(* Title: HOL/Nat.thy
Author: Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
Type "nat" is a linear order, and a datatype; arithmetic operators + -
and * (for div and mod, see theory Divides).
*)
section \<open>Natural numbers\<close>
theory Nat
imports Inductive Typedef Fun Fields
begin
ML_file "~~/src/Tools/rat.ML"
named_theorems arith "arith facts -- only ground formulas"
ML_file "Tools/arith_data.ML"
ML_file "~~/src/Provers/Arith/fast_lin_arith.ML"
subsection \<open>Type \<open>ind\<close>\<close>
typedecl ind
axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where
\<comment> \<open>the axiom of infinity in 2 parts\<close>
Suc_Rep_inject: "Suc_Rep x = Suc_Rep y ==> x = y" and
Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
subsection \<open>Type nat\<close>
text \<open>Type definition\<close>
inductive Nat :: "ind \<Rightarrow> bool" where
Zero_RepI: "Nat Zero_Rep"
| Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
typedef nat = "{n. Nat n}"
morphisms Rep_Nat Abs_Nat
using Nat.Zero_RepI by auto
lemma Nat_Rep_Nat:
"Nat (Rep_Nat n)"
using Rep_Nat by simp
lemma Nat_Abs_Nat_inverse:
"Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"
using Abs_Nat_inverse by simp
lemma Nat_Abs_Nat_inject:
"Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"
using Abs_Nat_inject by simp
instantiation nat :: zero
begin
definition Zero_nat_def:
"0 = Abs_Nat Zero_Rep"
instance ..
end
definition Suc :: "nat \<Rightarrow> nat" where
"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
lemma Suc_not_Zero: "Suc m \<noteq> 0"
by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat)
lemma Zero_not_Suc: "0 \<noteq> Suc m"
by (rule not_sym, rule Suc_not_Zero not_sym)
lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"
by (rule iffI, rule Suc_Rep_inject) simp_all
lemma nat_induct0:
fixes n
assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
shows "P n"
using assms
apply (unfold Zero_nat_def Suc_def)
apply (rule Rep_Nat_inverse [THEN subst]) \<comment> \<open>types force good instantiation\<close>
apply (erule Nat_Rep_Nat [THEN Nat.induct])
apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst])
done
free_constructors case_nat for
"0 :: nat"
| Suc pred
where
"pred (0 :: nat) = (0 :: nat)"
apply atomize_elim
apply (rename_tac n, induct_tac n rule: nat_induct0, auto)
apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject'
Rep_Nat_inject)
apply (simp only: Suc_not_Zero)
done
\<comment> \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close>
setup \<open>Sign.mandatory_path "old"\<close>
old_rep_datatype "0 :: nat" Suc
apply (erule nat_induct0, assumption)
apply (rule nat.inject)
apply (rule nat.distinct(1))
done
setup \<open>Sign.parent_path\<close>
\<comment> \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close>
setup \<open>Sign.mandatory_path "nat"\<close>
declare
old.nat.inject[iff del]
old.nat.distinct(1)[simp del, induct_simp del]
lemmas induct = old.nat.induct
lemmas inducts = old.nat.inducts
lemmas rec = old.nat.rec
lemmas simps = nat.inject nat.distinct nat.case nat.rec
setup \<open>Sign.parent_path\<close>
abbreviation rec_nat :: "'a \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a" where
"rec_nat \<equiv> old.rec_nat"
declare nat.sel[code del]
hide_const (open) Nat.pred \<comment> \<open>hide everything related to the selector\<close>
hide_fact
nat.case_eq_if
nat.collapse
nat.expand
nat.sel
nat.exhaust_sel
nat.split_sel
nat.split_sel_asm
lemma nat_exhaust [case_names 0 Suc, cases type: nat]:
\<comment> \<open>for backward compatibility -- names of variables differ\<close>
"(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P"
by (rule old.nat.exhaust)
lemma nat_induct [case_names 0 Suc, induct type: nat]:
\<comment> \<open>for backward compatibility -- names of variables differ\<close>
fixes n
assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
shows "P n"
using assms by (rule nat.induct)
hide_fact
nat_exhaust
nat_induct0
ML \<open>
val nat_basic_lfp_sugar =
let
val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global @{theory} @{type_name nat});
val recx = Logic.varify_types_global @{term rec_nat};
val C = body_type (fastype_of recx);
in
{T = HOLogic.natT, fp_res_index = 0, C = C, fun_arg_Tsss = [[], [[HOLogic.natT, C]]],
ctr_sugar = ctr_sugar, recx = recx, rec_thms = @{thms nat.rec}}
end;
\<close>
setup \<open>
let
fun basic_lfp_sugars_of _ [@{typ nat}] _ _ ctxt =
([], [0], [nat_basic_lfp_sugar], [], [], [], TrueI (*dummy*), [], false, ctxt)
| basic_lfp_sugars_of bs arg_Ts callers callssss ctxt =
BNF_LFP_Rec_Sugar.default_basic_lfp_sugars_of bs arg_Ts callers callssss ctxt;
in
BNF_LFP_Rec_Sugar.register_lfp_rec_extension
{nested_simps = [], is_new_datatype = K (K true), basic_lfp_sugars_of = basic_lfp_sugars_of,
rewrite_nested_rec_call = NONE}
end
\<close>
text \<open>Injectiveness and distinctness lemmas\<close>
lemma inj_Suc[simp]: "inj_on Suc N"
by (simp add: inj_on_def)
lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
by (rule notE, rule Suc_not_Zero)
lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
by (rule Suc_neq_Zero, erule sym)
lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
by (rule inj_Suc [THEN injD])
lemma n_not_Suc_n: "n \<noteq> Suc n"
by (induct n) simp_all
lemma Suc_n_not_n: "Suc n \<noteq> n"
by (rule not_sym, rule n_not_Suc_n)
text \<open>A special form of induction for reasoning
about @{term "m < n"} and @{term "m - n"}\<close>
lemma diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
(!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
apply (rule_tac x = m in spec)
apply (induct n)
prefer 2
apply (rule allI)
apply (induct_tac x, iprover+)
done
subsection \<open>Arithmetic operators\<close>
instantiation nat :: comm_monoid_diff
begin
primrec plus_nat where
add_0: "0 + n = (n::nat)"
| add_Suc: "Suc m + n = Suc (m + n)"
lemma add_0_right [simp]: "m + 0 = (m::nat)"
by (induct m) simp_all
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
by (induct m) simp_all
declare add_0 [code]
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
by simp
primrec minus_nat where
diff_0 [code]: "m - 0 = (m::nat)"
| diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
declare diff_Suc [simp del]
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
by (induct n) (simp_all add: diff_Suc)
lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
by (induct n) (simp_all add: diff_Suc)
instance proof
fix n m q :: nat
show "(n + m) + q = n + (m + q)" by (induct n) simp_all
