(** Algebra2
author Hidetsune Kobayashi
Department of Mathematics
Nihon University
hikoba@math.cst.nihon-u.ac.jp
May 3, 2004.
April 6, 2007 (revised)
chapter 2. ordered set (continued)
section 2 Pre elements
section 3 Transfinite induction
section 4. ordered_set2. preliminary lemmas for Zorn's lemma
section 5. Zorn's lemma
chapter 3. Group Theory. Focused on Jordan Hoelder theorem
section 1. definition of a group
section 2. subgroups
section 3. cosets
section 4. normal subgroups and quotient groups
**)
theory Algebra2 imports Algebra1 begin
lemma (in Order) less_and_segment:"b ∈ carrier D ==>
(∀a.((a \<prec> b ∧ a ∈ carrier D) --> (Q a))) =
(∀a∈carrier (Iod D (segment D b)).(Q a))"
apply (rule iffI)
apply (rule ballI)
apply (cut_tac segment_sub[of "b"], simp add:Iod_carrier,
thin_tac "segment D b ⊆ carrier D",
simp add:segment_def)
apply (rule allI, rule impI, erule conjE)
apply (cut_tac segment_sub[of "b"], simp add:Iod_carrier,
thin_tac "segment D b ⊆ carrier D",
simp add:segment_def)
done
lemma (in Worder) Word_compare2:"[|Worder E;
¬ (∀a∈carrier D. ∃b∈carrier E. ord_equiv (Iod D (segment D a))
(Iod E (segment E b)))|] ==>
∃c∈carrier D. ord_equiv (Iod D (segment D c)) E"
apply simp
apply (frule bex_nonempty_set[of "carrier D"],
frule nonempty_set_sub[of "carrier D" _],
thin_tac "∃a∈carrier D. ∀b∈carrier E.
¬ ord_equiv (Iod D (segment D a)) (Iod E (segment E b))")
apply (insert ex_minimum,
frule_tac a = "{x ∈ carrier D. ∀b∈carrier E.
¬ ord_equiv (Iod D (segment D x)) (Iod E (segment E b))}" in
forall_spec, simp,
thin_tac "{x ∈ carrier D. ∀b∈carrier E.
¬ ord_equiv (Iod D (segment D x)) (Iod E (segment E b))} ≠ {}",
thin_tac "∀X. X ⊆ carrier D ∧ X ≠ {} --> (∃x. minimum_elem D X x)",
erule exE)
apply (frule_tac X = "{x ∈ carrier D. ∀b∈carrier E.
¬ ord_equiv (Iod D (segment D x)) (Iod E (segment E b))}" and a = x in
minimum_elem_mem, assumption, simp)
apply (rename_tac d)
apply (erule conjE, thin_tac "∀b∈carrier E.
¬ ord_equiv (Iod D (segment D d)) (Iod E (segment E b))")
apply (frule_tac d = d in less_minimum) apply simp
apply (simp add:less_and_segment)
apply (cut_tac a = d in segment_Worder)
apply (frule_tac D = "Iod D (segment D d)" in Worder.well_ord_compare1 [of _
"E"], assumption+)
apply (rule ballI)
apply (frule_tac b = a in forball_spec1, assumption+)
apply (simp add:Iod_segment_segment)
apply (erule disjE) apply blast
apply (frule_tac X = "{x ∈ carrier D. ∀b∈carrier E.
¬ ord_equiv (Iod D (segment D x)) (Iod E (segment E b))}" and
a = d in minimum_elem_mem, assumption, simp)
done
lemma (in Worder) Worder_equiv:"[|Worder E;
∀a∈carrier D. ∃b∈carrier E. ord_equiv (Iod D (segment D a))
(Iod E (segment E b));
∀c∈carrier E. ∃d∈carrier D. ord_equiv (Iod E (segment E c))
(Iod D (segment D d))|] ==> ord_equiv D E"
apply (frule well_ord_compare1 [of "E"], assumption+,
erule disjE, assumption)
apply (erule bexE,
insert Worder,
frule_tac b = c in forball_spec1, assumption+,
thin_tac "∀c∈carrier E. ∃d∈carrier D.
ord_equiv (Iod E (segment E c)) (Iod D (segment D d))",
erule bexE)
apply (cut_tac a = d in segment_Worder,
cut_tac D = E and a = c in Worder.segment_Worder, assumption+,
frule_tac D = "Iod D (segment D d)" in Worder.Order,
frule_tac D = "Iod E (segment E c)" in Worder.Order,
insert Order)
apply (frule_tac D = "D" and E = "Iod E (segment E c)" and
F = "Iod D (segment D d)" in Order.ord_equiv_trans, assumption+,
frule_tac a = d in nonequiv_segment)
apply simp
done
lemma (in Worder) Worder_equiv1:"[|Worder E; ¬ ord_equiv D E|] ==>
¬ ((∀a∈carrier D. ∃b∈carrier E.
ord_equiv (Iod D (segment D a)) (Iod E (segment E b))) ∧
(∀c∈carrier E. ∃d∈carrier D.
ord_equiv (Iod E (segment E c)) (Iod D (segment D d))))"
apply (rule contrapos_pp, simp+) apply (erule conjE)
apply (frule Worder_equiv [of "E"], assumption+)
apply simp
done
lemma (in Worder) Word_compare:"Worder E ==>
(∃a∈carrier D. ord_equiv (Iod D (segment D a)) E) ∨ ord_equiv D E ∨
(∃b∈carrier E. ord_equiv D (Iod E (segment E b)))"
apply (frule Worder.Order[of "E"],
case_tac "ord_equiv D E", simp)
apply (frule Worder_equiv1 [of "E"], assumption+)
apply simp
apply (erule disjE)
apply (frule Word_compare2 [of "E"], simp)
apply blast
apply (cut_tac Worder.Word_compare2 [of "E" "D"])
apply (thin_tac "∃c∈carrier E. ∀d∈carrier D.
¬ ord_equiv (Iod E (segment E c)) (Iod D (segment D d))")
apply (erule bexE,
cut_tac a = c in Worder.segment_Worder[of "E"], assumption+,
frule_tac D = "Iod E (segment E c)" in Worder.Order,
insert Order,
frule_tac D = "Iod E (segment E c)" and E = D in Order.ord_equiv_sym,
assumption+, blast)
apply assumption apply (simp add:Worder)
apply simp
done
lemma (in Worder) Word_compareTr1:"[|Worder E;
∃a∈carrier D. ord_equiv (Iod D (segment D a)) E; ord_equiv D E |] ==>
False"
apply (erule bexE,
cut_tac a = a in segment_Worder,
frule_tac D = E in Worder.Order,
frule_tac D = "Iod D (segment D a)" in Worder.Order,
frule_tac D = "Iod D (segment D a)" and E = E in Order.ord_equiv_sym,
assumption+)
apply (insert Order,
frule_tac D = D and E = E and F = "Iod D (segment D a)" in
Order.ord_equiv_trans, assumption+)
apply (frule_tac a = a in nonequiv_segment, simp)
done
lemma (in Worder) Word_compareTr2:"[|Worder E; ord_equiv D E;
∃b∈carrier E. ord_equiv D (Iod E (segment E b))|] ==> False"
apply (erule bexE)
apply (cut_tac a = b in Worder.segment_Worder [of "E"], assumption+)
apply (cut_tac Worder,
frule Worder.Order[of "E"])
apply (frule_tac D = "Iod E (segment E b)" in Worder.Order)
apply (frule_tac E = E in ord_equiv_sym, assumption)
apply (meson Worder Worder.Word_compareTr1 ord_equiv_sym)
done
lemma (in Worder) Word_compareTr3:"[|Worder E;
∃b∈carrier E. ord_equiv D (Iod E (segment E b));
∃a∈carrier D. ord_equiv (Iod D (segment D a)) E|] ==> False"
apply (erule bexE)+
apply (simp add:ord_equiv_def[of "D"], erule exE)
(** ord_isom_segment_segment **)
apply (frule Worder.Torder[of "E"])
apply (cut_tac a = b in Worder.segment_Worder [of "E"], assumption+,
frule_tac D = "Iod E (segment E b)" in Worder.Torder,
frule_tac D = "Iod E (segment E b)" in Worder.Order)
apply (frule_tac E = "Iod E (segment E b)" and f = f and a = a in
ord_isom_segment_segment, assumption+)
apply (frule_tac f = f and a = a and E = "Iod E (segment E b)" in
ord_isom_mem, assumption+,
frule Worder.Order[of "E"],
frule_tac b = b and a = "f a" in Order.Iod_segment_segment[of "E"],
assumption+, simp,
thin_tac "Iod (Iod E (segment E b)) (segment (Iod E (segment E b))
(f a)) = Iod E (segment E (f a))")
apply (frule_tac D = "E" and a = b in Order.segment_sub)
apply (simp add:Order.Iod_carrier[of "E"])
apply (frule_tac c = "f a" and A = "segment E b" and B = "carrier E"
in subsetD, assumption+)
apply (cut_tac a = a in segment_Worder,
frule_tac D = "Iod D (segment D a)" in Worder.Order,
cut_tac a = "f a" in Worder.segment_Worder [of "E"], assumption+,
frule_tac D = "Iod E (segment E (f a))" in Worder.Order)
apply (
frule_tac D = "Iod D (segment D a)" and E = "Iod E (segment E (f a))" in Order.ord_equiv, assumption, simp)
apply (frule_tac D = "Iod D (segment D a)" and E = E in
Order.ord_equiv_sym, assumption+)
apply (frule_tac D = E and E = "Iod D (segment D a)" and
F = "Iod E (segment E (f a))" in Order.ord_equiv_trans, assumption+)
apply (simp add:Worder.nonequiv_segment[of "E"])
done
lemma (in Worder) subset_equiv_segment:"S ⊆ carrier D ==>
ord_equiv D (Iod D S) ∨
(∃a∈carrier D. ord_equiv (Iod D S) (Iod D (segment D a)))"
apply (frule subset_Worder [of "S"])
apply (frule Word_compare [of "Iod D S"])
apply (erule disjE)
apply (erule bexE)
apply (cut_tac a = a in segment_Worder,
frule_tac D = "Iod D (segment D a)" in Worder.Order,
frule Worder.Order[of "Iod D S"],
frule_tac D = "Iod D (segment D a)" in Order.ord_equiv_sym[of _
"Iod D S"], assumption+, blast)
apply (erule disjE) apply simp
apply (erule bexE)
apply (frule Worder.Order[of "Iod D S"],
frule_tac D = "Iod D S" and a = b in Order.segment_sub)
apply (frule_tac S = "segment (Iod D S) b" in Order.Iod_sub_sub[of "Iod D S" _ "S"])
apply (simp add:Iod_carrier) apply (simp add:Iod_carrier)
apply (simp add:Iod_sub_sub[of "S" "S"])
apply (simp add:Iod_carrier)
apply (frule_tac S = "segment (Iod D S) b" in Iod_sub_sub[of _ "S"],
assumption+, simp,
thin_tac "Iod (Iod D S) (segment (Iod D S) b) =
Iod D (segment (Iod D S) b)")
(** to_subset **)
apply (simp add:ord_equiv_def, erule exE)
apply (frule_tac A = "segment (Iod D S) b" and B = S and C = "carrier D" in
subset_trans, assumption+,
frule_tac c = b in subsetD[of "S" "carrier D"], assumption+)
apply (frule_tac T = "segment (Iod D S) b" and f = f in to_subset,
assumption+,
frule_tac a = b in forall_spec, assumption,
thin_tac "∀a. a ∈ carrier D --> a \<preceq> f a",
frule_tac T = "segment (Iod D S) b" in Iod_Order,
cut_tac D = "Iod D S" and a = b in Worder.segment_Worder)
apply (simp add:Iod_carrier)
apply (frule_tac S = "segment (Iod D S) b" in Iod_sub_sub[of _ "S"],
assumption+, simp,
thin_tac "Iod (Iod D S) (segment (Iod D S) b) =
Iod D (segment (Iod D S) b)")
apply (insert Order,
frule_tac E = "Iod D (segment (Iod D S) b)" and f = f and a = b in
ord_isom_mem, assumption+,
simp add:Iod_carrier,
frule_tac c = "f b" and A = "segment (Iod D S) b" in
subsetD[of _ "S"], assumption+,
simp add:segment_def[of "Iod D S"],
simp add:Iod_carrier,
simp add:Iod_less)
apply (frule_tac c = b in subsetD[of "S" "carrier D"], assumption+,
frule_tac c = "f b" in subsetD[of "S" "carrier D"], assumption+,
frule_tac a1 = "f b" and b1 = b in not_less_le[THEN sym], assumption+)
apply simp
done
constdefs
ordinal_number :: "'a Order => 'a Order set"
"ordinal_number S == {X. Worder X ∧ ord_equiv X S}"
ODnums :: "'a Order set set"
"ODnums == {X. ∃S. Worder S ∧ X = ordinal_number S}"
ODord :: "['a Order set, 'a Order set] => bool" (infix "\<sqsubset>" 60)
"X \<sqsubset> Y == ∃x ∈ X. ∃y ∈ Y. (∃c∈carrier y. ord_equiv x (Iod y (segment y c)))"
ODord_le::"['a Order set, 'a Order set] => bool" (infix "\<sqsubseteq>" 60)
"X \<sqsubseteq> Y == X = Y ∨ ODord X Y"
ODrel :: "((('a Order) set) * (('a Order) set)) set"
"ODrel == {Z. Z ∈ ODnums × ODnums ∧ ODord_le (fst Z) (snd Z)}"
ODnods :: "('a Order set) Order"
"ODnods == (|carrier = ODnums, rel = ODrel |)),"
lemma Worder_ord_equivTr:"[|Worder S; Worder T|] ==>
ord_equiv S T = (∃f. ord_isom S T f)"
by (frule Worder.Order[of "S"], frule Worder.Order[of "T"],
simp add:ord_equiv_def)
lemma Worder_ord_isom_mem:"[|Worder S; Worder T; ord_isom S T f; a ∈ carrier S|]
==> f a ∈ carrier T"
by (frule Worder.Order[of "S"], frule Worder.Order[of "T"],
simp add:Order.ord_isom_mem)
lemma Worder_refl:"Worder S ==> ord_equiv S S"
apply (frule_tac Worder.Order [of "S"])
apply (rule Order.ord_equiv_reflex [of "S"], assumption+)
done
lemma Worder_sym:"[|Worder S; Worder T; ord_equiv S T |] ==> ord_equiv T S"
apply (frule_tac Worder.Order [of "S"])
apply (frule_tac Worder.Order [of "T"])
apply (rule Order.ord_equiv_sym [of "S" "T"], assumption+)
done
lemma Worder_trans:"[|Worder S; Worder T; Worder U; ord_equiv S T; ord_equiv T U|] ==> ord_equiv S U"
apply (frule Worder.Order [of "S"],
frule Worder.Order [of "T"],
frule Worder.Order [of "U"])
apply (rule Order.ord_equiv_trans [of "S" "T" "U"], assumption+)
done
lemma ordinal_inc_self:"Worder S ==> S ∈ ordinal_number S"
by (simp add:ordinal_number_def, simp add:Worder_refl)
lemma ordinal_number_eq:"[|Worder D; Worder E|] ==>
(ord_equiv D E) = (ordinal_number D = ordinal_number E)"
apply (rule iffI)
apply (simp add:ordinal_number_def)
apply (rule equalityI)
apply (rule subsetI) apply simp apply (erule conjE)
apply (rule_tac S = x and T = D and U = E in Worder_trans,
assumption+)
apply (rule subsetI, simp, erule conjE)
apply (rule_tac S = x and T = E and U = D in Worder_trans,
assumption+)
apply (rule Worder_sym, assumption+)
apply (simp add:ordinal_number_def)
apply (frule Worder_refl[of "D"],
frule Worder_refl[of "E"])
apply blast
done
lemma mem_ordinal_number_equiv:"[|Worder D;
X ∈ ordinal_number D|] ==> ord_equiv X D"
by (simp add:ordinal_number_def)
lemma mem_ordinal_number_Worder:"[|Worder D;
X ∈ ordinal_number D|] ==> Worder X"
by (simp add:ordinal_number_def)
lemma mem_ordinal_number_Worder1:"[|x ∈ ODnums; X ∈ x|] ==> Worder X"
apply (simp add:ODnums_def, erule exE, erule conjE, simp)
apply (rule mem_ordinal_number_Worder, assumption+)
done
lemma mem_ODnums_nonempty:"X ∈ ODnums ==> ∃x. x ∈ X"
apply (simp add:ODnums_def, simp add:ordinal_number_def,
erule exE, erule conjE)
apply (frule_tac S = S in Worder_refl, blast)
done
lemma carr_ODnods:"carrier ODnods = ODnums"
by (simp add:ODnods_def)
lemma ordinal_number_mem_carrier_ODnods:
"Worder D ==> ordinal_number D ∈ carrier ODnods"
by (simp add:ODnods_def ODnums_def, blast)
lemma ordinal_number_mem_ODnums:
"Worder D ==> ordinal_number D ∈ ODnums"
by (simp add:ODnums_def, blast)
lemma ODordTr1:" [|Worder D; Worder E|] ==>
(ODord (ordinal_number D) (ordinal_number E)) =
(∃b∈ carrier E. ord_equiv D (Iod E (segment E b)))"
apply (rule iffI)
apply (simp add:ODord_def)
apply ((erule bexE)+, simp add:ordinal_number_def, (erule conjE)+)
apply (rename_tac X Y c)
apply (frule_tac S = Y and T = E in Worder_ord_equivTr, assumption,
simp, erule exE)
apply (frule_tac D = Y in Worder.Order,
frule_tac D = E in Worder.Order,
frule_tac D = Y and E = E and f = f and a = c in
Order.ord_isom_segment_segment, assumption+,
frule_tac S = Y and T = E and f = f and a = c in
Worder_ord_isom_mem, assumption+)
apply (cut_tac D = Y and a = c in Worder.segment_Worder, assumption,
cut_tac D = E and a = "f c" in Worder.segment_Worder, assumption)
apply (frule_tac S1 = "Iod Y (segment Y c)" and T1 = "Iod E (segment E (f c))"
in Worder_ord_equivTr[THEN sym], assumption+)
apply (frule_tac S = X and T = D in Worder_sym, assumption+,
thin_tac "ord_equiv X D",
frule_tac S = D and T = X and U = "Iod Y (segment Y c)" in
Worder_trans, assumption+,
frule_tac S = D and T = "Iod Y (segment Y c)" and
U = "Iod E (segment E (f c))" in Worder_trans, assumption+)
apply blast apply blast
apply (simp add:ODord_def)
apply (frule ordinal_inc_self[of "D"],
frule ordinal_inc_self[of "E"], blast)
done
lemma ODord:" [|Worder D; d ∈ carrier D|] ==>
ODord (ordinal_number (Iod D (segment D d))) (ordinal_number D)"
apply (cut_tac Worder.segment_Worder[of "D" "d"],
subst ODordTr1[of "Iod D (segment D d)" "D"], assumption+,
frule Worder_refl[of "Iod D (segment D d)"], blast, assumption)
done
lemma ord_less_ODord:"[|Worder D; c ∈ carrier D; d ∈ carrier D;
a = ordinal_number (Iod D (segment D c));
b = ordinal_number (Iod D (segment D d))|] ==>
c \<prec>D d = a \<sqsubset> b"
apply (rule iffI)
apply (frule Worder.Order[of "D"])
apply (simp add:Order.segment_inc)
apply (frule Order.Iod_carr_segment[THEN sym, of "D" "d"],
frule eq_set_inc[of "c" "segment D d" "carrier (Iod D (segment D d))"],
assumption+,
thin_tac "segment D d = carrier (Iod D (segment D d))",
thin_tac "c ∈ segment D d")
apply (cut_tac Worder.segment_Worder[of "D" "d"],
frule ODord[of "Iod D (segment D d)" "c"], assumption+,
simp add:Order.Iod_segment_segment, assumption)
apply simp
apply (frule Worder.segment_Worder[of D c],
frule Worder.segment_Worder[of D d])
apply (simp add:ODordTr1[of "Iod D (segment D c)" "Iod D (segment D d)"],
thin_tac "a = ordinal_number (Iod D (segment D c))",
thin_tac "b = ordinal_number (Iod D (segment D d))")
apply (erule bexE)
apply (frule Worder.Order[of D])
apply (simp add:Order.Iod_segment_segment)
apply (simp add:Order.Iod_carr_segment,
frule Order.segment_sub[of D d],
frule_tac c = b in subsetD[of "segment D d" "carrier D"], assumption+)
apply (frule_tac b = b in Worder.segment_unique[of D c], assumption+, simp)
apply (simp add:Order.segment_inc[THEN sym])
done
lemma ODord_le_ref:"[|X ∈ ODnums; Y ∈ ODnums; ODord_le X Y; Y \<sqsubseteq> X |] ==>
X = Y"
apply (simp add:ODnums_def)
apply ((erule exE)+, (erule conjE)+, rename_tac S T)
apply (simp add:ODord_le_def)
apply (erule disjE, simp)
apply (erule disjE, simp)
apply (simp add:ODordTr1)
apply (frule_tac D = T and E = S in Worder.Word_compareTr3, assumption+)
apply (erule bexE)
apply (frule_tac D = T and a = b in Worder.segment_Worder)
apply (frule_tac S = S and T = "Iod T (segment T b)" in Worder_sym,
assumption+) apply blast
apply simp
done
lemma ODord_le_trans:"[|X ∈ ODnums; Y ∈ ODnums; Z ∈ ODnums; X \<sqsubseteq> Y; Y \<sqsubseteq> Z |]
==> X \<sqsubseteq> Z"
apply (simp add:ODord_le_def)
apply (erule disjE, simp)
apply (erule disjE, simp)
apply (simp add:ODnums_def, (erule exE)+, (erule conjE)+)
apply (rename_tac A B C, simp)
apply (simp add:ODordTr1,
thin_tac "X = ordinal_number A",
thin_tac "Y = ordinal_number B",
thin_tac "Z = ordinal_number C")
apply (erule bexE)+
(* going to apply ord_isom_segment_segment *)
apply (frule_tac D = B in Worder.Order,
frule_tac D = C in Worder.Order,
frule_tac D = C and a = ba in Worder.segment_Worder,
frule_tac D = "Iod C (segment C ba)" in Worder.Order,
frule_tac D = B and E = "Iod C (segment C ba)" in
Order.ord_equiv_isom, assumption+, erule exE)
apply (frule_tac D = B and E = "Iod C (segment C ba)" and f = f in
Order.ord_isom_segment_segment, assumption+,
frule_tac D = B and E = "Iod C (segment C ba)" and f = f and a = b in
Order.ord_isom_mem, assumption+)
(** ord_isom_segment_segment applied **)
apply (simp add:Order.Iod_segment_segment)
apply (frule_tac D = B and a = b in Worder.segment_Worder,
frule_tac D = "Iod B (segment B b)" in Worder.Order,
frule_tac D = C and a = "f b" in Worder.segment_Worder,
frule_tac D = "Iod C (segment C (f b))" in Worder.Order)
apply (frule_tac D = "Iod B (segment B b)" and E = "Iod C (segment C (f b))"
in Order.ord_equiv, assumption+)
apply (frule_tac S = A and T = "Iod B (segment B b)" and
U = "Iod C (segment C (f b))" in Worder_trans, assumption+)
apply (simp add:Order.Iod_carr_segment)
apply (frule_tac D = C and a = ba in Order.segment_sub,
frule_tac c = "f b" and A = "segment C ba" and B = "carrier C" in
subsetD, assumption+)
apply blast
done
lemma ordinal_numberTr1:" X ∈ carrier ODnods ==> ∃D. Worder D ∧ D ∈ X"
apply (simp add:ODnods_def, simp add:ODnums_def)
apply (erule exE, erule conjE)
apply (simp add:ordinal_number_def)
apply (frule_tac S = S in Worder_refl, blast)
done
lemma ordinal_numberTr1_1:" X ∈ ODnums ==> ∃D. Worder D ∧ D ∈ X"
apply (simp add:ODnums_def, erule exE, erule conjE,
simp add:ordinal_number_def)
apply (frule_tac S = S in Worder_refl, blast)
done
lemma ordinal_numberTr1_2:"[|x ∈ ODnums; S ∈ x; T ∈ x|] ==>
ord_equiv S T"
by (simp add:ODnums_def, erule exE, erule conjE,
simp add:ordinal_number_def, (erule conjE)+,
frule_tac S = T and T = Sa in Worder_sym, assumption, assumption,
rule_tac S = S and T = Sa and U = T in Worder_trans, assumption+)
lemma ordinal_numberTr2:"[|Worder D; x = ordinal_number D|] ==>
D ∈ x"
by (simp add:ordinal_inc_self)
lemma ordinal_numberTr3:"[|Worder D; Worder F; ord_equiv D F;
x = ordinal_number D|] ==> x = ordinal_number F"
apply (simp add:ordinal_number_def,
thin_tac "x = {X. Worder X ∧ ord_equiv X D}")
apply (rule equalityI)
apply (rule subsetI, simp, erule conjE)
apply (rule_tac S = x and T = D and U = F in Worder_trans, assumption+)
apply (rule subsetI, simp, erule conjE)
apply (frule Worder_sym [of "D" "F"], assumption+,
rule_tac S = x and T = F and U = D in Worder_trans, assumption+)
done
lemma ordinal_numberTr4:"[|Worder D; X ∈ carrier ODnods; D ∈ X |] ==>
X = ordinal_number D"
apply (simp add:ODnods_def ODnums_def, erule exE, erule conjE)
apply simp
apply (frule_tac D = S in mem_ordinal_number_equiv[of _ "D"], assumption+,
frule_tac D = D and E = S in ordinal_number_eq, assumption+)
apply simp
done
lemma ordinal_numberTr5:"[|x ∈ ODnums; D ∈ x|] ==> x = ordinal_number D"
apply (frule mem_ordinal_number_Worder1[of x D], assumption+)
apply (rule ordinal_numberTr4[of D x], assumption,
simp add:ODnods_def, assumption)
done
lemma ordinal_number_ord:"[| X ∈ carrier ODnods; Y ∈ carrier ODnods|] ==>
ODord X Y ∨ X = Y ∨ ODord Y X"
apply (simp add:ODord_def)
apply (frule ordinal_numberTr1 [of "X"],
frule ordinal_numberTr1 [of "Y"], (erule exE)+, rename_tac D E)
apply (erule conjE)+
apply (frule_tac D = D and E = E in Worder.Word_compare, assumption+)
apply (erule disjE)+
apply (erule bexE,
cut_tac D = D and a = a in Worder.segment_Worder, assumption+,
frule_tac S = "Iod D (segment D a)" and T = E in Worder_sym,
assumption+, blast)
apply (erule disjE)
apply (frule_tac D = D and X = X in ordinal_numberTr4, assumption+,
frule_tac D = E and X = Y in ordinal_numberTr4, assumption+,
frule_tac D = D and E = E in ordinal_number_eq, assumption+)
apply simp
apply blast
done
lemma ODnum_subTr:"[|Worder D; x = ordinal_number D; y ∈ODnums; y \<sqsubset> x; Y ∈ y|]
==> ∃c∈carrier D. ord_equiv Y (Iod D (segment D c))"
apply simp
apply (thin_tac "x = ordinal_number D")
apply (simp add:ODnums_def, erule exE, erule conjE, simp,
thin_tac "y = ordinal_number S")
apply (frule_tac D = S and X = Y in mem_ordinal_number_Worder,
assumption+)
apply (frule_tac D = Y and X = "ordinal_number S" in ordinal_numberTr4,
simp add:ODnods_def ODnums_def, blast, simp)
apply simp
apply (thin_tac "Y ∈ ordinal_number Y",
thin_tac "ordinal_number S = ordinal_number Y")
apply (simp add:ODordTr1[of "Y" "D"])
done
lemma ODnum_segmentTr:"[|Worder D; x = ordinal_number D; y ∈ODnums; y \<sqsubset> x|] ==>
∃c. c∈carrier D ∧ (∀Y∈y. ord_equiv Y (Iod D (segment D c)))"
apply (frule ordinal_numberTr1_1[of "y"], erule exE, erule conjE,
frule_tac x = x and y = y and Y = Da in ODnum_subTr[of "D" ],
assumption+, erule bexE)
apply (rule ex_conjI, simp+)
apply (rule ballI)
apply (frule_tac D = D and a = c in Worder.segment_Worder)
apply (frule_tac X = Y in mem_ordinal_number_Worder1[of y], assumption)
apply (subst ordinal_number_eq, assumption+)
apply (simp add:ordinal_number_eq)
apply (subst ordinal_numberTr5[THEN sym, of y], assumption+)
apply (frule_tac D = Da in ordinal_numberTr5[of y], assumption, simp)
done
lemma ODnum_segmentTr1:"[|Worder D; x = ordinal_number D; y ∈ ODnums; y \<sqsubset> x|]
==> ∃c. c ∈ carrier D ∧ (y = ordinal_number (Iod D (segment D c)))"
apply (frule ODnum_segmentTr[of D x y], assumption+, erule exE, erule conjE)
apply (frule mem_ODnums_nonempty[of y], erule exE,
frule_tac b = xa in forball_spec1, assumption,
thin_tac "∀Y∈y. ord_equiv Y (Iod D (segment D c))")
(** going to apply ordinal_number_eq **)
apply (frule_tac D = xa in ordinal_numberTr5[of y], assumption, simp,
frule_tac a = c in Worder.segment_Worder[of D],
rotate_tac -2, frule sym, thin_tac "y = ordinal_number xa", simp,
frule_tac X = xa in mem_ordinal_number_Worder1[of y], assumption+)
apply (simp add:ordinal_number_eq) apply blast
done
lemma ODnods_less:"[|x ∈ carrier ODnods; y ∈ carrier ODnods|] ==>
x \<prec>ODnods y = x \<sqsubset> y"
apply (simp add:ODnods_def ole_def oless_def ODrel_def ODord_le_def)
apply (rule iffI)
apply (erule conjE, erule disjE, simp, assumption, simp)
apply (simp add:ODord_def, (erule bexE)+)
apply (rule contrapos_pp, simp+,
frule_tac x = y and S = ya and T = xa in ordinal_numberTr1_2,
assumption+)
apply (simp add:ODnums_def, erule exE, erule conjE, simp,
frule_tac D = S and X = xa in mem_ordinal_number_Worder, assumption+,
frule_tac D = S and X = ya in mem_ordinal_number_Worder, assumption+,
cut_tac D = ya and a = c in Worder.segment_Worder, assumption+)
apply (frule_tac S = ya and T = xa and U = "Iod ya (segment ya c)" in
Worder_trans, assumption+,
frule_tac D = ya and a = c in Worder.nonequiv_segment, assumption+)
apply simp
done
lemma ODord_less_not_eq:"[|x ∈ carrier ODnods; y ∈ carrier ODnods; x \<sqsubset> y|] ==>
x ≠ y"
apply (rule contrapos_pp, simp+)
apply (frule ordinal_numberTr1[of y], erule exE, erule conjE,
simp add:ODnods_def)
(* going to see ord_equiv D (Iod D (segment D c)) for some c,
at first show ordinal_number D = ordinal_number (Iod D (segment D c)) *)
apply (frule_tac D = D in ordinal_numberTr5[of y], assumption+,
frule_tac D = D and x = y and y = y in ODnum_segmentTr1, assumption+,
erule exE, erule conjE, simp,
frule_tac a = c and D = D in Worder.segment_Worder) (* equality got *)
apply (rotate_tac -4, frule sym,
thin_tac "ordinal_number (Iod D (segment D c)) = ordinal_number D",
simp add:ordinal_number_eq[THEN sym]) (* ord_equiv got *)
apply (frule_tac D = D and a = c in Worder.nonequiv_segment, assumption)
(**** this is the key lemma ****)
apply simp
done
lemma not_ODord:"[|a ∈ ODnums; b ∈ ODnums; a \<sqsubset> b|] ==> ¬ (b \<sqsubseteq> a)"
apply (rule contrapos_pp, simp+)
apply (simp add:ODord_le_def)
apply (cut_tac x = a and y = b in ODord_less_not_eq,
simp add:ODnods_def, simp add:ODnods_def, assumption)
apply (erule disjE) apply simp
(** Show that ord_equiv to segment each other does not happen **)
apply (frule ordinal_numberTr1_1[of a],
frule ordinal_numberTr1_1[of b], (erule exE)+, (erule conjE)+)
apply (frule_tac D = D in ordinal_numberTr5[of a], assumption,
frule_tac D = Da in ordinal_numberTr5[of b], assumption, simp)
apply (frule_tac D = D and E = Da in ODordTr1, assumption+,
frule_tac D = Da and E = D in ODordTr1, assumption+, simp)
apply (frule_tac D = D and E = Da in Worder.Word_compareTr3, assumption+)
apply (erule bexE)+
apply (frule_tac D = D and a = baa in Worder.segment_Worder,
frule_tac S = Da and T = "Iod D (segment D baa)" in Worder_sym,
assumption+) apply blast
apply assumption
done
lemma Order_ODnods:"Order ODnods"
apply (rule Order.intro)
apply (simp add:ODnods_def ODrel_def)
apply (rule subsetI) apply simp
apply (simp add:ODnods_def ODrel_def, simp add:ODord_le_def)
apply (simp add:ODnods_def ODrel_def, simp add:ODord_le_def)
apply (erule disjE, assumption)
apply (erule disjE, rule sym, assumption)
apply (frule_tac a = a and b = b in not_ODord, assumption+)
apply (simp add:ODord_le_def)
apply (simp add:ODnods_def ODrel_def)
apply (rule_tac X = a and Y = b and Z = c in ODord_le_trans, assumption+)
done
lemma Torder_ODnods:"Torder ODnods"
apply (rule Torder.intro)
apply (cut_tac Order_ODnods, assumption)
(*** tordered set ***)
apply (simp add:Torder_axioms_def)
apply ((rule allI, rule impI)+)
apply (cut_tac Order_ODnods)
apply (subst Order.le_imp_less_or_eq[of "ODnods"], assumption+,
subst Order.le_imp_less_or_eq[of "ODnods"], assumption+)
apply (simp add:ODnods_less,
frule_tac X = a and Y = b in ordinal_number_ord, assumption+,
blast)
done
constdefs
ODNmap :: "'a Order => ('a Order) set => 'a"
"ODNmap D y == SOME z. (z ∈ carrier D ∧
(∀Y∈y. ord_equiv Y (Iod D (segment D z))))"
lemma ODNmap_mem:"[|Worder D; x = ordinal_number D; y ∈ ODnums; ODord y x|] ==>
ODNmap D y ∈ carrier D ∧
(∀Y∈y. ord_equiv Y (Iod D (segment D (ODNmap D y))))"
apply (frule ODnum_segmentTr [of "D" "x" "y"], assumption+)
apply (simp add:ODNmap_def)
apply (rule someI2_ex, assumption+)
done
lemma ODNmapTr1:"[|Worder D; x = ordinal_number D; y ∈ ODnums; ODord y x|] ==>
y = ordinal_number (Iod D (segment D (ODNmap D y)))"
apply (frule ODNmap_mem [of "D" "x" "y"], assumption+, erule conjE,
frule ODnum_segmentTr1 [of "D" "x" "y"], assumption+)
apply (erule exE, erule conjE,
cut_tac D = D and a = c in Worder.segment_Worder, assumption+,
frule_tac D = "Iod D (segment D c)" and x = y in ordinal_numberTr2,
assumption+,
frule_tac b = "Iod D (segment D c)" in forball_spec1, assumption,
thin_tac "∀Y∈y. ord_equiv Y (Iod D (segment D (ODNmap D y)))",
cut_tac a = "ODNmap D y" in Worder.segment_Worder[of "D"],
assumption+)
apply (frule_tac D = "Iod D (segment D c)" and E = "Iod D (segment D (ODNmap D y))" in ordinal_number_eq, assumption+) apply simp
done
lemma ODNmap_self:"[|Worder D; c ∈ carrier D;
a = ordinal_number (Iod D (segment D c))|] ==> ODNmap D a = c"
apply (simp add:ODNmap_def)
apply (rule someI2_ex, rule ex_conjI, simp)
apply (rule ballI,
cut_tac Worder.segment_Worder[of "D" "c"],
rule_tac X = Y in mem_ordinal_number_equiv[of "Iod D (segment D c)"],
assumption+)
apply (erule conjE)
apply (cut_tac Worder.segment_Worder[of "D" "c"],
frule ordinal_inc_self[of "Iod D (segment D c)"],
frule_tac b = "Iod D (segment D c)" in forball_spec1, assumption)
apply (frule_tac b = x in Worder.segment_unique[of "D" "c" _], assumption+,
rule sym, assumption, assumption)
done
lemma ODord_ODNmap_less:"[|Worder D; c ∈ carrier D;
a = ordinal_number (Iod D (segment D c)); d ∈ carrier D;
b = ordinal_number (Iod D (segment D d)); a \<sqsubset> b|] ==>
ODNmap D a \<prec>D (ODNmap D b)"
apply (frule ODNmap_self [of "D" "c" "a"], assumption+,
frule ODNmap_self [of "D" "d" "b"], assumption+)
apply simp
apply (subst ord_less_ODord[of D c d a b], assumption+)
apply simp
done
lemma ODNmap_mem1:" [|Worder D; y ∈ segment ODnods (ordinal_number D)|]
==> ODNmap D y ∈ carrier D"
apply (frule_tac D = D and x = "ordinal_number D" and y = y in ODNmap_mem,
simp,
frule ordinal_number_mem_carrier_ODnods[of "D"],
simp add:ODnods_def segment_def)
apply (simp add:segment_def,
frule ordinal_number_mem_carrier_ODnods[of "D"], simp,
erule conjE, simp add:ODnods_def oless_def ole_def ODrel_def)
apply (simp add:ODord_le_def, blast, simp)
done
lemma ODnods_segment_inc_ODord:"[|Worder D;
y ∈ segment ODnods (ordinal_number D)|] ==> ODord y (ordinal_number D)"
apply (simp add:segment_def,
frule ordinal_number_mem_carrier_ODnods[of "D"], simp,
erule conjE, simp add:ODnods_def oless_def ole_def ODrel_def)
apply ((erule conjE)+, simp add:ODord_le_def)
done
lemma restict_ODNmap_func:"[|Worder D; x = ordinal_number D|] ==>
restrict (ODNmap D) (segment ODnods (ordinal_number D))
∈ segment ODnods (ordinal_number D) -> carrier D"
apply (cut_tac Order_ODnods,
frule Order.Iod_carr_segment[of "ODnods" "ordinal_number D"],
frule Order.segment_sub[of "ODnods" "ordinal_number D"])
apply (rule univar_func_test, rule ballI, simp,
(frule Order.Iod_carr_segment[of ODnods "ordinal_number D"],
simp,
thin_tac "carrier (Iod ODnods (segment ODnods (ordinal_number D))) =
segment ODnods (ordinal_number D) "),
rule ODNmap_mem1[of D], assumption+)
done
lemma ODNmap_ord_isom:"[|Worder D; x = ordinal_number D|] ==>
ord_isom (Iod ODnods (segment ODnods x)) D
(λx∈(carrier (Iod ODnods (segment ODnods x))). (ODNmap D x))"
apply (cut_tac Order_ODnods,
frule Order.Iod_carr_segment[of "ODnods" "ordinal_number D"],
frule ordinal_number_mem_carrier_ODnods[of D],
frule Order.segment_sub[of "ODnods" "ordinal_number D"])
(* above items are preliminary *)
apply (simp add:ord_isom_def)
apply (rule conjI)
apply (simp add:ord_inj_def)
apply (simp add:restict_ODNmap_func[of D x])
apply (rule conjI)
apply (subst inj_on_def) (* show inj_on *)
apply ((rule ballI)+, rule impI) (* show xa and y is the ordinal_number of
segments of D *)
apply (thin_tac "carrier (Iod ODnods (segment ODnods (ordinal_number D))) =
segment ODnods (ordinal_number D)")
apply (frule_tac y = xa in ODnods_segment_inc_ODord[of D], assumption+,
frule_tac y = y in ODnods_segment_inc_ODord[of D], assumption+)
apply (frule_tac y = y in ODNmapTr1[of D x], assumption+)
apply (frule_tac c = y in subsetD[of "segment ODnods (ordinal_number D)"
"carrier ODnods"], assumption+, simp add:carr_ODnods,
simp)
apply (frule_tac y = xa in ODNmapTr1[of D x], assumption+)
apply (frule_tac c = xa in subsetD[of "segment ODnods (ordinal_number D)"
"carrier ODnods"], assumption+, simp add:carr_ODnods,
simp, simp)
apply (rule ballI)+
apply (thin_tac "carrier (Iod ODnods (segment ODnods (ordinal_number D))) =
segment ODnods (ordinal_number D)")
apply (frule_tac y = a in ODnods_segment_inc_ODord[of D], assumption+,
frule_tac y = b in ODnods_segment_inc_ODord[of D], assumption+)
apply (frule_tac y = a in ODNmapTr1[of D x], assumption+)
apply (frule_tac c = a in subsetD[of "segment ODnods (ordinal_number D)"
"carrier ODnods"], assumption+, simp add:carr_ODnods,
simp)
apply (frule_tac y = b in ODNmapTr1[of D x], assumption+)
apply (frule_tac c = b in subsetD[of "segment ODnods (ordinal_number D)"
"carrier ODnods"], assumption+, simp add:carr_ODnods,
simp)
apply (frule_tac c = "ODNmap D a" and d = "ODNmap D b" and a = a and b = b in
ord_less_ODord[of D],
frule_tac x = x and y = a in ODNmap_mem[of D], assumption,
frule_tac c = a in subsetD[of "segment ODnods (ordinal_number D)"
"carrier ODnods"], assumption+, simp add:carr_ODnods,
simp, simp)
apply (frule_tac x = x and y = b in ODNmap_mem[of D], assumption,
frule_tac c = b in subsetD[of "segment ODnods (ordinal_number D)"
"carrier ODnods"], assumption+, simp add:carr_ODnods,
simp, simp) apply simp+
apply (frule_tac c = a in subsetD[of "segment ODnods (ordinal_number D)"
"carrier ODnods"], assumption+,
frule_tac c = b in subsetD[of "segment ODnods (ordinal_number D)"
"carrier ODnods"], assumption+)
apply (simp add:Order.Iod_less[of "ODnods"])
apply (simp add:ODnods_less) (** order_preserving proved **)
apply (rule surj_to_test) (* show surj_to *)
apply (simp add:restict_ODNmap_func)
apply (rule ballI,
cut_tac D = D and a = b in Worder.segment_Worder, assumption+,
frule_tac D = "Iod D (segment D b)" in
ordinal_number_mem_carrier_ODnods,
frule_tac c = b and a = "ordinal_number (Iod D (segment D b))"
in ODNmap_self, assumption, simp,
frule_tac d = b in ODord[of D], assumption,
frule_tac a = "ordinal_number (Iod D (segment D b))" in
Order.segment_inc[of "ODnods" _ "ordinal_number D"], assumption+,
cut_tac x = "ordinal_number (Iod D (segment D b))" in
ODnods_less[of _ "ordinal_number D"], assumption+,
simp)
apply (frule_tac c = b and a = "ordinal_number (Iod D (segment D b))" in
ODNmap_self[of D], assumption)
apply simp
apply blast
done
lemma ODnum_equiv_segment:"[|Worder D; x = ordinal_number D|] ==>
ord_equiv (Iod ODnods (segment ODnods x)) D"
apply (simp add: ord_equiv_def)
apply (frule ODNmap_ord_isom[of "D" "x"], assumption+, blast)
done
lemma ODnods_sub_carrier:"S ⊆ ODnums ==> carrier (Iod ODnods S) = S"
by (simp add:Iod_def)
lemma ODnum_sub_well_ordered:"S ⊆ ODnums ==> Worder (Iod ODnods S)"
apply (cut_tac Torder_ODnods,
cut_tac Order_ODnods)
apply (frule Torder.Iod_Torder[of "ODnods" S],
simp add:carr_ODnods)
apply intro_locales
apply (simp add:Torder_def, simp add:Torder_def)
apply (simp add:Worder_axioms_def,
rule allI, rule impI, erule conjE)
(** show existence of the minimum_element of X **)
apply (frule Order.Iod_carrier[of ODnods S],
simp add:carr_ODnods, simp)
apply (frule_tac A = X in nonempty_ex, erule exE)
apply (frule_tac c = x and A = X and B = S in subsetD, assumption+,
frule_tac c = x and A = S and B = ODnums in subsetD, assumption+)
apply (frule_tac X = x in ordinal_numberTr1_1, erule exE, erule conjE)
apply (frule_tac D = D and x = x in ODnum_equiv_segment)
apply (rule ordinal_numberTr4, assumption+, simp add:carr_ODnods, assumption)
apply (frule_tac D = D and T = "Iod ODnods (segment ODnods x)" in
Worder.equiv_Worder1,
rule Order.Iod_Order[of ODnods], assumption,
rule Order.segment_sub, assumption+)
apply (frule_tac D = "Iod ODnods (segment ODnods x)" in Worder.ex_minimum)
apply (case_tac "(segment ODnods x) ∩ X ≠ {}")
apply (frule_tac a = "segment ODnods x ∩ X" in forall_spec)
apply (simp add:Order.Iod_carr_segment, simp add:Int_lower1,
thin_tac "∀X. X ⊆ carrier (Iod ODnods (segment ODnods x)) ∧ X ≠ {}
--> (∃xa. minimum_elem (Iod ODnods (segment ODnods x)) X xa)")
apply (erule exE)
apply (simp add:carr_ODnods[THEN sym])
apply (frule_tac A = X and B = S and C = "carrier ODnods" in subset_trans,
assumption+);
apply (frule_tac d = x and m = xa and X = X in
Torder.segment_minimum_minimum[of ODnods], assumption+,
simp add:Int_commute,
simp add:Order.minimum_elem_sub[of ODnods S], blast)
apply (simp,
thin_tac "∀X. X ⊆ carrier (Iod ODnods (segment ODnods x)) ∧ X ≠ {} -->
(∃xa. minimum_elem (Iod ODnods (segment ODnods x)) X xa)")
apply (frule_tac A = "segment ODnods x" and B = X in no_meet2)
apply (simp add:carr_ODnods[THEN sym])
apply (frule_tac A = X and B = S and C = "carrier ODnods" in subset_trans,
assumption+)
apply (frule_tac A = X and B = S and C = "carrier ODnods" in subset_trans,
assumption+)
apply (simp add:Order.segment_inc[THEN sym, of ODnods])
apply (rule contrapos_pp, simp+)
apply (frule_tac a = x in forall_spec1,
thin_tac "∀x. ¬ minimum_elem (Iod ODnods S) X x")
apply (simp add:minimum_elem_def, erule bexE)
apply (frule_tac b = xa in forball_spec1, assumption,
thin_tac "∀a∈X. a ∉ segment ODnods x",
frule_tac c = xa and A = X and B = "carrier ODnods" in subsetD,
assumption+)
apply (simp add:Order.segment_inc[of ODnods, THEN sym],
frule_tac c = xa and A = X and B = S in subsetD, assumption+)
apply (simp add:Order.Iod_le[of ODnods S])
apply (simp add:Torder.not_le_less)
done
section "2. Pre" (* pre elements *)
constdefs (structure D)
ExPre :: "_ => 'a => bool"
"ExPre D a == ∃x. (x ∈ carrier D ∧ x \<prec> a
∧ ¬ (∃y∈carrier D. (x \<prec> y ∧ y \<prec> a)))"
(* exists pre element *)
Pre :: "[_ , 'a] => 'a"
"Pre D a == SOME x. (x ∈ carrier D ∧
x \<prec> a ∧
¬ (∃y∈carrier D. (x \<prec> y ∧ y \<prec> a)))"
(* Pre element of a *)
lemma (in Order) Pre_mem:"[|a ∈ carrier D; ExPre D a|] ==>
Pre D a ∈ carrier D"
apply (simp add:ExPre_def)
apply (subst Pre_def, rule someI2_ex, blast, simp)
done
lemma (in Order) Not_ExPre:"a ∈ carrier D ==> ¬ ExPre (Iod D {a}) a"
apply (simp add:ExPre_def,
rule allI, rule impI, rule impI,
frule singleton_sub[of "a" "carrier D"])
apply (simp add:Iod_less Iod_carrier)
done
lemma (in Worder) UniquePre:"[|a ∈ carrier D; ExPre D a;
a1 ∈ carrier D ∧ a1 \<prec> a ∧ ¬ (∃y∈carrier D. (a1 \<prec> y ∧ y \<prec> a)) |] ==>
Pre D a = a1"
apply (simp add:ExPre_def)
apply (subst Pre_def) (* definition *)
apply (rule someI2_ex, blast)
apply (erule conjE)+
apply (thin_tac "∃x. x ∈ carrier D ∧ x \<prec> a ∧
(∀y∈carrier D. x \<prec> y --> ¬ y \<prec> a)",
rename_tac z)
apply (rule contrapos_pp, simp+)
apply (frule_tac a = z and b = a1 in less_linear, assumption+,
simp)
apply (erule disjE)
apply (rotate_tac -4,
frule_tac b = a1 in forball_spec1, assumption+,
thin_tac "∀y∈carrier D. z \<prec> y --> ¬ y \<prec> a",
thin_tac "∀y∈carrier D. a1 \<prec> y --> ¬ y \<prec> a", simp)
apply (frule_tac b = z in forball_spec1, assumption+, simp)
done
lemma (in Order) Pre_element:"[|a ∈ carrier D; ExPre D a|] ==>
Pre D a ∈ carrier D ∧ (Pre D a) \<prec> a ∧
¬ (∃y∈carrier D. ((Pre D a) \<prec> y ∧ y \<prec> a))"
apply (simp add:ExPre_def)
apply (subst Pre_def)+
apply (rule someI2_ex)
apply simp+
done
lemma (in Order) Pre_in_segment:"[|a ∈ carrier D; ExPre D a|] ==>
Pre D a ∈ segment D a"
by (frule Pre_element[of "a"], assumption+, (erule conjE)+,
simp add:segment_inc)
lemma (in Worder) segment_forall:"[|a ∈ segment D b; b ∈ carrier D;
x ∈ segment D b; x \<prec> a; ∀y∈segment D b. x \<prec> y --> ¬ y \<prec> a|] ==>
∀y∈carrier D. x \<prec> y --> ¬ y \<prec> a"
apply (cut_tac segment_sub[of b])
apply (rule ballI, rule impI)
apply (case_tac "y ∈ segment D b",
frule_tac b = y in forball_spec1, assumption+, simp)
(***
------------------a---------------b---------------y----------
***)
apply (thin_tac "∀y∈segment D b. x \<prec> y --> ¬ y \<prec> a",
frule subsetD[of "segment D b" "carrier D" "a"], assumption+,
frule_tac c = x in subsetD[of "segment D b" "carrier D"], assumption+,
thin_tac "segment D b ⊆ carrier D",
thin_tac "x ∈ segment D b")
apply (simp add:segment_inc[THEN sym, of _ "b"]) apply (
simp add:not_less_le)
apply (frule_tac c = y in less_le_trans[of a b], assumption+)
apply (simp add:less_imp_le)
done
lemma (in Worder) segment_Expre:"a ∈ segment D b ==>
ExPre (Iod D (segment D b)) a = ExPre D a"
apply (case_tac "b ∉ carrier D")
apply (simp add:segment_def Iod_self[THEN sym])
apply simp
apply (cut_tac segment_sub[of "b"],
frule subsetD[of "segment D b" "carrier D" a], assumption+)
apply (rule iffI)
apply (simp add:ExPre_def, erule exE, (erule conjE)+)
apply (simp add:Iod_carrier Iod_less)
apply (frule_tac x = x in segment_forall[of "a" "b"], assumption+)
apply blast
apply (simp add:ExPre_def, erule exE, (erule conjE)+)
apply (frule_tac a = a in segment_inc[of _ b], assumption, simp)
apply (rule contrapos_pp, simp+)
apply (frule_tac a = x in forall_spec1,
thin_tac "∀x. x \<prec>Iod D (segment D b) a -->
x ∈ carrier (Iod D (segment D b)) -->
(∃y∈carrier (Iod D (segment D b)).
