--- a/src/ZF/Nat_ZF.thy Sun Mar 04 23:20:43 2012 +0100
+++ b/src/ZF/Nat_ZF.thy Tue Mar 06 15:15:49 2012 +0000
@@ -9,7 +9,7 @@
definition
nat :: i where
- "nat == lfp(Inf, %X. {0} Un {succ(i). i:X})"
+ "nat == lfp(Inf, %X. {0} \<union> {succ(i). i:X})"
definition
quasinat :: "i => o" where
@@ -18,26 +18,26 @@
definition
(*Has an unconditional succ case, which is used in "recursor" below.*)
nat_case :: "[i, i=>i, i]=>i" where
- "nat_case(a,b,k) == THE y. k=0 & y=a | (EX x. k=succ(x) & y=b(x))"
+ "nat_case(a,b,k) == THE y. k=0 & y=a | (\<exists>x. k=succ(x) & y=b(x))"
definition
nat_rec :: "[i, i, [i,i]=>i]=>i" where
- "nat_rec(k,a,b) ==
+ "nat_rec(k,a,b) ==
wfrec(Memrel(nat), k, %n f. nat_case(a, %m. b(m, f`m), n))"
(*Internalized relations on the naturals*)
-
+
definition
Le :: i where
- "Le == {<x,y>:nat*nat. x le y}"
+ "Le == {<x,y>:nat*nat. x \<le> y}"
definition
Lt :: i where
"Lt == {<x, y>:nat*nat. x < y}"
-
+
definition
Ge :: i where
- "Ge == {<x,y>:nat*nat. y le x}"
+ "Ge == {<x,y>:nat*nat. y \<le> x}"
definition
Gt :: i where
@@ -51,33 +51,33 @@
predecessors!*}
-lemma nat_bnd_mono: "bnd_mono(Inf, %X. {0} Un {succ(i). i:X})"
+lemma nat_bnd_mono: "bnd_mono(Inf, %X. {0} \<union> {succ(i). i:X})"
apply (rule bnd_monoI)
-apply (cut_tac infinity, blast, blast)
+apply (cut_tac infinity, blast, blast)
done
-(* nat = {0} Un {succ(x). x:nat} *)
+(* @{term"nat = {0} \<union> {succ(x). x:nat}"} *)
lemmas nat_unfold = nat_bnd_mono [THEN nat_def [THEN def_lfp_unfold]]
(** Type checking of 0 and successor **)
-lemma nat_0I [iff,TC]: "0 : nat"
+lemma nat_0I [iff,TC]: "0 \<in> nat"
apply (subst nat_unfold)
apply (rule singletonI [THEN UnI1])
done
-lemma nat_succI [intro!,TC]: "n : nat ==> succ(n) : nat"
+lemma nat_succI [intro!,TC]: "n \<in> nat ==> succ(n) \<in> nat"
apply (subst nat_unfold)
apply (erule RepFunI [THEN UnI2])
done
-lemma nat_1I [iff,TC]: "1 : nat"
+lemma nat_1I [iff,TC]: "1 \<in> nat"
by (rule nat_0I [THEN nat_succI])
-lemma nat_2I [iff,TC]: "2 : nat"
+lemma nat_2I [iff,TC]: "2 \<in> nat"
by (rule nat_1I [THEN nat_succI])
-lemma bool_subset_nat: "bool <= nat"
+lemma bool_subset_nat: "bool \<subseteq> nat"
by (blast elim!: boolE)
lemmas bool_into_nat = bool_subset_nat [THEN subsetD]
@@ -92,15 +92,15 @@
lemma natE:
"[| n: nat; n=0 ==> P; !!x. [| x: nat; n=succ(x) |] ==> P |] ==> P"
-by (erule nat_unfold [THEN equalityD1, THEN subsetD, THEN UnE], auto)
+by (erule nat_unfold [THEN equalityD1, THEN subsetD, THEN UnE], auto)
lemma nat_into_Ord [simp]: "n: nat ==> Ord(n)"
by (erule nat_induct, auto)
-(* i: nat ==> 0 le i; same thing as 0<succ(i) *)
+(* @{term"i: nat ==> 0 \<le> i"}; same thing as @{term"0<succ(i)"} *)
lemmas nat_0_le = nat_into_Ord [THEN Ord_0_le]
-(* i: nat ==> i le i; same thing as i<succ(i) *)
+(* @{term"i: nat ==> i \<le> i"}; same thing as @{term"i<succ(i)"} *)
lemmas nat_le_refl = nat_into_Ord [THEN le_refl]
lemma Ord_nat [iff]: "Ord(nat)"
@@ -108,7 +108,7 @@
apply (erule_tac [2] nat_into_Ord [THEN Ord_is_Transset])
apply (unfold Transset_def)
apply (rule ballI)
-apply (erule nat_induct, auto)
+apply (erule nat_induct, auto)
done
lemma Limit_nat [iff]: "Limit(nat)"
@@ -126,12 +126,12 @@
lemma nat_succ_iff [iff]: "succ(n): nat <-> n: nat"
by (blast dest!: succ_natD)
