src/ZF/func.thy
author wenzelm
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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 \<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{*Function Application*}
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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{*Lambda Abstraction*}
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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{*Extensionality*}
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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{*Images of Functions*}
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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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124
858ab9a9b047 made pseudo theories for all ML files;
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subsection{*Properties of @{term "restrict(f,A)"}*}
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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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   295
e320a52ff711 converted Arith, Univ, func to Isar format!
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   296
lemma restrict_empty [simp]: "restrict(f,0) = 0"
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   297
by (unfold restrict_def, simp)
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diff changeset
   298
13172
03a5afa7b888 tidying up
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   299
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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   300
by (simp add: restrict_def)
13172
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parents: 13168
diff changeset
   301
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   302
lemma restrict_restrict [simp]:
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   303
     "restrict(restrict(r,A),B) = restrict(r, A \<inter> B)"
13163
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diff changeset
   304
by (unfold restrict_def, blast)
e320a52ff711 converted Arith, Univ, func to Isar format!
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parents: 2469
diff changeset
   305
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   306
lemma domain_restrict [simp]: "domain(restrict(f,C)) = domain(f) \<inter> C"
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paulson
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   307
apply (unfold restrict_def)
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paulson
parents: 2469
diff changeset
   308
apply (auto simp add: domain_def)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   309
done
e320a52ff711 converted Arith, Univ, func to Isar format!
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   310
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   311
lemma restrict_idem: "f \<subseteq> Sigma(A,B) ==> restrict(f,A) = f"
13163
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diff changeset
   312
by (simp add: restrict_def, blast)
e320a52ff711 converted Arith, Univ, func to Isar format!
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   313
13248
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   314
ae66c22ed52e new theorems
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   315
(*converse probably holds too*)
ae66c22ed52e new theorems
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   316
lemma domain_restrict_idem:
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   317
     "[| domain(r) \<subseteq> A; relation(r) |] ==> restrict(r,A) = r"
13248
ae66c22ed52e new theorems
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parents: 13221
diff changeset
   318
by (simp add: restrict_def relation_def, blast)
ae66c22ed52e new theorems
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parents: 13221
diff changeset
   319
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diff changeset
   320
lemma domain_restrict_lam [simp]: "domain(restrict(Lambda(A,f),C)) = A \<inter> C"
13248
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paulson
parents: 13221
diff changeset
   321
apply (unfold restrict_def lam_def)
ae66c22ed52e new theorems
paulson
parents: 13221
diff changeset
   322
apply (rule equalityI)
ae66c22ed52e new theorems
paulson
parents: 13221
diff changeset
   323
apply (auto simp add: domain_iff)
ae66c22ed52e new theorems
paulson
parents: 13221
diff changeset
   324
done
ae66c22ed52e new theorems
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parents: 13221
diff changeset
   325
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diff changeset
   326
lemma restrict_if [simp]: "restrict(f,A) ` a = (if a \<in> A then f`a else 0)"
13163
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paulson
parents: 2469
diff changeset
   327
by (simp add: restrict apply_0)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   328
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   329
lemma restrict_lam_eq:
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diff changeset
   330
    "A<=C ==> restrict(\<lambda>x\<in>C. b(x), A) = (\<lambda>x\<in>A. b(x))"
13163
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paulson
parents: 2469
diff changeset
   331
by (unfold restrict_def lam_def, auto)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   332
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   333
lemma fun_cons_restrict_eq:
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parents: 41229
diff changeset
   334
     "f \<in> cons(a, b) -> B ==> f = cons(<a, f ` a>, restrict(f, b))"
13163
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paulson
parents: 2469
diff changeset
   335
apply (rule equalityI)
13248
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paulson
parents: 13221
diff changeset
   336
 prefer 2 apply (blast intro: apply_Pair restrict_subset [THEN subsetD])
13163
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paulson
parents: 2469
diff changeset
   337
apply (auto dest!: Pi_memberD simp add: restrict_def lam_def)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   338
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   339
e320a52ff711 converted Arith, Univ, func to Isar format!
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parents: 2469
diff changeset
   340
13355
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   341
subsection{*Unions of Functions*}
13163
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paulson
parents: 2469
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   342
e320a52ff711 converted Arith, Univ, func to Isar format!
