src/ZF/Constructible/L_axioms.thy
author paulson
Wed Sep 11 16:55:37 2002 +0200 (2002-09-11)
changeset 13566 52a419210d5c
parent 13564 1500a2e48d44
child 13628 87482b5e3f2e
permissions -rw-r--r--
Streamlined proofs of instances of Separation
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(*  Title:      ZF/Constructible/L_axioms.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   2002  University of Cambridge
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*)
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header {* The ZF Axioms (Except Separation) in L *}
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theory L_axioms = Formula + Relative + Reflection + MetaExists:
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text {* The class L satisfies the premises of locale @{text M_trivial} *}
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lemma transL: "[| y\<in>x; L(x) |] ==> L(y)"
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apply (insert Transset_Lset)
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apply (simp add: Transset_def L_def, blast)
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done
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lemma nonempty: "L(0)"
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apply (simp add: L_def)
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apply (blast intro: zero_in_Lset)
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done
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theorem upair_ax: "upair_ax(L)"
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apply (simp add: upair_ax_def upair_def, clarify)
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apply (rule_tac x="{x,y}" in rexI)
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apply (simp_all add: doubleton_in_L)
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done
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theorem Union_ax: "Union_ax(L)"
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apply (simp add: Union_ax_def big_union_def, clarify)
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apply (rule_tac x="Union(x)" in rexI)
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apply (simp_all add: Union_in_L, auto)
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apply (blast intro: transL)
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done
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theorem power_ax: "power_ax(L)"
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apply (simp add: power_ax_def powerset_def Relative.subset_def, clarify)
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apply (rule_tac x="{y \<in> Pow(x). L(y)}" in rexI)
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apply (simp_all add: LPow_in_L, auto)
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apply (blast intro: transL)
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done
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text{*We don't actually need @{term L} to satisfy the foundation axiom.*}
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theorem foundation_ax: "foundation_ax(L)"
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apply (simp add: foundation_ax_def)
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apply (rule rallI) 
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apply (cut_tac A=x in foundation)
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apply (blast intro: transL)
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done
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subsection{*For L to satisfy Replacement *}
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(*Can't move these to Formula unless the definition of univalent is moved
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there too!*)
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lemma LReplace_in_Lset:
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     "[|X \<in> Lset(i); univalent(L,X,Q); Ord(i)|]
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      ==> \<exists>j. Ord(j) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Lset(j)"
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apply (rule_tac x="\<Union>y \<in> Replace(X, %x y. Q(x,y) & L(y)). succ(lrank(y))"
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       in exI)
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apply simp
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apply clarify
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apply (rule_tac a=x in UN_I)
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 apply (simp_all add: Replace_iff univalent_def)
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apply (blast dest: transL L_I)
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done
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lemma LReplace_in_L:
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     "[|L(X); univalent(L,X,Q)|]
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      ==> \<exists>Y. L(Y) & Replace(X, %x y. Q(x,y) & L(y)) \<subseteq> Y"
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apply (drule L_D, clarify)
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apply (drule LReplace_in_Lset, assumption+)
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apply (blast intro: L_I Lset_in_Lset_succ)
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done
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theorem replacement: "replacement(L,P)"
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apply (simp add: replacement_def, clarify)
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apply (frule LReplace_in_L, assumption+, clarify)
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apply (rule_tac x=Y in rexI)
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apply (simp_all add: Replace_iff univalent_def, blast)
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done
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subsection{*Instantiating the locale @{text M_trivial}*}
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text{*No instances of Separation yet.*}
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lemma Lset_mono_le: "mono_le_subset(Lset)"
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by (simp add: mono_le_subset_def le_imp_subset Lset_mono)
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lemma Lset_cont: "cont_Ord(Lset)"
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by (simp add: cont_Ord_def Limit_Lset_eq OUnion_def Limit_is_Ord)
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lemmas Pair_in_Lset = Formula.Pair_in_LLimit
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lemmas L_nat = Ord_in_L [OF Ord_nat]
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theorem M_trivial_L: "PROP M_trivial(L)"
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  apply (rule M_trivial.intro)
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        apply (erule (1) transL)
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       apply (rule nonempty)
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      apply (rule upair_ax)
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     apply (rule Union_ax)
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    apply (rule power_ax)
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   apply (rule replacement)
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  apply (rule L_nat)
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  done
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lemmas rall_abs = M_trivial.rall_abs [OF M_trivial_L]
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  and rex_abs = M_trivial.rex_abs [OF M_trivial_L]
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  and ball_iff_equiv = M_trivial.ball_iff_equiv [OF M_trivial_L]
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  and M_equalityI = M_trivial.M_equalityI [OF M_trivial_L]
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  and empty_abs = M_trivial.empty_abs [OF M_trivial_L]
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  and subset_abs = M_trivial.subset_abs [OF M_trivial_L]
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  and upair_abs = M_trivial.upair_abs [OF M_trivial_L]
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  and upair_in_M_iff = M_trivial.upair_in_M_iff [OF M_trivial_L]
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  and singleton_in_M_iff = M_trivial.singleton_in_M_iff [OF M_trivial_L]
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  and pair_abs = M_trivial.pair_abs [OF M_trivial_L]
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  and pair_in_M_iff = M_trivial.pair_in_M_iff [OF M_trivial_L]
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  and pair_components_in_M = M_trivial.pair_components_in_M [OF M_trivial_L]
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  and cartprod_abs = M_trivial.cartprod_abs [OF M_trivial_L]
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  and union_abs = M_trivial.union_abs [OF M_trivial_L]
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  and inter_abs = M_trivial.inter_abs [OF M_trivial_L]
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  and setdiff_abs = M_trivial.setdiff_abs [OF M_trivial_L]
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  and Union_abs = M_trivial.Union_abs [OF M_trivial_L]
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  and Union_closed = M_trivial.Union_closed [OF M_trivial_L]
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  and Un_closed = M_trivial.Un_closed [OF M_trivial_L]
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  and cons_closed = M_trivial.cons_closed [OF M_trivial_L]
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  and successor_abs = M_trivial.successor_abs [OF M_trivial_L]
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  and succ_in_M_iff = M_trivial.succ_in_M_iff [OF M_trivial_L]
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  and separation_closed = M_trivial.separation_closed [OF M_trivial_L]
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  and strong_replacementI = 
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      M_trivial.strong_replacementI [OF M_trivial_L, rule_format]
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  and strong_replacement_closed = M_trivial.strong_replacement_closed [OF M_trivial_L]
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  and RepFun_closed = M_trivial.RepFun_closed [OF M_trivial_L]
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  and lam_closed = M_trivial.lam_closed [OF M_trivial_L]
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  and image_abs = M_trivial.image_abs [OF M_trivial_L]
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  and powerset_Pow = M_trivial.powerset_Pow [OF M_trivial_L]
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  and powerset_imp_subset_Pow = M_trivial.powerset_imp_subset_Pow [OF M_trivial_L]
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  and nat_into_M = M_trivial.nat_into_M [OF M_trivial_L]
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  and nat_case_closed = M_trivial.nat_case_closed [OF M_trivial_L]
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  and Inl_in_M_iff = M_trivial.Inl_in_M_iff [OF M_trivial_L]
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  and Inr_in_M_iff = M_trivial.Inr_in_M_iff [OF M_trivial_L]
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  and lt_closed = M_trivial.lt_closed [OF M_trivial_L]
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  and transitive_set_abs = M_trivial.transitive_set_abs [OF M_trivial_L]
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  and ordinal_abs = M_trivial.ordinal_abs [OF M_trivial_L]
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  and limit_ordinal_abs = M_trivial.limit_ordinal_abs [OF M_trivial_L]
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  and successor_ordinal_abs = M_trivial.successor_ordinal_abs [OF M_trivial_L]
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  and finite_ordinal_abs = M_trivial.finite_ordinal_abs [OF M_trivial_L]
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  and omega_abs = M_trivial.omega_abs [OF M_trivial_L]
