src/ZF/func.thy
author wenzelm
Thu Sep 02 00:48:07 2010 +0200 (2010-09-02)
changeset 38980 af73cf0dc31f
parent 32960 69916a850301
child 41229 d797baa3d57c
permissions -rw-r--r--
turned show_question_marks into proper configuration option;
show_question_marks only affects regular type/term pretty printing, not raw Term.string_of_vname;
tuned;
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(*  Title:      ZF/func.thy
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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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*)
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header{*Functions, Function Spaces, Lambda-Abstraction*}
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theory func imports equalities Sum begin
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subsection{*The Pi Operator: Dependent Function Space*}
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lemma subset_Sigma_imp_relation: "r <= Sigma(A,B) ==> relation(r)"
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by (simp add: relation_def, blast)
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lemma relation_converse_converse [simp]:
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     "relation(r) ==> converse(converse(r)) = r"
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by (simp add: relation_def, blast) 
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lemma relation_restrict [simp]:  "relation(restrict(r,A))"
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by (simp add: restrict_def relation_def, blast) 
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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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subsection{*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 rule_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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subsection{*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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subsection{*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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subsection{*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 \<Union>x\<in>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 image_eq_UN: 
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  assumes f: "f \<in> Pi(A,B)" "C \<subseteq> A" shows "f``C = (\<Union>x\<in>C. {f ` x})"
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by (auto simp add: image_fun [OF f]) 
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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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subsection{*Properties of @{term "restrict(f,A)"}*}
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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 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: "f <= Sigma(A,B) ==> restrict(f,A) = f"
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by (simp add: restrict_def, blast)
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(*converse probably holds too*)
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lemma domain_restrict_idem:
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     "[| domain(r) <= A; relation(r) |] ==> restrict(r,A) = r"
paulson@13248
   318
by (simp add: restrict_def relation_def, blast)
paulson@13248
   319
paulson@13248
   320
lemma domain_restrict_lam [simp]: "domain(restrict(Lambda(A,f),C)) = A Int C"
paulson@13248
   321
apply (unfold restrict_def lam_def)