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   343
(** The Union of a set of COMPATIBLE functions is a function **)
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   344
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
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   345
lemma function_Union:
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   346
    "[| \<forall>x\<in>S. function(x);
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parents: 41229
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   347
        \<forall>x\<in>S. \<forall>y\<in>S. x<=y | y<=x  |]
c656222c4dc1 mathematical symbols instead of ASCII
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parents: 41229
diff changeset
   348
     ==> function(\<Union>(S))"
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parents: 46821
diff changeset
   349
by (unfold function_def, blast)
13163
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paulson
parents: 2469
diff changeset
   350
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   351
lemma fun_Union:
46953
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parents: 46821
diff changeset
   352
    "[| \<forall>f\<in>S. \<exists>C D. f \<in> C->D;
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parents: 41229
diff changeset
   353
             \<forall>f\<in>S. \<forall>y\<in>S. f<=y | y<=f  |] ==>
c656222c4dc1 mathematical symbols instead of ASCII
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parents: 41229
diff changeset
   354
          \<Union>(S) \<in> domain(\<Union>(S)) -> range(\<Union>(S))"
13163
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paulson
parents: 2469
diff changeset
   355
apply (unfold Pi_def)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   356
apply (blast intro!: rel_Union function_Union)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   357
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   358
13174
85d3c0981a16 more tidying
paulson
parents: 13172
diff changeset
   359
lemma gen_relation_Union [rule_format]:
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parents: 41229
diff changeset
   360
     "\<forall>f\<in>F. relation(f) \<Longrightarrow> relation(\<Union>(F))"
46953
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paulson
parents: 46821
diff changeset
   361
by (simp add: relation_def)
13174
85d3c0981a16 more tidying
paulson
parents: 13172
diff changeset
   362
13163
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paulson
parents: 2469
diff changeset
   363
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   364
(** The Union of 2 disjoint functions is a function **)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   365
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   366
lemmas Un_rls = Un_subset_iff SUM_Un_distrib1 prod_Un_distrib2
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   367
                subset_trans [OF _ Un_upper1]
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   368
                subset_trans [OF _ Un_upper2]
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   369
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   370
lemma fun_disjoint_Un:
46953
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paulson
parents: 46821
diff changeset
   371
     "[| f \<in> A->B;  g \<in> C->D;  A \<inter> C = 0  |]
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paulson
parents: 41229
diff changeset
   372
      ==> (f \<union> g) \<in> (A \<union> C) -> (B \<union> D)"
13163
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paulson
parents: 2469
diff changeset
   373
(*Prove the product and domain subgoals using distributive laws*)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   374
apply (simp add: Pi_iff extension Un_rls)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   375
apply (unfold function_def, blast)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   376
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   377
46820
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paulson
parents: 41229
diff changeset
   378
lemma fun_disjoint_apply1: "a \<notin> domain(g) ==> (f \<union> g)`a = f`a"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   379
by (simp add: apply_def, blast)
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   380
46820
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paulson
parents: 41229
diff changeset
   381
lemma fun_disjoint_apply2: "c \<notin> domain(f) ==> (f \<union> g)`c = g`c"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   382
by (simp add: apply_def, blast)
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   383
13355
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   384
subsection{*Domain and Range of a Function or Relation*}
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   385
46820
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paulson
parents: 41229
diff changeset
   386
lemma domain_of_fun: "f \<in> Pi(A,B) ==> domain(f)=A"
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   387
by (unfold Pi_def, blast)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   388
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   389
lemma apply_rangeI: "[| f \<in> Pi(A,B);  a \<in> A |] ==> f`a \<in> range(f)"
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   390
by (erule apply_Pair [THEN rangeI], assumption)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   391
46820
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paulson
parents: 41229
diff changeset
   392
lemma range_of_fun: "f \<in> Pi(A,B) ==> f \<in> A->range(f)"
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   393
by (blast intro: Pi_type apply_rangeI)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   394
13355
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   395
subsection{*Extensions of Functions*}
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   396
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   397
lemma fun_extend:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   398
     "[| f \<in> A->B;  c\<notin>A |] ==> cons(<c,b>,f) \<in> cons(c,A) -> cons(b,B)"
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   399
apply (frule singleton_fun [THEN fun_disjoint_Un], blast)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   400
apply (simp add: cons_eq)
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   401
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   402
