src/ZF/func.thy
author paulson
Tue Jun 18 17:58:21 2002 +0200 (2002-06-18)
changeset 13219 7e44aa8a276e
parent 13179 3f6f00c6c56f
child 13221 e29378f347e4
permissions -rw-r--r--
new lemma
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(*  Title:      ZF/func.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1991  University of Cambridge
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Functions in Zermelo-Fraenkel Set Theory
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*)
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theory func = equalities:
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(*** The Pi operator -- dependent function space ***)
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lemma Pi_iff:
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    "f: Pi(A,B) <-> function(f) & f<=Sigma(A,B) & A<=domain(f)"
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by (unfold Pi_def, blast)
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(*For upward compatibility with the former definition*)
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lemma Pi_iff_old:
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    "f: Pi(A,B) <-> f<=Sigma(A,B) & (ALL x:A. EX! y. <x,y>: f)"
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by (unfold Pi_def function_def, blast)
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lemma fun_is_function: "f: Pi(A,B) ==> function(f)"
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by (simp only: Pi_iff)
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lemma function_imp_Pi:
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     "[|function(f); relation(f)|] ==> f \<in> domain(f) -> range(f)"
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by (simp add: Pi_iff relation_def, blast) 
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lemma functionI: 
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     "[| !!x y y'. [| <x,y>:r; <x,y'>:r |] ==> y=y' |] ==> function(r)"
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by (simp add: function_def, blast) 
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(*Functions are relations*)
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lemma fun_is_rel: "f: Pi(A,B) ==> f <= Sigma(A,B)"
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by (unfold Pi_def, blast)
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lemma Pi_cong:
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    "[| A=A';  !!x. x:A' ==> B(x)=B'(x) |] ==> Pi(A,B) = Pi(A',B')"
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by (simp add: Pi_def cong add: Sigma_cong)
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(*Sigma_cong, Pi_cong NOT given to Addcongs: they cause
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  flex-flex pairs and the "Check your prover" error.  Most
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  Sigmas and Pis are abbreviated as * or -> *)
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(*Weakening one function type to another; see also Pi_type*)
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lemma fun_weaken_type: "[| f: A->B;  B<=D |] ==> f: A->D"
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by (unfold Pi_def, best)
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(*** Function Application ***)
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lemma apply_equality2: "[| <a,b>: f;  <a,c>: f;  f: Pi(A,B) |] ==> b=c"
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by (unfold Pi_def function_def, blast)
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lemma function_apply_equality: "[| <a,b>: f;  function(f) |] ==> f`a = b"
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by (unfold apply_def function_def, blast)
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lemma apply_equality: "[| <a,b>: f;  f: Pi(A,B) |] ==> f`a = b"
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apply (unfold Pi_def)
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apply (blast intro: function_apply_equality)
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done
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(*Applying a function outside its domain yields 0*)
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lemma apply_0: "a ~: domain(f) ==> f`a = 0"
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by (unfold apply_def, blast)
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lemma Pi_memberD: "[| f: Pi(A,B);  c: f |] ==> EX x:A.  c = <x,f`x>"
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apply (frule fun_is_rel)
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apply (blast dest: apply_equality)
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done
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lemma function_apply_Pair: "[| function(f);  a : domain(f)|] ==> <a,f`a>: f"
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apply (simp add: function_def, clarify) 
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apply (subgoal_tac "f`a = y", blast) 
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apply (simp add: apply_def, blast) 
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done
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lemma apply_Pair: "[| f: Pi(A,B);  a:A |] ==> <a,f`a>: f"
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apply (simp add: Pi_iff)
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apply (blast intro: function_apply_Pair)
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done
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(*Conclusion is flexible -- use res_inst_tac or else apply_funtype below!*)
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lemma apply_type [TC]: "[| f: Pi(A,B);  a:A |] ==> f`a : B(a)"