show "n + m = m + n" by (induct n) simp_all
show "m + n - m = n" by (induct m) simp_all
show "n - m - q = n - (m + q)" by (induct q) (simp_all add: diff_Suc)
show "0 + n = n" by simp
show "0 - n = 0" by simp
qed
end
hide_fact (open) add_0 add_0_right diff_0
instantiation nat :: comm_semiring_1_cancel
begin
definition
One_nat_def [simp]: "1 = Suc 0"
primrec times_nat where
mult_0: "0 * n = (0::nat)"
| mult_Suc: "Suc m * n = n + (m * n)"
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
by (induct m) simp_all
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
by (induct m) (simp_all add: add.left_commute)
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
by (induct m) (simp_all add: add.assoc)
instance proof
fix n m q :: nat
show "0 \<noteq> (1::nat)" unfolding One_nat_def by simp
show "1 * n = n" unfolding One_nat_def by simp
show "n * m = m * n" by (induct n) simp_all
show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib)
show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib)
next
fix k m n :: nat
show "k * ((m::nat) - n) = (k * m) - (k * n)"
by (induct m n rule: diff_induct) simp_all
qed
end
subsubsection \<open>Addition\<close>
text \<open>Reasoning about \<open>m + 0 = 0\<close>, etc.\<close>
lemma add_is_0 [iff]:
fixes m n :: nat
shows "(m + n = 0) = (m = 0 & n = 0)"
by (cases m) simp_all
lemma add_is_1:
"(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
by (cases m) simp_all
lemma one_is_add:
"(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
by (rule trans, rule eq_commute, rule add_is_1)
lemma add_eq_self_zero:
fixes m n :: nat
shows "m + n = m \<Longrightarrow> n = 0"
by (induct m) simp_all
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
apply (induct k)
apply simp
apply(drule comp_inj_on[OF _ inj_Suc])
apply (simp add:o_def)
done
lemma Suc_eq_plus1: "Suc n = n + 1"
unfolding One_nat_def by simp
lemma Suc_eq_plus1_left: "Suc n = 1 + n"
unfolding One_nat_def by simp
subsubsection \<open>Difference\<close>
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
by (simp add: diff_diff_add)
lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
unfolding One_nat_def by simp
subsubsection \<open>Multiplication\<close>
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
by (induct m) auto
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"
apply (induct m)
apply simp
apply (induct n)
apply auto
done
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"
apply (rule trans)
apply (rule_tac [2] mult_eq_1_iff, fastforce)
done
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"
unfolding One_nat_def by (rule mult_eq_1_iff)
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
unfolding One_nat_def by (rule one_eq_mult_iff)
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
proof -
have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
proof (induct n arbitrary: m)
case 0 then show "m = 0" by simp
next
case (Suc n) then show "m = Suc n"
by (cases m) (simp_all add: eq_commute [of "0"])
qed
then show ?thesis by auto
qed
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
by (simp add: mult.commute)
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
by (subst mult_cancel1) simp
subsection \<open>Orders on @{typ nat}\<close>
subsubsection \<open>Operation definition\<close>
instantiation nat :: linorder
begin
primrec less_eq_nat where
"(0::nat) \<le> n \<longleftrightarrow> True"
| "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
declare less_eq_nat.simps [simp del]
lemma le0 [iff]: "0 \<le> (n::nat)" by (simp add: less_eq_nat.simps)
lemma [code]: "(0::nat) \<le> n \<longleftrightarrow> True" by simp
definition less_nat where
less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
by (simp add: less_eq_nat.simps(2))
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
unfolding less_eq_Suc_le ..
lemma le_0_eq [iff]: "(n::nat) \<le> 0 \<longleftrightarrow> n = 0"
by (induct n) (simp_all add: less_eq_nat.simps(2))
lemma not_less0 [iff]: "\<not> n < (0::nat)"
by (simp add: less_eq_Suc_le)
lemma less_nat_zero_code [code]: "n < (0::nat) \<longleftrightarrow> False"
by simp
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
by (simp add: less_eq_Suc_le)
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
by (simp add: less_eq_Suc_le)
lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
by (cases m) auto
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
by (induct m arbitrary: n)
(simp_all add: less_eq_nat.simps(2) split: nat.splits)
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
by (cases n) (auto intro: le_SucI)
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
by (simp add: less_eq_Suc_le) (erule Suc_leD)
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
by (simp add: less_eq_Suc_le) (erule Suc_leD)
instance
proof
fix n m :: nat
show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n"
proof (induct n arbitrary: m)
case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
next
case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
qed
next
fix n :: nat show "n \<le> n" by (induct n) simp_all
next
fix n m :: nat assume "n \<le> m" and "m \<le> n"
then show "n = m"
by (induct n arbitrary: m)
(simp_all add: less_eq_nat.simps(2) split: nat.splits)
next
fix n m q :: nat assume "n \<le> m" and "m \<le> q"
then show "n \<le> q"
proof (induct n arbitrary: m q)
case 0 show ?case by simp
next
case (Suc n) then show ?case
by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
qed
next
fix n m :: nat show "n \<le> m \<or> m \<le> n"
by (induct n arbitrary: m)
(simp_all add: less_eq_nat.simps(2) split: nat.splits)
qed
end
instantiation nat :: order_bot
begin
definition bot_nat :: nat where
"bot_nat = 0"
instance proof
qed (simp add: bot_nat_def)
end
instance nat :: no_top
by standard (auto intro: less_Suc_eq_le [THEN iffD2])
subsubsection \<open>Introduction properties\<close>
lemma lessI [iff]: "n < Suc n"
by (simp add: less_Suc_eq_le)
lemma zero_less_Suc [iff]: "0 < Suc n"
by (simp add: less_Suc_eq_le)
subsubsection \<open>Elimination properties\<close>
lemma less_not_refl: "~ n < (n::nat)"
by (rule order_less_irrefl)
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
by (rule not_sym) (rule less_imp_neq)
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
by (rule less_imp_neq)
lemma less_irrefl_nat: "(n::nat) < n ==> R"
by (rule notE, rule less_not_refl)
lemma less_zeroE: "(n::nat) < 0 ==> R"
by (rule notE) (rule not_less0)
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
unfolding less_Suc_eq_le le_less ..
