyn_expr R X k c = polyn_expr R X (fst c) c; ∃j≤k. (snd c) j ≠ \<zero>S|] ==>
c_max S (k, snd c) = c_max S c"
apply (simp add:polyn_expr_short[of c k],
frule pol_coeff_le[of c k], assumption+,
rule polyn_degree_unique[of "(k, snd c)" c], assumption+,
simp)
done
constdefs
cf_sol::"[('a, 'b) Ring_scheme, ('a, 'b1) Ring_scheme, 'a, 'a,
nat × (nat => 'a)] => bool"
"cf_sol R S X p c == pol_coeff S c ∧ (p = polyn_expr R X (fst c) c)"
constdefs
deg_n ::"[('a, 'b) Ring_scheme, ('a, 'b1) Ring_scheme, 'a, 'a] => nat"
"deg_n R S X p == c_max S (SOME c. cf_sol R S X p c)"
deg ::"[('a, 'b) Ring_scheme, ('a, 'b1) Ring_scheme, 'a, 'a] => ant"
"deg R S X p == if p = \<zero>R then -∞ else (an (deg_n R S X p))"
lemma (in PolynRg) ex_cf_sol:"p ∈ carrier R ==>
∃c. cf_sol R S X p c"
apply (unfold cf_sol_def)
apply (frule ex_polyn_expr[of p], (erule exE)+)
apply (cut_tac n = "fst c" in le_refl, blast)
done
lemma (in PolynRg) deg_in_aug_minf:"p ∈ carrier R ==>
deg R S X p ∈ Z-∞"
apply (simp add:aug_minf_def deg_def an_def)
done
lemma (in PolynRg) deg_noninf:"p ∈ carrier R ==>
deg R S X p ≠ ∞"
apply (cut_tac deg_in_aug_minf[of p], simp add:deg_def,
simp add:aug_minf_def)
apply (case_tac "p = \<zero>R", simp+)
done
lemma (in PolynRg) deg_ant_int:"[|p ∈ carrier R; p ≠ \<zero>|]
==> deg R S X p = ant (int (deg_n R S X p))"
by (simp add:deg_def an_def)
lemma (in PolynRg) deg_an:"[|p ∈ carrier R; p ≠ \<zero>|]
==> deg R S X p = an (deg_n R S X p)"
by (simp add:deg_def)
lemma (in PolynRg) pol_SOME_1:"p ∈ carrier R ==>
cf_sol R S X p (SOME f. cf_sol R S X p f)"
apply (frule ex_cf_sol[of p])
apply (rule_tac P = "cf_sol R S X p" in someI_ex, assumption)
done
lemma (in PolynRg) pol_SOME_2:"p ∈ carrier R ==>
pol_coeff S (SOME c. cf_sol R S X p c) ∧
p = polyn_expr R X (fst (SOME c. cf_sol R S X p c))
(SOME c. cf_sol R S X p c)"
apply (frule pol_SOME_1[of p])
apply (simp add:cf_sol_def)
done
lemma (in PolynRg) coeff_max_zeroTr:"pol_coeff S c ==>
∀j. j ≤ (fst c) ∧ (c_max S c) < j --> (snd c) j = \<zero>S"
apply (case_tac "∀j ≤ (fst c). (snd c) j = \<zero>S", rule allI, rule impI,
erule conjE, simp)
apply simp
apply (frule coeff_nonzero_polyn_nonzero[THEN sym, of c "fst c"], simp,
simp)
apply (rule allI, rule impI, erule conjE,
simp add:c_max_def,
simp add:polyn_degreeTr[of c "fst c"])
apply (subgoal_tac "{j. j ≤ (fst c) ∧ (snd c) j ≠ \<zero>S} ⊆ {j. j ≤ (fst c)}",
frule n_max[of "{j. j ≤ (fst c) ∧ (snd c) j ≠ \<zero>S}" "fst c"], blast)
apply (case_tac "∀x≤fst c. snd c x = \<zero>S ", blast, simp)
apply (erule conjE)
apply (rule contrapos_pp, simp+,
thin_tac "∃x≤fst c. snd c x ≠ \<zero>S",
thin_tac "{j. j ≤ fst c ∧ snd c j ≠ \<zero>S} ⊆ {j. j ≤ fst c}",
thin_tac "snd c (n_max {j. j ≤ fst c ∧ snd c j ≠ \<zero>S}) ≠ \<zero>S",
drule_tac a = j in forall_spec, simp)
apply simp
apply (rule subsetI, simp)
done
lemma (in PolynRg) coeff_max_nonzeroTr:"[|pol_coeff S c;
∃j ≤ (fst c). (snd c) j ≠ \<zero>S|] ==> (snd c) (c_max S c) ≠ \<zero>S"
apply (simp add:c_max_def)
apply (subgoal_tac "{j. j ≤ (fst c) ∧ (snd c) j ≠ \<zero>S} ⊆ {j. j ≤ (fst c)}",
frule n_max[of "{j. j ≤ (fst c) ∧ (snd c) j ≠ \<zero>S}" "fst c"], blast)
apply (erule conjE, simp)
apply (rule subsetI, simp)
done
lemma (in PolynRg) coeff_max_bddTr:"pol_coeff S c ==> c_max S c ≤ (fst c)"
apply (case_tac "∀j≤(fst c). (snd c) j = \<zero>S", simp add:c_max_def)
apply (simp add:c_max_def,
frule polyn_degreeTr[of c "fst c"], simp, simp,
subgoal_tac "{j. j ≤ (fst c) ∧ (snd c) j ≠ \<zero>S} ⊆ {j. j ≤ (fst c)}",
frule n_max[of "{j. j ≤ (fst c) ∧ (snd c) j ≠ \<zero>S}" "fst c"],
blast, erule conjE, simp)
apply (rule subsetI, simp)
done
lemma (in PolynRg) pol_coeff_max:"pol_coeff S c ==>
pol_coeff S ((c_max S c), snd c)"
apply (rule pol_coeff_le[of c "c_max S c"], assumption)
apply (simp add:coeff_max_bddTr)
done
lemma (in PolynRg) polyn_c_max:"pol_coeff S c ==>
polyn_expr R X (fst c) c = polyn_expr R X (c_max S c) c"
apply (case_tac "(c_max S c) = (fst c)", simp)
apply (frule coeff_max_bddTr[of c],
frule noteq_le_less[of "c_max S c" "fst c"], assumption)
apply (subst polyn_n_m1[of c "c_max S c" "fst c"], assumption+, simp)
apply (frule_tac polyn_mem[of c "c_max S c"], assumption+)
apply (subst higher_part_zero[of c "c_max S c"], assumption+)
apply (frule coeff_max_zeroTr[of c],
rule ballI, simp add:nset_def)
apply (cut_tac ring_is_ag, simp add:aGroup.ag_r_zero)
done
lemma (in PolynRg) pol_deg_eq_c_max:"[|p ∈ carrier R;
pol_coeff S c; p = polyn_expr R X (fst c) c|] ==>
deg_n R S X p = c_max S c"
apply (cut_tac subring, frule subring_Ring)
apply (frule polyn_c_max[of c])
apply (frule pol_SOME_2[of p])
apply (subst deg_n_def, erule conjE)
apply (rule polyn_degree_unique[of "Eps (cf_sol R S X p)" "c"], simp,
assumption)
apply simp
done
lemma (in PolynRg) pol_deg_le_n:"[|p ∈ carrier R; pol_coeff S c;
p = polyn_expr R X (fst c) c|] ==> deg_n R S X p ≤ (fst c)"
apply (frule pol_deg_eq_c_max[of p c], assumption+,
frule coeff_max_bddTr[of c])
apply simp
done
lemma (in PolynRg) pol_deg_le_n1:"[|p ∈ carrier R; pol_coeff S c; k ≤ (fst c);
p = polyn_expr R X k c|] ==> deg_n R S X p ≤ k"
apply (simp add:deg_n_def, drule sym, simp)
apply (frule pol_SOME_2[of p], erule conjE)
apply (frule pol_coeff_le[of c k], assumption)
apply (simp only:polyn_expr_short[of c k])
apply (drule sym)
apply (subst polyn_degree_unique[of "SOME c. cf_sol R S X p c" "(k, snd c)"],
assumption+, simp)
apply (frule coeff_max_bddTr[of "(k, snd c)"], simp)
done
lemma (in PolynRg) pol_len_gt_deg:"[|p ∈ carrier R; pol_coeff S c;
p = polyn_expr R X (fst c) c; deg R S X p < (an j); j ≤ (fst c)|]
==> (snd c) j = \<zero>S"
apply (case_tac "p = \<zero>R", simp, drule sym)
apply (simp add:coeff_0_pol_0[THEN sym, of c "fst c"])
apply (simp add:deg_def, simp add:aless_natless)
apply (drule sym, simp)
apply (frule coeff_max_zeroTr[of c])
apply (simp add:pol_deg_eq_c_max)
