--- a/src/HOL/Fun.thy Tue Aug 02 21:04:52 2016 +0200
+++ b/src/HOL/Fun.thy Tue Aug 02 21:05:34 2016 +0200
@@ -410,8 +410,8 @@
by (auto simp: bij_betw_def)
qed
-lemma bij_betw_cong: "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
- unfolding bij_betw_def inj_on_def by force
+lemma bij_betw_cong: "(\<And>a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
+ unfolding bij_betw_def inj_on_def by force (* slow *)
lemma bij_betw_id[intro, simp]: "bij_betw id A A"
unfolding bij_betw_def id_def by auto
--- a/src/HOL/Groups.thy Tue Aug 02 21:04:52 2016 +0200
+++ b/src/HOL/Groups.thy Tue Aug 02 21:05:34 2016 +0200
@@ -1,48 +1,46 @@
-(* Title: HOL/Groups.thy
- Author: Gertrud Bauer, Steven Obua, Lawrence C Paulson, Markus Wenzel, Jeremy Avigad
+(* Title: HOL/Groups.thy
+ Author: Gertrud Bauer
+ Author: Steven Obua
+ Author: Lawrence C Paulson
+ Author: Markus Wenzel
+ Author: Jeremy Avigad
*)
section \<open>Groups, also combined with orderings\<close>
theory Groups
-imports Orderings
+ imports Orderings
begin
subsection \<open>Dynamic facts\<close>
named_theorems ac_simps "associativity and commutativity simplification rules"
-
+ and algebra_simps "algebra simplification rules"
+ and field_simps "algebra simplification rules for fields"
text \<open>
- The rewrites accumulated in \<open>algebra_simps\<close> deal with the
- classical algebraic structures of groups, rings and family. They simplify
- terms by multiplying everything out (in case of a ring) and bringing sums and
- products into a canonical form (by ordered rewriting). As a result it decides
- group and ring equalities but also helps with inequalities.
-
- Of course it also works for fields, but it knows nothing about multiplicative
- inverses or division. This is catered for by \<open>field_simps\<close>.
-\<close>
+ The rewrites accumulated in \<open>algebra_simps\<close> deal with the classical
+ algebraic structures of groups, rings and family. They simplify terms by
+ multiplying everything out (in case of a ring) and bringing sums and
+ products into a canonical form (by ordered rewriting). As a result it
+ decides group and ring equalities but also helps with inequalities.
-named_theorems algebra_simps "algebra simplification rules"
-
+ Of course it also works for fields, but it knows nothing about
+ multiplicative inverses or division. This is catered for by \<open>field_simps\<close>.
-text \<open>
- Lemmas \<open>field_simps\<close> multiply with denominators in (in)equations
- if they can be proved to be non-zero (for equations) or positive/negative
- (for inequations). Can be too aggressive and is therefore separate from the
- more benign \<open>algebra_simps\<close>.
+ Facts in \<open>field_simps\<close> multiply with denominators in (in)equations if they
+ can be proved to be non-zero (for equations) or positive/negative (for
+ inequalities). Can be too aggressive and is therefore separate from the more
+ benign \<open>algebra_simps\<close>.
\<close>
-named_theorems field_simps "algebra simplification rules for fields"
-
subsection \<open>Abstract structures\<close>
text \<open>
- These locales provide basic structures for interpretation into
- bigger structures; extensions require careful thinking, otherwise
- undesired effects may occur due to interpretation.
+ These locales provide basic structures for interpretation into bigger
+ structures; extensions require careful thinking, otherwise undesired effects
+ may occur due to interpretation.
\<close>
locale semigroup =
@@ -114,16 +112,13 @@
by (simp add: assoc [symmetric])
qed
-lemma inverse_neutral [simp]:
- "inverse \<^bold>1 = \<^bold>1"
+lemma inverse_neutral [simp]: "inverse \<^bold>1 = \<^bold>1"
by (rule inverse_unique) simp
-lemma inverse_inverse [simp]:
- "inverse (inverse a) = a"
+lemma inverse_inverse [simp]: "inverse (inverse a) = a"
by (rule inverse_unique) simp
-lemma right_inverse [simp]:
- "a \<^bold>* inverse a = \<^bold>1"
+lemma right_inverse [simp]: "a \<^bold>* inverse a = \<^bold>1"
proof -
have "a \<^bold>* inverse a = inverse (inverse a) \<^bold>* inverse a"
by simp
@@ -132,8 +127,7 @@
then show ?thesis by simp
qed
-lemma inverse_distrib_swap:
- "inverse (a \<^bold>* b) = inverse b \<^bold>* inverse a"
+lemma inverse_distrib_swap: "inverse (a \<^bold>* b) = inverse b \<^bold>* inverse a"
proof (rule inverse_unique)
have "a \<^bold>* b \<^bold>* (inverse b \<^bold>* inverse a) =
a \<^bold>* (b \<^bold>* inverse b) \<^bold>* inverse a"
@@ -143,8 +137,7 @@
finally show "a \<^bold>* b \<^bold>* (inverse b \<^bold>* inverse a) = \<^bold>1" .
qed
-lemma right_cancel:
- "b \<^bold>* a = c \<^bold>* a \<longleftrightarrow> b = c"
+lemma right_cancel: "b \<^bold>* a = c \<^bold>* a \<longleftrightarrow> b = c"
proof
assume "b \<^bold>* a = c \<^bold>* a"
then have "b \<^bold>* a \<^bold>* inverse a= c \<^bold>* a \<^bold>* inverse a"
@@ -435,10 +428,8 @@
sublocale add: group plus 0 uminus
by standard (simp_all add: left_minus)
-lemma minus_unique:
- assumes "a + b = 0"
- shows "- a = b"
- using assms by (fact add.inverse_unique)
+lemma minus_unique: "a + b = 0 \<Longrightarrow> - a = b"
+ by (fact add.inverse_unique)
lemma minus_zero: "- 0 = 0"
by (fact add.inverse_neutral)
@@ -701,8 +692,8 @@
lemma add_le_cancel_left [simp]: "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
apply auto
- apply (drule add_le_imp_le_left)
- apply (simp_all add: add_left_mono)
+ apply (drule add_le_imp_le_left)
+ apply (simp_all add: add_left_mono)
done
lemma add_le_cancel_right [simp]: "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
@@ -803,36 +794,28 @@
class ordered_ab_semigroup_monoid_add_imp_le = monoid_add + ordered_ab_semigroup_add_imp_le
begin
-lemma add_less_same_cancel1 [simp]:
- "b + a < b \<longleftrightarrow> a < 0"
+lemma add_less_same_cancel1 [simp]: "b + a < b \<longleftrightarrow> a < 0"
using add_less_cancel_left [of _ _ 0] by simp
-lemma add_less_same_cancel2 [simp]:
- "a + b < b \<longleftrightarrow> a < 0"
+lemma add_less_same_cancel2 [simp]: "a + b < b \<longleftrightarrow> a < 0"
using add_less_cancel_right [of _ _ 0] by simp
-lemma less_add_same_cancel1 [simp]:
- "a < a + b \<longleftrightarrow> 0 < b"
+lemma less_add_same_cancel1 [simp]: "a < a + b \<longleftrightarrow> 0 < b"
using add_less_cancel_left [of _ 0] by simp
-lemma less_add_same_cancel2 [simp]:
- "a < b + a \<longleftrightarrow> 0 < b"
+lemma less_add_same_cancel2 [simp]: "a < b + a \<longleftrightarrow> 0 < b"
using add_less_cancel_right [of 0] by simp
-lemma add_le_same_cancel1 [simp]:
- "b + a \<le> b \<longleftrightarrow> a \<le> 0"
+lemma add_le_same_cancel1 [simp]: "b + a \<le> b \<longleftrightarrow> a \<le> 0"
using add_le_cancel_left [of _ _ 0] by simp
-lemma add_le_same_cancel2 [simp]:
- "a + b \<le> b \<longleftrightarrow> a \<le> 0"
+lemma add_le_same_cancel2 [simp]: "a + b \<le> b \<longleftrightarrow> a \<le> 0"
using add_le_cancel_right [of _ _ 0] by simp
-lemma le_add_same_cancel1 [simp]:
- "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
+lemma le_add_same_cancel1 [simp]: "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
using add_le_cancel_left [of _ 0] by simp
-lemma le_add_same_cancel2 [simp]:
- "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
+lemma le_add_same_cancel2 [simp]: "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
using add_le_cancel_right [of 0] by simp
subclass cancel_comm_monoid_add
@@ -911,7 +894,7 @@
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
proof -
- have "- (-a) < - b \<longleftrightarrow> b < - a"
+ have "- (- a) < - b \<longleftrightarrow> b < - a"
by (rule neg_less_iff_less)
then show ?thesis by simp
qed
@@ -925,12 +908,12 @@
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
proof -
- have mm: "- (- a) < -b \<Longrightarrow> - (- b) < -a" for a b :: 'a
+ have mm: "- (- a) < - b \<Longrightarrow> - (- b) < -a" for a b :: 'a
by (simp only: minus_less_iff)
- have "- (- a) \<le> -b \<longleftrightarrow> b \<le> - a"
+ have "- (- a) \<le> - b \<longleftrightarrow> b \<le> - a"
apply (auto simp only: le_less)
- apply (drule mm)
- apply (simp_all)
+ apply (drule mm)
+ apply (simp_all)
apply (drule mm[simplified], assumption)
done
then show ?thesis by simp
@@ -1039,11 +1022,11 @@
proof (rule ccontr)
assume *: "\<not> ?thesis"
then have "b \<le> a" by (simp add: linorder_not_le)
- then have le2: "c + b \<le> c + a" by (rule add_left_mono)
- have "a = b"
- apply (insert le1 le2)
+ then have "c + b \<le> c + a" by (rule add_left_mono)
+ with le1 have "a = b"
+ apply -
apply (drule antisym)
- apply simp_all
+ apply simp_all
done
with * show False
by (simp add: linorder_not_le [symmetric])
@@ -1117,8 +1100,8 @@
lemma double_zero_sym [simp]: "0 = a + a \<longleftrightarrow> a = 0"
apply (rule iffI)
- apply (drule sym)
- apply simp_all
+ apply (drule sym)
+ apply simp_all
done
lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"
@@ -1342,7 +1325,7 @@
fixes x::"'a::{dense_linorder,ordered_ab_group_add_abs}"
shows "(\<And>e. 0 < e \<Longrightarrow> \<bar>x\<bar> \<le> e) \<Longrightarrow> x = 0"
apply (cases "\<bar>x\<bar> = 0")
- apply simp
+ apply simp
apply (simp only: zero_less_abs_iff [symmetric])
apply (drule dense)
apply (auto simp add: not_less [symmetric])
@@ -1402,7 +1385,7 @@
begin
context
- fixes a b
+ fixes a b :: 'a
assumes le: "a \<le> b"
begin
--- a/src/HOL/Inductive.thy Tue Aug 02 21:04:52 2016 +0200
+++ b/src/HOL/Inductive.thy Tue Aug 02 21:05:34 2016 +0200
@@ -5,13 +5,13 @@
section \<open>Knaster-Tarski Fixpoint Theorem and inductive definitions\<close>
theory Inductive
-imports Complete_Lattices Ctr_Sugar
-keywords
- "inductive" "coinductive" "inductive_cases" "inductive_simps" :: thy_decl and
- "monos" and
- "print_inductives" :: diag and
- "old_rep_datatype" :: thy_goal and
- "primrec" :: thy_decl
+ imports Complete_Lattices Ctr_Sugar
+ keywords
+ "inductive" "coinductive" "inductive_cases" "inductive_simps" :: thy_decl and
+ "monos" and
+ "print_inductives" :: diag and
+ "old_rep_datatype" :: thy_goal and
+ "primrec" :: thy_decl
begin
subsection \<open>Least and greatest fixed points\<close>
@@ -50,8 +50,8 @@
lemma lfp_const: "lfp (\<lambda>x. t) = t"
by (rule lfp_unfold) (simp add: mono_def)
-lemma lfp_eqI: "\<lbrakk> mono F; F x = x; \<And>z. F z = z \<Longrightarrow> x \<le> z \<rbrakk> \<Longrightarrow> lfp F = x"
-by (rule antisym) (simp_all add: lfp_lowerbound lfp_unfold[symmetric])
+lemma lfp_eqI: "mono F \<Longrightarrow> F x = x \<Longrightarrow> (\<And>z. F z = z \<Longrightarrow> x \<le> z) \<Longrightarrow> lfp F = x"
+ by (rule antisym) (simp_all add: lfp_lowerbound lfp_unfold[symmetric])
subsection \<open>General induction rules for least fixed points\<close>
@@ -64,10 +64,11 @@
shows "P (lfp f)"
proof -
let ?M = "{S. S \<le> lfp f \<and> P S}"
- have "P (Sup ?M)" using P_Union by simp
+ from P_Union have "P (Sup ?M)" by simp
also have "Sup ?M = lfp f"
proof (rule antisym)
- show "Sup ?M \<le> lfp f" by (blast intro: Sup_least)
+ show "Sup ?M \<le> lfp f"
+ by (blast intro: Sup_least)
then have "f (Sup ?M) \<le> f (lfp f)"
by (rule mono [THEN monoD])
then have "f (Sup ?M) \<le> lfp f"
@@ -86,11 +87,18 @@
assumes mono: "mono f"
and ind: "f (inf (lfp f) P) \<le> P"
shows "lfp f \<le> P"
-proof (induction rule: lfp_ordinal_induct)
+proof (induct rule: lfp_ordinal_induct)
+ case mono
+ show ?case by fact
+next
case (step S)
then show ?case
by (intro order_trans[OF _ ind] monoD[OF mono]) auto
-qed (auto intro: mono Sup_least)
+next
+ case (union M)
+ then show ?case
+ by (auto intro: Sup_least)
+qed
lemma lfp_induct_set:
assumes lfp: "a \<in> lfp f"
@@ -144,10 +152,10 @@
by (iprover intro: order_antisym gfp_lemma2 gfp_lemma3)
lemma gfp_const: "gfp (\<lambda>x. t) = t"
-by (rule gfp_unfold) (simp add: mono_def)
+ by (rule gfp_unfold) (simp add: mono_def)
-lemma gfp_eqI: "\<lbrakk> mono F; F x = x; \<And>z. F z = z \<Longrightarrow> z \<le> x \<rbrakk> \<Longrightarrow> gfp F = x"
-by (rule antisym) (simp_all add: gfp_upperbound gfp_unfold[symmetric])
+lemma gfp_eqI: "mono F \<Longrightarrow> F x = x \<Longrightarrow> (\<And>z. F z = z \<Longrightarrow> z \<le> x) \<Longrightarrow> gfp F = x"
+ by (rule antisym) (simp_all add: gfp_upperbound gfp_unfold[symmetric])
subsection \<open>Coinduction rules for greatest fixed points\<close>
@@ -165,11 +173,11 @@
apply (frule gfp_lemma2)
apply (drule mono_sup)
apply (rule le_supI)
- apply assumption
- apply (rule order_trans)
+ apply assumption
apply (rule order_trans)
- apply assumption
- apply (rule sup_ge2)
+ apply (rule order_trans)
+ apply assumption
+ apply (rule sup_ge2)
apply assumption
done
@@ -188,7 +196,7 @@
shows "P (gfp f)"
proof -
let ?M = "{S. gfp f \<le> S \<and> P S}"
- have "P (Inf ?M)" using P_Union by simp
+ from P_Union have "P (Inf ?M)" by simp
also have "Inf ?M = gfp f"
proof (rule antisym)
show "gfp f \<le> Inf ?M"
@@ -211,10 +219,18 @@
assumes mono: "mono f"
and ind: "X \<le> f (sup X (gfp f))"
shows "X \<le> gfp f"
-proof (induction rule: gfp_ordinal_induct)
- case (step S) then show ?case
+proof (induct rule: gfp_ordinal_induct)
+ case mono
+ then show ?case by fact
+next
+ case (step S)
+ then show ?case
by (intro order_trans[OF ind _] monoD[OF mono]) auto
-qed (auto intro: mono Inf_greatest)
+next
+ case (union M)
+ then show ?case
+ by (auto intro: mono Inf_greatest)
+qed
subsection \<open>Even Stronger Coinduction Rule, by Martin Coen\<close>
@@ -228,9 +244,9 @@
"X \<subseteq> f (lfp (\<lambda>x. f x \<union> X \<union> gfp f)) \<Longrightarrow> mono f \<Longrightarrow>
lfp (\<lambda>x. f x \<union> X \<union> gfp f) \<subseteq> f (lfp (\<lambda>x. f x \<union> X \<union> gfp f))"
apply (rule subset_trans)
- apply (erule coinduct3_mono_lemma [THEN lfp_lemma3])
