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theory Satisfies_absolute(* Title: ZF/Constructible/Satisfies_absolute.thy ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory *) header {*Absoluteness for the Satisfies Relation on Formulas*} theory Satisfies_absolute imports Datatype_absolute Rec_Separation begin subsection {*More Internalization*} subsubsection{*The Formula @{term is_depth}, Internalized*} (* "is_depth(M,p,n) == ∃sn[M]. ∃formula_n[M]. ∃formula_sn[M]. 2 1 0 is_formula_N(M,n,formula_n) & p ∉ formula_n & successor(M,n,sn) & is_formula_N(M,sn,formula_sn) & p ∈ 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 ∈ nat; y ∈ nat |] ==> depth_fm(x,y) ∈ formula" by (simp add: depth_fm_def) lemma sats_depth_fm [simp]: "[| x ∈ nat; y < length(env); env ∈ list(A)|] ==> sats(A, depth_fm(x,y), env) <-> 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 ∈ nat; j < length(env); env ∈ list(A)|] ==> is_depth(##A, x, y) <-> sats(A, depth_fm(i,j), env)" by (simp add: sats_depth_fm) theorem depth_reflection: "REFLECTS[λx. is_depth(L, f(x), g(x)), λi 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{*The Operator @{term is_formula_case}*} text{*The arguments of @{term is_a} are always 2, 1, 0, and the formula will be enclosed by three quantifiers.*} (* 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) == (∀x[M]. ∀y[M]. x∈nat --> y∈nat --> is_Member(M,x,y,v) --> is_a(x,y,z)) & (∀x[M]. ∀y[M]. x∈nat --> y∈nat --> is_Equal(M,x,y,v) --> is_b(x,y,z)) & (∀x[M]. ∀y[M]. x∈formula --> y∈formula --> is_Nand(M,x,y,v) --> is_c(x,y,z)) & (∀x[M]. x∈formula --> is_Forall(M,x,v) --> 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 ∈ formula; is_b ∈ formula; is_c ∈ formula; is_d ∈ formula; x ∈ nat; y ∈ nat |] ==> formula_case_fm(is_a, is_b, is_c, is_d, x, y) ∈ formula" by (simp add: formula_case_fm_def) lemma sats_formula_case_fm: assumes is_a_iff_sats: "!!a0 a1 a2. [|a0∈A; a1∈A; a2∈A|] ==> ISA(a2, a1, a0) <-> sats(A, is_a, Cons(a0,Cons(a1,Cons(a2,env))))" and is_b_iff_sats: "!!a0 a1 a2. [|a0∈A; a1∈A; a2∈A|] ==> ISB(a2, a1, a0) <-> sats(A, is_b, Cons(a0,Cons(a1,Cons(a2,env))))" and is_c_iff_sats: "!!a0 a1 a2. [|a0∈A; a1∈A; a2∈A|] ==> ISC(a2, a1, a0) <-> sats(A, is_c, Cons(a0,Cons(a1,Cons(a2,env))))" and is_d_iff_sats: "!!a0 a1. [|a0∈A; a1∈A|] ==> ISD(a1, a0) <-> sats(A, is_d, Cons(a0,Cons(a1,env)))" shows "[|x ∈ nat; y < length(env); env ∈ list(A)|] ==> sats(A, formula_case_fm(is_a,is_b,is_c,is_d,x,y), env) <-> 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∈A; a1∈A; a2∈A|] ==> ISA(a2, a1, a0) <-> sats(A, is_a, Cons(a0,Cons(a1,Cons(a2,env))))" and is_b_iff_sats: "!!a0 a1 a2. [|a0∈A; a1∈A; a2∈A|] ==> ISB(a2, a1, a0) <-> sats(A, is_b, Cons(a0,Cons(a1,Cons(a2,env))))" and is_c_iff_sats: "!!a0 a1 a2. [|a0∈A; a1∈A; a2∈A|] ==> ISC(a2, a1, a0) <-> sats(A, is_c, Cons(a0,Cons(a1,Cons(a2,env))))" and is_d_iff_sats: "!!a0 a1. [|a0∈A; a1∈A|] ==> ISD(a1, a0) <-> sats(A, is_d, Cons(a0,Cons(a1,env)))" shows "[|nth(i,env) = x; nth(j,env) = y; i ∈ nat; j < length(env); env ∈ list(A)|] ==> is_formula_case(##A, ISA, ISB, ISC, ISD, x, y) <-> 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{*The second argument of @{term is_a} gives it direct access to @{term x}, which is essential for handling free variable references. Treatment is based on that of @{text is_nat_case_reflection}.