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   403
lemma fun_extend3:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   404
     "[| f \<in> A->B;  c\<notin>A;  b \<in> B |] ==> cons(<c,b>,f) \<in> cons(c,A) -> B"
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   405
by (blast intro: fun_extend [THEN fun_weaken_type])
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   406
13176
312bd350579b conversion of Perm to Isar. Strengthening of comp_fun_apply
paulson
parents: 13175
diff changeset
   407
lemma extend_apply:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   408
     "c \<notin> domain(f) ==> cons(<c,b>,f)`a = (if a=c then b else f`a)"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   409
by (auto simp add: apply_def)
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   410
13176
312bd350579b conversion of Perm to Isar. Strengthening of comp_fun_apply
paulson
parents: 13175
diff changeset
   411
lemma fun_extend_apply [simp]:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   412
     "[| f \<in> A->B;  c\<notin>A |] ==> cons(<c,b>,f)`a = (if a=c then b else f`a)"
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   413
apply (rule extend_apply)
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   414
apply (simp add: Pi_def, blast)
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   415
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   416
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   417
lemmas singleton_apply = apply_equality [OF singletonI singleton_fun, simp]
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   418
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   419
(*For Finite.ML.  Inclusion of right into left is easy*)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   420
lemma cons_fun_eq:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   421
     "c \<notin> A ==> cons(c,A) -> B = (\<Union>f \<in> A->B. \<Union>b\<in>B. {cons(<c,b>, f)})"
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   422
apply (rule equalityI)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   423
apply (safe elim!: fun_extend3)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   424
(*Inclusion of left into right*)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   425
apply (subgoal_tac "restrict (x, A) \<in> A -> B")
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   426
 prefer 2 apply (blast intro: restrict_type2)
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   427
apply (rule UN_I, assumption)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   428
apply (rule apply_funtype [THEN UN_I])
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   429
  apply assumption
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   430
 apply (rule consI1)
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   431
apply (simp (no_asm))
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   432
apply (rule fun_extension)
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   433
  apply assumption
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   434
 apply (blast intro: fun_extend)
13176
312bd350579b conversion of Perm to Isar. Strengthening of comp_fun_apply
paulson
parents: 13175
diff changeset
   435
apply (erule consE, simp_all)
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   436
done
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   437
13269
3ba9be497c33 Tidying and introduction of various new theorems
paulson
parents: 13248
diff changeset
   438
lemma succ_fun_eq: "succ(n) -> B = (\<Union>f \<in> n->B. \<Union>b\<in>B. {cons(<n,b>, f)})"
3ba9be497c33 Tidying and introduction of various new theorems
paulson
parents: 13248
diff changeset
   439
by (simp add: succ_def mem_not_refl cons_fun_eq)
3ba9be497c33 Tidying and introduction of various new theorems
paulson
parents: 13248
diff changeset
   440
13355
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   441
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   442
subsection{*Function Updates*}
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   443
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 16417
diff changeset
   444
definition
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 16417
diff changeset
   445
  update  :: "[i,i,i] => i"  where
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   446
   "update(f,a,b) == \<lambda>x\<in>cons(a, domain(f)). if(x=a, b, f`x)"
13355
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   447
41229
d797baa3d57c replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents: 32960
diff changeset
   448
nonterminal updbinds and updbind
13355
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   449
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   450
syntax
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   451
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   452
  (* Let expressions *)
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   453
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   454
  "_updbind"    :: "[i, i] => updbind"               ("(2_ :=/ _)")
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   455
  ""            :: "updbind => updbinds"             ("_")
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   456
  "_updbinds"   :: "[updbind, updbinds] => updbinds" ("_,/ _")
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   457
  "_Update"     :: "[i, updbinds] => i"              ("_/'((_)')" [900,0] 900)
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   458
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   459
translations
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   460
  "_Update (f, _updbinds(b,bs))"  == "_Update (_Update(f,b), bs)"
24893
b8ef7afe3a6b modernized specifications;
wenzelm
parents: 16417
diff changeset
   461
  "f(x:=y)"                       == "CONST update(f,x,y)"
13355
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   462
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   463
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   464
lemma update_apply [simp]: "f(x:=y) ` z = (if z=x then y else f`z)"
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   465
apply (simp add: update_def)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   466
apply (case_tac "z \<in> domain(f)")
13355
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   467
apply (simp_all add: apply_0)