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by (blast intro: apply_Pair dest: fun_is_rel)
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(*This version is acceptable to the simplifier*)
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lemma apply_funtype: "[| f: A->B;  a:A |] ==> f`a : B"
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by (blast dest: apply_type)
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lemma apply_iff: "f: Pi(A,B) ==> <a,b>: f <-> a:A & f`a = b"
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apply (frule fun_is_rel)
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apply (blast intro!: apply_Pair apply_equality)
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done
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(*Refining one Pi type to another*)
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lemma Pi_type: "[| f: Pi(A,C);  !!x. x:A ==> f`x : B(x) |] ==> f : Pi(A,B)"
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apply (simp only: Pi_iff)
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apply (blast dest: function_apply_equality)
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done
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(*Such functions arise in non-standard datatypes, ZF/ex/Ntree for instance*)
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lemma Pi_Collect_iff:
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     "(f : Pi(A, %x. {y:B(x). P(x,y)}))
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      <->  f : Pi(A,B) & (ALL x: A. P(x, f`x))"
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by (blast intro: Pi_type dest: apply_type)
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lemma Pi_weaken_type:
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        "[| f : Pi(A,B);  !!x. x:A ==> B(x)<=C(x) |] ==> f : Pi(A,C)"
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by (blast intro: Pi_type dest: apply_type)
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(** Elimination of membership in a function **)
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lemma domain_type: "[| <a,b> : f;  f: Pi(A,B) |] ==> a : A"
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by (blast dest: fun_is_rel)
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lemma range_type: "[| <a,b> : f;  f: Pi(A,B) |] ==> b : B(a)"
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by (blast dest: fun_is_rel)
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lemma Pair_mem_PiD: "[| <a,b>: f;  f: Pi(A,B) |] ==> a:A & b:B(a) & f`a = b"
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by (blast intro: domain_type range_type apply_equality)
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(*** Lambda Abstraction ***)
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lemma lamI: "a:A ==> <a,b(a)> : (lam x:A. b(x))"
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apply (unfold lam_def)
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apply (erule RepFunI)
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done
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lemma lamE:
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    "[| p: (lam x:A. b(x));  !!x.[| x:A; p=<x,b(x)> |] ==> P
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     |] ==>  P"
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by (simp add: lam_def, blast)
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lemma lamD: "[| <a,c>: (lam x:A. b(x)) |] ==> c = b(a)"
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by (simp add: lam_def)
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lemma lam_type [TC]:
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    "[| !!x. x:A ==> b(x): B(x) |] ==> (lam x:A. b(x)) : Pi(A,B)"
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by (simp add: lam_def Pi_def function_def, blast)
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lemma lam_funtype: "(lam x:A. b(x)) : A -> {b(x). x:A}"
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by (blast intro: lam_type)
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lemma function_lam: "function (lam x:A. b(x))"
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by (simp add: function_def lam_def) 
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lemma relation_lam: "relation (lam x:A. b(x))"  
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by (simp add: relation_def lam_def) 
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lemma beta_if [simp]: "(lam x:A. b(x)) ` a = (if a : A then b(a) else 0)"
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by (simp add: apply_def lam_def, blast)
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lemma beta: "a : A ==> (lam x:A. b(x)) ` a = b(a)"
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by (simp add: apply_def lam_def, blast)
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lemma lam_empty [simp]: "(lam x:0. b(x)) = 0"
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by (simp add: lam_def)
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lemma domain_lam [simp]: "domain(Lambda(A,b)) = A"
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by (simp add: lam_def, blast)
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(*congruence rule for lambda abstraction*)
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lemma lam_cong [cong]:
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    "[| A=A';  !!x. x:A' ==> b(x)=b'(x) |] ==> Lambda(A,b) = Lambda(A',b')"
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by (simp only: lam_def cong add: RepFun_cong)
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lemma lam_theI:
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    "(!!x. x:A ==> EX! y. Q(x,y)) ==> EX f. ALL x:A. Q(x, f`x)"