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  and number1_abs = M_trivial.number1_abs [OF M_trivial_L]
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  and number2_abs = M_trivial.number2_abs [OF M_trivial_L]
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  and number3_abs = M_trivial.number3_abs [OF M_trivial_L]
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declare rall_abs [simp]
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declare rex_abs [simp]
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declare empty_abs [simp]
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declare subset_abs [simp]
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declare upair_abs [simp]
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declare upair_in_M_iff [iff]
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declare singleton_in_M_iff [iff]
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declare pair_abs [simp]
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declare pair_in_M_iff [iff]
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declare cartprod_abs [simp]
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declare union_abs [simp]
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declare inter_abs [simp]
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declare setdiff_abs [simp]
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declare Union_abs [simp]
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declare Union_closed [intro, simp]
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declare Un_closed [intro, simp]
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declare cons_closed [intro, simp]
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declare successor_abs [simp]
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declare succ_in_M_iff [iff]
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declare separation_closed [intro, simp]
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declare strong_replacement_closed [intro, simp]
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declare RepFun_closed [intro, simp]
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declare lam_closed [intro, simp]
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declare image_abs [simp]
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declare nat_into_M [intro]
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declare Inl_in_M_iff [iff]
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declare Inr_in_M_iff [iff]
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declare transitive_set_abs [simp]
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declare ordinal_abs [simp]
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declare limit_ordinal_abs [simp]
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declare successor_ordinal_abs [simp]
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declare finite_ordinal_abs [simp]
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declare omega_abs [simp]
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declare number1_abs [simp]
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declare number2_abs [simp]
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declare number3_abs [simp]
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subsection{*Instantiation of the locale @{text reflection}*}
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text{*instances of locale constants*}
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constdefs
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  L_F0 :: "[i=>o,i] => i"
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    "L_F0(P,y) == \<mu>b. (\<exists>z. L(z) \<and> P(<y,z>)) --> (\<exists>z\<in>Lset(b). P(<y,z>))"
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  L_FF :: "[i=>o,i] => i"
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    "L_FF(P)   == \<lambda>a. \<Union>y\<in>Lset(a). L_F0(P,y)"
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  L_ClEx :: "[i=>o,i] => o"
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    "L_ClEx(P) == \<lambda>a. Limit(a) \<and> normalize(L_FF(P),a) = a"
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text{*We must use the meta-existential quantifier; otherwise the reflection
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      terms become enormous!*}
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constdefs
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  L_Reflects :: "[i=>o,[i,i]=>o] => prop"      ("(3REFLECTS/ [_,/ _])")
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    "REFLECTS[P,Q] == (??Cl. Closed_Unbounded(Cl) &
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                           (\<forall>a. Cl(a) --> (\<forall>x \<in> Lset(a). P(x) <-> Q(a,x))))"
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theorem Triv_reflection:
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     "REFLECTS[P, \<lambda>a x. P(x)]"
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apply (simp add: L_Reflects_def)
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apply (rule meta_exI)
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apply (rule Closed_Unbounded_Ord)
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done
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theorem Not_reflection:
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     "REFLECTS[P,Q] ==> REFLECTS[\<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x)]"
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apply (unfold L_Reflects_def)
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apply (erule meta_exE)
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apply (rule_tac x=Cl in meta_exI, simp)
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done
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theorem And_reflection:
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     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
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      ==> REFLECTS[\<lambda>x. P(x) \<and> P'(x), \<lambda>a x. Q(a,x) \<and> Q'(a,x)]"
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apply (unfold L_Reflects_def)
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apply (elim meta_exE)
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
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apply (simp add: Closed_Unbounded_Int, blast)
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done
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theorem Or_reflection:
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     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
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      ==> REFLECTS[\<lambda>x. P(x) \<or> P'(x), \<lambda>a x. Q(a,x) \<or> Q'(a,x)]"
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apply (unfold L_Reflects_def)
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apply (elim meta_exE)
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
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apply (simp add: Closed_Unbounded_Int, blast)
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done
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theorem Imp_reflection:
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     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
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      ==> REFLECTS[\<lambda>x. P(x) --> P'(x), \<lambda>a x. Q(a,x) --> Q'(a,x)]"
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apply (unfold L_Reflects_def)
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apply (elim meta_exE)
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
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apply (simp add: Closed_Unbounded_Int, blast)
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done
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theorem Iff_reflection:
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     "[| REFLECTS[P,Q]; REFLECTS[P',Q'] |]
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      ==> REFLECTS[\<lambda>x. P(x) <-> P'(x), \<lambda>a x. Q(a,x) <-> Q'(a,x)]"
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apply (unfold L_Reflects_def)
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apply (elim meta_exE)
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apply (rule_tac x="\<lambda>a. Cl(a) \<and> Cla(a)" in meta_exI)
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apply (simp add: Closed_Unbounded_Int, blast)
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done
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lemma reflection_Lset: "reflection(Lset)"
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apply (blast intro: reflection.intro Lset_mono_le Lset_cont Pair_in_Lset) +
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done
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theorem Ex_reflection:
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     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
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      ==> REFLECTS[\<lambda>x. \<exists>z. L(z) \<and> P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
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apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
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apply (elim meta_exE)
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apply (rule meta_exI)
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apply (erule reflection.Ex_reflection [OF reflection_Lset])
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done
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theorem All_reflection:
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     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
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      ==> REFLECTS[\<lambda>x. \<forall>z. L(z) --> P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]"
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apply (unfold L_Reflects_def L_ClEx_def L_FF_def L_F0_def L_def)
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apply (elim meta_exE)
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apply (rule meta_exI)
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apply (erule reflection.All_reflection [OF reflection_Lset])
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done
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theorem Rex_reflection:
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     "REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
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      ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z\<in>Lset(a). Q(a,x,z)]"
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apply (unfold rex_def)
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apply (intro And_reflection Ex_reflection, assumption)
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done
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theorem Rall_reflection:
paulson@13314
   294
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
wenzelm@13429
   295
      ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z\<in>Lset(a). Q(a,x,z)]"
wenzelm@13429
   296
apply (unfold rall_def)
paulson@13314
   297
apply (intro Imp_reflection All_reflection, assumption)
paulson@13314
   298
done
paulson@13314
   299
paulson@13440
   300
text{*This version handles an alternative form of the bounded quantifier
paulson@13440
   301
      in the second argument of @{text REFLECTS}.*}
paulson@13440
   302
theorem Rex_reflection':
paulson@13440
   303
     "REFLECTS[ \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
paulson@13440
   304
      ==> REFLECTS[\<lambda>x. \<exists>z[L]. P(x,z), \<lambda>a x. \<exists>z[**Lset(a)]. Q(a,x,z)]"
paulson@13440
   305
apply (unfold setclass_def rex_def)
paulson@13440
   306
apply (erule Rex_reflection [unfolded rex_def Bex_def]) 
paulson@13440
   307
done
paulson@13440
   308
paulson@13440
   309
text{*As above.*}
paulson@13440
   310
theorem Rall_reflection':
paulson@13440
   311
     "REFLECTS[\<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x))]
paulson@13440
   312
      ==> REFLECTS[\<lambda>x. \<forall>z[L]. P(x,z), \<lambda>a x. \<forall>z[**Lset(a)]. Q(a,x,z)]"