+ apply (erule coinduct3_mono_lemma [THEN lfp_lemma3])
apply (rule Un_least [THEN Un_least])
- apply (rule subset_refl, assumption)
+ apply (rule subset_refl, assumption)
apply (rule gfp_unfold [THEN equalityD1, THEN subset_trans], assumption)
apply (rule monoD, assumption)
apply (subst coinduct3_mono_lemma [THEN lfp_unfold], auto)
@@ -238,8 +254,8 @@
lemma coinduct3: "mono f \<Longrightarrow> a \<in> X \<Longrightarrow> X \<subseteq> f (lfp (\<lambda>x. f x \<union> X \<union> gfp f)) \<Longrightarrow> a \<in> gfp f"
apply (rule coinduct3_lemma [THEN [2] weak_coinduct])
- apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst])
- apply simp_all
+ apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst])
+ apply simp_all
done
text \<open>Definition forms of \<open>gfp_unfold\<close> and \<open>coinduct\<close>, to control unfolding.\<close>
--- a/src/HOL/Lattices.thy Tue Aug 02 21:04:52 2016 +0200
+++ b/src/HOL/Lattices.thy Tue Aug 02 21:05:34 2016 +0200
@@ -47,12 +47,10 @@
sublocale ordering less_eq less
proof
- fix a b
show "a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b" for a b
by (simp add: order_iff strict_order_iff)
next
- fix a
- show "a \<^bold>\<le> a"
+ show "a \<^bold>\<le> a" for a
by (simp add: order_iff)
next
fix a b
@@ -83,8 +81,10 @@
assumes "a \<^bold>\<le> b" and "a \<^bold>\<le> c"
shows "a \<^bold>\<le> b \<^bold>* c"
proof (rule orderI)
- from assms obtain "a \<^bold>* b = a" and "a \<^bold>* c = a" by (auto elim!: orderE)
- then show "a = a \<^bold>* (b \<^bold>* c)" by (simp add: assoc [symmetric])
+ from assms obtain "a \<^bold>* b = a" and "a \<^bold>* c = a"
+ by (auto elim!: orderE)
+ then show "a = a \<^bold>* (b \<^bold>* c)"
+ by (simp add: assoc [symmetric])
qed
lemma boundedE:
@@ -108,7 +108,8 @@
lemma strict_coboundedI1: "a \<^bold>< c \<Longrightarrow> a \<^bold>* b \<^bold>< c"
using irrefl
- by (auto intro: not_eq_order_implies_strict coboundedI1 strict_implies_order elim: strict_boundedE)
+ by (auto intro: not_eq_order_implies_strict coboundedI1 strict_implies_order
+ elim: strict_boundedE)
lemma strict_coboundedI2: "b \<^bold>< c \<Longrightarrow> a \<^bold>* b \<^bold>< c"
using strict_coboundedI1 [of b c a] by (simp add: commute)
@@ -117,10 +118,10 @@
by (blast intro: boundedI coboundedI1 coboundedI2)
lemma absorb1: "a \<^bold>\<le> b \<Longrightarrow> a \<^bold>* b = a"
- by (rule antisym) (auto simp add: refl)
+ by (rule antisym) (auto simp: refl)
lemma absorb2: "b \<^bold>\<le> a \<Longrightarrow> a \<^bold>* b = b"
- by (rule antisym) (auto simp add: refl)
+ by (rule antisym) (auto simp: refl)
lemma absorb_iff1: "a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>* b = a"
using order_iff by auto
@@ -165,8 +166,7 @@
and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
begin
-text \<open>Dual lattice\<close>
-
+text \<open>Dual lattice.\<close>
lemma dual_semilattice: "class.semilattice_inf sup greater_eq greater"
by (rule class.semilattice_inf.intro, rule dual_order)
(unfold_locales, simp_all add: sup_least)
@@ -330,7 +330,10 @@
begin
lemma dual_lattice: "class.lattice sup (op \<ge>) (op >) inf"
- by (rule class.lattice.intro, rule dual_semilattice, rule class.semilattice_sup.intro, rule dual_order)
+ by (rule class.lattice.intro,
+ rule dual_semilattice,
+ rule class.semilattice_sup.intro,
+ rule dual_order)
(unfold_locales, auto)
lemma inf_sup_absorb [simp]: "x \<sqinter> (x \<squnion> y) = x"
@@ -343,7 +346,7 @@
lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
-text\<open>Towards distributivity\<close>
+text \<open>Towards distributivity.\<close>
lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
@@ -533,7 +536,9 @@
lemma dual_boolean_algebra:
"class.boolean_algebra (\<lambda>x y. x \<squnion> - y) uminus sup greater_eq greater inf \<top> \<bottom>"
- by (rule class.boolean_algebra.intro, rule dual_bounded_lattice, rule dual_distrib_lattice)
+ by (rule class.boolean_algebra.intro,
+ rule dual_bounded_lattice,
+ rule dual_distrib_lattice)
(unfold_locales, auto simp add: inf_compl_bot sup_compl_top diff_eq)
lemma compl_inf_bot [simp]: "- x \<sqinter> x = \<bottom>"
@@ -645,7 +650,8 @@
assumes "- y \<sqsubset> x"
shows "- x \<sqsubset> y"
proof -
- from assms have "- x \<sqsubset> - (- y)" by (simp only: compl_less_compl_iff)
+ from assms have "- x \<sqsubset> - (- y)"
+ by (simp only: compl_less_compl_iff)
then show ?thesis by simp
qed
@@ -661,7 +667,8 @@
lemma inf_cancel_left2: "inf (inf (- x) a) (inf x b) = bot"
by (simp add: inf_sup_aci inf_compl_bot)
-declare inf_compl_bot [simp] and sup_compl_top [simp]
+declare inf_compl_bot [simp]
+ and sup_compl_top [simp]
lemma sup_compl_top_left1 [simp]: "sup (- x) (sup x y) = top"
by (simp add: sup_assoc[symmetric])
@@ -821,7 +828,8 @@
lemma sup_apply [simp, code]: "(f \<squnion> g) x = f x \<squnion> g x"
by (simp add: sup_fun_def)
-instance by standard (simp_all add: le_fun_def)
+instance
+ by standard (simp_all add: le_fun_def)
end
--- a/src/HOL/Nat.thy Tue Aug 02 21:04:52 2016 +0200
+++ b/src/HOL/Nat.thy Tue Aug 02 21:05:34 2016 +0200
@@ -1,5 +1,7 @@
(* Title: HOL/Nat.thy
- Author: Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
+ Author: Tobias Nipkow
+ Author: Lawrence C Paulson
+ Author: Markus Wenzel
Type "nat" is a linear order, and a datatype; arithmetic operators + -
and * (for div and mod, see theory Divides).
@@ -8,7 +10,7 @@
section \<open>Natural numbers\<close>
theory Nat
-imports Inductive Typedef Fun Rings
+ imports Inductive Typedef Fun Rings
begin
named_theorems arith "arith facts -- only ground formulas"
@@ -21,75 +23,70 @@
axiomatization Zero_Rep :: ind and Suc_Rep :: "ind \<Rightarrow> ind"
\<comment> \<open>The axiom of infinity in 2 parts:\<close>
-where Suc_Rep_inject: "Suc_Rep x = Suc_Rep y \<Longrightarrow> x = y"
- and Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
+ where Suc_Rep_inject: "Suc_Rep x = Suc_Rep y \<Longrightarrow> 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)"
+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)"
+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"
+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"
+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"
+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))"
+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)
+ 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)
+ by (rule not_sym) (rule Suc_not_Zero)
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)"
+ 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)"
+ 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 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
@@ -97,19 +94,19 @@
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
+ apply (erule nat_induct0)
+ apply 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]
+declare old.nat.inject[iff del]
+ and old.nat.distinct(1)[simp del, induct_simp del]
lemmas induct = old.nat.induct
lemmas inducts = old.nat.inducts
@@ -134,16 +131,16 @@
nat.split_sel_asm
lemma nat_exhaust [case_names 0 Suc, cases type: nat]:
+ "(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P"
\<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)
+ 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)
+ \<comment> \<open>for backward compatibility -- names of variables differ\<close>
+ using assms by (rule nat.induct)
hide_fact
nat_exhaust
@@ -180,35 +177,40 @@
by (simp add: inj_on_def)
lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
-by (rule notE, rule Suc_not_Zero)
+ by (rule notE) (rule Suc_not_Zero)
lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
-by (rule Suc_neq_Zero, erule sym)
+ by (rule Suc_neq_Zero) (erule sym)
lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
-by (rule inj_Suc [THEN injD])
+ by (rule inj_Suc [THEN injD])
lemma n_not_Suc_n: "n \<noteq> Suc n"
-by (induct n) simp_all
+ 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>
-
+ 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:
assumes "\<And>x. P x 0"
and "\<And>y. P 0 (Suc y)"
and "\<And>x y. P x y \<Longrightarrow> P (Suc x) (Suc y)"
shows "P m n"
- using assms
- apply (rule_tac x = m in spec)
- apply (induct n)
- prefer 2
- apply (rule allI)
- apply (induct_tac x, iprover+)
- done
+proof (induct n arbitrary: m)
+ case 0
+ show ?case by (rule assms(1))
+next
+ case (Suc n)
+ show ?case
+ proof (induct m)
+ case 0
+ show ?case by (rule assms(2))
+ next
+ case (Suc m)
+ from \<open>P m n\<close> show ?case by (rule assms(3))
+ qed
+qed
subsection \<open>Arithmetic operators\<close>
@@ -216,11 +218,13 @@
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" for m :: nat
+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"
+ for m :: nat
by (induct m) simp_all
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
@@ -231,13 +235,15 @@
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 \<Rightarrow> 0 | Suc k \<Rightarrow> k)"
+primrec minus_nat
+ where
+ diff_0 [code]: "m - 0 = (m::nat)"
+ | diff_Suc: "m - Suc n = (case m - n of 0 \<Rightarrow> 0 | Suc k \<Rightarrow> k)"
declare diff_Suc [simp del]
-lemma diff_0_eq_0 [simp, code]: "0 - n = 0" for n :: nat
+lemma diff_0_eq_0 [simp, code]: "0 - n = 0"
+ for n :: nat
by (induct n) (simp_all add: diff_Suc)
lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
@@ -261,30 +267,37 @@
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 * 0 = 0" for m :: nat
+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 * 0 = 0"
+ for m :: nat
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)" for m n k :: nat
+lemma add_mult_distrib: "(m + n) * k = (m * k) + (n * k)"
+ for m n k :: nat
by (induct m) (simp_all add: add.assoc)
instance
proof
fix k 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)
+ show "0 \<noteq> (1::nat)"
+ by simp
+ show "1 * n = n"
+ 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)
show "k * (m - n) = (k * m) - (k * n)"
by (induct m n rule: diff_induct) simp_all
qed
@@ -296,7 +309,8 @@
text \<open>Reasoning about \<open>m + 0 = 0\<close>, etc.\<close>
-lemma add_is_0 [iff]: "(m + n = 0) = (m = 0 \<and> n = 0)" for m n :: nat
+lemma add_is_0 [iff]: "m + n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
+ for m n :: nat
by (cases m) simp_all
lemma add_is_1: "m + n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = 0 | m = 0 \<and> n = Suc 0"
@@ -305,21 +319,26 @@
lemma one_is_add: "Suc 0 = m + n \<longleftrightarrow> m = Suc 0 \<and> n = 0 | m = 0 \<and> n = Suc 0"
by (rule trans, rule eq_commute, rule add_is_1)
-lemma add_eq_self_zero: "m + n = m \<Longrightarrow> n = 0" for m n :: nat
+lemma add_eq_self_zero: "m + n = m \<Longrightarrow> n = 0"
+ for m n :: nat
by (induct m) simp_all
-lemma inj_on_add_nat[simp]: "inj_on (\<lambda>n. n + k) N" for k :: nat
- apply (induct k)
- apply simp
- apply (drule comp_inj_on[OF _ inj_Suc])
- apply (simp add: o_def)
- done
+lemma inj_on_add_nat [simp]: "inj_on (\<lambda>n. n + k) N"
+ for k :: nat
+proof (induct k)
+ case 0
+ then show ?case by simp
+next
+ case (Suc k)
+ show ?case
+ using comp_inj_on [OF Suc inj_Suc] by (simp add: o_def)
+qed
lemma Suc_eq_plus1: "Suc n = n + 1"
- unfolding One_nat_def by simp
+ by simp
lemma Suc_eq_plus1_left: "Suc n = 1 + n"
- unfolding One_nat_def by simp
+ by simp
subsubsection \<open>Difference\<close>
@@ -328,7 +347,8 @@
by (simp add: diff_diff_add)
lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
- unfolding One_nat_def by simp
+ by simp
+
subsubsection \<open>Multiplication\<close>
@@ -336,24 +356,30 @@
by (induct m) auto
lemma mult_eq_1_iff [simp]: "m * n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
- apply (induct m)
- apply simp
- apply (induct n)
- apply auto
- done
+proof (induct m)
+ case 0
+ then show ?case by simp
+next
+ case (Suc m)
+ then show ?case by (induct n) auto
+qed
lemma one_eq_mult_iff [simp]: "Suc 0 = m * n \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
apply (rule trans)
- apply (rule_tac [2] mult_eq_1_iff, fastforce)
+ apply (rule_tac [2] mult_eq_1_iff)
+ apply fastforce
done
-lemma nat_mult_eq_1_iff [simp]: "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1" for m n :: nat
+lemma nat_mult_eq_1_iff [simp]: "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
+ for m n :: nat
unfolding One_nat_def by (rule mult_eq_1_iff)
-lemma nat_1_eq_mult_iff [simp]: "1 = m * n \<longleftrightarrow> m = 1 \<and> n = 1" for m n :: nat
+lemma nat_1_eq_mult_iff [simp]: "1 = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
+ for m n :: nat
unfolding One_nat_def by (rule one_eq_mult_iff)
-lemma mult_cancel1 [simp]: "k * m = k * n \<longleftrightarrow> m = n \<or> k = 0" for k m n :: nat
+lemma mult_cancel1 [simp]: "k * m = k * n \<longleftrightarrow> m = n \<or> k = 0"
+ for k m n :: nat
proof -
have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
proof (induct n arbitrary: m)
@@ -367,7 +393,8 @@
then show ?thesis by auto
qed
-lemma mult_cancel2 [simp]: "m * k = n * k \<longleftrightarrow> m = n \<or> k = 0" for k m n :: nat
+lemma mult_cancel2 [simp]: "m * k = n * k \<longleftrightarrow> m = n \<or> k = 0"
+ for k m n :: nat
by (simp add: mult.commute)
lemma Suc_mult_cancel1: "Suc k * m = Suc k * n \<longleftrightarrow> m = n"
@@ -381,20 +408,23 @@
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)"
+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" for n :: nat
+lemma le0 [iff]: "0 \<le> n" for
+ n :: nat
by (simp add: less_eq_nat.simps)
-lemma [code]: "0 \<le> n \<longleftrightarrow> True" for n :: nat
+lemma [code]: "0 \<le> n \<longleftrightarrow> True"
+ for n :: nat
by simp
-definition less_nat where
- less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
+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))
@@ -402,13 +432,16 @@
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
unfolding less_eq_Suc_le ..