paulson@13248
   322
apply (rule equalityI)
paulson@13248
   323
apply (auto simp add: domain_iff)
paulson@13248
   324
done
paulson@13248
   325
paulson@13163
   326
lemma restrict_if [simp]: "restrict(f,A) ` a = (if a : A then f`a else 0)"
paulson@13163
   327
by (simp add: restrict apply_0)
paulson@13163
   328
paulson@13163
   329
lemma restrict_lam_eq:
paulson@13163
   330
    "A<=C ==> restrict(lam x:C. b(x), A) = (lam x:A. b(x))"
paulson@13163
   331
by (unfold restrict_def lam_def, auto)
paulson@13163
   332
paulson@13163
   333
lemma fun_cons_restrict_eq:
paulson@13163
   334
     "f : cons(a, b) -> B ==> f = cons(<a, f ` a>, restrict(f, b))"
paulson@13163
   335
apply (rule equalityI)
paulson@13248
   336
 prefer 2 apply (blast intro: apply_Pair restrict_subset [THEN subsetD])
paulson@13163
   337
apply (auto dest!: Pi_memberD simp add: restrict_def lam_def)
paulson@13163
   338
done
paulson@13163
   339
paulson@13163
   340
paulson@13355
   341
subsection{*Unions of Functions*}
paulson@13163
   342
paulson@13163
   343
(** The Union of a set of COMPATIBLE functions is a function **)
paulson@13163
   344
paulson@13163
   345
lemma function_Union:
paulson@13163
   346
    "[| ALL x:S. function(x);
paulson@13163
   347
        ALL x:S. ALL y:S. x<=y | y<=x  |]
paulson@13163
   348
     ==> function(Union(S))"
paulson@13163
   349
by (unfold function_def, blast) 
paulson@13163
   350
paulson@13163
   351
lemma fun_Union:
paulson@13163
   352
    "[| ALL f:S. EX C D. f:C->D;
paulson@13163
   353
             ALL f:S. ALL y:S. f<=y | y<=f  |] ==>
paulson@13163
   354
          Union(S) : domain(Union(S)) -> range(Union(S))"
paulson@13163
   355
apply (unfold Pi_def)
paulson@13163
   356
apply (blast intro!: rel_Union function_Union)
paulson@13163
   357
done
paulson@13163
   358
paulson@13174
   359
lemma gen_relation_Union [rule_format]:
paulson@13174
   360
     "\<forall>f\<in>F. relation(f) \<Longrightarrow> relation(Union(F))"
paulson@13174
   361
by (simp add: relation_def) 
paulson@13174
   362
paulson@13163
   363
paulson@13163
   364
(** The Union of 2 disjoint functions is a function **)
paulson@13163
   365
paulson@13163
   366
lemmas Un_rls = Un_subset_iff SUM_Un_distrib1 prod_Un_distrib2
paulson@13163
   367
                subset_trans [OF _ Un_upper1]
paulson@13163
   368
                subset_trans [OF _ Un_upper2]
paulson@13163
   369
paulson@13163
   370
lemma fun_disjoint_Un:
paulson@13163
   371
     "[| f: A->B;  g: C->D;  A Int C = 0  |]
paulson@13163
   372
      ==> (f Un g) : (A Un C) -> (B Un D)"
paulson@13163
   373
(*Prove the product and domain subgoals using distributive laws*)
paulson@13163
   374
apply (simp add: Pi_iff extension Un_rls)
paulson@13163
   375
apply (unfold function_def, blast)
paulson@13163
   376
done
paulson@13163
   377
paulson@13179
   378
lemma fun_disjoint_apply1: "a \<notin> domain(g) ==> (f Un g)`a = f`a"
paulson@13179
   379
by (simp add: apply_def, blast) 
paulson@13163
   380
paulson@13179
   381
lemma fun_disjoint_apply2: "c \<notin> domain(f) ==> (f Un g)`c = g`c"
paulson@13179
   382
by (simp add: apply_def, blast) 
paulson@13163
   383
paulson@13355
   384
subsection{*Domain and Range of a Function or Relation*}
paulson@13163
   385
paulson@13163
   386
lemma domain_of_fun: "f : Pi(A,B) ==> domain(f)=A"
paulson@13163
   387
by (unfold Pi_def, blast)
paulson@13163
   388
paulson@13163
   389
lemma apply_rangeI: "[| f : Pi(A,B);  a: A |] ==> f`a : range(f)"
paulson@13163
   390
by (erule apply_Pair [THEN rangeI], assumption)
paulson@13163
   391
paulson@13163
   392