*} theorem is_formula_case_reflection: assumes is_a_reflection: "!!h f g g'. REFLECTS[λx. is_a(L, h(x), f(x), g(x), g'(x)), λi x. is_a(##Lset(i), h(x), f(x), g(x), g'(x))]" and is_b_reflection: "!!h f g g'. REFLECTS[λx. is_b(L, h(x), f(x), g(x), g'(x)), λi x. is_b(##Lset(i), h(x), f(x), g(x), g'(x))]" and is_c_reflection: "!!h f g g'. REFLECTS[λx. is_c(L, h(x), f(x), g(x), g'(x)), λi x. is_c(##Lset(i), h(x), f(x), g(x), g'(x))]" and is_d_reflection: "!!h f g g'. REFLECTS[λx. is_d(L, h(x), f(x), g(x)), λi x. is_d(##Lset(i), h(x), f(x), g(x))]" shows "REFLECTS[λx. is_formula_case(L, is_a(L,x), is_b(L,x), is_c(L,x), is_d(L,x), g(x), h(x)), λi x. is_formula_case(##Lset(i), is_a(##Lset(i), x), is_b(##Lset(i), x), is_c(##Lset(i), x), is_d(##Lset(i), x), g(x), h(x))]" 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 {*Absoluteness for the Function @{term satisfies}*} definition is_depth_apply :: "[i=>o,i,i,i] => o" where --{*Merely a useful abbreviation for the sequel.*} "is_depth_apply(M,h,p,z) == ∃dp[M]. ∃sdp[M]. ∃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 ∈ formula; M(z)|] ==> is_depth_apply(M,h,p,z) <-> z = h ` succ(depth(p)) ` p" by (simp add: is_depth_apply_def formula_into_M depth_type eq_commute) text{*There is at present some redundancy between the relativizations in e.g. @{text satisfies_is_a} and those in e.g. @{text Member_replacement}.*} text{*These constants let us instantiate the parameters @{term a}, @{term b}, @{term c}, @{term d}, etc., of the locale @{text Formula_Rec}.*} definition satisfies_a :: "[i,i,i]=>i" where "satisfies_a(A) == λx y. λenv ∈ list(A). bool_of_o (nth(x,env) ∈ nth(y,env))" definition satisfies_is_a :: "[i=>o,i,i,i,i]=>o" where "satisfies_is_a(M,A) == λx y zz. ∀lA[M]. is_list(M,A,lA) --> is_lambda(M, lA, λenv z. is_bool_of_o(M, ∃nx[M]. ∃ny[M]. is_nth(M,x,env,nx) & is_nth(M,y,env,ny) & nx ∈ ny, z), zz)" definition satisfies_b :: "[i,i,i]=>i" where "satisfies_b(A) == λx y. λenv ∈ list(A). bool_of_o (nth(x,env) = nth(y,env))" definition satisfies_is_b :: "[i=>o,i,i,i,i]=>o" where --{*We simplify the formula to have just @{term nx} rather than introducing @{term ny} with @{term "nx=ny"} *} "satisfies_is_b(M,A) == λx y zz. ∀lA[M]. is_list(M,A,lA) --> is_lambda(M, lA, λenv z. is_bool_of_o(M, ∃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) == λp q rp rq. λenv ∈ list(A). not(rp ` env and rq ` env)" definition satisfies_is_c :: "[i=>o,i,i,i,i,i]=>o" where "satisfies_is_c(M,A,h) == λp q zz. ∀lA[M]. is_list(M,A,lA) --> is_lambda(M, lA, λenv z. ∃hp[M]. ∃hq[M]. (∃rp[M]. is_depth_apply(M,h,p,rp) & fun_apply(M,rp,env,hp)) & (∃rq[M]. is_depth_apply(M,h,q,rq) & fun_apply(M,rq,env,hq)) & (∃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) == λp rp. λenv ∈ list(A). bool_of_o (∀x∈A. rp ` (Cons(x,env)) = 1)" definition satisfies_is_d :: "[i=>o,i,i,i,i]=>o" where "satisfies_is_d(M,A,h) == λp zz. ∀lA[M]. is_list(M,A,lA) --> is_lambda(M, lA, λenv z. ∃rp[M]. is_depth_apply(M,h,p,rp) & is_bool_of_o(M, ∀x[M]. ∀xenv[M]. ∀hp[M]. x∈A --> is_Cons(M,x,env,xenv) --> fun_apply(M,rp,xenv,hp) --> number1(M,hp), z), zz)" definition satisfies_MH :: "[i=>o,i,i,i,i]=>o" where --{*The variable @{term u} is unused, but gives @{term satisfies_MH} the correct arity.*} "satisfies_MH == λM A u f z. ∀fml[M]. is_formula(M,fml) --> 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{*This lemma relates the fragments defined above to the original primitive recursion in @{term satisfies}. Induction is not required: the definitions are directly equal!