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Quellcode-Bibliothek© Kompilation durch diese Firma[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]Datei: implicit_binders.v Sprache: IsabelleOriginal von: Isabelle© |
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(* Title: ZF/Constructible/Satisfies_absolute.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory *) section \<open>Absoluteness for the Satisfies Relation on Formulas\<close> theory Satisfies_absolute imports Datatype_absolute Rec_Separation begin subsection \<open>More Internalization\<close> subsubsection\<open>The Formula \<^term>\<open>is_depth\<close>, Internalized\<close> (* "is_depth(M,p,n) == \<exists>sn[M]. \<exists>formula_n[M]. \<exists>formula_sn[M]. 2 1 0 is_formula_N(M,n,formula_n) & p \<notin> formula_n & successor(M,n,sn) & is_formula_N(M,sn,formula_sn) & p \<in> formula_sn" *) definition depth_fm :: "[i,i]=>i" where "depth_fm(p,n) == Exists(Exists(Exists( And(formula_N_fm(n#+3,1), And(Neg(Member(p#+3,1)), And(succ_fm(n#+3,2), And(formula_N_fm(2,0), Member(p#+3,0))))))))" lemma depth_fm_type [TC]: "[| x \ by (simp add: depth_fm_def) lemma sats_depth_fm [simp]: "[| x \ ==> sats(A, depth_fm(x,y), env) \<longleftrightarrow> is_depth(##A, nth(x,env), nth(y,env))" apply (frule_tac x=y in lt_length_in_nat, assumption) apply (simp add: depth_fm_def is_depth_def) done lemma depth_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; i \<in> nat; j < length(env); env \<in> list(A)|] ==> is_depth(##A, x, y) \<longleftrightarrow> sats(A, depth_fm(i,j), env)" by (simp) theorem depth_reflection: "REFLECTS[\ \<lambda>i x. is_depth(##Lset(i), f(x), g(x))]" apply (simp only: is_depth_def) apply (intro FOL_reflections function_reflections formula_N_reflection) done subsubsection\<open>The Operator \<^term>\<open>is_formula_case\<close>\<close> text\<open>The arguments of \<^term>\<open>is_a\<close> are always 2, 1, 0, and the formula will be enclosed by three quantifiers.\<close> (* is_formula_case :: "[i=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i]=>o, i, i] => o" "is_formula_case(M, is_a, is_b, is_c, is_d, v, z) == (\<forall>x[M]. \<forall>y[M]. x\<in>nat \<longrightarrow> y\<in>nat \<longrightarrow> is_Member(M (\<forall>x[M]. \<forall>y[M]. x\<in>nat \<longrightarrow> y\<in>nat \<longrightarrow> is_Equal(M, (\<forall>x[M]. \<forall>y[M]. x\<in>formula \<longrightarrow> y\<in>formula \<longrightarrow> is_Nand(M,x,y,v) \<longrightarrow> is_c(x,y,z)) & (\<forall>x[M]. x\<in>formula \<longrightarrow> is_Forall(M,x,v) \<longrightarrow> is_d(x,z)) definition formula_case_fm :: "[i, i, i, i, i, i]=>i" where "formula_case_fm(is_a, is_b, is_c, is_d, v, z) == And(Forall(Forall(Implies(finite_ordinal_fm(1), Implies(finite_ordinal_fm(0), Implies(Member_fm(1,0,v#+2), Forall(Implies(Equal(0,z#+3), is_a))))))), And(Forall(Forall(Implies(finite_ordinal_fm(1), Implies(finite_ordinal_fm(0), Implies(Equal_fm(1,0,v#+2), Forall(Implies(Equal(0,z#+3), is_b))))))), And(Forall(Forall(Implies(mem_formula_fm(1), Implies(mem_formula_fm(0), Implies(Nand_fm(1,0,v#+2), Forall(Implies(Equal(0,z#+3), is_c))))))), Forall(Implies(mem_formula_fm(0), Implies(Forall_fm(0,succ(v)), Forall(Implies(Equal(0,z#+2), is_d))))))))" lemma is_formula_case_type [TC]: "[| is_a \ x \<in> nat; y \<in> nat |] ==> formula_case_fm(is_a, is_b, is_c, is_d, x, y) \<in> formula" by (simp add: formula_case_fm_def) lemma sats_formula_case_fm: assumes is_a_iff_sats: "!!a0 a1 a2. [|a0\<in>A; a1\<in>A; a2\<in>A|] ==> ISA(a2, a1, a0) \<longleftrightarrow> sats(A, is_a, Cons(a0,Cons(a1,Cons(a2,env))))" and is_b_iff_sats: "!!a0 a1 a2. [|a0\<in>A; a1\<in>A; a2\<in>A|] ==> ISB(a2, a1, a0) \<longleftrightarrow> sats(A, is_b, Cons(a0,Cons(a1,Cons(a2,env))))" and is_c_iff_sats: "!!a0 a1 a2. [|a0\<in>A; a1\<in>A; a2\<in>A|] ==> ISC(a2, a1, a0) \<longleftrightarrow> sats(A, is_c, Cons(a0,Cons(a1,Cons(a2,env))))" and is_d_iff_sats: "!!a0 a1. [|a0\<in>A; a1\<in>A|] ==> ISD(a1, a0) \<longleftrightarrow> sats(A, is_d, Cons(a0,Cons(a1,env)))" shows "[|x \ ==> sats(A, formula_case_fm(is_a,is_b,is_c,is_d,x,y), env) \<longleftrightarrow> is_formula_case(##A, ISA, ISB, ISC, ISD, nth(x,env), nth(y,env))" apply (frule_tac x=y in lt_length_in_nat, assumption) apply (simp add: formula_case_fm_def is_formula_case_def is_a_iff_sats [THEN iff_sym] is_b_iff_sats [THEN iff_sym] is_c_iff_sats [THEN