lemma range_of_fun: "f : Pi(A,B) ==> f : A->range(f)"
paulson@13163
   393
by (blast intro: Pi_type apply_rangeI)
paulson@13163
   394
paulson@13355
   395
subsection{*Extensions of Functions*}
paulson@13163
   396
paulson@13163
   397
lemma fun_extend:
paulson@13163
   398
     "[| f: A->B;  c~:A |] ==> cons(<c,b>,f) : cons(c,A) -> cons(b,B)"
paulson@13163
   399
apply (frule singleton_fun [THEN fun_disjoint_Un], blast)
paulson@13163
   400
apply (simp add: cons_eq) 
paulson@13163
   401
done
paulson@13163
   402
paulson@13163
   403
lemma fun_extend3:
paulson@13163
   404
     "[| f: A->B;  c~:A;  b: B |] ==> cons(<c,b>,f) : cons(c,A) -> B"
paulson@13163
   405
by (blast intro: fun_extend [THEN fun_weaken_type])
paulson@13163
   406
paulson@13176
   407
lemma extend_apply:
paulson@13176
   408
     "c ~: domain(f) ==> cons(<c,b>,f)`a = (if a=c then b else f`a)"
paulson@13176
   409
by (auto simp add: apply_def) 
paulson@13163
   410
paulson@13176
   411
lemma fun_extend_apply [simp]:
paulson@13176
   412
     "[| f: A->B;  c~:A |] ==> cons(<c,b>,f)`a = (if a=c then b else f`a)" 
paulson@13176
   413
apply (rule extend_apply) 
paulson@13176
   414
apply (simp add: Pi_def, blast) 
paulson@13163
   415
done
paulson@13163
   416
paulson@13163
   417
lemmas singleton_apply = apply_equality [OF singletonI singleton_fun, simp]
paulson@13163
   418
paulson@13163
   419
(*For Finite.ML.  Inclusion of right into left is easy*)
paulson@13163
   420
lemma cons_fun_eq:
paulson@13269
   421
     "c ~: A ==> cons(c,A) -> B = (\<Union>f \<in> A->B. \<Union>b\<in>B. {cons(<c,b>, f)})"
paulson@13163
   422
apply (rule equalityI)
paulson@13163
   423
apply (safe elim!: fun_extend3)
paulson@13163
   424
(*Inclusion of left into right*)
paulson@13163
   425
apply (subgoal_tac "restrict (x, A) : A -> B")
paulson@13163
   426
 prefer 2 apply (blast intro: restrict_type2)
paulson@13163
   427
apply (rule UN_I, assumption)
paulson@13163
   428
apply (rule apply_funtype [THEN UN_I]) 
paulson@13163
   429
  apply assumption
paulson@13163
   430
 apply (rule consI1) 
paulson@13163
   431
apply (simp (no_asm))
paulson@13163
   432
apply (rule fun_extension) 
paulson@13163
   433
  apply assumption
paulson@13163
   434
 apply (blast intro: fun_extend) 
paulson@13176
   435
apply (erule consE, simp_all)
paulson@13163
   436
done
paulson@13163
   437
paulson@13269
   438
lemma succ_fun_eq: "succ(n) -> B = (\<Union>f \<in> n->B. \<Union>b\<in>B. {cons(<n,b>, f)})"
paulson@13269
   439
by (simp add: succ_def mem_not_refl cons_fun_eq)
paulson@13269
   440
paulson@13355
   441
paulson@13355
   442
subsection{*Function Updates*}
paulson@13355
   443
wenzelm@24893
   444
definition
wenzelm@24893
   445
  update  :: "[i,i,i] => i"  where
paulson@13355
   446
   "update(f,a,b) == lam x: cons(a, domain(f)). if(x=a, b, f`x)"
paulson@13355
   447
paulson@13355
   448
nonterminals
paulson@13355
   449
  updbinds  updbind
paulson@13355
   450
paulson@13355
   451
syntax
paulson@13355
   452
paulson@13355
   453
  (* Let expressions *)
paulson@13355
   454
paulson@13355
   455
  "_updbind"    :: "[i, i] => updbind"               ("(2_ :=/ _)")
paulson@13355
   456
  ""            :: "updbind => updbinds"             ("_")
paulson@13355
   457
  "_updbinds"   :: "[updbind, updbinds] => updbinds" ("_,/ _")
paulson@13355
   458
  "_Update"     :: "[i, updbinds] => i"              ("_/'((_)')" [900,0] 900)
paulson@13355
   459
paulson@13355
   460
translations
paulson@13355
   461
  "_Update (f, _updbinds(b,bs))"  == "_Update (_Update(f,b), bs)"
wenzelm@24893
   462
