(** Algebra7
author Hidetsune Kobayashi
Group You Santo
Department of Mathematics
Nihon University
h_koba@math.cst.nihon-u.ac.jp
May 3, 2004.
April 6, 2007 (revised)
chapter 5. Modules
section 3. a module over two rings
section 4. eSum and Generators
subsection 4-1. sum up coefficients
subsection 4-2. free generators
**)
theory Algebra7 imports Algebra6 begin
chapter "5. Modules"
section "1. Basic properties of Modules"
record ('a, 'b) Module = "'a aGroup" +
sprod :: "'b => 'a => 'a" (infixl "·s\<index>" 76)
locale Module = aGroup M +
fixes R (structure)
assumes sc_Ring: "Ring R"
and sprod_closed :
"[| a ∈ carrier R; m ∈ carrier M|] ==> a ·s m ∈ carrier M"
and sprod_l_distr:
"[|a ∈ carrier R; b ∈ carrier R; m ∈ carrier M|] ==>
(a ±R b) ·s m = a ·s m ±M b ·s m"
and sprod_r_distr:
"[| a ∈ carrier R; m ∈ carrier M; n ∈ carrier M |] ==>
a ·s (m ±M n) = a ·s m ±M a ·s n"
and sprod_assoc:
"[| a ∈ carrier R; b ∈ carrier R; m ∈ carrier M |] ==>
(a ·rR b) ·s m = a ·s (b ·s m)"
and sprod_one:
"m ∈ carrier M ==> (1rR) ·s m = m"
constdefs
submodule :: "[('b, 'm) Ring_scheme, ('a, 'b, 'c) Module_scheme, 'a set] =>
bool"
"submodule R A H == H ⊆ carrier A ∧ A +> H ∧ (∀a. ∀m.
(a ∈ carrier R ∧ m ∈ H) --> (sprod A a m) ∈ H)"
constdefs
mdl :: "[('a, 'b, 'm) Module_scheme, 'a set] => ('a, 'b) Module"
"mdl M H == (|carrier = H, pop = pop M, mop = mop M, zero = zero M,
sprod = λa. λx∈H. sprod M a x|)),"
syntax
"@MODULE"::"('b, 'd) Ring_scheme => ('a, 'b, 'c) Module_scheme => bool"
(infixl "module" 58)
translations
"R module M" == "Module M R"
lemma (in Module) module_is_ag:"aGroup M"
apply unfold_locales
done
lemma (in Module) module_inc_zero:" \<zero>M ∈ carrier M"
apply (simp add:ag_inc_zero) (** type of M is ('c, 'a, 'd) Module_scheme **)
done (** type of M is (?'b, ?'b, ?'z) Module_scheme **)
lemma (in Module) submodule_subset:"submodule R M H ==> H ⊆ carrier M"
apply (simp add:submodule_def)
done
lemma (in Module) submodule_asubg:"submodule R M H ==> M +> H"
by (simp add:submodule_def)
lemma (in Module) submodule_subset1:"[|submodule R M H; h ∈ H|] ==>
h ∈ carrier M"
apply (simp add:submodule_def)
apply (erule conjE)+
apply (simp add:subsetD)
done
lemma (in Module) submodule_inc_0:"submodule R M H ==>
\<zero>M ∈ H"
apply (simp add:submodule_def, (erule conjE)+)
apply (rule asubg_inc_zero, assumption+)
done
lemma (in Module) sc_un:" m ∈ carrier M ==> 1rR ·s m = m"
apply (simp add:sprod_one)
done
lemma (in Module) sc_mem:"[|a ∈ carrier R; m ∈ carrier M|] ==>
a ·s m ∈ carrier M"
apply (simp add:sprod_closed)
done
lemma (in Module) submodule_sc_closed:"[|submodule R M H;
a ∈ carrier R; h ∈ H|] ==> a ·s h ∈ H"
apply (simp add:submodule_def)
done
lemma (in Module) sc_assoc:"[|a ∈ carrier R; b ∈ carrier R;
m ∈ carrier M|] ==> (a ·rR b) ·s m = a ·s ( b ·s m)"
apply (simp add:sprod_assoc)
done
lemma (in Module) sc_l_distr:"[|a ∈ carrier R; b ∈ carrier R;
m ∈ carrier M|] ==> (a ±R b)·s m = a ·s m ± b ·s m"
apply (simp add:sprod_l_distr)
done
lemma (in Module) sc_r_distr:"[|a ∈ carrier R; m ∈ carrier M; n ∈ carrier M|] ==>
a ·s (m ± n) = a ·s m ± a ·s n"
apply (simp add:sprod_r_distr)
done
lemma (in Module) sc_0_m:"m ∈ carrier M ==> \<zero>R·s m = \<zero>M"
apply (cut_tac sc_Ring,
frule Ring.ring_is_ag,
frule aGroup.ag_inc_zero [of "R"],
frule sc_l_distr[of "\<zero>R" "\<zero>R" "m"], assumption+,
frule sc_mem [of "\<zero>R" m], assumption+)
apply (simp add:aGroup.ag_l_zero, frule sym,
thin_tac "\<zero>R ·s m = \<zero>R ·s m ± \<zero>R ·s m")
apply (frule ag_eq_sol1 [of "\<zero>R ·s m" "\<zero>R ·s m" "\<zero>R ·s m"], assumption+,
simp add:ag_l_inv1)
done
lemma (in Module) sc_a_0:"a ∈ carrier R ==> a ·s \<zero> = \<zero>"
apply (cut_tac ag_inc_zero,
frule sc_r_distr[of a \<zero> \<zero>], assumption+,
frule sc_mem [of a \<zero>], assumption+)
apply (simp add:ag_l_zero, frule sym,
thin_tac "a ·s \<zero> = a ·s \<zero> ± a ·s \<zero>")
apply (frule ag_eq_sol1 [of "a ·s \<zero>" "a ·s \<zero>" "a ·s \<zero>"], assumption+,
simp add:ag_l_inv1)
done
lemma (in Module) sc_minus_am:"[|a ∈ carrier R; m ∈ carrier M|]
==> -a (a ·s m) = a ·s (-a m)"
apply (frule ag_mOp_closed [of m],
frule sc_r_distr[of a m "-a m"], assumption+,
simp add:ag_r_inv1,
simp add:sc_a_0, frule sym,
thin_tac "\<zero> = a ·s m ± a ·s (-a m)")
apply (frule sc_mem [of a m], assumption+,
frule sc_mem [of a "-a m"], assumption+,
frule ag_eq_sol1 [of "a ·s m" "a ·s (-a m)" "\<zero>"], assumption+,
simp add:ag_inc_zero, assumption)
apply (frule ag_mOp_closed [of "a ·s m"],
simp add:ag_r_zero)
done
lemma (in Module) sc_minus_am1:"[|a ∈ carrier R; m ∈ carrier M|]
==> -a (a ·s m) = (-aR a) ·s m"
apply (cut_tac sc_Ring, frule Ring.ring_is_ag,
frule aGroup.ag_mOp_closed [of R a], assumption+,
frule sc_l_distr[of a "-aR a" m], assumption+,
simp add:aGroup.ag_r_inv1 [of "R"],
simp add:sc_0_m, frule sym) apply (
thin_tac "\<zero> = a ·s m ± (-aR a) ·s m")
apply (frule sc_mem [of a m], assumption+,
frule sc_mem [of "-aR a" m], assumption+)
apply (frule ag_eq_sol1 [of "a ·s m" "(-aR a) ·s m" \<zero>], assumption+,
simp add:ag_inc_zero, assumption)
apply (frule ag_mOp_closed [of "a ·s m"])
apply (thin_tac "a ·s m ± (-aR a) ·s m = \<zero>",
simp add:ag_r_zero)
done
lemma (in Module) submodule_0:"submodule R M {\<zero>}"
apply (simp add:submodule_def)
apply (simp add:ag_inc_zero)
apply (simp add:asubg_zero)
apply (rule allI, rule impI)
apply (simp add:sc_a_0)
done
lemma (in Module) submodule_whole:"submodule R M (carrier M)"
apply (simp add:submodule_def)
apply (simp add:asubg_whole)
apply ((rule allI)+, rule impI, erule conjE)
apply (simp add:sc_mem)
done
lemma (in Module) submodule_pOp_closed:"[|submodule R M H; h ∈ H; k ∈ H|] ==>
h ± k ∈ H"
apply (simp add:submodule_def)
apply (erule conjE)+
apply (thin_tac "∀a m. a ∈ carrier R ∧ m ∈ H --> a ·s m ∈ H")
apply (simp add:asubg_pOp_closed)
done
lemma (in Module) submodule_mOp_closed:"[|submodule R M H; h ∈ H|]
==> -a h ∈ H"
apply (simp add:submodule_def,
(erule conjE)+,
thin_tac "∀a m. a ∈ carrier R ∧ m ∈ H --> a ·s m ∈ H")
apply (rule asubg_mOp_closed, assumption+)
done
constdefs
mHom :: "[('b, 'm) Ring_scheme, ('a, 'b, 'm1) Module_scheme,
('c, 'b, 'm2) Module_scheme] => ('a => 'c) set"
(* ("(3HOM_/ _/ _)" [90, 90, 91]90 ) *)
"mHom R M N == {f. f ∈ aHom M N ∧
(∀a∈carrier R. ∀m∈carrier M. f (a ·sM m) = a ·sN (f m))}"
mimg ::"[('b, 'm) Ring_scheme, ('a, 'b, 'm1) Module_scheme,
('c, 'b, 'm2) Module_scheme, 'a => 'c] => ('c, 'b) Module"
("(4mimg_ _,_/ _)" [88,88,88,89]88)
"mimgR M,N f == mdl N (f ` (carrier M))"
mzeromap::"[('a, 'b, 'm1) Module_scheme, ('c, 'b, 'm2) Module_scheme]
=> ('a => 'c)"
"mzeromap M N == λx∈carrier M. \<zero>N"
lemma (in Ring) mHom_func:"f ∈ mHom R M N ==> f ∈ carrier M -> carrier N"
by (simp add:mHom_def aHom_def)
lemma (in Module) mHom_test:"[|R module N; f ∈ carrier M -> carrier N ∧
f ∈ extensional (carrier M) ∧
(∀m∈carrier M. ∀n∈carrier M. f (m ±M n) = f m ±N (f n)) ∧
(∀a∈carrier R. ∀m∈carrier M. f (a ·sM m) = a ·sN (f m))|] ==>
f ∈ mHom R M N"
apply (simp add:mHom_def)
apply (simp add:aHom_def)
done
lemma (in Module) mHom_mem:"[|R module N; f ∈ mHom R M N; m ∈ carrier M|]
==> f m ∈ carrier N"
apply (simp add:mHom_def aHom_def) apply (erule conjE)+
apply (simp add:funcset_mem)
done
lemma (in Module) mHom_add:"[|R module N; f ∈ mHom R M N; m ∈ carrier M;
n ∈ carrier M|] ==> f (m ± n) = f m ±N (f n)"
apply (simp add:mHom_def) apply (erule conjE)+
apply (frule Module.module_is_ag [of N R],
cut_tac module_is_ag)
apply (simp add:aHom_add)
done
lemma (in Module) mHom_0:"[|R module N; f ∈ mHom R M N|] ==> f (\<zero>) = \<zero>N"
apply (simp add:mHom_def, (erule conjE)+,
frule Module.module_is_ag [of N],
cut_tac module_is_ag)
apply (simp add:aHom_0_0)
done
lemma (in Module) mHom_inv:"[|R module N; m ∈ carrier M; f ∈ mHom R M N|] ==>
f (-a m) = -aN (f m)"
apply (cut_tac module_is_ag,
frule Module.module_is_ag [of N])
apply (simp add:mHom_def, (erule conjE)+)
apply (rule aHom_inv_inv, assumption+)
done
lemma (in Module) mHom_lin:"[|R module N; m ∈ carrier M; f ∈ mHom R M N;
a ∈ carrier R|] ==> f (a ·s m) = a ·sN (f m)"
apply (simp add:mHom_def)
done
lemma (in Module) mker_inc_zero:
"[|R module N; f ∈ mHom R M N |] ==> \<zero> ∈ (kerM,N f)"
apply (simp add:ker_def)
apply (simp add:module_inc_zero)
apply (simp add:mHom_0)
done
lemma (in Module) mHom_eq_ker:"[|R module N; f ∈ mHom R M N; a ∈ carrier M;
b∈ carrier M; a ± (-a b) ∈ kerM,N f|] ==> f a = f b"
apply (simp add:ker_def, erule conjE)
apply (cut_tac module_is_ag,
frule aGroup.ag_mOp_closed [of "M" "b"], assumption+,
simp add:mHom_add, simp add:mHom_inv,
thin_tac "aGroup M")
apply (frule mHom_mem [of N f a], assumption+,
frule mHom_mem [of N f b], assumption+,
frule Module.module_is_ag[of N])
apply (subst aGroup.ag_eq_diffzero[of N], assumption+)
done
lemma (in Module) mHom_ker_eq:"[|R module N; f ∈ mHom R M N; a ∈ carrier M;
b∈ carrier M; f a = f b|] ==> a ± (-a b) ∈ kerM,N f"
apply (simp add:ker_def)
apply (frule ag_mOp_closed[of b])
apply (simp add:ag_pOp_closed)
apply (simp add:mHom_add mHom_inv)
apply (frule mHom_mem [of N f b], assumption+)
apply (frule_tac R = R and M = N in Module.module_is_ag,
simp add:aGroup.ag_r_inv1)
done
lemma (in Module) mker_submodule:"[|R module N; f ∈ mHom R M N|] ==>
submodule R M (kerM,N f)"
apply (cut_tac module_is_ag,
frule Module.module_is_ag [of N])
apply (simp add:submodule_def)
apply (rule conjI)
apply (rule subsetI, simp add:ker_def)
apply (rule conjI)
apply (simp add:mHom_def, (erule conjE)+, simp add:ker_subg)
apply ((rule allI)+, rule impI, erule conjE)
apply (simp add:ker_def, erule conjE)
apply (simp add:sc_mem)
apply (subst mHom_lin [of N _ f], assumption+, simp) (* key *)
apply (simp add:Module.sc_a_0[of N])
done
lemma (in Module) mker_mzeromap:"R module N ==>
kerM,N (mzeromap M N) = carrier M"
apply (simp add:ker_def mzeromap_def)
done
lemma (in Module) mdl_carrier:"submodule R M H ==> carrier (mdl M H) = H"
apply (simp add:mdl_def)
done
lemma (in Module) mdl_is_ag:"submodule R M H ==> aGroup (mdl M H)"
apply (cut_tac module_is_ag)
apply (rule aGroup.intro)
apply (simp add:mdl_def)
apply (rule bivar_func_test)
apply (rule ballI)+
apply (simp add:submodule_def, (erule conjE)+)
apply (simp add:asubg_pOp_closed)
apply (simp add:mdl_def)
apply (simp add:submodule_def, (erule conjE)+,
frule_tac c = a in subsetD[of H "carrier M"], assumption+,
frule_tac c = b in subsetD[of H "carrier M"], assumption+,
frule_tac c = c in subsetD[of H "carrier M"], assumption+,
simp add:aGroup.ag_pOp_assoc)
apply (simp add:submodule_def, (erule conjE)+,
simp add:mdl_def,
frule_tac c = a in subsetD[of H "carrier M"], assumption+,
frule_tac c = b in subsetD[of H "carrier M"], assumption+,
simp add:aGroup.ag_pOp_commute)
apply (simp add:mdl_def)
apply (rule univar_func_test, rule ballI,
simp add:submodule_def aGroup.asubg_mOp_closed)
apply (simp add:mdl_def,
simp add:submodule_def, (erule conjE)+,
frule_tac c = a in subsetD[of H "carrier M"], assumption+,
rule aGroup.ag_l_inv1, assumption+)
apply (simp add:mdl_def,
simp add:submodule_def, (erule conjE)+,
simp add:asubg_inc_zero)
apply (simp add:mdl_def, simp add:submodule_def, (erule conjE)+,
frule_tac c = a in subsetD[of H "carrier M"], assumption+)
apply (simp add:ag_l_zero)
done
lemma (in Module) mdl_is_module:"submodule R M H ==> R module (mdl M H)"
apply (rule Module.intro)
apply (simp add:mdl_is_ag)
apply (rule Module_axioms.intro)
apply (simp add:sc_Ring)
apply (simp add:mdl_def)
apply (simp add:submodule_def)
apply (simp add:mdl_def)
apply (simp add:submodule_def, (erule conjE)+,
frule_tac c = m in subsetD[of H "carrier M"], assumption+,
simp add:sc_l_distr)
apply (simp add:mdl_def submodule_def, (erule conjE)+,
simp add:asubg_pOp_closed,
frule_tac c = m in subsetD[of H "carrier M"], assumption+,
frule_tac c = n in subsetD[of H "carrier M"], assumption+,
simp add:sc_r_distr)
apply (simp add:mdl_def submodule_def, (erule conjE)+,
frule_tac c = m in subsetD[of H "carrier M"], assumption+,
simp add:sc_assoc)
apply (simp add:mdl_def submodule_def, (erule conjE)+,
frule_tac c = m in subsetD[of H "carrier M"], assumption+,
simp add:sprod_one)
done
lemma (in Module) submodule_of_mdl:"[|submodule R M H; submodule R M N; H ⊆ N|]
==> submodule R (mdl M N) H"
apply (subst submodule_def)
apply (rule conjI, simp add:mdl_def)
apply (rule conjI)
apply (rule aGroup.asubg_test[of "mdl M N" H])
apply (frule mdl_is_module[of N],
simp add:Module.module_is_ag, simp add:mdl_def)
apply (simp add:submodule_def[of R M H], (erule conjE)+)
apply (frule asubg_inc_zero[of H], simp add:nonempty)
apply ((rule ballI)+, simp add:mdl_def)
apply (simp add:submodule_def[of R M H], (erule conjE)+)
apply (frule_tac x = b in asubg_mOp_closed[of H], assumption+)
apply (rule asubg_pOp_closed[of H], assumption+)
apply ((rule allI)+, rule impI, erule conjE)
apply (simp add:mdl_def subsetD)
apply (simp add:submodule_def[of R M H])
done
lemma (in Module) img_set_submodule:"[|R module N; f ∈ mHom R M N|] ==>
submodule R N (f ` (carrier M))"
apply (simp add:submodule_def)
apply (rule conjI)
apply (rule subsetI)
apply (simp add:image_def)
apply (erule bexE, simp, thin_tac "x = f xa")
apply (simp add:mHom_mem)
apply (rule conjI)
apply (frule Module.module_is_ag [of N])
apply (rule aGroup.asubg_test, assumption+)
apply (rule subsetI) apply (simp add:image_def)
apply (erule bexE) apply (simp add:mHom_mem)
apply (cut_tac ag_inc_zero,
simp add:mHom_mem, simp add:nonempty)
apply ((rule ballI)+, simp add:image_def)
apply ((erule bexE)+, simp)
apply (simp add:mHom_inv[THEN sym],
frule_tac x = xa in ag_mOp_closed,
simp add:mHom_add[THEN sym, of N f],
frule_tac x = "x" and y = "-a xa" in ag_pOp_closed, assumption+)
apply blast
apply ((rule allI)+, rule impI, erule conjE)
apply (simp add:image_def, erule bexE, simp)
apply (simp add:mHom_lin[THEN sym, of N _ f])
apply (frule_tac a = a and m = x in sc_mem, assumption)
apply blast
done
lemma (in Module) mimg_module:"[|R module N; f ∈ mHom R M N|] ==>
R module (mimg R M N f)"
apply (simp add:mimg_def)
apply (rule Module.mdl_is_module[of N R "f ` (carrier M)"], assumption)
apply (simp add:img_set_submodule)
done
lemma (in Module) surjec_to_mimg:"[|R module N; f ∈ mHom R M N|] ==>
surjecM, (mimg R M N f) f"
apply (simp add:surjec_def)
apply (rule conjI)
apply (simp add:aHom_def)
apply (rule conjI)
apply (rule univar_func_test, rule ballI, simp add:mimg_def mdl_def)
apply (rule conjI)
apply (simp add:mHom_def aHom_def restrict_def extensional_def)
apply ((rule ballI)+, simp add:mimg_def mdl_def, simp add:mHom_add)
apply (simp add:mimg_def mdl_def)
apply (simp add:surj_to_def image_def)
done
constdefs
tOp_mHom :: "[('b, 'm) Ring_scheme, ('a, 'b, 'm1) Module_scheme,
('c, 'b, 'm2) Module_scheme] => ('a => 'c) => ('a => 'c) => ('a => 'c)"
"tOp_mHom R M N f g == λx ∈ carrier M. (f x ±N (g x))"
iOp_mHom :: "[('b, 'm) Ring_scheme, ('a, 'b, 'm1) Module_scheme,
('c, 'b, 'm2) Module_scheme] => ('a => 'c) => ('a => 'c)"
"iOp_mHom R M N f == λx ∈ carrier M. (-aN (f x))"
sprod_mHom ::"[('b, 'm) Ring_scheme, ('a, 'b, 'm1) Module_scheme,
('c, 'b, 'm2) Module_scheme] => 'b => ('a => 'c) => ('a => 'c)"
"sprod_mHom R M N a f == λx ∈ carrier M. a ·sN (f x)"
HOM :: "[('b, 'more) Ring_scheme, ('a, 'b, 'more1) Module_scheme,
('c, 'b, 'more2) Module_scheme] => ('a => 'c, 'b) Module"
("(3HOM_ _/ _)" [90, 90, 91]90 )
"HOMR M N == (|carrier = mHom R M N, pop = tOp_mHom R M N,
mop = iOp_mHom R M N, zero = mzeromap M N, sprod =sprod_mHom R M N |)),"
lemma (in Module) zero_HOM:"R module N ==>
mzeromap M N = \<zero>HOMR M N"
apply (simp add:HOM_def)
done
lemma (in Module) tOp_mHom_closed:"[|R module N; f ∈ mHom R M N; g ∈ mHom R M N|]
==> tOp_mHom R M N f g ∈ mHom R M N"
apply (rule mHom_test, assumption+)
apply (rule conjI)
apply (rule univar_func_test, rule ballI)
apply (simp add:tOp_mHom_def)
apply (frule_tac f = f and m = x in mHom_mem [of N], assumption+,
frule_tac f = g and m = x in mHom_mem [of N], assumption+,
frule Module.module_is_ag [of N],
simp add:aGroup.ag_pOp_closed[of N])
apply (rule conjI)
apply (simp add:tOp_mHom_def restrict_def extensional_def)
apply (rule conjI)
apply (rule ballI)+
apply (simp add:tOp_mHom_def)
apply (simp add:ag_pOp_closed)
apply (frule_tac f = f and m = m in mHom_mem [of N], assumption+,
frule_tac f = f and m = n in mHom_mem [of N], assumption+,
frule_tac f = g and m = m in mHom_mem [of N], assumption+,
frule_tac f = g and m = n in mHom_mem [of N], assumption+,
simp add:mHom_add,
frule Module.module_is_ag [of N],
subst aGroup.pOp_assocTr43[of "N"], assumption+,
frule_tac x = "f n" and y = "g m" in aGroup.ag_pOp_commute [of "N"],
assumption+)
apply simp
apply (subst aGroup.pOp_assocTr43[of "N"], assumption+, simp)
apply (rule ballI)+
apply (simp add:tOp_mHom_def)
apply (frule_tac a = a and m = m in sc_mem, assumption, simp)
apply (frule_tac f = f and m = m in mHom_mem [of N], assumption+,
frule_tac f = g and m = m in mHom_mem [of N], assumption+,
frule_tac a = a and m = "f m" and n = "g m" in
Module.sc_r_distr[of N R], assumption+,
simp)
apply (simp add:mHom_lin)
done
lemma (in Module) iOp_mHom_closed:"[|R module N; f ∈ mHom R M N|]
==> iOp_mHom R M N f ∈ mHom R M N"
apply (rule mHom_test, assumption+)
apply (rule conjI)
apply (rule univar_func_test, rule ballI)
apply (simp add:iOp_mHom_def)
apply (frule_tac f = f and m = x in mHom_mem [of N], assumption+)
apply (frule Module.module_is_ag [of N])
apply (simp add:aGroup.ag_mOp_closed)
apply (rule conjI)
apply (simp add:iOp_mHom_def restrict_def extensional_def)
apply (rule conjI) apply (rule ballI)+
apply (simp add:iOp_mHom_def)
apply (simp add:ag_pOp_closed)
apply (simp add:mHom_add)
apply (frule_tac f = f and m = m in mHom_mem [of N], assumption+,
frule_tac f = f and m = n in mHom_mem [of N], assumption+)
apply (frule Module.module_is_ag [of N])
apply (simp add:aGroup.ag_p_inv)
apply (rule ballI)+
apply (simp add:iOp_mHom_def)
apply (simp add:sc_mem)
apply (simp add:mHom_lin)
apply (frule_tac f = f and m = m in mHom_mem [of N], assumption+)
apply (simp add:Module.sc_minus_am[of N])
done
lemma (in Module) mHom_ex_zero:"R module N ==> mzeromap M N ∈ mHom R M N"
apply (simp add:mHom_def)
apply (rule conjI)
apply (simp add:aHom_def,
rule conjI, rule univar_func_test, rule ballI,
simp add:mzeromap_def, simp add:Module.module_inc_zero)
apply (simp add:mzeromap_def extensional_def)
apply ((rule ballI)+,
simp add:ag_pOp_closed,
frule Module.module_is_ag [of N],
frule aGroup.ag_inc_zero [of "N"],
simp add:aGroup.ag_l_zero)
apply ((rule ballI)+,
simp add:mzeromap_def,
simp add:sc_mem)
apply (simp add:Module.sc_a_0)
done
lemma (in Module) mHom_eq:"[|R module N; f ∈ mHom R M N; g ∈ mHom R M N;
∀m∈carrier M. f m = g m|] ==> f = g"
apply (simp add:mHom_def aHom_def)
apply (erule conjE)+
apply (rule funcset_eq, assumption+)
done
lemma (in Module) mHom_l_zero:"[|R module N; f ∈ mHom R M N|]
==> tOp_mHom R M N (mzeromap M N) f = f"
apply (frule mHom_ex_zero [of N])
apply (frule tOp_mHom_closed [of N "mzeromap M N" f], assumption+)
apply (rule mHom_eq, assumption+)
apply (rule ballI)
apply (simp add:tOp_mHom_def, simp add:mzeromap_def)
apply (frule_tac f = f and m = m in mHom_mem [of N], assumption+)
apply (frule Module.module_is_ag [of N])
apply (simp add:aGroup.ag_l_zero[of N])
done
lemma (in Module) mHom_l_inv:"[|R module N; f ∈ mHom R M N|]
==> tOp_mHom R M N (iOp_mHom R M N f) f = mzeromap M N"
apply (frule mHom_ex_zero [of N])
apply (frule_tac f = f in iOp_mHom_closed [of N], assumption,
frule_tac f = "iOp_mHom R M N f" and g = f in tOp_mHom_closed [of N],
assumption+,
frule mHom_ex_zero [of N])
apply (rule mHom_eq, assumption+, rule ballI)
apply (simp add:tOp_mHom_def iOp_mHom_def, simp add:mzeromap_def)
apply (frule_tac f = f and m = m in mHom_mem [of N], assumption+)
apply (frule Module.module_is_ag [of N])
apply (simp add:aGroup.ag_l_inv1)
done
lemma (in Module) mHom_tOp_assoc:"[|R module N; f ∈ mHom R M N; g ∈ mHom R M N;
h ∈ mHom R M N|] ==> tOp_mHom R M N (tOp_mHom R M N f g) h =
tOp_mHom R M N f (tOp_mHom R M N g h)"
apply (frule_tac f = f and g = g in tOp_mHom_closed [of N], assumption+,
frule_tac f = "tOp_mHom R M N f g" and g = h in
tOp_mHom_closed [of N], assumption+,
frule_tac f = g and g = h in tOp_mHom_closed [of N], assumption+,
frule_tac f = f and g = "tOp_mHom R M N g h" in
tOp_mHom_closed [of N], assumption+)
apply (rule mHom_eq, assumption+, rule ballI,
thin_tac "tOp_mHom R M N f g ∈ mHom R M N",
thin_tac "tOp_mHom R M N (tOp_mHom R M N f g) h ∈ mHom R M N",
thin_tac "tOp_mHom R M N g h ∈ mHom R M N",
thin_tac "tOp_mHom R M N f (tOp_mHom R M N g h) ∈ mHom R M N")
apply (simp add:tOp_mHom_def)
apply (frule_tac f = f and m = m in mHom_mem [of N], assumption+,
frule_tac f = g and m = m in mHom_mem [of N], assumption+,
frule_tac f = h and m = m in mHom_mem [of N], assumption+)
apply (frule Module.module_is_ag [of N])
apply (simp add:aGroup.ag_pOp_assoc)
done
lemma (in Module) mHom_tOp_commute:"[|R module N; f ∈ mHom R M N;
g ∈ mHom R M N|] ==> tOp_mHom R M N f g = tOp_mHom R M N g f"
apply (frule_tac f = f and g = g in tOp_mHom_closed [of N], assumption+,
frule_tac f = g and g = f in tOp_mHom_closed [of N], assumption+)
apply (rule mHom_eq, assumption+)
apply (rule ballI)
apply (thin_tac "tOp_mHom R M N f g ∈ mHom R M N",
thin_tac "tOp_mHom R M N g f ∈ mHom R M N")
apply (simp add:tOp_mHom_def)
apply (frule_tac f = f and m = m in mHom_mem [of N], assumption+,
frule_tac f = g and m = m in mHom_mem [of N], assumption+,
frule Module.module_is_ag [of N])
apply (simp add:aGroup.ag_pOp_commute)
done
lemma (in Module) HOM_is_ag:"R module N ==> aGroup (HOMR M N)"
apply (rule aGroup.intro)
apply (simp add:HOM_def)
apply (rule bivar_func_test)
apply (rule ballI)+
apply (simp add:tOp_mHom_closed)
apply (simp add:HOM_def)
apply (simp add:mHom_tOp_assoc)
apply (simp add:HOM_def)
apply (simp add:mHom_tOp_commute)
apply (simp add:HOM_def)
apply (rule univar_func_test, rule ballI)
apply (simp add:iOp_mHom_closed)
apply (simp add:HOM_def,
simp add:mHom_l_inv)
apply (simp add:HOM_def)
apply (simp add:mHom_ex_zero)
apply (simp add:HOM_def,
simp add:mHom_l_zero)
done
lemma (in Module) sprod_mHom_closed:"[|R module N; a ∈ carrier R;
f ∈ mHom R M N|] ==> sprod_mHom R M N a f ∈ mHom R M N"
apply (rule mHom_test, assumption+)
apply (rule conjI)
apply (simp add:Pi_def)
apply (rule allI, rule impI, simp add:sprod_mHom_def,
frule_tac f = f and m = x in mHom_mem [of N], assumption+,
simp add:Module.sc_mem [of N R a])
apply (rule conjI)
apply (simp add:sprod_mHom_def restrict_def extensional_def)
apply (rule conjI)
apply (rule ballI)+
apply (frule_tac x = m and y = n in ag_pOp_closed, assumption+)
apply (simp add:sprod_mHom_def)
apply (subst mHom_add [of N f], assumption+)
apply (frule_tac f = f and m = m in mHom_mem [of N], assumption+,
frule_tac f = f and m = n in mHom_mem [of N], assumption+)
apply (simp add:Module.sc_r_distr)
apply (rule ballI)+
apply (simp add:sprod_mHom_def)
apply (frule_tac a = aa and m = m in sc_mem, assumption+, simp)
apply (simp add:mHom_lin)
apply (frule_tac f = f and m = m in mHom_mem [of N], assumption+)
apply (simp add:Module.sc_assoc[THEN sym, of N R])
apply (cut_tac sc_Ring, simp add:Ring.ring_tOp_commute)
done
lemma (in Module) HOM_is_module:"R module N ==> R module (HOMR M N)"
apply (rule Module.intro)
apply (simp add:HOM_is_ag)
apply (rule Module_axioms.intro)
apply (simp add:sc_Ring)
apply (simp add:HOM_def)
apply (simp add:sprod_mHom_closed)
apply (simp add:HOM_def)
apply (cut_tac sc_Ring,
frule Ring.ring_is_ag[of R],
frule_tac x = a and y = b in aGroup.ag_pOp_closed[of R], assumption+,
frule_tac a = "a ±R b" and f = m in sprod_mHom_closed[of N],
assumption+)
apply(frule_tac a = a and f = m in sprod_mHom_closed[of N], assumption+,
frule_tac a = b and f = m in sprod_mHom_closed[of N], assumption+,
frule_tac f = "sprod_mHom R M N a m" and g = "sprod_mHom R M N b m" in
tOp_mHom_closed[of N], assumption+)
apply (rule mHom_eq[of N], assumption+, rule ballI,
simp add:sprod_mHom_def tOp_mHom_def)
apply (rename_tac a b f m)
apply (frule_tac f = f and m = m in mHom_mem[of N], assumption+)
apply (simp add:Module.sc_l_distr[of N])
apply (simp add:HOM_def)
apply (rename_tac a f g,
frule_tac f = f and g = g in tOp_mHom_closed[of N], assumption+,
frule_tac a = a and f = "tOp_mHom R M N f g" in
sprod_mHom_closed[of N], assumption+,
frule_tac a = a and f = f in sprod_mHom_closed[of N], assumption+,
frule_tac a = a and f = g in sprod_mHom_closed[of N], assumption+,
frule_tac f = "sprod_mHom R M N a f" and g = "sprod_mHom R M N a g"
in tOp_mHom_closed[of N], assumption+)
apply (rule mHom_eq[of N], assumption+, rule ballI,
simp add:sprod_mHom_def tOp_mHom_def,
frule_tac f = f and m = m in mHom_mem[of N], assumption+,
frule_tac f = g and m = m in mHom_mem[of N], assumption+)
apply (simp add:Module.sc_r_distr)
apply (simp add:HOM_def)
apply (rename_tac a b f)
apply (cut_tac sc_Ring,
frule_tac x = a and y = b in Ring.ring_tOp_closed, assumption+,
frule_tac a = "a ·rR b" and f = f in sprod_mHom_closed[of N],
assumption+,
frule_tac a = b and f = f in sprod_mHom_closed[of N], assumption+,
frule_tac a = a and f = "sprod_mHom R M N b f" in
sprod_mHom_closed[of N], assumption+)
apply (rule mHom_eq[of N], assumption+, rule ballI,
simp add:sprod_mHom_def,
frule_tac f = f and m = m in mHom_mem[of N], assumption+,
simp add:Module.sc_assoc)
apply (simp add:HOM_def)
apply (cut_tac sc_Ring,
frule Ring.ring_one,
frule_tac a = "1rR" and f = m in sprod_mHom_closed[of N], assumption+)
apply (rule mHom_eq, assumption+, rule ballI, rename_tac f m,
simp add:sprod_mHom_def,
frule_tac f = f and m = m in mHom_mem[of N], assumption+,
simp add:Module.sprod_one)
done
section "2. injective hom, surjective hom, bijective hom and iverse hom"
constdefs
invmfun :: "[('b, 'm) Ring_scheme, ('a, 'b, 'm1) Module_scheme,
('c, 'b, 'm2) Module_scheme, 'a => 'c] => 'c => 'a"
"invmfun R M N (f :: 'a => 'c) ==
λy∈(carrier N). SOME x. (x ∈ (carrier M) ∧ f x = y)"
misomorphic :: "[('b, 'm) Ring_scheme, ('a, 'b, 'm1) Module_scheme,
('c, 'b, 'm2) Module_scheme] => bool"
"misomorphic R M N == ∃f. f ∈ mHom R M N ∧ bijecM,N f"
mId :: "('a, 'b, 'm1) Module_scheme => 'a => 'a" ("(mId_/ )" [89]88)
"mIdM == λm∈carrier M. m"
mcompose::"[('a, 'r, 'm1) Module_scheme, 'b => 'c, 'a => 'b] => 'a => 'c"
"mcompose M g f == compose (carrier M) g f"
syntax
"@MISOM" ::"[('a, 'b, 'm1) Module_scheme, ('b, 'm) Ring_scheme,
('c, 'b, 'm2) Module_scheme] => bool"
("(3_ ≅_ _)" [82,82,83]82)
translations
"M ≅R N" == "misomorphic R M N"
lemma (in Module) minjec_inj:"[|R module N; injecM,N f|] ==>
inj_on f (carrier M)"
apply (simp add:inj_on_def, (rule ballI)+, rule impI)
apply (simp add:injec_def, erule conjE)
apply (frule Module.module_is_ag[of N])
apply (cut_tac module_is_ag)
apply (frule_tac a = x in aHom_mem[of M N f], assumption+,
frule_tac a = y in aHom_mem[of M N f], assumption+)
apply (simp add:aGroup.ag_eq_diffzero[of N])
apply (simp add:aHom_inv_inv[THEN sym, of M N f],
frule_tac x = y in aGroup.ag_mOp_closed, assumption+,
simp add:aHom_add[THEN sym, of M N f])
apply (simp add:ker_def)
apply (frule_tac x = x and y = "-a y" in ag_pOp_closed, assumption+)
apply (subgoal_tac "(x ± -a y) ∈ {a ∈ carrier M. f a = \<zero>N}", simp)
apply (simp add:ag_eq_diffzero)
apply blast
done
lemma (in Module) invmfun_l_inv:"[|R module N; bijecM,N f; m ∈ carrier M|] ==>
(invmfun R M N f) (f m) = m"
apply (simp add:bijec_def, erule conjE)
apply (frule minjec_inj [of N f], assumption+)
apply (simp add:surjec_def, erule conjE, simp add:aHom_def)
apply (frule conjunct1)
apply (thin_tac "f ∈ carrier M -> carrier N ∧
f ∈ extensional (carrier M) ∧
(∀a∈carrier M. ∀b∈carrier M. f (a ± b) = f a ±N f b)")
apply (frule invfun_l [of "f" "carrier M" "carrier N" "m"], assumption+)
apply (simp add:surj_to_def)
apply (simp add:invfun_def invmfun_def)
done
lemma (in Module) invmfun_mHom:"[|R module N; bijecM,N f; f ∈ mHom R M N |] ==>
invmfun R M N f ∈ mHom R N M"
apply (frule minjec_inj [of N f])
apply (simp add:bijec_def)
apply (subgoal_tac "surjecM,N f") prefer 2 apply (simp add:bijec_def)
apply (rule Module.mHom_test) apply assumption apply (rule Module_axioms)
apply (rule conjI)
apply (simp add:surjec_def, erule conjE)
apply (simp add:aHom_def, frule conjunct1)
apply (thin_tac "f ∈ carrier M -> carrier N ∧
f ∈ extensional (carrier M) ∧
(∀a∈carrier M. ∀b∈carrier M. f (a ± b) = f a ±N f b)")
apply (frule inv_func [of "f" "carrier M" "carrier N"], assumption+)
apply (simp add:invmfun_def invfun_def)
apply (rule conjI)
apply (simp add:invmfun_def restrict_def extensional_def)
apply (rule conjI)
apply (rule ballI)+
apply (simp add:surjec_def)
apply (erule conjE, simp add:surj_to_def)
apply (frule sym, thin_tac "f ` carrier M = carrier N", simp,
thin_tac "carrier N = f ` carrier M")
apply (simp add:image_def, (erule bexE)+, simp)
apply (simp add:mHom_add[THEN sym])
apply (frule_tac x = x and y = xa in ag_pOp_closed, assumption+)
apply (simp add:invmfun_l_inv)
apply (rule ballI)+
apply (simp add:surjec_def, erule conjE)
apply (simp add:surj_to_def, frule sym, thin_tac "f ` carrier M = carrier N")
apply (simp add:image_def, (erule bexE)+, simp)
apply (simp add:mHom_lin[THEN sym])
apply (frule_tac a = a and m = x in sc_mem, assumption+)
apply (simp add:invmfun_l_inv)
done
lemma (in Module) invmfun_r_inv:"[|R module N; bijecM,N f; n ∈ carrier N|] ==>
f ((invmfun R M N f) n) = n"
apply (frule minjec_inj[of N f])
apply (simp add:bijec_def)
apply (unfold bijec_def, frule conjunct2, fold bijec_def)
apply (simp add:surjec_def, erule conjE, simp add:surj_to_def)
apply (frule sym, thin_tac "f ` carrier M = carrier N", simp,
thin_tac "carrier N = f ` carrier M")
apply (simp add:image_def, erule bexE, simp)
apply (simp add:invmfun_l_inv)
done
lemma (in Module) mHom_compos:"[|R module L; R module N; f ∈ mHom R L M;
g ∈ mHom R M N |] ==> compos L g f ∈ mHom R L N"
apply (simp add:mHom_def [of "R" "L" "N"])
apply (frule Module.module_is_ag [of L],
frule Module.module_is_ag [of N])
apply (rule conjI)
apply (simp add:mHom_def, (erule conjE)+)
apply (rule aHom_compos[of L M N f], assumption+)
apply (cut_tac module_is_ag, assumption+)
apply (rule ballI)+
apply (simp add:compos_def compose_def)
apply (simp add:Module.sc_mem)
apply (subst Module.mHom_lin[of L R M _ f], assumption, rule Module_axioms, assumption+) (*apply (
simp add:Module_def, rule conjI, assumption+) *)
apply (subst Module.mHom_lin[of M R N _ g], rule Module_axioms, assumption) (*apply (
simp add:Module_def, rule conjI)*) (** ordering **)
apply (rule Module.mHom_mem[of L R M f], assumption, rule Module_axioms, assumption+)
apply simp
done
lemma (in Module) mcompos_inj_inj:"[|R module L; R module N; f ∈ mHom R L M;
g ∈ mHom R M N; injecL,M f; injecM,N g |] ==> injecL,N (compos L g f)"
apply (frule Module.module_is_ag [of L],
frule Module.module_is_ag [of N])
apply (simp add:injec_def [of "L" "N"])
apply (rule conjI)
apply (simp add:injec_def, (erule conjE)+,
rule_tac aHom_compos[of L M N], assumption+,
rule module_is_ag)
apply assumption+
apply (simp add:compos_def compose_def)
apply (rule equalityI)
apply (rule subsetI, simp)
apply (simp add:injec_def [of _ _ "g"], erule conjE, simp add:ker_def)
apply (subgoal_tac "f x ∈ {a. a ∈ carrier M ∧ g a = \<zero>N}")
apply simp
apply (simp add:injec_def [of _ _ "f"], erule conjE)
apply (subgoal_tac "x ∈ kerL,M f", simp, thin_tac "kerL,M f = {\<zero>L}")
apply (simp add:ker_def)
apply (thin_tac "{a ∈ carrier M. g a = \<zero>N} = {\<zero>}")
apply (simp, erule conjE, simp)
apply (rule Module.mHom_mem[of L R M f], assumption, rule Module_axioms, assumption+)
apply (rule subsetI, simp)
apply (frule Module.module_inc_zero [of L R])
apply (frule Module.mHom_0[of L R M f], rule Module_axioms, assumption+)
apply (simp add:ker_def)
apply (subst mHom_0[of N], assumption+, simp)
done
lemma (in Module) mcompos_surj_surj:"[|R module L; R module N; surjecL,M f;
surjecM,N g; f ∈ mHom R L M; g ∈ mHom R M N |] ==>
surjecL,N (compos L g f)"
apply (frule Module.module_is_ag [of L],
frule Module.module_is_ag [of N],
cut_tac module_is_ag)
apply (simp add:surjec_def [of "L" "N"])
apply (rule conjI)
apply (simp add:mHom_def, (erule conjE)+)
apply (rule aHom_compos[of L M N f g], assumption+)
apply (rule surj_to_test)
apply (cut_tac Module.mHom_compos [of M R L N f g])
apply (simp add:mHom_def aHom_def)
apply (rule Module_axioms, assumption+)
apply (rule ballI)
apply (simp add: compos_def compose_def)
apply (simp add:surjec_def [of _ _ "g"])
apply (erule conjE) apply (simp add:surj_to_def)
apply (frule sym, thin_tac "g ` carrier M = carrier N", simp add:image_def,
thin_tac "carrier N = {y. ∃x∈carrier M. y = g x}",
erule bexE, simp)
apply (simp add:surjec_def [of _ _ "f"], erule conjE, simp add:surj_to_def,
rotate_tac -1, frule sym, thin_tac "f ` carrier L = carrier M",
simp add:image_def, erule bexE, simp)
apply blast
done
lemma (in Module) mId_mHom:"mIdM ∈ mHom R M M"
apply (simp add:mHom_def)
apply (rule conjI)
apply (simp add:aHom_def)
apply (rule conjI)
apply (rule univar_func_test) apply (rule ballI)
apply (simp add:mId_def)
apply (simp add:mId_def extensional_def)
apply (rule ballI)+
apply (simp add:ag_pOp_closed)
apply (rule ballI)+
apply (simp add:mId_def)
apply (simp add:sc_mem)
done
lemma (in Module) mHom_mId_bijec:"[|R module N; f ∈ mHom R M N; g ∈ mHom R N M;
compose (carrier M) g f = mIdM; compose (carrier N) f g = mIdN|] ==>
bijecM,N f"
apply (simp add:bijec_def)
apply (rule conjI)
apply (simp add:injec_def)
apply (rule conjI)
apply (simp add:mHom_def)
apply (simp add:ker_def)
apply (rule equalityI)
apply (rule subsetI, simp, erule conjE)
apply (frule_tac x = "f x" and y = "\<zero>N" and f = g in eq_elems_eq_val)
apply (frule_tac f = "compose (carrier M) g f" and g = "mIdM" and x = x in
eq_fun_eq_val, thin_tac "compose (carrier M) g f = mIdM",
simp add:compose_def)
apply (cut_tac Module.mHom_0[of N R M g], simp add:mId_def, assumption,
rule Module_axioms, assumption)
apply (rule subsetI, simp,
simp add:ag_inc_zero, simp add:mHom_0)
apply (simp add:surjec_def)
apply (rule conjI, simp add:mHom_def)
apply (rule surj_to_test)
apply (simp add:mHom_def aHom_def)
apply (rule ballI)
apply (frule_tac f = "compose (carrier N) f g" and g = "mIdN" and x = b in
eq_fun_eq_val, thin_tac "compose (carrier M) g f = mIdM",
thin_tac "compose (carrier N) f g = mIdN",
simp add:compose_def)
apply (simp add:mId_def)
apply (frule_tac m = b in Module.mHom_mem [of N R M g], rule Module_axioms, assumption+)
apply blast
done
constdefs
sup_sharp::"[('r, 'n) Ring_scheme, ('b, 'r, 'm1) Module_scheme,
('c, 'r, 'm2) Module_scheme, ('a, 'r, 'm) Module_scheme, 'b => 'c]
=> ('c => 'a) => ('b => 'a)"
"sup_sharp R M N L u == λf∈mHom R N L. compos M f u"
sub_sharp::"[('r, 'n) Ring_scheme, ('a, 'r, 'm) Module_scheme,
('b, 'r, 'm1) Module_scheme, ('c, 'r, 'm2) Module_scheme, 'b => 'c]
=> ('a => 'b) => ('a => 'c)"
"sub_sharp R L M N u == λf∈mHom R L M. compos L u f"
(* L
f| u
M -> N, f -> u o f *)
lemma (in Module) sup_sharp_homTr:"[|R module N; R module L; u ∈ mHom R M N;
f ∈ mHom R N L |] ==> sup_sharp R M N L u f ∈ mHom R M L"
apply (simp add:sup_sharp_def)
apply (rule Module.mHom_compos, assumption, rule Module_axioms, assumption+)
done
lemma (in Module) sup_sharp_hom:"[|R module N; R module L; u ∈ mHom R M N|] ==>
sup_sharp R M N L u ∈ mHom R (HOMR N L) (HOMR M L)"
apply (simp add:mHom_def [of "R" "HOMR N L"])
apply (rule conjI)
apply (simp add:aHom_def)
apply (rule conjI)
apply (rule univar_func_test) apply (rule ballI)
apply (simp add:HOM_def)
apply (simp add:sup_sharp_homTr)
apply (rule conjI)
apply (simp add:sup_sharp_def extensional_def,
rule allI, rule impI, simp add:HOM_def)
apply (rule ballI)+
apply (simp add:HOM_def)
apply (frule_tac f = a and g = b in Module.tOp_mHom_closed, assumption+)
apply (subgoal_tac "R module M")
apply (frule_tac f = a in Module.sup_sharp_homTr [of M R N L u], assumption+)
apply (frule_tac f = b in Module.sup_sharp_homTr [of M R N L u], assumption+)
apply (frule_tac f = "tOp_mHom R N L a b" in
Module.sup_sharp_homTr[of M R N L u], assumption+)
apply (rule Module.mHom_eq, assumption+)
apply (rule Module.tOp_mHom_closed, assumption+)
apply (rule ballI)
apply (simp add:sup_sharp_def tOp_mHom_def compose_def compos_def)
apply (simp add:mHom_mem, rule Module_axioms)
apply (rule ballI)+
apply (simp add:HOM_def)
apply (frule_tac a = a and f = m in Module.sprod_mHom_closed [of N R L],
assumption+)
apply (subgoal_tac "R module M",
frule_tac f = "sprod_mHom R N L a m" in
Module.sup_sharp_homTr [of M R N L u], assumption+)
apply (frule_tac f = m in Module.sup_sharp_homTr [of M R N L u], assumption+)
apply (frule_tac a = a and f = "sup_sharp R M N L u m" in
Module.sprod_mHom_closed [of M R L], assumption+)
apply (rule mHom_eq, assumption+)
apply (rule ballI)
apply (simp add:sprod_mHom_def sup_sharp_def compose_def compos_def)
apply (simp add:Module.mHom_mem, rule Module_axioms)
done
lemma (in Module) sub_sharp_homTr:"[|R module N; R module L; u ∈ mHom R M N;
f ∈ mHom R L M|] ==> sub_sharp R L M N u f ∈ mHom R L N"
apply (simp add:sub_sharp_def)
apply (simp add:mHom_compos)
done
lemma (in Module) sub_sharp_hom:"[|R module N; R module L; u ∈ mHom R M N|] ==>
sub_sharp R L M N u ∈ mHom R (HOMR L M) (HOMR L N)"
apply (simp add:mHom_def [of _ "HOMR L M"])
apply (rule conjI)
apply (simp add:aHom_def)
apply (rule conjI)
apply (simp add:HOM_def)
apply (rule univar_func_test) apply (rule ballI)
apply (simp add:sub_sharp_homTr)
apply (rule conjI)
apply (simp add:sub_sharp_def extensional_def)
apply (simp add:HOM_def)
apply (rule ballI)+
apply (simp add:HOM_def)
apply (frule_tac f = a and g = b in Module.tOp_mHom_closed [of L R M],
rule Module_axioms, assumption+)
apply (subgoal_tac "R module M")
apply (frule_tac f = "tOp_mHom R L M a b" in Module.sub_sharp_homTr
[of M R N L u], assumption+)
apply (frule_tac f = b in Module.sub_sharp_homTr[of M R N L u],
assumption+,
frule_tac f = a in Module.sub_sharp_homTr[of M R N L u], assumption+)
apply (frule_tac f = "sub_sharp R L M N u a" and
g = "sub_sharp R L M N u b" in Module.tOp_mHom_closed [of L R N],assumption+)
apply (rule Module.mHom_eq, assumption+)
apply (rule ballI)
apply (simp add:tOp_mHom_def sub_sharp_def mcompose_def compose_def,
simp add:compos_def compose_def)
apply (rule Module.mHom_add [of M R], assumption+)
apply (simp add:Module.mHom_mem, simp add:Module.mHom_mem)
apply (rule Module_axioms)
apply (rule ballI)+
apply (simp add:HOM_def)
apply (subgoal_tac "R module M")
apply (frule_tac a = a and f = m in Module.sprod_mHom_closed [of L R M],
assumption+)
apply (frule_tac f = "sprod_mHom R L M a m" in Module.sub_sharp_homTr
[of M R N L u], assumption+)
apply (frule_tac f = m in Module.sub_sharp_homTr
[of M R N L u], assumption+)
apply (frule_tac a = a and f = "sub_sharp R L M N u m" in
Module.sprod_mHom_closed [of L R N], assumption+)
apply (rule Module.mHom_eq, assumption+)
apply (rule ballI)
apply (simp add:sprod_mHom_def sub_sharp_def mcompose_def compose_def)
apply (frule_tac f = m and m = ma in Module.mHom_mem [of L R M], assumption+)
apply (simp add:compos_def compose_def)
apply (simp add:mHom_lin)
apply (rule Module_axioms)
done
lemma (in Module) mId_bijec:"bijecM,M (mIdM)"
apply (simp add:bijec_def)
apply (cut_tac mId_mHom)
apply (rule conjI)
apply (simp add:injec_def)
apply (rule conjI) apply (simp add:mHom_def)
apply (simp add:ker_def) apply (simp add:mId_def)
apply (rule equalityI) apply (rule subsetI, simp)
apply (rule subsetI, simp, simp add:ag_inc_zero)
apply (simp add:surjec_def)
apply (rule conjI, simp add:mHom_def)
apply (rule surj_to_test)
apply (simp add:mHom_def aHom_def)
apply (rule ballI)
apply (simp add:mId_def)
done
lemma (in Module) invmfun_bijec:"[|R module N; f ∈ mHom R M N; bijecM,N f|] ==>
bijecN,M (invmfun R M N f)"
apply (frule invmfun_mHom [of N f], assumption+)
apply (simp add:bijec_def [of N M])
apply (rule conjI)
apply (simp add:injec_def)
apply (simp add:mHom_def [of "R" "N" "M"]) apply (erule conjE)+
apply (thin_tac "∀a∈carrier R.
