author | paulson |
Wed, 09 Oct 2002 11:07:13 +0200 | |
changeset 13634 | 99a593b49b04 |
parent 13506 | acc18a5d2b1a |
child 13651 | ac80e101306a |
permissions | -rw-r--r-- |
13505 | 1 |
(* Title: ZF/Constructible/Internalize.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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*) |
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||
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theory Internalize = L_axioms + Datatype_absolute: |
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subsection{*Internalized Forms of Data Structuring Operators*} |
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subsubsection{*The Formula @{term is_Inl}, Internalized*} |
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(* is_Inl(M,a,z) == \<exists>zero[M]. empty(M,zero) & pair(M,zero,a,z) *) |
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constdefs Inl_fm :: "[i,i]=>i" |
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"Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))" |
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lemma Inl_type [TC]: |
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"[| x \<in> nat; z \<in> nat |] ==> Inl_fm(x,z) \<in> formula" |
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by (simp add: Inl_fm_def) |
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|
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lemma sats_Inl_fm [simp]: |
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"[| x \<in> nat; z \<in> nat; env \<in> list(A)|] |
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==> sats(A, Inl_fm(x,z), env) <-> is_Inl(**A, nth(x,env), nth(z,env))" |
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by (simp add: Inl_fm_def is_Inl_def) |
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|
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lemma Inl_iff_sats: |
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"[| nth(i,env) = x; nth(k,env) = z; |
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i \<in> nat; k \<in> nat; env \<in> list(A)|] |
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==> is_Inl(**A, x, z) <-> sats(A, Inl_fm(i,k), env)" |
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by simp |
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theorem Inl_reflection: |
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"REFLECTS[\<lambda>x. is_Inl(L,f(x),h(x)), |
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\<lambda>i x. is_Inl(**Lset(i),f(x),h(x))]" |
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apply (simp only: is_Inl_def setclass_simps) |
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apply (intro FOL_reflections function_reflections) |
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done |
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|
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parents:
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subsubsection{*The Formula @{term is_Inr}, Internalized*} |
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(* is_Inr(M,a,z) == \<exists>n1[M]. number1(M,n1) & pair(M,n1,a,z) *) |
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constdefs Inr_fm :: "[i,i]=>i" |
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"Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))" |
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|
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lemma Inr_type [TC]: |
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"[| x \<in> nat; z \<in> nat |] ==> Inr_fm(x,z) \<in> formula" |
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by (simp add: Inr_fm_def) |
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|
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lemma sats_Inr_fm [simp]: |
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"[| x \<in> nat; z \<in> nat; env \<in> list(A)|] |
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==> sats(A, Inr_fm(x,z), env) <-> is_Inr(**A, nth(x,env), nth(z,env))" |
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by (simp add: Inr_fm_def is_Inr_def) |
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|
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lemma Inr_iff_sats: |
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"[| nth(i,env) = x; nth(k,env) = z; |
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i \<in> nat; k \<in> nat; env \<in> list(A)|] |
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==> is_Inr(**A, x, z) <-> sats(A, Inr_fm(i,k), env)" |
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by simp |
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|
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theorem Inr_reflection: |
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"REFLECTS[\<lambda>x. is_Inr(L,f(x),h(x)), |
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\<lambda>i x. is_Inr(**Lset(i),f(x),h(x))]" |
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apply (simp only: is_Inr_def setclass_simps) |
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apply (intro FOL_reflections function_reflections) |
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done |
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|
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|
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subsubsection{*The Formula @{term is_Nil}, Internalized*} |
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|
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(* is_Nil(M,xs) == \<exists>zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *) |
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|
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constdefs Nil_fm :: "i=>i" |
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"Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))" |
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|
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lemma Nil_type [TC]: "x \<in> nat ==> Nil_fm(x) \<in> formula" |
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by (simp add: Nil_fm_def) |
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|
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lemma sats_Nil_fm [simp]: |
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"[| x \<in> nat; env \<in> list(A)|] |
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==> sats(A, Nil_fm(x), env) <-> is_Nil(**A, nth(x,env))" |
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by (simp add: Nil_fm_def is_Nil_def) |
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|
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lemma Nil_iff_sats: |
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"[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|] |
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==> is_Nil(**A, x) <-> sats(A, Nil_fm(i), env)" |
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by simp |
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|
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theorem Nil_reflection: |
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"REFLECTS[\<lambda>x. is_Nil(L,f(x)), |
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\<lambda>i x. is_Nil(**Lset(i),f(x))]" |
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apply (simp only: is_Nil_def setclass_simps) |
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apply (intro FOL_reflections function_reflections Inl_reflection) |
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done |
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|
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|
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96 |
subsubsection{*The Formula @{term is_Cons}, Internalized*} |
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|
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|
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(* "is_Cons(M,a,l,Z) == \<exists>p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *) |
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constdefs Cons_fm :: "[i,i,i]=>i" |
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"Cons_fm(a,l,Z) == |
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102 |
Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))" |
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|
103 |
|
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|
104 |
lemma Cons_type [TC]: |
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parents:
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105 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Cons_fm(x,y,z) \<in> formula" |
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parents:
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106 |
by (simp add: Cons_fm_def) |
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107 |
|
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lemma sats_Cons_fm [simp]: |
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109 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
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|
110 |
==> sats(A, Cons_fm(x,y,z), env) <-> |
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|
111 |
is_Cons(**A, nth(x,env), nth(y,env), nth(z,env))" |
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|
112 |
by (simp add: Cons_fm_def is_Cons_def) |
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|
113 |
|
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|
114 |
lemma Cons_iff_sats: |
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paulson
parents:
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changeset
|
115 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
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paulson
parents:
diff
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|
116 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
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In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
117 |
==>is_Cons(**A, x, y, z) <-> sats(A, Cons_fm(i,j,k), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
118 |
by simp |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
119 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
120 |
theorem Cons_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
121 |
"REFLECTS[\<lambda>x. is_Cons(L,f(x),g(x),h(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
122 |
\<lambda>i x. is_Cons(**Lset(i),f(x),g(x),h(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
123 |
apply (simp only: is_Cons_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
124 |
apply (intro FOL_reflections pair_reflection Inr_reflection) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
125 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
126 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
127 |
subsubsection{*The Formula @{term is_quasilist}, Internalized*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