  "f(x:=y)"                       == "CONST update(f,x,y)"
paulson@13355
   463
paulson@13355
   464
paulson@13355
   465
lemma update_apply [simp]: "f(x:=y) ` z = (if z=x then y else f`z)"
paulson@13355
   466
apply (simp add: update_def)
paulson@14153
   467
apply (case_tac "z \<in> domain(f)")   
paulson@13355
   468
apply (simp_all add: apply_0)
paulson@13355
   469
done
paulson@13355
   470
paulson@13355
   471
lemma update_idem: "[| f`x = y;  f: Pi(A,B);  x: A |] ==> f(x:=y) = f"
paulson@13355
   472
apply (unfold update_def)
paulson@13355
   473
apply (simp add: domain_of_fun cons_absorb)
paulson@13355
   474
apply (rule fun_extension)
paulson@13355
   475
apply (best intro: apply_type if_type lam_type, assumption, simp)
paulson@13355
   476
done
paulson@13355
   477
paulson@13355
   478
paulson@13355
   479
(* [| f: Pi(A, B); x:A |] ==> f(x := f`x) = f *)
paulson@13355
   480
declare refl [THEN update_idem, simp]
paulson@13355
   481
paulson@13355
   482
lemma domain_update [simp]: "domain(f(x:=y)) = cons(x, domain(f))"
paulson@13355
   483
by (unfold update_def, simp)
paulson@13355
   484
paulson@14046
   485
lemma update_type: "[| f:Pi(A,B);  x : A;  y: B(x) |] ==> f(x:=y) : Pi(A, B)"
paulson@13355
   486
apply (unfold update_def)
paulson@13355
   487
apply (simp add: domain_of_fun cons_absorb apply_funtype lam_type)
paulson@13355
   488
done
paulson@13355
   489
paulson@13355
   490
paulson@13357
   491
subsection{*Monotonicity Theorems*}
paulson@13357
   492
paulson@13357
   493
subsubsection{*Replacement in its Various Forms*}
paulson@13357
   494
paulson@13357
   495
(*Not easy to express monotonicity in P, since any "bigger" predicate
paulson@13357
   496
  would have to be single-valued*)
paulson@13357
   497
lemma Replace_mono: "A<=B ==> Replace(A,P) <= Replace(B,P)"
paulson@13357
   498
by (blast elim!: ReplaceE)
paulson@13357
   499
paulson@13357
   500
lemma RepFun_mono: "A<=B ==> {f(x). x:A} <= {f(x). x:B}"
paulson@13357
   501
by blast
paulson@13357
   502
paulson@13357
   503
lemma Pow_mono: "A<=B ==> Pow(A) <= Pow(B)"
paulson@13357
   504
by blast
paulson@13357
   505
paulson@13357
   506
lemma Union_mono: "A<=B ==> Union(A) <= Union(B)"
paulson@13357
   507
by blast
paulson@13357
   508
paulson@13357
   509
lemma UN_mono:
paulson@13615
   510
    "[| A<=C;  !!x. x:A ==> B(x)<=D(x) |] ==> (\<Union>x\<in>A. B(x)) <= (\<Union>x\<in>C. D(x))"
paulson@13357
   511
by blast  
paulson@13357
   512
paulson@13357
   513
(*Intersection is ANTI-monotonic.  There are TWO premises! *)
paulson@14095
   514
lemma Inter_anti_mono: "[| A<=B;  A\<noteq>0 |] ==> Inter(B) <= Inter(A)"
paulson@13357
   515
by blast
paulson@13357
   516
paulson@13357
   517
lemma cons_mono: "C<=D ==> cons(a,C) <= cons(a,D)"
paulson@13357
   518
by blast
paulson@13357
   519
paulson@13357
   520
lemma Un_mono: "[| A<=C;  B<=D |] ==> A Un B <= C Un D"
paulson@13357
   521
by blast
paulson@13357
   522
paulson@13357
   523
lemma Int_mono: "[| A<=C;  B<=D |] ==> A Int B <= C Int D"
paulson@13357
   524
by blast
paulson@13357
   525
paulson@13357
   526
lemma Diff_mono: "[| A<=C;  D<=B |] ==> A-B <= C-D"
paulson@13357
   527
by blast
paulson@13357
   528
paulson@13357
   529
subsubsection{*Standard Products, Sums and Function Spaces *}
paulson@13357
   530
paulson@13357
   531
lemma Sigma_mono [rule_format]:
paulson@13357
   532
     "[| A<=C;  !!x. x:A --> B(x) <= D(x) |] ==> Sigma(A,B) <= Sigma(C,D)"
paulson@13357
   533
by blast
paulson@13357
   534
paulson@13357
   535
lemma sum_mono: "[| A<=C;  B<=D |] ==> A+B <= C+D"
paulson@13357
   536