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apply (rule_tac x = "lam x: A. THE y. Q (x,y)" in exI)
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apply simp 
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apply (blast intro: theI)
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done
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lemma lam_eqE: "[| (lam x:A. f(x)) = (lam x:A. g(x));  a:A |] ==> f(a)=g(a)"
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by (fast intro!: lamI elim: equalityE lamE)
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(*Empty function spaces*)
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lemma Pi_empty1 [simp]: "Pi(0,A) = {0}"
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by (unfold Pi_def function_def, blast)
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(*The singleton function*)
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lemma singleton_fun [simp]: "{<a,b>} : {a} -> {b}"
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by (unfold Pi_def function_def, blast)
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lemma Pi_empty2 [simp]: "(A->0) = (if A=0 then {0} else 0)"
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by (unfold Pi_def function_def, force)
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lemma  fun_space_empty_iff [iff]: "(A->X)=0 \<longleftrightarrow> X=0 & (A \<noteq> 0)"
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apply auto
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apply (fast intro!: equals0I intro: lam_type)
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done
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(** Extensionality **)
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(*Semi-extensionality!*)
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lemma fun_subset:
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    "[| f : Pi(A,B);  g: Pi(C,D);  A<=C;
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        !!x. x:A ==> f`x = g`x       |] ==> f<=g"
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by (force dest: Pi_memberD intro: apply_Pair)
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lemma fun_extension:
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    "[| f : Pi(A,B);  g: Pi(A,D);
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        !!x. x:A ==> f`x = g`x       |] ==> f=g"
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by (blast del: subsetI intro: subset_refl sym fun_subset)
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lemma eta [simp]: "f : Pi(A,B) ==> (lam x:A. f`x) = f"
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apply (rule fun_extension)
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apply (auto simp add: lam_type apply_type beta)
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done
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lemma fun_extension_iff:
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     "[| f:Pi(A,B); g:Pi(A,C) |] ==> (ALL a:A. f`a = g`a) <-> f=g"
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by (blast intro: fun_extension)
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(*thm by Mark Staples, proof by lcp*)
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lemma fun_subset_eq: "[| f:Pi(A,B); g:Pi(A,C) |] ==> f <= g <-> (f = g)"
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by (blast dest: apply_Pair
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	  intro: fun_extension apply_equality [symmetric])
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(*Every element of Pi(A,B) may be expressed as a lambda abstraction!*)
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lemma Pi_lamE:
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  assumes major: "f: Pi(A,B)"
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      and minor: "!!b. [| ALL x:A. b(x):B(x);  f = (lam x:A. b(x)) |] ==> P"
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  shows "P"
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apply (rule minor)
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apply (rule_tac [2] eta [symmetric])
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apply (blast intro: major apply_type)+
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done
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(** Images of functions **)
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lemma image_lam: "C <= A ==> (lam x:A. b(x)) `` C = {b(x). x:C}"
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by (unfold lam_def, blast)
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lemma Repfun_function_if:
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     "function(f) 
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      ==> {f`x. x:C} = (if C <= domain(f) then f``C else cons(0,f``C))";
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apply simp
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apply (intro conjI impI)  
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 apply (blast dest: function_apply_equality intro: function_apply_Pair) 
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apply (rule equalityI)
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 apply (blast intro!: function_apply_Pair apply_0) 
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apply (blast dest: function_apply_equality intro: apply_0 [symmetric]) 
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done
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(*For this lemma and the next, the right-hand side could equivalently 
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  be written UN x:C. {f`x} *)
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lemma image_function:
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     "[| function(f);  C <= domain(f) |] ==> f``C = {f`x. x:C}";
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by (simp add: Repfun_function_if) 
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lemma image_fun: "[| f : Pi(A,B);  C <= A |] ==> f``C = {f`x. x:C}"
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apply (simp add: Pi_iff) 
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apply (blast intro: image_function) 
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done
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lemma Pi_image_cons:
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     "[| f: Pi(A,B);  x: A |] ==> f `` cons(x,y) = cons(f`x, f``y)"
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by (blast dest: apply_equality apply_Pair)
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(*** properties of "restrict" ***)
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lemma restrict_subset: "restrict(f,A) <= f"
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by (unfold restrict_def, blast)
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lemma function_restrictI:
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    "function(f) ==> function(restrict(f,A))"
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by (unfold restrict_def function_def, blast)
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lemma restrict_type2: "[| f: Pi(C,B);  A<=C |] ==> restrict(f,A) : Pi(A,B)"
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by (simp add: Pi_iff function_def restrict_def, blast)
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lemma restrict: "restrict(f,A) ` a = (if a : A then f`a else 0)"
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by (simp add: apply_def restrict_def, blast)
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lemma restrict_empty [simp]: "restrict(f,0) = 0"
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by (unfold restrict_def, simp)
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lemma restrict_iff: "z \<in> restrict(r,A) \<longleftrightarrow> z \<in> r & (\<exists>x\<in>A. \<exists>y. z = \<langle>x, y\<rangle>)"
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by (simp add: restrict_def) 
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lemma domain_restrict_lam [simp]: "domain(restrict(Lambda(A,f),C)) = A Int C"
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apply (unfold restrict_def lam_def)
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apply (rule equalityI)
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apply (auto simp add: domain_iff)
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done
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lemma restrict_restrict [simp]:
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     "restrict(restrict(r,A),B) = restrict(r, A Int B)"
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by (unfold restrict_def, blast)
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lemma domain_restrict [simp]: "domain(restrict(f,C)) = domain(f) Int C"
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apply (unfold restrict_def)
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apply (auto simp add: domain_def)
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done
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lemma restrict_idem [simp]: "f <= Sigma(A,B) ==> restrict(f,A) = f"
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by (simp add: restrict_def, blast)
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lemma restrict_if [simp]: "restrict(f,A) ` a = (if a : A then f`a else 0)"
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by (simp add: restrict apply_0)
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lemma restrict_lam_eq:
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    "A<=C ==> restrict(lam x:C. b(x), A) = (lam x:A. b(x))"
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by (unfold restrict_def lam_def, auto)
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lemma fun_cons_restrict_eq:
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     "f : cons(a, b) -> B ==> f = cons(<a, f ` a>, restrict(f, b))"
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apply (rule equalityI)
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prefer 2 apply (blast intro: apply_Pair restrict_subset [THEN subsetD])
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apply (auto dest!: Pi_memberD simp add: restrict_def lam_def)
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done
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(*** Unions of functions ***)
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(** The Union of a set of COMPATIBLE functions is a function **)
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lemma function_Union:
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    "[| ALL x:S. function(x);
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        ALL x:S. ALL y:S. x<=y | y<=x  |]
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     ==> function(Union(S))"
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by (unfold function_def, blast) 
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lemma fun_Union:
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    "[| ALL f:S. EX C D. f:C->D;
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             ALL f:S. ALL y:S. f<=y | y<=f  |] ==>
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          Union(S) : domain(Union(S)) -> range(Union(S))"
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apply (unfold Pi_def)
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apply (blast intro!: rel_Union function_Union)
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done