*} 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{*Further constraints on the class @{term M} in order to prove absoluteness for the constants defined above. The ultimate goal is the absoluteness of the function @{term satisfies}. *} locale M_satisfies = M_eclose + assumes Member_replacement: "[|M(A); x ∈ nat; y ∈ nat|] ==> strong_replacement (M, λenv z. ∃bo[M]. ∃nx[M]. ∃ny[M]. env ∈ 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 Equal_replacement: "[|M(A); x ∈ nat; y ∈ nat|] ==> strong_replacement (M, λenv z. ∃bo[M]. ∃nx[M]. ∃ny[M]. env ∈ 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, λenv z. ∃rpe[M]. ∃rqe[M]. ∃andpq[M]. ∃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 ∈ list(A) & pair(M, env, notpq, z))" and Forall_replacement: "[|M(A); M(rp)|] ==> strong_replacement (M, λenv z. ∃bo[M]. env ∈ list(A) & is_bool_of_o (M, ∀a[M]. ∀co[M]. ∀rpco[M]. a∈A --> is_Cons(M,a,env,co) --> fun_apply(M,rp,co,rpco) --> number1(M, rpco), bo) & pair(M,env,bo,z))" and formula_rec_replacement: --{*For the @{term transrec}*} "[|n ∈ nat; M(A)|] ==> transrec_replacement(M, satisfies_MH(M,A), n)" and formula_rec_lambda_replacement: --{*For the @{text "λ-abstraction"} in the @{term transrec} body*} "[|M(g); M(A)|] ==> strong_replacement (M, λx y. mem_formula(M,x) & (∃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 ∈ nat; y ∈ nat|] ==> strong_replacement (M, λenv z. env ∈ list(A) & z = 〈env, bool_of_o(nth(x, env) ∈ nth(y, env))〉)" by (insert Member_replacement, simp) lemma (in M_satisfies) Equal_replacement': "[|M(A); x ∈ nat; y ∈ nat|] ==> strong_replacement (M, λenv z. env ∈ list(A) & z = 〈env, bool_of_o(nth(x, env) = nth(y, env))〉)" by (insert Equal_replacement, simp) lemma (in M_satisfies) Nand_replacement': "[|M(A); M(rp); M(rq)|] ==> strong_replacement (M, λenv z. env ∈ list(A) & z = 〈env, not(rp`env and rq`env)〉)" by (insert Nand_replacement, simp) lemma (in M_satisfies) Forall_replacement': "[|M(A); M(rp)|] ==> strong_replacement (M, λenv z. env ∈ list(A) & z = 〈env, bool_of_o (∀a∈A. rp ` Cons(a,env) = 1)〉)" by (insert Forall_replacement, simp) lemma (in M_satisfies) a_closed: "[|M(A); x∈nat; y∈nat|] ==> M(satisfies_a(A,x,y))" 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∈nat; y∈nat|] ==> M(satisfies_b(A,x,y))" 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 ∈ formula; y ∈ formula; M(rx); M(ry)|] ==> 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), λu 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 ∈ formula; M(rx)|] ==> M(satisfies_d(A,x,rx))" 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), λu. 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 ∈ nat; M(A)|] ==> transrec_replacement(M,satisfies_MH(M,A),n)" by (blast intro: formula_rec_replacement) lemma (in M_satisfies) formula_case_satisfies_closed: "[|M(g); M(A); x ∈ formula|] ==> M(formula_case (satisfies_a(A), satisfies_b(A), λu v. satisfies_c(A, u, v, g ` succ(depth(u)) ` u, g ` succ(depth(v)) ` v), λu. satisfies_d (A, u, g ` succ(depth(u)) ` u), x))" by (blast intro: formula_case_closed a_closed b_closed c_closed d_closed) lemma (in M_satisfies) fr_lam_replace: "[|M(g); M(A)|] ==> strong_replacement (M, λx y. x ∈ formula & y = 〈x, formula_rec_case(satisfies_a(A), satisfies_b(A), satisfies_c(A), satisfies_d(A), g, x)〉)" 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{*Instantiate