1.1 --- a/src/HOL/Equiv_Relations.thy Thu Aug 18 17:42:35 2011 +0200
1.2 +++ b/src/HOL/Equiv_Relations.thy Thu Aug 18 22:50:28 2011 +0200
1.3 @@ -164,7 +164,7 @@
1.4
1.5 text{*A congruence-preserving function*}
1.6
1.7 -definition congruent :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" where
1.8 +definition congruent :: "('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" where
1.9 "congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"
1.10
1.11 lemma congruentI:
1.12 @@ -229,7 +229,7 @@
1.13
1.14 text{*A congruence-preserving function of two arguments*}
1.15
1.16 -definition congruent2 :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool" where
1.17 +definition congruent2 :: "('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool" where
1.18 "congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"
1.19
1.20 lemma congruent2I':
2.1 --- a/src/HOL/Fun.thy Thu Aug 18 17:42:35 2011 +0200
2.2 +++ b/src/HOL/Fun.thy Thu Aug 18 22:50:28 2011 +0200
2.3 @@ -10,15 +10,6 @@
2.4 uses ("Tools/enriched_type.ML")
2.5 begin
2.6
2.7 -text{*As a simplification rule, it replaces all function equalities by
2.8 - first-order equalities.*}
2.9 -lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
2.10 -apply (rule iffI)
2.11 -apply (simp (no_asm_simp))
2.12 -apply (rule ext)
2.13 -apply (simp (no_asm_simp))
2.14 -done
2.15 -
2.16 lemma apply_inverse:
2.17 "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
2.18 by auto
2.19 @@ -26,26 +17,22 @@
2.20
2.21 subsection {* The Identity Function @{text id} *}
2.22
2.23 -definition
2.24 - id :: "'a \<Rightarrow> 'a"
2.25 -where
2.26 +definition id :: "'a \<Rightarrow> 'a" where
2.27 "id = (\<lambda>x. x)"
2.28
2.29 lemma id_apply [simp]: "id x = x"
2.30 by (simp add: id_def)
2.31
2.32 lemma image_id [simp]: "id ` Y = Y"
2.33 -by (simp add: id_def)
2.34 + by (simp add: id_def)
2.35
2.36 lemma vimage_id [simp]: "id -` A = A"
2.37 -by (simp add: id_def)
2.38 + by (simp add: id_def)
2.39
2.40
2.41 subsection {* The Composition Operator @{text "f \<circ> g"} *}
2.42
2.43 -definition
2.44 - comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)
2.45 -where
2.46 +definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55) where
2.47 "f o g = (\<lambda>x. f (g x))"
2.48
2.49 notation (xsymbols)
2.50 @@ -54,9 +41,6 @@
2.51 notation (HTML output)
2.52 comp (infixl "\<circ>" 55)
2.53
2.54 -text{*compatibility*}
2.55 -lemmas o_def = comp_def
2.56 -
2.57 lemma o_apply [simp]: "(f o g) x = f (g x)"
2.58 by (simp add: comp_def)
2.59
2.60 @@ -71,7 +55,7 @@
2.61
2.62 lemma o_eq_dest:
2.63 "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
2.64 - by (simp only: o_def) (fact fun_cong)
2.65 + by (simp only: comp_def) (fact fun_cong)
2.66
2.67 lemma o_eq_elim:
2.68 "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
2.69 @@ -89,9 +73,7 @@
2.70
2.71 subsection {* The Forward Composition Operator @{text fcomp} *}
2.72
2.73 -definition
2.74 - fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60)
2.75 -where
2.76 +definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) where
2.77 "f \<circ>> g = (\<lambda>x. g (f x))"
2.78
2.79 lemma fcomp_apply [simp]: "(f \<circ>> g) x = g (f x)"
2.80 @@ -569,8 +551,7 @@
2.81
2.82 subsection{*Function Updating*}
2.83
2.84 -definition
2.85 - fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
2.86 +definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
2.87 "fun_upd f a b == % x. if x=a then b else f x"
2.88
2.89 nonterminal updbinds and updbind
2.90 @@ -634,9 +615,7 @@
2.91
2.92 subsection {* @{text override_on} *}
2.93
2.94 -definition
2.95 - override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
2.96 -where
2.97 +definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where
2.98 "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
2.99
2.100 lemma override_on_emptyset[simp]: "override_on f g {} = f"
2.101 @@ -651,9 +630,7 @@
2.102
2.103 subsection {* @{text swap} *}
2.104
2.105 -definition
2.106 - swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
2.107 -where
2.108 +definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" where
2.109 "swap a b f = f (a := f b, b:= f a)"
2.110
2.111 lemma swap_self [simp]: "swap a a f = f"
2.112 @@ -716,7 +693,7 @@
2.113 subsection {* Inversion of injective functions *}
2.114
2.115 definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
2.116 -"the_inv_into A f == %x. THE y. y : A & f y = x"
2.117 + "the_inv_into A f == %x. THE y. y : A & f y = x"
2.118
2.119 lemma the_inv_into_f_f:
2.120 "[| inj_on f A; x : A |] ==> the_inv_into A f (f x) = x"
2.121 @@ -775,6 +752,11 @@
2.122 shows "the_inv f (f x) = x" using assms UNIV_I
2.123 by (rule the_inv_into_f_f)