done
lemma (in PolynRg) pol_diff_deg_less:"[|p ∈ carrier R; pol_coeff S c;
p = polyn_expr R X (fst c) c; pol_coeff S d;
fst c = fst d; (snd c) (fst c) = (snd d) (fst d)|] ==>
p ± (-a (polyn_expr R X (fst d) d)) = \<zero> ∨
deg_n R S X (p ± (-a (polyn_expr R X (fst d) d))) < (fst c)"
apply (cut_tac ring_is_ag,
cut_tac subring, frule subring_Ring)
apply (case_tac "p ±R (-aR (polyn_expr R X (fst d) d)) = \<zero>R", simp)
apply simp
apply (simp add:polyn_minus_m_cf[of d "fst d"],
frule m_cf_pol_coeff[of d])
apply (cut_tac polyn_add1[of c "m_cf S d"], simp add:m_cf_len,
thin_tac "polyn_expr R X (fst d) c ± polyn_expr R X (fst d) (m_cf S d)
= polyn_expr R X (fst d) (add_cf S c (m_cf S d))")
apply (frule add_cf_pol_coeff[of c "m_cf S d"], assumption+)
apply (cut_tac polyn_mem[of "add_cf S c (m_cf S d)" "fst d"],
frule pol_deg_le_n[of "polyn_expr R X (fst d) (add_cf S c (m_cf S d))"
"add_cf S c (m_cf S d)"], assumption+,
simp add:add_cf_len m_cf_len,
simp add:add_cf_len m_cf_len)
apply (rule noteq_le_less[of "deg_n R S X (polyn_expr R X (fst d)
(add_cf S c (m_cf S d)))" "fst d"], assumption)
apply (rule contrapos_pp, simp+)
apply (cut_tac pol_deg_eq_c_max[of "polyn_expr R X (fst d)
(add_cf S c (m_cf S d))" "add_cf S c (m_cf S d)"],
simp,
thin_tac "deg_n R S X (polyn_expr R X (fst d) (add_cf S c (m_cf S d)))
= fst d")
apply (frule coeff_nonzero_polyn_nonzero[of "add_cf S c (m_cf S d)" "fst d"],
simp add:add_cf_len m_cf_len, simp,
thin_tac "polyn_expr R X (fst d) (add_cf S c (m_cf S d)) ≠ \<zero>",
frule coeff_max_nonzeroTr[of "add_cf S c (m_cf S d)"],
simp add:add_cf_len m_cf_len,
thin_tac "∃j≤fst d. snd (add_cf S c (m_cf S d)) j ≠ \<zero>S",
thin_tac "pol_coeff S (m_cf S d)",
thin_tac "pol_coeff S (add_cf S c (m_cf S d))",
thin_tac "polyn_expr R X (fst d) (add_cf S c (m_cf S d)) ∈
carrier R", simp,
thin_tac "c_max S (add_cf S c (m_cf S d)) = fst d")
apply (simp add:add_cf_def m_cf_def,
frule pol_coeff_mem[of d "fst d"], simp,
frule Ring.ring_is_ag[of S],
simp add:aGroup.ag_r_inv1, assumption+,
simp add:add_cf_len m_cf_len, assumption,
simp add:add_cf_len m_cf_len, assumption+)
done
lemma (in PolynRg) pol_pre_lt_deg:"[|p ∈ carrier R; pol_coeff S c;
deg_n R S X p ≤ (fst c); (deg_n R S X p) ≠ 0;
p = polyn_expr R X (deg_n R S X p) c |] ==>
(deg_n R S X (polyn_expr R X ((deg_n R S X p) - Suc 0) c)) < (deg_n R S X p)"
apply (frule polyn_expr_short[of c "deg_n R S X p"], assumption)
apply (cut_tac pol_deg_le_n[of "polyn_expr R X (deg_n R S X p - Suc 0) c"
"(deg_n R S X p - Suc 0, snd c)"], simp)
apply (rule polyn_mem[of c "deg_n R S X p - Suc 0"], assumption+,
arith,
rule pol_coeff_le[of c "deg_n R S X p - Suc 0"], assumption,
arith, simp)
apply (subst polyn_expr_short[of c "deg_n R S X p - Suc 0"],
assumption+, arith, simp)
done
lemma (in PolynRg) pol_deg_n:"[|p ∈ carrier R; pol_coeff S c;
n ≤ fst c; p = polyn_expr R X n c; (snd c) n ≠ \<zero>S|] ==>
deg_n R S X p = n"
apply (simp add:polyn_expr_short[of c n])
apply (frule pol_coeff_le[of c n], assumption+,
cut_tac pol_deg_eq_c_max[of p "(n, snd c)"],
drule sym, simp, simp add:c_max_def)
apply (rule conjI, rule impI, cut_tac le_refl[of n],
thin_tac "deg_n R S X p =
(if ∀x≤n. snd c x = \<zero>S then 0
else n_max {j. j ≤ fst (n, snd c) ∧ snd (n, snd c) j ≠ \<zero>S})",
drule_tac a = n in forall_spec, assumption, simp)
apply (rule impI)
apply (cut_tac n_max[of "{j. j ≤ n ∧ snd c j ≠ \<zero>S}" n], erule conjE,
drule_tac b = n in forball_spec1, simp, simp)
apply (rule subsetI, simp, blast,
drule sym, simp, assumption)
apply simp
done
lemma (in PolynRg) pol_expr_deg:"[|p ∈ carrier R; p ≠ \<zero>|]
==> ∃c. pol_coeff S c ∧ deg_n R S X p ≤ (fst c) ∧
p = polyn_expr R X (deg_n R S X p) c ∧
(snd c) (deg_n R S X p) ≠ \<zero>S"
apply (cut_tac subring,
frule subring_Ring)
apply (frule ex_polyn_expr[of p], erule exE, erule conjE)
apply (frule_tac c = c in polyn_c_max)
apply (frule_tac c = c in pol_deg_le_n[of p], assumption+)
apply (frule_tac c1 = c and k1 ="fst c" in coeff_0_pol_0[THEN sym], simp)
apply (subgoal_tac "p = polyn_expr R X (deg_n R S X p) c ∧
snd c (deg_n R S X p) ≠ \<zero>S", blast)
apply (subst pol_deg_eq_c_max, assumption+)+
apply simp
apply (cut_tac c = c in coeff_max_nonzeroTr, simp+)
done
lemma (in PolynRg) deg_n_pos:"p ∈ carrier R ==> 0 ≤ deg_n R S X p"
by simp
lemma (in PolynRg) pol_expr_deg1:"[|p ∈ carrier R; d = na (deg R S X p)|] ==>
∃c. (pol_coeff S c ∧ p = polyn_expr R X d c)"
apply (cut_tac subring, frule subring_Ring)
apply (case_tac "p = \<zero>R",
simp add:deg_def na_minf,
subgoal_tac "pol_coeff S (0, (λj. \<zero>S))",
subgoal_tac "\<zero> = polyn_expr R X d (0, (λj. \<zero>S))", blast,
cut_tac coeff_0_pol_0[of "(d, λj. \<zero>S)" d], simp+,
simp add:pol_coeff_def,
simp add:Ring.ring_zero)
apply (simp add:deg_def na_an,
frule pol_expr_deg[of p], assumption,
erule exE, (erule conjE)+,
unfold split_paired_all, simp, blast)
done
end
lemma ideal_sumTr1:
[| ideal R A; ideal R B |]
==> A \<minusplus> B = Inter {J. ideal R J ∧ A ∪ B ⊆ J}
lemma sum_ideals_commute:
[| ideal R A; ideal R B |] ==> A \<minusplus> B = B \<minusplus> A
lemma ideal_prod_mono1:
[| ideal R A; ideal R B; ideal R C; A ⊆ B |] ==> A ♦r C ⊆ B ♦r C
lemma ideal_prod_mono2:
[| ideal R A; ideal R B; ideal R C; A ⊆ B |] ==> C ♦r A ⊆ C ♦r B
lemma cont_ideal_prod:
[| ideal R A; ideal R B; ideal R C; A ⊆ C; B ⊆ C |] ==> A ♦r B ⊆ C
lemma ideal_distrib:
[| ideal R A; ideal R B; ideal R C |]
==> A ♦r (B \<minusplus> C) = A ♦r B \<minusplus> A ♦r C
lemma coprimeTr:
[| ideal R A; ideal R B |] ==> coprime_ideals R A B = (∃a∈A. ∃b∈B. a ± b = 1r)
lemma coprime_int_prod:
[| ideal R A; ideal R B; coprime_ideals R A B |] ==> A ∩ B = A ♦r B
lemma coprime_elems:
[| ideal R A; ideal R B; coprime_ideals R A B |] ==> ∃a∈A. ∃b∈B. a ± b = 1r
lemma coprime_elemsTr:
[| ideal R A; ideal R B; a ∈ A; b ∈ B; a ± b = 1r |]
