--- a/src/HOL/UNITY/SubstAx.thy Mon Feb 03 11:45:05 2003 +0100
+++ b/src/HOL/UNITY/SubstAx.thy Tue Feb 04 18:12:40 2003 +0100
@@ -12,10 +12,10 @@
constdefs
Ensures :: "['a set, 'a set] => 'a program set" (infixl "Ensures" 60)
- "A Ensures B == {F. F : (reachable F Int A) ensures B}"
+ "A Ensures B == {F. F \<in> (reachable F \<inter> A) ensures B}"
LeadsTo :: "['a set, 'a set] => 'a program set" (infixl "LeadsTo" 60)
- "A LeadsTo B == {F. F : (reachable F Int A) leadsTo B}"
+ "A LeadsTo B == {F. F \<in> (reachable F \<inter> A) leadsTo B}"
syntax (xsymbols)
"op LeadsTo" :: "['a set, 'a set] => 'a program set" (infixl " \<longmapsto>w " 60)
@@ -23,7 +23,7 @@
(*Resembles the previous definition of LeadsTo*)
lemma LeadsTo_eq_leadsTo:
- "A LeadsTo B = {F. F : (reachable F Int A) leadsTo (reachable F Int B)}"
+ "A LeadsTo B = {F. F \<in> (reachable F \<inter> A) leadsTo (reachable F \<inter> B)}"
apply (unfold LeadsTo_def)
apply (blast dest: psp_stable2 intro: leadsTo_weaken)
done
@@ -34,35 +34,37 @@
(** Conjoining an Always property **)
lemma Always_LeadsTo_pre:
- "F : Always INV ==> (F : (INV Int A) LeadsTo A') = (F : A LeadsTo A')"
-by (simp add: LeadsTo_def Always_eq_includes_reachable Int_absorb2 Int_assoc [symmetric])
+ "F \<in> Always INV ==> (F \<in> (INV \<inter> A) LeadsTo A') = (F \<in> A LeadsTo A')"
+by (simp add: LeadsTo_def Always_eq_includes_reachable Int_absorb2
+ Int_assoc [symmetric])
lemma Always_LeadsTo_post:
- "F : Always INV ==> (F : A LeadsTo (INV Int A')) = (F : A LeadsTo A')"
-by (simp add: LeadsTo_eq_leadsTo Always_eq_includes_reachable Int_absorb2 Int_assoc [symmetric])
+ "F \<in> Always INV ==> (F \<in> A LeadsTo (INV \<inter> A')) = (F \<in> A LeadsTo A')"
+by (simp add: LeadsTo_eq_leadsTo Always_eq_includes_reachable Int_absorb2
+ Int_assoc [symmetric])
-(* [| F : Always C; F : (C Int A) LeadsTo A' |] ==> F : A LeadsTo A' *)
+(* [| F \<in> Always C; F \<in> (C \<inter> A) LeadsTo A' |] ==> F \<in> A LeadsTo A' *)
lemmas Always_LeadsToI = Always_LeadsTo_pre [THEN iffD1, standard]
-(* [| F : Always INV; F : A LeadsTo A' |] ==> F : A LeadsTo (INV Int A') *)
+(* [| F \<in> Always INV; F \<in> A LeadsTo A' |] ==> F \<in> A LeadsTo (INV \<inter> A') *)
lemmas Always_LeadsToD = Always_LeadsTo_post [THEN iffD2, standard]
subsection{*Introduction rules: Basis, Trans, Union*}
-lemma leadsTo_imp_LeadsTo: "F : A leadsTo B ==> F : A LeadsTo B"
+lemma leadsTo_imp_LeadsTo: "F \<in> A leadsTo B ==> F \<in> A LeadsTo B"
apply (simp add: LeadsTo_def)
apply (blast intro: leadsTo_weaken_L)
done
lemma LeadsTo_Trans:
- "[| F : A LeadsTo B; F : B LeadsTo C |] ==> F : A LeadsTo C"
+ "[| F \<in> A LeadsTo B; F \<in> B LeadsTo C |] ==> F \<in> A LeadsTo C"
apply (simp add: LeadsTo_eq_leadsTo)