∀m∈carrier N. invmfun R M N f (a ·sN m) = a ·s invmfun R M N f m")
apply (rule equalityI) apply (rule subsetI) apply (simp add:ker_def CollectI)
apply (erule conjE)
apply (frule_tac x = "invmfun R M N f x" and y = "\<zero>" and f = f in
eq_elems_eq_val,
thin_tac "invmfun R M N f x = \<zero>")
apply (simp add:invmfun_r_inv)
apply (simp add:mHom_0)
apply (rule subsetI, simp)
apply (simp add:ker_def)
apply (simp add:Module.module_inc_zero)
apply (cut_tac ag_inc_zero,
frule invmfun_l_inv[of N f \<zero>], assumption+)
apply (simp add:mHom_0)
apply (simp add:surjec_def,
frule invmfun_mHom[of N f], assumption+)
apply (rule conjI, simp add:mHom_def)
apply (simp add:surj_to_def)
apply (rule equalityI, rule subsetI, simp add:image_def, erule bexE,
simp) thm Module.mHom_mem[of N R M "invmfun R M N f"]
apply (rule Module.mHom_mem[of N R M "invmfun R M N f"], assumption,
rule Module_axioms, assumption+)
apply (rule subsetI, simp add:image_def)
apply (frule_tac m = x in invmfun_l_inv[of N f], assumption+)
apply (frule_tac m = x in mHom_mem[of N f], assumption+)
apply (frule sym, thin_tac "invmfun R M N f (f x) = x", blast)
done
lemma (in Module) misom_self:"M ≅R M"
apply (cut_tac mId_bijec)
apply (cut_tac mId_mHom)
apply (simp add:misomorphic_def)
apply blast
done
lemma (in Module) misom_sym:"[|R module N; M ≅R N|] ==> N ≅R M"
apply (simp add:misomorphic_def [of "R" "M" "N"])
apply (erule exE, erule conjE)
apply (frule_tac f = f in invmfun_mHom [of N], assumption+)
apply (frule_tac f = f in invmfun_bijec [of N], assumption+)
apply (simp add:misomorphic_def)
apply blast
done
lemma (in Module) misom_trans:"[|R module L; R module N; L ≅R M; M ≅R N|] ==>
L ≅R N"
apply (simp add:misomorphic_def)
apply ((erule exE)+, (erule conjE)+)
apply (subgoal_tac "bijecL,N (compos L fa f)")
apply (subgoal_tac "(compos L fa f) ∈ mHom R L N")
apply blast
apply (rule Module.mHom_compos[of M R L N], rule Module_axioms, assumption+)
apply (simp add:bijec_def) apply (erule conjE)+
apply (simp add:mcompos_inj_inj)
apply (simp add:mcompos_surj_surj)
done
constdefs
mr_coset :: "['a, ('a, 'b, 'more) Module_scheme, 'a set] => 'a set"
"mr_coset a M H == a \<uplus>M H"
constdefs
set_mr_cos:: "[('a, 'b, 'more) Module_scheme, 'a set] => 'a set set"
"set_mr_cos M H == {X. ∃a∈carrier M. X = a \<uplus>M H}"
constdefs
mr_cos_sprod ::"[('a, 'b, 'more) Module_scheme, 'a set] =>
'b => 'a set => 'a set"
"mr_cos_sprod M H a X == {z. ∃x∈X. ∃h∈H. z = h ±M (a ·sM x)}"
constdefs
mr_cospOp ::"[('a, 'b, 'more) Module_scheme, 'a set] =>
'a set => 'a set => 'a set"
"mr_cospOp M H == λX. λY. c_top (b_ag M) H X Y"
mr_cosmOp ::"[('a, 'b, 'more) Module_scheme, 'a set] =>
'a set => 'a set"
"mr_cosmOp M H == λX. c_iop (b_ag M) H X"
constdefs
qmodule :: "[('a, 'r, 'more) Module_scheme, 'a set] =>
('a set, 'r) Module"
"qmodule M H == (| carrier = set_mr_cos M H, pop = mr_cospOp M H,
mop = mr_cosmOp M H, zero = H, sprod = mr_cos_sprod M H|)),"
sub_mr_set_cos:: "[('a, 'r, 'more) Module_scheme, 'a set, 'a set] =>
'a set set"
"sub_mr_set_cos M H N == {X. ∃n∈N. X = n \<uplus>M H}"
(* N/H, where N is a submodule *)
syntax
"@QMODULE" :: "[('a, 'r, 'more) Module_scheme, 'a set] =>
('a set, 'r) Module" (infixl "'/'m" 200)
syntax
"@SUBMRSET" ::"['a set, ('a, 'r, 'more) Module_scheme, 'a set] =>
'a set set" ("(3_/ s'/'_/ _)" [82,82,83]82)
translations
"M /m H" == "qmodule M H"
"N s/M H" == "sub_mr_set_cos M H N"
lemma (in Module) qmodule_carr:"submodule R M H ==>
carrier (qmodule M H) = set_mr_cos M H"
apply (simp add:qmodule_def)
done
lemma (in Module) set_mr_cos_mem:"[|submodule R M H; m ∈ carrier M|] ==>
m \<uplus>M H ∈ set_mr_cos M H"
apply (simp add:set_mr_cos_def)
apply blast
done
lemma (in Module) mem_set_mr_cos:"[|submodule R M N; x ∈ set_mr_cos M N|] ==>
∃m ∈ carrier M. x = m \<uplus>M N"
by (simp add:set_mr_cos_def)
lemma (in Module) m_in_mr_coset:"[|submodule R M H; m ∈ carrier M|] ==>
m ∈ m \<uplus>M H"
apply (cut_tac module_is_ag)
apply (frule aGroup.b_ag_group)
apply (simp add:ar_coset_def)
apply (simp add:aGroup.ag_carrier_carrier [THEN sym])
apply (simp add:submodule_def) apply (erule conjE)+
apply (simp add:asubGroup_def)
apply (rule Group.a_in_rcs [of "b_ag M" "H" "m"], assumption+)
done
lemma (in Module) mr_cos_h_stable:"[|submodule R M H; h ∈ H|] ==>
H = h \<uplus>M H"
apply (cut_tac module_is_ag)
apply (frule aGroup.b_ag_group [of "M"])
apply (simp add:ar_coset_def)
apply (rule Group.rcs_Unit2[THEN sym], assumption+,
simp add:submodule_def, (erule conjE)+,
simp add:asubGroup_def)
apply assumption
done
lemma (in Module) mr_cos_h_stable1:"[|submodule R M H; m ∈ carrier M; h ∈ H|]
==> (m ± h) \<uplus>M H = m \<uplus>M H"
apply (cut_tac module_is_ag)
apply (subst aGroup.ag_pOp_commute, assumption+)
apply (simp add:submodule_def, (erule conjE)+, simp add:subsetD)
apply (frule aGroup.b_ag_group [of "M"])
apply (simp add:ar_coset_def)
apply (simp add:aGroup.agop_gop [THEN sym])
apply (simp add:aGroup.ag_carrier_carrier [THEN sym])
apply (simp add:submodule_def, (erule conjE)+, simp add:asubGroup_def)
apply (rule Group.rcs_fixed1 [THEN sym, of "b_ag M" "H" "m" "h"], assumption+)
done
lemma (in Module) x_in_mr_coset:"[|submodule R M H; m ∈ carrier M; x ∈ m \<uplus>M H|]
==> ∃h∈H. m ± h = x"
apply (cut_tac module_is_ag)
apply (frule aGroup.b_ag_group [of "M"])
apply (simp add:submodule_def, (erule conjE)+,
simp add:asubGroup_def)
apply (simp add:aGroup.ag_carrier_carrier [THEN sym])
apply (simp add:aGroup.agop_gop [THEN sym])
apply (simp add:ar_coset_def)
apply (frule Group.rcs_tool2[of "b_ag M" H m x], assumption+,
erule bexE)
apply (frule sym, thin_tac "h ·b_ag M m = x", simp)
apply (simp add:aGroup.agop_gop)
apply (simp add:aGroup.ag_carrier_carrier)
apply (frule_tac c = h in subsetD[of H "carrier M"], assumption+)
apply (subst ag_pOp_commute[of _ m], assumption+)
apply blast
done
lemma (in Module) mr_cos_sprodTr:"[|submodule R M H; a ∈ carrier R;
m ∈ carrier M|] ==> mr_cos_sprod M H a (m \<uplus>M H) = (a ·s m) \<uplus>M H"
apply (cut_tac module_is_ag,
frule aGroup.b_ag_group,
frule sc_mem[of a m], assumption)
apply (simp add:ar_coset_def,
simp add:mr_cos_sprod_def)
apply (simp add:submodule_def, (erule conjE)+)
apply (simp add:aGroup.ag_carrier_carrier [THEN sym],
simp add:aGroup.agop_gop [THEN sym])
apply (simp add:asubGroup_def)
apply (rule equalityI)
apply (rule subsetI, simp)
apply (erule bexE)+
apply (frule_tac x = xa in Group.rcs_tool2[of "b_ag M" H m], assumption+)
apply (erule bexE, rotate_tac -1, frule sym, thin_tac "ha ·b_ag M m = xa",
simp)
apply (simp add:aGroup.agop_gop, simp add:aGroup.ag_carrier_carrier)
apply (frule_tac c = ha in subsetD[of H "carrier M"], assumption+,
simp add:sc_r_distr,
drule_tac a = a in forall_spec1,
drule_tac a = ha in forall_spec, simp,
frule_tac c = "a ·s ha" in subsetD[of H "carrier M"], assumption+,
frule_tac c = h in subsetD[of H "carrier M"], assumption+,
subst ag_pOp_assoc[THEN sym], assumption+)
apply (simp add:aGroup.agop_gop[THEN sym],
simp add:aGroup.ag_carrier_carrier[THEN sym])
apply (frule_tac x = h and y = "a ·s ha" in
Group.sg_mult_closed[of "b_ag M" H], assumption+)
apply (frule_tac a = "a ·s m" and h = "h ·b_ag M (a ·s ha)" in
Group.rcs_fixed1[of "b_ag M" H], assumption+)
apply simp
apply (rule Group.a_in_rcs [of "b_ag M" "H"], assumption+)
apply (simp add:aGroup.agop_gop, simp add:aGroup.ag_carrier_carrier)
apply (rule ag_pOp_closed, simp add:subsetD, assumption)
apply (rule subsetI, simp,
frule_tac x = x in Group.rcs_tool2[of "b_ag M" H "a ·s m"], assumption+,
erule bexE,
rotate_tac -1, frule sym, thin_tac "h ·b_ag M (a ·s m) = x",
frule Group.a_in_rcs[of "b_ag M" H m], assumption+)
apply blast
done
lemma (in Module) mr_cos_sprod_mem:"[|submodule R M H; a ∈ carrier R;
X ∈ set_mr_cos M H|] ==> mr_cos_sprod M H a X ∈ set_mr_cos M H"
apply (simp add:set_mr_cos_def)
apply (erule bexE, rename_tac m, simp)
apply (subst mr_cos_sprodTr, assumption+)
apply (frule_tac m = m in sc_mem [of a], assumption)
apply blast
done
lemma (in Module) mr_cos_sprod_assoc:"[|submodule R M H; a ∈ carrier R;
b ∈ carrier R; X ∈ set_mr_cos M H|] ==> mr_cos_sprod M H (a ·rR b) X =
mr_cos_sprod M H a (mr_cos_sprod M H b X)"
apply (simp add:set_mr_cos_def, erule bexE, simp)
apply (frule_tac m = aa in sc_mem [of b], assumption)
apply (cut_tac sc_Ring,
frule Ring.ring_tOp_closed [of "R" "a" "b"], assumption+)
apply (subst mr_cos_sprodTr, assumption+)+
apply (simp add: sc_assoc)
done
lemma (in Module) mr_cos_sprod_one:"[|submodule R M H; X ∈ set_mr_cos M H|] ==>
mr_cos_sprod M H (1rR) X = X"
apply (simp add:set_mr_cos_def, erule bexE, simp,
thin_tac "X = a \<uplus>M H")
apply (cut_tac sc_Ring,
frule Ring.ring_one[of "R"])
apply (subst mr_cos_sprodTr, assumption+)
apply (simp add:sprod_one)
done
lemma (in Module) mr_cospOpTr:"[|submodule R M H; m ∈ carrier M; n ∈ carrier M|]
==> mr_cospOp M H (m \<uplus>M H) (n \<uplus>M H) = (m ± n) \<uplus>M H"
apply (cut_tac module_is_ag, frule aGroup.b_ag_group)
apply (simp add:mr_cospOp_def mr_coset_def agop_gop [THEN sym])
apply (simp add:ag_carrier_carrier [THEN sym])
apply (simp add:submodule_def, (erule conjE)+,
frule aGroup.asubg_nsubg, assumption+, simp add:ar_coset_def)
apply (simp add:Group.c_top_welldef[THEN sym, of "b_ag M" H m n])
done
lemma(in Module) mr_cos_sprod_distrib1:"[|submodule R M H; a ∈ carrier R;
b ∈ carrier R; X ∈ set_mr_cos M H|] ==>
mr_cos_sprod M H (a ±R b) X =
mr_cospOp M H (mr_cos_sprod M H a X) (mr_cos_sprod M H b X)"
apply (simp add:set_mr_cos_def, erule bexE, rename_tac m)
apply simp
apply (cut_tac sc_Ring,
frule Ring.ring_is_ag[of R])
apply (frule aGroup.ag_pOp_closed [of R a b], assumption+)
apply (subst mr_cos_sprodTr [of H], assumption+)+
apply (subst mr_cospOpTr, assumption+)
apply (simp add:sc_mem, simp add:sc_mem)
apply (simp add:sc_l_distr)
done
lemma (in Module) mr_cos_sprod_distrib2:"[|submodule R M H;
a ∈ carrier R; X ∈ set_mr_cos M H; Y ∈ set_mr_cos M H|] ==>
mr_cos_sprod M H a (mr_cospOp M H X Y) =
mr_cospOp M H (mr_cos_sprod M H a X) (mr_cos_sprod M H a Y)"
apply (simp add:set_mr_cos_def, (erule bexE)+, rename_tac m n, simp,
thin_tac "X = m \<uplus>M H", thin_tac "Y = n \<uplus>M H")
apply (subst mr_cos_sprodTr [of H], assumption+)+
apply (subst mr_cospOpTr, assumption+)
apply (subst mr_cospOpTr, assumption+)
apply (simp add:sc_mem)+
apply (subst mr_cos_sprodTr [of H], assumption+)
apply (rule ag_pOp_closed, assumption+)
apply (simp add:sc_r_distr)
done
lemma (in Module) mr_cosmOpTr:"[|submodule R M H; m ∈ carrier M|] ==>
mr_cosmOp M H (m \<uplus>M H) = (-a m) \<uplus>M H"
apply (simp add:ar_coset_def)
apply (cut_tac module_is_ag)
apply (frule aGroup.b_ag_group)
apply (simp add:ag_carrier_carrier [THEN sym])
apply (simp add:agiop_giop [THEN sym])
apply (simp add:mr_cosmOp_def)
apply (simp add:submodule_def, (erule conjE)+,
frule aGroup.asubg_nsubg[of M H], assumption)
apply (simp add:Group.c_iop_welldef[of "b_ag M" H m])
done
lemma (in Module) mr_cos_oneTr:"submodule R M H ==> H = \<zero> \<uplus>M H"
apply (cut_tac module_is_ag,
cut_tac ag_inc_zero)
apply (simp add:ar_coset_def)
apply (frule aGroup.b_ag_group)
apply (simp add:ag_carrier_carrier [THEN sym])
apply (subst aGroup.agunit_gone[THEN sym, of M], assumption)
apply (subst Group.rcs_Unit1, assumption)
apply (simp add:submodule_def, (erule conjE)+, simp add:asubGroup_def)
apply simp
done
lemma (in Module) mr_cos_oneTr1:"[|submodule R M H; m ∈ carrier M|] ==>
mr_cospOp M H H (m \<uplus>M H) = m \<uplus>M H"
apply (subgoal_tac "mr_cospOp M H (\<zero> \<uplus>M H) (m \<uplus>M H) = m \<uplus>M H")
apply (simp add:mr_cos_oneTr [THEN sym, of H])
apply (subst mr_cospOpTr, assumption+)
apply (simp add:ag_inc_zero)
apply assumption
apply (simp add:ag_l_zero)
done
lemma (in Module) qmodule_is_ag:"submodule R M H ==> aGroup (M /m H)"
apply (cut_tac sc_Ring)
apply (rule aGroup.intro)
apply (simp add:qmodule_def)
apply (rule bivar_func_test)
apply (rule ballI)+
apply (rename_tac X Y)
apply (simp add:set_mr_cos_def, (erule bexE)+, rename_tac n m, simp)
apply (subst mr_cospOpTr, assumption+,
frule_tac x = n and y = m in ag_pOp_closed, assumption+, blast)
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def, (erule bexE)+, rename_tac a b c m n n')
apply (simp add:mr_cospOpTr,
frule_tac x = m and y = n in ag_pOp_closed, assumption+,
frule_tac x = n and y = n' in ag_pOp_closed, assumption+,
simp add:mr_cospOpTr, simp add:ag_pOp_assoc)
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def, (erule bexE)+, rename_tac a b m n, simp)
apply (simp add:mr_cospOpTr,
simp add:ag_pOp_commute)
apply (simp add:qmodule_def,
rule univar_func_test, rule ballI,
simp add:set_mr_cos_def, erule bexE, simp)
apply (subst mr_cosmOpTr, assumption+,
frule_tac x = a in ag_mOp_closed, blast)
apply (simp add:qmodule_def,
simp add:set_mr_cos_def, erule bexE, simp,
simp add:mr_cosmOpTr,
frule_tac x = aa in ag_mOp_closed)
apply (simp add:mr_cospOpTr,
frule_tac x = "-a aa" and y = aa in ag_pOp_closed, assumption+,
simp add:ag_l_inv1, simp add:mr_cos_oneTr[THEN sym])
apply (simp add:qmodule_def,
simp add:set_mr_cos_def,
cut_tac mr_cos_oneTr[of H],
cut_tac ag_inc_zero, blast, assumption)
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def, erule bexE, simp)
apply (subgoal_tac "mr_cospOp M H (\<zero> \<uplus>M H) (aa \<uplus>M H) = aa \<uplus>M H")
apply (simp add:mr_cos_oneTr[THEN sym, of H])
apply (subst mr_cospOpTr, assumption+,
simp add:ag_inc_zero, assumption, simp add:ag_l_zero)
done
lemma (in Module) qmodule_module:"submodule R M H ==> R module (M /m H)"
apply (rule Module.intro)
apply (simp add:qmodule_is_ag)
apply (rule Module_axioms.intro)
apply (cut_tac sc_Ring, simp)
apply (simp add:qmodule_def)
apply (simp add:mr_cos_sprod_mem)
apply (simp add:qmodule_def)
apply (simp add:mr_cos_sprod_distrib1[of H])
apply (simp add:qmodule_def)
apply (simp add:mr_cos_sprod_distrib2[of H])
apply (simp add:qmodule_def)
apply (simp add:mr_cos_sprod_assoc)
apply (simp add:qmodule_def)
apply (simp add:mr_cos_sprod_one)
done
constdefs
indmhom :: "[('b, 'm) Ring_scheme, ('a, 'b, 'm1) Module_scheme,
('c, 'b, 'm2) Module_scheme, 'a => 'c] => 'a set => 'c"
"indmhom R M N f == λX∈ (set_mr_cos M (kerM,N f)). f ( SOME x. x ∈ X)"
syntax
"@INDMHOM"::"['a => 'b, ('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme,
('b, 'r, 'm2) Module_scheme] => ('a set => 'b )"
("(4_\<flat>_ _, _)" [92,92,92,93]92)
translations
"f\<flat>R M,N" == "indmhom R M N f"
lemma (in Module) indmhom_someTr:"[|R module N; f ∈ mHom R M N;
X ∈ set_mr_cos M (kerM,N f)|] ==> f (SOME xa. xa ∈ X) ∈ f `(carrier M)"
apply (simp add:set_mr_cos_def)
apply (erule bexE, simp)
apply (frule mker_submodule [of N f], assumption+)
apply (simp add:submodule_def) apply (erule conjE)+
apply (simp add:asubGroup_def)
apply (thin_tac "∀a m. a ∈ carrier R ∧ m ∈ kerM,N f --> a ·s m ∈ kerM,N f")
apply (cut_tac module_is_ag)
apply (frule aGroup.b_ag_group)
apply (rule someI2_ex)
apply (simp add:ar_coset_def)
apply (frule_tac a = a in Group.a_in_rcs[of "b_ag M" "kerM,N f"],
assumption+, simp add:ag_carrier_carrier [THEN sym], blast)
apply (simp add:ar_coset_def)
apply (frule_tac a = a and x = x in
Group.rcs_subset_elem[of "b_ag M" "kerM,N f"], assumption+)
apply (simp add:ag_carrier_carrier, assumption+)
apply (simp add:image_def,
simp add:ag_carrier_carrier, blast)
done
lemma (in Module) indmhom_someTr1:"[|R module N; f ∈ mHom R M N; m ∈ carrier M|]
==> f (SOME xa. xa ∈ (ar_coset m M (kerM,N f))) = f m"
apply (rule someI2_ex)
apply (frule mker_submodule[of N f], assumption)
apply (frule_tac m_in_mr_coset[of "kerM,N f" m], assumption+,
blast)
apply (frule mker_submodule [of N f], assumption+)
apply (frule_tac x = x in x_in_mr_coset [of "kerM,N f" "m"],
assumption+, erule bexE,
frule sym , thin_tac "m ± h = x", simp)
apply (simp add:ker_def, erule conjE)
apply (subst mHom_add[of N f ], assumption+, simp)
apply (frule Module.module_is_ag [of N R])
apply (frule mHom_mem [of "N" "f" "m"], assumption+)
apply (simp add:aGroup.ag_r_zero)
done
lemma (in Module) indmhom_someTr2:"[|R module N; f ∈ mHom R M N;
submodule R M H; m ∈ carrier M; H ⊆ kerM,N f|] ==>
f (SOME xa. xa ∈ m \<uplus>M H) = f m"
apply (rule someI2_ex)
apply (frule_tac m_in_mr_coset[of "H" m], assumption+, blast)
apply (frule_tac x = x in x_in_mr_coset [of H m],
assumption+, erule bexE,
frule sym , thin_tac "m ± h = x", simp)
apply (frule_tac c = h in subsetD[of H "kerM,N f"], assumption+)
apply (frule mker_submodule [of N f], assumption+,
simp add:submodule_def[of R M "kerM,N f"], (erule conjE)+,
frule_tac c = h in subsetD[of "kerM,N f" "carrier M"], assumption+)
apply (simp add:ker_def mHom_add,
frule_tac m = m in mHom_mem[of "N" "f"], assumption+)
apply (frule Module.module_is_ag[of N R])
apply (simp add:aGroup.ag_r_zero)
done
lemma (in Module) indmhomTr1:"[|R module N; f ∈ mHom R M N; m ∈ carrier M|] ==>
(f\<flat>R M,N) (m \<uplus>M (kerM,N f)) = f m"
apply (simp add:indmhom_def)
apply (subgoal_tac "m \<uplus>M kerM,N f ∈ set_mr_cos M (kerM,N f)", simp)
apply (rule indmhom_someTr1, assumption+)
apply (rule set_mr_cos_mem)
apply (rule mker_submodule, assumption+)
done
lemma (in Module) indmhomTr2:"[|R module N; f ∈ mHom R M N|]
==> (f\<flat>R M,N) ∈ set_mr_cos M (kerM,N f) -> carrier N"
apply (rule univar_func_test)
apply (rule ballI)
apply (simp add:set_mr_cos_def)
apply (erule bexE)
apply (frule_tac m = a in indmhomTr1 [of N f], assumption+)
apply (simp add:mHom_mem)
done
lemma (in Module) indmhom:"[|R module N; f ∈ mHom R M N|]
==> (f\<flat>R M,N) ∈ mHom R (M /m (kerM,N f)) N"
apply (simp add:mHom_def [of R "M /m (kerM,N f)" N])
apply (rule conjI)
apply (simp add:aHom_def)
apply (rule conjI)
apply (simp add:qmodule_def)
apply (simp add:indmhomTr2)
apply (rule conjI)
apply (simp add:qmodule_def indmhom_def extensional_def)
apply (rule ballI)+
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def, (erule bexE)+, simp, rename_tac m n)
apply (frule mker_submodule [of N f], assumption+,
simp add:mr_cospOpTr,
frule_tac x = m and y = n in ag_pOp_closed, assumption+)
apply (simp add:indmhomTr1, simp add:mHom_add)
apply (rule ballI)+
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def, (erule bexE)+, simp)
apply (frule mker_submodule [of N f], assumption+,
subst mr_cos_sprodTr [of "kerM,N f"], assumption+,
frule_tac a = a and m = aa in sc_mem, assumption)
apply (simp add:indmhomTr1)
apply (simp add:mHom_lin)
done
lemma (in Module) indmhom_injec:"[|R module N; f ∈ mHom R M N|] ==>
injec(M /m (kerM,N f)),N (f\<flat>R M,N)"
apply (simp add:injec_def)
apply (frule indmhom [of N f], assumption+)
apply (rule conjI)
apply (simp add:mHom_def)
apply (simp add:ker_def [of _ _ "f\<flat>R M, N"])
apply (simp add:qmodule_def) apply (fold qmodule_def)
apply (rule equalityI)
apply (rule subsetI) apply (simp add:CollectI) apply (erule conjE)
apply (simp add:set_mr_cos_def, erule bexE, simp)
apply (simp add:indmhomTr1)
apply (frule mker_submodule [of N f], assumption+)
apply (rule_tac h1 = a in mr_cos_h_stable [THEN sym, of "kerM,N f"],
assumption+)
apply (simp add:ker_def)
apply (rule subsetI) apply (simp add:CollectI)
apply (rule conjI)
apply (simp add:set_mr_cos_def)
apply (frule mker_submodule [of N f], assumption+)
apply (frule mr_cos_oneTr [of "kerM,N f"])
apply (cut_tac ag_inc_zero)
apply blast
apply (frule mker_submodule [of N f], assumption+)
apply (subst mr_cos_oneTr [of "kerM,N f"], assumption)
apply (cut_tac ag_inc_zero)
apply (subst indmhomTr1, assumption+)
apply (simp add:mHom_0)
done
lemma (in Module) indmhom_surjec1:"[|R module N; surjecM,N f;
f ∈ mHom R M N|] ==> surjec(M /m (kerM,N f)),N (f\<flat>R M,N)"
apply (simp add:surjec_def)
apply (frule indmhom [of N f], assumption+)
apply (rule conjI)
apply (simp add:mHom_def)
apply (rule surj_to_test)
apply (simp add:mHom_def aHom_def)
apply (rule ballI)
apply (erule conjE)
apply (simp add:surj_to_def, frule sym , thin_tac "f ` carrier M = carrier N",
simp,
thin_tac "carrier N = f ` carrier M")
apply (simp add:image_def, erule bexE, simp)
apply (frule_tac m = x in indmhomTr1 [of N f], assumption+)
apply (frule mker_submodule [of N f], assumption+)
apply (simp add:qmodule_carr)
apply (frule_tac m = x in set_mr_cos_mem [of "kerM,N f"], assumption+)
apply blast
done
lemma (in Module) module_homTr:"[|R module N; f ∈ mHom R M N|] ==>
f ∈ mHom R M (mimgR M,N f)"
apply (subst mHom_def, simp add:CollectI)
apply (rule conjI)
apply (simp add:aHom_def)
apply (rule conjI)
apply (simp add:mimg_def mdl_def)
apply (rule univar_func_test, rule ballI)
apply (simp add:image_def, blast)
apply (rule conjI)
apply (simp add:mHom_def aHom_def extensional_def)
apply (rule ballI)+
apply (simp add:mimg_def mdl_def)
apply (simp add:mHom_add)
apply (rule ballI)+
apply (simp add:mimg_def mdl_def)
apply (simp add:mHom_lin)
done
lemma (in Module) ker_to_mimg:"[|R module N; f ∈ mHom R M N|] ==>
kerM,mimgR M,N f f = kerM,N f"
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:ker_def mimg_def mdl_def)
apply (rule subsetI)
apply (simp add:ker_def mimg_def mdl_def)
done
lemma (in Module) module_homTr1:"[|R module N; f ∈ mHom R M N|] ==>
(mimgR (M /m (kerM,N f)),N (f\<flat>R M,N)) = mimgR M,N f" apply (simp add:mimg_def)
apply (subgoal_tac "f\<flat>R M, N ` carrier (M /m (kerM,N f)) = f ` carrier M ",
simp)
apply (simp add:qmodule_def)
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:image_def set_mr_cos_def)
apply (erule exE, erule conjE, erule bexE, simp)
apply (simp add:indmhomTr1, blast)
apply (rule subsetI,
simp add:image_def set_mr_cos_def, erule bexE, simp)
apply (frule_tac m1 = xa in indmhomTr1 [THEN sym, of N f],
assumption+)
apply blast
done
lemma (in Module) module_Homth_1:"[|R module N; f ∈ mHom R M N|] ==>
M /m (kerM,N f) ≅R mimgR M,N f"
apply (frule surjec_to_mimg[of N f], assumption,
frule module_homTr[of N f], assumption,
frule mimg_module[of N f], assumption,
frule indmhom_surjec1[of "mimgR M,N f" f], assumption+,
frule indmhom_injec[of "mimgR M,N f" f], assumption+,
frule indmhom[of "mimgR M,N f" f], assumption+)
apply (simp add:misomorphic_def,
simp add:bijec_def)
apply (simp add:ker_to_mimg)
apply blast
done
constdefs
mpj :: "[('a, 'r, 'm) Module_scheme, 'a set] => ('a => 'a set)"
"mpj M H == λx∈carrier M. x \<uplus>M H"
lemma (in Module) elem_mpj:"[|m ∈ carrier M; submodule R M H|] ==>
mpj M H m = m \<uplus>M H"
by (simp add:mpj_def)
lemma (in Module) mpj_mHom:"submodule R M H ==> mpj M H ∈ mHom R M (M /m H)"
apply (simp add:mHom_def)
apply (rule conjI)
apply (simp add:aHom_def)
apply (rule conjI)
apply (rule univar_func_test) apply (rule ballI)
apply (simp add:mpj_def qmodule_carr)
apply (simp add:set_mr_cos_mem)
apply (rule conjI)
apply (simp add:mpj_def extensional_def)
apply (rule ballI)+
apply (simp add:qmodule_def)
apply (simp add:mpj_def, simp add:ag_pOp_closed)
apply (simp add:mr_cospOpTr)
apply (rule ballI)+
apply (simp add:mpj_def sc_mem)
apply (simp add:qmodule_def)
apply (simp add:mr_cos_sprodTr)
done
lemma (in Module) mpj_mem:"[|submodule R M H; m ∈ carrier M|] ==>
mpj M H m ∈ carrier (M /m H)"
apply (frule mpj_mHom[of H])
apply (rule mHom_mem [of "M /m H" "mpj M H" "m"])
apply (simp add:qmodule_module) apply assumption+
done
lemma (in Module) mpj_surjec:"submodule R M H ==>
surjecM,(M /m H) (mpj M H)"
apply (simp add:surjec_def)
apply (frule mpj_mHom [of H])
apply (rule conjI, simp add:mHom_def)
apply (rule surj_to_test,
simp add:mHom_def aHom_def)
apply (rule ballI)
apply (thin_tac "mpj M H ∈ mHom R M (M /m H)")
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def, erule bexE, simp)
apply (frule_tac m = a in elem_mpj[of _ H], assumption, blast)
done
lemma (in Module) mpj_0:"[|submodule R M H; h ∈ H|] ==>
mpj M H h = \<zero>(M /m H)"
apply (simp add:submodule_def, (erule conjE)+)
apply (frule_tac c = h in subsetD[of H "carrier M"], assumption+)
apply (subst elem_mpj[of _ H], assumption+,
simp add:submodule_def)
apply (simp add:qmodule_def)
apply (rule mr_cos_h_stable[THEN sym],
simp add:submodule_def, assumption)
done
lemma (in Module) mker_of_mpj:"submodule R M H ==>
kerM,(M /m H) (mpj M H) = H"
apply (simp add:ker_def)
apply (rule equalityI)
apply (rule subsetI, simp, erule conjE)
apply (simp add:elem_mpj, simp add:qmodule_def)
apply (frule_tac m = x in m_in_mr_coset [of H], assumption+)
apply simp
apply (rule subsetI)
apply simp
apply (simp add:submodule_def, (erule conjE)+)
apply (simp add:subsetD)
apply (subst elem_mpj,
simp add:subsetD, simp add:submodule_def)
apply (simp add:qmodule_def)
apply (rule mr_cos_h_stable[THEN sym],
simp add:submodule_def, assumption)
done
lemma (in Module) indmhom1:"[|submodule R M H; R module N; f ∈ mHom R M N; H ⊆ kerM,N f|] ==> ∃!g. g ∈ (mHom R (M /m H) N) ∧ (compos M g (mpj M H)) = f"
apply (rule ex_ex1I)
apply (subgoal_tac "(λX∈set_mr_cos M H. f (SOME x. x ∈ X)) ∈ mHom R (M /m H) N ∧ compos M (λX∈set_mr_cos M H. f (SOME x. x ∈ X)) (mpj M H) = f")
apply blast
apply (rule conjI)
apply (rule Module.mHom_test)
apply (simp add:qmodule_module, assumption+)
apply (rule conjI)
apply (rule univar_func_test, rule ballI)
apply (simp add:qmodule_def, simp add:set_mr_cos_def, erule bexE, simp)
apply (simp add:indmhom_someTr2, simp add:mHom_mem)
apply (rule conjI)
apply (simp add:qmodule_def)
apply (rule conjI, (rule ballI)+)
apply (simp add:qmodule_def, simp add:set_mr_cos_def, (erule bexE)+, simp)
apply (simp add:mr_cospOpTr,
frule_tac x = a and y = aa in ag_pOp_closed, assumption+)
apply (simp add:indmhom_someTr2, simp add:mHom_add)
apply (rule impI)
apply (frule_tac b = "a ± aa" in forball_spec1, assumption+, simp)
apply ((rule ballI)+,
simp add:qmodule_def, simp add:set_mr_cos_def, erule bexE, simp,
simp add:mr_cos_sprodTr,
frule_tac a = a and m = aa in sc_mem, assumption)
apply (simp add:indmhom_someTr2, simp add:mHom_lin,
rule impI,
frule_tac b = "a ·s aa" in forball_spec1, assumption, simp)
apply (rule mHom_eq[of N _ f], assumption)
apply (rule Module.mHom_compos[of "M /m H" R M N "mpj M H"
"λX∈set_mr_cos M H. f (SOME x. x ∈ X)"]) apply (
simp add:qmodule_module, rule Module_axioms, assumption,
simp add:mpj_mHom)
apply (rule Module.mHom_test,
simp add:qmodule_module, assumption)
apply (rule conjI,
rule univar_func_test, rule ballI, simp add:qmodule_def,
simp add:set_mr_cos_def, erule bexE, simp add:indmhom_someTr2,
simp add:mHom_mem)
apply (rule conjI,
simp add:qmodule_def)
apply (rule conjI,
(rule ballI)+, simp add:qmodule_def, simp add:set_mr_cos_def,
(erule bexE)+, simp add:mr_cospOpTr,
frule_tac x = a and y = aa in ag_pOp_closed, assumption+,
simp add:indmhom_someTr2 mHom_add,
rule impI,
frule_tac b = "a ± aa" in forball_spec1, assumption, simp)
apply ((rule ballI)+, simp add:qmodule_def set_mr_cos_def, erule bexE, simp,
simp add:mr_cos_sprodTr,
frule_tac a = a and m = aa in sc_mem, assumption,
simp add:indmhom_someTr2 mHom_lin,
rule impI,
frule_tac b = "a ·s aa" in forball_spec1, assumption, simp,
assumption+)
apply (rule ballI, simp add:compos_def compose_def elem_mpj,
simp add:indmhom_someTr2,
rule impI, simp add:set_mr_cos_def,
frule_tac b = m in forball_spec1, assumption, simp)
apply (erule conjE)+
apply (rule_tac f = g and g = y in Module.mHom_eq[of "M /m H" R N],
simp add:qmodule_module, assumption+)
apply (rule ballI, simp add:qmodule_def, fold qmodule_def,
simp add:set_mr_cos_def, erule bexE, simp)
apply (rotate_tac -3, frule sym, thin_tac "compos M y (mpj M H) = f",
simp)
apply (frule_tac f = "compos M g (mpj M H)" and g = "compos M y (mpj M H)"
and x = a in eq_fun_eq_val,
thin_tac "compos M g (mpj M H) = compos M y (mpj M H)")
apply (simp add:compos_def compose_def elem_mpj)
done
constdefs
mQmp :: "[('a, 'r, 'm) Module_scheme, 'a set, 'a set] =>
('a set => 'a set)"
"mQmp M H N == λX∈ set_mr_cos M H. {z. ∃ x ∈ X. ∃ y ∈ N. (y ±M x = z)}"
(* H ⊆ N *)
syntax
"@MQP" :: "[('a, 'b) Module, 'a set, 'a set] => ('a set => 'a set)"
("(3Mp_ _,_)" [82,82,83]82)
translations