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
by (simp add: less_Suc_eq)
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
unfolding One_nat_def by (rule less_Suc0)
lemma Suc_mono: "m < n ==> Suc m < Suc n"
by simp
text \<open>"Less than" is antisymmetric, sort of\<close>
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
unfolding not_less less_Suc_eq_le by (rule antisym)
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
by (rule linorder_neq_iff)
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
shows "P n m"
apply (rule less_linear [THEN disjE])
apply (erule_tac [2] disjE)
apply (erule lessCase)
apply (erule sym [THEN eqCase])
apply (erule major)
done
subsubsection \<open>Inductive (?) properties\<close>
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
unfolding less_eq_Suc_le [of m] le_less by simp
lemma lessE:
assumes major: "i < k"
and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
shows P
proof -
from major have "\<exists>j. i \<le> j \<and> k = Suc j"
unfolding less_eq_Suc_le by (induct k) simp_all
then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
by (clarsimp simp add: less_le)
with p1 p2 show P by auto
qed
lemma less_SucE: assumes major: "m < Suc n"
and less: "m < n ==> P" and eq: "m = n ==> P" shows P
apply (rule major [THEN lessE])
apply (rule eq, blast)
apply (rule less, blast)
done
lemma Suc_lessE: assumes major: "Suc i < k"
and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
apply (rule major [THEN lessE])
apply (erule lessI [THEN minor])
apply (erule Suc_lessD [THEN minor], assumption)
done
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
by simp
lemma less_trans_Suc:
assumes le: "i < j" shows "j < k ==> Suc i < k"
apply (induct k, simp_all)
apply (insert le)
apply (simp add: less_Suc_eq)
apply (blast dest: Suc_lessD)
done
text \<open>Can be used with \<open>less_Suc_eq\<close> to get @{term "n = m | n < m"}\<close>
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
unfolding not_less less_Suc_eq_le ..
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
unfolding not_le Suc_le_eq ..
text \<open>Properties of "less than or equal"\<close>
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
unfolding less_Suc_eq_le .
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
unfolding not_le less_Suc_eq_le ..
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
by (drule le_Suc_eq [THEN iffD1], iprover+)
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
unfolding Suc_le_eq .
text \<open>Stronger version of \<open>Suc_leD\<close>\<close>
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
unfolding Suc_le_eq .
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
unfolding less_eq_Suc_le by (rule Suc_leD)
text \<open>For instance, \<open>(Suc m < Suc n) = (Suc m \<le> n) = (m < n)\<close>\<close>
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
text \<open>Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"}\<close>
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
unfolding le_less .
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
by (rule le_less)
text \<open>Useful with \<open>blast\<close>.\<close>
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
by auto
lemma le_refl: "n \<le> (n::nat)"
by simp
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
by (rule order_trans)
lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
by (rule antisym)
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
by (rule less_le)
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
unfolding less_le ..
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
by (rule linear)
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
unfolding less_Suc_eq_le by auto
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
unfolding not_less by (rule le_less_Suc_eq)
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
by (cases n) simp_all
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
by (cases n) simp_all
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
by (cases n) simp_all
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
by (cases n) simp_all
text \<open>This theorem is useful with \<open>blast\<close>\<close>
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
by (rule neq0_conv[THEN iffD1], iprover)
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
by (fast intro: not0_implies_Suc)
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
using neq0_conv by blast
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
by (induct m') simp_all
text \<open>Useful in certain inductive arguments\<close>
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
by (cases m) simp_all
subsubsection \<open>Monotonicity of Addition\<close>
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
by (simp add: diff_Suc split: nat.split)
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
unfolding One_nat_def by (rule Suc_pred)
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
by (induct k) simp_all
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
by (induct k) simp_all
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
by(auto dest:gr0_implies_Suc)
text \<open>strict, in 1st argument\<close>
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
by (induct k) simp_all
text \<open>strict, in both arguments\<close>
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
apply (rule add_less_mono1 [THEN less_trans], assumption+)
apply (induct j, simp_all)
done
text \<open>Deleted \<open>less_natE\<close>; use \<open>less_imp_Suc_add RS exE\<close>\<close>
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
apply (induct n)
apply (simp_all add: order_le_less)
apply (blast elim!: less_SucE
intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
done
lemma le_Suc_ex: "(k::nat) \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)
text \<open>strict, in 1st argument; proof is by induction on \<open>k > 0\<close>\<close>
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
apply(auto simp: gr0_conv_Suc)
apply (induct_tac m)
apply (simp_all add: add_less_mono)
done
text \<open>Addition is the inverse of subtraction:
if @{term "n \<le> m"} then @{term "n + (m - n) = m"}.\<close>
lemma add_diff_inverse_nat: "~ m < n ==> n + (m - n) = (m::nat)"
by (induct m n rule: diff_induct) simp_all
lemma nat_le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
using nat_add_left_cancel_le[of m 0] by (auto dest: le_Suc_ex)
text\<open>The naturals form an ordered \<open>semidom\<close> and a \<open>dioid\<close>\<close>
instance nat :: linordered_semidom
proof
show "0 < (1::nat)" by simp
show "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp
show "\<And>m n q :: nat. m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)
show "\<And>m n :: nat. m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0" by simp
show "\<And>m n :: nat. n \<le> m ==> (m - n) + n = (m::nat)"
by (simp add: add_diff_inverse_nat add.commute linorder_not_less)
qed
instance nat :: dioid
proof qed (rule nat_le_iff_add)
declare le0[simp del] -- \<open>This is now @{thm zero_le}\<close>
declare le_0_eq[simp del] -- \<open>This is now @{thm le_zero_eq}\<close>
declare not_less0[simp del] -- \<open>This is now @{thm not_less_zero}\<close>
declare not_gr0[simp del] -- \<open>This is now @{thm not_gr_zero}\<close>
instance nat :: ordered_cancel_comm_monoid_add
proof qed
instance nat :: ordered_cancel_comm_monoid_diff
proof qed
subsubsection \<open>@{term min} and @{term max}\<close>
lemma mono_Suc: "mono Suc"
by (rule monoI) simp
lemma min_0L [simp]: "min 0 n = (0::nat)"
by (rule min_absorb1) simp
lemma min_0R [simp]: "min n 0 = (0::nat)"
by (rule min_absorb2) simp
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