apply (blast intro: leadsTo_Trans)
done
lemma LeadsTo_Union:
- "(!!A. A : S ==> F : A LeadsTo B) ==> F : (Union S) LeadsTo B"
+ "(!!A. A \<in> S ==> F \<in> A LeadsTo B) ==> F \<in> (Union S) LeadsTo B"
apply (simp add: LeadsTo_def)
apply (subst Int_Union)
apply (blast intro: leadsTo_UN)
@@ -71,37 +73,37 @@
subsection{*Derived rules*}
-lemma LeadsTo_UNIV [simp]: "F : A LeadsTo UNIV"
+lemma LeadsTo_UNIV [simp]: "F \<in> A LeadsTo UNIV"
by (simp add: LeadsTo_def)
(*Useful with cancellation, disjunction*)
lemma LeadsTo_Un_duplicate:
- "F : A LeadsTo (A' Un A') ==> F : A LeadsTo A'"
+ "F \<in> A LeadsTo (A' \<union> A') ==> F \<in> A LeadsTo A'"
by (simp add: Un_ac)
lemma LeadsTo_Un_duplicate2:
- "F : A LeadsTo (A' Un C Un C) ==> F : A LeadsTo (A' Un C)"
+ "F \<in> A LeadsTo (A' \<union> C \<union> C) ==> F \<in> A LeadsTo (A' \<union> C)"
by (simp add: Un_ac)
lemma LeadsTo_UN:
- "(!!i. i : I ==> F : (A i) LeadsTo B) ==> F : (UN i:I. A i) LeadsTo B"
+ "(!!i. i \<in> I ==> F \<in> (A i) LeadsTo B) ==> F \<in> (\<Union>i \<in> I. A i) LeadsTo B"
apply (simp only: Union_image_eq [symmetric])
apply (blast intro: LeadsTo_Union)
done
(*Binary union introduction rule*)
lemma LeadsTo_Un:
- "[| F : A LeadsTo C; F : B LeadsTo C |] ==> F : (A Un B) LeadsTo C"
+ "[| F \<in> A LeadsTo C; F \<in> B LeadsTo C |] ==> F \<in> (A \<union> B) LeadsTo C"
apply (subst Un_eq_Union)
apply (blast intro: LeadsTo_Union)
done
(*Lets us look at the starting state*)
lemma single_LeadsTo_I:
- "(!!s. s : A ==> F : {s} LeadsTo B) ==> F : A LeadsTo B"
+ "(!!s. s \<in> A ==> F \<in> {s} LeadsTo B) ==> F \<in> A LeadsTo B"
by (subst UN_singleton [symmetric], rule LeadsTo_UN, blast)
-lemma subset_imp_LeadsTo: "A <= B ==> F : A LeadsTo B"
+lemma subset_imp_LeadsTo: "A \<subseteq> B ==> F \<in> A LeadsTo B"
apply (simp add: LeadsTo_def)
apply (blast intro: subset_imp_leadsTo)
done
@@ -109,73 +111,73 @@
lemmas empty_LeadsTo = empty_subsetI [THEN subset_imp_LeadsTo, standard, simp]
lemma LeadsTo_weaken_R [rule_format]:
- "[| F : A LeadsTo A'; A' <= B' |] ==> F : A LeadsTo B'"
-apply (simp (no_asm_use) add: LeadsTo_def)
+ "[| F \<in> A LeadsTo A'; A' \<subseteq> B' |] ==> F \<in> A LeadsTo B'"
+apply (simp add: LeadsTo_def)
apply (blast intro: leadsTo_weaken_R)
done
lemma LeadsTo_weaken_L [rule_format]:
- "[| F : A LeadsTo A'; B <= A |]
- ==> F : B LeadsTo A'"
-apply (simp (no_asm_use) add: LeadsTo_def)
+ "[| F \<in> A LeadsTo A'; B \<subseteq> A |]
+ ==> F \<in> B LeadsTo A'"
+apply (simp add: LeadsTo_def)
apply (blast intro: leadsTo_weaken_L)
done
lemma LeadsTo_weaken:
- "[| F : A LeadsTo A';
- B <= A; A' <= B' |]
- ==> F : B LeadsTo B'"
+ "[| F \<in> A LeadsTo A';
+ B \<subseteq> A; A' \<subseteq> B' |]