paulson@13440
   313
apply (unfold setclass_def rall_def)
paulson@13440
   314
apply (erule Rall_reflection [unfolded rall_def Ball_def]) 
paulson@13440
   315
done
paulson@13440
   316
wenzelm@13429
   317
lemmas FOL_reflections =
paulson@13314
   318
        Triv_reflection Not_reflection And_reflection Or_reflection
paulson@13314
   319
        Imp_reflection Iff_reflection Ex_reflection All_reflection
paulson@13440
   320
        Rex_reflection Rall_reflection Rex_reflection' Rall_reflection'
paulson@13291
   321
paulson@13291
   322
lemma ReflectsD:
wenzelm@13429
   323
     "[|REFLECTS[P,Q]; Ord(i)|]
paulson@13291
   324
      ==> \<exists>j. i<j & (\<forall>x \<in> Lset(j). P(x) <-> Q(j,x))"
wenzelm@13429
   325
apply (unfold L_Reflects_def Closed_Unbounded_def)
wenzelm@13429
   326
apply (elim meta_exE, clarify)
wenzelm@13429
   327
apply (blast dest!: UnboundedD)
paulson@13291
   328
done
paulson@13291
   329
paulson@13291
   330
lemma ReflectsE:
paulson@13314
   331
     "[| REFLECTS[P,Q]; Ord(i);
paulson@13291
   332
         !!j. [|i<j;  \<forall>x \<in> Lset(j). P(x) <-> Q(j,x)|] ==> R |]
paulson@13291
   333
      ==> R"
wenzelm@13429
   334
apply (drule ReflectsD, assumption, blast)
paulson@13314
   335
done
paulson@13291
   336
wenzelm@13428
   337
lemma Collect_mem_eq: "{x\<in>A. x\<in>B} = A \<inter> B"
paulson@13291
   338
by blast
paulson@13291
   339
paulson@13291
   340
paulson@13339
   341
subsection{*Internalized Formulas for some Set-Theoretic Concepts*}
paulson@13298
   342
paulson@13306
   343
lemmas setclass_simps = rall_setclass_is_ball rex_setclass_is_bex
paulson@13306
   344
paulson@13306
   345
subsubsection{*Some numbers to help write de Bruijn indices*}
paulson@13306
   346
paulson@13306
   347
syntax
paulson@13306
   348
    "3" :: i   ("3")
paulson@13306
   349
    "4" :: i   ("4")
paulson@13306
   350
    "5" :: i   ("5")
paulson@13306
   351
    "6" :: i   ("6")
paulson@13306
   352
    "7" :: i   ("7")
paulson@13306
   353
    "8" :: i   ("8")
paulson@13306
   354
    "9" :: i   ("9")
paulson@13306
   355
paulson@13306
   356
translations
paulson@13306
   357
   "3"  == "succ(2)"
paulson@13306
   358
   "4"  == "succ(3)"
paulson@13306
   359
   "5"  == "succ(4)"
paulson@13306
   360
   "6"  == "succ(5)"
paulson@13306
   361
   "7"  == "succ(6)"
paulson@13306
   362
   "8"  == "succ(7)"
paulson@13306
   363
   "9"  == "succ(8)"
paulson@13306
   364
paulson@13323
   365
paulson@13339
   366
subsubsection{*The Empty Set, Internalized*}
paulson@13323
   367
paulson@13323
   368
constdefs empty_fm :: "i=>i"
paulson@13323
   369
    "empty_fm(x) == Forall(Neg(Member(0,succ(x))))"
paulson@13323
   370
paulson@13323
   371
lemma empty_type [TC]:
paulson@13323
   372
     "x \<in> nat ==> empty_fm(x) \<in> formula"
wenzelm@13429
   373
by (simp add: empty_fm_def)
paulson@13323
   374
paulson@13323
   375
lemma arity_empty_fm [simp]:
paulson@13323
   376
     "x \<in> nat ==> arity(empty_fm(x)) = succ(x)"
wenzelm@13429
   377
by (simp add: empty_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13323
   378
paulson@13323
   379
lemma sats_empty_fm [simp]:
paulson@13323
   380
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13323
   381
    ==> sats(A, empty_fm(x), env) <-> empty(**A, nth(x,env))"
paulson@13323
   382
by (simp add: empty_fm_def empty_def)
paulson@13323
   383
paulson@13323
   384
lemma empty_iff_sats:
wenzelm@13429
   385
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13323
   386
          i \<in> nat; env \<in> list(A)|]
paulson@13323
   387
       ==> empty(**A, x) <-> sats(A, empty_fm(i), env)"
paulson@13323
   388
by simp
paulson@13323
   389
paulson@13323
   390
theorem empty_reflection:
wenzelm@13429
   391
     "REFLECTS[\<lambda>x. empty(L,f(x)),
paulson@13323
   392
               \<lambda>i x. empty(**Lset(i),f(x))]"
paulson@13323
   393
apply (simp only: empty_def setclass_simps)
wenzelm@13429
   394
apply (intro FOL_reflections)
paulson@13323
   395
done
paulson@13323
   396
paulson@13385
   397
text{*Not used.  But maybe useful?*}
paulson@13385
   398
lemma Transset_sats_empty_fm_eq_0:
paulson@13385
   399
   "[| n \<in> nat; env \<in> list(A); Transset(A)|]
paulson@13385
   400
    ==> sats(A, empty_fm(n), env) <-> nth(n,env) = 0"
paulson@13385
   401
apply (simp add: empty_fm_def empty_def Transset_def, auto)
wenzelm@13429
   402
apply (case_tac "n < length(env)")
wenzelm@13429
   403
apply (frule nth_type, assumption+, blast)
wenzelm@13429
   404
apply (simp_all add: not_lt_iff_le nth_eq_0)
paulson@13385
   405
done
paulson@13385
   406
paulson@13323
   407
paulson@13339
   408
subsubsection{*Unordered Pairs, Internalized*}
paulson@13298
   409
paulson@13298
   410
constdefs upair_fm :: "[i,i,i]=>i"
wenzelm@13429
   411
    "upair_fm(x,y,z) ==
wenzelm@13429
   412
       And(Member(x,z),
paulson@13298
   413
           And(Member(y,z),
wenzelm@13429
   414
               Forall(Implies(Member(0,succ(z)),
paulson@13298
   415
                              Or(Equal(0,succ(x)), Equal(0,succ(y)))))))"
paulson@13298
   416
paulson@13298
   417
lemma upair_type [TC]:
paulson@13298
   418
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> upair_fm(x,y,z) \<in> formula"
wenzelm@13429
   419
by (simp add: upair_fm_def)
paulson@13298
   420
paulson@13298
   421
lemma arity_upair_fm [simp]:
wenzelm@13429
   422
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13298
   423
      ==> arity(upair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
   424
by (simp add: upair_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13298
   425
paulson@13298
   426
lemma sats_upair_fm [simp]:
paulson@13298
   427
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
   428
    ==> sats(A, upair_fm(x,y,z), env) <->
paulson@13298
   429
            upair(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13298
   430
by (simp add: upair_fm_def upair_def)
paulson@13298
   431
paulson@13298
   432
lemma upair_iff_sats:
wenzelm@13429
   433
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13298
   434
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13298
   435
       ==> upair(**A, x, y, z) <-> sats(A, upair_fm(i,j,k), env)"
paulson@13298
   436
by (simp add: sats_upair_fm)
paulson@13298
   437
paulson@13298
   438
text{*Useful? At least it refers to "real" unordered pairs*}
paulson@13298
   439
lemma sats_upair_fm2 [simp]:
paulson@13298
   440
   "[| x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A); Transset(A)|]
wenzelm@13429
   441
    ==> sats(A, upair_fm(x,y,z), env) <->
paulson@13298
   442
        nth(z,env) = {nth(x,env), nth(y,env)}"
wenzelm@13429
   443
apply (frule lt_length_in_nat, assumption)
wenzelm@13429
   444
apply (simp add: upair_fm_def Transset_def, auto)
wenzelm@13429
   445
apply (blast intro: nth_type)
paulson@13298
   446
done
paulson@13298
   447
paulson@13314
   448
theorem upair_reflection:
wenzelm@13429
   449
     "REFLECTS[\<lambda>x. upair(L,f(x),g(x),h(x)),
wenzelm@13429
   450
               \<lambda>i x. upair(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   451
apply (simp add: upair_def)
wenzelm@13429
   452
apply (intro FOL_reflections)
paulson@13314
   453
done
paulson@13306
   454
paulson@13339
   455
subsubsection{*Ordered pairs, Internalized*}
paulson@13298
   456
paulson@13298
   457
constdefs pair_fm :: "[i,i,i]=>i"
wenzelm@13429
   458
    "pair_fm(x,y,z) ==
paulson@13298
   459
       Exists(And(upair_fm(succ(x),succ(x),0),
paulson@13298
   460
              Exists(And(upair_fm(succ(succ(x)),succ(succ(y)),0),
paulson@13298
   461
                         upair_fm(1,0,succ(succ(z)))))))"
paulson@13298
   462
paulson@13298
   463
lemma pair_type [TC]:
paulson@13298
   464
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pair_fm(x,y,z) \<in> formula"
wenzelm@13429
   465
by (simp add: pair_fm_def)
paulson@13298
   466
paulson@13298
   467
lemma arity_pair_fm [simp]:
wenzelm@13429
   468
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13298
   469
      ==> arity(pair_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
   470
by (simp add: pair_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13298
   471
paulson@13298
   472
lemma sats_pair_fm [simp]:
paulson@13298
   473
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
   474
    ==> sats(A, pair_fm(x,y,z), env) <->
paulson@13298
   475
        pair(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13298
   476
by (simp add: pair_fm_def pair_def)
paulson@13298
   477
paulson@13298
   478
lemma pair_iff_sats:
wenzelm@13429
   479
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13298
   480
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13298
   481
       ==> pair(**A, x, y, z) <-> sats(A, pair_fm(i,j,k), env)"
paulson@13298
   482
by (simp add: sats_pair_fm)
paulson@13298
   483
paulson@13314
   484
theorem pair_reflection:
wenzelm@13429
   485
     "REFLECTS[\<lambda>x. pair(L,f(x),g(x),h(x)),
paulson@13314
   486
               \<lambda>i x. pair(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   487
apply (simp only: pair_def setclass_simps)
wenzelm@13429
   488
apply (intro FOL_reflections upair_reflection)
paulson@13314
   489
done
paulson@13306
   490
paulson@13306
   491
paulson@13339
   492
subsubsection{*Binary Unions, Internalized*}
paulson@13298
   493
paulson@13306
   494
constdefs union_fm :: "[i,i,i]=>i"
wenzelm@13429
   495
    "union_fm(x,y,z) ==
paulson@13306
   496
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   497
                  Or(Member(0,succ(x)),Member(0,succ(y)))))"
paulson@13306
   498
paulson@13306
   499
lemma union_type [TC]:
paulson@13306
   500
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> union_fm(x,y,z) \<in> formula"
wenzelm@13429
   501
by (simp add: union_fm_def)
paulson@13306
   502
paulson@13306
   503
lemma arity_union_fm [simp]:
wenzelm@13429
   504
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13306
   505
      ==> arity(union_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
   506
by (simp add: union_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13298
   507
paulson@13306
   508
lemma sats_union_fm [simp]:
paulson@13306
   509
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
   510
    ==> sats(A, union_fm(x,y,z), env) <->
paulson@13306
   511
        union(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13306
   512
by (simp add: union_fm_def union_def)
paulson@13306
   513
paulson@13306
   514
lemma union_iff_sats:
wenzelm@13429
   515
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13306
   516
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   517
       ==> union(**A, x, y, z) <-> sats(A, union_fm(i,j,k), env)"
paulson@13306
   518
by (simp add: sats_union_fm)
paulson@13298
   519
paulson@13314
   520
theorem union_reflection:
wenzelm@13429
   521
     "REFLECTS[\<lambda>x. union(L,f(x),g(x),h(x)),
paulson@13314
   522
               \<lambda>i x. union(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   523
apply (simp only: union_def setclass_simps)
wenzelm@13429
   524
apply (intro FOL_reflections)
paulson@13314
   525
done
paulson@13306
   526
paulson@13298
   527
paulson@13339
   528
subsubsection{*Set ``Cons,'' Internalized*}
paulson@13306
   529
paulson@13306
   530
constdefs cons_fm :: "[i,i,i]=>i"
wenzelm@13429
   531
    "cons_fm(x,y,z) ==
paulson@13306
   532
       Exists(And(upair_fm(succ(x),succ(x),0),
paulson@13306
   533
                  union_fm(0,succ(y),succ(z))))"
paulson@13298
   534
paulson@13298
   535
paulson@13306
   536
lemma cons_type [TC]:
paulson@13306
   537
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cons_fm(x,y,z) \<in> formula"
wenzelm@13429
   538
by (simp add: cons_fm_def)
paulson@13306
   539
paulson@13306
   540
lemma arity_cons_fm [simp]:
wenzelm@13429
   541
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13306
   542
      ==> arity(cons_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
   543
by (simp add: cons_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13306
   544
paulson@13306
   545
lemma sats_cons_fm [simp]:
paulson@13306
   546
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
   547
    ==> sats(A, cons_fm(x,y,z), env) <->
paulson@13306
   548
        is_cons(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13306
   549