x \<prec>Iod D (segment D b) y ∧ y \<prec>Iod D (segment D b) a)",
frule_tac a = x in less_trans[of _ a b], assumption+,
frule_tac a = x in segment_inc[of _ b], assumption+, simp)
apply (simp add:Iod_carrier Iod_less)
apply (erule bexE, erule conjE,
frule_tac c = y in subsetD[of "segment D b" "carrier D"], assumption+)
apply (frule_tac b = y in forball_spec1, assumption,
thin_tac "∀y∈carrier D. x \<prec> y --> ¬ y \<prec> a", simp)
done
lemma (in Worder) Pre_segment:"[|a ∈ segment D b;
ExPre (Iod D (segment D b)) a|] ==>
ExPre D a ∧ Pre D a = Pre (Iod D (segment D b)) a"
apply (frule segment_Expre[of "a" "b"], simp)
apply (case_tac "b ∉ carrier D")
apply (simp add:segment_def, simp add:Iod_self[THEN sym])
apply simp
apply (cut_tac segment_sub[of "b"],
frule subsetD[of "segment D b" "carrier D" "a"], assumption+)
apply (frule_tac a = a and ?a1.0 = "Pre (Iod D (segment D b)) a" in
UniquePre, assumption+)
apply simp
apply (cut_tac segment_Worder[of "b"])
apply (frule_tac D = "Iod D (segment D b)" in Worder.Order)
apply (frule Order.Pre_element[of "Iod D (segment D b)" a],
simp add:Iod_carrier, assumption, erule conjE,
simp add:Iod_carrier, simp add:subsetD, simp add:Iod_less)
apply (erule conjE, rule ballI)
apply (case_tac "y ∈ segment D b",
frule_tac b = y in forball_spec1, assumption,
thin_tac "∀y∈segment D b. Pre (Iod D (segment D b)) a \<prec> y
--> ¬ y \<prec> a",
simp)
apply (rule impI)
apply (frule subsetD[of "segment D b" "carrier D" a], assumption+)
apply (frule_tac a = y in segment_inc[of _ b], assumption+, simp,
frule segment_inc[of a b], assumption+, simp)
apply (simp add:not_less_le)
apply (frule_tac c = y in less_le_trans[of a b], assumption+,
simp add:less_imp_le)
apply assumption
done
lemma (in Worder) Pre2segment:"[|a ∈ carrier D; b ∈ carrier D; b \<prec> a;
ExPre D b|] ==> ExPre (Iod D (segment D a)) b"
apply (frule segment_inc [of b a], assumption+, simp)
apply (simp add:segment_Expre[of b a])
done
lemma (in Worder) ord_isom_Pre1:"[|Worder E; a ∈ carrier D; ExPre D a;
ord_isom D E f|] ==> ExPre E (f a)"
apply (simp add:ExPre_def)
apply (erule exE,
frule Worder.Order[of "E"],
erule conjE,
frule_tac a = x in ord_isom_mem[of "E" "f"], assumption+,
frule_tac a = a in ord_isom_mem[of "E" "f"], assumption+,
erule conjE)
apply (frule_tac a = x in ord_isom_less[of "E" "f" _ "a"], assumption+, simp)
apply (frule ord_isom_less_forall[of "E" "f"], assumption+)
apply (frule_tac x = x and a = a in ord_isom_convert[of "E" "f"],
assumption+, simp) apply blast
done (** Here, ord_isom_convert transforms the inequality **)
lemma (in Worder) ord_isom_Pre11:"[|Worder E; a ∈ carrier D; ord_isom D E f|]
==> ExPre D a = ExPre E (f a)"
apply (rule iffI)
apply (simp add:ord_isom_Pre1)
apply (frule Worder.Order[of "E"],
frule ord_isom_sym[of "E" "f"], assumption+)
apply (cut_tac Worder)
apply (frule Worder_ord_isom_mem[of "D" "E" "f" "a"], assumption+,
frule Worder.ord_isom_Pre1[of "E" "D" "f a"
"invfun (carrier D) (carrier E) f"], assumption+)
apply (frule ord_isom_func[of "E" "f"], assumption+)
apply (simp add:ord_isom_def[of "D" "E" "f"] ord_inj_def, (erule conjE)+,
thin_tac "∀a∈carrier D. ∀b∈carrier D. a \<prec> b = f a \<prec>E f b")
apply (simp add:invfun_l)
done
lemma (in Worder) ord_isom_Pre2:"[|Worder E; a ∈ carrier D; ExPre D a;
ord_isom D E f|] ==> f (Pre D a) = Pre E (f a)"
apply (frule_tac E = E and a = a and f = f in ord_isom_Pre1, assumption+,
frule_tac a = a in Pre_element, assumption+, (erule conjE)+)
apply (frule Worder.Order[of "E"],
frule ord_isom_mem[of "E" "f" "a"], assumption+,
frule ord_isom_mem[of "E" "f" "Pre D a"], assumption+,
simp add:ord_isom_less[of "E" "f" "Pre D a" "a"])
apply (simp add:ord_isom_convert[of E f "Pre D a" a])
apply (rule Worder.UniquePre[THEN sym, of "E" "f a" "f (Pre D a)"],
assumption+, simp)
done
section "3. transfinite induction"
lemma (in Worder) transfinite_induction:"[|minimum_elem D (carrier D) s0; P s0; ∀t∈carrier D. ((∀u∈ segment D t. P u) --> P t)|] ==> ∀x∈carrier D. P x"
apply (rule contrapos_pp, simp+)
apply (frule bex_nonempty_set[of "carrier D"],
frule nonempty_set_sub[of "carrier D"])
apply (cut_tac ex_minimum)
apply (frule_tac a = "{x ∈ carrier D. ¬ P x}" in forall_spec,
simp,
thin_tac "∀X. X ⊆ carrier D ∧ X ≠ {} --> (∃x. minimum_elem D X x)")
apply (thin_tac "∃x∈carrier D. ¬ P x")
apply (erule exE)
apply (frule_tac d = x in less_minimum)
apply (frule_tac X = "{x ∈ carrier D. ¬ P x}" and a = x in minimum_elem_mem,
assumption+)
apply (frule_tac c = x and A = "{x ∈ carrier D. ¬ P x}" and B = "carrier D"
in subsetD, assumption+)
apply (frule_tac b = x in forball_spec1, assumption+,
thin_tac "∀t∈carrier D. (∀u∈segment D t. P u) --> P t")
apply (simp add:minimum_elem_def, (erule conjE)+)
apply (erule bexE, simp add:segment_def)
done
section "4. ordered_set2. Lemmas to prove Zorn's lemma "
constdefs (structure D)
adjunct_ord ::"[_ , 'a] => _"
"adjunct_ord D a == D (|carrier := carrier D ∪ {a},
rel := {(x,y). (x, y) ∈ rel D ∨
(x ∈ (carrier D ∪ {a}) ∧ y = a)}|)), "
lemma (in Order) carrier_adjunct_ord:
"carrier (adjunct_ord D a) = carrier D ∪ {a}"
by (simp add:adjunct_ord_def)
lemma (in Order) Order_adjunct_ord:"a ∉ carrier D ==>
Order (adjunct_ord D a)"
apply (cut_tac closed)
apply (rule Order.intro)
apply (rule subsetI)
apply (unfold split_paired_all)
apply simp
apply (simp add:adjunct_ord_def insert_absorb)
apply blast
apply (simp add:carrier_adjunct_ord)
apply (erule disjE)
apply (simp add:adjunct_ord_def)
apply (simp add:adjunct_ord_def)
apply (simp add:refl)
apply (simp add:adjunct_ord_def)
apply (erule disjE)+
apply simp+
apply (erule disjE)
apply (frule_tac c = "(a, b)" in subsetD[of "rel D" "carrier D × carrier D"],
assumption+)
apply blast
apply simp
apply (erule disjE)+
apply (cut_tac closed, simp+)
apply (frule_tac c = "(a, aa)" in subsetD[of "rel D"
"carrier D × carrier D"], assumption+, simp)
apply (frule_tac c = "(aa, b)" in subsetD[of "rel D"
"carrier D × carrier D"], assumption+, simp)
apply (erule disjE)
apply (frule_tac c = "(b, aa)" in subsetD[of "rel D"
"carrier D × carrier D"], assumption+)
apply simp+
apply (erule disjE)+
apply (rule antisym, assumption+)
apply (frule_tac c = "(aa, b)" in subsetD[of "rel D"
"carrier D × carrier D"], assumption+)
apply simp
apply (erule disjE)
apply (frule_tac c = "(b, aa)" in subsetD[of "rel D"
"carrier D × carrier D"], assumption+,
simp, simp)
apply (simp add:adjunct_ord_def)
apply (erule disjE)+
apply blast
apply blast
apply blast
apply (erule disjE)
apply simp
apply (erule disjE)
apply (frule_tac c = "(b, c)" in subsetD[of "rel D"
"carrier D × carrier D"], assumption+)
apply simp apply blast
apply (erule disjE, blast)
apply (erule disjE)
apply simp
apply (erule disjE)
apply (frule_tac c = "(a, b)" in subsetD[of "rel D"
"carrier D × carrier D"], assumption+, simp)
apply (frule_tac c = "(a, b)" in subsetD[of "rel D"
"carrier D × carrier D"], assumption+, simp)
apply (erule disjE, simp) apply blast
apply (erule disjE) apply simp
apply (erule disjE) apply blast
apply (erule disjE)
apply (frule_tac c = "(a, c)" in subsetD[of "rel D"
"carrier D × carrier D"], assumption+, simp)
apply blast
apply (erule disjE)+
apply blast apply blast apply blast
apply (erule disjE)
apply (erule disjE)
apply (frule_tac a = aa and b = b and c = c in trans, assumption+)
apply simp
apply blast
apply (erule disjE) apply simp apply blast
done
lemma (in Order) adjunct_ord_large_a:"[|Order D; a ∉ carrier D|] ==>
∀x∈carrier D. x \<prec>adjunct_ord D a a"
apply (rule ballI)
apply (subst oless_def)
apply (rule conjI)
apply (simp add:ole_def adjunct_ord_def)
apply (rule contrapos_pp, simp+)
done
lemma carr_Ssegment_adjunct_ord:"[|Order D; a ∉ carrier D|] ==>
carrier D = (Ssegment (adjunct_ord D a) a)"
apply (rule equalityI)
apply (rule subsetI)
apply (simp add:Ssegment_def Order.carrier_adjunct_ord)
apply (simp add:Order.adjunct_ord_large_a)
apply (rule subsetI)
apply (simp add:Ssegment_def Order.carrier_adjunct_ord)
apply (erule conjE)
apply (erule disjE, simp add:oless_def)
apply assumption
done
lemma (in Order) adjunct_ord_selfD:"a ∉ carrier D ==>
D = Iod (adjunct_ord D a) (carrier D)"
apply (simp add:Iod_def)
apply (simp add:adjunct_ord_def)
apply (subgoal_tac "rel D = {(aa, b).
((aa, b) ∈ rel D ∨ (aa = a ∨ aa ∈ carrier D) ∧ b = a) ∧
aa ∈ carrier D ∧ b ∈ carrier D}")
apply simp
apply (rule equalityI)
apply (rule subsetI)
apply (cut_tac closed,
frule_tac c = x in subsetD[of "rel D" "carrier D × carrier D"],
assumption+)
apply auto
done
lemma Ssegment_adjunct_ord:"[|Order D; a ∉ carrier D|] ==>
D = SIod (adjunct_ord D a) (Ssegment (adjunct_ord D a) a)"
apply (simp add: carr_Ssegment_adjunct_ord[THEN sym, of "D" "a"])
apply (frule Order.Order_adjunct_ord[of "D" "a"], assumption+)
apply (cut_tac Order.carrier_adjunct_ord[THEN sym, of "D" "a"])
apply (cut_tac Un_upper1[of "carrier D" "{a}"], simp)
apply (subst SIod_self_le[THEN sym, of "adjunct_ord D a" "D"],
assumption+)
apply (rule ballI)+
apply (frule_tac c = aa in subsetD[of "carrier D" "carrier (adjunct_ord D a)"],
assumption,
frule_tac c = b in subsetD[of "carrier D" "carrier (adjunct_ord D a)"],
assumption)
apply (simp add:Order.le_rel[of "adjunct_ord D a"])
apply (subst adjunct_ord_def, simp)
apply (case_tac "b = a", simp) apply simp
apply (simp add:Order.le_rel[of "D"])
apply simp+
done
lemma (in Order) Torder_adjunction:"[|X ⊆ carrier D; a ∈ carrier D;
∀x∈X. x \<preceq> a; Torder (Iod D X)|] ==> Torder (Iod D (X ∪ {a}))"
apply (frule insert_sub[of "X" "carrier D" "a"], assumption)
apply (subst Torder_def)
apply (frule singleton_sub[of "a" "carrier D"],
frule Un_least[of "X" "carrier D" "{a}"], assumption)
apply (rule conjI)
apply (simp add:Iod_Order[of "insert a X"])
apply (subst Torder_axioms_def)
apply ((rule allI)+, (rule impI)+)
apply (simp only:Iod_carrier, simp add:Iod_le)
apply (erule disjE, simp) apply (erule disjE, simp)
apply (simp add:le_refl) apply blast
apply (erule disjE, simp)
apply (simp add:Torder_def, simp add:Torder_axioms_def)
apply (simp add:Iod_carrier, erule conjE)
apply (frule_tac a = aa in forall_spec, assumption,
thin_tac "∀a. a ∈ X --> (∀b. b ∈ X --> a \<preceq>Iod D X b ∨ b \<preceq>Iod D X a)",
frule_tac a = b in forall_spec, assumption,
thin_tac "∀b. b ∈ X --> aa \<preceq>Iod D X b ∨ b \<preceq>Iod D X aa",
simp add:Iod_le)
done
lemma Torder_Sadjunction:"[|Order D; X ⊆ carrier D; a ∈ carrier D;
∀x∈X. x \<preceq>D a; Torder (SIod D X)|] ==> Torder (SIod D (X ∪ {a}))"
apply (frule insert_sub[of "X" "carrier D" "a"], assumption)
apply (subst Torder_def)
apply (frule singleton_sub[of "a" "carrier D"],
frule Un_least[of "X" "carrier D" "{a}"], assumption)
apply (rule conjI)
apply (simp add:SIod_Order[of _ "insert a X"])
apply (subst Torder_axioms_def)
apply ((rule allI)+, (rule impI)+)
apply (simp only:SIod_carrier, simp add:SIod_le)
apply (erule disjE, simp) apply (erule disjE, simp)
apply (simp add:Order.le_refl) apply blast
apply (erule disjE, simp)
apply (simp add:Torder_def, simp add:Torder_axioms_def)
apply (simp add:SIod_carrier, erule conjE)
apply (frule_tac a = aa in forall_spec, assumption,
thin_tac "∀a. a ∈ X --> (∀b. b ∈ X --> a \<preceq>SIod D X b ∨ b \<preceq>SIod D X a)",
frule_tac a = b in forall_spec, assumption,
thin_tac "∀b. b ∈ X --> aa \<preceq>SIod D X b ∨ b \<preceq>SIod D X aa",
simp add:SIod_le)
done
lemma (in Torder) Torder_adjunct_ord:"a ∉ carrier D ==>
Torder (adjunct_ord D a)"
apply (frule Order_adjunct_ord[of "a"],
cut_tac carrier_adjunct_ord[THEN sym, of a],
cut_tac Un_upper1[of "carrier D" "{a}"], simp,
frule Order.Torder_adjunction[of "adjunct_ord D a" "carrier D" a],
assumption+)
apply (simp add:carrier_adjunct_ord)
apply (cut_tac adjunct_ord_large_a[of a],
rule ballI,
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<prec>adjunct_ord D a a",
cut_tac insertI1[of a "carrier D"], simp,
frule_tac c = x in subsetD[of "carrier D" "carrier (adjunct_ord D a)"],
assumption+,
simp add:Order.less_imp_le, rule Order,
assumption+)
apply (simp add:adjunct_ord_selfD[THEN sym])
prefer 2
apply (simp add:Order.Iod_self[THEN sym, of "adjunct_ord D a"])
apply (unfold_locales)
done
lemma (in Order) well_ord_adjunction:"[|X ⊆ carrier D; a ∈ carrier D;
∀x∈X. x \<preceq> a; Worder (Iod D X)|] ==> Worder (Iod D (X ∪ {a}))"
apply (frule insert_sub[of "X" "carrier D" "a"], assumption)
apply (subst Worder_def)
apply (simp add:Iod_Order)
apply (frule_tac Torder_adjunction[of X a], assumption+)
apply (simp add:Worder.Torder)
apply (simp add:Torder_def)
apply (subst Worder_axioms_def)
apply (rule allI, rule impI, erule conjE)
apply (simp add:Iod_carrier)
apply (cut_tac insert_inc1[of "X" "a"])
apply (case_tac "Xa ⊆ X")
apply (simp only:Worder_def, (erule conjE)+) apply (
simp only:Worder_axioms_def)
apply (frule_tac a = Xa in forall_spec,
thin_tac "∀Xa. Xa ⊆ carrier (Iod D X) ∧ Xa ≠ {} -->
(∃x. minimum_elem (Iod D X) Xa x)",
simp add:Iod_carrier)
apply (thin_tac "∀Xa. Xa ⊆ carrier (Iod D X) ∧ Xa ≠ {} -->
(∃x. minimum_elem (Iod D X) Xa x)",
erule exE)
apply (frule_tac X = Xa and a = x in Order.minimum_elem_sub[of
"Iod D (insert a X)" "X"])
apply (cut_tac insert_inc1[of "X" "a"], simp add:Iod_carrier, assumption+)
apply (cut_tac insert_inc1[of "X" "a"],
simp add:Iod_sub_sub) apply blast
apply (erule conjE,
frule_tac A = Xa in insert_diff[of _ "a" "X"])
apply (case_tac "Xa - {a} = {}")
apply (frule_tac A = Xa in nonempty_ex, erule exE, simp,
frule_tac c = x and A = Xa and B = "{a}" in subsetD, assumption+,
simp,
frule_tac A = Xa in singleton_sub[of "a"],
frule_tac A = Xa and B = "{a}" in equalityI, assumption+, simp)
apply (simp add:minimum_elem_def)
apply (cut_tac insert_inc2[of "a" "X"],
simp add:Iod_le le_refl)
apply (simp only:Worder_def, (erule conjE)+,
simp only:Worder_axioms_def)
apply (frule_tac a = "Xa - {a}" in forall_spec,
thin_tac "∀Xa. Xa ⊆ carrier (Iod D X) ∧ Xa ≠ {} -->
(∃x. minimum_elem (Iod D X) Xa x)", simp add:Iod_carrier)
apply (thin_tac "∀Xa. Xa ⊆ carrier (Iod D X) ∧ Xa ≠ {} -->
(∃x. minimum_elem (Iod D X) Xa x)", erule exE)
apply (frule_tac Y = Xa and x = x in augmented_set_minimum[of "X" "a"],
assumption+, blast)
done
lemma well_ord_Sadjunction:"[|Order D; X ⊆ carrier D; a ∈ carrier D;
∀x∈X. x \<preceq>D a; Worder (SIod D X)|] ==> Worder (SIod D (X ∪ {a}))"
apply (frule insert_sub[of "X" "carrier D" "a"], assumption)
apply (subst Worder_def)
apply (simp add:SIod_Order)
apply (frule Torder_Sadjunction[of D X a], assumption+)
apply (simp add:Worder.Torder)
apply (simp add:Torder_def)
apply (subst Worder_axioms_def)
apply (rule allI, rule impI, erule conjE)
apply (simp add:SIod_carrier)
apply (cut_tac insert_inc1[of "X" "a"])
apply (case_tac "Xa ⊆ X")
apply (simp only:Worder_def, (erule conjE)+) apply (
simp only:Worder_axioms_def)
apply (frule_tac a = Xa in forall_spec,
thin_tac "∀Xa. Xa ⊆ carrier (SIod D X) ∧ Xa ≠ {} -->
(∃x. minimum_elem (SIod D X) Xa x)",
simp add:SIod_carrier)
apply (thin_tac "∀Xa. Xa ⊆ carrier (SIod D X) ∧ Xa ≠ {} -->
(∃x. minimum_elem (SIod D X) Xa x)",
erule exE)
apply (frule_tac X = Xa and a = x in minimum_elem_Ssub[of
"SIod D (insert a X)" "X"])
apply (cut_tac insert_inc1[of "X" "a"], simp add:SIod_carrier, assumption+)
apply (cut_tac insert_inc1[of "X" "a"],
simp add:SIod_sub_sub) apply blast
apply (erule conjE, frule_tac A = Xa in insert_diff[of _ "a" "X"])
apply (case_tac "Xa - {a} = {}")
apply (frule_tac A = Xa in nonempty_ex, erule exE, simp,
frule_tac c = x and A = Xa and B = "{a}" in subsetD, assumption+,
simp,
frule_tac A = Xa in singleton_sub[of "a"],
frule_tac A = Xa and B = "{a}" in equalityI, assumption+, simp)
apply (simp add:minimum_elem_def)
apply (cut_tac insert_inc2[of "a" "X"],
simp add:SIod_le Order.le_refl)
apply (simp only:Worder_def, (erule conjE)+,
simp only:Worder_axioms_def)
apply (frule_tac a = "Xa - {a}" in forall_spec,
thin_tac "∀Xa. Xa ⊆ carrier (SIod D X) ∧ Xa ≠ {} -->
(∃x. minimum_elem (SIod D X) Xa x)", simp add:SIod_carrier)
apply (thin_tac "∀Xa. Xa ⊆ carrier (SIod D X) ∧ Xa ≠ {} -->
(∃x. minimum_elem (SIod D X) Xa x)", erule exE)
apply (frule_tac Y = Xa and x = x in augmented_Sset_minimum[of "D" "X" "a"],
assumption+, blast)
done
lemma (in Worder) Worder_adjunct_ord:"a ∉ carrier D ==>
Worder (adjunct_ord D a)"
apply (frule Torder_adjunct_ord[of a])
apply (intro_locales)
apply (simp add:Torder_def)
apply (simp add:Torder_def)
apply (cut_tac carrier_adjunct_ord[THEN sym, of a],
cut_tac Un_upper1[of "carrier D" "{a}"], simp)
apply (simp add:Torder_def, erule conjE)
apply (cut_tac insertI1[of a "carrier D" ])
apply (frule Order.well_ord_adjunction[of "adjunct_ord D a" "carrier D" a],
assumption+)
apply (frule sym, thin_tac "insert a (carrier D) = carrier (adjunct_ord D a)",
simp)
apply (cut_tac adjunct_ord_large_a[of a],
rule ballI,
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<prec>adjunct_ord D a a",
cut_tac insertI1[of a "carrier D"], simp,
frule_tac c = x in subsetD[of "carrier D" "carrier (adjunct_ord D a)"],
assumption+,
simp add:Order.less_imp_le, rule Order,
assumption+)
apply (simp add:adjunct_ord_selfD[THEN sym])
prefer 2
apply (simp add:Order.Iod_self[THEN sym, of "adjunct_ord D a"] Worder_def)
apply unfold_locales
done
section "5. Zorn's lemma"
constdefs (structure D)
Chain :: "_ => 'a set => bool"
"Chain D C == C ⊆ carrier D ∧ Torder (Iod D C)"
upper_bound :: "[_, 'a set, 'a] => bool"
("(3ub\<index>/ _/ _)" [100,101]100)
"ub S b == b ∈ (carrier D) ∧ (∀s∈S. s \<preceq> b)"
"inductive_set":: "_ => bool"
"inductive_set D == ∀C. (Chain D C --> (∃b. ub C b))"
maximal_element::"[_, 'a] => bool"
("(maximal\<index>/ _)" [101]100)
"maximal m == m ∈ carrier D ∧ (∀b∈carrier D. m \<preceq> b --> m = b)"
upper_bounds::"[_, 'a set] => 'a set"
"upper_bounds D H == {x. ub H x}"
Sup::"[_, 'a set] => 'a"
"Sup D X == THE x. minimum_elem D (upper_bounds D X) x"
S_inductive_set::"_ => bool"
"S_inductive_set D == (∀C. Chain D C -->
(∃x∈carrier D. minimum_elem D (upper_bounds D C) x))"
lemma (in Order) mem_upper_bounds:"[|X ⊆ carrier D; b ∈ carrier D;
∀x∈X. x \<preceq> b|] ==> ub X b"
apply (simp add:upper_bounds_def upper_bound_def)
done
lemma (in Order) Torder_Chain:"[|X ⊆ carrier D; Torder (Iod D X)|]
==> Chain D X"
apply (simp add:Chain_def Torder_def)
done
lemma (in Order) Chain_Torder:"Chain D X ==>
Torder (Iod D X)"
apply (simp add:Chain_def)
done
lemma (in Order) Chain_sub:"Chain D X ==> X ⊆ carrier D"
apply (simp add:Chain_def)
done
lemma (in Order) Chain_sub_Chain:"[|Chain D X; Y ⊆ X |] ==> Chain D Y"
apply (frule Chain_sub[of "X"],
frule Chain_Torder[of "X"],
frule Torder.Iod_Torder[of "Iod D X" "Y"], simp add:Iod_carrier)
apply (simp add:Iod_sub_sub[of "Y" "X"],
frule subset_trans[of "Y" "X" "carrier D"], assumption+)
apply (simp add:Torder_Chain[of "Y"])
done
lemma (in Order) upper_bounds_sub:"X ⊆ carrier D ==>
upper_bounds D X ⊆ carrier D"
by (rule subsetI, simp add:upper_bounds_def upper_bound_def)
lemma (in Order) Sup:"[|X ⊆ carrier D; minimum_elem D (upper_bounds D X) a|] ==> Sup D X = a"
apply (subst Sup_def)
apply (rule the_equality, assumption,
rule_tac ?a1.0 = x in minimum_elem_unique[of "upper_bounds D X" _ "a"])
apply (rule subsetI, thin_tac "minimum_elem D (upper_bounds D X) a",
thin_tac "minimum_elem D (upper_bounds D X) x",
simp add:upper_bounds_def upper_bound_def)
apply assumption+
done
lemma (in Worder) Sup_mem:"[|X ⊆ carrier D; ∃b. ub X b|] ==>
Sup D X ∈ carrier D"
apply (frule upper_bounds_sub[of "X"],
frule minimum_elem_mem[of "upper_bounds D X" "Sup D X"],
simp add:Sup_def, rule theI')
apply (rule ex_ex1I);
apply (cut_tac local.D.ex_minimum);
apply (frule_tac a = "upper_bounds D X" in forall_spec,
thin_tac "∀X. X ⊆ carrier D ∧ X ≠ {} --> (∃x. minimum_elem D X x)",
simp, erule exE)
apply (rule_tac x = b in nonempty[of _ "upper_bounds D X"],
simp add:upper_bounds_def, assumption,
rule_tac ?a1.0 = x and ?a2.0 = y in minimum_elem_unique[of
"upper_bounds D X"], assumption+,
simp add:upper_bounds_def upper_bound_def)
done
lemma (in Order) S_inductive_sup:"[|S_inductive_set D; Chain D X|] ==>
minimum_elem D (upper_bounds D X) (Sup D X)"
apply (simp add:S_inductive_set_def)
apply (frule_tac a = X in forall_spec, assumption,
thin_tac "∀C. Chain D C --> (∃x∈carrier D. minimum_elem D
(upper_bounds D C) x)")
apply (erule bexE)
apply (frule Chain_sub[of "X"])
apply (frule_tac a = x in Sup[of "X" ], assumption+)
apply simp
done
lemma (in Order) adjunct_Chain:"[|Chain D X; b ∈ carrier D; ∀x∈X. x \<preceq> b|] ==>
Chain D (insert b X)"
apply (simp add:Chain_def, erule conjE)
apply (frule Torder_adjunction[of X b], assumption+)
apply simp
done
lemma (in Order) S_inductive_sup_mem:"[|S_inductive_set D; Chain D X|] ==>
Sup D X ∈ carrier D"
apply (frule_tac X = X in S_inductive_sup, assumption)
apply (simp add:minimum_elem_def, (erule conjE)+,
simp add:upper_bounds_def, simp add:upper_bound_def)
done
lemma (in Order) S_inductive_Sup_min_bounds:"[|S_inductive_set D; Chain D X;
ub X b|] ==> Sup D X \<preceq> b"
apply (frule S_inductive_sup[of "X"], assumption+,
simp add:minimum_elem_def, erule conjE)
apply (frule_tac b = b in forball_spec1,
simp add:upper_bounds_def, assumption)
done
lemma (in Order) S_inductive_sup_bound:"[|S_inductive_set D; Chain D X|] ==>
∀x∈X. x \<preceq> (Sup D X)"
apply (frule_tac X = X in S_inductive_sup, assumption+)
apply (rule ballI)
apply (simp add:minimum_elem_def) apply (erule conjE)
apply (thin_tac "∀x∈upper_bounds D X. Sup D X \<preceq> x")
apply (simp add:upper_bounds_def upper_bound_def)
done
lemma (in Order) S_inductive_Sup_in_ChainTr:
"[|S_inductive_set D; Chain D X; c ∈ carrier (Iod D (insert (Sup D X) X));
Sup D X ∉ X;
∀y∈carrier (Iod D (insert (Sup D X) X)).
c \<prec>Iod D (insert (Sup D X) X) y --> ¬ y \<prec>Iod D (insert (Sup D X) X) Sup D X|] ==>
c ∈ upper_bounds D X"
apply (subst upper_bounds_def, simp add:upper_bound_def)
apply (frule Chain_sub[of X],
frule S_inductive_sup_mem[of X], assumption+,
frule insert_sub[of X "carrier D" "Sup D X"], assumption)
apply (rule conjI)
apply (simp add:Iod_carrier,
frule Chain_sub[of X],
frule insert_sub[of X "carrier D" "Sup D X"], assumption,
erule disjE, simp, simp add:subsetD)
apply (rule ballI)
apply (simp add:Iod_carrier, (erule conjE)+,
frule S_inductive_sup_bound[of X], assumption+)
apply (erule disjE, simp)
apply (frule_tac b = s in forball_spec1, assumption,
thin_tac "∀y∈X. c \<prec>Iod D (insert (Sup D X) X) y -->
¬ y \<prec>Iod D (insert (Sup D X) X) Sup D X",
frule_tac b = s in forball_spec1, assumption,
thin_tac "∀x∈X. x \<preceq> Sup D X")
apply (frule insert_sub[of X "carrier D" "Sup D X"], assumption+,
cut_tac subset_insertI[of X "Sup D X"],
frule_tac c = s in subsetD[of X "insert (Sup D X) X"], assumption+,
frule_tac c = c in subsetD[of X "insert (Sup D X) X"], assumption+,
frule Iod_Order[of "insert (Sup D X) X"])
apply (subst Iod_le[THEN sym, of "insert (Sup D X) X"], assumption+,
rule contrapos_pp, (simp del:insert_subset)+)
apply (frule Torder_adjunction [of "X" "Sup D X"], assumption+,
rule S_inductive_sup_bound[of X], assumption+, simp add:Chain_Torder)
apply (frule_tac a = s and b = c in
Torder.not_le_less[of "Iod D (X ∪ {Sup D X})"])
apply (simp add:Iod_carrier, simp add:subsetD,
simp add:Iod_carrier)
apply (thin_tac "c \<prec>Iod D (insert (Sup D X) X) Sup D X -->
¬ Sup D X \<prec>Iod D (insert (Sup D X) X) Sup D X")
apply (simp del:insert_sub,
frule_tac a = s in
Torder.not_less_le[of "Iod D (insert (Sup D X) X)" _ "Sup D X"])
apply (frule insert_sub[of X "carrier D" "Sup D X"], assumption,
simp add:Iod_carrier subsetD,
frule insert_sub[of X "carrier D" "Sup D X"], assumption,
simp add:Iod_carrier)
apply simp
apply (simp add:Iod_le)
apply (frule_tac c = s in subsetD[of X "carrier D"], assumption+,
frule_tac a = s and b = "Sup D X" in le_antisym, assumption+)
apply simp
done
lemma (in Order) S_inductive_Sup_in_Chain:"[|S_inductive_set D; Chain D X;
ExPre (Iod D (insert (Sup D X) X)) (Sup D X)|] ==> Sup D X ∈ X"
apply (frule S_inductive_sup_mem[of X], assumption+)
apply (frule Chain_sub[of X],
frule insert_sub[of X "carrier D" "Sup D X"], assumption)
apply (rule contrapos_pp, (simp del:insert_subset)+)
apply (frule Iod_Order[of "insert (Sup D X) X"])
apply (frule Order.Pre_element[of "Iod D (insert (Sup D X) X)" "Sup D X"])
apply (simp add:Iod_carrier) apply assumption
apply ((erule conjE)+, simp del:insert_subset)
apply (frule S_inductive_Sup_in_ChainTr[of X
"Pre (Iod D (insert (Sup D X) X)) (Sup D X)"], assumption+)
apply (simp add:upper_bounds_def)
apply (frule S_inductive_Sup_min_bounds[of X
"Pre (Iod D (insert (Sup D X) X)) (Sup D X)"], assumption+,
thin_tac "∀y∈carrier (Iod D (insert (Sup D X) X)).