128 |
|
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In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
129 |
(* is_quasilist(M,xs) == is_Nil(M,z) | (\<exists>x[M]. \<exists>l[M]. is_Cons(M,x,l,z))" *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
130 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
131 |
constdefs quasilist_fm :: "i=>i" |
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In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
132 |
"quasilist_fm(x) == |
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In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
133 |
Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
134 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
135 |
lemma quasilist_type [TC]: "x \<in> nat ==> quasilist_fm(x) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
136 |
by (simp add: quasilist_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
137 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
138 |
lemma sats_quasilist_fm [simp]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
139 |
"[| x \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
140 |
==> sats(A, quasilist_fm(x), env) <-> is_quasilist(**A, nth(x,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
141 |
by (simp add: quasilist_fm_def is_quasilist_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
142 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
143 |
lemma quasilist_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
144 |
"[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
145 |
==> is_quasilist(**A, x) <-> sats(A, quasilist_fm(i), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
146 |
by simp |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
147 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
148 |
theorem quasilist_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
149 |
"REFLECTS[\<lambda>x. is_quasilist(L,f(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
150 |
\<lambda>i x. is_quasilist(**Lset(i),f(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
151 |
apply (simp only: is_quasilist_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
152 |
apply (intro FOL_reflections Nil_reflection Cons_reflection) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
153 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
154 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
155 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
156 |
subsection{*Absoluteness for the Function @{term nth}*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
157 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
158 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
159 |
subsubsection{*The Formula @{term is_hd}, Internalized*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
160 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
161 |
(* "is_hd(M,xs,H) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
162 |
(is_Nil(M,xs) --> empty(M,H)) & |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
163 |
(\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | H=x) & |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
164 |
(is_quasilist(M,xs) | empty(M,H))" *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
165 |
constdefs hd_fm :: "[i,i]=>i" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
166 |
"hd_fm(xs,H) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
167 |
And(Implies(Nil_fm(xs), empty_fm(H)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
168 |
And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(H#+2,1)))), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
169 |
Or(quasilist_fm(xs), empty_fm(H))))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
170 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
171 |
lemma hd_type [TC]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
172 |
"[| x \<in> nat; y \<in> nat |] ==> hd_fm(x,y) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
173 |
by (simp add: hd_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
174 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
175 |
lemma sats_hd_fm [simp]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
176 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
177 |
==> sats(A, hd_fm(x,y), env) <-> is_hd(**A, nth(x,env), nth(y,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
178 |
by (simp add: hd_fm_def is_hd_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
179 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
180 |
lemma hd_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
181 |
"[| nth(i,env) = x; nth(j,env) = y; |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
182 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
183 |
==> is_hd(**A, x, y) <-> sats(A, hd_fm(i,j), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
184 |
by simp |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
185 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
186 |
theorem hd_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
187 |
"REFLECTS[\<lambda>x. is_hd(L,f(x),g(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
188 |
\<lambda>i x. is_hd(**Lset(i),f(x),g(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
189 |
apply (simp only: is_hd_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
190 |
apply (intro FOL_reflections Nil_reflection Cons_reflection |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
191 |
quasilist_reflection empty_reflection) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
192 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
193 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
194 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
195 |
subsubsection{*The Formula @{term is_tl}, Internalized*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
196 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
197 |
(* "is_tl(M,xs,T) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
198 |
(is_Nil(M,xs) --> T=xs) & |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
199 |
(\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | T=l) & |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
200 |
(is_quasilist(M,xs) | empty(M,T))" *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
201 |
constdefs tl_fm :: "[i,i]=>i" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
202 |
"tl_fm(xs,T) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
203 |
And(Implies(Nil_fm(xs), Equal(T,xs)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
204 |
And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
205 |
Or(quasilist_fm(xs), empty_fm(T))))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
206 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
207 |
lemma tl_type [TC]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
208 |
"[| x \<in> nat; y \<in> nat |] ==> tl_fm(x,y) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
209 |
by (simp add: tl_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
210 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
211 |
lemma sats_tl_fm [simp]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
212 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
213 |
==> sats(A, tl_fm(x,y), env) <-> is_tl(**A, nth(x,env), nth(y,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
214 |
by (simp add: tl_fm_def is_tl_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
215 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
216 |
lemma tl_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
217 |
"[| nth(i,env) = x; nth(j,env) = y; |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
218 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
219 |
==> is_tl(**A, x, y) <-> sats(A, tl_fm(i,j), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
220 |
by simp |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
221 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
222 |
theorem tl_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
223 |
"REFLECTS[\<lambda>x. is_tl(L,f(x),g(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
224 |
\<lambda>i x. is_tl(**Lset(i),f(x),g(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
225 |
apply (simp only: is_tl_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
226 |
apply (intro FOL_reflections Nil_reflection Cons_reflection |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
227 |
quasilist_reflection empty_reflection) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
228 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
229 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
230 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
231 |
subsubsection{*The Operator @{term is_bool_of_o}*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
232 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
233 |
(* is_bool_of_o :: "[i=>o, o, i] => o" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
234 |
"is_bool_of_o(M,P,z) == (P & number1(M,z)) | (~P & empty(M,z))" *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
235 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
236 |
text{*The formula @{term p} has no free variables.*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
237 |
constdefs bool_of_o_fm :: "[i, i]=>i" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
238 |
"bool_of_o_fm(p,z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
239 |
Or(And(p,number1_fm(z)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
240 |
And(Neg(p),empty_fm(z)))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
241 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
242 |
lemma is_bool_of_o_type [TC]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
243 |
"[| p \<in> formula; z \<in> nat |] ==> bool_of_o_fm(p,z) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
244 |
by (simp add: bool_of_o_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
245 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
246 |
lemma sats_bool_of_o_fm: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
247 |
assumes p_iff_sats: "P <-> sats(A, p, env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
248 |
shows |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
249 |
"[|z \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
250 |
==> sats(A, bool_of_o_fm(p,z), env) <-> |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
251 |
is_bool_of_o(**A, P, nth(z,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
252 |
by (simp add: bool_of_o_fm_def is_bool_of_o_def p_iff_sats [THEN iff_sym]) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
253 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
254 |
lemma is_bool_of_o_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
255 |
"[| P <-> sats(A, p, env); nth(k,env) = z; k \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
256 |
==> is_bool_of_o(**A, P, z) <-> sats(A, bool_of_o_fm(p,k), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
