src/ZF/func.thy
author wenzelm
Tue Sep 01 22:32:58 2015 +0200 (2015-09-01)
changeset 61076 bdc1e2f0a86a
parent 60770 240563fbf41d
child 63901 4ce989e962e0
permissions -rw-r--r--
eliminated \<Colon>;
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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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section\<open>Functions, Function Spaces, Lambda-Abstraction\<close>
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theory func imports equalities Sum begin
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subsection\<open>The Pi Operator: Dependent Function Space\<close>
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lemma subset_Sigma_imp_relation: "r \<subseteq> 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 \<in> Pi(A,B) \<longleftrightarrow> 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 \<in> Pi(A,B) \<longleftrightarrow> f<=Sigma(A,B) & (\<forall>x\<in>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 \<in> 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 \<in> Pi(A,B) ==> f \<subseteq> 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 \<in> 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 \<in> A->B;  B<=D |] ==> f \<in> A->D"
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by (unfold Pi_def, best)
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subsection\<open>Function Application\<close>
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lemma apply_equality2: "[| <a,b>: f;  <a,c>: f;  f \<in> 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 \<in> 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 \<notin> domain(f) ==> f`a = 0"
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by (unfold apply_def, blast)
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lemma Pi_memberD: "[| f \<in> Pi(A,B);  c \<in> f |] ==> \<exists>x\<in>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 \<in> 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 \<in> Pi(A,B);  a \<in> 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 \<in> Pi(A,B);  a \<in> A |] ==> f`a \<in> 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 \<in> A->B;  a \<in> A |] ==> f`a \<in> B"
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by (blast dest: apply_type)
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lemma apply_iff: "f \<in> Pi(A,B) ==> <a,b>: f \<longleftrightarrow> a \<in> 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 \<in> Pi(A,C);  !!x. x \<in> A ==> f`x \<in> B(x) |] ==> f \<in> 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 \<in> Pi(A, %x. {y \<in> B(x). P(x,y)}))
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      \<longleftrightarrow>  f \<in> Pi(A,B) & (\<forall>x\<in>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 \<in> Pi(A,B);  !!x. x \<in> A ==> B(x)<=C(x) |] ==> f \<in> 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> \<in> f;  f \<in> Pi(A,B) |] ==> a \<in> A"
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by (blast dest: fun_is_rel)
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lemma range_type: "[| <a,b> \<in> f;  f \<in> Pi(A,B) |] ==> b \<in> B(a)"
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by (blast dest: fun_is_rel)
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lemma Pair_mem_PiD: "[| <a,b>: f;  f \<in> Pi(A,B) |] ==> a \<in> A & b \<in> B(a) & f`a = b"
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by (blast intro: domain_type range_type apply_equality)
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subsection\<open>Lambda Abstraction\<close>
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lemma lamI: "a \<in> A ==> <a,b(a)> \<in> (\<lambda>x\<in>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: (\<lambda>x\<in>A. b(x));  !!x.[| x \<in> 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>: (\<lambda>x\<in>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 \<in> A ==> b(x): B(x) |] ==> (\<lambda>x\<in>A. b(x)) \<in> Pi(A,B)"
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by (simp add: lam_def Pi_def function_def, blast)
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lemma lam_funtype: "(\<lambda>x\<in>A. b(x)) \<in> A -> {b(x). x \<in> A}"