2.124
2.125 +
2.126 +text{*compatibility*}
2.127 +lemmas o_def = comp_def
2.128 +
2.129 +
2.130 subsection {* Cantor's Paradox *}
2.131
2.132 lemma Cantors_paradox [no_atp]:
3.1 --- a/src/HOL/GCD.thy Thu Aug 18 17:42:35 2011 +0200
3.2 +++ b/src/HOL/GCD.thy Thu Aug 18 22:50:28 2011 +0200
3.3 @@ -531,11 +531,8 @@
3.4
3.5 lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = abs n"
3.6 apply(rule antisym)
3.7 - apply(rule Max_le_iff[THEN iffD2])
3.8 - apply simp
3.9 - apply fastsimp
3.10 - apply (metis Collect_def abs_ge_self dvd_imp_le_int mem_def zle_trans)
3.11 -apply simp
3.12 + apply(rule Max_le_iff [THEN iffD2])
3.13 + apply (auto intro: abs_le_D1 dvd_imp_le_int)
3.14 done
3.15
3.16 lemma gcd_is_Max_divisors_nat:
4.1 --- a/src/HOL/HOL.thy Thu Aug 18 17:42:35 2011 +0200
4.2 +++ b/src/HOL/HOL.thy Thu Aug 18 22:50:28 2011 +0200
4.3 @@ -1394,6 +1394,11 @@
4.4 "f (if c then x else y) = (if c then f x else f y)"
4.5 by simp
4.6
4.7 +text{*As a simplification rule, it replaces all function equalities by
4.8 + first-order equalities.*}
4.9 +lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
4.10 + by auto
4.11 +
4.12
4.13 subsubsection {* Generic cases and induction *}
4.14
5.1 --- a/src/HOL/IsaMakefile Thu Aug 18 17:42:35 2011 +0200
5.2 +++ b/src/HOL/IsaMakefile Thu Aug 18 22:50:28 2011 +0200
5.3 @@ -1039,32 +1039,30 @@
5.4
5.5 $(LOG)/HOL-ex.gz: $(OUT)/HOL Decision_Procs/Commutative_Ring.thy \
5.6 Number_Theory/Primes.thy ex/Abstract_NAT.thy ex/Antiquote.thy \
5.7 - ex/Arith_Examples.thy ex/Arithmetic_Series_Complex.thy \
5.8 - ex/BT.thy ex/BinEx.thy ex/Binary.thy ex/Birthday_Paradox.thy \
5.9 - ex/CTL.thy ex/Case_Product.thy \
5.10 - ex/Chinese.thy ex/Classical.thy ex/CodegenSML_Test.thy \
5.11 - ex/Coercion_Examples.thy ex/Coherent.thy ex/Dedekind_Real.thy \
5.12 - ex/Efficient_Nat_examples.thy ex/Eval_Examples.thy ex/Fundefs.thy \
5.13 - ex/Gauge_Integration.thy ex/Groebner_Examples.thy ex/Guess.thy \
5.14 - ex/HarmonicSeries.thy ex/Hebrew.thy ex/Hex_Bin_Examples.thy \
5.15 - ex/Higher_Order_Logic.thy ex/Iff_Oracle.thy ex/Induction_Schema.thy \
5.16 + ex/Arith_Examples.thy ex/Arithmetic_Series_Complex.thy ex/BT.thy \
5.17 + ex/BinEx.thy ex/Binary.thy ex/Birthday_Paradox.thy ex/CTL.thy \
5.18 + ex/Case_Product.thy ex/Chinese.thy ex/Classical.thy \
5.19 + ex/CodegenSML_Test.thy ex/Coercion_Examples.thy ex/Coherent.thy \
5.20 + ex/Dedekind_Real.thy ex/Efficient_Nat_examples.thy \
5.21 + ex/Eval_Examples.thy ex/Fundefs.thy ex/Gauge_Integration.thy \
5.22 + ex/Groebner_Examples.thy ex/Guess.thy ex/HarmonicSeries.thy \
5.23 + ex/Hebrew.thy ex/Hex_Bin_Examples.thy ex/Higher_Order_Logic.thy \
5.24 + ex/Iff_Oracle.thy ex/Induction_Schema.thy \
5.25 ex/Interpretation_with_Defs.thy ex/Intuitionistic.thy ex/Lagrange.thy \
5.26 ex/List_to_Set_Comprehension_Examples.thy ex/LocaleTest2.thy \
5.27 ex/MT.thy ex/MergeSort.thy ex/Meson_Test.thy ex/MonoidGroup.thy \
5.28 ex/Multiquote.thy ex/NatSum.thy ex/Normalization_by_Evaluation.thy \
5.29 ex/Numeral.thy ex/PER.thy ex/PresburgerEx.thy ex/Primrec.thy \
5.30 ex/Quickcheck_Examples.thy ex/Quickcheck_Lattice_Examples.thy \
5.31 - ex/Quickcheck_Narrowing_Examples.thy \
5.32 - ex/Quicksort.thy ex/ROOT.ML ex/Records.thy \
5.33 - ex/ReflectionEx.thy ex/Refute_Examples.thy ex/SAT_Examples.thy \
5.34 - ex/SVC_Oracle.thy ex/Serbian.thy ex/Set_Algebras.thy \
5.35 - ex/sledgehammer_tactics.ML ex/Sqrt.thy \
5.36 - ex/Sqrt_Script.thy ex/Sudoku.thy ex/Tarski.thy ex/Termination.thy \
5.37 - ex/Transfer_Ex.thy ex/Tree23.thy \
5.38 + ex/Quickcheck_Narrowing_Examples.thy ex/Quicksort.thy ex/ROOT.ML \
5.39 + ex/Records.thy ex/ReflectionEx.thy ex/Refute_Examples.thy \
5.40 + ex/SAT_Examples.thy ex/Serbian.thy ex/Set_Theory.thy \
5.41 + ex/Set_Algebras.thy ex/SVC_Oracle.thy ex/sledgehammer_tactics.ML \
5.42 + ex/Sqrt.thy ex/Sqrt_Script.thy ex/Sudoku.thy ex/Tarski.thy \
5.43 + ex/Termination.thy ex/Transfer_Ex.thy ex/Tree23.thy \
5.44 ex/Unification.thy ex/While_Combinator_Example.thy \
5.45 - ex/document/root.bib ex/document/root.tex ex/set.thy ex/svc_funcs.ML \
5.46 - ex/svc_test.thy \
5.47 - ../Tools/interpretation_with_defs.ML
5.48 + ex/document/root.bib ex/document/root.tex ex/svc_funcs.ML \
5.49 + ex/svc_test.thy ../Tools/interpretation_with_defs.ML
5.50 @$(ISABELLE_TOOL) usedir $(OUT)/HOL ex
5.51
5.52
6.1 --- a/src/HOL/Nat.thy Thu Aug 18 17:42:35 2011 +0200
6.2 +++ b/src/HOL/Nat.thy Thu Aug 18 22:50:28 2011 +0200
6.3 @@ -39,11 +39,20 @@
6.4 Zero_RepI: "Nat Zero_Rep"
6.5 | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
6.6
6.7 -typedef (open Nat) nat = Nat
6.8 - by (rule exI, unfold mem_def, rule Nat.Zero_RepI)
6.9 +typedef (open Nat) nat = "{n. Nat n}"
6.10 + using Nat.Zero_RepI by auto
