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theory Internalize(* Title: ZF/Constructible/Internalize.thy ID: $Id: Internalize.thy,v 1.10 2005/06/17 14:15:10 haftmann Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory *) theory Internalize imports L_axioms Datatype_absolute begin subsection{*Internalized Forms of Data Structuring Operators*} subsubsection{*The Formula @{term is_Inl}, Internalized*} (* is_Inl(M,a,z) == ∃zero[M]. empty(M,zero) & pair(M,zero,a,z) *) constdefs Inl_fm :: "[i,i]=>i" "Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))" lemma Inl_type [TC]: "[| x ∈ nat; z ∈ nat |] ==> Inl_fm(x,z) ∈ formula" by (simp add: Inl_fm_def) lemma sats_Inl_fm [simp]: "[| x ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, Inl_fm(x,z), env) <-> is_Inl(##A, nth(x,env), nth(z,env))" by (simp add: Inl_fm_def is_Inl_def) lemma Inl_iff_sats: "[| nth(i,env) = x; nth(k,env) = z; i ∈ nat; k ∈ nat; env ∈ list(A)|] ==> is_Inl(##A, x, z) <-> sats(A, Inl_fm(i,k), env)" by simp theorem Inl_reflection: "REFLECTS[λx. is_Inl(L,f(x),h(x)), λi x. is_Inl(##Lset(i),f(x),h(x))]" apply (simp only: is_Inl_def) apply (intro FOL_reflections function_reflections) done subsubsection{*The Formula @{term is_Inr}, Internalized*} (* is_Inr(M,a,z) == ∃n1[M]. number1(M,n1) & pair(M,n1,a,z) *) constdefs Inr_fm :: "[i,i]=>i" "Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))" lemma Inr_type [TC]: "[| x ∈ nat; z ∈ nat |] ==> Inr_fm(x,z) ∈ formula" by (simp add: Inr_fm_def) lemma sats_Inr_fm [simp]: "[| x ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, Inr_fm(x,z), env) <-> is_Inr(##A, nth(x,env), nth(z,env))" by (simp add: Inr_fm_def is_Inr_def) lemma Inr_iff_sats: "[| nth(i,env) = x; nth(k,env) = z; i ∈ nat; k ∈ nat; env ∈ list(A)|] ==> is_Inr(##A, x, z) <-> sats(A, Inr_fm(i,k), env)" by simp theorem Inr_reflection: "REFLECTS[λx. is_Inr(L,f(x),h(x)), λi x. is_Inr(##Lset(i),f(x),h(x))]" apply (simp only: is_Inr_def) apply (intro FOL_reflections function_reflections) done subsubsection{*The Formula @{term is_Nil}, Internalized*} (* is_Nil(M,xs) == ∃zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *) constdefs Nil_fm :: "i=>i" "Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))" lemma Nil_type [TC]: "x ∈ nat ==> Nil_fm(x) ∈ formula" by (simp add: Nil_fm_def) lemma sats_Nil_fm [simp]: "[| x ∈ nat; env ∈ list(A)|] ==> sats(A, Nil_fm(x), env) <-> is_Nil(##A, nth(x,env))" by (simp add: Nil_fm_def is_Nil_def) lemma Nil_iff_sats: "[| nth(i,env) = x; i ∈ nat; env ∈ list(A)|] ==> is_Nil(##A, x) <-> sats(A, Nil_fm(i), env)" by simp theorem Nil_reflection: "REFLECTS[λx. is_Nil(L,f(x)), λi x. is_Nil(##Lset(i),f(x))]" apply (simp only: is_Nil_def) apply (intro FOL_reflections function_reflections Inl_reflection) done subsubsection{*The Formula @{term is_Cons}, Internalized*} (* "is_Cons(M,a,l,Z) == ∃p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *) constdefs Cons_fm :: "[i,i,i]=>i" "Cons_fm(a,l,Z) == Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))" lemma Cons_type [TC]: "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Cons_fm(x,y,z) ∈ formula" by (simp add: Cons_fm_def) lemma sats_Cons_fm [simp]: "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, Cons_fm(x,y,z), env) <-> is_Cons(##A, nth(x,env), nth(y,env), nth(z,env))" by (simp add: Cons_fm_def is_Cons_def) lemma Cons_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|] ==>is_Cons(##A, x, y, z) <-> sats(A, Cons_fm(i,j,k), env)" by simp theorem Cons_reflection: "REFLECTS[λx. is_Cons(L,f(x),g(x),h(x)), λi x. is_Cons(##Lset(i),f(x),g(x),h(x))]" apply (simp only: is_Cons_def) apply (intro FOL_reflections pair_reflection Inr_reflection) done subsubsection{*The Formula @{term is_quasilist}, Internalized*} (* is_quasilist(M,xs) == is_Nil(M,z) | (∃x[M]. ∃l[M]. is_Cons(M,x,l,z))" *) constdefs quasilist_fm :: "i=>i" "quasilist_fm(x) == Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))" lemma quasilist_type [TC]: "x ∈ nat ==> quasilist_fm(x) ∈ formula" by (simp add: quasilist_fm_def) lemma sats_quasilist_fm [simp]: "[| x ∈ nat; env ∈ list(A)|] ==> sats(A, quasilist_fm(x), env) <-> is_quasilist(##A, nth(x,env))" by (simp add: quasilist_fm_def is_quasilist_def) lemma quasilist_iff_sats: "[| nth(i,env) = x; i ∈ nat; env ∈ list(A)|] ==> is_quasilist(##A, x) <-> sats(A, quasilist_fm(i), env)" by simp theorem quasilist_reflection: "REFLECTS[λx. is_quasilist(L,f(x)), λi x. is_quasilist(##Lset(i),f(x))]" apply (simp only: is_quasilist_def) apply (intro FOL_reflections Nil_reflection Cons_reflection) done subsection{*Absoluteness for the Function @{term nth}*} subsubsection{*The Formula @{term is_hd}, Internalized*} (* "is_hd(M,xs,H) == (is_Nil(M,xs) --> empty(M,H)) & (∀x[M]. ∀l[M]. ~ is_Cons(M,x,l,xs) | H=x) & (is_quasilist(M,xs) | empty(M,H))" *) constdefs hd_fm :: "[i,i]=>i" "hd_fm(xs,H) == And(Implies(Nil_fm(xs), empty_fm(H)), And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(H#+2,1)))), Or(quasilist_fm(xs), empty_fm(H))))" lemma hd_type [TC]: "[| x ∈ nat; y ∈ nat |] ==> hd_fm(x,y) ∈ formula" by (simp add: hd_fm_def) lemma sats_hd_fm [simp]: "[| x ∈ nat; y ∈ nat; env ∈ list(A)|] ==> sats(A, hd_fm(x,y), env) <-> is_hd(##A, nth(x,env), nth(y,env))" by (simp add: hd_fm_def is_hd_def) lemma hd_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; i ∈ nat; j ∈ nat; env ∈ list(A)|] ==> is_hd(##A, x, y) <-> sats(A, hd_fm(i,j), env)" by simp theorem hd_reflection: "REFLECTS[λx. is_hd(L,f(x),g(x)), λi x. is_hd(##Lset(i),f(x),g(x))]" apply (simp only: is_hd_def) apply (intro FOL_reflections Nil_reflection Cons_reflection quasilist_reflection empty_reflection) done subsubsection{*The Formula @{term is_tl}, Internalized*} (* "is_tl(M,xs,T) == (is_Nil(M,xs) --> T=xs) & (∀x[M]. ∀l[M]. ~ is_Cons(M,x,l,xs) | T=l) & (is_quasilist(M,xs) | empty(M,T))" *) constdefs tl_fm :: "[i,i]=>i" "tl_fm(xs,T) == And(Implies(Nil_fm(xs), Equal(T,xs)), And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))), Or(quasilist_fm(xs), empty_fm(T))))" lemma tl_type [TC]: "[| x ∈ nat; y ∈ nat |] ==> tl_fm(x,y) ∈ formula" by (simp add: tl_fm_def) lemma sats_tl_fm [simp]: "[| x ∈ nat; y ∈ nat; env ∈ list(A)|] ==> sats(A, tl_fm(x,y), env) <-> is_tl(##A, nth(x,env), nth(y,env))" by (simp add: tl_fm_def is_tl_def) lemma tl_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; i ∈ nat; j ∈ nat; env ∈ list(A)|] ==> is_tl(##A, x, y) <-> sats(A, tl_fm(i,j), env)" by simp theorem tl_reflection: "REFLECTS[λx. is_tl(L,f(x),g(x)), λi x. is_tl(##Lset(i),f(x),g(x))]" apply (simp only: is_tl_def) apply (intro FOL_reflections Nil_reflection Cons_reflection quasilist_reflection empty_reflection) done subsubsection{*The Operator @{term is_bool_of_o}*} (* is_bool_of_o :: "[i=>o, o, i] => o" "is_bool_of_o(M,P,z) == (P & number1(M,z)) | (~P & empty(M,z))" *) text{*The formula @{term p} has no free variables.*} constdefs bool_of_o_fm :: "[i, i]=>i" "bool_of_o_fm(p,z) == Or(And(p,number1_fm(z)), And(Neg(p),empty_fm(z)))" lemma is_bool_of_o_type [TC]: "[| p ∈ formula; z ∈ nat |] ==> bool_of_o_fm(p,z) ∈ formula" by (simp add: bool_of_o_fm_def) lemma sats_bool_of_o_fm: assumes p_iff_sats: "P <-> sats(A, p, env)" shows "[|z ∈ nat; env ∈ list(A)|] ==> sats(A, bool_of_o_fm(p,z), env) <-> is_bool_of_o(##A, P, nth(z,env))" by (simp add: bool_of_o_fm_def is_bool_of_o_def p_iff_sats [THEN iff_sym]) lemma is_bool_of_o_iff_sats: "[| P <-> sats(A, p, env); nth(k,env) = z; k ∈ nat; env ∈ list(A)|] ==> is_bool_of_o(##A, P, z) <-> sats(A, bool_of_o_fm(p,k), env)" by (simp add: sats_bool_of_o_fm) theorem bool_of_o_reflection: "REFLECTS [P(L), λi. P(##Lset(i))] ==> REFLECTS[λx. is_bool_of_o(L, P(L,x), f(x)), λi x. is_bool_of_o(##Lset(i), P(##Lset(i),x), f(x))]" apply (simp (no_asm) only: is_bool_of_o_def) apply (intro FOL_reflections function_reflections, assumption+) done subsection{*More Internalizations*} subsubsection{*The Operator @{term is_lambda}*} text{*The two arguments of @{term p} are always 1, 0. Remember that @{term p} will be enclosed by three quantifiers.*} (* is_lambda :: "[i=>o, i, [i,i]=>o, i] => o" "is_lambda(M, A, is_b, z) == ∀p[M]. p ∈ z <-> (∃u[M]. ∃v[M]. u∈A & pair(M,u,v,p) & is_b(u,v))" *) constdefs lambda_fm :: "[i, i, i]=>i" "lambda_fm(p,A,z) == Forall(Iff(Member(0,succ(z)), Exists(Exists(And(Member(1,A#+3), And(pair_fm(1,0,2), p))))))" text{*We call @{term p} with arguments x, y by equating them with the corresponding quantified variables with de Bruijn indices 1, 0.