-lemma nat_le_Limit: "Limit(i) ==> nat le i"
+lemma nat_le_Limit: "Limit(i) ==> nat \<le> i"
apply (rule subset_imp_le)
-apply (simp_all add: Limit_is_Ord)
+apply (simp_all add: Limit_is_Ord)
apply (rule subsetI)
apply (erule nat_induct)
- apply (erule Limit_has_0 [THEN ltD])
+ apply (erule Limit_has_0 [THEN ltD])
apply (blast intro: Limit_has_succ [THEN ltD] ltI Limit_is_Ord)
done
@@ -140,10 +140,10 @@
lemma lt_nat_in_nat: "[| m<n; n: nat |] ==> m: nat"
apply (erule ltE)
-apply (erule Ord_trans, assumption, simp)
+apply (erule Ord_trans, assumption, simp)
done
-lemma le_in_nat: "[| m le n; n:nat |] ==> m:nat"
+lemma le_in_nat: "[| m \<le> n; n:nat |] ==> m:nat"
by (blast dest!: lt_nat_in_nat)
@@ -153,59 +153,59 @@
lemmas complete_induct = Ord_induct [OF _ Ord_nat, case_names less, consumes 1]
-lemmas complete_induct_rule =
+lemmas complete_induct_rule =
complete_induct [rule_format, case_names less, consumes 1]
-lemma nat_induct_from_lemma [rule_format]:
- "[| n: nat; m: nat;
- !!x. [| x: nat; m le x; P(x) |] ==> P(succ(x)) |]
- ==> m le n --> P(m) --> P(n)"
-apply (erule nat_induct)
+lemma nat_induct_from_lemma [rule_format]:
+ "[| n: nat; m: nat;
+ !!x. [| x: nat; m \<le> x; P(x) |] ==> P(succ(x)) |]
+ ==> m \<le> n \<longrightarrow> P(m) \<longrightarrow> P(n)"
+apply (erule nat_induct)
apply (simp_all add: distrib_simps le0_iff le_succ_iff)
done
(*Induction starting from m rather than 0*)
-lemma nat_induct_from:
- "[| m le n; m: nat; n: nat;
- P(m);
- !!x. [| x: nat; m le x; P(x) |] ==> P(succ(x)) |]
+lemma nat_induct_from:
+ "[| m \<le> n; m: nat; n: nat;
+ P(m);
+ !!x. [| x: nat; m \<le> x; P(x) |] ==> P(succ(x)) |]
==> P(n)"
apply (blast intro: nat_induct_from_lemma)
done
(*Induction suitable for subtraction and less-than*)
lemma diff_induct [case_names 0 0_succ succ_succ, consumes 2]:
- "[| m: nat; n: nat;
- !!x. x: nat ==> P(x,0);
- !!y. y: nat ==> P(0,succ(y));
+ "[| m: nat; n: nat;
+ !!x. x: nat ==> P(x,0);
+ !!y. y: nat ==> P(0,succ(y));
!!x y. [| x: nat; y: nat; P(x,y) |] ==> P(succ(x),succ(y)) |]
==> P(m,n)"
apply (erule_tac x = m in rev_bspec)
-apply (erule nat_induct, simp)
+apply (erule nat_induct, simp)
apply (rule ballI)
apply (rename_tac i j)
-apply (erule_tac n=j in nat_induct, auto)
+apply (erule_tac n=j in nat_induct, auto)
done
(** Induction principle analogous to trancl_induct **)
lemma succ_lt_induct_lemma [rule_format]:
- "m: nat ==> P(m,succ(m)) --> (ALL x: nat. P(m,x) --> P(m,succ(x))) -->
- (ALL n:nat. m<n --> P(m,n))"
+ "m: nat ==> P(m,succ(m)) \<longrightarrow> (\<forall>x\<in>nat. P(m,x) \<longrightarrow> P(m,succ(x))) \<longrightarrow>
+ (\<forall>n\<in>nat. m<n \<longrightarrow> P(m,n))"
apply (erule nat_induct)
apply (intro impI, rule nat_induct [THEN ballI])
prefer 4 apply (intro impI, rule nat_induct [THEN ballI])
-apply (auto simp add: le_iff)
+apply (auto simp add: le_iff)
done
lemma succ_lt_induct:
- "[| m<n; n: nat;
- P(m,succ(m));
+ "[| m<n; n: nat;
+ P(m,succ(m));
!!x. [| x: nat; P(m,x) |] ==> P(m,succ(x)) |]
==> P(m,n)"
-by (blast intro: succ_lt_induct_lemma lt_nat_in_nat)
+by (blast intro: succ_lt_induct_lemma lt_nat_in_nat)
subsection{*quasinat: to allow a case-split rule for @{term nat_case}*}
@@ -219,36 +219,36 @@