by (simp add: cons_fm_def is_cons_def)
paulson@13306
   550
paulson@13306
   551
lemma cons_iff_sats:
wenzelm@13429
   552
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13306
   553
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   554
       ==> is_cons(**A, x, y, z) <-> sats(A, cons_fm(i,j,k), env)"
paulson@13306
   555
by simp
paulson@13306
   556
paulson@13314
   557
theorem cons_reflection:
wenzelm@13429
   558
     "REFLECTS[\<lambda>x. is_cons(L,f(x),g(x),h(x)),
paulson@13314
   559
               \<lambda>i x. is_cons(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
   560
apply (simp only: is_cons_def setclass_simps)
wenzelm@13429
   561
apply (intro FOL_reflections upair_reflection union_reflection)
paulson@13323
   562
done
paulson@13323
   563
paulson@13323
   564
paulson@13339
   565
subsubsection{*Successor Function, Internalized*}
paulson@13323
   566
paulson@13323
   567
constdefs succ_fm :: "[i,i]=>i"
paulson@13323
   568
    "succ_fm(x,y) == cons_fm(x,x,y)"
paulson@13323
   569
paulson@13323
   570
lemma succ_type [TC]:
paulson@13323
   571
     "[| x \<in> nat; y \<in> nat |] ==> succ_fm(x,y) \<in> formula"
wenzelm@13429
   572
by (simp add: succ_fm_def)
paulson@13323
   573
paulson@13323
   574
lemma arity_succ_fm [simp]:
wenzelm@13429
   575
     "[| x \<in> nat; y \<in> nat |]
paulson@13323
   576
      ==> arity(succ_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13323
   577
by (simp add: succ_fm_def)
paulson@13323
   578
paulson@13323
   579
lemma sats_succ_fm [simp]:
paulson@13323
   580
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
wenzelm@13429
   581
    ==> sats(A, succ_fm(x,y), env) <->
paulson@13323
   582
        successor(**A, nth(x,env), nth(y,env))"
paulson@13323
   583
by (simp add: succ_fm_def successor_def)
paulson@13323
   584
paulson@13323
   585
lemma successor_iff_sats:
wenzelm@13429
   586
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13323
   587
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13323
   588
       ==> successor(**A, x, y) <-> sats(A, succ_fm(i,j), env)"
paulson@13323
   589
by simp
paulson@13323
   590
paulson@13323
   591
theorem successor_reflection:
wenzelm@13429
   592
     "REFLECTS[\<lambda>x. successor(L,f(x),g(x)),
paulson@13323
   593
               \<lambda>i x. successor(**Lset(i),f(x),g(x))]"
paulson@13323
   594
apply (simp only: successor_def setclass_simps)
wenzelm@13429
   595
apply (intro cons_reflection)
paulson@13314
   596
done
paulson@13298
   597
paulson@13298
   598
paulson@13363
   599
subsubsection{*The Number 1, Internalized*}
paulson@13363
   600
paulson@13363
   601
(* "number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" *)
paulson@13363
   602
constdefs number1_fm :: "i=>i"
paulson@13363
   603
    "number1_fm(a) == Exists(And(empty_fm(0), succ_fm(0,succ(a))))"
paulson@13363
   604
paulson@13363
   605
lemma number1_type [TC]:
paulson@13363
   606
     "x \<in> nat ==> number1_fm(x) \<in> formula"
wenzelm@13429
   607
by (simp add: number1_fm_def)
paulson@13363
   608
paulson@13363
   609
lemma arity_number1_fm [simp]:
paulson@13363
   610
     "x \<in> nat ==> arity(number1_fm(x)) = succ(x)"
wenzelm@13429
   611
by (simp add: number1_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13363
   612
paulson@13363
   613
lemma sats_number1_fm [simp]:
paulson@13363
   614
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13363
   615
    ==> sats(A, number1_fm(x), env) <-> number1(**A, nth(x,env))"
paulson@13363
   616
by (simp add: number1_fm_def number1_def)
paulson@13363
   617
paulson@13363
   618
lemma number1_iff_sats:
wenzelm@13429
   619
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13363
   620
          i \<in> nat; env \<in> list(A)|]
paulson@13363
   621
       ==> number1(**A, x) <-> sats(A, number1_fm(i), env)"
paulson@13363
   622
by simp
paulson@13363
   623
paulson@13363
   624
theorem number1_reflection:
wenzelm@13429
   625
     "REFLECTS[\<lambda>x. number1(L,f(x)),
paulson@13363
   626
               \<lambda>i x. number1(**Lset(i),f(x))]"
paulson@13363
   627
apply (simp only: number1_def setclass_simps)
paulson@13363
   628
apply (intro FOL_reflections empty_reflection successor_reflection)
paulson@13363
   629
done
paulson@13363
   630
paulson@13363
   631
paulson@13352
   632
subsubsection{*Big Union, Internalized*}
paulson@13306
   633
paulson@13352
   634
(*  "big_union(M,A,z) == \<forall>x[M]. x \<in> z <-> (\<exists>y[M]. y\<in>A & x \<in> y)" *)
paulson@13352
   635
constdefs big_union_fm :: "[i,i]=>i"
wenzelm@13429
   636
    "big_union_fm(A,z) ==
paulson@13352
   637
       Forall(Iff(Member(0,succ(z)),
paulson@13352
   638
                  Exists(And(Member(0,succ(succ(A))), Member(1,0)))))"
paulson@13298
   639
paulson@13352
   640
lemma big_union_type [TC]:
paulson@13352
   641
     "[| x \<in> nat; y \<in> nat |] ==> big_union_fm(x,y) \<in> formula"
wenzelm@13429
   642
by (simp add: big_union_fm_def)
paulson@13306
   643
paulson@13352
   644
lemma arity_big_union_fm [simp]:
wenzelm@13429
   645
     "[| x \<in> nat; y \<in> nat |]
paulson@13352
   646
      ==> arity(big_union_fm(x,y)) = succ(x) \<union> succ(y)"
paulson@13352
   647
by (simp add: big_union_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13298
   648
paulson@13352
   649
lemma sats_big_union_fm [simp]:
paulson@13352
   650
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
wenzelm@13429
   651
    ==> sats(A, big_union_fm(x,y), env) <->
paulson@13352
   652
        big_union(**A, nth(x,env), nth(y,env))"
paulson@13352
   653
by (simp add: big_union_fm_def big_union_def)
paulson@13306
   654
paulson@13352
   655
lemma big_union_iff_sats:
wenzelm@13429
   656
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13352
   657
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13352
   658
       ==> big_union(**A, x, y) <-> sats(A, big_union_fm(i,j), env)"
paulson@13306
   659
by simp
paulson@13306
   660
paulson@13352
   661
theorem big_union_reflection:
wenzelm@13429
   662
     "REFLECTS[\<lambda>x. big_union(L,f(x),g(x)),
paulson@13352
   663
               \<lambda>i x. big_union(**Lset(i),f(x),g(x))]"
paulson@13352
   664
apply (simp only: big_union_def setclass_simps)
wenzelm@13429
   665
apply (intro FOL_reflections)
paulson@13314
   666
done
paulson@13298
   667
paulson@13298
   668
paulson@13306
   669
subsubsection{*Variants of Satisfaction Definitions for Ordinals, etc.*}
paulson@13306
   670
paulson@13306
   671
text{*Differs from the one in Formula by using "ordinal" rather than "Ord"*}
paulson@13306
   672
paulson@13306
   673
paulson@13306
   674
lemma sats_subset_fm':
paulson@13306
   675
   "[|x \<in> nat; y \<in> nat; env \<in> list(A)|]
wenzelm@13429
   676
    ==> sats(A, subset_fm(x,y), env) <-> subset(**A, nth(x,env), nth(y,env))"
wenzelm@13429
   677
by (simp add: subset_fm_def Relative.subset_def)
paulson@13298
   678
paulson@13314
   679
theorem subset_reflection:
wenzelm@13429
   680
     "REFLECTS[\<lambda>x. subset(L,f(x),g(x)),
wenzelm@13429
   681
               \<lambda>i x. subset(**Lset(i),f(x),g(x))]"
paulson@13323
   682
apply (simp only: Relative.subset_def setclass_simps)
wenzelm@13429
   683
apply (intro FOL_reflections)
paulson@13314
   684
done
paulson@13306
   685
paulson@13306
   686
lemma sats_transset_fm':
paulson@13306
   687
   "[|x \<in> nat; env \<in> list(A)|]
paulson@13306
   688
    ==> sats(A, transset_fm(x), env) <-> transitive_set(**A, nth(x,env))"
wenzelm@13429
   689
by (simp add: sats_subset_fm' transset_fm_def transitive_set_def)
paulson@13298
   690
paulson@13314
   691
theorem transitive_set_reflection:
paulson@13314
   692
     "REFLECTS[\<lambda>x. transitive_set(L,f(x)),
paulson@13314
   693
               \<lambda>i x. transitive_set(**Lset(i),f(x))]"
paulson@13314
   694
apply (simp only: transitive_set_def setclass_simps)
wenzelm@13429
   695
apply (intro FOL_reflections subset_reflection)
paulson@13314
   696
done
paulson@13306
   697
paulson@13306
   698
lemma sats_ordinal_fm':
paulson@13306
   699
   "[|x \<in> nat; env \<in> list(A)|]
paulson@13306
   700
    ==> sats(A, ordinal_fm(x), env) <-> ordinal(**A,nth(x,env))"
paulson@13306
   701
by (simp add: sats_transset_fm' ordinal_fm_def ordinal_def)
paulson@13306
   702
paulson@13306
   703
lemma ordinal_iff_sats:
paulson@13306
   704
      "[| nth(i,env) = x;  i \<in> nat; env \<in> list(A)|]
paulson@13306
   705
       ==> ordinal(**A, x) <-> sats(A, ordinal_fm(i), env)"
paulson@13306
   706
by (simp add: sats_ordinal_fm')
paulson@13306
   707
paulson@13314
   708
theorem ordinal_reflection:
paulson@13314
   709
     "REFLECTS[\<lambda>x. ordinal(L,f(x)), \<lambda>i x. ordinal(**Lset(i),f(x))]"
paulson@13314
   710
apply (simp only: ordinal_def setclass_simps)
wenzelm@13429
   711
apply (intro FOL_reflections transitive_set_reflection)
paulson@13314
   712
done
paulson@13298
   713
paulson@13298
   714
paulson@13339
   715
subsubsection{*Membership Relation, Internalized*}
paulson@13298
   716
paulson@13306
   717
constdefs Memrel_fm :: "[i,i]=>i"
wenzelm@13429
   718
    "Memrel_fm(A,r) ==
paulson@13306
   719
       Forall(Iff(Member(0,succ(r)),
paulson@13306
   720
                  Exists(And(Member(0,succ(succ(A))),
paulson@13306
   721
                             Exists(And(Member(0,succ(succ(succ(A)))),
paulson@13306
   722
                                        And(Member(1,0),
paulson@13306
   723
                                            pair_fm(1,0,2))))))))"
paulson@13306
   724
paulson@13306
   725
lemma Memrel_type [TC]:
paulson@13306
   726
     "[| x \<in> nat; y \<in> nat |] ==> Memrel_fm(x,y) \<in> formula"
wenzelm@13429
   727
by (simp add: Memrel_fm_def)
paulson@13298
   728
paulson@13306
   729
lemma arity_Memrel_fm [simp]:
wenzelm@13429
   730
     "[| x \<in> nat; y \<in> nat |]
paulson@13306
   731
      ==> arity(Memrel_fm(x,y)) = succ(x) \<union> succ(y)"
wenzelm@13429
   732
by (simp add: Memrel_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13306
   733
paulson@13306
   734
lemma sats_Memrel_fm [simp]:
paulson@13306
   735
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
wenzelm@13429
   736
    ==> sats(A, Memrel_fm(x,y), env) <->
paulson@13306
   737
        membership(**A, nth(x,env), nth(y,env))"
paulson@13306
   738
by (simp add: Memrel_fm_def membership_def)
paulson@13298
   739
paulson@13306
   740
lemma Memrel_iff_sats:
wenzelm@13429
   741
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13306
   742
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   743
       ==> membership(**A, x, y) <-> sats(A, Memrel_fm(i,j), env)"
paulson@13306
   744
by simp
paulson@13304
   745
paulson@13314
   746
theorem membership_reflection:
wenzelm@13429
   747
     "REFLECTS[\<lambda>x. membership(L,f(x),g(x)),
paulson@13314
   748
               \<lambda>i x. membership(**Lset(i),f(x),g(x))]"
paulson@13314
   749
apply (simp only: membership_def setclass_simps)
wenzelm@13429
   750
apply (intro FOL_reflections pair_reflection)
paulson@13314
   751
done
paulson@13304
   752
paulson@13339
   753
subsubsection{*Predecessor Set, Internalized*}
paulson@13304
   754
paulson@13306
   755
constdefs pred_set_fm :: "[i,i,i,i]=>i"
wenzelm@13429
   756
    "pred_set_fm(A,x,r,B) ==
paulson@13306
   757
       Forall(Iff(Member(0,succ(B)),
paulson@13306
   758
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   759
                             And(Member(1,succ(succ(A))),
paulson@13306
   760
                                 pair_fm(1,succ(succ(x)),0))))))"
paulson@13306
   761
paulson@13306
   762
paulson@13306
   763
lemma pred_set_type [TC]:
wenzelm@13429
   764
     "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |]
paulson@13306
   765
      ==> pred_set_fm(A,x,r,B) \<in> formula"
wenzelm@13429
   766
by (simp add: pred_set_fm_def)
paulson@13304
   767
paulson@13306
   768
lemma arity_pred_set_fm [simp]:
wenzelm@13429
   769
   "[| A \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat |]
paulson@13306
   770
    ==> arity(pred_set_fm(A,x,r,B)) = succ(A) \<union> succ(x) \<union> succ(r) \<union> succ(B)"
wenzelm@13429
   771
by (simp add: pred_set_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13306
   772
paulson@13306
   773
lemma sats_pred_set_fm [simp]:
paulson@13306
   774
   "[| U \<in> nat; x \<in> nat; r \<in> nat; B \<in> nat; env \<in> list(A)|]
wenzelm@13429