+ ==> F \<in> B LeadsTo B'"
by (blast intro: LeadsTo_weaken_R LeadsTo_weaken_L LeadsTo_Trans)
lemma Always_LeadsTo_weaken:
- "[| F : Always C; F : A LeadsTo A';
- C Int B <= A; C Int A' <= B' |]
- ==> F : B LeadsTo B'"
+ "[| F \<in> Always C; F \<in> A LeadsTo A';
+ C \<inter> B \<subseteq> A; C \<inter> A' \<subseteq> B' |]
+ ==> F \<in> B LeadsTo B'"
by (blast dest: Always_LeadsToI intro: LeadsTo_weaken intro: Always_LeadsToD)
(** Two theorems for "proof lattices" **)
-lemma LeadsTo_Un_post: "F : A LeadsTo B ==> F : (A Un B) LeadsTo B"
+lemma LeadsTo_Un_post: "F \<in> A LeadsTo B ==> F \<in> (A \<union> B) LeadsTo B"
by (blast intro: LeadsTo_Un subset_imp_LeadsTo)
lemma LeadsTo_Trans_Un:
- "[| F : A LeadsTo B; F : B LeadsTo C |]
- ==> F : (A Un B) LeadsTo C"
+ "[| F \<in> A LeadsTo B; F \<in> B LeadsTo C |]
+ ==> F \<in> (A \<union> B) LeadsTo C"
by (blast intro: LeadsTo_Un subset_imp_LeadsTo LeadsTo_weaken_L LeadsTo_Trans)
(** Distributive laws **)
lemma LeadsTo_Un_distrib:
- "(F : (A Un B) LeadsTo C) = (F : A LeadsTo C & F : B LeadsTo C)"
+ "(F \<in> (A \<union> B) LeadsTo C) = (F \<in> A LeadsTo C & F \<in> B LeadsTo C)"
by (blast intro: LeadsTo_Un LeadsTo_weaken_L)
lemma LeadsTo_UN_distrib:
- "(F : (UN i:I. A i) LeadsTo B) = (ALL i : I. F : (A i) LeadsTo B)"
+ "(F \<in> (\<Union>i \<in> I. A i) LeadsTo B) = (\<forall>i \<in> I. F \<in> (A i) LeadsTo B)"
by (blast intro: LeadsTo_UN LeadsTo_weaken_L)
lemma LeadsTo_Union_distrib:
- "(F : (Union S) LeadsTo B) = (ALL A : S. F : A LeadsTo B)"
+ "(F \<in> (Union S) LeadsTo B) = (\<forall>A \<in> S. F \<in> A LeadsTo B)"
by (blast intro: LeadsTo_Union LeadsTo_weaken_L)
(** More rules using the premise "Always INV" **)
-lemma LeadsTo_Basis: "F : A Ensures B ==> F : A LeadsTo B"
+lemma LeadsTo_Basis: "F \<in> A Ensures B ==> F \<in> A LeadsTo B"
by (simp add: Ensures_def LeadsTo_def leadsTo_Basis)
lemma EnsuresI:
- "[| F : (A-B) Co (A Un B); F : transient (A-B) |]
- ==> F : A Ensures B"
+ "[| F \<in> (A-B) Co (A \<union> B); F \<in> transient (A-B) |]
+ ==> F \<in> A Ensures B"
apply (simp add: Ensures_def Constrains_eq_constrains)
apply (blast intro: ensuresI constrains_weaken transient_strengthen)
done
lemma Always_LeadsTo_Basis:
- "[| F : Always INV;
- F : (INV Int (A-A')) Co (A Un A');
- F : transient (INV Int (A-A')) |]
- ==> F : A LeadsTo A'"
+ "[| F \<in> Always INV;
+ F \<in> (INV \<inter> (A-A')) Co (A \<union> A');
+ F \<in> transient (INV \<inter> (A-A')) |]
+ ==> F \<in> A LeadsTo A'"
apply (rule Always_LeadsToI, assumption)
apply (blast intro: EnsuresI LeadsTo_Basis Always_ConstrainsD [THEN Constrains_weaken] transient_strengthen)
done
@@ -183,14 +185,14 @@
(*Set difference: maybe combine with leadsTo_weaken_L??