-lemma le_0_eq [iff]: "n \<le> 0 \<longleftrightarrow> n = 0" for n :: nat
+lemma le_0_eq [iff]: "n \<le> 0 \<longleftrightarrow> n = 0"
+ for n :: nat
by (induct n) (simp_all add: less_eq_nat.simps(2))
-lemma not_less0 [iff]: "\<not> n < 0" for n :: nat
+lemma not_less0 [iff]: "\<not> n < 0"
+ for n :: nat
by (simp add: less_eq_Suc_le)
-lemma less_nat_zero_code [code]: "n < 0 \<longleftrightarrow> False" for n :: nat
+lemma less_nat_zero_code [code]: "n < 0 \<longleftrightarrow> False"
+ for n :: nat
by simp
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
@@ -438,12 +471,15 @@
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)
+ 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)
+ then show ?case
+ by (cases m) (simp_all add: less_eq_Suc_le)
qed
- show "n \<le> n" by (induct n) simp_all
+ show "n \<le> n"
+ by (induct n) simp_all
then show "n = m" if "n \<le> m" and "m \<le> n"
using that by (induct n arbitrary: m)
(simp_all add: less_eq_nat.simps(2) split: nat.splits)
@@ -469,9 +505,11 @@
instantiation nat :: order_bot
begin
-definition bot_nat :: nat where "bot_nat = 0"
-
-instance by standard (simp add: bot_nat_def)
+definition bot_nat :: nat
+ where "bot_nat = 0"
+
+instance
+ by standard (simp add: bot_nat_def)
end
@@ -490,19 +528,24 @@
subsubsection \<open>Elimination properties\<close>
-lemma less_not_refl: "\<not> n < n" for n :: nat
+lemma less_not_refl: "\<not> n < n"
+ for n :: nat
by (rule order_less_irrefl)
-lemma less_not_refl2: "n < m \<Longrightarrow> m \<noteq> n" for m n :: nat
+lemma less_not_refl2: "n < m \<Longrightarrow> m \<noteq> n"
+ for m n :: nat
by (rule not_sym) (rule less_imp_neq)
-lemma less_not_refl3: "s < t \<Longrightarrow> s \<noteq> t" for s t :: nat
+lemma less_not_refl3: "s < t \<Longrightarrow> s \<noteq> t"
+ for s t :: nat
by (rule less_imp_neq)
-lemma less_irrefl_nat: "n < n \<Longrightarrow> R" for n :: nat
+lemma less_irrefl_nat: "n < n \<Longrightarrow> R"
+ for n :: nat
by (rule notE, rule less_not_refl)
-lemma less_zeroE: "n < 0 \<Longrightarrow> R" for n :: nat
+lemma less_zeroE: "n < 0 \<Longrightarrow> R"
+ for n :: nat
by (rule notE) (rule not_less0)
lemma less_Suc_eq: "m < Suc n \<longleftrightarrow> m < n \<or> m = n"
@@ -511,17 +554,19 @@
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
by (simp add: less_Suc_eq)
-lemma less_one [iff]: "n < 1 \<longleftrightarrow> n = 0" for n :: nat
+lemma less_one [iff]: "n < 1 \<longleftrightarrow> n = 0"
+ for n :: nat
unfolding One_nat_def by (rule less_Suc0)
lemma Suc_mono: "m < n \<Longrightarrow> 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"
+text \<open>"Less than" is antisymmetric, sort of.\<close>
+lemma less_antisym: "\<not> n < m \<Longrightarrow> n < Suc m \<Longrightarrow> m = n"
unfolding not_less less_Suc_eq_le by (rule antisym)
-lemma nat_neq_iff: "m \<noteq> n \<longleftrightarrow> m < n \<or> n < m" for m n :: nat
+lemma nat_neq_iff: "m \<noteq> n \<longleftrightarrow> m < n \<or> n < m"
+ for m n :: nat
by (rule linorder_neq_iff)
@@ -549,8 +594,10 @@
and eq: "m = n \<Longrightarrow> P"
shows P
apply (rule major [THEN lessE])
- apply (rule eq, blast)
- apply (rule less, blast)
+ apply (rule eq)
+ apply blast
+ apply (rule less)
+ apply blast
done
lemma Suc_lessE:
@@ -558,8 +605,9 @@
and minor: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P"
shows P
apply (rule major [THEN lessE])
- apply (erule lessI [THEN minor])
- apply (erule Suc_lessD [THEN minor], assumption)
+ apply (erule lessI [THEN minor])
+ apply (erule Suc_lessD [THEN minor])
+ apply assumption
done
lemma Suc_less_SucD: "Suc m < Suc n \<Longrightarrow> m < n"
@@ -568,39 +616,42 @@
lemma less_trans_Suc:
assumes le: "i < j"
shows "j < k \<Longrightarrow> Suc i < k"
- apply (induct k, simp_all)
- using 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 @{prop "n = m \<or> n < m"}\<close>
+proof (induct k)
+ case 0
+ then show ?case by simp
+next
+ case (Suc k)
+ with le show ?case
+ by simp (auto simp add: less_Suc_eq dest: Suc_lessD)
+qed
+
+text \<open>Can be used with \<open>less_Suc_eq\<close> to get @{prop "n = m \<or> n < m"}.\<close>
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
- unfolding not_less less_Suc_eq_le ..
+ by (simp only: 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>
+ by (simp only: not_le Suc_le_eq)
+
+text \<open>Properties of "less than or equal".\<close>
lemma le_imp_less_Suc: "m \<le> n \<Longrightarrow> m < Suc n"
- unfolding less_Suc_eq_le .
+ by (simp only: less_Suc_eq_le)
lemma Suc_n_not_le_n: "\<not> 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: not_le less_Suc_eq_le)
+
+lemma le_Suc_eq: "m \<le> Suc n \<longleftrightarrow> m \<le> n \<or> m = Suc n"
by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
lemma le_SucE: "m \<le> Suc n \<Longrightarrow> (m \<le> n \<Longrightarrow> R) \<Longrightarrow> (m = Suc n \<Longrightarrow> R) \<Longrightarrow> R"
by (drule le_Suc_eq [THEN iffD1], iprover+)
-lemma Suc_leI: "m < n \<Longrightarrow> Suc(m) \<le> n"
- unfolding Suc_le_eq .
-
-text \<open>Stronger version of \<open>Suc_leD\<close>\<close>
+lemma Suc_leI: "m < n \<Longrightarrow> Suc m \<le> n"
+ by (simp only: Suc_le_eq)
+
+text \<open>Stronger version of \<open>Suc_leD\<close>.\<close>
lemma Suc_le_lessD: "Suc m \<le> n \<Longrightarrow> m < n"
- unfolding Suc_le_eq .
+ by (simp only: Suc_le_eq)
lemma less_imp_le_nat: "m < n \<Longrightarrow> m \<le> n" for m n :: nat
unfolding less_eq_Suc_le by (rule Suc_leD)
@@ -611,32 +662,41 @@
text \<open>Equivalence of \<open>m \<le> n\<close> and \<open>m < n \<or> m = n\<close>\<close>
-lemma less_or_eq_imp_le: "m < n \<or> m = n \<Longrightarrow> m \<le> n" for m n :: nat
+lemma less_or_eq_imp_le: "m < n \<or> m = n \<Longrightarrow> m \<le> n"
+ for m n :: nat
unfolding le_less .
-lemma le_eq_less_or_eq: "m \<le> n \<longleftrightarrow> m < n \<or> m = n" for m n :: nat
+lemma le_eq_less_or_eq: "m \<le> n \<longleftrightarrow> m < n \<or> m = n"
+ for m n :: nat
by (rule le_less)
text \<open>Useful with \<open>blast\<close>.\<close>
-lemma eq_imp_le: "m = n \<Longrightarrow> m \<le> n" for m n :: nat
+lemma eq_imp_le: "m = n \<Longrightarrow> m \<le> n"
+ for m n :: nat
by auto
-lemma le_refl: "n \<le> n" for n :: nat
+lemma le_refl: "n \<le> n"
+ for n :: nat
by simp
-lemma le_trans: "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k" for i j k :: nat
+lemma le_trans: "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
+ for i j k :: nat
by (rule order_trans)
-lemma le_antisym: "m \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> m = n" for m n :: nat
+lemma le_antisym: "m \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> m = n"
+ for m n :: nat
by (rule antisym)
-lemma nat_less_le: "m < n \<longleftrightarrow> m \<le> n \<and> m \<noteq> n" for m n :: nat
+lemma nat_less_le: "m < n \<longleftrightarrow> m \<le> n \<and> m \<noteq> n"
+ for m n :: nat
by (rule less_le)
-lemma le_neq_implies_less: "m \<le> n \<Longrightarrow> m \<noteq> n \<Longrightarrow> m < n" for m n :: nat
+lemma le_neq_implies_less: "m \<le> n \<Longrightarrow> m \<noteq> n \<Longrightarrow> m < n"
+ for m n :: nat
unfolding less_le ..