by (simp add: mono_Suc min_of_mono)
lemma min_Suc1:
"min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
by (simp split: nat.split)
lemma min_Suc2:
"min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
by (simp split: nat.split)
lemma max_0L [simp]: "max 0 n = (n::nat)"
by (rule max_absorb2) simp
lemma max_0R [simp]: "max n 0 = (n::nat)"
by (rule max_absorb1) simp
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
by (simp add: mono_Suc max_of_mono)
lemma max_Suc1:
"max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
by (simp split: nat.split)
lemma max_Suc2:
"max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
by (simp split: nat.split)
lemma nat_mult_min_left:
fixes m n q :: nat
shows "min m n * q = min (m * q) (n * q)"
by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
lemma nat_mult_min_right:
fixes m n q :: nat
shows "m * min n q = min (m * n) (m * q)"
by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
lemma nat_add_max_left:
fixes m n q :: nat
shows "max m n + q = max (m + q) (n + q)"
by (simp add: max_def)
lemma nat_add_max_right:
fixes m n q :: nat
shows "m + max n q = max (m + n) (m + q)"
by (simp add: max_def)
lemma nat_mult_max_left:
fixes m n q :: nat
shows "max m n * q = max (m * q) (n * q)"
by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
lemma nat_mult_max_right:
fixes m n q :: nat
shows "m * max n q = max (m * n) (m * q)"
by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
subsubsection \<open>Additional theorems about @{term "op \<le>"}\<close>
text \<open>Complete induction, aka course-of-values induction\<close>
instance nat :: wellorder proof
fix P and n :: nat
assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
have "\<And>q. q \<le> n \<Longrightarrow> P q"
proof (induct n)
case (0 n)
have "P 0" by (rule step) auto
thus ?case using 0 by auto
next
case (Suc m n)
then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
thus ?case
proof
assume "n \<le> m" thus "P n" by (rule Suc(1))
next
assume n: "n = Suc m"
show "P n"
by (rule step) (rule Suc(1), simp add: n le_simps)
qed
qed
then show "P n" by auto
qed
lemma Least_eq_0[simp]: "P(0::nat) \<Longrightarrow> Least P = 0"
by (rule Least_equality[OF _ le0])
lemma Least_Suc:
"[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
apply (cases n, auto)
apply (frule LeastI)
apply (drule_tac P = "%x. P (Suc x) " in LeastI)
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
apply (erule_tac [2] Least_le)
apply (cases "LEAST x. P x", auto)
apply (drule_tac P = "%x. P (Suc x) " in Least_le)
apply (blast intro: order_antisym)
done
lemma Least_Suc2:
"[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
apply (erule (1) Least_Suc [THEN ssubst])
apply simp
done
lemma ex_least_nat_le: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not>P i) & P(k)"
apply (cases n)
apply blast
apply (rule_tac x="LEAST k. P(k)" in exI)
apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
done
lemma ex_least_nat_less: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not>P i) & P(k+1)"
unfolding One_nat_def
apply (cases n)
apply blast
apply (frule (1) ex_least_nat_le)
apply (erule exE)
apply (case_tac k)
apply simp
apply (rename_tac k1)
apply (rule_tac x=k1 in exI)
apply (auto simp add: less_eq_Suc_le)
done
lemma nat_less_induct:
assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
using assms less_induct by blast
lemma measure_induct_rule [case_names less]:
fixes f :: "'a \<Rightarrow> nat"
assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
shows "P a"
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
text \<open>old style induction rules:\<close>
lemma measure_induct:
fixes f :: "'a \<Rightarrow> nat"
shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
by (rule measure_induct_rule [of f P a]) iprover
lemma full_nat_induct:
assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
shows "P n"
by (rule less_induct) (auto intro: step simp:le_simps)
text\<open>An induction rule for estabilishing binary relations\<close>
lemma less_Suc_induct:
assumes less: "i < j"
and step: "!!i. P i (Suc i)"
and trans: "!!i j k. i < j ==> j < k ==> P i j ==> P j k ==> P i k"
shows "P i j"
proof -
from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
have "P i (Suc (i + k))"
proof (induct k)
case 0
show ?case by (simp add: step)
next
case (Suc k)
have "0 + i < Suc k + i" by (rule add_less_mono1) simp
hence "i < Suc (i + k)" by (simp add: add.commute)
from trans[OF this lessI Suc step]
show ?case by simp
qed
thus "P i j" by (simp add: j)
qed
text \<open>The method of infinite descent, frequently used in number theory.
Provided by Roelof Oosterhuis.
$P(n)$ is true for all $n\in\mathbb{N}$ if
\begin{itemize}
\item case ``0'': given $n=0$ prove $P(n)$,
\item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
a smaller integer $m$ such that $\neg P(m)$.
\end{itemize}\<close>
text\<open>A compact version without explicit base case:\<close>
lemma infinite_descent:
"\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m \<rbrakk> \<Longrightarrow> P n"
by (induct n rule: less_induct) auto
lemma infinite_descent0[case_names 0 smaller]:
"\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
by (rule infinite_descent) (case_tac "n>0", auto)
text \<open>
Infinite descent using a mapping to $\mathbb{N}$:
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
\begin{itemize}
\item case ``0'': given $V(x)=0$ prove $P(x)$,
\item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$.
\end{itemize}
NB: the proof also shows how to use the previous lemma.\<close>
corollary infinite_descent0_measure [case_names 0 smaller]:
assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
and A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
shows "P x"
proof -
obtain n where "n = V x" by auto
moreover have "\<And>x. V x = n \<Longrightarrow> P x"
proof (induct n rule: infinite_descent0)
case 0 \<comment> "i.e. $V(x) = 0$"
with A0 show "P x" by auto
next \<comment> "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
case (smaller n)
then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
then show ?case by auto
qed
ultimately show "P x" by auto
qed
text\<open>Again, without explicit base case:\<close>
lemma infinite_descent_measure:
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
proof -
from assms obtain n where "n = V x" by auto
moreover have "!!x. V x = n \<Longrightarrow> P x"
proof (induct n rule: infinite_descent, auto)
fix x assume "\<not> P x"
with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
qed
ultimately show "P x" by auto
qed
text \<open>A [clumsy] way of lifting \<open><\<close>
monotonicity to \<open>\<le>\<close> monotonicity\<close>
lemma less_mono_imp_le_mono:
"\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
by (simp add: order_le_less) (blast)
text \<open>non-strict, in 1st argument\<close>
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
by (rule add_right_mono)
text \<open>non-strict, in both arguments\<close>
lemma add_le_mono: "[| i \<le> j; k \<le> l |] ==> i + k \<le> j + (l::nat)"
by (rule add_mono)
lemma le_add2: "n \<le> ((m + n)::nat)"
by (insert add_right_mono [of 0 m n], simp)
lemma le_add1: "n \<le> ((n + m)::nat)"
by (simp add: add.commute, rule le_add2)
lemma less_add_Suc1: "i < Suc (i + m)"
by (rule le_less_trans, rule le_add1, rule lessI)
lemma less_add_Suc2: "i < Suc (m + i)"
by (rule le_less_trans, rule le_add2, rule lessI)
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
by (rule le_trans, assumption, rule le_add1)
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
by (rule le_trans, assumption, rule le_add2)
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