==> pj R A b = 1rR /r A ∧ pj R B a = 1rR /r B
lemma partition_of_unity:
[| ideal R A; a ∈ A; b ∈ carrier R; a ± b = 1r; u ∈ carrier R; v ∈ carrier R |]
==> pj R A (a ·r v ± b ·r u) = pj R A u
lemma coprimes_commute:
[| ideal R A; ideal R B; coprime_ideals R A B |] ==> coprime_ideals R B A
lemma coprime_surjTr:
[| ideal R A; ideal R B; coprime_ideals R A B; X ∈ carrier (R /r A);
Y ∈ carrier (R /r B) |]
==> ∃r∈carrier R. pj R A r = X ∧ pj R B r = Y
lemma coprime_n_idealsTr0:
[| ideal R A; ideal R B; ideal R C; coprime_ideals R A C;
coprime_ideals R B C |]
==> coprime_ideals R (A ♦r B) C
lemma coprime_n_idealsTr1:
ideal R C
==> (∀k≤n. ideal R (J k)) ∧ (∀i≤n. coprime_ideals R (J i) C) -->
coprime_ideals R (iΠR,n J) C
lemma coprime_n_idealsTr2:
[| ideal R C; ∀k≤n. ideal R (J k); ∀i≤n. coprime_ideals R (J i) C |]
==> coprime_ideals R (iΠR,n J) C
lemma coprime_n_idealsTr3:
(∀k≤Suc n. ideal R (J k)) ∧
(∀i≤Suc n. ∀j≤Suc n. i ≠ j --> coprime_ideals R (J i) (J j)) -->
coprime_ideals R (iΠR,n J) (J (Suc n))
lemma coprime_n_idealsTr4:
(∀k≤Suc n. ideal R (J k)) ∧
(∀i≤Suc n. ∀j≤Suc n. i ≠ j --> coprime_ideals R (J i) (J j))
==> coprime_ideals R (iΠR,n J) (J (Suc n))
lemma carr_prodrg_mem_eq:
[| f ∈ carrier (rΠI A); g ∈ carrier (rΠI A); ∀i∈I. f i = g i |] ==> f = g
lemma prod_tOp_mem:
[| ∀k∈I. Ring (A k); X ∈ carr_prodag I A; Y ∈ carr_prodag I A |]
==> prod_tOp I A X Y ∈ carr_prodag I A
lemma prod_tOp_func:
∀k∈I. Ring (A k)
==> prod_tOp I A ∈ carr_prodag I A -> carr_prodag I A -> carr_prodag I A
lemma prod_one_func:
∀k∈I. Ring (A k) ==> prod_one I A ∈ carr_prodag I A
lemma prodrg_carrier:
∀k∈I. Ring (A k) ==> carrier (rΠI A) = carrier (aΠI A)
lemma prodrg_ring:
∀k∈I. Ring (A k) ==> Ring (rΠI A)
lemma prodrg_elem_extensional:
[| ∀k∈I. Ring (A k); f ∈ carrier (rΠI A) |] ==> f ∈ extensional I
lemma prodrg_pOp:
∀k∈I. Ring (A k) ==> op ±rΠI A = prod_pOp I A
lemma prodrg_mOp:
∀k∈I. Ring (A k) ==> mop (rΠI A) = prod_mOp I A
lemma prodrg_zero:
∀k∈I. Ring (A k) ==> \<zero>rΠI A = prod_zero I A
lemma prodrg_tOp:
∀k∈I. Ring (A k) ==> op ·rrΠI A = prod_tOp I A
lemma prodrg_one:
∀k∈I. Ring (A k) ==> 1rrΠI A = prod_one I A
lemma prodrg_sameTr5:
[| ∀k∈I. Ring (A k); ∀k∈I. A k = B k |] ==> prod_tOp I A = prod_tOp I B
lemma prodrg_sameTr6:
[| ∀k∈I. Ring (A k); ∀k∈I. A k = B k |] ==> prod_one I A = prod_one I B
lemma prodrg_same:
[| ∀k∈I. Ring (A k); ∀k∈I. A k = B k |] ==> rΠI A = rΠI B
lemma prodrg_component:
[| f ∈ carrier (rΠI A); i ∈ I |] ==> f i ∈ carrier (A i)
lemma project_rhom:
[| ∀k∈I. Ring (A k); j ∈ I |] ==> πI,A,j ∈ rHom (rΠI A) (A j)
lemma augm_funcTr:
[| ∀k≤Suc n. Ring (B k); f ∈ carr_prodag {i. i ≤ Suc n} B |]
==> f = augm_func n (restrict f {i. i ≤ n}) (Un_carrier {i. i ≤ n} B) (Suc 0)
(λx∈{0}. f (x + Suc n))
(Un_carrier {0} (compose {0} B (slide (Suc n))))
lemma A_to_prodag_mem:
[| Ring A; ∀k∈I. Ring (B k); ∀k∈I. S k ∈ rHom A (B k); x ∈ carrier A |]
==> A_to_prodag A I S B x ∈ carr_prodag I B
lemma A_to_prodag_rHom:
[| Ring A; ∀k∈I. Ring (B k); ∀k∈I. S k ∈ rHom A (B k) |]
==> A_to_prodag A I S B ∈ rHom A (rΠI B)
lemma ac_fProd_ProdTr1:
∀k≤Suc n. Ring (B k)
==> ag_setfunc n B (Suc 0) (compose {0} B (slide (Suc n)))
(carr_prodag {i. i ≤ n} B)
(carr_prodag {0} (compose {0} B (slide (Suc n))))
⊆ carr_prodag {i. i ≤ Suc n} B
lemma ac_fProd_Prod:
∀k≤n. Ring (B k) ==> ac_fProd_Rg n B = carr_prodag {j. j ≤ n} B
lemma fprodrg_ring:
∀k≤n. Ring (B k) ==> Ring (frΠ\<^sub>n B)
lemma Chinese_remTr1:
[| Ring A; ∀k≤n. ideal A (J k); ∀k≤n. B k = A /r J k; ∀k≤n. S k = pj A (J k) |]
==> kerA,(rΠ{j. j ≤ n} B) A_to_prodag A {j. j ≤ n} S B =
Inter {I. ∃k∈{j. j ≤ n}. I = J k}
lemma coprime_prod_int2Tr:
(∀k≤Suc n. ideal R (J k)) ∧
(∀i≤Suc n. ∀j≤Suc n. i ≠ j --> coprime_ideals R (J i) (J j)) -->
Inter {I. ∃k≤Suc n. I = J k} = iΠR,Suc n J
lemma coprime_prod_int2:
[| ∀k≤Suc n. ideal R (J k);
∀i≤Suc n. ∀j≤Suc n. i ≠ j --> coprime_ideals R (J i) (J j) |]
==> Inter {I. ∃k≤Suc n. I = J k} = iΠR,Suc n J
lemma coprime_2_n:
[| ideal R A; ideal R B |]
==> R /r A \<Oplus>r R /r B = rΠ{j. j ≤ Suc 0} prodB1 (R /r A) (R /r B)
lemma A_to_prodag2_hom:
[| ideal R A; ideal R B; S 0 = pj R A; S (Suc 0) = pj R B |]
==> A_to_prodag R {j. j ≤ Suc 0} S (prodB1 (R /r A) (R /r B))
∈ rHom R (R /r A \<Oplus>r R /r B)
lemma A2coprime_rsurjecTr:
[| ideal R A; ideal R B; S 0 = pj R A; S (Suc 0) = pj R B |]
==> carrier (R /r A \<Oplus>r R /r B) =
carr_prodag {j. j ≤ Suc 0} (prodB1 (R /r A) (R /r B))
lemma A2coprime_rsurjec:
[| ideal R A; ideal R B; S 0 = pj R A; S (Suc 0) = pj R B;
coprime_ideals R A B |]
==> surjecR,(R /r A \<Oplus>r
R /r B) A_to_prodag R {j. j ≤ Suc 0} S (prodB1 (R /r A) (R /r B))
lemma prod2_n_Tr1:
[| ∀k≤Suc 0. ideal R (J k); ∀k≤Suc 0. B k = R /r J k;
∀k≤Suc 0. S k = pj R (J k) |]
==> A_to_prodag R {j. j ≤ Suc 0} S (prodB1 (R /r J 0) (R /r J (Suc 0))) =
A_to_prodag R {j. j ≤ Suc 0} S B
lemma restrict_prod_Suc:
[| ∀k≤Suc (Suc n). ideal R (J k); ∀k≤Suc (Suc n). B k = R /r J k;
∀k≤Suc (Suc n). S k = pj R (J k); f ∈ carrier (rΠ{j. j ≤ Suc (Suc n)} B) |]
==> restrict f {j. j ≤ Suc n} ∈ carrier (rΠ{j. j ≤ Suc n} B)
lemma Chinese_remTr2:
(∀k≤Suc n. ideal R (J k)) ∧
(∀k≤Suc n. B k = R /r J k) ∧
(∀k≤Suc n. S k = pj R (J k)) ∧
(∀i≤Suc n. ∀j≤Suc n. i ≠ j --> coprime_ideals R (J i) (J j)) -->
surjecR,(rΠ{j. j ≤ Suc n} B) A_to_prodag R {j. j ≤ Suc n} S B
lemma Chinese_remTr3:
[| ∀k≤Suc n. ideal R (J k); ∀k≤Suc n. B k = R /r J k;
∀k≤Suc n. S k = pj R (J k);
∀i≤Suc n. ∀j≤Suc n. i ≠ j --> coprime_ideals R (J i) (J j) |]
==> surjecR,(rΠ{j. j ≤ Suc n} B) A_to_prodag R {j. j ≤ Suc n} S B
lemma imset:
∀k≤Suc n. ideal R (J k)
==> {I. ∃k≤Suc n. I = J k} = {J k |k. k ∈ {j. j ≤ Suc n}}
theorem Chinese_remThm:
[| ∀k≤Suc n. ideal R (J k); ∀k≤Suc n. B k = R /r J k;
∀k≤Suc n. S k = pj R (J k);
∀i≤Suc n. ∀j≤Suc n. i ≠ j --> coprime_ideals R (J i) (J j) |]
==> bijecR /r
Inter
{J k |k.