This is the most useful form of the "disjunction" rule*)
lemma LeadsTo_Diff:
- "[| F : (A-B) LeadsTo C; F : (A Int B) LeadsTo C |]
- ==> F : A LeadsTo C"
+ "[| F \<in> (A-B) LeadsTo C; F \<in> (A \<inter> B) LeadsTo C |]
+ ==> F \<in> A LeadsTo C"
by (blast intro: LeadsTo_Un LeadsTo_weaken)
lemma LeadsTo_UN_UN:
- "(!! i. i:I ==> F : (A i) LeadsTo (A' i))
- ==> F : (UN i:I. A i) LeadsTo (UN i:I. A' i)"
+ "(!! i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i))
+ ==> F \<in> (\<Union>i \<in> I. A i) LeadsTo (\<Union>i \<in> I. A' i)"
apply (simp only: Union_image_eq [symmetric])
apply (blast intro: LeadsTo_Union LeadsTo_weaken_R)
done
@@ -198,48 +200,47 @@
(*Version with no index set*)
lemma LeadsTo_UN_UN_noindex:
- "(!! i. F : (A i) LeadsTo (A' i))
- ==> F : (UN i. A i) LeadsTo (UN i. A' i)"
+ "(!!i. F \<in> (A i) LeadsTo (A' i)) ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"
by (blast intro: LeadsTo_UN_UN)
(*Version with no index set*)
lemma all_LeadsTo_UN_UN:
- "ALL i. F : (A i) LeadsTo (A' i)
- ==> F : (UN i. A i) LeadsTo (UN i. A' i)"
+ "\<forall>i. F \<in> (A i) LeadsTo (A' i)
+ ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"
by (blast intro: LeadsTo_UN_UN)
(*Binary union version*)
lemma LeadsTo_Un_Un:
- "[| F : A LeadsTo A'; F : B LeadsTo B' |]
- ==> F : (A Un B) LeadsTo (A' Un B')"
+ "[| F \<in> A LeadsTo A'; F \<in> B LeadsTo B' |]
+ ==> F \<in> (A \<union> B) LeadsTo (A' \<union> B')"
by (blast intro: LeadsTo_Un LeadsTo_weaken_R)
(** The cancellation law **)
lemma LeadsTo_cancel2:
- "[| F : A LeadsTo (A' Un B); F : B LeadsTo B' |]
- ==> F : A LeadsTo (A' Un B')"
+ "[| F \<in> A LeadsTo (A' \<union> B); F \<in> B LeadsTo B' |]
+ ==> F \<in> A LeadsTo (A' \<union> B')"
by (blast intro: LeadsTo_Un_Un subset_imp_LeadsTo LeadsTo_Trans)
lemma LeadsTo_cancel_Diff2:
- "[| F : A LeadsTo (A' Un B); F : (B-A') LeadsTo B' |]
- ==> F : A LeadsTo (A' Un B')"
+ "[| F \<in> A LeadsTo (A' \<union> B); F \<in> (B-A') LeadsTo B' |]
+ ==> F \<in> A LeadsTo (A' \<union> B')"
apply (rule LeadsTo_cancel2)
prefer 2 apply assumption
apply (simp_all (no_asm_simp))
done
lemma LeadsTo_cancel1:
- "[| F : A LeadsTo (B Un A'); F : B LeadsTo B' |]
- ==> F : A LeadsTo (B' Un A')"
+ "[| F \<in> A LeadsTo (B \<union> A'); F \<in> B LeadsTo B' |]
+ ==> F \<in> A LeadsTo (B' \<union> A')"
apply (simp add: Un_commute)
apply (blast intro!: LeadsTo_cancel2)
done
lemma LeadsTo_cancel_Diff1:
- "[| F : A LeadsTo (B Un A'); F : (B-A') LeadsTo B' |]
- ==> F : A LeadsTo (B' Un A')"
+ "[| F \<in> A LeadsTo (B \<union> A'); F \<in> (B-A') LeadsTo B' |]
+ ==> F \<in> A LeadsTo (B' \<union> A')"
apply (rule LeadsTo_cancel1)