Pre (Iod D (insert (Sup D X) X)) (Sup D X) \<prec>Iod D (insert (Sup D X) X)
y --> ¬ y \<prec>Iod D (insert (Sup D X) X) Sup D X")
apply (frule Order.less_le_trans[of "Iod D (insert (Sup D X) X)"
"Pre (Iod D (insert (Sup D X) X)) (Sup D X)"
"Sup D X" "Pre (Iod D (insert (Sup D X) X)) (Sup D X)"])
apply assumption+
apply (frule insert_sub[of X "carrier D" "Sup D X"], assumption,
simp add:Iod_carrier) apply assumption+
apply (frule insert_sub[of X "carrier D" "Sup D X"], assumption,
simp add:Iod_carrier Iod_le)
apply (simp add:oless_def)
done
lemma (in Order) S_inductive_bounds_compare:"[|S_inductive_set D; Chain D X1;
Chain D X2; X1 ⊆ X2|] ==> upper_bounds D X2 ⊆ upper_bounds D X1 "
apply (rule subsetI,
simp add:upper_bounds_def upper_bound_def,
erule conjE, rule ballI,
frule_tac c = s in subsetD[of "X1" "X2"], assumption+)
apply simp
done
lemma (in Order) S_inductive_sup_compare:"[|S_inductive_set D; Chain D X1;
Chain D X2; X1 ⊆ X2|] ==> Sup D X1 \<preceq> Sup D X2"
apply (frule S_inductive_bounds_compare[of "X1" "X2"], assumption+,
frule Chain_sub[of "X1"], frule Chain_sub[of "X2"],
frule upper_bounds_sub[of "X1"], frule upper_bounds_sub[of "X2"])
apply (rule_tac s = "Sup D X2" and t = "Sup D X1" in
compare_minimum_elements[of "upper_bounds D X2" "upper_bounds D X1"],
assumption+,
simp add:S_inductive_sup, simp add:S_inductive_sup)
done
constdefs (structure D)
Wa :: "[_, 'a set, 'a => 'a, 'a] => bool"
"Wa D W g a == W ⊆ carrier D ∧ Worder (Iod D W) ∧ a ∈ W ∧ (∀x∈W. a \<preceq> x) ∧
(∀x∈W. (if (ExPre (Iod D W) x) then g (Pre (Iod D W) x) = x else
(if a ≠ x then Sup D (segment (Iod D W) x) = x else a = a)))"
WWa :: "[_, 'a => 'a, 'a] => 'a set set"
"WWa D g a == {W. Wa D W g a}"
lemma (in Order) mem_of_WWa:"[|W ⊆ carrier D; Worder (Iod D W); a ∈ W;
(∀x∈W. a \<preceq> x);
(∀x∈W. (if (ExPre (Iod D W) x) then g (Pre (Iod D W) x) = x else
(if a ≠ x then Sup D (segment (Iod D W) x) = x else a = a)))|] ==>
W ∈ WWa D g a"
by (simp add:WWa_def, simp add:Wa_def)
lemma (in Order) mem_WWa_then:"W ∈ WWa D g a ==> W ⊆ carrier D ∧
Worder (Iod D W) ∧ a ∈ W ∧ (∀x∈W. a \<preceq> x) ∧
(∀x∈W. (if (ExPre (Iod D W) x) then g (Pre (Iod D W) x) = x else
(if a ≠ x then Sup D (segment (Iod D W) x) = x else a = a)))"
by (simp add:WWa_def Wa_def)
lemma (in Order) mem_wwa_Worder:"W ∈ WWa D g a ==> Worder (Iod D W)"
by (simp add:WWa_def Wa_def)
lemma (in Order) mem_WWa_sub_carrier:"W ∈ WWa D g a ==> W ⊆ carrier D"
by (simp add:WWa_def Wa_def)
lemma (in Order) Union_WWa_sub_carrier:"\<Union> WWa D g a ⊆ carrier D"
by (rule Union_least[of "WWa D g a" "carrier D"], simp add:mem_WWa_sub_carrier)
lemma (in Order) mem_WWa_inc_a:"W ∈ WWa D g a ==> a ∈ W"
by (simp add:WWa_def Wa_def)
lemma (in Order) mem_WWa_Chain:"W ∈ WWa D g a ==> Chain D W"
apply (simp add:Chain_def)
apply (simp add:mem_WWa_sub_carrier)
apply (frule mem_wwa_Worder[of "W"])
apply (simp add:Worder.Torder)
done
lemma (in Order) Sup_adjunct_Sup:"[|S_inductive_set D;
f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
W ∈ WWa D f a; Sup D W ∉ W|]
==> Sup D (insert (Sup D W) W) = Sup D W"
(** use
le_antisym
At first applying le_antisym, we see following four items should be proved.
Sup D W ∈ carrier D
Sup D (insert (Sup D W) W) ∈ carrier D
Sup D W \<preceq> Sup D (insert (Sup D W) W)
Sup D (insert (Sup D W) W) \<preceq> Sup D W
To show Sup D W ∈ carrier D, we apply S_inductive_Sup_mem.
**)
apply (frule mem_WWa_Chain[of "W"],
frule S_inductive_sup_bound[of "W"], assumption,
frule mem_wwa_Worder[of "W"],
frule mem_WWa_sub_carrier[of "W"],
frule S_inductive_sup_mem[of "W"], assumption+)
apply (frule well_ord_adjunction[of "W" "Sup D W"], assumption+, simp,
frule insert_sub[of "W" "carrier D" "Sup D W"], assumption+,
frule Worder.Torder[of "Iod D (insert (Sup D W) W)"],
frule Torder_Chain[of "insert (Sup D W) W"], assumption+,
frule S_inductive_sup_mem[of "insert (Sup D W) W"], assumption+)
apply (rule le_antisym[of "Sup D (insert (Sup D W) W)" "Sup D W"], assumption+,
rule S_inductive_Sup_min_bounds[of "insert (Sup D W) W" "Sup D W"],
assumption+,
simp add:upper_bound_def, simp add:le_refl)
apply (rule S_inductive_sup_compare[of "W" "insert (Sup D W) W"], assumption+)
apply (simp add:subset_insertI[of "W" "Sup D W"])
done
lemma (in Order) BNTr1:"a ∈ carrier D ==> Worder (Iod D {a})"
apply intro_locales
apply (frule singleton_sub[of "a" "carrier D"],
rule Iod_Order[of "{a}"], assumption)
apply (simp add:Torder_axioms_def)
apply (rule allI, rule impI)+
apply (simp add:Iod_carrier, simp add:Iod_le le_refl)
apply (simp add:Worder_axioms_def)
apply (rule allI, rule impI, erule conjE, simp add:Iod_carrier)
apply (frule_tac A = X in nonempty_ex, erule exE,
frule_tac c = x and A = X and B = "{a}" in subsetD, assumption+,
simp,
frule_tac A = X in singleton_sub[of "a"],
frule_tac A = X and B = "{a}" in equalityI, assumption+, simp)
apply (simp add:minimum_elem_def Iod_le le_refl)
done
lemma (in Order) BNTr2:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> (f x)|] ==> {a} ∈ WWa D f a"
apply (simp add:WWa_def Wa_def)
apply (simp add:Not_ExPre[of "a"])
apply (simp add:BNTr1 le_refl)
done
lemma (in Order) BNTr2_1:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> (f x); W ∈ WWa D f a|] ==> ∀x∈W. a \<preceq> x"
by (rule ballI, simp add:WWa_def Wa_def)
lemma (in Order) BNTr3:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> (f x); W ∈ WWa D f a|] ==> minimum_elem (Iod D W) W a"
(** a ∈ W is required by minimum_elem_def and Iod_le *)
apply (frule mem_WWa_inc_a[of W])
apply (subst minimum_elem_def)
apply simp
apply (rule ballI)
apply (frule mem_WWa_sub_carrier[of W f a])
apply (frule BNTr2_1[of f a W], assumption+)
apply (simp add:Iod_le)
done
lemma (in Order) Adjunct_segment_sub:"[|S_inductive_set D; Chain D X|] ==>
segment (Iod D (insert (Sup D X) X)) (Sup D X) ⊆ X"
apply (frule S_inductive_sup_mem[of "X"], assumption+,
frule Chain_sub[of "X"],
frule insert_sub[of "X" "carrier D" "Sup D X"], assumption)
apply (rule subsetI)
apply (simp add:segment_def)
apply (case_tac "Sup D X ∉ carrier (Iod D (insert (Sup D X) X))", simp)
apply (simp add:Iod_carrier)
apply (simp add:Iod_carrier, erule conjE, simp add:oless_def)
done
lemma (in Order) Adjunct_segment_eq:"[|S_inductive_set D; Chain D X;
Sup D X ∉ X|] ==>
segment (Iod D (insert (Sup D X) X)) (Sup D X) = X"
apply (frule Chain_sub[of "X"],
frule Adjunct_segment_sub[of "X"], assumption)
apply (rule equalityI, assumption)
apply (frule S_inductive_sup_mem[of "X"], assumption+,
frule insert_sub[of "X" "carrier D" "Sup D X"], assumption+,
rule subsetI,
simp add:segment_def Iod_carrier,
cut_tac subset_insertI[of "X" "Sup D X"],
cut_tac insertI1[of "Sup D X" "X"],
frule_tac c = x in subsetD[of "X" "insert (Sup D X) X"], assumption+)
apply (simp add:Iod_less[of "insert (Sup D X) X"],
frule S_inductive_sup_bound[of "X"], assumption+,
frule_tac b = x in forball_spec1, assumption+,
thin_tac "∀x∈X. x \<preceq> Sup D X",
simp add:oless_def)
apply (rule contrapos_pp, simp+)
done
constdefs
fixp :: "['a => 'a, 'a] => bool"
"fixp f a == f a = a" (** a is a fixed point of f **)
lemma (in Order) fixp_same:"[|W1 ⊆ carrier D; W2 ⊆ carrier D; t ∈ W1;
b ∈ carrier D; ord_isom (Iod D W1) (Iod (Iod D W2) (segment (Iod D W2) b)) g;
∀u∈segment (Iod D W1) t. fixp g u|] ==>
segment (Iod D W1) t = segment (Iod D W2) (g t)"
apply (frule Iod_Order[of "W1"],
frule Iod_Order[of "W2"],
frule Order.segment_sub[of "Iod D W1" "t"],
frule Order.segment_sub[of "Iod D W2" "b"],
frule Order.Iod_Order[of "Iod D W2" "segment (Iod D W2) b"],
assumption+)
apply (frule Order.ord_isom_segment_segment[of "Iod D W1"
"Iod (Iod D W2) (segment (Iod D W2) b)" g t], assumption+)
apply (simp add:Iod_carrier, simp add:Iod_carrier)
apply (frule Order.ord_isom_mem[of "Iod D W1"
"Iod (Iod D W2) (segment (Iod D W2) b)" g t], assumption+,
simp add:Iod_carrier)
apply (frule Order.Iod_segment_segment[of "Iod D W2" "g t" b],
assumption, simp)
apply (simp add:Iod_sub_sub[of "segment (Iod D W1) t" W1])
apply (frule Order.segment_sub [of "Iod (Iod D W2)
(segment (Iod D W2) b)" "g t"])
apply (simp add:Iod_sub_sub)
apply (frule subset_trans[of "segment (Iod D W2) b" W2 "carrier D"],
assumption+)
apply (simp add:Iod_carrier)
apply (frule Order.segment_sub[of "Iod D W2" "g t"],
simp add:Iod_carrier[of W2])
apply (simp add:Iod_sub_sub[of "segment (Iod D W2) (g t)" W2],
thin_tac "Iod (Iod D (segment (Iod D W2) b))
(segment (Iod D (segment (Iod D W2) b)) (g t)) =
Iod D (segment (Iod D W2) (g t))")
apply (frule subset_trans[of "segment (Iod D W1) t" W1 "carrier D"],
assumption+,
frule subset_trans[of "segment (Iod D W2) (g t)" W2 "carrier D"],
assumption+,
frule Iod_Order[of "segment (Iod D W1) t"],
frule Iod_Order[of "segment (Iod D W2) (g t)"])
apply (thin_tac "segment (Iod D (segment (Iod D W2) b)) (g t) ⊆
segment (Iod D W2) b",
thin_tac "segment (Iod D W2) b ⊆ carrier D",
thin_tac "segment (Iod D W2) (g t) ⊆ W2",
thin_tac "ord_isom (Iod D W1) (Iod D (segment (Iod D W2) b)) g")
apply (rule equalityI)
apply (rule subsetI)
apply (frule_tac a = x in Order.ord_isom_mem[of "Iod D (segment (Iod D W1) t)"
"Iod D (segment (Iod D W2) (g t))"
"restrict g (carrier (Iod D (segment (Iod D W1) t)))"], assumption+,
simp add:Iod_carrier, simp add:Iod_carrier, simp add:fixp_def)
apply (metis Order.Iod_carrier Order_axioms)
apply (rule subsetI)
apply (frule_tac b = x in Order.ord_isom_surj[of
"Iod D (segment (Iod D W1) t)" "Iod D (segment (Iod D W2) (g t))"
"restrict g (carrier (Iod D (segment (Iod D W1) t)))"]);
apply assumption
apply (metis Order.Iod_carrier Order_axioms)
apply (metis Order.Iod_carrier Order.segment_free Order_axioms)
apply (metis Order.Iod_carrier Order_axioms fixp_def restrict_apply)
done
lemma (in Order) BNTr4_1:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
b ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x); W1 ∈ WWa D f a; W2 ∈ WWa D f a;
ord_isom (Iod D W1) (Iod D (segment (Iod D W2) b)) g|] ==>
∀x∈W1. g x = x"
apply (frule mem_wwa_Worder[of "W1" "f" "a"],
frule mem_wwa_Worder[of "W2" "f" "a"],
frule Worder.Order[of "Iod D W2"],
frule mem_WWa_sub_carrier[of "W2" "f" "a"],
frule mem_WWa_sub_carrier[of "W1" "f" "a"],
cut_tac Worder.segment_Worder[of "Iod D W2" "b"],
simp add:Worder.Order)
apply (cut_tac Order.segment_sub[of "Iod D W2" "b"],
simp add:Iod_carrier,
frule subset_trans[of "segment (Iod D W2) b" "W2" "carrier D"],
assumption+,
frule Iod_Order[of "segment (Iod D W2) b"])
apply (frule Worder.Order[of "Iod D W1"],
frule Order.ord_isom_onto[of "Iod D W1"
"Iod D (segment (Iod D W2) b)" "g"], assumption+)
(** transfinite induction **)
apply (frule Order.ord_isom_minimum[of "Iod D W1"
"Iod D (segment (Iod D W2) b)" "g" "W1" "a"], assumption+,
simp add:Iod_carrier, simp add:Iod_carrier mem_WWa_inc_a,
simp add:BNTr3)
apply (frule Order.ord_isom_onto[of "Iod D W1"
"Iod D (segment (Iod D W2) b)" "g"], assumption+,
simp add:Iod_carrier, frule Worder.Torder[of "Iod D W2"])
apply (simp add:minimum_elem_sub[THEN sym, of "segment (Iod D W2) b"
"segment (Iod D W2) b"])
apply (simp add:minimum_elem_sub[of "W2" "segment (Iod D W2) b"])
apply (frule Torder.minimum_segment_of_sub[of "Iod D W2" "W2" "b" "g a"],
simp add:Iod_carrier, cut_tac subset_self[of "W2"],
simp add:Iod_sub_sub[of "W2" "W2"],
thin_tac "minimum_elem (Iod D W2) (segment (Iod D W2) b) (g a)",
frule BNTr3[of "f" "a" "W2"], assumption+)
apply (frule Worder.Order[of "Iod D W2"],
frule Order.minimum_elem_unique[of "Iod D W2" "W2" "g a" "a"],
simp add:Iod_carrier, assumption+)
apply (simp add:Iod_sub_sub[THEN sym, of "segment (Iod D W2) b" "W2"],
frule Worder.transfinite_induction[of "Iod D W1" "a" "fixp g"],
simp add:Iod_carrier, simp add:BNTr3, simp add:fixp_def,
rule ballI, rule impI)
apply (frule_tac a = t in Worder_ord_isom_mem[of "Iod D W1"
"Iod (Iod D W2) (segment (Iod D W2) b)" "g"], assumption+,
frule Iod_carrier[THEN sym, of "W2"],
frule subset_trans[of "segment (Iod D W2) b" "W2"
"carrier (Iod D W2)"], simp,
thin_tac "W2 = carrier (Iod D W2)",
simp add:Order.Iod_carrier[of "Iod D W2" "segment (Iod D W2) b"],
frule_tac c = "g t" in subsetD[of "segment (Iod D W2) b"
"carrier (Iod D W2)"], assumption+,
simp add:Iod_carrier)
apply (case_tac "t = a")
apply (simp add:fixp_def)
apply (case_tac "ExPre (Iod D W1) t",
frule_tac a = t in Worder.ord_isom_Pre1[of "Iod D W1"
"Iod (Iod D W2) (segment (Iod D W2) b)" _ "g"], assumption+,
simp add:Iod_carrier, assumption+)
apply (frule_tac a = t in Worder.ord_isom_Pre2[of "Iod D W1"
"Iod (Iod D W2) (segment (Iod D W2) b)" _ "g"], assumption+,
simp add:Iod_carrier, assumption+)
apply (frule_tac a = t in Order.Pre_in_segment[of "Iod D W1"],
simp add:Iod_carrier, assumption)
apply (frule_tac b = "Pre (Iod D W1) t" in forball_spec1, assumption,
thin_tac "∀u∈segment (Iod D W1) t. fixp g u")
apply (simp add:fixp_def)
apply (frule_tac a = "g t" in Worder.Pre_segment[of "Iod D W2" _ "b"],
simp add:Iod_carrier, assumption+)
apply (rotate_tac -2, frule sym, thin_tac "Pre (Iod D W1) t =
Pre (Iod (Iod D W2) (segment (Iod D W2) b)) (g t)", simp,
thin_tac "Pre (Iod (Iod D W2) (segment (Iod D W2) b)) (g t) =
Pre (Iod D W1) t")
apply (erule conjE)
apply (simp add:WWa_def Wa_def, (erule conjE)+,
thin_tac "∀x∈W1. a \<preceq> x", thin_tac "∀x∈W2. a \<preceq> x",
thin_tac "∀x∈carrier D. x \<preceq> f x")
apply (frule_tac b = t in forball_spec1, assumption+,
thin_tac "∀x∈W1.
if ExPre (Iod D W1) x then f (Pre (Iod D W1) x) = x
else if a ≠ x then Sup D (segment (Iod D W1) x) = x else a = a",
frule_tac b = "g t" in forball_spec1, assumption+,
thin_tac "∀x∈W2.
if ExPre (Iod D W2) x then f (Pre (Iod D W2) x) = x
else if a ≠ x then Sup D (segment (Iod D W2) x) = x else a = a")
apply simp
apply (frule_tac a = t in Worder.ord_isom_Pre11[of "Iod D W1"
"Iod (Iod D W2) (segment (Iod D W2) b)" _ "g"], assumption+,
simp add:Iod_carrier, assumption, simp)
apply (frule_tac a = "g t" in Worder.segment_Expre[of "Iod D W2" _ "b"],
assumption, simp,
thin_tac "¬ ExPre (Iod (Iod D W2) (segment (Iod D W2) b)) (g t)")
apply (simp add:WWa_def Wa_def, (erule conjE)+)
apply (rotate_tac -3,
frule_tac b = t in forball_spec1, assumption,
thin_tac "∀x∈W1.
if ExPre (Iod D W1) x then f (Pre (Iod D W1) x) = x
else if a ≠ x then Sup D (segment (Iod D W1) x) = x else a = a")
apply (rotate_tac 1,
frule_tac b = "g t" in forball_spec1, assumption,
thin_tac "∀x∈W2.
if ExPre (Iod D W2) x then f (Pre (Iod D W2) x) = x
else if a ≠ x then Sup D (segment (Iod D W2) x) = x else a = a",
simp)
apply (frule_tac t = t and s = a in not_sym, thin_tac " t ≠ a")
apply (frule_tac b = t in Order.ord_isom_inj[of "Iod D W1"
"Iod (Iod D W2) (segment (Iod D W2) b)" "g" "a"], assumption+,
simp add:Iod_carrier, simp add:Iod_carrier, simp)
apply (frule_tac t1 = t in fixp_same[THEN sym, of "W1" "W2" _ "b" "g"],
assumption+, simp, simp add:fixp_def)
apply (rule ballI,
frule_tac b = x in forball_spec1,
simp add:subsetD[of "W1" "carrier D"],
simp add:Iod_carrier fixp_def)
apply (simp add:Worder.Order, assumption)
done
lemma (in Order) BNTr4_2:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
b ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x); W1 ∈ WWa D f a; W2 ∈ WWa D f a;
ord_equiv (Iod D W1) (Iod D (segment (Iod D W2) b))|] ==>
W1 = segment (Iod D W2) b"
apply (simp add:ord_equiv_def,
erule exE)
apply (rename_tac g)
apply (frule mem_wwa_Worder[of "W1" "f" "a"],
frule mem_wwa_Worder[of "W2" "f" "a"],
frule Worder.Order[of "Iod D W2"],
frule mem_WWa_sub_carrier[of "W2" "f" "a"],
frule mem_WWa_sub_carrier[of "W1" "f" "a"],
cut_tac a = b in Worder.segment_Worder[of "Iod D W2"], assumption)
apply (frule Worder.Order[of "Iod D W1"])
apply (frule_tac D = "Iod (Iod D W2) (segment (Iod D W2) b)" in Worder.Order)
apply (frule Worder.Order[of "Iod D W2"])
apply (frule_tac a = b in Order.segment_sub[of "Iod D W2"],
simp add:Iod_carrier)
apply (frule_tac A = "segment (Iod D W2) b" in subset_trans[of _ "W2"
"carrier D"], assumption+)
apply (frule_tac T = "segment (Iod D W2) b" in Iod_Order)
apply (frule_tac E = "Iod D (segment (Iod D W2) b)" and f = g in
Order.ord_isom_func[of "Iod D W1" ], assumption+)
apply (frule_tac f = g in Order.ord_isom_onto[of "Iod D W1"
"Iod D (segment (Iod D W2) b)"], assumption+)
apply (simp only:Iod_carrier)
apply (frule_tac b = b and g = g in BNTr4_1[of "f" "a" _ "W1" "W2"],
assumption+)
apply (simp add:image_def)
done
lemma (in Order) BNTr4:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> (f x); W1 ∈ WWa D f a; W2 ∈ WWa D f a;
∃b∈carrier D. ord_equiv (Iod D W1) (Iod D (segment (Iod D W2) b))|] ==>
W1 ⊆ W2"
apply (erule bexE)
apply (rename_tac b)
apply (frule_tac b = b in BNTr4_2[of "f" "a" _ "W1" "W2"], assumption+)
apply (frule mem_WWa_sub_carrier[of "W2" "f" "a"],
frule Iod_Order[of "W2"])
apply (frule_tac a = b in Order.segment_sub[of "Iod D W2"],
simp add:Iod_carrier)
done
lemma (in Order) Iod_same:"A = B ==> Iod D A = Iod D B"
by simp
lemma (in Order) eq_ord_equivTr:"[|ord_equiv D E; E = F|] ==> ord_equiv D F"
by simp
lemma (in Order) BNTr5:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> (f x); W1 ∈ WWa D f a; W2 ∈ WWa D f a;
ord_equiv (Iod D W1) (Iod D W2)|] ==> W1 ⊆ W2 "
apply (frule mem_WWa_sub_carrier[of "W1" "f" "a"],
frule mem_WWa_sub_carrier[of "W2" "f" "a"])
apply (case_tac "W2 = carrier D")
apply simp
apply (frule not_sym, thin_tac "W2 ≠ carrier D")
apply (frule sets_not_eq[of "carrier D" "W2"], assumption, erule bexE)
apply (frule Iod_Order[of "W2"],
frule Iod_Order[of "W1"],
frule Iod_carrier[THEN sym, of "W2"])
apply (frule_tac a = aa and A = W2 and B = "carrier (Iod D W2)" in
eq_set_not_inc, assumption)
apply (thin_tac "W2 = carrier (Iod D W2)")
apply (frule_tac a = aa in Order.segment_free[of "Iod D W2"],
assumption, simp add:Iod_carrier)
apply (rule BNTr4[of f a W1 W2], assumption+) (** key **)
apply (frule_tac a = aa in Order.segment_free[of "Iod D W2"])
apply (simp add:Iod_carrier)
apply (simp only:Iod_carrier,
rotate_tac -1, frule sym, thin_tac "segment (Iod D W2) aa = W2")
apply (frule_tac B = "segment (Iod D W2) aa" in
Iod_same[of W2])
apply (frule_tac F = "Iod D (segment (Iod D W2) aa)" in
Order.eq_ord_equivTr[of "Iod D W1" "Iod D W2"], assumption+)
apply blast
done
lemma (in Order) BNTr6:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> (f x); W1 ∈ WWa D f a; W2 ∈ WWa D f a; W1 ⊂ W2|] ==>
(∃b∈carrier (Iod D W2). ord_equiv (Iod D W1) (Iod D (segment (Iod D W2) b)))"
apply (frule mem_wwa_Worder[of "W1" "f" "a"],
frule mem_wwa_Worder[of "W2" "f" "a"])
apply (frule_tac D = "Iod D W1" and E = "Iod D W2" in Worder.Word_compare,
assumption+) (** key 1 **)
apply (erule disjE)
apply (erule bexE,
frule_tac a = aa in Worder.segment_Worder[of "Iod D W1"],
frule_tac S = "Iod (Iod D W1) (segment (Iod D W1) aa)" in
Worder_sym[of _ "Iod D W2"], assumption+,
thin_tac "ord_equiv (Iod (Iod D W1) (segment (Iod D W1) aa)) (Iod D W2)",
frule BNTr4[of "f" "a" "W2" "W1"], assumption+,
frule mem_WWa_sub_carrier[of "W1"],
simp add:Iod_carrier,
frule_tac c = aa in subsetD[of "W1" "carrier D"], assumption+,
frule Iod_Order[of "W1"],
frule_tac a = aa in Order.segment_sub[of "Iod D W1"],
simp add:Iod_carrier)
apply (frule_tac S = "segment (Iod D W1) aa" and T = W1 in Iod_sub_sub,
assumption+, simp, blast)
apply (simp add:subset_contr[of "W1" "W2"])
apply (erule disjE)
apply (frule Worder_sym[of "Iod D W1" "Iod D W2"], assumption+,
thin_tac "ord_equiv (Iod D W1) (Iod D W2)",
frule BNTr5[of "f" "a" "W2" "W1"], assumption+)
apply (simp add:subset_contr[of "W1" "W2"])
apply (erule bexE,
frule mem_WWa_sub_carrier[of "W2"],
simp add:Iod_carrier,
frule_tac c = b in subsetD[of "W2" "carrier D"], assumption+,
frule Iod_Order[of "W2"],
frule_tac a = b in Order.segment_sub[of "Iod D W2"],
simp add:Iod_carrier,
frule_tac S = "segment (Iod D W2) b" and T = W2 in Iod_sub_sub,
assumption+, simp)
apply blast
done
lemma (in Order) BNTr6_1:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> (f x); W1 ∈ WWa D f a; W2 ∈ WWa D f a; W1 ⊂ W2|] ==>
(∃b∈carrier (Iod D W2). W1 = (segment (Iod D W2) b))"
by (frule_tac BNTr6[of "f" "a" "W1" "W2"], assumption+, erule bexE,
frule mem_WWa_sub_carrier[of "W2"], simp add:Iod_carrier,
frule_tac c = b in subsetD[of "W2" "carrier D"], assumption+,
frule_tac b = b in BNTr4_2[of "f" "a" _ "W1" "W2"], assumption+,
blast)
lemma (in Order) BNTr7:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> (f x); W1 ∈ WWa D f a; W2 ∈ WWa D f a|] ==>
W1 ⊆ W2 ∨ W2 ⊆ W1"
apply (frule mem_wwa_Worder[of "W1" "f" "a"],
frule mem_wwa_Worder[of "W2" "f" "a"])
apply (frule_tac D = "Iod D W1" and E = "Iod D W2" in Worder.Word_compare,
assumption+)
apply (erule disjE, erule bexE)
apply (frule mem_WWa_sub_carrier[of "W1"],
frule mem_WWa_sub_carrier[of "W2"],
frule Iod_Order[of "W1"],
frule Iod_Order[of "W2"],
frule_tac a = aa in Order.segment_sub[of "Iod D W1"],
simp add:Iod_carrier,
frule_tac S = "segment (Iod D W1) aa" and T = W1 in Iod_sub_sub,
assumption+, simp,
thin_tac "Iod (Iod D W1) (segment (Iod D W1) aa) =
Iod D (segment (Iod D W1) aa)",
frule_tac A = "segment (Iod D W1) aa" in subset_trans[of _ "W1"
"carrier D"], assumption+,
frule_tac T = "segment (Iod D W1) aa" in Iod_Order)
apply (frule_tac D = "Iod D (segment (Iod D W1) aa)" in
Order.ord_equiv_sym[of _ "Iod D W2"], assumption+)
apply (frule_tac c = aa in subsetD[of "W1" "carrier D"], assumption+)
apply (frule BNTr4[of "f" "a" "W2" "W1"], assumption+, blast, simp)
apply (erule disjE)
apply (simp add:BNTr5)
apply (frule mem_WWa_sub_carrier[of "W2"], simp add:Iod_carrier)
apply (frule BNTr4[of "f" "a" "W1" "W2"], assumption+)
apply (erule bexE,
frule mem_WWa_sub_carrier[of "W2"],
simp add:Iod_carrier,
frule_tac c = b in subsetD[of "W2" "carrier D"], assumption+,
frule Iod_Order[of "W2"],
frule_tac a = b in Order.segment_sub[of "Iod D W2"],
simp add:Iod_carrier,
frule_tac S = "segment (Iod D W2) b" and T = W2 in Iod_sub_sub,
assumption+, simp,
frule_tac c = b in subsetD[of "W2" "carrier D"], assumption+)
apply blast
apply simp
done
lemma (in Order) BNTr7_1:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; x ∈ W; W ∈ WWa D f a; xa ∈ \<Union> WWa D f a;
xa \<prec>Iod D (\<Union> WWa D f a) x|] ==> xa ∈ W"
apply (cut_tac Union_WWa_sub_carrier[of "f" "a"],
frule_tac X = W and C = "WWa D f a" and A = x in UnionI, assumption+,
simp del:Union_iff add:Iod_less)
apply (simp only:Union_iff[of "xa" "WWa D f a"], erule bexE, rename_tac W')
apply (frule_tac ?W1.0 = W and ?W2.0 = W' in BNTr7[of "f" "a"],
assumption+)
apply (case_tac "W' ⊆ W", simp add:subsetD[of _ "W"])
apply (simp del:Union_iff)
apply (frule_tac B = W' in psubsetI[of "W"])
apply (rule not_sym, assumption)
apply (thin_tac "W ⊆ W'", thin_tac "W' ≠ W")
apply (frule_tac ?W2.0 = W' in BNTr6_1[of "f" "a" "W"], assumption+)
apply (erule bexE)
apply (frule_tac W = W' in mem_WWa_sub_carrier)
apply (simp add:Iod_carrier)
apply (frule_tac c = b and A = W' in subsetD[of _ "carrier D"],
assumption+)
apply (frule_tac W = W' and a = b and y = xa and x = x in segment_inc_less,
assumption+)
done
lemma (in Order) BNTr7_1_1:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; x ∈ W; W ∈ WWa D f a; xa ∈ \<Union> WWa D f a;
xa \<prec> x|] ==> xa ∈ W"
apply (cut_tac Union_WWa_sub_carrier[of "f" "a"],
frule Iod_Order[of "\<Union> WWa D f a"],
frule Iod_less[THEN sym, of "\<Union> WWa D f a" "xa" "x"], assumption+,
rule UnionI[of "W" "WWa D f a" "x"], assumption+)
apply (simp del:Union_iff, rule BNTr7_1, assumption+)
done
lemma (in Order) BNTr7_2:" [|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; x ∈ \<Union>WWa D f a; ExPre (Iod D (\<Union>WWa D f a)) x |]
==> ∀W∈WWa D f a. (x ∈ W --> ExPre (Iod D W) x )"
apply (cut_tac Union_WWa_sub_carrier[of "f" "a"])
apply (rule ballI, rule impI)
apply (simp only:ExPre_def)
apply (erule exE, (erule conjE)+)
apply (simp only:Iod_carrier)
apply (frule_tac X = W and C = "WWa D f a" and A = x in UnionI, assumption+)
apply (simp del:Union_iff)
apply (frule_tac b = W in forball_spec1, assumption,
thin_tac "∀y∈WWa D f a.
∀y∈y. xa \<prec>Iod D (\<Union>WWa D f a) y --> ¬ y \<prec>Iod D (\<Union>WWa D f a) x")
apply (frule_tac W = W and xa = xa in BNTr7_1[of "f" "a" "x"], assumption+,
frule_tac W = W in mem_WWa_sub_carrier)
apply (simp del:Union_iff add:Iod_less)
apply (subgoal_tac "∀y∈W. xa \<prec>Iod D W y --> ¬ y \<prec>Iod D W x")
apply (simp only:Iod_carrier)
apply (subgoal_tac "xa \<prec>Iod D W x", blast)
apply (simp add:Iod_less)
apply (rule ballI, rule impI)
apply (frule_tac b = y in forball_spec1, assumption,
thin_tac "∀y∈W. xa \<prec>Iod D (\<Union>WWa D f a) y --> ¬ y \<prec>Iod D (\<Union>WWa D f a) x",
frule_tac X = W and C = "WWa D f a" and A = y in UnionI, assumption+)
apply (simp add:Iod_less)
done
lemma (in Order) BNTr7_3:" [|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; x ∈ \<Union>WWa D f a; ExPre (Iod D (\<Union>WWa D f a)) x |]
==> ∀W∈WWa D f a. (x ∈ W --> Pre (Iod D (\<Union>WWa D f a)) x = Pre (Iod D W) x)"
apply (rule ballI)
apply (rule impI)
apply (rule_tac s = "Pre (Iod D W) x" in
sym[of _ "Pre (Iod D (\<Union>WWa D f a)) x"],
frule_tac W = W in mem_wwa_Worder,
frule_tac W = W in mem_WWa_sub_carrier,
frule BNTr7_2[of "f" "a" "x"], assumption+,
frule_tac b = W in forball_spec1, assumption,
thin_tac "∀W∈WWa D f a. x ∈ W --> ExPre (Iod D W) x",
simp)
apply (rule_tac D = "Iod D W" and a = x and
?a1.0 = "Pre (Iod D (\<Union>WWa D f a)) x" in Worder.UniquePre,
assumption,
simp add:Iod_carrier, assumption, simp add:Iod_carrier)
apply (frule_tac D = "Iod D W" in Worder.Order,
frule_tac D = "Iod D W" and a = x in Order.Pre_element,
simp add:Iod_carrier, assumption,
(erule conjE)+, simp add:Iod_carrier)
apply (cut_tac Union_WWa_sub_carrier[of "f" "a"],
frule Iod_Order[of "\<Union>WWa D f a"],
frule_tac X = W and A = x in UnionI[of _ "WWa D f a"], assumption+,
frule_tac D = "Iod D (\<Union>WWa D f a)" and a = x in Order.Pre_element,
simp del:Union_iff add:Iod_carrier, assumption)
apply (erule conjE)+
apply (frule_tac W = W and xa = "Pre (Iod D (\<Union>WWa D f a)) x" in
BNTr7_1[of "f" "a" "x"], assumption+,
simp only:Iod_carrier, assumption,
simp del:Union_iff)
apply (rule conjI,
simp del:Union_iff add:Iod_carrier Iod_less,
rule ballI,
frule_tac X = W and A = y in UnionI[of _ "WWa D f a"], assumption+,
thin_tac "∀y∈W. Pre (Iod D W) x \<prec>Iod D W y --> ¬ y \<prec>Iod D W x")
apply (simp only:Iod_carrier,
frule_tac b = y in forball_spec1, assumption,
thin_tac "∀y∈\<Union>WWa D f a.