k ∈ {j. j ≤ Suc n}},(rΠ{j. j ≤ Suc n}
B) (A_to_prodag R {j. j ≤ Suc n} S
B°R,(rΠ{j. j ≤ Suc n} B))
lemma prod_prime:
[| ideal R A; ∀k≤Suc n. prime_ideal R (P k); ∀l≤Suc n. ¬ A ⊆ P l;
∀k≤Suc n. ∀l≤Suc n. k = l ∨ ¬ P k ⊆ P l |]
==> ∀i≤Suc n.
eΠR,n ppa R P A i ∈ A ∧
(∀l∈{j. j ≤ Suc n} - {i}. eΠR,n ppa R P A i ∈ P l) ∧
eΠR,n ppa R P A i ∉ P i
lemma skip_im1:
[| i ≤ Suc n; P ∈ {j. j ≤ Suc n} -> Collect (prime_ideal R) |]
==> compose {j. j ≤ n} P (skip i) ` {j. j ≤ n} = P ` ({j. j ≤ Suc n} - {i})
lemma mutch_aux1:
[| ideal R A; i ≤ Suc n; P ∈ {j. j ≤ Suc n} -> Collect (prime_ideal R) |]
==> compose {j. j ≤ n} P (skip i) ∈ {j. j ≤ n} -> Collect (prime_ideal R)
lemma prime_ideal_cont1Tr:
ideal R A
==> ∀P. P ∈ {j. j ≤ n} -> {X. prime_ideal R X} ∧ A ⊆ Union (P ` {j. j ≤ n}) -->
(∃i≤n. A ⊆ P i)
lemma prime_ideal_cont1:
[| ideal R A; ∀i≤n. prime_ideal R (P i); A ⊆ Union {X. ∃i≤n. X = P i} |]
==> ∃i≤n. A ⊆ P i
lemma prod_n_ideal_contTr0:
(∀l≤n. ideal R (J l)) --> iΠR,n J ⊆ Inter {X. ∃k≤n. X = J k}
lemma prod_n_ideal_contTr:
∀l≤n. ideal R (J l) ==> iΠR,n J ⊆ Inter {X. ∃k≤n. X = J k}
lemma prod_n_ideal_cont2:
[| ∀l≤n. ideal R (J l); prime_ideal R P; Inter {X. ∃k≤n. X = J k} ⊆ P |]
==> ∃l≤n. J l ⊆ P
lemma prod_n_ideal_cont3:
[| ∀l≤n. ideal R (J l); prime_ideal R P; Inter {X. ∃k≤n. X = J k} = P |]
==> ∃l≤n. J l = P
lemma ideal_quotient_is_ideal:
[| ideal R A; ideal R B |] ==> ideal R (A \<dagger>\<^sub>R B)
lemma nsum_mem1Tr:
@ASubG A J ==> (∀j≤n. f j ∈ J) --> Σe A f n ∈ J
lemma fSum_mem:
[| ∀j∈nset (Suc n) m. f j ∈ carrier A; n < m |] ==> Σf A f Suc n m ∈ carrier A
lemma nsum_mem1:
[| @ASubG A J; ∀j≤n. f j ∈ J |] ==> Σe A f n ∈ J
lemma nsum_eq_i:
[| ∀j≤n. f j ∈ carrier A; ∀j≤n. g j ∈ carrier A; i ≤ n; ∀l≤i. f l = g l |]
==> Σe A f i = Σe A g i
lemma nsum_cmp_eq:
[| f ∈ {j. j ≤ n} -> carrier A; h1.0 ∈ {j. j ≤ n} -> {j. j ≤ n};
h2.0 ∈ {j. j ≤ n} -> {j. j ≤ n}; i ≤ n |]
==> Σe A cmp f (cmp h2.0 h1.0) i = Σe A cmp (cmp f h2.0) h1.0 i
lemma nsum_cmp_eq_transpos:
[| ∀j≤Suc n. f j ∈ carrier A; i ≤ n; i ≠ n |]
==> Σe A cmp f
(cmp (transpos i n) (cmp (transpos n (Suc n)) (transpos i n))) Suc n =
Σe A cmp f (transpos i (Suc n)) Suc n
lemma transpos_Tr_n1:
Suc (Suc 0) ≤ n ==> transpos (n - Suc 0) n n = n - Suc 0
lemma transpos_Tr_n2:
Suc (Suc 0) ≤ n ==> transpos (n - Suc 0) n (n - Suc 0) = n
lemma additionTr0:
[| 0 < n; ∀j≤n. f j ∈ carrier A |]
==> Σe A cmp f (transpos (n - 1) n) n = Σe A f n
lemma additionTr1:
[| ∀f h. f ∈ {j. j ≤ Suc n} -> carrier A ∧
h ∈ {j. j ≤ Suc n} -> {j. j ≤ Suc n} ∧ inj_on h {j. j ≤ Suc n} -->
Σe A cmp f h Suc n = Σe A f Suc n;
f ∈ {j. j ≤ Suc (Suc n)} -> carrier A;
h ∈ {j. j ≤ Suc (Suc n)} -> {j. j ≤ Suc (Suc n)};
inj_on h {j. j ≤ Suc (Suc n)}; h (Suc (Suc n)) = Suc (Suc n) |]
==> Σe A cmp f h Suc (Suc n) = Σe A f Suc (Suc n)
lemma additionTr1_1:
[| ∀f h. f ∈ {j. j ≤ Suc n} -> carrier A ∧
h ∈ {j. j ≤ Suc n} -> {j. j ≤ Suc n} ∧ inj_on h {j. j ≤ Suc n} -->
Σe A cmp f h Suc n = Σe A f Suc n;
f ∈ {j. j ≤ Suc (Suc n)} -> carrier A; i ≤ n |]
==> Σe A cmp f (transpos i (Suc n)) Suc (Suc n) = Σe A f Suc (Suc n)
lemma additionTr1_2:
[| ∀f h. f ∈ {j. j ≤ Suc n} -> carrier A ∧
h ∈ {j. j ≤ Suc n} -> {j. j ≤ Suc n} ∧ inj_on h {j. j ≤ Suc n} -->
Σe A cmp f h Suc n = Σe A f Suc n;
f ∈ {j. j ≤ Suc (Suc n)} -> carrier A; i ≤ Suc n |]
==> Σe A cmp f (transpos i (Suc (Suc n))) Suc (Suc n) = Σe A f Suc (Suc n)
lemma additionTr2:
∀f h. f ∈ {j. j ≤ Suc n} -> carrier A ∧
h ∈ {j. j ≤ Suc n} -> {j. j ≤ Suc n} ∧ inj_on h {j. j ≤ Suc n} -->
Σe A cmp f h Suc n = Σe A f Suc n
lemma addition2:
[| f ∈ {j. j ≤ Suc n} -> carrier A; h ∈ {j. j ≤ Suc n} -> {j. j ≤ Suc n};
inj_on h {j. j ≤ Suc n} |]
==> Σe A cmp f h Suc n = Σe A f Suc n
lemma addition21:
[| f ∈ {j. j ≤ n} -> carrier A; h ∈ {j. j ≤ n} -> {j. j ≤ n};
inj_on h {j. j ≤ n} |]
==> Σe A cmp f h n = Σe A f n
lemma addition3:
[| ∀j≤Suc n. f j ∈ carrier A; j ≤ Suc n; j ≠ Suc n |]
==> Σe A f Suc n = Σe A cmp f (transpos j (Suc n)) Suc n
lemma nsum_splitTr:
(∀j≤Suc (n + m). f j ∈ carrier A) -->
Σe A f Suc (n + m) = Σe A f n ± Σe A cmp f (slide (Suc n)) m
lemma nsum_split:
∀j≤Suc (n + m). f j ∈ carrier A
==> Σe A f Suc (n + m) = Σe A f n ± Σe A cmp f (slide (Suc n)) m
lemma nsum_split1:
[| ∀j≤m. f j ∈ carrier A; n < m |] ==> Σe A f m = Σe A f n ± Σf A f Suc n m
lemma nsum_minusTr:
(∀j≤n. f j ∈ carrier A) --> -a Σe A f n = Σe A (λx∈{j. j ≤ n}. -a f x) n
lemma nsum_minus:
∀j≤n. f j ∈ carrier A ==> -a Σe A f n = Σe A (λx∈{j. j ≤ n}. -a f x) n
lemma ring_nsum_zeroTr:
(∀j≤n. f j ∈ carrier A) ∧ (∀j≤n. f j = \<zero>) --> Σe A f n = \<zero>
lemma ring_nsum_zero:
∀j≤n. f j = \<zero> ==> Σe A f n = \<zero>
lemma ag_nsum_1_nonzeroTr:
∀f. (∀j≤n. f j ∈ carrier A) ∧ l ≤ n ∧ (∀j∈{j. j ≤ n} - {l}. f j = \<zero>) -->
Σe A f n = f l
lemma ag_nsum_1_nonzero:
[| ∀j≤n. f j ∈ carrier A; l ≤ n; ∀j∈{j. j ≤ n} - {l}. f j = \<zero> |]
==> Σe A f n = f l