   775
    ==> sats(A, pred_set_fm(U,x,r,B), env) <->
paulson@13306
   776
        pred_set(**A, nth(U,env), nth(x,env), nth(r,env), nth(B,env))"
paulson@13306
   777
by (simp add: pred_set_fm_def pred_set_def)
paulson@13306
   778
paulson@13306
   779
lemma pred_set_iff_sats:
wenzelm@13429
   780
      "[| nth(i,env) = U; nth(j,env) = x; nth(k,env) = r; nth(l,env) = B;
paulson@13306
   781
          i \<in> nat; j \<in> nat; k \<in> nat; l \<in> nat; env \<in> list(A)|]
paulson@13306
   782
       ==> pred_set(**A,U,x,r,B) <-> sats(A, pred_set_fm(i,j,k,l), env)"
paulson@13306
   783
by (simp add: sats_pred_set_fm)
paulson@13306
   784
paulson@13314
   785
theorem pred_set_reflection:
wenzelm@13429
   786
     "REFLECTS[\<lambda>x. pred_set(L,f(x),g(x),h(x),b(x)),
wenzelm@13429
   787
               \<lambda>i x. pred_set(**Lset(i),f(x),g(x),h(x),b(x))]"
paulson@13314
   788
apply (simp only: pred_set_def setclass_simps)
wenzelm@13429
   789
apply (intro FOL_reflections pair_reflection)
paulson@13314
   790
done
paulson@13304
   791
paulson@13304
   792
paulson@13298
   793
paulson@13339
   794
subsubsection{*Domain of a Relation, Internalized*}
paulson@13306
   795
wenzelm@13429
   796
(* "is_domain(M,r,z) ==
wenzelm@13429
   797
        \<forall>x[M]. (x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. pair(M,x,y,w))))" *)
paulson@13306
   798
constdefs domain_fm :: "[i,i]=>i"
wenzelm@13429
   799
    "domain_fm(r,z) ==
paulson@13306
   800
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   801
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   802
                             Exists(pair_fm(2,0,1))))))"
paulson@13306
   803
paulson@13306
   804
lemma domain_type [TC]:
paulson@13306
   805
     "[| x \<in> nat; y \<in> nat |] ==> domain_fm(x,y) \<in> formula"
wenzelm@13429
   806
by (simp add: domain_fm_def)
paulson@13306
   807
paulson@13306
   808
lemma arity_domain_fm [simp]:
wenzelm@13429
   809
     "[| x \<in> nat; y \<in> nat |]
paulson@13306
   810
      ==> arity(domain_fm(x,y)) = succ(x) \<union> succ(y)"
wenzelm@13429
   811
by (simp add: domain_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13306
   812
paulson@13306
   813
lemma sats_domain_fm [simp]:
paulson@13306
   814
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
wenzelm@13429
   815
    ==> sats(A, domain_fm(x,y), env) <->
paulson@13306
   816
        is_domain(**A, nth(x,env), nth(y,env))"
paulson@13306
   817
by (simp add: domain_fm_def is_domain_def)
paulson@13306
   818
paulson@13306
   819
lemma domain_iff_sats:
wenzelm@13429
   820
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13306
   821
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   822
       ==> is_domain(**A, x, y) <-> sats(A, domain_fm(i,j), env)"
paulson@13306
   823
by simp
paulson@13306
   824
paulson@13314
   825
theorem domain_reflection:
wenzelm@13429
   826
     "REFLECTS[\<lambda>x. is_domain(L,f(x),g(x)),
paulson@13314
   827
               \<lambda>i x. is_domain(**Lset(i),f(x),g(x))]"
paulson@13314
   828
apply (simp only: is_domain_def setclass_simps)
wenzelm@13429
   829
apply (intro FOL_reflections pair_reflection)
paulson@13314
   830
done
paulson@13306
   831
paulson@13306
   832
paulson@13339
   833
subsubsection{*Range of a Relation, Internalized*}
paulson@13306
   834
wenzelm@13429
   835
(* "is_range(M,r,z) ==
wenzelm@13429
   836
        \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. pair(M,x,y,w))))" *)
paulson@13306
   837
constdefs range_fm :: "[i,i]=>i"
wenzelm@13429
   838
    "range_fm(r,z) ==
paulson@13306
   839
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   840
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   841
                             Exists(pair_fm(0,2,1))))))"
paulson@13306
   842
paulson@13306
   843
lemma range_type [TC]:
paulson@13306
   844
     "[| x \<in> nat; y \<in> nat |] ==> range_fm(x,y) \<in> formula"
wenzelm@13429
   845
by (simp add: range_fm_def)
paulson@13306
   846
paulson@13306
   847
lemma arity_range_fm [simp]:
wenzelm@13429
   848
     "[| x \<in> nat; y \<in> nat |]
paulson@13306
   849
      ==> arity(range_fm(x,y)) = succ(x) \<union> succ(y)"
wenzelm@13429
   850
by (simp add: range_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13306
   851
paulson@13306
   852
lemma sats_range_fm [simp]:
paulson@13306
   853
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
wenzelm@13429
   854
    ==> sats(A, range_fm(x,y), env) <->
paulson@13306
   855
        is_range(**A, nth(x,env), nth(y,env))"
paulson@13306
   856
by (simp add: range_fm_def is_range_def)
paulson@13306
   857
paulson@13306
   858
lemma range_iff_sats:
wenzelm@13429
   859
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13306
   860
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13306
   861
       ==> is_range(**A, x, y) <-> sats(A, range_fm(i,j), env)"
paulson@13306
   862
by simp
paulson@13306
   863
paulson@13314
   864
theorem range_reflection:
wenzelm@13429
   865
     "REFLECTS[\<lambda>x. is_range(L,f(x),g(x)),
paulson@13314
   866
               \<lambda>i x. is_range(**Lset(i),f(x),g(x))]"
paulson@13314
   867
apply (simp only: is_range_def setclass_simps)
wenzelm@13429
   868
apply (intro FOL_reflections pair_reflection)
paulson@13314
   869
done
paulson@13306
   870
wenzelm@13429
   871
paulson@13339
   872
subsubsection{*Field of a Relation, Internalized*}
paulson@13323
   873
wenzelm@13429
   874
(* "is_field(M,r,z) ==
wenzelm@13429
   875
        \<exists>dr[M]. is_domain(M,r,dr) &
paulson@13323
   876
            (\<exists>rr[M]. is_range(M,r,rr) & union(M,dr,rr,z))" *)
paulson@13323
   877
constdefs field_fm :: "[i,i]=>i"
wenzelm@13429
   878
    "field_fm(r,z) ==
wenzelm@13429
   879
       Exists(And(domain_fm(succ(r),0),
wenzelm@13429
   880
              Exists(And(range_fm(succ(succ(r)),0),
paulson@13323
   881
                         union_fm(1,0,succ(succ(z)))))))"
paulson@13323
   882
paulson@13323
   883
lemma field_type [TC]:
paulson@13323
   884
     "[| x \<in> nat; y \<in> nat |] ==> field_fm(x,y) \<in> formula"
wenzelm@13429
   885
by (simp add: field_fm_def)
paulson@13323
   886
paulson@13323
   887
lemma arity_field_fm [simp]:
wenzelm@13429
   888
     "[| x \<in> nat; y \<in> nat |]
paulson@13323
   889
      ==> arity(field_fm(x,y)) = succ(x) \<union> succ(y)"
wenzelm@13429
   890
by (simp add: field_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13323
   891
paulson@13323
   892
lemma sats_field_fm [simp]:
paulson@13323
   893
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
wenzelm@13429
   894
    ==> sats(A, field_fm(x,y), env) <->
paulson@13323
   895
        is_field(**A, nth(x,env), nth(y,env))"
paulson@13323
   896
by (simp add: field_fm_def is_field_def)
paulson@13323
   897
paulson@13323
   898
lemma field_iff_sats:
wenzelm@13429
   899
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13323
   900
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13323
   901
       ==> is_field(**A, x, y) <-> sats(A, field_fm(i,j), env)"
paulson@13323
   902
by simp
paulson@13323
   903
paulson@13323
   904
theorem field_reflection:
wenzelm@13429
   905
     "REFLECTS[\<lambda>x. is_field(L,f(x),g(x)),
paulson@13323
   906
               \<lambda>i x. is_field(**Lset(i),f(x),g(x))]"
paulson@13323
   907
apply (simp only: is_field_def setclass_simps)
paulson@13323
   908
apply (intro FOL_reflections domain_reflection range_reflection
paulson@13323
   909
             union_reflection)
paulson@13323
   910
done
paulson@13323
   911
paulson@13323
   912
paulson@13339
   913
subsubsection{*Image under a Relation, Internalized*}
paulson@13306
   914
wenzelm@13429
   915
(* "image(M,r,A,z) ==
paulson@13306
   916
        \<forall>y[M]. (y \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,w))))" *)
paulson@13306
   917
constdefs image_fm :: "[i,i,i]=>i"
wenzelm@13429
   918
    "image_fm(r,A,z) ==
paulson@13306
   919
       Forall(Iff(Member(0,succ(z)),
paulson@13306
   920
                  Exists(And(Member(0,succ(succ(r))),
paulson@13306
   921
                             Exists(And(Member(0,succ(succ(succ(A)))),
wenzelm@13429
   922
                                        pair_fm(0,2,1)))))))"
paulson@13306
   923
paulson@13306
   924
lemma image_type [TC]:
paulson@13306
   925
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> image_fm(x,y,z) \<in> formula"
wenzelm@13429
   926
by (simp add: image_fm_def)
paulson@13306
   927
paulson@13306
   928
lemma arity_image_fm [simp]:
wenzelm@13429
   929
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13306
   930
      ==> arity(image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
   931
by (simp add: image_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13306
   932
paulson@13306
   933
lemma sats_image_fm [simp]:
paulson@13306
   934
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
   935
    ==> sats(A, image_fm(x,y,z), env) <->
paulson@13306
   936
        image(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13323
   937
by (simp add: image_fm_def Relative.image_def)
paulson@13306
   938
paulson@13306
   939
lemma image_iff_sats:
wenzelm@13429
   940
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13306
   941
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13306
   942
       ==> image(**A, x, y, z) <-> sats(A, image_fm(i,j,k), env)"
paulson@13306
   943
by (simp add: sats_image_fm)
paulson@13306
   944
paulson@13314
   945
theorem image_reflection:
wenzelm@13429
   946
     "REFLECTS[\<lambda>x. image(L,f(x),g(x),h(x)),
paulson@13314
   947
               \<lambda>i x. image(**Lset(i),f(x),g(x),h(x))]"
paulson@13323
   948
apply (simp only: Relative.image_def setclass_simps)
wenzelm@13429
   949
apply (intro FOL_reflections pair_reflection)
paulson@13314
   950
done
paulson@13306
   951
paulson@13306
   952
paulson@13348
   953
subsubsection{*Pre-Image under a Relation, Internalized*}
paulson@13348
   954
wenzelm@13429
   955
(* "pre_image(M,r,A,z) ==
wenzelm@13429
   956
        \<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *)
paulson@13348
   957
constdefs pre_image_fm :: "[i,i,i]=>i"
wenzelm@13429
   958
    "pre_image_fm(r,A,z) ==
paulson@13348
   959
       Forall(Iff(Member(0,succ(z)),
paulson@13348
   960
                  Exists(And(Member(0,succ(succ(r))),
paulson@13348
   961
                             Exists(And(Member(0,succ(succ(succ(A)))),
wenzelm@13429
   962
                                        pair_fm(2,0,1)))))))"
paulson@13348
   963
paulson@13348
   964
lemma pre_image_type [TC]:
paulson@13348
   965
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
wenzelm@13429
   966
by (simp add: pre_image_fm_def)
paulson@13348
   967
paulson@13348
   968
lemma arity_pre_image_fm [simp]:
wenzelm@13429
   969
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13348
   970
      ==> arity(pre_image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
   971
by (simp add: pre_image_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13348
   972
paulson@13348
   973
lemma sats_pre_image_fm [simp]:
paulson@13348
   974
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
   975
    ==> sats(A, pre_image_fm(x,y,z), env) <->
paulson@13348
   976
        pre_image(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13348
   977
by (simp add: pre_image_fm_def Relative.pre_image_def)
paulson@13348
   978
paulson@13348
   979
lemma pre_image_iff_sats:
wenzelm@13429
   980
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13348
   981
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13348
   982
       ==> pre_image(**A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)"
paulson@13348
   983
by (simp add: sats_pre_image_fm)
paulson@13348
   984
paulson@13348
   985
theorem pre_image_reflection:
wenzelm@13429
   986
     "REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)),
paulson@13348
   987
               \<lambda>i x. pre_image(**Lset(i),f(x),g(x),h(x))]"
paulson@13348
   988
apply (simp only: Relative.pre_image_def setclass_simps)
wenzelm@13429
   989
apply (intro FOL_reflections pair_reflection)
paulson@13348
   990
done
paulson@13348
   991
paulson@13348
   992
paulson@13352
   993
subsubsection{*Function Application, Internalized*}
paulson@13352
   994
wenzelm@13429
   995
(* "fun_apply(M,f,x,y) ==
wenzelm@13429
   996
        (\<exists>xs[M]. \<exists>fxs[M].
