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(* Title: HOL/Bali/Basis.thy

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ID: $Id$


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Author: David von Oheimb


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Copyright 1997 Technische Universitaet Muenchen


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*)


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header {* Definitions extending HOL as logical basis of Bali *}


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theory Basis = Main:


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ML_setup {*


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Unify.search_bound := 40;


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Unify.trace_bound := 40;


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quick_and_dirty:=true;


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Pretty.setmargin 77;


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goals_limit:=2;


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*}


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(*print_depth 100;*)


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(*Syntax.ambiguity_level := 1;*)


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section "misc"


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declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)


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(* ###TO HOL/???.ML?? *)


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ML {*


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fun make_simproc name pat pred thm = Simplifier.mk_simproc name


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[Thm.read_cterm (Thm.sign_of_thm thm) (pat, HOLogic.typeT)]


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(K (K (fn s => if pred s then None else Some (standard (mk_meta_eq thm)))))


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*}


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declare split_if_asm [split] option.split [split] option.split_asm [split]


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ML {*


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simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac)


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*}


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declare if_weak_cong [cong del] option.weak_case_cong [cong del]


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declare length_Suc_conv [iff];


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(*###to be phased out *)


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ML {*


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bind_thm ("make_imp", rearrange_prems [1,0] mp)


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*}


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lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"


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apply auto


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done


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lemma subset_insertD:


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"A <= insert x B ==> A <= B & x ~: A  (EX B'. A = insert x B' & B' <= B)"


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apply (case_tac "x:A")


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apply (rule disjI2)


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apply (rule_tac x = "A{x}" in exI)


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apply fast+


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done


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syntax


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"3" :: nat ("3")


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"4" :: nat ("4")


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translations


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"3" == "Suc 2"


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"4" == "Suc 3"


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(*unused*)


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lemma range_bool_domain: "range f = {f True, f False}"


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apply auto


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apply (case_tac "xa")


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apply auto


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done


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(* context (theory "Transitive_Closure") *)


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lemma irrefl_tranclI': "r^1 Int r^+ = {} ==> !x. (x, x) ~: r^+"


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apply (rule allI)


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apply (erule irrefl_tranclI)


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done


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lemma trancl_rtrancl_trancl:


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"\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"


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by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)


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lemma rtrancl_into_trancl3:


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"\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+"


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apply (drule rtranclD)


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apply auto


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done


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lemma rtrancl_into_rtrancl2:


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"\<lbrakk> (a, b) \<in> r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in> r^*"


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by (auto intro: r_into_rtrancl rtrancl_trans)


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lemma triangle_lemma:


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"\<lbrakk> \<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c; (a,x)\<in>r\<^sup>*; (a,y)\<in>r\<^sup>*\<rbrakk>


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\<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"


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proof 


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note converse_rtrancl_induct = converse_rtrancl_induct [consumes 1]


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note converse_rtranclE = converse_rtranclE [consumes 1]


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assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"


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assume "(a,x)\<in>r\<^sup>*"


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then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"


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proof (induct rule: converse_rtrancl_induct)


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assume "(x,y)\<in>r\<^sup>*"


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then show ?thesis


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by blast


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next


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fix a v


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assume a_v_r: "(a, v) \<in> r" and


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v_x_rt: "(v, x) \<in> r\<^sup>*" and


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a_y_rt: "(a, y) \<in> r\<^sup>*" and


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hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"


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from a_y_rt


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show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"


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proof (cases rule: converse_rtranclE)


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assume "a=y"


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with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"


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by (auto intro: r_into_rtrancl rtrancl_trans)


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then show ?thesis


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by blast


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next


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fix w


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assume a_w_r: "(a, w) \<in> r" and


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w_y_rt: "(w, y) \<in> r\<^sup>*"


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from a_v_r a_w_r unique


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have "v=w"


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by auto


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with w_y_rt hyp


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show ?thesis


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by blast


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qed


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qed


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qed


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lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:


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"\<lbrakk>(a,b)\<in>r\<^sup>*; a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"


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apply (erule rtranclE)


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apply (auto dest: rtrancl_into_trancl1)


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done


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(* ### To Transitive_Closure *)


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theorems converse_rtrancl_induct


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= converse_rtrancl_induct [consumes 1,case_names Id Step]


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theorems converse_trancl_induct


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= converse_trancl_induct [consumes 1,case_names Single Step]


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(* context (theory "Set") *)


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lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"


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by auto


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(* context (theory "Finite") *)


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lemma finite_SetCompr2: "[ finite (Collect P); !y. P y > finite (range (f y)) ] ==>


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finite {f y x x y. P y}"


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apply (subgoal_tac "{f y x x y. P y} = UNION (Collect P) (%y. range (f y))")


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prefer 2 apply fast


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apply (erule ssubst)


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apply (erule finite_UN_I)


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apply fast


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done


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(* ### TO theory "List" *)


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lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>


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\<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"


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apply (induct_tac "xs1")


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apply simp


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apply (rule allI)


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apply (induct_tac "xs2")


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apply simp


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apply (rule allI)


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apply (induct_tac "xs3")


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apply auto


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done


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section "pairs"


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lemma surjective_pairing5: "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),


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snd (snd (snd (snd p))))"


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apply auto


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done


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lemma fst_splitE [elim!]:


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"[ fst s' = x'; !!x s. [ s' = (x,s); x = x' ] ==> Q ] ==> Q"