locale @{text Formula_Rec} for the Function @{term satisfies}*} 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) ==> PROP 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 ∈ formula|] ==> M(satisfies(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 ∈ formula|] ==> is_satisfies(M,A,p,z) <-> 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{*Internalizations Needed to Instantiate @{text "M_satisfies"}*} subsubsection{*The Operator @{term is_depth_apply}, Internalized*} (* is_depth_apply(M,h,p,z) == ∃dp[M]. ∃sdp[M]. ∃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 ∈ nat; y ∈ nat; z ∈ nat |] ==> depth_apply_fm(x,y,z) ∈ formula" by (simp add: depth_apply_fm_def) lemma sats_depth_apply_fm [simp]: "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, depth_apply_fm(x,y,z), env) <-> 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 ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|] ==> is_depth_apply(##A, x, y, z) <-> sats(A, depth_apply_fm(i,j,k), env)" by simp lemma depth_apply_reflection: "REFLECTS[λx. is_depth_apply(L,f(x),g(x),h(x)), λi 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{*The Operator @{term satisfies_is_a}, Internalized*} (* satisfies_is_a(M,A) == λx y zz. ∀lA[M]. is_list(M,A,lA) --> is_lambda(M, lA, λenv z. is_bool_of_o(M, ∃nx[M]. ∃ny[M]. is_nth(M,x,env,nx) & is_nth(M,y,env,ny) & nx ∈ 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 ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat |] ==> satisfies_is_a_fm(A,x,y,z) ∈ formula" by (simp add: satisfies_is_a_fm_def) lemma sats_satisfies_is_a_fm [simp]: "[| u ∈ nat; x < length(env); y < length(env); z ∈ nat; env ∈ list(A)|] ==> sats(A, satisfies_is_a_fm(u,x,y,z), env) <-> 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 ∈ nat; x < length(env); y < length(env); z ∈ nat; env ∈ list(A)|] ==> satisfies_is_a(##A,nu,nx,ny,nz) <-> sats(A, satisfies_is_a_fm(u,x,y,z), env)" by simp theorem satisfies_is_a_reflection: "REFLECTS[λx. satisfies_is_a(L,f(x),g(x),h(x),g'(x)), λi 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{*The Operator @{term satisfies_is_b}, Internalized*} (* satisfies_is_b(M,A) == λx y zz. ∀lA[M]. is_list(M,A,lA) --> is_lambda(M, lA, λenv z. is_bool_of_o(M, ∃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 ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat |] ==> satisfies_is_b_fm(A,x,y,z) ∈ formula" by (simp add: satisfies_is_b_fm_def) lemma sats_satisfies_is_b_fm [simp]: "[| u ∈ nat; x < length(env); y < length(env); z ∈ nat; env ∈ list(A)|] ==> sats(A, satisfies_is_b_fm(u,x,y,z), env) <-> 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 ∈ nat; x < length(env); y < length(env); z ∈ nat; env ∈ list(A)|] ==> satisfies_is_b(##A,nu,nx,ny,nz) <-> sats(A, satisfies_is_b_fm(u,x,y,z), env)" by simp theorem satisfies_is_b_reflection: "REFLECTS[λx. satisfies_is_b(L,f(x),g(x),h(x),g'(x)), λi 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{*The Operator @{term satisfies_is_c}, Internalized*} (* satisfies_is_c(M,A,h) == λp q zz. ∀lA[M]. is_list(M,A,lA) --> is_lambda(M, lA, λenv z. ∃hp[M]. ∃hq[M]. (∃rp[M]. is_depth_apply(M,h,p,rp) & fun_apply(M,rp,env,hp)) & (∃rq[M]. is_depth_apply(M,h,q,rq) & fun_apply(M,rq,env,hq)) & (∃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 ∈ nat; h ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat |] ==> satisfies_is_c_fm(A,h,x,y,z) ∈ formula" by (simp add: satisfies_is_c_fm_def) lemma sats_satisfies_is_c_fm [simp]: "[| u ∈ nat; v ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, satisfies_is_c_fm(u,v,x,y,z), env) <-> 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 ∈ nat; v ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> satisfies_is_c(##A,nu,nv,nx,ny,nz) <-> sats(A, satisfies_is_c_fm(u,v,x,y,z), env)" by simp theorem satisfies_is_c_reflection: "REFLECTS[λx. satisfies_is_c(L,f(x),g(x),h(x),g'(x),h'(x)), λi 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{*The Operator @{term satisfies_is_d}, Internalized*} (* satisfies_is_d(M,A,h) == λp zz. ∀lA[M]. is_list(M,A,lA) --> is_lambda(M, lA, λenv z. ∃rp[M]. is_depth_apply(M,h,p,rp) & is_bool_of_o(M, ∀x[M]. ∀xenv[M]. ∀hp[M]. x∈A --> is_Cons(M,x,env,xenv) --> fun_apply(M,rp,xenv,hp) --> 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 ∈ nat; h ∈ nat; x ∈ nat; z ∈ nat |] ==> satisfies_is_d_fm(A,h,x,z) ∈ formula" by (simp add: satisfies_is_d_fm_def) lemma sats_satisfies_is_d_fm [simp]: "[| u ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, satisfies_is_d_fm(u,x,y,z), env) <-> 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 ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> satisfies_is_d(##A,nu,nx,ny,nz) <-> sats(A, satisfies_is_d_fm(u,x,y,z), env)" by simp theorem satisfies_is_d_reflection: "REFLECTS[λx. satisfies_is_d(L,f(x),g(x),h(x),g'(x)), λi 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{*The Operator @{term satisfies_MH}, Internalized*} (* satisfies_MH == λM A u f zz. ∀fml[M]. is_formula(M,fml) --> 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 ∈ nat; u ∈ nat; x ∈ nat; z ∈ nat |] ==> satisfies_MH_fm(A,u,x,z) ∈ formula" by (simp add: satisfies_MH_fm_def) lemma sats_satisfies_MH_fm [simp]: "[| u ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, satisfies_MH_fm(u,x,y,z), env) <-> 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 ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> satisfies_MH(##A,nu,nx,ny,nz) <-> 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[λx. satisfies_MH(L,f(x),g(x),h(x),g'(x)), λi 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{*Lemmas for Instantiating the Locale @{text "M_satisfies"}*} subsubsection{*The @{term "Member"} Case*} lemma Member_Reflects: "REFLECTS[λu. ∃v[L]. v ∈ B ∧ (∃bo[L]. ∃nx[L]. ∃ny[L]. v ∈ lstA ∧ is_nth(L,x,v,nx) ∧ is_nth(L,y,v,ny) ∧ is_bool_of_o(L, nx ∈ ny, bo) ∧ pair(L,v,bo,u)), λi u. ∃v ∈ Lset(i). v ∈ B ∧ (∃bo ∈ Lset(i). ∃nx ∈ Lset(i). ∃ny ∈ Lset(i). v ∈ lstA ∧ is_nth(##Lset(i), x, v, nx) ∧ is_nth(##Lset(i), y, v, ny) ∧ is_bool_of_o(##Lset(i), nx ∈ ny, bo) ∧ pair(##Lset(i), v, bo, u))]" by (intro FOL_reflections function_reflections nth_reflection bool_of_o_reflection) lemma Member_replacement: "[|L(A); x ∈ nat; y ∈ nat|] ==> strong_replacement (L, λenv z. ∃bo[L]. ∃nx[L]. ∃ny[L]. env ∈ 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 Member_Reflects], auto simp add: nat_into_M list_closed) 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{*The @{term "Equal"} Case*} lemma Equal_Reflects: "REFLECTS[λu. ∃v[L]. v ∈ B ∧ (∃bo[L]. ∃nx[L]. ∃ny[L]. v ∈ lstA ∧ is_nth(L, x, v, nx) ∧ is_nth(L, y, v, ny) ∧ is_bool_of_o(L, nx = ny, bo) ∧ pair(L, v, bo, u)), λi u. ∃v ∈ Lset(i). v ∈ B ∧ (∃bo ∈ Lset(i). ∃nx ∈ Lset(i). ∃ny ∈ Lset(i). v ∈ lstA ∧ is_nth(##Lset(i), x, v, nx) ∧ is_nth(##Lset(i), y, v, ny) ∧ is_bool_of_o(##Lset(i), nx = ny, bo) ∧ pair(##Lset(i), v, bo, u))]" by (intro FOL_reflections function_reflections nth_reflection bool_of_o_reflection) lemma Equal_replacement: "[|L(A); x ∈ nat; y ∈ nat|] ==> strong_replacement (L, λenv z. ∃bo[L]. ∃nx[L]. ∃ny[L]. env ∈ 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 simp add: nat_into_M list_closed) 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{*The @{term "Nand"} Case*} lemma Nand_Reflects: "REFLECTS [λx. ∃u[L]. u ∈ B ∧ (∃rpe[L]. ∃rqe[L]. ∃andpq[L]. ∃notpq[L]. fun_apply(L, rp, u, rpe) ∧ fun_apply(L, rq, u, rqe) ∧ is_and(L, rpe, rqe, andpq) ∧ is_not(L, andpq, notpq) ∧ u ∈ list(A) ∧ pair(L, u, notpq, x)), λi x. ∃u ∈ Lset(i). u ∈ B ∧ (∃rpe ∈ Lset(i). ∃rqe ∈ Lset(i). ∃andpq ∈ Lset(i). ∃notpq ∈ Lset(i). fun_apply(##Lset(i), rp, u, rpe) ∧ fun_apply(##Lset(i), rq, u, rqe) ∧ is_and(##Lset(i), rpe, rqe, andpq) ∧ is_not(##Lset(i), andpq, notpq) ∧ u ∈ list(A) ∧ 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, λenv z. ∃rpe[L]. ∃rqe[L]. ∃andpq[L]. ∃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 ∈ 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 simp add: list_closed) 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{*The @{term "Forall"} Case*} lemma Forall_Reflects: "REFLECTS [λx. ∃u[L]. u ∈ B ∧ (∃bo[L]. u ∈ list(A) ∧ is_bool_of_o (L, ∀a[L]. ∀co[L]. ∀rpco[L]. a ∈ A --> is_Cons(L,a,u,co) --> fun_apply(L,rp,co,rpco) --> number1(L,rpco), bo) ∧ pair(L,u,bo,x)), λi x. ∃u ∈ Lset(i). u ∈ B ∧ (∃bo ∈ Lset(i). u ∈ list(A) ∧ is_bool_of_o (##Lset(i), ∀a ∈ Lset(i). ∀co ∈ Lset(i). ∀rpco ∈ Lset(i). a ∈ A --> is_Cons(##Lset(i),a,u,co) --> fun_apply(##Lset(i),rp,co,rpco) --> number1(##Lset(i),rpco), bo) ∧ 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, λenv z. ∃bo[L]. env ∈ list(A) & is_bool_of_o (L, ∀a[L]. ∀co[L]. ∀rpco[L]. a∈A --> is_Cons(L,a,env,co) --> fun_apply(L,rp,co,rpco) --> 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 simp add: list_closed) 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{*The @{term "transrec_replacement"} Case*} lemma formula_rec_replacement_Reflects: "REFLECTS [λx. ∃u[L]. u ∈ B ∧ (∃y[L]. pair(L, u, y, x) ∧ is_wfrec (L, satisfies_MH(L,A), mesa, u, y)), λi x. ∃u ∈ Lset(i). u ∈ B ∧ (∃y ∈ 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: --{*For the @{term transrec}*} "[|n ∈ nat; L(A)|] ==> transrec_replacement(L, satisfies_MH(L,A), n)" apply (rule transrec_replacementI, simp add: 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: 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{*The Lambda Replacement Case*} lemma formula_rec_lambda_replacement_Reflects: "REFLECTS [λx. ∃u[L]. u ∈ B & mem_formula(L,u) & (∃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)), λi x. ∃u ∈ Lset(i). u ∈ B & mem_formula(##Lset(i),u) & (∃c ∈ 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: --{*For the @{term transrec}*} "[|L(g); L(A)|] ==> strong_replacement (L, λx y. mem_formula(L,x) & (∃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{*Instantiating @{text M_satisfies}*} 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: "PROP M_satisfies(L)" apply (rule M_satisfies.intro) apply (rule M_eclose_L) apply (rule M_satisfies_axioms_L) done text{*Finally: the point of the whole theory!*} lemmas satisfies_closed = M_satisfies.satisfies_closed [OF M_satisfies_L] and satisfies_abs = M_satisfies.satisfies_abs [OF M_satisfies_L] end