paulson@13352
   997
         upair(M,x,x,xs) & image(M,f,xs,fxs) & big_union(M,fxs,y))" *)
paulson@13352
   998
constdefs fun_apply_fm :: "[i,i,i]=>i"
wenzelm@13429
   999
    "fun_apply_fm(f,x,y) ==
paulson@13352
  1000
       Exists(Exists(And(upair_fm(succ(succ(x)), succ(succ(x)), 1),
wenzelm@13429
  1001
                         And(image_fm(succ(succ(f)), 1, 0),
paulson@13352
  1002
                             big_union_fm(0,succ(succ(y)))))))"
paulson@13352
  1003
paulson@13352
  1004
lemma fun_apply_type [TC]:
paulson@13352
  1005
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> fun_apply_fm(x,y,z) \<in> formula"
wenzelm@13429
  1006
by (simp add: fun_apply_fm_def)
paulson@13352
  1007
paulson@13352
  1008
lemma arity_fun_apply_fm [simp]:
wenzelm@13429
  1009
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13352
  1010
      ==> arity(fun_apply_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
  1011
by (simp add: fun_apply_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13352
  1012
paulson@13352
  1013
lemma sats_fun_apply_fm [simp]:
paulson@13352
  1014
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
  1015
    ==> sats(A, fun_apply_fm(x,y,z), env) <->
paulson@13352
  1016
        fun_apply(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13352
  1017
by (simp add: fun_apply_fm_def fun_apply_def)
paulson@13352
  1018
paulson@13352
  1019
lemma fun_apply_iff_sats:
wenzelm@13429
  1020
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13352
  1021
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13352
  1022
       ==> fun_apply(**A, x, y, z) <-> sats(A, fun_apply_fm(i,j,k), env)"
paulson@13352
  1023
by simp
paulson@13352
  1024
paulson@13352
  1025
theorem fun_apply_reflection:
wenzelm@13429
  1026
     "REFLECTS[\<lambda>x. fun_apply(L,f(x),g(x),h(x)),
wenzelm@13429
  1027
               \<lambda>i x. fun_apply(**Lset(i),f(x),g(x),h(x))]"
paulson@13352
  1028
apply (simp only: fun_apply_def setclass_simps)
paulson@13352
  1029
apply (intro FOL_reflections upair_reflection image_reflection
wenzelm@13429
  1030
             big_union_reflection)
paulson@13352
  1031
done
paulson@13352
  1032
paulson@13352
  1033
paulson@13339
  1034
subsubsection{*The Concept of Relation, Internalized*}
paulson@13306
  1035
wenzelm@13429
  1036
(* "is_relation(M,r) ==
paulson@13306
  1037
        (\<forall>z[M]. z\<in>r --> (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,z)))" *)
paulson@13306
  1038
constdefs relation_fm :: "i=>i"
wenzelm@13429
  1039
    "relation_fm(r) ==
paulson@13306
  1040
       Forall(Implies(Member(0,succ(r)), Exists(Exists(pair_fm(1,0,2)))))"
paulson@13306
  1041
paulson@13306
  1042
lemma relation_type [TC]:
paulson@13306
  1043
     "[| x \<in> nat |] ==> relation_fm(x) \<in> formula"
wenzelm@13429
  1044
by (simp add: relation_fm_def)
paulson@13306
  1045
paulson@13306
  1046
lemma arity_relation_fm [simp]:
paulson@13306
  1047
     "x \<in> nat ==> arity(relation_fm(x)) = succ(x)"
wenzelm@13429
  1048
by (simp add: relation_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13306
  1049
paulson@13306
  1050
lemma sats_relation_fm [simp]:
paulson@13306
  1051
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13306
  1052
    ==> sats(A, relation_fm(x), env) <-> is_relation(**A, nth(x,env))"
paulson@13306
  1053
by (simp add: relation_fm_def is_relation_def)
paulson@13306
  1054
paulson@13306
  1055
lemma relation_iff_sats:
wenzelm@13429
  1056
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13306
  1057
          i \<in> nat; env \<in> list(A)|]
paulson@13306
  1058
       ==> is_relation(**A, x) <-> sats(A, relation_fm(i), env)"
paulson@13306
  1059
by simp
paulson@13306
  1060
paulson@13314
  1061
theorem is_relation_reflection:
wenzelm@13429
  1062
     "REFLECTS[\<lambda>x. is_relation(L,f(x)),
paulson@13314
  1063
               \<lambda>i x. is_relation(**Lset(i),f(x))]"
paulson@13314
  1064
apply (simp only: is_relation_def setclass_simps)
wenzelm@13429
  1065
apply (intro FOL_reflections pair_reflection)
paulson@13314
  1066
done
paulson@13306
  1067
paulson@13306
  1068
paulson@13339
  1069
subsubsection{*The Concept of Function, Internalized*}
paulson@13306
  1070
wenzelm@13429
  1071
(* "is_function(M,r) ==
wenzelm@13429
  1072
        \<forall>x[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M].
paulson@13306
  1073
           pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r --> y=y'" *)
paulson@13306
  1074
constdefs function_fm :: "i=>i"
wenzelm@13429
  1075
    "function_fm(r) ==
paulson@13306
  1076
       Forall(Forall(Forall(Forall(Forall(
paulson@13306
  1077
         Implies(pair_fm(4,3,1),
paulson@13306
  1078
                 Implies(pair_fm(4,2,0),
paulson@13306
  1079
                         Implies(Member(1,r#+5),
paulson@13306
  1080
                                 Implies(Member(0,r#+5), Equal(3,2))))))))))"
paulson@13306
  1081
paulson@13306
  1082
lemma function_type [TC]:
paulson@13306
  1083
     "[| x \<in> nat |] ==> function_fm(x) \<in> formula"
wenzelm@13429
  1084
by (simp add: function_fm_def)
paulson@13306
  1085
paulson@13306
  1086
lemma arity_function_fm [simp]:
paulson@13306
  1087
     "x \<in> nat ==> arity(function_fm(x)) = succ(x)"
wenzelm@13429
  1088
by (simp add: function_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13306
  1089
paulson@13306
  1090
lemma sats_function_fm [simp]:
paulson@13306
  1091
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13306
  1092
    ==> sats(A, function_fm(x), env) <-> is_function(**A, nth(x,env))"
paulson@13306
  1093
by (simp add: function_fm_def is_function_def)
paulson@13306
  1094
paulson@13505
  1095
lemma is_function_iff_sats:
wenzelm@13429
  1096
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13306
  1097
          i \<in> nat; env \<in> list(A)|]
paulson@13306
  1098
       ==> is_function(**A, x) <-> sats(A, function_fm(i), env)"
paulson@13306
  1099
by simp
paulson@13306
  1100
paulson@13314
  1101
theorem is_function_reflection:
wenzelm@13429
  1102
     "REFLECTS[\<lambda>x. is_function(L,f(x)),
paulson@13314
  1103
               \<lambda>i x. is_function(**Lset(i),f(x))]"
paulson@13314
  1104
apply (simp only: is_function_def setclass_simps)
wenzelm@13429
  1105
apply (intro FOL_reflections pair_reflection)
paulson@13314
  1106
done
paulson@13298
  1107
paulson@13298
  1108
paulson@13339
  1109
subsubsection{*Typed Functions, Internalized*}
paulson@13309
  1110
wenzelm@13429
  1111
(* "typed_function(M,A,B,r) ==
paulson@13309
  1112
        is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
paulson@13309
  1113
        (\<forall>u[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>B))" *)
paulson@13309
  1114
paulson@13309
  1115
constdefs typed_function_fm :: "[i,i,i]=>i"
wenzelm@13429
  1116
    "typed_function_fm(A,B,r) ==
paulson@13309
  1117
       And(function_fm(r),
paulson@13309
  1118
         And(relation_fm(r),
paulson@13309
  1119
           And(domain_fm(r,A),
paulson@13309
  1120
             Forall(Implies(Member(0,succ(r)),
paulson@13309
  1121
                  Forall(Forall(Implies(pair_fm(1,0,2),Member(0,B#+3)))))))))"
paulson@13309
  1122
paulson@13309
  1123
lemma typed_function_type [TC]:
paulson@13309
  1124
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> typed_function_fm(x,y,z) \<in> formula"
wenzelm@13429
  1125
by (simp add: typed_function_fm_def)
paulson@13309
  1126
paulson@13309
  1127
lemma arity_typed_function_fm [simp]:
wenzelm@13429
  1128
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13309
  1129
      ==> arity(typed_function_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
  1130
by (simp add: typed_function_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13309
  1131
paulson@13309
  1132
lemma sats_typed_function_fm [simp]:
paulson@13309
  1133
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
  1134
    ==> sats(A, typed_function_fm(x,y,z), env) <->
paulson@13309
  1135
        typed_function(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1136
by (simp add: typed_function_fm_def typed_function_def)
paulson@13309
  1137
paulson@13309
  1138
lemma typed_function_iff_sats:
wenzelm@13429
  1139
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13309
  1140
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1141
   ==> typed_function(**A, x, y, z) <-> sats(A, typed_function_fm(i,j,k), env)"
paulson@13309
  1142
by simp
paulson@13309
  1143
wenzelm@13429
  1144
lemmas function_reflections =
paulson@13363
  1145
        empty_reflection number1_reflection
wenzelm@13429
  1146
        upair_reflection pair_reflection union_reflection
wenzelm@13429
  1147
        big_union_reflection cons_reflection successor_reflection
paulson@13323
  1148
        fun_apply_reflection subset_reflection
wenzelm@13429
  1149
        transitive_set_reflection membership_reflection
wenzelm@13429
  1150
        pred_set_reflection domain_reflection range_reflection field_reflection
paulson@13348
  1151
        image_reflection pre_image_reflection
wenzelm@13429
  1152
        is_relation_reflection is_function_reflection
paulson@13309
  1153
wenzelm@13429
  1154
lemmas function_iff_sats =
wenzelm@13429
  1155
        empty_iff_sats number1_iff_sats
wenzelm@13429
  1156
        upair_iff_sats pair_iff_sats union_iff_sats
paulson@13505
  1157
        big_union_iff_sats cons_iff_sats successor_iff_sats
paulson@13323
  1158
        fun_apply_iff_sats  Memrel_iff_sats
wenzelm@13429
  1159
        pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
wenzelm@13429
  1160
        image_iff_sats pre_image_iff_sats
paulson@13505
  1161
        relation_iff_sats is_function_iff_sats
paulson@13323
  1162
paulson@13309
  1163
paulson@13314
  1164
theorem typed_function_reflection:
wenzelm@13429
  1165
     "REFLECTS[\<lambda>x. typed_function(L,f(x),g(x),h(x)),
paulson@13314
  1166
               \<lambda>i x. typed_function(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1167
apply (simp only: typed_function_def setclass_simps)
wenzelm@13429
  1168
apply (intro FOL_reflections function_reflections)
paulson@13323
  1169
done
paulson@13323
  1170
paulson@13323
  1171
paulson@13339
  1172
subsubsection{*Composition of Relations, Internalized*}
paulson@13323
  1173
wenzelm@13429
  1174
(* "composition(M,r,s,t) ==
wenzelm@13429
  1175
        \<forall>p[M]. p \<in> t <->
wenzelm@13429
  1176
               (\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M].
wenzelm@13429
  1177
                pair(M,x,z,p) & pair(M,x,y,xy) & pair(M,y,z,yz) &
paulson@13323
  1178
                xy \<in> s & yz \<in> r)" *)
paulson@13323
  1179
constdefs composition_fm :: "[i,i,i]=>i"
wenzelm@13429
  1180
  "composition_fm(r,s,t) ==
paulson@13323
  1181
     Forall(Iff(Member(0,succ(t)),
wenzelm@13429
  1182
             Exists(Exists(Exists(Exists(Exists(
paulson@13323
  1183
              And(pair_fm(4,2,5),
paulson@13323
  1184
               And(pair_fm(4,3,1),
paulson@13323
  1185
                And(pair_fm(3,2,0),
paulson@13323
  1186
                 And(Member(1,s#+6), Member(0,r#+6))))))))))))"
paulson@13323
  1187
paulson@13323
  1188
lemma composition_type [TC]:
paulson@13323
  1189
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> composition_fm(x,y,z) \<in> formula"
wenzelm@13429
  1190
by (simp add: composition_fm_def)
paulson@13323
  1191
paulson@13323
  1192
lemma arity_composition_fm [simp]:
wenzelm@13429
  1193
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13323
  1194
      ==> arity(composition_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
  1195
by (simp add: composition_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13323
  1196
paulson@13323
  1197
lemma sats_composition_fm [simp]:
paulson@13323
  1198
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
  1199
    ==> sats(A, composition_fm(x,y,z), env) <->
paulson@13323
  1200
        composition(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13323
  1201
by (simp add: composition_fm_def composition_def)
paulson@13323
  1202
paulson@13323
  1203
lemma composition_iff_sats:
wenzelm@13429
  1204
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13323
  1205
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13323
  1206
       ==> composition(**A, x, y, z) <-> sats(A, composition_fm(i,j,k), env)"
paulson@13323
  1207
by simp
paulson@13323
  1208
paulson@13323
  1209
theorem composition_reflection:
wenzelm@13429
  1210
     "REFLECTS[\<lambda>x. composition(L,f(x),g(x),h(x)),
paulson@13323
  1211
               \<lambda>i x. composition(**Lset(i),f(x),g(x),h(x))]"
paulson@13323
  1212
apply (simp only: composition_def setclass_simps)
wenzelm@13429
  1213
apply (intro FOL_reflections pair_reflection)
paulson@13314
  1214
done
paulson@13314
  1215
paulson@13309
  1216
paulson@13339
  1217
subsubsection{*Injections, Internalized*}
paulson@13309
  1218
wenzelm@13429
  1219
(* "injection(M,A,B,f) ==
wenzelm@13429
  1220
        typed_function(M,A,B,f) &
wenzelm@13429
  1221
        (\<forall>x[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M].