prefer 2 apply assumption
apply (simp_all (no_asm_simp))
@@ -249,8 +250,8 @@
(** The impossibility law **)
(*The set "A" may be non-empty, but it contains no reachable states*)
-lemma LeadsTo_empty: "F : A LeadsTo {} ==> F : Always (-A)"
-apply (simp (no_asm_use) add: LeadsTo_def Always_eq_includes_reachable)
+lemma LeadsTo_empty: "F \<in> A LeadsTo {} ==> F \<in> Always (-A)"
+apply (simp add: LeadsTo_def Always_eq_includes_reachable)
apply (drule leadsTo_empty, auto)
done
@@ -259,33 +260,33 @@
(*Special case of PSP: Misra's "stable conjunction"*)
lemma PSP_Stable:
- "[| F : A LeadsTo A'; F : Stable B |]
- ==> F : (A Int B) LeadsTo (A' Int B)"
-apply (simp (no_asm_use) add: LeadsTo_eq_leadsTo Stable_eq_stable)
+ "[| F \<in> A LeadsTo A'; F \<in> Stable B |]
+ ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B)"
+apply (simp add: LeadsTo_eq_leadsTo Stable_eq_stable)
apply (drule psp_stable, assumption)
apply (simp add: Int_ac)
done
lemma PSP_Stable2:
- "[| F : A LeadsTo A'; F : Stable B |]
- ==> F : (B Int A) LeadsTo (B Int A')"
+ "[| F \<in> A LeadsTo A'; F \<in> Stable B |]
+ ==> F \<in> (B \<inter> A) LeadsTo (B \<inter> A')"
by (simp add: PSP_Stable Int_ac)
lemma PSP:
- "[| F : A LeadsTo A'; F : B Co B' |]
- ==> F : (A Int B') LeadsTo ((A' Int B) Un (B' - B))"
-apply (simp (no_asm_use) add: LeadsTo_def Constrains_eq_constrains)
+ "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]
+ ==> F \<in> (A \<inter> B') LeadsTo ((A' \<inter> B) \<union> (B' - B))"
+apply (simp add: LeadsTo_def Constrains_eq_constrains)
apply (blast dest: psp intro: leadsTo_weaken)
done
lemma PSP2:
- "[| F : A LeadsTo A'; F : B Co B' |]
- ==> F : (B' Int A) LeadsTo ((B Int A') Un (B' - B))"
+ "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]
+ ==> F \<in> (B' \<inter> A) LeadsTo ((B \<inter> A') \<union> (B' - B))"
by (simp add: PSP Int_ac)
lemma PSP_Unless:
- "[| F : A LeadsTo A'; F : B Unless B' |]
- ==> F : (A Int B) LeadsTo ((A' Int B) Un B')"
+ "[| F \<in> A LeadsTo A'; F \<in> B Unless B' |]
+ ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B) \<union> B')"
apply (unfold Unless_def)
apply (drule PSP, assumption)
apply (blast intro: LeadsTo_Diff LeadsTo_weaken subset_imp_LeadsTo)
@@ -293,8 +294,8 @@
lemma Stable_transient_Always_LeadsTo:
- "[| F : Stable A; F : transient C;
- F : Always (-A Un B Un C) |] ==> F : A LeadsTo B"
+ "[| F \<in> Stable A; F \<in> transient C;
+ F \<in> Always (-A \<union> B \<union> C) |] ==> F \<in> A LeadsTo B"
apply (erule Always_LeadsTo_weaken)
apply (rule LeadsTo_Diff)
prefer 2
@@ -309,10 +310,10 @@
(** Meta or object quantifier ????? **)
lemma LeadsTo_wf_induct:
"[| wf r;