-lemma nat_le_linear: "m \<le> n | n \<le> m" for m n :: nat
+lemma nat_le_linear: "m \<le> n | n \<le> m"
+ for m n :: nat
by (rule linear)
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
@@ -655,20 +715,24 @@
lemma gr0_implies_Suc: "n > 0 \<Longrightarrow> \<exists>m. n = Suc m"
by (cases n) simp_all
-lemma gr_implies_not0: "m < n \<Longrightarrow> n \<noteq> 0" for m n :: nat
+lemma gr_implies_not0: "m < n \<Longrightarrow> n \<noteq> 0"
+ for m n :: nat
by (cases n) simp_all
-lemma neq0_conv[iff]: "n \<noteq> 0 \<longleftrightarrow> 0 < n" for n :: nat
+lemma neq0_conv[iff]: "n \<noteq> 0 \<longleftrightarrow> 0 < n"
+ for n :: nat
by (cases n) simp_all
text \<open>This theorem is useful with \<open>blast\<close>\<close>
-lemma gr0I: "(n = 0 \<Longrightarrow> False) \<Longrightarrow> 0 < n" for n :: nat
- by (rule neq0_conv[THEN iffD1], iprover)
+lemma gr0I: "(n = 0 \<Longrightarrow> False) \<Longrightarrow> 0 < n"
+ for n :: nat
+ by (rule neq0_conv[THEN iffD1]) iprover
lemma gr0_conv_Suc: "0 < n \<longleftrightarrow> (\<exists>m. n = Suc m)"
by (fast intro: not0_implies_Suc)
-lemma not_gr0 [iff]: "\<not> 0 < n \<longleftrightarrow> n = 0" for n :: nat
+lemma not_gr0 [iff]: "\<not> 0 < n \<longleftrightarrow> n = 0"
+ for n :: nat
using neq0_conv by blast
lemma Suc_le_D: "Suc n \<le> m' \<Longrightarrow> \<exists>m. m' = Suc m"
@@ -687,34 +751,45 @@
lemma Suc_diff_1 [simp]: "0 < n \<Longrightarrow> Suc (n - 1) = n"
unfolding One_nat_def by (rule Suc_pred)
-lemma nat_add_left_cancel_le [simp]: "k + m \<le> k + n \<longleftrightarrow> m \<le> n" for k m n :: nat
+lemma nat_add_left_cancel_le [simp]: "k + m \<le> k + n \<longleftrightarrow> m \<le> n"
+ for k m n :: nat
by (induct k) simp_all
-lemma nat_add_left_cancel_less [simp]: "k + m < k + n \<longleftrightarrow> m < n" for k m n :: nat
+lemma nat_add_left_cancel_less [simp]: "k + m < k + n \<longleftrightarrow> m < n"
+ for k m n :: nat
by (induct k) simp_all
-lemma add_gr_0 [iff]: "m + n > 0 \<longleftrightarrow> m > 0 \<or> n > 0" for m n :: nat
+lemma add_gr_0 [iff]: "m + n > 0 \<longleftrightarrow> m > 0 \<or> n > 0"
+ for m n :: nat
by (auto dest: gr0_implies_Suc)
text \<open>strict, in 1st argument\<close>
-lemma add_less_mono1: "i < j \<Longrightarrow> i + k < j + k" for i j k :: nat
+lemma add_less_mono1: "i < j \<Longrightarrow> i + k < j + k"
+ for i j k :: nat
by (induct k) simp_all
text \<open>strict, in both arguments\<close>
-lemma add_less_mono: "i < j \<Longrightarrow> k < l \<Longrightarrow> i + k < j + l" for i j k l :: nat
+lemma add_less_mono: "i < j \<Longrightarrow> k < l \<Longrightarrow> i + k < j + l"
+ for i j k l :: nat
apply (rule add_less_mono1 [THEN less_trans], assumption+)
- apply (induct j, simp_all)
+ apply (induct j)
+ apply 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 \<Longrightarrow> \<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 \<le> l \<Longrightarrow> (\<exists>n. l = k + n)" for k l :: nat
+proof (induct n)
+ case 0
+ then show ?case by simp
+next
+ case Suc
+ then show ?case
+ by (simp add: order_le_less)
+ (blast elim!: less_SucE intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
+qed
+
+lemma le_Suc_ex: "k \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
+ for k l :: nat
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>
@@ -734,27 +809,34 @@
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: "\<not> m < n \<Longrightarrow> n + (m - n) = m" for m n :: nat
+lemma add_diff_inverse_nat: "\<not> m < n \<Longrightarrow> n + (m - n) = m"
+ for m n :: nat
by (induct m n rule: diff_induct) simp_all
-lemma nat_le_iff_add: "m \<le> n \<longleftrightarrow> (\<exists>k. n = m + k)" for m n :: nat
+lemma nat_le_iff_add: "m \<le> n \<longleftrightarrow> (\<exists>k. n = m + k)"
+ for m n :: nat
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>
+text \<open>The naturals form an ordered \<open>semidom\<close> and a \<open>dioid\<close>.\<close>
instance nat :: linordered_semidom
proof
fix m n q :: nat
- show "0 < (1::nat)" by simp
- show "m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp
- show "m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)
- show "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0" by simp
+ show "0 < (1::nat)"
+ by simp
+ show "m \<le> n \<Longrightarrow> q + m \<le> q + n"
+ by simp
+ show "m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n"
+ by (simp add: mult_less_mono2)
+ show "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0"
+ by simp
show "n \<le> m \<Longrightarrow> (m - n) + n = m"
by (simp add: add_diff_inverse_nat add.commute linorder_not_less)
qed
instance nat :: dioid
by standard (rule nat_le_iff_add)
+
declare le0[simp del] \<comment> \<open>This is now @{thm zero_le}\<close>
declare le_0_eq[simp del] \<comment> \<open>This is now @{thm le_zero_eq}\<close>
declare not_less0[simp del] \<comment> \<open>This is now @{thm not_less_zero}\<close>
@@ -769,10 +851,12 @@
lemma mono_Suc: "mono Suc"
by (rule monoI) simp
-lemma min_0L [simp]: "min 0 n = 0" for n :: nat
+lemma min_0L [simp]: "min 0 n = 0"
+ for n :: nat
by (rule min_absorb1) simp
-lemma min_0R [simp]: "min n 0 = 0" for n :: nat
+lemma min_0R [simp]: "min n 0 = 0"
+ for n :: nat
by (rule min_absorb2) simp
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
@@ -784,10 +868,12 @@
lemma min_Suc2: "min m (Suc n) = (case m of 0 \<Rightarrow> 0 | Suc m' \<Rightarrow> Suc(min m' n))"
by (simp split: nat.split)
-lemma max_0L [simp]: "max 0 n = n" for n :: nat
+lemma max_0L [simp]: "max 0 n = n"
+ for n :: nat
by (rule max_absorb2) simp
-lemma max_0R [simp]: "max n 0 = n" for n :: nat
+lemma max_0R [simp]: "max n 0 = n"
+ for n :: nat
by (rule max_absorb1) simp
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc (max m n)"
@@ -799,25 +885,31 @@
lemma max_Suc2: "max m (Suc n) = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max m' n))"
by (simp split: nat.split)
-lemma nat_mult_min_left: "min m n * q = min (m * q) (n * q)" for m n q :: nat
+lemma nat_mult_min_left: "min m n * q = min (m * q) (n * q)"
+ for m n q :: nat
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: "m * min n q = min (m * n) (m * q)" for m n q :: nat
+lemma nat_mult_min_right: "m * min n q = min (m * n) (m * q)"
+ for m n q :: nat
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: "max m n + q = max (m + q) (n + q)" for m n q :: nat
+lemma nat_add_max_left: "max m n + q = max (m + q) (n + q)"
+ for m n q :: nat
by (simp add: max_def)
-lemma nat_add_max_right: "m + max n q = max (m + n) (m + q)" for m n q :: nat
+lemma nat_add_max_right: "m + max n q = max (m + n) (m + q)"
+ for m n q :: nat
by (simp add: max_def)
-lemma nat_mult_max_left: "max m n * q = max (m * q) (n * q)" for m n q :: nat
+lemma nat_mult_max_left: "max m n * q = max (m * q) (n * q)"
+ for m n q :: nat
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: "m * max n q = max (m * n) (m * q)" for m n q :: nat
+lemma nat_mult_max_right: "m * max n q = max (m * n) (m * q)"
+ for m n q :: nat
by (simp add: max_def not_le)
(auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
@@ -834,10 +926,11 @@
proof (induct n)
case (0 n)
have "P 0" by (rule step) auto
- then show ?case using 0 by auto
+ with 0 show ?case by auto
next
case (Suc m n)
- then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
+ then have "n \<le> m \<or> n = Suc m"
+ by (simp add: le_Suc_eq)
then show ?case
proof
assume "n \<le> m"
@@ -851,34 +944,36 @@
qed
-lemma Least_eq_0[simp]: "P 0 \<Longrightarrow> Least P = 0" for P :: "nat \<Rightarrow> bool"
+lemma Least_eq_0[simp]: "P 0 \<Longrightarrow> Least P = 0"
+ for P :: "nat \<Rightarrow> bool"
by (rule Least_equality[OF _ le0])
lemma Least_Suc: "P n \<Longrightarrow> \<not> P 0 \<Longrightarrow> (LEAST n. P n) = Suc (LEAST m. P (Suc m))"
- apply (cases n, auto)
+ apply (cases n)
+ apply auto
apply (frule LeastI)
- apply (drule_tac P = "\<lambda>x. P (Suc x) " in LeastI)
+ apply (drule_tac P = "\<lambda>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 = "\<lambda>x. P (Suc x) " in Least_le)
+ apply (erule_tac [2] Least_le)
+ apply (cases "LEAST x. P x")
+ apply auto
+ apply (drule_tac P = "\<lambda>x. P (Suc x)" in Least_le)
apply (blast intro: order_antisym)
done
lemma Least_Suc2: "P n \<Longrightarrow> Q m \<Longrightarrow> \<not> P 0 \<Longrightarrow> \<forall>k. P (Suc k) = Q k \<Longrightarrow> 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 \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not> P i) \<and> P k" for P :: "nat \<Rightarrow> bool"
+ by (erule (1) Least_Suc [THEN ssubst]) simp
+
+lemma ex_least_nat_le: "\<not> P 0 \<Longrightarrow> P n \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not> P i) \<and> P k"
+ for P :: "nat \<Rightarrow> bool"
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 \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not> P i) \<and> P (k + 1)" for P :: "nat \<Rightarrow> bool"
- unfolding One_nat_def
+lemma ex_least_nat_less: "\<not> P 0 \<Longrightarrow> P n \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not> P i) \<and> P (k + 1)"
+ for P :: "nat \<Rightarrow> bool"
apply (cases n)
apply blast
apply (frule (1) ex_least_nat_le)
@@ -957,7 +1052,7 @@
apply (rule infinite_descent)
using assms
apply (case_tac "n > 0")
- apply auto
+ apply auto
done
text \<open>
@@ -988,7 +1083,7 @@
ultimately show "P x" by auto
qed
-text\<open>Again, without explicit base case:\<close>
+text \<open>Again, without explicit base case:\<close>
lemma infinite_descent_measure:
fixes V :: "'a \<Rightarrow> nat"
assumes "\<And>x. \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y"
@@ -1014,17 +1109,21 @@
text \<open>non-strict, in 1st argument\<close>
-lemma add_le_mono1: "i \<le> j \<Longrightarrow> i + k \<le> j + k" for i j k :: nat
+lemma add_le_mono1: "i \<le> j \<Longrightarrow> i + k \<le> j + k"
+ for i j k :: nat
by (rule add_right_mono)
text \<open>non-strict, in both arguments\<close>
-lemma add_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i + k \<le> j + l" for i j k l :: nat
+lemma add_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i + k \<le> j + l"
+ for i j k l :: nat
by (rule add_mono)
-lemma le_add2: "n \<le> m + n" for m n :: nat
+lemma le_add2: "n \<le> m + n"
+ for m n :: nat
by simp
-lemma le_add1: "n \<le> n + m" for m n :: nat
+lemma le_add1: "n \<le> n + m"
+ for m n :: nat
by simp
lemma less_add_Suc1: "i < Suc (i + m)"
@@ -1036,43 +1135,54 @@
lemma less_iff_Suc_add: "m < n \<longleftrightarrow> (\<exists>k. n = Suc (m + k))"
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
-lemma trans_le_add1: "i \<le> j \<Longrightarrow> i \<le> j + m" for i j m :: nat
+lemma trans_le_add1: "i \<le> j \<Longrightarrow> i \<le> j + m"
+ for i j m :: nat
by (rule le_trans, assumption, rule le_add1)
-lemma trans_le_add2: "i \<le> j \<Longrightarrow> i \<le> m + j" for i j m :: nat
+lemma trans_le_add2: "i \<le> j \<Longrightarrow> i \<le> m + j"
+ for i j m :: nat
by (rule le_trans, assumption, rule le_add2)
-lemma trans_less_add1: "i < j \<Longrightarrow> i < j + m" for i j m :: nat
+lemma trans_less_add1: "i < j \<Longrightarrow> i < j + m"
+ for i j m :: nat
by (rule less_le_trans, assumption, rule le_add1)
-lemma trans_less_add2: "i < j \<Longrightarrow> i < m + j" for i j m :: nat
+lemma trans_less_add2: "i < j \<Longrightarrow> i < m + j"
+ for i j m :: nat
by (rule less_le_trans, assumption, rule le_add2)
-lemma add_lessD1: "i + j < k \<Longrightarrow> i < k" for i j k :: nat
+lemma add_lessD1: "i + j < k \<Longrightarrow> i < k"
+ for i j k :: nat
by (rule le_less_trans [of _ "i+j"]) (simp_all add: le_add1)
-lemma not_add_less1 [iff]: "\<not> i + j < i" for i j :: nat
+lemma not_add_less1 [iff]: "\<not> i + j < i"
+ for i j :: nat
apply (rule notI)
apply (drule add_lessD1)
apply (erule less_irrefl [THEN notE])
done
-lemma not_add_less2 [iff]: "\<not> j + i < i" for i j :: nat
+lemma not_add_less2 [iff]: "\<not> j + i < i"
+ for i j :: nat
by (simp add: add.commute)
-lemma add_leD1: "m + k \<le> n \<Longrightarrow> m \<le> n" for k m n :: nat
- by (rule order_trans [of _ "m+k"]) (simp_all add: le_add1)
-
-lemma add_leD2: "m + k \<le> n \<Longrightarrow> k \<le> n" for k m n :: nat
+lemma add_leD1: "m + k \<le> n \<Longrightarrow> m \<le> n"
+ for k m n :: nat
+ by (rule order_trans [of _ "m + k"]) (simp_all add: le_add1)
+
+lemma add_leD2: "m + k \<le> n \<Longrightarrow> k \<le> n"
+ for k m n :: nat
apply (simp add: add.commute)
apply (erule add_leD1)
done
-lemma add_leE: "m + k \<le> n \<Longrightarrow> (m \<le> n \<Longrightarrow> k \<le> n \<Longrightarrow> R) \<Longrightarrow> R" for k m n :: nat
+lemma add_leE: "m + k \<le> n \<Longrightarrow> (m \<le> n \<Longrightarrow> k \<le> n \<Longrightarrow> R) \<Longrightarrow> R"
+ for k m n :: nat
by (blast dest: add_leD1 add_leD2)
text \<open>needs \<open>\<And>k\<close> for \<open>ac_simps\<close> to work\<close>
-lemma less_add_eq_less: "\<And>k::nat. k < l \<Longrightarrow> m + l = k + n \<Longrightarrow> m < n"
+lemma less_add_eq_less: "\<And>k. k < l \<Longrightarrow> m + l = k + n \<Longrightarrow> m < n"
+ for l m n :: nat
by (force simp del: add_Suc_right simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)
@@ -1082,42 +1192,52 @@
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" for m n :: nat
+ 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"
+ for m n :: nat
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
-lemma less_imp_diff_less: "j < k \<Longrightarrow> j - n < k" for j k n :: nat
+lemma less_imp_diff_less: "j < k \<Longrightarrow> j - n < k"
+ for j k n :: nat
by (rule le_less_trans, rule diff_le_self)
lemma diff_Suc_less [simp]: "0 < n \<Longrightarrow> n - Suc i < n"
by (cases n) (auto simp add: le_simps)
-lemma diff_add_assoc: "k \<le> j \<Longrightarrow> (i + j) - k = i + (j - k)" for i j k :: nat
+lemma diff_add_assoc: "k \<le> j \<Longrightarrow> (i + j) - k = i + (j - k)"
+ for i j k :: nat
by (induct j k rule: diff_induct) simp_all
-lemma add_diff_assoc [simp]: "k \<le> j \<Longrightarrow> i + (j - k) = i + j - k" for i j k :: nat
+lemma add_diff_assoc [simp]: "k \<le> j \<Longrightarrow> i + (j - k) = i + j - k"
+ for i j k :: nat
by (fact diff_add_assoc [symmetric])
-lemma diff_add_assoc2: "k \<le> j \<Longrightarrow> (j + i) - k = (j - k) + i" for i j k :: nat
+lemma diff_add_assoc2: "k \<le> j \<Longrightarrow> (j + i) - k = (j - k) + i"