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by (blast intro: lam_type)
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lemma function_lam: "function (\<lambda>x\<in>A. b(x))"
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by (simp add: function_def lam_def)
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lemma relation_lam: "relation (\<lambda>x\<in>A. b(x))"
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by (simp add: relation_def lam_def)
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lemma beta_if [simp]: "(\<lambda>x\<in>A. b(x)) ` a = (if a \<in> 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 \<in> A ==> (\<lambda>x\<in>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]: "(\<lambda>x\<in>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 \<in> 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 \<in> A ==> EX! y. Q(x,y)) ==> \<exists>f. \<forall>x\<in>A. Q(x, f`x)"
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apply (rule_tac x = "\<lambda>x\<in>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: "[| (\<lambda>x\<in>A. f(x)) = (\<lambda>x\<in>A. g(x));  a \<in> 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>} \<in> {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\<open>Extensionality\<close>
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(*Semi-extensionality!*)
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lemma fun_subset:
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    "[| f \<in> Pi(A,B);  g \<in> Pi(C,D);  A<=C;
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        !!x. x \<in> 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 \<in> Pi(A,B);  g \<in> Pi(A,D);
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        !!x. x \<in> 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 \<in> Pi(A,B) ==> (\<lambda>x\<in>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 \<in> Pi(A,B); g \<in> Pi(A,C) |] ==> (\<forall>a\<in>A. f`a = g`a) \<longleftrightarrow> 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 \<in> Pi(A,B); g \<in> Pi(A,C) |] ==> f \<subseteq> g \<longleftrightarrow> (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 \<in> Pi(A,B)"
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      and minor: "!!b. [| \<forall>x\<in>A. b(x):B(x);  f = (\<lambda>x\<in>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\<open>Images of Functions\<close>
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lemma image_lam: "C \<subseteq> A ==> (\<lambda>x\<in>A. b(x)) `` C = {b(x). x \<in> 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 \<in> C} = (if C \<subseteq> 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 \<subseteq> domain(f) |] ==> f``C = {f`x. x \<in> C}"
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by (simp add: Repfun_function_if)
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lemma image_fun: "[| f \<in> Pi(A,B);  C \<subseteq> A |] ==> f``C = {f`x. x \<in> 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 \<in> Pi(A,B);  x \<in> 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\<open>Properties of @{term "restrict(f,A)"}\<close>
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lemma restrict_subset: "restrict(f,A) \<subseteq> 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 \<in> Pi(C,B);  A<=C |] ==> restrict(f,A) \<in> 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 \<in> 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 \<inter> B)"
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by (unfold restrict_def, blast)
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lemma domain_restrict [simp]: "domain(restrict(f,C)) = domain(f) \<inter> C"
paulson@13163
   307
apply (unfold restrict_def)
paulson@13163
   308
apply (auto simp add: domain_def)
paulson@13163
   309
done
paulson@13163
   310
paulson@46820
   311
lemma restrict_idem: "f \<subseteq> Sigma(A,B) ==> restrict(f,A) = f"
paulson@13163
   312
by (simp add: restrict_def, blast)
paulson@13163
   313
paulson@13248
   314
paulson@13248
   315
(*converse probably holds too*)
paulson@13248
   316
lemma domain_restrict_idem:
paulson@46820
   317
     "[| domain(r) \<subseteq> A; relation(r) |] ==> restrict(r,A) = r"
paulson@13248
   318
by (simp add: restrict_def relation_def, blast)