iff_sym] is_d_iff_sats [THEN iff_sym]) done lemma formula_case_iff_sats: assumes is_a_iff_sats: "!!a0 a1 a2. [|a0\<in>A; a1\<in>A; a2\<in>A|] ==> ISA(a2, a1, a0) \<longleftrightarrow> sats(A, is_a, Cons(a0,Cons(a1,Cons(a2,env))))" and is_b_iff_sats: "!!a0 a1 a2. [|a0\<in>A; a1\<in>A; a2\<in>A|] ==> ISB(a2, a1, a0) \<longleftrightarrow> sats(A, is_b, Cons(a0,Cons(a1,Cons(a2,env))))" and is_c_iff_sats: "!!a0 a1 a2. [|a0\<in>A; a1\<in>A; a2\<in>A|] ==> ISC(a2, a1, a0) \<longleftrightarrow> sats(A, is_c, Cons(a0,Cons(a1,Cons(a2,env))))" and is_d_iff_sats: "!!a0 a1. [|a0\<in>A; a1\<in>A|] ==> ISD(a1, a0) \<longleftrightarrow> sats(A, is_d, Cons(a0,Cons(a1,env)))" shows "[|nth(i,env) = x; nth(j,env) = y; i \<in> nat; j < length(env); env \<in> list(A)|] ==> is_formula_case(##A, ISA, ISB, ISC, ISD, x, y) \<longleftrightarrow> sats(A, formula_case_fm(is_a,is_b,is_c,is_d,i,j), env)" by (simp add: sats_formula_case_fm [OF is_a_iff_sats is_b_iff_sats is_c_iff_sats is_d_iff_sats]) text\<open>The second argument of \<^term>\<open>is_a\<close> gives it direct access to \<^term>\<ope which is essential for handling free variable references. Treatment is based on that of \<open>is_nat_case_reflection\<close>.\<close> theorem is_formula_case_reflection: assumes is_a_reflection: "!!h f g g'. REFLECTS[\ \<lambda>i x. is_a(##Lset(i), h(x), f(x), g(x), g'(x))]" and is_b_reflection: "!!h f g g'. REFLECTS[\ \<lambda>i x. is_b(##Lset(i), h(x), f(x), g(x), g'(x))]" and is_c_reflection: "!!h f g g'. REFLECTS[\ \<lambda>i x. is_c(##Lset(i), h(x), f(x), g(x), g'(x))]" and is_d_reflection: "!!h f g g'. REFLECTS[\ \<lambda>i x. is_d(##Lset(i), h(x), f(x), g(x))]" shows "REFLECTS[\ \<lambda>i x. is_formula_case(##Lset(i), is_a(##Lset(i), x), is_b(##Lset(i), x), is_c(##L apply (simp (no_asm_use) only: is_formula_case_def) apply (intro FOL_reflections function_reflections finite_ordinal_reflection mem_formula_reflection Member_reflection Equal_reflection Nand_reflection Forall_reflection is_a_reflection is_b_reflection is_c_reflection is_d_reflection) done subsection \<open>Absoluteness for the Function \<^term>\<open>satisfies\<close>\<close> definition is_depth_apply :: "[i=>o,i,i,i] => o" where \<comment> \<open>Merely a useful abbreviation for the sequel.\<close> "is_depth_apply(M,h,p,z) == \<exists>dp[M]. \<exists>sdp[M]. \<exists>hsdp[M]. finite_ordinal(M,dp) & is_depth(M,p,dp) & successor(M,dp,sdp) & fun_apply(M,h,sdp,hsdp) & fun_apply(M,hsdp,p,z)" lemma (in M_datatypes) is_depth_apply_abs [simp]: "[|M(h); p \ ==> is_depth_apply(M,h,p,z) \<longleftrightarrow> z = h ` succ(depth(p)) ` p" by (simp add: is_depth_apply_def formula_into_M depth_type eq_commute) text\<open>There is at present some redundancy between the relativizations in e.g. \<open>satisfies_is_a\<close> and those in e.g. \<open>Member_replacement\<close>.\<close> text\<open>These constants let us instantiate the parameters \<^term>\<open>a\<close>, \<^term>\<op \<^term>\<open>c\<close>, \<^term>\<open>d\<close>, etc., of the locale \<open>Formula_Rec\<close>.\<cl definition satisfies_a :: "[i,i,i]=>i" where "satisfies_a(A) == \<lambda>x y. \<lambda>env \<in> list(A). bool_of_o (nth(x,env) \<in> nth(y,env))" definition satisfies_is_a :: "[i=>o,i,i,i,i]=>o" where "satisfies_is_a(M,A) == \<lambda>x y zz. \<forall>lA[M]. is_list(M,A,lA) \<longrightarrow> is_lambda(M, lA, \<lambda>env z. is_bool_of_o(M, \<exists>nx[M]. \<exists>ny[M]. is_nth(M,x,env,nx) & is_nth(M,y,env,ny) & nx \<in> ny, z), zz)" definition satisfies_b :: "[i,i,i]=>i" where "satisfies_b(A) == \<lambda>x y. \<lambda>env \<in> list(A). bool_of_o (nth(x,env) = nth(y,env))" definition satisfies_is_b :: "[i=>o,i,i,i,i]=>o" where \<comment> \<open>We simplify the formula to have just \<^term>\<open>nx\<close> rather than introducing \<^term>\<open>ny\<close> with \<^term>\<open>nx=ny\<close>\<close> "satisfies_is_b(M,A) == \<lambda>x y zz. \<forall>lA[M]. is_list(M,A,lA) \<longrightarrow> is_lambda(M, lA, \<lambda>env z. is_bool_of_o(M, \<exists>nx[M]. is_nth(M,x,env,nx) & is_nth(M,y,env,nx), z), zz)" definition satisfies_c :: "[i,i,i,i,i]=>i" where "satisfies_c(A) == \ definition satisfies_is_c :: "[i=>o,i,i,i,i,i]=>o" where "satisfies_is_c(M,A,h) == \<lambda>p q zz. \<forall>lA[M]. is_list(M,A,lA) \<longrightarrow> is_lambda(M, lA, \<lambda>env