*} lemma is_lambda_type [TC]: "[| p ∈ formula; x ∈ nat; y ∈ nat |] ==> lambda_fm(p,x,y) ∈ formula" by (simp add: lambda_fm_def) lemma sats_lambda_fm: assumes is_b_iff_sats: "!!a0 a1 a2. [|a0∈A; a1∈A; a2∈A|] ==> is_b(a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,env))))" shows "[|x ∈ nat; y ∈ nat; env ∈ list(A)|] ==> sats(A, lambda_fm(p,x,y), env) <-> is_lambda(##A, nth(x,env), is_b, nth(y,env))" by (simp add: lambda_fm_def is_lambda_def is_b_iff_sats [THEN iff_sym]) theorem is_lambda_reflection: assumes is_b_reflection: "!!f g h. REFLECTS[λx. is_b(L, f(x), g(x), h(x)), λi x. is_b(##Lset(i), f(x), g(x), h(x))]" shows "REFLECTS[λx. is_lambda(L, A(x), is_b(L,x), f(x)), λi x. is_lambda(##Lset(i), A(x), is_b(##Lset(i),x), f(x))]" apply (simp (no_asm_use) only: is_lambda_def) apply (intro FOL_reflections is_b_reflection pair_reflection) done subsubsection{*The Operator @{term is_Member}, Internalized*} (* "is_Member(M,x,y,Z) == ∃p[M]. ∃u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inl(M,u,Z)" *) constdefs Member_fm :: "[i,i,i]=>i" "Member_fm(x,y,Z) == Exists(Exists(And(pair_fm(x#+2,y#+2,1), And(Inl_fm(1,0), Inl_fm(0,Z#+2)))))" lemma is_Member_type [TC]: "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Member_fm(x,y,z) ∈ formula" by (simp add: Member_fm_def) lemma sats_Member_fm [simp]: "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, Member_fm(x,y,z), env) <-> is_Member(##A, nth(x,env), nth(y,env), nth(z,env))" by (simp add: Member_fm_def is_Member_def) lemma Member_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|] ==> is_Member(##A, x, y, z) <-> sats(A, Member_fm(i,j,k), env)" by (simp add: sats_Member_fm) theorem Member_reflection: "REFLECTS[λx. is_Member(L,f(x),g(x),h(x)), λi x. is_Member(##Lset(i),f(x),g(x),h(x))]" apply (simp only: is_Member_def) apply (intro FOL_reflections pair_reflection Inl_reflection) done subsubsection{*The Operator @{term is_Equal}, Internalized*} (* "is_Equal(M,x,y,Z) == ∃p[M]. ∃u[M]. pair(M,x,y,p) & is_Inr(M,p,u) & is_Inl(M,u,Z)" *) constdefs Equal_fm :: "[i,i,i]=>i" "Equal_fm(x,y,Z) == Exists(Exists(And(pair_fm(x#+2,y#+2,1), And(Inr_fm(1,0), Inl_fm(0,Z#+2)))))" lemma is_Equal_type [TC]: "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Equal_fm(x,y,z) ∈ formula" by (simp add: Equal_fm_def) lemma sats_Equal_fm [simp]: "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, Equal_fm(x,y,z), env) <-> is_Equal(##A, nth(x,env), nth(y,env), nth(z,env))" by (simp add: Equal_fm_def is_Equal_def) lemma Equal_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|] ==> is_Equal(##A, x, y, z) <-> sats(A, Equal_fm(i,j,k), env)" by (simp add: sats_Equal_fm) theorem Equal_reflection: "REFLECTS[λx. is_Equal(L,f(x),g(x),h(x)), λi x. is_Equal(##Lset(i),f(x),g(x),h(x))]" apply (simp only: is_Equal_def) apply (intro FOL_reflections pair_reflection Inl_reflection Inr_reflection) done subsubsection{*The Operator @{term is_Nand}, Internalized*} (* "is_Nand(M,x,y,Z) == ∃p[M]. ∃u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inr(M,u,Z)" *) constdefs Nand_fm :: "[i,i,i]=>i" "Nand_fm(x,y,Z) == Exists(Exists(And(pair_fm(x#+2,y#+2,1), And(Inl_fm(1,0), Inr_fm(0,Z#+2)))))" lemma is_Nand_type [TC]: "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Nand_fm(x,y,z) ∈ formula" by (simp add: Nand_fm_def) lemma sats_Nand_fm [simp]: "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, Nand_fm(x,y,z), env) <-> is_Nand(##A, nth(x,env), nth(y,env), nth(z,env))" by (simp add: Nand_fm_def is_Nand_def) lemma Nand_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|] ==> is_Nand(##A, x, y, z) <-> sats(A, Nand_fm(i,j,k), env)" by (simp add: sats_Nand_fm) theorem Nand_reflection: "REFLECTS[λx. is_Nand(L,f(x),g(x),h(x)), λi x. is_Nand(##Lset(i),f(x),g(x),h(x))]" apply (simp only: is_Nand_def) apply (intro FOL_reflections pair_reflection Inl_reflection Inr_reflection) done subsubsection{*The Operator @{term is_Forall}, Internalized*} (* "is_Forall(M,p,Z) == ∃u[M]. is_Inr(M,p,u) & is_Inr(M,u,Z)" *) constdefs Forall_fm :: "[i,i]=>i" "Forall_fm(x,Z) == Exists(And(Inr_fm(succ(x),0), Inr_fm(0,succ(Z))))" lemma is_Forall_type [TC]: "[| x ∈ nat; y ∈ nat |] ==> Forall_fm(x,y) ∈ formula" by (simp add: Forall_fm_def) lemma sats_Forall_fm [simp]: "[| x ∈ nat; y ∈ nat; env ∈ list(A)|] ==> sats(A, Forall_fm(x,y), env) <-> is_Forall(##A, nth(x,env), nth(y,env))" by (simp add: Forall_fm_def is_Forall_def) lemma Forall_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; i ∈ nat; j ∈ nat; env ∈ list(A)|] ==> is_Forall(##A, x, y) <-> sats(A, Forall_fm(i,j), env)" by (simp add: sats_Forall_fm) theorem Forall_reflection: "REFLECTS[λx. is_Forall(L,f(x),g(x)), λi x. is_Forall(##Lset(i),f(x),g(x))]" apply (simp only: is_Forall_def) apply (intro FOL_reflections pair_reflection Inr_reflection) done subsubsection{*The Operator @{term is_and}, Internalized*} (* is_and(M,a,b,z) == (number1(M,a) & z=b) | (~number1(M,a) & empty(M,z)) *) constdefs and_fm :: "[i,i,i]=>i" "and_fm(a,b,z) == Or(And(number1_fm(a), Equal(z,b)), And(Neg(number1_fm(a)),empty_fm(z)))" lemma is_and_type [TC]: "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> and_fm(x,y,z) ∈ formula" by (simp add: and_fm_def) lemma sats_and_fm [simp]: "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, and_fm(x,y,z), env) <-> is_and(##A, nth(x,env), nth(y,env), nth(z,env))" by (simp add: and_fm_def is_and_def) lemma is_and_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|] ==> is_and(##A, x, y, z) <-> sats(A, and_fm(i,j,k), env)" by simp theorem is_and_reflection: "REFLECTS[λx. is_and(L,f(x),g(x),h(x)), λi x. is_and(##Lset(i),f(x),g(x),h(x))]" apply (simp only: is_and_def) apply (intro FOL_reflections function_reflections) done subsubsection{*The Operator @{term is_or}, Internalized*} (* is_or(M,a,b,z) == (number1(M,a) & number1(M,z)) | (~number1(M,a) & z=b) *) constdefs or_fm :: "[i,i,i]=>i" "or_fm(a,b,z) == Or(And(number1_fm(a), number1_fm(z)), And(Neg(number1_fm(a)), Equal(z,b)))" lemma is_or_type [TC]: "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> or_fm(x,y,z) ∈ formula" by (simp add: or_fm_def) lemma sats_or_fm [simp]: "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, or_fm(x,y,z), env) <-> is_or(##A, nth(x,env), nth(y,env), nth(z,env))" by (simp add: or_fm_def is_or_def) lemma is_or_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|] ==> is_or(##A, x, y, z) <-> sats(A, or_fm(i,j,k), env)" by simp theorem is_or_reflection: "REFLECTS[λx. is_or(L,f(x),g(x),h(x)), λi x. is_or(##Lset(i),f(x),g(x),h(x))]" apply (simp only: is_or_def) apply (intro FOL_reflections function_reflections) done subsubsection{*The Operator @{term is_not}, Internalized*} (* is_not(M,a,z) == (number1(M,a) & empty(M,z)) | (~number1(M,a) & number1(M,z)) *) constdefs not_fm :: "[i,i]=>i" "not_fm(a,z) == Or(And(number1_fm(a), empty_fm(z)), And(Neg(number1_fm(a)), number1_fm(z)))" lemma is_not_type [TC]: "[| x ∈ nat; z ∈ nat |] ==> not_fm(x,z) ∈ formula" by (simp add: not_fm_def) lemma sats_is_not_fm [simp]: "[| x ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, not_fm(x,z), env) <-> is_not(##A, nth(x,env), nth(z,env))" by (simp add: not_fm_def is_not_def) lemma is_not_iff_sats: "[| nth(i,env) = x; nth(k,env) = z; i ∈ nat; k ∈ nat; env ∈ list(A)|] ==> is_not(##A, x, z) <-> sats(A, not_fm(i,k), env)" by simp theorem is_not_reflection: "REFLECTS[λx. is_not(L,f(x),g(x)), λi x. is_not(##Lset(i),f(x),g(x))]" apply (simp only: is_not_def) apply (intro FOL_reflections function_reflections) done lemmas extra_reflections = Inl_reflection Inr_reflection Nil_reflection Cons_reflection quasilist_reflection hd_reflection tl_reflection bool_of_o_reflection is_lambda_reflection Member_reflection Equal_reflection Nand_reflection Forall_reflection is_and_reflection is_or_reflection is_not_reflection subsection{*Well-Founded Recursion!*} subsubsection{*The Operator @{term M_is_recfun}*} text{*Alternative definition, minimizing nesting of quantifiers around MH*} lemma M_is_recfun_iff: "M_is_recfun(M,MH,r,a,f) <-> (∀z[M]. z ∈ f <-> (∃x[M]. ∃f_r_sx[M]. ∃y[M]. MH(x, f_r_sx, y) & pair(M,x,y,z) & (∃xa[M]. ∃sx[M]. ∃r_sx[M]. pair(M,x,a,xa) & upair(M,x,x,sx) & pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) & xa ∈ r)))" apply (simp add: M_is_recfun_def) apply (rule rall_cong, blast) done (* M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o" "M_is_recfun(M,MH,r,a,f) == ∀z[M]. z ∈ f <-> 2 1 0 new def (∃x[M]. ∃f_r_sx[M]. ∃y[M]. MH(x, f_r_sx, y) & pair(M,x,y,z) & (∃xa[M]. ∃sx[M]. ∃r_sx[M]. pair(M,x,a,xa) & upair(M,x,x,sx) & pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) & xa ∈ r)" *) text{*The three arguments of @{term p} are always 2, 1, 0 and z*} constdefs is_recfun_fm :: "[i, i, i, i]=>i" "is_recfun_fm(p,r,a,f) == Forall(Iff(Member(0,succ(f)), Exists(Exists(Exists( And(p, And(pair_fm(2,0,3), Exists(Exists(Exists( And(pair_fm(5,a#+7,2), And(upair_fm(5,5,1), And(pre_image_fm(r#+7,1,0), And(restriction_fm(f#+7,0,4), Member(2,r#+7)))))))))))))))" lemma is_recfun_type [TC]: "[| p ∈ formula; x ∈ nat; y ∈ nat; z ∈ nat |] ==> is_recfun_fm(p,x,y,z) ∈ formula" by (simp add: is_recfun_fm_def) lemma sats_is_recfun_fm: assumes MH_iff_sats: "!!a0 a1 a2 a3. [|a0∈A; a1∈A; a2∈A; a3∈A|] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))" shows "[|x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, is_recfun_fm(p,x,y,z), env) <-> M_is_recfun(##A, MH, nth(x,env), nth(y,env), nth(z,env))" by (simp add: is_recfun_fm_def M_is_recfun_iff MH_iff_sats [THEN iff_sym]) lemma is_recfun_iff_sats: assumes MH_iff_sats: "!!a0 a1 a2 a3. [|a0∈A; a1∈A; a2∈A; a3∈A|] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))" shows "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|] ==> M_is_recfun(##A, MH, x, y, z) <-> sats(A, is_recfun_fm(p,i,j,k), env)" by (simp add: sats_is_recfun_fm [OF MH_iff_sats]) text{*The additional variable in the premise, namely @{term f'}, is essential. It lets @{term MH} depend upon @{term x}, which seems often necessary. The same thing occurs in @{text is_wfrec_reflection}.