by (rule less_le_trans, assumption, rule le_add1)
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
by (rule less_le_trans, assumption, rule le_add2)
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
apply (rule le_less_trans [of _ "i+j"])
apply (simp_all add: le_add1)
done
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
apply (rule notI)
apply (drule add_lessD1)
apply (erule less_irrefl [THEN notE])
done
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
by (simp add: add.commute)
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
apply (rule order_trans [of _ "m+k"])
apply (simp_all add: le_add1)
done
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
apply (simp add: add.commute)
apply (erule add_leD1)
done
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
by (blast dest: add_leD1 add_leD2)
text \<open>needs \<open>!!k\<close> for \<open>ac_simps\<close> to work\<close>
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
by (force simp del: add_Suc_right
simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)
subsubsection \<open>More results about difference\<close>
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
by (induct m n rule: diff_induct) simp_all
lemma diff_less_Suc: "m - n < Suc m"
apply (induct m n rule: diff_induct)
apply (erule_tac [3] less_SucE)
apply (simp_all add: less_Suc_eq)
done
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
by (rule le_less_trans, rule diff_le_self)
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
by (cases n) (auto simp add: le_simps)
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
by (induct j k rule: diff_induct) simp_all
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
by (simp add: add.commute diff_add_assoc)
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
by auto
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
by (induct m n rule: diff_induct) simp_all
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
by (rule iffD2, rule diff_is_0_eq)
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
by (induct m n rule: diff_induct) simp_all
lemma less_imp_add_positive:
assumes "i < j"
shows "\<exists>k::nat. 0 < k & i + k = j"
proof
from assms show "0 < j - i & i + (j - i) = j"
by (simp add: order_less_imp_le)
qed
text \<open>a nice rewrite for bounded subtraction\<close>
lemma nat_minus_add_max:
fixes n m :: nat
shows "n - m + m = max n m"
by (simp add: max_def not_le order_less_imp_le)
lemma nat_diff_split:
fixes a b :: nat
shows "P (a - b) \<longleftrightarrow> (a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d)"
\<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close>\<close>
by (cases "a < b")
(auto simp add: not_less le_less dest!: add_eq_self_zero [OF sym])
lemma nat_diff_split_asm:
fixes a b :: nat
shows "P (a - b) \<longleftrightarrow> \<not> (a < b \<and> \<not> P 0 \<or> (\<exists>d. a = b + d \<and> \<not> P d))"
\<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close> in assumptions\<close>
by (auto split: nat_diff_split)
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
by simp
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
unfolding One_nat_def by (cases m) simp_all
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
unfolding One_nat_def by (cases m) simp_all
lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"
unfolding One_nat_def by (cases n) simp_all
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
unfolding One_nat_def by (cases m) simp_all
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
by (fact Let_def)
subsubsection \<open>Monotonicity of multiplication\<close>
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
by (simp add: mult_right_mono)
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
by (simp add: mult_left_mono)
text \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close>
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
by (simp add: mult_mono)
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
by (simp add: mult_strict_right_mono)
text\<open>Differs from the standard \<open>zero_less_mult_iff\<close> in that
there are no negative numbers.\<close>
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
apply (induct m)
apply simp
apply (case_tac n)
apply simp_all
done
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
apply (induct m)
apply simp
apply (case_tac n)
apply simp_all
done
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
apply (safe intro!: mult_less_mono1)
apply (cases k, auto)
apply (simp del: le_0_eq add: linorder_not_le [symmetric])
apply (blast intro: mult_le_mono1)
done
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
by (simp add: mult.commute [of k])
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
by (simp add: linorder_not_less [symmetric], auto)
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
by (simp add: linorder_not_less [symmetric], auto)
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
by (subst mult_less_cancel1) simp
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
by (subst mult_le_cancel1) simp
lemma le_square: "m \<le> m * (m::nat)"
by (cases m) (auto intro: le_add1)
lemma le_cube: "(m::nat) \<le> m * (m * m)"
by (cases m) (auto intro: le_add1)
text \<open>Lemma for \<open>gcd\<close>\<close>
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
apply (drule sym)
apply (rule disjCI)
apply (rule nat_less_cases, erule_tac [2] _)
apply (drule_tac [2] mult_less_mono2)
apply (auto)
done
lemma mono_times_nat:
fixes n :: nat
assumes "n > 0"
shows "mono (times n)"
proof
fix m q :: nat
assume "m \<le> q"
with assms show "n * m \<le> n * q" by simp
qed
text \<open>the lattice order on @{typ nat}\<close>
instantiation nat :: distrib_lattice
begin
definition
"(inf :: nat \<Rightarrow> nat \<Rightarrow> nat) = min"
definition
"(sup :: nat \<Rightarrow> nat \<Rightarrow> nat) = max"
instance by intro_classes
(auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
end
subsection \<open>Natural operation of natural numbers on functions\<close>
text \<open>
We use the same logical constant for the power operations on
functions and relations, in order to share the same syntax.
\<close>
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where
"f ^^ n \<equiv> compow n f"
notation (latex output)
compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
text \<open>\<open>f ^^ n = f o ... o f\<close>, the n-fold composition of \<open>f\<close>\<close>
overloading
funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
begin
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
"funpow 0 f = id"
| "funpow (Suc n) f = f o funpow n f"
end
lemma funpow_0 [simp]: "(f ^^ 0) x = x"
by simp
lemma funpow_Suc_right:
"f ^^ Suc n = f ^^ n \<circ> f"
proof (induct n)
case 0 then show ?case by simp
next
fix n
assume "f ^^ Suc n = f ^^ n \<circ> f"
then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"
by (simp add: o_assoc)
qed
lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right
text \<open>for code generation\<close>
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
funpow_code_def [code_abbrev]: "funpow = compow"
lemma [code]:
"funpow (Suc n) f = f o funpow n f"
"funpow 0 f = id"
by (simp_all add: funpow_code_def)
hide_const (open) funpow
lemma funpow_add:
"f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
by (induct m) simp_all
lemma funpow_mult:
fixes f :: "'a \<Rightarrow> 'a"
shows "(f ^^ m) ^^ n = f ^^ (m * n)"
by (induct n) (simp_all add: funpow_add)
lemma funpow_swap1:
"f ((f ^^ n) x) = (f ^^ n) (f x)"
proof -
have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
also have "\<dots> = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
also have "\<dots> = (f ^^ n) (f x)" by simp
finally show ?thesis .