lemma set_mult_sub:
[| A ⊆ carrier R; B ⊆ carrier R |] ==> set_mult R A B ⊆ carrier R
lemma set_mult_mono:
[| A1.0 ⊆ carrier R; A2.0 ⊆ carrier R; A1.0 ⊆ A2.0; B ⊆ carrier R |]
==> set_mult R A1.0 B ⊆ set_mult R A2.0 B
lemma sum_mult_Tr1:
[| A ⊆ carrier R; B ⊆ carrier R |]
==> (∀j≤n. f j ∈ set_mult R A B) --> Σe R f n ∈ carrier R
lemma sum_mult_mem:
[| A ⊆ carrier R; B ⊆ carrier R; ∀j≤n. f j ∈ set_mult R A B |]
==> Σe R f n ∈ carrier R
lemma sum_mult_mem1:
[| A ⊆ carrier R; B ⊆ carrier R; x ∈ sum_mult R A B |]
==> ∃n. ∃f∈{j. j ≤ n} -> set_mult R A B. Σe R f n = x
lemma sum_mult_subR:
[| A ⊆ carrier R; B ⊆ carrier R |] ==> sum_mult R A B ⊆ carrier R
lemma times_mem_sum_mult:
[| A ⊆ carrier R; B ⊆ carrier R; a ∈ A; b ∈ B |] ==> a ·r b ∈ sum_mult R A B
lemma mem_minus_sum_multTr2:
[| A ⊆ carrier R; B ⊆ carrier R; ∀j≤n. f j ∈ set_mult R A B; i ≤ n |]
==> f i ∈ carrier R
lemma nsum_jointfun:
[| ∀j≤n. f j ∈ carrier A; ∀j≤m. g j ∈ carrier A |]
==> Σe A jointfun n f m g Suc (n + m) = Σe A f n ± Σe A g m
lemma sum_mult_pOp_closed:
[| A ⊆ carrier R; B ⊆ carrier R; a ∈ sum_mult R A B; b ∈ sum_mult R A B |]
==> a ± b ∈ sum_mult R A B
lemma set_mult_mOp_closed:
[| A ⊆ carrier R; ideal R B; x ∈ set_mult R A B |] ==> -a x ∈ set_mult R A B
lemma set_mult_ring_times_closed:
[| A ⊆ carrier R; ideal R B; x ∈ set_mult R A B; r ∈ carrier R |]
==> r ·r x ∈ set_mult R A B
lemma set_mult_sub_sum_mult:
[| A ⊆ carrier R; ideal R B |] ==> set_mult R A B ⊆ sum_mult R A B
lemma sum_mult_pOp_closedn:
[| A ⊆ carrier R; ideal R B |]
==> (∀j≤n. f j ∈ set_mult R A B) --> Σe R f n ∈ sum_mult R A B
lemma mem_minus_sum_multTr4:
[| A ⊆ carrier R; ideal R B |]
==> (∀j≤n. f j ∈ set_mult R A B) --> -a Σe R f n ∈ sum_mult R A B
lemma sum_mult_iOp_closed:
[| A ⊆ carrier R; ideal R B; x ∈ sum_mult R A B |] ==> -a x ∈ sum_mult R A B
lemma sum_mult_ring_multiplicationTr:
[| A ⊆ carrier R; ideal R B; r ∈ carrier R |]
==> (∀j≤n. f j ∈ set_mult R A B) --> r ·r Σe R f n ∈ sum_mult R A B
lemma sum_mult_ring_multiplication:
[| A ⊆ carrier R; ideal R B; r ∈ carrier R; a ∈ sum_mult R A B |]
==> r ·r a ∈ sum_mult R A B
lemma ideal_sum_mult:
[| A ⊆ carrier R; A ≠ {}; ideal R B |] ==> ideal R (sum_mult R A B)
lemma ideal_inc_set_multTr:
[| A ⊆ carrier R; ideal R B; ideal R C; set_mult R A B ⊆ C |]
==> ∀f∈{j. j ≤ n} -> set_mult R A B. Σe R f n ∈ C
lemma ideal_inc_set_mult:
[| A ⊆ carrier R; ideal R B; ideal R C; set_mult R A B ⊆ C |]
==> sum_mult R A B ⊆ C
lemma AB_inc_sum_mult:
[| ideal R A; ideal R B |] ==> sum_mult R A B ⊆ A ∩ B
lemma sum_mult_is_ideal_prod:
[| ideal R A; ideal R B |] ==> sum_mult R A B = A ♦r B
lemma ideal_prod_assocTr0:
[| ideal R A; ideal R B; ideal R C; y ∈ C; z ∈ set_mult R A B |]
==> z ·r y ∈ sum_mult R A (B ♦r C)
lemma ideal_prod_assocTr1:
[| ideal R A; ideal R B; ideal R C; y ∈ C |]
==> ∀f∈{j. j ≤ n} -> set_mult R A B. Σe R f n ·r y ∈ A ♦r (B ♦r C)
lemma ideal_quotient_idealTr:
[| ideal R A; ideal R B; ideal R C; x ∈ carrier R;
∀c∈C. x ·r c ∈ A \<dagger>\<^sub>R B |]
==> f ∈ {j. j ≤ n} -> set_mult R B C --> x ·r Σe R f n ∈ A
lemma ideal_quotient_ideal:
[| ideal R A; ideal R B; ideal R C |]
==> A \<dagger>\<^sub>R B \<dagger>\<^sub>R C = A \<dagger>\<^sub>R B ♦r C
lemma ideal_prod_assocTr:
[| ideal R A; ideal R B; ideal R C |]
==> ∀f. f ∈ {j. j ≤ n} -> set_mult R (A ♦r B) C --> Σe R f n ∈ A ♦r (B ♦r C)
lemma ideal_prod_assoc:
[| ideal R A; ideal R B; ideal R C |] ==> (A ♦r B) ♦r C = A ♦r (B ♦r C)
lemma prod_principal_idealTr0:
[| a ∈ carrier R; b ∈ carrier R; z ∈ set_mult R (R ♦p a) (R ♦p b) |]
==> z ∈ R ♦p (a ·r b)
lemma prod_principal_idealTr1:
[| a ∈ carrier R; b ∈ carrier R |]
==> ∀f∈{j. j ≤ n} -> set_mult R (R ♦p a) (R ♦p b). Σe R f n ∈ R ♦p (a ·r b)
lemma prod_principal_ideal:
[| a ∈ carrier R; b ∈ carrier R |] ==> R ♦p a ♦r R ♦p b = R ♦p (a ·r b)
lemma principal_ideal_n_pow1:
a ∈ carrier R ==> R ♦p a ♦R n = R ♦p (a^R n)
lemma principal_ideal_n_pow:
[| a ∈ carrier R; I = R ♦p a |] ==> I ♦R n = R ♦p (a^R n)
lemma nprod_eqTr:
f ∈ {j. j ≤ n} -> carrier R ∧
g ∈ {j. j ≤ n} -> carrier R ∧ (∀j≤n. f j = g j) -->
eΠR,n f = eΠR,n g
lemma nprod_eq:
[| ∀j≤n. f j ∈ carrier R; ∀j≤n. g j ∈ carrier R; ∀j≤n. f j = g j |]
==> eΠR,n f = eΠR,n g
lemma mprodR_Suc:
[| e ∈ {j. j ≤ Suc n} -> {j. 0 ≤ j}; f ∈ {j. j ≤ Suc n} -> carrier R |]
==> mprod_expR R e f (Suc n) = mprod_expR R e f n ·r f (Suc n)^R e (Suc n)
lemma mprod_expR_memTr:
e ∈ {j. j ≤ n} -> {j. 0 ≤ j} ∧ f ∈ {j. j ≤ n} -> carrier R -->
mprod_expR R e f n ∈ carrier R
lemma mprod_expR_mem:
[| e ∈ {j. j ≤ n} -> {j. 0 ≤ j}; f ∈ {j. j ≤ n} -> carrier R |]
==> mprod_expR R e f n ∈ carrier R
lemma prod_n_principal_idealTr:
e ∈ {j. j ≤ n} -> {j. 0 ≤ j} ∧
f ∈ {j. j ≤ n} -> carrier R ∧ (∀k≤n. J k = R ♦p f k ♦R e k) -->
iΠR,n J = R ♦p mprod_expR R e f n
lemma prod_n_principal_ideal:
[| e ∈ {j. j ≤ n} -> {j. 0 ≤ j}; f ∈ {j. j ≤ n} -> carrier R;