"MpM H,N" == "mQmp M H N"
(* "[| R Module M; H ⊆ N |] ==> MpM H,N ∈ rHom (M / m H) (M /m N)" *)
lemma (in Module) mQmpTr0:"[|submodule R M H; submodule R M N; H ⊆ N;
m ∈ carrier M|] ==> mQmp M H N (m \<uplus>M H) = m \<uplus>M N"
apply (frule set_mr_cos_mem [of H m], assumption+)
apply (simp add:mQmp_def)
apply (rule equalityI)
apply (rule subsetI, simp, (erule bexE)+, rotate_tac -1, frule sym,
thin_tac "y ± xa = x", simp)
apply (frule_tac x = xa in x_in_mr_coset[of H m], assumption+, erule bexE,
rotate_tac -1, frule sym, thin_tac "m ± h = xa", simp)
apply (unfold submodule_def, frule conjunct1, rotate_tac 1, frule conjunct1,
fold submodule_def,
frule_tac c = y in subsetD[of N "carrier M"], assumption+,
frule_tac c = h in subsetD[of H "carrier M"], assumption+,
simp add:ag_pOp_assoc[THEN sym],
simp add:ag_pOp_commute[of _ m], simp add:ag_pOp_assoc,
frule_tac c = h in subsetD[of H N], assumption+,
frule_tac h = y and k = h in submodule_pOp_closed[of N], assumption+,
frule_tac h1 = "y ± h" in mr_cos_h_stable1[THEN sym, of N m],
assumption+, simp)
apply (rule m_in_mr_coset, assumption+,
rule ag_pOp_closed, assumption+, simp add:subsetD)
apply (rule subsetI, simp,
frule_tac x = x in x_in_mr_coset[of N m], assumption+,
erule bexE, frule sym, thin_tac "m ± h = x", simp,
simp add:submodule_def[of R M N], frule conjunct1, fold submodule_def,
frule_tac c = h in subsetD[of N "carrier M"], assumption+)
apply (frule_tac m_in_mr_coset[of H m], assumption+,
subst ag_pOp_commute[of m], assumption+)
apply blast
done
(* show mQmp M H N is a welldefined map from M/H to M/N. step2 *)
lemma (in Module) mQmpTr1:"[|submodule R M H; submodule R M N; H ⊆ N;
m ∈ carrier M; n ∈ carrier M; m \<uplus>M H = n \<uplus>M H|] ==> m \<uplus>M N = n \<uplus>M N"
apply (frule_tac m_in_mr_coset [of H m], assumption+)
apply simp
apply (frule_tac x_in_mr_coset [of H n m], assumption+)
apply (erule bexE, rotate_tac -1, frule sym, thin_tac "n ± h = m", simp)
apply (frule_tac c = h in subsetD [of "H" "N"], assumption+)
apply (rule mr_cos_h_stable1[of N n], assumption+)
done
lemma (in Module) mQmpTr2:"[|submodule R M H; submodule R M N; H ⊆ N ;
X ∈ carrier (M /m H)|] ==> (mQmp M H N) X ∈ carrier (M /m N)"
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def)
apply (erule bexE, simp)
apply (frule_tac m = a in mQmpTr0 [of H N], assumption+)
apply blast
done
lemma (in Module) mQmpTr2_1:"[|submodule R M H; submodule R M N; H ⊆ N |]
==> mQmp M H N ∈ carrier (M /m H) -> carrier (M /m N)"
apply (rule univar_func_test, rule ballI)
apply (simp add:mQmpTr2)
done
lemma (in Module) mQmpTr3:"[|submodule R M H; submodule R M N; H ⊆ N ;
X ∈ carrier (M /m H); Y ∈ carrier (M /m H)|] ==> (mQmp M H N) (mr_cospOp M H X Y) = mr_cospOp M N ((mQmp M H N) X) ((mQmp M H N) Y)"
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def)
apply ((erule bexE)+, simp)
apply (simp add:mr_cospOpTr)
apply (frule_tac x = a and y = aa in ag_pOp_closed, assumption+)
apply (subst mQmpTr0, assumption+)+
apply (subst mr_cospOpTr, assumption+)
apply simp
done
lemma (in Module) mQmpTr4:"[|submodule R M H; submodule R M N; H ⊆ N;
a ∈ N|] ==> mr_coset a (mdl M N) H = mr_coset a M H"
apply (simp add:mr_coset_def)
apply (unfold submodule_def[of R M N], frule conjunct1, fold submodule_def,
frule subsetD[of N "carrier M" a], assumption+)
apply (rule equalityI)
apply (rule subsetI)
apply (frule mdl_is_module[of N])
apply (frule_tac x = x in Module.x_in_mr_coset[of "mdl M N" R H a])
apply (simp add:submodule_of_mdl)
apply (simp add:mdl_carrier)
apply assumption+
apply (erule bexE)
apply (unfold submodule_def[of R M H], frule conjunct1, fold submodule_def)
apply (frule_tac c = h in subsetD[of H "carrier M"], assumption+)
apply (thin_tac "x ∈ a \<uplus>mdl M N H", thin_tac "R module mdl M N",
simp add:mdl_def)
apply (frule sym, thin_tac "a ± h = x", simp)
apply (subst mr_cos_h_stable1[THEN sym, of H a], assumption+)
apply (frule_tac x = a and y = h in ag_pOp_closed, assumption+)
apply (rule m_in_mr_coset, assumption+)
apply (rule subsetI)
apply (frule_tac x = x in x_in_mr_coset[of H a], assumption+)
apply (erule bexE, frule sym, thin_tac "a ± h = x", simp)
apply (frule mdl_is_module[of N])
apply (frule submodule_of_mdl[of H N], assumption+)
apply (subst Module.mr_cos_h_stable1[THEN sym, of "mdl M N" R H a],
assumption+, simp add:mdl_carrier, simp)
apply (subgoal_tac "a ± h = a ±mdl M N h", simp)
apply (rule Module.m_in_mr_coset[of "mdl M N" R H], assumption+)
apply (frule Module.module_is_ag[of "mdl M N" R])
apply (rule aGroup.ag_pOp_closed, assumption,
simp add:mdl_carrier, simp add:mdl_carrier subsetD)
apply (subst mdl_def, simp)
done
lemma (in Module) mQmp_mHom:"[|submodule R M H; submodule R M N; H ⊆ N|] ==>
(MpM H,N) ∈ mHom R (M /m H) (M /m N)"
apply (simp add:mHom_def)
apply (rule conjI)
apply (simp add:aHom_def)
apply (simp add:mQmpTr2_1)
apply (rule conjI)
apply (simp add:mQmp_def extensional_def qmodule_def)
apply (rule ballI)+
apply (frule_tac X1 = a and Y1 = b in mQmpTr3 [THEN sym, of H N],
assumption+)
apply (simp add:qmodule_def)
apply (rule ballI)+
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def)
apply (erule bexE, simp)
apply (subst mr_cos_sprodTr, assumption+)
apply (frule_tac a = a and m = aa in sc_mem, assumption)
apply (simp add:mQmpTr0)
apply (subst mr_cos_sprodTr, assumption+)
apply simp
done
lemma (in Module) Mp_surjec:"[|submodule R M H; submodule R M N; H ⊆ N|] ==>
surjec(M /m H),(M /m N) (MpM H,N)"
apply (simp add:surjec_def)
apply (frule mQmp_mHom [of H N], assumption+)
apply (rule conjI)
apply (simp add:mHom_def)
apply (rule surj_to_test)
apply (simp add:mHom_def aHom_def)
apply (rule ballI)
apply (thin_tac "MpM H,N ∈ mHom R (M /m H) (M /m N)")
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def, erule bexE, simp)
apply (frule_tac m = a in mQmpTr0 [of H N], assumption+)
apply blast
done
lemma (in Module) kerQmp:"[|submodule R M H; submodule R M N; H ⊆ N|]
==> ker(M /m H),(M /m N) (MpM H,N) = carrier ((mdl M N) /m H)"
apply (simp add:ker_def)
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:CollectI, erule conjE)
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def [of "mdl M N" "H"])
apply (simp add:set_mr_cos_def)
apply (erule bexE, simp)
apply (simp add:mQmpTr0)
apply (frule_tac m = a in m_in_mr_coset[of N], assumption+, simp)
apply (frule_tac a = a in mQmpTr4[of H N], assumption+,
simp add:mr_coset_def,
rotate_tac -1, frule sym,thin_tac "a \<uplus>mdl M N H = a \<uplus>M H",
simp only:mdl_carrier, blast)
apply (rule subsetI)
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def [of "mdl M N" "H"])
apply (erule bexE, simp)
apply (simp add:mdl_carrier)
apply (frule_tac a = a in mQmpTr4[of H N], assumption+,
simp add:mr_coset_def)
apply (thin_tac "a \<uplus>mdl M N H = a \<uplus>M H")
apply (unfold submodule_def[of R M N], frule conjunct1, fold submodule_def,
frule_tac c = a in subsetD[of N "carrier M"], assumption+)
apply (rule conjI)
apply (simp add:set_mr_cos_def, blast)
apply (simp add:mQmpTr0)
apply (simp add:mr_cos_h_stable [THEN sym])
done
lemma (in Module) misom2Tr:"[|submodule R M H; submodule R M N; H ⊆ N|] ==>
(M /m H) /m (carrier ((mdl M N) /m H)) ≅R (M /m N)"
apply (frule mQmp_mHom [of H N], assumption+)
apply (frule qmodule_module [of H])
apply (frule qmodule_module [of N]) thm Module.indmhom
apply (frule Module.indmhom [of "M /m H" R "M /m N" "MpM H,N"], assumption+)
apply (simp add:kerQmp)
apply (subgoal_tac "bijec((M /m H) /m (carrier((mdl M N) /m H))),(M /m N)
(indmhom R (M /m H) (M /m N) (mQmp M H N))")
apply (simp add:misomorphic_def) apply blast
apply (simp add:bijec_def)
apply (rule conjI)
apply (simp add:kerQmp [THEN sym])
apply (rule Module.indmhom_injec [of "M /m H" R "M /m N" "MpM H,N"], assumption+)
apply (frule Mp_surjec [of H N], assumption+)
apply (simp add:kerQmp [THEN sym])
apply (rule Module.indmhom_surjec1, assumption+)
done
lemma (in Module) eq_class_of_Submodule:"[|submodule R M H; submodule R M N;
H ⊆ N|] ==> carrier ((mdl M N) /m H) = N s/M H"
apply (rule equalityI)
apply (rule subsetI) apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def) apply (erule bexE, simp)
apply (frule_tac a = a in mQmpTr4 [of H N], assumption+)
apply (simp add:mdl_def) apply (simp add:mr_coset_def)
apply (simp add:sub_mr_set_cos_def)
apply (simp add:mdl_carrier, blast)
apply (rule subsetI)
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def)
apply (simp add:sub_mr_set_cos_def)
apply (erule bexE, simp add:mdl_carrier)
apply (frule_tac a1 = n in mQmpTr4[THEN sym, of H N], assumption+)
apply (simp add:mr_coset_def)
apply blast
done
theorem (in Module) misom2:"[|submodule R M H; submodule R M N; H ⊆ N|] ==>
(M /m H) /m (N s/M H) ≅R (M /m N)"
apply (frule misom2Tr [of H N], assumption+)
apply (simp add:eq_class_of_Submodule)
done
consts
natm :: "('a, 'm) aGroup_scheme => nat => 'a => 'a"
primrec
natm_0: "natm M 0 x = \<zero>M"
natm_Suc: "natm M (Suc n) x = (natm M n x) ±M x"
constdefs
finitesum_base::"[('a, 'r, 'm) Module_scheme, 'b set, 'b => 'a set]
=> 'a set "
"finitesum_base M I f == \<Union>{f i | i. i ∈ I}"
constdefs
finitesum ::"[('a, 'r, 'm) Module_scheme, 'b set, 'b => 'a set]
=> 'a set "
"finitesum M I f == {x. ∃n. ∃g. g ∈ {j. j ≤ (n::nat)} -> finitesum_base M I f
∧ x = nsum M g n}"
lemma (in Module) finitesumbase_sub_carrier:"f ∈ I -> {X. submodule R M X} ==>
finitesum_base M I f ⊆ carrier M"
apply (simp add:finitesum_base_def)
apply (rule subsetI)
apply (simp add:CollectI)
apply (erule exE, erule conjE, erule exE, erule conjE)
apply (frule_tac x = i in funcset_mem[of f I "{X. submodule R M X}"],
assumption+, simp)
apply (thin_tac "f ∈ I -> {X. submodule R M X}", unfold submodule_def,
frule conjunct1, fold submodule_def, simp add:subsetD)
done
lemma (in Module) finitesum_sub_carrier:"f ∈ I -> {X. submodule R M X} ==>
finitesum M I f ⊆ carrier M"
apply (rule subsetI, simp add:finitesum_def)
apply ((erule exE)+, erule conjE, simp)
apply (frule finitesumbase_sub_carrier)
apply (rule nsum_mem, rule allI, rule impI)
apply (frule_tac x = j and f = g and A = "{j. j ≤ n}" and
B = "finitesum_base M I f" in funcset_mem, simp)
apply (simp add:subsetD)
done
lemma (in Module) finitesum_inc_zero:"[|f ∈ I -> {X. submodule R M X}; I ≠ {}|]
==> \<zero> ∈ finitesum M I f"
apply (simp add:finitesum_def)
apply (frule nonempty_ex)
apply (subgoal_tac "∀i. i∈I --> (∃n g. g ∈ {j. j ≤ (n::nat)} ->
finitesum_base M I f ∧ \<zero>M = Σe M g n)")
apply blast
apply (rule allI, rule impI)
apply (subgoal_tac "(λx∈{j. j ≤ (0::nat)}. \<zero>) ∈
{j. j ≤ (0::nat)} -> finitesum_base M I f ∧
\<zero>M = Σe M (λx∈{j. j ≤ (0::nat)}. \<zero>) 0")
apply blast
apply (rule conjI)
apply (rule univar_func_test)
apply (rule ballI)
apply (simp add:finitesum_base_def, thin_tac "∃x. x ∈ I")
apply (frule_tac x = i in funcset_mem[of f I "{X. submodule R M X}"],
assumption+)
apply (frule_tac x = i in funcset_mem [of "f" "I" "{X. submodule R M X}"],
assumption+, simp)
apply (frule_tac H = "f i" in submodule_inc_0)
apply blast
apply simp
done
lemma (in Module) finitesum_mOp_closed:
"[|f ∈ I -> {X. submodule R M X}; I ≠ {}; a ∈ finitesum M I f|] ==>
-a a ∈ finitesum M I f"
apply (simp add:finitesum_def)
apply ((erule exE)+, erule conjE)
apply (frule finitesumbase_sub_carrier [of f I])
apply (frule_tac f = g and A = "{j. j ≤ n}" and B = "finitesum_base M I f"
and ?B1.0 = "carrier M" in extend_fun, assumption+)
apply (frule sym, thin_tac "a = Σe M g n")
apply (cut_tac n = n and f = g in nsum_minus,
rule allI, simp add:funcset_mem, simp)
apply (subgoal_tac "(λx∈{j. j ≤ n}. -a (g x)) ∈ {j. j ≤ n} ->
finitesum_base M I f")
apply blast
apply (rule univar_func_test)
apply (rule ballI, simp)
apply (frule_tac f = g and A = "{j. j ≤ n}" and B = "finitesum_base M I f"
and x = x in funcset_mem, simp)
apply (simp add:finitesum_base_def)
apply (erule exE, erule conjE, erule exE, erule conjE)
apply (frule_tac f = f and A = I and B = "{X. submodule R M X}" and
x = i in funcset_mem, assumption+, simp add:CollectI)
apply (thin_tac "f ∈ I -> {X. submodule R M X}")
apply (simp add:submodule_def, (erule conjE)+,
frule_tac H = "f i" and x = "g x" in asubg_mOp_closed, assumption+)
apply blast
done
lemma (in Module) finitesum_pOp_closed:
"[|f ∈ I -> {X. submodule R M X}; a ∈ finitesum M I f; b ∈ finitesum M I f|]
==> a ± b ∈ finitesum M I f"
apply (simp add:finitesum_def)
apply ((erule exE)+, (erule conjE)+)
apply (frule_tac f = g and n = n and A = "finitesum_base M I f" and
g = ga and m = na and B = "finitesum_base M I f" in jointfun_hom0,
assumption+, simp)
apply (cut_tac finitesumbase_sub_carrier[of f I],
cut_tac n1 = n and f1 = g and m1 = na and g1 = ga in
nsum_add_nm[THEN sym], rule allI, rule impI,
frule_tac x = j and f = g and A = "{j. j ≤ n}" and
B = "finitesum_base M I f" in funcset_mem, simp,
simp add:subsetD,
rule allI, rule impI,
frule_tac x = j and f = ga and A = "{j. j ≤ na}" and
B = "finitesum_base M I f" in funcset_mem, simp,
simp add:subsetD)
apply blast
apply assumption
done
lemma (in Module) finitesum_sprodTr:"[|f ∈ I -> {X. submodule R M X}; I ≠ {};
r ∈ carrier R|] ==> g ∈{j. j ≤ (n::nat)} -> (finitesum_base M I f)
--> r ·s (nsum M g n) = nsum M (λx. r ·s (g x)) n"
apply (induct_tac n)
apply (rule impI)
apply simp
apply (rule impI)
apply (frule func_pre) apply simp
apply (frule finitesumbase_sub_carrier [of f I])
apply (frule_tac f = g and A = "{j. j ≤ Suc n}" in extend_fun [of _ _ "finitesum_base M I f" "carrier M"], assumption+)
apply (thin_tac "g ∈ {j. j ≤ Suc n} -> finitesum_base M I f",
thin_tac "g ∈ {j. j ≤ n} -> finitesum_base M I f",
frule func_pre)
apply (cut_tac n = n in nsum_mem [of _ g])
apply (rule allI, simp add:funcset_mem)
apply (frule_tac x = "Suc n" in funcset_mem [of "g" _ "carrier M"], simp)
apply (subst sc_r_distr, assumption+)
apply simp
done
lemma (in Module) finitesum_sprod:"[|f ∈ I -> {X. submodule R M X}; I ≠ {};
r ∈ carrier R; g ∈{j. j ≤ (n::nat)} -> (finitesum_base M I f) |] ==>
r ·s (nsum M g n) = nsum M (λx. r ·s (g x)) n"
apply (simp add:finitesum_sprodTr)
done
lemma (in Module) finitesum_subModule:"[|f ∈ I -> {X. submodule R M X}; I ≠ {}|]
==> submodule R M (finitesum M I f)"
apply (simp add:submodule_def [of _ _ "(finitesum M I f)"])
apply (simp add:finitesum_sub_carrier)
apply (rule conjI)
apply (rule asubg_test)
apply (simp add:finitesum_sub_carrier)
apply (frule finitesum_inc_zero, assumption, blast)
apply (rule ballI)+
apply (rule finitesum_pOp_closed, assumption+,
rule finitesum_mOp_closed, assumption+)
apply ((rule allI)+, rule impI, erule conjE)
apply (simp add:finitesum_def, (erule exE)+, erule conjE, simp)
apply (simp add:finitesum_sprod)
apply (subgoal_tac "(λx. a ·s g x) ∈ {j. j ≤ n} -> finitesum_base M I f",
blast)
apply (rule univar_func_test, rule ballI)
apply (frule_tac x = x and f = g and A = "{j. j ≤ n}" in
funcset_mem[of _ _ "finitesum_base M I f"], assumption+,
thin_tac "g ∈ {j. j ≤ n} -> finitesum_base M I f",
simp add:finitesum_base_def, erule exE, erule conjE, erule exE,
erule conjE, simp)
apply (frule_tac x = i and f = f and A = I in
funcset_mem[of _ _ "{X. submodule R M X}"], assumption+, simp,
frule_tac H = "f i" and a = a and h = "g x" in submodule_sc_closed,
assumption+)
apply blast
done
(*
constdefs
sSum ::"[('a, 'r, 'm1) Module_scheme, 'a set, 'a set] => 'a set"
"sSum M H1 H2 == {x. ∃h1∈H1. ∃h2∈H2. x = h1 ±M h2}"
syntax
"@SSUM":: "['a set, ('a, 'r, 'm1) Module_scheme, 'a set] => 'a set"
("(3_/ ±_/ _)" [60,60,61]60)
translations
"H1 ±M H2" == "sSum M H1 H2" *)
lemma (in Module) sSum_cont_H:"[|submodule R M H; submodule R M K|] ==>
H ⊆ H \<minusplus> K"
apply (rule subsetI)
apply (unfold submodule_def[of R M H], frule conjunct1, fold submodule_def,
unfold submodule_def[of R M K], frule conjunct1, fold submodule_def)
apply (simp add:set_sum)
apply (frule submodule_inc_0 [of K])
apply (cut_tac t = x in ag_r_zero [THEN sym],
rule submodule_subset1, assumption+)
apply blast
done
lemma (in Module) sSum_commute:"[|submodule R M H; submodule R M K|] ==>
H \<minusplus> K = K \<minusplus> H"
apply (unfold submodule_def[of R M H], frule conjunct1, fold submodule_def,
unfold submodule_def[of R M K], frule conjunct1, fold submodule_def)
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:set_sum)
apply ((erule bexE)+, simp)
apply (frule_tac c = h in subsetD[of H "carrier M"], assumption+,
frule_tac c = k in subsetD[of K "carrier M"], assumption+)
apply (subst ag_pOp_commute, assumption+)
apply blast
apply (rule subsetI)
apply (simp add:set_sum)
apply ((erule bexE)+, simp)
apply (frule_tac h = h in submodule_subset1[of K ], assumption+,
frule_tac h = k in submodule_subset1[of H ], assumption+)
apply (subst ag_pOp_commute, assumption+)
apply blast
done
lemma (in Module) Sum_of_SubmodulesTr:"[|submodule R M H; submodule R M K|] ==>
g ∈ {j. j ≤ (n::nat)} -> H ∪ K --> Σe M g n ∈ H \<minusplus> K"
apply (induct_tac n)
apply (rule impI)
apply simp
apply (frule funcset_mem [of "g" "{0}" "H ∪ K" "0"], simp)
apply (frule submodule_subset[of H],
frule submodule_subset[of K])
apply (simp add:set_sum)
apply (erule disjE) thm ag_r_zero[THEN sym]
apply (frule_tac c = "g 0" in subsetD[of H "carrier M"], assumption+,
frule_tac t = "g 0" in ag_r_zero[THEN sym]) apply (
frule submodule_inc_0[of K], blast)
apply (frule_tac c = "g 0" in subsetD[of K "carrier M"], assumption+,
frule_tac t = "g 0" in ag_l_zero[THEN sym]) apply (
frule submodule_inc_0[of H], blast)
apply simp
apply (rule impI, frule func_pre, simp)
apply (frule submodule_subset[of H],
frule submodule_subset[of K])
apply (simp add:set_sum[of H K], (erule bexE)+, simp)
apply (frule_tac x = "Suc n" and f = g and A = "{j. j ≤ Suc n}" and
B = "H ∪ K" in funcset_mem, simp,
thin_tac "g ∈ {j. j ≤ n} -> H ∪ K",
thin_tac "g ∈ {j. j ≤ Suc n} -> H ∪ K",
thin_tac "Σe M g n = h ± k", simp)
apply (erule disjE)
apply (frule_tac h = h in submodule_subset1[of H], assumption,
frule_tac h = "g (Suc n)" in submodule_subset1[of H], assumption,
frule_tac h = k in submodule_subset1[of K], assumption)
apply (subst ag_pOp_assoc, assumption+)
apply (frule_tac x = k and y = "g (Suc n)" in ag_pOp_commute, assumption+,
simp, subst ag_pOp_assoc[THEN sym], assumption+)
apply (frule_tac h = h and k = "g (Suc n)" in submodule_pOp_closed[of H],
assumption+, blast)
apply (frule_tac h = h in submodule_subset1[of H], assumption,
frule_tac h = "g (Suc n)" in submodule_subset1[of K], assumption,
frule_tac h = k in submodule_subset1[of K], assumption)
apply (subst ag_pOp_assoc, assumption+,
frule_tac h = k and k = "g (Suc n)" in submodule_pOp_closed[of K],
assumption+, blast)
done
lemma (in Module) sSum_two_Submodules:"[|submodule R M H; submodule R M K|] ==>
submodule R M (H \<minusplus> K)"
apply (subst submodule_def)
apply (frule submodule_asubg[of H],
frule submodule_asubg[of K])
apply (frule plus_subgs[of H K], assumption, simp add:asubg_subset)
apply (rule allI)+
apply (rule impI, erule conjE, frule asubg_subset[of H],
frule asubg_subset[of K])
apply (simp add:set_sum[of H K], (erule bexE)+, simp)
apply (frule_tac H = H and a = a and h = h in submodule_sc_closed,
assumption+,
frule_tac H = K and a = a and h = k in submodule_sc_closed,
assumption+)
apply (frule_tac c = h in subsetD[of H "carrier M"], assumption+,
frule_tac c = k in subsetD[of K "carrier M"], assumption+,
simp add:sc_r_distr)
apply blast
done
constdefs
iotam::"[('a, 'r, 'm) Module_scheme, 'a set, 'a set] => ('a => 'a)"
("(3ιm_ _,_)" [82, 82, 83]82)
"ιmM H,K == λx∈H. (x ±M \<zero>M)" (** later we define miota. This is not
equal to iotam **)
lemma (in Module) iotam_mHom:"[|submodule R M H; submodule R M K|]
==> ιmM H,K ∈ mHom R (mdl M H) (mdl M (H \<minusplus> K))"
apply (simp add:mHom_def)
apply (rule conjI)
apply (simp add:aHom_def)
apply (simp add:mdl_def)
apply (rule conjI)
apply (rule univar_func_test)
apply (rule ballI)
apply (simp add:iotam_def)
apply (frule submodule_subset[of H], frule submodule_subset[of K],
simp add:set_sum)
apply (frule submodule_inc_0 [of K])
apply blast
apply (rule conjI)
apply (simp add:iotam_def extensional_def mdl_def)
apply (rule ballI)+
apply (simp add:mdl_def iotam_def)
apply (frule_tac h = a and k = b in submodule_pOp_closed [of H],
assumption+, simp)
apply (frule submodule_subset[of H],
frule_tac c = a in subsetD[of H "carrier M"], assumption) apply (
simp add:ag_r_zero)
apply ( frule_tac c = b in subsetD[of H "carrier M"], assumption,
subst ag_pOp_assoc, assumption+,
simp add:ag_inc_zero, simp)
apply (rule ballI)+
apply (simp add:iotam_def mdl_def)
apply (simp add:submodule_sc_closed)
apply (frule submodule_inc_0[of K])
apply (frule submodule_asubg[of H], frule submodule_asubg[of K],
simp add:mem_sum_subgs)
apply (frule_tac a = a and h = m in submodule_sc_closed, assumption+,
frule submodule_subset[of H],
frule_tac c = m in subsetD[of H "carrier M"], assumption+,
frule_tac c = "a ·s m" in subsetD[of H "carrier M"], assumption+)
apply (simp add:ag_r_zero)
done
lemma (in Module) mhomom3Tr:"[|submodule R M H; submodule R M K|] ==>
submodule R (mdl M (H \<minusplus> K)) K"
apply (subst submodule_def)
apply (rule conjI)
apply (simp add:mdl_def)
apply (subst sSum_commute, assumption+)
apply (simp add:sSum_cont_H)
apply (rule conjI)
apply (rule aGroup.asubg_test)
apply (frule sSum_two_Submodules [of H K], assumption+)
apply (frule mdl_is_module [of "(H \<minusplus> K)"])
apply (rule Module.module_is_ag, assumption+)
apply (simp add:mdl_def)
apply (subst sSum_commute, assumption+)
apply (simp add:sSum_cont_H)
apply (frule submodule_inc_0 [of K])
apply (simp add:nonempty)
apply (rule ballI)+
apply (simp add:mdl_def)
apply (rule submodule_pOp_closed, assumption+)
apply (rule submodule_mOp_closed, assumption+)
apply ((rule allI)+, rule impI)
apply (simp add:mdl_def, erule conjE)
apply (frule sSum_cont_H[of K H], assumption,
simp add:sSum_commute[of K H])
apply (simp add:subsetD submodule_sc_closed)
done
lemma (in Module) mhomom3Tr0:"[|submodule R M H; submodule R M K|]
==> compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (ιmM H,K)
∈ mHom R (mdl M H) (mdl M (H \<minusplus> K) /m K)"
apply (frule mdl_is_module [of H])
apply (frule mhomom3Tr[of H K], assumption+)
apply (frule sSum_two_Submodules [of H K], assumption+)
apply (frule mdl_is_module [of "H \<minusplus> K"])
apply (frule iotam_mHom [of H K], assumption+) thm Module.mpj_mHom
apply (frule Module.mpj_mHom [of "mdl M (H \<minusplus> K)" R "K"], assumption+)
apply (rule Module.mHom_compos[of "mdl M (H \<minusplus> K)" R "mdl M H"], assumption+)
apply (simp add:Module.qmodule_module, assumption)
apply (simp add:mpj_mHom)
done
lemma (in Module) mhomom3Tr1:"[|submodule R M H; submodule R M K|] ==>
surjec(mdl M H),((mdl M (H \<minusplus> K))/m K)
(compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (ιmM H,K))"
apply (simp add:surjec_def)
apply (frule mhomom3Tr0 [of H K], assumption+)
apply (rule conjI)
apply (simp add:mHom_def)
apply (rule surj_to_test)
apply (simp add:mHom_def aHom_def)
apply (rule ballI)
apply (simp add:compos_def compose_def)
apply (thin_tac "(λx∈carrier (mdl M H). mpj (mdl M (H \<minusplus> K)) K ((ιmM H,K) x))
∈ mHom R (mdl M H) (mdl M (H \<minusplus> K) /m K)")
apply (simp add:qmodule_def)
apply (simp add:set_mr_cos_def)
apply (erule bexE, simp)
apply (simp add:mdl_carrier)
apply (simp add:iotam_def)
apply (simp add:mpj_def)
apply (frule sSum_two_Submodules[of H K], assumption+)
apply (simp add:mdl_carrier)
apply (subgoal_tac "∀aa∈H. aa ± \<zero> ∈ H \<minusplus> K", simp)
apply (frule submodule_subset[of H], frule submodule_subset[of K],
thin_tac "∀aa∈H. aa ± \<zero> ∈ H \<minusplus> K",
simp add:set_sum, (erule bexE)+)
apply (simp add:set_sum[THEN sym])
apply (frule mdl_is_module[of "H \<minusplus> K"],
frule mhomom3Tr[of H K], assumption+)
apply (frule_tac m = h and h = k in Module.mr_cos_h_stable1[of "mdl M (H \<minusplus> K)"
R K], assumption+)
apply (simp add:mdl_carrier)
apply (frule sSum_cont_H[of H K], assumption+, simp add:subsetD, assumption)
apply (simp add:mdl_def, fold mdl_def)
apply (subgoal_tac "∀a∈H. a ± \<zero> = a", simp, blast)
apply (rule ballI)
apply (frule_tac c = aa in subsetD[of H "carrier M"], assumption+,
simp add:ag_r_zero)
apply (rule ballI)
apply (frule submodule_inc_0[of K])
apply (rule mem_sum_subgs,
simp add:submodule_def, simp add:submodule_def, assumption+)
done
lemma (in Module) mhomom3Tr2:"[|submodule R M H; submodule R M K|] ==>
ker(mdl M H),((mdl M (H \<minusplus> K)) /m K)
(compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (ιmM H,K)) = H ∩ K"
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:ker_def, erule conjE)
apply (simp add:qmodule_def)
apply (simp add:mdl_carrier)
apply (simp add:compos_def compose_def mdl_def iotam_def)
apply (fold mdl_def)
apply (simp add:iotam_def mpj_def)
apply (frule sSum_two_Submodules[of H K], assumption+, simp add:mdl_carrier)
apply (frule submodule_asubg[of H], frule submodule_asubg[of K])
apply (frule_tac h = x and k = \<zero> in mem_sum_subgs[of H K], assumption+)
apply (simp add:submodule_inc_0)
apply simp apply (frule mhomom3Tr[of H K], assumption+)
(*thm Module.m_in_mr_coset[of "mdl M (H \<minusplus> K)" R K]
apply (frule_tac m = "x ± \<zero>" in Module.m_in_mr_coset[of "mdl M (H \<minusplus> K)" R K])*)
apply (frule sSum_two_Submodules[of H K], assumption,
frule mdl_is_module [of "H \<minusplus> K"])
apply (frule_tac m = "x ± \<zero>" in Module.m_in_mr_coset[of "mdl M (H \<minusplus> K)" R K],
assumption+)
apply (simp add:mdl_carrier, simp)
apply (frule submodule_subset[of H],
frule_tac c = x in subsetD[of H "carrier M"], assumption+)
apply (simp add:ag_r_zero)
apply (rule subsetI)
apply (simp add:ker_def)
apply (simp add:mdl_carrier)
apply (simp add:qmodule_def)
apply (simp add:compos_def compose_def)
apply (simp add:mdl_carrier)
apply (simp add:iotam_def mpj_def)
apply (frule sSum_two_Submodules[of H K], assumption+)
apply (simp add:mdl_carrier)
apply (erule conjE,
frule submodule_inc_0[of K],
frule submodule_asubg[of H], frule submodule_asubg[of K],
simp add:mem_sum_subgs)
apply (frule submodule_subset[of K]) apply (
frule_tac c = x in subsetD[of K "carrier M"], assumption+)
apply (simp add:ag_r_zero,
frule mdl_is_module [of "H \<minusplus> K"],
frule mhomom3Tr[of H K], assumption+)
apply (frule_tac h1 = x in Module.mr_cos_h_stable[THEN sym, of "mdl M (H \<minusplus> K)"
R K], assumption+)
done
lemma (in Module) mhomom_3:"[|submodule R M H; submodule R M K|] ==>
(mdl M H) /m (H ∩ K) ≅R (mdl M (H \<minusplus> K)) /m K"
apply (frule sSum_two_Submodules [of H K], assumption+)
apply (frule mdl_is_module [of H])
apply (frule mdl_is_module [of K])
apply (frule mdl_is_module [of "H \<minusplus> K"])
apply (frule mhomom3Tr [of H K], assumption+)
apply (frule Module.qmodule_module [of "mdl M (H \<minusplus> K)" R K], assumption+)
apply (simp add:misomorphic_def)
apply (frule mhomom3Tr0[of H K], assumption+)
apply (frule mhomom3Tr1[of H K], assumption+)
apply (frule Module.indmhom [of "mdl M H" R "mdl M (H \<minusplus> K) /m K" "compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (ιmM H,K)"], assumption+)
apply (frule Module.indmhom_injec[of "mdl M H" R "mdl M (H \<minusplus> K) /m K"
"compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (ιmM H,K)"], assumption+)
apply (frule Module.indmhom_surjec1[of "mdl M H" R "mdl M (H \<minusplus> K) /m K" "compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (ιmM H,K)"], assumption+)
apply (simp add:bijec_def)
apply (simp add:mhomom3Tr2[of H K])
apply blast
done
constdefs
l_comb::"[('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme, nat] => (nat => 'r) => (nat => 'a) => 'a"
"l_comb R M n s m == nsum M (λj. (s j) ·sM (m j)) n"
linear_span::"[('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme, 'r set,
'a set] => 'a set"
"linear_span R M A H == if H = {} then {\<zero>M} else
{x. ∃n. ∃f ∈ {j. j ≤ (n::nat)} -> H.