paulson@13248
   319
paulson@46820
   320
lemma domain_restrict_lam [simp]: "domain(restrict(Lambda(A,f),C)) = A \<inter> 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@46820
   326
lemma restrict_if [simp]: "restrict(f,A) ` a = (if a \<in> 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@46820
   330
    "A<=C ==> restrict(\<lambda>x\<in>C. b(x), A) = (\<lambda>x\<in>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@46820
   334
     "f \<in> 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
wenzelm@60770
   341
subsection\<open>Unions of Functions\<close>
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@46820
   346
    "[| \<forall>x\<in>S. function(x);
paulson@46820
   347
        \<forall>x\<in>S. \<forall>y\<in>S. x<=y | y<=x  |]
paulson@46820
   348
     ==> function(\<Union>(S))"
paulson@46953
   349
by (unfold function_def, blast)
paulson@13163
   350
paulson@13163
   351
lemma fun_Union:
paulson@46953
   352
    "[| \<forall>f\<in>S. \<exists>C D. f \<in> C->D;
paulson@46820
   353
             \<forall>f\<in>S. \<forall>y\<in>S. f<=y | y<=f  |] ==>
paulson@46820
   354
          \<Union>(S) \<in> 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@46820
   360
     "\<forall>f\<in>F. relation(f) \<Longrightarrow> relation(\<Union>(F))"
paulson@46953
   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@46953
   371
     "[| f \<in> A->B;  g \<in> C->D;  A \<inter> C = 0  |]
paulson@46820
   372
      ==> (f \<union> g) \<in> (A \<union> C) -> (B \<union> 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@46820
   378
lemma fun_disjoint_apply1: "a \<notin> domain(g) ==> (f \<union> g)`a = f`a"
paulson@46953
   379
by (simp add: apply_def, blast)
paulson@13163
   380
paulson@46820
   381
lemma fun_disjoint_apply2: "c \<notin> domain(f) ==> (f \<union> g)`c = g`c"
paulson@46953
   382
by (simp add: apply_def, blast)
paulson@13163
   383
wenzelm@60770
   384
subsection\<open>Domain and Range of a Function or Relation\<close>
paulson@13163
   385
paulson@46820
   386
lemma domain_of_fun: "f \<in> Pi(A,B) ==> domain(f)=A"
paulson@13163
   387
by (unfold Pi_def, blast)
paulson@13163
   388
paulson@46953
   389
lemma apply_rangeI: "[| f \<in> Pi(A,B);  a \<in> A |] ==> f`a \<in> range(f)"
paulson@13163
   390
by (erule apply_Pair [THEN rangeI], assumption)
paulson@13163
   391
paulson@46820
   392
lemma range_of_fun: "f \<in> Pi(A,B) ==> f \<in> A->range(f)"
paulson@13163
   393
by (blast intro: Pi_type apply_rangeI)
paulson@13163
   394
wenzelm@60770
   395
subsection\<open>Extensions of Functions\<close>
paulson@13163
   396
paulson@13163
   397
lemma fun_extend:
paulson@46953
   398
     "[| f \<in> A->B;  c\<notin>A |] ==> cons(<c,b>,f) \<in> cons(c,A) -> cons(b,B)"
paulson@13163
   399
apply (frule singleton_fun [THEN fun_disjoint_Un], blast)
paulson@46953
   400
apply (simp add: cons_eq)
paulson@13163
   401
done
paulson@13163
   402
paulson@13163
   403
lemma fun_extend3:
paulson@46953
   404
     "[| f \<in> A->B;  c\<notin>A;  b \<in> B |] ==> cons(<c,b>,f) \<in> 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@46820
   408
     "c \<notin> domain(f) ==> cons(<c,b>,f)`a = (if a=c then b else f`a)"
paulson@46953
   409
by (auto simp add: apply_def)
paulson@13163
   410
paulson@13176
   411
lemma fun_extend_apply [simp]:
paulson@46953
   412
     "[| f \<in> A->B;  c\<notin>A |] ==> cons(<c,b>,f)`a = (if a=c then b else f`a)"
paulson@46953
   413
apply (rule extend_apply)
paulson@46953
   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@46820
   421
     "c \<notin> 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@46820
   425
apply (subgoal_tac "restrict (x, A) \<in> A -> B")
paulson@13163
   426
 prefer 2 apply (blast intro: restrict_type2)
paulson@13163
   427
apply (rule UN_I, assumption)
paulson@46953
   428
apply (rule apply_funtype [THEN UN_I])
paulson@13163
   429
  apply assumption
paulson@46953
   430
 apply (rule consI1)
paulson@13163
   431
apply (simp (no_asm))
paulson@46953
   432
apply (rule fun_extension)
paulson@13163
   433
  apply assumption
paulson@46953
   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
wenzelm@60770
   442
subsection\<open>Function Updates\<close>
paulson@13355
   443
wenzelm@24893