∀k≤n. J k = R ♦p f k ♦R e k |]
==> iΠR,n J = R ♦p mprod_expR R e f n
lemma a_notin_n_pow1:
[| a ∈ carrier R; ¬ Unit R a; a ≠ \<zero>; 0 < n |] ==> a ∉ R ♦p a ♦R Suc n
lemma a_notin_n_pow2:
[| a ∈ carrier R; ¬ Unit R a; a ≠ \<zero>; 0 < n |] ==> a^R n ∉ R ♦p a ♦R Suc n
lemma n_pow_not_prime:
[| a ∈ carrier R; a ≠ \<zero>; 0 < n |] ==> ¬ prime_ideal R (R ♦p a ♦R Suc n)
lemma principal_pow_prime_condTr:
[| a ∈ carrier R; a ≠ \<zero>; prime_ideal R (R ♦p a ♦R Suc n) |] ==> n = 0
lemma principal_pow_prime_cond:
[| a ∈ carrier R; a ≠ \<zero>; prime_ideal R (R ♦p a ♦R n) |] ==> n = Suc 0
lemma i_contract_sub:
[| f ∈ rHom R R'; ideal R' J |] ==> i_contract f R R' J ⊆ carrier R
lemma i_contract_ideal:
[| f ∈ rHom R R'; ideal R' J |] ==> ideal R (i_contract f R R' J)
lemma i_contract_mono:
[| f ∈ rHom R R'; ideal R' J1.0; ideal R' J2.0; J1.0 ⊆ J2.0 |]
==> i_contract f R R' J1.0 ⊆ i_contract f R R' J2.0
lemma i_contract_prime:
[| f ∈ rHom R R'; prime_ideal R' P |] ==> prime_ideal R (i_contract f R R' P)
lemma i_extension_ideal:
[| f ∈ rHom R R'; ideal R I |] ==> ideal R' (i_extension f R R' I)
lemma i_extension_mono:
[| f ∈ rHom R R'; ideal R I1.0; ideal R I2.0; I1.0 ⊆ I2.0 |]
==> i_extension f R R' I1.0 ⊆ i_extension f R R' I2.0
lemma e_c_inc_self:
[| f ∈ rHom R R'; ideal R I |] ==> I ⊆ i_contract f R R' (i_extension f R R' I)
lemma c_e_incd_self:
[| f ∈ rHom R R'; ideal R' J |] ==> i_extension f R R' (i_contract f R R' J) ⊆ J
lemma c_e_c_eq_c:
[| f ∈ rHom R R'; ideal R' J |]
==> i_contract f R R' (i_extension f R R' (i_contract f R R' J)) =
i_contract f R R' J
lemma e_c_e_eq_e:
[| f ∈ rHom R R'; ideal R I |]
==> i_extension f R R' (i_contract f R R' (i_extension f R R' I)) =
i_extension f R R' I
lemma csrp_mem:
[| ideal R I; a ∈ carrier R |] ==> csrp_fn R I (a \<uplus>R I) ∈ a \<uplus>R I
lemma csrp_same:
[| ideal R I; a ∈ carrier R |]
==> csrp_fn R I (a \<uplus>R I) \<uplus>R I = a \<uplus>R I
lemma csrp_mem1:
[| ideal R I; x ∈ carrier (R /r I) |] ==> csrp_fn R I x ∈ x
lemma csrp_fn_mem:
[| ideal R I; x ∈ carrier (R /r I) |] ==> csrp_fn R I x ∈ carrier R
lemma csrp_eq_coset:
[| ideal R I; x ∈ carrier (R /r I) |] ==> csrp_fn R I x \<uplus>R I = x
lemma csrp_nz_nz:
[| ideal R I; x ∈ carrier (R /r I); x ≠ \<zero>R /r I |]
==> csrp_fn R I x ≠ \<zero>
lemma csrp_diff_in_vpr:
[| ideal R I; x ∈ carrier R |] ==> x ± -a csrp_fn R I (pj R I x) ∈ I
lemma csrp_pj:
[| ideal R I; x ∈ carrier (R /r I) |] ==> pj R I (csrp_fn R I x) = x
lemma Subring_subset:
Subring R S ==> carrier S ⊆ carrier R
lemma subring_Ring:
Subring R S ==> Ring S
lemma mem_subring_mem_ring:
[| Subring R S; x ∈ carrier S |] ==> x ∈ carrier R
lemma Subring_pOp_ring_pOp:
[| Subring R S; a ∈ carrier S; b ∈ carrier S |] ==> a ±S b = a ± b
lemma Subring_tOp_ring_tOp:
[| Subring R S; a ∈ carrier S; b ∈ carrier S |] ==> a ·rS b = a ·r b
lemma Subring_one_ring_one:
Subring R S ==> 1rS = 1r
lemma Subring_zero_ring_zero:
Subring R S ==> \<zero>S = \<zero>
lemma Subring_minus_ring_minus:
[| Subring R S; x ∈ carrier S |] ==> -aS x = -a x
lemma Subring_pow_ring_pow:
x ∈ carrier S ==> x^S n = x^R n
lemma is_Ring:
Ring R
lemma polyn_ring_nonzero:
1r ≠ \<zero>
lemma polyn_ring_S_nonzero:
1rS ≠ \<zero>S
lemma polyn_ring_X_nonzero:
X ≠ \<zero>
lemma pol_coeff_split:
pol_coeff S f = pol_coeff S (fst f, snd f)
lemma pol_coeff_cartesian:
pol_coeff S c ==> (fst c, snd c) = c
lemma split_pol_coeff:
[| pol_coeff S c; k ≤ fst c |] ==> pol_coeff S (k, snd c)
lemma pol_coeff_pre:
pol_coeff S (Suc n, f) ==> pol_coeff S (n, f)
lemma pol_coeff_le:
[| pol_coeff S c; n ≤ fst c |] ==> pol_coeff S (n, snd c)
lemma pol_coeff_mem:
[| pol_coeff S c; j ≤ fst c |] ==> snd c j ∈ carrier S
lemma pol_coeff_mem_R:
[| pol_coeff S c; j ≤ fst c |] ==> snd c j ∈ carrier R
lemma Slide_pol_coeff:
[| pol_coeff S c; n < fst c |]
==> pol_coeff S (fst c - Suc n, λx. snd c (Suc (n + x)))
lemma monomial_mem:
pol_coeff S c ==> ∀j≤fst c. snd c j ·r X^R j ∈ carrier R
lemma polyn_mem:
[| pol_coeff S c; k ≤ fst c |] ==> polyn_expr R X k c ∈ carrier R
lemma polyn_exprs_eq:
[| pol_coeff S c; pol_coeff S d; k ≤ min (fst c) (fst d);
∀j≤k. snd c j = snd d j |]
==> polyn_expr R X k c = polyn_expr R X k d
lemma polyn_expr_restrict:
pol_coeff S (Suc n, f) ==> polyn_expr R X n (Suc n, f) = polyn_expr R X n (n, f)
lemma polyn_expr_short:
[| pol_coeff S c; k ≤ fst c |]
==> polyn_expr R X k c = polyn_expr R X k (k, snd c)
lemma polyn_expr0:
pol_coeff S c ==> polyn_expr R X 0 c = snd c 0
lemma polyn_expr_split:
polyn_expr R X k f = polyn_expr R X k (fst f, snd f)
lemma polyn_Suc:
Suc n ≤ fst c
==> polyn_expr R X (Suc n) (Suc n, snd c) =
polyn_expr R X n c ± snd c (Suc n) ·r X^R Suc n
lemma polyn_Suc_split:
pol_coeff S (Suc n, f)
==> polyn_expr R X (Suc n) (Suc n, f) =
polyn_expr R X n (n, f) ± f (Suc n) ·r X^R Suc n
lemma polyn_n_m:
[| pol_coeff S c; n < m; m ≤ fst c |]