6.11
6.12 -definition Suc :: "nat => nat" where
6.13 - "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
6.14 +lemma Nat_Rep_Nat:
6.15 + "Nat (Rep_Nat n)"
6.16 + using Rep_Nat by simp
6.17 +
6.18 +lemma Nat_Abs_Nat_inverse:
6.19 + "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"
6.20 + using Abs_Nat_inverse by simp
6.21 +
6.22 +lemma Nat_Abs_Nat_inject:
6.23 + "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"
6.24 + using Abs_Nat_inject by simp
6.25
6.26 instantiation nat :: zero
6.27 begin
6.28 @@ -55,9 +64,11 @@
6.29
6.30 end
6.31
6.32 +definition Suc :: "nat \<Rightarrow> nat" where
6.33 + "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
6.34 +
6.35 lemma Suc_not_Zero: "Suc m \<noteq> 0"
6.36 - by (simp add: Zero_nat_def Suc_def Abs_Nat_inject [unfolded mem_def]
6.37 - Rep_Nat [unfolded mem_def] Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep [unfolded mem_def])
6.38 + by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat)
6.39
6.40 lemma Zero_not_Suc: "0 \<noteq> Suc m"
6.41 by (rule not_sym, rule Suc_not_Zero not_sym)
6.42 @@ -67,12 +78,12 @@
6.43
6.44 rep_datatype "0 \<Colon> nat" Suc
6.45 apply (unfold Zero_nat_def Suc_def)
6.46 - apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
6.47 - apply (erule Rep_Nat [unfolded mem_def, THEN Nat.induct])
6.48 - apply (iprover elim: Abs_Nat_inverse [unfolded mem_def, THEN subst])
6.49 - apply (simp_all add: Abs_Nat_inject [unfolded mem_def] Rep_Nat [unfolded mem_def]
6.50 - Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep [unfolded mem_def]
6.51 - Suc_Rep_not_Zero_Rep [unfolded mem_def, symmetric]
6.52 + apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
6.53 + apply (erule Nat_Rep_Nat [THEN Nat.induct])
6.54 + apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst])
6.55 + apply (simp_all add: Nat_Abs_Nat_inject Nat_Rep_Nat
6.56 + Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep
6.57 + Suc_Rep_not_Zero_Rep [symmetric]
6.58 Suc_Rep_inject' Rep_Nat_inject)
6.59 done
6.60
7.1 --- a/src/HOL/Nitpick.thy Thu Aug 18 17:42:35 2011 +0200
7.2 +++ b/src/HOL/Nitpick.thy Thu Aug 18 22:50:28 2011 +0200
7.3 @@ -76,19 +76,19 @@
7.4 definition prod :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
7.5 "prod A B = {(a, b). a \<in> A \<and> b \<in> B}"
7.6
7.7 -definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
7.8 +definition refl' :: "('a \<times> 'a) set \<Rightarrow> bool" where
7.9 "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
7.10
7.11 -definition wf' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
7.12 +definition wf' :: "('a \<times> 'a) set \<Rightarrow> bool" where
7.13 "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
7.14
7.15 -definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where
7.16 +definition card' :: "'a set \<Rightarrow> nat" where
7.17 "card' A \<equiv> if finite A then length (SOME xs. set xs = A \<and> distinct xs) else 0"
7.18
7.19 -definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where
7.20 +definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> 'a set \<Rightarrow> 'b" where
7.21 "setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
7.22
7.23 -inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where
7.24 +inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool" where
7.25 "fold_graph' f z {} z" |
7.26 "\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
7.27
8.1 --- a/src/HOL/RealDef.thy Thu Aug 18 17:42:35 2011 +0200
8.2 +++ b/src/HOL/RealDef.thy Thu Aug 18 22:50:28 2011 +0200
8.3 @@ -121,7 +121,7 @@
8.4 subsection {* Cauchy sequences *}
8.5
8.6 definition
8.7 - cauchy :: "(nat \<Rightarrow> rat) set"
8.8 + cauchy :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
8.9 where
8.10 "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
8.11
9.1 --- a/src/HOL/Relation.thy Thu Aug 18 17:42:35 2011 +0200
9.2 +++ b/src/HOL/Relation.thy Thu Aug 18 22:50:28 2011 +0200
9.3 @@ -133,9 +133,8 @@
9.4 by blast
9.5
9.6 lemma Id_on_def' [nitpick_unfold, code]:
9.7 - "(Id_on (A :: 'a => bool)) = (%(x, y). x = y \<and> A x)"
9.8 -by (auto simp add: fun_eq_iff
9.9 - elim: Id_onE[unfolded mem_def] intro: Id_onI[unfolded mem_def])
9.10 + "Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"
9.11 +by auto
9.12
9.13 lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
9.14 by blast
10.1 --- a/src/HOL/String.thy Thu Aug 18 17:42:35 2011 +0200
10.2 +++ b/src/HOL/String.thy Thu Aug 18 22:50:28 2011 +0200
10.3 @@ -155,7 +155,7 @@
10.4
10.5 subsection {* Strings as dedicated type *}
10.6
10.7 -typedef (open) literal = "UNIV :: string \<Rightarrow> bool"
10.8 +typedef (open) literal = "UNIV :: string set"
10.9 morphisms explode STR ..