Pre (Iod D (\<Union>WWa D f a)) x \<prec>Iod D (\<Union>WWa D f a) y -->
¬ y \<prec>Iod D (\<Union>WWa D f a) x")
apply (simp del:Union_iff add:Iod_less)
done
lemma (in Order) BNTr7_4:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; x ∈ W; W ∈ WWa D f a|] ==>
ExPre (Iod D (\<Union>WWa D f a)) x = ExPre (Iod D W) x"
apply (rule iffI)
apply (frule BNTr7_2[of "f" "a" "x"], assumption+)
apply (frule_tac X = W and A = x in UnionI[of _ "WWa D f a"], assumption+,
simp)
apply (simp only:ExPre_def, erule exE, (erule conjE)+)
apply (cut_tac Union_WWa_sub_carrier[of "f" "a"])
apply (frule mem_WWa_sub_carrier[of "W"], simp only:Iod_carrier,
frule Iod_Order[of "W"]) apply (
frule Iod_Order[of "\<Union>WWa D f a"]) apply (
simp only:Iod_less)
apply (frule_tac X = W and A = xa in UnionI[of _ "WWa D f a"], assumption+,
frule_tac X = W and A = x in UnionI[of _ "WWa D f a"], assumption+)
apply (frule_tac T1 = "\<Union>WWa D f a" and a1 = xa and b1 = x in
Iod_less[THEN sym], assumption+)
apply (subgoal_tac " ¬ (∃y∈\<Union>WWa D f a.
xa \<prec>Iod D (\<Union>WWa D f a) y ∧ y \<prec>Iod D (\<Union>WWa D f a) x)")
apply blast
apply (rule contrapos_pp, (simp del:Union_iff)+)
apply (erule bexE, rename_tac xa W', erule bexE, erule conjE,
frule_tac X = W' and A = y in UnionI[of _ "WWa D f a"], assumption+)
apply (frule_tac xa = y in BNTr7_1[of "f" "a" "x" "W"], assumption+)
apply (frule_tac b = y in forball_spec1, assumption+,
thin_tac "∀y∈W. xa \<prec>Iod D W y --> ¬ y \<prec>Iod D W x")
apply (simp add:Iod_less)
done
lemma (in Order) BNTr7_5:" [|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; x ∈ W; W ∈ WWa D f a|]
==> (segment (Iod D (\<Union>WWa D f a)) x) = segment (Iod D W) x"
apply (cut_tac Union_WWa_sub_carrier[of "f" "a"])
apply (frule_tac W = W in mem_WWa_sub_carrier)
apply (frule Iod_Order[of "\<Union>WWa D f a"],
frule Iod_Order[of "W"])
apply (rule equalityI)
apply (rule subsetI,
frule Order.segment_sub[of "Iod D (\<Union>WWa D f a)" "x"],
frule_tac c = xa in subsetD[of "segment (Iod D (\<Union>WWa D f a)) x"
"carrier (Iod D (\<Union>WWa D f a))"], assumption+,
frule_tac X = W and A = x in UnionI[of _ "WWa D f a"], assumption+)
apply (frule_tac a1 = xa in Order.segment_inc[THEN sym,
of "Iod D (\<Union>WWa D f a)" _ "x"],
assumption, simp add:Iod_carrier, simp del:Union_iff,
frule_tac xa = xa in BNTr7_1[of "f" "a" "x"], assumption+,
simp only:Iod_carrier, assumption, simp only:Iod_carrier,
simp only:Iod_less,
frule_tac a1 = xa in Iod_less[THEN sym, of "W" _ "x"],
assumption+, simp del:Union_iff,
subst Order.segment_inc[THEN sym, of "Iod D W" _ "x"], assumption+,
simp add:Iod_carrier, simp add:Iod_carrier, assumption)
apply (rule subsetI,
frule_tac a1 = xa in Order.segment_inc[THEN sym, of "Iod D W" _ "x"])
apply (frule Order.segment_sub[of "Iod D W" "x"],
rule_tac c = xa in subsetD[of "segment (Iod D W) x" "carrier (Iod D W)"],
assumption+, simp only:Iod_carrier,
frule_tac W = W in mem_WWa_sub_carrier, frule Iod_Order[of "W"],
frule Order.segment_sub[of "Iod D W" "x"], simp only:Iod_carrier,
frule_tac c = xa in subsetD[of "segment (Iod D W) x" "W"],
simp add:Iod_less)
apply (frule_tac X = W and A = xa in UnionI[of _ "WWa D f a"], assumption+,
frule_tac X = W and A = x in UnionI[of _ "WWa D f a"], assumption+,
subst Order.segment_inc[THEN sym, of "Iod D (\<Union>WWa D f a)" _ "x"],
assumption+, simp only:Iod_carrier, simp only:Iod_carrier,
simp only:Iod_less)
done
lemma (in Order) BNTr7_6:"[|f ∈ carrier D -> carrier D;
a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x)|] ==> a ∈ \<Union>WWa D f a"
apply (frule BNTr2[of "f" "a"], assumption+,
frule UnionI[of "{a}" "WWa D f a" "a"])
apply (simp, assumption)
done
lemma (in Order) BNTr7_7:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x); ∃xa. Wa D xa f a ∧ x ∈ xa|] ==>
x ∈ \<Union>WWa D f a"
apply (subst Union_iff[of "x" "WWa D f a"])
apply (subst WWa_def, blast)
done
lemma (in Order) BNTr7_8:"[|S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x); ∃xa. Wa D xa f a ∧ x ∈ xa|] ==> x ∈ carrier D"
apply (erule exE) apply (rename_tac W, erule conjE)
apply (rule_tac A = W and B = "carrier D" and c = x in subsetD,
rule_tac W = W in mem_WWa_sub_carrier[of _ "f" "a"])
apply (simp add:WWa_def, assumption)
done
lemma (in Order) BNTr7_9:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> (f x); x ∈ \<Union>WWa D f a |] ==> x ∈ carrier D"
by (cut_tac Union_WWa_sub_carrier[of "f" "a"],
rule subsetD[of "\<Union>WWa D f a" "carrier D" "x"], assumption+)
lemma (in Order) BNTr7_10:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x); W ∈ WWa D f a; Sup D W ∉ W|] ==>
¬ ExPre (Iod D (insert (Sup D W) W)) (Sup D W)"
apply (frule mem_WWa_sub_carrier[of "W" "f" "a"])
apply (frule mem_WWa_Chain[of "W" "f" "a"],
frule S_inductive_sup_mem[of "W"], assumption+,
frule insert_sub[of "W" "carrier D" "Sup D W"], assumption+,
cut_tac insertI1[of "Sup D W" "W"],
cut_tac subset_insertI[of "W" "Sup D W"],
frule insert_sub[of "W" "carrier D" "Sup D W"], assumption+)
apply (rule contrapos_pp, simp+)
apply (simp add:ExPre_def)
apply (erule exE, (erule conjE)+)
apply (frule forball_contra[of "carrier (Iod D (insert (Sup D W) W))" _ _ _
"op = (Sup D W)"],
thin_tac "∀y∈carrier (Iod D (insert (Sup D W) W)).
x \<prec>Iod D (insert (Sup D W) W) y -->
¬ y \<prec>Iod D (insert (Sup D W) W) Sup D W")
apply (frule well_ord_adjunction[of "W" "Sup D W"], assumption+,
simp add:S_inductive_sup_bound[of "W"],
simp add:mem_wwa_Worder[of "W"],
frule insert_sub[of "W" "carrier D" "Sup D W"], assumption+,
frule Worder.Torder[of "Iod D (W ∪ {Sup D W})"], simp,
frule insert_sub[of "W" "carrier D" "Sup D W"], assumption+,
frule Torder_Chain[of "insert (Sup D W) W"], assumption+,
frule S_inductive_sup_bound[of "insert (Sup D W) W"], assumption+)
apply (simp only:Iod_carrier)
apply (rule ballI)
apply (rotate_tac -2,
frule_tac b = y in forball_spec1, assumption+,
thin_tac "∀x∈insert (Sup D W) W. x \<preceq> Sup D (insert (Sup D W) W)",
cut_tac insertI1[of "Sup D W" "W"],
simp only:Iod_less)
apply (simp add:Sup_adjunct_Sup, erule disjE)
apply (frule sym, thin_tac "y = Sup D W", simp)
apply (frule_tac c = y in subsetD[of "W" "carrier D"], assumption+)
apply (frule_tac a = y and b = "Sup D W" in le_imp_less_or_eq, assumption+,
simp) apply (
thin_tac "y \<preceq> Sup D W",
thin_tac "x = Sup D W ∨ x ∈ W")
apply (erule disjE, simp, simp)
apply (thin_tac "∀y∈carrier (Iod D (insert (Sup D W) W)).
x \<prec>Iod D (insert (Sup D W) W) y -->
¬ y \<prec>Iod D (insert (Sup D W) W) Sup D W")
apply (frule insert_sub[of "W" "carrier D" "Sup D W"], assumption+,
cut_tac insertI1[of "Sup D W" "W"],
simp only:Iod_carrier Iod_less)
apply (frule mem_wwa_Worder[of "W"],
frule S_inductive_sup_bound[of "W"], assumption+,
frule well_ord_adjunction[of "W" "Sup D W"], assumption+,
frule Worder.Torder[of "Iod D (W ∪ {Sup D W})"], simp)
apply (frule forball_contra1,
thin_tac "∀y∈W. x \<prec>Iod D (insert (Sup D W) W) y --> Sup D W = y")
apply (rule ballI)
apply (rule contrapos_pp, simp+)
apply (frule_tac b = x in S_inductive_Sup_min_bounds[of "W"], assumption+)
apply (simp add:upper_bound_def)
apply (erule disjE, simp add:oless_def)
apply (simp add:subsetD[of "W" "carrier D"])
apply (rule ballI)
apply (thin_tac "∀x∈carrier D. x \<preceq> f x",
thin_tac "∀x∈W. x \<preceq> Sup D W")
apply (frule_tac b = s in forball_spec1, assumption+,
thin_tac "∀y∈W. ¬ x \<prec>Iod D (insert (Sup D W) W) y")
apply (frule_tac c = x in subsetD[of "W" "insert (Sup D W) W"], assumption+,
frule_tac c = s in subsetD[of "W" "insert (Sup D W) W"], assumption+)
apply (simp add:Iod_not_less_le Iod_le)
apply (erule disjE, simp add:oless_def,
frule_tac c = x in subsetD[of "W" "insert (Sup D W) W"], assumption+,
cut_tac insertI1[of "Sup D W" "W"],
frule insert_sub[of "W" "carrier D" "Sup D W"], assumption+)
apply (frule_tac a1 = "Sup D W" and b1 = x in Iod_le[THEN sym,
of "insert (Sup D W) W"], assumption+, simp)
apply (frule_tac c = x in subsetD[of "W" "insert (Sup D W) W"], assumption+,
cut_tac insertI1[of "Sup D W" "W"],
frule insert_sub[of "W" "carrier D" "Sup D W"], assumption+)
apply (simp only:Iod_not_less_le[THEN sym])
apply (frule_tac c = x in subsetD[of "W" "insert (Sup D W) W"], assumption+,
cut_tac insertI1[of "Sup D W" "W"],
frule insert_sub[of "W" "carrier D" "Sup D W"], assumption+)
apply (simp add:Iod_less)
done
lemma (in Order) BNTr7_11:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; b ∈ carrier D; ∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a;
∀x∈W. x \<preceq> b; x ∈ W|] ==>
ExPre (Iod D (insert b W)) x = ExPre (Iod D W) x"
apply (case_tac "b ∈ W",
simp add:insert_absorb[of "b" "W"])
apply (frule mem_WWa_sub_carrier[of "W"],
frule subsetD[of "W" "carrier D" "x"], assumption+,
frule mem_wwa_Worder[of "W"],
frule mem_WWa_Chain[of "W" "f" "a"],
frule well_ord_adjunction[of "W" "b"], assumption+,
frule insert_sub[of "W" "carrier D" "b"], assumption+,
cut_tac insertI1[of "b" "W"],
cut_tac subset_insertI[of "W" "b"])
apply (simp del:insert_iff insert_subset add:Un_commute,
frule Worder.Torder[of "Iod D (insert b W)"])
apply (rule iffI)
apply (frule subsetD[of "W" "insert b W" "x"], assumption+,
frule Iod_Order[of "insert b W"],
unfold ExPre_def)
apply (erule exE, (erule conjE)+)
apply (rule contrapos_pp, (simp del:insert_iff insert_subset)+)
apply (frule_tac a = xa in forall_spec,
thin_tac "∀xa. xa \<prec>Iod D W x --> xa ∈ carrier (Iod D W) -->
(∃y∈carrier (Iod D W). xa \<prec>Iod D W y ∧ y \<prec>Iod D W x)")
apply (simp only:Iod_carrier)
apply (simp del:insert_subset) apply (cut_tac insertI1[of "b" "W"])
apply (erule disjE) apply (simp del:insert_iff insert_subset)
apply (thin_tac "∀x∈carrier D. x \<preceq> f x")
apply (frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈W. x \<preceq> b")
apply (frule subsetD[of "W" "insert b W" "x"], assumption+,
frule Iod_carrier[THEN sym, of "insert b W"],
frule eq_set_inc[of "x" "insert b W" "carrier (Iod D (insert b W))"],
assumption,
frule eq_set_inc[of "b" "insert b W" "carrier (Iod D (insert b W))"],
assumption,
frule Torder.not_le_less[THEN sym, of "Iod D (insert b W)" "x" "b"],
assumption+,
thin_tac "insert b W = carrier (Iod D (insert b W))")
apply (simp del:insert_iff insert_subset add:Iod_le)
apply (frule_tac c = xa in subsetD[of "W" "insert b W"], assumption+,
frule subsetD[of "W" "insert b W" "x"], assumption+,
simp only:Iod_less, simp only:Iod_carrier)
apply (cut_tac a = xa in insert_iff[of _ "b" "W"],
simp del:insert_iff insert_subset)
apply (erule disjE)
apply (frule Iod_carrier[THEN sym, of "insert b W"])
apply (frule_tac a = b in eq_set_inc[of _ "insert b W"
"carrier (Iod D (insert b W))"], assumption+,
frule_tac a = x in eq_set_inc[of _ "insert b W"
"carrier (Iod D (insert b W))"], assumption+,
thin_tac "insert b W = carrier (Iod D (insert b W))")
apply (simp del:insert_iff insert_subset add:Torder.not_le_less[THEN sym,
of "Iod D (insert b W)" "x" "b"])
apply (rotate_tac 5,
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈W. x \<preceq> b", simp add:Iod_le)
apply (thin_tac "xa ∈ insert b W", erule conjE,
simp del:insert_iff insert_subset,
thin_tac "∀xa. xa \<prec>Iod D W x -->
xa ∈ W --> (∃y∈W. xa \<prec>Iod D W y ∧ y \<prec>Iod D W x)")
apply (erule bexE, erule conjE)
apply (thin_tac "∀x∈carrier D. x \<preceq> f x",
thin_tac "∀x∈W. x \<preceq> b",
frule_tac b = y in forball_spec1, assumption,
thin_tac "∀y∈W. xa \<prec>Iod D (insert b W) y --> ¬ y \<prec>Iod D (insert b W) x",
frule_tac c = xa in subsetD[of "W" "insert b W"], assumption+,
frule_tac c = y in subsetD[of "W" "insert b W"], assumption+)
apply (simp del:insert_iff insert_subset add:Iod_less)
apply (erule exE, (erule conjE)+)
apply (rule contrapos_pp, (simp del:insert_iff insert_subset)+)
apply (frule_tac a = xa in forall_spec,
simp only:Iod_carrier,
frule_tac c = xa in subsetD[of "W" "insert b W"], assumption+,
frule_tac c = x in subsetD[of "W" "insert b W"], assumption+,
simp add:Iod_less)
apply (simp only:Iod_carrier,
frule_tac c = xa in subsetD[of "W" "insert b W"], assumption,
simp del:insert_iff insert_subset add:Iod_less)
apply (erule disjE, erule conjE) apply (
frule_tac a = xa in forall_spec,
thin_tac "∀xa. xa \<prec>Iod D (insert b W) x --> xa ∈ insert b W -->
xa \<prec> b ∧ b \<prec>Iod D (insert b W) x ∨ (∃y∈W. xa \<prec> y ∧ y \<prec>Iod D (insert b W) x)")
apply (thin_tac "∀x∈carrier D. x \<preceq> f x",
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈W. x \<preceq> b",
frule_tac c = xa in subsetD[of "W" "insert b W"], assumption+,
frule_tac c = x in subsetD[of "W" "insert b W"], assumption+,
simp only:Iod_less)
apply (thin_tac "∀xa. xa \<prec>Iod D (insert b W) x --> xa ∈ insert b W -->
xa \<prec> b ∧ b \<prec>Iod D (insert b W) x ∨ (∃y∈W. xa \<prec> y ∧ y \<prec>Iod D (insert b W) x)")
apply (simp del:insert_iff insert_subset)
apply (thin_tac "∀x∈carrier D. x \<preceq> f x",
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈W. x \<preceq> b",
frule_tac subsetD[of "W" "insert b W" "x"], assumption+,
frule_tac c = xa in subsetD[of "W" "insert b W"], assumption+,
frule Iod_carrier[THEN sym, of "insert b W"],
frule eq_set_inc[of "x" "insert b W" "carrier (Iod D (insert b W))"],
assumption,
frule eq_set_inc[of "b" "insert b W" "carrier (Iod D (insert b W))"],
assumption,
frule Torder.not_le_less[THEN sym, of "Iod D (insert b W)" "x" "b"],
assumption+,
thin_tac "insert b W = carrier (Iod D (insert b W))",
simp add:Iod_le,
thin_tac "∀xa. xa \<prec>Iod D (insert b W) x --> xa ∈ insert b W -->
xa \<prec> b ∧ b \<prec>Iod D (insert b W) x ∨ (∃y∈W. xa \<prec> y ∧ y \<prec>Iod D (insert b W) x)")
apply (erule bexE, erule conjE,
thin_tac "∀x∈W. x \<preceq> b",
frule_tac b = y in forball_spec1, assumption,
thin_tac "∀y∈W. xa \<prec> y --> ¬ y \<prec> x",
frule_tac c = y in subsetD[of "W" "insert b W"], assumption+,
frule_tac c = x in subsetD[of "W" "insert b W"], assumption+,
simp add:Iod_less)
done
lemma (in Order) BNTr7_12:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; b ∈ carrier D; ∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a;
∀x∈W. x \<preceq> b; x ∈ W; ExPre (Iod D W) x|] ==>
Pre (Iod D (insert b W)) x = Pre (Iod D W) x"
apply (case_tac "b ∈ W", simp only:insert_absorb)
apply (frule mem_WWa_Chain[of "W"],
frule mem_wwa_Worder[of "W"],
frule mem_WWa_sub_carrier[of "W"],
frule well_ord_adjunction[of "W" "b"], assumption+,
simp add:Un_commute[of "W" "{b}"],
frule insert_sub[of "W" "carrier D" "b"], assumption+,
cut_tac subset_insertI[of "W" "b"],
frule subsetD[of "W" "insert b W" "x"], assumption+,
cut_tac insertI1[of "b" "W"])
apply (rule Worder.UniquePre[of "Iod D (insert b W)" "x"
"Pre (Iod D W) x"], assumption+,
simp add:Iod_carrier,
subst BNTr7_11[of "f" "a" "b" "W" "x"], assumption+,
frule Worder.Order[of "Iod D W"])
apply (frule Order.Pre_element[of "Iod D W" "x"],
simp add:Iod_carrier, assumption)
apply (erule conjE)+
apply (rule conjI)
apply (simp add:Iod_carrier)
apply (rule conjI)
apply (simp only:Iod_carrier,
frule subsetD[of "W" "insert b W" "x"], assumption+,
frule subsetD[of "W" "insert b W" "Pre (Iod D W) x"],
assumption+)
apply (simp only:Iod_less)
apply (rule contrapos_pp, (simp del:insert_iff insert_subset)+)
apply (erule bexE)
apply (simp only:Iod_carrier,
frule subsetD[of "W" "insert b W" "Pre (Iod D W) x"],
assumption+)
apply (cut_tac a = y in insert_iff[of _ "b" "W"])
apply (frule_tac P = "y ∈ insert b W" and Q = "y = b ∨ y ∈ W" in
eq_prop, assumption+,
thin_tac "(y ∈ insert b W) = (y = b ∨ y ∈ W)",
thin_tac "y ∈ insert b W")
apply (erule disjE,
thin_tac "∀x∈carrier D. x \<preceq> f x",
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈W. x \<preceq> b", erule conjE,
frule Iod_carrier[THEN sym, of "insert b W"],
frule eq_set_inc[of "x" "insert b W" "carrier (Iod D (insert b W))"],
assumption,
frule eq_set_inc[of "b" "insert b W" "carrier (Iod D (insert b W))"],
assumption,
thin_tac "insert b W = carrier (Iod D (insert b W))",
simp del:insert_iff insert_subset,
frule Worder.Torder[of "Iod D (insert b W)"],
frule Torder.not_le_less[THEN sym, of "Iod D (insert b W)" "x" "b"],
assumption+, simp add:Iod_le)
apply (rotate_tac -4,
frule_tac b = y in forball_spec1, assumption,
thin_tac "∀y∈W. Pre (Iod D W) x \<prec>Iod D W y --> ¬ y \<prec>Iod D W x",
frule_tac c = y in subsetD[of "W" "insert b W"], assumption,
simp add:Iod_less)
done
lemma (in Order) BNTr7_13:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; b ∈ carrier D; ∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a;
∀x∈W. x \<preceq> b; x ∈ W|] ==>
(segment (Iod D (insert b W)) x) = segment (Iod D W) x"
apply (case_tac "b ∈ W", simp add:insert_absorb)
apply (frule mem_wwa_Worder[of "W"],
frule mem_WWa_sub_carrier[of "W"],
frule mem_WWa_Chain[of "W"],
frule insert_sub[of "W" "carrier D" "b"], assumption+,
frule well_ord_adjunction[of "W" "b"], assumption+,
simp del:insert_subset add:Un_commute,
cut_tac subset_insertI[of "W" "b"],
cut_tac insertI1[of "b" "W"],
frule Worder.Torder[of "Iod D (insert b W)"])
apply (rule equalityI)
apply (rule subsetI)
apply (simp del:insert_iff insert_subset add:segment_def Iod_carrier)
apply (frule subsetD[of "W" "insert b W" "x"], assumption+,
simp del:insert_iff insert_subset)
apply (erule conjE, simp only:Iod_carrier)
apply (cut_tac a = xa in insert_iff[of _ "b" "W"],
frule_tac P = "xa ∈ insert b W" and Q = "xa = b ∨ xa ∈ W" in
eq_prop, assumption+)
apply (thin_tac "(xa ∈ insert b W) = (xa = b ∨ xa ∈ W)",
erule disjE, simp del:insert_iff insert_subset,
frule Iod_carrier[THEN sym, of "insert b W"],
frule eq_set_inc[of "x" "insert b W" "carrier (Iod D (insert b W))"],
assumption,
frule eq_set_inc[of "b" "insert b W" "carrier (Iod D (insert b W))"],
assumption,
thin_tac "insert b W = carrier (Iod D (insert b W))",
frule Torder.not_le_less[THEN sym, of "Iod D (insert b W)" "x" "b"],
assumption+, simp add:Iod_le)
apply (frule_tac c = xa in subsetD[of "W" "insert b W"], assumption+,
frule_tac c = x in subsetD[of "W" "insert b W"], assumption+,
simp add:Iod_less)
apply (rule subsetI)
apply (simp del:insert_iff insert_subset add:segment_def,
simp only:Iod_carrier)
apply (frule_tac subsetD[of "W" "insert b W" "x"], assumption+,
simp del:insert_iff insert_subset)
apply (erule conjE,
frule_tac c = xa in subsetD[of "W" "insert b W"], assumption+,
simp add:Iod_less)
done
lemma (in Order) BNTr7_14:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x); W ∈ WWa D f a|] ==>
(insert (Sup D W) W) ∈ WWa D f a"
apply (case_tac "Sup D W ∈ W",
simp add:insert_absorb[of "Sup D W" "W"])
apply (frule mem_WWa_sub_carrier[of "W" "f" "a"],
frule mem_WWa_Chain[of "W" "f" "a"],
frule S_inductive_sup_mem[of "W"], assumption+,
frule insert_sub[of "W" "carrier D" "Sup D W"], assumption+,
rule mem_of_WWa [of "insert (Sup D W) W" "a" "f"], assumption+,
frule S_inductive_sup_bound[of "W"], assumption+)
apply (frule well_ord_adjunction[of "W" "Sup D W"], assumption+,
simp add:mem_wwa_Worder, simp,
frule mem_WWa_inc_a[of "W" "f" "a"], simp)
apply (rule ballI)
apply (simp add:WWa_def Wa_def, (erule conjE)+, erule disjE,
frule S_inductive_sup_bound[of "W"], assumption+,
simp, simp)
apply (rule ballI)
apply (simp only:insert_iff)
apply (erule disjE)
apply (frule mem_WWa_inc_a[of "W" "f" "a"])
apply (frule not_eq_outside [of "Sup D W" "W"])
apply (rotate_tac -1,
frule_tac b = a in forball_spec1, assumption+,
thin_tac "∀b∈W. b ≠ Sup D W")
apply (frule BNTr7_10[of "f" "a" "W"], assumption+)
apply (simp del:insert_iff insert_subset add:Adjunct_segment_eq)
apply (frule S_inductive_sup_bound[of "W"], assumption+)
apply (subst BNTr7_11[of "f" "a" "Sup D W" "W"], assumption+)
apply (case_tac "ExPre (Iod D W) x",
subst BNTr7_12[of "f" "a" "Sup D W" "W"], assumption+)
apply (simp del:insert_iff insert_subset add:WWa_def)
apply (unfold Wa_def, simp)
apply (simp del:insert_iff insert_subset)
apply (rule impI)
apply (subst BNTr7_13[of "f" "a" "Sup D W" "W"], assumption+)
apply (simp add:WWa_def Wa_def)
done
lemma (in Order) BNTr7_15:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x); W ∈ WWa D f a;
f (Sup D W) ≠ Sup D W|] ==>
ExPre (Iod D (insert (f (Sup D W)) (insert (Sup D W) W))) (f (Sup D W))"
apply (simp add:ExPre_def)
apply (rule contrapos_pp, simp+)
apply (frule BNTr7_14[of "f" "a" "W"], assumption+)
apply (frule mem_WWa_sub_carrier[of "insert (Sup D W) W" "f" "a"],
frule mem_WWa_Chain[of "insert (Sup D W) W" "f" "a"],
frule mem_wwa_Worder[of "insert (Sup D W) W" "f" "a"],
frule mem_WWa_Chain[of "W" "f" "a"],
frule S_inductive_sup_mem[of "W"], assumption+,
frule funcset_mem[of "f" "carrier D" "carrier D" "Sup D W"], assumption+,
frule insert_sub[of "insert (Sup D W) W" "carrier D" "f (Sup D W)"],
assumption+,
frule S_inductive_sup_bound[of "W"], assumption+)
apply (frule well_ord_adjunction[of "insert (Sup D W) W" "f (Sup D W)"],
assumption+,
rule ballI,
simp only:insert_iff, erule disjE, simp del:insert_iff insert_subset)
apply (frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x") apply (
rotate_tac -3,
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈W. x \<preceq> Sup D W") apply (
frule mem_WWa_sub_carrier[of "W"],
frule_tac c = x in subsetD[of "W" "carrier D"], assumption+,
rule_tac a = x in le_trans[of _ "Sup D W" "f (Sup D W)"], assumption+,
cut_tac insertI1[of "Sup D W" "W"],
cut_tac insertI1[of "f (Sup D W)" "insert (Sup D W) W"],
cut_tac subset_insertI[of "insert (Sup D W) W" "f (Sup D W)"],
frule subsetD[of "insert (Sup D W) W" "insert (f (Sup D W))
(insert (Sup D W) W)" "Sup D W"], assumption+)
apply (frule_tac a = "Sup D W" in forall_spec)
apply (frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x",
frule not_sym,
thin_tac "f (Sup D W) ≠ Sup D W",
simp del:insert_iff insert_subset
add:le_imp_less_or_eq[of "Sup D W" "f (Sup D W)"],
simp only:Iod_less)
apply (thin_tac "∀x. x \<prec>Iod D (insert (f (Sup D W)) (insert (Sup D W) W)) f (Sup D W) -->
x ∈ carrier (Iod D (insert (f (Sup D W)) (insert (Sup D W) W))) -->
(∃y∈carrier (Iod D (insert (f (Sup D W)) (insert (Sup D W) W))).
x \<prec>Iod D (insert (f (Sup D W)) (insert (Sup D W) W)) y ∧
y \<prec>Iod D (insert (f (Sup D W)) (insert (Sup D W) W))
f (Sup D W))",
simp only:Iod_carrier,
frule True_then[of "∃y∈insert (f (Sup D W)) (insert (Sup D W) W).
Sup D W \<prec>Iod D (insert (f (Sup D W)) (insert (Sup D W) W)) y ∧
y \<prec>Iod D (insert (f (Sup D W)) (insert (Sup D W) W)) f (Sup D W)"],
thin_tac "True -->
(∃y∈insert (f (Sup D W)) (insert (Sup D W) W).
Sup D W \<prec>Iod D (insert (f (Sup D W)) (insert (Sup D W) W)) y ∧
y \<prec>Iod D (insert (f (Sup D W)) (insert (Sup D W) W)) f (Sup D W))",
erule bexE, erule conjE)
apply (cut_tac a = y in insert_iff[of _ "f (Sup D W)" "insert (Sup D W) W"],
frule_tac P = "y ∈ insert (f (Sup D W)) (insert (Sup D W) W)" and
Q = "y = f (Sup D W) ∨ y ∈ insert (Sup D W) W" in eq_prop,
assumption+,
thin_tac "y ∈ insert (f (Sup D W)) (insert (Sup D W) W)",
thin_tac "(y ∈ insert (f (Sup D W)) (insert (Sup D W) W)) =
(y = f (Sup D W) ∨ y ∈ insert (Sup D W) W)")
apply (erule disjE, simp add:oless_def)
apply (cut_tac a = y in insert_iff[of _ "Sup D W" "W"],
frule_tac P = "y ∈ insert (Sup D W) W" and
Q = "y = (Sup D W) ∨ y ∈ W" in eq_prop, assumption+,
thin_tac "y ∈ insert (Sup D W) W",
thin_tac "y ∈ insert (Sup D W) W = (y = (Sup D W) ∨ y ∈ W)",
erule disjE, simp add:oless_def,
simp del:insert_iff insert_subset)
apply (frule_tac b = y in forball_spec1, assumption,
thin_tac "∀x∈W. x \<preceq> Sup D W")
apply (frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x",
cut_tac subset_insertI[of "W" "Sup D W"],
frule_tac c = y in subsetD[of "W" "insert (Sup D W) W"], assumption+,
frule_tac c = y in subsetD[of "insert (Sup D W) W"
"insert (f (Sup D W)) (insert (Sup D W) W)"], assumption+,
frule_tac Worder.Torder[of "Iod D (insert (f (Sup D W))
(insert (Sup D W) W))"])
apply (frule_tac a1 = y in Torder.not_le_less[THEN sym, of
"Iod D (insert (f (Sup D W)) (insert (Sup D W) W))" _ "Sup D W"],
simp add:Iod_carrier, simp add:Iod_carrier)
apply (simp add:Iod_le)
done
lemma (in Order) BNTr7_16:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x); W ∈ WWa D f a;
f (Sup D W) ≠ (Sup D W)|] ==>
Pre (Iod D (insert (f (Sup D W)) (insert (Sup D W) W))) (f (Sup D W)) =
(Sup D W)"
apply (frule BNTr7_14[of "f" "a" "W"], assumption+,
frule mem_WWa_sub_carrier[of "insert (Sup D W) W" "f" "a"],
frule mem_WWa_Chain[of "insert (Sup D W) W" "f" "a"],
frule mem_wwa_Worder[of "insert (Sup D W) W" "f" "a"],
frule mem_WWa_Chain[of "W" "f" "a"],
frule S_inductive_sup_mem[of "W"], assumption+,
frule funcset_mem[of "f" "carrier D" "carrier D" "Sup D W"], assumption+,
frule insert_sub[of "insert (Sup D W) W" "carrier D" "f (Sup D W)"],
assumption+,
frule S_inductive_sup_bound[of "W"], assumption+,
frule well_ord_adjunction[of "insert (Sup D W) W" "f (Sup D W)"],
assumption+, rule ballI,
simp only:insert_iff, erule disjE, simp del:insert_iff insert_subset,
frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x",
rotate_tac -3,
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈W. x \<preceq> Sup D W",
frule mem_WWa_sub_carrier[of "W"],
frule_tac c = x in subsetD[of "W" "carrier D"], assumption+,
rule_tac a = x in le_trans[of _ "Sup D W" "f (Sup D W)"], assumption+,
cut_tac insertI1[of "Sup D W" "W"],
cut_tac insertI1[of "f (Sup D W)" "insert (Sup D W) W"],
cut_tac subset_insertI[of "insert (Sup D W) W" "f (Sup D W)"],
frule subsetD[of "insert (Sup D W) W" "insert (f (Sup D W))
(insert (Sup D W) W)" "Sup D W"], assumption+)
apply (simp only:Un_commute[of "insert (Sup D W) W" "{f (Sup D W)}"],
simp only:insert_is_Un[THEN sym])
apply (rule Worder.UniquePre[of "Iod D (insert (f (Sup D W))
(insert (Sup D W) W))" "f (Sup D W)" "Sup D W"], assumption+,
simp only:Iod_carrier,
rule BNTr7_15, assumption+,
rule conjI, simp only:Iod_carrier, rule conjI, simp only:Iod_less,
frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x", frule not_sym,
thin_tac "f (Sup D W) ≠ Sup D W", simp add:oless_def)
apply (rule contrapos_pp, (simp del:insert_iff insert_subset)+,
erule bexE, erule conjE)
apply (simp only:Iod_carrier) apply (
cut_tac a = y in insert_iff[of _ "f (Sup D W)" "insert (Sup D W) W"],
frule_tac P = "y ∈ insert (f (Sup D W)) (insert (Sup D W) W)" and
Q = "y = f (Sup D W) ∨ y ∈ insert (Sup D W) W" in eq_prop,
assumption,
thin_tac "y ∈ insert (f (Sup D W)) (insert (Sup D W) W)",
thin_tac "y ∈ insert (f (Sup D W)) (insert (Sup D W) W) =
(y = f (Sup D W) ∨ y ∈ insert (Sup D W) W)",
erule disjE, simp add:oless_def) apply (
cut_tac a = y in insert_iff[of _ "Sup D W" "W"],
frule_tac P = "y ∈ insert (Sup D W) W" and
Q = "y = (Sup D W) ∨ y ∈ W" in eq_prop, assumption,
thin_tac "y ∈ insert (Sup D W) W",
thin_tac "y ∈ (insert (Sup D W) W) = (y = (Sup D W) ∨ y ∈ W)",
erule disjE, simp add:oless_def)
apply (frule_tac b = y in forball_spec1, assumption,
thin_tac "∀x∈W. x \<preceq> Sup D W",
cut_tac subset_insertI[of "W" "Sup D W"],
frule_tac c = y in subsetD[of "W" "insert (Sup D W) W"], assumption,
frule Worder.Torder[of "Iod D (insert (f (Sup D W))
(insert (Sup D W) W))"])
apply (frule mem_WWa_sub_carrier[of "W"],
frule_tac c = y in subsetD[of "W" "carrier D"], assumption+,
frule_tac c = y in subsetD[of "insert (Sup D W) W"
"insert (f (Sup D W)) (insert (Sup D W) W)"], assumption+,
frule_tac a1 = y in Torder.not_le_less[THEN sym,
of "Iod D (insert (f (Sup D W)) (insert (Sup D W) W))" _ "Sup D W"],
simp add:Iod_carrier, simp add:Iod_carrier)
apply (simp add:Iod_le)
done
lemma (in Order) BNTr7_17:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x); W ∈ WWa D f a|] ==>
(insert (f (Sup D W)) (insert (Sup D W) W)) ∈ WWa D f a"
apply (frule mem_WWa_Chain[of "W"],
frule S_inductive_sup_mem[of "W"], assumption+)
apply (case_tac "f (Sup D W) = Sup D W",
simp add:insert_absorb, simp add: BNTr7_14,
frule not_sym, thin_tac "f (Sup D W) ≠ Sup D W")
apply (frule BNTr7_14[of "f" "a" "W"], assumption+,
frule mem_WWa_sub_carrier[of "insert (Sup D W) W" "f" "a"],
frule mem_WWa_Chain[of "insert (Sup D W) W" "f" "a"],
frule funcset_mem[of "f" "carrier D" "carrier D" "Sup D W"], assumption+,
frule mem_wwa_Worder[of "insert (Sup D W) W"])
apply (frule insert_sub[of "insert (Sup D W) W" "carrier D" "f (Sup D W)"],
assumption+,
rule mem_of_WWa [of "(insert (f (Sup D W)) (insert (Sup D W) W))" "a"
"f"], assumption+)
apply (frule well_ord_adjunction[of "insert (Sup D W) W" "f (Sup D W)"],
assumption+,
frule S_inductive_sup_bound[of "W"], assumption+,
rule ballI,
simp only:insert_iff, erule disjE, simp,
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈W. x \<preceq> Sup D W",
frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x",
frule mem_WWa_sub_carrier[of "W"],
frule_tac c = x in subsetD[of "W" "carrier D"], assumption+,
rule_tac a = x in le_trans[of _ "Sup D W" "f (Sup D W)"],
assumption+, simp)
apply (frule mem_WWa_inc_a[of "insert (Sup D W) W" "f" "a"],
simp)
apply (rule ballI)
apply (frule BNTr2_1[of "f" "a" "insert (Sup D W) W"], assumption+,
cut_tac a = x in insert_iff[of _ "f (Sup D W)" "insert (Sup D W) W"],
frule_tac P = "x ∈ insert (f (Sup D W)) (insert (Sup D W) W)" and
Q = "x = (f (Sup D W)) ∨ x ∈ (insert (Sup D W) W)" in eq_prop,
assumption+,
thin_tac "x ∈ insert (f (Sup D W)) (insert (Sup D W) W)",
thin_tac "(x ∈ insert (f (Sup D W)) (insert (Sup D W) W)) =
(x = f (Sup D W) ∨ x ∈ insert (Sup D W) W)",
erule disjE)
apply (frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x",
frule_tac b = "Sup D W" in forball_spec1, simp)
apply (
thin_tac "∀x∈insert (Sup D W) W. a \<preceq> x",
simp add:le_trans[of "a" "Sup D W" "f (Sup D W)"])
apply (frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈insert (Sup D W) W. a \<preceq> x",
simp)
apply (rule ballI)
apply (cut_tac a = x in insert_iff[of _ "f (Sup D W)" "insert (Sup D W) W"],
frule_tac P = "x ∈ insert (f (Sup D W)) (insert (Sup D W) W)" and
Q = "x = (f (Sup D W)) ∨ x ∈ (insert (Sup D W) W)" in eq_prop,
assumption+,
thin_tac "x ∈ insert (f (Sup D W)) (insert (Sup D W) W)",
thin_tac "(x ∈ insert (f (Sup D W)) (insert (Sup D W) W)) =
(x = f (Sup D W) ∨ x ∈ insert (Sup D W) W)",
erule disjE)
apply (frule not_sym, thin_tac "Sup D W ≠ f (Sup D W)",
frule BNTr7_15[of "f" "a" "W"], assumption+,
simp del:insert_iff insert_subset)
apply (subst BNTr7_16[of "f" "a" "W"], assumption+, simp,
frule S_inductive_sup_bound[of "W"], assumption+)
apply (subst BNTr7_11[of "f" "a" "f (Sup D W)"
"insert (Sup D W) W"], assumption+,
rule ballI,
thin_tac "x ∈ insert (Sup D W) W", simp only:insert_iff,
erule disjE,
frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x",
simp add:le_antisym,
frule_tac b = xa in forball_spec1, assumption+,
thin_tac "∀x∈W. x \<preceq> Sup D W",
frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x",
frule mem_WWa_sub_carrier[of "W"],
frule_tac c = xa in subsetD[of "W" "carrier D"], assumption+,
rule_tac a = xa in le_trans[of _ "Sup D W" "f (Sup D W)"],
assumption+)
apply (case_tac "ExPre (Iod D (insert (Sup D W) W)) x",
simp del:insert_iff insert_subset,
subst BNTr7_12[of "f" "a" "f (Sup D W)" "insert (Sup D W) W"],
assumption+,
rule ballI, thin_tac "x ∈ insert (Sup D W) W",
simp only:insert_iff, erule disjE,
frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x",
simp add:le_antisym,
frule_tac b = xa in forball_spec1, assumption+,
thin_tac "∀x∈W. x \<preceq> Sup D W",
frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x",
frule mem_WWa_sub_carrier[of "W"],
frule_tac c = xa in subsetD[of "W" "carrier D"], assumption+,
rule_tac a = xa in le_trans[of _ "Sup D W" "f (Sup D W)"],
assumption+)
apply (frule mem_WWa_then[of "insert (Sup D W) W" "f" "a"],
(erule conjE)+,
thin_tac "∀x∈insert (Sup D W) W. a \<preceq> x",
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈insert (Sup D W) W.
if ExPre (Iod D (insert (Sup D W) W)) x
then f (Pre (Iod D (insert (Sup D W) W)) x) = x
else if a ≠ x
then Sup D (segment (Iod D (insert (Sup D W) W)) x) = x
else a = a", simp)
apply (simp del:insert_iff insert_subset, rule impI)
apply (subst BNTr7_13[of "f" "a" "f (Sup D W)" "insert (Sup D W) W"],
assumption+,
rule ballI, thin_tac "x ∈ insert (Sup D W) W",
simp only:insert_iff,
erule disjE,
frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x",
simp add:le_antisym,
frule_tac b = xa in forball_spec1, assumption+,
thin_tac "∀x∈W. x \<preceq> Sup D W",
frule_tac b = "Sup D W" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x",
frule mem_WWa_sub_carrier[of "W"],
frule_tac c = xa in subsetD[of "W" "carrier D"], assumption+,
rule_tac a = xa in le_trans[of _ "Sup D W" "f (Sup D W)"],
assumption+)
apply (frule mem_WWa_then[of "insert (Sup D W) W" "f" "a"],
(erule conjE)+,
thin_tac "∀x∈insert (Sup D W) W. a \<preceq> x",
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀x∈insert (Sup D W) W.
if ExPre (Iod D (insert (Sup D W) W)) x
then f (Pre (Iod D (insert (Sup D W) W)) x) = x
else if a ≠ x
then Sup D (segment (Iod D (insert (Sup D W) W)) x) = x
else a = a", simp)
done
lemma (in Order) BNTr8:"[|f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> (f x)|] ==> \<Union> (WWa D f a) ∈ (WWa D f a)"
apply (cut_tac Union_WWa_sub_carrier[of "f" "a"])
apply (rule mem_of_WWa[of "\<Union>WWa D f a" "a" "f"], simp)
apply (simp add:Worder_def Torder_def)
apply (simp add:Iod_Order[of "\<Union>WWa D f a"])
apply (rule conjI)
apply (simp add:Torder_axioms_def)
apply ((rule allI, rule impI)+, simp add:Iod_carrier,
(erule bexE)+)
apply (rename_tac b c X1 X2)
apply (frule_tac ?W1.0 = X1 and ?W2.0 = X2 in BNTr7[of "f" "a"],
assumption+)
apply (erule disjE)
apply ((frule_tac c = b and A = X1 and B = X2 in subsetD, assumption+),
(frule_tac W = X2 in mem_wwa_Worder[of _ "f" "a"]),
(simp add:Worder_def Torder_def, (erule conjE)+,
simp add:Torder_axioms_def),
(frule_tac W = X1 in mem_WWa_sub_carrier[of _ "f" "a"],
frule_tac W = X2 in mem_WWa_sub_carrier[of _ "f" "a"],
simp add:Iod_carrier),
(frule_tac a = b in forall_spec, assumption,
thin_tac "∀a. a ∈ X2 --> (∀b. b ∈ X2 -->
a \<preceq>Iod D X2 b ∨ b \<preceq>Iod D X2 a)",
frule_tac a = c in forall_spec, assumption,
thin_tac "∀ba. ba ∈ X2 --> b \<preceq>Iod D X2 ba ∨ ba \<preceq>Iod D X2 b",
frule_tac X = X1 and A = b in UnionI[of _ "WWa D f a"], assumption+,
frule_tac X = X2 and A = c in UnionI[of _ "WWa D f a"], assumption+,
simp add:Iod_le))
apply (frule_tac c = c and A = X2 and B = X1 in subsetD, assumption+,
frule_tac W = X1 in mem_wwa_Worder[of _ "f" "a"],
(simp add:Worder_def Torder_def, (erule conjE)+,
simp add:Torder_axioms_def),
(frule_tac W = X2 in mem_WWa_sub_carrier[of _ "f" "a"],
frule_tac W = X1 in mem_WWa_sub_carrier[of _ "f" "a"],
simp add:Iod_carrier,
frule_tac a = b in forall_spec, assumption),
thin_tac "∀a. a ∈ X1 --> (∀b. b ∈ X1 -->
a \<preceq>Iod D X1 b ∨ b \<preceq>Iod D X1 a)",
(frule_tac a = c in forall_spec, assumption,
thin_tac "∀ba. ba ∈ X1 --> b \<preceq>Iod D X1 ba ∨ ba \<preceq>Iod D X1 b",
frule_tac X = X1 and A = b in UnionI[of _ "WWa D f a"], assumption+,
frule_tac X = X1 and A = c in UnionI[of _ "WWa D f a"], assumption+),
simp add:Iod_le)
apply (subst Worder_axioms_def)
apply (rule allI, rule impI, erule conjE)
apply (frule_tac A = X in nonempty_ex, erule exE)
apply (simp only:Iod_carrier,
frule_tac c = x and A = X and B = "\<Union>WWa D f a" in subsetD,
assumption, simp, erule bexE)
apply (rename_tac X x W)
apply (frule_tac W = W in mem_wwa_Worder[of _ "f" "a"],
simp add:Worder_def Torder_def, erule conjE,
simp add:Worder_axioms_def, erule conjE)
apply (frule_tac a = "X ∩ W" in forall_spec,
thin_tac "∀X. X ⊆ carrier (Iod D W) ∧ X ≠ {} -->
(∃x. minimum_elem (Iod D W) X x)",
frule_tac W = W in mem_WWa_sub_carrier[of _ "f" "a"],
simp only:Iod_carrier, simp add:Int_lower2, blast,
thin_tac "∀X. X ⊆ carrier (Iod D W) ∧ X ≠ {} -->
(∃x. minimum_elem (Iod D W) X x)", erule exE)
apply (frule_tac W = W in mem_WWa_sub_carrier,
frule_tac D = "Iod D W" and X = "X ∩ W" and a = xa in
Order.minimum_elem_mem,
simp add:Iod_carrier, rule subsetI, simp, assumption+)
apply (rule contrapos_pp, (simp del:Union_iff)+, erule conjE)
apply (simp add:minimum_elem_def)
apply (frule_tac a = xa in forall_spec, assumption+,
thin_tac "∀x. x ∈ X --> (∃xa∈X. ¬ x \<preceq>Iod D (\<Union>WWa D f a) xa)",
erule bexE)
apply (frule_tac c = xb and A = X and B = "\<Union>WWa D f a" in subsetD,
assumption+)
apply (cut_tac A = xb and C = "WWa D f a" in Union_iff, simp del:Union_iff,
erule bexE, rename_tac X x W xa xb W',
frule_tac ?W1.0 = W and ?W2.0 = W' in BNTr7[of "f" "a"], assumption+)
apply (case_tac "W' ⊆ W",
frule_tac c = xb and A = W' and B = W in subsetD, assumption+,
rotate_tac 4,
frule_tac b = xb in forball_spec1, simp,
thin_tac "∀x∈X ∩ W. xa \<preceq>Iod D W x")
apply (frule Iod_Order[of "\<Union>WWa D f a"],
frule_tac X = W and A = xa in UnionI[of _ "WWa D f a"], assumption+,
simp del:Union_iff add:Iod_le)
apply (simp del:Union_iff)
apply (frule_tac c = xa and A = W and B = W' in subsetD, assumption+,
frule_tac X = W' and A = xa in UnionI[of _ "WWa D f a"], assumption+,
frule_tac X = W' and A = xb in UnionI[of _ "WWa D f a"], assumption+)
apply (simp only:Iod_le)
apply (frule_tac W = W' in mem_wwa_Worder,
frule_tac D = "Iod D W'" in Worder.Torder,
frule_tac D = "Iod D W'" in Worder.Order)
apply (frule_tac W = W' in mem_WWa_sub_carrier,
frule_tac T1 = W' and a1 = xa and b1 = xb in Iod_le[THEN sym],
assumption+, simp del:Union_iff)
apply (frule_tac D = "Iod D W'" and a = xa and b = xb in Torder.not_le_less)
apply (simp add:Iod_carrier) apply (simp add:Iod_carrier)
apply (simp del:Union_iff add:Iod_less, thin_tac "¬ xa \<preceq> xb",
thin_tac "¬ xa \<preceq>Iod D W' xb")
apply (frule Iod_Order[of "\<Union>WWa D f a"])
apply (frule_tac a1 = xb and b1 = xa in Iod_less[THEN sym, of "\<Union> WWa D f a"],
assumption+, simp del:Union_iff,
frule_tac x = xa and xa = xb and W = W in BNTr7_1[of "f" "a"],
assumption+,
frule_tac a1 = xb and b1 = xa and T1 = W in Iod_less[THEN sym],
assumption+, simp del:Union_iff,
frule_tac W = W in mem_wwa_Worder,
frule_tac D = "Iod D W" in Worder.Torder)
apply (frule_tac D1 = "Iod D W" and a1 = xa and b1 = xb in
Torder.not_le_less[THEN sym],
simp add:Iod_carrier, simp add:Iod_carrier, simp del:Union_iff)
apply (rule BNTr7_6, assumption+)
apply (rule ballI)
apply (cut_tac A = x in Union_iff[of _ "WWa D f a"], simp del:Union_iff)
apply (erule bexE, rename_tac x W)
apply (simp add:WWa_def Wa_def[of "D" _ "f" "a"])
apply (rule ballI)
apply (case_tac "ExPre (Iod D (\<Union>WWa D f a)) x", simp,
erule bexE, rename_tac x W,
frule_tac X = W and A = x in UnionI[of _ "WWa D f a"], assumption+,
frule_tac x = x in BNTr7_2[of "f" "a"], assumption+,
frule_tac b = W in forball_spec1, assumption,
thin_tac "∀W∈WWa D f a. x ∈ W --> ExPre (Iod D W) x",
simp del:Union_iff)
apply (frule_tac x = x in BNTr7_3[of "f" "a"], assumption+,
frule_tac b = W in forball_spec1, assumption,
thin_tac "∀W∈WWa D f a.