*} theorem is_recfun_reflection: assumes MH_reflection: "!!f' f g h. REFLECTS[λx. MH(L, f'(x), f(x), g(x), h(x)), λi x. MH(##Lset(i), f'(x), f(x), g(x), h(x))]" shows "REFLECTS[λx. M_is_recfun(L, MH(L,x), f(x), g(x), h(x)), λi x. M_is_recfun(##Lset(i), MH(##Lset(i),x), f(x), g(x), h(x))]" apply (simp (no_asm_use) only: M_is_recfun_def) apply (intro FOL_reflections function_reflections restriction_reflection MH_reflection) done subsubsection{*The Operator @{term is_wfrec}*} text{*The three arguments of @{term p} are always 2, 1, 0; @{term p} is enclosed by 5 quantifiers.*} (* is_wfrec :: "[i=>o, i, [i,i,i]=>o, i, i] => o" "is_wfrec(M,MH,r,a,z) == ∃f[M]. M_is_recfun(M,MH,r,a,f) & MH(a,f,z)" *) constdefs is_wfrec_fm :: "[i, i, i, i]=>i" "is_wfrec_fm(p,r,a,z) == Exists(And(is_recfun_fm(p, succ(r), succ(a), 0), Exists(Exists(Exists(Exists( And(Equal(2,a#+5), And(Equal(1,4), And(Equal(0,z#+5), p)))))))))" text{*We call @{term p} with arguments a, f, z by equating them with the corresponding quantified variables with de Bruijn indices 2, 1, 0.*} text{*There's an additional existential quantifier to ensure that the environments in both calls to MH have the same length.*} lemma is_wfrec_type [TC]: "[| p ∈ formula; x ∈ nat; y ∈ nat; z ∈ nat |] ==> is_wfrec_fm(p,x,y,z) ∈ formula" by (simp add: is_wfrec_fm_def) lemma sats_is_wfrec_fm: assumes MH_iff_sats: "!!a0 a1 a2 a3 a4. [|a0∈A; a1∈A; a2∈A; a3∈A; a4∈A|] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))" shows "[|x ∈ nat; y < length(env); z < length(env); env ∈ list(A)|] ==> sats(A, is_wfrec_fm(p,x,y,z), env) <-> is_wfrec(##A, MH, nth(x,env), nth(y,env), nth(z,env))" apply (frule_tac x=z in lt_length_in_nat, assumption) apply (frule lt_length_in_nat, assumption) apply (simp add: is_wfrec_fm_def sats_is_recfun_fm is_wfrec_def MH_iff_sats [THEN iff_sym], blast) done lemma is_wfrec_iff_sats: assumes MH_iff_sats: "!!a0 a1 a2 a3 a4. [|a0∈A; a1∈A; a2∈A; a3∈A; a4∈A|] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))" shows "[|nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j < length(env); k < length(env); env ∈ list(A)|] ==> is_wfrec(##A, MH, x, y, z) <-> sats(A, is_wfrec_fm(p,i,j,k), env)" by (simp add: sats_is_wfrec_fm [OF MH_iff_sats]) theorem is_wfrec_reflection: assumes MH_reflection: "!!f' f g h. REFLECTS[λx. MH(L, f'(x), f(x), g(x), h(x)), λi x. MH(##Lset(i), f'(x), f(x), g(x), h(x))]" shows "REFLECTS[λx. is_wfrec(L, MH(L,x), f(x), g(x), h(x)), λi x. is_wfrec(##Lset(i), MH(##Lset(i),x), f(x), g(x), h(x))]" apply (simp (no_asm_use) only: is_wfrec_def) apply (intro FOL_reflections MH_reflection is_recfun_reflection) done subsection{*For Datatypes*} subsubsection{*Binary Products, Internalized*} constdefs cartprod_fm :: "[i,i,i]=>i" (* "cartprod(M,A,B,z) == ∀u[M]. u ∈ z <-> (∃x[M]. x∈A & (∃y[M]. y∈B & pair(M,x,y,u)))" *) "cartprod_fm(A,B,z) == Forall(Iff(Member(0,succ(z)), Exists(And(Member(0,succ(succ(A))), Exists(And(Member(0,succ(succ(succ(B)))), pair_fm(1,0,2)))))))" lemma cartprod_type [TC]: "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> cartprod_fm(x,y,z) ∈ formula" by (simp add: cartprod_fm_def) lemma sats_cartprod_fm [simp]: "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, cartprod_fm(x,y,z), env) <-> cartprod(##A, nth(x,env), nth(y,env), nth(z,env))" by (simp add: cartprod_fm_def cartprod_def) lemma cartprod_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|] ==> cartprod(##A, x, y, z) <-> sats(A, cartprod_fm(i,j,k), env)" by (simp add: sats_cartprod_fm) theorem cartprod_reflection: "REFLECTS[λx. cartprod(L,f(x),g(x),h(x)), λi x. cartprod(##Lset(i),f(x),g(x),h(x))]" apply (simp only: cartprod_def) apply (intro FOL_reflections pair_reflection) done subsubsection{*Binary Sums, Internalized*} (* "is_sum(M,A,B,Z) == ∃A0[M]. ∃n1[M]. ∃s1[M]. ∃B1[M]. 3 2 1 0 number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) & cartprod(M,s1,B,B1) & union(M,A0,B1,Z)" *) constdefs sum_fm :: "[i,i,i]=>i" "sum_fm(A,B,Z) == Exists(Exists(Exists(Exists( And(number1_fm(2), And(cartprod_fm(2,A#+4,3), And(upair_fm(2,2,1), And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))" lemma sum_type [TC]: "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> sum_fm(x,y,z) ∈ formula" by (simp add: sum_fm_def) lemma sats_sum_fm [simp]: "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, sum_fm(x,y,z), env) <-> is_sum(##A, nth(x,env), nth(y,env), nth(z,env))" by (simp add: sum_fm_def is_sum_def) lemma sum_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|] ==> is_sum(##A, x, y, z) <-> sats(A, sum_fm(i,j,k), env)" by simp theorem sum_reflection: "REFLECTS[λx. is_sum(L,f(x),g(x),h(x)), λi x. is_sum(##Lset(i),f(x),g(x),h(x))]" apply (simp only: is_sum_def) apply (intro FOL_reflections function_reflections cartprod_reflection) done subsubsection{*The Operator @{term quasinat}*} (* "is_quasinat(M,z) == empty(M,z) | (∃m[M]. successor(M,m,z))" *) constdefs quasinat_fm :: "i=>i" "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))" lemma quasinat_type [TC]: "x ∈ nat ==> quasinat_fm(x) ∈ formula" by (simp add: quasinat_fm_def) lemma sats_quasinat_fm [simp]: "[| x ∈ nat; env ∈ list(A)|] ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(##A, nth(x,env))" by (simp add: quasinat_fm_def is_quasinat_def) lemma quasinat_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; i ∈ nat; env ∈ list(A)|] ==> is_quasinat(##A, x) <-> sats(A, quasinat_fm(i), env)" by simp theorem quasinat_reflection: "REFLECTS[λx. is_quasinat(L,f(x)), λi x. is_quasinat(##Lset(i),f(x))]" apply (simp only: is_quasinat_def) apply (intro FOL_reflections function_reflections) done subsubsection{*The Operator @{term is_nat_case}*} text{*I could not get it to work with the more natural assumption that @{term is_b} takes two arguments. Instead it must be a formula where 1 and 0 stand for @{term m} and @{term b}, respectively.*} (* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o" "is_nat_case(M, a, is_b, k, z) == (empty(M,k) --> z=a) & (∀m[M]. successor(M,m,k) --> is_b(m,z)) & (is_quasinat(M,k) | empty(M,z))" *) text{*The formula @{term is_b} has free variables 1 and 0.*} constdefs is_nat_case_fm :: "[i, i, i, i]=>i" "is_nat_case_fm(a,is_b,k,z) == And(Implies(empty_fm(k), Equal(z,a)), And(Forall(Implies(succ_fm(0,succ(k)), Forall(Implies(Equal(0,succ(succ(z))), is_b)))), Or(quasinat_fm(k), empty_fm(z))))" lemma is_nat_case_type [TC]: "[| is_b ∈ formula; x ∈ nat; y ∈ nat; z ∈ nat |] ==> is_nat_case_fm(x,is_b,y,z) ∈ formula" by (simp add: is_nat_case_fm_def) lemma sats_is_nat_case_fm: assumes is_b_iff_sats: "!!a. a ∈ A ==> is_b(a,nth(z, env)) <-> sats(A, p, Cons(nth(z,env), Cons(a, env)))" shows "[|x ∈ nat; y ∈ nat; z < length(env); env ∈ list(A)|] ==> sats(A, is_nat_case_fm(x,p,y,z), env) <-> is_nat_case(##A, nth(x,env), is_b, nth(y,env), nth(z,env))" apply (frule lt_length_in_nat, assumption) apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym]) done lemma is_nat_case_iff_sats: "[| (!!a. a ∈ A ==> is_b(a,z) <-> sats(A, p, Cons(z, Cons(a,env)))); nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k < length(env); env ∈ list(A)|] ==> is_nat_case(##A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i,p,j,k), env)" by (simp add: sats_is_nat_case_fm [of A is_b]) text{*The second argument of @{term is_b} gives it direct access to @{term x}, which is essential for handling free variable references. Without this argument, we cannot prove reflection for @{term iterates_MH}.*} theorem is_nat_case_reflection: assumes is_b_reflection: "!!h f g. REFLECTS[λx. is_b(L, h(x), f(x), g(x)), λi x. is_b(##Lset(i), h(x), f(x), g(x))]" shows "REFLECTS[λx. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)), λi x. is_nat_case(##Lset(i), f(x), is_b(##Lset(i), x), g(x), h(x))]" apply (simp (no_asm_use) only: is_nat_case_def) apply (intro FOL_reflections function_reflections restriction_reflection is_b_reflection quasinat_reflection) done subsection{*The Operator @{term iterates_MH}, Needed for Iteration*} (* iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o" "iterates_MH(M,isF,v,n,g,z) == is_nat_case(M, v, λm u. ∃gm[M]. fun_apply(M,g,m,gm) & isF(gm,u), n, z)" *) constdefs iterates_MH_fm :: "[i, i, i, i, i]=>i" "iterates_MH_fm(isF,v,n,g,z) == is_nat_case_fm(v, Exists(And(fun_apply_fm(succ(succ(succ(g))),2,0), Forall(Implies(Equal(0,2), isF)))), n, z)" lemma iterates_MH_type [TC]: "[| p ∈ formula; v ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat |] ==> iterates_MH_fm(p,v,x,y,z) ∈ formula" by (simp add: iterates_MH_fm_def) lemma sats_iterates_MH_fm: assumes is_F_iff_sats: "!!a b c d. [| a ∈ A; b ∈ A; c ∈ A; d ∈ A|] ==> is_F(a,b) <-> sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env)))))" shows "[|v ∈ nat; x ∈ nat; y ∈ nat; z < length(env); env ∈ list(A)|] ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) <-> iterates_MH(##A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))" apply (frule lt_length_in_nat, assumption) apply (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm is_F_iff_sats [symmetric]) apply (rule is_nat_case_cong) apply (simp_all add: setclass_def) done lemma iterates_MH_iff_sats: assumes is_F_iff_sats: "!!a b c d. [| a ∈ A; b ∈ A; c ∈ A; d ∈ A|] ==> is_F(a,b) <-> sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env)))))" shows "[| nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i' ∈ nat; i ∈ nat; j ∈ nat; k < length(env); env ∈ list(A)|] ==> iterates_MH(##A, is_F, v, x, y, z) <-> sats(A, iterates_MH_fm(p,i',i,j,k), env)" by (simp add: sats_iterates_MH_fm [OF is_F_iff_sats]) text{*The second argument of @{term p} gives it direct access to @{term x}, which is essential for handling free variable references. Without this argument, we cannot prove reflection for @{term list_N}.*} theorem iterates_MH_reflection: assumes p_reflection: "!!f g h. REFLECTS[λx. p(L, h(x), f(x), g(x)), λi x. p(##Lset(i), h(x), f(x), g(x))]" shows "REFLECTS[λx. iterates_MH(L, p(L,x), e(x), f(x), g(x), h(x)), λi x. iterates_MH(##Lset(i), p(##Lset(i),x), e(x), f(x), g(x), h(x))]" apply (simp (no_asm_use) only: iterates_MH_def) apply (intro FOL_reflections function_reflections is_nat_case_reflection restriction_reflection p_reflection) done subsubsection{*The Operator @{term is_iterates}*} text{*The three arguments of @{term p} are always 2, 1, 0; @{term p} is enclosed by 9 (??) quantifiers.*} (* "is_iterates(M,isF,v,n,Z) == ∃sn[M]. ∃msn[M]. successor(M,n,sn) & membership(M,sn,msn) & 1 0 is_wfrec(M, iterates_MH(M,isF,v), msn, n, Z)"*) constdefs is_iterates_fm :: "[i, i, i, i]=>i" "is_iterates_fm(p,v,n,Z) == Exists(Exists( And(succ_fm(n#+2,1), And(Memrel_fm(1,0), is_wfrec_fm(iterates_MH_fm(p, v#+7, 2, 1, 0), 0, n#+2, Z#+2)))))" text{*We call @{term p} with arguments a, f, z by equating them with the corresponding quantified variables with de Bruijn indices 2, 1, 0.*} lemma is_iterates_type [TC]: "[| p ∈ formula; x ∈ nat; y ∈ nat; z ∈ nat |] ==> is_iterates_fm(p,x,y,z) ∈ formula" by (simp add: is_iterates_fm_def) lemma sats_is_iterates_fm: assumes is_F_iff_sats: "!!a b c d e f g h i j k. [| a ∈ A; b ∈ A; c ∈ A; d ∈ A; e ∈ A; f ∈ A; g ∈ A; h ∈ A; i ∈ A; j ∈ A; k ∈ A|] ==> is_F(a,b) <-> sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d, Cons(e, Cons(f, Cons(g, Cons(h, Cons(i, Cons(j, Cons(k, env))))))))))))" shows "[|x ∈ nat; y < length(env); z < length(env); env ∈ list(A)|] ==> sats(A, is_iterates_fm(p,x,y,z), env) <-> is_iterates(##A, is_F, nth(x,env), nth(y,env), nth(z,env))" apply (frule_tac x=z in lt_length_in_nat, assumption) apply (frule lt_length_in_nat, assumption) apply (simp add: is_iterates_fm_def is_iterates_def sats_is_nat_case_fm is_F_iff_sats [symmetric] sats_is_wfrec_fm sats_iterates_MH_fm) done lemma is_iterates_iff_sats: assumes is_F_iff_sats: "!!a b c d e f g h i j k. [| a ∈ A; b ∈ A; c ∈ A; d ∈ A; e ∈ A; f ∈ A; g ∈ A; h ∈ A; i ∈ A; j ∈ A; k ∈ A|] ==> is_F(a,b) <-> sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d, Cons(e, Cons(f, Cons(g, Cons(h, Cons(i, Cons(j, Cons(k, env))))))))))))" shows "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j < length(env); k < length(env); env ∈ list(A)|] ==> is_iterates(##A, is_F, x, y, z) <-> sats(A, is_iterates_fm(p,i,j,k), env)" by (simp add: sats_is_iterates_fm [OF is_F_iff_sats]) text{*The second argument of @{term p} gives it direct access to @{term x}, which is essential for handling free variable references. Without this argument, we cannot prove reflection for @{term list_N}.*} theorem is_iterates_reflection: assumes p_reflection: "!!f g h. REFLECTS[λx. p(L, h(x), f(x), g(x)), λi x. p(##Lset(i), h(x), f(x), g(x))]" shows "REFLECTS[λx. is_iterates(L, p(L,x), f(x), g(x), h(x)), λi x. is_iterates(##Lset(i), p(##Lset(i),x), f(x), g(x), h(x))]" apply (simp (no_asm_use) only: is_iterates_def) apply (intro FOL_reflections function_reflections p_reflection is_wfrec_reflection iterates_MH_reflection) done subsubsection{*The Formula @{term is_eclose_n}, Internalized*} (* is_eclose_n(M,A,n,Z) == is_iterates(M, big_union(M), A, n, Z) *) constdefs eclose_n_fm :: "[i,i,i]=>i" "eclose_n_fm(A,n,Z) == is_iterates_fm(big_union_fm(1,0), A, n, Z)" lemma eclose_n_fm_type [TC]: "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> eclose_n_fm(x,y,z) ∈ formula" by (simp add: eclose_n_fm_def) lemma sats_eclose_n_fm [simp]: "[| x ∈ nat; y < length(env); z < length(env); env ∈ list(A)|] ==> sats(A, eclose_n_fm(x,y,z), env) <-> is_eclose_n(##A, nth(x,env), nth(y,env), nth(z,env))" apply (frule_tac x=z in lt_length_in_nat, assumption) apply (frule_tac x=y in lt_length_in_nat, assumption) apply (simp add: eclose_n_fm_def is_eclose_n_def sats_is_iterates_fm) done lemma eclose_n_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j < length(env); k < length(env); env ∈ list(A)|] ==> is_eclose_n(##A, x, y, z) <-> sats(A, eclose_n_fm(i,j,k), env)" by (simp add: sats_eclose_n_fm) theorem eclose_n_reflection: "REFLECTS[λx. is_eclose_n(L, f(x), g(x), h(x)), λi x. is_eclose_n(##Lset(i), f(x), g(x), h(x))]" apply (simp only: is_eclose_n_def) apply (intro FOL_reflections function_reflections is_iterates_reflection) done subsubsection{*Membership in @{term "eclose(A)"}*} (* mem_eclose(M,A,l) == ∃n[M]. ∃eclosen[M]. finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l ∈ eclosen *) constdefs mem_eclose_fm :: "[i,i]=>i" "mem_eclose_fm(x,y) == Exists(Exists( And(finite_ordinal_fm(1), And(eclose_n_fm(x#+2,1,0), Member(y#+2,0)))))" lemma mem_eclose_type [TC]: "[| x ∈ nat; y ∈ nat |] ==> mem_eclose_fm(x,y) ∈ formula" by (simp add: mem_eclose_fm_def) lemma sats_mem_eclose_fm [simp]: "[| x ∈ nat; y ∈ nat; env ∈ list(A)|] ==> sats(A, mem_eclose_fm(x,y), env) <-> mem_eclose(##A, nth(x,env), nth(y,env))" by (simp add: mem_eclose_fm_def mem_eclose_def) lemma mem_eclose_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; i ∈ nat; j ∈ nat; env ∈ list(A)|] ==> mem_eclose(##A, x, y) <-> sats(A, mem_eclose_fm(i,j), env)" by simp theorem mem_eclose_reflection: "REFLECTS[λx. mem_eclose(L,f(x),g(x)), λi x. mem_eclose(##Lset(i),f(x),g(x))]" apply (simp only: mem_eclose_def) apply (intro FOL_reflections finite_ordinal_reflection eclose_n_reflection) done subsubsection{*The Predicate ``Is @{term "eclose(A)"}''*} (* is_eclose(M,A,Z) == ∀l[M]. l ∈ Z <-> mem_eclose(M,A,l) *) constdefs is_eclose_fm :: "[i,i]=>i" "is_eclose_fm(A,Z) == Forall(Iff(Member(0,succ(Z)), mem_eclose_fm(succ(A),0)))" lemma is_eclose_type [TC]: "[| x ∈ nat; y ∈ nat |] ==> is_eclose_fm(x,y) ∈ formula" by (simp add: is_eclose_fm_def) lemma sats_is_eclose_fm [simp]: "[| x ∈ nat; y ∈ nat; env ∈ list(A)|] ==> sats(A, is_eclose_fm(x,y), env) <-> is_eclose(##A, nth(x,env), nth(y,env))" by (simp add: is_eclose_fm_def is_eclose_def) lemma is_eclose_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; i ∈ nat; j ∈ nat; env ∈ list(A)|] ==> is_eclose(##A, x, y) <-> sats(A, is_eclose_fm(i,j), env)" by simp theorem is_eclose_reflection: "REFLECTS[λx. is_eclose(L,f(x),g(x)), λi x. is_eclose(##Lset(i),f(x),g(x))]" apply (simp only: is_eclose_def) apply (intro FOL_reflections mem_eclose_reflection) done subsubsection{*The List Functor, Internalized*} constdefs list_functor_fm :: "[i,i,i]=>i" (* "is_list_functor(M,A,X,Z) == ∃n1[M]. ∃AX[M]. number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *) "list_functor_fm(A,X,Z) == Exists(Exists( And(number1_fm(1), And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))" lemma list_functor_type [TC]: "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> list_functor_fm(x,y,z) ∈ formula" by (simp add: list_functor_fm_def) lemma sats_list_functor_fm [simp]: "[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A)|] ==> sats(A, list_functor_fm(x,y,z), env) <-> is_list_functor(##A, nth(x,env), nth(y,env), nth(z,env))" by (simp add: list_functor_fm_def is_list_functor_def) lemma list_functor_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A)|] ==> is_list_functor(##A, x, y, z) <-> sats(A, list_functor_fm(i,j,k), env)" by simp theorem list_functor_reflection: "REFLECTS[λx. is_list_functor(L,f(x),g(x),h(x)), λi x. is_list_functor(##Lset(i),f(x),g(x),h(x))]" apply (simp only: is_list_functor_def) apply (intro FOL_reflections number1_reflection cartprod_reflection sum_reflection) done subsubsection{*The Formula @{term is_list_N}, Internalized*} (* "is_list_N(M,A,n,Z) == ∃zero[M]. empty(M,zero) & is_iterates(M, is_list_functor(M,A), zero, n, Z)" *) constdefs list_N_fm :: "[i,i,i]=>i" "list_N_fm(A,n,Z) == Exists( And(empty_fm(0), is_iterates_fm(list_functor_fm(A#+9#+3,1,0), 0, n#+1, Z#+1)))" lemma list_N_fm_type [TC]: "[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> list_N_fm(x,y,z) ∈ formula" by (simp add: list_N_fm_def) lemma sats_list_N_fm [simp]: "[| x ∈ nat; y < length(env); z < length(env); env ∈ list(A)|] ==> sats(A, list_N_fm(x,y,z), env) <-> is_list_N(##A, nth(x,env), nth(y,env), nth(z,env))" apply (frule_tac x=z in lt_length_in_nat, assumption) apply (frule_tac x=y in lt_length_in_nat, assumption) apply (simp add: list_N_fm_def is_list_N_def sats_is_iterates_fm) done lemma list_N_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; i ∈ nat; j < length(env); k < length(env); env ∈ list(A)|] ==> is_list_N(##A, x, y, z) <-> sats(A, list_N_fm(i,j,k), env)" by (simp add: sats_list_N_fm) theorem list_N_reflection: "REFLECTS[λx. is_list_N(L, f(x), g(x), h(x)), λi x. is_list_N(##Lset(i), f(x), g(x), h(x))]" apply (simp only: is_list_N_def) apply (intro FOL_reflections function_reflections is_iterates_reflection list_functor_reflection) done subsubsection{*The Predicate ``Is A List''*} (* mem_list(M,A,l) == ∃n[M]. ∃listn[M]. finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l ∈ listn *) constdefs mem_list_fm :: "[i,i]=>i" "mem_list_fm(x,y) == Exists(Exists( And(finite_ordinal_fm(1), And(list_N_fm(x#+2,1,0), Member(y#+2,0)))))" lemma mem_list_type [TC]: "[| x ∈ nat; y ∈ nat |] ==> mem_list_fm(x,y) ∈ formula" by (simp add: mem_list_fm_def) lemma sats_mem_list_fm [simp]: "[| x ∈ nat; y ∈ nat; env ∈ list(A)|] ==> sats(A, mem_list_fm(x,y), env) <-> mem_list(##A, nth(x,env), nth(y,env))" by (simp add: mem_list_fm_def mem_list_def) lemma mem_list_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; i ∈ nat; j ∈ nat; env ∈ list(A)|] ==> mem_list(##A, x, y) <-> sats(A, mem_list_fm(i,j), env)" by simp theorem mem_list_reflection: "REFLECTS[λx. mem_list(L,f(x),g(x)), λi x. mem_list(##Lset(i),f(x),g(x))]" apply (simp only: mem_list_def) apply (intro FOL_reflections finite_ordinal_reflection list_N_reflection) done subsubsection{*The Predicate ``Is @{term "list(A)"}''*} (* is_list(M,A,Z) == ∀l[M]. l ∈ Z <-> mem_list(M,A,l) *) constdefs is_list_fm :: "[i,i]=>i" "is_list_fm(A,Z) == Forall(Iff(Member(0,succ(Z)), mem_list_fm(succ(A),0)))" lemma is_list_type [TC]: "[| x ∈ nat; y ∈ nat |] ==> is_list_fm(x,y) ∈ formula" by (simp add: is_list_fm_def) lemma sats_is_list_fm [simp]: "[| x ∈ nat; y ∈ nat; env ∈ list(A)|] ==> sats(A, is_list_fm(x,y), env) <-> is_list(##A, nth(x,env), nth(y,env))" by (simp add: is_list_fm_def is_list_def) lemma is_list_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; i ∈ nat; j ∈ nat; env ∈ list(A)|] ==> is_list(##A, x, y) <-> sats(A, is_list_fm(i,j), env)" by simp theorem is_list_reflection: "REFLECTS[λx. is_list(L,f(x),g(x)), λi x. is_list(##Lset(i),f(x),g(x))]" apply (simp only: is_list_def) apply (intro FOL_reflections mem_list_reflection) done subsubsection{*The Formula Functor, Internalized*} constdefs formula_functor_fm :: "[i,i]=>i" (* "is_formula_functor(M,X,Z) == ∃nat'[M]. ∃natnat[M]. ∃natnatsum[M]. ∃XX[M]. ∃X3[M]. 4 3 2 1 0 omega(M,nat') & cartprod(M,nat',nat',natnat) & is_sum(M,natnat,natnat,natnatsum) & cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) & is_sum(M,natnatsum,X3,Z)" *) "formula_functor_fm(X,Z) == Exists(Exists(Exists(Exists(Exists( And(omega_fm(4), And(cartprod_fm(4,4,3), And(sum_fm(3,3,2), And(cartprod_fm(X#+5,X#+5,1), And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))" lemma formula_functor_type [TC]: "[| x ∈ nat; y ∈ nat |] ==> formula_functor_fm(x,y) ∈ formula" by (simp add: formula_functor_fm_def) lemma sats_formula_functor_fm [simp]: "[| x ∈ nat; y ∈ nat; env ∈ list(A)|] ==> sats(A, formula_functor_fm(x,y), env) <-> is_formula_functor(##A, nth(x,env), nth(y,env))" by (simp add: formula_functor_fm_def is_formula_functor_def) lemma formula_functor_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; i ∈ nat; j ∈ nat; env ∈ list(A)|] ==> is_formula_functor(##A, x, y) <-> sats(A, formula_functor_fm(i,j), env)" by simp theorem formula_functor_reflection: "REFLECTS[λx. is_formula_functor(L,f(x),g(x)), λi x. is_formula_functor(##Lset(i),f(x),g(x))]" apply (simp only: is_formula_functor_def) apply (intro FOL_reflections omega_reflection cartprod_reflection sum_reflection) done subsubsection{*The Formula @{term is_formula_N}, Internalized*} (* "is_formula_N(M,n,Z) == ∃zero[M]. empty(M,zero) & is_iterates(M, is_formula_functor(M), zero, n, Z)" *) constdefs formula_N_fm :: "[i,i]=>i" "formula_N_fm(n,Z) == Exists( And(empty_fm(0), is_iterates_fm(formula_functor_fm(1,0), 0, n#+1, Z#+1)))" lemma formula_N_fm_type [TC]: "[| x ∈ nat; y ∈ nat |] ==> formula_N_fm(x,y) ∈ formula" by (simp add: formula_N_fm_def) lemma sats_formula_N_fm [simp]: "[| x < length(env); y < length(env); env ∈ list(A)|] ==> sats(A, formula_N_fm(x,y), env) <-> is_formula_N(##A, nth(x,env), nth(y,env))" apply (frule_tac x=y in lt_length_in_nat, assumption) apply (frule lt_length_in_nat, assumption) apply (simp add: formula_N_fm_def is_formula_N_def sats_is_iterates_fm) done lemma formula_N_iff_sats: "[| nth(i,env) = x; nth(j,env) = y; i < length(env); j < length(env); env ∈ list(A)|] ==> is_formula_N(##A, x, y) <-> sats(A, formula_N_fm(i,j), env)" by (simp add: sats_formula_N_fm) theorem formula_N_reflection: "REFLECTS[λx. is_formula_N(L, f(x), g(x)), λi x. is_formula_N(##Lset(i), f(x), g(x))]" apply (simp only: is_formula_N_def) apply (intro FOL_reflections function_reflections is_iterates_reflection formula_functor_reflection) done subsubsection{*The Predicate ``Is A Formula''*} (* mem_formula(M,p) == ∃n[M]. ∃formn[M]. finite_ordinal(M,n) & is_formula_N(M,n,formn) & p ∈ formn *) constdefs mem_formula_fm :: "i=>i" "mem_formula_fm(x) == Exists(Exists( And(finite_ordinal_fm(1), And(formula_N_fm(1,0), Member(x#+2,0)))))" lemma mem_formula_type [TC]: "x ∈ nat ==> mem_formula_fm(x) ∈ formula" by (simp add: mem_formula_fm_def) lemma sats_mem_formula_fm [simp]: "[| x ∈ nat; env ∈ list(A)|] ==> sats(A, mem_formula_fm(x), env) <-> mem_formula(##A, nth(x,env))" by (simp add: mem_formula_fm_def mem_formula_def) lemma mem_formula_iff_sats: "[| nth(i,env) = x; i ∈ nat; env ∈ list(A)|] ==> mem_formula(##A, x) <-> sats(A, mem_formula_fm(i), env)" by simp theorem mem_formula_reflection: "REFLECTS[λx. mem_formula(L,f(x)), λi x. mem_formula(##Lset(i),f(x))]" apply (simp only: mem_formula_def) apply (intro FOL_reflections finite_ordinal_reflection formula_N_reflection) done subsubsection{*The Predicate ``Is @{term "formula"}''*} (* is_formula(M,Z) == ∀p[M]. p ∈ Z <-> mem_formula(M,p) *) constdefs is_formula_fm :: "i=>i" "is_formula_fm(Z) == Forall(Iff(Member(0,succ(Z)), mem_formula_fm(0)))" lemma is_formula_type [TC]: "x ∈ nat ==> is_formula_fm(x) ∈ formula" by (simp add: is_formula_fm_def) lemma sats_is_formula_fm [simp]: "[| x ∈ nat; env ∈ list(A)|] ==> sats(A, is_formula_fm(x), env) <-> is_formula(##A, nth(x,env))" by (simp add: is_formula_fm_def is_formula_def) lemma is_formula_iff_sats: "[| nth(i,env) = x; i ∈ nat; env ∈ list(A)|] ==> is_formula(##A, x) <-> sats(A, is_formula_fm(i), env)" by simp theorem is_formula_reflection: "REFLECTS[λx. is_formula(L,f(x)), λi x. is_formula(##Lset(i),f(x))]" apply (simp only: is_formula_def) apply (intro FOL_reflections mem_formula_reflection) done subsubsection{*The Operator @{term is_transrec}*} text{*The three arguments of @{term p} are always 2, 1, 0. It is buried within eight quantifiers! We call @{term p} with arguments a, f, z by equating them with the corresponding quantified variables with de Bruijn indices 2, 1, 0.*} (* is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o" "is_transrec(M,MH,a,z) == ∃sa[M]. ∃esa[M]. ∃mesa[M]. 