- ALL m. F : (A Int f-`{m}) LeadsTo
- ((A Int f-`(r^-1 `` {m})) Un B) |]
- ==> F : A LeadsTo B"
-apply (simp (no_asm_use) add: LeadsTo_eq_leadsTo)
+ \<forall>m. F \<in> (A \<inter> f-`{m}) LeadsTo
+ ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]
+ ==> F \<in> A LeadsTo B"
+apply (simp add: LeadsTo_eq_leadsTo)
apply (erule leadsTo_wf_induct)
apply (blast intro: leadsTo_weaken)
done
@@ -320,28 +321,27 @@
lemma Bounded_induct:
"[| wf r;
- ALL m:I. F : (A Int f-`{m}) LeadsTo
- ((A Int f-`(r^-1 `` {m})) Un B) |]
- ==> F : A LeadsTo ((A - (f-`I)) Un B)"
+ \<forall>m \<in> I. F \<in> (A \<inter> f-`{m}) LeadsTo
+ ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]
+ ==> F \<in> A LeadsTo ((A - (f-`I)) \<union> B)"
apply (erule LeadsTo_wf_induct, safe)
-apply (case_tac "m:I")
+apply (case_tac "m \<in> I")
apply (blast intro: LeadsTo_weaken)
apply (blast intro: subset_imp_LeadsTo)
done
lemma LessThan_induct:
- "(!!m::nat. F : (A Int f-`{m}) LeadsTo ((A Int f-`(lessThan m)) Un B))
- ==> F : A LeadsTo B"
-apply (rule wf_less_than [THEN LeadsTo_wf_induct], auto)
-done
+ "(!!m::nat. F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B))
+ ==> F \<in> A LeadsTo B"
+by (rule wf_less_than [THEN LeadsTo_wf_induct], auto)
(*Integer version. Could generalize from 0 to any lower bound*)
lemma integ_0_le_induct:
- "[| F : Always {s. (0::int) <= f s};
- !! z. F : (A Int {s. f s = z}) LeadsTo
- ((A Int {s. f s < z}) Un B) |]
- ==> F : A LeadsTo B"
+ "[| F \<in> Always {s. (0::int) \<le> f s};
+ !! z. F \<in> (A \<inter> {s. f s = z}) LeadsTo
+ ((A \<inter> {s. f s < z}) \<union> B) |]
+ ==> F \<in> A LeadsTo B"
apply (rule_tac f = "nat o f" in LessThan_induct)
apply (simp add: vimage_def)
apply (rule Always_LeadsTo_weaken, assumption+)
@@ -349,42 +349,42 @@
done
lemma LessThan_bounded_induct:
- "!!l::nat. [| ALL m:(greaterThan l). F : (A Int f-`{m}) LeadsTo
- ((A Int f-`(lessThan m)) Un B) |]
- ==> F : A LeadsTo ((A Int (f-`(atMost l))) Un B)"
-apply (simp only: Diff_eq [symmetric] vimage_Compl Compl_greaterThan [symmetric])
-apply (rule wf_less_than [THEN Bounded_induct])
-apply (simp (no_asm_simp))
+ "!!l::nat. \<forall>m \<in> greaterThan l.
+ F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B)
+ ==> F \<in> A LeadsTo ((A \<inter> (f-`(atMost l))) \<union> B)"
+apply (simp only: Diff_eq [symmetric] vimage_Compl
+ Compl_greaterThan [symmetric])
+apply (rule wf_less_than [THEN Bounded_induct], simp)
done
lemma GreaterThan_bounded_induct:
- "!!l::nat. [| ALL m:(lessThan l). F : (A Int f-`{m}) LeadsTo
- ((A Int f-`(greaterThan m)) Un B) |]
- ==> F : A LeadsTo ((A Int (f-`(atLeast l))) Un B)"
+ "!!l::nat. \<forall>m \<in> lessThan l.