by (unfold sum_def, blast)
paulson@13357
   537
paulson@13357
   538
(*Note that B->A and C->A are typically disjoint!*)
paulson@13357
   539
lemma Pi_mono: "B<=C ==> A->B <= A->C"
paulson@13357
   540
by (blast intro: lam_type elim: Pi_lamE)
paulson@13357
   541
paulson@13357
   542
lemma lam_mono: "A<=B ==> Lambda(A,c) <= Lambda(B,c)"
paulson@13357
   543
apply (unfold lam_def)
paulson@13357
   544
apply (erule RepFun_mono)
paulson@13357
   545
done
paulson@13357
   546
paulson@13357
   547
subsubsection{*Converse, Domain, Range, Field*}
paulson@13357
   548
paulson@13357
   549
lemma converse_mono: "r<=s ==> converse(r) <= converse(s)"
paulson@13357
   550
by blast
paulson@13357
   551
paulson@13357
   552
lemma domain_mono: "r<=s ==> domain(r)<=domain(s)"
paulson@13357
   553
by blast
paulson@13357
   554
paulson@13357
   555
lemmas domain_rel_subset = subset_trans [OF domain_mono domain_subset]
paulson@13357
   556
paulson@13357
   557
lemma range_mono: "r<=s ==> range(r)<=range(s)"
paulson@13357
   558
by blast
paulson@13357
   559
paulson@13357
   560
lemmas range_rel_subset = subset_trans [OF range_mono range_subset]
paulson@13357
   561
paulson@13357
   562
lemma field_mono: "r<=s ==> field(r)<=field(s)"
paulson@13357
   563
by blast
paulson@13357
   564
paulson@13357
   565
lemma field_rel_subset: "r <= A*A ==> field(r) <= A"
paulson@13357
   566
by (erule field_mono [THEN subset_trans], blast)
paulson@13357
   567
paulson@13357
   568
paulson@13357
   569
subsubsection{*Images*}
paulson@13357
   570
paulson@13357
   571
lemma image_pair_mono:
paulson@13357
   572
    "[| !! x y. <x,y>:r ==> <x,y>:s;  A<=B |] ==> r``A <= s``B"
paulson@13357
   573
by blast 
paulson@13357
   574
paulson@13357
   575
lemma vimage_pair_mono:
paulson@13357
   576
    "[| !! x y. <x,y>:r ==> <x,y>:s;  A<=B |] ==> r-``A <= s-``B"
paulson@13357
   577
by blast 
paulson@13357
   578
paulson@13357
   579
lemma image_mono: "[| r<=s;  A<=B |] ==> r``A <= s``B"
paulson@13357
   580
by blast
paulson@13357
   581
paulson@13357
   582
lemma vimage_mono: "[| r<=s;  A<=B |] ==> r-``A <= s-``B"
paulson@13357
   583
by blast
paulson@13357
   584
paulson@13357
   585
lemma Collect_mono:
paulson@13357
   586
    "[| A<=B;  !!x. x:A ==> P(x) --> Q(x) |] ==> Collect(A,P) <= Collect(B,Q)"
paulson@13357
   587
by blast
paulson@13357
   588
paulson@13357
   589
(*Used in intr_elim.ML and in individual datatype definitions*)
paulson@13357
   590
lemmas basic_monos = subset_refl imp_refl disj_mono conj_mono ex_mono 
paulson@13357
   591
                     Collect_mono Part_mono in_mono
paulson@13357
   592
ballarin@27702
   593
(* Useful with simp; contributed by Clemens Ballarin. *)
ballarin@27702
   594
ballarin@27702
   595
lemma bex_image_simp:
ballarin@27702
   596
  "[| f : Pi(X, Y); A \<subseteq> X |]  ==> (EX x : f``A. P(x)) <-> (EX x:A. P(f`x))"
ballarin@27702
   597
  apply safe
ballarin@27702
   598
   apply rule
ballarin@27702
   599
    prefer 2 apply assumption
ballarin@27702
   600
   apply (simp add: apply_equality)
ballarin@27702
   601
  apply (blast intro: apply_Pair)
ballarin@27702
   602
  done
ballarin@27702
   603
ballarin@27702
   604
lemma ball_image_simp:
ballarin@27702
   605
  "[| f : Pi(X, Y); A \<subseteq> X |]  ==> (ALL x : f``A. P(x)) <-> (ALL x:A. P(f`x))"
ballarin@27702
   606
  apply safe
ballarin@27702
   607
   apply (blast intro: apply_Pair)
ballarin@27702
   608
  apply (drule bspec) apply assumption
ballarin@27702
   609
  apply (simp add: apply_equality)
ballarin@27702
   610
  done
ballarin@27702
   611
paulson@13163
   612
end