257 |
by (simp add: sats_bool_of_o_fm) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
258 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
259 |
theorem bool_of_o_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
260 |
"REFLECTS [P(L), \<lambda>i. P(**Lset(i))] ==> |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
261 |
REFLECTS[\<lambda>x. is_bool_of_o(L, P(L,x), f(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
262 |
\<lambda>i x. is_bool_of_o(**Lset(i), P(**Lset(i),x), f(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
263 |
apply (simp (no_asm) only: is_bool_of_o_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
264 |
apply (intro FOL_reflections function_reflections, assumption+) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
265 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
266 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
267 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
268 |
subsection{*More Internalizations*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
269 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
270 |
subsubsection{*The Operator @{term is_lambda}*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
271 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
272 |
text{*The two arguments of @{term p} are always 1, 0. Remember that |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
273 |
@{term p} will be enclosed by three quantifiers.*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
274 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
275 |
(* is_lambda :: "[i=>o, i, [i,i]=>o, i] => o" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
276 |
"is_lambda(M, A, is_b, z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
277 |
\<forall>p[M]. p \<in> z <-> |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
278 |
(\<exists>u[M]. \<exists>v[M]. u\<in>A & pair(M,u,v,p) & is_b(u,v))" *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
279 |
constdefs lambda_fm :: "[i, i, i]=>i" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
280 |
"lambda_fm(p,A,z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
281 |
Forall(Iff(Member(0,succ(z)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
282 |
Exists(Exists(And(Member(1,A#+3), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
283 |
And(pair_fm(1,0,2), p))))))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
284 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
285 |
text{*We call @{term p} with arguments x, y by equating them with |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
286 |
the corresponding quantified variables with de Bruijn indices 1, 0.*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
287 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
288 |
lemma is_lambda_type [TC]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
289 |
"[| p \<in> formula; x \<in> nat; y \<in> nat |] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
290 |
==> lambda_fm(p,x,y) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
291 |
by (simp add: lambda_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
292 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
293 |
lemma sats_lambda_fm: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
294 |
assumes is_b_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
295 |
"!!a0 a1 a2. |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
296 |
[|a0\<in>A; a1\<in>A; a2\<in>A|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
297 |
==> is_b(a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,env))))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
298 |
shows |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
299 |
"[|x \<in> nat; y \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
300 |
==> sats(A, lambda_fm(p,x,y), env) <-> |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
301 |
is_lambda(**A, nth(x,env), is_b, nth(y,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
302 |
by (simp add: lambda_fm_def is_lambda_def is_b_iff_sats [THEN iff_sym]) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
303 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
304 |
lemma is_lambda_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
305 |
assumes is_b_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
306 |
"!!a0 a1 a2. |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
307 |
[|a0\<in>A; a1\<in>A; a2\<in>A|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
308 |
==> is_b(a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,env))))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
309 |
shows |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
310 |
"[|nth(i,env) = x; nth(j,env) = y; |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
311 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
312 |
==> is_lambda(**A, x, is_b, y) <-> sats(A, lambda_fm(p,i,j), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
313 |
by (simp add: sats_lambda_fm [OF is_b_iff_sats]) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
314 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
315 |
theorem is_lambda_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
316 |
assumes is_b_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
317 |
"!!f' f g h. REFLECTS[\<lambda>x. is_b(L, f'(x), f(x), g(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
318 |
\<lambda>i x. is_b(**Lset(i), f'(x), f(x), g(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
319 |
shows "REFLECTS[\<lambda>x. is_lambda(L, A(x), is_b(L,x), f(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
320 |
\<lambda>i x. is_lambda(**Lset(i), A(x), is_b(**Lset(i),x), f(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
321 |
apply (simp (no_asm_use) only: is_lambda_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
322 |
apply (intro FOL_reflections is_b_reflection pair_reflection) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
323 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
324 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
325 |
subsubsection{*The Operator @{term is_Member}, Internalized*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
326 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
327 |
(* "is_Member(M,x,y,Z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
328 |
\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inl(M,u,Z)" *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
329 |
constdefs Member_fm :: "[i,i,i]=>i" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
330 |
"Member_fm(x,y,Z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
331 |
Exists(Exists(And(pair_fm(x#+2,y#+2,1), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
332 |
And(Inl_fm(1,0), Inl_fm(0,Z#+2)))))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
333 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
334 |
lemma is_Member_type [TC]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
335 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Member_fm(x,y,z) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
336 |
by (simp add: Member_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
337 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
338 |
lemma sats_Member_fm [simp]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
339 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
340 |
==> sats(A, Member_fm(x,y,z), env) <-> |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
341 |
is_Member(**A, nth(x,env), nth(y,env), nth(z,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
342 |
by (simp add: Member_fm_def is_Member_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
343 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
344 |
lemma Member_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
345 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
346 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
347 |
==> is_Member(**A, x, y, z) <-> sats(A, Member_fm(i,j,k), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
348 |
by (simp add: sats_Member_fm) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
349 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
350 |
theorem Member_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
351 |
"REFLECTS[\<lambda>x. is_Member(L,f(x),g(x),h(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
352 |
\<lambda>i x. is_Member(**Lset(i),f(x),g(x),h(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
353 |
apply (simp only: is_Member_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
354 |
apply (intro FOL_reflections pair_reflection Inl_reflection) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
355 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
356 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
357 |
subsubsection{*The Operator @{term is_Equal}, Internalized*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
358 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
359 |
(* "is_Equal(M,x,y,Z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
360 |
\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inr(M,p,u) & is_Inl(M,u,Z)" *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
361 |
constdefs Equal_fm :: "[i,i,i]=>i" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
362 |
"Equal_fm(x,y,Z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
363 |
Exists(Exists(And(pair_fm(x#+2,y#+2,1), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
364 |
And(Inr_fm(1,0), Inl_fm(0,Z#+2)))))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
365 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
366 |
lemma is_Equal_type [TC]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
367 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Equal_fm(x,y,z) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
368 |
by (simp add: Equal_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
369 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
370 |
lemma sats_Equal_fm [simp]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
371 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
372 |
==> sats(A, Equal_fm(x,y,z), env) <-> |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
373 |
is_Equal(**A, nth(x,env), nth(y,env), nth(z,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
374 |
by (simp add: Equal_fm_def is_Equal_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
375 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
376 |
lemma Equal_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
377 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
378 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
379 |
==> is_Equal(**A, x, y, z) <-> sats(A, Equal_fm(i,j,k), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
380 |
by (simp add: sats_Equal_fm) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
381 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
382 |
theorem Equal_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