10.10
10.11 instantiation literal :: size
11.1 --- a/src/HOL/ex/ROOT.ML Thu Aug 18 17:42:35 2011 +0200
11.2 +++ b/src/HOL/ex/ROOT.ML Thu Aug 18 22:50:28 2011 +0200
11.3 @@ -48,7 +48,7 @@
11.4 "Primrec",
11.5 "Tarski",
11.6 "Classical",
11.7 - "set",
11.8 + "Set_Theory",
11.9 "Meson_Test",
11.10 "Termination",
11.11 "Coherent",
12.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
12.2 +++ b/src/HOL/ex/Set_Theory.thy Thu Aug 18 22:50:28 2011 +0200
12.3 @@ -0,0 +1,227 @@
12.4 +(* Title: HOL/ex/Set_Theory.thy
12.5 + Author: Tobias Nipkow and Lawrence C Paulson
12.6 + Copyright 1991 University of Cambridge
12.7 +*)
12.8 +
12.9 +header {* Set Theory examples: Cantor's Theorem, Schröder-Bernstein Theorem, etc. *}
12.10 +
12.11 +theory Set_Theory
12.12 +imports Main
12.13 +begin
12.14 +
12.15 +text{*
12.16 + These two are cited in Benzmueller and Kohlhase's system description
12.17 + of LEO, CADE-15, 1998 (pages 139-143) as theorems LEO could not
12.18 + prove.
12.19 +*}
12.20 +
12.21 +lemma "(X = Y \<union> Z) =
12.22 + (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"
12.23 + by blast
12.24 +
12.25 +lemma "(X = Y \<inter> Z) =
12.26 + (X \<subseteq> Y \<and> X \<subseteq> Z \<and> (\<forall>V. V \<subseteq> Y \<and> V \<subseteq> Z \<longrightarrow> V \<subseteq> X))"
12.27 + by blast
12.28 +
12.29 +text {*
12.30 + Trivial example of term synthesis: apparently hard for some provers!
12.31 +*}
12.32 +
12.33 +schematic_lemma "a \<noteq> b \<Longrightarrow> a \<in> ?X \<and> b \<notin> ?X"
12.34 + by blast
12.35 +
12.36 +
12.37 +subsection {* Examples for the @{text blast} paper *}
12.38 +
12.39 +lemma "(\<Union>x \<in> C. f x \<union> g x) = \<Union>(f ` C) \<union> \<Union>(g ` C)"
12.40 + -- {* Union-image, called @{text Un_Union_image} in Main HOL *}
12.41 + by blast
12.42 +
12.43 +lemma "(\<Inter>x \<in> C. f x \<inter> g x) = \<Inter>(f ` C) \<inter> \<Inter>(g ` C)"
12.44 + -- {* Inter-image, called @{text Int_Inter_image} in Main HOL *}
12.45 + by blast
12.46 +
12.47 +lemma singleton_example_1:
12.48 + "\<And>S::'a set set. \<forall>x \<in> S. \<forall>y \<in> S. x \<subseteq> y \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
12.49 + by blast
12.50 +
12.51 +lemma singleton_example_2:
12.52 + "\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
12.53 + -- {*Variant of the problem above. *}
12.54 + by blast
12.55 +
12.56 +lemma "\<exists>!x. f (g x) = x \<Longrightarrow> \<exists>!y. g (f y) = y"
12.57 + -- {* A unique fixpoint theorem --- @{text fast}/@{text best}/@{text meson} all fail. *}
12.58 + by metis
12.59 +
12.60 +
12.61 +subsection {* Cantor's Theorem: There is no surjection from a set to its powerset *}
12.62 +
12.63 +lemma cantor1: "\<not> (\<exists>f:: 'a \<Rightarrow> 'a set. \<forall>S. \<exists>x. f x = S)"
12.64 + -- {* Requires best-first search because it is undirectional. *}
12.65 + by best
12.66 +
12.67 +schematic_lemma "\<forall>f:: 'a \<Rightarrow> 'a set. \<forall>x. f x \<noteq> ?S f"
12.68 + -- {*This form displays the diagonal term. *}
12.69 + by best
12.70 +
12.71 +schematic_lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
12.72 + -- {* This form exploits the set constructs. *}
12.73 + by (rule notI, erule rangeE, best)
12.74 +
12.75 +schematic_lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
12.76 + -- {* Or just this! *}
12.77 + by best
12.78 +
12.79 +
12.80 +subsection {* The Schröder-Berstein Theorem *}
12.81 +
12.82 +lemma disj_lemma: "- (f ` X) = g ` (-X) \<Longrightarrow> f a = g b \<Longrightarrow> a \<in> X \<Longrightarrow> b \<in> X"
12.83 + by blast
12.84 +
12.85 +lemma surj_if_then_else:
12.86 + "-(f ` X) = g ` (-X) \<Longrightarrow> surj (\<lambda>z. if z \<in> X then f z else g z)"