z. \<exists>hp[M]. \<exists>hq[M]. (\<exists>rp[M]. is_depth_apply(M,h,p,rp) & fun_apply(M,rp,env,hp)) & (\<exists>rq[M]. is_depth_apply(M,h,q,rq) & fun_apply(M,rq,env,hq)) & (\<exists>pq[M]. is_and(M,hp,hq,pq) & is_not(M,pq,z)), zz)" definition satisfies_d :: "[i,i,i]=>i" where "satisfies_d(A) == \<lambda>p rp. \<lambda>env \<in> list(A). bool_of_o (\<forall>x\<in>A. rp ` (Cons(x,env)) = 1)" definition satisfies_is_d :: "[i=>o,i,i,i,i]=>o" where "satisfies_is_d(M,A,h) == \<lambda>p zz. \<forall>lA[M]. is_list(M,A,lA) \<longrightarrow> is_lambda(M, lA, \<lambda>env z. \<exists>rp[M]. is_depth_apply(M,h,p,rp) & is_bool_of_o(M, \<forall>x[M]. \<forall>xenv[M]. \<forall>hp[M]. x\<in>A \<longrightarrow> is_Cons(M,x,env,xenv) \<longrightarrow> fun_apply(M,rp,xenv,hp) \<longrightarrow> number1(M,hp), z), zz)" definition satisfies_MH :: "[i=>o,i,i,i,i]=>o" where \<comment> \<open>The variable \<^term>\<open>u\<close> is unused, but gives \<^term>\<open>satisfies_ the correct arity.\<close> "satisfies_MH == \<lambda>M A u f z. \<forall>fml[M]. is_formula(M,fml) \<longrightarrow> is_lambda (M, fml, is_formula_case (M, satisfies_is_a(M,A), satisfies_is_b(M,A), satisfies_is_c(M,A,f), satisfies_is_d(M,A,f)), z)" definition is_satisfies :: "[i=>o,i,i,i]=>o" where "is_satisfies(M,A) == is_formula_rec (M, satisfies_MH(M,A))" text\<open>This lemma relates the fragments defined above to the original primitive recursion in \<^term>\<open>satisfies\<close>. Induction is not required: the definitions are directly equal!\<close> lemma satisfies_eq: "satisfies(A,p) = formula_rec (satisfies_a(A), satisfies_b(A), satisfies_c(A), satisfies_d(A), p)" by (simp add: satisfies_formula_def satisfies_a_def satisfies_b_def satisfies_c_def satisfies_d_def) text\<open>Further constraints on the class \<^term>\<open>M\<close> in order to prove absoluteness for the constants defined above. The ultimate goal is the absoluteness of the function \<^term>\<open>satisfies\<close>.\<close> locale M_satisfies = M_eclose + assumes Member_replacement: "[|M(A); x \ ==> strong_replacement (M, \<lambda>env z. \<exists>bo[M]. \<exists>nx[M]. \<exists>ny[M]. env \<in> list(A) & is_nth(M,x,env,nx) & is_nth(M,y,env,ny) & is_bool_of_o(M, nx \<in> ny, bo) & pair(M, env, bo, z))" and Equal_replacement: "[|M(A); x \ ==> strong_replacement (M, \<lambda>env z. \<exists>bo[M]. \<exists>nx[M]. \<exists>ny[M]. env \<in> list(A) & is_nth(M,x,env,nx) & is_nth(M,y,env,ny) & is_bool_of_o(M, nx = ny, bo) & pair(M, env, bo, z))" and Nand_replacement: "[|M(A); M(rp); M(rq)|] ==> strong_replacement (M, \<lambda>env z. \<exists>rpe[M]. \<exists>rqe[M]. \<exists>andpq[M]. \<exists>notpq[M]. fun_apply(M,rp,env,rpe) & fun_apply(M,rq,env,rqe) & is_and(M,rpe,rqe,andpq) & is_not(M,andpq,notpq) & env \<in> list(A) & pair(M, env, notpq, z))" and Forall_replacement: "[|M(A); M(rp)|] ==> strong_replacement (M, \<lambda>env z. \<exists>bo[M]. env \<in> list(A) & is_bool_of_o (M, \<forall>a[M]. \<forall>co[M]. \<forall>rpco[M]. a\<in>A \<longrightarrow> is_Cons(M,a,env,co) \<longrightarrow> fun_apply(M,rp,co,rpco) \<longrightarrow> number1(M, rpco), bo) & pair(M,env,bo,z))" and formula_rec_replacement: \<comment> \<open>For the \<^term>\<open>transrec\<close>\<close> "[|n \ and formula_rec_lambda_replacement: \<comment> \<open>For the \<open>\<lambda>-abstraction\<close> in the \<^term>\<open>transrec\<close> b "[|M(g); M(A)|] ==> strong_replacement (M, \<lambda>x y. mem_formula(M,x) & (\<exists>c[M]. is_formula_case(M, satisfies_is_a(M,A), satisfies_is_b(M,A), satisfies_is_c(M,A,g), satisfies_is_d(M,A,g), x, c) & pair(M, x, c, y)))" lemma (in M_satisfies) Member_replacement': "[|M(A); x \ ==> strong_replacement (M, \<lambda>env z. env \<in> list(A) & z = \<langle>env, bool_of_o(nth(x, env) \<in> nth(y, env))\<rangle>)" by (insert Member_replacement, simp) lemma (in M_satisfies) Equal_replacement': "[|M(A); x \ ==> strong_replacement (M, \<lambda>env z. env \<in> list(A) & z = \<langle>env, bool_of_o(nth(x, env) = nth(y, env))\<rangle>)" by (insert Equal_replacement, simp) lemma (in M_satisfies) Nand_replacement': "[|M(A); M(rp); M(rq)|] ==> strong_replacement (M, \<lambda>env z. env \<in> list(A) & z = \<langle>env, not(rp`env and rq`env)\<rangle>)" by (insert Nand_replacement, simp) lemma (in M_satisfies) Forall_replacement': "[|M(A); M(rp)|] ==> strong_replacement (M, \<lambda>env z. env \<in> list(A) & z = \<langle>env, bool_of_o (\<forall>a\<in>A. rp ` Cons(a,env) = 1)\<rangle>)" by (insert Forall_replacement, simp) lemma (in M_satisfies) a_closed: "[|M(A); x\ apply (simp add: satisfies_a_def) apply (blast intro: lam_closed2 Member_replacement') done lemma (in M_satisfies) a_rel: "M(A) ==> Relation2(M, nat, nat, satisfies_is_a(M,A), satisfies_a(A))" apply (simp add: Relation2_def satisfies_is_a_def satisfies_a_def) apply (auto del: iffI intro!: lambda_abs2 simp add: Relation1_def) done lemma (in M_satisfies) b_closed: "[|M(A); x\ apply (simp add: satisfies_b_def) apply (blast intro: lam_closed2 Equal_replacement') done lemma (in M_satisfies) b_rel: "M(A) ==> Relation2(M, nat, nat, satisfies_is_b(M,A), satisfies_b(A))" apply (simp add: Relation2_def satisfies_is_b_def satisfies_b_def) apply (auto del: iffI intro!: lambda_abs2 simp add: Relation1_def) done lemma (in M_satisfies) c_closed: "[|M(A); x \ ==> M(satisfies_c(A,x,y,rx,ry))" apply (simp add: satisfies_c_def) apply (rule lam_closed2) apply (rule Nand_replacement') apply (simp_all add: formula_into_M list_into_M [of _ A]) done lemma (in M_satisfies) c_rel: "[|M(A); M(f)|] ==> Relation2 (M, formula, formula, satisfies_is_c(M,A,f), \<lambda>u v. satisfies_c(A, u, v, f ` succ(depth(u)) ` u, f ` succ(depth(v)) ` v))" apply (simp add: Relation2_def satisfies_is_c_def satisfies_c_def) apply (auto del: iffI intro!: lambda_abs2 simp add: Relation1_def formula_into_M) done lemma (in M_satisfies) d_closed: "[|M(A); x \ apply (simp add: satisfies_d_def) apply (rule lam_closed2) apply (rule Forall_replacement') apply (simp_all add: formula_into_M list_into_M [of _ A]) done lemma (in M_satisfies) d_rel: "[|M(A); M(f)|] ==> Relation1(M, formula, satisfies_is_d(M,A,f), \<lambda>u. satisfies_d(A, u, f ` succ(depth(u)) ` u))" apply (simp del: rall_abs add: Relation1_def satisfies_is_d_def satisfies_d_def) apply (auto del: iffI intro!: lambda_abs2 simp add: Relation1_def) done lemma (in M_satisfies) fr_replace: "[|n \ by (blast intro: formula_rec_replacement) lemma (in M_satisfies) formula_case_satisfies_closed: "[|M(g); M(A); x \ M(formula_case (satisfies_a(A), satisfies_b(A), \<lambda>u v. satisfies_c(A, u, v, g ` succ(depth(u)) ` u, g ` succ(depth(v)) ` v), \<lambda>u. satisfies_d (A, u, g ` succ(depth(u)) ` u), x))" by (blast intro: a_closed b_closed c_closed d_closed) lemma (in M_satisfies) fr_lam_replace: "[|M(g); M(A)|] ==> strong_replacement (M, \<lambda>x y. x \<in> formula & y = \<langle>x, formula_rec_case(satisfies_a(A), satisfies_b(A), satisfies_c(A), satisfies_d(A), g, x)\<rangle>)" apply (insert formula_rec_lambda_replacement) apply (simp add: formula_rec_case_def formula_case_satisfies_closed formula_case_abs [OF a_rel b_rel c_rel d_rel]) done text\<open>Instantiate locale \<open>Formula_Rec\<close> for the Function \<^term>\<open>satisfies\<close>\<close> lemma (in M_satisfies) Formula_Rec_axioms_M: "M(A) ==> Formula_Rec_axioms(M, satisfies_a(A), satisfies_is_a(M,A), satisfies_b(A), satisfies_is_b(M,A), satisfies_c(A), satisfies_is_c(M,A), satisfies_d(A), satisfies_is_d(M,A))" apply (rule Formula_Rec_axioms.intro) apply (assumption | rule a_closed a_rel b_closed b_rel c_closed c_rel d_closed d_rel fr_replace [unfolded satisfies_MH_def] fr_lam_replace) + done theorem (in M_satisfies) Formula_Rec_M: "M(A) ==> Formula_Rec(M, satisfies_a(A), satisfies_is_a(M,A), satisfies_b(A), satisfies_is_b(M,A), satisfies_c(A), satisfies_is_c(M,A), satisfies_d(A), satisfies_is_d(M,A))" apply (rule Formula_Rec.intro) apply (rule M_satisfies.axioms, rule M_satisfies_axioms) apply (erule Formula_Rec_axioms_M) done lemmas (in M_satisfies) satisfies_closed' = Formula_Rec.formula_rec_closed [OF Formula_Rec_M] and satisfies_abs' = Formula_Rec.formula_rec_abs [OF Formula_Rec_M] lemma (in M_satisfies) satisfies_closed: "[|M(A); p \ by (simp add: Formula_Rec.formula_rec_closed [OF Formula_Rec_M] satisfies_eq) lemma (in M_satisfies) satisfies_abs: "[|M(A); M(z); p \ ==> is_satisfies(M,A,p,z) \<longleftrightarrow> z = satisfies(A,p)" by (simp only: Formula_Rec.formula_rec_abs [OF Formula_Rec_M] satisfies_eq is_satisfies_def satisfies_MH_def) subsection\<open>Internalizations Needed to Instantiate \<open>M_satisfies\<close>\<close> subsubsection\<open>The Operator \<^term>\<open>is_depth_apply\<close>, Internalized\<close> (* is_depth_apply(M,h,p,z) == \<exists>dp[M]. \<exists>sdp[M]. \<exists>hsdp[M]. 