+ F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(greaterThan m)) \<union> B)
+ ==> F \<in> A LeadsTo ((A \<inter> (f-`(atLeast l))) \<union> B)"
apply (rule_tac f = f and f1 = "%k. l - k"
in wf_less_than [THEN wf_inv_image, THEN LeadsTo_wf_induct])
apply (simp add: inv_image_def Image_singleton, clarify)
apply (case_tac "m<l")
- prefer 2 apply (blast intro: not_leE subset_imp_LeadsTo)
-apply (blast intro: LeadsTo_weaken_R diff_less_mono2)
+ apply (blast intro: LeadsTo_weaken_R diff_less_mono2)
+apply (blast intro: not_leE subset_imp_LeadsTo)
done
subsection{*Completion: Binary and General Finite versions*}
lemma Completion:
- "[| F : A LeadsTo (A' Un C); F : A' Co (A' Un C);
- F : B LeadsTo (B' Un C); F : B' Co (B' Un C) |]
- ==> F : (A Int B) LeadsTo ((A' Int B') Un C)"
-apply (simp (no_asm_use) add: LeadsTo_eq_leadsTo Constrains_eq_constrains Int_Un_distrib)
+ "[| F \<in> A LeadsTo (A' \<union> C); F \<in> A' Co (A' \<union> C);
+ F \<in> B LeadsTo (B' \<union> C); F \<in> B' Co (B' \<union> C) |]
+ ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B') \<union> C)"
+apply (simp add: LeadsTo_eq_leadsTo Constrains_eq_constrains Int_Un_distrib)
apply (blast intro: completion leadsTo_weaken)
done
lemma Finite_completion_lemma:
"finite I
- ==> (ALL i:I. F : (A i) LeadsTo (A' i Un C)) -->
- (ALL i:I. F : (A' i) Co (A' i Un C)) -->
- F : (INT i:I. A i) LeadsTo ((INT i:I. A' i) Un C)"
+ ==> (\<forall>i \<in> I. F \<in> (A i) LeadsTo (A' i \<union> C)) -->
+ (\<forall>i \<in> I. F \<in> (A' i) Co (A' i \<union> C)) -->
+ F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
apply (erule finite_induct, auto)
apply (rule Completion)
prefer 4
@@ -394,15 +394,15 @@
lemma Finite_completion:
"[| finite I;
- !!i. i:I ==> F : (A i) LeadsTo (A' i Un C);
- !!i. i:I ==> F : (A' i) Co (A' i Un C) |]
- ==> F : (INT i:I. A i) LeadsTo ((INT i:I. A' i) Un C)"
+ !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i \<union> C);
+ !!i. i \<in> I ==> F \<in> (A' i) Co (A' i \<union> C) |]
+ ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
by (blast intro: Finite_completion_lemma [THEN mp, THEN mp])
lemma Stable_completion:
- "[| F : A LeadsTo A'; F : Stable A';
- F : B LeadsTo B'; F : Stable B' |]
- ==> F : (A Int B) LeadsTo (A' Int B')"
+ "[| F \<in> A LeadsTo A'; F \<in> Stable A';
+ F \<in> B LeadsTo B'; F \<in> Stable B' |]
+ ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B')"
apply (unfold Stable_def)
apply (rule_tac C1 = "{}" in Completion [THEN LeadsTo_weaken_R])
apply (force+)
@@ -410,14 +410,12 @@
lemma Finite_stable_completion:
"[| finite I;
- !!i. i:I ==> F : (A i) LeadsTo (A' i);
- !!i. i:I ==> F : Stable (A' i) |]
- ==> F : (INT i:I. A i) LeadsTo (INT i:I. A' i)"
+ !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i);
+ !!i. i \<in> I ==> F \<in> Stable (A' i) |]
+ ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo (\<Inter>i \<in> I. A' i)"
apply (unfold Stable_def)
apply (rule_tac C1 = "{}" in Finite_completion [THEN LeadsTo_weaken_R])
-apply (simp_all (no_asm_simp))
-apply blast+
+apply (simp_all, blast+)
done
-
end