qed
lemma comp_funpow:
fixes f :: "'a \<Rightarrow> 'a"
shows "comp f ^^ n = comp (f ^^ n)"
by (induct n) simp_all
lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)"
by (induct n) simp_all
lemma id_funpow[simp]: "id ^^ n = id"
by (induct n) simp_all
lemma funpow_mono:
fixes f :: "'a \<Rightarrow> ('a::lattice)"
shows "mono f \<Longrightarrow> A \<le> B \<Longrightarrow> (f ^^ n) A \<le> (f ^^ n) B"
by (induct n arbitrary: A B)
(auto simp del: funpow.simps(2) simp add: funpow_Suc_right mono_def)
subsection \<open>Kleene iteration\<close>
lemma Kleene_iter_lpfp:
assumes "mono f" and "f p \<le> p" shows "(f^^k) (bot::'a::order_bot) \<le> p"
proof(induction k)
case 0 show ?case by simp
next
case Suc
from monoD[OF assms(1) Suc] assms(2)
show ?case by simp
qed
lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"
shows "lfp f = (f^^k) bot"
proof(rule antisym)
show "lfp f \<le> (f^^k) bot"
proof(rule lfp_lowerbound)
show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp
qed
next
show "(f^^k) bot \<le> lfp f"
using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
qed
lemma mono_pow:
fixes f :: "'a \<Rightarrow> 'a::complete_lattice"
shows "mono f \<Longrightarrow> mono (f ^^ n)"
by (induction n) (auto simp: mono_def)
lemma lfp_funpow:
assumes f: "mono f" shows "lfp (f ^^ Suc n) = lfp f"
proof (rule antisym)
show "lfp f \<le> lfp (f ^^ Suc n)"
proof (rule lfp_lowerbound)
have "f (lfp (f ^^ Suc n)) = lfp (\<lambda>x. f ((f ^^ n) x))"
unfolding funpow_Suc_right by (simp add: lfp_rolling f mono_pow comp_def)
then show "f (lfp (f ^^ Suc n)) \<le> lfp (f ^^ Suc n)"
by (simp add: comp_def)
qed
have "(f^^n) (lfp f) = lfp f" for n
by (induction n) (auto intro: f lfp_unfold[symmetric])
then show "lfp (f^^Suc n) \<le> lfp f"
by (intro lfp_lowerbound) (simp del: funpow.simps)
qed
lemma gfp_funpow:
assumes f: "mono f" shows "gfp (f ^^ Suc n) = gfp f"
proof (rule antisym)
show "gfp f \<ge> gfp (f ^^ Suc n)"
proof (rule gfp_upperbound)
have "f (gfp (f ^^ Suc n)) = gfp (\<lambda>x. f ((f ^^ n) x))"
unfolding funpow_Suc_right by (simp add: gfp_rolling f mono_pow comp_def)
then show "f (gfp (f ^^ Suc n)) \<ge> gfp (f ^^ Suc n)"
by (simp add: comp_def)
qed
have "(f^^n) (gfp f) = gfp f" for n
by (induction n) (auto intro: f gfp_unfold[symmetric])
then show "gfp (f^^Suc n) \<ge> gfp f"
by (intro gfp_upperbound) (simp del: funpow.simps)
qed
subsection \<open>Embedding of the naturals into any \<open>semiring_1\<close>: @{term of_nat}\<close>
context semiring_1
begin
definition of_nat :: "nat \<Rightarrow> 'a" where
"of_nat n = (plus 1 ^^ n) 0"
lemma of_nat_simps [simp]:
shows of_nat_0: "of_nat 0 = 0"
and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
by (simp_all add: of_nat_def)
lemma of_nat_1 [simp]: "of_nat 1 = 1"
by (simp add: of_nat_def)
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
by (induct m) (simp_all add: ac_simps)
lemma of_nat_mult [simp]: "of_nat (m * n) = of_nat m * of_nat n"
by (induct m) (simp_all add: ac_simps distrib_right)
lemma mult_of_nat_commute: "of_nat x * y = y * of_nat x"
by (induction x) (simp_all add: algebra_simps)
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
"of_nat_aux inc 0 i = i"
| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" \<comment> \<open>tail recursive\<close>
lemma of_nat_code:
"of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
proof (induct n)
case 0 then show ?case by simp
next
case (Suc n)
have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"
by (induct n) simp_all
from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"
by simp
with Suc show ?case by (simp add: add.commute)
qed
end
declare of_nat_code [code]
text\<open>Class for unital semirings with characteristic zero.
Includes non-ordered rings like the complex numbers.\<close>
class semiring_char_0 = semiring_1 +
assumes inj_of_nat: "inj of_nat"
begin
lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
by (auto intro: inj_of_nat injD)
text\<open>Special cases where either operand is zero\<close>
lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
lemma of_nat_neq_0 [simp]:
"of_nat (Suc n) \<noteq> 0"
unfolding of_nat_eq_0_iff by simp
lemma of_nat_0_neq [simp]:
"0 \<noteq> of_nat (Suc n)"
unfolding of_nat_0_eq_iff by simp
end
context linordered_semidom
begin
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
by (induct n) simp_all
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
by (simp add: not_less)
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
by (induct m n rule: diff_induct) (simp_all add: add_pos_nonneg)
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
by (simp add: not_less [symmetric] linorder_not_less [symmetric])
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
by simp
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
by simp
text\<open>Every \<open>linordered_semidom\<close> has characteristic zero.\<close>
subclass semiring_char_0 proof
qed (auto intro!: injI simp add: eq_iff)
text\<open>Special cases where either operand is zero\<close>
lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
by (rule of_nat_le_iff [of _ 0, simplified])
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
by (rule of_nat_less_iff [of 0, simplified])
end
context ring_1
begin
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
by (simp add: algebra_simps of_nat_add [symmetric])
end
context linordered_idom
begin
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
unfolding abs_if by auto
end
lemma of_nat_id [simp]: "of_nat n = n"
by (induct n) simp_all
lemma of_nat_eq_id [simp]: "of_nat = id"
by (auto simp add: fun_eq_iff)
subsection \<open>The set of natural numbers\<close>
context semiring_1
begin
definition Nats :: "'a set" ("\<nat>")
where "\<nat> = range of_nat"
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
by (simp add: Nats_def)
lemma Nats_0 [simp]: "0 \<in> \<nat>"
apply (simp add: Nats_def)
apply (rule range_eqI)
apply (rule of_nat_0 [symmetric])
done
lemma Nats_1 [simp]: "1 \<in> \<nat>"
apply (simp add: Nats_def)
apply (rule range_eqI)
apply (rule of_nat_1 [symmetric])
done
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
apply (auto simp add: Nats_def)
apply (rule range_eqI)
apply (rule of_nat_add [symmetric])
done
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
apply (auto simp add: Nats_def)
apply (rule range_eqI)
apply (rule of_nat_mult [symmetric])
done
lemma Nats_cases [cases set: Nats]:
assumes "x \<in> \<nat>"
obtains (of_nat) n where "x = of_nat n"
unfolding Nats_def
proof -
from \<open>x \<in> \<nat>\<close> have "x \<in> range of_nat" unfolding Nats_def .