2 1 0 finite_ordinal(M,dp) & is_depth(M,p,dp) & successor(M,dp,sdp) & fun_apply(M,h,sdp,hsdp) & fun_apply(M,hsdp,p,z) *) definition depth_apply_fm :: "[i,i,i]=>i" where "depth_apply_fm(h,p,z) == Exists(Exists(Exists( And(finite_ordinal_fm(2), And(depth_fm(p#+3,2), And(succ_fm(2,1), And(fun_apply_fm(h#+3,1,0), fun_apply_fm(0,p#+3,z#+3))))))))" lemma depth_apply_type [TC]: "[| x \ by (simp add: depth_apply_fm_def) lemma sats_depth_apply_fm [simp]: "[| x \ ==> sats(A, depth_apply_fm(x,y,z), env) \<longleftrightarrow> is_depth_apply(##A, nth(x,env), nth(y,env), nth(z,env))" by (simp add: depth_apply_fm_def is_depth_apply_def) lemma depth_apply_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] ==> is_depth_apply(##A, x, y, z) \<longleftrightarrow> sats(A, depth_apply_fm(i,j,k), env)" by simp lemma depth_apply_reflection: "REFLECTS[\ \<lambda>i x. is_depth_apply(##Lset(i),f(x),g(x),h(x))]" apply (simp only: is_depth_apply_def) apply (intro FOL_reflections function_reflections depth_reflection finite_ordinal_reflection) done subsubsection\<open>The Operator \<^term>\<open>satisfies_is_a\<close>, Internalized\<close> (* satisfies_is_a(M,A) == \<lambda>x y zz. \<forall>lA[M]. is_list(M,A,lA) \<longrightarrow> is_lambda(M, lA, \<lambda>env z. is_bool_of_o(M, \<exists>nx[M]. \<exists>ny[M]. is_nth(M,x,env,nx) & is_nth(M,y,env,ny) & nx \<in> ny, z), zz) *) definition satisfies_is_a_fm :: "[i,i,i,i]=>i" where "satisfies_is_a_fm(A,x,y,z) == Forall( Implies(is_list_fm(succ(A),0), lambda_fm( bool_of_o_fm(Exists( Exists(And(nth_fm(x#+6,3,1), And(nth_fm(y#+6,3,0), Member(1,0))))), 0), 0, succ(z))))" lemma satisfies_is_a_type [TC]: "[| A \ ==> satisfies_is_a_fm(A,x,y,z) \<in> formula" by (simp add: satisfies_is_a_fm_def) lemma sats_satisfies_is_a_fm [simp]: "[| u \ ==> sats(A, satisfies_is_a_fm(u,x,y,z), env) \<longleftrightarrow> satisfies_is_a(##A, nth(u,env), nth(x,env), nth(y,env), nth(z,env))" apply (frule_tac x=x in lt_length_in_nat, assumption) apply (frule_tac x=y in lt_length_in_nat, assumption) apply (simp add: satisfies_is_a_fm_def satisfies_is_a_def sats_lambda_fm sats_bool_of_o_fm) done lemma satisfies_is_a_iff_sats: "[| nth(u,env) = nu; nth(x,env) = nx; nth(y,env) = ny; nth(z,env) = nz; u \<in> nat; x < length(env); y < length(env); z \<in> nat; env \<in> list(A)|] ==> satisfies_is_a(##A,nu,nx,ny,nz) \<longleftrightarrow> sats(A, satisfies_is_a_fm(u,x,y,z), env)" by simp theorem satisfies_is_a_reflection: "REFLECTS[\ \<lambda>i x. satisfies_is_a(##Lset(i),f(x),g(x),h(x),g'(x))]" apply (unfold satisfies_is_a_def) apply (intro FOL_reflections is_lambda_reflection bool_of_o_reflection nth_reflection is_list_reflection) done subsubsection\<open>The Operator \<^term>\<open>satisfies_is_b\<close>, Internalized\<close> (* satisfies_is_b(M,A) == \<lambda>x y zz. \<forall>lA[M]. is_list(M,A,lA) \<longrightarrow> is_lambda(M, lA, \<lambda>env z. is_bool_of_o(M, \<exists>nx[M]. is_nth(M,x,env,nx) & is_nth(M,y,env,nx), z), zz) *) definition satisfies_is_b_fm :: "[i,i,i,i]=>i" where "satisfies_is_b_fm(A,x,y,z) == Forall( Implies(is_list_fm(succ(A),0), lambda_fm( bool_of_o_fm(Exists(And(nth_fm(x#+5,2,0), nth_fm(y#+5,2,0))), 0), 0, succ(z))))" lemma satisfies_is_b_type [TC]: "[| A \ ==> satisfies_is_b_fm(A,x,y,z) \<in> formula" by (simp add: satisfies_is_b_fm_def) lemma sats_satisfies_is_b_fm [simp]: "[| u \ ==> sats(A, satisfies_is_b_fm(u,x,y,z), env) \<longleftrightarrow> satisfies_is_b(##A, nth(u,env), nth(x,env), nth(y,env), nth(z,env))" apply (frule_tac x=x in lt_length_in_nat, assumption) apply (frule_tac x=y in lt_length_in_nat, assumption) apply (simp add: satisfies_is_b_fm_def satisfies_is_b_def sats_lambda_fm sats_bool_of_o_fm) done lemma satisfies_is_b_iff_sats: "[| nth(u,env) = nu; nth(x,env) = nx; nth(y,env) = ny; nth(z,env) = nz; u \<in> nat; x < length(env); y < length(env); z \<in> nat; env \<in> list(A)|] ==> satisfies_is_b(##A,nu,nx,ny,nz) \<longleftrightarrow> sats(A, satisfies_is_b_fm(u,x,y,z), env)" by simp theorem satisfies_is_b_reflection: "REFLECTS[\ \<lambda>i x. satisfies_is_b(##Lset(i),f(x),g(x),h(x),g'(x))]" apply (unfold satisfies_is_b_def) apply (intro FOL_reflections is_lambda_reflection bool_of_o_reflection nth_reflection is_list_reflection) done subsubsection\<open>The Operator \<^term>\<open>satisfies_is_c\<close>, Internalized\<close> (* satisfies_is_c(M,A,h) == \<lambda>p q