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lemma gen_relation_Union [rule_format]:
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     "\<forall>f\<in>F. relation(f) \<Longrightarrow> relation(Union(F))"
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by (simp add: relation_def) 
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(** The Union of 2 disjoint functions is a function **)
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lemmas Un_rls = Un_subset_iff SUM_Un_distrib1 prod_Un_distrib2
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                subset_trans [OF _ Un_upper1]
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                subset_trans [OF _ Un_upper2]
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lemma fun_disjoint_Un:
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     "[| f: A->B;  g: C->D;  A Int C = 0  |]
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      ==> (f Un g) : (A Un C) -> (B Un D)"
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(*Prove the product and domain subgoals using distributive laws*)
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apply (simp add: Pi_iff extension Un_rls)
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apply (unfold function_def, blast)
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done
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lemma fun_disjoint_apply1: "a \<notin> domain(g) ==> (f Un g)`a = f`a"
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by (simp add: apply_def, blast) 
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lemma fun_disjoint_apply2: "c \<notin> domain(f) ==> (f Un g)`c = g`c"
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by (simp add: apply_def, blast) 
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(** Domain and range of a function/relation **)
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lemma domain_of_fun: "f : Pi(A,B) ==> domain(f)=A"
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by (unfold Pi_def, blast)
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lemma apply_rangeI: "[| f : Pi(A,B);  a: A |] ==> f`a : range(f)"
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by (erule apply_Pair [THEN rangeI], assumption)
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lemma range_of_fun: "f : Pi(A,B) ==> f : A->range(f)"
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by (blast intro: Pi_type apply_rangeI)
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(*** Extensions of functions ***)
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lemma fun_extend:
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     "[| f: A->B;  c~:A |] ==> cons(<c,b>,f) : cons(c,A) -> cons(b,B)"
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apply (frule singleton_fun [THEN fun_disjoint_Un], blast)
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apply (simp add: cons_eq) 
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done
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lemma fun_extend3:
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     "[| f: A->B;  c~:A;  b: B |] ==> cons(<c,b>,f) : cons(c,A) -> B"
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by (blast intro: fun_extend [THEN fun_weaken_type])
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lemma extend_apply:
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     "c ~: domain(f) ==> cons(<c,b>,f)`a = (if a=c then b else f`a)"
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by (auto simp add: apply_def) 
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lemma fun_extend_apply [simp]:
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     "[| f: A->B;  c~:A |] ==> cons(<c,b>,f)`a = (if a=c then b else f`a)" 
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apply (rule extend_apply) 
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apply (simp add: Pi_def, blast) 
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done
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lemmas singleton_apply = apply_equality [OF singletonI singleton_fun, simp]
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(*For Finite.ML.  Inclusion of right into left is easy*)
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lemma cons_fun_eq:
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     "c ~: A ==> cons(c,A) -> B = (UN f: A->B. UN b:B. {cons(<c,b>, f)})"
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apply (rule equalityI)
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   404
apply (safe elim!: fun_extend3)
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   405
(*Inclusion of left into right*)
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apply (subgoal_tac "restrict (x, A) : A -> B")
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   407
 prefer 2 apply (blast intro: restrict_type2)
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   408
apply (rule UN_I, assumption)
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apply (rule apply_funtype [THEN UN_I]) 
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   410
  apply assumption
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   411
 apply (rule consI1) 
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   412
apply (simp (no_asm))
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   413
apply (rule fun_extension) 
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   414
  apply assumption
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   415
 apply (blast intro: fun_extend) 
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   416
apply (erule consE, simp_all)
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   417
done
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   418
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   419