x ∈ W --> Pre (Iod D (\<Union>WWa D f a)) x = Pre (Iod D W) x",
simp del:Union_iff)
apply (simp add:WWa_def Wa_def)
apply (simp del:Union_iff, rule impI)
apply (cut_tac A = x in Union_iff[of _ "WWa D f a"], simp del:Union_iff,
erule bexE, rename_tac x W,
frule_tac x = x and W = W in BNTr7_5[of "f" "a" _], assumption+,
simp)
apply (frule_tac x = x and W = W in BNTr7_4[of "f" "a"], assumption+,
simp del:Union_iff,
thin_tac "\<Union>WWa D f a ⊆ carrier D",
thin_tac "∃X∈WWa D f a. x ∈ X",
thin_tac "segment (Iod D (\<Union>WWa D f a)) x = segment (Iod D W) x",
thin_tac "¬ ExPre (Iod D (\<Union>WWa D f a)) x")
apply (simp add:WWa_def Wa_def)
done
(*
lemma (in Order) BNTr9:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x)|] ==>
insert (Sup D (\<Union>WWa D f a)) (\<Union>WWa D f a) ∈ WWa D f a"
apply (frule BNTr8[of "f" "a"], assumption+,
rule BNTr7_14[of "f" "a" "\<Union>WWa D f a"], assumption+)
done
lemma (in Order) S_inductive_postSup_outside:" [|S_inductive_set D;
Chain D X; b ∈ carrier D; Sup D X \<prec> b|] ==> b ∉ X"
apply (frule S_inductive_sup_mem[of "X"], assumption+)
apply (rule contrapos_pp, simp+)
apply (frule S_inductive_sup_bound[of "X"], assumption,
frule_tac a = b in forball_spec1, assumption,
thin_tac "∀x∈X. x \<preceq> Sup D X",
frule less_imp_le[of "Sup D X" "b"], assumption+)
apply (frule le_antisym[of "Sup D X" "b"], assumption+, simp add:oless_def)
done
lemma (in Order) S_inductive_adjunct_postSup_mem_WWa:"[|S_inductive_set D;
∀x∈carrier D. x \<preceq> (f x); W ∈ WWa D f a; b ∈ carrier D; (Sup D W) \<prec> b|]
==> insert b (insert (Sup D W) W) ∈ WWa D f a"
apply (frule mem_WWa_sub_carrier[of "W"],
frule mem_WWa_Chain[of "W"],
frule mem_wwa_Worder[of "W"],
frule S_inductive_sup_mem[of "W"], assumption+)
apply (rule mem_of_WWa[of "insert b (insert (Sup D W) W)" "a" "f"],
rule insert_sub[of "insert (Sup D W) W" "carrier D" "b"],
rule insert_sub[of "W" "carrier D" "Sup D W"], assumption+)
apply (frule S_inductive_sup_bound[of "W"], assumption+,
frule insert_sub[of "W" "carrier D" "Sup D W"], assumption+)
apply (frule well_ord_adjunction[of "W" "Sup D W"], assumption+)
apply (frule well_ord_adjunction[of "insert (Sup D W) W" "b"], assumption+)
apply (rule ballI, simp, erule disjE, simp add:less_imp_le,
frule_tac a = x in forball_spec1, assumption,
thin_tac "∀x∈W. x \<preceq> Sup D W")
apply (frule_tac c = x in subsetD[of "W" "carrier D"], assumption+,
frule less_imp_le[of "Sup D W" "b"], assumption+,
frule_tac a = x in le_trans[of _ "Sup D W" "b"], assumption+,
simp add:Un_commute[of "W" "{Sup D W}"],
simp add:Un_commute[of "insert (Sup D W) W" "{b}"])
apply (cut_tac subset_insertI[of "W" "Sup D W"],
cut_tac subset_insertI[of "insert (Sup D W) W" "b"],
frule mem_WWa_inc_a[of "W"],
frule subsetD[of "W" "insert (Sup D W) W" "a"], assumption+,
frule subsetD[of "insert (Sup D W) W" "insert b (insert (Sup D W) W)"
"a"], assumption+)
apply (rule ballI) apply simp
apply (erule disjE, simp,
frule less_imp_le[of "Sup D W" "b"], assumption+,
frule S_inductive_sup_bound[of "W"], assumption,
frule mem_WWa_inc_a[of "W"],
frule_tac a = a in forball_spec1, assumption,
frule subsetD[of "W" "carrier D" "a"], assumption+,
rule le_trans[of "a" "Sup D W" "b"], assumption+)
apply (erule disjE, simp,
frule mem_WWa_inc_a[of "W"],
frule S_inductive_sup_bound[of "W"], assumption,
simp, simp add:WWa_def Wa_def)
apply (rule ballI)
apply (simp only:insert_iff)
apply (erule disjE)*)
lemma (in Order) BNTr10:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x)|] ==>
(Sup D (\<Union>WWa D f a)) ∈ (\<Union>WWa D f a)"
apply (frule_tac f = f and a = a in BNTr8, assumption+,
frule BNTr7_14[of "f" "a" "\<Union>WWa D f a"], assumption+,
frule mem_family_sub_Un[of "insert (Sup D (\<Union>WWa D f a)) (\<Union>WWa D f a)"
"WWa D f a"],
cut_tac insertI1[of "Sup D (\<Union>WWa D f a)" "\<Union>WWa D f a"])
apply (simp add:subsetD)
done
lemma (in Order) BNTr11:"[|S_inductive_set D; f ∈ carrier D -> carrier D;
a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x)|] ==>
f (Sup D (\<Union>WWa D f a)) = (Sup D (\<Union>WWa D f a))"
apply (frule_tac f = f and a = a in BNTr8, assumption+,
frule mem_WWa_Chain[of "\<Union>WWa D f a"],
frule BNTr10[of "f" "a"], assumption+,
frule S_inductive_sup_mem[of "\<Union>WWa D f a"], assumption)
apply (frule BNTr7_17[of "f" "a" "\<Union>WWa D f a"], assumption+)
apply (cut_tac insertI1[of "f (Sup D (\<Union>WWa D f a) )" "insert (Sup D (\<Union>WWa D f a)) (\<Union>WWa D f a)"],
frule mem_family_sub_Un[of "insert (f (Sup D (\<Union>WWa D f a)))
(insert (Sup D (\<Union>WWa D f a)) (\<Union>WWa D f a))" "WWa D f a"])
apply (frule subsetD[of "insert (f (Sup D (\<Union>WWa D f a)))
(insert (Sup D (\<Union>WWa D f a)) (\<Union>WWa D f a))" "\<Union>WWa D f a" "f (Sup D (\<Union>WWa D f a))"],
assumption+)
apply (frule S_inductive_sup_bound[of "\<Union>WWa D f a"], assumption)
apply (frule_tac b = "f (Sup D (\<Union>WWa D f a))" in forball_spec1,
assumption,
thin_tac "∀x∈\<Union>WWa D f a. x \<preceq> Sup D (\<Union>WWa D f a)")
apply (frule_tac b = "Sup D (\<Union>WWa D f a)" in forball_spec1, assumption,
thin_tac "∀x∈carrier D. x \<preceq> f x")
apply (frule funcset_mem[of "f" "carrier D" "carrier D" "Sup D (\<Union>WWa D f a)"],
assumption+,
rule le_antisym[of "f (Sup D (\<Union>WWa D f a))" "Sup D (\<Union>WWa D f a)"],
assumption+)
done
lemma (in Order) Bourbaki_Nakayama:"[|S_inductive_set D;
f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> (f x)|] ==>
∃x0∈carrier D. f x0 = x0"
apply (frule BNTr8[of "f" "a"], assumption+,
frule mem_WWa_Chain[of "\<Union>WWa D f a" "f" "a"],
frule S_inductive_sup_mem[of "\<Union>WWa D f a"], assumption+,
frule BNTr11[of "f" "a"], assumption+)
apply blast
done
constdefs (structure D)
maxl_fun::" _ => 'a => 'a"
"maxl_fun D == λx∈carrier D. if ∃y. y∈(upper_bounds D {x}) ∧ y ≠ x then
SOME z. z ∈ (upper_bounds D {x}) ∧ z ≠ x else x"
lemma (in Order) maxl_funTr:"[|x ∈ carrier D;
∃y. y ∈ upper_bounds D {x} ∧ y ≠ x|] ==>
(SOME z. z ∈ upper_bounds D {x} ∧ z ≠ x) ∈ carrier D"
apply (rule someI2_ex, assumption+,
simp add:upper_bounds_def upper_bound_def)
done
lemma (in Order) maxl_fun_func:"maxl_fun D ∈ carrier D -> carrier D"
apply (rule univar_func_test, rule ballI)
apply (simp add:maxl_fun_def, rule impI,
rule maxl_funTr, assumption+)
done
lemma (in Order) maxl_fun_gt:"[|x ∈ carrier D;
∃y∈ carrier D. x \<preceq> y ∧ x ≠ y |] ==>
x \<preceq> (maxl_fun D x) ∧ (maxl_fun D x) ≠ x"
apply (simp add:maxl_fun_def upper_bounds_def upper_bound_def,
rule conjI, rule impI)
apply (rule someI2_ex, assumption+, simp,
erule bexE, blast)
done
lemma (in Order) maxl_fun_maxl:"[|x ∈ carrier D; maxl_fun D x = x |]
==> maximal x"
apply (rule contrapos_pp, simp+, simp add:maximal_element_def)
apply (frule maxl_fun_gt[of "x"], assumption, erule conjE, simp)
done
lemma (in Order) maxl_fun_asc:"∀x∈carrier D. x \<preceq> (maxl_fun D x)"
apply (rule ballI)
apply (simp add:maxl_fun_def, rule conjI, rule impI)
apply (rule someI2_ex, assumption, simp add:upper_bounds_def upper_bound_def,
rule impI, rule le_refl, assumption)
done
lemma (in Order) S_inductive_maxl:"[|S_inductive_set D; carrier D ≠ {}|] ==>
∃m. maximal m"
apply (frule nonempty_ex [of "carrier D"],
erule exE, rename_tac a)
apply (cut_tac maxl_fun_asc, cut_tac maxl_fun_func,
frule_tac a = a in Bourbaki_Nakayama[of "maxl_fun D" _], assumption+)
apply (erule bexE,
frule_tac x = x0 in maxl_fun_maxl, assumption+)
apply blast
done
lemma (in Order) maximal_mem:"maximal m ==> m ∈ carrier D"
by (simp add:maximal_element_def)
constdefs (structure D)
Chains :: " _ => ('a set) set"
"Chains D == {C. Chain D C}"
family_Torder::" _ => ('a set) Order"
("(fTo _)" [999]1000)
"fTo D == (|carrier = Chains D , rel = {Z. Z ∈ (Chains D) × (Chains D) ∧ (fst Z) ⊆ (snd Z)}|)),"
lemma (in Order) Chain_mem_fTo:"Chain D C ==> C ∈ carrier (fTo D)"
by (simp add:family_Torder_def Chains_def)
lemma (in Order) fTOrder:"Order (fTo D)"
apply (subst Order_def)
apply (simp add:family_Torder_def)
apply auto
done
lemma (in Order) fTo_Order_sub:"[|A ∈ carrier (fTo D); B ∈ carrier (fTo D)|]
==> (A \<preceq>(fTo D) B) = (A ⊆ B)"
by (subst ole_def, simp add:family_Torder_def)
lemma (in Order) mem_fTo_Chain:"X ∈ carrier (fTo D) ==> Chain D X"
by (simp add:family_Torder_def Chains_def)
lemma (in Order) mem_fTo_sub_carrier:"X ∈ carrier (fTo D) ==> X ⊆ carrier D"
by (frule mem_fTo_Chain[of "X"], simp add:Chain_sub)
lemma (in Order) Un_fTo_Chain:"Chain (fTo D) CC ==> Chain D (\<Union> CC)"
apply (cut_tac fTOrder,
frule Order.Chain_sub[of "fTo D" "CC"], assumption+,
cut_tac family_subset_Un_sub[of "CC" "carrier D"],
subst Chain_def, simp, simp add:Torder_def, simp add:Iod_Order,
simp add:Torder_axioms_def)
apply ((rule allI, rule impI)+, simp add:Iod_carrier)
apply ((erule bexE)+,
frule_tac A = X in mem_family_sub_Un[of _ "CC"],
frule_tac A = Xa in mem_family_sub_Un[of _ "CC"],
frule_tac c = a and A = X in subsetD[of _ "\<Union>CC"], assumption+,
frule_tac c = b and A = Xa in subsetD[of _ "\<Union>CC"], assumption+,
simp only:Iod_le,
frule_tac c = X in subsetD[of "CC" "carrier fTo D"], assumption+,
frule_tac c = Xa in subsetD[of "CC" "carrier fTo D"], assumption+)
apply (simp add:Chain_def Torder_def Torder_axioms_def,
thin_tac "∃X∈CC. a ∈ X", thin_tac "∃X∈CC. b ∈ X",
(erule conjE)+,
frule_tac a = X in forall_spec,
simp only:Order.Iod_carrier[of "fTo D" "CC"],
thin_tac "∀a. a ∈ carrier (Iod (fTo D) CC) -->
(∀b. b ∈ carrier (Iod (fTo D) CC) -->
a \<preceq>Iod (fTo D) CC b ∨ b \<preceq>Iod (fTo D) CC a)",
frule_tac a = Xa in forall_spec,
simp only:Order.Iod_carrier[of "fTo D" "CC"],
thin_tac "∀b. b ∈ carrier (Iod (fTo D) CC) -->
X \<preceq>Iod (fTo D) CC b ∨ b \<preceq>Iod (fTo D) CC X")
apply (simp add:Order.Iod_le) apply (simp add:fTo_Order_sub)
apply (frule_tac X = X in mem_fTo_Chain,
frule_tac X = Xa in mem_fTo_Chain,
frule_tac X = X in Chain_Torder,
frule_tac X = Xa in Chain_Torder,
frule_tac X = X in Chain_sub,
frule_tac X = Xa in Chain_sub)
apply (simp add:Torder_def Torder_axioms_def, (erule conjE)+,
simp only:Iod_carrier)
apply (erule disjE,
frule_tac c = a and A = X and B = Xa in subsetD, assumption+,
thin_tac "∀a. a ∈ X --> (∀b. b ∈ X --> a \<preceq>Iod D X b ∨ b \<preceq>Iod D X a)",
frule_tac a = a in forall_spec,
thin_tac "∀a. a ∈ Xa --> (∀b. b ∈ Xa --> a \<preceq>Iod D Xa b ∨ b \<preceq>Iod D Xa a)",
assumption,
thin_tac "∀a. a ∈ Xa --> (∀b. b ∈ Xa --> a \<preceq>Iod D Xa b ∨ b \<preceq>Iod D Xa a)",
frule_tac a = b in forall_spec, assumption,
thin_tac "∀b. b ∈ Xa --> a \<preceq>Iod D Xa b ∨ b \<preceq>Iod D Xa a",
simp add:Iod_le)
apply (
frule_tac c = b and A = Xa and B = X in subsetD, assumption+,
thin_tac "∀a. a ∈ Xa --> (∀b. b ∈ Xa --> a \<preceq>Iod D Xa b ∨ b \<preceq>Iod D Xa a)",
frule_tac a = a in forall_spec,
thin_tac "∀a. a ∈ X --> (∀b. b ∈ X --> a \<preceq>Iod D X b ∨ b \<preceq>Iod D X a)",
assumption,
thin_tac "∀a. a ∈ X --> (∀b. b ∈ X --> a \<preceq>Iod D X b ∨ b \<preceq>Iod D X a)",
frule_tac a = b in forall_spec, assumption,
thin_tac "∀b. b ∈ X --> a \<preceq>Iod D X b ∨ b \<preceq>Iod D X a",
simp add:Iod_le)
apply (rule ballI,
frule_tac c = A in subsetD[of "CC" "carrier (fTo D)"], assumption+,
rule mem_fTo_sub_carrier, assumption+)
done
lemma (in Order) Un_fTo_Chain_mem_fTo:"Chain (fTo D) CC ==>
(\<Union> CC) ∈ carrier (fTo D)"
apply (frule Un_fTo_Chain[of "CC"], thin_tac "Chain (fTo D) CC")
apply (simp add:family_Torder_def Chains_def)
done
lemma (in Order) Un_upper_bound:"Chain (fTo D) C ==>
\<Union>C ∈ upper_bounds (fTo D) C"
apply (simp add:upper_bounds_def upper_bound_def,
simp add:Un_fTo_Chain_mem_fTo)
apply (rule ballI,
simp add:ole_def,
subst family_Torder_def, simp)
apply (cut_tac fTOrder,
frule Order.Chain_sub[of "fTo D" "C"], assumption,
frule_tac c = s in subsetD[of "C" "carrier (fTo D)"], assumption+,
cut_tac Un_fTo_Chain_mem_fTo[of "C"],
simp add:mem_fTo_Chain Chains_def,
rule_tac A = s in mem_family_sub_Un[of _ "C"], assumption)
apply assumption
done
lemma (in Order) fTo_conditional_inc_C:"C ∈ carrier (fTo D) ==>
C ∈ carrier (Iod (fTo D) {S ∈ carrier fTo D. C ⊆ S})"
apply (cut_tac fTOrder,
cut_tac conditional_subset[of "carrier (fTo D)" "op ⊆ C"])
apply (simp add:Order.Iod_carrier)
done
lemma (in Order) fTo_conditional_Un_Chain_mem1:" [|C ∈ carrier (fTo D);
Chain (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}) Ca; Ca ≠ {}|] ==>
\<Union>Ca ∈ upper_bounds (Iod (fTo D) {S ∈ carrier fTo D. C ⊆ S}) Ca"
apply (cut_tac fTOrder,
cut_tac conditional_subset[of "carrier (fTo D)" "op ⊆ C"])
apply (simp add:upper_bounds_def upper_bound_def)
apply (subgoal_tac "\<Union>Ca ∈carrier (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S})")
apply simp
apply (rule ballI)
apply (simp only:Order.Iod_carrier)
apply (frule Order.Iod_Order[of "fTo D" "{x ∈ carrier fTo D. C ⊆ x}"],
assumption+,
frule Order.Chain_sub[of "Iod (fTo D) {S ∈ carrier fTo D. C ⊆ S}"
"Ca"], assumption+, simp only:Order.Iod_carrier,
frule_tac c = s in subsetD[of "Ca" "{S ∈ carrier fTo D. C ⊆ S}"],
assumption+,
frule_tac c = s in subsetD[of "{S ∈ carrier fTo D. C ⊆ S}"
"carrier (fTo D)"], assumption+)
apply (subst Order.Iod_le[of "fTo D" "{x ∈ carrier fTo D. C ⊆ x}"],
assumption+)
apply (subst Order.fTo_Order_sub, rule Order_axioms, assumption, simp)
apply (rule_tac A = s in mem_family_sub_Un[of _ "Ca"], assumption)
apply (simp add:Order.Iod_carrier[of "fTo D"])
apply (rule conjI)
apply (rule Un_fTo_Chain_mem_fTo[of "Ca"])
apply (simp add:Chain_def, erule conjE)
apply (simp add:Order.Iod_carrier,
frule subset_trans[of "Ca" "{S ∈ carrier (fTo D). C ⊆ S}"
"carrier (fTo D)"], assumption+, simp)
apply (frule_tac T = "{S ∈ carrier (fTo D). C ⊆ S}" in
Order.Iod_sub_sub[of "fTo D" "Ca"], assumption+,
simp)
apply (rule sub_Union[of "Ca" "C"])
apply (frule nonempty_ex[of "Ca"], erule exE)
apply (frule Order.Iod_Order[of "fTo D" "{x ∈ carrier (fTo D). C ⊆ x}"],
assumption+,
frule Order.Chain_sub[of "Iod (fTo D) {S ∈ carrier fTo D. C ⊆ S}" "Ca"],
assumption+) apply (simp add:Order.Iod_carrier)
apply (frule_tac c = x in subsetD[of "Ca" "{S ∈ carrier fTo D. C ⊆ S}"],
assumption+, simp, erule conjE, blast)
done
lemma (in Order) fTo_conditional_min1:" [|C ∈ carrier (fTo D);
Chain (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}) Ca; Ca ≠ {}|] ==>
minimum_elem (Iod (fTo D) {S ∈ carrier fTo D. C ⊆ S})
(upper_bounds (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}) Ca) (\<Union>Ca)"
apply (frule fTo_conditional_Un_Chain_mem1[of "C" "Ca"], assumption+,
simp add:minimum_elem_def)
apply (rule ballI)
apply (simp add:upper_bounds_def upper_bound_def, (erule conjE)+,
cut_tac fTOrder,
cut_tac conditional_subset[of "carrier (fTo D)" "op ⊆ C"])
apply (simp only:Order.Iod_carrier,
simp only:Order.Iod_le[of "fTo D"])
apply (frule_tac c = "\<Union>Ca" in subsetD[of "{S ∈ carrier fTo D. C ⊆ S}"
"carrier (fTo D)"], assumption+,
frule_tac c = x in subsetD[of "{S ∈ carrier fTo D. C ⊆ S}"
"carrier (fTo D)"], assumption+)
apply (subst Order.fTo_Order_sub, rule Order_axioms, assumption, simp,
rule_tac C = Ca and B = x in family_subset_Un_sub)
apply (rule ballI)
apply (thin_tac "∀s∈Ca. s \<preceq>Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S} \<Union>Ca",
frule_tac b = A in forball_spec1, assumption,
thin_tac "∀s∈Ca. s \<preceq>Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S} x")
apply (frule Order.Iod_Order[of "fTo D" "{S ∈ carrier fTo D. C ⊆ S}"],
assumption,
frule Order.Chain_sub[of "Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}"
"Ca"], assumption,
simp only:Order.Iod_carrier[of "fTo D" "{S ∈ carrier fTo D. C ⊆ S}"])
apply (frule_tac c = x in subsetD[of "{x ∈ carrier fTo D. C ⊆ x}"
"carrier (fTo D)"], simp,
frule_tac c = A in subsetD[of "Ca" "{x ∈ carrier fTo D. C ⊆ x}"],
assumption+,
frule_tac a = A and b = x in Order.Iod_le[of "fTo D"
"{x ∈ carrier fTo D. C ⊆ x}"], assumption+)
apply simp apply (simp add:fTo_Order_sub)
done
lemma (in Order) fTo_conditional_Un_Chain_mem2:" [|C ∈ carrier (fTo D);
Chain (Iod (fTo D) {S ∈ carrier fTo D. C ⊆ S}) Ca; Ca = {}|] ==>
C ∈ upper_bounds (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}) Ca"
apply (cut_tac fTOrder,
cut_tac conditional_subset[of "carrier (fTo D)" "op ⊆ C"])
apply (simp add:upper_bounds_def upper_bound_def, simp add:Order.Iod_carrier)
done
lemma (in Order) fTo_conditional_min2:" [|C ∈ carrier (fTo D);
Chain (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}) Ca; Ca = {}|] ==>
minimum_elem (Iod (fTo D) {S ∈ carrier fTo D. C ⊆ S})
(upper_bounds (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}) Ca) C"
apply (simp add:minimum_elem_def upper_bounds_def upper_bound_def)
apply (cut_tac fTOrder,
cut_tac conditional_subset[of "carrier (fTo D)" "op ⊆ C"],
simp add:Order.Iod_carrier)
apply (rule allI, rule impI, erule conjE,
subst Order.Iod_le[of "fTo D" "{x ∈ carrier (fTo D). C ⊆ x}"],
assumption+, simp, simp)
apply (subst fTo_Order_sub, assumption+)
done
lemma (in Order) fTo_S_inductive:"S_inductive_set (fTo D)"
apply (simp add:S_inductive_set_def,
rule allI, rule impI)
apply (rule contrapos_pp, simp+)
apply (simp add:minimum_elem_def,
frule_tac CC = C in Un_fTo_Chain_mem_fTo,
frule_tac b = "\<Union>C" in forball_spec1, assumption,
thin_tac "∀x∈carrier (fTo D).
x ∈ upper_bounds (fTo D) C -->
(∃xa∈upper_bounds (fTo D) C. ¬ x \<preceq>(fTo D) xa)")
apply (frule_tac C = C in Un_upper_bound, simp,
erule bexE,
thin_tac "\<Union>C ∈ upper_bounds (fTo D) C")
apply (simp add:upper_bounds_def upper_bound_def,
erule conjE)
apply (cut_tac C = C and B = x in family_subset_Un_sub)
apply (rule ballI)
apply (frule_tac b = A in forball_spec1, assumption,
thin_tac "∀s∈C. s \<preceq>fTo D x",
cut_tac fTOrder,
frule_tac X = C in Order.Chain_sub[of "fTo D"], assumption+,
frule_tac c = A and A = C in subsetD[of _ "carrier (fTo D)"],
assumption+,
simp add:fTo_Order_sub)
apply (simp add:fTo_Order_sub)
done
lemma (in Order) conditional_min_upper_bound:" [|C ∈ carrier (fTo D);
Chain (Iod (fTo D) {S ∈ carrier fTo D. C ⊆ S}) Ca|] ==>
∃X. minimum_elem (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S})
(upper_bounds (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}) Ca) X"
apply (case_tac "Ca = {}",
frule fTo_conditional_min2[of "C"], assumption+, blast,
frule fTo_conditional_min1[of "C"], assumption+, blast)
done
lemma (in Order) Hausdorff_acTr:"C ∈ carrier (fTo D) ==>
S_inductive_set (Iod (fTo D) {S. S ∈ (carrier (fTo D)) ∧ C ⊆ S})"
apply (simp add:S_inductive_set_def)
apply (rule allI, rule impI)
apply (frule_tac Ca = Ca in conditional_min_upper_bound[of "C"],
assumption+)
apply (erule exE, cut_tac fTOrder)
apply (cut_tac conditional_subset[of "carrier (fTo D)" "op ⊆ C"])
apply (frule Order.Iod_Order[of "fTo D" "{S ∈ carrier fTo D. C ⊆ S}"],
assumption+)
apply (frule_tac X = "upper_bounds (Iod (fTo D)
{S ∈ carrier (fTo D). C ⊆ S}) Ca"
and a = X in
Order.minimum_elem_mem[of "Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}"])
apply (simp only:Order.Iod_carrier)
apply (rule subsetI,
thin_tac "minimum_elem (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S})
(upper_bounds (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}) Ca) X")
apply (simp add:upper_bounds_def upper_bound_def, erule conjE)
apply (simp add:Order.Iod_carrier) apply assumption
apply (subgoal_tac "X∈carrier (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}) ")
apply blast
apply (frule_tac X = Ca in Order.Chain_sub[of "Iod (fTo D)
{S ∈ carrier (fTo D). C ⊆ S}"], assumption,
frule_tac X = Ca in Order.upper_bounds_sub[of
"Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}"], assumption)
apply (thin_tac "minimum_elem (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S})
(upper_bounds (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S}) Ca) X")
apply (rule_tac A = "upper_bounds (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S})
Ca" and B = "carrier (Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S})" and
c = X in subsetD, assumption+)
done
lemma satisfy_cond_mem_set:"[| x ∈ A; P x |] ==> x ∈ {y ∈ A. P y}"
by blast
lemma (in Order) maximal_conditional_maximal:" [|C ∈ carrier (fTo D);
maximalIod (fTo D) {S ∈ carrier (fTo D). C ⊆ S} m|] ==> maximal(fTo D) m"
apply (unfold maximal_element_def, erule conjE)
apply (cut_tac conditional_subset[of "carrier (fTo D)" "op ⊆ C"],
cut_tac fTOrder,
frule Order.Iod_Order[of "fTo D" "{x ∈ carrier fTo D. C ⊆ x}"],
assumption+,
simp only:Order.Iod_carrier,
frule subsetD[of "{S ∈ carrier fTo D. C ⊆ S}"
"carrier (fTo D)" "m"], assumption+, simp)
apply (rule ballI, rule impI)
apply (simp only:fTo_Order_sub,
frule_tac a = b in forall_spec, simp,
thin_tac "∀b. b ∈ carrier (fTo D) ∧ C ⊆ b -->
m \<preceq>Iod (fTo D) {S ∈ carrier (fTo D). C ⊆ S} b --> m = b")
apply (frule_tac x = m in
satisfy_cond_mem_set[of _ "carrier (fTo D)" "op ⊆ C"], assumption+,
frule_tac A = C and B = m and C = b in subset_trans, assumption+,
frule_tac x = b in
satisfy_cond_mem_set[of _ "carrier (fTo D)" "op ⊆ C"], assumption+,
simp only:Order.Iod_le, simp only:fTo_Order_sub)
done
lemma (in Order) Hausdorff_ac:"C ∈ carrier (fTo D) ==>
∃M∈carrier (fTo D). C ⊆ M ∧ maximal(fTo D) M"
apply (frule_tac Hausdorff_acTr[of "C"],
cut_tac conditional_subset[of "carrier (fTo D)" "op ⊆ C"],
cut_tac fTOrder,
frule Order.Iod_Order[of "fTo D" "{x ∈ carrier fTo D. C ⊆ x}"],
assumption+)
apply (frule Order.S_inductive_maxl[of "Iod (fTo D)
{S ∈ carrier (fTo D). C ⊆ S}"], assumption+,
frule fTo_conditional_inc_C[of "C"], simp add:nonempty,
erule exE,
frule_tac m = m in Order.maximal_mem[of
"Iod (fTo D) {x ∈ carrier (fTo D). C ⊆ x}"], assumption+,
simp add:Order.Iod_carrier, erule conjE,
frule_tac m = m in maximal_conditional_maximal[of "C"], assumption+)
apply blast
done
(** g_Zorn_lemma is the Zorn's lemma in general ordered set **)
lemma (in Order) Zorn_lemmaTr:"[|Chain D C; M ∈ carrier fTo D; C ⊆ M;
maximalfTo D M; b ∈ carrier D; ∀s∈M. s \<preceq> b |] ==>
maximal b ∧ b ∈ upper_bounds D C"
apply (simp add:upper_bounds_def upper_bound_def)
apply (rule conjI)
prefer 2
apply (rule ballI, simp add:subsetD,
rule contrapos_pp, simp+, simp add:maximal_element_def,
erule bexE, erule conjE)
apply (frule_tac X = M in mem_fTo_Chain,
frule_tac X = M and b = ba in adjunct_Chain, assumption+,
rule ballI)
apply (frule_tac b = x in forball_spec1, assumption+,
thin_tac "∀s∈M. s \<preceq> b",
frule_tac X = M in Chain_sub,
frule_tac c = x in subsetD[of "M" "carrier D"], assumption+,
rule_tac a = x and b = b and c = ba in le_trans, assumption+)
apply (cut_tac B = M and a = ba in subset_insertI,
cut_tac a = ba in insertI1[of _ "M"],
cut_tac C = "insert ba M" in Chain_mem_fTo, assumption)
apply (frule_tac b = "insert ba M" in forball_spec1, assumption,
thin_tac "∀b∈carrier fTo D. M \<preceq>fTo D b --> M = b",
simp del:insert_iff insert_subset add:fTo_Order_sub)
apply (frule_tac b = ba in forball_spec1, assumption,
thin_tac "∀s∈M. s \<preceq> b")
apply (frule_tac a = b and b = ba in le_antisym, assumption+, simp)
done
lemma (in Order) g_Zorn_lemma1:"[|inductive_set D; Chain D C|] ==> ∃m. maximal m ∧ m ∈ upper_bounds D C"
apply (frule Chain_mem_fTo [of "C"],
frule Hausdorff_ac[of "C"])
apply (erule bexE, erule conjE)
apply (frule_tac X = M in mem_fTo_Chain)
apply (simp add:inductive_set_def)
apply (frule_tac a = M in forall_spec, assumption,
thin_tac "∀C. Chain D C --> (∃b. ub C b)",
erule exE, simp add:upper_bound_def, erule conjE)
apply (frule_tac M = M and b = b in Zorn_lemmaTr[of "C"], assumption+)
apply blast
done
lemma (in Order) g_Zorn_lemma2:"[|inductive_set D; a ∈ carrier D|] ==>
∃m∈carrier D. maximal m ∧ a \<preceq> m"
apply (frule BNTr1 [of "a"],
frule singleton_sub[of "a" "carrier D"],
frule_tac X = "{a}" in Torder_Chain,
simp add:Worder.Torder)
apply (frule_tac C = "{a}" in g_Zorn_lemma1, assumption+,
erule exE, erule conjE,
simp add:upper_bounds_def upper_bound_def)
apply blast
done
lemma (in Order) g_Zorn_lemma3:"inductive_set D ==> ∃m∈carrier D. maximal m"
apply (cut_tac Iod_empty_Torder,
cut_tac empty_subsetI[of "carrier D"],
frule Torder_Chain[of "{}"], assumption+)
apply (frule_tac C = "{}" in g_Zorn_lemma1, assumption+,
simp add:upper_bounds_def upper_bound_def,
blast)
done
chapter "3. Group Theory. Focused on Jordan Hoelder theorem"
section "1. Definition of a Group"
record 'a Group = "'a carrier" +
top :: "['a, 'a ] => 'a" (infixl "·\<index>" 70)
iop :: "'a => 'a" ("ρ\<index> _" [81] 80)
one :: "'a" ("\<one>\<index>")
locale Group =
fixes G (structure)
assumes top_closed: "top G ∈ carrier G -> carrier G -> carrier G"
and tassoc : "[|a ∈ carrier G; b ∈ carrier G; c ∈ carrier G|] ==>
(a · b) · c = a · (b · c)"
and iop_closed:"iop G ∈ carrier G -> carrier G"
and l_i :"a ∈ carrier G ==> (ρ a) · a = \<one>"
and unit_closed: "\<one> ∈ carrier G"
and l_unit:"a ∈ carrier G ==> \<one> · a = a"
lemma (in Group) mult_closed:"[|a ∈ carrier G; b ∈ carrier G|] ==>
a · b ∈ carrier G"
apply (cut_tac top_closed)
apply (frule funcset_mem[of "op ·" "carrier G" "carrier G -> carrier G" "a"],
assumption+,
frule funcset_mem[of "op · a" "carrier G" "carrier G" "b"],
assumption+ )
done
lemma (in Group) i_closed:"a ∈ carrier G ==> (ρ a) ∈ carrier G"
apply (cut_tac iop_closed,
frule funcset_mem[of "iop G" "carrier G" "carrier G" "a"])
apply assumption+
done
lemma (in Group) r_mult_eqn:"[|a ∈ carrier G; b ∈ carrier G;
c ∈ carrier G; a = b|] ==> a · c = b · c"
by simp
lemma (in Group) l_mult_eqn:"[|a ∈ carrier G; b ∈ carrier G;
c ∈ carrier G; a = b|] ==> c · a = c · b"
by simp
lemma (in Group) r_i:"a ∈ carrier G ==>
a · (ρ a) = \<one> "
apply (frule mult_closed[of "a" "ρ a"],
simp add:i_closed,
cut_tac l_unit[of "a"],
cut_tac unit_closed,
frule mult_closed[of "\<one>" "a"], assumption+,
frule i_closed[of "a"],
frule mult_closed[of "ρ a" "a"], assumption+)
apply (frule r_mult_eqn[of "(ρ a) · a" "\<one>" "ρ a"], assumption+,
simp add:l_i,
simp add:l_unit[of "ρ a"])
(* (i i a) from the left *)
apply (frule i_closed[of "a"],
frule i_closed[of "ρ a"],
frule mult_closed[of "ρ a" "a"], assumption+,
frule mult_closed[of "(ρ a) · a" "ρ a"], assumption+,
frule l_mult_eqn[of "(ρ a) · a · (ρ a)" "ρ a" "ρ (ρ a)"],
assumption+)
apply (thin_tac "(ρ a) · a · (ρ a) = (ρ a)",
simp add:l_i[of "ρ a"],
simp add:tassoc[THEN sym, of "ρ (ρ a)" "(ρ a) · a" "ρ a"],
simp add:tassoc[THEN sym, of "ρ (ρ a)" "ρ a" "a"])
apply (simp add:l_i[of "ρ a"])
apply assumption
done
lemma (in Group) r_unit:"a ∈ carrier G ==> a · \<one> = a"
apply (cut_tac unit_closed,
frule i_closed[of "a"],
frule mult_closed[of "a" "\<one>"], assumption+)
apply (cut_tac l_i[THEN sym, of "a"],
simp,
thin_tac "\<one> = ρ a · a")
apply (simp add:tassoc[THEN sym] r_i l_unit)
apply assumption
done
lemma (in Group) l_i_unique:"[|a ∈ carrier G; b ∈ carrier G;
b · a = \<one> |] ==> (ρ a) = b "
apply (cut_tac unit_closed,
frule i_closed[of "a"],
frule mult_closed[of "b" "a"], assumption+)
apply (frule r_mult_eqn[of "b · a" "\<one>" "ρ a"], assumption+)
apply (thin_tac "b · a = \<one>",
simp add:tassoc[of "b" "a" "ρ a"] r_i)
apply (simp add:l_unit r_unit)
done
lemma (in Group) l_i_i:"a ∈ carrier G ==> (ρ (ρ a)) · (ρ a) = \<one>"
by (frule i_closed[of "a"],
cut_tac l_i[of "ρ a"], assumption+)
lemma (in Group) l_div_eqn:"[|a ∈ carrier G; x ∈ carrier G; y ∈ carrier G;
a · x = a · y |] ==> x = y"
apply (frule mult_closed[of "a" "x"], assumption+,
frule mult_closed[of "a" "y"], assumption+,
frule i_closed[of "a"],
frule l_mult_eqn[of "a · x" "a · y" "ρ a"], assumption+)
apply (thin_tac "a · x = a · y",
simp add:tassoc[THEN sym])
apply (simp add:l_i l_unit)
done
lemma (in Group) r_div_eqn:"[|a ∈ carrier G; x ∈ carrier G; y ∈ carrier G;
x · a = y · a |] ==> x = y "
apply (frule mult_closed[of "x" "a"], assumption+,
frule mult_closed[of "y" "a"], assumption+,
frule i_closed[of "a"],
frule r_mult_eqn[of "x · a" "y · a" "ρ a"], assumption+)
apply (thin_tac "x · a = y · a",
simp add:tassoc, simp add:r_i r_unit)
done
lemma (in Group) l_mult_eqn1:"[|a ∈ carrier G; x ∈ carrier G; y ∈ carrier G;
(ρ a) · x = (ρ a) · y|] ==> x = y "
by (frule i_closed[of "a"], rule l_div_eqn[of "ρ a" "x" "y"], assumption+)
lemma (in Group) tOp_assocTr41:"[|a ∈ carrier G; b ∈ carrier G; c ∈ carrier G;
d ∈ carrier G|] ==> a · b · c · d = a · b · (c · d)"
by (frule mult_closed[of "a" "b"], assumption+,
simp add:tassoc[of "a · b" "c" "d"])
lemma (in Group) tOp_assocTr42:"[|a ∈ carrier G; b ∈ carrier G; c ∈ carrier G;
d ∈ carrier G|] ==> a · b · c · d = a · (b · c)· d"
by (simp add:tassoc[of "a" "b" "c"])
lemma (in Group) tOp_assocTr44:"[|a ∈ carrier G; b ∈ carrier G; c ∈ carrier G;
d ∈ carrier G |] ==> (ρ a) · b · ((ρ c) · d) =
(ρ a) · ((b · (ρ c)) · d)"
apply (frule i_closed[of "a"],
frule i_closed[of "c"])
apply (simp add:tassoc[of "b" "ρ c" "d"],
frule mult_closed[of "ρ c" "d"], assumption+,
simp add:tassoc[THEN sym, of "ρ a" "b" "(ρ c) · d"])
done
lemma (in Group) tOp_assocTr45:"[|a ∈ carrier G; b ∈ carrier G; c ∈ carrier G;
d ∈ carrier G|] ==> a · b · c · d = a · (b · (c · d))"
apply (frule mult_closed[of "c" "d"], assumption+)
apply (simp add:tassoc[of "a" "b" "c · d", THEN sym])
apply (simp add:tOp_assocTr41)
done
lemma (in Group) one_unique:"[|a ∈ carrier G; x ∈ carrier G; x · a = x|] ==>
a = \<one>"
apply (frule mult_closed[of "x" "a"], assumption+,
frule i_closed[of "x"],
frule l_mult_eqn[of "x · a" "x" "ρ x"], assumption+)
apply (thin_tac "x · a = x",
simp add:tassoc[THEN sym, of "ρ x" "x" "a"],
simp add:l_i l_unit)
done
lemma (in Group) i_one:"ρ \<one> = \<one>"
by (cut_tac unit_closed, frule l_i[of "\<one>"],
frule i_closed[of "\<one>"], simp add:r_unit)
lemma (in Group) eqn_inv1:"[|a ∈ carrier G; x ∈ carrier G; a = (ρ x) |] ==>
x = (ρ a)"
apply (frule i_closed[of "x"],
frule l_mult_eqn[of "a" "ρ x" "x"], assumption+,
thin_tac "a = ρ x", simp add:r_i,
simp add:l_i_unique[of "a" "x"])
done
lemma (in Group) eqn_inv2:"[|a ∈ carrier G; x ∈ carrier G; x · a = x · (ρ x)|] ==>
x = (ρ a)"
apply (rule eqn_inv1, assumption+)
apply (rule l_div_eqn[of "x" "a" "ρ x"], assumption+,
simp add:i_closed, assumption)
done
lemma (in Group) r_one_unique:"[|a ∈ carrier G; x ∈ carrier G; a · x = a|] ==>
x = \<one>"
apply (frule mult_closed[of "a" "x"], assumption+,
frule i_closed[of "a"],
frule l_mult_eqn[of "a · x" "a" "ρ a"], assumption+,
thin_tac "a · x = a",
simp add:tassoc[THEN sym] l_i l_unit)
done
lemma (in Group) r_i_unique:"[|a ∈ carrier G; x ∈ carrier G; a · x = \<one>|] ==>
x = (ρ a)"
apply (cut_tac unit_closed,
frule mult_closed[of "a" "x"], assumption+,
frule i_closed[of "x"],
frule r_mult_eqn[of "a · x" "\<one>" "ρ x"], assumption+,
thin_tac "a · x = \<one>",
simp add:tassoc[of "a" "x" "ρ x"] r_i r_unit l_unit)
apply (simp add:eqn_inv1)
done
lemma (in Group) iop_i_i:"a ∈ carrier G ==> ρ (ρ a) = a"
apply (frule i_closed[of "a"], frule i_closed[of "ρ a"],
frule l_i[of "ρ a"],
frule mult_closed[of "ρ (ρ a)" "ρ a"], assumption+)
apply (cut_tac unit_closed,
frule r_mult_eqn[of "ρ (ρ a) · ρ a" "\<one>" "a"], assumption+,
thin_tac "ρ (ρ a) · ρ a = \<one>",
simp only:tassoc)
apply (simp add:l_i r_unit l_unit)
done
lemma (in Group) i_ab:"[|a ∈ carrier G; b ∈ carrier G|] ==>
ρ (a · b) = (ρ b) · (ρ a)"
apply (frule mult_closed[of "a" "b"], assumption+,
frule i_closed[of "a · b"],
frule i_closed[of "a"], frule i_closed[of "b"],
frule l_div_eqn[of "a · b" "ρ (a · b)" "(ρ b) · (ρ a)"], assumption+,
simp add:mult_closed, simp add:r_i[of "a · b"],
simp add:tOp_assocTr41[THEN sym], simp add:tOp_assocTr42,
simp add:r_i r_unit)
apply assumption
done
lemma (in Group) sol_eq_l:"[|a ∈ carrier G; b ∈ carrier G; x ∈ carrier G;
a · x = b|] ==> x = (ρ a) · b"
apply (frule mult_closed[of "a" "x"], assumption+,
frule i_closed[of "a"],
frule l_mult_eqn[of "a · x" "b" "ρ a"], assumption+)
apply (thin_tac "a · x = b",
simp add:tassoc[THEN sym],
simp add:l_i l_unit)
done
lemma (in Group) sol_eq_r:"[|a ∈ carrier G; b ∈ carrier G; x ∈ carrier G;
x · a = b |] ==> x = b · (ρ a)"
apply (frule mult_closed[of "x" "a"], assumption+,
frule i_closed[of "a"],
frule r_mult_eqn[of "x · a" "b" "ρ a"], assumption+)
apply (thin_tac "x · a = b",
simp add:tassoc,
simp add:r_i r_unit)
done
lemma (in Group) r_div_eq:"[|a ∈ carrier G; b ∈ carrier G; a · (ρ b) = \<one>|] ==>
a = b"
apply (frule i_closed[of "b"],
frule mult_closed[of "a" "ρ b"], assumption+,
cut_tac unit_closed,
frule r_mult_eqn[of "a · (ρ b)" "\<one>" "b"], assumption+)
apply (thin_tac "a · ρ b = \<one>",
simp add:tassoc l_i r_i, simp add:l_unit r_unit)
done
lemma (in Group) l_div_eq:"[|a ∈ carrier G; b ∈ carrier G; (ρ a) · b = \<one>|] ==>
a = b"
apply (frule i_closed[of "a"],
frule mult_closed[of "ρ a" "b"], assumption+,
cut_tac unit_closed,
frule l_mult_eqn[of "(ρ a) · b" "\<one>" "a"], assumption+)
apply (thin_tac "ρ a · b = \<one>",
simp add:tassoc[THEN sym] r_i l_unit r_unit)
done
lemma (in Group) i_m_closed:"[|a ∈ carrier G ; b ∈ carrier G|] ==>
(ρ a) · b ∈ carrier G "
by (rule mult_closed,
simp add:i_closed, assumption)
section "2. Subgroups"
constdefs (structure G)
sg ::"[_ , 'a set ] => bool" ("_ » _ " [60, 61]60)
"G » H == H ≠ {} ∧ H ⊆ carrier G ∧ (∀a ∈ H. ∀b ∈ H. a · (ρ b) ∈ H)"
(** SG is a set satisfying the condition above. In math textbooks this is
called a subgroup and it is also taken as a group with induced structure.**)
Gp::"_ => 'a set => _" ("(\<natural>\<index>_)" 70)
"\<natural>H ≡ G (| carrier := H, top := top G, iop := iop G, one := one G|)),"
(** Gp G H is a group with carrier H, where H is sg **)
constdefs (structure G)
rcs :: "[_ , 'a set, 'a] => 'a set" (infix "•\<index>" 70)
"H • a ≡ {b. ∃ h ∈ H. h · a = b}"
lcs:: "[_ , 'a, 'a set] => 'a set" (infix "♦\<index>" 70)
"a ♦ H ≡ {b. ∃ h ∈ H. a · h = b}"
nsg :: "_ => 'a set => bool" ("_ \<triangleright> _" [60,61]60)
"G \<triangleright> H == (G » H) ∧
(∀x ∈ carrier G. H • x = x ♦ H)"
set_rcs :: "[_ , 'a set] => 'a set set"
"set_rcs G H == {C. ∃ a ∈ carrier G. C = H • a}"
c_iop :: "[_ , 'a set] => 'a set => 'a set"
"c_iop G H ==λX∈set_rcs G H. {z. ∃ x ∈ X . ∃h ∈ H. h · (ρ x) = z}"
c_top :: "[_, 'a set] => (['a set, 'a set] => 'a set)"
"c_top G H == λX∈set_rcs G H. λY∈set_rcs G H.