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   468
done
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   469
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   470
lemma update_idem: "[| f`x = y;  f \<in> Pi(A,B);  x \<in> A |] ==> f(x:=y) = f"
13355
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   471
apply (unfold update_def)
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   472
apply (simp add: domain_of_fun cons_absorb)
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   473
apply (rule fun_extension)
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   474
apply (best intro: apply_type if_type lam_type, assumption, simp)
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   475
done
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   476
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   477
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   478
(* [| f \<in> Pi(A, B); x \<in> A |] ==> f(x := f`x) = f *)
13355
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   479
declare refl [THEN update_idem, simp]
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   480
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   481
lemma domain_update [simp]: "domain(f(x:=y)) = cons(x, domain(f))"
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   482
by (unfold update_def, simp)
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   483
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   484
lemma update_type: "[| f \<in> Pi(A,B);  x \<in> A;  y \<in> B(x) |] ==> f(x:=y) \<in> Pi(A, B)"
13355
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   485
apply (unfold update_def)
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   486
apply (simp add: domain_of_fun cons_absorb apply_funtype lam_type)
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   487
done
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   488
d19cdbd8b559 merged Update with func
paulson
parents: 13269
diff changeset
   489
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   490
subsection{*Monotonicity Theorems*}
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   491
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   492
subsubsection{*Replacement in its Various Forms*}
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   493
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   494
(*Not easy to express monotonicity in P, since any "bigger" predicate
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   495
  would have to be single-valued*)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   496
lemma Replace_mono: "A<=B ==> Replace(A,P) \<subseteq> Replace(B,P)"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   497
by (blast elim!: ReplaceE)
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   498
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   499
lemma RepFun_mono: "A<=B ==> {f(x). x \<in> A} \<subseteq> {f(x). x \<in> B}"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   500
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   501
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   502
lemma Pow_mono: "A<=B ==> Pow(A) \<subseteq> Pow(B)"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   503
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   504
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   505
lemma Union_mono: "A<=B ==> \<Union>(A) \<subseteq> \<Union>(B)"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   506
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   507
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   508
lemma UN_mono:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   509
    "[| A<=C;  !!x. x \<in> A ==> B(x)<=D(x) |] ==> (\<Union>x\<in>A. B(x)) \<subseteq> (\<Union>x\<in>C. D(x))"
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   510
by blast
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   511
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   512
(*Intersection is ANTI-monotonic.  There are TWO premises! *)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   513
lemma Inter_anti_mono: "[| A<=B;  A\<noteq>0 |] ==> \<Inter>(B) \<subseteq> \<Inter>(A)"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   514
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   515
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   516
lemma cons_mono: "C<=D ==> cons(a,C) \<subseteq> cons(a,D)"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   517
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   518
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   519
lemma Un_mono: "[| A<=C;  B<=D |] ==> A \<union> B \<subseteq> C \<union> D"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   520
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   521
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   522
lemma Int_mono: "[| A<=C;  B<=D |] ==> A \<inter> B \<subseteq> C \<inter> D"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   523
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   524
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   525
lemma Diff_mono: "[| A<=C;  D<=B |] ==> A-B \<subseteq> C-D"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   526
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   527
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   528
subsubsection{*Standard Products, Sums and Function Spaces *}
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   529
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   530
lemma Sigma_mono [rule_format]:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   531
     "[| A<=C;  !!x. x \<in> A \<longrightarrow> B(x) \<subseteq> D(x) |] ==> Sigma(A,B) \<subseteq> Sigma(C,D)"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   532
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   533
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   534
lemma sum_mono: "[| A<=C;  B<=D |] ==> A+B \<subseteq> C+D"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   535
by (unfold sum_def, blast)
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   536
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   537
(*Note that B->A and C->A are typically disjoint!*)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   538
lemma Pi_mono: "B<=C ==> A->B \<subseteq> A->C"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   539
by (blast intro: lam_type elim: Pi_lamE)