   444
definition
wenzelm@24893
   445
  update  :: "[i,i,i] => i"  where
paulson@46820
   446
   "update(f,a,b) == \<lambda>x\<in>cons(a, domain(f)). if(x=a, b, f`x)"
paulson@13355
   447
wenzelm@41229
   448
nonterminal updbinds and updbind
paulson@13355
   449
paulson@13355
   450
syntax
paulson@13355
   451
paulson@13355
   452
  (* Let expressions *)
paulson@13355
   453
paulson@13355
   454
  "_updbind"    :: "[i, i] => updbind"               ("(2_ :=/ _)")
paulson@13355
   455
  ""            :: "updbind => updbinds"             ("_")
paulson@13355
   456
  "_updbinds"   :: "[updbind, updbinds] => updbinds" ("_,/ _")
paulson@13355
   457
  "_Update"     :: "[i, updbinds] => i"              ("_/'((_)')" [900,0] 900)
paulson@13355
   458
paulson@13355
   459
translations
paulson@13355
   460
  "_Update (f, _updbinds(b,bs))"  == "_Update (_Update(f,b), bs)"
wenzelm@24893
   461
  "f(x:=y)"                       == "CONST update(f,x,y)"
paulson@13355
   462
paulson@13355
   463
paulson@13355
   464
lemma update_apply [simp]: "f(x:=y) ` z = (if z=x then y else f`z)"
paulson@13355
   465
apply (simp add: update_def)
paulson@46953
   466
apply (case_tac "z \<in> domain(f)")
paulson@13355
   467
apply (simp_all add: apply_0)
paulson@13355
   468
done
paulson@13355
   469
paulson@46953
   470
lemma update_idem: "[| f`x = y;  f \<in> Pi(A,B);  x \<in> A |] ==> f(x:=y) = f"
paulson@13355
   471
apply (unfold update_def)
paulson@13355
   472
apply (simp add: domain_of_fun cons_absorb)
paulson@13355
   473
apply (rule fun_extension)
paulson@13355
   474
apply (best intro: apply_type if_type lam_type, assumption, simp)
paulson@13355
   475
done
paulson@13355
   476
paulson@13355
   477
paulson@46953
   478
(* [| f \<in> Pi(A, B); x \<in> A |] ==> f(x := f`x) = f *)
paulson@13355
   479
declare refl [THEN update_idem, simp]
paulson@13355
   480
paulson@13355
   481
lemma domain_update [simp]: "domain(f(x:=y)) = cons(x, domain(f))"
paulson@13355
   482
by (unfold update_def, simp)
paulson@13355
   483
paulson@46953
   484
lemma update_type: "[| f \<in> Pi(A,B);  x \<in> A;  y \<in> B(x) |] ==> f(x:=y) \<in> Pi(A, B)"
paulson@13355
   485
apply (unfold update_def)
paulson@13355
   486
apply (simp add: domain_of_fun cons_absorb apply_funtype lam_type)
paulson@13355
   487
done
paulson@13355
   488
paulson@13355
   489
wenzelm@60770
   490
subsection\<open>Monotonicity Theorems\<close>
paulson@13357
   491
wenzelm@60770
   492
subsubsection\<open>Replacement in its Various Forms\<close>
paulson@13357
   493
paulson@13357
   494
(*Not easy to express monotonicity in P, since any "bigger" predicate
paulson@13357
   495
  would have to be single-valued*)
paulson@46820
   496
lemma Replace_mono: "A<=B ==> Replace(A,P) \<subseteq> Replace(B,P)"
paulson@13357
   497
by (blast elim!: ReplaceE)
paulson@13357
   498
paulson@46953
   499
lemma RepFun_mono: "A<=B ==> {f(x). x \<in> A} \<subseteq> {f(x). x \<in> B}"
paulson@13357
   500
by blast
paulson@13357
   501
paulson@46820
   502
lemma Pow_mono: "A<=B ==> Pow(A) \<subseteq> Pow(B)"
paulson@13357
   503
by blast
paulson@13357
   504
paulson@46820
   505
lemma Union_mono: "A<=B ==> \<Union>(A) \<subseteq> \<Union>(B)"
paulson@13357
   506
by blast
paulson@13357
   507
paulson@13357
   508
lemma UN_mono:
paulson@46953
   509
    "[| A<=C;  !!x. x \<in> A ==> B(x)<=D(x) |] ==> (\<Union>x\<in>A. B(x)) \<subseteq> (\<Union>x\<in>C. D(x))"
paulson@46953
   510
by blast
paulson@13357
   511
paulson@13357
   512
(*Intersection is ANTI-monotonic.  There are TWO premises! *)
paulson@46820
   513
lemma Inter_anti_mono: "[| A<=B;  A\<noteq>0 |] ==> \<Inter>(B) \<subseteq> \<Inter>(A)"
paulson@13357
   514
by blast
paulson@13357
   515
paulson@46820
   516
lemma cons_mono: "C<=D ==> cons(a,C) \<subseteq> cons(a,D)"
paulson@13357
   517
by blast
paulson@13357
   518
paulson@46820
   519
lemma Un_mono: "[| A<=C;  B<=D |] ==> A \<union> B \<subseteq> C \<union> D"
paulson@13357
   520
by blast
paulson@13357
   521
paulson@46820
   522
lemma Int_mono: "[| A<=C;  B<=D |] ==> A \<inter> B \<subseteq> C \<inter> D"
paulson@13357
   523
by blast
paulson@13357
   524
paulson@46820
   525
lemma Diff_mono: "[| A<=C;  D<=B |] ==> A-B \<subseteq> C-D"
paulson@13357
   526
by blast