paulson@13309
  1222
          pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f --> x=x')" *)
paulson@13309
  1223
constdefs injection_fm :: "[i,i,i]=>i"
wenzelm@13429
  1224
 "injection_fm(A,B,f) ==
paulson@13309
  1225
    And(typed_function_fm(A,B,f),
paulson@13309
  1226
       Forall(Forall(Forall(Forall(Forall(
paulson@13309
  1227
         Implies(pair_fm(4,2,1),
paulson@13309
  1228
                 Implies(pair_fm(3,2,0),
paulson@13309
  1229
                         Implies(Member(1,f#+5),
paulson@13309
  1230
                                 Implies(Member(0,f#+5), Equal(4,3)))))))))))"
paulson@13309
  1231
paulson@13309
  1232
paulson@13309
  1233
lemma injection_type [TC]:
paulson@13309
  1234
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> injection_fm(x,y,z) \<in> formula"
wenzelm@13429
  1235
by (simp add: injection_fm_def)
paulson@13309
  1236
paulson@13309
  1237
lemma arity_injection_fm [simp]:
wenzelm@13429
  1238
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13309
  1239
      ==> arity(injection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
  1240
by (simp add: injection_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13309
  1241
paulson@13309
  1242
lemma sats_injection_fm [simp]:
paulson@13309
  1243
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
  1244
    ==> sats(A, injection_fm(x,y,z), env) <->
paulson@13309
  1245
        injection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1246
by (simp add: injection_fm_def injection_def)
paulson@13309
  1247
paulson@13309
  1248
lemma injection_iff_sats:
wenzelm@13429
  1249
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13309
  1250
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1251
   ==> injection(**A, x, y, z) <-> sats(A, injection_fm(i,j,k), env)"
paulson@13309
  1252
by simp
paulson@13309
  1253
paulson@13314
  1254
theorem injection_reflection:
wenzelm@13429
  1255
     "REFLECTS[\<lambda>x. injection(L,f(x),g(x),h(x)),
paulson@13314
  1256
               \<lambda>i x. injection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1257
apply (simp only: injection_def setclass_simps)
wenzelm@13429
  1258
apply (intro FOL_reflections function_reflections typed_function_reflection)
paulson@13314
  1259
done
paulson@13309
  1260
paulson@13309
  1261
paulson@13339
  1262
subsubsection{*Surjections, Internalized*}
paulson@13309
  1263
paulson@13309
  1264
(*  surjection :: "[i=>o,i,i,i] => o"
wenzelm@13429
  1265
    "surjection(M,A,B,f) ==
paulson@13309
  1266
        typed_function(M,A,B,f) &
paulson@13309
  1267
        (\<forall>y[M]. y\<in>B --> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" *)
paulson@13309
  1268
constdefs surjection_fm :: "[i,i,i]=>i"
wenzelm@13429
  1269
 "surjection_fm(A,B,f) ==
paulson@13309
  1270
    And(typed_function_fm(A,B,f),
paulson@13309
  1271
       Forall(Implies(Member(0,succ(B)),
paulson@13309
  1272
                      Exists(And(Member(0,succ(succ(A))),
paulson@13309
  1273
                                 fun_apply_fm(succ(succ(f)),0,1))))))"
paulson@13309
  1274
paulson@13309
  1275
lemma surjection_type [TC]:
paulson@13309
  1276
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> surjection_fm(x,y,z) \<in> formula"
wenzelm@13429
  1277
by (simp add: surjection_fm_def)
paulson@13309
  1278
paulson@13309
  1279
lemma arity_surjection_fm [simp]:
wenzelm@13429
  1280
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13309
  1281
      ==> arity(surjection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
  1282
by (simp add: surjection_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13309
  1283
paulson@13309
  1284
lemma sats_surjection_fm [simp]:
paulson@13309
  1285
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
  1286
    ==> sats(A, surjection_fm(x,y,z), env) <->
paulson@13309
  1287
        surjection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1288
by (simp add: surjection_fm_def surjection_def)
paulson@13309
  1289
paulson@13309
  1290
lemma surjection_iff_sats:
wenzelm@13429
  1291
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13309
  1292
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1293
   ==> surjection(**A, x, y, z) <-> sats(A, surjection_fm(i,j,k), env)"
paulson@13309
  1294
by simp
paulson@13309
  1295
paulson@13314
  1296
theorem surjection_reflection:
wenzelm@13429
  1297
     "REFLECTS[\<lambda>x. surjection(L,f(x),g(x),h(x)),
paulson@13314
  1298
               \<lambda>i x. surjection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1299
apply (simp only: surjection_def setclass_simps)
wenzelm@13429
  1300
apply (intro FOL_reflections function_reflections typed_function_reflection)
paulson@13314
  1301
done
paulson@13309
  1302
paulson@13309
  1303
paulson@13309
  1304
paulson@13339
  1305
subsubsection{*Bijections, Internalized*}
paulson@13309
  1306
paulson@13309
  1307
(*   bijection :: "[i=>o,i,i,i] => o"
paulson@13309
  1308
    "bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)" *)
paulson@13309
  1309
constdefs bijection_fm :: "[i,i,i]=>i"
paulson@13309
  1310
 "bijection_fm(A,B,f) == And(injection_fm(A,B,f), surjection_fm(A,B,f))"
paulson@13309
  1311
paulson@13309
  1312
lemma bijection_type [TC]:
paulson@13309
  1313
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> bijection_fm(x,y,z) \<in> formula"
wenzelm@13429
  1314
by (simp add: bijection_fm_def)
paulson@13309
  1315
paulson@13309
  1316
lemma arity_bijection_fm [simp]:
wenzelm@13429
  1317
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13309
  1318
      ==> arity(bijection_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
  1319
by (simp add: bijection_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13309
  1320
paulson@13309
  1321
lemma sats_bijection_fm [simp]:
paulson@13309
  1322
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
  1323
    ==> sats(A, bijection_fm(x,y,z), env) <->
paulson@13309
  1324
        bijection(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13309
  1325
by (simp add: bijection_fm_def bijection_def)
paulson@13309
  1326
paulson@13309
  1327
lemma bijection_iff_sats:
wenzelm@13429
  1328
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13309
  1329
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13309
  1330
   ==> bijection(**A, x, y, z) <-> sats(A, bijection_fm(i,j,k), env)"
paulson@13309
  1331
by simp
paulson@13309
  1332
paulson@13314
  1333
theorem bijection_reflection:
wenzelm@13429
  1334
     "REFLECTS[\<lambda>x. bijection(L,f(x),g(x),h(x)),
paulson@13314
  1335
               \<lambda>i x. bijection(**Lset(i),f(x),g(x),h(x))]"
paulson@13314
  1336
apply (simp only: bijection_def setclass_simps)
wenzelm@13429
  1337
apply (intro And_reflection injection_reflection surjection_reflection)
paulson@13314
  1338
done
paulson@13309
  1339
paulson@13309
  1340
paulson@13348
  1341
subsubsection{*Restriction of a Relation, Internalized*}
paulson@13348
  1342
paulson@13348
  1343
wenzelm@13429
  1344
(* "restriction(M,r,A,z) ==
wenzelm@13429
  1345
        \<forall>x[M]. x \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *)
paulson@13348
  1346
constdefs restriction_fm :: "[i,i,i]=>i"
wenzelm@13429
  1347
    "restriction_fm(r,A,z) ==
paulson@13348
  1348
       Forall(Iff(Member(0,succ(z)),
paulson@13348
  1349
                  And(Member(0,succ(r)),
paulson@13348
  1350
                      Exists(And(Member(0,succ(succ(A))),
paulson@13348
  1351
                                 Exists(pair_fm(1,0,2)))))))"
paulson@13348
  1352
paulson@13348
  1353
lemma restriction_type [TC]:
paulson@13348
  1354
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula"
wenzelm@13429
  1355
by (simp add: restriction_fm_def)
paulson@13348
  1356
paulson@13348
  1357
lemma arity_restriction_fm [simp]:
wenzelm@13429
  1358
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13348
  1359
      ==> arity(restriction_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13429
  1360
by (simp add: restriction_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13348
  1361
paulson@13348
  1362
lemma sats_restriction_fm [simp]:
paulson@13348
  1363
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13429
  1364
    ==> sats(A, restriction_fm(x,y,z), env) <->
paulson@13348
  1365
        restriction(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13348
  1366
by (simp add: restriction_fm_def restriction_def)
paulson@13348
  1367
paulson@13348
  1368
lemma restriction_iff_sats:
wenzelm@13429
  1369
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13348
  1370
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13348
  1371
       ==> restriction(**A, x, y, z) <-> sats(A, restriction_fm(i,j,k), env)"
paulson@13348
  1372
by simp
paulson@13348
  1373
paulson@13348
  1374
theorem restriction_reflection:
wenzelm@13429
  1375
     "REFLECTS[\<lambda>x. restriction(L,f(x),g(x),h(x)),
paulson@13348
  1376
               \<lambda>i x. restriction(**Lset(i),f(x),g(x),h(x))]"
paulson@13348
  1377
apply (simp only: restriction_def setclass_simps)
wenzelm@13429
  1378
apply (intro FOL_reflections pair_reflection)
paulson@13348
  1379
done
paulson@13348
  1380
paulson@13339
  1381
subsubsection{*Order-Isomorphisms, Internalized*}
paulson@13309
  1382
paulson@13309
  1383
(*  order_isomorphism :: "[i=>o,i,i,i,i,i] => o"
wenzelm@13429
  1384
   "order_isomorphism(M,A,r,B,s,f) ==
wenzelm@13429
  1385
        bijection(M,A,B,f) &
paulson@13309
  1386
        (\<forall>x[M]. x\<in>A --> (\<forall>y[M]. y\<in>A -->
paulson@13309
  1387
          (\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M].