{z. ∃ x ∈ X. ∃ y ∈ Y. x · y = z}"
lemma (in Group) sg_subset:"G » H ==> H ⊆ carrier G"
by (simp add:sg_def)
lemma (in Group) one_Gp_one:"G » H ==> \<one>(Gp G H) = \<one>"
by (simp add:Gp_def)
lemma (in Group) carrier_Gp:"G » H ==> (carrier (\<natural>H)) = H"
by (simp add:Gp_def)
lemma (in Group) sg_subset_elem:"[|G » H; h ∈ H |] ==> h ∈ carrier G"
by (frule sg_subset [of "H"], simp only:subsetD)
lemma (in Group) sg_mult_closedr:"[|G » H; x ∈ carrier G; h ∈ H|] ==>
x · h ∈ carrier G"
apply (frule sg_subset_elem [of "H" "h"], assumption+)
apply (simp add:mult_closed)
done
lemma (in Group) sg_mult_closedl:"[|G » H; x ∈ carrier G; h ∈ H|] ==>
h · x ∈ carrier G"
apply (frule sg_subset_elem[of "H" "h"], assumption+)
apply (simp add:mult_closed)
done
lemma (in Group) sg_condTr1:"[|H ⊆ carrier G; H ≠ {};
∀a. ∀b. a ∈ H ∧ b ∈ H --> a · (ρ b) ∈ H|] ==> \<one> ∈ H"
apply (frule nonempty_ex [of "H"], erule exE)
apply (frule_tac a = x in forall_spec1,
thin_tac "∀a b. a ∈ H ∧ b ∈ H --> a · ρ b ∈ H",
frule_tac a = x in forall_spec1,
thin_tac "∀b. x ∈ H ∧ b ∈ H --> x · ρ b ∈ H")
apply (frule_tac c = x in subsetD[of "H" "carrier G"], assumption+,
simp add:r_i)
done
lemma (in Group) sg_unit_closed:"G » H ==> \<one> ∈ H"
apply (simp add:sg_def, (erule conjE)+,
rule sg_condTr1, assumption+, blast)
done
lemma (in Group) sg_i_closed:"[|G » H; x ∈ H|] ==> (ρ x) ∈ H"
apply (frule sg_unit_closed,
frule sg_subset_elem[of "H" "x"], assumption,
simp add:sg_def, (erule conjE)+)
apply (frule_tac b = \<one> in forball_spec1, assumption,
rotate_tac -1,
frule_tac b = x in forball_spec1, assumption,
thin_tac "∀b∈H. \<one> · ρ b ∈ H",
thin_tac "∀a∈H. ∀b∈H. a · ρ b ∈ H")
apply (frule i_closed[of "x"],
simp add:l_unit)
done
lemma (in Group) sg_mult_closed:"[|G » H; x ∈ H; y ∈ H|] ==>
x · y ∈ H"
apply (frule sg_i_closed[of "H" "y"], assumption,
simp add:sg_def, (erule conjE)+)
apply (frule_tac b = x in forball_spec1, assumption,
rotate_tac -1,
frule_tac b = "ρ y" in forball_spec1, assumption,
thin_tac "∀b∈H. x · ρ b ∈ H",
thin_tac "∀a∈H. ∀b∈H. a · ρ b ∈ H")
apply (frule subsetD[of "H" "carrier G"], assumption+,
simp add:iop_i_i)
done
lemma (in Group) nsg_sg: "G \<triangleright> H ==> G » H"
by (simp add:nsg_def)
lemma (in Group) nsg_subset:"G \<triangleright> N ==> N ⊆ carrier G"
apply (simp add:nsg_def, (erule conjE)+)
apply (simp add:sg_subset)
done
lemma (in Group) nsg_lr_cst_eq:"[|G \<triangleright> N; a ∈ carrier G|] ==>
a ♦ N = N • a"
by (simp add: nsg_def)
lemma (in Group) sg_i_m_closed:"[|G » H; a ∈ H ; b ∈ H|] ==> (ρ a) · b ∈ H"
apply (rule sg_mult_closed, assumption+,
simp add:sg_i_closed, assumption)
done
lemma (in Group) sg_m_i_closed:"[|G » H; a ∈ H ; b ∈ H |] ==> a · (ρ b) ∈ H"
apply (simp add:sg_def)
done
constdefs (structure G)
sg_gen :: "[_ , 'a set] => 'a set"
"sg_gen G A == \<Inter> {H. ((G » H ) ∧ (A ⊆ H))}"
lemma (in Group) smallest_sg_gen:"[|A ⊆ carrier G; G » H; A ⊆ H|] ==>
sg_gen G A ⊆ H"
apply (simp add:sg_gen_def)
apply auto
done
lemma (in Group) special_sg_G: "G » (carrier G)"
apply (simp add:sg_def,
cut_tac unit_closed, simp add:nonempty)
apply ((rule ballI)+, simp add:mult_closed i_closed)
done
lemma (in Group) special_sg_self: "G = \<natural>(carrier G)"
by (simp add:Gp_def)
lemma (in Group) special_sg_e: "G » {\<one>}"
apply (simp add:sg_def)
apply (simp add:unit_closed i_one l_unit)
done
lemma (in Group) inter_sgs:"[|G » H; G » K|] ==> G » (H ∩ K)"
apply (frule sg_unit_closed[of "H"],
frule sg_unit_closed[of "K"])
apply (simp add:sg_def)
apply auto
done
lemma (in Group) subg_generated:"A ⊆ carrier G ==> G » (sg_gen G A)"
apply (simp add:sg_def)
apply (rule conjI,
simp add:sg_gen_def,
rule ex_nonempty, simp)
apply (rule contrapos_pp, simp+,
frule_tac a = \<one> in forall_spec1,
thin_tac "∀x. ∃xa. G » xa ∧ A ⊆ xa ∧ x ∉ xa",
erule exE, (erule conjE)+,
frule_tac H = x in sg_unit_closed, simp)
apply (rule conjI)
apply (cut_tac special_sg_G,
simp add:sg_gen_def,
rule subsetI,
blast)
apply ((rule ballI)+,
simp add:sg_gen_def,
rule allI, rule impI,
frule_tac a = x in forall_spec, assumption,
thin_tac "∀x. G » x ∧ A ⊆ x --> a ∈ x",
frule_tac a = x in forall_spec, assumption,
thin_tac "∀x. G » x ∧ A ⊆ x --> b ∈ x")
apply (simp add:sg_m_i_closed)
done
constdefs (structure G)
Qg::"[_ , 'a set] =>
(|carrier:: 'a set set, top:: ['a set, 'a set] => 'a set,
iop:: 'a set => 'a set, one:: 'a set |)),"
"Qg G H == (|carrier = set_rcs G H, top = c_top G H, iop = c_iop G H,
one = H |)),"
constdefs (structure G)
Pj :: "[_ , 'a set] => ( 'a => 'a set)"
"Pj G H == λx ∈ carrier G. H • x"
syntax
"@QGRP" :: "([('a, 'more) Group_scheme, 'a set] => ('a set) Group)"
(infixl "'/'" 200)
translations
"G / H" == "Qg G H"
constdefs
gHom ::"[('a, 'more) Group_scheme, ('b, 'more1) Group_scheme] =>
('a => 'b) set"
"gHom G F =={f. (f ∈ extensional (carrier G) ∧ f ∈ carrier G -> carrier F) ∧
(∀x ∈ carrier G. ∀y ∈ carrier G. f (x ·G y) = (f x) ·F (f y))}"
constdefs
gkernel ::"[('a, 'more) Group_scheme , ('b, 'more1) Group_scheme, 'a => 'b]
=> 'a set"
"gkernel G F f == {x. (x ∈ carrier G) ∧ (f x = \<one>F)}"
iim::"[('a, 'more) Group_scheme, ('b, 'more1) Group_scheme, 'a => 'b,
'b set] => 'a set"
"iim G F f K == {x. (x ∈ carrier G) ∧ (f x ∈ K)}"
syntax
"@GKER"::"[('a, 'more) Group_scheme, ('b, 'more1) Group_scheme, 'a => 'b ] =>
'a set" ("(3gker_,_ _)" [88,88,89]88)
translations
"gkerG,F f" == "gkernel G F f"
constdefs
gsurjec ::"[('a, 'more) Group_scheme, ('b, 'more1) Group_scheme,
'a => 'b] => bool" ("(3gsurj_,_ _)" [88,88,89]88)
"gsurjF,G f == (f ∈ gHom F G) ∧ (surj_to f (carrier F) (carrier G))"
ginjec ::"[('a, 'more) Group_scheme, ('b, 'more1) Group_scheme,
'a => 'b] => bool" ("(3ginj_,_ _)" [88,88,89]88)
"ginjF,G f == (f ∈ gHom F G) ∧ (inj_on f (carrier F))"
gbijec :: "[('a, 'm) Group_scheme, ('b, 'm1) Group_scheme, 'a => 'b]
=> bool" ("(3gbij_,_ _)" [88,88,89]88)
"gbijF,G f == (gsurjF,G f) ∧ (ginjF,G f)"
constdefs (structure G)
Ug :: "_ => ('a, 'more) Group_scheme"
"Ug G == \<natural> {\<one>}"
Ugp:: "_ => bool"
"Ugp G == Group G ∧ carrier G = {\<one>}"
constdefs
isomorphic::"[('a, 'm) Group_scheme, ('b, 'm1) Group_scheme]
=> bool" (infix "≅" 100)
"F ≅ G == ∃f. (gbijF,G f)"
constdefs
constghom::"[('a, 'm) Group_scheme, ('b, 'm1) Group_scheme]
=> ('a => 'b)" ("('1'_,_)" [88,89]88)
"1F,G == λx∈carrier F. (\<one>G)"
constdefs
cmpghom:: "[('a, 'm) Group_scheme, 'b => 'c, 'a => 'b] => 'a => 'c"
"cmpghom F g f == compose (carrier F) g f"
syntax
"@GCOMP" :: "['b => 'c, ('a, 'm) Group_scheme, 'a => 'b] => 'a => 'c"
("(3_ o_ _)" [88, 88, 89]88)
translations
"g oF f"=="cmpghom F g f"
lemma Group_Ugp:"Ugp G ==> Group G"
by (simp add:Ugp_def)
lemma (in Group) r_mult_in_sg:"[|G » H; a ∈ carrier G; x ∈ carrier G; x · a ∈ H|]
==> ∃h ∈ H. h · (ρ a) = x"
apply (frule inEx[of "x · a" "H"],
erule bexE)
apply (rotate_tac -1, frule sym, thin_tac "y = x · a",
frule_tac h = y in sg_subset_elem[of "H"], assumption+,
frule_tac b1 = y in sol_eq_r[THEN sym, of "a" _ "x"], assumption+)
apply blast
done
lemma (in Group) r_unit_sg:"[|G » H; h ∈ H|] ==> h · \<one> = h"
by (frule sg_subset_elem [of "H" "h"], assumption,
simp add:r_unit)
lemma (in Group) sg_l_unit:"[|G » H; h ∈ H|] ==> \<one> · h = h"
by (frule sg_subset_elem [of "H" "h"], assumption+, simp add:l_unit)
lemma (in Group) sg_l_i:"[|G » H; x ∈ H |] ==> (ρ x) · x = \<one>"
by (frule sg_subset_elem[of "H" "x"], assumption+,
simp add:l_i)
lemma (in Group) sg_tassoc:"[|G » H; x ∈ H; y ∈ H; z ∈ H|] ==>
x · y · z = x · (y · z)"
apply (frule sg_subset_elem[of "H" "x"], assumption+,
frule sg_subset_elem[of "H" "y"], assumption+,
frule sg_subset_elem[of "H" "z"], assumption+)
apply (simp add:tassoc)
done
lemma (in Group) sg_condition:"[|H ⊆ carrier G; H ≠ {};
∀a. ∀b. a ∈ H ∧ b ∈ H --> a · (ρ b) ∈ H|] ==> G » H"
by (simp add:sg_def)
constdefs
Gimage :: "[('a, 'm) Group_scheme, ('b, 'm1) Group_scheme, 'a => 'b] =>
('b, 'm1) Group_scheme"
"Gimage F G f == Gp G (f `(carrier F))"
syntax
"@GIMAGE" :: "[('a, 'm) Group_scheme, ('b, 'm1) Group_scheme,
'a => 'b ] => ('b, 'm1) Group_scheme" ("(3Img_,_ _)" [88,88,89]88)
translations
"ImgF,G f" == "Gimage F G f"
lemma (in Group) Group_Gp:"G » H ==> Group (\<natural> H)"
apply (simp add:Group_def Gp_def)
apply (simp add:sg_tassoc sg_l_i sg_unit_closed sg_l_unit)
apply (rule conjI,
rule univar_func_test, rule ballI, rule univar_func_test, rule ballI,
simp add:sg_mult_closed)
apply (rule univar_func_test, rule ballI, simp add:sg_i_closed)
done
lemma (in Group) Gp_carrier:"G » H ==> carrier (Gp G H) = H"
by (simp add:Gp_def)
lemma (in Group) sg_sg:"[|G » K; G » H; H ⊆ K|] ==> Gp G K » H"
apply (simp add:sg_def [of "Gp G K" "H"])
apply (rule conjI, simp add:sg_def)
apply (simp add:Gp_def)
apply ((rule ballI)+, simp add:sg_m_i_closed)
done
lemma (in Group) sg_subset_of_subG:"[|G » K; Gp G K » H|] ==> H ⊆ K"
by (simp add:sg_def[of "\<natural> K"], simp add:Gp_def)
lemma const_ghom:"[|Group F; Group G|] ==> 1F,G ∈ gHom F G"
apply (simp add:gHom_def constghom_def)
apply (simp add:Pi_def Group.unit_closed)
apply ((rule ballI)+,
cut_tac Group.unit_closed[of "G"],
simp add:Group.mult_closed Group.l_unit)
apply assumption
done
lemma (in Group) const_gbij:"gbij(\<natural> {\<one>}),(\<natural> {\<one>}) (1(\<natural>{\<one>}),(\<natural>{\<one>}))"
apply (cut_tac special_sg_e,
frule Group_Gp[of "{\<one>}"],
frule const_ghom[of "\<natural> {\<one>}" "\<natural> {\<one>}"], assumption)
apply (simp add:gbijec_def)
apply (rule conjI, simp add:gsurjec_def,
simp add:surj_to_def Gp_def constghom_def)
apply (simp add:ginjec_def inj_on_def Gp_def)
done
lemma (in Group) unit_Groups_isom:" (\<natural> {\<one>}) ≅ (\<natural> {\<one>})"
apply (unfold isomorphic_def,
cut_tac const_gbij, blast)
done
lemma Ugp_const_gHom:"[|Ugp G; Ugp E|] ==> (λx∈carrier G. \<one>E) ∈ gHom G E"
apply (simp add:gHom_def)
apply (rule conjI)
apply (rule univar_func_test, rule ballI)
apply (simp add:Group.unit_closed[of "E"] Ugp_def)
apply (simp add:Ugp_def, (erule conjE)+)
apply (cut_tac Group.unit_closed[of "G"], cut_tac Group.unit_closed[of "E"])
apply (simp add:Group.l_unit) apply assumption+
done
lemma Ugp_const_gbij:"[|Ugp G; Ugp E|] ==> gbijG,E (λx∈carrier G. \<one>E)"
apply (simp add:gbijec_def)
apply (simp add:gsurjec_def ginjec_def)
apply (simp add:Ugp_const_gHom)
apply (rule conjI)
apply (simp add:surj_to_def, simp add:Ugp_def)
apply (simp add:inj_on_def)
apply ((rule ballI)+, simp add:Ugp_def)
done
lemma Ugps_isomorphic:"[|Ugp G; Ugp E|] ==> G ≅ E"
apply (simp add:isomorphic_def)
apply (frule_tac Ugp_const_gbij[of "G" "E"], assumption+)
apply blast
done
lemma (in Group) Gp_mult_induced:"[|G » L; a ∈ L; b ∈ L|] ==>
a ·(Gp G L) b = a · b"
by (simp add:Gp_def)
lemma (in Group) sg_i_induced:"[|G » L; a ∈ L|] ==> ρ(Gp G L) a = ρ a"
by (simp add:Gp_def)
lemma (in Group) Gp_mult_induced1:"[|G » H ; G » K; a ∈ H ∩ K; b ∈ H ∩ K|]
==> a ·\<natural>(H ∩ K) b = a ·(\<natural>H) b"
by (simp add:Gp_def)
lemma (in Group) Gp_mult_induced2:"[|G » H ; G » K; a ∈ H ∩ K; b ∈ H ∩ K|]
==> a ·\<natural>(H ∩ K) b = a ·(\<natural>K) b"
by (simp add:Gp_def)
lemma (in Group) sg_i_induced1:"[|G » H ; G » K; a ∈ H ∩ K|]
==> ρ\<natural>(H ∩ K) a = ρ(\<natural>H) a"
by (simp add:Gp_def)
lemma (in Group) sg_i_induced2:"[|G » H ; G » K; a ∈ H ∩ K|]
==> ρ\<natural>(H ∩ K) a = ρ\<natural>K a"
by (simp add:Gp_def)
lemma (in Group) subg_sg_sg:"[|G » K; (Gp G K) » H |] ==> G » H"
apply (frule sg_subset_of_subG[of "K" "H"], assumption+,
simp add:sg_def [of _ "H"])
apply (simp add:Gp_def[of _ "K"],
frule sg_subset[of "K"], simp add:subset_trans[of "H" "K" "carrier G"])
done
lemma (in Group) Gp_inherited:"[|G » K; G » L; K ⊆ L|] ==>
Gp (Gp G L) K = Gp G K"
by (simp add:Gp_def)
section "3. Cosets"
(* left cosets *)
lemma (in Group) mem_lcs:"[|G » H; a ∈ carrier G; x ∈ a ♦ H|] ==>
x ∈ carrier G"
by (simp add: lcs_def, erule bexE,
frule_tac h = h in sg_mult_closedr[of "H" "a"], assumption+, simp)
lemma (in Group) lcs_subset:"[|G » H; a ∈ carrier G|] ==> a ♦ H ⊆ carrier G"
apply (simp add:lcs_def,
rule subsetI, simp, erule bexE)
apply (frule_tac h = h in sg_subset_elem[of "H"], assumption+,
frule_tac a = a and b = h in mult_closed, assumption+)
apply simp
done
lemma (in Group) a_in_lcs:"[|G » H; a ∈ carrier G|] ==> a ∈ a ♦ H"
apply (simp add: lcs_def,
rule bexI [of _ "\<one>"],
subst r_unit, assumption+, simp)
apply (simp add:sg_unit_closed)
done
lemma (in Group) eq_lcs1:"[|G » H; a ∈ carrier G; b ∈ carrier G;
x ∈ a ♦ H; a ♦ H = b ♦ H|] ==> x ∈ b ♦ H"
by simp
lemma (in Group) eq_lcs2:"[|G » H; a ∈ carrier G; b ∈ carrier G;
a ♦ H = b ♦ H|] ==> a ∈ b ♦ H"
by (frule a_in_lcs[of "H" "a"], assumption+, simp)
lemma (in Group) lcs_mem_ldiv:"[|G » H; a ∈ carrier G; b ∈ carrier G|] ==>
(a ∈ b ♦ H) = ((ρ b) · a ∈ H)"
apply (rule iffI)
apply (simp add: lcs_def, erule bexE)
apply (frule_tac x = h in sol_eq_l[of "b" "a"], assumption+,
simp add:sg_subset_elem[of "H"], assumption+)
apply (thin_tac "b · h = a", simp)
apply (frule_tac x = "(ρ b) · a" and A = H in inEx,
erule bexE,
frule_tac h = y in sg_subset_elem[of "H"], assumption+,
frule sg_subset_elem[of "H" "(ρ b) · a"], assumption+)
apply (frule_tac a = y in l_mult_eqn[of _ "(ρ b) · a" "b"], assumption+,
frule i_closed[of "b"],
thin_tac "y = ρ b · a",
simp add:tassoc[THEN sym], simp add:r_i l_unit)
apply (rotate_tac -2, frule sym, thin_tac "b · y = a", simp add:lcs_def,
blast)
done
(*
lemma (in Group) lcsTr4:"[|G » H; a ∈ carrier G; b ∈ carrier G;
x ∈ carrier G|] ==> (i ((i a) · b)) · ((i a) · x) = (i b) · x"
apply (frule i_closed[of "a"], frule i_closed[of "b"],
simp add:i_ab[of "i a" "b"])
apply (frule mult_closed[of "i a" "x"], assumption+,
simp add:iop_i_i,
simp add:tassoc[of "i b" "a" "(i a) · x"],
simp add:tassoc[THEN sym, of "a" "i a" "x"] r_i l_unit)
done *)
lemma (in Group) lcsTr5:"[|G » H; a ∈ carrier G; b ∈ carrier G;
(ρ a) · b ∈ H; x ∈ a ♦ H|] ==> ((ρ b) · x) ∈ H"
apply (frule mem_lcs[of "H" "a" "x"], assumption+,
subst lcs_mem_ldiv[THEN sym, of "H" "x" "b"], assumption+,
simp add:lcs_def, erule bexE)
apply (frule_tac h = h in sg_subset_elem[of "H"], assumption+,
frule_tac x = h in sol_eq_l[of "a" "x"], assumption+,
frule sg_i_m_closed[of "H" "(ρ a) · b" "(ρ a) · x"], assumption+,
rotate_tac -1, frule sym, thin_tac "h = ρ a · x", simp)
apply (frule i_closed[of "a"],
simp add:i_ab iop_i_i, frule i_closed[of "b"],
simp add:tassoc[of "ρ b" "a"],
simp add:tassoc[THEN sym, of "a" "ρ a" "x"] r_i l_unit)
apply (frule inEx[of "(ρ b) · x"], erule bexE,
rotate_tac -1, frule sym, thin_tac "y = ρ b · x",
frule_tac b = y in sol_eq_l[of "ρ b" _ "x"],
simp add:sg_subset_elem, assumption+,
simp add:iop_i_i, blast)
done
lemma (in Group) lcsTr6:"[|G » H; a ∈ carrier G; b ∈ carrier G;
(ρ a) · b ∈ H; x ∈ a ♦ H|] ==> x ∈ b ♦ H"
by (frule lcsTr5[of "H" "a" "b" "x"], assumption+,
subst lcs_mem_ldiv, assumption+, rule mem_lcs, assumption+)
lemma (in Group) lcs_Unit1:"G » H ==> \<one> ♦ H = H"
apply (rule equalityI)
apply (rule subsetI, simp add:lcs_def, erule bexE,
frule_tac h = h in sg_subset_elem[of "H"], assumption+,
simp add:l_unit)
apply (rule subsetI,
simp add:lcs_def,
frule_tac h = x in sg_subset_elem[of "H"], assumption+,
frule_tac a = x in l_unit)
apply blast
done
lemma (in Group) lcs_Unit2:"[|G » H; h ∈ H|] ==> h ♦ H = H"
apply (rule equalityI)
apply (rule subsetI, simp add:lcs_def, erule bexE,
frule_tac x = h and y = ha in sg_mult_closed, assumption+, simp)
apply (rule subsetI,
simp add:lcs_def,
rule_tac x = "(ρ h) · x" in bexI[of _ _ "H"])
apply (frule sg_subset_elem[of "H" "h"], assumption+,
frule i_closed[of "h"],
frule_tac h = x in sg_subset_elem[of "H"], assumption+,
simp add:tassoc[THEN sym, of "h" "ρ h"] r_i l_unit)
apply (simp add:sg_i_m_closed[of "H" "h"])
done
lemma (in Group) lcsTr7:"[|G » H; a ∈ carrier G; b ∈ carrier G; (ρ a) · b ∈ H|]
==> a ♦ H ⊆ b ♦ H"
apply (rule subsetI)
apply (simp add:lcsTr6 [of "H" "a" "b" _])
done
lemma (in Group) lcsTr8:"[|G » H; a ∈ carrier G; h ∈ H|] ==> a · h ∈ a ♦ H"
apply (simp add:lcs_def)
apply blast
done
lemma (in Group) lcs_tool1:"[|G » H; a ∈ carrier G; b ∈ carrier G;
(ρ a) · b ∈ H|] ==> (ρ b) · a ∈ H"
by (frule sg_i_closed [of "H" "(ρ a) · b"], assumption+,
frule i_closed[of "a"], simp add:i_ab iop_i_i)
theorem (in Group) lcs_eq:"[|G » H; a ∈ carrier G; b ∈ carrier G|] ==>
((ρ a) · b ∈ H) = (a ♦ H = b ♦ H)"
apply (rule iffI)
apply ((rule equalityI),
(rule lcsTr7 [of "H" "a" "b"], assumption+),
(frule lcs_tool1 [of "H" "a" "b"], assumption+),
(rule lcsTr7 [of "H" "b" "a"], assumption+))
apply (subst lcs_mem_ldiv[THEN sym, of "H" "b" "a"], assumption+,
simp add:a_in_lcs[of "H" "b"])
done
lemma (in Group) rcs_subset:"[|G » H; a ∈ carrier G|] ==> H • a ⊆ carrier G"
apply (rule subsetI,
simp add:rcs_def, erule bexE,
frule_tac h = h in sg_mult_closedl[of "H" "a"], assumption+)
apply simp
done
lemma (in Group) mem_rcs:"[|G » H; x ∈ H • a|] ==> ∃h∈H. h · a = x"
by (simp add:rcs_def)
lemma (in Group) rcs_subset_elem:"[|G » H; a ∈ carrier G; x ∈ H • a|] ==>
x ∈ carrier G"
apply (simp add:rcs_def, erule bexE)
apply (frule_tac h = h in sg_mult_closedl[of "H" "a"], assumption+,
simp)
done
lemma (in Group) rcs_in_set_rcs:"[|G » H; a ∈ carrier G|] ==>
H • a ∈ set_rcs G H"
apply (simp add:set_rcs_def)
apply blast
done
lemma (in Group) rcsTr0:"[|G » H; a ∈ carrier G; b ∈ carrier G|] ==>
H • (a · b) ∈ set_rcs G H"
apply (rule rcs_in_set_rcs [of "H" "a · b"], assumption)
apply (simp add:mult_closed)
done
lemma (in Group) a_in_rcs:"[|G » H; a ∈ carrier G|] ==> a ∈ H • a"
apply (simp add: rcs_def)
apply (frule l_unit[of "a"],
frule sg_unit_closed[of "H"], blast)
done
lemma (in Group) rcs_nonempty:"[|G » H; X ∈ set_rcs G H|] ==> X ≠ {}"
apply (simp add:set_rcs_def, erule bexE)
apply (frule_tac a = a in a_in_rcs[of "H"], assumption+, simp)
apply (simp add:nonempty)
done
lemma (in Group) rcs_tool0:"[|G » H; a ∈ carrier G; b ∈ carrier G;
a · (ρ b) ∈ H|] ==> b · (ρ a) ∈ H"
by (frule sg_i_closed [of "H" "a · ( ρ b)"], assumption+,
frule i_closed[of "b"], simp add:i_ab iop_i_i)
lemma (in Group) rcsTr1:"[|G » H; a ∈ carrier G; b ∈ carrier G;
x ∈ H • a; H • a = H • b|] ==> x ∈ H • b"
by simp
lemma (in Group) rcs_eqTr:"[|G » H; a ∈ carrier G; b ∈ carrier G;
H • a = H • b|] ==> a ∈ H • b"
apply (rule rcsTr1, assumption+)
apply (rule a_in_rcs, assumption+)
done
lemma (in Group) rcs_eqTr1:"[|G » H; a ∈ carrier G; b ∈ carrier G|] ==>
(a ∈ H • b) = (a · (ρ b) ∈ H)"
apply (rule iffI)
apply (simp add:rcs_def, erule bexE,
frule_tac h = h in sg_subset_elem[of "H"], assumption+)
apply (frule_tac x = h in sol_eq_r[of "b" "a" _], assumption+, simp)
apply (frule inEx[of "a · (ρ b)" "H"], erule bexE)
apply (frule_tac h = y in sg_subset_elem[of "H"], assumption+,
frule i_closed[of "b"])
apply (rotate_tac -3, frule sym, thin_tac "y = a · ρ b",
frule_tac b = y in sol_eq_r[of "ρ b" _ "a"], assumption+,
simp add:iop_i_i)
apply (simp add:rcs_def, blast)
done
lemma (in Group) rcsTr2:"[|G » H; a ∈ carrier G; b ∈ H • (ρ a)|] ==>
b · a ∈ H"
apply (frule i_closed[of "a"],
frule_tac rcs_subset_elem[of "H" "ρ a" "b"], assumption+,
frule rcs_eqTr1[of "H" "b" "ρ a"], assumption+)
apply (simp add:iop_i_i)
done
lemma (in Group) rcsTr5:"[|G » H; a ∈ carrier G; b ∈ carrier G;
b · (ρ a) ∈ H; x ∈ H • a|] ==> x · (ρ b) ∈ H"
apply (frule rcs_subset_elem[of "H" "a" "x"], assumption+,
simp add:rcs_def, erule bexE)
apply (frule_tac h = h in sg_subset_elem[of "H"], assumption,
frule_tac a = h and b = a in mult_closed, assumption+,
frule i_closed[of "b"])
apply (frule_tac a = "h · a" in r_mult_eqn[of _ "x" "ρ b"], assumption+,
thin_tac "h · a = x",
simp add:tassoc[of _ "a" "ρ b"],
frule_tac x = h in sg_mult_closed[of "H" _ "a · (ρ b)"], assumption+,
rule rcs_tool0[of "H" "b" "a"], assumption+)
apply simp
done
lemma (in Group) rcsTr6:"[|G » H; a ∈ carrier G; b ∈ carrier G;
b · (ρ a) ∈ H ; x ∈ H • a|] ==> x ∈ H • b"
apply (frule rcsTr5 [of "H" "a" "b" "x"], assumption+)
apply (subst rcs_eqTr1[of "H" "x" "b"], assumption+)
apply (rule rcs_subset_elem[of "H" "a" "x"], assumption+)
done
lemma (in Group) rcs_Unit1:"G » H ==> H • \<one> = H"
apply (rule equalityI)
apply (rule subsetI,
simp add:rcs_def, erule bexE,
frule_tac h = h in sg_subset_elem[of "H"], assumption+,
simp add:r_unit)
apply (rule subsetI)
apply (simp add:rcs_def,
frule_tac h = x in sg_subset_elem[of "H"], assumption+,
frule_tac a = x in r_unit, blast)
done
lemma (in Group) unit_rcs_in_set_rcs:"G » H ==> H ∈ set_rcs G H"
apply (cut_tac unit_closed)
apply (frule rcs_in_set_rcs[of "H" "\<one>"], assumption+)
apply (simp add:rcs_Unit1)
done
lemma (in Group) rcs_Unit2:"[|G » H; h ∈ H|] ==> H • h = H"
apply (rule equalityI)
apply (rule subsetI,
simp add:rcs_def, erule bexE,
frule_tac x = ha and y = h in sg_mult_closed[of "H"], assumption+,
simp)
apply (rule subsetI,
simp add:rcs_def)
apply (frule_tac h = h in sg_subset_elem[of "H"], assumption+,
frule_tac h = x in sg_subset_elem[of "H"], assumption+,
frule i_closed[of "h"],
frule_tac a = x in tassoc[of _ "ρ h" "h"], assumption+,
simp add:l_i r_unit)
apply (frule_tac a = x in sg_m_i_closed[of "H" _ "h"], assumption+,
blast)
done
lemma (in Group) rcsTr7:"[|G » H; a ∈ carrier G; b ∈ carrier G; b · (ρ a) ∈ H|]
==> H • a ⊆ H • b"
apply (rule subsetI)
apply (rule rcsTr6[of "H" "a" "b"], assumption+)
done
lemma (in Group) rcs_tool1:"[|G » H; a ∈ carrier G; b ∈ carrier G;
b · (ρ a) ∈ H|] ==> a · (ρ b) ∈ H "
apply (frule sg_i_closed[of "H" "b · (ρ a)"], assumption+)
apply (frule i_closed[of "a"], simp add:i_ab iop_i_i)
done
lemma (in Group) rcs_tool2:"[|G » H; a ∈ carrier G; x ∈ H • a|] ==>
∃ h ∈ H. h · a = x"
apply (simp add:rcs_def)
done
theorem (in Group) rcs_eq:"[|G » H; a ∈ carrier G; b ∈ carrier G|] ==>
(b · (ρ a) ∈ H) = (H • a = H • b)"
apply (rule iffI)
apply (rule equalityI)
apply (frule rcsTr7[of "H" "a" "b"], assumption+)
apply (frule rcs_tool1[of "H" "a" "b"], assumption+)
apply (rule rcsTr7[of "H" "b" "a"], assumption+)
apply (frule a_in_rcs[of "H" "a"], assumption, simp)
apply (simp add:rcs_eqTr1[of "H" "a" "b"])
apply (rule rcs_tool1, assumption+)
done
lemma (in Group) rcs_eq1:"[|G » H; a ∈ carrier G; x ∈ H • a|] ==>
H • a = H • x"
apply (frule rcs_subset_elem[of "H" "a" "x"], assumption+)
apply (subst rcs_eq[THEN sym, of "H" "a" "x"], assumption+)
apply (subst rcs_eqTr1[THEN sym, of "H" "x" "a"], assumption+)
done
lemma (in Group) rcs_eq2:"[|G » H; a ∈ carrier G; b ∈ carrier G;
(H • a) ∩ (H • b) ≠ {}|] ==> (H • a) = (H • b)"
apply (frule nonempty_int [of "H • a" "H • b"], erule exE)
apply (simp, erule conjE)
apply (frule_tac x = x in rcs_eq1[of "H" "a"], assumption+,
frule_tac x = x in rcs_eq1[of "H" "b"], assumption+)
apply simp
done
lemma (in Group) rcs_meet:"[|G » H; X ∈ set_rcs G H; Y ∈ set_rcs G H;
X ∩ Y ≠ {}|] ==> X = Y"
apply (simp add:set_rcs_def, (erule bexE)+, simp)
apply (rule_tac a = a and b = aa in rcs_eq2[of "H"], assumption+)
done
lemma (in Group) rcsTr8:"[|G » H; a ∈ carrier G; h ∈ H; x ∈ H • a|] ==>
h · x ∈ H • a"
apply (frule rcs_subset_elem[of "H" "a" "x"], assumption+,
frule sg_subset_elem[of "H" "h"], assumption+)
apply (simp add:rcs_def, erule bexE,
frule_tac h = ha in sg_subset_elem[of "H"], assumption+,
frule_tac a = ha and b = a in mult_closed, assumption,
frule_tac a = "ha · a" and b = x and c = h in l_mult_eqn, assumption+,
thin_tac "ha · a = x", simp add:tassoc[THEN sym, of "h" _ "a"])
apply (frule_tac x = h and y = ha in sg_mult_closed[of "H"], assumption+,
blast)
done
lemma (in Group) rcsTr9:"[|G » H; a ∈ carrier G; h ∈ H; (ρ x) ∈ H • a|] ==>
h · (ρ x) ∈ H • a"
by (insert rcsTr8 [of "H" "a" "h" "ρ x"], simp)
lemma (in Group) rcsTr10:"[|G » H; a ∈ carrier G; x ∈ H • a; y ∈ H • a|] ==>
x · (ρ y) ∈ H"
apply (simp add:rcs_def)
apply (erule bexE)+
apply (rotate_tac -1, frule sym, thin_tac "ha · a = y",
frule sym, thin_tac "h · a = x", simp,
thin_tac "y = ha · a", thin_tac "x = h · a")
apply (frule_tac h = ha in sg_subset_elem[of "H"], assumption+,
frule_tac h = h in sg_subset_elem[of "H"], assumption+,
simp add:i_ab)
apply (frule_tac a = a in i_closed, frule_tac a = ha in i_closed,
simp add:tOp_assocTr41[THEN sym], simp add:tOp_assocTr42,
simp add:r_i r_unit)
apply (simp add:sg_m_i_closed[of "H"])
done
lemma (in Group) PrSubg4_2:"[|G » H; a ∈ carrier G; x ∈ H • (ρ a)|] ==>
x ∈ {z. ∃v∈(H • a). ∃h∈H. h · (ρ v) = z}"
apply (simp add:rcs_def[of _ "H" "ρ a"], erule bexE,
frule_tac h = h in sg_subset_elem[of "H"], assumption+,
frule i_closed[of "a"])
apply (frule a_in_rcs[of "H" "a"], assumption, blast)
done
lemma (in Group) rcs_fixed:"[|G » H; a ∈ carrier G; H • a = H|] ==> a ∈ H"
by (frule a_in_rcs[of "H" "a"], assumption+, simp)
lemma (in Group) rcs_fixed1:"[|G » H; a ∈ carrier G; h ∈ H|] ==>
H • a = (H • (h · a))"
apply (rule rcs_eq1[of "H" "a" "h · a"], assumption+)
apply (simp add:rcs_def, blast)
done
lemma (in Group) rcs_fixed2:"G » H ==> ∀h∈H. H • h = H"
apply (rule ballI)
apply (simp add:rcs_Unit2)
done
lemma (in Group) Gp_rcs:"[|G » H; G » K; H ⊆ K; x ∈ K|] ==>
H •(Gp G K) x = (H • x)"
by (simp add:rcs_def, simp add:Gp_def)
lemma (in Group) subg_lcs:"[|G » H; G » K; H ⊆ K; x ∈ K|] ==>
x ♦(Gp G K) H = x ♦ H"
by (simp add:lcs_def, simp add:Gp_def)
section "4. Normal subgroups and Quotient groups"
lemma (in Group) nsg1:"[|G » H; b ∈ carrier G; h ∈ H;
∀a∈ carrier G. ∀h∈H. (a · h)· (ρ a) ∈ H|] ==> b · h · (ρ b) ∈ H"
by blast
lemma (in Group) nsg2:"[|G » H; b ∈ carrier G; h ∈ H;
∀a∈carrier G. ∀h∈H. (a · h) · (ρ a) ∈ H|] ==> (ρ b) · h · b ∈ H"
apply (frule i_closed[of "b"],
frule_tac b = "ρ b" in forball_spec1, assumption,
thin_tac "∀a∈carrier G. ∀h∈H. a · h · ρ a ∈ H",
frule_tac b = h in forball_spec1, assumption,
thin_tac "∀h∈H. ρ b · h · ρ (ρ b) ∈ H")
apply (simp add:iop_i_i)
done
lemma (in Group) nsg_subset_elem:"[|G \<triangleright> H; h ∈ H|] ==> h ∈ carrier G"
by (insert nsg_sg[of "H"], simp add:sg_subset_elem)
lemma (in Group) nsg_l_rcs_eq:"[|G \<triangleright> N; a ∈ carrier G|] ==> a ♦ N = N • a"
by (simp add: nsg_def)
lemma (in Group) sg_nsg1:"[|G » H; ∀a∈ carrier G. ∀h∈H. (a · h)· (ρ a) ∈ H;
b ∈ carrier G |] ==> H • b = b ♦ H"
apply (rule equalityI)
(* H • b ⊆ b ♦ H *)
apply (rule subsetI, simp add:rcs_def, erule bexE, frule i_closed[of "b"])
apply (frule_tac b = "ρ b" in forball_spec1, assumption,
thin_tac "∀a∈carrier G. ∀h∈H. a · h · ρ a ∈ H",
frule_tac b = h in forball_spec1, assumption,
thin_tac "∀h∈H. ρ b · h · ρ (ρ b) ∈ H",
simp add:iop_i_i)
apply (frule_tac h = h in sg_mult_closedl[of "H" "b"], assumption+, simp,
frule_tac h = h in sg_subset_elem[of "H"], assumption+,
simp only:tassoc[of "ρ b" _ "b"], thin_tac "h · b = x")
apply (subst lcs_mem_ldiv[of "H" _ "b"], assumption+)
apply (rule subsetI, simp add:lcs_def, erule bexE)
apply (frule_tac b = b in forball_spec1, assumption,
thin_tac "∀a∈carrier G. ∀h∈H. a · h · ρ a ∈ H",