∃s∈{j. j ≤ (n::nat)} -> A. x = l_comb R M n s f}"
coefficient::"[('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme,
nat, nat => 'r, nat => 'a] => nat => 'r"
"coefficient R M n s m j == s j"
body::"[('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme, nat, nat => 'r,
nat => 'a] => nat => 'a"
"body R M n s m j == m j"
lemma (in Module) l_comb_mem_linear_span:"[|ideal R A; H ⊆ carrier M;
s ∈ {j. j ≤ (n::nat)} -> A; f ∈ {j. j ≤ n} -> H|] ==>
l_comb R M n s f ∈ linear_span R M A H"
apply (frule_tac x = 0 in funcset_mem[of f "{j. j ≤ n}" H], simp)
apply (frule nonempty[of "f 0" H])
apply (simp add:linear_span_def)
apply blast
done
lemma (in Module) linear_comb_eqTr:"H ⊆ carrier M ==>
s ∈ {j. j ≤ (n::nat)} -> carrier R ∧
f ∈ {j. j ≤ n} -> H ∧
g ∈ {j. j ≤ n} -> H ∧
(∀j∈{j. j ≤ n}. f j = g j) -->
l_comb R M n s f = l_comb R M n s g"
apply (induct_tac n)
apply (rule impI) apply (erule conjE)+ apply (simp add:l_comb_def)
apply (rule impI) apply (erule conjE)+
apply (frule_tac f = s in func_pre)
apply (frule_tac f = f in func_pre)
apply (frule_tac f = g in func_pre)
apply (cut_tac n = n in Nsetn_sub_mem1, simp)
apply (thin_tac "s ∈ {j. j ≤ n} -> carrier R",
thin_tac "f ∈ {j. j ≤ n} -> H",
thin_tac "g ∈ {j. j ≤ n} -> H")
apply (simp add:l_comb_def)
done
lemma (in Module) linear_comb_eq:"[|H ⊆ carrier M;
s ∈ {j. j ≤ (n::nat)} -> carrier R; f ∈ {j. j ≤ n} -> H;
g ∈ {j. j ≤ n} -> H; ∀j∈{j. j ≤ n}. f j = g j|] ==>
l_comb R M n s f = l_comb R M n s g"
apply (simp add:linear_comb_eqTr)
done
lemma (in Module) l_comb_Suc:"[|H ⊆ carrier M; ideal R A;
s ∈ {j. j ≤ (Suc n)} -> carrier R; f ∈ {j. j ≤ (Suc n)} -> H|] ==>
l_comb R M (Suc n) s f = l_comb R M n s f ± s (Suc n) ·s f (Suc n)"
apply (simp add:l_comb_def)
done
lemma (in Module) l_comb_jointfun_jj:"[|H ⊆ carrier M; ideal R A;
s ∈ {j. j ≤ (n::nat)} -> A; f ∈ {j. j ≤ (n::nat)} -> H;
t ∈ {j. j ≤ (m::nat)} -> A; g ∈ {j. j ≤ (m::nat)} -> H|] ==>
nsum M (λj. (jointfun n s m t) j ·s (jointfun n f m g) j) n =
nsum M (λj. s j ·s f j) n"
apply (cut_tac sc_Ring)
apply (rule nsum_eq)
apply (rule allI, rule impI, simp add:jointfun_def,
rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI,
rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, simp add:jointfun_def)
done
lemma (in Module) l_comb_jointfun_jj1:"[|H ⊆ carrier M; ideal R A;
s ∈ {j. j ≤ (n::nat)} -> A; f ∈ {j. j ≤ (n::nat)} -> H;
t ∈ {j. j ≤ (m::nat)} -> A; g ∈ {j. j ≤ (m::nat)} -> H|] ==>
l_comb R M n (jointfun n s m t) (jointfun n f m g) =
l_comb R M n s f"
by (simp add:l_comb_def, simp add:l_comb_jointfun_jj)
lemma (in Module) l_comb_jointfun_jf:"[|H ⊆ carrier M; ideal R A;
s ∈ {j. j ≤ (n::nat)} -> A; f ∈ {j. j ≤ Suc (n + m)} -> H;
t ∈ {j. j ≤ (m::nat)} -> A|] ==>
nsum M (λj. (jointfun n s m t) j ·s f j) n =
nsum M (λj. s j ·s f j) n"
apply (cut_tac sc_Ring)
apply (rule nsum_eq)
apply (rule allI, rule impI, simp add:jointfun_def,
rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI,
rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, simp add:jointfun_def)
done
lemma (in Module) l_comb_jointfun_jf1:"[|H ⊆ carrier M; ideal R A;
s ∈ {j. j ≤ (n::nat)} -> A; f ∈ {j. j ≤ Suc (n + m)} -> H;
t ∈ {j. j ≤ (m::nat)} -> A|] ==>
l_comb R M n (jointfun n s m t) f = l_comb R M n s f"
by (simp add:l_comb_def l_comb_jointfun_jf)
lemma (in Module) l_comb_jointfun_fj:"[|H ⊆ carrier M; ideal R A;
s ∈ {j. j ≤ Suc (n + m)} -> A; f ∈ {j. j ≤ (n::nat)} -> H;
g ∈ {j. j ≤ (m::nat)} -> H|] ==>
nsum M (λj. s j ·s (jointfun n f m g) j) n =
nsum M (λj. s j ·s f j) n"
apply (cut_tac sc_Ring)
apply (rule nsum_eq)
apply (rule allI, rule impI, simp add:jointfun_def,
rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI,
rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, simp add:jointfun_def)
done
lemma (in Module) l_comb_jointfun_fj1:"[|H ⊆ carrier M; ideal R A;
s ∈ {j. j ≤ Suc (n + m)} -> A; f ∈ {j. j ≤ (n::nat)} -> H;
g ∈ {j. j ≤ (m::nat)} -> H|] ==>
l_comb R M n s (jointfun n f m g) = l_comb R M n s f"
by (simp add:l_comb_def l_comb_jointfun_fj)
lemma (in Module) linear_comb0_1Tr:"H ⊆ carrier M ==>
s ∈ {j. j ≤ (n::nat)} -> {\<zero>R} ∧
m ∈ {j. j ≤ n} -> H --> l_comb R M n s m = \<zero>M"
apply (induct_tac n)
apply (rule impI) apply (erule conjE)
apply (simp add:l_comb_def)
apply (frule_tac x = 0 and f = s in funcset_mem[of _ "{0}" "{\<zero>R}"], simp+,
frule_tac x = 0 and f = m in funcset_mem[of _ "{0}" H], simp+,
frule_tac c = "m 0" in subsetD[of H "carrier M"], assumption+,
simp add:sc_0_m)
apply (rule impI) apply (erule conjE)
apply (frule func_pre [of _ _ "{\<zero>R}"])
apply (frule func_pre [of _ _ "H"])
apply simp
apply (thin_tac "s ∈ {j. j ≤ n} -> {\<zero>R}",
thin_tac "m ∈ {j. j ≤ n} -> H")
apply (simp add:l_comb_def)
apply (frule_tac x = "Suc n" and f = s and A = "{j. j ≤ Suc n}" in
funcset_mem[of _ _ "{\<zero>R}"], simp, simp,
frule_tac x = "Suc n" and f = m and A = "{j. j ≤ Suc n}" in
funcset_mem[of _ _ H], simp,
frule_tac c = "m (Suc n)" in subsetD[of H "carrier M"], assumption+,
simp add:sc_0_m)
apply (cut_tac ag_inc_zero)
apply (simp add:ag_l_zero)
done
lemma (in Module) linear_comb0_1:"[|H ⊆ carrier M;
s ∈ {j. j ≤ (n::nat)} -> {\<zero>R}; m ∈ {j. j ≤ n} -> H |] ==>
l_comb R M n s m = \<zero>M"
apply (simp add:linear_comb0_1Tr)
done
lemma (in Module) linear_comb0_2Tr:"ideal R A ==> s ∈ {j. j ≤ (n::nat)} -> A
∧ m ∈ {j. j ≤ n} -> {\<zero>M} --> l_comb R M n s m = \<zero>M"
apply (induct_tac n )
apply (rule impI) apply (erule conjE)
apply (simp add:l_comb_def)
apply (frule funcset_mem [of "m" "{0}" "{\<zero>M}" "0"], simp+,
frule funcset_mem [of "s" "{0}" A "0"], simp+,
cut_tac sc_Ring,
frule_tac h = "s 0" in Ring.ideal_subset, assumption+,
rule sc_a_0, assumption+)
apply (rule impI)
apply (erule conjE)+
apply (frule func_pre [of "s"],
frule func_pre [of "m"], simp)
apply (thin_tac "s ∈ {j. j ≤ n} -> A",
thin_tac "m ∈ {j. j ≤ n} -> {\<zero>}")
apply (simp add:l_comb_def)
apply (frule_tac A = "{j. j ≤ Suc n}" and x = "Suc n" in
funcset_mem [of "m" _ "{\<zero>}"], simp+,
frule_tac A = "{j. j ≤ Suc n}" and x = "Suc n" in
funcset_mem[of s _ A], simp+,
cut_tac sc_Ring,
frule_tac h = "s (Suc n)" in Ring.ideal_subset[of R A], assumption+)
apply (cut_tac ag_inc_zero, simp add:sc_a_0)
apply (simp add:ag_l_zero)
done
lemma (in Module) linear_comb0_2:"[|ideal R A; s ∈ {j. j ≤ (n::nat)} -> A;
m ∈ {j. j ≤ n} -> {\<zero>M} |] ==> l_comb R M n s m = \<zero>M"
apply (simp add:linear_comb0_2Tr)
done
lemma (in Module) liear_comb_memTr:"[|ideal R A; H ⊆ carrier M|] ==>
∀s. ∀m. s ∈ {j. j ≤ (n::nat)} -> A ∧
m ∈ {j. j ≤ n} -> H --> l_comb R M n s m ∈ carrier M"
apply (induct_tac n)
apply (rule allI)+ apply (rule impI) apply (erule conjE)
apply (simp add:l_comb_def)
apply (rule sc_mem)
apply (frule_tac x = 0 and f = s in funcset_mem[of _ "{0}" A], simp,
cut_tac sc_Ring,
rule_tac h = "s 0" in Ring.ideal_subset[of R A], assumption+,
frule_tac x = 0 and f = m in funcset_mem[of _ "{0}" H], simp,
simp add:subsetD)
apply (rule allI)+ apply (rule impI) apply (erule conjE)
apply (frule func_pre [of _ _ "A"],
frule func_pre [of _ _ "H"],
drule_tac a = s in forall_spec1,
drule_tac a = m in forall_spec1)
apply (simp add:l_comb_def)
apply (rule ag_pOp_closed, assumption+)
apply (rule sc_mem)
apply (cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset subsetD)
apply (simp add:funcset_mem subsetD)
done
lemma (in Module) l_comb_mem:"[|ideal R A; H ⊆ carrier M;
s ∈ {j. j ≤ (n::nat)} -> A; m ∈ {j. j ≤ n} -> H|] ==>
l_comb R M n s m ∈ carrier M"
apply (simp add:liear_comb_memTr)
done
lemma (in Module) l_comb_transpos:" [|ideal R A; H ⊆ carrier M;
s ∈ {l. l ≤ Suc n} -> A; f ∈ {l. l ≤ Suc n} -> H;
j < Suc n |] ==>
Σe M (cmp (λk. s k ·s f k) (transpos j (Suc n))) (Suc n) =
Σe M (λk. (cmp s (transpos j (Suc n))) k ·s
(cmp f (transpos j (Suc n))) k) (Suc n)"
apply (cut_tac sc_Ring)
apply (rule nsum_eq)
apply (rule allI, rule impI, simp add:cmp_def)
apply (cut_tac l = ja in transpos_mem[of j "Suc n" "Suc n"],
simp add:less_imp_le, simp, simp, assumption)
apply (rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI, simp add:cmp_def)
apply (frule less_imp_le[of j "Suc n"],
frule_tac l = ja in transpos_mem[of j "Suc n" "Suc n"], simp,
simp, assumption+)
apply (rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI,
simp add:cmp_def)
done
lemma (in Module) l_comb_transpos1:" [|ideal R A; H ⊆ carrier M;
s ∈ {l. l ≤ Suc n} -> A; f ∈ {l. l ≤ Suc n} -> H; j < Suc n |] ==>
l_comb R M (Suc n) s f =
l_comb R M (Suc n) (cmp s (transpos j (Suc n))) (cmp f (transpos j (Suc n)))"
apply (cut_tac sc_Ring)
apply (frule l_comb_transpos[THEN sym, of A H s n f j], assumption+)
apply (simp del:nsum_suc add:l_comb_def,
thin_tac "Σe M (λk. (cmp s (transpos j (Suc n))) k ·s
(cmp f (transpos j (Suc n))) k) (Suc n) =
Σe M (cmp (λk. s k ·s f k) (transpos j (Suc n))) (Suc n)")
apply (cut_tac addition2[of "λj. s j ·s f j" n "transpos j (Suc n)"],
simp)
apply (rule univar_func_test, rule ballI, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule_tac i = j and n = "Suc n" and j = "Suc n" in transpos_hom,
simp add:less_imp_le, simp, simp)
apply (rule_tac i = j and n = "Suc n" and j = "Suc n" in transpos_inj,
simp add:less_imp_le, simp, simp)
done
lemma (in Module) sc_linear_span:"[|ideal R A; H ⊆ carrier M; a ∈ A;
h ∈ H|] ==> a ·s h ∈ linear_span R M A H"
apply (simp add:linear_span_def)
apply (simp add:nonempty)
apply (simp add:l_comb_def)
apply (subgoal_tac "(λk∈{j. j ≤ (0::nat)}. a) ∈{j. j ≤ 0} -> A")
apply (subgoal_tac "(λk∈{j. j ≤ 0}. h) ∈ {j. j ≤ (0::nat)} -> H")
apply (subgoal_tac "a ·s h =
Σe M (λj. (λk∈{j. j ≤ (0::nat)}. a) j ·s (λk∈{j. j ≤ (0::nat)}. h) j) 0")
apply blast
apply simp
apply (rule univar_func_test, rule ballI, simp)
apply (rule univar_func_test, rule ballI, simp)
done
lemma (in Module) l_span_cont_H:"H ⊆ carrier M ==>
H ⊆ linear_span R M (carrier R) H"
apply (rule subsetI)
apply (cut_tac sc_Ring,
cut_tac Ring.whole_ideal[of R])
apply (frule_tac A = "carrier R" and H = H and a = "1rR"
and h = x in sc_linear_span, assumption+)
apply (simp add:Ring.ring_one, assumption+)
apply (frule_tac c = x in subsetD[of H "carrier M"], assumption+,
simp add:sprod_one, assumption)
done
lemma (in Module) linear_span_inc_0:"[|ideal R A; H ⊆ carrier M|] ==>
\<zero> ∈ linear_span R M A H"
apply (case_tac "H = {}")
apply (simp add:linear_span_def)
apply (frule nonempty_ex[of H], erule exE)
apply (frule_tac h = x in sc_linear_span[of A H "\<zero>R"], assumption)
apply (cut_tac sc_Ring, simp add:Ring.ideal_zero, assumption)
apply (frule_tac c = x in subsetD[of H "carrier M"], assumption,
simp add:sc_0_m)
done
lemma (in Module) linear_span_iOp_closedTr1:"[|ideal R A;
s ∈ {j. j ≤ (n::nat)} -> A|] ==>
(λx∈{j. j ≤ n}. -aR (s x)) ∈ {j. j ≤ n} -> A"
apply (rule univar_func_test) apply (rule ballI)
apply simp
apply (cut_tac sc_Ring,
rule Ring.ideal_inv1_closed, assumption+)
apply (simp add:funcset_mem)
done
lemma (in Module) l_span_gen_mono:"[|K ⊆ H; H ⊆ carrier M; ideal R A|] ==>
linear_span R M A K ⊆ linear_span R M A H"
apply (rule subsetI)
apply (case_tac "K = {}", simp add:linear_span_def[of _ _ _ "{}"],
simp add:linear_span_inc_0)
apply (frule nonempty_ex[of K], erule exE,
frule_tac c = xa in subsetD[of K H], assumption+,
frule nonempty[of _ H])
apply (simp add:linear_span_def[of _ _ _ K],
erule exE, (erule bexE)+, simp,
frule extend_fun[of _ _ K H], assumption+)
apply (simp add: l_comb_mem_linear_span)
done
lemma (in Module) l_comb_add:"[|ideal R A; H ⊆ carrier M;
s ∈ {j. j ≤ (n::nat)} -> A; f ∈ {j. j ≤ n} -> H;
t ∈ {j. j ≤ (m::nat)} -> A; g ∈ {j. j ≤ m} -> H|] ==>
l_comb R M (Suc (n + m)) (jointfun n s m t) (jointfun n f m g) =
l_comb R M n s f ± l_comb R M m t g"
apply (cut_tac sc_Ring)
apply (simp del:nsum_suc add:l_comb_def)
apply (subst nsum_split)
apply (rule allI, rule impI)
apply (case_tac "j ≤ n", simp add:jointfun_def,
rule sc_mem, simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (simp add:jointfun_def sliden_def)
apply (frule_tac m = j and n = "Suc (n + m)" and l = "Suc n" in diff_le_mono,
thin_tac "j ≤ Suc (n + m)", simp,
rule sc_mem, simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (simp add:l_comb_jointfun_jj[of H A s n f t m g])
apply (cut_tac nsum_eq[of m "cmp (λj. jointfun n s m t j ·s
jointfun n f m g j) (slide (Suc n))" "λj. t j ·s g j"], simp)
apply (rule allI, rule impI, simp add:cmp_def,
simp add:jointfun_def sliden_def slide_def,
rule sc_mem, simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI,
rule sc_mem, simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, simp add:cmp_def jointfun_def sliden_def slide_def)
done
lemma (in Module) l_comb_add1Tr:"[|ideal R A; H ⊆ carrier M|] ==>
f ∈ {j. j ≤ (n::nat)} -> H ∧ s ∈ {j. j ≤ n} -> A ∧ t ∈ {j. j ≤ n} -> A -->
l_comb R M n (λx∈{j. j ≤ n}. (s x) ±R (t x)) f =
l_comb R M n s f ± l_comb R M n t f"
apply (induct_tac n)
apply (simp add:l_comb_def)
apply (rule impI, (erule conjE)+)
apply (frule_tac x = 0 in funcset_mem[of s "{0}" A], simp,
frule_tac x = 0 in funcset_mem[of t "{0}" A], simp,
frule_tac x = 0 in funcset_mem[of f "{0}" H], simp,
cut_tac sc_Ring,
frule_tac h = "s 0" in Ring.ideal_subset, assumption+,
frule_tac h = "t 0" in Ring.ideal_subset, assumption+,
frule_tac c = "f 0" in subsetD[of H "carrier M"], assumption+)
apply (simp add:sc_l_distr)
apply (rule impI, (erule conjE)+)
apply (frule func_pre[of f], frule func_pre[of s], frule func_pre[of t],
simp)
apply (simp add:l_comb_def, cut_tac sc_Ring)
apply (cut_tac n = n and f = "λj. (if j ≤ n then s j ±R t j else arbitrary) ·s f j" and g = "λj. (if j ≤ Suc n then s j ±R t j else arbitrary) ·s
f j" in nsum_eq)
apply (rule allI, rule impI, simp,
rule sc_mem, frule Ring.ring_is_ag,
rule aGroup.ag_pOp_closed[of R], assumption,
simp add:funcset_mem[of s _ A] Ring.ideal_subset,
simp add:funcset_mem[of t _ A] Ring.ideal_subset,
simp add:funcset_mem[of f _ H] subsetD)
apply (rule allI, rule impI, simp,
rule sc_mem, frule Ring.ring_is_ag,
rule aGroup.ag_pOp_closed[of R], assumption,
simp add:funcset_mem[of s _ A] Ring.ideal_subset,
simp add:funcset_mem[of t _ A] Ring.ideal_subset,
simp add:funcset_mem[of f _ H] subsetD)
apply (rule allI, simp)
apply simp
apply (thin_tac "Σe M (λj. (if j ≤ n then s j ±R t j else arbitrary) ·s f j)
n = Σe M (λj. s j ·s f j) n ± Σe M (λj. t j ·s f j) n",
thin_tac "Σe M (λj. (if j ≤ Suc n then s j ±R t j else arbitrary) ·s
f j) n = Σe M (λj. s j ·s f j) n ± Σe M (λj. t j ·s f j) n")
apply (frule_tac x = "Suc n" and A = "{j. j ≤ Suc n}" in
funcset_mem[of s _ A], simp,
frule_tac x = "Suc n" and A = "{j. j ≤ Suc n}" in
funcset_mem[of t _ A], simp,
frule_tac x = "Suc n" and A = "{j. j ≤ Suc n}" in
funcset_mem[of f _ H], simp,
cut_tac sc_Ring,
frule_tac h = "s (Suc n)" in Ring.ideal_subset, assumption+,
frule_tac h = "t (Suc n)" in Ring.ideal_subset, assumption+,
frule_tac c = "f (Suc n)" in subsetD[of H "carrier M"], assumption+)
apply (simp add:sc_l_distr)
apply (cut_tac n = n and f = "λj. s j ·s f j" in nsum_mem,
rule allI, rule impI, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (cut_tac n = n and f = "λj. t j ·s f j" in nsum_mem,
rule allI, rule impI, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (cut_tac a = "s (Suc n)" and m = "f (Suc n)" in sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (cut_tac a = "t (Suc n)" and m = "f (Suc n)" in sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (subst pOp_assocTr41[THEN sym], assumption+,
subst pOp_assocTr42, assumption+)
apply (frule_tac x = "Σe M (λj. t j ·s f j) n" and
y = "s (Suc n) ·s f (Suc n)" in ag_pOp_commute, assumption+, simp)
apply (subst pOp_assocTr42[THEN sym], assumption+,
subst pOp_assocTr41, assumption+, simp)
done
lemma (in Module) l_comb_add1:"[|ideal R A; H ⊆ carrier M;
f ∈ {j. j ≤ (n::nat)} -> H; s ∈ {j. j ≤ n} -> A; t ∈ {j. j ≤ n} -> A |] ==>
l_comb R M n (λx∈{j. j ≤ n}. (s x) ±R (t x)) f =
l_comb R M n s f ± l_comb R M n t f"
apply (simp add:l_comb_add1Tr)
done
lemma (in Module) linear_span_iOp_closedTr2:"[|ideal R A; H ⊆ carrier M;
f ∈ {j. j ≤ (n::nat)} -> H; s ∈ {j. j ≤ n} -> A|] ==>
-a (l_comb R M n s f) =
l_comb R M n (λx∈{j. j ≤ n}. -aR (s x)) f"
apply (frule_tac f = f and A = "{j. j ≤ n}" and B = H and x = 0 in
funcset_mem, simp)
apply (frule_tac A = A and s = s in linear_span_iOp_closedTr1, assumption+)
apply (frule l_comb_add1[of A H f n s "λx∈{j. j ≤ n}. -aR (s x)"],
assumption+)
apply (cut_tac linear_comb0_1[of H "λx∈{j. j ≤ n}. s x ±R
(λx∈{j. j ≤ n}. -aR (s x)) x" n f])
apply (simp,
thin_tac "l_comb R M n
(λx∈{j. j ≤ n}. s x ±R (if x ≤ n then -aR (s x) else arbitrary)) f = \<zero>")
apply (frule l_comb_mem[of A H s n f], assumption+,
frule l_comb_mem[of A H "λx∈{j. j ≤ n}. -aR (s x)" n f], assumption+)
apply (frule ag_mOp_closed[of "l_comb R M n s f"])
apply (frule ag_pOp_assoc[of "-a (l_comb R M n s f)" "l_comb R M n s f" "l_comb R M n (λx∈{j. j ≤ n}. -aR (s x)) f"], assumption+)
apply (simp, simp add:ag_l_inv1, simp add:ag_l_zero, simp add:ag_r_zero)
apply assumption+
apply (rule univar_func_test, rule ballI, simp)
apply (frule_tac x = x in funcset_mem[of s "{j. j ≤ n}" A], simp,
cut_tac sc_Ring,
frule_tac h = "s x" in Ring.ideal_subset[of R A], assumption+)
apply (frule Ring.ring_is_ag[of R],
simp add:aGroup.ag_r_inv1[of R])
apply assumption
done
lemma (in Module) linear_span_iOp_closed:"[|ideal R A; H ⊆ carrier M;
a ∈ linear_span R M A H|] ==> -a a ∈ linear_span R M A H"
apply (case_tac "H = {}")
apply (simp add:linear_span_def)
apply (simp add:ag_inv_zero)
apply (simp add:linear_span_def, erule exE, (erule bexE)+)
apply simp
apply (frule_tac f = f and n = n and s = s in
linear_span_iOp_closedTr2[of A H], assumption+)
apply (subgoal_tac "(λx∈{j. j ≤ n}. -aR (s x)) ∈ {j. j ≤ n} -> A")
apply blast
apply (rule univar_func_test, rule ballI, simp)
apply(cut_tac sc_Ring,
rule Ring.ideal_inv1_closed, assumption+,
simp add:funcset_mem)
done
lemma (in Module) linear_span_pOp_closed:
"[|ideal R A; H ⊆ carrier M; a ∈ linear_span R M A H; b ∈ linear_span R M A H|]
==> a ± b ∈ linear_span R M A H"
apply (case_tac "H = {}")
apply (simp add:linear_span_def)
apply (cut_tac ag_inc_zero, simp add:ag_r_zero)
apply (simp add:linear_span_def)
apply ((erule exE)+, (erule bexE)+)
apply (rename_tac n m f g s t)
apply (simp add:l_comb_def)
apply (cut_tac n = n and f = "λj. s j ·s f j" and m = m and
g = "λj. t j ·s g j" in nsum_add_nm)
apply (rule allI, rule impI, rule sc_mem,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI, rule sc_mem,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rotate_tac -1, frule sym,
thin_tac "Σe M (jointfun n (λj. s j ·s f j) m (λj. t j ·s g j))
(Suc (n + m)) =
Σe M (λj. s j ·s f j) n ± Σe M (λj. t j ·s g j) m",
simp del:nsum_suc)
apply (cut_tac n = "Suc (n + m)" and f = "jointfun n (λj. s j ·s f j) m
(λj. t j ·s g j)" and g = "λj. (jointfun n s m t) j ·s (jointfun n f m g) j"
in nsum_eq)
apply (rule allI, rule impI)
apply (simp add:jointfun_def)
apply (case_tac "j ≤ n", simp)
apply (rule sc_mem,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (simp, rule sc_mem)
apply (simp add:sliden_def,
frule_tac m = j and n = "Suc (n + m)" and l = "Suc n" in diff_le_mono,
thin_tac "j ≤ Suc (n + m)", simp,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset)
apply (simp add:sliden_def,
frule_tac m = j and n = "Suc (n + m)" and l = "Suc n" in diff_le_mono,
thin_tac "j ≤ Suc (n + m)", simp,
cut_tac sc_Ring,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI)
apply (simp add:jointfun_def)
apply (case_tac "j ≤ n", simp)
apply (rule sc_mem,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (simp, simp add:sliden_def,
rule sc_mem,
frule_tac m = j and n = "Suc (n + m)" and l = "Suc n" in diff_le_mono,
thin_tac "j ≤ Suc (n + m)", simp,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset)
apply (frule_tac m = j and n = "Suc (n + m)" and l = "Suc n" in diff_le_mono,
thin_tac "j ≤ Suc (n + m)", simp,
cut_tac sc_Ring,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI,
simp add:jointfun_def)
apply (simp del:nsum_suc,
thin_tac "Σe M (λj. s j ·s f j) n ± Σe M (λj. t j ·s g j) m =
Σe M (λj. jointfun n s m t j ·s jointfun n f m g j) (Suc (n + m))",
thin_tac "Σe M (jointfun n (λj. s j ·s f j) m (λj. t j ·s g j))
(Suc (n + m)) =
Σe M (λj. jointfun n s m t j ·s jointfun n f m g j) (Suc (n + m))")
apply (frule_tac f = s and n = n and A = A and g = t and m = m and B = A in
jointfun_hom0, assumption+, simp del:nsum_suc,
frule_tac f = f and n = n and A = H and g = g and m = m and B = H in
jointfun_hom0, assumption+, simp del:nsum_suc)
apply blast
done
lemma (in Module) l_comb_scTr:"[|ideal R A; H ⊆ carrier M;
r ∈ carrier R; H ≠ {}|] ==> s ∈ {j. j ≤ (n::nat)} -> A ∧
g ∈ {j. j ≤ n} -> H --> r ·s (nsum M (λk. (s k) ·s (g k)) n) =
nsum M (λk. r ·s ((s k) ·s (g k))) n"
apply (induct_tac n)
apply (rule impI, (erule conjE)+, simp)
apply (rule impI) apply (erule conjE)
apply (frule func_pre [of _ _ "A"]) apply (frule func_pre [of _ _ "H"])
apply (simp)
apply (cut_tac n = n and f = "λk. s k ·s g k" in nsum_mem,
rule allI, rule impI,
cut_tac sc_Ring, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (cut_tac a = "s (Suc n)" and m = "g (Suc n)" in sc_mem,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (simp add:sc_r_distr)
done
lemma (in Module) l_comb_sc1Tr:"[|ideal R A; H ⊆ carrier M;
r ∈ carrier R; H ≠ {}|] ==> s ∈ {j. j ≤ (n::nat)} -> A ∧
g ∈ {j. j ≤ n} -> H --> r ·s (nsum M (λk. (s k) ·s (g k)) n) =
nsum M (λk. (r ·rR (s k)) ·s (g k)) n"
apply (cut_tac sc_Ring)
apply (induct_tac n)
apply (rule impI, (erule conjE)+, simp)
apply (subst sc_assoc, assumption+,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD, simp)
apply (rule impI) apply (erule conjE)
apply (frule func_pre [of _ _ "A"], frule func_pre [of _ _ "H"])
apply simp
apply (cut_tac n = n and f = "λk. s k ·s g k" in nsum_mem,
rule allI, rule impI,
cut_tac sc_Ring, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (cut_tac a = "s (Suc n)" and m = "g (Suc n)" in sc_mem,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (simp add:sc_r_distr)
apply (subst sc_assoc, assumption+,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD, simp)
done
lemma (in Module) l_comb_sc:"[|ideal R A; H ⊆ carrier M; r ∈ carrier R;
s ∈ {j. j ≤ (n::nat)} -> A; g ∈ {j. j ≤ n} -> H|] ==>
r ·s (nsum M (λk. (s k) ·s (g k)) n) = nsum M (λk. r ·s ((s k) ·s (g k))) n"
apply (case_tac "H ≠ {}")
apply (simp add:l_comb_scTr)
apply simp
apply (frule_tac x = 0 in funcset_mem[of g " {j. j ≤ n}" "{}"], simp)
apply blast
done
lemma (in Module) l_comb_sc1:"[|ideal R A; H ⊆ carrier M; r ∈ carrier R;
s ∈ {j. j ≤ (n::nat)} -> A; g ∈ {j. j ≤ n} -> H|] ==>
r ·s (nsum M (λk. (s k) ·s (g k)) n) = nsum M (λk. (r ·rR (s k)) ·s (g k)) n"
apply (case_tac "H ≠ {}")
apply (simp add:l_comb_sc1Tr)
apply simp
apply (frule_tac x = 0 in funcset_mem[of g " {j. j ≤ n}" "{}"], simp)
apply blast
done
lemma (in Module) linear_span_sc_closed:"[|ideal R A; H ⊆ carrier M;
r ∈ carrier R; x ∈ linear_span R M A H|] ==> r ·s x ∈ linear_span R M A H"
apply (case_tac "H = {}")
apply (simp add:linear_span_def)
apply (simp add:sc_a_0)
apply (simp add:linear_span_def)
apply (erule exE, (erule bexE)+)
apply (simp add:l_comb_def)
apply (simp add:l_comb_sc)
apply (cut_tac n = n and f = "λj. r ·s (s j ·s f j)" and
g = "λj. (r ·rR (s j)) ·s f j" in nsum_eq)
apply (rule allI, rule impI,
rule sc_mem, assumption, rule sc_mem,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI,
rule sc_mem,
cut_tac sc_Ring,
rule Ring.ring_tOp_closed, assumption+,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI,
subst sc_assoc, assumption,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD, simp,
simp,
thin_tac "Σe M (λj. r ·s (s j ·s f j)) n =
Σe M (λj. (r ·rR s j) ·s f j) n",
thin_tac "x = Σe M (λj. s j ·s f j) n")
apply (cut_tac n = n and f = "λj. (r ·rR s j) ·s f j" and
g = "λj. (λx∈{j. j ≤ n}. r ·rR (s x)) j ·s f j" in nsum_eq)
apply (rule allI, rule impI,
rule sc_mem,
cut_tac sc_Ring,
rule Ring.ring_tOp_closed, assumption+,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI,
rule sc_mem, simp) apply (
cut_tac sc_Ring,
rule Ring.ring_tOp_closed, assumption+,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI)
apply simp
apply (subgoal_tac "(λx∈{j. j ≤ n}. r ·rR s x) ∈ {j. j ≤ n} -> A",
blast)
apply (rule univar_func_test, simp)
apply (thin_tac "Σe M (λj. (r ·rR s j) ·s f j) n =
Σe M (λj. (if j ≤ n then r ·rR s j else arbitrary) ·s f j) n",
rule allI, rule impI, cut_tac sc_Ring,
rule Ring.ideal_ring_multiple, assumption+, simp add:funcset_mem,
assumption)
done
lemma (in Module) mem_single_l_spanTr:"[|ideal R A; h ∈ carrier M|] ==>
s ∈ {j. j ≤ (n::nat)} -> A ∧
f ∈ {j. j ≤ n} -> {h} ∧ l_comb R M n s f ∈ linear_span R M A {h}
--> (∃a ∈ A. l_comb R M n s f = a ·s h)"
apply (cut_tac sc_Ring)
apply (induct_tac n)
apply (rule impI, (erule conjE)+, simp add:l_comb_def)
apply (frule funcset_mem[of f "{0}" "{h}" 0], simp, simp,
frule funcset_mem[of s "{0}" A 0], simp,
frule_tac h = "s 0" in Ring.ideal_subset[of R A], assumption+,
blast)
apply (rule impI, (erule conjE)+,
frule func_pre[of _ _ A], frule func_pre[of _ _ "{h}"],
frule_tac n = n in l_comb_mem_linear_span[of A "{h}" s _ f],
rule subsetI, simp, assumption+, simp,
erule bexE)
apply (frule singleton_sub[of h "carrier M"])
apply (frule Ring.ideal_subset1[of R A], assumption)
apply (frule extend_fun[of s _ A "carrier R"], assumption)
apply (frule_tac n = n in l_comb_Suc[of "{h}" A s _ f], assumption+,
simp)
apply (frule_tac A = "{j. j ≤ Suc n}" and x = "Suc n" in
funcset_mem[of f _ "{h}"], simp, simp,
frule_tac A = "{j. j ≤ Suc n}" and x = "Suc n" in
funcset_mem[of s _ A], simp,
frule_tac h = "s (Suc n)" in Ring.ideal_subset[of R A], assumption+)
apply (frule_tac h = a in Ring.ideal_subset[of R A], assumption+,
frule_tac h = "s (Suc n)" in Ring.ideal_subset[of R A], assumption+,
simp add:sc_l_distr[THEN sym],
frule_tac x = a and y = "s (Suc n)" in Ring.ideal_pOp_closed[of R A],
assumption+, blast)
done
lemma (in Module) mem_single_l_span:"[|ideal R A; h ∈ carrier M;
s ∈ {j. j ≤ (n::nat)} -> A; f ∈ {j. j ≤ n} -> {h};
l_comb R M n s f ∈ linear_span R M A {h}|] ==>
∃a ∈ A. l_comb R M n s f = a ·s h"
apply (simp add:mem_single_l_spanTr)
done
lemma (in Module) mem_single_l_span1:"[|ideal R A; h ∈ carrier M;
x ∈ linear_span R M A {h}|] ==> ∃a ∈ A. x = a ·s h"
apply (simp add:linear_span_def, erule exE, (erule bexE)+, simp)
apply (frule_tac s = s and n = n and f = f in mem_single_l_span[of A h],
assumption+)
apply (frule singleton_sub[of h "carrier M"],
rule_tac s = s and f = f in l_comb_mem_linear_span[of A "{h}"],
assumption+)
done
lemma (in Module) linear_span_subModule:"[|ideal R A; H ⊆ carrier M|] ==>
submodule R M (linear_span R M A H)"
apply (case_tac "H = {}")
apply (simp add:linear_span_def)
apply (simp add:submodule_0)
apply (simp add:submodule_def)
apply (rule conjI)
apply (simp add:linear_span_def)
apply (rule subsetI)
apply (simp add:CollectI)
apply (erule exE, (erule bexE)+)
apply simp
apply (simp add:l_comb_mem)
apply (rule conjI)
apply (rule asubg_test)
apply (rule subsetI) apply (simp add:linear_span_def)
apply (erule exE, (erule bexE)+)
apply (simp add:l_comb_mem)
apply (frule linear_span_inc_0[of A H], assumption, blast)
apply (rule ballI)+
apply (rule linear_span_pOp_closed, assumption+)
apply (rule linear_span_iOp_closed, assumption+)
apply (rule allI)+
apply (simp add:linear_span_sc_closed)
done
lemma (in Module) l_comb_mem_submoduleTr:"[|ideal R A; submodule R M N|] ==>
(s ∈ {j. j ≤ (n::nat)} -> A ∧ f ∈ {j. j ≤ n} -> carrier M ∧
(∀j ≤ n.(s j) ·s (f j) ∈ N)) --> l_comb R M n s f ∈ N"
apply (induct_tac n)
apply (simp add:l_comb_def, rule impI, (erule conjE)+)
apply (frule func_pre[of _ _ A], frule func_pre[of _ _ "carrier M"], simp)
apply (simp add:l_comb_def)
apply (frule_tac a = "Suc n" in forall_spec, simp)
apply (rule submodule_pOp_closed, assumption+)
done
lemma (in Module) l_span_sub_submodule:"[|ideal R A; submodule R M N; H ⊆ N|] ==>
linear_span R M A H ⊆ N"
apply (cut_tac sc_Ring)
apply (rule subsetI, simp add:linear_span_def)
apply (case_tac "H = {}", simp)
apply (simp add:submodule_inc_0)
apply simp
apply (erule exE, (erule bexE)+)
apply (cut_tac s = s and A = A and f = f and N = N and n = n in
l_comb_mem_submoduleTr, assumption+,
frule submodule_subset[of N],
frule subset_trans[of H N "carrier M"], assumption+,
frule_tac f = f and A = "{j. j ≤ n}" and B = H and ?B1.0 = "carrier M"
in extend_fun, assumption+)
apply (subgoal_tac "∀j≤n. s j ·s f j ∈ N", simp)
apply (rule allI, rule impI)
apply (rule submodule_sc_closed[of N], assumption,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
done
lemma (in Module) linear_span_sub:"[|ideal R A; H ⊆ carrier M|] ==>
(linear_span R M A H) ⊆ carrier M"
apply (frule linear_span_subModule[of A H], assumption+)
apply (simp add:submodule_subset)
done
constdefs
smodule_ideal_coeff::"[('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme,
'r set] => 'a set"
"smodule_ideal_coeff R M A == linear_span R M A (carrier M)"
syntax
"@SMLIDEALCOEFF" ::"['r set, ('r, 'm) Ring_scheme,
('a, 'r, 'm1) Module_scheme] => 'a set" ("(3_/ \<odot>_ _)" [64,64,65]64)
translations
"A \<odot>R M" == "smodule_ideal_coeff R M A"
lemma (in Module) smodule_ideal_coeff_is_Submodule:"ideal R A ==>
submodule R M (A \<odot>R M)"
apply (simp add:smodule_ideal_coeff_def)
apply (simp add:linear_span_subModule)
done
lemma (in Module) mem_smodule_ideal_coeff:"[|ideal R A; x ∈ A \<odot>R M|] ==>
∃n. ∃s ∈ {j. j ≤ n} -> A. ∃g ∈ {j. j ≤ n} -> carrier M.
x = l_comb R M n s g"
apply (cut_tac ag_inc_zero,
frule nonempty[of "\<zero>" "carrier M"])
apply (simp add:smodule_ideal_coeff_def linear_span_def,
erule exE, (erule bexE)+, blast)
done
constdefs
quotient_of_submodules::"[('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme,
'a set, 'a set] => 'r set"
"quotient_of_submodules R M N P == {x | x. x∈carrier R ∧
(linear_span R M (Rxa R x) P) ⊆ N}"
Annihilator::"[('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme]
=> 'r set" ("(Ann_ _)" [82,83]82)
"AnnR M == quotient_of_submodules R M {\<zero>M} (carrier M)"
syntax
"@QOFSUBMDS" :: "['a set, ('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme,
'a set] => 'r set" ("(4_ _\<ddagger>_ _)" [82,82,82,83]82)
translations
"N R\<ddagger>M P" == "quotient_of_submodules R M N P"
lemma (in Module) quotient_of_submodules_inc_0:
"[|submodule R M P; submodule R M Q|] ==> \<zero>R ∈ (P R\<ddagger>M Q)"
apply (simp add:quotient_of_submodules_def)
apply (cut_tac sc_Ring, simp add:Ring.ring_zero)
apply (simp add:linear_span_def)
apply (frule submodule_inc_0[of Q], simp add:nonempty)
apply (rule subsetI)
apply (simp, erule exE, (erule bexE)+)
apply (simp, thin_tac "x = l_comb R M n s f", simp add:l_comb_def)
apply (cut_tac n = n and f = "λj. s j ·s f j" in nsum_zeroA)
apply (rule allI, rule impI,
frule_tac x = j and f = s and A = "{j. j ≤ n}" in
funcset_mem[of _ _ "R ♦p \<zero>R"], simp)
apply (simp add:Rxa_def, erule bexE, simp) apply (
simp add:Ring.ring_times_x_0,
rule sc_0_m) apply (
frule submodule_subset[of Q],
simp add:funcset_mem subsetD)
apply (simp add:submodule_inc_0)
done
lemma (in Module) quotient_of_submodules_is_ideal:
"[|submodule R M P; submodule R M Q|] ==> ideal R (P R\<ddagger>M Q)"
apply (frule quotient_of_submodules_inc_0 [of P Q], assumption+)
apply (cut_tac sc_Ring,
rule Ring.ideal_condition[of R], assumption+)
apply (simp add:quotient_of_submodules_def)
apply (rule subsetI)
apply (simp add:CollectI)
apply (simp add:nonempty) apply (thin_tac "\<zero>R ∈ P R\<ddagger>M Q")
apply (rule ballI)+
apply (simp add:quotient_of_submodules_def)
apply (erule conjE)+
apply (rule conjI)
apply (frule Ring.ring_is_ag,
rule aGroup.ag_pOp_closed[of R], assumption+)
apply (rule aGroup.ag_mOp_closed, assumption+)
apply (subst linear_span_def)
apply (frule submodule_inc_0 [of Q], simp add:nonempty)
apply (rule subsetI, simp,
erule exE, (erule bexE)+, simp add:l_comb_def,
thin_tac "xa = Σe M (λj. s j ·s f j) n")
apply (cut_tac s = s and n = n and f = f in
l_comb_mem_submoduleTr[of "carrier R" P])
apply (simp add:Ring.whole_ideal, assumption+)
apply (frule Ring.ring_is_ag[of R],
frule_tac x = y in aGroup.ag_mOp_closed[of R], assumption+,
frule_tac x = x and y = "-aR y" in aGroup.ag_pOp_closed, assumption+,
frule_tac a = "x ±R -aR y" in Ring.principal_ideal[of R], assumption+,
frule_tac I = "R ♦p (x ±R -aR y)" in Ring.ideal_subset1, assumption+)
apply (frule_tac f = s and A = "{j. j ≤ n}" and B = "R ♦p (x ±R -aR y)"
and ?B1.0 = "carrier R" in extend_fun, assumption+,
frule_tac submodule_subset[of Q],
frule_tac f = f and A = "{j. j ≤ n}" and B = Q
and ?B1.0 = "carrier M" in extend_fun, assumption+)
apply (subgoal_tac "∀j≤n. s j ·s f j ∈ P", simp add:l_comb_def,
thin_tac "s ∈ {j. j ≤ n} -> carrier R ∧
f ∈ {j. j ≤ n} -> carrier M ∧ (∀j≤n. s j ·s f j ∈ P) -->
l_comb R M n s f ∈ P",
thin_tac "s ∈ {j. j ≤ n} -> carrier R",
thin_tac "f ∈ {j. j ≤ n} -> carrier M")
apply (rule allI, rule impI,
frule_tac x = j and f = s and A = "{j. j ≤ n}" and
B = "R ♦p (x ±R -aR y)" in funcset_mem, simp,
thin_tac "s ∈ {j. j ≤ n} -> R ♦p (x ±R -aR y)",
thin_tac "ideal R (R ♦p (x ±R -aR y))")
apply (simp add:Rxa_def, fold Rxa_def, erule bexE, simp,
thin_tac "s j = r ·rR (x ±R -aR y)")
apply (simp add:Ring.ring_distrib1,
frule_tac x = r and y = x in Ring.ring_tOp_closed, assumption+,
frule_tac x = r and y = "-aR y" in Ring.ring_tOp_closed, assumption+,
frule_tac x = j and A = "{j. j ≤ n}" and B = Q in funcset_mem,
simp,
frule_tac c = "f j" in subsetD[of Q "carrier M"], assumption+,
simp add:sc_l_distr)
apply (subst Ring.ring_inv1_2[THEN sym], assumption+,
subst Ring.ring_inv1_1, assumption+)
apply (frule_tac a = x in Ring.principal_ideal[of R], assumption+,
frule_tac a = x in Ring.principal_ideal[of R], assumption+,
frule_tac A = "R ♦p x" and H = Q and a = "r ·rR x" and h = "f j" in
sc_linear_span, assumption+, simp add:Rxa_def, blast,
simp add:funcset_mem)
apply (frule_tac x = r in aGroup.ag_mOp_closed[of R], assumption+,
frule_tac a = y in Ring.principal_ideal[of R], assumption+,
frule_tac a = y in Ring.principal_ideal[of R], assumption+,
frule_tac A = "R ♦p y" and H = Q and a = "(-aR r) ·rR y" and
h = "f j" in sc_linear_span, assumption+, simp add:Rxa_def,
blast,
simp add:funcset_mem)
apply (frule_tac c = "(r ·rR x) ·s f j" and A = "linear_span R M (R ♦p x) Q"
and B = P in subsetD, assumption+) apply (
frule_tac c = "((-aR r) ·rR y) ·s f j" and
A = "linear_span R M (R ♦p y) Q" and B = P in subsetD, assumption+)
apply (rule submodule_pOp_closed, assumption+)
apply ((rule ballI)+,
thin_tac "\<zero>R ∈ P R\<ddagger>M Q",
simp add:quotient_of_submodules_def, erule conjE)
apply (simp add:Ring.ring_tOp_closed)
apply (rule subsetI)
apply (frule submodule_inc_0[of Q],
simp add:linear_span_def nonempty)
apply (erule exE, (erule bexE)+)
apply (rule_tac c = xa and A = "{xa. ∃n. ∃f∈{j. j ≤ n} -> Q.
∃s∈{j. j ≤ n} -> R ♦p x. xa = l_comb R M n s f}" in
subsetD[of _ P], assumption+,
thin_tac "{xa. ∃n. ∃f∈{j. j ≤ n} -> Q.