383 |
"REFLECTS[\<lambda>x. is_Equal(L,f(x),g(x),h(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
384 |
\<lambda>i x. is_Equal(**Lset(i),f(x),g(x),h(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
385 |
apply (simp only: is_Equal_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
386 |
apply (intro FOL_reflections pair_reflection Inl_reflection Inr_reflection) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
387 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
388 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
389 |
subsubsection{*The Operator @{term is_Nand}, Internalized*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
390 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
391 |
(* "is_Nand(M,x,y,Z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
392 |
\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inr(M,u,Z)" *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
393 |
constdefs Nand_fm :: "[i,i,i]=>i" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
394 |
"Nand_fm(x,y,Z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
395 |
Exists(Exists(And(pair_fm(x#+2,y#+2,1), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
396 |
And(Inl_fm(1,0), Inr_fm(0,Z#+2)))))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
397 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
398 |
lemma is_Nand_type [TC]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
399 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Nand_fm(x,y,z) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
400 |
by (simp add: Nand_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
401 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
402 |
lemma sats_Nand_fm [simp]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
403 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
404 |
==> sats(A, Nand_fm(x,y,z), env) <-> |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
405 |
is_Nand(**A, nth(x,env), nth(y,env), nth(z,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
406 |
by (simp add: Nand_fm_def is_Nand_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
407 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
408 |
lemma Nand_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
409 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
410 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
411 |
==> is_Nand(**A, x, y, z) <-> sats(A, Nand_fm(i,j,k), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
412 |
by (simp add: sats_Nand_fm) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
413 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
414 |
theorem Nand_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
415 |
"REFLECTS[\<lambda>x. is_Nand(L,f(x),g(x),h(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
416 |
\<lambda>i x. is_Nand(**Lset(i),f(x),g(x),h(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
417 |
apply (simp only: is_Nand_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
418 |
apply (intro FOL_reflections pair_reflection Inl_reflection Inr_reflection) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
419 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
420 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
421 |
subsubsection{*The Operator @{term is_Forall}, Internalized*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
422 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
423 |
(* "is_Forall(M,p,Z) == \<exists>u[M]. is_Inr(M,p,u) & is_Inr(M,u,Z)" *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
424 |
constdefs Forall_fm :: "[i,i]=>i" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
425 |
"Forall_fm(x,Z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
426 |
Exists(And(Inr_fm(succ(x),0), Inr_fm(0,succ(Z))))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
427 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
428 |
lemma is_Forall_type [TC]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
429 |
"[| x \<in> nat; y \<in> nat |] ==> Forall_fm(x,y) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
430 |
by (simp add: Forall_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
431 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
432 |
lemma sats_Forall_fm [simp]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
433 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
434 |
==> sats(A, Forall_fm(x,y), env) <-> |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
435 |
is_Forall(**A, nth(x,env), nth(y,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
436 |
by (simp add: Forall_fm_def is_Forall_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
437 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
438 |
lemma Forall_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
439 |
"[| nth(i,env) = x; nth(j,env) = y; |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
440 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
441 |
==> is_Forall(**A, x, y) <-> sats(A, Forall_fm(i,j), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
442 |
by (simp add: sats_Forall_fm) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
443 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
444 |
theorem Forall_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
445 |
"REFLECTS[\<lambda>x. is_Forall(L,f(x),g(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
446 |
\<lambda>i x. is_Forall(**Lset(i),f(x),g(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
447 |
apply (simp only: is_Forall_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
448 |
apply (intro FOL_reflections pair_reflection Inr_reflection) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
449 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
450 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
451 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
452 |
subsubsection{*The Operator @{term is_and}, Internalized*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
453 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
454 |
(* is_and(M,a,b,z) == (number1(M,a) & z=b) | |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
455 |
(~number1(M,a) & empty(M,z)) *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
456 |
constdefs and_fm :: "[i,i,i]=>i" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
457 |
"and_fm(a,b,z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
458 |
Or(And(number1_fm(a), Equal(z,b)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
459 |
And(Neg(number1_fm(a)),empty_fm(z)))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
460 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
461 |
lemma is_and_type [TC]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
462 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> and_fm(x,y,z) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
463 |
by (simp add: and_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
464 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
465 |
lemma sats_and_fm [simp]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
466 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
467 |
==> sats(A, and_fm(x,y,z), env) <-> |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
468 |
is_and(**A, nth(x,env), nth(y,env), nth(z,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
469 |
by (simp add: and_fm_def is_and_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
470 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
471 |
lemma is_and_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
472 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
473 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
474 |
==> is_and(**A, x, y, z) <-> sats(A, and_fm(i,j,k), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
475 |
by simp |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
476 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
477 |
theorem is_and_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
478 |
"REFLECTS[\<lambda>x. is_and(L,f(x),g(x),h(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
479 |
\<lambda>i x. is_and(**Lset(i),f(x),g(x),h(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
480 |
apply (simp only: is_and_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
481 |
apply (intro FOL_reflections function_reflections) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
482 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
483 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
484 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
485 |
subsubsection{*The Operator @{term is_or}, Internalized*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
486 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
487 |
(* is_or(M,a,b,z) == (number1(M,a) & number1(M,z)) | |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
488 |
(~number1(M,a) & z=b) *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
489 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
490 |
constdefs or_fm :: "[i,i,i]=>i" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
491 |
"or_fm(a,b,z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
492 |
Or(And(number1_fm(a), number1_fm(z)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
493 |
And(Neg(number1_fm(a)), Equal(z,b)))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
494 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
495 |
lemma is_or_type [TC]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
496 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> or_fm(x,y,z) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
497 |
by (simp add: or_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
498 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
499 |
lemma sats_or_fm [simp]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
500 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
501 |
==> sats(A, or_fm(x,y,z), env) <-> |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
502 |
is_or(**A, nth(x,env), nth(y,env), nth(z,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
503 |
by (simp add: or_fm_def is_or_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
504 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
505 |
lemma is_or_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
506 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
507 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
508 |
==> is_or(**A, x, y, z) <-> sats(A, or_fm(i,j,k), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
509 |
by simp |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
510 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
511 |
theorem is_or_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
512 |
"REFLECTS[\<lambda>x. is_or(L,f(x),g(x),h(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
513 |
\<lambda>i x. is_or(**Lset(i),f(x),g(x),h(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
514 |
apply (simp only: is_or_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