+ for i j k :: nat
by (simp add: ac_simps)
-lemma add_diff_assoc2 [simp]: "k \<le> j \<Longrightarrow> j - k + i = j + i - k" for i j k :: nat
+lemma add_diff_assoc2 [simp]: "k \<le> j \<Longrightarrow> j - k + i = j + i - k"
+ for i j k :: nat
by (fact diff_add_assoc2 [symmetric])
-lemma le_imp_diff_is_add: "i \<le> j \<Longrightarrow> (j - i = k) = (j = k + i)" for i j k :: nat
+lemma le_imp_diff_is_add: "i \<le> j \<Longrightarrow> (j - i = k) = (j = k + i)"
+ for i j k :: nat
by auto
-lemma diff_is_0_eq [simp]: "m - n = 0 \<longleftrightarrow> m \<le> n" for m n :: nat
+lemma diff_is_0_eq [simp]: "m - n = 0 \<longleftrightarrow> m \<le> n"
+ for m n :: nat
by (induct m n rule: diff_induct) simp_all
-lemma diff_is_0_eq' [simp]: "m \<le> n \<Longrightarrow> m - n = 0" for m n :: nat
+lemma diff_is_0_eq' [simp]: "m \<le> n \<Longrightarrow> m - n = 0"
+ for m n :: nat
by (rule iffD2, rule diff_is_0_eq)
-lemma zero_less_diff [simp]: "0 < n - m \<longleftrightarrow> m < n" for m n :: nat
+lemma zero_less_diff [simp]: "0 < n - m \<longleftrightarrow> m < n"
+ for m n :: nat
by (induct m n rule: diff_induct) simp_all
lemma less_imp_add_positive:
@@ -1129,18 +1249,18 @@
qed
text \<open>a nice rewrite for bounded subtraction\<close>
-lemma nat_minus_add_max: "n - m + m = max n m" for m n :: nat
- by (simp add: max_def not_le order_less_imp_le)
+lemma nat_minus_add_max: "n - m + m = max n m"
+ for m n :: nat
+ by (simp add: max_def not_le order_less_imp_le)
lemma nat_diff_split: "P (a - b) \<longleftrightarrow> (a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d)"
for a b :: nat
- \<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])
+ \<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: "P (a - b) \<longleftrightarrow> \<not> (a < b \<and> \<not> P 0 \<or> (\<exists>d. a = b + d \<and> \<not> P d))"
for a b :: nat
- \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close> in assumptions\<close>
+ \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close> in assumptions\<close>
by (auto split: nat_diff_split)
lemma Suc_pred': "0 < n \<Longrightarrow> n = Suc(n - 1)"
@@ -1149,64 +1269,78 @@
lemma add_eq_if: "m + 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 * n = (if m = 0 then 0 else n + ((m - 1) * n))" for m n :: nat
- unfolding One_nat_def by (cases m) simp_all
+lemma mult_eq_if: "m * n = (if m = 0 then 0 else n + ((m - 1) * n))"
+ for m n :: nat
+ by (cases m) simp_all
lemma Suc_diff_eq_diff_pred: "0 < n \<Longrightarrow> Suc m - n = m - (n - 1)"
- unfolding One_nat_def by (cases n) simp_all
+ 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 (cases m) simp_all
+
+lemma Let_Suc [simp]: "Let (Suc n) f \<equiv> f (Suc n)"
by (fact Let_def)
subsubsection \<open>Monotonicity of multiplication\<close>
-lemma mult_le_mono1: "i \<le> j \<Longrightarrow> i * k \<le> j * k" for i j k :: nat
+lemma mult_le_mono1: "i \<le> j \<Longrightarrow> i * k \<le> j * k"
+ for i j k :: nat
by (simp add: mult_right_mono)
-lemma mult_le_mono2: "i \<le> j \<Longrightarrow> k * i \<le> k * j" for i j k :: nat
+lemma mult_le_mono2: "i \<le> j \<Longrightarrow> k * i \<le> k * j"
+ for i j k :: nat
by (simp add: mult_left_mono)
text \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close>
-lemma mult_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i * k \<le> j * l" for i j k l :: nat
+lemma mult_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i * k \<le> j * l"
+ for i j k l :: nat
by (simp add: mult_mono)
-lemma mult_less_mono1: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> i * k < j * k" for i j k :: nat
+lemma mult_less_mono1: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> i * k < j * k"
+ for i j k :: nat
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 * n \<longleftrightarrow> 0 < m \<and> 0 < n" for m n :: nat
- apply (induct m)
- apply simp
- apply (case_tac n)
- apply simp_all
- done
+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 * n \<longleftrightarrow> 0 < m \<and> 0 < n"
+ for m n :: nat
+proof (induct m)
+ case 0
+ then show ?case by simp
+next
+ case (Suc m)
+ then show ?case by (cases n) simp_all
+qed
lemma one_le_mult_iff [simp]: "Suc 0 \<le> m * n \<longleftrightarrow> Suc 0 \<le> m \<and> Suc 0 \<le> n"
- apply (induct m)
- apply simp
- apply (case_tac n)
- apply simp_all
- done
-
-lemma mult_less_cancel2 [simp]: "m * k < n * k \<longleftrightarrow> 0 < k \<and> m < n" for k m n :: nat
+proof (induct m)
+ case 0
+ then show ?case by simp
+next
+ case (Suc m)
+ then show ?case by (cases n) simp_all
+qed
+
+lemma mult_less_cancel2 [simp]: "m * k < n * k \<longleftrightarrow> 0 < k \<and> m < n"
+ for k m n :: nat
apply (safe intro!: mult_less_mono1)
- apply (cases k, auto)
+ apply (cases k)
+ apply auto
apply (simp add: linorder_not_le [symmetric])
apply (blast intro: mult_le_mono1)
done
-lemma mult_less_cancel1 [simp]: "k * m < k * n \<longleftrightarrow> 0 < k \<and> m < n" for k m n :: nat
+lemma mult_less_cancel1 [simp]: "k * m < k * n \<longleftrightarrow> 0 < k \<and> m < n"
+ for k m n :: nat
by (simp add: mult.commute [of k])
-lemma mult_le_cancel1 [simp]: "k * m \<le> k * n \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)" for k m n :: nat
+lemma mult_le_cancel1 [simp]: "k * m \<le> k * n \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)"
+ for k m n :: nat
by (simp add: linorder_not_less [symmetric], auto)
-lemma mult_le_cancel2 [simp]: "m * k \<le> n * k \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)" for k m n :: nat
+lemma mult_le_cancel2 [simp]: "m * k \<le> n * k \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)"
+ for k m n :: nat
by (simp add: linorder_not_less [symmetric], auto)
lemma Suc_mult_less_cancel1: "Suc k * m < Suc k * n \<longleftrightarrow> m < n"
@@ -1215,19 +1349,24 @@
lemma Suc_mult_le_cancel1: "Suc k * m \<le> Suc k * n \<longleftrightarrow> m \<le> n"
by (subst mult_le_cancel1) simp
-lemma le_square: "m \<le> m * m" for m :: nat
+lemma le_square: "m \<le> m * m"
+ for m :: nat
by (cases m) (auto intro: le_add1)
-lemma le_cube: "m \<le> m * (m * m)" for m :: nat
+lemma le_cube: "m \<le> m * (m * m)"
+ for m :: nat
by (cases m) (auto intro: le_add1)
text \<open>Lemma for \<open>gcd\<close>\<close>
-lemma mult_eq_self_implies_10: "m = m * n \<Longrightarrow> n = 1 \<or> m = 0" for m n :: nat
+lemma mult_eq_self_implies_10: "m = m * n \<Longrightarrow> n = 1 \<or> m = 0"
+ for m n :: nat
apply (drule sym)
apply (rule disjCI)
- apply (rule linorder_cases, erule_tac [2] _)
- apply (drule_tac [2] mult_less_mono2)
- apply (auto)
+ apply (rule linorder_cases)
+ defer
+ apply assumption
+ apply (drule mult_less_mono2)
+ apply auto
done
lemma mono_times_nat:
@@ -1240,7 +1379,7 @@
with assms show "n * m \<le> n * q" by simp
qed
-text \<open>the lattice order on @{typ nat}\<close>
+text \<open>The lattice order on @{typ nat}.\<close>
instantiation nat :: distrib_lattice
begin
@@ -1272,15 +1411,16 @@
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>
+text \<open>\<open>f ^^ n = f \<circ> \<dots> \<circ> f\<close>, the \<open>n\<close>-fold composition of \<open>f\<close>\<close>
overloading
funpow \<equiv> "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"
+primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
+ where
+ "funpow 0 f = id"
+ | "funpow (Suc n) f = f \<circ> funpow n f"
end
@@ -1300,7 +1440,7 @@
lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right
-text \<open>for code generation\<close>
+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"
@@ -1315,18 +1455,20 @@
lemma funpow_add: "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
by (induct m) simp_all
-lemma funpow_mult: "(f ^^ m) ^^ n = f ^^ (m * n)" for f :: "'a \<Rightarrow> 'a"
+lemma funpow_mult: "(f ^^ m) ^^ n = f ^^ (m * n)"
+ for f :: "'a \<Rightarrow> 'a"
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 \<circ> f ^^ 1) x" by (simp only: funpow_add)
also have "\<dots> = (f ^^ n) (f x)" by simp
finally show ?thesis .
qed
-lemma comp_funpow: "comp f ^^ n = comp (f ^^ n)" for f :: "'a \<Rightarrow> 'a"
+lemma comp_funpow: "comp f ^^ n = comp (f ^^ n)"
+ for f :: "'a \<Rightarrow> 'a"
by (induct n) simp_all
lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)"
@@ -1342,12 +1484,15 @@
lemma funpow_mono2:
assumes "mono f"
- assumes "i \<le> j"
- assumes "x \<le> y"
- assumes "x \<le> f x"
+ and "i \<le> j"
+ and "x \<le> y"
+ and "x \<le> f x"
shows "(f ^^ i) x \<le> (f ^^ j) y"
-using assms(2,3)
-proof(induct j arbitrary: y)
+ using assms(2,3)
+proof (induct j arbitrary: y)
+ case 0
+ then show ?case by simp
+next
case (Suc j)
show ?case
proof(cases "i = Suc j")
@@ -1358,40 +1503,43 @@
case False
with assms(1,4) Suc show ?thesis
by (simp del: funpow.simps add: funpow_simps_right le_eq_less_or_eq less_Suc_eq_le)
- (simp add: Suc.hyps monoD order_subst1)
+ (simp add: Suc.hyps monoD order_subst1)
qed
-qed simp
+qed
subsection \<open>Kleene iteration\<close>
lemma Kleene_iter_lpfp:
+ fixes f :: "'a::order_bot \<Rightarrow> 'a"
assumes "mono f"
and "f p \<le> p"
- shows "(f^^k) (bot::'a::order_bot) \<le> p"
-proof(induction k)
+ shows "(f ^^ k) bot \<le> p"
+proof (induct k)
case 0
show ?case by simp
next
case Suc
- from monoD[OF assms(1) Suc] assms(2) show ?case by simp
+ show ?case
+ using monoD[OF assms(1) Suc] assms(2) by simp
qed
lemma lfp_Kleene_iter:
assumes "mono f"
- and "(f^^Suc k) bot = (f^^k) bot"
- shows "lfp f = (f^^k) bot"
+ and "(f ^^ Suc k) bot = (f ^^ k) bot"
+ shows "lfp f = (f ^^ k) bot"
proof (rule antisym)
- show "lfp f \<le> (f^^k) bot"
+ show "lfp f \<le> (f ^^ k) bot"
proof (rule lfp_lowerbound)
- show "f ((f^^k) bot) \<le> (f^^k) bot"
+ show "f ((f ^^ k) bot) \<le> (f ^^ k) bot"
using assms(2) by simp
qed
- show "(f^^k) bot \<le> lfp f"
+ 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: "mono f \<Longrightarrow> mono (f ^^ n)" for f :: "'a \<Rightarrow> 'a::complete_lattice"
+lemma mono_pow: "mono f \<Longrightarrow> mono (f ^^ n)"
+ for f :: "'a \<Rightarrow> 'a::complete_lattice"
by (induct n) (auto simp: mono_def)
lemma lfp_funpow:
@@ -1405,9 +1553,9 @@
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
+ have "(f ^^ n) (lfp f) = lfp f" for n
by (induct n) (auto intro: f lfp_unfold[symmetric])
- then show "lfp (f^^Suc n) \<le> lfp f"
+ then show "lfp (f ^^ Suc n) \<le> lfp f"
by (intro lfp_lowerbound) (simp del: funpow.simps)
qed
@@ -1422,32 +1570,36 @@
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
+ have "(f ^^ n) (gfp f) = gfp f" for n
by (induct n) (auto intro: f gfp_unfold[symmetric])
- then show "gfp (f^^Suc n) \<ge> gfp f"
+ then show "gfp (f ^^ Suc n) \<ge> gfp f"
by (intro gfp_upperbound) (simp del: funpow.simps)
qed
lemma Kleene_iter_gpfp:
+ fixes f :: "'a::order_top \<Rightarrow> 'a"
assumes "mono f"
- and "p \<le> f p"
- shows "p \<le> (f ^^ k) (top::'a::order_top)"
-proof(induction k)
- case 0 show ?case by simp
+ and "p \<le> f p"
+ shows "p \<le> (f ^^ k) top"
+proof (induct k)
+ case 0
+ show ?case by simp
next
case Suc
- from monoD[OF assms(1) Suc] assms(2)
- show ?case by simp
+ show ?case
+ using monoD[OF assms(1) Suc] assms(2) by simp
qed
lemma gfp_Kleene_iter:
assumes "mono f"
- and "(f ^^ Suc k) top = (f ^^ k) top"
- shows "gfp f = (f ^^ k) top" (is "?lhs = ?rhs")
-proof(rule antisym)
- have "?rhs \<le> f ?rhs" using assms(2) by simp
- then show "?rhs \<le> ?lhs" by(rule gfp_upperbound)
-
+ and "(f ^^ Suc k) top = (f ^^ k) top"
+ shows "gfp f = (f ^^ k) top"
+ (is "?lhs = ?rhs")
+proof (rule antisym)
+ have "?rhs \<le> f ?rhs"
+ using assms(2) by simp
+ then show "?rhs \<le> ?lhs"
+ by (rule gfp_upperbound)
show "?lhs \<le> ?rhs"
using Kleene_iter_gpfp[OF assms(1)] gfp_unfold[OF assms(1)] by simp
qed
@@ -1478,9 +1630,10 @@
lemma mult_of_nat_commute: "of_nat x * y = y * of_nat x"
by (induct 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>
+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)
@@ -1492,7 +1645,8 @@
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)
+ with Suc show ?case
+ by (simp add: add.commute)
qed
end
@@ -1525,12 +1679,10 @@
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"
+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)"
+lemma of_nat_0_neq [simp]: "0 \<noteq> of_nat (Suc n)"
unfolding of_nat_0_eq_iff by simp
end
@@ -1600,28 +1752,28 @@
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
+ 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
+ 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
+ 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
+ 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>"
@@ -1633,8 +1785,7 @@
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"
+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
@@ -1669,7 +1820,8 @@
begin
lemma lift_Suc_mono_le:
- assumes mono: "\<And>n. f n \<le> f (Suc n)" and "n \<le> n'"
+ 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
@@ -1681,7 +1833,8 @@
qed
lemma lift_Suc_antimono_le:
- assumes mono: "\<And>n. f n \<ge> f (Suc n)" and "n \<le> n'"
+ 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
@@ -1693,7 +1846,8 @@
qed
lemma lift_Suc_mono_less:
- assumes mono: "\<And>n. f n < f (Suc n)" and "n < n'"
+ 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) (auto intro: mono)
@@ -1737,56 +1891,71 @@
then show ?thesis by simp
qed
-lemma less_diff_conv: "i < j - k \<longleftrightarrow> i + k < j" for i j k :: nat
+lemma less_diff_conv: "i < j - k \<longleftrightarrow> i + k < j"
+ for i j k :: nat
by (cases "k \<le> j") (auto simp add: not_le dest: less_imp_Suc_add le_Suc_ex)
-lemma less_diff_conv2: "k \<le> j \<Longrightarrow> j - k < i \<longleftrightarrow> j < i + k" for j k i :: nat
+lemma less_diff_conv2: "k \<le> j \<Longrightarrow> j - k < i \<longleftrightarrow> j < i + k"
+ for j k i :: nat
by (auto dest: le_Suc_ex)
-lemma le_diff_conv: "j - k \<le> i \<longleftrightarrow> j \<le> i + k" for j k i :: nat
+lemma le_diff_conv: "j - k \<le> i \<longleftrightarrow> j \<le> i + k"
+ for j k i :: nat
by (cases "k \<le> j") (auto simp add: not_le dest!: less_imp_Suc_add le_Suc_ex)
-lemma diff_diff_cancel [simp]: "i \<le> n \<Longrightarrow> n - (n - i) = i" for i n :: nat
+lemma diff_diff_cancel [simp]: "i \<le> n \<Longrightarrow> n - (n - i) = i"
+ for i n :: nat
by (auto dest: le_Suc_ex)
-lemma diff_less [simp]: "0 < n \<Longrightarrow> 0 < m \<Longrightarrow> m - n < m" for i n :: nat
