author | krauss |
Mon, 11 Feb 2008 15:40:21 +0100 | |
changeset 26056 | 6a0801279f4c |
child 32960 | 69916a850301 |
permissions | -rw-r--r-- |
(* Title: ZF/Nat.thy ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1994 University of Cambridge *) header{*The Natural numbers As a Least Fixed Point*} theory Nat_ZF imports OrdQuant Bool begin definition nat :: i where "nat == lfp(Inf, %X. {0} Un {succ(i). i:X})" definition quasinat :: "i => o" where "quasinat(n) == n=0 | (\<exists>m. n = succ(m))" 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))" definition nat_rec :: "[i, i, [i,i]=>i]=>i" where "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}" definition Lt :: i where "Lt == {<x, y>:nat*nat. x < y}" definition Ge :: i where "Ge == {<x,y>:nat*nat. y le x}" definition Gt :: i where "Gt == {<x,y>:nat*nat. y < x}" definition greater_than :: "i=>i" where "greater_than(n) == {i:nat. n < i}" text{*No need for a less-than operator: a natural number is its list of predecessors!*} lemma nat_bnd_mono: "bnd_mono(Inf, %X. {0} Un {succ(i). i:X})" apply (rule bnd_monoI) apply (cut_tac infinity, blast, blast) done (* nat = {0} Un {succ(x). x:nat} *) lemmas nat_unfold = nat_bnd_mono [THEN nat_def [THEN def_lfp_unfold], standard] (** Type checking of 0 and successor **) lemma nat_0I [iff,TC]: "0 : nat" apply (subst nat_unfold) apply (rule singletonI [THEN UnI1]) done lemma nat_succI [intro!,TC]: "n : nat ==> succ(n) : nat" apply (subst nat_unfold) apply (erule RepFunI [THEN UnI2]) done lemma nat_1I [iff,TC]: "1 : nat" by (rule nat_0I [THEN nat_succI]) lemma nat_2I [iff,TC]: "2 : nat" by (rule nat_1I [THEN nat_succI]) lemma bool_subset_nat: "bool <= nat" by (blast elim!: boolE) lemmas bool_into_nat = bool_subset_nat [THEN subsetD, standard] subsection{*Injectivity Properties and Induction*} (*Mathematical induction*) lemma nat_induct [case_names 0 succ, induct set: nat]: "[| n: nat; P(0); !!x. [| x: nat; P(x) |] ==> P(succ(x)) |] ==> P(n)" by (erule def_induct [OF nat_def nat_bnd_mono], blast) 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) 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) *) lemmas nat_0_le = nat_into_Ord [THEN Ord_0_le, standard] (* i: nat ==> i le i; same thing as i<succ(i) *) lemmas nat_le_refl = nat_into_Ord [THEN le_refl, standard] lemma Ord_nat [iff]: "Ord(nat)" apply (rule OrdI) apply (erule_tac [2] nat_into_Ord [THEN Ord_is_Transset]) apply (unfold Transset_def) apply (rule ballI) apply (erule nat_induct, auto) done lemma Limit_nat [iff]: "Limit(nat)" apply (unfold Limit_def) apply (safe intro!: ltI Ord_nat) apply (erule ltD) done lemma naturals_not_limit: "a \<in> nat ==> ~ Limit(a)" by (induct a rule: nat_induct, auto) lemma succ_natD: "succ(i): nat ==> i: nat" by (rule Ord_trans [OF succI1], auto) lemma nat_succ_iff [iff]: "succ(n): nat <-> n: nat" by (blast dest!: succ_natD) lemma nat_le_Limit: "Limit(i) ==> nat le i" apply (rule subset_imp_le) apply (simp_all add: Limit_is_Ord) apply (rule subsetI) apply (erule nat_induct) apply (erule Limit_has_0 [THEN ltD]) apply (blast intro: Limit_has_succ [THEN ltD] ltI Limit_is_Ord) done (* [| succ(i): k; k: nat |] ==> i: k *) lemmas succ_in_naturalD = Ord_trans [OF succI1 _ nat_into_Ord] lemma lt_nat_in_nat: "[| m<n; n: nat |] ==> m: nat" apply (erule ltE) apply (erule Ord_trans, assumption, simp) done lemma le_in_nat: "[| m le n; n:nat |] ==> m:nat" by (blast dest!: lt_nat_in_nat) subsection{*Variations on Mathematical Induction*} (*complete induction*) lemmas complete_induct = Ord_induct [OF _ Ord_nat, case_names less, consumes 1] 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) 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)) |] ==> 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)); !!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 (rule ballI) apply (rename_tac i j) 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))" 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) done lemma succ_lt_induct: "[| 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) subsection{*quasinat: to allow a case-split rule for @{term nat_case}*} text{*True if the argument is zero or any successor*} lemma [iff]: "quasinat(0)" by (simp add: quasinat_def) lemma [iff]: "quasinat(succ(x))" by (simp add: quasinat_def) 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 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) done lemma nat_cases: "[|k=0 ==> P; !!y. k = succ(y) ==> P; ~ quasinat(k) ==> P|] ==> P" 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)" 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) 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]) apply (auto simp add: non_nat_case) done subsection{*Recursion on the Natural Numbers*} (** nat_rec is used to define eclose and transrec, then becomes obsolete. The operator rec, from arith.thy, has fewer typing conditions **) 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 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 (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" apply (rule Un_least_lt [THEN ltD]) apply (simp_all add: lt_def) done lemma Int_nat_type [TC]: "[| i: nat; j: nat |] ==> i Int j: nat" apply (rule Int_greatest_lt [THEN ltD]) apply (simp_all add: lt_def) done (*needed to simplify unions over nat*) lemma nat_nonempty [simp]: "nat ~= 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 (auto simp add: Ord_mem_iff_lt) apply (blast intro: lt_trans) done lemma Le_iff [iff]: "<x,y> : Le <-> x le y & x : nat & y : nat" by (force simp add: Le_def) end