∃s∈{j. j ≤ n} -> R ♦p x. xa = l_comb R M n s f} ⊆ P")
apply simp
apply (frule_tac a = r and b = x in Ring.Rxa_mult_smaller[of R], assumption+)
apply (frule_tac f = s and A = "{j. j ≤ n}" and B = "R ♦p (r ·rR x)" and
?B1.0 = "R ♦p x" in extend_fun, assumption+)
apply blast
done
lemma (in Module) Ann_is_ideal:"ideal R (AnnR M)"
apply (simp add:Annihilator_def)
apply (rule quotient_of_submodules_is_ideal)
apply (simp add:submodule_0)
apply (simp add:submodule_whole)
done
lemma (in Module) linmap_im_of_lincombTr:"[|ideal R A; R module N;
f ∈ mHom R M N; H ⊆ carrier M|] ==>
s ∈ {j. j ≤ (n::nat)} -> A ∧ g ∈ {j. j ≤ n} -> H -->
f (l_comb R M n s g) = l_comb R N n s (cmp f g)"
apply (induct_tac n)
apply (rule impI) apply (erule conjE)
apply (simp add:l_comb_def)
apply (cut_tac m = "g 0" and f = f and a = "s 0" in mHom_lin [of N],
assumption+,
simp add:funcset_mem subsetD, assumption,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset, simp add:cmp_def)
apply (rule impI, erule conjE)
apply (frule_tac f = s in func_pre,
frule_tac f = g in func_pre, simp)
apply (simp add:l_comb_def)
apply (subst mHom_add[of N f], assumption+)
apply (rule nsum_mem,
rule allI, rule impI, rule sc_mem,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (rule sc_mem,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD, simp,
frule_tac x = "Suc n" and A = "{j. j ≤ Suc n}" and f = s and
B = A in funcset_mem, simp,
cut_tac sc_Ring,
frule_tac h = "s (Suc n)" in Ring.ideal_subset[of R A], assumption+,
frule_tac x = "Suc n" and A = "{j. j ≤ Suc n}" and f = g and
B = H in funcset_mem, simp,
frule_tac c = "g (Suc n)" in subsetD[of H "carrier M"], assumption+)
apply (simp add:mHom_lin cmp_def)
done
lemma (in Module) linmap_im_lincomb:"[|ideal R A; R module N; f ∈ mHom R M N;
H ⊆ carrier M; s ∈ {j. j ≤ (n::nat)} -> A; g ∈ {j. j ≤ n} -> H |] ==>
f (l_comb R M n s g) = l_comb R N n s (cmp f g)"
apply (simp add:linmap_im_of_lincombTr)
done
lemma (in Module) linmap_im_linspan:"[|ideal R A; R module N; f ∈ mHom R M N;
H ⊆ carrier M; s ∈ {j. j ≤ (n::nat)} -> A; g ∈ {j. j ≤ n} -> H |] ==>
f (l_comb R M n s g) ∈ linear_span R N A (f ` H)"
apply (frule l_comb_mem_linear_span[of A H s n g], assumption+)
apply (simp add:linmap_im_lincomb)
apply (rule Module.l_comb_mem_linear_span[of N R A "f ` H" s n "cmp f g"],
assumption+,
rule subsetI,
simp add:image_def, erule bexE, simp,
frule_tac c = xa in subsetD[of H "carrier M"], assumption+,
simp add:mHom_mem[of N f], assumption+)
apply (rule univar_func_test, rule ballI, simp add:cmp_def)
apply (frule_tac f = g and A = "{j. j ≤ n}" and B = H and x = x in
funcset_mem, simp, simp add:image_def)
apply blast
done
lemma (in Module) linmap_im_linspan1:"[|ideal R A; R module N; f ∈ mHom R M N;
H ⊆ carrier M; h ∈ linear_span R M A H|] ==>
f h ∈ linear_span R N A (f ` H)"
apply (simp add:linear_span_def [of "R" "M"])
apply (case_tac "H = {}", simp add:linear_span_def)
apply (simp add:mHom_0, simp)
apply (erule exE, (erule bexE)+)
apply (simp add:linmap_im_linspan)
done
(*
section "3. a module over two rings"
record ('a, 'r, 's) bModule = "'a aGroup" +
sc_l :: "'r => 'a => 'a" (infixl "·sl\<index>" 70)
sc_r :: "'a => 's => 'a" (infixl "·sr\<index>" 70)
locale bModule = aGroup M +
fixes R (structure)
fixes S (structure)
assumes scl_Ring: "Ring R"
and scr_Ring: "Ring S"
and scl_closed :
"[| a ∈ carrier R; m ∈ carrier M|] ==> a ·sl m ∈ carrier M"
and scr_closed :
"[| b ∈ carrier S; m ∈ carrier M|] ==> m ·sr b ∈ carrier M"
and scl_l_distr:
"[|a ∈ carrier R; b ∈ carrier R; m ∈ carrier M|] ==>
(a ±R b) ·sl m = a ·sl m ± b ·sl m"
and scr_l_distr:
"[|a ∈ carrier S; m ∈ carrier M; n ∈ carrier M |] ==>
(m ± n) ·sr a = m ·sr a ± n ·sr a"
and scl_r_distr:
"[| a ∈ carrier R; m ∈ carrier M; n ∈ carrier M |] ==>
a ·sl (m ± n) = a ·sl m ± a ·sl n"
and scr_r_distr:
"[|a ∈ carrier S; b ∈ carrier S; m ∈ carrier M|] ==>
m ·sr (a ±S b) = m ·sr a ± m ·sr b"
and scl_assoc:
"[| a ∈ carrier R; b ∈ carrier R; m ∈ carrier M |] ==>
(a ·rR b) ·sl m = a ·sl (b ·sl m)"
and scr_assoc:
"[|a ∈ carrier S; b ∈ carrier S; m ∈ carrier M |] ==>
m ·sr (a ·rS b) = (m ·sr a) ·sr b"
and scl_one:
"m ∈ carrier M ==> (1rR) ·sl m = m"
and scr_one:
"m ∈ carrier M ==> m ·sr (1rS) = m"
constdefs
lModule::"('a, 'r, 's, 'more) bModule_scheme => ('a, 'r) Module"
("(_l)" [1000]999)
"Ml == (|carrier = carrier M, pop = pop M, mop = mop M,
zero = zero M, sprod = sc_l M |)),"
constdefs
scr_re :: "('a, 'b, 'c, 'more) bModule_scheme => 'c => 'a => 'a"
"scr_re M r m == sc_r M m r"
constdefs
rModule::"('a, 'r, 's, 'more) bModule_scheme => ('a, 's) Module"
("(_r)" [1000]999)
"Mr == (|carrier = carrier M, pop = pop M, mop = mop M,
zero = zero M, sprod = scr_re M |)),"
lemma (in bModule) bmodule_is_ag:"aGroup M"
apply assumption
done
lemma (in bModule) lModule_is_Module:"R module Ml"
apply (subgoal_tac "aGroup M")
apply (rule Module.intro)
apply (rule aGroup.intro)
apply (simp add:lModule_def, simp add:aGroup.pop_closed[of M])
apply (simp add:lModule_def, simp add:aGroup.ag_pOp_assoc)
apply (simp add:lModule_def, simp add:aGroup.ag_pOp_commute)
apply (simp add:lModule_def, rule mop_closed)
apply (simp add:lModule_def, rule l_m, assumption+)
apply (simp add:lModule_def, rule ex_zero)
apply (simp add:lModule_def, rule l_zero, assumption)
apply (rule Module_axioms.intro)
apply (simp add:scl_Ring)
apply (simp add:lModule_def, rule scl_closed, assumption+)
apply (simp add:lModule_def, rule scl_l_distr, assumption+)
apply (simp add:lModule_def, rule scl_r_distr, assumption+)
apply (simp add:lModule_def, rule scl_assoc, assumption+)
apply (simp add:lModule_def, rule scl_one, assumption+)
done
lemma (in bModule) rModule_is_Module:"S module Mr"
apply (subgoal_tac "aGroup M")
apply (rule Module.intro)
apply (rule aGroup.intro)
apply (simp add:rModule_def, simp add:aGroup.pop_closed[of M])
apply (simp add:rModule_def, simp add:aGroup.ag_pOp_assoc)
apply (simp add:rModule_def, simp add:aGroup.ag_pOp_commute)
apply (simp add:rModule_def, rule mop_closed)
apply (simp add:rModule_def, rule l_m, assumption+)
apply (simp add:rModule_def, rule ex_zero)
apply (simp add:rModule_def, rule l_zero, assumption)
apply (rule Module_axioms.intro,
simp add:scr_Ring)
apply (simp add:rModule_def, simp add:scr_re_def scr_closed)
apply (simp add:rModule_def, simp add:scr_re_def, simp add:scr_r_distr)
apply (simp add:rModule_def, simp add:scr_re_def, rule scr_l_distr,
assumption+)
apply (simp add:rModule_def scr_re_def,
subst scr_assoc[THEN sym], assumption+,
cut_tac scr_Ring,
simp add:Ring.ring_tOp_commute)
apply (simp add:rModule_def scr_re_def)
apply (cut_tac m = m in scr_one, simp)
apply assumption+
done
lemma (in Module) sprodr_welldefTr1:"[|ideal R A; A ⊆ AnnR M; a ∈ A;
m ∈ carrier M|] ==> a ·s m = \<zero>"
apply (simp add:Annihilator_def quotient_of_submodules_def)
apply (frule subsetD, assumption+)
apply (simp add:CollectI, erule conjE,
thin_tac "A ⊆ {u ∈ carrier R.
linear_span R M (R ♦p u) (carrier M) ⊆ {\<zero>}}")
apply (cut_tac sc_Ring,
cut_tac a = a and A = "Rxa R a" in
sc_linear_span[of _ "carrier M" _ "m"],
simp add:Ring.principal_ideal, simp,
simp add:Ring.a_in_principal, assumption)
apply (frule subsetD[of "linear_span R M (R ♦p a) (carrier M)" "{\<zero>}"
"a ·s m"], assumption)
apply simp
done
lemma (in Module) sprodr_welldefTr2:"[|ideal R A; A ⊆ AnnR M; a ∈ carrier R;
x ∈ a \<uplus>R A; m ∈ carrier M|] ==> a ·s m = x ·s m"
apply (cut_tac sc_Ring,
frule Ring.mem_ar_coset1 [of R A a x], assumption+, erule bexE,
rotate_tac -1, frule sym, thin_tac "h ±R a = x", simp)
apply (subst sc_l_distr)
apply (simp add:Ring.ideal_subset, assumption+)
apply (simp add:sprodr_welldefTr1)
apply (frule sc_mem [of a m], assumption+)
apply (simp add:ag_l_zero)
done
constdefs
cos_scr::"[('r, 'm) Ring_scheme, 'r set, ('a, 'r, 'm1) Module_scheme] =>
'a => 'r set => 'a"
"cos_scr R A M == λm. λX. (SOME x. x ∈ X) ·sM m"
lemma (in Module) cos_scr_welldef:"[|ideal R A; A ⊆ AnnR M; a ∈ carrier R;
X = a \<uplus>R A; m ∈ carrier M|] ==> cos_scr R A M m X = a ·s m"
apply (cut_tac sc_Ring,
frule Ring.a_in_ar_coset [of R A a], assumption+)
apply (simp add:cos_scr_def,
rule sprodr_welldefTr2[THEN sym], assumption+)
prefer 2 apply simp
apply (rule someI2_ex, blast, assumption)
done
constdefs
r_qr_bmod::"[('r, 'm) Ring_scheme, 'r set, ('a, 'r, 'm1) Module_scheme] =>
('a, 'r, 'r set) bModule"
"r_qr_bmod R A M == (|carrier = carrier M, pop = pop M, mop = mop M,
zero = zero M, sc_l = sprod M, sc_r = cos_scr R A M |))," *)
(* Remark. A should be an ideal contained in AnnR M. *) (*
syntax
"@RQBMOD" :: "[('a, 'r, 'm1) Module_scheme, ('r, 'm) Ring_scheme,
'r set] => ('a, 'r, 'r set) bModule" ("(3__ _)" [84,84,85]84)
translations
"MR A" == "r_qr_bmod R A M"
lemma r_qr_Mmodule:"[|Ring R; R module M; A ⊆ AnnR M; ideal R A|] ==>
bModule (r_qr_bmod R A M) R (R /r A)"
apply (simp add:bModule_def)
apply (simp add:r_qr_bmod_def)
apply (simp add:qring_ring)
apply (subgoal_tac " agroup
(|carrier = carrier M, pOp = pOp M, mOp = mOp M, zero = 0M,
sprodl = sprod M, sprodr = cos_scr R A M|)),") apply simp
prefer 2 apply (frule module_is_ag [of "R" "M"], assumption+)
apply (simp add:agroup_def) apply (fold agroup_def)
apply (rule impI) apply (rule ballI) apply (simp add:ag_r_zero)
apply (thin_tac " agroup
(|carrier = carrier M, pOp = pOp M, mOp = mOp M, zero = 0M,
sprodl = sprod M, sprodr = cos_scr R A M|)),")
apply (rule conjI)
apply (simp add:Module_def)
apply (rule conjI)
apply (simp add:Module_def)
apply (rule conjI) apply (simp add:qring_def)
apply (subgoal_tac "set_r_cos (b_ag R) A = set_ar_cos R A") apply simp
apply (rule bivar_func_test) apply (rule ballI)+
apply (thin_tac "set_r_cos (b_ag R) A = set_ar_cos R A")
apply (simp add:set_ar_cos_def)
apply (subgoal_tac "∀aa∈carrier R. a = aa \<uplus>R A -->
cos_scr R A M a b ∈ carrier M")
apply blast apply (thin_tac "∃aa∈carrier R. a = aa \<uplus>R A")
apply (rule ballI) apply (rule impI) apply simp
apply (rename_tac X m a)
apply (frule_tac X = "a \<uplus>R A" and a = a and m = m in
cos_scr_welldef[of "R" "M" "A"], assumption+)
apply (simp add:set_ar_cos_def) apply blast
apply assumption apply simp apply assumption apply simp
apply (simp add:sprod_mem)
apply (simp add:set_ar_cos_def)
apply (frule ring_is_ag)
apply (frule b_ag_group)
apply (simp add:ag_carrier_carrier [THEN sym])
apply (simp add:ar_coset_def set_r_cos_def)
apply (rule ballI)+
apply (frule ring_is_ag)
apply (frule b_ag_group)
apply (simp add:qring_def)
apply (subgoal_tac "set_r_cos (b_ag R) A = set_ar_cos R A") apply simp
apply (rename_tac X Y m n)
apply (subgoal_tac "∃x∈carrier R. X = x \<uplus>R A")
apply (subgoal_tac "∃y∈carrier R. Y = y \<uplus>R A")
apply (subgoal_tac "∀x ∈ carrier R. ∀y∈ carrier R. X = x \<uplus>R A ∧ Y = y \<uplus>R A
--> cos_scr R A M (rcostOp R A X Y) m =
cos_scr R A M X (cos_scr R A M Y m) ∧
cos_scr R A M (costOp (b_ag R) A X Y) m =
cos_scr R A M X m +M (cos_scr R A M Y m) ∧
cos_scr R A M X ( m +M n) =
cos_scr R A M X m +M (cos_scr R A M X n) ∧
cos_scr R A M (1R \<uplus>R A) m = m")
apply blast
apply (thin_tac "∃x∈carrier R. X = x \<uplus>R A")
apply (thin_tac "∃y∈carrier R. Y = y \<uplus>R A")
apply (rule ballI)+
apply (rule impI) apply (erule conjE) apply simp
apply (subst rcostOp, assumption+)
apply (frule_tac x = x and y = y in ring_tOp_closed, assumption+)
apply (simp add:cos_scr_welldef)
apply (subgoal_tac "costOp (b_ag R) A (x \<uplus>R A) (y \<uplus>R A) = (x +R y) \<uplus>R A")
apply simp
prefer 3 apply (simp add:set_ar_cos_def)
prefer 3 apply (simp add:set_ar_cos_def)
prefer 3 apply (simp add:set_ar_cos_def)
apply (simp add:ag_carrier_carrier [THEN sym])
apply (simp add:set_r_cos_def ar_coset_def)
prefer 2
apply (simp add:ag_carrier_carrier [THEN sym])
apply (simp add:ar_coset_def) apply (simp add:agop_gop [THEN sym])
apply (rule costOpwelldef [THEN sym], assumption+)
apply (simp add:ideal_def) apply (erule conjE)
apply (simp add:asubg_nsubg) apply assumption+
apply (frule_tac x = x and y = y in ag_pOp_closed[of "R"], assumption+)
apply (frule module_is_ag [of "R" "M"], assumption)
apply (frule_tac x = m and y = n in ag_pOp_closed [of "M"], assumption+)
apply (frule_tac a = y and m = m in sprod_mem [of "R" "M"], assumption+)
apply (frule ring_one [of "R"])
apply (simp add:cos_scr_welldef)
apply (frule_tac X = "x ·R y \<uplus>R A" and a = "x ·R y" and m = m in
cos_scr_welldef [of "R" "M" "A"], assumption+)
apply (simp add:set_ar_cos_def) apply blast apply assumption apply simp
apply assumption apply simp
apply (simp add:sprod_assoc)
apply (frule ring_one [of "R"])
apply (frule_tac X = "(x +R y) \<uplus>R A" and a = "(x +R y)" and m = m in
cos_scr_welldef [of "R" "M" "A"], assumption+)
apply (simp add:set_ar_cos_def) apply blast
apply simp+
apply (simp add:sprod_distrib1)
apply (simp add:sprod_distrib2)
apply (frule_tac X = "1R \<uplus>R A" and a = "1R" and m = m in
cos_scr_welldef [of "R" "M" "A"], assumption+)
apply (simp add:set_ar_cos_def) apply blast apply assumption apply simp+
apply (simp add:sprod_one)
done *)
constdefs
faithful::"[('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme]
=> bool"
"faithful R M == AnnR M = {\<zero>R}"
section "4. nsum and Generators"
constdefs
generator ::"[('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme,
'a set] => bool"
"generator R M H == H ⊆ carrier M ∧
linear_span R M (carrier R) H = carrier M"
finite_generator::"[('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme,
'a set] => bool"
"finite_generator R M H == finite H ∧ generator R M H"
fGOver :: "[('a, 'r, 'm1) Module_scheme, ('r, 'm) Ring_scheme] => bool"
(*(infixl 70)*)
"fGOver M R == ∃H. finite_generator R M H"
syntax
"@FGENOVER"::"[('a, 'r, 'm1) Module_scheme, ('r, 'm) Ring_scheme] => bool"
(infixl "fgover" 70)
translations
"M fgover R" == "fGOver M R"
lemma (in Module) h_in_linear_span:"[|H ⊆ carrier M; h ∈ H|] ==>
h ∈ linear_span R M (carrier R) H"
apply (subst sprod_one [THEN sym, of h])
apply (simp add:subsetD)
apply (cut_tac sc_Ring)
apply (frule Ring.ring_one)
apply (rule sc_linear_span [of "carrier R" "H" "1rR" "h"])
apply (simp add:Ring.whole_ideal) apply assumption+
done
lemma (in Module) generator_sub_carrier:"generator R M H ==>
H ⊆ carrier M"
apply (simp add:generator_def)
done
lemma (in Module) lin_span_sub_carrier:"[|ideal R A;
H ⊆ carrier M|] ==> linear_span R M A H ⊆ carrier M"
apply (cut_tac sc_Ring)
apply (rule subsetI)
apply (simp add:linear_span_def)
apply (case_tac "H = {}") apply simp
apply (simp add:module_inc_zero)
apply simp
apply (erule exE, (erule bexE)+, simp,
thin_tac "x = l_comb R M n s f")
apply (simp add:l_comb_def)
apply (rule_tac n = n in nsum_mem)
apply (rule allI, rule impI)
apply (rule sc_mem)
apply (simp add:funcset_mem Ring.ideal_subset)
apply (simp add:funcset_mem subsetD)
done
lemma (in Module) lin_span_coeff_mono:"[|ideal R A; H ⊆ carrier M|]==>
linear_span R M A H ⊆ linear_span R M (carrier R) H"
apply (cut_tac sc_Ring)
apply (rule subsetI)
apply (simp add:linear_span_def)
apply (case_tac "H = {}") apply simp apply simp
apply (erule exE, (erule bexE)+)
apply (frule Ring.ideal_subset1 [of R A], assumption+)
apply (frule_tac f = s in extend_fun, assumption+)
apply blast
done
lemma (in Module) l_span_sum_closedTr:"[|ideal R A; H ⊆ carrier M|]==>
∀s. ∀f. s∈{j. j ≤ (n::nat)} -> A ∧
f ∈ {j. j ≤ n} -> linear_span R M A H -->
(nsum M (λj. s j ·s (f j)) n ∈ linear_span R M A H)"
apply (cut_tac sc_Ring)
apply (induct_tac n)
apply ((rule allI)+, rule impI, simp)
apply (erule conjE)
apply (rule linear_span_sc_closed, assumption+)
apply (simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem)
apply ((rule allI)+, rule impI, erule conjE)
apply (frule func_pre [of _ _ "A"],
frule func_pre [of _ _ "linear_span R M A H"])
apply (drule_tac a = s in forall_spec1,
drule_tac a = f in forall_spec1)
apply simp
apply (rule linear_span_pOp_closed, assumption+)
apply (rule linear_span_sc_closed, assumption+,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
done
lemma (in Module) l_span_closed:"[|ideal R A; H ⊆ carrier M;
s ∈ {j. j ≤ (n::nat)} -> A; f ∈ {j. j ≤ n} -> linear_span R M A H |] ==>
l_comb R M n s f ∈ linear_span R M A H"
apply (simp add:l_comb_def)
apply (simp add: l_span_sum_closedTr)
done
lemma (in Module) l_span_closed1:"[|H ⊆ carrier M;
s ∈ {j. j ≤ (n::nat)} -> carrier R;
f ∈ {j. j ≤ n} -> linear_span R M (carrier R) H |] ==>
Σe M (λj. s j ·s (f j)) n ∈ linear_span R M (carrier R) H"
apply (cut_tac sc_Ring,
frule Ring.whole_ideal [of "R"])
apply (frule l_span_sum_closedTr[of "carrier R" H n], assumption+)
apply (drule_tac a = s in forall_spec1,
drule_tac a = f in forall_spec1,
simp)
done
lemma (in Module) l_span_closed2Tr0:"[|ideal R A; H ⊆ carrier M; Ring R; s ∈ A;
f ∈ linear_span R M (carrier R) H |] ==> s ·s f ∈ linear_span R M A H"
apply (cut_tac sc_Ring)
apply (case_tac "H = {}")
apply (simp add:linear_span_def)
apply (rule sc_a_0,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset)
apply (simp add:linear_span_def)
apply (erule exE, (erule bexE)+, simp,
thin_tac "f = l_comb R M n sa fa")
apply (frule Ring.whole_ideal[of R])
apply (frule_tac h = s in Ring.ideal_subset[of R A], assumption+)
apply (frule_tac s = sa and g = f in l_comb_sc1[of "carrier R" H s],
assumption+, simp add:l_comb_def,
thin_tac "s ·s Σe M (λk. sa k ·s f k) n =
Σe M (λk. (s ·rR sa k) ·s f k) n")
apply (cut_tac n = n and f = "λj. (s ·rR sa j) ·s f j" and
g = "λj. ((λx∈{j. j ≤ n}. (s ·rR sa x)) j) ·s f j" in nsum_eq)
apply (rule allI, rule impI, rule sc_mem,
rule Ring.ring_tOp_closed, assumption+,
simp add:funcset_mem,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI, simp,
rule sc_mem,
rule Ring.ring_tOp_closed, assumption+,
simp add:funcset_mem,
simp add:funcset_mem subsetD)
apply (rule allI, rule impI, simp)
apply (subgoal_tac "(λx∈{j. j ≤ n}. (s ·rR sa x)) ∈ {j. j ≤ n} -> A",
blast,
thin_tac "Σe M (λj. (s ·rR sa j) ·s f j) n =
Σe M (λj. (λx∈{j. j ≤ n}. s ·rR sa x) j ·s f j) n")
apply (rule univar_func_test, rule ballI, simp,
rule_tac x = s and r = "sa x" in
Ring.ideal_ring_multiple1[of R A], assumption+)
apply (simp add:funcset_mem)
done
lemma (in Module) l_span_closed2Tr:"[|ideal R A; H ⊆ carrier M|] ==>
s ∈ {j. j ≤ (n::nat)} -> A ∧
f ∈ {j. j ≤ n} -> linear_span R M (carrier R) H -->
l_comb R M n s f ∈ linear_span R M A H"
apply (cut_tac sc_Ring)
apply (induct_tac n)
apply (rule impI, (erule conjE)+)
apply (case_tac "H = {}")
apply (simp add:linear_span_def)
apply (simp add:l_comb_def)
apply (frule_tac f = f and A = "{0}" and B = "{\<zero>}" and x = 0 in
funcset_mem, simp+)
apply (rule sc_a_0,
cut_tac sc_Ring,
simp add:funcset_mem Ring.ideal_subset)
apply (simp add:l_comb_def)
apply (rule l_span_closed2Tr0[of A H "s 0" "f 0"], assumption+,
simp add:funcset_mem, simp add:funcset_mem)
apply (rule impI, erule conjE,
frule func_pre[of s], frule func_pre[of f], simp)
apply (simp add:l_comb_def)
apply (rule linear_span_pOp_closed, assumption+)
apply (rule_tac s = "s (Suc n)" and f = "f (Suc n)" in
l_span_closed2Tr0[of A H], assumption+,
(simp add:funcset_mem)+)
done
lemma (in Module) l_span_closed2:"[|ideal R A; H ⊆ carrier M;
s ∈ {j. j ≤ (n::nat)} -> A ;
f ∈ {j. j ≤ n} -> linear_span R M (carrier R) H|] ==>
l_comb R M n s f ∈ linear_span R M A H"
apply (simp add:l_span_closed2Tr)
done
lemma (in Module) l_span_l_span:"H ⊆ carrier M ==>
linear_span R M (carrier R) (linear_span R M (carrier R) H) =
linear_span R M (carrier R) H"
apply (cut_tac sc_Ring, frule Ring.whole_ideal[of R])
apply (rule equalityI)
apply (rule subsetI)
apply (frule linear_span_inc_0[of "carrier R" H], assumption+,
frule nonempty[of _ "linear_span R M (carrier R) H"],
simp add:linear_span_def[of R M "carrier R"
"linear_span R M (carrier R) H"],
erule exE, (erule bexE)+, simp)
apply (frule_tac s = s and n = n and f = f in l_span_closed2[of "carrier R"],
assumption+,
frule lin_span_sub_carrier[of "carrier R" "H"], assumption+,
rule subsetI)
apply (rule_tac h = x in h_in_linear_span[of "linear_span R M (carrier R) H"],
assumption+)
done
lemma (in Module) l_spanA_l_span:"[|ideal R A; H ⊆ carrier M|] ==>
linear_span R M A (linear_span R M (carrier R) H) =
linear_span R M A H"
apply (cut_tac sc_Ring, frule Ring.whole_ideal[of R])
apply (rule equalityI)
apply (rule subsetI)
apply (frule linear_span_inc_0[of "carrier R" H], assumption+,
frule nonempty[of _ "linear_span R M (carrier R) H"],
simp add:linear_span_def[of R M A
"linear_span R M (carrier R) H"],
erule exE, (erule bexE)+, simp)
apply (frule_tac s = s and n = n and f = f in l_span_closed2[of A],
assumption+)
apply (frule l_span_cont_H[of H])
apply (frule l_span_gen_mono[of "H" "linear_span R M (carrier R) H" A],
simp add:lin_span_sub_carrier[of "carrier R" H], assumption)
apply assumption
done
lemma (in Module) l_span_zero:"ideal R A ==> linear_span R M A {\<zero>} = {\<zero>}"
apply (cut_tac sc_Ring)
apply (rule equalityI)
apply (rule subsetI,
frule_tac x = x in mem_single_l_span1[of A \<zero>],
simp add:ag_inc_zero, assumption,
erule bexE, frule_tac h = a in Ring.ideal_subset[of R A], assumption+,
simp add:sc_a_0)
apply (rule subsetI, simp, rule linear_span_inc_0, assumption,
rule subsetI, simp add:ag_inc_zero)
done
lemma (in Module) l_span_closed3:"[|ideal R A; generator R M H;
A \<odot>R M = carrier M|] ==> linear_span R M A H = carrier M"
apply (cut_tac sc_Ring)
apply (rule equalityI)
apply (cut_tac linear_span_subModule[of A H],
simp add:submodule_subset, assumption,
simp add:generator_def)
apply (rule subsetI)
apply (simp add:generator_def)
apply (erule conjE)
apply (case_tac "H = {}", simp, simp add:linear_span_def)
apply (simp add:smodule_ideal_coeff_def)
apply (rotate_tac -2, frule sym,
thin_tac "linear_span R M (carrier R) H = carrier M")
apply simp
apply (frule sym,
thin_tac "linear_span R M A (linear_span R M (carrier R) H) =
linear_span R M (carrier R) H")
apply (frule_tac a = x in eq_set_inc[of _ "linear_span R M (carrier R) H"
"linear_span R M A (linear_span R M (carrier R) H)"], assumption+,
thin_tac "x ∈ linear_span R M (carrier R) H",
thin_tac "linear_span R M (carrier R) H =
linear_span R M A (linear_span R M (carrier R) H)")
apply (frule sym,
thin_tac "carrier M = linear_span R M (carrier R) H",
frule subset_trans[of H "linear_span R M (carrier R) H" "carrier M"],
simp,
thin_tac "linear_span R M (carrier R) H = carrier M")
apply (frule Ring.whole_ideal,
frule linear_span_inc_0 [of "carrier R" "H"], assumption+,
frule nonempty [of "\<zero>" "linear_span R M (carrier R) H"])
apply (simp add:linear_span_def [of _ _ _ "linear_span R M (carrier R) H"])
apply (erule exE, (erule bexE)+)
apply (simp add:l_span_closed2)
done
lemma (in Module) generator_generator:"[|generator R M H; H1 ⊆ carrier M;
H ⊆ linear_span R M (carrier R) H1|] ==> generator R M H1"
apply (cut_tac sc_Ring,
frule Ring.whole_ideal[of R],
frule linear_span_subModule[of "carrier R" H1], assumption,
frule l_span_sub_submodule[of "carrier R"
"linear_span R M (carrier R) H1" H], assumption+)
apply (simp add:generator_def)
apply (rule equalityI,
simp add:submodule_subset, assumption)
done
lemma (in Module) generator_elimTr:
"f ∈ {j. j ≤ (n::nat)} -> carrier M ∧ generator R M (f ` {j. j ≤ n}) ∧
(∀i∈nset (Suc 0) n. f i ∈
linear_span R M (carrier R) (f ` {j. j ≤ (i - Suc 0)})) -->
linear_span R M (carrier R) {f 0} = carrier M"
apply (induct_tac n)
apply (rule impI, (erule conjE)+)
apply (simp add:nset_def generator_def)
apply (rule impI)
apply (erule conjE)+
apply (frule func_pre [of _ _ "carrier M"], simp)
apply (subgoal_tac "generator R M (f ` {j. j ≤ n})")
apply (subgoal_tac "∀i∈nset (Suc 0) n.
f i ∈ linear_span R M (carrier R) (f ` {j. j ≤ (i - Suc 0)})")
apply simp
apply (thin_tac "generator R M (f ` {j. j ≤ n}) ∧
(∀i∈nset (Suc 0) n. f i ∈ linear_span R M (carrier R)
(f ` {j. j ≤ i - Suc 0})) -->
linear_span R M (carrier R) {f 0} = carrier M")
apply (rule ballI)
apply (frule_tac b = i in forball_spec1, simp add:nset_def, assumption)
apply (thin_tac "generator R M (f ` {j. j ≤ n}) ∧
(∀i∈nset (Suc 0) n.
f i ∈ linear_span R M (carrier R) (f ` {j. j ≤ i - Suc 0})) -->
linear_span R M (carrier R) {f 0} = carrier M")
apply (frule_tac b = "Suc n" in forball_spec1, simp add:nset_def,
thin_tac "∀i∈nset (Suc 0) (Suc n).
f i ∈ linear_span R M (carrier R) (f ` {j. j ≤ i - Suc 0})",
simp)
apply (subgoal_tac "f ` {j. j ≤ Suc n} ⊆ linear_span R M (carrier R) (f ` {j. j ≤ n})")
apply (frule_tac H = "f ` {j. j ≤ Suc n}" and ?H1.0 = "f ` {j. j ≤ n}"
in generator_generator,
rule subsetI, simp add:image_def, erule exE, erule conjE, simp,
simp add:funcset_mem)
apply assumption+
apply (rule subsetI, simp add:image_def, erule exE, erule conjE)
apply (case_tac "xa = Suc n", simp)
apply (frule_tac m = xa and n = "Suc n" in noteq_le_less, assumption,
thin_tac "xa ≤ Suc n",
frule_tac x = xa and n = "Suc n" in less_le_diff,
thin_tac "xa < Suc n", simp)
apply (rule_tac H = "{y. ∃x≤n. y = f x}" and h = "f xa" in
h_in_linear_span,
rule subsetI, simp add:image_def, erule exE, erule conjE,
simp add:funcset_mem)
apply (simp, blast)
done
lemma (in Module) generator_generator_elim:
"[|f ∈ {j. j ≤ (n::nat)} -> carrier M; generator R M (f ` {j. j ≤ n});
(∀i∈nset (Suc 0) n. f i ∈ linear_span R M (carrier R)
(f ` {j. j ≤ (i - Suc 0)}))|] ==>
linear_span R M (carrier R) {f 0} = carrier M"
apply (simp add:generator_elimTr [of f n])
done
lemma (in Module) surjec_generator:"[|R module N; f ∈ mHom R M N;
surjecM,N f; generator R M H|] ==> generator R N (f ` H)"
apply (cut_tac sc_Ring, frule Ring.whole_ideal)
apply (simp add:generator_def, erule conjE)
apply (simp add:surjec_def, (erule conjE)+)
apply (simp add:aHom_def, (erule conjE)+)
apply (simp add:image_sub [of "f" "carrier M" "carrier N" "H"])
apply (frule Module.lin_span_sub_carrier[of N R "carrier R" "f ` H"],
assumption,
simp add:image_sub [of "f" "carrier M" "carrier N" "H"])
apply (rule equalityI, assumption+)
apply (rule subsetI)
apply (simp add:surj_to_def,
thin_tac "f ∈ extensional (carrier M)",
thin_tac "∀a∈carrier M. ∀b∈carrier M. f (a ± b) = f a ±N f b")
apply (frule sym, rotate_tac 6, frule sym,
thin_tac "f ` carrier M = carrier N",
frule_tac a = x and A = "carrier N" and B = "f ` carrier M" in
eq_set_inc, assumption,
thin_tac "carrier N = f ` carrier M",
thin_tac "carrier M = linear_span R M (carrier R) H")
apply (simp add:image_def[of f "carrier M"], erule bexE)
apply (frule sym, thin_tac "linear_span R M (carrier R) H = carrier M",
frule_tac a = xa in eq_set_inc[of _ "carrier M"
"linear_span R M (carrier R) H"], assumption,
thin_tac "carrier M = linear_span R M (carrier R) H",
thin_tac "linear_span R N (carrier R) (f ` H) ⊆ carrier N")
apply (simp add:linear_span_def)
apply (case_tac "H = {}", simp)
apply (simp add:mHom_0, simp,
erule exE, (erule bexE)+)
apply (cut_tac sc_Ring, frule Ring.whole_ideal[of R],
frule_tac s = s and n = n and g = fa in
linmap_im_linspan[of "carrier R" N f H], assumption+,
rotate_tac -5, frule sym,
thin_tac "xa = l_comb R M n s fa", simp,
thin_tac "l_comb R M n s fa = xa")
apply (simp add:linear_span_def)
done
lemma (in Module) surjec_finitely_gen:"[|R module N; f ∈ mHom R M N;
surjecM,N f; M fgover R|] ==> N fgover R"
apply (simp add:fGOver_def)
apply (erule exE)
apply (simp add:finite_generator_def [of "R" "M"],erule conjE)
apply (frule_tac H = H in surjec_generator[of N f], assumption+)
apply (simp add:finite_generator_def [of "R" "N"])
apply (frule_tac F = H and h = f in finite_imageI)
apply blast
done
subsection "4-1. sum up coefficients"
text{* Symbolic calculation. *}
lemma (in Module) similar_termTr:"[|ideal R A; a ∈ A|] ==>
∀s. ∀f. s ∈ {j. j ≤ (n::nat)} -> A ∧
f ∈ {j. j ≤ n} -> carrier M ∧
m ∈ f ` {j. j ≤ n} -->
(∃t∈{j. j ≤ n} -> A. nsum M (λj. s j ·s (f j)) n ± a ·s m =
nsum M (λj. t j ·s (f j)) n )"
apply (cut_tac sc_Ring)
apply (induct_tac n)
apply (rule allI)+ apply (rule impI) apply (erule conjE)+
apply simp
apply (frule_tac x = 0 and f = s in funcset_mem[of _ "{0}" A], simp,
frule_tac h = "s 0" in Ring.ideal_subset[of R A], assumption+,
frule_tac x = 0 and f = f in funcset_mem[of _ "{0}" "carrier M"], simp,
frule_tac h = a in Ring.ideal_subset[of R A], assumption+,
subst sc_l_distr[THEN sym], assumption+)
apply (subgoal_tac "(λk∈{0::nat}. (s 0 ±R a)) ∈ {0} -> A")
apply (subgoal_tac "(s 0 ±R a) ·s f 0 = (λk∈{0::nat}. s 0 ±R a) 0 ·s f 0")
apply blast
apply (simp, rule univar_func_test, rule ballI,
simp add:Ring.ideal_pOp_closed)
(** n **)
apply ((rule allI)+, rule impI, (erule conjE)+)
apply (simp del:nsum_suc add:image_def)
apply (cut_tac n = n and f = "λj. s j ·s f j" in nsum_mem,
rule allI, rule impI, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem,
frule_tac x = "Suc n" and f = s and A = "{j. j ≤ Suc n}" and
B = A in funcset_mem, simp,
frule_tac h = "s (Suc n)" in Ring.ideal_subset, assumption+,
frule_tac x = "Suc n" and f = f and A = "{j. j ≤ Suc n}" and
B = "carrier M" in funcset_mem, simp,
frule_tac a = "s (Suc n)" and m = "f (Suc n)" in sc_mem, assumption+,
cut_tac a = a and m = m in sc_mem,
simp add:Ring.ideal_subset, erule exE, simp add:funcset_mem,
erule exE, erule conjE)
apply (case_tac "x = Suc n", simp) (***** case x = Suc n ********)
apply (subst ag_pOp_assoc, assumption+)
apply (thin_tac "Σe M (λj. s j ·s f j) n ∈ carrier M",
thin_tac "s (Suc n) ·s f (Suc n) ∈ carrier M",
thin_tac "a ·s f (Suc n) ∈ carrier M",
thin_tac "∀s fa.
s ∈ {j. j ≤ n} -> A ∧
fa ∈ {j. j ≤ n} -> carrier M ∧ (∃x≤n. f (Suc n) = fa x) -->
(∃t∈{j. j ≤ n} -> A.
Σe M (λj. s j ·s fa j) n ± a ·s f (Suc n) =
Σe M (λj. t j ·s fa j) n)")
apply (subst sc_l_distr[THEN sym], assumption+,
simp add:Ring.ideal_subset, assumption+)
apply (frule func_pre[of _ _ A],
frule_tac f = s and n = n and g = "λk∈{0::nat}. (s (Suc n) ±R a)" and
m = 0 and A = A and B = A in jointfun_hom0,
rule univar_func_test, rule ballI, simp,
rule Ring.ideal_pOp_closed, assumption+, simp)
apply (subgoal_tac "Σe M (λj. s j ·s f j) n ± (s (Suc n) ±R a) ·s f (Suc n) =
Σe M (λj. (jointfun n s 0 (λk∈{0}. s (Suc n) ±R a)) j ·s f j) (Suc n)",
simp,
thin_tac "Σe M (λj. s j ·s f j) n ± (s (Suc n) ±R a) ·s f (Suc n) =
Σe M (λj. jointfun n s 0 (λk∈{0}. s (Suc n) ±R a) j ·s f j) n ±
jointfun n s 0 (λk∈{0}. s (Suc n) ±R a) (Suc n) ·s f (Suc n)")
apply blast
apply simp
apply (simp add:jointfun_def sliden_def)
apply (cut_tac n = n and f = "λj. s j ·s f j" and g = "λj. (if j ≤ n then s j
else (λk∈{0}. s (Suc n) ±R a) (sliden (Suc n) j)) ·s f j" in
nsum_eq)
apply (rule allI, rule impI, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem)
apply (rule allI, rule impI, simp, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem)
apply (rule allI, rule impI, simp)
apply simp
apply (frule_tac m = x and n = "Suc n" in noteq_le_less, assumption,
thin_tac "x ≤ Suc n",
frule_tac x = x and n = "Suc n" in less_le_diff,
thin_tac "x < Suc n", simp)
apply (frule func_pre[of _ _ A], frule func_pre[of _ _ "carrier M"])
apply (drule_tac a = s in forall_spec1,
drule_tac a = f in forall_spec1)
apply (subgoal_tac "∃xa≤n. f x = f xa", simp,
thin_tac "∃xa≤n. f x = f xa", erule bexE)
apply (subst ag_pOp_assoc, assumption+,
frule_tac x = "s (Suc n) ·s f (Suc n)" and y = "a ·s f x" in
ag_pOp_commute, assumption+, simp,
thin_tac "s (Suc n) ·s f (Suc n) ± a ·s f x =
a ·s f x ± s (Suc n) ·s f (Suc n)",
subst ag_pOp_assoc[THEN sym], assumption+, simp,
thin_tac "Σe M (λj. s j ·s f j) n ± a ·s f x = Σe M (λj. t j ·s f j) n")
apply (frule_tac f = t and n = n and g = "λk∈{0::nat}. s (Suc n)" and
m = 0 and A = A and B = A in jointfun_hom0,
rule univar_func_test, rule ballI, simp, simp)
apply (subgoal_tac "Σe M (λj. t j ·s f j) n ± s (Suc n) ·s f (Suc n) =
Σe M (λj. (jointfun n t 0 (λk∈{0}. s (Suc n))) j ·s f j) (Suc n)",
simp,
thin_tac "Σe M (λj. t j ·s f j) n ± s (Suc n) ·s f (Suc n) =
Σe M (λj. jointfun n t 0 (λk∈{0}. s (Suc n)) j ·s f j) n ±
jointfun n t 0 (λk∈{0}. s (Suc n)) (Suc n) ·s f (Suc n)")
apply blast
apply (simp add:jointfun_def sliden_def)
apply (cut_tac n = n and f = "λj. t j ·s f j" and
g = "λj. (if j ≤ n then t j else (λk∈{0}. s (Suc n))
(sliden (Suc n) j)) ·s f j" in nsum_eq)
apply (rule allI, rule impI, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem)
apply (rule allI, rule impI, simp, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem)
apply (rule allI, rule impI, simp, simp)
apply blast
done
lemma (in Module) similar_term1:"[|ideal R A; a ∈ A; s ∈ {j. j≤(n::nat)} -> A;
f ∈ {j. j ≤ n} -> carrier M; m ∈ f ` {j. j ≤ n}|] ==>
∃t∈{j. j ≤ n} -> A. Σe M (λj. s j ·s (f j)) n ± a ·s m =
Σe M (λj. t j ·s (f j)) n"
apply (simp add:similar_termTr)
done
lemma (in Module) same_togetherTr:"[|ideal R A; H ⊆ carrier M |] ==>
∀s. ∀f. s∈{j. j ≤ (n::nat)} -> A ∧ f ∈ {j. j ≤ n} -> H -->
(∃t ∈ {j. j ≤ (card (f ` {j. j ≤ n}) - Suc 0)} -> A.
∃g ∈ {j. j ≤ (card (f ` {j. j ≤ n}) - Suc 0)} -> f ` {j. j ≤ n}.
surj_to g {j. j ≤ (card (f ` {j. j ≤ n}) - Suc 0)} (f ` {j. j ≤ n}) ∧
nsum M (λj. s j ·s (f j)) n = nsum M (λk. t k ·s (g k))
(card (f ` {j. j ≤ n}) - Suc 0))"
apply (induct_tac n)
apply ((rule allI)+, rule impI, erule conjE)
apply simp
apply (frule_tac f = f and A = "{0}" and B= H in func_to_img,
frule_tac f = f and A = "{0}" and B= H in surj_to_image,
simp add:image_def, blast)
apply ((rule allI)+, rule impI, erule conjE)
apply (frule func_pre [of _ _ "A"], frule func_pre [of _ _ "H"])
apply (drule_tac a = s in forall_spec1,
drule_tac a = f in forall_spec1,
simp, (erule bexE)+ , (erule conjE)+, simp,
thin_tac "Σe M (λj. s j ·s f j) n =
Σe M (λk. t k ·s g k) (card (f ` {j. j ≤ n}) - Suc 0)")
apply (case_tac "f (Suc n) ∈ f ` {j. j ≤ n}")
apply (frule_tac a = "s (Suc n)" and s = t and
n = "card (f ` {j. j ≤ n}) - Suc 0" and f = g and m = "f (Suc n)" in
similar_term1[of A],
simp add:funcset_mem,
assumption,
frule_tac f = f and A = "{j. j ≤ n}" and B = H in image_sub0,
frule_tac A = "f ` {j. j ≤ n}" and B = H and C = "carrier M"
in subset_trans, assumption,
rule_tac f = g and A = "{j. j ≤ card (f ` {j. j ≤ n}) - Suc 0}" and
B = "f ` {j. j ≤ n}" and ?B1.0 = "carrier M" in extend_fun,
assumption+)
apply (simp add:surj_to_def)
apply (erule bexE, simp,
thin_tac "Σe M (λj. t j ·s g j) (card (f ` {j. j ≤ n}) - Suc 0) ±
s (Suc n) ·s f (Suc n) =
Σe M (λj. ta j ·s g j) (card (f ` {j. j ≤ n}) - Suc 0)")
apply (simp add:Nset_img0)
apply blast
apply (frule_tac f = t and n = "card (f ` {j. j ≤ n}) - Suc 0" and A = A and
g = "λk∈{0::nat}. s (Suc n)" and m = 0 and B = A in jointfun_hom0)
apply (rule univar_func_test, rule ballI, simp add:funcset_mem,
simp)
apply (frule_tac f = g and n = "card (f ` {j. j ≤ n}) - Suc 0" and
A = "f ` {j. j ≤ n}" and g = "λk∈{0::nat}. f (Suc n)" and m = 0 and
B = "{f (Suc n)}" in jointfun_hom0)
apply (rule univar_func_test, rule ballI, simp add:funcset_mem,
simp)
apply (subgoal_tac "Σe M (λk. t k ·s g k) (card (f ` {j. j ≤ n}) - Suc 0) ±
s (Suc n) ·s f (Suc n) =
Σe M (λj. (jointfun (card (f ` {j. j ≤ n}) - Suc 0) t 0 (λk∈{0}.
s (Suc n))) j ·s (jointfun (card (f ` {j. j ≤ n}) - Suc 0) g 0
(λk∈{0}. f (Suc n))) j) (card (f ` {j. j ≤ (Suc n)}) - Suc 0)", simp,
thin_tac "Σe M (λk. t k ·s g k) (card (f ` {j. j ≤ n}) - Suc 0) ±
s (Suc n) ·s f (Suc n) =
Σe M (λj. jointfun (card (f ` {j. j ≤ n}) - Suc 0) t 0
(λk∈{0}. s (Suc n)) j ·s
jointfun (card (f ` {j. j ≤ n}) - Suc 0) g 0
(λk∈{0}. f (Suc n))
j) (card (f ` {j. j ≤ Suc n}) - Suc 0)")
apply (simp del:nsum_suc add:card_image_Nsetn_Suc)
apply (simp del:nsum_suc add:image_Nset_Suc[THEN sym])
apply (subgoal_tac "surj_to (jointfun (card (f ` {j. j ≤ n}) - Suc 0) g 0
(λk∈{0}. f (Suc n))) {l. l ≤ Suc (card (f ` {j. j ≤ n}) - Suc 0)}
(f ` {j. j ≤ Suc n})", blast)
apply (simp add:surj_to_def)
apply (frule_tac f = g and n = "card (f ` {j. j ≤ n}) - Suc 0" and A = "f ` {j. j ≤ n}" and g = "λk∈{0}. f (Suc n)" and m = 0 and B = "{f (Suc n)}" in
im_jointfun)
apply (rule univar_func_test, rule ballI, simp add:funcset_mem)
apply simp
apply (simp add:image_Nset_Suc[THEN sym])
apply (simp add:card_image_Nsetn_Suc)
apply (simp add:Nset_img)
apply (frule_tac f = f and A = "{j. j ≤ Suc n}" and B = H in image_sub0)
apply (frule_tac A = "f ` {j. j ≤ Suc n}" and B = H and C = "carrier M" in
subset_trans, assumption+)
apply (cut_tac H = H and s = t and n = "card (f ` {j. j ≤ n}) - Suc 0"
and f = g and t = "λk∈{0}. s (Suc n)" and m = 0 and
g = "λk∈{0}. f (Suc n)" in
l_comb_jointfun_jj[of _ A], assumption+)
apply (frule_tac f = f and A = "{j. j ≤ n}" and B = H in image_sub0)
apply (rule_tac f = g and A = "{j. j ≤ card (f ` {j. j ≤ n}) - Suc 0}" and
B = "f ` {j. j ≤ n}" in extend_fun[of _ _ _ H], assumption+,
rule univar_func_test, simp add:funcset_mem,
rule univar_func_test, simp add:funcset_mem)
apply simp
apply (simp add:jointfun_def sliden_def)
done
(* H shall a generator *)
lemma (in Module) same_together:"[|ideal R A; H ⊆ carrier M;
s ∈ {j. j ≤ (n::nat)} -> A; f ∈ {j. j ≤ n} -> H|] ==>
∃t ∈ {j. j ≤ (card (f ` {j. j ≤ (n::nat)}) - Suc 0)} -> A.