+lemma diff_less [simp]: "0 < n \<Longrightarrow> 0 < m \<Longrightarrow> m - n < m"
+ for i n :: nat
by (auto dest: less_imp_Suc_add)
text \<open>Simplification of relational expressions involving subtraction\<close>
-lemma diff_diff_eq: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k - (n - k) = m - n" for m n k :: nat
+lemma diff_diff_eq: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k - (n - k) = m - n"
+ for m n k :: nat
by (auto dest!: le_Suc_ex)
hide_fact (open) diff_diff_eq
-lemma eq_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k = n - k \<longleftrightarrow> m = n" for m n k :: nat
+lemma eq_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k = n - k \<longleftrightarrow> m = n"
+ for m n k :: nat
by (auto dest: le_Suc_ex)
-lemma less_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k < n - k \<longleftrightarrow> m < n" for m n k :: nat
+lemma less_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k < n - k \<longleftrightarrow> m < n"
+ for m n k :: nat
by (auto dest!: le_Suc_ex)
-lemma le_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k \<le> n - k \<longleftrightarrow> m \<le> n" for m n k :: nat
+lemma le_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k \<le> n - k \<longleftrightarrow> m \<le> n"
+ for m n k :: nat
by (auto dest!: le_Suc_ex)
-lemma le_diff_iff': "a \<le> c \<Longrightarrow> b \<le> c \<Longrightarrow> c - a \<le> c - b \<longleftrightarrow> b \<le> a" for a b c :: nat
+lemma le_diff_iff': "a \<le> c \<Longrightarrow> b \<le> c \<Longrightarrow> c - a \<le> c - b \<longleftrightarrow> b \<le> a"
+ for a b c :: nat
by (force dest: le_Suc_ex)
text \<open>(Anti)Monotonicity of subtraction -- by Stephan Merz\<close>
-lemma diff_le_mono: "m \<le> n \<Longrightarrow> m - l \<le> n - l" for m n l :: nat
+lemma diff_le_mono: "m \<le> n \<Longrightarrow> m - l \<le> n - l"
+ for m n l :: nat
by (auto dest: less_imp_le less_imp_Suc_add split add: nat_diff_split)
-lemma diff_le_mono2: "m \<le> n \<Longrightarrow> l - n \<le> l - m" for m n l :: nat
+lemma diff_le_mono2: "m \<le> n \<Longrightarrow> l - n \<le> l - m"
+ for m n l :: nat
by (auto dest: less_imp_le le_Suc_ex less_imp_Suc_add less_le_trans split add: nat_diff_split)
-lemma diff_less_mono2: "m < n \<Longrightarrow> m < l \<Longrightarrow> l - n < l - m" for m n l :: nat
+lemma diff_less_mono2: "m < n \<Longrightarrow> m < l \<Longrightarrow> l - n < l - m"
+ for m n l :: nat
by (auto dest: less_imp_Suc_add split add: nat_diff_split)
-lemma diffs0_imp_equal: "m - n = 0 \<Longrightarrow> n - m = 0 \<Longrightarrow> m = n" for m n :: nat
+lemma diffs0_imp_equal: "m - n = 0 \<Longrightarrow> n - m = 0 \<Longrightarrow> m = n"
+ for m n :: nat
by (simp split add: nat_diff_split)
-lemma min_diff: "min (m - i) (n - i) = min m n - i" for m n i :: nat
+lemma min_diff: "min (m - i) (n - i) = min m n - i"
+ for m n i :: nat
by (cases m n rule: le_cases)
(auto simp add: not_le min.absorb1 min.absorb2 min.absorb_iff1 [symmetric] diff_le_mono)
@@ -1803,7 +1972,8 @@
text \<open>Rewriting to pull differences out\<close>
-lemma diff_diff_right [simp]: "k \<le> j \<Longrightarrow> i - (j - k) = i + k - j" for i j k :: nat
+lemma diff_diff_right [simp]: "k \<le> j \<Longrightarrow> i - (j - k) = i + k - j"
+ for i j k :: nat
by (fact diff_diff_right)
lemma diff_Suc_diff_eq1 [simp]:
@@ -1870,12 +2040,9 @@
from \<open>Suc d = j - n\<close> have "d + 1 = j - n" by simp
then have "d + 1 - 1 = j - n - 1" by simp
then have "d = j - n - 1" by simp
- then have "d = j - (n + 1)"
- by (simp add: diff_diff_eq)
- then have "d = j - Suc n"
- by simp
- moreover from \<open>n < j\<close> have "Suc n \<le> j"
- by (simp add: Suc_le_eq)
+ then have "d = j - (n + 1)" by (simp add: diff_diff_eq)
+ then have "d = j - Suc n" by simp
+ moreover from \<open>n < j\<close> have "Suc n \<le> j" by (simp add: Suc_le_eq)
ultimately have "P (Suc n)"
proof (rule Suc.hyps)
fix q
@@ -1883,11 +2050,9 @@
then have "n \<le> q" by (simp add: Suc_le_eq less_imp_le)
moreover assume "q < j"
moreover assume "P (Suc q)"
- ultimately show "P q"
- by (rule Suc.prems)
+ ultimately show "P q" by (rule Suc.prems)
qed
- with order_refl \<open>n < j\<close> show "P n"
- by (rule Suc.prems)
+ with order_refl \<open>n < j\<close> show "P n" by (rule Suc.prems)
qed
lemma strict_inc_induct [consumes 1, case_names base step]:
@@ -1906,16 +2071,11 @@
case (Suc d i)
from \<open>Suc d = j - i - 1\<close> have *: "Suc d = j - Suc i"
by (simp add: diff_diff_add)
- then have "Suc d - 1 = j - Suc i - 1"
- by simp
- then have "d = j - Suc i - 1"
- by simp
- moreover from * have "j - Suc i \<noteq> 0"
- by auto
- then have "Suc i < j"
- by (simp add: not_le)
- ultimately have "P (Suc i)"
- by (rule Suc.hyps)
+ then have "Suc d - 1 = j - Suc i - 1" by simp
+ then have "d = j - Suc i - 1" by simp
+ moreover from * have "j - Suc i \<noteq> 0" by auto
+ then have "Suc i < j" by (simp add: not_le)
+ ultimately have "P (Suc i)" by (rule Suc.hyps)
with \<open>i < j\<close> show "P i" by (rule step)
qed
@@ -1925,7 +2085,7 @@
lemma zero_induct: "P k \<Longrightarrow> (\<And>n. P (Suc n) \<Longrightarrow> P n) \<Longrightarrow> P 0"
using inc_induct[of 0 k P] by blast
-text \<open>Further induction rule similar to @{thm inc_induct}\<close>
+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"
@@ -1947,11 +2107,9 @@
moreover assume "q < j" then have "q < Suc j"
by (simp add: less_Suc_eq)
moreover assume "P q"
- ultimately show "P (Suc q)"
- by (rule Suc.prems)
+ ultimately show "P (Suc q)" by (rule Suc.prems)
qed
- ultimately show "P (Suc j)"
- by (rule Suc.prems)
+ ultimately show "P (Suc j)" by (rule Suc.prems)
next
case 2
with \<open>P i\<close> show "P (Suc j)" by simp
@@ -1961,28 +2119,24 @@
subsection \<open>Monotonicity of \<open>funpow\<close>\<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>"
+lemma funpow_increasing: "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ n) \<top> \<le> (f ^^ m) \<top>"
+ for f :: "'a::{lattice,order_top} \<Rightarrow> 'a"
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>"
+ (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
+ intro: order_trans[OF _ funpow_mono])
+
+lemma funpow_decreasing: "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ m) \<bottom> \<le> (f ^^ n) \<bottom>"
+ for f :: "'a::{lattice,order_bot} \<Rightarrow> 'a"
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>)"
+ (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
+ intro: order_trans[OF _ funpow_mono])
+
+lemma mono_funpow: "mono Q \<Longrightarrow> mono (\<lambda>i. (Q ^^ i) \<bottom>)"
+ for Q :: "'a::{lattice,order_bot} \<Rightarrow> 'a"
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>)"
+lemma antimono_funpow: "mono Q \<Longrightarrow> antimono (\<lambda>i. (Q ^^ i) \<top>)"
+ for Q :: "'a::{lattice,order_top} \<Rightarrow> 'a"
by (auto intro!: funpow_increasing simp: antimono_def)
@@ -1994,21 +2148,26 @@
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 \<longleftrightarrow> m = 1" for m :: nat
+lemma nat_dvd_1_iff_1 [simp]: "m dvd 1 \<longleftrightarrow> m = 1"
+ for m :: nat
by (simp add: dvd_def)
-lemma dvd_antisym: "m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n" for m n :: nat
+lemma dvd_antisym: "m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
+ for 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)" for k m n :: nat
+lemma dvd_diff_nat [simp]: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m - n)"
+ for k 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" for k m n :: nat
+lemma dvd_diffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> n \<le> m \<Longrightarrow> k dvd m"
+ for k m n :: 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" for k m n :: nat
+lemma dvd_diffD1: "k dvd m - n \<Longrightarrow> k dvd m \<Longrightarrow> n \<le> m \<Longrightarrow> k dvd n"
+ for k m n :: nat
by (drule_tac m = m in dvd_diff_nat) auto
lemma dvd_mult_cancel:
@@ -2022,19 +2181,24 @@
then show ?thesis ..
qed
-lemma dvd_mult_cancel1: "0 < m \<Longrightarrow> m * n dvd m \<longleftrightarrow> n = 1" for m n :: nat
+lemma dvd_mult_cancel1: "0 < m \<Longrightarrow> m * n dvd m \<longleftrightarrow> n = 1"
+ for m n :: nat
apply auto
- apply (subgoal_tac "m * n dvd m * 1")
- apply (drule dvd_mult_cancel, auto)
+ apply (subgoal_tac "m * n dvd m * 1")
+ apply (drule dvd_mult_cancel)
+ apply auto
done
-lemma dvd_mult_cancel2: "0 < m \<Longrightarrow> n * m dvd m \<longleftrightarrow> n = 1" for m n :: nat
+lemma dvd_mult_cancel2: "0 < m \<Longrightarrow> n * m dvd m \<longleftrightarrow> n = 1"
+ for m n :: 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" for k n :: nat
+lemma dvd_imp_le: "k dvd n \<Longrightarrow> 0 < n \<Longrightarrow> k \<le> n"
+ for k n :: nat
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
-lemma nat_dvd_not_less: "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m" for m n :: nat
+lemma nat_dvd_not_less: "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
+ for m n :: nat
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
lemma less_eq_dvd_minus:
@@ -2047,7 +2211,8 @@
then show ?thesis by (simp add: add.commute [of m])
qed
-lemma dvd_minus_self: "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n" for m n :: nat
+lemma dvd_minus_self: "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"
+ for m n :: nat
by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add dest: less_imp_le)
lemma dvd_minus_add:
@@ -2066,55 +2231,72 @@
subsection \<open>Aliasses\<close>
-lemma nat_mult_1: "1 * n = n" for n :: nat
+lemma nat_mult_1: "1 * n = n"
+ for n :: nat
by (fact mult_1_left)
-lemma nat_mult_1_right: "n * 1 = n" for n :: nat
+lemma nat_mult_1_right: "n * 1 = n"
+ for n :: nat
by (fact mult_1_right)
-lemma nat_add_left_cancel: "k + m = k + n \<longleftrightarrow> m = n" for k m n :: nat
+lemma nat_add_left_cancel: "k + m = k + n \<longleftrightarrow> m = n"
+ for k m n :: nat
by (fact add_left_cancel)
-lemma nat_add_right_cancel: "m + k = n + k \<longleftrightarrow> m = n" for k m n :: nat
+lemma nat_add_right_cancel: "m + k = n + k \<longleftrightarrow> m = n"
+ for k m n :: nat
by (fact add_right_cancel)
-lemma diff_mult_distrib: "(m - n) * k = (m * k) - (n * k)" for k m n :: nat
+lemma diff_mult_distrib: "(m - n) * k = (m * k) - (n * k)"
+ for k m n :: nat
by (fact left_diff_distrib')
-lemma diff_mult_distrib2: "k * (m - n) = (k * m) - (k * n)" for k m n :: nat
+lemma diff_mult_distrib2: "k * (m - n) = (k * m) - (k * n)"
+ for k m n :: nat
by (fact right_diff_distrib')
-lemma le_add_diff: "k \<le> n \<Longrightarrow> m \<le> n + m - k" for k m n :: nat
+lemma le_add_diff: "k \<le> n \<Longrightarrow> m \<le> n + m - k"
+ for k m n :: nat
by (fact le_add_diff) (* FIXME delete *)
-lemma le_diff_conv2: "k \<le> j \<Longrightarrow> (i \<le> j - k) = (i + k \<le> j)" for i j k :: nat
+lemma le_diff_conv2: "k \<le> j \<Longrightarrow> (i \<le> j - k) = (i + k \<le> j)"
+ for i j k :: nat
by (fact le_diff_conv2) (* FIXME delete *)
-lemma diff_self_eq_0 [simp]: "m - m = 0" for m :: nat
+lemma diff_self_eq_0 [simp]: "m - m = 0"
+ for m :: nat
by (fact diff_cancel)
-lemma diff_diff_left [simp]: "i - j - k = i - (j + k)" for i j k :: nat
+lemma diff_diff_left [simp]: "i - j - k = i - (j + k)"
+ for i j k :: nat
by (fact diff_diff_add)
-lemma diff_commute: "i - j - k = i - k - j" for i j k :: nat
+lemma diff_commute: "i - j - k = i - k - j"
+ for i j k :: nat
by (fact diff_right_commute)
-lemma diff_add_inverse: "(n + m) - n = m" for m n :: nat
+lemma diff_add_inverse: "(n + m) - n = m"
+ for m n :: nat
by (fact add_diff_cancel_left')
-lemma diff_add_inverse2: "(m + n) - n = m" for m n :: nat
+lemma diff_add_inverse2: "(m + n) - n = m"
+ for m n :: nat
by (fact add_diff_cancel_right')
-lemma diff_cancel: "(k + m) - (k + n) = m - n" for k m n :: nat
+lemma diff_cancel: "(k + m) - (k + n) = m - n"
+ for k m n :: nat
by (fact add_diff_cancel_left)
-lemma diff_cancel2: "(m + k) - (n + k) = m - n" for k m n :: nat
+lemma diff_cancel2: "(m + k) - (n + k) = m - n"
+ for k m n :: nat
by (fact add_diff_cancel_right)
-lemma diff_add_0: "n - (n + m) = 0" for m n :: nat
+lemma diff_add_0: "n - (n + m) = 0"
+ for m n :: nat
by (fact diff_add_zero)
-lemma add_mult_distrib2: "k * (m + n) = (k * m) + (k * n)" for k m n :: nat
+lemma add_mult_distrib2: "k * (m + n) = (k * m) + (k * n)"
+ for k m n :: nat
by (fact distrib_left)
lemmas nat_distrib =
--- a/src/HOL/Rings.thy Tue Aug 02 21:04:52 2016 +0200
+++ b/src/HOL/Rings.thy Tue Aug 02 21:05:34 2016 +0200
@@ -10,7 +10,7 @@
section \<open>Rings\<close>
theory Rings
-imports Groups Set
+ imports Groups Set
begin
class semiring = ab_semigroup_add + semigroup_mult +
@@ -42,10 +42,14 @@
subclass semiring_0
proof
fix a :: 'a
- have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])
- then show "0 * a = 0" by (simp only: add_left_cancel)
- have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
- then show "a * 0 = 0" by (simp only: add_left_cancel)
+ have "0 * a + 0 * a = 0 * a + 0"
+ by (simp add: distrib_right [symmetric])
+ then show "0 * a = 0"
+ by (simp only: add_left_cancel)
+ have "a * 0 + a * 0 = a * 0 + 0"
+ by (simp add: distrib_left [symmetric])
+ then show "a * 0 = 0"
+ by (simp only: add_left_cancel)
qed
end
@@ -57,11 +61,16 @@
subclass semiring
proof
fix a b c :: 'a
- show "(a + b) * c = a * c + b * c" by (simp add: distrib)
- have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)
- also have "\<dots> = b * a + c * a" by (simp only: distrib)
- also have "\<dots> = a * b + a * c" by (simp add: ac_simps)
- finally show "a * (b + c) = a * b + a * c" by blast
+ show "(a + b) * c = a * c + b * c"
+ by (simp add: distrib)
+ have "a * (b + c) = (b + c) * a"
+ by (simp add: ac_simps)
+ also have "\<dots> = b * a + c * a"
+ by (simp only: distrib)
+ also have "\<dots> = a * b + a * c"
+ by (simp add: ac_simps)
+ finally show "a * (b + c) = a * b + a * c"
+ by blast
qed
end
@@ -140,9 +149,12 @@
assumes "a dvd b" and "b dvd c"
shows "a dvd c"
proof -
- from assms obtain v where "b = a * v" by (auto elim!: dvdE)
- moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
- ultimately have "c = a * (v * w)" by (simp add: mult.assoc)
+ from assms obtain v where "b = a * v"
+ by (auto elim!: dvdE)
+ moreover from assms obtain w where "c = b * w"
+ by (auto elim!: dvdE)
+ ultimately have "c = a * (v * w)"
+ by (simp add: mult.assoc)
then show ?thesis ..