==> polyn_expr R X m (m, snd c) =
polyn_expr R X n (n, snd c) ± Σf R (λj. snd c j ·r X^R j) Suc n m
lemma polyn_n_m1:
[| pol_coeff S c; n < m; m ≤ fst c |]
==> polyn_expr R X m c =
polyn_expr R X n c ± Σf R (λj. snd c j ·r X^R j) Suc n m
lemma polyn_n_m_mem:
[| pol_coeff S c; n < m; m ≤ fst c |]
==> Σf R (λj. snd c j ·r X^R j) Suc n m ∈ carrier R
lemma polyn_n_ms_eq:
[| pol_coeff S c; pol_coeff S d; m ≤ min (fst c) (fst d); n < m;
∀j∈nset (Suc n) m. snd c j = snd d j |]
==> Σf R (λj. snd c j ·r X^R j) Suc n m = Σf R (λj. snd d j ·r X^R j) Suc n m
lemma polyn_addTr:
pol_coeff S (n, f) ∧ pol_coeff S (n, g) -->
polyn_expr R X n (n, f) ± polyn_expr R X n (n, g) =
Σe R (λj. (f j ±S g j) ·r X^R j) n
lemma polyn_add_n:
[| pol_coeff S (n, f); pol_coeff S (n, g) |]
==> polyn_expr R X n (n, f) ± polyn_expr R X n (n, g) =
Σe R (λj. (f j ±S g j) ·r X^R j) n
lemma add_cf_pol_coeff:
[| pol_coeff S c; pol_coeff S d |] ==> pol_coeff S (add_cf S c d)
lemma add_cf_len:
[| pol_coeff S c; pol_coeff S d |] ==> fst (add_cf S c d) = max (fst c) (fst d)
lemma polyn_expr_restrict1:
[| pol_coeff S (n, f); pol_coeff S (Suc (m + n), g) |]
==> polyn_expr R X (m + n) (add_cf S (n, f) (m + n, g)) =
polyn_expr R X (m + n) (m + n, snd (add_cf S (n, f) (Suc (m + n), g)))
lemma polyn_add_n1:
[| pol_coeff S (n, f); pol_coeff S (n, g) |]
==> polyn_expr R X n (n, f) ± polyn_expr R X n (n, g) =
polyn_expr R X n (add_cf S (n, f) (n, g))
lemma add_cf_val_hi:
fst c < fst d ==> snd (add_cf S c d) (fst d) = snd d (fst d)
lemma add_cf_commute:
[| pol_coeff S c; pol_coeff S d |]
==> ∀j≤max (fst c) (fst d). snd (add_cf S c d) j = snd (add_cf S d c) j
lemma polyn_addTr1:
pol_coeff S (n, f)
==> ∀g. pol_coeff S (n + m, g) -->
polyn_expr R X n (n, f) ± polyn_expr R X (n + m) (n + m, g) =
polyn_expr R X (n + m) (add_cf S (n, f) (n + m, g))
lemma polyn_add:
[| pol_coeff S (n, f); pol_coeff S (m, g) |]
==> polyn_expr R X n (n, f) ± polyn_expr R X m (m, g) =
polyn_expr R X (max n m) (add_cf S (n, f) (m, g))
lemma polyn_add1:
[| pol_coeff S c; pol_coeff S d |]
==> polyn_expr R X (fst c) c ± polyn_expr R X (fst d) d =
polyn_expr R X (max (fst c) (fst d)) (add_cf S c d)
lemma polyn_minus_nsum:
[| pol_coeff S c; k ≤ fst c |]
==> -a polyn_expr R X k c = Σe R (λj. (-aS snd c j) ·r X^R j) k
lemma minus_pol_coeff:
pol_coeff S c ==> pol_coeff S (fst c, λj. -aS snd c j)
lemma polyn_minus:
[| pol_coeff S c; k ≤ fst c |]
==> -a polyn_expr R X k c = polyn_expr R X k (fst c, λj. -aS snd c j)
lemma m_cf_pol_coeff:
pol_coeff S c ==> pol_coeff S (m_cf S c)
lemma m_cf_len:
pol_coeff S c ==> fst (m_cf S c) = fst c
lemma polyn_minus_m_cf:
[| pol_coeff S c; k ≤ fst c |]
==> -a polyn_expr R X k c = polyn_expr R X k (m_cf S c)
lemma polyn_zero_minus_zero:
[| pol_coeff S c; k ≤ fst c |]
==> (polyn_expr R X k c = \<zero>) = (polyn_expr R X k (m_cf S c) = \<zero>)
lemma coeff_0_pol_0:
[| pol_coeff S c; k ≤ fst c |]
==> (∀j≤k. snd c j = \<zero>S) = (polyn_expr R X k c = \<zero>)
lemma special_cf_pol_coeff:
pol_coeff S (C0 S)
lemma special_cf_len:
fst (C0 S) = 0
lemma ext_cf_pol_coeff:
pol_coeff S c ==> pol_coeff S (ext_cf S n c)
lemma ext_cf_len:
pol_coeff S c ==> fst (ext_cf S m c) = m + fst c
lemma ext_special_cf_len:
fst (ext_cf S m (C0 S)) = m
lemma ext_cf_self:
pol_coeff S c ==> ∀j≤fst c. snd (ext_cf S 0 c) j = snd c j
lemma ext_cf_hi:
pol_coeff S c ==> snd c (fst c) = snd (ext_cf S n c) (n + fst c)
lemma ext_special_cf_hi:
snd (ext_cf S n (C0 S)) n = 1rS
lemma ext_cf_lo_zero:
[| pol_coeff S c; 0 < n; x ≤ n - Suc 0 |] ==> snd (ext_cf S n c) x = \<zero>S
lemma ext_special_cf_lo_zero:
[| 0 < n; x ≤ n - Suc 0 |] ==> snd (ext_cf S n (C0 S)) x = \<zero>S
lemma sp_cf_pol_coeff:
[| pol_coeff S c; a ∈ carrier S |] ==> pol_coeff S (sp_cf S a c)
lemma sp_cf_len:
[| pol_coeff S c; a ∈ carrier S |] ==> fst (sp_cf S a c) = fst c
lemma sp_cf_val:
[| pol_coeff S c; j ≤ fst c; a ∈ carrier S |]
==> snd (sp_cf S a c) j = a ·rS snd c j
lemma polyn_ext_cf_lo_zero:
[| pol_coeff S c; 0 < j |]
==> polyn_expr R X (j - Suc 0) (ext_cf S j c) = \<zero>
lemma monomial_d:
pol_coeff S c ==> polyn_expr R X d (ext_cf S d c) = snd c 0 ·r X^R d
lemma X_to_d:
X^R d = polyn_expr R X d (ext_cf S d (C0 S))
lemma c_max_ext_special_cf:
c_max S (ext_cf S n (C0 S)) = n
lemma scalar_times_polynTr:
a ∈ carrier S
==> ∀f. pol_coeff S (n, f) -->
a ·r polyn_expr R X n (n, f) = polyn_expr R X n (sp_cf S a (n, f))
lemma scalar_times_pol_expr:
[| a ∈ carrier S; pol_coeff S c; n ≤ fst c |]
==> a ·r polyn_expr R X n c = polyn_expr R X n (sp_cf S a c)
lemma sp_coeff_nonzero:
[| Idomain S; a ∈ carrier S; a ≠ \<zero>S; pol_coeff S c; snd c j ≠ \<zero>S;
j ≤ fst c |]
==> snd (sp_cf S a c) j ≠ \<zero>S
lemma ext_cf_inductTl:
pol_coeff S (Suc n, f)
==> polyn_expr R X (n + j) (ext_cf S j (Suc n, f)) =
polyn_expr R X (n + j) (ext_cf S j (n, f))
lemma low_deg_terms_zeroTr:
pol_coeff S (n, f) -->