2 1 0 upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) & is_wfrec(M,MH,mesa,a,z)" *) constdefs is_transrec_fm :: "[i, i, i]=>i" "is_transrec_fm(p,a,z) == Exists(Exists(Exists( And(upair_fm(a#+3,a#+3,2), And(is_eclose_fm(2,1), And(Memrel_fm(1,0), is_wfrec_fm(p,0,a#+3,z#+3)))))))" lemma is_transrec_type [TC]: "[| p ∈ formula; x ∈ nat; z ∈ nat |] ==> is_transrec_fm(p,x,z) ∈ formula" by (simp add: is_transrec_fm_def) lemma sats_is_transrec_fm: assumes MH_iff_sats: "!!a0 a1 a2 a3 a4 a5 a6 a7. [|a0∈A; a1∈A; a2∈A; a3∈A; a4∈A; a5∈A; a6∈A; a7∈A|] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3, Cons(a4,Cons(a5,Cons(a6,Cons(a7,env)))))))))" shows "[|x < length(env); z < length(env); env ∈ list(A)|] ==> sats(A, is_transrec_fm(p,x,z), env) <-> is_transrec(##A, MH, nth(x,env), nth(z,env))" apply (frule_tac x=z in lt_length_in_nat, assumption) apply (frule_tac x=x in lt_length_in_nat, assumption) apply (simp add: is_transrec_fm_def sats_is_wfrec_fm is_transrec_def MH_iff_sats [THEN iff_sym]) done lemma is_transrec_iff_sats: assumes MH_iff_sats: "!!a0 a1 a2 a3 a4 a5 a6 a7. [|a0∈A; a1∈A; a2∈A; a3∈A; a4∈A; a5∈A; a6∈A; a7∈A|] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3, Cons(a4,Cons(a5,Cons(a6,Cons(a7,env)))))))))" shows "[|nth(i,env) = x; nth(k,env) = z; i < length(env); k < length(env); env ∈ list(A)|] ==> is_transrec(##A, MH, x, z) <-> sats(A, is_transrec_fm(p,i,k), env)" by (simp add: sats_is_transrec_fm [OF MH_iff_sats]) theorem is_transrec_reflection: assumes MH_reflection: "!!f' f g h. REFLECTS[λx. MH(L, f'(x), f(x), g(x), h(x)), λi x. MH(##Lset(i), f'(x), f(x), g(x), h(x))]" shows "REFLECTS[λx. is_transrec(L, MH(L,x), f(x), h(x)), λi x. is_transrec(##Lset(i), MH(##Lset(i),x), f(x), h(x))]" apply (simp (no_asm_use) only: is_transrec_def) apply (intro FOL_reflections function_reflections MH_reflection is_wfrec_reflection is_eclose_reflection) done end
lemma Inl_type:
[| x ∈ nat; z ∈ nat |] ==> Inl_fm(x, z) ∈ formula
lemma sats_Inl_fm:
[| x ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, Inl_fm(x, z), env) <-> is_Inl(##A, nth(x, env), nth(z, env))
lemma Inl_iff_sats:
[| nth(i, env) = x; nth(k, env) = z; i ∈ nat; k ∈ nat; env ∈ list(A) |] ==> is_Inl(##A, x, z) <-> sats(A, Inl_fm(i, k), env)
theorem Inl_reflection:
REFLECTS [%x. is_Inl(L, f(x), h(x)), %i x. is_Inl(##Lset(i), f(x), h(x))]
lemma Inr_type:
[| x ∈ nat; z ∈ nat |] ==> Inr_fm(x, z) ∈ formula
lemma sats_Inr_fm:
[| x ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, Inr_fm(x, z), env) <-> is_Inr(##A, nth(x, env), nth(z, env))
lemma Inr_iff_sats:
[| nth(i, env) = x; nth(k, env) = z; i ∈ nat; k ∈ nat; env ∈ list(A) |] ==> is_Inr(##A, x, z) <-> sats(A, Inr_fm(i, k), env)
theorem Inr_reflection:
REFLECTS [%x. is_Inr(L, f(x), h(x)), %i x. is_Inr(##Lset(i), f(x), h(x))]
lemma Nil_type:
x ∈ nat ==> Nil_fm(x) ∈ formula
lemma sats_Nil_fm:
[| x ∈ nat; env ∈ list(A) |] ==> sats(A, Nil_fm(x), env) <-> is_Nil(##A, nth(x, env))
lemma Nil_iff_sats:
[| nth(i, env) = x; i ∈ nat; env ∈ list(A) |] ==> is_Nil(##A, x) <-> sats(A, Nil_fm(i), env)
theorem Nil_reflection:
REFLECTS [%x. is_Nil(L, f(x)), %i x. is_Nil(##Lset(i), f(x))]
lemma Cons_type:
[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Cons_fm(x, y, z) ∈ formula
lemma sats_Cons_fm:
[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, Cons_fm(x, y, z), env) <-> is_Cons(##A, nth(x, env), nth(y, env), nth(z, env))
lemma Cons_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A) |] ==> is_Cons(##A, x, y, z) <-> sats(A, Cons_fm(i, j, k), env)
theorem Cons_reflection:
REFLECTS [%x. is_Cons(L, f(x), g(x), h(x)), %i x. is_Cons(##Lset(i), f(x), g(x), h(x))]
lemma quasilist_type:
x ∈ nat ==> quasilist_fm(x) ∈ formula
lemma sats_quasilist_fm:
[| x ∈ nat; env ∈ list(A) |] ==> sats(A, quasilist_fm(x), env) <-> is_quasilist(##A, nth(x, env))
lemma quasilist_iff_sats:
[| nth(i, env) = x; i ∈ nat; env ∈ list(A) |] ==> is_quasilist(##A, x) <-> sats(A, quasilist_fm(i), env)
theorem quasilist_reflection:
REFLECTS [%x. is_quasilist(L, f(x)), %i x. is_quasilist(##Lset(i), f(x))]
lemma hd_type:
[| x ∈ nat; y ∈ nat |] ==> hd_fm(x, y) ∈ formula
lemma sats_hd_fm:
[| x ∈ nat; y ∈ nat; env ∈ list(A) |] ==> sats(A, hd_fm(x, y), env) <-> is_hd(##A, nth(x, env), nth(y, env))
lemma hd_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |] ==> is_hd(##A, x, y) <-> sats(A, hd_fm(i, j), env)
theorem hd_reflection:
REFLECTS [%x. is_hd(L, f(x), g(x)), %i x. is_hd(##Lset(i), f(x), g(x))]
lemma tl_type:
[| x ∈ nat; y ∈ nat |] ==> tl_fm(x, y) ∈ formula
lemma sats_tl_fm:
[| x ∈ nat; y ∈ nat; env ∈ list(A) |] ==> sats(A, tl_fm(x, y), env) <-> is_tl(##A, nth(x, env), nth(y, env))
lemma tl_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |] ==> is_tl(##A, x, y) <-> sats(A, tl_fm(i, j), env)
theorem tl_reflection:
REFLECTS [%x. is_tl(L, f(x), g(x)), %i x. is_tl(##Lset(i), f(x), g(x))]
lemma is_bool_of_o_type:
[| p ∈ formula; z ∈ nat |] ==> bool_of_o_fm(p, z) ∈ formula
lemma sats_bool_of_o_fm:
[| P <-> sats(A, p, env); z ∈ nat; env ∈ list(A) |] ==> sats(A, bool_of_o_fm(p, z), env) <-> is_bool_of_o(##A, P, nth(z, env))
lemma is_bool_of_o_iff_sats:
[| P <-> sats(A, p, env); nth(k, env) = z; k ∈ nat; env ∈ list(A) |] ==> is_bool_of_o(##A, P, z) <-> sats(A, bool_of_o_fm(p, k), env)
theorem bool_of_o_reflection:
REFLECTS [P(L), %i. P(##Lset(i))] ==> REFLECTS [%x. is_bool_of_o(L, P(L, x), f(x)), %i x. is_bool_of_o(##Lset(i), P(##Lset(i), x), f(x))]
lemma is_lambda_type:
[| p ∈ formula; x ∈ nat; y ∈ nat |] ==> lambda_fm(p, x, y) ∈ formula
lemma sats_lambda_fm:
[| !!a0 a1 a2. [| a0 ∈ A; a1 ∈ A; a2 ∈ A |] ==> is_b(a1, a0) <-> sats(A, p, Cons(a0, Cons(a1, Cons(a2, env)))); x ∈ nat; y ∈ nat; env ∈ list(A) |] ==> sats(A, lambda_fm(p, x, y), env) <-> is_lambda(##A, nth(x, env), is_b, nth(y, env))
theorem is_lambda_reflection:
(!!f g h. REFLECTS [%x. is_b(L, f(x), g(x), h(x)), %i x. is_b(##Lset(i), f(x), g(x), h(x))]) ==> REFLECTS [%x. is_lambda(L, A(x), is_b(L, x), f(x)), %i x. is_lambda(##Lset(i), A(x), is_b(##Lset(i), x), f(x))]
lemma is_Member_type:
[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Member_fm(x, y, z) ∈ formula
lemma sats_Member_fm:
[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, Member_fm(x, y, z), env) <-> is_Member(##A, nth(x, env), nth(y, env), nth(z, env))
lemma Member_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A) |] ==> is_Member(##A, x, y, z) <-> sats(A, Member_fm(i, j, k), env)
theorem Member_reflection:
REFLECTS [%x. is_Member(L, f(x), g(x), h(x)), %i x. is_Member(##Lset(i), f(x), g(x), h(x))]
lemma is_Equal_type:
[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Equal_fm(x, y, z) ∈ formula
lemma sats_Equal_fm:
[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, Equal_fm(x, y, z), env) <-> is_Equal(##A, nth(x, env), nth(y, env), nth(z, env))
lemma Equal_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A) |] ==> is_Equal(##A, x, y, z) <-> sats(A, Equal_fm(i, j, k), env)
theorem Equal_reflection:
REFLECTS [%x. is_Equal(L, f(x), g(x), h(x)), %i x. is_Equal(##Lset(i), f(x), g(x), h(x))]
lemma is_Nand_type:
[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> Nand_fm(x, y, z) ∈ formula
lemma sats_Nand_fm:
[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, Nand_fm(x, y, z), env) <-> is_Nand(##A, nth(x, env), nth(y, env), nth(z, env))
lemma Nand_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A) |] ==> is_Nand(##A, x, y, z) <-> sats(A, Nand_fm(i, j, k), env)
theorem Nand_reflection:
REFLECTS [%x. is_Nand(L, f(x), g(x), h(x)), %i x. is_Nand(##Lset(i), f(x), g(x), h(x))]
lemma is_Forall_type:
[| x ∈ nat; y ∈ nat |] ==> Forall_fm(x, y) ∈ formula
lemma sats_Forall_fm:
[| x ∈ nat; y ∈ nat; env ∈ list(A) |] ==> sats(A, Forall_fm(x, y), env) <-> is_Forall(##A, nth(x, env), nth(y, env))
lemma Forall_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |] ==> is_Forall(##A, x, y) <-> sats(A, Forall_fm(i, j), env)
theorem Forall_reflection:
REFLECTS [%x. is_Forall(L, f(x), g(x)), %i x. is_Forall(##Lset(i), f(x), g(x))]
lemma is_and_type:
[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> and_fm(x, y, z) ∈ formula
lemma sats_and_fm:
[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, and_fm(x, y, z), env) <-> is_and(##A, nth(x, env), nth(y, env), nth(z, env))
lemma is_and_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A) |] ==> is_and(##A, x, y, z) <-> sats(A, and_fm(i, j, k), env)
theorem is_and_reflection:
REFLECTS [%x. is_and(L, f(x), g(x), h(x)), %i x. is_and(##Lset(i), f(x), g(x), h(x))]
lemma is_or_type:
[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> or_fm(x, y, z) ∈ formula
lemma sats_or_fm:
[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, or_fm(x, y, z), env) <-> is_or(##A, nth(x, env), nth(y, env), nth(z, env))
lemma is_or_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A) |] ==> is_or(##A, x, y, z) <-> sats(A, or_fm(i, j, k), env)
theorem is_or_reflection:
REFLECTS [%x. is_or(L, f(x), g(x), h(x)), %i x. is_or(##Lset(i), f(x), g(x), h(x))]
lemma is_not_type:
[| x ∈ nat; z ∈ nat |] ==> not_fm(x, z) ∈ formula
lemma sats_is_not_fm:
[| x ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, not_fm(x, z), env) <-> is_not(##A, nth(x, env), nth(z, env))
lemma is_not_iff_sats:
[| nth(i, env) = x; nth(k, env) = z; i ∈ nat; k ∈ nat; env ∈ list(A) |] ==> is_not(##A, x, z) <-> sats(A, not_fm(i, k), env)
theorem is_not_reflection:
REFLECTS [%x. is_not(L, f(x), g(x)), %i x. is_not(##Lset(i), f(x), g(x))]
lemmas extra_reflections:
REFLECTS [%x. is_Inl(L, f(x), h(x)), %i x. is_Inl(##Lset(i), f(x), h(x))]
REFLECTS [%x. is_Inr(L, f(x), h(x)), %i x. is_Inr(##Lset(i), f(x), h(x))]
REFLECTS [%x. is_Nil(L, f(x)), %i x. is_Nil(##Lset(i), f(x))]
REFLECTS [%x. is_Cons(L, f(x), g(x), h(x)), %i x. is_Cons(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_quasilist(L, f(x)), %i x. is_quasilist(##Lset(i), f(x))]
REFLECTS [%x. is_hd(L, f(x), g(x)), %i x. is_hd(##Lset(i), f(x), g(x))]
REFLECTS [%x. is_tl(L, f(x), g(x)), %i x. is_tl(##Lset(i), f(x), g(x))]
REFLECTS [P(L), %i. P(##Lset(i))] ==> REFLECTS [%x. is_bool_of_o(L, P(L, x), f(x)), %i x. is_bool_of_o(##Lset(i), P(##Lset(i), x), f(x))]
(!!f g h. REFLECTS [%x. is_b(L, f(x), g(x), h(x)), %i x. is_b(##Lset(i), f(x), g(x), h(x))]) ==> REFLECTS [%x. is_lambda(L, A(x), is_b(L, x), f(x)), %i x. is_lambda(##Lset(i), A(x), is_b(##Lset(i), x), f(x))]
REFLECTS [%x. is_Member(L, f(x), g(x), h(x)), %i x. is_Member(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_Equal(L, f(x), g(x), h(x)), %i x. is_Equal(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_Nand(L, f(x), g(x), h(x)), %i x. is_Nand(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_Forall(L, f(x), g(x)), %i x. is_Forall(##Lset(i), f(x), g(x))]
REFLECTS [%x. is_and(L, f(x), g(x), h(x)), %i x. is_and(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_or(L, f(x), g(x), h(x)), %i x. is_or(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_not(L, f(x), g(x)), %i x. is_not(##Lset(i), f(x), g(x))]
lemmas extra_reflections:
REFLECTS [%x. is_Inl(L, f(x), h(x)), %i x. is_Inl(##Lset(i), f(x), h(x))]
REFLECTS [%x. is_Inr(L, f(x), h(x)), %i x. is_Inr(##Lset(i), f(x), h(x))]
REFLECTS [%x. is_Nil(L, f(x)), %i x. is_Nil(##Lset(i), f(x))]
REFLECTS [%x. is_Cons(L, f(x), g(x), h(x)), %i x. is_Cons(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_quasilist(L, f(x)), %i x. is_quasilist(##Lset(i), f(x))]
REFLECTS [%x. is_hd(L, f(x), g(x)), %i x. is_hd(##Lset(i), f(x), g(x))]
REFLECTS [%x. is_tl(L, f(x), g(x)), %i x. is_tl(##Lset(i), f(x), g(x))]
REFLECTS [P(L), %i. P(##Lset(i))] ==> REFLECTS [%x. is_bool_of_o(L, P(L, x), f(x)), %i x. is_bool_of_o(##Lset(i), P(##Lset(i), x), f(x))]
(!!f g h. REFLECTS [%x. is_b(L, f(x), g(x), h(x)), %i x. is_b(##Lset(i), f(x), g(x), h(x))]) ==> REFLECTS [%x. is_lambda(L, A(x), is_b(L, x), f(x)), %i x. is_lambda(##Lset(i), A(x), is_b(##Lset(i), x), f(x))]
REFLECTS [%x. is_Member(L, f(x), g(x), h(x)), %i x. is_Member(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_Equal(L, f(x), g(x), h(x)), %i x. is_Equal(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_Nand(L, f(x), g(x), h(x)), %i x. is_Nand(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_Forall(L, f(x), g(x)), %i x. is_Forall(##Lset(i), f(x), g(x))]
REFLECTS [%x. is_and(L, f(x), g(x), h(x)), %i x. is_and(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_or(L, f(x), g(x), h(x)), %i x. is_or(##Lset(i), f(x), g(x), h(x))]
REFLECTS [%x. is_not(L, f(x), g(x)), %i x. is_not(##Lset(i), f(x), g(x))]
lemma M_is_recfun_iff:
M_is_recfun(M, MH, r, a, f) <-> (∀z[M]. z ∈ f <-> (∃x[M]. ∃f_r_sx[M]. ∃y[M]. MH(x, f_r_sx, y) ∧ pair(M, x, y, z) ∧ (∃xa[M]. ∃sx[M]. ∃r_sx[M]. pair(M, x, a, xa) ∧ upair(M, x, x, sx) ∧ pre_image(M, r, sx, r_sx) ∧ restriction(M, f, r_sx, f_r_sx) ∧ xa ∈ r)))
lemma is_recfun_type:
[| p ∈ formula; x ∈ nat; y ∈ nat; z ∈ nat |] ==> is_recfun_fm(p, x, y, z) ∈ formula
lemma sats_is_recfun_fm:
[| !!a0 a1 a2 a3. [| a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A |] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0, Cons(a1, Cons(a2, Cons(a3, env))))); x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, is_recfun_fm(p, x, y, z), env) <-> M_is_recfun(##A, MH, nth(x, env), nth(y, env), nth(z, env))
lemma is_recfun_iff_sats:
[| !!a0 a1 a2 a3. [| a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A |] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0, Cons(a1, Cons(a2, Cons(a3, env))))); nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A) |] ==> M_is_recfun(##A, MH, x, y, z) <-> sats(A, is_recfun_fm(p, i, j, k), env)
theorem is_recfun_reflection:
(!!f' f g h. REFLECTS [%x. MH(L, f'(x), f(x), g(x), h(x)), %i x. MH(##Lset(i), f'(x), f(x), g(x), h(x))]) ==> REFLECTS [%x. M_is_recfun(L, MH(L, x), f(x), g(x), h(x)), %i x. M_is_recfun(##Lset(i), MH(##Lset(i), x), f(x), g(x), h(x))]
lemma is_wfrec_type:
[| p ∈ formula; x ∈ nat; y ∈ nat; z ∈ nat |] ==> is_wfrec_fm(p, x, y, z) ∈ formula
lemma sats_is_wfrec_fm:
[| !!a0 a1 a2 a3 a4. [| a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A |] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, env)))))); x ∈ nat; y < length(env); z < length(env); env ∈ list(A) |] ==> sats(A, is_wfrec_fm(p, x, y, z), env) <-> is_wfrec(##A, MH, nth(x, env), nth(y, env), nth(z, env))
lemma is_wfrec_iff_sats:
[| !!a0 a1 a2 a3 a4. [| a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A |] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, env)))))); nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j < length(env); k < length(env); env ∈ list(A) |] ==> is_wfrec(##A, MH, x, y, z) <-> sats(A, is_wfrec_fm(p, i, j, k), env)
theorem is_wfrec_reflection:
(!!f' f g h. REFLECTS [%x. MH(L, f'(x), f(x), g(x), h(x)), %i x. MH(##Lset(i), f'(x), f(x), g(x), h(x))]) ==> REFLECTS [%x. is_wfrec(L, MH(L, x), f(x), g(x), h(x)), %i x. is_wfrec(##Lset(i), MH(##Lset(i), x), f(x), g(x), h(x))]
lemma cartprod_type:
[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> cartprod_fm(x, y, z) ∈ formula
lemma sats_cartprod_fm:
[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, cartprod_fm(x, y, z), env) <-> cartprod(##A, nth(x, env), nth(y, env), nth(z, env))
lemma cartprod_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A) |] ==> cartprod(##A, x, y, z) <-> sats(A, cartprod_fm(i, j, k), env)
theorem cartprod_reflection:
REFLECTS [%x. cartprod(L, f(x), g(x), h(x)), %i x. cartprod(##Lset(i), f(x), g(x), h(x))]
lemma sum_type:
[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> sum_fm(x, y, z) ∈ formula
lemma sats_sum_fm:
[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, sum_fm(x, y, z), env) <-> is_sum(##A, nth(x, env), nth(y, env), nth(z, env))
lemma sum_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A) |] ==> is_sum(##A, x, y, z) <-> sats(A, sum_fm(i, j, k), env)
theorem sum_reflection:
REFLECTS [%x. is_sum(L, f(x), g(x), h(x)), %i x. is_sum(##Lset(i), f(x), g(x), h(x))]
lemma quasinat_type:
x ∈ nat ==> quasinat_fm(x) ∈ formula
lemma sats_quasinat_fm:
[| x ∈ nat; env ∈ list(A) |] ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(##A, nth(x, env))
lemma quasinat_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; i ∈ nat; env ∈ list(A) |] ==> is_quasinat(##A, x) <-> sats(A, quasinat_fm(i), env)
theorem quasinat_reflection:
REFLECTS [%x. is_quasinat(L, f(x)), %i x. is_quasinat(##Lset(i), f(x))]
lemma is_nat_case_type:
[| is_b ∈ formula; x ∈ nat; y ∈ nat; z ∈ nat |] ==> is_nat_case_fm(x, is_b, y, z) ∈ formula
lemma sats_is_nat_case_fm:
[| !!a. a ∈ A ==> is_b(a, nth(z, env)) <-> sats(A, p, Cons(nth(z, env), Cons(a, env))); x ∈ nat; y ∈ nat; z < length(env); env ∈ list(A) |] ==> sats(A, is_nat_case_fm(x, p, y, z), env) <-> is_nat_case(##A, nth(x, env), is_b, nth(y, env), nth(z, env))
lemma is_nat_case_iff_sats:
[| !!a. a ∈ A ==> is_b(a, z) <-> sats(A, p, Cons(z, Cons(a, env))); nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k < length(env); env ∈ list(A) |] ==> is_nat_case(##A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i, p, j, k), env)
theorem is_nat_case_reflection:
(!!h f g. REFLECTS [%x. is_b(L, h(x), f(x), g(x)), %i x. is_b(##Lset(i), h(x), f(x), g(x))]) ==> REFLECTS [%x. is_nat_case(L, f(x), is_b(L, x), g(x), h(x)), %i x. is_nat_case(##Lset(i), f(x), is_b(##Lset(i), x), g(x), h(x))]
lemma iterates_MH_type:
[| p ∈ formula; v ∈ nat; x ∈ nat; y ∈ nat; z ∈ nat |] ==> iterates_MH_fm(p, v, x, y, z) ∈ formula
lemma sats_iterates_MH_fm:
[| !!a b c d. [| a ∈ A; b ∈ A; c ∈ A; d ∈ A |] ==> is_F(a, b) <-> sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d, env))))); v ∈ nat; x ∈ nat; y ∈ nat; z < length(env); env ∈ list(A) |] ==> sats(A, iterates_MH_fm(p, v, x, y, z), env) <-> iterates_MH(##A, is_F, nth(v, env), nth(x, env), nth(y, env), nth(z, env))