qed
@@ -174,7 +186,8 @@
proof -
from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
moreover from \<open>c dvd d\<close> obtain d' where "d = c * d'" ..
- ultimately have "b * d = (a * c) * (b' * d')" by (simp add: ac_simps)
+ ultimately have "b * d = (a * c) * (b' * d')"
+ by (simp add: ac_simps)
then show ?thesis ..
qed
@@ -208,7 +221,8 @@
proof -
from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
moreover from \<open>a dvd c\<close> obtain c' where "c = a * c'" ..
- ultimately have "b + c = a * (b' + c')" by (simp add: distrib_left)
+ ultimately have "b + c = a * (b' + c')"
+ by (simp add: distrib_left)
then show ?thesis ..
qed
@@ -571,13 +585,13 @@
then show ?thesis by simp
next
case False
- {
- assume "a * c = b * c"
- then have "a * c div c = b * c div c"
+ have "a = b" if "a * c = b * c"
+ proof -
+ from that have "a * c div c = b * c div c"
by simp
- with False have "a = b"
+ with False show ?thesis
by simp
- }
+ qed
then show ?thesis by auto
qed
show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" for a b c
@@ -617,22 +631,23 @@
lemma dvd_times_left_cancel_iff [simp]:
assumes "a \<noteq> 0"
- shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
+ shows "a * b dvd a * c \<longleftrightarrow> b dvd c"
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof
- assume ?P
+ assume ?lhs
then obtain d where "a * c = a * b * d" ..
with assms have "c = b * d" by (simp add: ac_simps)
- then show ?Q ..
+ then show ?rhs ..
next
- assume ?Q
+ assume ?rhs
then obtain d where "c = b * d" ..
then have "a * c = a * b * d" by (simp add: ac_simps)
- then show ?P ..
+ then show ?lhs ..
qed
lemma dvd_times_right_cancel_iff [simp]:
assumes "a \<noteq> 0"
- shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
+ shows "b * a dvd c * a \<longleftrightarrow> b dvd c"
using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
lemma div_dvd_iff_mult:
@@ -698,15 +713,16 @@
lemma dvd_div_eq_mult:
assumes "a \<noteq> 0" and "a dvd b"
shows "b div a = c \<longleftrightarrow> b = c * a"
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof
- assume "b = c * a"
- then show "b div a = c" by (simp add: assms)
+ assume ?rhs
+ then show ?lhs by (simp add: assms)
next
- assume "b div a = c"
+ assume ?lhs
then have "b div a * a = c * a" by simp
moreover from assms have "b div a * a = b"
by (auto elim!: dvdE simp add: ac_simps)
- ultimately show "b = c * a" by simp
+ ultimately show ?rhs by simp
qed
lemma dvd_div_mult_self [simp]: "a dvd b \<Longrightarrow> b div a * a = b"
@@ -743,16 +759,17 @@
lemma dvd_div_div_eq_mult:
assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
- shows "b div a = d div c \<longleftrightarrow> b * c = a * d" (is "?P \<longleftrightarrow> ?Q")
+ shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof -
from assms have "a * c \<noteq> 0" by simp
- then have "?P \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)"
+ then have "?lhs \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)"
by simp
also have "\<dots> \<longleftrightarrow> (a * (b div a)) * c = (c * (d div c)) * a"
by (simp add: ac_simps)
also have "\<dots> \<longleftrightarrow> (a * b div a) * c = (c * d div c) * a"
using assms by (simp add: div_mult_swap)
- also have "\<dots> \<longleftrightarrow> ?Q"
+ also have "\<dots> \<longleftrightarrow> ?rhs"
using assms by (simp add: ac_simps)
finally show ?thesis .
qed
@@ -765,7 +782,8 @@
next
case False
from \<open>a * c dvd b\<close> obtain d where "b = a * c * d" ..
- with False show ?thesis by (simp add: mult.commute [of a] mult.assoc)
+ with False show ?thesis
+ by (simp add: mult.commute [of a] mult.assoc)
qed
@@ -943,23 +961,20 @@
fixes normalize :: "'a \<Rightarrow> 'a"
and unit_factor :: "'a \<Rightarrow> 'a"
assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a"
- assumes normalize_0 [simp]: "normalize 0 = 0"
+ and normalize_0 [simp]: "normalize 0 = 0"
and unit_factor_0 [simp]: "unit_factor 0 = 0"
- assumes is_unit_normalize:
- "is_unit a \<Longrightarrow> normalize a = 1"
- assumes unit_factor_is_unit [iff]:
- "a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)"
- assumes unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b"
+ and is_unit_normalize: "is_unit a \<Longrightarrow> normalize a = 1"
+ and unit_factor_is_unit [iff]: "a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)"
+ and unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b"
begin
text \<open>
- Class @{class normalization_semidom} cultivates the idea that
- each integral domain can be split into equivalence classes
- whose representants are associated, i.e. divide each other.
- @{const normalize} specifies a canonical representant for each equivalence
- class. The rationale behind this is that it is easier to reason about equality
- than equivalences, hence we prefer to think about equality of normalized
- values rather than associated elements.
+ Class @{class normalization_semidom} cultivates the idea that each integral
+ domain can be split into equivalence classes whose representants are
+ associated, i.e. divide each other. @{const normalize} specifies a canonical
+ representant for each equivalence class. The rationale behind this is that
+ it is easier to reason about equality than equivalences, hence we prefer to
+ think about equality of normalized values rather than associated elements.
\<close>
lemma unit_factor_dvd [simp]: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
@@ -972,25 +987,25 @@
using unit_factor_mult_normalize [of a] by (simp add: ac_simps)
lemma normalize_eq_0_iff [simp]: "normalize a = 0 \<longleftrightarrow> a = 0"
- (is "?P \<longleftrightarrow> ?Q")
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof
- assume ?P
+ assume ?lhs
moreover have "unit_factor a * normalize a = a" by simp
- ultimately show ?Q by simp
+ ultimately show ?rhs by simp
next
- assume ?Q
- then show ?P by simp
+ assume ?rhs
+ then show ?lhs by simp
qed
lemma unit_factor_eq_0_iff [simp]: "unit_factor a = 0 \<longleftrightarrow> a = 0"
- (is "?P \<longleftrightarrow> ?Q")
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof
- assume ?P
+ assume ?lhs
moreover have "unit_factor a * normalize a = a" by simp
- ultimately show ?Q by simp
+ ultimately show ?rhs by simp
next
- assume ?Q
- then show ?P by simp
+ assume ?rhs
+ then show ?lhs by simp
qed
lemma is_unit_unit_factor:
@@ -1009,20 +1024,20 @@
by (rule is_unit_normalize) simp
lemma normalize_1_iff: "normalize a = 1 \<longleftrightarrow> is_unit a"
- (is "?P \<longleftrightarrow> ?Q")
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof
- assume ?Q
- then show ?P by (rule is_unit_normalize)
+ assume ?rhs
+ then show ?lhs by (rule is_unit_normalize)
next
- assume ?P
- then have "a \<noteq> 0" by auto
- from \<open>?P\<close> have "unit_factor a * normalize a = unit_factor a * 1"
+ assume ?lhs
+ then have "unit_factor a * normalize a = unit_factor a * 1"
by simp
then have "unit_factor a = a"
by simp
- moreover have "is_unit (unit_factor a)"
- using \<open>a \<noteq> 0\<close> by simp
- ultimately show ?Q by simp
+ moreover
+ from \<open>?lhs\<close> have "a \<noteq> 0" by auto
+ then have "is_unit (unit_factor a)" by simp
+ ultimately show ?rhs by simp
qed
lemma div_normalize [simp]: "a div normalize a = unit_factor a"
@@ -1045,7 +1060,8 @@
case False
then have "unit_factor a \<noteq> 0" by simp
with nonzero_mult_divide_cancel_left
- have "unit_factor a * normalize a div unit_factor a = normalize a" by blast
+ have "unit_factor a * normalize a div unit_factor a = normalize a"
+ by blast
then show ?thesis by simp
qed
@@ -1071,7 +1087,8 @@
then show ?thesis by auto
next
case False
- from unit_factor_mult_normalize have "unit_factor (a * b) * normalize (a * b) = a * b" .
+ have "unit_factor (a * b) * normalize (a * b) = a * b"
+ by (rule unit_factor_mult_normalize)
then have "normalize (a * b) = a * b div unit_factor (a * b)"
by simp
also have "\<dots> = a * b div unit_factor (b * a)"
@@ -1163,11 +1180,11 @@
qed
text \<open>
- We avoid an explicit definition of associated elements but prefer
- explicit normalisation instead. In theory we could define an abbreviation
- like @{prop "associated a b \<longleftrightarrow> normalize a = normalize b"} but this is
- counterproductive without suggestive infix syntax, which we do not want
- to sacrifice for this purpose here.
+ We avoid an explicit definition of associated elements but prefer explicit
+ normalisation instead. In theory we could define an abbreviation like @{prop
+ "associated a b \<longleftrightarrow> normalize a = normalize b"} but this is counterproductive
+ without suggestive infix syntax, which we do not want to sacrifice for this
+ purpose here.
\<close>
lemma associatedI:
@@ -1202,20 +1219,20 @@
using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
lemma associated_iff_dvd: "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a"
- (is "?P \<longleftrightarrow> ?Q")
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof
- assume ?Q
- then show ?P by (auto intro!: associatedI)
+ assume ?rhs
+ then show ?lhs by (auto intro!: associatedI)
next
- assume ?P
+ assume ?lhs
then have "unit_factor a * normalize a = unit_factor a * normalize b"
by simp
then have *: "normalize b * unit_factor a = a"
by (simp add: ac_simps)
- show ?Q
+ show ?rhs
proof (cases "a = 0 \<or> b = 0")
case True
- with \<open>?P\<close> show ?thesis by auto
+ with \<open>?lhs\<close> show ?thesis by auto
next
case False
then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b"
@@ -1231,8 +1248,10 @@
proof -
from assms have "normalize a = normalize b"
unfolding associated_iff_dvd by simp
- with \<open>normalize a = a\<close> have "a = normalize b" by simp
- with \<open>normalize b = b\<close> show "a = b" by simp
+ with \<open>normalize a = a\<close> have "a = normalize b"
+ by simp
+ with \<open>normalize b = b\<close> show "a = b"
+ by simp
qed
end
@@ -1248,9 +1267,7 @@
done
lemma mult_mono': "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
- apply (rule mult_mono)
- apply (fast intro: order_trans)+
- done
+ by (rule mult_mono) (fast intro: order_trans)+
end
@@ -1266,11 +1283,9 @@
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
using mult_right_mono [of a 0 b] by simp
-text \<open>Legacy - use \<open>mult_nonpos_nonneg\<close>\<close>
+text \<open>Legacy -- use @{thm [source] mult_nonpos_nonneg}.\<close>
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
- apply (drule mult_right_mono [of b 0])
- apply auto
- done
+ by (drule mult_right_mono [of b 0]) auto
lemma split_mult_neg_le: "(0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
@@ -1281,6 +1296,7 @@
begin
subclass semiring_0_cancel ..