frule_tac b = h in forball_spec1, assumption,
thin_tac "∀h∈H. b · h · (ρ b) ∈ H",
simp add:iop_i_i)
apply (frule_tac h = h in sg_mult_closedr[of "H" "b"], assumption+, simp)
apply (subst rcs_eqTr1[of "H" _ "b"], assumption+)
done
lemma (in Group) cond_nsg:"[|G » H; ∀a∈carrier G. ∀h∈H. a · h · (ρ a) ∈ H |]
==> G \<triangleright> H"
apply (subst nsg_def, simp)
apply (rule ballI, rule sg_nsg1, assumption+)
done
lemma (in Group) special_nsg_e:"G » H ==> Gp G H \<triangleright> {\<one>}"
apply (simp add:nsg_def,
frule Group_Gp[of "H"])
apply (rule conjI)
apply (frule Group.special_sg_e[of "\<natural>H"],
simp add:one_Gp_one[THEN sym])
apply (rule ballI,
simp add:lcs_def rcs_def, simp add:Gp_def,
frule_tac h = x in sg_subset_elem[of "H"], assumption+,
simp add:l_unit r_unit)
done
lemma (in Group) special_nsg_G:"G \<triangleright> (carrier G)"
apply (simp add:nsg_def,
simp add:special_sg_G)
apply (rule ballI, rule equalityI)
apply (rule subsetI,
simp add:rcs_def lcs_def, erule bexE)
apply (frule_tac a = h and b = x in mult_closed, assumption+,
frule_tac a = x in i_closed,
frule_tac a1 = x and b1 = "ρ x" and c1 = "h · x" in tassoc[THEN sym],
assumption+, simp add:r_i l_unit,
frule_tac a = "ρ x" and b = xa in mult_closed, assumption+, blast)
apply (rule subsetI, simp add:rcs_def lcs_def, erule bexE,
frule_tac a = x and b = h in mult_closed, assumption+,
frule_tac a = x in i_closed,
frule_tac a = "x · h" and b = "ρ x" and c = x in tassoc,
assumption+, simp add:l_i r_unit,
frule_tac a = xa and b = "ρ x" in mult_closed, assumption+,
blast)
done
lemma (in Group) special_nsg_G1:"G » H ==> Gp G H \<triangleright> H"
apply (frule Group_Gp[of "H"], frule Group.special_nsg_G[of "\<natural>H"])
apply (simp add:Gp_carrier)
done
lemma (in Group) nsgTr0:"[|G \<triangleright> N; a ∈ carrier G; b ∈ carrier G; b ∈ N • a |]
==> (a · (ρ b) ∈ N) ∧ ((ρ a) · b ∈ N)"
apply (frule nsg_sg[of "N"],
frule rcs_eqTr1[of "N" "b" "a"], assumption+, simp,
frule i_closed[of "a"],
frule sg_i_closed[of "N" "b · ρ a"], assumption, simp add:i_ab iop_i_i)
apply (simp add:nsg_l_rcs_eq[THEN sym, of "N" "a"],
subst lcs_mem_ldiv[THEN sym, of "N" "b" "a"], assumption+)
done
lemma (in Group) nsgTr1:"[|G \<triangleright> N; a ∈ carrier G; b ∈ carrier G; b · (ρ a) ∈ N|]
==> (ρ b) · a ∈ N"
apply (frule nsg_sg[of "N"],
simp add:rcs_eqTr1[THEN sym, of "N" "b" "a"],
simp add:nsg_l_rcs_eq[THEN sym])
apply (simp add:lcs_mem_ldiv[of "N" "b" "a"],
rule lcs_tool1, assumption+)
done
lemma (in Group) nsgTr2:"[|a ∈ carrier G; b ∈ carrier G; a1 ∈ carrier G;
b1 ∈ carrier G |] ==> (a · b) · (ρ (a1 · b1)) =
a · (((b · (ρ b1)) · ((ρ a1) · a)) · (ρ a))"
apply (frule i_closed[of "a1"], frule i_closed[of "b1"],
frule i_closed[of "a"],
frule mult_closed[of "b" "ρ b1"], assumption+,
frule mult_closed[of "ρ a1" "a"], assumption+,
subst tassoc[of "b · (ρ b1)" "(ρ a1) · a" "ρ a"], assumption+,
simp add:tassoc[of "ρ a1" "a" "ρ a"] r_i r_unit)
apply (simp add:i_ab,
subst tassoc[of "a" "b" "(ρ b1) · (ρ a1)"], assumption+,
rule mult_closed, assumption+,
simp add:tassoc[THEN sym, of "b" "ρ b1" "ρ a1"])
done
lemma (in Group) nsgPr1:"[|G \<triangleright> N; a ∈ carrier G; h ∈ N|] ==>
a · (h · (ρ a)) ∈ N"
apply (frule nsg_sg[of "N"],
frule sg_subset_elem[of "N" "h"], assumption+,
frule i_closed[of "a"],
frule rcs_fixed1[THEN sym, of "N" "ρ a" "h"], assumption+,
frule mult_closed[of "h" "ρ a"], assumption+)
apply (frule a_in_rcs[of "N" "h · (ρ a)"], assumption+, simp,
thin_tac "N • (h · ρ a) = N • ρ a")
apply (simp add:nsg_l_rcs_eq[THEN sym, of "N" "ρ a"],
simp add:lcs_mem_ldiv[of "N" "h · (ρ a)" "ρ a"] iop_i_i)
done
lemma (in Group) nsgPr1_1:"[|G \<triangleright> N; a ∈ carrier G ; h ∈ N|] ==>
(a · h) · (ρ a) ∈ N"
apply (frule nsgPr1[of "N" "a" "h"], assumption+)
apply (frule nsg_sg[of "N"],
frule sg_subset_elem[of "N" "h"], assumption+,
frule i_closed[of "a"],
simp add:tassoc[THEN sym, of "a" "h" "ρ a"])
done
lemma (in Group) nsgPr2:"[|G \<triangleright> N; a ∈ carrier G; h ∈ N|] ==>
(ρ a) · (h · a) ∈ N"
apply (frule i_closed[of "a"],
frule nsgPr1[of "N" "ρ a" "h"], assumption+)
apply (simp add:iop_i_i)
done
lemma (in Group) nsgPr2_1:"[|G \<triangleright> N; a ∈ carrier G; h ∈ N|] ==>
(ρ a) · h · a ∈ N"
apply (frule i_closed[of "a"],
frule nsgPr1_1[of "N" "ρ a" "h"], assumption+, simp add:iop_i_i)
done
lemma (in Group) nsgTr3:"[|G \<triangleright> N; a ∈ carrier G; b ∈ carrier G;
a1 ∈ carrier G; b1 ∈ carrier G; a · (ρ a1) ∈ N; b · (ρ b1) ∈ N|] ==>
(a · b) · (ρ (a1 · b1)) ∈ N"
apply (frule nsg_sg[of "N"])
apply (frule nsgTr2 [of "a" "b" "a1" "b1"], assumption+)
apply (frule nsgTr1 [of "N" "a1" "a"], assumption+)
apply (frule i_closed[of "a"],
frule sg_i_closed [of "N" "(ρ a) · a1"], assumption+,
simp add:i_ab[of "ρ a" "a1"] iop_i_i,
frule sg_mult_closed[of "N" "b · (ρ b1)" "(ρ a1) · a"], assumption+)
apply (rule nsgPr1[of "N" "a" "b · (ρ b1) · ((ρ a1) · a)"], assumption+)
done
lemma (in Group) nsg_in_Gp:"[|G \<triangleright> N; G » H; N ⊆ H|] ==> (Gp G H) \<triangleright> N"
apply (frule Group_Gp [of "H"],
frule nsg_sg[of "N"])
apply (rule Group.cond_nsg [of "\<natural>H" "N"], assumption+,
simp add:sg_sg[of "H" "N"])
apply (rule ballI, rule ballI,
(subst Gp_def)+, simp add:Gp_carrier)
apply (frule_tac h = a in sg_subset_elem[of "H"], assumption+,
rule_tac a = a and h = h in nsgPr1_1[of "N"], assumption+)
done
lemma (in Group) nsgTr4:"[|G \<triangleright> N; a ∈ carrier G; x ∈ N • a|] ==>
(ρ x) ∈ N • (ρ a)"
apply (frule nsgTr0 [of "N"], assumption+)
apply (frule nsg_sg[of "N"], rule a_in_rcs[of "N"], assumption+,
thin_tac "a · ρ a ∈ N ∧ ρ a · a ∈ N",
frule nsg_sg[of "N"],
frule rcs_subset_elem[of "N" "a" "x"], assumption+,
simp add:rcs_eqTr1[of "N" "x" "a"])
apply (frule nsgTr1[of "N" "a" "x"], assumption+,
frule i_closed[of "x"],
subst rcs_eqTr1[of "N" "ρ x" "ρ a"], assumption+,
simp add:i_closed, simp add:iop_i_i)
done
lemma (in Group) c_topTr1:"[|G \<triangleright> N; a ∈ carrier G; b ∈ carrier G;
a1 ∈ carrier G; b1 ∈ carrier G; N • a = N • a1; N • b = N • b1|] ==>
N • (a · b) = N • (a1 · b1)"
apply (frule nsg_sg[of "N"],
frule mult_closed[of "a" "b"], assumption+,
frule mult_closed[of "a1" "b1"], assumption+,
simp add:rcs_eq[THEN sym, of "N" "a" "a1"],
simp add:rcs_eq[THEN sym, of "N" "b" "b1"])
apply (subst rcs_eq[THEN sym, of "N" "a · b" "a1 · b1"], assumption+)
apply (rule nsgTr3[of "N" "a1" "b1" "a" "b"], assumption+)
done
lemma (in Group) c_topTr2:"[|G \<triangleright> N; a ∈ carrier G; a1 ∈ carrier G;
N • a = N • a1 |] ==> N • (ρ a) = N • (ρ a1)"
apply (frule nsg_sg[of "N"],
simp add:rcs_eq[THEN sym, of "N" "a" "a1"])
apply (subst rcs_eq[THEN sym, of "N" "ρ a" "ρ a1"], assumption+,
simp add:i_closed, simp add:i_closed, simp add:iop_i_i)
apply (rule nsgTr1[of "N" "a" "a1"], assumption+)
done
lemma (in Group) c_iop_welldefTr1:"[|G \<triangleright> N; a ∈ carrier G|] ==>
c_iop G N (N • a) ⊆ N • (ρ a)"
apply (frule nsg_sg[of "N"],
frule i_closed[of "a"])
apply (rule subsetI)
apply (simp add:c_iop_def rcs_in_set_rcs)
apply (erule bexE)+
apply (simp add:rcs_def[of _ "N" "a"], erule bexE,
rotate_tac -1, frule sym, thin_tac "ha · a = xa", simp,
thin_tac "xa = ha · a")
apply (frule_tac h = ha in sg_subset_elem[of "N"], assumption+,
frule_tac a = ha and b = a in mult_closed, assumption+,
frule_tac a = "ha · a" in i_closed,
frule_tac h = h in sg_subset_elem[of "N"], assumption+,
frule_tac a = h and b = "ρ (ha · a)" in mult_closed, assumption+)
apply (frule_tac a = "h · ρ (ha · a)" and b = x in r_mult_eqn[of _ _
"a · (ρ a)"], simp, simp add:r_i unit_closed, assumption)
apply (frule_tac a = "h · ρ (ha · a)" in tassoc[of _ "a" "ρ a"], assumption+,
frule_tac a = "h · ρ (ha · a)" and A = "carrier G" and b = x in
eq_elem_in, assumption+,
thin_tac "h · ρ (ha · a) = x", simp,
thin_tac "h · ρ (ha · a) · (a · ρ a) = x · (a · ρ a)")
apply (frule_tac a = h and b = "ρ (ha · a)" and c = a in tassoc, assumption+,
simp, thin_tac "h · ρ (ha · a) · a = h · (ρ (ha · a) · a)")
apply (simp add:i_ab r_i r_unit)
apply (frule_tac x = ha in sg_i_closed[of "N"], assumption+,
frule sym, thin_tac "h · (ρ a · ρ ha · a) · ρ a = x", simp,
thin_tac "x = h · (ρ a · ρ ha · a) · ρ a")
apply (frule_tac h = "ρ ha" in nsgPr2_1[of "N" "a"], assumption+,
frule_tac x = h and y = "ρ a · ρ ha · a" in sg_mult_closed, assumption+)
apply (simp add:rcs_def, blast)
done
lemma (in Group) c_iop_welldefTr2:"[|G \<triangleright> N; a ∈ carrier G|] ==>
N • (ρ a) ⊆ c_iop G N (N • a)"
apply (rule subsetI)
apply (frule nsg_sg[of "N"],
frule i_closed[of "a"],
frule_tac x = x in rcs_subset_elem[of "N" "ρ a"], assumption+)
apply (simp add:c_iop_def,
simp add:rcs_in_set_rcs[of "N" "a"],
simp add:rcs_def[of _ "N" "ρ a"])
apply (frule a_in_rcs[of "N" "a"], assumption, blast)
done
lemma (in Group) c_iop_welldef:"[|G \<triangleright> N; a ∈ carrier G|] ==>
c_iop G N (N • a) = N • (ρ a)"
apply (rule equalityI)
apply (simp only:c_iop_welldefTr1[of "N" "a"])
apply (simp only:c_iop_welldefTr2[of "N" "a"])
done
lemma (in Group) c_top_welldefTr1:"[|G \<triangleright> N; a ∈ carrier G;
b ∈ carrier G; x ∈ N • a; y ∈ N • b|] ==> x · y ∈ N • (a · b)"
apply (frule nsg_sg[of "N"])
apply (frule_tac x = x in rcs_subset_elem[of "N" "a"], assumption+,
frule_tac x = y in rcs_subset_elem[of "N" "b"], assumption+,
frule rcs_eqTr1[of "N" "x" "a"], assumption+,
frule rcs_eqTr1[of "N" "y" "b"], assumption+, simp)
apply (frule_tac mult_closed[of "a" "b"], assumption+,
frule_tac mult_closed[of "x" "y"], assumption+)
apply (subst rcs_eqTr1[of "N" "x · y" "a · b"], assumption+,
rule nsgTr3[of "N" "x" "y" "a" "b"], assumption+)
done
lemma (in Group) c_top_welldefTr2:"[|G \<triangleright> N; a ∈ carrier G; b ∈ carrier G |]
==> c_top G N (N • a) (N • b) ⊆ N • (a · b)"
apply (frule nsg_sg[of "N"],
simp add:c_top_def, simp add:rcs_in_set_rcs)
apply (rule subsetI, simp, (erule bexE)+,
frule_tac x = xa and y = y in c_top_welldefTr1[of "N" "a" "b"],
assumption+, simp)
done
lemma (in Group) c_top_welldefTr4:"[|G \<triangleright> N; a ∈ carrier G; b ∈ carrier G;
x ∈ N • (a · b)|] ==> x ∈ c_top G N (N • a) (N • b)"
apply (frule nsg_sg[of "N"])
apply (simp add:c_top_def, simp add:rcs_in_set_rcs,
simp add:rcs_def[of _ "N" "a · b"], erule bexE,
frule_tac h = h in sg_subset_elem[of "N"], assumption+,
simp add:tassoc[THEN sym, of _ "a" "b"])
apply (frule a_in_rcs[of "N" "b"], assumption,
frule_tac h1 = h in rcs_fixed1[THEN sym, of "N" "a"], assumption+,
frule_tac a = h and b = a in mult_closed, assumption+,
frule_tac a = "h · a" in a_in_rcs[of "N"], assumption+, simp)
apply blast
done
lemma (in Group) c_top_welldefTr5:"[|G \<triangleright> N; a ∈ carrier G; b ∈ carrier G|] ==>
N • (a · b) ⊆ c_top G N (N • a) (N • b)"
by (rule subsetI,
rule c_top_welldefTr4[of "N" "a" "b" _], assumption+)
lemma (in Group) c_top_welldef:"[|G \<triangleright> N; a ∈ carrier G; b ∈ carrier G|] ==>
N • (a · b) = c_top G N (N • a) (N • b)"
by (rule equalityI, simp only:c_top_welldefTr5, simp only:c_top_welldefTr2)
lemma (in Group) Qg_unitTr:"[|G \<triangleright> N; a ∈ carrier G|] ==>
c_top G N N (N • a) = N • a"
apply (frule nsg_sg[of "N"])
apply (rule equalityI)
apply (rule subsetI, simp add:c_top_def)
apply (simp add:unit_rcs_in_set_rcs rcs_in_set_rcs)
apply (erule bexE)+
apply (simp add:rcs_def, erule bexE)
apply (frule sym, thin_tac "xa · y = x", frule sym, thin_tac "h · a = y",
simp, thin_tac "x = xa · (h · a)", thin_tac "y = h · a",
frule_tac h = xa in sg_subset_elem[of "N"], assumption+,
frule_tac h = h in sg_subset_elem[of "N"], assumption+,
simp add:tassoc[THEN sym],
frule_tac x = xa and y = h in sg_mult_closed[of "N"], assumption+,
blast)
apply (rule subsetI,
simp add:c_top_def, simp add:unit_rcs_in_set_rcs rcs_in_set_rcs)
apply (frule_tac x = x in mem_rcs [of "N" _ "a"], assumption, erule bexE,
frule a_in_rcs[of "N" "a"], assumption, blast)
done
lemma (in Group) Qg_unit:"G \<triangleright> N ==> ∀x∈set_rcs G N. c_top G N N x = x"
by (rule ballI,
simp add:set_rcs_def, erule bexE, simp,
simp add:Qg_unitTr)
lemma (in Group) Qg_iTr:"[|G \<triangleright> N; a ∈ carrier G|] ==>
c_top G N (c_iop G N (N • a)) (N • a) = N"
apply (simp add:c_iop_welldef [of "N" "a"])
apply (frule i_closed[of "a"],
simp add:c_top_welldef[THEN sym, of "N" "ρ a" "a"],
simp add:l_i)
apply (frule nsg_sg[of "N"],
simp add:rcs_Unit1[of "N"])
done
lemma (in Group) Qg_i:"G \<triangleright> N ==>
∀x ∈ set_rcs G N. c_top G N (c_iop G N x) x = N"
apply (rule ballI, simp add:set_rcs_def, erule bexE)
apply (simp add:Qg_iTr)
done
lemma (in Group) Qg_tassocTr:
"[|G \<triangleright> N; a ∈ carrier G; b ∈ carrier G; c ∈ carrier G |] ==>
c_top G N (N • a) (c_top G N (N • b) (N • c)) =
c_top G N (c_top G N (N • a) (N • b)) (N • c)"
apply (frule mult_closed[of "b" "c"], assumption+,
frule mult_closed[of "a" "b"], assumption+,
simp add:c_top_welldef[THEN sym])
apply (simp add:tassoc)
done
lemma (in Group) Qg_tassoc: "G \<triangleright> N ==>
∀X∈set_rcs G N. ∀Y∈set_rcs G N. ∀Z∈set_rcs G N. c_top G N X (c_top G N Y Z)
= c_top G N (c_top G N X Y) Z"
apply (rule ballI)+ apply (simp add:set_rcs_def, (erule bexE)+)
apply (simp add:Qg_tassocTr)
done
lemma (in Group) Qg_top:"G \<triangleright> N ==>
c_top G N : set_rcs G N -> set_rcs G N -> set_rcs G N"
apply (rule bivar_func_test,
(rule ballI)+, simp add:set_rcs_def, (erule bexE)+,
simp add:c_top_welldef[THEN sym])
apply (frule_tac a = aa and b = ab in mult_closed, assumption+, blast)
done
lemma (in Group) Qg_top_closed:"[|G \<triangleright> N; A ∈ set_rcs G N; B ∈ set_rcs G N|] ==>
c_top G N A B ∈ set_rcs G N"
apply (frule Qg_top[of "N"])
apply (frule funcset_mem [of "c_top G N" "set_rcs G N" _ "A"], assumption)
apply (rule funcset_mem[of "c_top G N A" "set_rcs G N" "set_rcs G N" "B"],
assumption, assumption)
done
lemma (in Group) Qg_iop: "G \<triangleright> N ==>
c_iop G N :set_rcs G N -> set_rcs G N"
apply (rule univar_func_test)
apply (rule ballI)
apply (simp add:set_rcs_def, erule bexE)
apply (simp add:c_iop_welldef)
apply (frule_tac a = a in i_closed, blast)
done
lemma (in Group) Qg_iop_closed:"[|G \<triangleright> N; A ∈ set_rcs G N|] ==>
c_iop G N A ∈ set_rcs G N"
by (frule Qg_iop[of "N"],
erule funcset_mem, assumption)
lemma (in Group) Qg_unit_closed: "G \<triangleright> N ==> N ∈ set_rcs G N"
by (frule nsg_sg[of "N"],
simp only:unit_rcs_in_set_rcs)
theorem (in Group) Group_Qg:"G \<triangleright> N ==> Group (Qg G N)"
apply (frule Qg_top [of "N"], frule Qg_iop [of "N"],
frule Qg_unit[of "N"], frule Qg_i[of "N" ],
frule Qg_tassoc[of "N"], frule Qg_unit_closed[of "N" ])
apply (simp add:Qg_def Group_def)
done
lemma (in Group) Qg_one:"G \<triangleright> N ==> one (G / N) = N"
by (simp add:Qg_def)
lemma (in Group) Qg_carrier:"carrier (G / (N::'a set)) = set_rcs G N"
by (simp add:Qg_def) (** only a trick *)
lemma (in Group) Qg_unit_group:"G \<triangleright> N ==>
(set_rcs G N = {N}) = (carrier G = N)"
apply (rule iffI)
apply (rule contrapos_pp, simp+,
frule nsg_sg[of "N"],
frule sg_subset[of "N"],
frule sets_not_eq[of "carrier G" "N"], assumption, erule bexE,
frule_tac a = a in rcs_in_set_rcs[of "N"], assumption+,
simp)
apply (frule_tac a = a in a_in_rcs[of "N"], assumption+,
simp)
apply (simp add:set_rcs_def, frule nsg_sg[of "N"],
frule rcs_fixed2[of "N"], frule_tac sg_unit_closed[of "N"], blast)
done
lemma (in Group) Gp_Qg:"G \<triangleright> N ==> Gp(G / N) (carrier(G / N)) = G / N"
by (simp add:Gp_def Qg_def)
lemma (in Group) Pj_hom0:"[|G \<triangleright> N; x ∈ carrier G; y ∈ carrier G|]
==> Pj G N (x · y) = (Pj G N x) ·(G / N) (Pj G N y)"
apply (simp add:Pj_def mult_closed)
apply (simp add:Qg_def, simp add:c_top_welldef)
done
lemma (in Group) Pj_ghom:"G \<triangleright> N ==> (Pj G N) ∈ gHom G (G / N)"
apply (simp add:gHom_def)
apply (rule conjI,
simp add:restrict_def Pj_def extensional_def)
apply (rule conjI, simp add:Pi_def,
rule allI, rule impI,
simp add:Qg_def set_rcs_def, simp add:Pj_def, blast)
apply ((rule ballI)+, simp add:Pj_hom0)
done
lemma (in Group) Pj_mem:"[|G \<triangleright> N; x ∈ carrier G|] ==> (Pj G N) x = N • x"
by (simp add:Pj_def)
lemma (in Group) Pj_gsurjec:"G \<triangleright> N ==> gsurjec G (G/N) (Pj G N)"
apply (simp add:gsurjec_def)
apply (simp add:Pj_ghom)
apply (rule surj_to_test[of "Pj G N" "carrier G" "carrier (G / N)"],
frule Pj_ghom[of "N"], simp add:gHom_def,
rule ballI,
simp add:Qg_def set_rcs_def, erule bexE)
apply (frule_tac x = a in Pj_mem[of "N"], assumption, simp, blast)
done
lemma (in Group) lcs_in_Gp:"[|G » H; G » K; K ⊆ H; a ∈ H|] ==>
a ♦ K = a ♦(Gp G H) K"
by (simp add:lcs_def, simp add:Gp_def)
lemma (in Group) rcs_in_Gp:"[|G » H; G » K; K ⊆ H; a ∈ H |] ==>
K • a = K •(Gp G H) a"
apply (simp add:rcs_def)
apply (simp add:Gp_def)
done
end
lemma less_and_segment:
b ∈ carrier D
==> (∀a. a \<prec> b ∧ a ∈ carrier D --> Q a) =
(∀a∈carrier (Iod D (segment D b)). Q a)
lemma Word_compare2:
[| Worder E;
¬ (∀a∈carrier D.
∃b∈carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b))) |]
==> ∃c∈carrier D. ord_equiv (Iod D (segment D c)) E
lemma Worder_equiv:
[| Worder E;
∀a∈carrier D.
∃b∈carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b));
∀c∈carrier E.
∃d∈carrier D. ord_equiv (Iod E (segment E c)) (Iod D (segment D d)) |]
==> ord_equiv D E
lemma Worder_equiv1:
[| Worder E; ¬ ord_equiv D E |]
==> ¬ ((∀a∈carrier D.
∃b∈carrier E.
ord_equiv (Iod D (segment D a)) (Iod E (segment E b))) ∧
(∀c∈carrier E.
∃d∈carrier D. ord_equiv (Iod E (segment E c)) (Iod D (segment D d))))
lemma Word_compare:
Worder E
==> (∃a∈carrier D. ord_equiv (Iod D (segment D a)) E) ∨
ord_equiv D E ∨ (∃b∈carrier E. ord_equiv D (Iod E (segment E b)))
lemma Word_compareTr1:
[| Worder E; ∃a∈carrier D. ord_equiv (Iod D (segment D a)) E; ord_equiv D E |]
==> False
lemma Word_compareTr2:
[| Worder E; ord_equiv D E; ∃b∈carrier E. ord_equiv D (Iod E (segment E b)) |]
==> False
lemma Word_compareTr3:
[| Worder E; ∃b∈carrier E. ord_equiv D (Iod E (segment E b));
∃a∈carrier D. ord_equiv (Iod D (segment D a)) E |]
==> False
lemma subset_equiv_segment:
S ⊆ carrier D
==> ord_equiv D (Iod D S) ∨
(∃a∈carrier D. ord_equiv (Iod D S) (Iod D (segment D a)))
lemma Worder_ord_equivTr:
[| Worder S; Worder T |] ==> ord_equiv S T = (∃f. ord_isom S T f)
lemma Worder_ord_isom_mem:
[| Worder S; Worder T; ord_isom S T f; a ∈ carrier S |] ==> f a ∈ carrier T
lemma Worder_refl:
Worder S ==> ord_equiv S S
lemma Worder_sym:
[| Worder S; Worder T; ord_equiv S T |] ==> ord_equiv T S
lemma Worder_trans:
[| Worder S; Worder T; Worder U; ord_equiv S T; ord_equiv T U |]
==> ord_equiv S U
lemma ordinal_inc_self:
Worder S ==> S ∈ ordinal_number S
lemma ordinal_number_eq:
[| Worder D; Worder E |]
==> ord_equiv D E = (ordinal_number D = ordinal_number E)
lemma mem_ordinal_number_equiv:
[| Worder D; X ∈ ordinal_number D |] ==> ord_equiv X D
lemma mem_ordinal_number_Worder:
[| Worder D; X ∈ ordinal_number D |] ==> Worder X
lemma mem_ordinal_number_Worder1:
[| x ∈ ODnums; X ∈ x |] ==> Worder X
lemma mem_ODnums_nonempty:
X ∈ ODnums ==> ∃x. x ∈ X
lemma carr_ODnods:
carrier ODnods = ODnums
lemma ordinal_number_mem_carrier_ODnods:
Worder D ==> ordinal_number D ∈ carrier ODnods
lemma ordinal_number_mem_ODnums:
Worder D ==> ordinal_number D ∈ ODnums
lemma ODordTr1:
[| Worder D; Worder E |]
==> ordinal_number D \<sqsubset> ordinal_number E =
(∃b∈carrier E. ord_equiv D (Iod E (segment E b)))
lemma ODord:
[| Worder D; d ∈ carrier D |]
==> ordinal_number (Iod D (segment D d)) \<sqsubset> ordinal_number D
lemma ord_less_ODord:
[| Worder D; c ∈ carrier D; d ∈ carrier D;
a = ordinal_number (Iod D (segment D c));
b = ordinal_number (Iod D (segment D d)) |]
==> c \<prec>D d = a \<sqsubset> b
lemma ODord_le_ref:
[| X ∈ ODnums; Y ∈ ODnums; X \<sqsubseteq> Y; Y \<sqsubseteq> X |] ==> X = Y
lemma ODord_le_trans:
[| X ∈ ODnums; Y ∈ ODnums; Z ∈ ODnums; X \<sqsubseteq> Y; Y \<sqsubseteq> Z |]
==> X \<sqsubseteq> Z
lemma ordinal_numberTr1:
X ∈ carrier ODnods ==> ∃D. Worder D ∧ D ∈ X
lemma ordinal_numberTr1_1:
X ∈ ODnums ==> ∃D. Worder D ∧ D ∈ X
lemma ordinal_numberTr1_2:
[| x ∈ ODnums; S ∈ x; T ∈ x |] ==> ord_equiv S T
lemma ordinal_numberTr2:
[| Worder D; x = ordinal_number D |] ==> D ∈ x
lemma ordinal_numberTr3:
[| Worder D; Worder F; ord_equiv D F; x = ordinal_number D |]
==> x = ordinal_number F
lemma ordinal_numberTr4:
[| Worder D; X ∈ carrier ODnods; D ∈ X |] ==> X = ordinal_number D
lemma ordinal_numberTr5:
[| x ∈ ODnums; D ∈ x |] ==> x = ordinal_number D
lemma ordinal_number_ord:
[| X ∈ carrier ODnods; Y ∈ carrier ODnods |]
==> X \<sqsubset> Y ∨ X = Y ∨ Y \<sqsubset> X
lemma ODnum_subTr:
[| Worder D; x = ordinal_number D; y ∈ ODnums; y \<sqsubset> x; Y ∈ y |]
==> ∃c∈carrier D. ord_equiv Y (Iod D (segment D c))
lemma ODnum_segmentTr:
[| Worder D; x = ordinal_number D; y ∈ ODnums; y \<sqsubset> x |]
==> ∃c. c ∈ carrier D ∧ (∀Y∈y. ord_equiv Y (Iod D (segment D c)))
lemma ODnum_segmentTr1:
[| Worder D; x = ordinal_number D; y ∈ ODnums; y \<sqsubset> x |]
==> ∃c. c ∈ carrier D ∧ y = ordinal_number (Iod D (segment D c))
lemma ODnods_less:
[| x ∈ carrier ODnods; y ∈ carrier ODnods |]
==> x \<prec>ODnods y = x \<sqsubset> y
lemma ODord_less_not_eq:
[| x ∈ carrier ODnods; y ∈ carrier ODnods; x \<sqsubset> y |] ==> x ≠ y
lemma not_ODord:
[| a ∈ ODnums; b ∈ ODnums; a \<sqsubset> b |] ==> ¬ b \<sqsubseteq> a
lemma Order_ODnods:
Order ODnods
lemma Torder_ODnods:
Torder ODnods
lemma ODNmap_mem:
[| Worder D; x = ordinal_number D; y ∈ ODnums; y \<sqsubset> x |]
==> ODNmap D y ∈ carrier D ∧
(∀Y∈y. ord_equiv Y (Iod D (segment D (ODNmap D y))))
lemma ODNmapTr1:
[| Worder D; x = ordinal_number D; y ∈ ODnums; y \<sqsubset> x |]
==> y = ordinal_number (Iod D (segment D (ODNmap D y)))
lemma ODNmap_self:
[| Worder D; c ∈ carrier D; a = ordinal_number (Iod D (segment D c)) |]
==> ODNmap D a = c
lemma ODord_ODNmap_less:
[| Worder D; c ∈ carrier D; a = ordinal_number (Iod D (segment D c));
d ∈ carrier D; b = ordinal_number (Iod D (segment D d)); a \<sqsubset> b |]
==> ODNmap D a \<prec>D ODNmap D b
lemma ODNmap_mem1:
[| Worder D; y ∈ segment ODnods (ordinal_number D) |] ==> ODNmap D y ∈ carrier D
lemma ODnods_segment_inc_ODord:
[| Worder D; y ∈ segment ODnods (ordinal_number D) |]
==> y \<sqsubset> ordinal_number D
lemma restict_ODNmap_func:
[| Worder D; x = ordinal_number D |]
==> restrict (ODNmap D) (segment ODnods (ordinal_number D))
∈ segment ODnods (ordinal_number D) -> carrier D
lemma ODNmap_ord_isom:
[| Worder D; x = ordinal_number D |]
==> ord_isom (Iod ODnods (segment ODnods x)) D
(restrict (ODNmap D) (carrier (Iod ODnods (segment ODnods x))))
lemma ODnum_equiv_segment:
[| Worder D; x = ordinal_number D |]
==> ord_equiv (Iod ODnods (segment ODnods x)) D
lemma ODnods_sub_carrier:
S ⊆ ODnums ==> carrier (Iod ODnods S) = S
lemma ODnum_sub_well_ordered:
S ⊆ ODnums ==> Worder (Iod ODnods S)
lemma Pre_mem:
[| a ∈ carrier D; ExPre D a |] ==> Pre D a ∈ carrier D
lemma Not_ExPre:
a ∈ carrier D ==> ¬ ExPre (Iod D {a}) a
lemma UniquePre:
[| a ∈ carrier D; ExPre D a;
a1.0 ∈ carrier D ∧
a1.0 \<prec> a ∧ ¬ (∃y∈carrier D. a1.0 \<prec> y ∧ y \<prec> a) |]
==> Pre D a = a1.0
lemma Pre_element:
[| a ∈ carrier D; ExPre D a |]
==> Pre D a ∈ carrier D ∧
Pre D a \<prec> a ∧ ¬ (∃y∈carrier D. Pre D a \<prec> y ∧ y \<prec> a)
lemma Pre_in_segment:
[| a ∈ carrier D; ExPre D a |] ==> Pre D a ∈ segment D a
lemma segment_forall:
[| a ∈ segment D b; b ∈ carrier D; x ∈ segment D b; x \<prec> a;
∀y∈segment D b. x \<prec> y --> ¬ y \<prec> a |]
==> ∀y∈carrier D. x \<prec> y --> ¬ y \<prec> a
lemma segment_Expre:
a ∈ segment D b ==> ExPre (Iod D (segment D b)) a = ExPre D a
lemma Pre_segment:
[| a ∈ segment D b; ExPre (Iod D (segment D b)) a |]
==> ExPre D a ∧ Pre D a = Pre (Iod D (segment D b)) a
lemma Pre2segment:
[| a ∈ carrier D; b ∈ carrier D; b \<prec> a; ExPre D b |]
==> ExPre (Iod D (segment D a)) b
lemma ord_isom_Pre1:
[| Worder E; a ∈ carrier D; ExPre D a; ord_isom D E f |] ==> ExPre E (f a)
lemma ord_isom_Pre11:
[| Worder E; a ∈ carrier D; ord_isom D E f |] ==> ExPre D a = ExPre E (f a)
lemma ord_isom_Pre2:
[| Worder E; a ∈ carrier D; ExPre D a; ord_isom D E f |]
==> f (Pre D a) = Pre E (f a)
lemma transfinite_induction:
[| minimum_elem D (carrier D) s0.0; P s0.0;
∀t∈carrier D. (∀u∈segment D t. P u) --> P t |]
==> ∀x∈carrier D. P x
lemma carrier_adjunct_ord:
carrier (adjunct_ord D a) = carrier D ∪ {a}
lemma Order_adjunct_ord:
a ∉ carrier D ==> Order (adjunct_ord D a)
lemma adjunct_ord_large_a:
[| Order D; a ∉ carrier D |] ==> ∀x∈carrier D. x \<prec>adjunct_ord D a a
lemma carr_Ssegment_adjunct_ord:
[| Order D; a ∉ carrier D |] ==> carrier D = Ssegment (adjunct_ord D a) a
lemma adjunct_ord_selfD:
a ∉ carrier D ==> D = Iod (adjunct_ord D a) (carrier D)
lemma Ssegment_adjunct_ord:
[| Order D; a ∉ carrier D |]
==> D = SIod (adjunct_ord D a) (Ssegment (adjunct_ord D a) a)
lemma Torder_adjunction:
[| X ⊆ carrier D; a ∈ carrier D; ∀x∈X. x \<preceq> a; Torder (Iod D X) |]
==> Torder (Iod D (X ∪ {a}))
lemma Torder_Sadjunction:
[| Order D; X ⊆ carrier D; a ∈ carrier D; ∀x∈X. x \<preceq>D a;
Torder (SIod D X) |]
==> Torder (SIod D (X ∪ {a}))
lemma Torder_adjunct_ord:
a ∉ carrier D ==> Torder (adjunct_ord D a)
lemma well_ord_adjunction:
[| X ⊆ carrier D; a ∈ carrier D; ∀x∈X. x \<preceq> a; Worder (Iod D X) |]
==> Worder (Iod D (X ∪ {a}))
lemma well_ord_Sadjunction:
[| Order D; X ⊆ carrier D; a ∈ carrier D; ∀x∈X. x \<preceq>D a;
Worder (SIod D X) |]
==> Worder (SIod D (X ∪ {a}))
lemma Worder_adjunct_ord:
a ∉ carrier D ==> Worder (adjunct_ord D a)
lemma mem_upper_bounds:
[| X ⊆ carrier D; b ∈ carrier D; ∀x∈X. x \<preceq> b |] ==> ub X b
lemma Torder_Chain:
[| X ⊆ carrier D; Torder (Iod D X) |] ==> Chain D X
lemma Chain_Torder:
Chain D X ==> Torder (Iod D X)
lemma Chain_sub:
Chain D X ==> X ⊆ carrier D
lemma Chain_sub_Chain:
[| Chain D X; Y ⊆ X |] ==> Chain D Y
lemma upper_bounds_sub:
X ⊆ carrier D ==> upper_bounds D X ⊆ carrier D
lemma Sup:
[| X ⊆ carrier D; minimum_elem D (upper_bounds D X) a |]
==> Algebra2.Sup D X = a
lemma Sup_mem:
[| X ⊆ carrier D; ∃b. ub X b |] ==> Algebra2.Sup D X ∈ carrier D
lemma S_inductive_sup:
[| S_inductive_set D; Chain D X |]
==> minimum_elem D (upper_bounds D X) (Algebra2.Sup D X)
lemma adjunct_Chain:
[| Chain D X; b ∈ carrier D; ∀x∈X. x \<preceq> b |] ==> Chain D (insert b X)
lemma S_inductive_sup_mem:
[| S_inductive_set D; Chain D X |] ==> Algebra2.Sup D X ∈ carrier D
lemma S_inductive_Sup_min_bounds:
[| S_inductive_set D; Chain D X; ub X b |] ==> Algebra2.Sup D X \<preceq> b
lemma S_inductive_sup_bound:
[| S_inductive_set D; Chain D X |] ==> ∀x∈X. x \<preceq> Algebra2.Sup D X
lemma S_inductive_Sup_in_ChainTr:
[| S_inductive_set D; Chain D X;
c ∈ carrier (Iod D (insert (Algebra2.Sup D X) X)); Algebra2.Sup D X ∉ X;
∀y∈carrier (Iod D (insert (Algebra2.Sup D X) X)).