12.87 + by (simp add: surj_def) blast
12.88 +
12.89 +lemma bij_if_then_else:
12.90 + "inj_on f X \<Longrightarrow> inj_on g (-X) \<Longrightarrow> -(f ` X) = g ` (-X) \<Longrightarrow>
12.91 + h = (\<lambda>z. if z \<in> X then f z else g z) \<Longrightarrow> inj h \<and> surj h"
12.92 + apply (unfold inj_on_def)
12.93 + apply (simp add: surj_if_then_else)
12.94 + apply (blast dest: disj_lemma sym)
12.95 + done
12.96 +
12.97 +lemma decomposition: "\<exists>X. X = - (g ` (- (f ` X)))"
12.98 + apply (rule exI)
12.99 + apply (rule lfp_unfold)
12.100 + apply (rule monoI, blast)
12.101 + done
12.102 +
12.103 +theorem Schroeder_Bernstein:
12.104 + "inj (f :: 'a \<Rightarrow> 'b) \<Longrightarrow> inj (g :: 'b \<Rightarrow> 'a)
12.105 + \<Longrightarrow> \<exists>h:: 'a \<Rightarrow> 'b. inj h \<and> surj h"
12.106 + apply (rule decomposition [where f=f and g=g, THEN exE])
12.107 + apply (rule_tac x = "(\<lambda>z. if z \<in> x then f z else inv g z)" in exI)
12.108 + --{*The term above can be synthesized by a sufficiently detailed proof.*}
12.109 + apply (rule bij_if_then_else)
12.110 + apply (rule_tac [4] refl)
12.111 + apply (rule_tac [2] inj_on_inv_into)
12.112 + apply (erule subset_inj_on [OF _ subset_UNIV])
12.113 + apply blast
12.114 + apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric])
12.115 + done
12.116 +
12.117 +
12.118 +subsection {* A simple party theorem *}
12.119 +
12.120 +text{* \emph{At any party there are two people who know the same
12.121 +number of people}. Provided the party consists of at least two people
12.122 +and the knows relation is symmetric. Knowing yourself does not count
12.123 +--- otherwise knows needs to be reflexive. (From Freek Wiedijk's talk
12.124 +at TPHOLs 2007.) *}
12.125 +
12.126 +lemma equal_number_of_acquaintances:
12.127 +assumes "Domain R <= A" and "sym R" and "card A \<ge> 2"
12.128 +shows "\<not> inj_on (%a. card(R `` {a} - {a})) A"
12.129 +proof -
12.130 + let ?N = "%a. card(R `` {a} - {a})"
12.131 + let ?n = "card A"
12.132 + have "finite A" using `card A \<ge> 2` by(auto intro:ccontr)
12.133 + have 0: "R `` A <= A" using `sym R` `Domain R <= A`
12.134 + unfolding Domain_def sym_def by blast
12.135 + have h: "ALL a:A. R `` {a} <= A" using 0 by blast
12.136 + hence 1: "ALL a:A. finite(R `` {a})" using `finite A`
12.137 + by(blast intro: finite_subset)
12.138 + have sub: "?N ` A <= {0..<?n}"
12.139 + proof -
12.140 + have "ALL a:A. R `` {a} - {a} < A" using h by blast
12.141 + thus ?thesis using psubset_card_mono[OF `finite A`] by auto
12.142 + qed
12.143 + show "~ inj_on ?N A" (is "~ ?I")
12.144 + proof
12.145 + assume ?I
12.146 + hence "?n = card(?N ` A)" by(rule card_image[symmetric])
12.147 + with sub `finite A` have 2[simp]: "?N ` A = {0..<?n}"
12.148 + using subset_card_intvl_is_intvl[of _ 0] by(auto)
12.149 + have "0 : ?N ` A" and "?n - 1 : ?N ` A" using `card A \<ge> 2` by simp+
12.150 + then obtain a b where ab: "a:A" "b:A" and Na: "?N a = 0" and Nb: "?N b = ?n - 1"
12.151 + by (auto simp del: 2)
12.152 + have "a \<noteq> b" using Na Nb `card A \<ge> 2` by auto
12.153 + have "R `` {a} - {a} = {}" by (metis 1 Na ab card_eq_0_iff finite_Diff)
12.154 + hence "b \<notin> R `` {a}" using `a\<noteq>b` by blast
12.155 + hence "a \<notin> R `` {b}" by (metis Image_singleton_iff assms(2) sym_def)
12.156 + hence 3: "R `` {b} - {b} <= A - {a,b}" using 0 ab by blast
12.157 + have 4: "finite (A - {a,b})" using `finite A` by simp
12.158 + have "?N b <= ?n - 2" using ab `a\<noteq>b` `finite A` card_mono[OF 4 3] by simp
12.159 + then show False using Nb `card A \<ge> 2` by arith
12.160 + qed
12.161 +qed
12.162 +
12.163 +text {*
12.164 + From W. W. Bledsoe and Guohui Feng, SET-VAR. JAR 11 (3), 1993, pages
12.165 + 293-314.