zz. \<forall>lA[M]. is_list(M,A,lA) \<longrightarrow> is_lambda(M, lA, \<lambda>env z. \<exists>hp[M]. \<exists>hq[M]. (\<exists>rp[M]. is_depth_apply(M,h,p,rp) & fun_apply(M,rp,env,hp)) & (\<exists>rq[M]. is_depth_apply(M,h,q,rq) & fun_apply(M,rq,env,hq)) & (\<exists>pq[M]. is_and(M,hp,hq,pq) & is_not(M,pq,z)), zz) *) definition satisfies_is_c_fm :: "[i,i,i,i,i]=>i" where "satisfies_is_c_fm(A,h,p,q,zz) == Forall( Implies(is_list_fm(succ(A),0), lambda_fm( Exists(Exists( And(Exists(And(depth_apply_fm(h#+7,p#+7,0), fun_apply_fm(0,4,2))), And(Exists(And(depth_apply_fm(h#+7,q#+7,0), fun_apply_fm(0,4,1))), Exists(And(and_fm(2,1,0), not_fm(0,3))))))), 0, succ(zz))))" lemma satisfies_is_c_type [TC]: "[| A \ ==> satisfies_is_c_fm(A,h,x,y,z) \<in> formula" by (simp add: satisfies_is_c_fm_def) lemma sats_satisfies_is_c_fm [simp]: "[| u \ ==> sats(A, satisfies_is_c_fm(u,v,x,y,z), env) \<longleftrightarrow> satisfies_is_c(##A, nth(u,env), nth(v,env), nth(x,env), nth(y,env), nth(z,env))" by (simp add: satisfies_is_c_fm_def satisfies_is_c_def sats_lambda_fm) lemma satisfies_is_c_iff_sats: "[| nth(u,env) = nu; nth(v,env) = nv; nth(x,env) = nx; nth(y,env) = ny; nth(z,env) = nz; u \<in> nat; v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] ==> satisfies_is_c(##A,nu,nv,nx,ny,nz) \<longleftrightarrow> sats(A, satisfies_is_c_fm(u,v,x,y,z), env)" by simp theorem satisfies_is_c_reflection: "REFLECTS[\ \<lambda>i x. satisfies_is_c(##Lset(i),f(x),g(x),h(x),g'(x),h'(x))]" apply (unfold satisfies_is_c_def) apply (intro FOL_reflections function_reflections is_lambda_reflection extra_reflections nth_reflection depth_apply_reflection is_list_reflection) done subsubsection\<open>The Operator \<^term>\<open>satisfies_is_d\<close>, Internalized\<close> (* satisfies_is_d(M,A,h) == \<lambda>p zz. \<forall>lA[M]. is_list(M,A,lA) \<longrightarrow> is_lambda(M, lA, \<lambda>env z. \<exists>rp[M]. is_depth_apply(M,h,p,rp) & is_bool_of_o(M, \<forall>x[M]. \<forall>xenv[M]. \<forall>hp[M]. x\<in>A \<longrightarrow> is_Cons(M,x,env,xenv) \<longrightarrow> fun_apply(M,rp,xenv,hp) \<longrightarrow> number1(M,hp), z), zz) *) definition satisfies_is_d_fm :: "[i,i,i,i]=>i" where "satisfies_is_d_fm(A,h,p,zz) == Forall( Implies(is_list_fm(succ(A),0), lambda_fm( Exists( And(depth_apply_fm(h#+5,p#+5,0), bool_of_o_fm( Forall(Forall(Forall( Implies(Member(2,A#+8), Implies(Cons_fm(2,5,1), Implies(fun_apply_fm(3,1,0), number1_fm(0))))))), 1))), 0, succ(zz))))" lemma satisfies_is_d_type [TC]: "[| A \ ==> satisfies_is_d_fm(A,h,x,z) \<in> formula" by (simp add: satisfies_is_d_fm_def) lemma sats_satisfies_is_d_fm [simp]: "[| u \ ==> sats(A, satisfies_is_d_fm(u,x,y,z), env) \<longleftrightarrow> satisfies_is_d(##A, nth(u,env), nth(x,env), nth(y,env), nth(z,env))" by (simp add: satisfies_is_d_fm_def satisfies_is_d_def sats_lambda_fm sats_bool_of_o_fm) lemma satisfies_is_d_iff_sats: "[| nth(u,env) = nu; nth(x,env) = nx; nth(y,env) = ny; nth(z,env) = nz; u \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] ==> satisfies_is_d(##A,nu,nx,ny,nz) \<longleftrightarrow> sats(A, satisfies_is_d_fm(u,x,y,z), env)" by simp theorem satisfies_is_d_reflection: "REFLECTS[\ \<lambda>i x. satisfies_is_d(##Lset(i),f(x),g(x),h(x),g'(x))]" apply (unfold satisfies_is_d_def) apply (intro FOL_reflections function_reflections is_lambda_reflection extra_reflections nth_reflection depth_apply_reflection is_list_reflection) done subsubsection\<open>The Operator \<^term>\<open>satisfies_MH\<close>, Internalized\<close> (* satisfies_MH == \<lambda>M A u f zz. \<forall>fml[M]. is_formula(M,fml) \<longrightarrow> is_lambda (M, fml, is_formula_case (M, satisfies_is_a(M,A), satisfies_is_b(M,A), satisfies_is_c(M,A,f), satisfies_is_d(M,A,f)), zz) *) definition satisfies_MH_fm :: "[i,i,i,i]=>i" where "satisfies_MH_fm(A,u,f,zz) == Forall( Implies(is_formula_fm(0), lambda_fm( formula_case_fm(satisfies_is_a_fm(A#+7,2,1,0), satisfies_is_b_fm(A#+7,2,1,0), satisfies_is_c_fm(A#+7,f#+7,2,1,0), satisfies_is_d_fm(A#+6,f#+6,1,0), 1, 0), 0, succ(zz))))" lemma satisfies_MH_type [TC]: "[| A \ ==> satisfies_MH_fm(A,u,x,z) \<in> formula" by (simp add: satisfies_MH_fm_def) lemma sats_satisfies_MH_fm [simp]: "[| u \ ==> sats(A, satisfies_MH_fm(u,x,y,z), env) \<longleftrightarrow> satisfies_MH(##A, nth(u,env), nth(x,env), nth(y,env), nth(z,env))" by (simp add: satisfies_MH_fm_def satisfies_MH_def sats_lambda_fm sats_formula_case_fm) lemma