c \<prec>Iod D (insert (Algebra2.Sup D X) X) y -->
¬ y \<prec>Iod D (insert (Algebra2.Sup D X) X) Algebra2.Sup D X |]
==> c ∈ upper_bounds D X
lemma S_inductive_Sup_in_Chain:
[| S_inductive_set D; Chain D X;
ExPre (Iod D (insert (Algebra2.Sup D X) X)) (Algebra2.Sup D X) |]
==> Algebra2.Sup D X ∈ X
lemma S_inductive_bounds_compare:
[| S_inductive_set D; Chain D X1.0; Chain D X2.0; X1.0 ⊆ X2.0 |]
==> upper_bounds D X2.0 ⊆ upper_bounds D X1.0
lemma S_inductive_sup_compare:
[| S_inductive_set D; Chain D X1.0; Chain D X2.0; X1.0 ⊆ X2.0 |]
==> Algebra2.Sup D X1.0 \<preceq> Algebra2.Sup D X2.0
lemma mem_of_WWa:
[| W ⊆ carrier D; Worder (Iod D W); a ∈ W; ∀x∈W. a \<preceq> x;
∀x∈W. if ExPre (Iod D W) x then g (Pre (Iod D W) x) = x
else if a ≠ x then Algebra2.Sup D (segment (Iod D W) x) = x
else a = a |]
==> W ∈ WWa D g a
lemma mem_WWa_then:
W ∈ WWa D g a
==> W ⊆ carrier D ∧
Worder (Iod D W) ∧
a ∈ W ∧
(∀x∈W. a \<preceq> x) ∧
(∀x∈W. if ExPre (Iod D W) x then g (Pre (Iod D W) x) = x
else if a ≠ x then Algebra2.Sup D (segment (Iod D W) x) = x
else a = a)
lemma mem_wwa_Worder:
W ∈ WWa D g a ==> Worder (Iod D W)
lemma mem_WWa_sub_carrier:
W ∈ WWa D g a ==> W ⊆ carrier D
lemma Union_WWa_sub_carrier:
Union (WWa D g a) ⊆ carrier D
lemma mem_WWa_inc_a:
W ∈ WWa D g a ==> a ∈ W
lemma mem_WWa_Chain:
W ∈ WWa D g a ==> Chain D W
lemma Sup_adjunct_Sup:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a; Algebra2.Sup D W ∉ W |]
==> Algebra2.Sup D (insert (Algebra2.Sup D W) W) = Algebra2.Sup D W
lemma BNTr1:
a ∈ carrier D ==> Worder (Iod D {a})
lemma BNTr2:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x |]
==> {a} ∈ WWa D f a
lemma BNTr2_1:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
W ∈ WWa D f a |]
==> ∀x∈W. a \<preceq> x
lemma BNTr3:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
W ∈ WWa D f a |]
==> minimum_elem (Iod D W) W a
lemma Adjunct_segment_sub:
[| S_inductive_set D; Chain D X |]
==> segment (Iod D (insert (Algebra2.Sup D X) X)) (Algebra2.Sup D X) ⊆ X
lemma Adjunct_segment_eq:
[| S_inductive_set D; Chain D X; Algebra2.Sup D X ∉ X |]
==> segment (Iod D (insert (Algebra2.Sup D X) X)) (Algebra2.Sup D X) = X
lemma fixp_same:
[| W1.0 ⊆ carrier D; W2.0 ⊆ carrier D; t ∈ W1.0; b ∈ carrier D;
ord_isom (Iod D W1.0) (Iod (Iod D W2.0) (segment (Iod D W2.0) b)) g;
∀u∈segment (Iod D W1.0) t. fixp g u |]
==> segment (Iod D W1.0) t = segment (Iod D W2.0) (g t)
lemma BNTr4_1:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; b ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; W1.0 ∈ WWa D f a; W2.0 ∈ WWa D f a;
ord_isom (Iod D W1.0) (Iod D (segment (Iod D W2.0) b)) g |]
==> ∀x∈W1.0. g x = x
lemma BNTr4_2:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; b ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; W1.0 ∈ WWa D f a; W2.0 ∈ WWa D f a;
ord_equiv (Iod D W1.0) (Iod D (segment (Iod D W2.0) b)) |]
==> W1.0 = segment (Iod D W2.0) b
lemma BNTr4:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
W1.0 ∈ WWa D f a; W2.0 ∈ WWa D f a;
∃b∈carrier D. ord_equiv (Iod D W1.0) (Iod D (segment (Iod D W2.0) b)) |]
==> W1.0 ⊆ W2.0
lemma Iod_same:
A = B ==> Iod D A = Iod D B
lemma eq_ord_equivTr:
[| ord_equiv D E; E = F |] ==> ord_equiv D F
lemma BNTr5:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
W1.0 ∈ WWa D f a; W2.0 ∈ WWa D f a; ord_equiv (Iod D W1.0) (Iod D W2.0) |]
==> W1.0 ⊆ W2.0
lemma BNTr6:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
W1.0 ∈ WWa D f a; W2.0 ∈ WWa D f a; W1.0 ⊂ W2.0 |]
==> ∃b∈carrier (Iod D W2.0).
ord_equiv (Iod D W1.0) (Iod D (segment (Iod D W2.0) b))
lemma BNTr6_1:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
W1.0 ∈ WWa D f a; W2.0 ∈ WWa D f a; W1.0 ⊂ W2.0 |]
==> ∃b∈carrier (Iod D W2.0). W1.0 = segment (Iod D W2.0) b
lemma BNTr7:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
W1.0 ∈ WWa D f a; W2.0 ∈ WWa D f a |]
==> W1.0 ⊆ W2.0 ∨ W2.0 ⊆ W1.0
lemma BNTr7_1:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
x ∈ W; W ∈ WWa D f a; xa ∈ Union (WWa D f a);
xa \<prec>Iod D (Union (WWa D f a)) x |]
==> xa ∈ W
lemma BNTr7_1_1:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
x ∈ W; W ∈ WWa D f a; xa ∈ Union (WWa D f a); xa \<prec> x |]
==> xa ∈ W
lemma BNTr7_2:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
x ∈ Union (WWa D f a); ExPre (Iod D (Union (WWa D f a))) x |]
==> ∀W∈WWa D f a. x ∈ W --> ExPre (Iod D W) x
lemma BNTr7_3:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
x ∈ Union (WWa D f a); ExPre (Iod D (Union (WWa D f a))) x |]
==> ∀W∈WWa D f a. x ∈ W --> Pre (Iod D (Union (WWa D f a))) x = Pre (Iod D W) x
lemma BNTr7_4:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
x ∈ W; W ∈ WWa D f a |]
==> ExPre (Iod D (Union (WWa D f a))) x = ExPre (Iod D W) x
lemma BNTr7_5:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
x ∈ W; W ∈ WWa D f a |]
==> segment (Iod D (Union (WWa D f a))) x = segment (Iod D W) x
lemma BNTr7_6:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x |]
==> a ∈ Union (WWa D f a)
lemma BNTr7_7:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; ∃xa. Wa D xa f a ∧ x ∈ xa |]
==> x ∈ Union (WWa D f a)
lemma BNTr7_8:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; ∃xa. Wa D xa f a ∧ x ∈ xa |]
==> x ∈ carrier D
lemma BNTr7_9:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x;
x ∈ Union (WWa D f a) |]
==> x ∈ carrier D
lemma BNTr7_10:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a; Algebra2.Sup D W ∉ W |]
==> ¬ ExPre (Iod D (insert (Algebra2.Sup D W) W)) (Algebra2.Sup D W)
lemma BNTr7_11:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D; b ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a; ∀x∈W. x \<preceq> b; x ∈ W |]
==> ExPre (Iod D (insert b W)) x = ExPre (Iod D W) x
lemma BNTr7_12:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D; b ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a; ∀x∈W. x \<preceq> b; x ∈ W;
ExPre (Iod D W) x |]
==> Pre (Iod D (insert b W)) x = Pre (Iod D W) x
lemma BNTr7_13:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D; b ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a; ∀x∈W. x \<preceq> b; x ∈ W |]
==> segment (Iod D (insert b W)) x = segment (Iod D W) x
lemma BNTr7_14:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a |]
==> insert (Algebra2.Sup D W) W ∈ WWa D f a
lemma BNTr7_15:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a;
f (Algebra2.Sup D W) ≠ Algebra2.Sup D W |]
==> ExPre (Iod D (insert (f (Algebra2.Sup D W)) (insert (Algebra2.Sup D W) W)))
(f (Algebra2.Sup D W))
lemma BNTr7_16:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a;
f (Algebra2.Sup D W) ≠ Algebra2.Sup D W |]
==> Pre (Iod D (insert (f (Algebra2.Sup D W)) (insert (Algebra2.Sup D W) W)))
(f (Algebra2.Sup D W)) =
Algebra2.Sup D W
lemma BNTr7_17:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x; W ∈ WWa D f a |]
==> insert (f (Algebra2.Sup D W)) (insert (Algebra2.Sup D W) W) ∈ WWa D f a
lemma BNTr8:
[| f ∈ carrier D -> carrier D; a ∈ carrier D; ∀x∈carrier D. x \<preceq> f x |]
==> Union (WWa D f a) ∈ WWa D f a
lemma BNTr10:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x |]
==> Algebra2.Sup D (Union (WWa D f a)) ∈ Union (WWa D f a)
lemma BNTr11:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x |]
==> f (Algebra2.Sup D (Union (WWa D f a))) = Algebra2.Sup D (Union (WWa D f a))
lemma Bourbaki_Nakayama:
[| S_inductive_set D; f ∈ carrier D -> carrier D; a ∈ carrier D;
∀x∈carrier D. x \<preceq> f x |]
==> ∃x0∈carrier D. f x0 = x0
lemma maxl_funTr:
[| x ∈ carrier D; ∃y. y ∈ upper_bounds D {x} ∧ y ≠ x |]
==> (SOME z. z ∈ upper_bounds D {x} ∧ z ≠ x) ∈ carrier D
lemma maxl_fun_func:
maxl_fun D ∈ carrier D -> carrier D
lemma maxl_fun_gt:
[| x ∈ carrier D; ∃y∈carrier D. x \<preceq> y ∧ x ≠ y |]
==> x \<preceq> maxl_fun D x ∧ maxl_fun D x ≠ x
lemma maxl_fun_maxl:
[| x ∈ carrier D; maxl_fun D x = x |] ==> maximal x
lemma maxl_fun_asc:
∀x∈carrier D. x \<preceq> maxl_fun D x
lemma S_inductive_maxl:
[| S_inductive_set D; carrier D ≠ {} |] ==> ∃m. maximal m
lemma maximal_mem:
maximal m ==> m ∈ carrier D
lemma Chain_mem_fTo:
Chain D C ==> C ∈ carrier fTo D
lemma fTOrder:
Order fTo D
lemma fTo_Order_sub:
[| A ∈ carrier fTo D; B ∈ carrier fTo D |] ==> A \<preceq>fTo D B = (A ⊆ B)
lemma mem_fTo_Chain:
X ∈ carrier fTo D ==> Chain D X
lemma mem_fTo_sub_carrier:
X ∈ carrier fTo D ==> X ⊆ carrier D
lemma Un_fTo_Chain:
Chain fTo D CC ==> Chain D (Union CC)
lemma Un_fTo_Chain_mem_fTo:
Chain fTo D CC ==> Union CC ∈ carrier fTo D
lemma Un_upper_bound:
Chain fTo D C ==> Union C ∈ upper_bounds fTo D C
lemma fTo_conditional_inc_C:
C ∈ carrier fTo D ==> C ∈ carrier (Iod fTo D {S : carrier fTo D. C ⊆ S})
lemma fTo_conditional_Un_Chain_mem1:
[| C ∈ carrier fTo D; Chain (Iod fTo D {S : carrier fTo D. C ⊆ S}) Ca;
Ca ≠ {} |]
==> Union Ca ∈ upper_bounds (Iod fTo D {S : carrier fTo D. C ⊆ S}) Ca
lemma fTo_conditional_min1:
[| C ∈ carrier fTo D; Chain (Iod fTo D {S : carrier fTo D. C ⊆ S}) Ca;
Ca ≠ {} |]
==> minimum_elem (Iod fTo D {S : carrier fTo D. C ⊆ S})
(upper_bounds (Iod fTo D {S : carrier fTo D. C ⊆ S}) Ca) (Union Ca)
lemma fTo_conditional_Un_Chain_mem2:
[| C ∈ carrier fTo D; Chain (Iod fTo D {S : carrier fTo D. C ⊆ S}) Ca;
Ca = {} |]
==> C ∈ upper_bounds (Iod fTo D {S : carrier fTo D. C ⊆ S}) Ca
lemma fTo_conditional_min2:
[| C ∈ carrier fTo D; Chain (Iod fTo D {S : carrier fTo D. C ⊆ S}) Ca;
Ca = {} |]
==> minimum_elem (Iod fTo D {S : carrier fTo D. C ⊆ S})
(upper_bounds (Iod fTo D {S : carrier fTo D. C ⊆ S}) Ca) C
lemma fTo_S_inductive:
S_inductive_set fTo D
lemma conditional_min_upper_bound:
[| C ∈ carrier fTo D; Chain (Iod fTo D {S : carrier fTo D. C ⊆ S}) Ca |]
==> ∃X. minimum_elem (Iod fTo D {S : carrier fTo D. C ⊆ S})
(upper_bounds (Iod fTo D {S : carrier fTo D. C ⊆ S}) Ca) X
lemma Hausdorff_acTr:
C ∈ carrier fTo D ==> S_inductive_set (Iod fTo D {S : carrier fTo D. C ⊆ S})
lemma satisfy_cond_mem_set:
[| x ∈ A; P x |] ==> x ∈ {y : A. P y}
lemma maximal_conditional_maximal:
[| C ∈ carrier fTo D; maximalIod fTo D {S : carrier fTo D. C ⊆ S} m |]
==> maximalfTo D m
lemma Hausdorff_ac:
C ∈ carrier fTo D ==> ∃M∈carrier fTo D. C ⊆ M ∧ maximalfTo D M
lemma Zorn_lemmaTr:
[| Chain D C; M ∈ carrier fTo D; C ⊆ M; maximalfTo D M; b ∈ carrier D;
∀s∈M. s \<preceq> b |]
==> maximal b ∧ b ∈ upper_bounds D C
lemma g_Zorn_lemma1:
[| inductive_set D; Chain D C |] ==> ∃m. maximal m ∧ m ∈ upper_bounds D C
lemma g_Zorn_lemma2:
[| inductive_set D; a ∈ carrier D |] ==> ∃m∈carrier D. maximal m ∧ a \<preceq> m
lemma g_Zorn_lemma3:
inductive_set D ==> ∃m∈carrier D. maximal m
lemma mult_closed:
[| a ∈ carrier G; b ∈ carrier G |] ==> Group.top G a b ∈ carrier G
lemma i_closed:
a ∈ carrier G ==> ρ a ∈ carrier G
lemma r_mult_eqn:
[| a ∈ carrier G; b ∈ carrier G; c ∈ carrier G; a = b |]
==> Group.top G a c = Group.top G b c
lemma l_mult_eqn:
[| a ∈ carrier G; b ∈ carrier G; c ∈ carrier G; a = b |]
==> Group.top G c a = Group.top G c b
lemma r_i:
a ∈ carrier G ==> Group.top G a (ρ a) = \<one>
lemma r_unit:
a ∈ carrier G ==> Group.top G a \<one> = a
lemma l_i_unique:
[| a ∈ carrier G; b ∈ carrier G; Group.top G b a = \<one> |] ==> ρ a = b
lemma l_i_i:
a ∈ carrier G ==> Group.top G (ρ (ρ a)) (ρ a) = \<one>
lemma l_div_eqn:
[| a ∈ carrier G; x ∈ carrier G; y ∈ carrier G;
Group.top G a x = Group.top G a y |]
==> x = y
lemma r_div_eqn:
[| a ∈ carrier G; x ∈ carrier G; y ∈ carrier G;
Group.top G x a = Group.top G y a |]
==> x = y
lemma l_mult_eqn1:
[| a ∈ carrier G; x ∈ carrier G; y ∈ carrier G;
Group.top G (ρ a) x = Group.top G (ρ a) y |]
==> x = y
lemma tOp_assocTr41:
[| a ∈ carrier G; b ∈ carrier G; c ∈ carrier G; d ∈ carrier G |]
==> Group.top G (Group.top G (Group.top G a b) c) d =
Group.top G (Group.top G a b) (Group.top G c d)
lemma tOp_assocTr42:
[| a ∈ carrier G; b ∈ carrier G; c ∈ carrier G; d ∈ carrier G |]
==> Group.top G (Group.top G (Group.top G a b) c) d =
Group.top G (Group.top G a (Group.top G b c)) d
lemma tOp_assocTr44:
[| a ∈ carrier G; b ∈ carrier G; c ∈ carrier G; d ∈ carrier G |]
==> Group.top G (Group.top G (ρ a) b) (Group.top G (ρ c) d) =
Group.top G (ρ a) (Group.top G (Group.top G b (ρ c)) d)
lemma tOp_assocTr45:
[| a ∈ carrier G; b ∈ carrier G; c ∈ carrier G; d ∈ carrier G |]
==> Group.top G (Group.top G (Group.top G a b) c) d =
Group.top G a (Group.top G b (Group.top G c d))
lemma one_unique:
[| a ∈ carrier G; x ∈ carrier G; Group.top G x a = x |] ==> a = \<one>
lemma i_one:
ρ \<one> = \<one>
lemma eqn_inv1:
[| a ∈ carrier G; x ∈ carrier G; a = ρ x |] ==> x = ρ a
lemma eqn_inv2:
[| a ∈ carrier G; x ∈ carrier G; Group.top G x a = Group.top G x (ρ x) |]
==> x = ρ a
lemma r_one_unique:
[| a ∈ carrier G; x ∈ carrier G; Group.top G a x = a |] ==> x = \<one>
lemma r_i_unique:
[| a ∈ carrier G; x ∈ carrier G; Group.top G a x = \<one> |] ==> x = ρ a
lemma iop_i_i:
a ∈ carrier G ==> ρ (ρ a) = a
lemma i_ab:
[| a ∈ carrier G; b ∈ carrier G |]
==> ρ Group.top G a b = Group.top G (ρ b) (ρ a)
lemma sol_eq_l:
[| a ∈ carrier G; b ∈ carrier G; x ∈ carrier G; Group.top G a x = b |]
==> x = Group.top G (ρ a) b
lemma sol_eq_r:
[| a ∈ carrier G; b ∈ carrier G; x ∈ carrier G; Group.top G x a = b |]
==> x = Group.top G b (ρ a)
lemma r_div_eq:
[| a ∈ carrier G; b ∈ carrier G; Group.top G a (ρ b) = \<one> |] ==> a = b
lemma l_div_eq:
[| a ∈ carrier G; b ∈ carrier G; Group.top G (ρ a) b = \<one> |] ==> a = b
lemma i_m_closed:
[| a ∈ carrier G; b ∈ carrier G |] ==> Group.top G (ρ a) b ∈ carrier G
lemma sg_subset:
G » H ==> H ⊆ carrier G
lemma one_Gp_one:
G » H ==> \<one>\<natural>H = \<one>
lemma carrier_Gp:
G » H ==> carrier (\<natural>H) = H
lemma sg_subset_elem:
[| G » H ; h ∈ H |] ==> h ∈ carrier G
lemma sg_mult_closedr:
[| G » H ; x ∈ carrier G; h ∈ H |] ==> Group.top G x h ∈ carrier G
lemma sg_mult_closedl:
[| G » H ; x ∈ carrier G; h ∈ H |] ==> Group.top G h x ∈ carrier G
lemma sg_condTr1:
[| H ⊆ carrier G; H ≠ {}; ∀a b. a ∈ H ∧ b ∈ H --> Group.top G a (ρ b) ∈ H |]
==> \<one> ∈ H
lemma sg_unit_closed:
G » H ==> \<one> ∈ H
lemma sg_i_closed:
[| G » H ; x ∈ H |] ==> ρ x ∈ H
lemma sg_mult_closed:
[| G » H ; x ∈ H; y ∈ H |] ==> Group.top G x y ∈ H
lemma nsg_sg:
G \<triangleright> H ==> G » H
lemma nsg_subset:
G \<triangleright> N ==> N ⊆ carrier G
lemma nsg_lr_cst_eq:
[| G \<triangleright> N; a ∈ carrier G |] ==> a ♦ N = N • a
lemma sg_i_m_closed:
[| G » H ; a ∈ H; b ∈ H |] ==> Group.top G (ρ a) b ∈ H
lemma sg_m_i_closed:
[| G » H ; a ∈ H; b ∈ H |] ==> Group.top G a (ρ b) ∈ H
lemma smallest_sg_gen:
[| A ⊆ carrier G; G » H ; A ⊆ H |] ==> sg_gen G A ⊆ H
lemma special_sg_G:
G » carrier G
lemma special_sg_self:
G = \<natural>carrier G
lemma special_sg_e:
G » {\<one>}
lemma inter_sgs:
[| G » H ; G » K |] ==> G » H ∩ K
lemma subg_generated:
A ⊆ carrier G ==> G » sg_gen G A
lemma Group_Ugp:
Ugp G ==> Group G
lemma r_mult_in_sg:
[| G » H ; a ∈ carrier G; x ∈ carrier G; Group.top G x a ∈ H |]
==> ∃h∈H. Group.top G h (ρ a) = x
lemma r_unit_sg:
[| G » H ; h ∈ H |] ==> Group.top G h \<one> = h
lemma sg_l_unit:
[| G » H ; h ∈ H |] ==> Group.top G \<one> h = h
lemma sg_l_i:
[| G » H ; x ∈ H |] ==> Group.top G (ρ x) x = \<one>
lemma sg_tassoc:
[| G » H ; x ∈ H; y ∈ H; z ∈ H |]
==> Group.top G (Group.top G x y) z = Group.top G x (Group.top G y z)
lemma sg_condition:
[| H ⊆ carrier G; H ≠ {}; ∀a b. a ∈ H ∧ b ∈ H --> Group.top G a (ρ b) ∈ H |]
==> G » H
lemma Group_Gp:
G » H ==> Group (\<natural>H)
lemma Gp_carrier:
G » H ==> carrier (\<natural>H) = H
lemma sg_sg:
[| G » K ; G » H ; H ⊆ K |] ==> \<natural>K » H
lemma sg_subset_of_subG:
[| G » K ; \<natural>K » H |] ==> H ⊆ K
lemma const_ghom:
[| Group F; Group G |] ==> 1F,G ∈ gHom F G
lemma const_gbij:
gbij(\<natural>{\<one>}),(\<natural>{\<one>}) (1(\<natural>{\<one>}),(\<natural>{\<one>}))
lemma unit_Groups_isom:
(\<natural>{\<one>}) ≅ (\<natural>{\<one>})
lemma Ugp_const_gHom:
[| Ugp G; Ugp E |] ==> (λx∈carrier G. \<one>E) ∈ gHom G E
lemma Ugp_const_gbij:
[| Ugp G; Ugp E |] ==> gbijG,E (λx∈carrier G. \<one>E)
lemma Ugps_isomorphic:
[| Ugp G; Ugp E |] ==> G ≅ E
lemma Gp_mult_induced:
[| G » L ; a ∈ L; b ∈ L |] ==> Group.top (\<natural>L) a b = Group.top G a b
lemma sg_i_induced:
[| G » L ; a ∈ L |] ==> ρ\<natural>L a = ρ a
lemma Gp_mult_induced1:
[| G » H ; G » K ; a ∈ H ∩ K; b ∈ H ∩ K |]
==> Group.top (\<natural>H ∩ K) a b = Group.top (\<natural>H) a b
lemma Gp_mult_induced2:
[| G » H ; G » K ; a ∈ H ∩ K; b ∈ H ∩ K |]
==> Group.top (\<natural>H ∩ K) a b = Group.top (\<natural>K) a b
lemma sg_i_induced1:
[| G » H ; G » K ; a ∈ H ∩ K |] ==> ρ\<natural>H ∩ K a = ρ\<natural>H a
lemma sg_i_induced2:
[| G » H ; G » K ; a ∈ H ∩ K |] ==> ρ\<natural>H ∩ K a = ρ\<natural>K a
lemma subg_sg_sg:
[| G » K ; \<natural>K » H |] ==> G » H
lemma Gp_inherited:
[| G » K ; G » L ; K ⊆ L |] ==> \<natural>\<natural>LK = \<natural>K
lemma mem_lcs:
[| G » H ; a ∈ carrier G; x ∈ a ♦ H |] ==> x ∈ carrier G
lemma lcs_subset:
[| G » H ; a ∈ carrier G |] ==> a ♦ H ⊆ carrier G
lemma a_in_lcs:
[| G » H ; a ∈ carrier G |] ==> a ∈ a ♦ H
lemma eq_lcs1:
[| G » H ; a ∈ carrier G; b ∈ carrier G; x ∈ a ♦ H; a ♦ H = b ♦ H |]
==> x ∈ b ♦ H
lemma eq_lcs2:
[| G » H ; a ∈ carrier G; b ∈ carrier G; a ♦ H = b ♦ H |] ==> a ∈ b ♦ H
lemma lcs_mem_ldiv:
[| G » H ; a ∈ carrier G; b ∈ carrier G |]
==> (a ∈ b ♦ H) = (Group.top G (ρ b) a ∈ H)
lemma lcsTr5:
[| G » H ; a ∈ carrier G; b ∈ carrier G; Group.top G (ρ a) b ∈ H; x ∈ a ♦ H |]
==> Group.top G (ρ b) x ∈ H
lemma lcsTr6:
[| G » H ; a ∈ carrier G; b ∈ carrier G; Group.top G (ρ a) b ∈ H; x ∈ a ♦ H |]
==> x ∈ b ♦ H
lemma lcs_Unit1:
G » H ==> \<one> ♦ H = H
lemma lcs_Unit2:
[| G » H ; h ∈ H |] ==> h ♦ H = H
lemma lcsTr7:
[| G » H ; a ∈ carrier G; b ∈ carrier G; Group.top G (ρ a) b ∈ H |]
==> a ♦ H ⊆ b ♦ H
lemma lcsTr8:
[| G » H ; a ∈ carrier G; h ∈ H |] ==> Group.top G a h ∈ a ♦ H
lemma lcs_tool1:
[| G » H ; a ∈ carrier G; b ∈ carrier G; Group.top G (ρ a) b ∈ H |]
==> Group.top G (ρ b) a ∈ H
theorem lcs_eq:
[| G » H ; a ∈ carrier G; b ∈ carrier G |]
==> (Group.top G (ρ a) b ∈ H) = (a ♦ H = b ♦ H)
lemma rcs_subset:
[| G » H ; a ∈ carrier G |] ==> H • a ⊆ carrier G
lemma mem_rcs:
[| G » H ; x ∈ H • a |] ==> ∃h∈H. Group.top G h a = x
lemma rcs_subset_elem:
[| G » H ; a ∈ carrier G; x ∈ H • a |] ==> x ∈ carrier G
lemma rcs_in_set_rcs:
[| G » H ; a ∈ carrier G |] ==> H • a ∈ set_rcs G H
lemma rcsTr0:
[| G » H ; a ∈ carrier G; b ∈ carrier G |] ==> H • Group.top G a b ∈ set_rcs G H
lemma a_in_rcs:
[| G » H ; a ∈ carrier G |] ==> a ∈ H • a
lemma rcs_nonempty:
[| G » H ; X ∈ set_rcs G H |] ==> X ≠ {}
lemma rcs_tool0:
[| G » H ; a ∈ carrier G; b ∈ carrier G; Group.top G a (ρ b) ∈ H |]
==> Group.top G b (ρ a) ∈ H
lemma rcsTr1:
[| G » H ; a ∈ carrier G; b ∈ carrier G; x ∈ H • a; H • a = H • b |]
==> x ∈ H • b
lemma rcs_eqTr:
[| G » H ; a ∈ carrier G; b ∈ carrier G; H • a = H • b |] ==> a ∈ H • b
lemma rcs_eqTr1:
[| G » H ; a ∈ carrier G; b ∈ carrier G |]
==> (a ∈ H • b) = (Group.top G a (ρ b) ∈ H)
lemma rcsTr2:
[| G » H ; a ∈ carrier G; b ∈ H • ρ a |] ==> Group.top G b a ∈ H
lemma rcsTr5:
[| G » H ; a ∈ carrier G; b ∈ carrier G; Group.top G b (ρ a) ∈ H; x ∈ H • a |]
==> Group.top G x (ρ b) ∈ H
lemma rcsTr6:
[| G » H ; a ∈ carrier G; b ∈ carrier G; Group.top G b (ρ a) ∈ H; x ∈ H • a |]
==> x ∈ H • b
lemma rcs_Unit1:
G » H ==> H • \<one> = H
lemma unit_rcs_in_set_rcs:
G » H ==> H ∈ set_rcs G H
lemma rcs_Unit2:
[| G » H ; h ∈ H |] ==> H • h = H
lemma rcsTr7:
[| G » H ; a ∈ carrier G; b ∈ carrier G; Group.top G b (ρ a) ∈ H |]
==> H • a ⊆ H • b
lemma rcs_tool1:
[| G » H ; a ∈ carrier G; b ∈ carrier G; Group.top G b (ρ a) ∈ H |]
==> Group.top G a (ρ b) ∈ H
lemma rcs_tool2:
[| G » H ; a ∈ carrier G; x ∈ H • a |] ==> ∃h∈H. Group.top G h a = x
theorem rcs_eq:
[| G » H ; a ∈ carrier G; b ∈ carrier G |]
==> (Group.top G b (ρ a) ∈ H) = (H • a = H • b)
lemma rcs_eq1:
[| G » H ; a ∈ carrier G; x ∈ H • a |] ==> H • a = H • x
lemma rcs_eq2:
[| G » H ; a ∈ carrier G; b ∈ carrier G; H • a ∩ (H • b) ≠ {} |]
==> H • a = H • b
lemma rcs_meet:
[| G » H ; X ∈ set_rcs G H; Y ∈ set_rcs G H; X ∩ Y ≠ {} |] ==> X = Y
lemma rcsTr8:
[| G » H ; a ∈ carrier G; h ∈ H; x ∈ H • a |] ==> Group.top G h x ∈ H • a
lemma rcsTr9:
[| G » H ; a ∈ carrier G; h ∈ H; ρ x ∈ H • a |] ==> Group.top G h (ρ x) ∈ H • a
lemma rcsTr10:
[| G » H ; a ∈ carrier G; x ∈ H • a; y ∈ H • a |] ==> Group.top G x (ρ y) ∈ H
lemma PrSubg4_2:
[| G » H ; a ∈ carrier G; x ∈ H • ρ a |]
==> x ∈ {z. ∃v∈H • a. ∃h∈H. Group.top G h (ρ v) = z}
lemma rcs_fixed:
[| G » H ; a ∈ carrier G; H • a = H |] ==> a ∈ H
lemma rcs_fixed1:
[| G » H ; a ∈ carrier G; h ∈ H |] ==> H • a = H • Group.top G h a
lemma rcs_fixed2:
G » H ==> ∀h∈H. H • h = H
lemma Gp_rcs:
[| G » H ; G » K ; H ⊆ K; x ∈ K |] ==> H •\<natural>K x = H • x
lemma subg_lcs:
[| G » H ; G » K ; H ⊆ K; x ∈ K |] ==> x ♦\<natural>K H = x ♦ H
lemma nsg1:
[| G » H ; b ∈ carrier G; h ∈ H;
∀a∈carrier G. ∀h∈H. Group.top G (Group.top G a h) (ρ a) ∈ H |]
==> Group.top G (Group.top G b h) (ρ b) ∈ H
lemma nsg2:
[| G » H ; b ∈ carrier G; h ∈ H;
∀a∈carrier G. ∀h∈H. Group.top G (Group.top G a h) (ρ a) ∈ H |]
==> Group.top G (Group.top G (ρ b) h) b ∈ H
lemma nsg_subset_elem:
[| G \<triangleright> H; h ∈ H |] ==> h ∈ carrier G
lemma nsg_l_rcs_eq:
[| G \<triangleright> N; a ∈ carrier G |] ==> a ♦ N = N • a
lemma sg_nsg1:
[| G » H ; ∀a∈carrier G. ∀h∈H. Group.top G (Group.top G a h) (ρ a) ∈ H;
b ∈ carrier G |]
==> H • b = b ♦ H
lemma cond_nsg:
[| G » H ; ∀a∈carrier G. ∀h∈H. Group.top G (Group.top G a h) (ρ a) ∈ H |]
==> G \<triangleright> H
lemma special_nsg_e:
G » H ==> \<natural>H \<triangleright> {\<one>}
lemma special_nsg_G:
G \<triangleright> carrier G
lemma special_nsg_G1:
G » H ==> \<natural>H \<triangleright> H
lemma nsgTr0:
[| G \<triangleright> N; a ∈ carrier G; b ∈ carrier G; b ∈ N • a |]
==> Group.top G a (ρ b) ∈ N ∧ Group.top G (ρ a) b ∈ N
lemma nsgTr1:
[| G \<triangleright> N; a ∈ carrier G; b ∈ carrier G;
Group.top G b (ρ a) ∈ N |]
==> Group.top G (ρ b) a ∈ N
lemma nsgTr2:
[| a ∈ carrier G; b ∈ carrier G; a1.0 ∈ carrier G; b1.0 ∈ carrier G |]
==> Group.top G (Group.top G a b) (ρ Group.top G a1.0 b1.0) =
Group.top G a
(Group.top G
(Group.top G (Group.top G b (ρ b1.0)) (Group.top G (ρ a1.0) a)) (ρ a))
lemma nsgPr1:
[| G \<triangleright> N; a ∈ carrier G; h ∈ N |]
==> Group.top G a (Group.top G h (ρ a)) ∈ N
lemma nsgPr1_1:
[| G \<triangleright> N; a ∈ carrier G; h ∈ N |]
==> Group.top G (Group.top G a h) (ρ a) ∈ N
lemma nsgPr2:
[| G \<triangleright> N; a ∈ carrier G; h ∈ N |]
==> Group.top G (ρ a) (Group.top G h a) ∈ N
lemma nsgPr2_1:
[| G \<triangleright> N; a ∈ carrier G; h ∈ N |]
==> Group.top G (Group.top G (ρ a) h) a ∈ N
lemma nsgTr3:
[| G \<triangleright> N; a ∈ carrier G; b ∈ carrier G; a1.0 ∈ carrier G;
b1.0 ∈ carrier G; Group.top G a (ρ a1.0) ∈ N; Group.top G b (ρ b1.0) ∈ N |]
==> Group.top G (Group.top G a b) (ρ Group.top G a1.0 b1.0) ∈ N
lemma nsg_in_Gp:
[| G \<triangleright> N; G » H ; N ⊆ H |] ==> \<natural>H \<triangleright> N
lemma nsgTr4:
[| G \<triangleright> N; a ∈ carrier G; x ∈ N • a |] ==> ρ x ∈ N • ρ a
lemma c_topTr1:
[| G \<triangleright> N; a ∈ carrier G; b ∈ carrier G; a1.0 ∈ carrier G;
b1.0 ∈ carrier G; N • a = N • a1.0; N • b = N • b1.0 |]
==> N • Group.top G a b = N • Group.top G a1.0 b1.0
lemma c_topTr2:
[| G \<triangleright> N; a ∈ carrier G; a1.0 ∈ carrier G; N • a = N • a1.0 |]
==> N • ρ a = N • ρ a1.0
lemma c_iop_welldefTr1:
[| G \<triangleright> N; a ∈ carrier G |] ==> c_iop G N (N • a) ⊆ N • ρ a
lemma c_iop_welldefTr2:
[| G \<triangleright> N; a ∈ carrier G |] ==> N • ρ a ⊆ c_iop G N (N • a)
lemma c_iop_welldef:
[| G \<triangleright> N; a ∈ carrier G |] ==> c_iop G N (N • a) = N • ρ a
lemma c_top_welldefTr1:
[| G \<triangleright> N; a ∈ carrier G; b ∈ carrier G; x ∈ N • a; y ∈ N • b |]
==> Group.top G x y ∈ N • Group.top G a b
lemma c_top_welldefTr2:
[| G \<triangleright> N; a ∈ carrier G; b ∈ carrier G |]
==> c_top G N (N • a) (N • b) ⊆ N • Group.top G a b
lemma c_top_welldefTr4:
[| G \<triangleright> N; a ∈ carrier G; b ∈ carrier G;
x ∈ N • Group.top G a b |]
==> x ∈ c_top G N (N • a) (N • b)
lemma c_top_welldefTr5:
[| G \<triangleright> N; a ∈ carrier G; b ∈ carrier G |]
==> N • Group.top G a b ⊆ c_top G N (N • a) (N • b)
lemma c_top_welldef:
[| G \<triangleright> N; a ∈ carrier G; b ∈ carrier G |]
==> N • Group.top G a b = c_top G N (N • a) (N • b)
lemma Qg_unitTr:
[| G \<triangleright> N; a ∈ carrier G |] ==> c_top G N N (N • a) = N • a
lemma Qg_unit:
G \<triangleright> N ==> ∀x∈set_rcs G N. c_top G N N x = x
lemma Qg_iTr:
[| G \<triangleright> N; a ∈ carrier G |]
==> c_top G N (c_iop G N (N • a)) (N • a) = N
lemma Qg_i:
G \<triangleright> N ==> ∀x∈set_rcs G N. c_top G N (c_iop G N x) x = N
lemma Qg_tassocTr:
[| G \<triangleright> N; a ∈ carrier G; b ∈ carrier G; c ∈ carrier G |]
==> c_top G N (N • a) (c_top G N (N • b) (N • c)) =
c_top G N (c_top G N (N • a) (N • b)) (N • c)
lemma Qg_tassoc:
G \<triangleright> N
==> ∀X∈set_rcs G N.
∀Y∈set_rcs G N.
∀Z∈set_rcs G N.
c_top G N X (c_top G N Y Z) = c_top G N (c_top G N X Y) Z
lemma Qg_top:
G \<triangleright> N ==> c_top G N ∈ set_rcs G N -> set_rcs G N -> set_rcs G N
lemma Qg_top_closed:
[| G \<triangleright> N; A ∈ set_rcs G N; B ∈ set_rcs G N |]
==> c_top G N A B ∈ set_rcs G N
lemma Qg_iop:
G \<triangleright> N ==> c_iop G N ∈ set_rcs G N -> set_rcs G N
lemma Qg_iop_closed:
[| G \<triangleright> N; A ∈ set_rcs G N |] ==> c_iop G N A ∈ set_rcs G N
lemma Qg_unit_closed:
G \<triangleright> N ==> N ∈ set_rcs G N
theorem Group_Qg:
G \<triangleright> N ==> Group (G / N)
lemma Qg_one:
G \<triangleright> N ==> \<one>G / N = N
lemma Qg_carrier:
carrier (G / N) = set_rcs G N
lemma Qg_unit_group:
G \<triangleright> N ==> (set_rcs G N = {N}) = (carrier G = N)
lemma Gp_Qg:
G \<triangleright> N ==> \<natural>G / Ncarrier (G / N) = G / N
lemma Pj_hom0:
[| G \<triangleright> N; x ∈ carrier G; y ∈ carrier G |]
==> Pj G N (Group.top G x y) = Group.top (G / N) (Pj G N x) (Pj G N y)
lemma Pj_ghom:
G \<triangleright> N ==> Pj G N ∈ gHom G (G / N)
lemma Pj_mem:
[| G \<triangleright> N; x ∈ carrier G |] ==> Pj G N x = N • x
lemma Pj_gsurjec:
G \<triangleright> N ==> gsurjG,(G / N) Pj G N
lemma lcs_in_Gp:
[| G » H ; G » K ; K ⊆ H; a ∈ H |] ==> a ♦ K = a ♦\<natural>H K
lemma rcs_in_Gp:
[| G » H ; G » K ; K ⊆ H; a ∈ H |] ==> K • a = K •\<natural>H a