12.166 +
12.167 + Isabelle can prove the easy examples without any special mechanisms,
12.168 + but it can't prove the hard ones.
12.169 +*}
12.170 +
12.171 +lemma "\<exists>A. (\<forall>x \<in> A. x \<le> (0::int))"
12.172 + -- {* Example 1, page 295. *}
12.173 + by force
12.174 +
12.175 +lemma "D \<in> F \<Longrightarrow> \<exists>G. \<forall>A \<in> G. \<exists>B \<in> F. A \<subseteq> B"
12.176 + -- {* Example 2. *}
12.177 + by force
12.178 +
12.179 +lemma "P a \<Longrightarrow> \<exists>A. (\<forall>x \<in> A. P x) \<and> (\<exists>y. y \<in> A)"
12.180 + -- {* Example 3. *}
12.181 + by force
12.182 +
12.183 +lemma "a < b \<and> b < (c::int) \<Longrightarrow> \<exists>A. a \<notin> A \<and> b \<in> A \<and> c \<notin> A"
12.184 + -- {* Example 4. *}
12.185 + by force
12.186 +
12.187 +lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
12.188 + -- {*Example 5, page 298. *}
12.189 + by force
12.190 +
12.191 +lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
12.192 + -- {* Example 6. *}
12.193 + by force
12.194 +
12.195 +lemma "\<exists>A. a \<notin> A"
12.196 + -- {* Example 7. *}
12.197 + by force
12.198 +
12.199 +lemma "(\<forall>u v. u < (0::int) \<longrightarrow> u \<noteq> abs v)
12.200 + \<longrightarrow> (\<exists>A::int set. (\<forall>y. abs y \<notin> A) \<and> -2 \<in> A)"
12.201 + -- {* Example 8 now needs a small hint. *}
12.202 + by (simp add: abs_if, force)
12.203 + -- {* not @{text blast}, which can't simplify @{text "-2 < 0"} *}
12.204 +
12.205 +text {* Example 9 omitted (requires the reals). *}
12.206 +
12.207 +text {* The paper has no Example 10! *}
12.208 +
12.209 +lemma "(\<forall>A. 0 \<in> A \<and> (\<forall>x \<in> A. Suc x \<in> A) \<longrightarrow> n \<in> A) \<and>
12.210 + P 0 \<and> (\<forall>x. P x \<longrightarrow> P (Suc x)) \<longrightarrow> P n"
12.211 + -- {* Example 11: needs a hint. *}
12.212 +by(metis nat.induct)
12.213 +
12.214 +lemma
12.215 + "(\<forall>A. (0, 0) \<in> A \<and> (\<forall>x y. (x, y) \<in> A \<longrightarrow> (Suc x, Suc y) \<in> A) \<longrightarrow> (n, m) \<in> A)
12.216 + \<and> P n \<longrightarrow> P m"
12.217 + -- {* Example 12. *}
12.218 + by auto
12.219 +
12.220 +lemma
12.221 + "(\<forall>x. (\<exists>u. x = 2 * u) = (\<not> (\<exists>v. Suc x = 2 * v))) \<longrightarrow>
12.222 + (\<exists>A. \<forall>x. (x \<in> A) = (Suc x \<notin> A))"
12.223 + -- {* Example EO1: typo in article, and with the obvious fix it seems
12.224 + to require arithmetic reasoning. *}
12.225 + apply clarify
12.226 + apply (rule_tac x = "{x. \<exists>u. x = 2 * u}" in exI, auto)
12.227 + apply metis+
12.228 + done
12.229 +
12.230 +end
13.1 --- a/src/HOL/ex/set.thy Thu Aug 18 17:42:35 2011 +0200
13.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
13.3 @@ -1,225 +0,0 @@
13.4 -(* Title: HOL/ex/set.thy
13.5 - Author: Tobias Nipkow and Lawrence C Paulson
13.6 - Copyright 1991 University of Cambridge
13.7 -*)
13.8 -
13.9 -header {* Set Theory examples: Cantor's Theorem, Schröder-Bernstein Theorem, etc. *}
13.10 -
13.11 -theory set imports Main begin
13.12 -
13.13 -text{*
13.14 - These two are cited in Benzmueller and Kohlhase's system description
13.15 - of LEO, CADE-15, 1998 (pages 139-143) as theorems LEO could not
13.16 - prove.
13.17 -*}
13.18 -
13.19 -lemma "(X = Y \<union> Z) =
13.20 - (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"
13.21 - by blast
13.22 -
13.23 -lemma "(X = Y \<inter> Z) =
13.24 - (X \<subseteq> Y \<and> X \<subseteq> Z \<and> (\<forall>V. V \<subseteq> Y \<and> V \<subseteq> Z \<longrightarrow> V \<subseteq> X))"
13.25 - by blast
13.26 -
13.27 -text {*
13.28 - Trivial example of term synthesis: apparently hard for some provers!