ML
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   420
{*
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val Pi_iff = thm "Pi_iff";
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   422
val Pi_iff_old = thm "Pi_iff_old";
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   423
val fun_is_function = thm "fun_is_function";
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   424
val fun_is_rel = thm "fun_is_rel";
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   425
val Pi_cong = thm "Pi_cong";
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   426
val fun_weaken_type = thm "fun_weaken_type";
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   427
val apply_equality2 = thm "apply_equality2";
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   428
val function_apply_equality = thm "function_apply_equality";
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   429
val apply_equality = thm "apply_equality";
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   430
val apply_0 = thm "apply_0";
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   431
val Pi_memberD = thm "Pi_memberD";
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   432
val function_apply_Pair = thm "function_apply_Pair";
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   433
val apply_Pair = thm "apply_Pair";
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   434
val apply_type = thm "apply_type";
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   435
val apply_funtype = thm "apply_funtype";
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   436
val apply_iff = thm "apply_iff";
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   437
val Pi_type = thm "Pi_type";
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   438
val Pi_Collect_iff = thm "Pi_Collect_iff";
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   439
val Pi_weaken_type = thm "Pi_weaken_type";
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   440
val domain_type = thm "domain_type";
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   441
val range_type = thm "range_type";
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   442
val Pair_mem_PiD = thm "Pair_mem_PiD";
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   443
val lamI = thm "lamI";
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   444
val lamE = thm "lamE";
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   445
val lamD = thm "lamD";
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   446
val lam_type = thm "lam_type";
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   447
val lam_funtype = thm "lam_funtype";
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   448
val beta = thm "beta";
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   449
val lam_empty = thm "lam_empty";
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   450
val domain_lam = thm "domain_lam";
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   451
val lam_cong = thm "lam_cong";
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   452
val lam_theI = thm "lam_theI";
paulson@13163
   453
val lam_eqE = thm "lam_eqE";
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   454
val Pi_empty1 = thm "Pi_empty1";
paulson@13163
   455
val singleton_fun = thm "singleton_fun";
paulson@13163
   456
val Pi_empty2 = thm "Pi_empty2";
paulson@13163
   457
val fun_space_empty_iff = thm "fun_space_empty_iff";
paulson@13163
   458
val fun_subset = thm "fun_subset";
paulson@13163
   459
val fun_extension = thm "fun_extension";
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   460
val eta = thm "eta";
paulson@13163
   461
val fun_extension_iff = thm "fun_extension_iff";
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   462
val fun_subset_eq = thm "fun_subset_eq";
paulson@13163
   463
val Pi_lamE = thm "Pi_lamE";
paulson@13163
   464
val image_lam = thm "image_lam";
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   465
val image_fun = thm "image_fun";
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   466
val Pi_image_cons = thm "Pi_image_cons";
paulson@13163
   467
val restrict_subset = thm "restrict_subset";
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   468
val function_restrictI = thm "function_restrictI";
paulson@13163
   469
val restrict_type2 = thm "restrict_type2";
paulson@13163
   470
val restrict = thm "restrict";
paulson@13163
   471
val restrict_empty = thm "restrict_empty";
paulson@13163
   472
val domain_restrict_lam = thm "domain_restrict_lam";
paulson@13163
   473
val restrict_restrict = thm "restrict_restrict";
paulson@13163
   474
val domain_restrict = thm "domain_restrict";
paulson@13163
   475
val restrict_idem = thm "restrict_idem";
paulson@13163
   476
val restrict_if = thm "restrict_if";
paulson@13163
   477
val restrict_lam_eq = thm "restrict_lam_eq";
paulson@13163
   478
val fun_cons_restrict_eq = thm "fun_cons_restrict_eq";
paulson@13163
   479
val function_Union = thm "function_Union";
paulson@13163
   480
val fun_Union = thm "fun_Union";
paulson@13163
   481
val fun_disjoint_Un = thm "fun_disjoint_Un";
paulson@13163
   482
val fun_disjoint_apply1 = thm "fun_disjoint_apply1";
paulson@13163
   483
val fun_disjoint_apply2 = thm "fun_disjoint_apply2";
paulson@13163
   484
val domain_of_fun = thm "domain_of_fun";
paulson@13163
   485
val apply_rangeI = thm "apply_rangeI";
paulson@13163
   486
val range_of_fun = thm "range_of_fun";
paulson@13163
   487
val fun_extend = thm "fun_extend";
paulson@13163
   488
val fun_extend3 = thm "fun_extend3";
paulson@13176
   489
val fun_extend_apply = thm "fun_extend_apply";
paulson@13163
   490
val singleton_apply = thm "singleton_apply";
paulson@13163
   491
val cons_fun_eq = thm "cons_fun_eq";
paulson@13163
   492
*}
paulson@13163
   493
paulson@13163
   494
end