satisfies_MH_iff_sats: "[| nth(u,env) = nu; nth(x,env) = nx; nth(y,env) = ny; nth(z,env) = nz; u \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] ==> satisfies_MH(##A,nu,nx,ny,nz) \<longleftrightarrow> sats(A, satisfies_MH_fm(u,x,y,z), env)" by simp lemmas satisfies_reflections = is_lambda_reflection is_formula_reflection is_formula_case_reflection satisfies_is_a_reflection satisfies_is_b_reflection satisfies_is_c_reflection satisfies_is_d_reflection theorem satisfies_MH_reflection: "REFLECTS[\ \<lambda>i x. satisfies_MH(##Lset(i),f(x),g(x),h(x),g'(x))]" apply (unfold satisfies_MH_def) apply (intro FOL_reflections satisfies_reflections) done subsection\<open>Lemmas for Instantiating the Locale \<open>M_satisfies\<close>\<close> subsubsection\<open>The \<^term>\<open>Member\<close> Case\<close> lemma Member_Reflects: "REFLECTS[\ v \<in> lstA \<and> is_nth(L,x,v,nx) \<and> is_nth(L,y,v,ny) \<and> is_bool_of_o(L, nx \<in> ny, bo) \<and> pair(L,v,bo,u)), \<lambda>i u. \<exists>v \<in> Lset(i). v \<in> B \<and> (\<exists>bo \<in> Lset(i). \<exists>nx \<in> Lset(i). \< v \<in> lstA \<and> is_nth(##Lset(i), x, v, nx) \<and> is_nth(##Lset(i), y, v, ny) \<and> is_bool_of_o(##Lset(i), nx \<in> ny, bo) \<and> pair(##Lset(i), v, bo, u))]" by (intro FOL_reflections function_reflections nth_reflection bool_of_o_reflection) lemma Member_replacement: "[|L(A); x \ ==> strong_replacement (L, \<lambda>env z. \<exists>bo[L]. \<exists>nx[L]. \<exists>ny[L]. env \<in> list(A) & is_nth(L,x,env,nx) & is_nth(L,y,env,ny) & is_bool_of_o(L, nx \<in> ny, bo) & pair(L, env, bo, z))" apply (rule strong_replacementI) apply (rule_tac u="{list(A),B,x,y}" in gen_separation_multi [OF Member_Reflects], auto) apply (rule_tac env="[list(A),B,x,y]" in DPow_LsetI) apply (rule sep_rules nth_iff_sats is_bool_of_o_iff_sats | simp)+ done subsubsection\<open>The \<^term>\<open>Equal\<close> Case\<close> lemma Equal_Reflects: "REFLECTS[\ v \<in> lstA \<and> is_nth(L, x, v, nx) \<and> is_nth(L, y, v, ny) \<and> is_bool_of_o(L, nx = ny, bo) \<and> pair(L, v, bo, u)), \<lambda>i u. \<exists>v \<in> Lset(i). v \<in> B \<and> (\<exists>bo \<in> Lset(i). \<exists>nx \<in> Lset(i). \< v \<in> lstA \<and> is_nth(##Lset(i), x, v, nx) \<and> is_nth(##Lset(i), y, v, ny) \<and> is_bool_of_o(##Lset(i), nx = ny, bo) \<and> pair(##Lset(i), v, bo, u))]" by (intro FOL_reflections function_reflections nth_reflection bool_of_o_reflection) lemma Equal_replacement: "[|L(A); x \ ==> strong_replacement (L, \<lambda>env z. \<exists>bo[L]. \<exists>nx[L]. \<exists>ny[L]. env \<in> list(A) & is_nth(L,x,env,nx) & is_nth(L,y,env,ny) & is_bool_of_o(L, nx = ny, bo) & pair(L, env, bo, z))" apply (rule strong_replacementI) apply (rule_tac u="{list(A),B,x,y}" in gen_separation_multi [OF Equal_Reflects], auto) apply (rule_tac env="[list(A),B,x,y]" in DPow_LsetI) apply (rule sep_rules nth_iff_sats is_bool_of_o_iff_sats | simp)+ done subsubsection\<open>The \<^term>\<open>Nand\<close> Case\<close> lemma Nand_Reflects: "REFLECTS [\ (\<exists>rpe[L]. \<exists>rqe[L]. \<exists>andpq[L]. \<exists>notpq[L]. fun_apply(L, rp, u, rpe) \<and> fun_apply(L, rq, u, rqe) \<and> is_and(L, rpe, rqe, andpq) \<and> is_not(L, andpq, notpq) \<and> u \<in> list(A) \<and> pair(L, u, notpq, x)), \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>rpe \<in> Lset(i). \<exists>rqe \<in> Lset(i). \<exists>andpq \<in> Lset(i). \<exists>notpq \<i fun_apply(##Lset(i), rp, u, rpe) \<and> fun_apply(##Lset(i), rq, u, rqe) \<and> is_and(##Lset(i), rpe, rqe, andpq) \<and> is_not(##Lset(i), andpq, notpq) \<and> u \<in> list(A) \<and> pair(##Lset(i), u, notpq, x))]" apply (unfold is_and_def is_not_def) apply (intro FOL_reflections function_reflections) done lemma Nand_replacement: "[|L(A); L(rp); L(rq)|] ==> strong_replacement (L, \<lambda>env z. \<exists>rpe[L]. \<exists>rqe[L]. \<exists>andpq[L]. \<exists>notpq[L]. fun_apply(L,rp,env,rpe) & fun_apply(L,rq,env,rqe) & is_and(L,rpe,rqe,andpq) & is_not(L,andpq,notpq) & env \<in> list(A) & pair(L, env, notpq, z))" apply (rule strong_replacementI) apply (rule_tac u="{list(A),B,rp,rq}" in gen_separation_multi [OF Nand_Reflects], auto) apply (rule_tac env="[list(A),B,rp,rq]" in DPow_LsetI) apply (rule sep_rules is_and_iff_sats is_not_iff_sats | simp)+ done subsubsection\<open>The \<^term>\<open>Forall\<close> Case\<close> lemma Forall_Reflects: "REFLECTS [\ is_bool_of_o (L, \<forall>a[L]. \<forall>co[L]. \<forall>rpco[L]. a \<in> A \<longrightarrow> is_Cons(L,a,u,co) \<longrightarrow> fun_apply(L,rp,co,rpco) \<longrightarrow> number1(L,rpco), bo) \<and> pair(L,u,bo,x)), \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>bo \<in> Lset(i). u \<in> list(A) \<and> is_bool_of_o (##Lset(i), \<forall>a \<in> Lset(i). \<forall>co \<in> Lset(i). \<forall>rpco \<in> Lset(i). a \<in> A \<longrightarro is_Cons(##Lset(i),a,u,co) \<longrightarrow> fun_apply(##Lset(i),rp,co,rpco) \<longrigh number1(##Lset(i),rpco), bo) \<and> pair(##Lset(i),u,bo,x))]" apply (unfold is_bool_of_o_def) apply (intro FOL_reflections function_reflections Cons_reflection) done lemma Forall_replacement: "[|L(A); L(rp)|] ==> strong_replacement (L, \<lambda>env z. \<exists>bo[L]. env \<in> list(A) & is_bool_of_o (L, \<forall>a[L]. \<forall>co[L]. \<forall>rpco[L]. a\<in>A \<longrightarrow> is_Cons(L,a,env,co) \<longrightarrow> fun_apply(L,rp,co,rpco) \<longrightarrow> number1(L, rpco), bo) & pair(L,env,bo,z))" apply (rule strong_replacementI) apply (rule_tac u="{A,list(A),B,rp}" in gen_separation_multi [OF Forall_Reflects], auto) apply (rule_tac env="[A,list(A),B,rp]" in DPow_LsetI) apply (rule sep_rules is_bool_of_o_iff_sats Cons_iff_sats | simp)+ done subsubsection\<open>The \<^term>\<open>transrec_replacement\<close> Case\<close> lemma formula_rec_replacement_Reflects: "REFLECTS [\ is_wfrec (L, satisfies_MH(L,A), mesa, u, y)), \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(##Lset(i), u, y, x) \< is_wfrec (##Lset(i), satisfies_MH(##Lset(i),A), mesa, u, y))]" by (intro FOL_reflections function_reflections satisfies_MH_reflection is_wfrec_reflection) lemma formula_rec_replacement: \<comment> \<open>For the \<^term>\<open>transrec\<close>\<close> "[|n \ apply (rule L.transrec_replacementI, simp add: L.nat_into_M) apply (rule strong_replacementI) apply (rule_tac u="{B,A,n,Memrel(eclose({n}))}" in gen_separation_multi [OF formula_rec_replacement_Reflects], auto simp add: L.nat_into_M) apply (rule_tac env="[B,A,n,Memrel(eclose({n}))]" in DPow_LsetI) apply (rule sep_rules satisfies_MH_iff_sats is_wfrec_iff_sats | simp)+ done subsubsection\<open>The Lambda Replacement Case\<close> lemma formula_rec_lambda_replacement_Reflects: "REFLECTS [\ mem_formula(L,u) & (\<exists>c[L]. is_formula_case (L, satisfies_is_a(L,A), satisfies_is_b(L,A), satisfies_is_c(L,A,g), satisfies_is_d(L,A,g), u, c) & pair(L,u,c,x)), \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B & mem_formula(##Lset(i),u) & (\<exists>c \<in> Lset(i). is_formula_case (##Lset(i), satisfies_is_a(##Lset(i),A), satisfies_is_b(##Lset(i),A), satisfies_is_c(##Lset(i),A,g), satisfies_is_d(##Lset(i),A,g), u, c) & pair(##Lset(i),u,c,x))]" by (intro FOL_reflections function_reflections mem_formula_reflection is_formula_case_reflection satisfies_is_a_reflection satisfies_is_b_reflection satisfies_is_c_reflection satisfies_is_d_reflection) lemma formula_rec_lambda_replacement: \<comment> \<open>For the \<^term>\<open>transrec\<close>\<close> "[|L(g); L(A)|] ==> strong_replacement (L, \<lambda>x y. mem_formula(L,x) & (\<exists>c[L]. is_formula_case(L, satisfies_is_a(L,A), satisfies_is_b(L,A), satisfies_is_c(L,A,g), satisfies_is_d(L,A,g), x, c) & pair(L, x, c, y)))" apply (rule strong_replacementI) apply (rule_tac u="{B,A,g}" in gen_separation_multi [OF formula_rec_lambda_replacement_Reflects], auto) apply (rule_tac env="[A,g,B]" in DPow_LsetI) apply (rule sep_rules mem_formula_iff_sats formula_case_iff_sats satisfies_is_a_iff_sats satisfies_is_b_iff_sats satisfies_is_c_iff_sats satisfies_is_d_iff_sats | simp)+ done subsection\<open>Instantiating \<open>M_satisfies\<close>\<close> lemma M_satisfies_axioms_L: "M_satisfies_axioms(L)" apply (rule M_satisfies_axioms.intro) apply (assumption | rule Member_replacement Equal_replacement Nand_replacement Forall_replacement formula_rec_replacement formula_rec_lambda_replacement)+ done theorem M_satisfies_L: "M_satisfies(L)" apply (rule M_satisfies.intro) apply (rule M_eclose_L) apply (rule M_satisfies_axioms_L) done text\<open>Finally: the point of the whole theory!\<close> lemmas satisfies_closed = M_satisfies.satisfies_closed [OF M_satisfies_L] and satisfies_abs = M_satisfies.satisfies_abs [OF M_satisfies_L] end ¤ Dauer der Verarbeitung: 0.46 Sekunden (vorverarbeitet) ¤ |
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