515 |
apply (intro FOL_reflections function_reflections) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
516 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
517 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
518 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
519 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
520 |
subsubsection{*The Operator @{term is_not}, Internalized*} |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
521 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
522 |
(* is_not(M,a,z) == (number1(M,a) & empty(M,z)) | |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
523 |
(~number1(M,a) & number1(M,z)) *) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
524 |
constdefs not_fm :: "[i,i]=>i" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
525 |
"not_fm(a,z) == |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
526 |
Or(And(number1_fm(a), empty_fm(z)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
527 |
And(Neg(number1_fm(a)), number1_fm(z)))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
528 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
529 |
lemma is_not_type [TC]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
530 |
"[| x \<in> nat; z \<in> nat |] ==> not_fm(x,z) \<in> formula" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
531 |
by (simp add: not_fm_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
532 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
533 |
lemma sats_is_not_fm [simp]: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
534 |
"[| x \<in> nat; z \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
535 |
==> sats(A, not_fm(x,z), env) <-> is_not(**A, nth(x,env), nth(z,env))" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
536 |
by (simp add: not_fm_def is_not_def) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
537 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
538 |
lemma is_not_iff_sats: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
539 |
"[| nth(i,env) = x; nth(k,env) = z; |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
540 |
i \<in> nat; k \<in> nat; env \<in> list(A)|] |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
541 |
==> is_not(**A, x, z) <-> sats(A, not_fm(i,k), env)" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
542 |
by simp |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
543 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
544 |
theorem is_not_reflection: |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
545 |
"REFLECTS[\<lambda>x. is_not(L,f(x),g(x)), |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
546 |
\<lambda>i x. is_not(**Lset(i),f(x),g(x))]" |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
547 |
apply (simp only: is_not_def setclass_simps) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
548 |
apply (intro FOL_reflections function_reflections) |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
549 |
done |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
550 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
551 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
552 |
lemmas extra_reflections = |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
553 |
Inl_reflection Inr_reflection Nil_reflection Cons_reflection |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
554 |
quasilist_reflection hd_reflection tl_reflection bool_of_o_reflection |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
555 |
is_lambda_reflection Member_reflection Equal_reflection Nand_reflection |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
556 |
Forall_reflection is_and_reflection is_or_reflection is_not_reflection |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
557 |
|
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
558 |
lemmas extra_iff_sats = |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
559 |
Inl_iff_sats Inr_iff_sats Nil_iff_sats Cons_iff_sats quasilist_iff_sats |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
560 |
hd_iff_sats tl_iff_sats is_bool_of_o_iff_sats is_lambda_iff_sats |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
561 |
Member_iff_sats Equal_iff_sats Nand_iff_sats Forall_iff_sats |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
562 |
is_and_iff_sats is_or_iff_sats is_not_iff_sats |
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
563 |
|
13503 | 564 |
|
565 |
subsection{*Well-Founded Recursion!*} |
|
566 |
||
13506 | 567 |
subsubsection{*The Operator @{term M_is_recfun}*} |
13503 | 568 |
|
569 |
text{*Alternative definition, minimizing nesting of quantifiers around MH*} |
|
570 |
lemma M_is_recfun_iff: |
|
571 |
"M_is_recfun(M,MH,r,a,f) <-> |
|
572 |
(\<forall>z[M]. z \<in> f <-> |
|
573 |
(\<exists>x[M]. \<exists>f_r_sx[M]. \<exists>y[M]. |
|
574 |
MH(x, f_r_sx, y) & pair(M,x,y,z) & |
|
575 |
(\<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. |
|
576 |
pair(M,x,a,xa) & upair(M,x,x,sx) & |
|
577 |
pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) & |
|
578 |
xa \<in> r)))" |
|
579 |
apply (simp add: M_is_recfun_def) |
|
580 |
apply (rule rall_cong, blast) |
|
581 |
done |
|
582 |
||
583 |
||
584 |
(* M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o" |
|
585 |
"M_is_recfun(M,MH,r,a,f) == |
|
586 |
\<forall>z[M]. z \<in> f <-> |
|
587 |
2 1 0 |
|
588 |
new def (\<exists>x[M]. \<exists>f_r_sx[M]. \<exists>y[M]. |
|
589 |
MH(x, f_r_sx, y) & pair(M,x,y,z) & |
|
590 |
(\<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. |
|
591 |
pair(M,x,a,xa) & upair(M,x,x,sx) & |
|
592 |
pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) & |
|
593 |
xa \<in> r)" |
|
594 |
*) |
|
595 |
||
596 |
text{*The three arguments of @{term p} are always 2, 1, 0 and z*} |
|
597 |
constdefs is_recfun_fm :: "[i, i, i, i]=>i" |
|
598 |
"is_recfun_fm(p,r,a,f) == |
|
599 |
Forall(Iff(Member(0,succ(f)), |
|
600 |
Exists(Exists(Exists( |
|
601 |
And(p, |
|
602 |
And(pair_fm(2,0,3), |
|
603 |
Exists(Exists(Exists( |
|
604 |
And(pair_fm(5,a#+7,2), |
|
605 |
And(upair_fm(5,5,1), |
|
606 |
And(pre_image_fm(r#+7,1,0), |
|
607 |
And(restriction_fm(f#+7,0,4), Member(2,r#+7)))))))))))))))" |
|
608 |
||
609 |
lemma is_recfun_type [TC]: |
|
610 |
"[| p \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |] |
|
611 |
==> is_recfun_fm(p,x,y,z) \<in> formula" |
|
612 |
by (simp add: is_recfun_fm_def) |
|
613 |
||
614 |
||
615 |
lemma sats_is_recfun_fm: |
|
616 |
assumes MH_iff_sats: |
|
617 |
"!!a0 a1 a2 a3. |
|
618 |
[|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A|] |
|
619 |
==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))" |
|
620 |
shows |
|
621 |
"[|x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
622 |
==> sats(A, is_recfun_fm(p,x,y,z), env) <-> |
|
623 |
M_is_recfun(**A, MH, nth(x,env), nth(y,env), nth(z,env))" |
|
624 |
by (simp add: is_recfun_fm_def M_is_recfun_iff MH_iff_sats [THEN iff_sym]) |
|
625 |
||
626 |
lemma is_recfun_iff_sats: |
|
627 |
assumes MH_iff_sats: |
|
628 |
"!!a0 a1 a2 a3. |
|
629 |
[|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A|] |
|
630 |
==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))" |
|
631 |
shows |
|
632 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
633 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
634 |
==> M_is_recfun(**A, MH, x, y, z) <-> sats(A, is_recfun_fm(p,i,j,k), env)" |
|
635 |
by (simp add: sats_is_recfun_fm [OF MH_iff_sats]) |
|
636 |
||
637 |
text{*The additional variable in the premise, namely @{term f'}, is essential. |
|
638 |
It lets @{term MH} depend upon @{term x}, which seems often necessary. |
|
639 |
The same thing occurs in @{text is_wfrec_reflection}.*} |
|
640 |
theorem is_recfun_reflection: |
|
641 |
assumes MH_reflection: |
|
642 |
"!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)), |
|
643 |
\<lambda>i x. MH(**Lset(i), f'(x), f(x), g(x), h(x))]" |
|
644 |
shows "REFLECTS[\<lambda>x. M_is_recfun(L, MH(L,x), f(x), g(x), h(x)), |
|
645 |
\<lambda>i x. M_is_recfun(**Lset(i), MH(**Lset(i),x), f(x), g(x), h(x))]" |
|
646 |
apply (simp (no_asm_use) only: M_is_recfun_def setclass_simps) |
|
647 |
apply (intro FOL_reflections function_reflections |
|
648 |
restriction_reflection MH_reflection) |
|
649 |
done |
|
650 |
||
651 |
subsubsection{*The Operator @{term is_wfrec}*} |
|
652 |
||
653 |
text{*The three arguments of @{term p} are always 2, 1, 0*} |
|
654 |
||
655 |
(* is_wfrec :: "[i=>o, i, [i,i,i]=>o, i, i] => o" |
|
656 |
"is_wfrec(M,MH,r,a,z) == |
|
657 |
\<exists>f[M]. M_is_recfun(M,MH,r,a,f) & MH(a,f,z)" *) |
|
658 |
constdefs is_wfrec_fm :: "[i, i, i, i]=>i" |
|
659 |
"is_wfrec_fm(p,r,a,z) == |
|
660 |
Exists(And(is_recfun_fm(p, succ(r), succ(a), 0), |
|
661 |
Exists(Exists(Exists(Exists( |
|
662 |
And(Equal(2,a#+5), And(Equal(1,4), And(Equal(0,z#+5), p)))))))))" |
|
663 |
||
664 |
text{*We call @{term p} with arguments a, f, z by equating them with |
|
665 |
the corresponding quantified variables with de Bruijn indices 2, 1, 0.*} |
|
666 |
||
667 |
text{*There's an additional existential quantifier to ensure that the |
|
668 |
environments in both calls to MH have the same length.*} |
|
669 |
||
670 |
lemma is_wfrec_type [TC]: |
|
671 |
"[| p \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |] |
|
672 |
==> is_wfrec_fm(p,x,y,z) \<in> formula" |
|
673 |
by (simp add: is_wfrec_fm_def) |
|
674 |
||
675 |
lemma sats_is_wfrec_fm: |
|
676 |
assumes MH_iff_sats: |
|
677 |
"!!a0 a1 a2 a3 a4. |
|
678 |
[|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A|] |
|
679 |
==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))" |
|
680 |
shows |
|
681 |
"[|x \<in> nat; y < length(env); z < length(env); env \<in> list(A)|] |
|
682 |
==> sats(A, is_wfrec_fm(p,x,y,z), env) <-> |
|
683 |
is_wfrec(**A, MH, nth(x,env), nth(y,env), nth(z,env))" |
|
684 |
apply (frule_tac x=z in lt_length_in_nat, assumption) |
|
685 |
apply (frule lt_length_in_nat, assumption) |
|
686 |
apply (simp add: is_wfrec_fm_def sats_is_recfun_fm is_wfrec_def MH_iff_sats [THEN iff_sym], blast) |
|
687 |
done |
|
688 |
||
689 |
||
690 |
lemma is_wfrec_iff_sats: |
|
691 |
assumes MH_iff_sats: |
|
692 |
"!!a0 a1 a2 a3 a4. |
|
693 |
[|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A|] |
|
694 |
==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))" |
|
695 |
shows |
|
696 |
"[|nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
697 |
i \<in> nat; j < length(env); k < length(env); env \<in> list(A)|] |
|
698 |
==> is_wfrec(**A, MH, x, y, z) <-> sats(A, is_wfrec_fm(p,i,j,k), env)" |
|
699 |
by (simp add: sats_is_wfrec_fm [OF MH_iff_sats]) |
|
700 |
||
701 |
theorem is_wfrec_reflection: |
|
702 |
assumes MH_reflection: |
|
703 |
"!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)), |
|
704 |
\<lambda>i x. MH(**Lset(i), f'(x), f(x), g(x), h(x))]" |
|
705 |
shows "REFLECTS[\<lambda>x. is_wfrec(L, MH(L,x), f(x), g(x), h(x)), |
|
706 |
\<lambda>i x. is_wfrec(**Lset(i), MH(**Lset(i),x), f(x), g(x), h(x))]" |