wenzelm@13429
  1388
            pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) -->
paulson@13309
  1389
            pair(M,fx,fy,q) --> (p\<in>r <-> q\<in>s))))"
paulson@13309
  1390
  *)
paulson@13309
  1391
paulson@13309
  1392
constdefs order_isomorphism_fm :: "[i,i,i,i,i]=>i"
wenzelm@13429
  1393
 "order_isomorphism_fm(A,r,B,s,f) ==
wenzelm@13429
  1394
   And(bijection_fm(A,B,f),
paulson@13309
  1395
     Forall(Implies(Member(0,succ(A)),
paulson@13309
  1396
       Forall(Implies(Member(0,succ(succ(A))),
paulson@13309
  1397
         Forall(Forall(Forall(Forall(
paulson@13309
  1398
           Implies(pair_fm(5,4,3),
paulson@13309
  1399
             Implies(fun_apply_fm(f#+6,5,2),
paulson@13309
  1400
               Implies(fun_apply_fm(f#+6,4,1),
wenzelm@13429
  1401
                 Implies(pair_fm(2,1,0),
paulson@13309
  1402
                   Iff(Member(3,r#+6), Member(0,s#+6)))))))))))))))"
paulson@13309
  1403
paulson@13309
  1404
lemma order_isomorphism_type [TC]:
wenzelm@13429
  1405
     "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]
paulson@13309
  1406
      ==> order_isomorphism_fm(A,r,B,s,f) \<in> formula"
wenzelm@13429
  1407
by (simp add: order_isomorphism_fm_def)
paulson@13309
  1408
paulson@13309
  1409
lemma arity_order_isomorphism_fm [simp]:
wenzelm@13429
  1410
     "[| A \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat |]
wenzelm@13429
  1411
      ==> arity(order_isomorphism_fm(A,r,B,s,f)) =
wenzelm@13429
  1412
          succ(A) \<union> succ(r) \<union> succ(B) \<union> succ(s) \<union> succ(f)"
wenzelm@13429
  1413
by (simp add: order_isomorphism_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13309
  1414
paulson@13309
  1415
lemma sats_order_isomorphism_fm [simp]:
paulson@13309
  1416
   "[| U \<in> nat; r \<in> nat; B \<in> nat; s \<in> nat; f \<in> nat; env \<in> list(A)|]
wenzelm@13429
  1417
    ==> sats(A, order_isomorphism_fm(U,r,B,s,f), env) <->
wenzelm@13429
  1418
        order_isomorphism(**A, nth(U,env), nth(r,env), nth(B,env),
paulson@13309
  1419
                               nth(s,env), nth(f,env))"
paulson@13309
  1420
by (simp add: order_isomorphism_fm_def order_isomorphism_def)
paulson@13309
  1421
paulson@13309
  1422
lemma order_isomorphism_iff_sats:
wenzelm@13429
  1423
  "[| nth(i,env) = U; nth(j,env) = r; nth(k,env) = B; nth(j',env) = s;
wenzelm@13429
  1424
      nth(k',env) = f;
paulson@13309
  1425
      i \<in> nat; j \<in> nat; k \<in> nat; j' \<in> nat; k' \<in> nat; env \<in> list(A)|]
wenzelm@13429
  1426
   ==> order_isomorphism(**A,U,r,B,s,f) <->
wenzelm@13429
  1427
       sats(A, order_isomorphism_fm(i,j,k,j',k'), env)"
paulson@13309
  1428
by simp
paulson@13309
  1429
paulson@13314
  1430
theorem order_isomorphism_reflection:
wenzelm@13429
  1431
     "REFLECTS[\<lambda>x. order_isomorphism(L,f(x),g(x),h(x),g'(x),h'(x)),
paulson@13314
  1432
               \<lambda>i x. order_isomorphism(**Lset(i),f(x),g(x),h(x),g'(x),h'(x))]"
paulson@13314
  1433
apply (simp only: order_isomorphism_def setclass_simps)
wenzelm@13429
  1434
apply (intro FOL_reflections function_reflections bijection_reflection)
paulson@13323
  1435
done
paulson@13323
  1436
paulson@13339
  1437
subsubsection{*Limit Ordinals, Internalized*}
paulson@13323
  1438
paulson@13323
  1439
text{*A limit ordinal is a non-empty, successor-closed ordinal*}
paulson@13323
  1440
wenzelm@13429
  1441
(* "limit_ordinal(M,a) ==
wenzelm@13429
  1442
        ordinal(M,a) & ~ empty(M,a) &
paulson@13323
  1443
        (\<forall>x[M]. x\<in>a --> (\<exists>y[M]. y\<in>a & successor(M,x,y)))" *)
paulson@13323
  1444
paulson@13323
  1445
constdefs limit_ordinal_fm :: "i=>i"
wenzelm@13429
  1446
    "limit_ordinal_fm(x) ==
paulson@13323
  1447
        And(ordinal_fm(x),
paulson@13323
  1448
            And(Neg(empty_fm(x)),
wenzelm@13429
  1449
                Forall(Implies(Member(0,succ(x)),
paulson@13323
  1450
                               Exists(And(Member(0,succ(succ(x))),
paulson@13323
  1451
                                          succ_fm(1,0)))))))"
paulson@13323
  1452
paulson@13323
  1453
lemma limit_ordinal_type [TC]:
paulson@13323
  1454
     "x \<in> nat ==> limit_ordinal_fm(x) \<in> formula"
wenzelm@13429
  1455
by (simp add: limit_ordinal_fm_def)
paulson@13323
  1456
paulson@13323
  1457
lemma arity_limit_ordinal_fm [simp]:
paulson@13323
  1458
     "x \<in> nat ==> arity(limit_ordinal_fm(x)) = succ(x)"
wenzelm@13429
  1459
by (simp add: limit_ordinal_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13323
  1460
paulson@13323
  1461
lemma sats_limit_ordinal_fm [simp]:
paulson@13323
  1462
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13323
  1463
    ==> sats(A, limit_ordinal_fm(x), env) <-> limit_ordinal(**A, nth(x,env))"
paulson@13323
  1464
by (simp add: limit_ordinal_fm_def limit_ordinal_def sats_ordinal_fm')
paulson@13323
  1465
paulson@13323
  1466
lemma limit_ordinal_iff_sats:
wenzelm@13429
  1467
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13323
  1468
          i \<in> nat; env \<in> list(A)|]
paulson@13323
  1469
       ==> limit_ordinal(**A, x) <-> sats(A, limit_ordinal_fm(i), env)"
paulson@13323
  1470
by simp
paulson@13323
  1471
paulson@13323
  1472
theorem limit_ordinal_reflection:
wenzelm@13429
  1473
     "REFLECTS[\<lambda>x. limit_ordinal(L,f(x)),
paulson@13323
  1474
               \<lambda>i x. limit_ordinal(**Lset(i),f(x))]"
paulson@13323
  1475
apply (simp only: limit_ordinal_def setclass_simps)
wenzelm@13429
  1476
apply (intro FOL_reflections ordinal_reflection
wenzelm@13429
  1477
             empty_reflection successor_reflection)
paulson@13314
  1478
done
paulson@13309
  1479
paulson@13493
  1480
subsubsection{*Finite Ordinals: The Predicate ``Is A Natural Number''*}
paulson@13493
  1481
paulson@13493
  1482
(*     "finite_ordinal(M,a) == 
paulson@13493
  1483
	ordinal(M,a) & ~ limit_ordinal(M,a) & 
paulson@13493
  1484
        (\<forall>x[M]. x\<in>a --> ~ limit_ordinal(M,x))" *)
paulson@13493
  1485
constdefs finite_ordinal_fm :: "i=>i"
paulson@13493
  1486
    "finite_ordinal_fm(x) ==
paulson@13493
  1487
       And(ordinal_fm(x),
paulson@13493
  1488
          And(Neg(limit_ordinal_fm(x)),
paulson@13493
  1489
           Forall(Implies(Member(0,succ(x)),
paulson@13493
  1490
                          Neg(limit_ordinal_fm(0))))))"
paulson@13493
  1491
paulson@13493
  1492
lemma finite_ordinal_type [TC]:
paulson@13493
  1493
     "x \<in> nat ==> finite_ordinal_fm(x) \<in> formula"
paulson@13493
  1494
by (simp add: finite_ordinal_fm_def)
paulson@13493
  1495
paulson@13493
  1496
lemma sats_finite_ordinal_fm [simp]:
paulson@13493
  1497
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13493
  1498
    ==> sats(A, finite_ordinal_fm(x), env) <-> finite_ordinal(**A, nth(x,env))"
paulson@13493
  1499
by (simp add: finite_ordinal_fm_def sats_ordinal_fm' finite_ordinal_def)
paulson@13493
  1500
paulson@13493
  1501
lemma finite_ordinal_iff_sats:
paulson@13493
  1502
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13493
  1503
          i \<in> nat; env \<in> list(A)|]
paulson@13493
  1504
       ==> finite_ordinal(**A, x) <-> sats(A, finite_ordinal_fm(i), env)"
paulson@13493
  1505
by simp
paulson@13493
  1506
paulson@13493
  1507
theorem finite_ordinal_reflection:
paulson@13493
  1508
     "REFLECTS[\<lambda>x. finite_ordinal(L,f(x)),
paulson@13493
  1509
               \<lambda>i x. finite_ordinal(**Lset(i),f(x))]"
paulson@13493
  1510
apply (simp only: finite_ordinal_def setclass_simps)
paulson@13493
  1511
apply (intro FOL_reflections ordinal_reflection limit_ordinal_reflection)
paulson@13493
  1512
done
paulson@13493
  1513
paulson@13493
  1514
paulson@13323
  1515
subsubsection{*Omega: The Set of Natural Numbers*}
paulson@13323
  1516
paulson@13323
  1517
(* omega(M,a) == limit_ordinal(M,a) & (\<forall>x[M]. x\<in>a --> ~ limit_ordinal(M,x)) *)
paulson@13323
  1518
constdefs omega_fm :: "i=>i"
wenzelm@13429
  1519
    "omega_fm(x) ==
paulson@13323
  1520
       And(limit_ordinal_fm(x),
paulson@13323
  1521
           Forall(Implies(Member(0,succ(x)),
paulson@13323
  1522
                          Neg(limit_ordinal_fm(0)))))"
paulson@13323
  1523
paulson@13323
  1524
lemma omega_type [TC]:
paulson@13323
  1525
     "x \<in> nat ==> omega_fm(x) \<in> formula"
wenzelm@13429
  1526
by (simp add: omega_fm_def)
paulson@13323
  1527
paulson@13323
  1528
lemma arity_omega_fm [simp]:
paulson@13323
  1529
     "x \<in> nat ==> arity(omega_fm(x)) = succ(x)"
wenzelm@13429
  1530
by (simp add: omega_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13323
  1531
paulson@13323
  1532
lemma sats_omega_fm [simp]:
paulson@13323
  1533
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13323
  1534
    ==> sats(A, omega_fm(x), env) <-> omega(**A, nth(x,env))"
paulson@13323
  1535
by (simp add: omega_fm_def omega_def)
paulson@13316
  1536
paulson@13323
  1537
lemma omega_iff_sats:
wenzelm@13429
  1538
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13323
  1539
          i \<in> nat; env \<in> list(A)|]
paulson@13323
  1540
       ==> omega(**A, x) <-> sats(A, omega_fm(i), env)"
paulson@13323
  1541
by simp
paulson@13323
  1542
paulson@13323
  1543
theorem omega_reflection:
wenzelm@13429
  1544
     "REFLECTS[\<lambda>x. omega(L,f(x)),
paulson@13323
  1545
               \<lambda>i x. omega(**Lset(i),f(x))]"
paulson@13323
  1546
apply (simp only: omega_def setclass_simps)
wenzelm@13429
  1547
apply (intro FOL_reflections limit_ordinal_reflection)
paulson@13323
  1548
done
paulson@13323
  1549
paulson@13323
  1550
paulson@13323
  1551
lemmas fun_plus_reflections =
paulson@13323
  1552
        typed_function_reflection composition_reflection
paulson@13323
  1553
        injection_reflection surjection_reflection
paulson@13348
  1554
        bijection_reflection restriction_reflection
paulson@13496
  1555
        order_isomorphism_reflection finite_ordinal_reflection 
paulson@13323
  1556
        ordinal_reflection limit_ordinal_reflection omega_reflection
paulson@13323
  1557
wenzelm@13429
  1558
lemmas fun_plus_iff_sats =
wenzelm@13429
  1559
        typed_function_iff_sats composition_iff_sats
wenzelm@13429
  1560
        injection_iff_sats surjection_iff_sats
wenzelm@13429
  1561
        bijection_iff_sats restriction_iff_sats
paulson@13496
  1562
        order_isomorphism_iff_sats finite_ordinal_iff_sats
paulson@13323
  1563
        ordinal_iff_sats limit_ordinal_iff_sats omega_iff_sats
paulson@13316
  1564
paulson@13223
  1565
end