polyn_expr R X (n + j) (ext_cf S j (n, f)) = X^R j ·r polyn_expr R X n (n, f)
lemma low_deg_terms_zero:
pol_coeff S (n, f)
==> polyn_expr R X (n + j) (ext_cf S j (n, f)) =
X^R j ·r polyn_expr R X n (n, f)
lemma low_deg_terms_zero1:
pol_coeff S c
==> polyn_expr R X (fst c + j) (ext_cf S j c) =
X^R j ·r polyn_expr R X (fst c) c
lemma polyn_expr_tOpTr:
pol_coeff S (n, f)
==> ∀g. pol_coeff S (m, g) -->
(∃h. pol_coeff S (n + m, h) ∧
h (n + m) = f n ·rS g m ∧
polyn_expr R X (n + m) (n + m, h) =
polyn_expr R X n (n, f) ·r polyn_expr R X m (m, g))
lemma polyn_expr_tOp:
[| pol_coeff S (n, f); pol_coeff S (m, g) |]
==> ∃e. pol_coeff S (n + m, e) ∧
e (n + m) = f n ·rS g m ∧
polyn_expr R X (n + m) (n + m, e) =
polyn_expr R X n (n, f) ·r polyn_expr R X m (m, g)
lemma polyn_expr_tOp_c:
[| pol_coeff S c; pol_coeff S d |]
==> ∃e. pol_coeff S e ∧
fst e = fst c + fst d ∧
snd e (fst e) = snd c (fst c) ·rS snd d (fst d) ∧
polyn_expr R X (fst e) e =
polyn_expr R X (fst c) c ·r polyn_expr R X (fst d) d
lemma polyn_degreeTr:
[| pol_coeff S c; k ≤ fst c |]
==> (polyn_expr R X k c = \<zero>) = ({j. j ≤ k ∧ snd c j ≠ \<zero>S} = {})
lemma higher_part_zero:
[| pol_coeff S c; k < fst c; ∀j∈nset (Suc k) (fst c). snd c j = \<zero>S |]
==> Σf R (λj. snd c j ·r X^R j) Suc k fst c = \<zero>
lemma coeff_nonzero_polyn_nonzero:
[| pol_coeff S c; k ≤ fst c |]
==> (polyn_expr R X k c ≠ \<zero>) = (∃j≤k. snd c j ≠ \<zero>S)
lemma pol_expr_unique:
[| p ∈ carrier R; p ≠ \<zero>; pol_coeff S c; p = polyn_expr R X (fst c) c;
snd c (fst c) ≠ \<zero>S; pol_coeff S d; p = polyn_expr R X (fst d) d;
snd d (fst d) ≠ \<zero>S |]
==> fst c = fst d ∧ (∀j≤fst c. snd c j = snd d j)
lemma pol_expr_unique2:
[| pol_coeff S c; pol_coeff S d; fst c = fst d |]
==> (polyn_expr R X (fst c) c = polyn_expr R X (fst d) d) =
(∀j≤fst c. snd c j = snd d j)
lemma pol_expr_unique3:
[| pol_coeff S c; pol_coeff S d; fst c < fst d |]
==> (polyn_expr R X (fst c) c = polyn_expr R X (fst d) d) =
((∀j≤fst c. snd c j = snd d j) ∧
(∀j∈nset (Suc (fst c)) (fst d). snd d j = \<zero>S))
lemma polyn_degree_unique:
[| pol_coeff S c; pol_coeff S d;
polyn_expr R X (fst c) c = polyn_expr R X (fst d) d |]
==> c_max S c = c_max S d
lemma ex_polyn_expr:
p ∈ carrier R ==> ∃c. pol_coeff S c ∧ p = polyn_expr R X (fst c) c
lemma c_max_eqTr0:
[| pol_coeff S c; k ≤ fst c; polyn_expr R X k c = polyn_expr R X (fst c) c;
∃j≤k. snd c j ≠ \<zero>S |]
==> c_max S (k, snd c) = c_max S c
lemma ex_cf_sol:
p ∈ carrier R ==> ∃c. cf_sol R S X p c
lemma deg_in_aug_minf:
p ∈ carrier R ==> deg R S X p ∈ Z-∞
lemma deg_noninf:
p ∈ carrier R ==> deg R S X p ≠ ∞
lemma deg_ant_int:
[| p ∈ carrier R; p ≠ \<zero> |] ==> deg R S X p = ant (int (deg_n R S X p))
lemma deg_an:
[| p ∈ carrier R; p ≠ \<zero> |] ==> deg R S X p = an (deg_n R S X p)
lemma pol_SOME_1:
p ∈ carrier R ==> cf_sol R S X p (SOME f. cf_sol R S X p f)
lemma pol_SOME_2:
p ∈ carrier R
==> pol_coeff S (SOME c. cf_sol R S X p c) ∧
p = polyn_expr R X (fst (SOME c. cf_sol R S X p c))
(SOME c. cf_sol R S X p c)
lemma coeff_max_zeroTr:
pol_coeff S c ==> ∀j. j ≤ fst c ∧ c_max S c < j --> snd c j = \<zero>S
lemma coeff_max_nonzeroTr:
[| pol_coeff S c; ∃j≤fst c. snd c j ≠ \<zero>S |]
==> snd c (c_max S c) ≠ \<zero>S
lemma coeff_max_bddTr:
pol_coeff S c ==> c_max S c ≤ fst c
lemma pol_coeff_max:
pol_coeff S c ==> pol_coeff S (c_max S c, snd c)
lemma polyn_c_max:
pol_coeff S c ==> polyn_expr R X (fst c) c = polyn_expr R X (c_max S c) c
lemma pol_deg_eq_c_max:
[| p ∈ carrier R; pol_coeff S c; p = polyn_expr R X (fst c) c |]
==> deg_n R S X p = c_max S c
lemma pol_deg_le_n:
[| p ∈ carrier R; pol_coeff S c; p = polyn_expr R X (fst c) c |]
==> deg_n R S X p ≤ fst c
lemma pol_deg_le_n1:
[| p ∈ carrier R; pol_coeff S c; k ≤ fst c; p = polyn_expr R X k c |]
==> deg_n R S X p ≤ k
lemma pol_len_gt_deg:
[| p ∈ carrier R; pol_coeff S c; p = polyn_expr R X (fst c) c;
deg R S X p < an j; j ≤ fst c |]
==> snd c j = \<zero>S
lemma pol_diff_deg_less:
[| p ∈ carrier R; pol_coeff S c; p = polyn_expr R X (fst c) c; pol_coeff S d;
fst c = fst d; snd c (fst c) = snd d (fst d) |]
==> p ± -a polyn_expr R X (fst d) d = \<zero> ∨
deg_n R S X (p ± -a polyn_expr R X (fst d) d) < fst c
lemma pol_pre_lt_deg:
[| p ∈ carrier R; pol_coeff S c; deg_n R S X p ≤ fst c; deg_n R S X p ≠ 0;
p = polyn_expr R X (deg_n R S X p) c |]
==> deg_n R S X (polyn_expr R X (deg_n R S X p - Suc 0) c) < deg_n R S X p
lemma pol_deg_n:
[| p ∈ carrier R; pol_coeff S c; n ≤ fst c; p = polyn_expr R X n c;
snd c n ≠ \<zero>S |]
==> deg_n R S X p = n
lemma pol_expr_deg:
[| p ∈ carrier R; p ≠ \<zero> |]
==> ∃c. pol_coeff S c ∧
deg_n R S X p ≤ fst c ∧
p = polyn_expr R X (deg_n R S X p) c ∧ snd c (deg_n R S X p) ≠ \<zero>S
lemma deg_n_pos:
p ∈ carrier R ==> 0 ≤ deg_n R S X p
lemma pol_expr_deg1:
[| p ∈ carrier R; d = na (deg R S X p) |]
==> ∃c. pol_coeff S c ∧ p = polyn_expr R X d c