lemma iterates_MH_iff_sats:
[| !!a b c d. [| a ∈ A; b ∈ A; c ∈ A; d ∈ A |] ==> is_F(a, b) <-> sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d, env))))); nth(i', env) = v; nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i' ∈ nat; i ∈ nat; j ∈ nat; k < length(env); env ∈ list(A) |] ==> iterates_MH(##A, is_F, v, x, y, z) <-> sats(A, iterates_MH_fm(p, i', i, j, k), env)
theorem iterates_MH_reflection:
(!!f g h. REFLECTS [%x. p(L, h(x), f(x), g(x)), %i x. p(##Lset(i), h(x), f(x), g(x))]) ==> REFLECTS [%x. iterates_MH(L, p(L, x), e(x), f(x), g(x), h(x)), %i x. iterates_MH(##Lset(i), p(##Lset(i), x), e(x), f(x), g(x), h(x))]
lemma is_iterates_type:
[| p ∈ formula; x ∈ nat; y ∈ nat; z ∈ nat |] ==> is_iterates_fm(p, x, y, z) ∈ formula
lemma sats_is_iterates_fm:
[| !!a b c d e f g h i j k. [| a ∈ A; b ∈ A; c ∈ A; d ∈ A; e ∈ A; f ∈ A; g ∈ A; h ∈ A; i ∈ A; j ∈ A; k ∈ A |] ==> is_F(a, b) <-> sats(A, p, Cons(b, Cons(a, Cons(c, Cons (d, Cons(e, Cons(f, Cons(g, Cons(h, Cons(i, Cons(j, Cons(k, env)))))))))))); x ∈ nat; y < length(env); z < length(env); env ∈ list(A) |] ==> sats(A, is_iterates_fm(p, x, y, z), env) <-> is_iterates(##A, is_F, nth(x, env), nth(y, env), nth(z, env))
lemma is_iterates_iff_sats:
[| !!a b c d e f g h i j k. [| a ∈ A; b ∈ A; c ∈ A; d ∈ A; e ∈ A; f ∈ A; g ∈ A; h ∈ A; i ∈ A; j ∈ A; k ∈ A |] ==> is_F(a, b) <-> sats(A, p, Cons(b, Cons(a, Cons(c, Cons (d, Cons(e, Cons(f, Cons(g, Cons(h, Cons(i, Cons(j, Cons(k, env)))))))))))); nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j < length(env); k < length(env); env ∈ list(A) |] ==> is_iterates(##A, is_F, x, y, z) <-> sats(A, is_iterates_fm(p, i, j, k), env)
theorem is_iterates_reflection:
(!!f g h. REFLECTS [%x. p(L, h(x), f(x), g(x)), %i x. p(##Lset(i), h(x), f(x), g(x))]) ==> REFLECTS [%x. is_iterates(L, p(L, x), f(x), g(x), h(x)), %i x. is_iterates(##Lset(i), p(##Lset(i), x), f(x), g(x), h(x))]
lemma eclose_n_fm_type:
[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> eclose_n_fm(x, y, z) ∈ formula
lemma sats_eclose_n_fm:
[| x ∈ nat; y < length(env); z < length(env); env ∈ list(A) |] ==> sats(A, eclose_n_fm(x, y, z), env) <-> is_eclose_n(##A, nth(x, env), nth(y, env), nth(z, env))
lemma eclose_n_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j < length(env); k < length(env); env ∈ list(A) |] ==> is_eclose_n(##A, x, y, z) <-> sats(A, eclose_n_fm(i, j, k), env)
theorem eclose_n_reflection:
REFLECTS [%x. is_eclose_n(L, f(x), g(x), h(x)), %i x. is_eclose_n(##Lset(i), f(x), g(x), h(x))]
lemma mem_eclose_type:
[| x ∈ nat; y ∈ nat |] ==> mem_eclose_fm(x, y) ∈ formula
lemma sats_mem_eclose_fm:
[| x ∈ nat; y ∈ nat; env ∈ list(A) |] ==> sats(A, mem_eclose_fm(x, y), env) <-> mem_eclose(##A, nth(x, env), nth(y, env))
lemma mem_eclose_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |] ==> mem_eclose(##A, x, y) <-> sats(A, mem_eclose_fm(i, j), env)
theorem mem_eclose_reflection:
REFLECTS [%x. mem_eclose(L, f(x), g(x)), %i x. mem_eclose(##Lset(i), f(x), g(x))]
lemma is_eclose_type:
[| x ∈ nat; y ∈ nat |] ==> is_eclose_fm(x, y) ∈ formula
lemma sats_is_eclose_fm:
[| x ∈ nat; y ∈ nat; env ∈ list(A) |] ==> sats(A, is_eclose_fm(x, y), env) <-> is_eclose(##A, nth(x, env), nth(y, env))
lemma is_eclose_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |] ==> is_eclose(##A, x, y) <-> sats(A, is_eclose_fm(i, j), env)
theorem is_eclose_reflection:
REFLECTS [%x. is_eclose(L, f(x), g(x)), %i x. is_eclose(##Lset(i), f(x), g(x))]
lemma list_functor_type:
[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> list_functor_fm(x, y, z) ∈ formula
lemma sats_list_functor_fm:
[| x ∈ nat; y ∈ nat; z ∈ nat; env ∈ list(A) |] ==> sats(A, list_functor_fm(x, y, z), env) <-> is_list_functor(##A, nth(x, env), nth(y, env), nth(z, env))
lemma list_functor_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j ∈ nat; k ∈ nat; env ∈ list(A) |] ==> is_list_functor(##A, x, y, z) <-> sats(A, list_functor_fm(i, j, k), env)
theorem list_functor_reflection:
REFLECTS [%x. is_list_functor(L, f(x), g(x), h(x)), %i x. is_list_functor(##Lset(i), f(x), g(x), h(x))]
lemma list_N_fm_type:
[| x ∈ nat; y ∈ nat; z ∈ nat |] ==> list_N_fm(x, y, z) ∈ formula
lemma sats_list_N_fm:
[| x ∈ nat; y < length(env); z < length(env); env ∈ list(A) |] ==> sats(A, list_N_fm(x, y, z), env) <-> is_list_N(##A, nth(x, env), nth(y, env), nth(z, env))
lemma list_N_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; nth(k, env) = z; i ∈ nat; j < length(env); k < length(env); env ∈ list(A) |] ==> is_list_N(##A, x, y, z) <-> sats(A, list_N_fm(i, j, k), env)
theorem list_N_reflection:
REFLECTS [%x. is_list_N(L, f(x), g(x), h(x)), %i x. is_list_N(##Lset(i), f(x), g(x), h(x))]
lemma mem_list_type:
[| x ∈ nat; y ∈ nat |] ==> mem_list_fm(x, y) ∈ formula
lemma sats_mem_list_fm:
[| x ∈ nat; y ∈ nat; env ∈ list(A) |] ==> sats(A, mem_list_fm(x, y), env) <-> mem_list(##A, nth(x, env), nth(y, env))
lemma mem_list_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |] ==> mem_list(##A, x, y) <-> sats(A, mem_list_fm(i, j), env)
theorem mem_list_reflection:
REFLECTS [%x. mem_list(L, f(x), g(x)), %i x. mem_list(##Lset(i), f(x), g(x))]
lemma is_list_type:
[| x ∈ nat; y ∈ nat |] ==> is_list_fm(x, y) ∈ formula
lemma sats_is_list_fm:
[| x ∈ nat; y ∈ nat; env ∈ list(A) |] ==> sats(A, is_list_fm(x, y), env) <-> is_list(##A, nth(x, env), nth(y, env))
lemma is_list_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |] ==> is_list(##A, x, y) <-> sats(A, is_list_fm(i, j), env)
theorem is_list_reflection:
REFLECTS [%x. is_list(L, f(x), g(x)), %i x. is_list(##Lset(i), f(x), g(x))]
lemma formula_functor_type:
[| x ∈ nat; y ∈ nat |] ==> formula_functor_fm(x, y) ∈ formula
lemma sats_formula_functor_fm:
[| x ∈ nat; y ∈ nat; env ∈ list(A) |] ==> sats(A, formula_functor_fm(x, y), env) <-> is_formula_functor(##A, nth(x, env), nth(y, env))
lemma formula_functor_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; i ∈ nat; j ∈ nat; env ∈ list(A) |] ==> is_formula_functor(##A, x, y) <-> sats(A, formula_functor_fm(i, j), env)
theorem formula_functor_reflection:
REFLECTS [%x. is_formula_functor(L, f(x), g(x)), %i x. is_formula_functor(##Lset(i), f(x), g(x))]
lemma formula_N_fm_type:
[| x ∈ nat; y ∈ nat |] ==> formula_N_fm(x, y) ∈ formula
lemma sats_formula_N_fm:
[| x < length(env); y < length(env); env ∈ list(A) |] ==> sats(A, formula_N_fm(x, y), env) <-> is_formula_N(##A, nth(x, env), nth(y, env))
lemma formula_N_iff_sats:
[| nth(i, env) = x; nth(j, env) = y; i < length(env); j < length(env); env ∈ list(A) |] ==> is_formula_N(##A, x, y) <-> sats(A, formula_N_fm(i, j), env)
theorem formula_N_reflection:
REFLECTS [%x. is_formula_N(L, f(x), g(x)), %i x. is_formula_N(##Lset(i), f(x), g(x))]
lemma mem_formula_type:
x ∈ nat ==> mem_formula_fm(x) ∈ formula
lemma sats_mem_formula_fm:
[| x ∈ nat; env ∈ list(A) |] ==> sats(A, mem_formula_fm(x), env) <-> mem_formula(##A, nth(x, env))
lemma mem_formula_iff_sats:
[| nth(i, env) = x; i ∈ nat; env ∈ list(A) |] ==> mem_formula(##A, x) <-> sats(A, mem_formula_fm(i), env)
theorem mem_formula_reflection:
REFLECTS [%x. mem_formula(L, f(x)), %i x. mem_formula(##Lset(i), f(x))]
lemma is_formula_type:
x ∈ nat ==> is_formula_fm(x) ∈ formula
lemma sats_is_formula_fm:
[| x ∈ nat; env ∈ list(A) |] ==> sats(A, is_formula_fm(x), env) <-> is_formula(##A, nth(x, env))
lemma is_formula_iff_sats:
[| nth(i, env) = x; i ∈ nat; env ∈ list(A) |] ==> is_formula(##A, x) <-> sats(A, is_formula_fm(i), env)
theorem is_formula_reflection:
REFLECTS [%x. is_formula(L, f(x)), %i x. is_formula(##Lset(i), f(x))]
lemma is_transrec_type:
[| p ∈ formula; x ∈ nat; z ∈ nat |] ==> is_transrec_fm(p, x, z) ∈ formula
lemma sats_is_transrec_fm:
[| !!a0 a1 a2 a3 a4 a5 a6 a7. [| a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A |] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0, Cons(a1, Cons(a2, Cons (a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, env))))))))); x < length(env); z < length(env); env ∈ list(A) |] ==> sats(A, is_transrec_fm(p, x, z), env) <-> is_transrec(##A, MH, nth(x, env), nth(z, env))
lemma is_transrec_iff_sats:
[| !!a0 a1 a2 a3 a4 a5 a6 a7. [| a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A |] ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0, Cons(a1, Cons(a2, Cons (a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, env))))))))); nth(i, env) = x; nth(k, env) = z; i < length(env); k < length(env); env ∈ list(A) |] ==> is_transrec(##A, MH, x, z) <-> sats(A, is_transrec_fm(p, i, k), env)
theorem is_transrec_reflection:
(!!f' f g h. REFLECTS [%x. MH(L, f'(x), f(x), g(x), h(x)), %i x. MH(##Lset(i), f'(x), f(x), g(x), h(x))]) ==> REFLECTS [%x. is_transrec(L, MH(L, x), f(x), h(x)), %i x. is_transrec(##Lset(i), MH(##Lset(i), x), f(x), h(x))]