∃g ∈ {j. j ≤ (card (f ` {j. j ≤ n}) - Suc 0)} -> f ` {j. j ≤ n}.
surj_to g {j. j ≤ (card (f ` {j. j ≤ n}) - Suc 0)} (f ` {j. j ≤ n}) ∧
Σe M (λj. s j ·s (f j)) n =
Σe M (λk. t k ·s (g k)) (card (f ` {j. j ≤ n}) - Suc 0)"
apply (simp add:same_togetherTr[of A H])
done
lemma (in Module) one_last:"[|ideal R A; H ⊆ carrier M;
s ∈ {j. j ≤ (Suc n)} -> A; f ∈ {j. j ≤ (Suc n)} -> H;
bij_to f {j. j ≤ (Suc n)} H; j ≤ (Suc n); j ≠ (Suc n)|] ==>
∃t ∈ {j. j ≤ (Suc n)} -> A. ∃g ∈ {j. j ≤ (Suc n)} -> H.
Σe M (λk. s k ·s (f k)) (Suc n) = Σe M (λk. t k ·s (g k)) (Suc n) ∧
g (Suc n) = f j ∧ t (Suc n) = s j ∧ bij_to g {j. j ≤ (Suc n)} H"
apply (cut_tac sc_Ring)
apply (subgoal_tac "(λk. s k ·s (f k)) ∈ {j. j ≤ Suc n} -> carrier M")
apply (frule transpos_hom[of j "Suc n" "Suc n"], simp, assumption,
frule transpos_inj[of j "Suc n" "Suc n"], simp, assumption,
frule_tac f1 = "λk. s k ·s (f k)" and n1 = n and h1 =
"transpos j (Suc n)" in addition2 [THEN sym], assumption+,
simp del:nsum_suc)
prefer 2
apply (rule univar_func_test, rule ballI, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem subsetD)
apply (frule cmp_fun[of "transpos j (Suc n)" "{j. j ≤ Suc n}"
"{j. j ≤ Suc n}" s A], assumption+,
frule cmp_fun[of "transpos j (Suc n)" "{j. j ≤ Suc n}"
"{j. j ≤ Suc n}" f H], assumption+)
apply (simp del:nsum_suc add:l_comb_transpos[of A H])
apply (subgoal_tac "bij_to (cmp f (transpos j (Suc n))) {j. j ≤ (Suc n)} H")
apply (subgoal_tac "(cmp f (transpos j (Suc n))) (Suc n) = f j")
apply (subgoal_tac "(cmp s (transpos j (Suc n))) (Suc n) = s j")
apply blast
apply (simp add:cmp_def, simp add:transpos_ij_2,
simp add:cmp_def, simp add:transpos_ij_2)
apply (simp add:bij_to_def, rule conjI,
rule cmp_surj[of "transpos j (Suc n)" "{j. j ≤ Suc n}"
"{j. j ≤ Suc n}" f H], assumption+,
simp add:transpos_surjec, assumption+, simp)
apply (rule cmp_inj[of "transpos j (Suc n)" "{j. j ≤ Suc n}"
"{j. j ≤ Suc n}" f H], assumption+, simp)
done
lemma (in Module) finite_lin_spanTr1:"[|ideal R A; z ∈ carrier M|] ==>
h ∈ {j. j ≤ (n::nat)} -> {z} ∧ t ∈ {j. j ≤ n} -> A -->
(∃s∈{0::nat} -> A. Σe M (λj. t j ·s (h j)) n = s 0 ·s z)"
apply (induct_tac n)
apply (rule impI)
apply ((erule conjE)+, simp)
apply (frule_tac f = h and A = "{0}" and B = "{z}" and x = 0 in funcset_mem,
simp, simp)
apply blast
apply (rule impI) apply (erule conjE)+
apply (frule func_pre [of _ _ "{z}"], frule func_pre [of _ _ "A"])
apply (simp del:nsum_suc, erule bexE, simp,
frule_tac f = h and A = "{j. j ≤ Suc n}" and B = "{z}" and x = "Suc n"
in funcset_mem, simp, simp,
frule_tac f = s and A = "{0}" and B = A and x = 0 in funcset_mem,
simp,
frule_tac f = t and A = "{j. j ≤ Suc n}" and B = A and x = "Suc n" in
funcset_mem, simp, cut_tac sc_Ring,
frule_tac h = "s 0" in Ring.ideal_subset[of R A], assumption+,
frule_tac h = "t (Suc n)" in Ring.ideal_subset[of R A], assumption+)
apply (simp add:sc_l_distr[THEN sym])
apply (subgoal_tac "(λl∈{0::nat}. (s 0 ±R (t (Suc n)))) ∈ {0} -> A")
apply (subgoal_tac "(s 0 ±R t (Suc n)) ·s z = (λl∈{0::nat}. (s 0 ±R (t (Suc n)))) 0 ·s z ") apply blast
apply simp
apply (rule univar_func_test) apply (rule ballI) apply simp
apply (rule Ring.ideal_pOp_closed, assumption+)
done
lemma (in Module) single_span:"[|ideal R A; z ∈ carrier M;
h ∈ {j. j ≤ (n::nat)} -> {z}; t ∈ {j. j ≤ n} -> A|] ==>
∃s∈{0::nat} -> A. Σe M (λj. t j ·s (h j)) n = s 0 ·s z"
apply (simp add:finite_lin_spanTr1)
done
(*
lemma (in Module) finite_lin_spanTr2:"[|ideal R A; ∀m.
(∃n1. ∃f∈{j. j ≤ n1} -> h ` {j. j ≤ n}. ∃s∈{j. j ≤ n1} -> A.
m = Σe M (λj. s j ·s (f j)) n1) -->
(∃s∈{j. j ≤ n} -> A. m = Σe M (λj. s j ·s (h j)) n);
h ∈ {j. j ≤ (Suc n)} -> carrier M; f ∈ {j. j ≤ n1} -> h ` {j. j ≤ n};
s ∈ {j. j ≤ n1} -> A; m = Σe M (λj. s j ·s (f j)) n1|] ==>
∃sa∈{j. j ≤ (Suc n)} -> A. Σe M (λj. s j ·s (f j)) n1 =
Σe M (λj. sa j ·s (h j)) n ± (sa (Suc n) ·s (h (Suc n)))"
apply (frule_tac
apply (subgoal_tac "∃l∈{j. j ≤ n} -> A. m = Σe M (λj. l j ·s (h j)) n")
prefer 2
apply (thin_tac "h ∈ {j. j ≤ (Suc n)} -> carrier M")
apply blast
apply (thin_tac " ∀m. (∃n1. ∃f∈Nset n1 -> h ` Nset n.
∃s∈Nset n1 -> A. m = eΣ M (λj. s j ∗M (f j)) n1) -->
(∃s∈Nset n -> A. m = eΣ M (λj. s j ∗M (h j)) n)")
apply (subgoal_tac "∀l∈Nset n -> A. m = eΣ M (λj. l j ∗M (h j)) n --> (∃sa∈Nset (Suc n) -> A. eΣ M (λj. s j ∗M (f j)) n1 = eΣ M (λj. sa j ∗M (h j)) n +M (sa (Suc n) ∗M (h (Suc n))))")
apply blast
apply (thin_tac "∃l∈Nset n -> A. m = eΣ M (λj. l j ∗M (h j)) n")
apply (rule ballI) apply (rule impI)
apply (frule sym) apply (thin_tac "m = eΣ M (λj. s j ∗M (f j)) n1")
apply simp
apply (thin_tac "m = eΣ M (λj. l j ∗M (h j)) n")
apply (thin_tac "eΣ M (λj. s j ∗M (f j)) n1 = eΣ M (λj. l j ∗M (h j)) n")
apply (subgoal_tac "jointfun n l 0 (λx∈Nset 0. (0R)) ∈ Nset (Suc n) -> A")
apply (subgoal_tac " eΣ M (λj. l j ∗M (h j)) n =
eΣ M (λj. (jointfun n l 0 (λx∈Nset 0. (0R))) j ∗M (h j)) n +M ((jointfun n l 0 (λx∈Nset 0. (0R))) (Suc n)) ∗M (h (Suc n))")
apply blast
apply (subgoal_tac "jointfun n l 0 (λx∈Nset 0. 0R) (Suc n) ∗M (h (Suc n)) =
0M") apply simp
apply (subgoal_tac "eΣ M (λj. jointfun n l 0 (λx∈Nset 0. 0R) j ∗M (h j)) n =
eΣ M (λj. l j ∗M (h j)) n ") apply simp
apply (frule module_is_ag [of "R" "M"], assumption+)
apply (subst ag_r_zero, assumption+)
apply (subgoal_tac "(λj. l j ∗M (h j)) ∈ Nset n -> carrier M")
apply (rule eSum_mem, assumption+) apply (simp add:n_in_Nsetn)
apply (rule univar_func_test) apply (rule ballI)
apply (rule sprod_mem, assumption+)
apply (simp add:funcset_mem ideal_subset)
apply (frule func_pre [of "h" _ "carrier M"])
apply (simp add:funcset_mem) apply simp
apply (rule eSum_eq)
apply (rule module_is_ag [of "R" "M"], assumption+)
apply (rule univar_func_test)
apply (rule ballI)
apply (frule_tac x = x and n = n in Nset_le)
apply (insert Nset_nonempty[of "0"])
apply (simp add:jointfun_def)
apply (rule sprod_mem, assumption+)
apply (simp add:funcset_mem ideal_subset)
apply (frule func_pre [of "h" _ "carrier M"])
apply (simp add:funcset_mem)
apply (rule univar_func_test) apply (rule ballI)
apply (rule sprod_mem, assumption+)
apply (simp add:funcset_mem ideal_subset)
apply (frule func_pre [of "h" _ "carrier M"])
apply (simp add:funcset_mem)
apply (rule ballI)
apply (frule_tac x = la and n = n in Nset_le)
apply (simp add:jointfun_def)
apply (subgoal_tac "0 ∈ Nset 0")
apply (simp add:jointfun_def sliden_def slide_def)
apply (rule sprod_0_m, assumption+)
apply (subgoal_tac "Suc n ∈ Nset (Suc n)")
apply (simp add:funcset_mem) apply (simp add:n_in_Nsetn)+
apply (frule_tac f = l and n = n and A = A and g = "λx∈Nset 0. 0R" and m = 0
and B = A in jointfun_hom0)
apply (rule univar_func_test) apply (rule ballI) apply (simp add:Nset_def)
apply (simp add:ideal_zero) apply simp
done *)
constdefs
coeff_at_k::"[('r, 'm) Ring_scheme, 'r, nat] => (nat => 'r)"
"coeff_at_k R a k == λj. if j = k then a else (\<zero>R)"
lemma card_Nset_im:"f ∈ {j. j ≤ (n::nat)} -> A ==>
(Suc 0) ≤ card (f `{j. j ≤ n})"
apply (cut_tac image_Nsetn_card_pos[of f n])
apply (frule_tac x = 0 and n = "card (f ` {i. i ≤ n})" in less_Suc_le1,
assumption+)
done
lemma (in Module) eSum_changeTr1:"[|ideal R A;
t ∈ {k. k ≤ (card (f ` {j. j ≤ (n1::nat)}) - Suc 0)} -> A;
g ∈ {k. k ≤ (card (f ` {j. j ≤ n1}) - Suc 0)} -> f `{j. j ≤ n1};
Suc 0 < card (f `{j. j ≤ n1}); g x = h (Suc n); x = Suc n;
card (f `{j. j ≤ n1}) - Suc 0 = Suc (card (f ` {j. j ≤ n1}) - Suc 0 - Suc 0)|]
==>
Σe M (λk. t k ·s (g k)) (card (f ` {j. j ≤ n1}) - Suc 0) =
Σe M (λk. t k ·s (g k)) (card (f ` {j. j ≤ n1}) - Suc 0 - Suc 0) ±
(t (Suc (card (f ` {j. j ≤ n1}) - Suc 0 - Suc 0)) ·s
(g ( Suc (card (f ` {j. j ≤ n1}) - Suc 0 - Suc 0))))"
apply simp
done
constdefs
zeroi::"[('r, 'm) Ring_scheme] => nat => 'r"
"zeroi R == λj. \<zero>R"
lemma zeroi_func:"[|Ring R; ideal R A|] ==> zeroi R ∈ {j. j ≤ 0} -> A"
apply (rule univar_func_test, rule ballI)
apply (simp add:zeroi_def Ring.ideal_zero)
done
lemma (in Module) prep_arrTr1:"[|ideal R A; h ∈ {j. j ≤ (Suc n)} -> carrier M;
f ∈ {j. j ≤ (n1::nat)} -> h ` {j. j ≤ (Suc n)}; s ∈ {j. j ≤ n1}-> A;
m = l_comb R M n1 s f|] ==>
∃l∈{j. j ≤ (Suc n)}. (∃s∈{j. j ≤ (l::nat)} -> A.
∃g∈ {j. j ≤ l} -> h `{j. j ≤ (Suc n)}. m = l_comb R M l s g ∧
bij_to g {j. j ≤ l} (f ` {j. j ≤ n1}))"
apply (cut_tac sc_Ring)
apply (frule_tac s = s and n = n1 and f = f in same_together[of A
"h ` {j. j ≤ (Suc n)}"])
apply (simp add:image_sub0, assumption+)
apply (erule bexE)+
apply (simp add:l_comb_def, erule conjE)
apply (thin_tac "Σe M (λj. s j ·s f j) n1 =
Σe M (λk. t k ·s g k) (card (f ` {j. j ≤ n1}) - Suc 0)")
apply (subgoal_tac "(card (f ` {j. j ≤ n1}) - Suc 0) ∈ {j. j ≤ Suc n}")
apply (subgoal_tac "g ∈ {k. k ≤ (card (f `{j. j ≤ n1}) - Suc 0)} ->
h ` {j. j ≤ Suc n}")
apply (subgoal_tac "bij_to g {k. k ≤ (card (f ` {j. j ≤ n1}) - Suc 0)} (f ` {j. j ≤ n1})")
apply blast
prefer 2
apply (frule_tac f = f and A = "{j. j ≤ n1}" and B = "h ` {j. j ≤ Suc n}"
in image_sub0, simp)
apply (rule extend_fun, assumption+)
apply (simp add:bij_to_def)
apply (rule_tac A = "f ` {j. j ≤ n1}" and n = "card (f `{j. j ≤ n1}) - Suc 0" and f = g in Nset2finite_inj)
apply (rule finite_imageI, simp add:finite_Nset)
apply (frule_tac f = f and n = n1 and A = "h ` {j. j ≤ (Suc n)}" in card_Nset_im)
apply (simp, assumption)
apply (subgoal_tac "finite (h ` {j. j ≤ (Suc n)})")
apply (frule_tac f = f and A = "{j. j ≤ n1}" and B = "h ` {j. j ≤ (Suc n)}"
in image_sub0, simp)
apply (frule_tac B = "h ` {j. j ≤ (Suc n)}" and A = "f ` {j. j ≤ n1}" in
card_mono, assumption+,
insert finite_Nset [of "Suc n"],
frule card_image_le [of "{j. j ≤ (Suc n)}" "h"],
frule_tac i = "card (f ` {j. j ≤ n1})" and
j = "card (h ` {j. j ≤ (Suc n)})" and k = "card {j. j ≤ (Suc n)}" in
le_trans, assumption+)
apply (simp add:card_Nset[of "Suc n"])
apply (rule finite_imageI, simp add:finite_Nset)
done
lemma two_func_imageTr:"[| h ∈ {j. j ≤ Suc n} -> B;
f ∈ {j. j ≤ (m::nat)} -> h ` {j. j ≤ Suc n}; h (Suc n) ∉ f ` {j. j ≤ m}|]
==> f ∈ {j. j ≤ m} -> h ` {j. j ≤ n}"
apply (rule univar_func_test, rule ballI)
apply (frule_tac x = x and f = f and A = "{j. j ≤ m}" and
B = "h ` {j. j ≤ Suc n}" in funcset_mem, assumption)
apply (thin_tac "h ∈ {j. j ≤ Suc n} -> B")
apply (rule contrapos_pp, simp+)
apply (simp add:image_def[of h])
apply (erule exE, erule conjE)
apply (case_tac "xa ≠ Suc n",
frule_tac m = xa and n = "Suc n" in noteq_le_less, assumption)
apply (
thin_tac "xa ≤ Suc n",
frule_tac x = xa and n = "Suc n" in less_le_diff,
thin_tac "xa < Suc n", simp) apply blast
apply simp
apply (subgoal_tac "(f x) ∈ f ` {j. j ≤ m}", simp)
apply (thin_tac "h (Suc n) ∉ f ` {j. j ≤ m}",
thin_tac "∀x≤n. h (Suc n) ≠ h x",
thin_tac "f x = h (Suc n)",
thin_tac "xa = Suc n")
apply (simp add:image_def, blast)
done
lemma (in Module) finite_lin_spanTr3_0:"[|bij_to g {j. j ≤ l} (g `{j. j ≤ l});
ideal R A;
∀na. ∀s∈{j. j ≤ na} -> A.
∀f∈{j. j ≤ na} -> h ` {j. j ≤ n}.
∃t∈{j. j ≤ n} -> A. l_comb R M na s f = l_comb R M n t h;
h ∈ {j. j ≤ Suc n} -> carrier M; s ∈ {j. j ≤ m} -> A;
f ∈ {j. j ≤ m} -> h ` {j. j ≤ Suc n};
l ≤ Suc n; sa ∈ {j. j ≤ l} -> A; g ∈ {j. j ≤ l} -> h ` {j. j ≤ Suc n};
0 < l; f ` {j. j ≤ m} = g ` {j. j ≤ l}; h (Suc n) = g l|]
==> ∃t∈{j. j ≤ Suc n} -> A. l_comb R M l sa g = l_comb R M (Suc n) t h"
apply (cut_tac sc_Ring)
apply (subgoal_tac "l_comb R M l sa g = l_comb R M (Suc (l - Suc 0)) sa g",
simp del:Suc_pred,
thin_tac "l_comb R M l sa g = l_comb R M (Suc (l - Suc 0)) sa g",
simp del:Suc_pred add:l_comb_def)
apply (drule_tac a = "l - Suc 0" in forall_spec1,
drule_tac b = sa in forball_spec1)
apply (rule univar_func_test, rule ballI, simp)
apply (rule_tac x = x and f = sa and A = "{j. j ≤ l}"and B = A
in funcset_mem, assumption, simp) (*
apply (rule_tac i = x and j = "l - Suc 0" and k = l in le_trans)
apply (
assumption, subst Suc_le_mono[THEN sym], simp) *)
apply (drule_tac b = g in forball_spec1,
thin_tac "f ∈ {j. j ≤ m} -> h ` {j. j ≤ Suc n}",
thin_tac "sa ∈ {j. j ≤ l} -> A",
thin_tac "f ` {j. j ≤ m} = g ` {j. j ≤ l}")
apply (rule univar_func_test, rule ballI, simp)
apply (frule_tac x = x and f = g and A = "{j. j ≤ l}" and
B = "h ` {j. j ≤ Suc n}" in funcset_mem)
apply simp (*
apply (rule_tac i = x and j = "l - Suc 0" and k = l in Nat.le_trans,
assumption, subst Suc_le_mono[THEN sym], simp) *)
apply (unfold bij_to_def, frule conjunct2, fold bij_to_def,
thin_tac "bij_to g {j. j ≤ l} (g ` {j. j ≤ l})",
thin_tac "g ∈ {j. j ≤ l} -> h ` {j. j ≤ Suc n}")
apply (simp add:image_def, erule exE, erule conjE)
apply (case_tac "xa = Suc n", simp add:inj_on_def,
drule_tac a = x in forall_spec) apply simp
(*
apply (frule_tac i = x and j = "l - Suc 0" and k = l in Nat.le_trans,
subst Suc_le_mono[THEN sym], simp, assumption) *)
apply(drule_tac a = l in forall_spec, simp)
apply (cut_tac n1 = l and m1 = "l - Suc 0" in Suc_le_mono[THEN sym])
apply simp
apply (frule_tac m = xa and n = "Suc n" in noteq_le_less, assumption,
thin_tac "xa ≤ Suc n",
frule_tac x = xa and n = "Suc n" in less_le_diff,
thin_tac "xa < Suc n", simp)
apply blast
apply (erule bexE, simp)
apply (rotate_tac -4, frule sym, thin_tac "h (Suc n) = g l", simp)
apply (frule_tac f = t and n = n and A = A and g = "λk∈{0::nat}. sa l"
and m = 0 and B = A in jointfun_hom0,
rule univar_func_test, rule ballI, simp add:funcset_mem, simp)
apply (subgoal_tac " Σe M (λj. t j ·s h j) n ± sa l ·s h (Suc n) =
Σe M (λj. (jointfun n t 0 (λk∈{0}. sa l)) j ·s h j) (Suc n)",
simp, blast)
apply (cut_tac H = "carrier M" and A = A and s = t and f = h and n = n and
m = 0 and t = "λk∈{0}. sa l" in l_comb_jointfun_jf)
apply simp+
apply (rule univar_func_test, rule ballI, simp add:funcset_mem)
apply simp
apply (simp add:jointfun_def sliden_def, simp)
done
lemma (in Module) finite_lin_spanTr3:"ideal R A ==>
h ∈ {j. j ≤ (n::nat)} -> carrier M -->
(∀na. ∀s ∈ {j. j ≤ (na::nat)} -> A.
∀f∈ {j. j ≤ na} -> (h ` {j. j ≤ n}). (∃t ∈ {j. j ≤ n} -> A.
l_comb R M na s f = l_comb R M n t h))"
apply (cut_tac sc_Ring)
apply (induct_tac n)
apply (rule impI, rule allI, (rule ballI)+)
apply (insert Nset_nonempty [of "0"])
apply (simp add:l_comb_def)
apply (frule_tac z = "h 0" and h = f and t = s and n = na in
single_span [of A])
apply (simp add:funcset_mem)
apply assumption+
(********** n = 0 done ***********)
apply (rule impI, rule allI, (rule ballI)+)
apply (frule func_pre, simp)
apply (case_tac "h (Suc n) ∉ f ` {j. j ≤ na}")
apply (frule_tac h = h and n = n and B = "carrier M" and f = f and
m = na in two_func_imageTr, assumption+)
apply (drule_tac a = na in forall_spec1,
drule_tac b = s in forball_spec1, assumption,
drule_tac b = f in forball_spec1, assumption)
apply (erule bexE, simp )
apply (thin_tac "l_comb R M na s f = l_comb R M n t h")
apply (simp add:l_comb_def)
apply (subgoal_tac "Σe M (λj. t j ·s (h j)) n =
Σe M (λj. (jointfun n t 0 (zeroi R)) j ·s (h j)) (Suc n)", simp,
thin_tac "Σe M (λj. t j ·s h j) n =
Σe M (λj. jointfun n t 0 (zeroi R) j ·s h j) n ±
jointfun n t 0 (zeroi R) (Suc n) ·s h (Suc n)")
apply (frule_tac f = t and n = n and g = "zeroi R" and m = 0 and A = A and
B = A in jointfun_hom)
apply (rule zeroi_func, assumption+, simp, blast)
apply (cut_tac H = "carrier M" and s = t and n = n and f = h and m = 0 and
t = "zeroi R" in l_comb_jointfun_jf[of _ A],
simp, assumption+, simp,
rule zeroi_func, assumption+)
apply (simp,
thin_tac "Σe M (λj. jointfun n t 0 (zeroi R) j ·s h j) n =
Σe M (λj. t j ·s h j) n",
simp add:jointfun_def sliden_def zeroi_def,
subst sc_0_m, simp add:funcset_mem,
subst ag_r_zero,
rule nsum_mem, rule allI, rule impI, rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem,
simp)
(*** case h (Suc n) ∉ f ` (Nset na) done ***)
apply simp
apply (frule_tac h = h and n = n and m = "l_comb R M na s f" in
prep_arrTr1 [of "A"], assumption+, simp)
apply (erule bexE)+
apply (simp, (erule conjE)+)
apply (case_tac "l = 0", simp)
apply (unfold bij_to_def, frule conjunct1, frule conjunct2, fold bij_to_def)
apply (thin_tac "l_comb R M na s f = l_comb R M 0 sa g")
apply (simp add:l_comb_def)
apply (simp add:surj_to_def, rotate_tac -1, frule sym,
thin_tac "{g 0} = f ` {j. j ≤ na}", simp,
rotate_tac -6, frule sym, thin_tac "h (Suc n) = g 0", simp)
apply (cut_tac f = "zeroi R" and n = n and g = "λj. sa 0" and m = 0 and
A = A and B = A in jointfun_hom0)
apply (rule univar_func_test, rule ballI,
simp add:zeroi_def Ring.ideal_zero)
apply (rule univar_func_test, rule ballI, simp add:funcset_mem)
apply simp
apply (subgoal_tac "sa 0 ·s h (Suc n) = nsum M (λj. (jointfun n (zeroi R) 0
(λj. sa 0) j ·s h j)) (Suc n)", simp,
thin_tac "sa 0 ·s h (Suc n) =
Σe M (λj. jointfun n (zeroi R) 0 (λj. sa 0) j ·s h j) n ±
jointfun n (zeroi R) 0 (λj. sa 0) (Suc n) ·s h (Suc n)",
blast)
apply simp
apply (cut_tac n = n and f = "λj. jointfun n (zeroi R) 0 (λj. sa 0) j ·s h j"
in nsum_zeroA)
apply (rule allI, rule impI,
simp add:jointfun_def zeroi_def,
rule sc_0_m, simp add:funcset_mem, simp,
thin_tac "Σe M (λj. jointfun n (zeroi R) 0 (λj. sa 0) j ·s h j) n = \<zero>")
apply (simp add:jointfun_def sliden_def,
subst ag_l_zero,
rule sc_mem, simp add:funcset_mem Ring.ideal_subset,
simp add:funcset_mem, simp)
(**** l = 0 done ***)
apply (simp)
apply (thin_tac "l_comb R M na s f = l_comb R M l sa g")
apply (unfold bij_to_def, frule conjunct1, frule conjunct2, fold bij_to_def)
apply (simp add:surj_to_def, rotate_tac -2, frule sym,
thin_tac "g ` {j. j ≤ l} = f ` {j. j ≤ na}", simp)
apply (subgoal_tac "∃x∈{j. j ≤ l}. h (Suc n) = g x")
prefer 2 apply (simp add:image_def)
apply (erule bexE)
apply (case_tac "x = l", simp)
apply (frule_tac g = g and l = l and A = A and h = h and n = n and s = s and
m = na and f = f and l = l and sa = sa in finite_lin_spanTr3_0,
assumption+)
apply (subgoal_tac "l_comb R M l sa g = l_comb R M (Suc (l - Suc 0)) sa g")
prefer 2 apply simp
apply (simp del:nsum_suc Suc_pred,
thin_tac "l_comb R M l sa g = l_comb R M (Suc (l - Suc 0)) sa g",
simp del:nsum_suc Suc_pred add:l_comb_def)
apply (cut_tac f1 = "λj. sa j ·s g j" and n1 = "l - Suc 0" and
h1 = "transpos x (Suc (l - Suc 0))" in addition2[THEN sym],
thin_tac "∀na. ∀s∈{j. j ≤ na} -> A.
∀f∈{j. j ≤ na} -> h ` {j. j ≤ n}.
∃t∈{j. j ≤ n} -> A.
Σe M (λj. s j ·s f j) na = Σe M (λj. t j ·s h j) n",
rule univar_func_test, rule ballI, simp)
apply (rule sc_mem,
simp add:funcset_mem Ring.ideal_subset,
frule_tac f = h and A = "{j. j ≤ Suc n}" and B = "carrier M" in
image_sub0,
frule_tac x = xa and f = g and A = "{j. j ≤ l}" and
B = "h ` {j. j ≤ Suc n}" in funcset_mem, simp, simp add:subsetD)
apply (simp,
rule_tac i = x and n = l and j = l in transpos_hom,
assumption+, simp, assumption+)
apply (simp,
rule_tac i = x and n = l and j = l in transpos_inj,
assumption+, simp, assumption+)
apply (simp del:Suc_pred nsum_suc)
apply (subst l_comb_transpos[of A "carrier M"], assumption, simp,
simp, simp,
rule univar_func_test, rule ballI,
frule_tac f = h and A = "{j. j ≤ Suc n}" and B = "carrier M" in
image_sub0,
frule_tac x = xa and f = g and A = "{j. j ≤ l}" and
B = "h ` {j. j ≤ Suc n}" in funcset_mem, simp, simp add:subsetD,
simp)
apply (simp del:Suc_pred, simp,
thin_tac "Σe M (λj. sa j ·s g j) (l - Suc 0) ± sa l ·s g l =
Σe M (cmp (λj. sa j ·s g j) (transpos x l)) (l - Suc 0) ±
cmp (λj. sa j ·s g j) (transpos x l) l")
apply (cut_tac g = "cmp g (transpos x l)" and l = l and A = A and
h = h and n = n and s = s and m = na and f = f and l = l and
sa = "cmp sa (transpos x l)" in finite_lin_spanTr3_0)
apply (frule_tac i = x and n = l and j = l in transpos_hom,
simp, assumption)
apply (cut_tac n = l in Nat.le_refl)
apply (frule_tac i = x and n = l and j = l in transpos_surjec, assumption+)
apply (frule_tac f = "transpos x l" and A = "{j. j ≤ l}" and
B = "{j. j ≤ l}" and g = g and C = "g ` {j. j ≤ l}" in cmp_surj,
assumption+)
apply (rule_tac f = g and A = "{j. j ≤ l}" and B = "h ` {j. j ≤ Suc n}"
in func_to_img, assumption)
apply (simp add:bij_to_def)
apply (subst bij_to_def, simp)
apply (subgoal_tac "cmp g (transpos x l) ` {j. j ≤ l} = g ` {j. j ≤ l}",
simp)
apply (frule_tac f = "transpos x l" and A = "{j. j ≤ l}" and
B = "{j. j ≤ l}" and g = g and C = "h ` {j. j ≤ Suc n}" in cmp_inj,
assumption+)
apply (rule_tac i = x and n = l and j = l in transpos_inj, assumption,
simp, assumption, simp add:bij_to_def,
assumption)
apply (simp add:cmp_fun_image, simp add:surj_to_def)
apply assumption+
apply (simp add:l_comb_def)
apply assumption+
apply (rule univar_func_test, rule ballI)
apply (simp add:cmp_def)
apply (cut_tac n = l in Nat.le_refl,
frule_tac i = x and n = l and j = l and l = xa in transpos_mem,
assumption+,
simp add:funcset_mem)
apply (rule univar_func_test, rule ballI,
simp add:cmp_def,
cut_tac n = l in Nat.le_refl,
frule_tac i = x and n = l and j = l and l = xa in transpos_mem,
assumption+,
simp add:funcset_mem)
apply simp
apply (cut_tac n = l in Nat.le_refl,
frule_tac i = x and n = l and j = l in transpos_surjec, assumption+)
apply (frule_tac i = x and n = l and j = l in transpos_hom,
simp, assumption)
apply (frule_tac f = "transpos x l" and A = "{i. i ≤ l}" and
B = "{i. i ≤ l}" and g = g and C = "h ` {j. j ≤ Suc n}" in
cmp_fun_image, assumption+)
apply (simp add:surj_to_def)
apply (simp add:cmp_def)
apply (simp add:transpos_ij_2)
apply (erule bexE)
apply (thin_tac "∀na. ∀s∈{j. j ≤ na} -> A.
∀f∈{j. j ≤ na} -> h ` {j. j ≤ n}.
∃t∈{j. j ≤ n} -> A.
Σe M (λj. s j ·s f j) na = Σe M (λj. t j ·s h j) n")
apply (rename_tac n na s f l sa g x sb)
apply (subgoal_tac "l_comb R M l (cmp sa (transpos x l))
(cmp g (transpos x l)) = l_comb R M (Suc (l - Suc 0))
(cmp sa (transpos x l)) (cmp g (transpos x l)) ",
simp del:Suc_pred,
thin_tac "l_comb R M l (cmp sa (transpos x l)) (cmp g (transpos x l)) =
l_comb R M (Suc n) sb h")
apply (simp del:Suc_pred add:l_comb_def, simp,
thin_tac " Σe M (λj. cmp sa (transpos x l) j ·s
cmp g (transpos x l) j) (l - Suc 0) ±
cmp sa (transpos x l) l ·s cmp g (transpos x l) l =
Σe M (λj. sb j ·s h j) n ± sb (Suc n) ·s g x")
apply (rotate_tac -3, frule sym, thin_tac "h (Suc n) = g x", simp)
apply blast
apply simp
done
lemma (in Module) finite_lin_span:
"[|ideal R A; h ∈ {j. j ≤ (n::nat)} -> carrier M; s ∈ {j. j ≤ (n1::nat)} -> A;
f ∈ {j. j ≤ n1} -> h ` {j. j ≤ n}|] ==> ∃t∈{j. j ≤ n} -> A.