13.29 -*}
13.30 -
13.31 -schematic_lemma "a \<noteq> b \<Longrightarrow> a \<in> ?X \<and> b \<notin> ?X"
13.32 - by blast
13.33 -
13.34 -
13.35 -subsection {* Examples for the @{text blast} paper *}
13.36 -
13.37 -lemma "(\<Union>x \<in> C. f x \<union> g x) = \<Union>(f ` C) \<union> \<Union>(g ` C)"
13.38 - -- {* Union-image, called @{text Un_Union_image} in Main HOL *}
13.39 - by blast
13.40 -
13.41 -lemma "(\<Inter>x \<in> C. f x \<inter> g x) = \<Inter>(f ` C) \<inter> \<Inter>(g ` C)"
13.42 - -- {* Inter-image, called @{text Int_Inter_image} in Main HOL *}
13.43 - by blast
13.44 -
13.45 -lemma singleton_example_1:
13.46 - "\<And>S::'a set set. \<forall>x \<in> S. \<forall>y \<in> S. x \<subseteq> y \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
13.47 - by blast
13.48 -
13.49 -lemma singleton_example_2:
13.50 - "\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
13.51 - -- {*Variant of the problem above. *}
13.52 - by blast
13.53 -
13.54 -lemma "\<exists>!x. f (g x) = x \<Longrightarrow> \<exists>!y. g (f y) = y"
13.55 - -- {* A unique fixpoint theorem --- @{text fast}/@{text best}/@{text meson} all fail. *}
13.56 - by metis
13.57 -
13.58 -
13.59 -subsection {* Cantor's Theorem: There is no surjection from a set to its powerset *}
13.60 -
13.61 -lemma cantor1: "\<not> (\<exists>f:: 'a \<Rightarrow> 'a set. \<forall>S. \<exists>x. f x = S)"
13.62 - -- {* Requires best-first search because it is undirectional. *}
13.63 - by best
13.64 -
13.65 -schematic_lemma "\<forall>f:: 'a \<Rightarrow> 'a set. \<forall>x. f x \<noteq> ?S f"
13.66 - -- {*This form displays the diagonal term. *}
13.67 - by best
13.68 -
13.69 -schematic_lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
13.70 - -- {* This form exploits the set constructs. *}
13.71 - by (rule notI, erule rangeE, best)
13.72 -
13.73 -schematic_lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
13.74 - -- {* Or just this! *}
13.75 - by best
13.76 -
13.77 -
13.78 -subsection {* The Schröder-Berstein Theorem *}
13.79 -
13.80 -lemma disj_lemma: "- (f ` X) = g ` (-X) \<Longrightarrow> f a = g b \<Longrightarrow> a \<in> X \<Longrightarrow> b \<in> X"
13.81 - by blast
13.82 -
13.83 -lemma surj_if_then_else:
13.84 - "-(f ` X) = g ` (-X) \<Longrightarrow> surj (\<lambda>z. if z \<in> X then f z else g z)"
13.85 - by (simp add: surj_def) blast
13.86 -
13.87 -lemma bij_if_then_else:
13.88 - "inj_on f X \<Longrightarrow> inj_on g (-X) \<Longrightarrow> -(f ` X) = g ` (-X) \<Longrightarrow>
13.89 - h = (\<lambda>z. if z \<in> X then f z else g z) \<Longrightarrow> inj h \<and> surj h"
13.90 - apply (unfold inj_on_def)
13.91 - apply (simp add: surj_if_then_else)
13.92 - apply (blast dest: disj_lemma sym)
13.93 - done
13.94 -
13.95 -lemma decomposition: "\<exists>X. X = - (g ` (- (f ` X)))"
13.96 - apply (rule exI)
13.97 - apply (rule lfp_unfold)
13.98 - apply (rule monoI, blast)
13.99 - done
13.100 -
13.101 -theorem Schroeder_Bernstein:
13.102 - "inj (f :: 'a \<Rightarrow> 'b) \<Longrightarrow> inj (g :: 'b \<Rightarrow> 'a)
13.103 - \<Longrightarrow> \<exists>h:: 'a \<Rightarrow> 'b. inj h \<and> surj h"
13.104 - apply (rule decomposition [where f=f and g=g, THEN exE])
13.105 - apply (rule_tac x = "(\<lambda>z. if z \<in> x then f z else inv g z)" in exI)
13.106 - --{*The term above can be synthesized by a sufficiently detailed proof.*}
13.107 - apply (rule bij_if_then_else)
13.108 - apply (rule_tac [4] refl)
13.109 - apply (rule_tac [2] inj_on_inv_into)
13.110 - apply (erule subset_inj_on [OF _ subset_UNIV])
13.111 - apply blast
13.112 - apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric])
13.113 - done
13.114 -
13.115 -
13.116 -subsection {* A simple party theorem *}
13.117 -
13.118 -text{* \emph{At any party there are two people who know the same
13.119 -number of people}. Provided the party consists of at least two people
13.120 -and the knows relation is symmetric. Knowing yourself does not count
13.121 ---- otherwise knows needs to be reflexive. (From Freek Wiedijk's talk
13.122 -at TPHOLs 2007.) *}
13.123 -
13.124 -lemma equal_number_of_acquaintances:
13.125 -assumes "Domain R <= A" and "sym R" and "card A \<ge> 2"
13.126 -shows "\<not> inj_on (%a. card(R `` {a} - {a})) A"
13.127 -proof -
13.128 - let ?N = "%a. card(R `` {a} - {a})"
13.129 - let ?n = "card A"