+
subclass ordered_semiring_0 ..
end
@@ -1327,11 +1343,11 @@
subclass linordered_semiring
proof
fix a b c :: 'a
- assume A: "a \<le> b" "0 \<le> c"
- from A show "c * a \<le> c * b"
+ assume *: "a \<le> b" "0 \<le> c"
+ then show "c * a \<le> c * b"
unfolding le_less
using mult_strict_left_mono by (cases "c = 0") auto
- from A show "a * c \<le> b * c"
+ from * show "a * c \<le> b * c"
unfolding le_less
using mult_strict_right_mono by (cases "c = 0") auto
qed
@@ -1351,11 +1367,9 @@
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
using mult_strict_right_mono [of a 0 b] by simp
-text \<open>Legacy - use \<open>mult_neg_pos\<close>\<close>
+text \<open>Legacy -- use @{thm [source] mult_neg_pos}.\<close>
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
- apply (drule mult_strict_right_mono [of b 0])
- apply auto
- done
+ by (drule mult_strict_right_mono [of b 0]) auto
lemma zero_less_mult_pos: "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
apply (cases "b \<le> 0")
@@ -1377,9 +1391,9 @@
shows "a * c < b * d"
using assms
apply (cases "c = 0")
- apply simp
+ apply simp
apply (erule mult_strict_right_mono [THEN less_trans])
- apply (auto simp add: le_less)
+ apply (auto simp add: le_less)
apply (erule (1) mult_strict_left_mono)
done
@@ -1394,9 +1408,9 @@
shows "a * c < b * d"
using assms
apply (subgoal_tac "a * c < b * c")
- apply (erule less_le_trans)
- apply (erule mult_left_mono)
- apply simp
+ apply (erule less_le_trans)
+ apply (erule mult_left_mono)
+ apply simp
apply (erule (1) mult_strict_right_mono)
done
@@ -1405,9 +1419,9 @@
shows "a * c < b * d"
using assms
apply (subgoal_tac "a * c \<le> b * c")
- apply (erule le_less_trans)
- apply (erule mult_strict_left_mono)
- apply simp
+ apply (erule le_less_trans)
+ apply (erule mult_strict_left_mono)
+ apply simp
apply (erule (1) mult_right_mono)
done
@@ -1461,8 +1475,10 @@
proof
fix a b c :: 'a
assume "a < b" "0 < c"
- then show "c * a < c * b" by (rule comm_mult_strict_left_mono)
- then show "a * c < b * c" by (simp only: mult.commute)
+ then show "c * a < c * b"
+ by (rule comm_mult_strict_left_mono)
+ then show "a * c < b * c"
+ by (simp only: mult.commute)
qed
subclass ordered_cancel_comm_semiring
@@ -1522,7 +1538,7 @@
show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
by (auto simp add: abs_if not_le not_less algebra_simps
simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
-qed (auto simp add: abs_if)
+qed (auto simp: abs_if)
lemma zero_le_square [simp]: "0 \<le> a * a"
using linear [of 0 a] by (auto simp add: mult_nonpos_nonpos)
@@ -1560,33 +1576,33 @@
proof
fix a b
assume "a \<noteq> 0"
- then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
+ then have a: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
assume "b \<noteq> 0"
- then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
+ then have b: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
have "a * b < 0 \<or> 0 < a * b"
proof (cases "a < 0")
- case A': True
+ case True
show ?thesis
proof (cases "b < 0")
case True
- with A' show ?thesis by (auto dest: mult_neg_neg)
+ with \<open>a < 0\<close> show ?thesis by (auto dest: mult_neg_neg)
next
case False
- with B have "0 < b" by auto
- with A' show ?thesis by (auto dest: mult_strict_right_mono)
+ with b have "0 < b" by auto
+ with \<open>a < 0\<close> show ?thesis by (auto dest: mult_strict_right_mono)
qed
next
case False
- with A have A': "0 < a" by auto
+ with a have "0 < a" by auto
show ?thesis
proof (cases "b < 0")
case True
- with A' show ?thesis
+ with \<open>0 < a\<close> show ?thesis
by (auto dest: mult_strict_right_mono_neg)
next
case False
- with B have "0 < b" by auto
- with A' show ?thesis by auto
+ with b have "0 < b" by auto
+ with \<open>0 < a\<close> show ?thesis by auto
qed
qed
then show "a * b \<noteq> 0"
@@ -1618,18 +1634,18 @@
lemma mult_less_cancel_right_disj: "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a"
apply (cases "c = 0")
- apply (auto simp add: neq_iff mult_strict_right_mono mult_strict_right_mono_neg)
- apply (auto simp add: not_less not_le [symmetric, of "a*c"] not_le [symmetric, of a])
- apply (erule_tac [!] notE)
- apply (auto simp add: less_imp_le mult_right_mono mult_right_mono_neg)
+ apply (auto simp add: neq_iff mult_strict_right_mono mult_strict_right_mono_neg)
+ apply (auto simp add: not_less not_le [symmetric, of "a*c"] not_le [symmetric, of a])
+ apply (erule_tac [!] notE)
+ apply (auto simp add: less_imp_le mult_right_mono mult_right_mono_neg)
done
lemma mult_less_cancel_left_disj: "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a"
apply (cases "c = 0")
- apply (auto simp add: neq_iff mult_strict_left_mono mult_strict_left_mono_neg)
- apply (auto simp add: not_less not_le [symmetric, of "c * a"] not_le [symmetric, of a])
- apply (erule_tac [!] notE)
- apply (auto simp add: less_imp_le mult_left_mono mult_left_mono_neg)
+ apply (auto simp add: neq_iff mult_strict_left_mono mult_strict_left_mono_neg)
+ apply (auto simp add: not_less not_le [symmetric, of "c * a"] not_le [symmetric, of a])
+ apply (erule_tac [!] notE)
+ apply (auto simp add: less_imp_le mult_left_mono mult_left_mono_neg)
done
text \<open>
@@ -1727,7 +1743,8 @@
apply (subst add_le_cancel_right [where c=k, symmetric])
apply (frule le_add_diff_inverse2)
apply (simp only: add.assoc [symmetric])
- using add_implies_diff apply fastforce
+ using add_implies_diff
+ apply fastforce
done
lemma add_le_add_imp_diff_le:
@@ -1765,8 +1782,7 @@
proof
have "0 \<le> 1 * 1" by (rule zero_le_square)
then show "0 < 1" by (simp add: le_less)
- show "b \<le> a \<Longrightarrow> a - b + b = a" for a b
- by simp
+ show "b \<le> a \<Longrightarrow> a - b + b = a" for a b by simp
qed
lemma linorder_neqE_linordered_idom:
@@ -1774,7 +1790,7 @@
obtains "x < y" | "y < x"
using assms by (rule neqE)
-text \<open>These cancellation simprules also produce two cases when the comparison is a goal.\<close>
+text \<open>These cancellation simp rules also produce two cases when the comparison is a goal.\<close>
lemma mult_le_cancel_right1: "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
using mult_le_cancel_right [of 1 c b] by simp
@@ -1928,7 +1944,7 @@
begin
subclass ordered_ring_abs
- by standard (auto simp add: abs_if not_less mult_less_0_iff)
+ by standard (auto simp: abs_if not_less mult_less_0_iff)
lemma abs_mult: "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
by (rule abs_eq_mult) auto
--- a/src/HOL/Set.thy Tue Aug 02 21:04:52 2016 +0200
+++ b/src/HOL/Set.thy Tue Aug 02 21:05:34 2016 +0200
@@ -1,9 +1,13 @@
-(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel *)
+(* Title: HOL/Set.thy
+ Author: Tobias Nipkow
+ Author: Lawrence C Paulson
+ Author: Markus Wenzel
+*)
section \<open>Set theory for higher-order logic\<close>
theory Set
-imports Lattices
+ imports Lattices
begin
subsection \<open>Sets as predicates\<close>
@@ -12,8 +16,8 @@
axiomatization Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set" \<comment> "comprehension"
and member :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" \<comment> "membership"
-where mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a"
- and Collect_mem_eq [simp]: "Collect (\<lambda>x. member x A) = A"
+ where mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a"
+ and Collect_mem_eq [simp]: "Collect (\<lambda>x. member x A) = A"
notation
member ("op \<in>") and
@@ -76,7 +80,8 @@
assumes "\<And>x. x \<in> A \<longleftrightarrow> x \<in> B"
shows "A = B"
proof -
- from assms have "{x. x \<in> A} = {x. x \<in> B}" by simp
+ from assms have "{x. x \<in> A} = {x. x \<in> B}"
+ by simp
then show ?thesis by simp
qed
@@ -347,7 +352,6 @@
by (simp add: Ball_def)
text \<open>Gives better instantiation for bound:\<close>
-
setup \<open>
map_theory_claset (fn ctxt =>
ctxt addbefore ("bspec", fn ctxt' => dresolve_tac ctxt' @{thms bspec} THEN' assume_tac ctxt'))
@@ -459,7 +463,7 @@
\<comment> \<open>Rule in Modus Ponens style.\<close>
lemma rev_subsetD [intro?]: "c \<in> A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> c \<in> B"
- \<comment> \<open>The same, with reversed premises for use with \<open>erule\<close> -- cf. \<open>rev_mp\<close>.\<close>
+ \<comment> \<open>The same, with reversed premises for use with @{method erule} -- cf. @{thm rev_mp}.\<close>
by (rule subsetD)
lemma subsetCE [elim]: "A \<subseteq> B \<Longrightarrow> (c \<notin> A \<Longrightarrow> P) \<Longrightarrow> (c \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
@@ -696,8 +700,7 @@
lemma UnI2 [elim?]: "c \<in> B \<Longrightarrow> c \<in> A \<union> B"
by simp
-text \<open>\<^medskip> Classical introduction rule: no commitment to @{prop A} vs @{prop B}.\<close>
-
+text \<open>\<^medskip> Classical introduction rule: no commitment to \<open>A\<close> vs. \<open>B\<close>.\<close>
lemma UnCI [intro!]: "(c \<notin> B \<Longrightarrow> c \<in> A) \<Longrightarrow> c \<in> A \<union> B"
by auto
@@ -960,7 +963,7 @@
text \<open>\<^medskip> Range of a function -- just an abbreviation for image!\<close>
-abbreviation range :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set" \<comment> "of function"
+abbreviation range :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set" \<comment> \<open>of function\<close>
where "range f \<equiv> f ` UNIV"
lemma range_eqI: "b = f x \<Longrightarrow> b \<in> range f"
@@ -1387,7 +1390,8 @@
lemma Compl_subset_Compl_iff [iff]: "- A \<subseteq> - B \<longleftrightarrow> B \<subseteq> A"
by (fact compl_le_compl_iff) (* FIXME: already simp *)
-lemma Compl_eq_Compl_iff [iff]: "- A = - B \<longleftrightarrow> A = B" for A B :: "'a set"
+lemma Compl_eq_Compl_iff [iff]: "- A = - B \<longleftrightarrow> A = B"
+ for A B :: "'a set"
by (fact compl_eq_compl_iff) (* FIXME: already simp *)
lemma Compl_insert: "- insert x A = (- A) - {x}"
@@ -1417,7 +1421,8 @@
lemma Diff_cancel [simp]: "A - A = {}"
by blast
-lemma Diff_idemp [simp]: "(A - B) - B = A - B" for A B :: "'a set"
+lemma Diff_idemp [simp]: "(A - B) - B = A - B"
+ for A B :: "'a set"
by blast
lemma Diff_triv: "A \<inter> B = {} \<Longrightarrow> A - B = A"
@@ -1526,7 +1531,7 @@
by (auto simp add: Pow_def)
lemma Pow_singleton_iff [simp]: "Pow X = {Y} \<longleftrightarrow> X = {} \<and> Y = {}"
- by blast
+ by blast (* somewhat slow *)
lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)"
by (blast intro: image_eqI [where ?x = "u - {a}" for u])
@@ -1612,9 +1617,7 @@
text \<open>\<^medskip> Monotonicity of implications.\<close>
lemma in_mono: "A \<subseteq> B \<Longrightarrow> x \<in> A \<longrightarrow> x \<in> B"
- apply (rule impI)
- apply (erule subsetD, assumption)
- done
+ by (rule impI) (erule subsetD)
lemma conj_mono: "P1 \<longrightarrow> Q1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<and> P2) \<longrightarrow> (Q1 \<and> Q2)"
by iprover
@@ -1730,7 +1733,7 @@
lemma vimage_ident [simp]: "(\<lambda>x. x) -` Y = Y"
by blast
-
+
subsubsection \<open>Singleton sets\<close>
definition is_singleton :: "'a set \<Rightarrow> bool"
@@ -1778,9 +1781,9 @@
\<comment> \<open>Courtesy of Stephan Merz\<close>
apply clarify
apply (erule_tac P = "\<lambda>x. x : S" in LeastI2_order)
- apply fast
+ apply fast
apply (rule LeastI2_order)
- apply (auto elim: monoD intro!: order_antisym)
+ apply (auto elim: monoD intro!: order_antisym)
done
@@ -1791,20 +1794,21 @@
hide_const (open) bind
-lemma bind_bind: "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)" for A :: "'a set"
- by (auto simp add: bind_def)
+lemma bind_bind: "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)"
+ for A :: "'a set"
+ by (auto simp: bind_def)
lemma empty_bind [simp]: "Set.bind {} f = {}"
by (simp add: bind_def)
lemma nonempty_bind_const: "A \<noteq> {} \<Longrightarrow> Set.bind A (\<lambda>_. B) = B"
- by (auto simp add: bind_def)
+ by (auto simp: bind_def)
lemma bind_const: "Set.bind A (\<lambda>_. B) = (if A = {} then {} else B)"
- by (auto simp add: bind_def)
+ by (auto simp: bind_def)
lemma bind_singleton_conv_image: "Set.bind A (\<lambda>x. {f x}) = f ` A"
- by(auto simp add: bind_def)
+ by (auto simp: bind_def)
subsubsection \<open>Operations for execution\<close>
@@ -1842,12 +1846,14 @@
text \<open>Misc\<close>
-definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x \<noteq> y \<longrightarrow> R x y)"
+definition pairwise :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
+ where "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x \<noteq> y \<longrightarrow> R x y)"
lemma pairwise_subset: "pairwise P S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> pairwise P T"
by (force simp: pairwise_def)
-definition disjnt where "disjnt A B \<longleftrightarrow> A \<inter> B = {}"
+definition disjnt :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
+ where "disjnt A B \<longleftrightarrow> A \<inter> B = {}"
lemma disjnt_iff: "disjnt A B \<longleftrightarrow> (\<forall>x. \<not> (x \<in> A \<and> x \<in> B))"
by (force simp: disjnt_def)