lemma nat_imp_quasinat: "n \<in> nat ==> quasinat(n)"
by (erule natE, simp_all)
-lemma non_nat_case: "~ quasinat(x) ==> nat_case(a,b,x) = 0"
-by (simp add: quasinat_def nat_case_def)
+lemma non_nat_case: "~ quasinat(x) ==> nat_case(a,b,x) = 0"
+by (simp add: quasinat_def nat_case_def)
lemma nat_cases_disj: "k=0 | (\<exists>y. k = succ(y)) | ~ quasinat(k)"
-apply (case_tac "k=0", simp)
-apply (case_tac "\<exists>m. k = succ(m)")
-apply (simp_all add: quasinat_def)
+apply (case_tac "k=0", simp)
+apply (case_tac "\<exists>m. k = succ(m)")
+apply (simp_all add: quasinat_def)
done
lemma nat_cases:
"[|k=0 ==> P; !!y. k = succ(y) ==> P; ~ quasinat(k) ==> P|] ==> P"
-by (insert nat_cases_disj [of k], blast)
+by (insert nat_cases_disj [of k], blast)
(** nat_case **)
lemma nat_case_0 [simp]: "nat_case(a,b,0) = a"
by (simp add: nat_case_def)
-lemma nat_case_succ [simp]: "nat_case(a,b,succ(n)) = b(n)"
+lemma nat_case_succ [simp]: "nat_case(a,b,succ(n)) = b(n)"
by (simp add: nat_case_def)
lemma nat_case_type [TC]:
- "[| n: nat; a: C(0); !!m. m: nat ==> b(m): C(succ(m)) |]
- ==> nat_case(a,b,n) : C(n)";
-by (erule nat_induct, auto)
+ "[| n: nat; a: C(0); !!m. m: nat ==> b(m): C(succ(m)) |]
+ ==> nat_case(a,b,n) \<in> C(n)";
+by (erule nat_induct, auto)
lemma split_nat_case:
- "P(nat_case(a,b,k)) <->
- ((k=0 --> P(a)) & (\<forall>x. k=succ(x) --> P(b(x))) & (~ quasinat(k) \<longrightarrow> P(0)))"
-apply (rule nat_cases [of k])
+ "P(nat_case(a,b,k)) <->
+ ((k=0 \<longrightarrow> P(a)) & (\<forall>x. k=succ(x) \<longrightarrow> P(b(x))) & (~ quasinat(k) \<longrightarrow> P(0)))"
+apply (rule nat_cases [of k])
apply (auto simp add: non_nat_case)
done
@@ -260,41 +260,41 @@
lemma nat_rec_0: "nat_rec(0,a,b) = a"
apply (rule nat_rec_def [THEN def_wfrec, THEN trans])
- apply (rule wf_Memrel)
+ apply (rule wf_Memrel)
apply (rule nat_case_0)
done
lemma nat_rec_succ: "m: nat ==> nat_rec(succ(m),a,b) = b(m, nat_rec(m,a,b))"
apply (rule nat_rec_def [THEN def_wfrec, THEN trans])
- apply (rule wf_Memrel)
+ apply (rule wf_Memrel)
apply (simp add: vimage_singleton_iff)
done
(** The union of two natural numbers is a natural number -- their maximum **)
-lemma Un_nat_type [TC]: "[| i: nat; j: nat |] ==> i Un j: nat"
+lemma Un_nat_type [TC]: "[| i: nat; j: nat |] ==> i \<union> j: nat"
apply (rule Un_least_lt [THEN ltD])
-apply (simp_all add: lt_def)
+apply (simp_all add: lt_def)
done
-lemma Int_nat_type [TC]: "[| i: nat; j: nat |] ==> i Int j: nat"
+lemma Int_nat_type [TC]: "[| i: nat; j: nat |] ==> i \<inter> j: nat"
apply (rule Int_greatest_lt [THEN ltD])
-apply (simp_all add: lt_def)
+apply (simp_all add: lt_def)
done
(*needed to simplify unions over nat*)
-lemma nat_nonempty [simp]: "nat ~= 0"
+lemma nat_nonempty [simp]: "nat \<noteq> 0"
by blast
text{*A natural number is the set of its predecessors*}
lemma nat_eq_Collect_lt: "i \<in> nat ==> {j\<in>nat. j<i} = i"
apply (rule equalityI)
-apply (blast dest: ltD)
+apply (blast dest: ltD)
apply (auto simp add: Ord_mem_iff_lt)
-apply (blast intro: lt_trans)
+apply (blast intro: lt_trans)
done
-lemma Le_iff [iff]: "<x,y> : Le <-> x le y & x : nat & y : nat"
+lemma Le_iff [iff]: "<x,y> \<in> Le <-> x \<le> y & x \<in> nat & y \<in> nat"
by (force simp add: Le_def)
end