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apply (cut_tac p = "s'" in surjective_pairing)


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apply auto


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done


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lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l > x : fst ` set l"


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apply (induct_tac "l")


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apply auto


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done


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section "quantifiers"


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(*###to be phased out *)


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ML {*


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fun noAll_simpset () = simpset() setmksimps


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mksimps (filter (fn (x,_) => x<>"All") mksimps_pairs)


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*}


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lemma All_Ex_refl_eq2 [simp]:


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"(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b > P (f b))"


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apply auto


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done


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lemma ex_ex_miniscope1 [simp]:


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"(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"


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apply auto


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done


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lemma ex_miniscope2 [simp]:


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"(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))"


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apply auto


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done


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lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"


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apply auto


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done


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lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) > P x) = (!b. P (f b))"


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apply auto


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done


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section "sums"


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hide const In0 In1


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syntax


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fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)


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translations


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"fun_sum" == "sum_case"


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consts the_Inl :: "'a + 'b \<Rightarrow> 'a"


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the_Inr :: "'a + 'b \<Rightarrow> 'b"


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primrec "the_Inl (Inl a) = a"


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primrec "the_Inr (Inr b) = b"


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datatype ('a, 'b, 'c) sum3 = In1 'a  In2 'b  In3 'c


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consts the_In1 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"


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the_In2 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"


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the_In3 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"


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primrec "the_In1 (In1 a) = a"


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primrec "the_In2 (In2 b) = b"


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primrec "the_In3 (In3 c) = c"


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syntax


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In1l :: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"


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In1r :: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"


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translations


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"In1l e" == "In1 (Inl e)"


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"In1r c" == "In1 (Inr c)"


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ML {*


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fun sum3_instantiate thm = map (fn s => simplify(simpset()delsimps[not_None_eq])


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(read_instantiate [("t","In"^s^" ?x")] thm)) ["1l","2","3","1r"]


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*}


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(* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)


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translations


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"option"<= (type) "Option.option"


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"list" <= (type) "List.list"


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"sum3" <= (type) "Basis.sum3"


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section "quantifiers for option type"


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syntax


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Oall :: "[pttrn, 'a option, bool] => bool" ("(3! _:_:/ _)" [0,0,10] 10)


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Oex :: "[pttrn, 'a option, bool] => bool" ("(3? _:_:/ _)" [0,0,10] 10)


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syntax (symbols)


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Oall :: "[pttrn, 'a option, bool] => bool" ("(3\<forall>_\<in>_:/ _)" [0,0,10] 10)


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Oex :: "[pttrn, 'a option, bool] => bool" ("(3\<exists>_\<in>_:/ _)" [0,0,10] 10)


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translations


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"! x:A: P" == "! x:o2s A. P"


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"? x:A: P" == "? x:o2s A. P"


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section "unique association lists"


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constdefs


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unique :: "('a \<times> 'b) list \<Rightarrow> bool"


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"unique \<equiv> nodups \<circ> map fst"


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lemma uniqueD [rule_format (no_asm)]:


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"unique l> (!x y. (x,y):set l > (!x' y'. (x',y'):set l > x=x'> y=y'))"


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apply (unfold unique_def o_def)


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apply (induct_tac "l")


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apply (auto dest: fst_in_set_lemma)


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done


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lemma unique_Nil [simp]: "unique []"


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apply (unfold unique_def)


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apply (simp (no_asm))


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done


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lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"


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apply (unfold unique_def)


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apply (auto dest: fst_in_set_lemma)


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done


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lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]


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lemma unique_single [simp]: "!!p. unique [p]"


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apply auto


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done


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lemma unique_ConsD: "unique (x#xs) ==> unique xs"


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apply (simp add: unique_def)


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done


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lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l >


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(!(x,y):set l. !(x',y'):set l'. x' ~= x) > unique (l @ l')"


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apply (induct_tac "l")


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apply (auto dest: fst_in_set_lemma)


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done


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lemma unique_map_inj [rule_format (no_asm)]: "unique l > inj f > unique (map (%(k,x). (f k, g k x)) l)"


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apply (induct_tac "l")


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apply (auto dest: fst_in_set_lemma simp add: inj_eq)


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done


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lemma map_of_SomeI [rule_format (no_asm)]: "unique l > (k, x) : set l > map_of l k = Some x"


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apply (induct_tac "l")


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apply auto


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done


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section "list patterns"


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consts


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lsplit :: "[['a, 'a list] => 'b, 'a list] => 'b"


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defs


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lsplit_def: "lsplit == %f l. f (hd l) (tl l)"


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(* list patterns  extends predefined type "pttrn" used in abstractions *)


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syntax


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"_lpttrn" :: "[pttrn,pttrn] => pttrn" ("_#/_" [901,900] 900)


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translations


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"%y#x#xs. b" == "lsplit (%y x#xs. b)"


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"%x#xs . b" == "lsplit (%x xs . b)"


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lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"


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apply (unfold lsplit_def)


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apply (simp (no_asm))


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done


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lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"


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apply (unfold lsplit_def)


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apply simp


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done


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section "dummy pattern for quantifiers, let, etc."


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syntax


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"@dummy_pat" :: pttrn ("'_")


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parse_translation {*


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let fun dummy_pat_tr [] = Free ("_",dummyT)


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 dummy_pat_tr ts = raise TERM ("dummy_pat_tr", ts);


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in [("@dummy_pat", dummy_pat_tr)]


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end


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*}


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end