l_comb R M n1 s f = l_comb R M n t h"
apply (simp add:finite_lin_spanTr3)
done
subsection "4-2. free generators"
constdefs
free_generator::"[('r, 'm) Ring_scheme, ('a, 'r, 'm1) Module_scheme, 'a set]
=> bool"
"free_generator R M H == generator R M H ∧
(∀n. (∀s f. (s ∈ {j. j ≤ (n::nat)} -> carrier R ∧
f ∈ {j. j ≤ n} -> H ∧ inj_on f {j. j ≤ n} ∧
l_comb R M n s f = \<zero>M) --> s ∈ {j. j ≤ n} -> {\<zero>R}))"
lemma (in Module) free_generator_generator:"free_generator R M H ==>
generator R M H"
by (simp add:free_generator_def)
lemma (in Module) free_generator_sub:"free_generator R M H ==>
H ⊆ carrier M"
by (simp add:free_generator_def generator_def)
lemma (in Module) free_generator_nonzero:"[|¬ (zeroring R);
free_generator R M H; h ∈ H|] ==> h ≠ \<zero>"
apply (cut_tac sc_Ring)
apply (rule contrapos_pp, simp+)
apply (simp add:free_generator_def, (erule conjE)+)
apply (subgoal_tac "(λt. 1rR) ∈ {j. j ≤ (0::nat)} -> carrier R")
apply (subgoal_tac "(λt. \<zero>) ∈ {j. j ≤ (0::nat)} -> H ∧
inj_on (λt. \<zero>) {j. j ≤ (0::nat)} ∧
l_comb R M 0 (λt. 1rR) (λt. \<zero>) = \<zero>")
apply (subgoal_tac "(λt. 1rR) ∈ {j. j ≤ (0::nat)} -> {\<zero>R}")
prefer 2 apply blast
apply (frule_tac f = "λt. 1rR" and A = "{j. j ≤ (0::nat)}" and B = "{\<zero>R}"
and x = 0 in funcset_mem, simp, simp)
apply (frule Ring.Zero_ring1 [of "R"], assumption+, simp)
apply simp
apply (thin_tac "∀n s. s ∈ {j. j ≤ n} -> carrier R ∧
(∃f. f ∈ {j. j ≤ n} -> H ∧
inj_on f {j. j ≤ n} ∧ l_comb R M n s f = \<zero>) -->
s ∈ {j. j ≤ n} -> {\<zero>R}")
apply (rule conjI)
apply (rule univar_func_test, rule ballI, simp)
apply (simp add:l_comb_def)
apply (rule sc_a_0)
apply (simp add:Ring.ring_one)
apply (rule univar_func_test, rule ballI)
apply (simp add:Ring.ring_one)
done
lemma (in Module) has_free_generator_nonzeroring:" [|free_generator R M H;
∃p ∈ linear_span R M (carrier R) H. p ≠ \<zero> |] ==> ¬ zeroring R"
apply (erule bexE, simp add:linear_span_def)
apply (case_tac "H = {}", simp, simp)
apply (erule exE, (erule bexE)+, simp,
thin_tac "p = l_comb R M n s f")
apply (rule contrapos_pp, simp+)
apply (simp add:zeroring_def, erule conjE)
apply (frule Ring.ring_one[of "R"], simp)
apply (simp add:l_comb_def)
apply (cut_tac n = n and f = "λj. s j ·s f j" in nsum_zeroA)
apply (rule allI, rule impI)
apply (simp add:free_generator_def generator_def, frule conjunct1,
frule_tac x = j and f = f and A = "{j. j ≤ n}" and B = H in
funcset_mem, simp,
frule_tac c = "f j" in subsetD[of H "carrier M"], assumption+,
frule_tac x = j and f = s and A = "{j. j ≤ n}" and B = "{\<zero>R}" in
funcset_mem, simp, simp add:sc_0_m)
apply simp
done
lemma (in Module) unique_expression1:"[|H ⊆ carrier M; free_generator R M H;
s ∈ {j. j ≤ (n::nat)} -> carrier R; m ∈ {j. j ≤ n} -> H;
inj_on m {j. j ≤ n}; l_comb R M n s m = \<zero>|] ==>
∀j∈{j. j ≤ n}. s j = \<zero>R"
apply (rule ballI)
apply (simp add:free_generator_def, (erule conjE)+)
apply (subgoal_tac "s ∈ {j. j ≤ n} -> {\<zero>R}")
apply (frule_tac f = s and A = "{j. j ≤ n}" and B = "{\<zero>R}" and x = j in
funcset_mem, simp, simp)
apply blast
done
lemma (in Module) free_gen_coeff_zero:"[|H ⊆ carrier M; free_generator R M H;
h ∈ H; a ∈ carrier R; a ·s h = \<zero>|] ==> a = \<zero>R"
apply (frule unique_expression1[of H "λx∈{0::nat}. a" 0 "λx∈{0::nat}. h"],
assumption+,
rule univar_func_test, rule ballI, simp,
rule univar_func_test, rule ballI, simp,
simp add:inj_on_def,
simp add:l_comb_def,
simp)
done
lemma (in Module) unique_expression2:"[|H ⊆ carrier M;
f ∈ {j. j ≤ (n::nat)} -> H; s ∈ {j. j ≤ n} -> carrier R|] ==>
∃m g t. g ∈ ({j. j ≤ (m::nat)} -> H) ∧
bij_to g {j. j ≤ (m::nat)} (f ` {j. j ≤ n}) ∧
t ∈ {j. j ≤ m} -> carrier R ∧
l_comb R M n s f = l_comb R M m t g"
apply (cut_tac sc_Ring)
apply (frule Ring.whole_ideal [of "R"])
apply (frule_tac A = "carrier R" and H = H and s = s and f = f in
same_together, assumption+)
apply ((erule bexE)+, erule conjE)
apply (frule_tac f = f and A = "{j. j ≤ n}" in image_sub0,
frule_tac f = g and A = "{j. j ≤ card (f ` {j. j ≤ n}) - Suc 0}"
and B = "f ` {j. j ≤ n}" in extend_fun[of _ _ _ "H"], assumption)
apply (subgoal_tac "bij_to g {j. j ≤ (card (f ` {j. j ≤ n}) - Suc 0)}
(f ` {j. j ≤ n})")
apply (simp add:l_comb_def, blast)
apply (simp add:bij_to_def)
apply (cut_tac finite_Nset[of n],
frule finite_imageI[of "{j. j ≤ n}" f])
apply (rule_tac A = "f ` {j. j ≤ n}" and n = "card (f ` {j. j ≤ n}) -
Suc 0" and f = g in Nset2finite_inj, assumption)
using image_Nsetn_card_pos[of f n] apply simp
apply assumption
done
lemma (in Module) unique_expression3_1:"[|H ⊆ carrier M;
f ∈ {l. l ≤ (Suc n)} -> H; s ∈ {l. l ≤ (Suc n)} -> carrier R;
(f (Suc n)) ∉ f `({l. l ≤ (Suc n)} - {Suc n})|] ==>
∃g m t. g ∈ {l. l ≤ (m::nat)} -> H ∧
inj_on g {l. l ≤ (m::nat)} ∧
t ∈ {l. l ≤ (m::nat)} -> carrier R ∧
l_comb R M (Suc n) s f =
l_comb R M m t g ∧ t m = s (Suc n) ∧ g m = f (Suc n)"
apply (cut_tac sc_Ring,
frule Ring.whole_ideal)
apply (simp add:Nset_pre1)
apply (subst l_comb_Suc[of H "carrier R" s n f], assumption+)
apply (frule func_pre[of _ _ H], frule func_pre[of _ _ "carrier R"])
apply (frule unique_expression2[of H f n s], assumption+)
apply ((erule exE)+, (erule conjE)+, simp,
thin_tac "l_comb R M n s f = l_comb R M m t g")
apply (frule_tac f = g and n = m and A = H and g = "λk∈{0::nat}. f (Suc n)"
and m = 0 and B = H in jointfun_hom0,
rule univar_func_test, rule ballI, simp add:funcset_mem, simp)
apply (frule_tac f = t and n = m and A = "carrier R" and
g = "λk∈{0::nat}. s (Suc n)" and m = 0 and B = "carrier R" in
jointfun_hom0,
rule univar_func_test, rule ballI, simp add:funcset_mem, simp)
apply (subgoal_tac "inj_on (jointfun m g 0 (λk∈{0}. f (Suc n)))
{l. l ≤ Suc m}",
subgoal_tac "l_comb R M m t g ± s (Suc n) ·s f (Suc n) =
l_comb R M (Suc m) (jointfun m t 0 (λk∈{0}. s (Suc n)))
(jointfun m g 0 (λk∈{0}. f (Suc n)))",
subgoal_tac "(jointfun m t 0 (λk∈{0}. s (Suc n))) (Suc m) = s (Suc n) ∧
(jointfun m g 0 (λk∈{0}. f (Suc n))) (Suc m) = f (Suc n)",
simp, blast)
apply (simp add:jointfun_def sliden_def)
apply (frule_tac s = t and n = m and f = g and t = "λk∈{0}. s (Suc n)" and
m = 0 and g = "λk∈{0}. f (Suc n)" in l_comb_jointfun_jj[of H
"carrier R"], assumption+,
rule univar_func_test, rule ballI, simp add:funcset_mem, simp,
rule univar_func_test, rule ballI, simp add:funcset_mem)
apply (simp add:l_comb_def, simp add:jointfun_def sliden_def)
apply (thin_tac "jointfun m g 0 (λk∈{0}. f (Suc n)) ∈ {l. l ≤ Suc m} -> H",
thin_tac "jointfun m t 0 (λk∈{0}. s (Suc n)) ∈ {l. l ≤ Suc m} -> carrier R",
thin_tac "t ∈ {j. j ≤ m} -> carrier R",
thin_tac "s ∈ {j. j ≤ n} -> carrier R")
apply (rule_tac f = g and n = m and b = "f (Suc n)" and B = H in jointfun_inj,
assumption+)
apply (simp add:bij_to_def)
apply (unfold bij_to_def, frule conjunct1, fold bij_to_def,
simp add:surj_to_def)
done
(*
lemma (in Module) unique_expression3_1:"[|H ⊆ carrier M;
f ∈ {l. l ≤ (Suc n)} -> H; s ∈ {l. l ≤ (Suc n)} -> carrier R;
(f (Suc n)) ∉ f `({l. l ≤ (Suc n)} - {Suc n})|] ==>
∃g m t. g ∈ {l. l ≤ (m::nat)} -> H ∧
inj_on g {l. l ≤ (m::nat)} ∧
t ∈ {l. l ≤ (m::nat)} -> carrier R ∧
l_comb R M (Suc n) s f = l_comb R M m t g ∧
t m = s (Suc n)"
apply (cut_tac sc_Ring,
frule Ring.whole_ideal)
apply (simp add:Nset_pre1)
apply (subst l_comb_Suc[of H "carrier R" s n f], assumption+)
apply (frule func_pre[of _ _ H], frule func_pre[of _ _ "carrier R"])
apply (frule unique_expression2[of H f n s], assumption+)
apply ((erule exE)+, (erule conjE)+, simp,
thin_tac "l_comb R M n s f = l_comb R M m t g")
apply (frule_tac f = g and n = m and A = H and g = "λk∈{0::nat}. f (Suc n)"
and m = 0 and B = H in jointfun_hom0,
rule univar_func_test, rule ballI, simp add:funcset_mem, simp)
apply (frule_tac f = t and n = m and A = "carrier R" and
g = "λk∈{0::nat}. s (Suc n)" and m = 0 and B = "carrier R" in
jointfun_hom0,
rule univar_func_test, rule ballI, simp add:funcset_mem, simp)
apply (subgoal_tac "inj_on (jointfun m g 0 (λk∈{0}. f (Suc n)))
{l. l ≤ Suc m}",
subgoal_tac "l_comb R M m t g ± s (Suc n) ·s f (Suc n) =
l_comb R M (Suc m) (jointfun m t 0 (λk∈{0}. s (Suc n)))
(jointfun m g 0 (λk∈{0}. f (Suc n)))",
subgoal_tac "(jointfun m t 0 (λk∈{0}. s (Suc n))) (Suc m) = s (Suc n)",
simp, blast)
apply (simp add:jointfun_def sliden_def)
apply (frule_tac s = t and n = m and f = g and t = "λk∈{0}. s (Suc n)" and
m = 0 and g = "λk∈{0}. f (Suc n)" in l_comb_jointfun_jj[of H
"carrier R"], assumption+,
rule univar_func_test, rule ballI, simp add:funcset_mem, simp,
rule univar_func_test, rule ballI, simp add:funcset_mem)
apply (simp add:l_comb_def, simp add:jointfun_def sliden_def)
apply (thin_tac "jointfun m g 0 (λk∈{0}. f (Suc n)) ∈ {l. l ≤ Suc m} -> H",
thin_tac "jointfun m t 0 (λk∈{0}. s (Suc n)) ∈ {l. l ≤ Suc m} -> carrier R",
thin_tac "t ∈ {j. j ≤ m} -> carrier R",
thin_tac "s ∈ {j. j ≤ n} -> carrier R")
apply (rule_tac f = g and n = m and b = "f (Suc n)" and B = H in jointfun_inj,
assumption+)
apply (simp add:bij_to_def)
apply (unfold bij_to_def, frule conjunct1, fold bij_to_def,
simp add:surj_to_def)
done *)
lemma (in Module) unique_expression3_2:"[|H ⊆ carrier M;
f ∈ {k. k ≤ (Suc n)} -> H; s ∈ {k. k ≤ (Suc n)} -> carrier R;
l ≤ (Suc n); (f l) ∉ f ` ({k. k ≤ (Suc n)} - {l}); l ≠ Suc n|] ==>
∃g m t. g ∈ {l. l ≤ (m::nat)} -> H ∧ inj_on g {l. l ≤ (m::nat)} ∧
t ∈ {l. l ≤ m} -> carrier R ∧
l_comb R M (Suc n) s f = l_comb R M m t g ∧
t m = s l ∧ g m = f l"
apply (cut_tac sc_Ring,
frule Ring.whole_ideal)
apply (subst l_comb_transpos1[of "carrier R" H s n f l], assumption+,
rule noteq_le_less[of l "Suc n"], assumption+)
apply (cut_tac unique_expression3_1[of H "cmp f (transpos l (Suc n))" n
"cmp s (transpos l (Suc n))"])
apply ((erule exE)+, (erule conjE)+, simp)
apply (subgoal_tac "t m = s l ∧ g m = f l", blast)
apply (thin_tac "l_comb R M (Suc n) (cmp s (transpos l (Suc n)))
(cmp f (transpos l (Suc n))) = l_comb R M m t g")
apply (simp add:cmp_def)
apply (subst transpos_ij_2[of l "Suc n" "Suc n"], simp+,
subst transpos_ij_2[of l "Suc n" "Suc n"], simp+)
apply (rule univar_func_test, rule ballI, simp add:cmp_def,
frule_tac l = x in transpos_mem[of l "Suc n" "Suc n"], simp,
assumption+, simp add:funcset_mem)
apply (rule univar_func_test, rule ballI, simp add:cmp_def,
frule_tac l = x in transpos_mem[of l "Suc n" "Suc n"], simp,
assumption+, simp add:funcset_mem)
apply (frule_tac i = l and n = "Suc n" and j = "Suc n" in transpos_hom,
simp, assumption)
apply (frule cmp_fun_sub_image[of "transpos l (Suc n)" "{i. i ≤ Suc n}"
"{i. i ≤ Suc n}" f H "{l. l ≤ Suc n} - {Suc n}"], assumption+)
apply (rule subsetI, simp)
apply simp
apply (frule_tac i = l and n = "Suc n" and j = "Suc n" in transpos_inj,
simp, assumption+)
apply (subst injfun_elim_image[of "transpos l (Suc n)" "{i. i ≤ Suc n}"
"{i. i ≤ Suc n}" "Suc n"], assumption+, simp)
apply (thin_tac "cmp f (transpos l (Suc n)) ` ({l. l ≤ Suc n} -
{Suc n}) = f ` transpos l (Suc n) ` ({l. l ≤ Suc n} - {Suc n})")
apply (frule_tac i = l and n = "Suc n" and j = "Suc n" in
transpos_surjec, simp, assumption+)
apply (simp add:surj_to_def cmp_def)
apply (simp add:transpos_ij_2)
done
(*
lemma (in Module) unique_expression3_2:"[|H ⊆ carrier M;
f ∈ {k. k ≤ (Suc n)} -> H; s ∈ {k. k ≤ (Suc n)} -> carrier R;
l ≤ (Suc n); (f l) ∉ f ` ({k. k ≤ (Suc n)} - {l}); l ≠ Suc n|] ==>
∃g m t. g ∈ {l. l ≤ (m::nat)} -> H ∧ inj_on g {l. l ≤ (m::nat)} ∧
t ∈ {l. l ≤ m} -> carrier R ∧
l_comb R M (Suc n) s f = l_comb R M m t g ∧ t m = s l"
apply (cut_tac sc_Ring,
frule Ring.whole_ideal)
apply (subst l_comb_transpos1[of "carrier R" H s n f l], assumption+,
rule noteq_le_less[of l "Suc n"], assumption+)
apply (cut_tac unique_expression3_1[of H "cmp f (transpos l (Suc n))" n
"cmp s (transpos l (Suc n))"])
apply ((erule exE)+, (erule conjE)+, simp)
apply (subgoal_tac "t m = s l", blast)
apply (thin_tac "l_comb R M (Suc n) (cmp s (transpos l (Suc n)))
(cmp f (transpos l (Suc n))) = l_comb R M m t g")
apply (simp add:cmp_def)
apply (subst transpos_ij_2[of l "Suc n" "Suc n"], assumption+,
simp, assumption, simp, assumption)
apply (rule univar_func_test, rule ballI, simp add:cmp_def,
frule_tac l = x in transpos_mem[of l "Suc n" "Suc n"], simp,
assumption+, simp add:funcset_mem)
apply (rule univar_func_test, rule ballI, simp add:cmp_def,
frule_tac l = x in transpos_mem[of l "Suc n" "Suc n"], simp,
assumption+, simp add:funcset_mem)
apply (frule_tac i = l and n = "Suc n" and j = "Suc n" in transpos_hom,
simp, assumption)
apply (frule cmp_fun_sub_image[of "transpos l (Suc n)" "{i. i ≤ Suc n}"
"{i. i ≤ Suc n}" f H "{l. l ≤ Suc n} - {Suc n}"], assumption+)
apply (rule subsetI, simp)
apply simp
apply (frule_tac i = l and n = "Suc n" and j = "Suc n" in transpos_inj,
simp, assumption+)
apply (subst injfun_elim_image[of "transpos l (Suc n)" "{i. i ≤ Suc n}"
"{i. i ≤ Suc n}" "Suc n"], assumption+, simp)
apply (thin_tac "cmp f (transpos l (Suc n)) ` ({l. l ≤ Suc n} -
{Suc n}) = f ` transpos l (Suc n) ` ({l. l ≤ Suc n} - {Suc n})")
apply (frule_tac i = l and n = "Suc n" and j = "Suc n" in
transpos_surjec, simp, assumption+)
apply (simp add:surj_to_def cmp_def)
apply (simp add:transpos_ij_2)
done *)
lemma (in Module) unique_expression3:
"[|H ⊆ carrier M; f ∈ {k. k ≤ (Suc n)} -> H;
s ∈ {k. k ≤ (Suc n)} -> carrier R; l ≤ (Suc n);
(f l) ∉ f ` ({k. k ≤ (Suc n)} - {l})|] ==>
∃g m t. g ∈ {k. k ≤ (m::nat)} -> H ∧
inj_on g {k. k ≤ m} ∧
t ∈ {k. k ≤ m} -> carrier R ∧
l_comb R M (Suc n) s f = l_comb R M m t g ∧ t m = s l ∧ g m = f l"
apply (case_tac "l = Suc n", simp)
apply (cut_tac unique_expression3_1[of H f n s], blast,
assumption+)
apply (rule unique_expression3_2[of H f n s l], assumption+)
done
lemma (in Module) unique_expression4:"free_generator R M H ==>
f ∈ {k. k ≤ (n::nat)} -> H ∧ inj_on f {k. k ≤ n} ∧
s ∈ {k. k ≤ n} -> carrier R ∧ l_comb R M n s f ≠ \<zero> -->
(∃m g t. (g ∈ {k. k ≤ m} -> H) ∧ inj_on g {k. k ≤ m} ∧
(g ` {k. k ≤ m} ⊆ f ` {k. k ≤ n}) ∧ (t ∈ {k. k ≤ m} -> carrier R) ∧
(∀l ∈ {k. k ≤ m}. t l ≠ \<zero>R) ∧ l_comb R M n s f = l_comb R M m t g)"
apply (cut_tac sc_Ring)
apply (frule free_generator_sub[of H])
apply (induct_tac n)
apply (rule impI, (erule conjE)+)
apply (frule has_free_generator_nonzeroring[of H])
apply (frule Ring.whole_ideal,
frule_tac s = s and n = 0 and f = f in
l_comb_mem_linear_span[of "carrier R" H], assumption+)
apply blast
apply (simp add:l_comb_def)
apply (subgoal_tac "f ∈ {j. j ≤ (0::nat)} -> H ∧
inj_on f {j. j ≤ 0} ∧ f ` {j. j ≤ 0} ⊆ f ` {0} ∧
s ∈ {j. j ≤ 0} -> carrier R ∧ (∀l ≤ 0. s l ≠ \<zero>R) ∧
s 0 ·s (f 0) = Σe M (λj. s j ·s (f j)) 0",
(erule conjE)+, blast)
apply simp
apply (rule contrapos_pp, simp+)
apply (cut_tac m = "f 0" in sc_0_m,
simp add:funcset_mem subsetD, simp)
apply (rule impI) apply (erule conjE)+
apply (frule func_pre[of _ _ H],
frule_tac f = f and A = "{k. k ≤ Suc n}" and ?A1.0 = "{k. k ≤ n}" in
restrict_inj, rule subsetI, simp,
frule func_pre[of _ _ "carrier R"], simp)
apply (frule Ring.whole_ideal)
apply (frule free_generator_sub[of H],
simp add:l_comb_Suc[of H "carrier R" s _ f])
apply (case_tac "s (Suc n) = \<zero>R", simp)
apply (frule_tac x = "Suc n" and f = f and A = "{k. k ≤ Suc n}" and
B = H in funcset_mem, simp,
frule_tac c = "f (Suc n)" in subsetD[of H "carrier M"], simp)
apply (frule_tac m = "f (Suc n)" in sc_0_m, simp)
apply (frule_tac n = n in l_comb_mem[of "carrier R" H s _ f],
assumption+, simp add:ag_r_zero)
apply ((erule exE)+, (erule conjE)+)
apply (frule_tac f = f and A = "{k. k ≤ Suc n}" and B = H and
?A1.0 = "{k. k ≤ n}" and ?A2.0 = "{k. k ≤ Suc n}" in im_set_mono,
rule subsetI, simp, simp,
frule_tac A = "g ` {k. k ≤ m}" and B = "f ` {k. k ≤ n}" and
C = "f ` {k. k ≤ Suc n}" in subset_trans, assumption+)
apply blast
apply (case_tac "l_comb R M n s f = \<zero>M", simp,
frule_tac x = "Suc n" and f = s and A = "{k. k ≤ Suc n}" and
B = "carrier R" in funcset_mem, simp,
frule_tac x = "Suc n" and f = f and A = "{k. k ≤ Suc n}" and
B = H in funcset_mem, simp,
frule_tac c = "f (Suc n)" in subsetD[of H "carrier M"], assumption+,
frule_tac a = "s (Suc n)" and m = "f (Suc n)" in sc_mem, assumption+,
simp add:ag_l_zero)
apply (subgoal_tac "(λj∈{0::nat}. f (Suc n)) ∈ {j. j ≤ (0::nat)} -> H ∧
inj_on (λj∈{0::nat}. f (Suc n)) {j. j ≤ (0::nat)} ∧
(λj∈{0::nat}. f (Suc n)) ` {j. j ≤ (0::nat)} ⊆ f ` {k. k ≤ (Suc n)} ∧
(λj∈{0::nat}. s (Suc n))∈ {k. k ≤ 0} -> carrier R ∧
(∀l≤0. (λj∈{0::nat}. s (Suc n)) l ≠ \<zero>R) ∧
s (Suc n) ·s f (Suc n) =
l_comb R M 0 (λj∈{0::nat}. s (Suc n)) (λj∈{0::nat}. f (Suc n))")
apply ((erule conjE)+, blast)
apply simp
apply (rule conjI, rule univar_func_test, simp)
apply (rule conjI, rule univar_func_test, simp)
apply (simp add:l_comb_def)
apply simp
apply ((erule exE)+, (erule conjE)+, erule exE, (erule conjE)+, simp)
apply (thin_tac "l_comb R M m t g ≠ \<zero>",
thin_tac "l_comb R M m t g ± s (Suc n) ·s f (Suc n) ≠ \<zero>",
thin_tac "l_comb R M n s f = l_comb R M m t g")
apply (frule_tac f = g and n = m and A = H and g = "λj∈{0::nat}. f (Suc n)"
and m = 0 and B = H in jointfun_hom,
rule univar_func_test, simp add:funcset_mem,
frule_tac f = t and n = m and A = "carrier R" and
g = "λj∈{0::nat}. s (Suc n)" and m = 0 and B = "carrier R" in
jointfun_hom, rule univar_func_test, simp add:funcset_mem, simp)
apply (subgoal_tac "inj_on (jointfun m g 0 (λj∈{0}. f (Suc n)))
{k. k ≤ Suc m} ∧
(jointfun m g 0 (λj∈{0}. f (Suc n))) ` {k. k ≤ Suc m} ⊆ f ` {k. k ≤ Suc n} ∧
(∀l ≤ (Suc m). (jointfun m t 0 (λj∈{0}. s (Suc n))) l ≠ \<zero>R) ∧
l_comb R M m t g ± s (Suc n) ·s f (Suc n) =
l_comb R M (Suc m) (jointfun m t 0 (λj∈{0}. s (Suc n)))
(jointfun m g 0 (λj∈{0}. f (Suc n)))")
apply (erule conjE)+ apply blast
apply (rule conjI)
apply (rule_tac f = g and n = m and b = "f (Suc n)" and B = H in
jointfun_inj, assumption+)
apply (rule contrapos_pp, simp+)
apply (frule_tac c = "f (Suc n)" and A = "g ` {k. k ≤ m}" and
B = "f ` {k. k ≤ n}" in subsetD, assumption+)
apply (thin_tac "inj_on f {k. k ≤ n}",
thin_tac "g ` {k. k ≤ m} ⊆ f ` {k. k ≤ n}",
thin_tac "f (Suc n) ∈ g ` {j. j ≤ m}", simp add:image_def,
erule exE, erule conjE)
apply (simp add:inj_on_def,
drule_tac a = "Suc n" in forall_spec, simp,
thin_tac "∀x≤m. ∀y≤m. g x = g y --> x = y",
thin_tac "∀l≤m. t l ≠ \<zero>R",
drule_tac a = x in forall_spec, simp, simp)
apply (rule conjI, rule subsetI)
apply (simp add:image_def, erule exE, erule conjE)
apply (case_tac "xa = Suc m", simp add:jointfun_def sliden_def)
apply (cut_tac n = "Suc n" in Nat.le_refl, blast)
apply (frule_tac m = xa and n = "Suc m" in noteq_le_less, assumption,
thin_tac "xa ≤ Suc m",
frule_tac x = xa and n = "Suc m" in less_le_diff,
thin_tac "xa < Suc m", simp,
thin_tac "jointfun m g 0 (λj∈{0}. f (Suc n)) ∈ {j. j ≤ Suc m} -> H",
thin_tac "jointfun m t 0 (λj∈{0}. s (Suc n)) ∈ {j. j ≤ Suc m} -> carrier R",
simp add:jointfun_def)
apply (subgoal_tac "g xa ∈ {y. ∃x≤n. y = f x}", simp, erule exE)
apply (erule conjE, frule_tac i = xb and j = n and k = "Suc n" in
le_trans, simp, blast)
apply (rule_tac c = "g xa" and A = "{y. ∃x≤m. y = g x}" and
B = "{y. ∃x≤n. y = f x}" in subsetD, assumption+,
simp, blast)
apply (rule conjI, rule allI, rule impI)
apply (case_tac "l = Suc m", simp add:jointfun_def sliden_def)
apply (frule_tac m = l and n = "Suc m" in noteq_le_less, assumption,
thin_tac "l ≤ Suc m",
frule_tac x = l and n = "Suc m" in less_le_diff,
thin_tac "l < Suc m", simp,
thin_tac "jointfun m g 0 (λj∈{0}. f (Suc n)) ∈ {j. j ≤ Suc m} -> H",
thin_tac "jointfun m t 0 (λj∈{0}. s (Suc n)) ∈ {j. j ≤ Suc m} -> carrier R",
simp add:jointfun_def)
apply (simp add:l_comb_def,
subst l_comb_jointfun_jj[of H "carrier R"], assumption+,
rule univar_func_test, rule ballI, simp add:funcset_mem,
rule univar_func_test, rule ballI, simp add:funcset_mem)
apply (simp add:jointfun_def sliden_def)
done
lemma (in Module) unique_prepression5_0:"[|free_generator R M H;
f ∈ {j. j ≤ n} -> H; inj_on f {j. j ≤ n};
s ∈ {j. j ≤ n} -> carrier R; g ∈ {j. j ≤ m} -> H;
inj_on g {j. j ≤ m}; t ∈ {j. j ≤ m} -> carrier R;
l_comb R M n s f = l_comb R M m t g;∀j≤n. s j ≠ \<zero>R; ∀k≤m. t k ≠ \<zero>R;
f n ∉ g ` {j. j ≤ m}; 0 < n|] ==> False"
apply (cut_tac sc_Ring,
frule Ring.ring_is_ag,
frule Ring.whole_ideal,
frule free_generator_sub[of H])
apply (cut_tac l_comb_Suc[of H "carrier R" s "n - Suc 0" f],
simp,
thin_tac "l_comb R M n s f = l_comb R M (n - Suc 0) s f ± s n ·s f n")
apply (frule free_generator_sub[of H],
frule l_comb_mem[of "carrier R" H t m g], assumption+,
frule l_comb_mem[of "carrier R" H s "n - Suc 0" f], assumption+,
rule func_pre, simp, rule func_pre, simp,
cut_tac sc_mem[of "s n" "f n"])
apply (frule ag_pOp_closed[of "l_comb R M (n - Suc 0) s f" "s n ·s f n"],
assumption+,
frule ag_mOp_closed[of "l_comb R M (n - Suc 0) s f"])
apply (frule ag_pOp_add_l[of "l_comb R M m t g" "l_comb R M (n - Suc 0) s f ± s n ·s f n" "-a (l_comb R M (n - Suc 0) s f)"], assumption+,
thin_tac "l_comb R M m t g = l_comb R M (n - Suc 0) s f ± s n ·s f n")
apply (simp add:ag_pOp_assoc[THEN sym, of "-a (l_comb R M (n - Suc 0) s f)"
"l_comb R M (n - Suc 0) s f" "s n ·s f n"],
simp add:ag_l_inv1 ag_l_zero)
apply (cut_tac func_pre[of f "n - Suc 0" H],
cut_tac func_pre[of s "n - Suc 0" "carrier R"])
apply (frule linear_span_iOp_closedTr2[of "carrier R" "H" f "n - Suc 0" s],
assumption+)
apply (simp,
thin_tac "-a (l_comb R M (n - Suc 0) s f) =
l_comb R M (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) f")
apply (subgoal_tac "(λx∈{j. j ≤ n - Suc 0}. -aR (s x))
∈ {j. j ≤ n - Suc 0} -> carrier R")
apply (simp add:l_comb_add[THEN sym, of "carrier R" H
"λx∈{j. j ≤ n - Suc 0}. -aR (s x)" "n - Suc 0" f t m g],
thin_tac "l_comb R M m t g ∈ carrier M",
thin_tac "l_comb R M (n - Suc 0) s f ∈ carrier M",
thin_tac "l_comb R M (n - Suc 0) s f ± s n ·s f n ∈ carrier M",
thin_tac "l_comb R M (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) f
∈ carrier M")
apply (frule jointfun_hom[of f "n - Suc 0" H g m H], assumption+,
frule jointfun_hom[of "λx∈{j. j ≤ n - Suc 0}. -aR (s x)" "n - Suc 0"
"carrier R" t m "carrier R"], assumption+, simp)
(* to apply unique_expression3_1, we show
f n ∉ (jointfun (n - Suc 0) f m g) ` {j. j ≤ n + m} *)
apply (frule im_jointfun[of f "n - Suc 0" H g m H], assumption+)
apply (frule unique_expression3_1[of H
"jointfun (n + m) (jointfun (n - Suc 0) f m g) 0 (λx∈{0::nat}. (f n))"
"n + m"
"jointfun (n + m) (jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x))
m t) 0 (λx∈{0::nat}. -aR (s n))"])
apply (rule univar_func_test, rule ballI,
case_tac "x ≤ (n + m)", simp,
simp add:jointfun_def[of "n+m"], simp add:funcset_mem,
simp add:jointfun_def[of "n+m"] sliden_def, simp add:funcset_mem)
apply (rule univar_func_test, rule ballI,
case_tac "x ≤ (n + m)", simp,
simp add:jointfun_def[of "n+m"], simp add:funcset_mem)
apply (simp add:jointfun_def[of "n+m"] sliden_def,
frule Ring.ring_is_ag[of R], rule aGroup.ag_mOp_closed, assumption,
simp add:funcset_mem)
apply (thin_tac "s ∈ {j. j ≤ n} -> carrier R",
thin_tac "t ∈ {j. j ≤ m} -> carrier R",
thin_tac "∀j≤n. s j ≠ \<zero>R",
thin_tac "∀k≤m. t k ≠ \<zero>R",
thin_tac "l_comb R M (n + m)
(jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t)
(jointfun (n - Suc 0) f m g) =
s n ·s f n",
thin_tac "s ∈ {j. j ≤ n - Suc 0} -> carrier R")
apply (thin_tac "(λx∈{j. j ≤ n - Suc 0}. -aR (s x))
∈ {j. j ≤ n - Suc 0} -> carrier R",
thin_tac "jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t
∈ {j. j ≤ n + m} -> carrier R")
apply (simp add:Nset_pre1,
simp add:im_jointfunTr1[of "n + m" "jointfun (n - Suc 0) f m g" 0
"λx∈{0}. f n"],
thin_tac "jointfun (n - Suc 0) f m g ∈ {j. j ≤ n + m} -> H",
thin_tac "jointfun (n - Suc 0) f m g ` {j. j ≤ n + m} =
f ` {j. j ≤ n - Suc 0} ∪ g ` {j. j ≤ m}",
simp add:jointfun_def[of "n+m"] sliden_def)
apply (rule contrapos_pp, simp+, simp add:image_def, erule exE,erule conjE,
simp add:inj_on_def[of f],
drule_tac a = n in forall_spec, simp,
thin_tac "∀xa≤m. f x ≠ g xa",
drule_tac a = x in forall_spec,
rule_tac i = x and j = "n - Suc 0" and k = n in Nat.le_trans,
assumption+, subst Suc_le_mono[THEN sym], simp,
simp,
cut_tac n1 = x and m1 = "x - Suc 0" in
Suc_le_mono[THEN sym], simp)
defer
apply (rule univar_func_test, rule ballI, simp,
rule aGroup.ag_mOp_closed, assumption,
cut_tac i = x and j = "n - Suc 0" and k = n in Nat.le_trans,
assumption, subst Suc_le_mono[THEN sym], simp,
simp add:funcset_mem, simp, simp, simp add:funcset_mem,
simp add:funcset_mem,
simp add:funcset_mem subsetD, assumption+, simp, simp)
apply ((erule exE)+, (erule conjE)+, erule exE, (erule conjE)+)
apply (cut_tac l_comb_Suc[of H "carrier R" "jointfun (n + m)
(jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t) 0
(λx∈{0}. -aR (s n))" "n + m"
"jointfun (n + m) (jointfun (n - Suc 0) f m g) 0 (λx∈{0}. f n)"],
simp) apply (
thin_tac "l_comb R M (Suc (n + m))
(jointfun (n + m)
(jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t) 0
(λx∈{0}. -aR (s n)))
(jointfun (n + m) (jointfun (n - Suc 0) f m g) 0 (λx∈{0}. f n)) =
l_comb R M ma ta ga")
apply (subgoal_tac "l_comb R M (n + m)
(jointfun (n + m)
(jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t) 0
(λx∈{0}. -aR (s n)))
(jointfun (n + m) (jointfun (n - Suc 0) f m g) 0 (λx∈{0}. f n)) ±
jointfun (n + m)
(jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t) 0
(λx∈{0}. -aR (s n)) (Suc (n + m)) ·s
jointfun (n + m) (jointfun (n - Suc 0) f m g) 0 (λx∈{0}. f n)
(Suc (n + m)) = \<zero>M", simp,
thin_tac "l_comb R M (n + m)
(jointfun (n + m)
(jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t) 0
(λx∈{0}. -aR (s n)))
(jointfun (n + m) (jointfun (n - Suc 0) f m g) 0 (λx∈{0}. f n)) ±
jointfun (n + m)
(jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t) 0
(λx∈{0}. -aR (s n)) (Suc (n + m)) ·s
jointfun (n + m) (jointfun (n - Suc 0) f m g) 0 (λx∈{0}. f n)
(Suc (n + m)) =
l_comb R M ma ta ga",
thin_tac "l_comb R M (n + m)
(jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t)
(jointfun (n - Suc 0) f m g) =
s n ·s f n",
thin_tac "jointfun (n - Suc 0) f m g ∈ {j. j ≤ n + m} -> H",
thin_tac "jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t
∈ {j. j ≤ n + m} -> carrier R",
thin_tac "jointfun (n - Suc 0) f m g ` {j. j ≤ n + m} =
f ` {j. j ≤ n - Suc 0} ∪ g ` {j. j ≤ m}")
apply (simp add:jointfun_def[of "n+m"] sliden_def)
apply (rotate_tac -3, frule sym, thin_tac "\<zero> = l_comb R M ma ta ga")
apply (frule_tac s = ta and n = ma and m = ga in unique_expression1[of H],
assumption+)
apply (rotate_tac -1,
drule_tac b = ma in forball_spec1, simp)
apply (frule_tac funcset_mem[of s "{j. j ≤ n}" "carrier R" n], simp,
frule sym, thin_tac "ta ma = -aR (s n)",
frule aGroup.ag_inv_inv[of R "s n"], assumption+, simp,
thin_tac " -aR (s n) = \<zero>R",
rotate_tac -1, frule sym, thin_tac " -aR \<zero>R = s n",
simp add:aGroup.ag_inv_zero[of R])
apply (thin_tac "l_comb R M (n + m)
(jointfun (n + m)
(jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t) 0
(λx∈{0}. -aR (s n)))
(jointfun (n + m) (jointfun (n - Suc 0) f m g) 0 (λx∈{0}. f n)) ±
jointfun (n + m)
(jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t) 0
(λx∈{0}. -aR (s n)) (Suc (n + m)) ·s
jointfun (n + m) (jointfun (n - Suc 0) f m g) 0 (λx∈{0}. f n)
(Suc (n + m)) =
l_comb R M ma ta ga",
thin_tac "ta ma =
jointfun (n + m)
(jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t) 0
(λx∈{0}. -aR (s n)) (Suc (n + m))")
apply (subst l_comb_jointfun_jj1[of H "carrier R"], assumption+,
rule univar_func_test, rule ballI, simp,
rule aGroup.ag_mOp_closed, assumption, simp add:funcset_mem,
rule univar_func_test, rule ballI, simp add:funcset_mem)
apply (simp,
thin_tac "l_comb R M (n + m) (jointfun (n - Suc 0)
(λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t) (jointfun (n - Suc 0) f m g) =
s n ·s f n",
thin_tac "jointfun (n - Suc 0) f m g ∈ {j. j ≤ n + m} -> H",
thin_tac "jointfun (n - Suc 0) (λx∈{j. j ≤ n - Suc 0}. -aR (s x)) m t
∈ {j. j ≤ n + m} -> carrier R",
thin_tac "jointfun (n - Suc 0) f m g ` {j. j ≤ n + m} =
f ` {j. j ≤ n - Suc 0} ∪ g ` {j. j ≤ m}")
apply (simp add:jointfun_def[of "n+m"] sliden_def,
subst sc_minus_am1[THEN sym],
simp add:funcset_mem, simp add:funcset_mem subsetD,
simp add:ag_r_inv1, simp add:free_generator_sub)
apply (assumption+,
rule univar_func_test, rule ballI,
case_tac "x ≤ n + m", simp add:jointfun_def[of "n+m"],
simp add:funcset_mem,
simp add:jointfun_def[of "n+m"] sliden_def,
rule aGroup.ag_mOp_closed, assumption, simp add:funcset_mem,
rule univar_func_test, rule ballI, simp,
case_tac "x ≤ n+m", simp add:jointfun_def[of "n+m"],
simp add:funcset_mem,
simp add:jointfun_def[of "n+m"] sliden_def,
simp add:funcset_mem)
done
lemma (in Module) unique_expression5:"[|free_generator R M H;
f ∈ {j. j ≤ (n::nat)} -> H; inj_on f {j. j ≤ n};
s ∈ {j. j ≤ n} -> carrier R; g ∈ {j. j ≤ (m::nat)} -> H;
inj_on g {j. j ≤ m}; t ∈ {j. j ≤ m} -> carrier R;
l_comb R M n s f = l_comb R M m t g;
∀j ∈ {j. j ≤ n}. s j ≠ \<zero>R; ∀k ∈ {j. j ≤ m}. t k ≠ \<zero>R|] ==>
f ` {j. j ≤ n} ⊆ g ` {j. j ≤ m}"
apply (cut_tac sc_Ring, frule Ring.ring_is_ag[of R],
frule Ring.whole_ideal,
frule free_generator_sub[of H])
apply (rule contrapos_pp, simp+, simp add:subset_eq)
apply (erule exE, erule conjE)
apply (case_tac "n = 0", simp)
apply (frule_tac f = t and n = m and A = "carrier R" and
g = "λk∈{0::nat}. -aR (s 0)" and m = 0 and B = "carrier R" in
jointfun_hom0,
rule univar_func_test, rule ballI, simp add:funcset_mem,
rule aGroup.ag_mOp_closed, assumption, simp add:funcset_mem,
frule_tac f = g and n = m and A = H and
g = "λk∈{0::nat}. (f 0)" and m = 0 and B = H in
jointfun_hom0,
rule univar_func_test, rule ballI, simp add:funcset_mem subsetD,
simp)
apply (frule sym, thin_tac "l_comb R M 0 s f = l_comb R M m t g")
apply (frule_tac n = 0 in l_comb_mem[of "carrier R" H s _ f],
simp add:free_generator_sub, simp+,
frule_tac n = m in l_comb_mem[of "carrier R" H t _ g],
simp add:free_generator_sub, assumption+)
apply (simp add:ag_eq_diffzero[of "l_comb R M m t g" "l_comb R M 0 s f"],
simp add:l_comb_def[of R M 0 s f],
frule_tac x = 0 in funcset_mem[of s "{0}" "carrier R"], simp,
frule_tac x = 0 in funcset_mem[of f "{0}" H], simp,
frule free_generator_sub[of H],
frule_tac c = "f 0" in subsetD[of H "carrier M"], assumption+,
simp add:sc_minus_am1)
apply (subgoal_tac "l_comb R M m t g ± (-aR (s 0)) ·s f 0 =
l_comb R M (Suc m) (jointfun m t 0 (λk∈{0}. (-aR (s 0))))
(jointfun m g 0 (λk∈{0}. f 0))", simp)
apply (frule_tac f = g and n = m and B = H and b = "f 0" in jointfun_inj,
assumption+)
apply (frule unique_expression1[of H "jointfun m t 0 (λk∈{0}. (-aR (s 0)))"
"Suc m" "jointfun m g 0 (λk∈{0}. f 0)"], assumption+)
apply (frule_tac b = "Suc m" in forball_spec1, simp,
thin_tac "∀j∈{j. j ≤ Suc m}. jointfun m t 0 (λk∈{0}. -aR (s 0)) j
= \<zero>R")
apply (simp add:jointfun_def sliden_def)
apply (frule aGroup.ag_inv_inv[THEN sym, of R "s 0"], assumption,
simp add:aGroup.ag_inv_zero)
apply (thin_tac "l_comb R M m t g ± (-aR (s 0)) ·s f 0 = \<zero>",
simp del:nsum_suc add:l_comb_def)
apply (cut_tac l_comb_jointfun_jj[of H "carrier R" t m g "λk∈{0}. -aR (s 0)"
0 "λk∈{0}. f 0"], simp,
thin_tac "Σe M (λj. jointfun m t 0 (λk∈{0}. -aR (s 0)) j ·s
jointfun m g 0 (λk∈{0}. f 0) j) m =
Σe M (λj. t j ·s g j) m",
simp add:jointfun_def sliden_def, simp add:free_generator_sub,
assumption+,
rule univar_func_test, rule ballI, simp,
rule aGroup.ag_mOp_closed, assumption+,
rule univar_func_test, rule ballI, simp)
apply (case_tac "x = n", simp,
rule unique_prepression5_0[of H f n s g m t], assumption+)
apply (frule_tac j = x in l_comb_transpos1[of "carrier R" H s "n - Suc 0" f],
rule subsetI, simp,
simp+,
rotate_tac -1, frule sym,
thin_tac "l_comb R M m t g =
l_comb R M n (cmp s (transpos x n)) (cmp f (transpos x n))",
frule_tac i = x and n = n and j = n in transpos_hom, simp,
assumption,
frule_tac i = x and n = n and j = n in transpos_inj, simp,
assumption+,
rule_tac f = "cmp f (transpos x n)" and s = "cmp s (transpos x n)" in
unique_prepression5_0[of H _ n _ g m t], assumption+,
simp add:cmp_fun, simp add:cmp_fun, simp add:cmp_inj,
simp add:cmp_fun, assumption+,
rule allI, rule impI, simp add:cmp_def,
frule_tac i = x and n = n and j = n and l = j in transpos_mem,
simp, assumption+, blast, assumption)
apply (simp add:cmp_def transpos_ij_2)
apply simp
done
lemma (in Module) unique_expression6:"[|free_generator R M H;
f ∈ {j. j ≤ (n::nat)} -> H; inj_on f {j. j ≤ n};
s ∈ {j. j ≤ n} -> carrier R;
g ∈ {j. j ≤ (m::nat)} -> H; inj_on g {j. j ≤ m};
t ∈ {j. j ≤ m} -> carrier R;
l_comb R M n s f = l_comb R M m t g;
∀j∈{j. j ≤ n}. s j ≠ \<zero>R; ∀k∈ {j. j ≤ m}. t k ≠ \<zero>R|] ==>
f `{j. j ≤ n} = g ` {j. j ≤ m}"
apply (rule equalityI)
apply (rule_tac H = H and f = f and n = n and s = s and g = g and m = m and
t = t in unique_expression5, assumption+)
apply (rule_tac H = H and f = g and n = m and s = t and g = f and m = n and
t = s in unique_expression5, assumption+)
apply (rule sym, assumption, blast, blast)
done
lemma (in Module) unique_expression7_1:"[|free_generator R M H;
f ∈ {j. j ≤ (n::nat)} -> H; inj_on f {j. j ≤ n};
s ∈ {j. j ≤ n} -> carrier R;
g ∈ {j. j ≤ (m::nat)} -> H; inj_on g {j. j ≤ m};
t ∈ {j. j ≤ m} -> carrier R;
l_comb R M n s f = l_comb R M m t g;
∀j ∈ {j. j ≤ n}. s j ≠ \<zero>R; ∀k∈{j. j ≤ m}. t k ≠ \<zero>R|] ==> n = m"
apply (cut_tac finite_Nset [of "n"], cut_tac finite_Nset [of "m"])