|
707 |
apply (simp (no_asm_use) only: is_wfrec_def setclass_simps) |
|
708 |
apply (intro FOL_reflections MH_reflection is_recfun_reflection) |
|
709 |
done |
|
710 |
||
711 |
||
712 |
subsection{*For Datatypes*} |
|
713 |
||
714 |
subsubsection{*Binary Products, Internalized*} |
|
715 |
||
716 |
constdefs cartprod_fm :: "[i,i,i]=>i" |
|
717 |
(* "cartprod(M,A,B,z) == |
|
718 |
\<forall>u[M]. u \<in> z <-> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,u)))" *) |
|
719 |
"cartprod_fm(A,B,z) == |
|
720 |
Forall(Iff(Member(0,succ(z)), |
|
721 |
Exists(And(Member(0,succ(succ(A))), |
|
722 |
Exists(And(Member(0,succ(succ(succ(B)))), |
|
723 |
pair_fm(1,0,2)))))))" |
|
724 |
||
725 |
lemma cartprod_type [TC]: |
|
726 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cartprod_fm(x,y,z) \<in> formula" |
|
727 |
by (simp add: cartprod_fm_def) |
|
728 |
||
729 |
lemma arity_cartprod_fm [simp]: |
|
730 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
731 |
==> arity(cartprod_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
732 |
by (simp add: cartprod_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
733 |
||
734 |
lemma sats_cartprod_fm [simp]: |
|
735 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
736 |
==> sats(A, cartprod_fm(x,y,z), env) <-> |
|
737 |
cartprod(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
738 |
by (simp add: cartprod_fm_def cartprod_def) |
|
739 |
||
740 |
lemma cartprod_iff_sats: |
|
741 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
742 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
743 |
==> cartprod(**A, x, y, z) <-> sats(A, cartprod_fm(i,j,k), env)" |
|
744 |
by (simp add: sats_cartprod_fm) |
|
745 |
||
746 |
theorem cartprod_reflection: |
|
747 |
"REFLECTS[\<lambda>x. cartprod(L,f(x),g(x),h(x)), |
|
748 |
\<lambda>i x. cartprod(**Lset(i),f(x),g(x),h(x))]" |
|
749 |
apply (simp only: cartprod_def setclass_simps) |
|
750 |
apply (intro FOL_reflections pair_reflection) |
|
751 |
done |
|
752 |
||
753 |
||
754 |
subsubsection{*Binary Sums, Internalized*} |
|
755 |
||
756 |
(* "is_sum(M,A,B,Z) == |
|
757 |
\<exists>A0[M]. \<exists>n1[M]. \<exists>s1[M]. \<exists>B1[M]. |
|
758 |
3 2 1 0 |
|
759 |
number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) & |
|
760 |
cartprod(M,s1,B,B1) & union(M,A0,B1,Z)" *) |
|
761 |
constdefs sum_fm :: "[i,i,i]=>i" |
|
762 |
"sum_fm(A,B,Z) == |
|
763 |
Exists(Exists(Exists(Exists( |
|
764 |
And(number1_fm(2), |
|
765 |
And(cartprod_fm(2,A#+4,3), |
|
766 |
And(upair_fm(2,2,1), |
|
767 |
And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))" |
|
768 |
||
769 |
lemma sum_type [TC]: |
|
770 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> sum_fm(x,y,z) \<in> formula" |
|
771 |
by (simp add: sum_fm_def) |
|
772 |
||
773 |
lemma arity_sum_fm [simp]: |
|
774 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
775 |
==> arity(sum_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
776 |
by (simp add: sum_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
777 |
||
778 |
lemma sats_sum_fm [simp]: |
|
779 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
780 |
==> sats(A, sum_fm(x,y,z), env) <-> |
|
781 |
is_sum(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
782 |
by (simp add: sum_fm_def is_sum_def) |
|
783 |
||
784 |
lemma sum_iff_sats: |
|
785 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
786 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
787 |
==> is_sum(**A, x, y, z) <-> sats(A, sum_fm(i,j,k), env)" |
|
788 |
by simp |
|
789 |
||
790 |
theorem sum_reflection: |
|
791 |
"REFLECTS[\<lambda>x. is_sum(L,f(x),g(x),h(x)), |
|
792 |
\<lambda>i x. is_sum(**Lset(i),f(x),g(x),h(x))]" |
|
793 |
apply (simp only: is_sum_def setclass_simps) |
|
794 |
apply (intro FOL_reflections function_reflections cartprod_reflection) |
|
795 |
done |
|
796 |
||
797 |
||
798 |
subsubsection{*The Operator @{term quasinat}*} |
|
799 |
||
800 |
(* "is_quasinat(M,z) == empty(M,z) | (\<exists>m[M]. successor(M,m,z))" *) |
|
801 |
constdefs quasinat_fm :: "i=>i" |
|
802 |
"quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))" |
|
803 |
||
804 |
lemma quasinat_type [TC]: |
|
805 |
"x \<in> nat ==> quasinat_fm(x) \<in> formula" |
|
806 |
by (simp add: quasinat_fm_def) |
|
807 |
||
808 |
lemma arity_quasinat_fm [simp]: |
|
809 |
"x \<in> nat ==> arity(quasinat_fm(x)) = succ(x)" |
|
810 |
by (simp add: quasinat_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
811 |
||
812 |
lemma sats_quasinat_fm [simp]: |
|
813 |
"[| x \<in> nat; env \<in> list(A)|] |
|
814 |
==> sats(A, quasinat_fm(x), env) <-> is_quasinat(**A, nth(x,env))" |
|
815 |
by (simp add: quasinat_fm_def is_quasinat_def) |
|
816 |
||
817 |
lemma quasinat_iff_sats: |
|
818 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
819 |
i \<in> nat; env \<in> list(A)|] |
|
820 |
==> is_quasinat(**A, x) <-> sats(A, quasinat_fm(i), env)" |
|
821 |
by simp |
|
822 |
||
823 |
theorem quasinat_reflection: |
|
824 |
"REFLECTS[\<lambda>x. is_quasinat(L,f(x)), |
|
825 |
\<lambda>i x. is_quasinat(**Lset(i),f(x))]" |
|
826 |
apply (simp only: is_quasinat_def setclass_simps) |
|
827 |
apply (intro FOL_reflections function_reflections) |
|
828 |
done |
|
829 |
||
830 |
||
831 |
subsubsection{*The Operator @{term is_nat_case}*} |
|
832 |
text{*I could not get it to work with the more natural assumption that |
|
833 |
@{term is_b} takes two arguments. Instead it must be a formula where 1 and 0 |
|
834 |
stand for @{term m} and @{term b}, respectively.*} |
|
835 |
||
836 |
(* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o" |
|
837 |
"is_nat_case(M, a, is_b, k, z) == |
|
838 |
(empty(M,k) --> z=a) & |
|
839 |
(\<forall>m[M]. successor(M,m,k) --> is_b(m,z)) & |
|
840 |
(is_quasinat(M,k) | empty(M,z))" *) |
|
841 |
text{*The formula @{term is_b} has free variables 1 and 0.*} |
|
842 |
constdefs is_nat_case_fm :: "[i, i, i, i]=>i" |
|
843 |
"is_nat_case_fm(a,is_b,k,z) == |
|
844 |
And(Implies(empty_fm(k), Equal(z,a)), |
|
845 |
And(Forall(Implies(succ_fm(0,succ(k)), |
|
846 |
Forall(Implies(Equal(0,succ(succ(z))), is_b)))), |
|
847 |
Or(quasinat_fm(k), empty_fm(z))))" |
|
848 |
||
849 |
lemma is_nat_case_type [TC]: |
|
850 |
"[| is_b \<in> formula; |
|
851 |
x \<in> nat; y \<in> nat; z \<in> nat |] |
|
852 |
==> is_nat_case_fm(x,is_b,y,z) \<in> formula" |
|
853 |
by (simp add: is_nat_case_fm_def) |
|
854 |
||
855 |
lemma sats_is_nat_case_fm: |
|
856 |
assumes is_b_iff_sats: |
|
857 |
"!!a. a \<in> A ==> is_b(a,nth(z, env)) <-> |
|
858 |
sats(A, p, Cons(nth(z,env), Cons(a, env)))" |
|
859 |
shows |
|
860 |
"[|x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|] |
|
861 |
==> sats(A, is_nat_case_fm(x,p,y,z), env) <-> |
|
862 |
is_nat_case(**A, nth(x,env), is_b, nth(y,env), nth(z,env))" |
|
863 |
apply (frule lt_length_in_nat, assumption) |
|
864 |
apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym]) |
|
865 |
done |
|
866 |
||
867 |
lemma is_nat_case_iff_sats: |
|
868 |
"[| (!!a. a \<in> A ==> is_b(a,z) <-> |
|
869 |
sats(A, p, Cons(z, Cons(a,env)))); |
|
870 |
nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
871 |
i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|] |
|
872 |
==> is_nat_case(**A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i,p,j,k), env)" |
|
873 |
by (simp add: sats_is_nat_case_fm [of A is_b]) |
|
874 |
||
875 |
||
876 |
text{*The second argument of @{term is_b} gives it direct access to @{term x}, |
|
877 |
which is essential for handling free variable references. Without this |
|
878 |
argument, we cannot prove reflection for @{term iterates_MH}.*} |
|
879 |
theorem is_nat_case_reflection: |
|
880 |
assumes is_b_reflection: |
|
881 |
"!!h f g. REFLECTS[\<lambda>x. is_b(L, h(x), f(x), g(x)), |
|
882 |
\<lambda>i x. is_b(**Lset(i), h(x), f(x), g(x))]" |
|
883 |
shows "REFLECTS[\<lambda>x. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)), |
|
884 |
\<lambda>i x. is_nat_case(**Lset(i), f(x), is_b(**Lset(i), x), g(x), h(x))]" |
|
885 |
apply (simp (no_asm_use) only: is_nat_case_def setclass_simps) |
|
886 |
apply (intro FOL_reflections function_reflections |
|
887 |
restriction_reflection is_b_reflection quasinat_reflection) |
|
888 |
done |
|
889 |
||
890 |
||
891 |
subsection{*The Operator @{term iterates_MH}, Needed for Iteration*} |
|
892 |
||
893 |
(* iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o" |
|
894 |
"iterates_MH(M,isF,v,n,g,z) == |
|
895 |
is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u), |
|
896 |
n, z)" *) |
|
897 |
constdefs iterates_MH_fm :: "[i, i, i, i, i]=>i" |
|
898 |
"iterates_MH_fm(isF,v,n,g,z) == |
|
899 |
is_nat_case_fm(v, |
|
900 |
Exists(And(fun_apply_fm(succ(succ(succ(g))),2,0), |
|
901 |
Forall(Implies(Equal(0,2), isF)))), |
|
902 |
n, z)" |
|
903 |
||
904 |
lemma iterates_MH_type [TC]: |
|
905 |
"[| p \<in> formula; |
|
906 |
v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |] |
|
907 |
==> iterates_MH_fm(p,v,x,y,z) \<in> formula" |
|
908 |
by (simp add: iterates_MH_fm_def) |
|
909 |
||
910 |
lemma sats_iterates_MH_fm: |
|
911 |
assumes is_F_iff_sats: |
|
912 |
"!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|] |
|
913 |
==> is_F(a,b) <-> |
|
914 |
sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env)))))" |
|
915 |
shows |
|
916 |
"[|v \<in> nat; x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|] |
|
917 |
==> sats(A, iterates_MH_fm(p,v,x,y,z), env) <-> |
|
918 |
iterates_MH(**A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))" |
|
919 |
apply (frule lt_length_in_nat, assumption) |
|
920 |
apply (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm |
|
921 |
is_F_iff_sats [symmetric]) |
|
922 |
apply (rule is_nat_case_cong) |
|
923 |
apply (simp_all add: setclass_def) |
|
924 |
done |
|
925 |
||
926 |
lemma iterates_MH_iff_sats: |
|
927 |
assumes is_F_iff_sats: |
|
928 |
"!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|] |
|
929 |
==> is_F(a,b) <-> |
|
930 |
sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env)))))" |
|
931 |
shows |
|
932 |
"[| nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
933 |
i' \<in> nat; i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|] |
|
934 |
==> iterates_MH(**A, is_F, v, x, y, z) <-> |
|
935 |
sats(A, iterates_MH_fm(p,i',i,j,k), env)" |
|
936 |
by (simp add: sats_iterates_MH_fm [OF is_F_iff_sats]) |
|
937 |
||
938 |
text{*The second argument of @{term p} gives it direct access to @{term x}, |
|
939 |
which is essential for handling free variable references. Without this |
|
940 |
argument, we cannot prove reflection for @{term list_N}.*} |
|
941 |
theorem iterates_MH_reflection: |
|
942 |
assumes p_reflection: |
|
943 |
"!!f g h. REFLECTS[\<lambda>x. p(L, h(x), f(x), g(x)), |
|
944 |
\<lambda>i x. p(**Lset(i), h(x), f(x), g(x))]" |
|
945 |
shows "REFLECTS[\<lambda>x. iterates_MH(L, p(L,x), e(x), f(x), g(x), h(x)), |
|
946 |
\<lambda>i x. iterates_MH(**Lset(i), p(**Lset(i),x), e(x), f(x), g(x), h(x))]" |
|
947 |
apply (simp (no_asm_use) only: iterates_MH_def) |