paulson@13357
   527
wenzelm@60770
   528
subsubsection\<open>Standard Products, Sums and Function Spaces\<close>
paulson@13357
   529
paulson@13357
   530
lemma Sigma_mono [rule_format]:
paulson@46953
   531
     "[| A<=C;  !!x. x \<in> A \<longrightarrow> B(x) \<subseteq> D(x) |] ==> Sigma(A,B) \<subseteq> Sigma(C,D)"
paulson@13357
   532
by blast
paulson@13357
   533
paulson@46820
   534
lemma sum_mono: "[| A<=C;  B<=D |] ==> A+B \<subseteq> C+D"
paulson@13357
   535
by (unfold sum_def, blast)
paulson@13357
   536
paulson@13357
   537
(*Note that B->A and C->A are typically disjoint!*)
paulson@46820
   538
lemma Pi_mono: "B<=C ==> A->B \<subseteq> A->C"
paulson@13357
   539
by (blast intro: lam_type elim: Pi_lamE)
paulson@13357
   540
paulson@46820
   541
lemma lam_mono: "A<=B ==> Lambda(A,c) \<subseteq> Lambda(B,c)"
paulson@13357
   542
apply (unfold lam_def)
paulson@13357
   543
apply (erule RepFun_mono)
paulson@13357
   544
done
paulson@13357
   545
wenzelm@60770
   546
subsubsection\<open>Converse, Domain, Range, Field\<close>
paulson@13357
   547
paulson@46820
   548
lemma converse_mono: "r<=s ==> converse(r) \<subseteq> converse(s)"
paulson@13357
   549
by blast
paulson@13357
   550
paulson@13357
   551
lemma domain_mono: "r<=s ==> domain(r)<=domain(s)"
paulson@13357
   552
by blast
paulson@13357
   553
paulson@13357
   554
lemmas domain_rel_subset = subset_trans [OF domain_mono domain_subset]
paulson@13357
   555
paulson@13357
   556
lemma range_mono: "r<=s ==> range(r)<=range(s)"
paulson@13357
   557
by blast
paulson@13357
   558
paulson@13357
   559
lemmas range_rel_subset = subset_trans [OF range_mono range_subset]
paulson@13357
   560
paulson@13357
   561
lemma field_mono: "r<=s ==> field(r)<=field(s)"
paulson@13357
   562
by blast
paulson@13357
   563
paulson@46820
   564
lemma field_rel_subset: "r \<subseteq> A*A ==> field(r) \<subseteq> A"
paulson@13357
   565
by (erule field_mono [THEN subset_trans], blast)
paulson@13357
   566
paulson@13357
   567
wenzelm@60770
   568
subsubsection\<open>Images\<close>
paulson@13357
   569
paulson@13357
   570
lemma image_pair_mono:
paulson@46820
   571
    "[| !! x y. <x,y>:r ==> <x,y>:s;  A<=B |] ==> r``A \<subseteq> s``B"
paulson@46953
   572
by blast
paulson@13357
   573
paulson@13357
   574
lemma vimage_pair_mono:
paulson@46820
   575
    "[| !! x y. <x,y>:r ==> <x,y>:s;  A<=B |] ==> r-``A \<subseteq> s-``B"
paulson@46953
   576
by blast
paulson@13357
   577
paulson@46820
   578
lemma image_mono: "[| r<=s;  A<=B |] ==> r``A \<subseteq> s``B"
paulson@13357
   579
by blast
paulson@13357
   580
paulson@46820
   581
lemma vimage_mono: "[| r<=s;  A<=B |] ==> r-``A \<subseteq> s-``B"
paulson@13357
   582
by blast
paulson@13357
   583
paulson@13357
   584
lemma Collect_mono:
paulson@46953
   585
    "[| A<=B;  !!x. x \<in> A ==> P(x) \<longrightarrow> Q(x) |] ==> Collect(A,P) \<subseteq> Collect(B,Q)"
paulson@13357
   586
by blast
paulson@13357
   587
paulson@13357
   588
(*Used in intr_elim.ML and in individual datatype definitions*)
paulson@46953
   589
lemmas basic_monos = subset_refl imp_refl disj_mono conj_mono ex_mono
paulson@13357
   590
                     Collect_mono Part_mono in_mono
paulson@13357
   591
ballarin@27702
   592
(* Useful with simp; contributed by Clemens Ballarin. *)
ballarin@27702
   593
ballarin@27702
   594
lemma bex_image_simp:
paulson@46821
   595
  "[| f \<in> Pi(X, Y); A \<subseteq> X |]  ==> (\<exists>x\<in>f``A. P(x)) \<longleftrightarrow> (\<exists>x\<in>A. P(f`x))"
ballarin@27702
   596
  apply safe
ballarin@27702
   597
   apply rule
ballarin@27702
   598
    prefer 2 apply assumption
ballarin@27702
   599
   apply (simp add: apply_equality)
ballarin@27702
   600
  apply (blast intro: apply_Pair)
ballarin@27702
   601
  done
ballarin@27702
   602
ballarin@27702
   603
lemma ball_image_simp:
paulson@46821
   604
  "[| f \<in> Pi(X, Y); A \<subseteq> X |]  ==> (\<forall>x\<in>f``A. P(x)) \<longleftrightarrow> (\<forall>x\<in>A. P(f`x))"
ballarin@27702
   605
  apply safe
ballarin@27702
   606
   apply (blast intro: apply_Pair)
ballarin@27702
   607
  apply (drule bspec) apply assumption
ballarin@27702
   608
  apply (simp add: apply_equality)
ballarin@27702
   609
  done
ballarin@27702
   610
paulson@13163
   611
end