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   540
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   541
lemma lam_mono: "A<=B ==> Lambda(A,c) \<subseteq> Lambda(B,c)"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   542
apply (unfold lam_def)
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   543
apply (erule RepFun_mono)
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   544
done
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   545
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   546
subsubsection{*Converse, Domain, Range, Field*}
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   547
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   548
lemma converse_mono: "r<=s ==> converse(r) \<subseteq> converse(s)"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   549
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   550
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   551
lemma domain_mono: "r<=s ==> domain(r)<=domain(s)"
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   552
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   553
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   554
lemmas domain_rel_subset = subset_trans [OF domain_mono domain_subset]
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   555
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   556
lemma range_mono: "r<=s ==> range(r)<=range(s)"
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   557
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   558
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   559
lemmas range_rel_subset = subset_trans [OF range_mono range_subset]
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   560
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   561
lemma field_mono: "r<=s ==> field(r)<=field(s)"
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   562
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   563
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   564
lemma field_rel_subset: "r \<subseteq> A*A ==> field(r) \<subseteq> A"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   565
by (erule field_mono [THEN subset_trans], blast)
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   566
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   567
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   568
subsubsection{*Images*}
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   569
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   570
lemma image_pair_mono:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   571
    "[| !! x y. <x,y>:r ==> <x,y>:s;  A<=B |] ==> r``A \<subseteq> s``B"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   572
by blast
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   573
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   574
lemma vimage_pair_mono:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   575
    "[| !! x y. <x,y>:r ==> <x,y>:s;  A<=B |] ==> r-``A \<subseteq> s-``B"
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   576
by blast
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   577
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   578
lemma image_mono: "[| r<=s;  A<=B |] ==> r``A \<subseteq> s``B"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   579
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   580
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 41229
diff changeset
   581
lemma vimage_mono: "[| r<=s;  A<=B |] ==> r-``A \<subseteq> s-``B"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   582
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   583
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   584
lemma Collect_mono:
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   585
    "[| A<=B;  !!x. x \<in> A ==> P(x) \<longrightarrow> Q(x) |] ==> Collect(A,P) \<subseteq> Collect(B,Q)"
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   586
by blast
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   587
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   588
(*Used in intr_elim.ML and in individual datatype definitions*)
46953
2b6e55924af3 replacing ":" by "\<in>"
paulson
parents: 46821
diff changeset
   589
lemmas basic_monos = subset_refl imp_refl disj_mono conj_mono ex_mono
13357
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   590
                     Collect_mono Part_mono in_mono
6f54e992777e Removal of mono.thy
paulson
parents: 13355
diff changeset
   591
27702
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   592
(* Useful with simp; contributed by Clemens Ballarin. *)
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   593
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   594
lemma bex_image_simp:
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   595
  "[| f \<in> Pi(X, Y); A \<subseteq> X |]  ==> (\<exists>x\<in>f``A. P(x)) \<longleftrightarrow> (\<exists>x\<in>A. P(f`x))"
27702
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   596
  apply safe
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   597
   apply rule
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   598
    prefer 2 apply assumption
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   599
   apply (simp add: apply_equality)
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   600
  apply (blast intro: apply_Pair)
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   601
  done
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   602
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   603
lemma ball_image_simp:
46821
ff6b0c1087f2 Using mathematical notation for <-> and cardinal arithmetic
paulson
parents: 46820
diff changeset
   604
  "[| f \<in> Pi(X, Y); A \<subseteq> X |]  ==> (\<forall>x\<in>f``A. P(x)) \<longleftrightarrow> (\<forall>x\<in>A. P(f`x))"
27702
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   605
  apply safe
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   606
   apply (blast intro: apply_Pair)
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   607
  apply (drule bspec) apply assumption
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   608
  apply (simp add: apply_equality)
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   609
  done
80608e96e760 Lemmas added
ballarin
parents: 27150
diff changeset
   610
13163
e320a52ff711 converted Arith, Univ, func to Isar format!
paulson
parents: 2469
diff changeset
   611
end