|
948 |
txt{*Must be careful: simplifying with @{text setclass_simps} above would |
|
949 |
change @{text "\<exists>gm[**Lset(i)]"} into @{text "\<exists>gm \<in> Lset(i)"}, when |
|
950 |
it would no longer match rule @{text is_nat_case_reflection}. *} |
|
951 |
apply (rule is_nat_case_reflection) |
|
952 |
apply (simp (no_asm_use) only: setclass_simps) |
|
953 |
apply (intro FOL_reflections function_reflections is_nat_case_reflection |
|
954 |
restriction_reflection p_reflection) |
|
955 |
done |
|
956 |
||
957 |
||
958 |
||
959 |
subsubsection{*The Formula @{term is_eclose_n}, Internalized*} |
|
960 |
||
961 |
(* is_eclose_n(M,A,n,Z) == |
|
962 |
\<exists>sn[M]. \<exists>msn[M]. |
|
963 |
1 0 |
|
964 |
successor(M,n,sn) & membership(M,sn,msn) & |
|
965 |
is_wfrec(M, iterates_MH(M, big_union(M), A), msn, n, Z) *) |
|
966 |
||
967 |
constdefs eclose_n_fm :: "[i,i,i]=>i" |
|
968 |
"eclose_n_fm(A,n,Z) == |
|
969 |
Exists(Exists( |
|
970 |
And(succ_fm(n#+2,1), |
|
971 |
And(Memrel_fm(1,0), |
|
972 |
is_wfrec_fm(iterates_MH_fm(big_union_fm(1,0), |
|
973 |
A#+7, 2, 1, 0), |
|
974 |
0, n#+2, Z#+2)))))" |
|
975 |
||
976 |
lemma eclose_n_fm_type [TC]: |
|
977 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> eclose_n_fm(x,y,z) \<in> formula" |
|
978 |
by (simp add: eclose_n_fm_def) |
|
979 |
||
980 |
lemma sats_eclose_n_fm [simp]: |
|
981 |
"[| x \<in> nat; y < length(env); z < length(env); env \<in> list(A)|] |
|
982 |
==> sats(A, eclose_n_fm(x,y,z), env) <-> |
|
983 |
is_eclose_n(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
984 |
apply (frule_tac x=z in lt_length_in_nat, assumption) |
|
985 |
apply (frule_tac x=y in lt_length_in_nat, assumption) |
|
986 |
apply (simp add: eclose_n_fm_def is_eclose_n_def sats_is_wfrec_fm |
|
987 |
sats_iterates_MH_fm) |
|
988 |
done |
|
989 |
||
990 |
lemma eclose_n_iff_sats: |
|
991 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
992 |
i \<in> nat; j < length(env); k < length(env); env \<in> list(A)|] |
|
993 |
==> is_eclose_n(**A, x, y, z) <-> sats(A, eclose_n_fm(i,j,k), env)" |
|
994 |
by (simp add: sats_eclose_n_fm) |
|
995 |
||
996 |
theorem eclose_n_reflection: |
|
997 |
"REFLECTS[\<lambda>x. is_eclose_n(L, f(x), g(x), h(x)), |
|
998 |
\<lambda>i x. is_eclose_n(**Lset(i), f(x), g(x), h(x))]" |
|
999 |
apply (simp only: is_eclose_n_def setclass_simps) |
|
1000 |
apply (intro FOL_reflections function_reflections is_wfrec_reflection |
|
1001 |
iterates_MH_reflection) |
|
1002 |
done |
|
1003 |
||
1004 |
||
1005 |
subsubsection{*Membership in @{term "eclose(A)"}*} |
|
1006 |
||
1007 |
(* mem_eclose(M,A,l) == |
|
1008 |
\<exists>n[M]. \<exists>eclosen[M]. |
|
1009 |
finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l \<in> eclosen *) |
|
1010 |
constdefs mem_eclose_fm :: "[i,i]=>i" |
|
1011 |
"mem_eclose_fm(x,y) == |
|
1012 |
Exists(Exists( |
|
1013 |
And(finite_ordinal_fm(1), |
|
1014 |
And(eclose_n_fm(x#+2,1,0), Member(y#+2,0)))))" |
|
1015 |
||
1016 |
lemma mem_eclose_type [TC]: |
|
1017 |
"[| x \<in> nat; y \<in> nat |] ==> mem_eclose_fm(x,y) \<in> formula" |
|
1018 |
by (simp add: mem_eclose_fm_def) |
|
1019 |
||
1020 |
lemma sats_mem_eclose_fm [simp]: |
|
1021 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
1022 |
==> sats(A, mem_eclose_fm(x,y), env) <-> mem_eclose(**A, nth(x,env), nth(y,env))" |
|
1023 |
by (simp add: mem_eclose_fm_def mem_eclose_def) |
|
1024 |
||
1025 |
lemma mem_eclose_iff_sats: |
|
1026 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
1027 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
|
1028 |
==> mem_eclose(**A, x, y) <-> sats(A, mem_eclose_fm(i,j), env)" |
|
1029 |
by simp |
|
1030 |
||
1031 |
theorem mem_eclose_reflection: |
|
1032 |
"REFLECTS[\<lambda>x. mem_eclose(L,f(x),g(x)), |
|
1033 |
\<lambda>i x. mem_eclose(**Lset(i),f(x),g(x))]" |
|
1034 |
apply (simp only: mem_eclose_def setclass_simps) |
|
1035 |
apply (intro FOL_reflections finite_ordinal_reflection eclose_n_reflection) |
|
1036 |
done |
|
1037 |
||
1038 |
||
1039 |
subsubsection{*The Predicate ``Is @{term "eclose(A)"}''*} |
|
1040 |
||
1041 |
(* is_eclose(M,A,Z) == \<forall>l[M]. l \<in> Z <-> mem_eclose(M,A,l) *) |
|
1042 |
constdefs is_eclose_fm :: "[i,i]=>i" |
|
1043 |
"is_eclose_fm(A,Z) == |
|
1044 |
Forall(Iff(Member(0,succ(Z)), mem_eclose_fm(succ(A),0)))" |
|
1045 |
||
1046 |
lemma is_eclose_type [TC]: |
|
1047 |
"[| x \<in> nat; y \<in> nat |] ==> is_eclose_fm(x,y) \<in> formula" |
|
1048 |
by (simp add: is_eclose_fm_def) |
|
1049 |
||
1050 |
lemma sats_is_eclose_fm [simp]: |
|
1051 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
1052 |
==> sats(A, is_eclose_fm(x,y), env) <-> is_eclose(**A, nth(x,env), nth(y,env))" |
|
1053 |
by (simp add: is_eclose_fm_def is_eclose_def) |
|
1054 |
||
1055 |
lemma is_eclose_iff_sats: |
|
1056 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
1057 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
|
1058 |
==> is_eclose(**A, x, y) <-> sats(A, is_eclose_fm(i,j), env)" |
|
1059 |
by simp |
|
1060 |
||
1061 |
theorem is_eclose_reflection: |
|
1062 |
"REFLECTS[\<lambda>x. is_eclose(L,f(x),g(x)), |
|
1063 |
\<lambda>i x. is_eclose(**Lset(i),f(x),g(x))]" |
|
1064 |
apply (simp only: is_eclose_def setclass_simps) |
|
1065 |
apply (intro FOL_reflections mem_eclose_reflection) |
|
1066 |
done |
|
1067 |
||
1068 |
||
1069 |
subsubsection{*The List Functor, Internalized*} |
|
1070 |
||
1071 |
constdefs list_functor_fm :: "[i,i,i]=>i" |
|
1072 |
(* "is_list_functor(M,A,X,Z) == |
|
1073 |
\<exists>n1[M]. \<exists>AX[M]. |
|
1074 |
number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *) |
|
1075 |
"list_functor_fm(A,X,Z) == |
|
1076 |
Exists(Exists( |
|
1077 |
And(number1_fm(1), |
|
1078 |
And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))" |
|
1079 |
||
1080 |
lemma list_functor_type [TC]: |
|
1081 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_functor_fm(x,y,z) \<in> formula" |
|
1082 |
by (simp add: list_functor_fm_def) |
|
1083 |
||
1084 |
lemma arity_list_functor_fm [simp]: |
|
1085 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] |
|
1086 |
==> arity(list_functor_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)" |
|
1087 |
by (simp add: list_functor_fm_def succ_Un_distrib [symmetric] Un_ac) |
|
1088 |
||
1089 |
lemma sats_list_functor_fm [simp]: |
|
1090 |
"[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|] |
|
1091 |
==> sats(A, list_functor_fm(x,y,z), env) <-> |
|
1092 |
is_list_functor(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
1093 |
by (simp add: list_functor_fm_def is_list_functor_def) |
|
1094 |
||
1095 |
lemma list_functor_iff_sats: |
|
1096 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
1097 |
i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|] |
|
1098 |
==> is_list_functor(**A, x, y, z) <-> sats(A, list_functor_fm(i,j,k), env)" |
|
1099 |
by simp |
|
1100 |
||
1101 |
theorem list_functor_reflection: |
|
1102 |
"REFLECTS[\<lambda>x. is_list_functor(L,f(x),g(x),h(x)), |
|
1103 |
\<lambda>i x. is_list_functor(**Lset(i),f(x),g(x),h(x))]" |
|
1104 |
apply (simp only: is_list_functor_def setclass_simps) |
|
1105 |
apply (intro FOL_reflections number1_reflection |
|
1106 |
cartprod_reflection sum_reflection) |
|
1107 |
done |
|
1108 |
||
1109 |
||
1110 |
subsubsection{*The Formula @{term is_list_N}, Internalized*} |
|
1111 |
||
1112 |
(* "is_list_N(M,A,n,Z) == |
|
1113 |
\<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M]. |
|
1114 |
empty(M,zero) & |
|
1115 |
successor(M,n,sn) & membership(M,sn,msn) & |
|
1116 |
is_wfrec(M, iterates_MH(M, is_list_functor(M,A),zero), msn, n, Z)" *) |
|
1117 |
||
1118 |
constdefs list_N_fm :: "[i,i,i]=>i" |
|
1119 |
"list_N_fm(A,n,Z) == |
|
1120 |
Exists(Exists(Exists( |
|
1121 |
And(empty_fm(2), |
|
1122 |
And(succ_fm(n#+3,1), |
|
1123 |
And(Memrel_fm(1,0), |
|
1124 |
is_wfrec_fm(iterates_MH_fm(list_functor_fm(A#+9#+3,1,0), |
|
1125 |
7,2,1,0), |
|
1126 |
0, n#+3, Z#+3)))))))" |
|
1127 |
||
1128 |
lemma list_N_fm_type [TC]: |
|
1129 |
"[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_N_fm(x,y,z) \<in> formula" |
|
1130 |
by (simp add: list_N_fm_def) |
|
1131 |
||
1132 |
lemma sats_list_N_fm [simp]: |
|
1133 |
"[| x \<in> nat; y < length(env); z < length(env); env \<in> list(A)|] |
|
1134 |
==> sats(A, list_N_fm(x,y,z), env) <-> |
|
1135 |
is_list_N(**A, nth(x,env), nth(y,env), nth(z,env))" |
|
1136 |
apply (frule_tac x=z in lt_length_in_nat, assumption) |
|
1137 |
apply (frule_tac x=y in lt_length_in_nat, assumption) |
|
1138 |
apply (simp add: list_N_fm_def is_list_N_def sats_is_wfrec_fm |
|
1139 |
sats_iterates_MH_fm) |
|
1140 |
done |
|
1141 |
||
1142 |
lemma list_N_iff_sats: |
|
1143 |
"[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; |
|
1144 |
i \<in> nat; j < length(env); k < length(env); env \<in> list(A)|] |
|
1145 |
==> is_list_N(**A, x, y, z) <-> sats(A, list_N_fm(i,j,k), env)" |
|
1146 |
by (simp add: sats_list_N_fm) |
|
1147 |
||
1148 |
theorem list_N_reflection: |
|
1149 |
"REFLECTS[\<lambda>x. is_list_N(L, f(x), g(x), h(x)), |
|
1150 |
\<lambda>i x. is_list_N(**Lset(i), f(x), g(x), h(x))]" |
|
1151 |
apply (simp only: is_list_N_def setclass_simps) |
|
1152 |
apply (intro FOL_reflections function_reflections is_wfrec_reflection |
|
1153 |
iterates_MH_reflection list_functor_reflection) |
|
1154 |
done |
|
1155 |
||
1156 |
||
1157 |
||
1158 |
subsubsection{*The Predicate ``Is A List''*} |
|
1159 |
||
1160 |
(* mem_list(M,A,l) == |
|
1161 |
\<exists>n[M]. \<exists>listn[M]. |
|
1162 |
finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l \<in> listn *) |
|
1163 |
constdefs mem_list_fm :: "[i,i]=>i" |
|
1164 |
"mem_list_fm(x,y) == |
|
1165 |
Exists(Exists( |
|
1166 |
And(finite_ordinal_fm(1), |
|
1167 |
And(list_N_fm(x#+2,1,0), Member(y#+2,0)))))" |
|
1168 |
||
1169 |
lemma mem_list_type [TC]: |
|
1170 |
"[| x \<in> nat; y \<in> nat |] ==> mem_list_fm(x,y) \<in> formula" |
|
1171 |
by (simp add: mem_list_fm_def) |
|
1172 |
||
1173 |
lemma sats_mem_list_fm [simp]: |
|
1174 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
1175 |
==> sats(A, mem_list_fm(x,y), env) <-> mem_list(**A, nth(x,env), nth(y,env))" |
|
1176 |
by (simp add: mem_list_fm_def mem_list_def) |
|
1177 |
||
1178 |
lemma mem_list_iff_sats: |
|
1179 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
1180 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
|
1181 |
==> mem_list(**A, x, y) <-> sats(A, mem_list_fm(i,j), env)" |
|
1182 |
by simp |
|
1183 |
||
1184 |
theorem mem_list_reflection: |
|
1185 |
"REFLECTS[\<lambda>x. mem_list(L,f(x),g(x)), |
|
1186 |
\<lambda>i x. mem_list(**Lset(i),f(x),g(x))]" |
|
1187 |
apply (simp only: mem_list_def setclass_simps) |
|
1188 |
apply (intro FOL_reflections finite_ordinal_reflection list_N_reflection) |
|
1189 |
done |
|
1190 |
||
1191 |
||
1192 |
subsubsection{*The Predicate ``Is @{term "list(A)"}''*} |
|
1193 |
||
1194 |
(* is_list(M,A,Z) == \<forall>l[M]. l \<in> Z <-> mem_list(M,A,l) *) |
|
1195 |
constdefs is_list_fm :: "[i,i]=>i" |
|