then obtain n where "x = of_nat n" ..
then show thesis ..
qed
lemma Nats_induct [case_names of_nat, induct set: Nats]:
"x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
by (rule Nats_cases) auto
end
subsection \<open>Further arithmetic facts concerning the natural numbers\<close>
lemma subst_equals:
assumes 1: "t = s" and 2: "u = t"
shows "u = s"
using 2 1 by (rule trans)
ML_file "Tools/nat_arith.ML"
simproc_setup nateq_cancel_sums
("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =
\<open>fn phi => try o Nat_Arith.cancel_eq_conv\<close>
simproc_setup natless_cancel_sums
("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =
\<open>fn phi => try o Nat_Arith.cancel_less_conv\<close>
simproc_setup natle_cancel_sums
("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =
\<open>fn phi => try o Nat_Arith.cancel_le_conv\<close>
simproc_setup natdiff_cancel_sums
("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
\<open>fn phi => try o Nat_Arith.cancel_diff_conv\<close>
ML_file "Tools/lin_arith.ML"
setup \<open>Lin_Arith.global_setup\<close>
declaration \<open>K Lin_Arith.setup\<close>
simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) \<le> n" | "(m::nat) = n") =
\<open>K Lin_Arith.simproc\<close>
(* Because of this simproc, the arithmetic solver is really only
useful to detect inconsistencies among the premises for subgoals which are
*not* themselves (in)equalities, because the latter activate
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
solver all the time rather than add the additional check. *)
lemmas [arith_split] = nat_diff_split split_min split_max
context order
begin
lemma lift_Suc_mono_le:
assumes mono: "\<And>n. f n \<le> f (Suc n)" and "n \<le> n'"
shows "f n \<le> f n'"
proof (cases "n < n'")
case True
then show ?thesis
by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
qed (insert \<open>n \<le> n'\<close>, auto) \<comment> \<open>trivial for @{prop "n = n'"}\<close>
lemma lift_Suc_antimono_le:
assumes mono: "\<And>n. f n \<ge> f (Suc n)" and "n \<le> n'"
shows "f n \<ge> f n'"
proof (cases "n < n'")
case True
then show ?thesis
by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
qed (insert \<open>n \<le> n'\<close>, auto) \<comment> \<open>trivial for @{prop "n = n'"}\<close>
lemma lift_Suc_mono_less:
assumes mono: "\<And>n. f n < f (Suc n)" and "n < n'"
shows "f n < f n'"
using \<open>n < n'\<close>
by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
lemma lift_Suc_mono_less_iff:
"(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m"
by (blast intro: less_asym' lift_Suc_mono_less [of f]
dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1])
end
lemma mono_iff_le_Suc:
"mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
lemma antimono_iff_le_Suc:
"antimono f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f])
lemma mono_nat_linear_lb:
fixes f :: "nat \<Rightarrow> nat"
assumes "\<And>m n. m < n \<Longrightarrow> f m < f n"
shows "f m + k \<le> f (m + k)"
proof (induct k)
case 0 then show ?case by simp
next
case (Suc k)
then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp
also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))"
by (simp add: Suc_le_eq)
finally show ?case by simp
qed
text\<open>Subtraction laws, mostly by Clemens Ballarin\<close>
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
by arith
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
by arith
lemma less_diff_conv2:
fixes j k i :: nat
assumes "k \<le> j"
shows "j - k < i \<longleftrightarrow> j < i + k"
using assms by arith
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
by arith
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
by arith
(*Replaces the previous diff_less and le_diff_less, which had the stronger
second premise n\<le>m*)
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
by arith
text \<open>Simplification of relational expressions involving subtraction\<close>
lemma diff_diff_eq: "[| k \<le> m; k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
by (simp split add: nat_diff_split)
hide_fact (open) diff_diff_eq
lemma eq_diff_iff: "[| k \<le> m; k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
by (auto split add: nat_diff_split)
lemma less_diff_iff: "[| k \<le> m; k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
by (auto split add: nat_diff_split)
lemma le_diff_iff: "[| k \<le> m; k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
by (auto split add: nat_diff_split)
text\<open>(Anti)Monotonicity of subtraction -- by Stephan Merz\<close>
(* Monotonicity of subtraction in first argument *)
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
by (simp split add: nat_diff_split)
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
by (simp split add: nat_diff_split)
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
by (simp split add: nat_diff_split)
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==> m=n"
by (simp split add: nat_diff_split)
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
by auto
lemma inj_on_diff_nat:
assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
shows "inj_on (\<lambda>n. n - k) N"
proof (rule inj_onI)
fix x y
assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
with k_le_n have "x - k + k = y - k + k" by auto
with a k_le_n show "x = y" by auto
qed
text\<open>Rewriting to pull differences out\<close>
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
by arith
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
by arith
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
by arith
lemma Suc_diff_Suc: "n < m \<Longrightarrow> Suc (m - Suc n) = m - n"
by simp
(*The others are
i - j - k = i - (j + k),
k \<le> j ==> j - k + i = j + i - k,
k \<le> j ==> i + (j - k) = i + j - k *)
lemmas add_diff_assoc = diff_add_assoc [symmetric]
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
declare add_diff_assoc [simp] add_diff_assoc2[simp]
text\<open>At present we prove no analogue of \<open>not_less_Least\<close> or \<open>Least_Suc\<close>, since there appears to be no need.\<close>
text\<open>Lemmas for ex/Factorization\<close>
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
by (cases m) auto
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
by (cases m) auto
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
by (cases m) auto
text \<open>Specialized induction principles that work "backwards":\<close>
lemma inc_induct[consumes 1, case_names base step]:
assumes less: "i \<le> j"
assumes base: "P j"
assumes step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n"
shows "P i"
using less step
proof (induct d\<equiv>"j - i" arbitrary: i)
case (0 i)
hence "i = j" by simp
with base show ?case by simp
next
case (Suc d n)
hence "n \<le> n" "n < j" "P (Suc n)"
by simp_all
then show "P n" by fact
qed
lemma strict_inc_induct[consumes 1, case_names base step]:
assumes less: "i < j"
assumes base: "!!i. j = Suc i ==> P i"
assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
shows "P i"
using less
proof (induct d=="j - i - 1" arbitrary: i)
case (0 i)
with \<open>i < j\<close> have "j = Suc i" by simp
with base show ?case by simp
next
case (Suc d i)
hence "i < j" "P (Suc i)"
by simp_all
thus "P i" by (rule step)
qed
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
using inc_induct[of "k - i" k P, simplified] by blast
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
using inc_induct[of 0 k P] by blast
text \<open>Further induction rule similar to @{thm inc_induct}\<close>
lemma dec_induct[consumes 1, case_names base step]:
"i \<le> j \<Longrightarrow> P i \<Longrightarrow> (\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n \<Longrightarrow> P (Suc n)) \<Longrightarrow> P j"
by (induct j arbitrary: i) (auto simp: le_Suc_eq)
subsection \<open> Monotonicity of funpow \<close>
lemma funpow_increasing:
fixes f :: "'a \<Rightarrow> ('a::{lattice, order_top})"
shows "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ n) \<top> \<le> (f ^^ m) \<top>"
by (induct rule: inc_induct)
(auto simp del: funpow.simps(2) simp add: funpow_Suc_right
intro: order_trans[OF _ funpow_mono])
lemma funpow_decreasing:
fixes f :: "'a \<Rightarrow> ('a::{lattice, order_bot})"
shows "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ m) \<bottom> \<le> (f ^^ n) \<bottom>"
by (induct rule: dec_induct)
(auto simp del: funpow.simps(2) simp add: funpow_Suc_right
intro: order_trans[OF _ funpow_mono])
lemma mono_funpow:
fixes Q :: "'a::{lattice, order_bot} \<Rightarrow> 'a"
shows "mono Q \<Longrightarrow> mono (\<lambda>i. (Q ^^ i) \<bottom>)"
by (auto intro!: funpow_decreasing simp: mono_def)
lemma antimono_funpow:
fixes Q :: "'a::{lattice, order_top} \<Rightarrow> 'a"
shows "mono Q \<Longrightarrow> antimono (\<lambda>i. (Q ^^ i) \<top>)"
by (auto intro!: funpow_increasing simp: antimono_def)
subsection \<open>The divides relation on @{typ nat}\<close>
lemma dvd_1_left [iff]:
"Suc 0 dvd k"
by (simp add: dvd_def)
lemma dvd_1_iff_1 [simp]:
"m dvd Suc 0 \<longleftrightarrow> m = Suc 0"
by (simp add: dvd_def)
lemma nat_dvd_1_iff_1 [simp]:
"m dvd (1::nat) \<longleftrightarrow> m = 1"
by (simp add: dvd_def)
lemma dvd_antisym:
"m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = (n::nat)"
unfolding dvd_def
by (force dest: mult_eq_self_implies_10 simp add: mult.assoc)
lemma dvd_diff_nat [simp]:
"k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m - n :: nat)"
unfolding dvd_def
by (blast intro: right_diff_distrib' [symmetric])
lemma dvd_diffD:
"k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> n \<le> m \<Longrightarrow> k dvd (m::nat)"
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
apply (blast intro: dvd_add)
done
lemma dvd_diffD1:
"k dvd m - n \<Longrightarrow> k dvd m \<Longrightarrow> n \<le> m \<Longrightarrow> k dvd (n::nat)"
by (drule_tac m = m in dvd_diff_nat) auto
lemma dvd_mult_cancel:
fixes m n k :: nat
assumes "k * m dvd k * n" and "0 < k"
shows "m dvd n"
proof -
from assms(1) obtain q where "k * n = (k * m) * q" ..
then have "k * n = k * (m * q)" by (simp add: ac_simps)
with \<open>0 < k\<close> have "n = m * q" by simp
then show ?thesis ..
qed
lemma dvd_mult_cancel1:
"0 < m \<Longrightarrow> m * n dvd m \<longleftrightarrow> n = (1::nat)"
apply auto
apply (subgoal_tac "m*n dvd m*1")
apply (drule dvd_mult_cancel, auto)
done
lemma dvd_mult_cancel2:
"0 < m \<Longrightarrow> n * m dvd m \<longleftrightarrow> n = (1::nat)"
using dvd_mult_cancel1 [of m n] by (simp add: ac_simps)
lemma dvd_imp_le:
"k dvd n \<Longrightarrow> 0 < n \<Longrightarrow> k \<le> (n::nat)"
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
lemma nat_dvd_not_less:
fixes m n :: nat
shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
lemma less_eq_dvd_minus:
fixes m n :: nat
assumes "m \<le> n"
shows "m dvd n \<longleftrightarrow> m dvd n - m"
proof -
from assms have "n = m + (n - m)" by simp
then obtain q where "n = m + q" ..
then show ?thesis by (simp add: add.commute [of m])
qed
lemma dvd_minus_self:
fixes m n :: nat
shows "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"
by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add)
lemma dvd_minus_add:
fixes m n q r :: nat
assumes "q \<le> n" "q \<le> r * m"
shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"
proof -
have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"
using dvd_add_times_triv_left_iff [of m r] by simp
also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp
also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp
also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute)
finally show ?thesis .
qed
subsection \<open>Aliasses\<close>
lemma nat_mult_1: "(1::nat) * n = n"
by (fact mult_1_left)
lemma nat_mult_1_right: "n * (1::nat) = n"
by (fact mult_1_right)
lemma nat_add_left_cancel:
fixes k m n :: nat
shows "k + m = k + n \<longleftrightarrow> m = n"
by (fact add_left_cancel)
lemma nat_add_right_cancel:
fixes k m n :: nat
shows "m + k = n + k \<longleftrightarrow> m = n"
by (fact add_right_cancel)
lemma diff_mult_distrib:
"((m::nat) - n) * k = (m * k) - (n * k)"
by (fact left_diff_distrib')
lemma diff_mult_distrib2:
"k * ((m::nat) - n) = (k * m) - (k * n)"
by (fact right_diff_distrib')
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
by (fact le_add_diff) \<comment> \<open>FIXME delete\<close>
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
by (fact le_diff_conv2) \<comment> \<open>FIXME delete\<close>
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
by (fact diff_cancel)
lemma diff_diff_left [simp]: "(i::nat) - j - k = i - (j + k)"
by (fact diff_diff_add)
lemma diff_commute: "(i::nat) - j - k = i - k - j"
by (fact diff_right_commute)
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
by (fact add_diff_cancel_left')
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
by (fact add_diff_cancel_right')
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
by (fact add_diff_cancel_left)
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
by (fact add_diff_cancel_right)
lemma diff_add_0: "n - (n + m) = (0::nat)"
by (fact diff_add_zero)
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
by (fact distrib_left)
lemmas nat_distrib =
add_mult_distrib distrib_left diff_mult_distrib diff_mult_distrib2
subsection \<open>Size of a datatype value\<close>
class size =
fixes size :: "'a \<Rightarrow> nat" \<comment> \<open>see further theory \<open>Wellfounded\<close>\<close>
instantiation nat :: size
begin
definition size_nat where
[simp, code]: "size (n::nat) = n"
instance ..
end
subsection \<open>Code module namespace\<close>
code_identifier
code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
hide_const (open) of_nat_aux
end