13.130 - have "finite A" using `card A \<ge> 2` by(auto intro:ccontr)
13.131 - have 0: "R `` A <= A" using `sym R` `Domain R <= A`
13.132 - unfolding Domain_def sym_def by blast
13.133 - have h: "ALL a:A. R `` {a} <= A" using 0 by blast
13.134 - hence 1: "ALL a:A. finite(R `` {a})" using `finite A`
13.135 - by(blast intro: finite_subset)
13.136 - have sub: "?N ` A <= {0..<?n}"
13.137 - proof -
13.138 - have "ALL a:A. R `` {a} - {a} < A" using h by blast
13.139 - thus ?thesis using psubset_card_mono[OF `finite A`] by auto
13.140 - qed
13.141 - show "~ inj_on ?N A" (is "~ ?I")
13.142 - proof
13.143 - assume ?I
13.144 - hence "?n = card(?N ` A)" by(rule card_image[symmetric])
13.145 - with sub `finite A` have 2[simp]: "?N ` A = {0..<?n}"
13.146 - using subset_card_intvl_is_intvl[of _ 0] by(auto)
13.147 - have "0 : ?N ` A" and "?n - 1 : ?N ` A" using `card A \<ge> 2` by simp+
13.148 - then obtain a b where ab: "a:A" "b:A" and Na: "?N a = 0" and Nb: "?N b = ?n - 1"
13.149 - by (auto simp del: 2)
13.150 - have "a \<noteq> b" using Na Nb `card A \<ge> 2` by auto
13.151 - have "R `` {a} - {a} = {}" by (metis 1 Na ab card_eq_0_iff finite_Diff)
13.152 - hence "b \<notin> R `` {a}" using `a\<noteq>b` by blast
13.153 - hence "a \<notin> R `` {b}" by (metis Image_singleton_iff assms(2) sym_def)
13.154 - hence 3: "R `` {b} - {b} <= A - {a,b}" using 0 ab by blast
13.155 - have 4: "finite (A - {a,b})" using `finite A` by simp
13.156 - have "?N b <= ?n - 2" using ab `a\<noteq>b` `finite A` card_mono[OF 4 3] by simp
13.157 - then show False using Nb `card A \<ge> 2` by arith
13.158 - qed
13.159 -qed
13.160 -
13.161 -text {*
13.162 - From W. W. Bledsoe and Guohui Feng, SET-VAR. JAR 11 (3), 1993, pages
13.163 - 293-314.
13.164 -
13.165 - Isabelle can prove the easy examples without any special mechanisms,
13.166 - but it can't prove the hard ones.
13.167 -*}
13.168 -
13.169 -lemma "\<exists>A. (\<forall>x \<in> A. x \<le> (0::int))"
13.170 - -- {* Example 1, page 295. *}
13.171 - by force
13.172 -
13.173 -lemma "D \<in> F \<Longrightarrow> \<exists>G. \<forall>A \<in> G. \<exists>B \<in> F. A \<subseteq> B"
13.174 - -- {* Example 2. *}
13.175 - by force
13.176 -
13.177 -lemma "P a \<Longrightarrow> \<exists>A. (\<forall>x \<in> A. P x) \<and> (\<exists>y. y \<in> A)"
13.178 - -- {* Example 3. *}
13.179 - by force
13.180 -
13.181 -lemma "a < b \<and> b < (c::int) \<Longrightarrow> \<exists>A. a \<notin> A \<and> b \<in> A \<and> c \<notin> A"
13.182 - -- {* Example 4. *}
13.183 - by force
13.184 -
13.185 -lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
13.186 - -- {*Example 5, page 298. *}
13.187 - by force
13.188 -
13.189 -lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
13.190 - -- {* Example 6. *}
13.191 - by force
13.192 -
13.193 -lemma "\<exists>A. a \<notin> A"
13.194 - -- {* Example 7. *}
13.195 - by force
13.196 -
13.197 -lemma "(\<forall>u v. u < (0::int) \<longrightarrow> u \<noteq> abs v)
13.198 - \<longrightarrow> (\<exists>A::int set. (\<forall>y. abs y \<notin> A) \<and> -2 \<in> A)"
13.199 - -- {* Example 8 now needs a small hint. *}
13.200 - by (simp add: abs_if, force)
13.201 - -- {* not @{text blast}, which can't simplify @{text "-2 < 0"} *}
13.202 -
13.203 -text {* Example 9 omitted (requires the reals). *}
13.204 -
13.205 -text {* The paper has no Example 10! *}
13.206 -
13.207 -lemma "(\<forall>A. 0 \<in> A \<and> (\<forall>x \<in> A. Suc x \<in> A) \<longrightarrow> n \<in> A) \<and>
13.208 - P 0 \<and> (\<forall>x. P x \<longrightarrow> P (Suc x)) \<longrightarrow> P n"
13.209 - -- {* Example 11: needs a hint. *}
13.210 -by(metis nat.induct)
13.211 -
13.212 -lemma
13.213 - "(\<forall>A. (0, 0) \<in> A \<and> (\<forall>x y. (x, y) \<in> A \<longrightarrow> (Suc x, Suc y) \<in> A) \<longrightarrow> (n, m) \<in> A)
13.214 - \<and> P n \<longrightarrow> P m"
13.215 - -- {* Example 12. *}
13.216 - by auto
13.217 -
13.218 -lemma
13.219 - "(\<forall>x. (\<exists>u. x = 2 * u) = (\<not> (\<exists>v. Suc x = 2 * v))) \<longrightarrow>
13.220 - (\<exists>A. \<forall>x. (x \<in> A) = (Suc x \<notin> A))"
13.221 - -- {* Example EO1: typo in article, and with the obvious fix it seems
13.222 - to require arithmetic reasoning. *}
13.223 - apply clarify
13.224 - apply (rule_tac x = "{x. \<exists>u. x = 2 * u}" in exI, auto)
13.225 - apply metis+
13.226 - done
13.227 -
13.228 -end