1196 |
"is_list_fm(A,Z) == |
|
1197 |
Forall(Iff(Member(0,succ(Z)), mem_list_fm(succ(A),0)))" |
|
1198 |
||
1199 |
lemma is_list_type [TC]: |
|
1200 |
"[| x \<in> nat; y \<in> nat |] ==> is_list_fm(x,y) \<in> formula" |
|
1201 |
by (simp add: is_list_fm_def) |
|
1202 |
||
1203 |
lemma sats_is_list_fm [simp]: |
|
1204 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
1205 |
==> sats(A, is_list_fm(x,y), env) <-> is_list(**A, nth(x,env), nth(y,env))" |
|
1206 |
by (simp add: is_list_fm_def is_list_def) |
|
1207 |
||
1208 |
lemma is_list_iff_sats: |
|
1209 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
1210 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
|
1211 |
==> is_list(**A, x, y) <-> sats(A, is_list_fm(i,j), env)" |
|
1212 |
by simp |
|
1213 |
||
1214 |
theorem is_list_reflection: |
|
1215 |
"REFLECTS[\<lambda>x. is_list(L,f(x),g(x)), |
|
1216 |
\<lambda>i x. is_list(**Lset(i),f(x),g(x))]" |
|
1217 |
apply (simp only: is_list_def setclass_simps) |
|
1218 |
apply (intro FOL_reflections mem_list_reflection) |
|
1219 |
done |
|
1220 |
||
1221 |
||
1222 |
subsubsection{*The Formula Functor, Internalized*} |
|
1223 |
||
1224 |
constdefs formula_functor_fm :: "[i,i]=>i" |
|
1225 |
(* "is_formula_functor(M,X,Z) == |
|
1226 |
\<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M]. |
|
1227 |
4 3 2 1 0 |
|
1228 |
omega(M,nat') & cartprod(M,nat',nat',natnat) & |
|
1229 |
is_sum(M,natnat,natnat,natnatsum) & |
|
1230 |
cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) & |
|
1231 |
is_sum(M,natnatsum,X3,Z)" *) |
|
1232 |
"formula_functor_fm(X,Z) == |
|
1233 |
Exists(Exists(Exists(Exists(Exists( |
|
1234 |
And(omega_fm(4), |
|
1235 |
And(cartprod_fm(4,4,3), |
|
1236 |
And(sum_fm(3,3,2), |
|
1237 |
And(cartprod_fm(X#+5,X#+5,1), |
|
1238 |
And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))" |
|
1239 |
||
1240 |
lemma formula_functor_type [TC]: |
|
1241 |
"[| x \<in> nat; y \<in> nat |] ==> formula_functor_fm(x,y) \<in> formula" |
|
1242 |
by (simp add: formula_functor_fm_def) |
|
1243 |
||
1244 |
lemma sats_formula_functor_fm [simp]: |
|
1245 |
"[| x \<in> nat; y \<in> nat; env \<in> list(A)|] |
|
1246 |
==> sats(A, formula_functor_fm(x,y), env) <-> |
|
1247 |
is_formula_functor(**A, nth(x,env), nth(y,env))" |
|
1248 |
by (simp add: formula_functor_fm_def is_formula_functor_def) |
|
1249 |
||
1250 |
lemma formula_functor_iff_sats: |
|
1251 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
1252 |
i \<in> nat; j \<in> nat; env \<in> list(A)|] |
|
1253 |
==> is_formula_functor(**A, x, y) <-> sats(A, formula_functor_fm(i,j), env)" |
|
1254 |
by simp |
|
1255 |
||
1256 |
theorem formula_functor_reflection: |
|
1257 |
"REFLECTS[\<lambda>x. is_formula_functor(L,f(x),g(x)), |
|
1258 |
\<lambda>i x. is_formula_functor(**Lset(i),f(x),g(x))]" |
|
1259 |
apply (simp only: is_formula_functor_def setclass_simps) |
|
1260 |
apply (intro FOL_reflections omega_reflection |
|
1261 |
cartprod_reflection sum_reflection) |
|
1262 |
done |
|
1263 |
||
1264 |
||
1265 |
subsubsection{*The Formula @{term is_formula_N}, Internalized*} |
|
1266 |
||
1267 |
(* "is_formula_N(M,n,Z) == |
|
1268 |
\<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M]. |
|
1269 |
2 1 0 |
|
1270 |
empty(M,zero) & |
|
1271 |
successor(M,n,sn) & membership(M,sn,msn) & |
|
1272 |
is_wfrec(M, iterates_MH(M, is_formula_functor(M),zero), msn, n, Z)" *) |
|
1273 |
constdefs formula_N_fm :: "[i,i]=>i" |
|
1274 |
"formula_N_fm(n,Z) == |
|
1275 |
Exists(Exists(Exists( |
|
1276 |
And(empty_fm(2), |
|
1277 |
And(succ_fm(n#+3,1), |
|
1278 |
And(Memrel_fm(1,0), |
|
1279 |
is_wfrec_fm(iterates_MH_fm(formula_functor_fm(1,0),7,2,1,0), |
|
1280 |
0, n#+3, Z#+3)))))))" |
|
1281 |
||
1282 |
lemma formula_N_fm_type [TC]: |
|
1283 |
"[| x \<in> nat; y \<in> nat |] ==> formula_N_fm(x,y) \<in> formula" |
|
1284 |
by (simp add: formula_N_fm_def) |
|
1285 |
||
1286 |
lemma sats_formula_N_fm [simp]: |
|
1287 |
"[| x < length(env); y < length(env); env \<in> list(A)|] |
|
1288 |
==> sats(A, formula_N_fm(x,y), env) <-> |
|
1289 |
is_formula_N(**A, nth(x,env), nth(y,env))" |
|
1290 |
apply (frule_tac x=y in lt_length_in_nat, assumption) |
|
1291 |
apply (frule lt_length_in_nat, assumption) |
|
1292 |
apply (simp add: formula_N_fm_def is_formula_N_def sats_is_wfrec_fm sats_iterates_MH_fm) |
|
1293 |
done |
|
1294 |
||
1295 |
lemma formula_N_iff_sats: |
|
1296 |
"[| nth(i,env) = x; nth(j,env) = y; |
|
1297 |
i < length(env); j < length(env); env \<in> list(A)|] |
|
1298 |
==> is_formula_N(**A, x, y) <-> sats(A, formula_N_fm(i,j), env)" |
|
1299 |
by (simp add: sats_formula_N_fm) |
|
1300 |
||
1301 |
theorem formula_N_reflection: |
|
1302 |
"REFLECTS[\<lambda>x. is_formula_N(L, f(x), g(x)), |
|
1303 |
\<lambda>i x. is_formula_N(**Lset(i), f(x), g(x))]" |
|
1304 |
apply (simp only: is_formula_N_def setclass_simps) |
|
1305 |
apply (intro FOL_reflections function_reflections is_wfrec_reflection |
|
1306 |
iterates_MH_reflection formula_functor_reflection) |
|
1307 |
done |
|
1308 |
||
1309 |
||
1310 |
||
1311 |
subsubsection{*The Predicate ``Is A Formula''*} |
|
1312 |
||
1313 |
(* mem_formula(M,p) == |
|
1314 |
\<exists>n[M]. \<exists>formn[M]. |
|
1315 |
finite_ordinal(M,n) & is_formula_N(M,n,formn) & p \<in> formn *) |
|
1316 |
constdefs mem_formula_fm :: "i=>i" |
|
1317 |
"mem_formula_fm(x) == |
|
1318 |
Exists(Exists( |
|
1319 |
And(finite_ordinal_fm(1), |
|
1320 |
And(formula_N_fm(1,0), Member(x#+2,0)))))" |
|
1321 |
||
1322 |
lemma mem_formula_type [TC]: |
|
1323 |
"x \<in> nat ==> mem_formula_fm(x) \<in> formula" |
|
1324 |
by (simp add: mem_formula_fm_def) |
|
1325 |
||
1326 |
lemma sats_mem_formula_fm [simp]: |
|
1327 |
"[| x \<in> nat; env \<in> list(A)|] |
|
1328 |
==> sats(A, mem_formula_fm(x), env) <-> mem_formula(**A, nth(x,env))" |
|
1329 |
by (simp add: mem_formula_fm_def mem_formula_def) |
|
1330 |
||
1331 |
lemma mem_formula_iff_sats: |
|
1332 |
"[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|] |
|
1333 |
==> mem_formula(**A, x) <-> sats(A, mem_formula_fm(i), env)" |
|
1334 |
by simp |
|
1335 |
||
1336 |
theorem mem_formula_reflection: |
|
1337 |
"REFLECTS[\<lambda>x. mem_formula(L,f(x)), |
|
1338 |
\<lambda>i x. mem_formula(**Lset(i),f(x))]" |
|
1339 |
apply (simp only: mem_formula_def setclass_simps) |
|
1340 |
apply (intro FOL_reflections finite_ordinal_reflection formula_N_reflection) |
|
1341 |
done |
|
1342 |
||
1343 |
||
1344 |
||
1345 |
subsubsection{*The Predicate ``Is @{term "formula"}''*} |
|
1346 |
||
1347 |
(* is_formula(M,Z) == \<forall>p[M]. p \<in> Z <-> mem_formula(M,p) *) |
|
1348 |
constdefs is_formula_fm :: "i=>i" |
|
1349 |
"is_formula_fm(Z) == Forall(Iff(Member(0,succ(Z)), mem_formula_fm(0)))" |
|
1350 |
||
1351 |
lemma is_formula_type [TC]: |
|
1352 |
"x \<in> nat ==> is_formula_fm(x) \<in> formula" |
|
1353 |
by (simp add: is_formula_fm_def) |
|
1354 |
||
1355 |
lemma sats_is_formula_fm [simp]: |
|
1356 |
"[| x \<in> nat; env \<in> list(A)|] |
|
1357 |
==> sats(A, is_formula_fm(x), env) <-> is_formula(**A, nth(x,env))" |
|
1358 |
by (simp add: is_formula_fm_def is_formula_def) |
|
1359 |
||
1360 |
lemma is_formula_iff_sats: |
|
1361 |
"[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|] |
|
1362 |
==> is_formula(**A, x) <-> sats(A, is_formula_fm(i), env)" |
|
1363 |
by simp |
|
1364 |
||
1365 |
theorem is_formula_reflection: |
|
1366 |
"REFLECTS[\<lambda>x. is_formula(L,f(x)), |
|
1367 |
\<lambda>i x. is_formula(**Lset(i),f(x))]" |
|
1368 |
apply (simp only: is_formula_def setclass_simps) |
|
1369 |
apply (intro FOL_reflections mem_formula_reflection) |
|
1370 |
done |
|
1371 |
||
1372 |
||
1373 |
subsubsection{*The Operator @{term is_transrec}*} |
|
1374 |
||
1375 |
text{*The three arguments of @{term p} are always 2, 1, 0. It is buried |
|
1376 |
within eight quantifiers! |
|
1377 |
We call @{term p} with arguments a, f, z by equating them with |
|
1378 |
the corresponding quantified variables with de Bruijn indices 2, 1, 0.*} |
|
1379 |
||
1380 |
(* is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o" |
|
1381 |
"is_transrec(M,MH,a,z) == |
|
1382 |
\<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M]. |
|
1383 |
2 1 0 |
|
1384 |
upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) & |
|
1385 |
is_wfrec(M,MH,mesa,a,z)" *) |
|
1386 |
constdefs is_transrec_fm :: "[i, i, i]=>i" |
|
1387 |
"is_transrec_fm(p,a,z) == |
|
1388 |
Exists(Exists(Exists( |
|
1389 |
And(upair_fm(a#+3,a#+3,2), |
|
1390 |
And(is_eclose_fm(2,1), |
|
1391 |
And(Memrel_fm(1,0), is_wfrec_fm(p,0,a#+3,z#+3)))))))" |
|
1392 |
||
1393 |
||
1394 |
lemma is_transrec_type [TC]: |
|
1395 |
"[| p \<in> formula; x \<in> nat; z \<in> nat |] |
|
1396 |
==> is_transrec_fm(p,x,z) \<in> formula" |
|
1397 |
by (simp add: is_transrec_fm_def) |
|
1398 |
||
1399 |
lemma sats_is_transrec_fm: |
|
1400 |
assumes MH_iff_sats: |
|
1401 |
"!!a0 a1 a2 a3 a4 a5 a6 a7. |
|
1402 |
[|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A; a5\<in>A; a6\<in>A; a7\<in>A|] |
|
1403 |
==> MH(a2, a1, a0) <-> |
|
1404 |
sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3, |
|
1405 |
Cons(a4,Cons(a5,Cons(a6,Cons(a7,env)))))))))" |
|
1406 |
shows |
|
1407 |
"[|x < length(env); z < length(env); env \<in> list(A)|] |
|
1408 |
==> sats(A, is_transrec_fm(p,x,z), env) <-> |
|
1409 |
is_transrec(**A, MH, nth(x,env), nth(z,env))" |
|
1410 |
apply (frule_tac x=z in lt_length_in_nat, assumption) |
|
1411 |
apply (frule_tac x=x in lt_length_in_nat, assumption) |
|
1412 |
apply (simp add: is_transrec_fm_def sats_is_wfrec_fm is_transrec_def MH_iff_sats [THEN iff_sym]) |
|
1413 |
done |
|
1414 |
||
1415 |
||
1416 |
lemma is_transrec_iff_sats: |
|
1417 |
assumes MH_iff_sats: |
|
1418 |
"!!a0 a1 a2 a3 a4 a5 a6 a7. |
|
1419 |
[|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A; a5\<in>A; a6\<in>A; a7\<in>A|] |
|
1420 |
==> MH(a2, a1, a0) <-> |
|
1421 |
sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3, |
|
1422 |
Cons(a4,Cons(a5,Cons(a6,Cons(a7,env)))))))))" |
|
1423 |
shows |
|
1424 |
"[|nth(i,env) = x; nth(k,env) = z; |
|
1425 |
i < length(env); k < length(env); env \<in> list(A)|] |
|
1426 |
==> is_transrec(**A, MH, x, z) <-> sats(A, is_transrec_fm(p,i,k), env)" |
|
1427 |
by (simp add: sats_is_transrec_fm [OF MH_iff_sats]) |
|
1428 |
||
1429 |
theorem is_transrec_reflection: |
|
1430 |
assumes MH_reflection: |
|
1431 |
"!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)), |
|
1432 |
\<lambda>i x. MH(**Lset(i), f'(x), f(x), g(x), h(x))]" |
|
1433 |
shows "REFLECTS[\<lambda>x. is_transrec(L, MH(L,x), f(x), h(x)), |
|
1434 |
\<lambda>i x. is_transrec(**Lset(i), MH(**Lset(i),x), f(x), h(x))]" |
|
1435 |
apply (simp (no_asm_use) only: is_transrec_def setclass_simps) |
|
1436 |
apply (intro FOL_reflections function_reflections MH_reflection |
|
1437 |
is_wfrec_reflection is_eclose_reflection) |
|
1438 |
done |
|
1439 |
||
13496
6f0c57def6d5
In ZF/Constructible, moved many results from Satisfies_absolute, etc., to
paulson
parents:
diff
changeset
|
1440 |
end |