apply (frule_tac A = "{j. j ≤ n}" and f = f in card_image,
frule_tac A = "{j. j ≤ m}" and f = g in card_image)
apply (frule_tac H = H and f = f and n = n and s = s and g = g and t = t and
m = m in unique_expression6, assumption+)
apply (rotate_tac -3, frule sym,
thin_tac "card (f ` {j. j ≤ n}) = card ({j. j ≤ n})")
apply simp
apply (simp add:card_Nset)
done
lemma (in Module) unique_expression7_2:"[|free_generator R M H;
f ∈ {j. j ≤ (n::nat)} -> H; inj_on f {j. j ≤ n};
s ∈ {j. j ≤ n} -> carrier R; t ∈ {j. j ≤ n} -> carrier R;
l_comb R M n s f = l_comb R M n t f|] ==> (∀l ∈ {j. j ≤ n}. s l = t l)"
apply (cut_tac sc_Ring, frule Ring.whole_ideal)
apply (frule free_generator_sub[of H])
apply (frule l_comb_mem[of "carrier R" H s n f], assumption+,
frule l_comb_mem[of "carrier R" H t n f], assumption+)
apply (simp add:ag_eq_diffzero[of "l_comb R M n s f" "l_comb R M n t f"])
apply (simp add:linear_span_iOp_closedTr2[of "carrier R" H f n t])
apply (frule l_comb_add1[THEN sym, of "carrier R" H f n s "λj∈{k. k ≤ n}. -aR (t j)"],
assumption+)
apply (rule univar_func_test, rule ballI)
apply (simp, frule Ring.ring_is_ag[of R],
rule aGroup.ag_mOp_closed[of R], simp add:funcset_mem)
apply (simp add:funcset_mem)
apply simp
apply (frule_tac s = "λx∈{x. x ≤ n}. s x ±R (if x ≤ n then -aR (t x) else
arbitrary)" in unique_expression1[of H _ n f], assumption+)
apply (rule univar_func_test, rule ballI, simp)
apply (frule Ring.ring_is_ag[of R], rule aGroup.ag_pOp_closed, assumption,
simp add:funcset_mem,
rule aGroup.ag_mOp_closed, assumption,
simp add:funcset_mem, assumption+)
apply (rule allI, rule impI)
apply (subst aGroup.ag_eq_diffzero[of R],
simp add:Ring.ring_is_ag,
simp add:funcset_mem, simp add:funcset_mem)
apply (drule_tac b = l in forball_spec1, simp)
apply simp
done
end
lemma module_is_ag:
aGroup M
lemma module_inc_zero:
\<zero> ∈ carrier M
lemma submodule_subset:
submodule R M H ==> H ⊆ carrier M
lemma submodule_asubg:
submodule R M H ==> @ASubG M H
lemma submodule_subset1:
[| submodule R M H; h ∈ H |] ==> h ∈ carrier M
lemma submodule_inc_0:
submodule R M H ==> \<zero> ∈ H
lemma sc_un:
m ∈ carrier M ==> 1rR ·s m = m
lemma sc_mem:
[| a ∈ carrier R; m ∈ carrier M |] ==> a ·s m ∈ carrier M
lemma submodule_sc_closed:
[| submodule R M H; a ∈ carrier R; h ∈ H |] ==> a ·s h ∈ H
lemma sc_assoc:
[| a ∈ carrier R; b ∈ carrier R; m ∈ carrier M |]
==> (a ·rR b) ·s m = a ·s (b ·s m)
lemma sc_l_distr:
[| a ∈ carrier R; b ∈ carrier R; m ∈ carrier M |]
==> (a ±R b) ·s m = a ·s m ± b ·s m
lemma sc_r_distr:
[| a ∈ carrier R; m ∈ carrier M; n ∈ carrier M |]
==> a ·s (m ± n) = a ·s m ± a ·s n
lemma sc_0_m:
m ∈ carrier M ==> \<zero>R ·s m = \<zero>
lemma sc_a_0:
a ∈ carrier R ==> a ·s \<zero> = \<zero>
lemma sc_minus_am:
[| a ∈ carrier R; m ∈ carrier M |] ==> -a a ·s m = a ·s (-a m)
lemma sc_minus_am1:
[| a ∈ carrier R; m ∈ carrier M |] ==> -a a ·s m = (-aR a) ·s m
lemma submodule_0:
submodule R M {\<zero>}
lemma submodule_whole:
submodule R M (carrier M)
lemma submodule_pOp_closed:
[| submodule R M H; h ∈ H; k ∈ H |] ==> h ± k ∈ H
lemma submodule_mOp_closed:
[| submodule R M H; h ∈ H |] ==> -a h ∈ H
lemma mHom_func:
f ∈ mHom R M N ==> f ∈ carrier M -> carrier N
lemma mHom_test:
[| Module N R;
f ∈ carrier M -> carrier N ∧
f ∈ extensional (carrier M) ∧
(∀m∈carrier M. ∀n∈carrier M. f (m ± n) = f m ±N f n) ∧
(∀a∈carrier R. ∀m∈carrier M. f (a ·s m) = a ·sN f m) |]
==> f ∈ mHom R M N
lemma mHom_mem:
[| Module N R; f ∈ mHom R M N; m ∈ carrier M |] ==> f m ∈ carrier N
lemma mHom_add:
[| Module N R; f ∈ mHom R M N; m ∈ carrier M; n ∈ carrier M |]
==> f (m ± n) = f m ±N f n
lemma mHom_0:
[| Module N R; f ∈ mHom R M N |] ==> f \<zero> = \<zero>N
lemma mHom_inv:
[| Module N R; m ∈ carrier M; f ∈ mHom R M N |] ==> f (-a m) = -aN f m
lemma mHom_lin:
[| Module N R; m ∈ carrier M; f ∈ mHom R M N; a ∈ carrier R |]
==> f (a ·s m) = a ·sN f m
lemma mker_inc_zero:
[| Module N R; f ∈ mHom R M N |] ==> \<zero> ∈ kerM,N f
lemma mHom_eq_ker:
[| Module N R; f ∈ mHom R M N; a ∈ carrier M; b ∈ carrier M;
a ± -a b ∈ kerM,N f |]
==> f a = f b
lemma mHom_ker_eq:
[| Module N R; f ∈ mHom R M N; a ∈ carrier M; b ∈ carrier M; f a = f b |]
==> a ± -a b ∈ kerM,N f
lemma mker_submodule:
[| Module N R; f ∈ mHom R M N |] ==> submodule R M (kerM,N f)
lemma mker_mzeromap:
Module N R ==> kerM,N mzeromap M N = carrier M
lemma mdl_carrier:
submodule R M H ==> carrier (mdl M H) = H
lemma mdl_is_ag:
submodule R M H ==> aGroup (mdl M H)
lemma mdl_is_module:
submodule R M H ==> Module (mdl M H) R
lemma submodule_of_mdl:
[| submodule R M H; submodule R M N; H ⊆ N |] ==> submodule R (mdl M N) H
lemma img_set_submodule:
[| Module N R; f ∈ mHom R M N |] ==> submodule R N (f ` carrier M)
lemma mimg_module:
[| Module N R; f ∈ mHom R M N |] ==> Module (mimgR M,N f) R
lemma surjec_to_mimg:
[| Module N R; f ∈ mHom R M N |] ==> surjecM,mimgR M,N f f
lemma zero_HOM:
Module N R ==> mzeromap M N = \<zero>HOMR M N
lemma tOp_mHom_closed:
[| Module N R; f ∈ mHom R M N; g ∈ mHom R M N |]
==> tOp_mHom R M N f g ∈ mHom R M N
lemma iOp_mHom_closed:
[| Module N R; f ∈ mHom R M N |] ==> iOp_mHom R M N f ∈ mHom R M N
lemma mHom_ex_zero:
Module N R ==> mzeromap M N ∈ mHom R M N
lemma mHom_eq:
[| Module N R; f ∈ mHom R M N; g ∈ mHom R M N; ∀m∈carrier M. f m = g m |]
==> f = g
lemma mHom_l_zero:
[| Module N R; f ∈ mHom R M N |] ==> tOp_mHom R M N (mzeromap M N) f = f
lemma mHom_l_inv:
[| Module N R; f ∈ mHom R M N |]
==> tOp_mHom R M N (iOp_mHom R M N f) f = mzeromap M N
lemma mHom_tOp_assoc:
[| Module N R; f ∈ mHom R M N; g ∈ mHom R M N; h ∈ mHom R M N |]
==> tOp_mHom R M N (tOp_mHom R M N f g) h =
tOp_mHom R M N f (tOp_mHom R M N g h)
lemma mHom_tOp_commute:
[| Module N R; f ∈ mHom R M N; g ∈ mHom R M N |]
==> tOp_mHom R M N f g = tOp_mHom R M N g f
lemma HOM_is_ag:
Module N R ==> aGroup (HOMR M N)
lemma sprod_mHom_closed:
[| Module N R; a ∈ carrier R; f ∈ mHom R M N |]
==> sprod_mHom R M N a f ∈ mHom R M N
lemma HOM_is_module:
Module N R ==> Module (HOMR M N) R
lemma minjec_inj:
[| Module N R; injecM,N f |] ==> inj_on f (carrier M)
lemma invmfun_l_inv:
[| Module N R; bijecM,N f; m ∈ carrier M |] ==> invmfun R M N f (f m) = m
lemma invmfun_mHom:
[| Module N R; bijecM,N f; f ∈ mHom R M N |] ==> invmfun R M N f ∈ mHom R N M
lemma invmfun_r_inv:
[| Module N R; bijecM,N f; n ∈ carrier N |] ==> f (invmfun R M N f n) = n
lemma mHom_compos:
[| Module L R; Module N R; f ∈ mHom R L M; g ∈ mHom R M N |]
==> compos L g f ∈ mHom R L N
lemma mcompos_inj_inj:
[| Module L R; Module N R; f ∈ mHom R L M; g ∈ mHom R M N; injecL,M f;
injecM,N g |]
==> injecL,N compos L g f
lemma mcompos_surj_surj:
[| Module L R; Module N R; surjecL,M f; surjecM,N g; f ∈ mHom R L M;
g ∈ mHom R M N |]
==> surjecL,N compos L g f
lemma mId_mHom:
mIdM ∈ mHom R M M
lemma mHom_mId_bijec:
[| Module N R; f ∈ mHom R M N; g ∈ mHom R N M; compose (carrier M) g f = mIdM ;
compose (carrier N) f g = mIdN |]
==> bijecM,N f
lemma sup_sharp_homTr:
[| Module N R; Module L R; u ∈ mHom R M N; f ∈ mHom R N L |]
==> sup_sharp R M N L u f ∈ mHom R M L
lemma sup_sharp_hom:
[| Module N R; Module L R; u ∈ mHom R M N |]
==> sup_sharp R M N L u ∈ mHom R (HOMR N L) (HOMR M L)
lemma sub_sharp_homTr:
[| Module N R; Module L R; u ∈ mHom R M N; f ∈ mHom R L M |]
==> sub_sharp R L M N u f ∈ mHom R L N
lemma sub_sharp_hom:
[| Module N R; Module L R; u ∈ mHom R M N |]
==> sub_sharp R L M N u ∈ mHom R (HOMR L M) (HOMR L N)
lemma mId_bijec:
bijecM,M mIdM
lemma invmfun_bijec:
[| Module N R; f ∈ mHom R M N; bijecM,N f |] ==> bijecN,M invmfun R M N f
lemma misom_self:
M ≅R M
lemma misom_sym:
[| Module N R; M ≅R N |] ==> N ≅R M
lemma misom_trans:
[| Module L R; Module N R; L ≅R M; M ≅R N |] ==> L ≅R N
lemma qmodule_carr:
submodule R M H ==> carrier (M /m H) = set_mr_cos M H
lemma set_mr_cos_mem:
[| submodule R M H; m ∈ carrier M |] ==> m \<uplus>M H ∈ set_mr_cos M H
lemma mem_set_mr_cos:
[| submodule R M N; x ∈ set_mr_cos M N |] ==> ∃m∈carrier M. x = m \<uplus>M N
lemma m_in_mr_coset:
[| submodule R M H; m ∈ carrier M |] ==> m ∈ m \<uplus>M H
lemma mr_cos_h_stable:
[| submodule R M H; h ∈ H |] ==> H = h \<uplus>M H
lemma mr_cos_h_stable1:
[| submodule R M H; m ∈ carrier M; h ∈ H |]
==> (m ± h) \<uplus>M H = m \<uplus>M H
lemma x_in_mr_coset:
[| submodule R M H; m ∈ carrier M; x ∈ m \<uplus>M H |] ==> ∃h∈H. m ± h = x
lemma mr_cos_sprodTr:
[| submodule R M H; a ∈ carrier R; m ∈ carrier M |]
==> mr_cos_sprod M H a (m \<uplus>M H) = a ·s m \<uplus>M H
lemma mr_cos_sprod_mem:
[| submodule R M H; a ∈ carrier R; X ∈ set_mr_cos M H |]
==> mr_cos_sprod M H a X ∈ set_mr_cos M H
lemma mr_cos_sprod_assoc:
[| submodule R M H; a ∈ carrier R; b ∈ carrier R; X ∈ set_mr_cos M H |]
==> mr_cos_sprod M H (a ·rR b) X = mr_cos_sprod M H a (mr_cos_sprod M H b X)
lemma mr_cos_sprod_one:
[| submodule R M H; X ∈ set_mr_cos M H |] ==> mr_cos_sprod M H 1rR X = X
lemma mr_cospOpTr:
[| submodule R M H; m ∈ carrier M; n ∈ carrier M |]
==> mr_cospOp M H (m \<uplus>M H) (n \<uplus>M H) = (m ± n) \<uplus>M H
lemma mr_cos_sprod_distrib1:
[| submodule R M H; a ∈ carrier R; b ∈ carrier R; X ∈ set_mr_cos M H |]
==> mr_cos_sprod M H (a ±R b) X =
mr_cospOp M H (mr_cos_sprod M H a X) (mr_cos_sprod M H b X)
lemma mr_cos_sprod_distrib2:
[| submodule R M H; a ∈ carrier R; X ∈ set_mr_cos M H; Y ∈ set_mr_cos M H |]
==> mr_cos_sprod M H a (mr_cospOp M H X Y) =
mr_cospOp M H (mr_cos_sprod M H a X) (mr_cos_sprod M H a Y)
lemma mr_cosmOpTr:
[| submodule R M H; m ∈ carrier M |]
==> mr_cosmOp M H (m \<uplus>M H) = (-a m) \<uplus>M H
lemma mr_cos_oneTr:
submodule R M H ==> H = \<zero> \<uplus>M H
lemma mr_cos_oneTr1:
[| submodule R M H; m ∈ carrier M |]
==> mr_cospOp M H H (m \<uplus>M H) = m \<uplus>M H
lemma qmodule_is_ag:
submodule R M H ==> aGroup (M /m H)
lemma qmodule_module:
submodule R M H ==> Module (M /m H) R
lemma indmhom_someTr:
[| Module N R; f ∈ mHom R M N; X ∈ set_mr_cos M (kerM,N f) |]
==> f (SOME xa. xa ∈ X) ∈ f ` carrier M
lemma indmhom_someTr1:
[| Module N R; f ∈ mHom R M N; m ∈ carrier M |]
==> f (SOME xa. xa ∈ m \<uplus>M kerM,N f) = f m
lemma indmhom_someTr2:
[| Module N R; f ∈ mHom R M N; submodule R M H; m ∈ carrier M; H ⊆ kerM,N f |]
==> f (SOME xa. xa ∈ m \<uplus>M H) = f m
lemma indmhomTr1:
[| Module N R; f ∈ mHom R M N; m ∈ carrier M |]
==> (f\<flat>R M, N) (m \<uplus>M kerM,N f) = f m
lemma indmhomTr2:
[| Module N R; f ∈ mHom R M N |]
==> f\<flat>R M, N ∈ set_mr_cos M (kerM,N f) -> carrier N
lemma indmhom:
[| Module N R; f ∈ mHom R M N |] ==> f\<flat>R M, N ∈ mHom R (M /m (kerM,N f)) N
lemma indmhom_injec:
[| Module N R; f ∈ mHom R M N |] ==> injecM /m (kerM,N f),N f\<flat>R M, N
lemma indmhom_surjec1:
[| Module N R; surjecM,N f; f ∈ mHom R M N |]
==> surjecM /m (kerM,N f),N f\<flat>R M, N
lemma module_homTr:
[| Module N R; f ∈ mHom R M N |] ==> f ∈ mHom R M (mimgR M,N f)
lemma ker_to_mimg:
[| Module N R; f ∈ mHom R M N |] ==> kerM,mimgR M,N f f = kerM,N f
lemma module_homTr1:
[| Module N R; f ∈ mHom R M N |]
==> mimgR M /m (kerM,N f),N f\<flat>R M, N = mimgR M,N f
lemma module_Homth_1:
[| Module N R; f ∈ mHom R M N |] ==> M /m (kerM,N f) ≅R mimgR M,N f
lemma elem_mpj:
[| m ∈ carrier M; submodule R M H |] ==> mpj M H m = m \<uplus>M H
lemma mpj_mHom:
submodule R M H ==> mpj M H ∈ mHom R M (M /m H)
lemma mpj_mem:
[| submodule R M H; m ∈ carrier M |] ==> mpj M H m ∈ carrier (M /m H)
lemma mpj_surjec:
submodule R M H ==> surjecM,M /m H mpj M H
lemma mpj_0:
[| submodule R M H; h ∈ H |] ==> mpj M H h = \<zero>M /m H
lemma mker_of_mpj:
submodule R M H ==> kerM,M /m H mpj M H = H
lemma indmhom1:
[| submodule R M H; Module N R; f ∈ mHom R M N; H ⊆ kerM,N f |]
==> ∃!g. g ∈ mHom R (M /m H) N ∧ compos M g (mpj M H) = f
lemma mQmpTr0:
[| submodule R M H; submodule R M N; H ⊆ N; m ∈ carrier M |]
==> (MpM H,N) (m \<uplus>M H) = m \<uplus>M N
lemma mQmpTr1:
[| submodule R M H; submodule R M N; H ⊆ N; m ∈ carrier M; n ∈ carrier M;
m \<uplus>M H = n \<uplus>M H |]
==> m \<uplus>M N = n \<uplus>M N
lemma mQmpTr2:
[| submodule R M H; submodule R M N; H ⊆ N; X ∈ carrier (M /m H) |]
==> (MpM H,N) X ∈ carrier (M /m N)
lemma mQmpTr2_1:
[| submodule R M H; submodule R M N; H ⊆ N |]
==> MpM H,N ∈ carrier (M /m H) -> carrier (M /m N)
lemma mQmpTr3:
[| submodule R M H; submodule R M N; H ⊆ N; X ∈ carrier (M /m H);
Y ∈ carrier (M /m H) |]
==> (MpM H,N) (mr_cospOp M H X Y) = mr_cospOp M N ((MpM H,N) X) ((MpM H,N) Y)
lemma mQmpTr4:
[| submodule R M H; submodule R M N; H ⊆ N; a ∈ N |]
==> mr_coset a (mdl M N) H = mr_coset a M H
lemma mQmp_mHom:
[| submodule R M H; submodule R M N; H ⊆ N |]
==> MpM H,N ∈ mHom R (M /m H) (M /m N)
lemma Mp_surjec:
[| submodule R M H; submodule R M N; H ⊆ N |] ==> surjecM /m H,M /m N (MpM H,N)
lemma kerQmp:
[| submodule R M H; submodule R M N; H ⊆ N |]
==> kerM /m H,M /m N (MpM H,N) = carrier (mdl M N /m H)
lemma misom2Tr:
[| submodule R M H; submodule R M N; H ⊆ N |]
==> M /m H /m carrier (mdl M N /m H) ≅R M /m N
lemma eq_class_of_Submodule:
[| submodule R M H; submodule R M N; H ⊆ N |]
==> carrier (mdl M N /m H) = N s/\<^sub>M H
theorem misom2:
[| submodule R M H; submodule R M N; H ⊆ N |]
==> M /m H /m (N s/\<^sub>M H) ≅R M /m N
lemma finitesumbase_sub_carrier:
f ∈ I -> {X. submodule R M X} ==> finitesum_base M I f ⊆ carrier M
lemma finitesum_sub_carrier:
f ∈ I -> {X. submodule R M X} ==> finitesum M I f ⊆ carrier M
lemma finitesum_inc_zero:
[| f ∈ I -> {X. submodule R M X}; I ≠ {} |] ==> \<zero> ∈ finitesum M I f
lemma finitesum_mOp_closed:
[| f ∈ I -> {X. submodule R M X}; I ≠ {}; a ∈ finitesum M I f |]
==> -a a ∈ finitesum M I f
lemma finitesum_pOp_closed:
[| f ∈ I -> {X. submodule R M X}; a ∈ finitesum M I f; b ∈ finitesum M I f |]
==> a ± b ∈ finitesum M I f
lemma finitesum_sprodTr:
[| f ∈ I -> {X. submodule R M X}; I ≠ {}; r ∈ carrier R |]
==> g ∈ {j. j ≤ n} -> finitesum_base M I f -->
r ·s Σe M g n = Σe M (λx. r ·s g x) n
lemma finitesum_sprod:
[| f ∈ I -> {X. submodule R M X}; I ≠ {}; r ∈ carrier R;
g ∈ {j. j ≤ n} -> finitesum_base M I f |]
==> r ·s Σe M g n = Σe M (λx. r ·s g x) n
lemma finitesum_subModule:
[| f ∈ I -> {X. submodule R M X}; I ≠ {} |] ==> submodule R M (finitesum M I f)
lemma sSum_cont_H:
[| submodule R M H; submodule R M K |] ==> H ⊆ H \<minusplus> K
lemma sSum_commute:
[| submodule R M H; submodule R M K |] ==> H \<minusplus> K = K \<minusplus> H
lemma Sum_of_SubmodulesTr:
[| submodule R M H; submodule R M K |]
==> g ∈ {j. j ≤ n} -> H ∪ K --> Σe M g n ∈ H \<minusplus> K
lemma sSum_two_Submodules:
[| submodule R M H; submodule R M K |] ==> submodule R M (H \<minusplus> K)
lemma iotam_mHom:
[| submodule R M H; submodule R M K |]
==> ιmM H,K ∈ mHom R (mdl M H) (mdl M (H \<minusplus> K))
lemma mhomom3Tr:
[| submodule R M H; submodule R M K |]
==> submodule R (mdl M (H \<minusplus> K)) K
lemma mhomom3Tr0:
[| submodule R M H; submodule R M K |]
==> compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (ιmM H,K)
∈ mHom R (mdl M H) (mdl M (H \<minusplus> K) /m K)
lemma mhomom3Tr1:
[| submodule R M H; submodule R M K |]
==> surjecmdl M
H,mdl M (H \<minusplus> K) /m
K compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (ιmM H,K)
lemma mhomom3Tr2:
[| submodule R M H; submodule R M K |]
==> kermdl M
H,mdl M (H \<minusplus> K) /m
K compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (ιmM H,K) =
H ∩ K
lemma mhomom_3:
[| submodule R M H; submodule R M K |]
==> mdl M H /m (H ∩ K) ≅R mdl M (H \<minusplus> K) /m K
lemma l_comb_mem_linear_span:
[| ideal R A; H ⊆ carrier M; s ∈ {j. j ≤ n} -> A; f ∈ {j. j ≤ n} -> H |]
==> l_comb R M n s f ∈ linear_span R M A H
lemma linear_comb_eqTr:
H ⊆ carrier M
==> s ∈ {j. j ≤ n} -> carrier R ∧
f ∈ {j. j ≤ n} -> H ∧ g ∈ {j. j ≤ n} -> H ∧ (∀j∈{j. j ≤ n}. f j = g j) -->
l_comb R M n s f = l_comb R M n s g
lemma linear_comb_eq:
[| H ⊆ carrier M; s ∈ {j. j ≤ n} -> carrier R; f ∈ {j. j ≤ n} -> H;
g ∈ {j. j ≤ n} -> H; ∀j∈{j. j ≤ n}. f j = g j |]
==> l_comb R M n s f = l_comb R M n s g
lemma l_comb_Suc:
[| H ⊆ carrier M; ideal R A; s ∈ {j. j ≤ Suc n} -> carrier R;
f ∈ {j. j ≤ Suc n} -> H |]
==> l_comb R M (Suc n) s f = l_comb R M n s f ± s (Suc n) ·s f (Suc n)
lemma l_comb_jointfun_jj:
[| H ⊆ carrier M; ideal R A; s ∈ {j. j ≤ n} -> A; f ∈ {j. j ≤ n} -> H;
t ∈ {j. j ≤ m} -> A; g ∈ {j. j ≤ m} -> H |]
==> Σe M (λj. jointfun n s m t j ·s jointfun n f m g j) n =
Σe M (λj. s j ·s f j) n
lemma l_comb_jointfun_jj1:
[| H ⊆ carrier M; ideal R A; s ∈ {j. j ≤ n} -> A; f ∈ {j. j ≤ n} -> H;
t ∈ {j. j ≤ m} -> A; g ∈ {j. j ≤ m} -> H |]
==> l_comb R M n (jointfun n s m t) (jointfun n f m g) = l_comb R M n s f
lemma l_comb_jointfun_jf:
[| H ⊆ carrier M; ideal R A; s ∈ {j. j ≤ n} -> A; f ∈ {j. j ≤ Suc (n + m)} -> H;
t ∈ {j. j ≤ m} -> A |]
==> Σe M (λj. jointfun n s m t j ·s f j) n = Σe M (λj. s j ·s f j) n
lemma l_comb_jointfun_jf1:
[| H ⊆ carrier M; ideal R A; s ∈ {j. j ≤ n} -> A; f ∈ {j. j ≤ Suc (n + m)} -> H;
t ∈ {j. j ≤ m} -> A |]
==> l_comb R M n (jointfun n s m t) f = l_comb R M n s f
lemma l_comb_jointfun_fj:
[| H ⊆ carrier M; ideal R A; s ∈ {j. j ≤ Suc (n + m)} -> A; f ∈ {j. j ≤ n} -> H;
g ∈ {j. j ≤ m} -> H |]
==> Σe M (λj. s j ·s jointfun n f m g j) n = Σe M (λj. s j ·s f j) n
lemma l_comb_jointfun_fj1:
[| H ⊆ carrier M; ideal R A; s ∈ {j. j ≤ Suc (n + m)} -> A; f ∈ {j. j ≤ n} -> H;
g ∈ {j. j ≤ m} -> H |]
==> l_comb R M n s (jointfun n f m g) = l_comb R M n s f
lemma linear_comb0_1Tr:
H ⊆ carrier M
==> s ∈ {j. j ≤ n} -> {\<zero>R} ∧ m ∈ {j. j ≤ n} -> H -->
l_comb R M n s m = \<zero>
lemma linear_comb0_1:
[| H ⊆ carrier M; s ∈ {j. j ≤ n} -> {\<zero>R}; m ∈ {j. j ≤ n} -> H |]
==> l_comb R M n s m = \<zero>
lemma linear_comb0_2Tr:
ideal R A
==> s ∈ {j. j ≤ n} -> A ∧ m ∈ {j. j ≤ n} -> {\<zero>} -->
l_comb R M n s m = \<zero>
lemma linear_comb0_2:
[| ideal R A; s ∈ {j. j ≤ n} -> A; m ∈ {j. j ≤ n} -> {\<zero>} |]
==> l_comb R M n s m = \<zero>
lemma liear_comb_memTr:
[| ideal R A; H ⊆ carrier M |]
==> ∀s m. s ∈ {j. j ≤ n} -> A ∧ m ∈ {j. j ≤ n} -> H -->
l_comb R M n s m ∈ carrier M
lemma l_comb_mem:
[| ideal R A; H ⊆ carrier M; s ∈ {j. j ≤ n} -> A; m ∈ {j. j ≤ n} -> H |]
==> l_comb R M n s m ∈ carrier M
lemma l_comb_transpos:
[| ideal R A; H ⊆ carrier M; s ∈ {l. l ≤ Suc n} -> A; f ∈ {l. l ≤ Suc n} -> H;
j < Suc n |]
==> Σe M cmp (λk. s k ·s f k) (transpos j (Suc n)) Suc n =
Σe M (λk. cmp s (transpos j (Suc n)) k ·s
cmp f (transpos j (Suc n)) k) Suc n
lemma l_comb_transpos1:
[| ideal R A; H ⊆ carrier M; s ∈ {l. l ≤ Suc n} -> A; f ∈ {l. l ≤ Suc n} -> H;
j < Suc n |]
==> l_comb R M (Suc n) s f =
l_comb R M (Suc n) (cmp s (transpos j (Suc n))) (cmp f (transpos j (Suc n)))
lemma sc_linear_span:
[| ideal R A; H ⊆ carrier M; a ∈ A; h ∈ H |] ==> a ·s h ∈ linear_span R M A H
lemma l_span_cont_H:
H ⊆ carrier M ==> H ⊆ linear_span R M (carrier R) H
lemma linear_span_inc_0:
[| ideal R A; H ⊆ carrier M |] ==> \<zero> ∈ linear_span R M A H
lemma linear_span_iOp_closedTr1:
[| ideal R A; s ∈ {j. j ≤ n} -> A |]
==> (λx∈{j. j ≤ n}. -aR s x) ∈ {j. j ≤ n} -> A
lemma l_span_gen_mono:
[| K ⊆ H; H ⊆ carrier M; ideal R A |]
==> linear_span R M A K ⊆ linear_span R M A H
lemma l_comb_add:
[| ideal R A; H ⊆ carrier M; s ∈ {j. j ≤ n} -> A; f ∈ {j. j ≤ n} -> H;
t ∈ {j. j ≤ m} -> A; g ∈ {j. j ≤ m} -> H |]
==> l_comb R M (Suc (n + m)) (jointfun n s m t) (jointfun n f m g) =
l_comb R M n s f ± l_comb R M m t g
lemma l_comb_add1Tr:
[| ideal R A; H ⊆ carrier M |]
==> f ∈ {j. j ≤ n} -> H ∧ s ∈ {j. j ≤ n} -> A ∧ t ∈ {j. j ≤ n} -> A -->
l_comb R M n (λx∈{j. j ≤ n}. s x ±R t x) f =
l_comb R M n s f ± l_comb R M n t f
lemma l_comb_add1:
[| ideal R A; H ⊆ carrier M; f ∈ {j. j ≤ n} -> H; s ∈ {j. j ≤ n} -> A;
t ∈ {j. j ≤ n} -> A |]
==> l_comb R M n (λx∈{j. j ≤ n}. s x ±R t x) f =
l_comb R M n s f ± l_comb R M n t f
lemma linear_span_iOp_closedTr2:
[| ideal R A; H ⊆ carrier M; f ∈ {j. j ≤ n} -> H; s ∈ {j. j ≤ n} -> A |]
==> -a l_comb R M n s f = l_comb R M n (λx∈{j. j ≤ n}. -aR s x) f
lemma linear_span_iOp_closed:
[| ideal R A; H ⊆ carrier M; a ∈ linear_span R M A H |]
==> -a a ∈ linear_span R M A H
lemma linear_span_pOp_closed:
[| ideal R A; H ⊆ carrier M; a ∈ linear_span R M A H; b ∈ linear_span R M A H |]
==> a ± b ∈ linear_span R M A H
lemma l_comb_scTr:
[| ideal R A; H ⊆ carrier M; r ∈ carrier R; H ≠ {} |]
==> s ∈ {j. j ≤ n} -> A ∧ g ∈ {j. j ≤ n} -> H -->
r ·s Σe M (λk. s k ·s g k) n = Σe M (λk. r ·s (s k ·s g k)) n
lemma l_comb_sc1Tr:
[| ideal R A; H ⊆ carrier M; r ∈ carrier R; H ≠ {} |]
==> s ∈ {j. j ≤ n} -> A ∧ g ∈ {j. j ≤ n} -> H -->
r ·s Σe M (λk. s k ·s g k) n = Σe M (λk. (r ·rR s k) ·s g k) n
lemma l_comb_sc:
[| ideal R A; H ⊆ carrier M; r ∈ carrier R; s ∈ {j. j ≤ n} -> A;
g ∈ {j. j ≤ n} -> H |]
==> r ·s Σe M (λk. s k ·s g k) n = Σe M (λk. r ·s (s k ·s g k)) n
lemma l_comb_sc1:
[| ideal R A; H ⊆ carrier M; r ∈ carrier R; s ∈ {j. j ≤ n} -> A;
g ∈ {j. j ≤ n} -> H |]
==> r ·s Σe M (λk. s k ·s g k) n = Σe M (λk. (r ·rR s k) ·s g k) n
lemma linear_span_sc_closed:
[| ideal R A; H ⊆ carrier M; r ∈ carrier R; x ∈ linear_span R M A H |]
==> r ·s x ∈ linear_span R M A H
lemma mem_single_l_spanTr:
[| ideal R A; h ∈ carrier M |]
==> s ∈ {j. j ≤ n} -> A ∧
f ∈ {j. j ≤ n} -> {h} ∧ l_comb R M n s f ∈ linear_span R M A {h} -->
(∃a∈A. l_comb R M n s f = a ·s h)
lemma mem_single_l_span:
[| ideal R A; h ∈ carrier M; s ∈ {j. j ≤ n} -> A; f ∈ {j. j ≤ n} -> {h};
l_comb R M n s f ∈ linear_span R M A {h} |]
==> ∃a∈A. l_comb R M n s f = a ·s h
lemma mem_single_l_span1:
[| ideal R A; h ∈ carrier M; x ∈ linear_span R M A {h} |] ==> ∃a∈A. x = a ·s h
lemma linear_span_subModule:
[| ideal R A; H ⊆ carrier M |] ==> submodule R M (linear_span R M A H)
lemma l_comb_mem_submoduleTr:
[| ideal R A; submodule R M N |]
==> s ∈ {j. j ≤ n} -> A ∧
f ∈ {j. j ≤ n} -> carrier M ∧ (∀j≤n. s j ·s f j ∈ N) -->
l_comb R M n s f ∈ N
lemma l_span_sub_submodule:
[| ideal R A; submodule R M N; H ⊆ N |] ==> linear_span R M A H ⊆ N
lemma linear_span_sub:
[| ideal R A; H ⊆ carrier M |] ==> linear_span R M A H ⊆ carrier M
lemma smodule_ideal_coeff_is_Submodule:
ideal R A ==> submodule R M (A \<odot>R M)
lemma mem_smodule_ideal_coeff:
[| ideal R A; x ∈ A \<odot>R M |]
==> ∃n. ∃s∈{j. j ≤ n} -> A. ∃g∈{j. j ≤ n} -> carrier M. x = l_comb R M n s g
lemma quotient_of_submodules_inc_0:
[| submodule R M P; submodule R M Q |] ==> \<zero>R ∈ P R\<ddagger>M Q
lemma quotient_of_submodules_is_ideal:
[| submodule R M P; submodule R M Q |] ==> ideal R (P R\<ddagger>M Q)
lemma Ann_is_ideal:
ideal R (AnnR M)
lemma linmap_im_of_lincombTr:
[| ideal R A; Module N R; f ∈ mHom R M N; H ⊆ carrier M |]
==> s ∈ {j. j ≤ n} -> A ∧ g ∈ {j. j ≤ n} -> H -->
f (l_comb R M n s g) = l_comb R N n s (cmp f g)
lemma linmap_im_lincomb:
[| ideal R A; Module N R; f ∈ mHom R M N; H ⊆ carrier M; s ∈ {j. j ≤ n} -> A;
g ∈ {j. j ≤ n} -> H |]
==> f (l_comb R M n s g) = l_comb R N n s (cmp f g)
lemma linmap_im_linspan:
[| ideal R A; Module N R; f ∈ mHom R M N; H ⊆ carrier M; s ∈ {j. j ≤ n} -> A;
g ∈ {j. j ≤ n} -> H |]
==> f (l_comb R M n s g) ∈ linear_span R N A (f ` H)
lemma linmap_im_linspan1:
[| ideal R A; Module N R; f ∈ mHom R M N; H ⊆ carrier M;
h ∈ linear_span R M A H |]
==> f h ∈ linear_span R N A (f ` H)
lemma h_in_linear_span:
[| H ⊆ carrier M; h ∈ H |] ==> h ∈ linear_span R M (carrier R) H
lemma generator_sub_carrier:
generator R M H ==> H ⊆ carrier M
lemma lin_span_sub_carrier:
[| ideal R A; H ⊆ carrier M |] ==> linear_span R M A H ⊆ carrier M
lemma lin_span_coeff_mono:
[| ideal R A; H ⊆ carrier M |]
==> linear_span R M A H ⊆ linear_span R M (carrier R) H
lemma l_span_sum_closedTr:
[| ideal R A; H ⊆ carrier M |]
==> ∀s f. s ∈ {j. j ≤ n} -> A ∧ f ∈ {j. j ≤ n} -> linear_span R M A H -->
Σe M (λj. s j ·s f j) n ∈ linear_span R M A H
lemma l_span_closed:
[| ideal R A; H ⊆ carrier M; s ∈ {j. j ≤ n} -> A;
f ∈ {j. j ≤ n} -> linear_span R M A H |]
==> l_comb R M n s f ∈ linear_span R M A H
lemma l_span_closed1:
[| H ⊆ carrier M; s ∈ {j. j ≤ n} -> carrier R;
f ∈ {j. j ≤ n} -> linear_span R M (carrier R) H |]
==> Σe M (λj. s j ·s f j) n ∈ linear_span R M (carrier R) H
lemma l_span_closed2Tr0:
[| ideal R A; H ⊆ carrier M; Ring R; s ∈ A; f ∈ linear_span R M (carrier R) H |]
==> s ·s f ∈ linear_span R M A H
lemma l_span_closed2Tr:
[| ideal R A; H ⊆ carrier M |]
==> s ∈ {j. j ≤ n} -> A ∧ f ∈ {j. j ≤ n} -> linear_span R M (carrier R) H -->
l_comb R M n s f ∈ linear_span R M A H
lemma l_span_closed2:
[| ideal R A; H ⊆ carrier M; s ∈ {j. j ≤ n} -> A;
f ∈ {j. j ≤ n} -> linear_span R M (carrier R) H |]
==> l_comb R M n s f ∈ linear_span R M A H
lemma l_span_l_span:
H ⊆ carrier M
==> linear_span R M (carrier R) (linear_span R M (carrier R) H) =
linear_span R M (carrier R) H
lemma l_spanA_l_span:
[| ideal R A; H ⊆ carrier M |]
==> linear_span R M A (linear_span R M (carrier R) H) = linear_span R M A H
lemma l_span_zero:
ideal R A ==> linear_span R M A {\<zero>} = {\<zero>}
lemma l_span_closed3:
[| ideal R A; generator R M H; A \<odot>R M = carrier M |]
==> linear_span R M A H = carrier M
lemma generator_generator:
[| generator R M H; H1.0 ⊆ carrier M; H ⊆ linear_span R M (carrier R) H1.0 |]
==> generator R M H1.0
lemma generator_elimTr:
f ∈ {j. j ≤ n} -> carrier M ∧
generator R M (f ` {j. j ≤ n}) ∧
(∀i∈nset (Suc 0) n.
f i ∈ linear_span R M (carrier R) (f ` {j. j ≤ i - Suc 0})) -->
linear_span R M (carrier R) {f 0} = carrier M
lemma generator_generator_elim:
[| f ∈ {j. j ≤ n} -> carrier M; generator R M (f ` {j. j ≤ n});
∀i∈nset (Suc 0) n.
f i ∈ linear_span R M (carrier R) (f ` {j. j ≤ i - Suc 0}) |]
==> linear_span R M (carrier R) {f 0} = carrier M
lemma surjec_generator:
[| Module N R; f ∈ mHom R M N; surjecM,N f; generator R M H |]
==> generator R N (f ` H)
lemma surjec_finitely_gen:
[| Module N R; f ∈ mHom R M N; surjecM,N f; M fgover R |] ==> N fgover R
lemma similar_termTr:
[| ideal R A; a ∈ A |]
==> ∀s f. s ∈ {j. j ≤ n} -> A ∧
f ∈ {j. j ≤ n} -> carrier M ∧ m ∈ f ` {j. j ≤ n} -->
(∃t∈{j. j ≤ n} -> A.
Σe M (λj. s j ·s f j) n ± a ·s m = Σe M (λj. t j ·s f j) n)
lemma similar_term1:
[| ideal R A; a ∈ A; s ∈ {j. j ≤ n} -> A; f ∈ {j. j ≤ n} -> carrier M;
m ∈ f ` {j. j ≤ n} |]
==> ∃t∈{j. j ≤ n} -> A.
Σe M (λj. s j ·s f j) n ± a ·s m = Σe M (λj. t j ·s f j) n
lemma same_togetherTr:
[| ideal R A; H ⊆ carrier M |]
==> ∀s f. s ∈ {j. j ≤ n} -> A ∧ f ∈ {j. j ≤ n} -> H -->
(∃t∈{j. j ≤ card (f ` {j. j ≤ n}) - Suc 0} -> A.
∃g∈{j. j ≤ card (f ` {j. j ≤ n}) - Suc 0} -> f ` {j. j ≤ n}.
surj_to g {j. j ≤ card (f ` {j. j ≤ n}) - Suc 0}
(f ` {j. j ≤ n}) ∧
Σe M (λj. s j ·s f j) n =
Σe M (λk. t k ·s g k) (card (f ` {j. j ≤ n}) - Suc 0))
lemma same_together:
[| ideal R A; H ⊆ carrier M; s ∈ {j. j ≤ n} -> A; f ∈ {j. j ≤ n} -> H |]
==> ∃t∈{j. j ≤ card (f ` {j. j ≤ n}) - Suc 0} -> A.
∃g∈{j. j ≤ card (f ` {j. j ≤ n}) - Suc 0} -> f ` {j. j ≤ n}.
surj_to g {j. j ≤ card (f ` {j. j ≤ n}) - Suc 0} (f ` {j. j ≤ n}) ∧
Σe M (λj. s j ·s f j) n =
Σe M (λk. t k ·s g k) (card (f ` {j. j ≤ n}) - Suc 0)
lemma one_last:
[| ideal R A; H ⊆ carrier M; s ∈ {j. j ≤ Suc n} -> A; f ∈ {j. j ≤ Suc n} -> H;
bij_to f {j. j ≤ Suc n} H; j ≤ Suc n; j ≠ Suc n |]
==> ∃t∈{j. j ≤ Suc n} -> A.
∃g∈{j. j ≤ Suc n} -> H.
Σe M (λk. s k ·s f k) Suc n = Σe M (λk. t k ·s g k) Suc n ∧
g (Suc n) = f j ∧ t (Suc n) = s j ∧ bij_to g {j. j ≤ Suc n} H
lemma finite_lin_spanTr1:
[| ideal R A; z ∈ carrier M |]
==> h ∈ {j. j ≤ n} -> {z} ∧ t ∈ {j. j ≤ n} -> A -->
(∃s∈{0} -> A. Σe M (λj. t j ·s h j) n = s 0 ·s z)
lemma single_span:
[| ideal R A; z ∈ carrier M; h ∈ {j. j ≤ n} -> {z}; t ∈ {j. j ≤ n} -> A |]
==> ∃s∈{0} -> A. Σe M (λj. t j ·s h j) n = s 0 ·s z
lemma card_Nset_im:
f ∈ {j. j ≤ n} -> A ==> Suc 0 ≤ card (f ` {j. j ≤ n})
lemma eSum_changeTr1:
[| ideal R A; t ∈ {k. k ≤ card (f ` {j. j ≤ n1.0}) - Suc 0} -> A;
g ∈ {k. k ≤ card (f ` {j. j ≤ n1.0}) - Suc 0} -> f ` {j. j ≤ n1.0};
Suc 0 < card (f ` {j. j ≤ n1.0}); g x = h (Suc n); x = Suc n;
card (f ` {j. j ≤ n1.0}) - Suc 0 =
Suc (card (f ` {j. j ≤ n1.0}) - Suc 0 - Suc 0) |]
==> Σe M (λk. t k ·s g k) (card (f ` {j. j ≤ n1.0}) - Suc 0) =
Σe M (λk. t k ·s g k) (card (f ` {j. j ≤ n1.0}) - Suc 0 - Suc 0) ±
t (Suc (card (f ` {j. j ≤ n1.0}) - Suc 0 - Suc 0)) ·s
g (Suc (card (f ` {j. j ≤ n1.0}) - Suc 0 - Suc 0))
lemma zeroi_func:
[| Ring R; ideal R A |] ==> zeroi R ∈ {j. j ≤ 0} -> A
lemma prep_arrTr1:
[| ideal R A; h ∈ {j. j ≤ Suc n} -> carrier M;
f ∈ {j. j ≤ n1.0} -> h ` {j. j ≤ Suc n}; s ∈ {j. j ≤ n1.0} -> A;
m = l_comb R M n1.0 s f |]
==> ∃l∈{j. j ≤ Suc n}.
∃s∈{j. j ≤ l} -> A.
∃g∈{j. j ≤ l} -> h ` {j. j ≤ Suc n}.
m = l_comb R M l s g ∧ bij_to g {j. j ≤ l} (f ` {j. j ≤ n1.0})
lemma two_func_imageTr:
[| h ∈ {j. j ≤ Suc n} -> B; f ∈ {j. j ≤ m} -> h ` {j. j ≤ Suc n};
h (Suc n) ∉ f ` {j. j ≤ m} |]
==> f ∈ {j. j ≤ m} -> h ` {j. j ≤ n}
lemma finite_lin_spanTr3_0:
[| bij_to g {j. j ≤ l} (g ` {j. j ≤ l}); ideal R A;
∀na. ∀s∈{j. j ≤ na} -> A.
∀f∈{j. j ≤ na} -> h ` {j. j ≤ n}.
∃t∈{j. j ≤ n} -> A. l_comb R M na s f = l_comb R M n t h;
h ∈ {j. j ≤ Suc n} -> carrier M; s ∈ {j. j ≤ m} -> A;
f ∈ {j. j ≤ m} -> h ` {j. j ≤ Suc n}; l ≤ Suc n; sa ∈ {j. j ≤ l} -> A;
g ∈ {j. j ≤ l} -> h ` {j. j ≤ Suc n}; 0 < l; f ` {j. j ≤ m} = g ` {j. j ≤ l};
h (Suc n) = g l |]
==> ∃t∈{j. j ≤ Suc n} -> A. l_comb R M l sa g = l_comb R M (Suc n) t h
lemma finite_lin_spanTr3:
ideal R A
==> h ∈ {j. j ≤ n} -> carrier M -->
(∀na. ∀s∈{j. j ≤ na} -> A.
∀f∈{j. j ≤ na} -> h ` {j. j ≤ n}.
∃t∈{j. j ≤ n} -> A. l_comb R M na s f = l_comb R M n t h)
lemma finite_lin_span:
[| ideal R A; h ∈ {j. j ≤ n} -> carrier M; s ∈ {j. j ≤ n1.0} -> A;
f ∈ {j. j ≤ n1.0} -> h ` {j. j ≤ n} |]
==> ∃t∈{j. j ≤ n} -> A. l_comb R M n1.0 s f = l_comb R M n t h
lemma free_generator_generator:
free_generator R M H ==> generator R M H
lemma free_generator_sub:
free_generator R M H ==> H ⊆ carrier M
lemma free_generator_nonzero:
[| ¬ zeroring R; free_generator R M H; h ∈ H |] ==> h ≠ \<zero>
lemma has_free_generator_nonzeroring:
[| free_generator R M H; ∃p∈linear_span R M (carrier R) H. p ≠ \<zero> |]
==> ¬ zeroring R
lemma unique_expression1:
[| H ⊆ carrier M; free_generator R M H; s ∈ {j. j ≤ n} -> carrier R;
m ∈ {j. j ≤ n} -> H; inj_on m {j. j ≤ n}; l_comb R M n s m = \<zero> |]
==> ∀j∈{j. j ≤ n}. s j = \<zero>R
lemma free_gen_coeff_zero:
[| H ⊆ carrier M; free_generator R M H; h ∈ H; a ∈ carrier R;
a ·s h = \<zero> |]
==> a = \<zero>R
lemma unique_expression2:
[| H ⊆ carrier M; f ∈ {j. j ≤ n} -> H; s ∈ {j. j ≤ n} -> carrier R |]
==> ∃m g t.
g ∈ {j. j ≤ m} -> H ∧
bij_to g {j. j ≤ m} (f ` {j. j ≤ n}) ∧
t ∈ {j. j ≤ m} -> carrier R ∧ l_comb R M n s f = l_comb R M m t g
lemma unique_expression3_1:
[| H ⊆ carrier M; f ∈ {l. l ≤ Suc n} -> H; s ∈ {l. l ≤ Suc n} -> carrier R;
f (Suc n) ∉ f ` ({l. l ≤ Suc n} - {Suc n}) |]
==> ∃g m t.
g ∈ {l. l ≤ m} -> H ∧
inj_on g {l. l ≤ m} ∧
t ∈ {l. l ≤ m} -> carrier R ∧
l_comb R M (Suc n) s f = l_comb R M m t g ∧
t m = s (Suc n) ∧ g m = f (Suc n)
lemma unique_expression3_2:
[| H ⊆ carrier M; f ∈ {k. k ≤ Suc n} -> H; s ∈ {k. k ≤ Suc n} -> carrier R;
l ≤ Suc n; f l ∉ f ` ({k. k ≤ Suc n} - {l}); l ≠ Suc n |]
==> ∃g m t.
g ∈ {l. l ≤ m} -> H ∧
inj_on g {l. l ≤ m} ∧
t ∈ {l. l ≤ m} -> carrier R ∧
l_comb R M (Suc n) s f = l_comb R M m t g ∧ t m = s l ∧ g m = f l
lemma unique_expression3:
[| H ⊆ carrier M; f ∈ {k. k ≤ Suc n} -> H; s ∈ {k. k ≤ Suc n} -> carrier R;
l ≤ Suc n; f l ∉ f ` ({k. k ≤ Suc n} - {l}) |]
==> ∃g m t.
g ∈ {k. k ≤ m} -> H ∧
inj_on g {k. k ≤ m} ∧
t ∈ {k. k ≤ m} -> carrier R ∧
l_comb R M (Suc n) s f = l_comb R M m t g ∧ t m = s l ∧ g m = f l
lemma unique_expression4:
free_generator R M H
==> f ∈ {k. k ≤ n} -> H ∧
inj_on f {k. k ≤ n} ∧
s ∈ {k. k ≤ n} -> carrier R ∧ l_comb R M n s f ≠ \<zero> -->
(∃m g t.
g ∈ {k. k ≤ m} -> H ∧
inj_on g {k. k ≤ m} ∧
g ` {k. k ≤ m} ⊆ f ` {k. k ≤ n} ∧
t ∈ {k. k ≤ m} -> carrier R ∧
(∀l∈{k. k ≤ m}. t l ≠ \<zero>R) ∧ l_comb R M n s f = l_comb R M m t g)
lemma unique_prepression5_0:
[| free_generator R M H; f ∈ {j. j ≤ n} -> H; inj_on f {j. j ≤ n};
s ∈ {j. j ≤ n} -> carrier R; g ∈ {j. j ≤ m} -> H; inj_on g {j. j ≤ m};
t ∈ {j. j ≤ m} -> carrier R; l_comb R M n s f = l_comb R M m t g;
∀j≤n. s j ≠ \<zero>R; ∀k≤m. t k ≠ \<zero>R; f n ∉ g ` {j. j ≤ m}; 0 < n |]
==> False
lemma unique_expression5:
[| free_generator R M H; f ∈ {j. j ≤ n} -> H; inj_on f {j. j ≤ n};
s ∈ {j. j ≤ n} -> carrier R; g ∈ {j. j ≤ m} -> H; inj_on g {j. j ≤ m};
t ∈ {j. j ≤ m} -> carrier R; l_comb R M n s f = l_comb R M m t g;
∀j∈{j. j ≤ n}. s j ≠ \<zero>R; ∀k∈{j. j ≤ m}. t k ≠ \<zero>R |]
==> f ` {j. j ≤ n} ⊆ g ` {j. j ≤ m}
lemma unique_expression6:
[| free_generator R M H; f ∈ {j. j ≤ n} -> H; inj_on f {j. j ≤ n};
s ∈ {j. j ≤ n} -> carrier R; g ∈ {j. j ≤ m} -> H; inj_on g {j. j ≤ m};
t ∈ {j. j ≤ m} -> carrier R; l_comb R M n s f = l_comb R M m t g;
∀j∈{j. j ≤ n}. s j ≠ \<zero>R; ∀k∈{j. j ≤ m}. t k ≠ \<zero>R |]
==> f ` {j. j ≤ n} = g ` {j. j ≤ m}
lemma unique_expression7_1:
[| free_generator R M H; f ∈ {j. j ≤ n} -> H; inj_on f {j. j ≤ n};
s ∈ {j. j ≤ n} -> carrier R; g ∈ {j. j ≤ m} -> H; inj_on g {j. j ≤ m};
t ∈ {j. j ≤ m} -> carrier R; l_comb R M n s f = l_comb R M m t g;
∀j∈{j. j ≤ n}. s j ≠ \<zero>R; ∀k∈{j. j ≤ m}. t k ≠ \<zero>R |]
==> n = m
lemma unique_expression7_2:
[| free_generator R M H; f ∈ {j. j ≤ n} -> H; inj_on f {j. j ≤ n};
s ∈ {j. j ≤ n} -> carrier R; t ∈ {j. j ≤ n} -> carrier R;
l_comb R M n s f = l_comb R M n t f |]
==> ∀l∈{j. j ≤ n}. s l = t l