--- a/NEWS Thu Apr 24 16:53:04 2008 +0200
+++ b/NEWS Fri Apr 25 15:30:33 2008 +0200
@@ -100,6 +100,9 @@
*** HOL ***
+* Merged theories Wellfounded_Recursion, Accessible_Part and Wellfounded_Relations
+ to "Wellfounded.thy"
+
* Explicit class "eq" for executable equality. INCOMPATIBILITY.
* Class finite no longer treats UNIV as class parameter. Use class enum from
--- a/src/HOL/Accessible_Part.thy Thu Apr 24 16:53:04 2008 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,167 +0,0 @@
-(* Title: HOL/Accessible_Part.thy
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1994 University of Cambridge
-*)
-
-header {* The accessible part of a relation *}
-
-theory Accessible_Part
-imports Wellfounded_Recursion
-begin
-
-subsection {* Inductive definition *}
-
-text {*
- Inductive definition of the accessible part @{term "acc r"} of a
- relation; see also \cite{paulin-tlca}.
-*}
-
-inductive_set
- acc :: "('a * 'a) set => 'a set"
- for r :: "('a * 'a) set"
- where
- accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
-
-abbreviation
- termip :: "('a => 'a => bool) => 'a => bool" where
- "termip r == accp (r\<inverse>\<inverse>)"
-
-abbreviation
- termi :: "('a * 'a) set => 'a set" where
- "termi r == acc (r\<inverse>)"
-
-lemmas accpI = accp.accI
-
-subsection {* Induction rules *}
-
-theorem accp_induct:
- assumes major: "accp r a"
- assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"
- shows "P a"
- apply (rule major [THEN accp.induct])
- apply (rule hyp)
- apply (rule accp.accI)
- apply fast
- apply fast
- done
-
-theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]
-
-theorem accp_downward: "accp r b ==> r a b ==> accp r a"
- apply (erule accp.cases)
- apply fast
- done
-
-lemma not_accp_down:
- assumes na: "\<not> accp R x"
- obtains z where "R z x" and "\<not> accp R z"
-proof -
- assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
-
- show thesis
- proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
- case True
- hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto
- hence "accp R x"
- by (rule accp.accI)
- with na show thesis ..
- next
- case False then obtain z where "R z x" and "\<not> accp R z"
- by auto
- with a show thesis .
- qed
-qed
-
-lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"
- apply (erule rtranclp_induct)
- apply blast
- apply (blast dest: accp_downward)
- done
-
-theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"
- apply (blast dest: accp_downwards_aux)
- done
-
-theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
- apply (rule wfPUNIVI)
- apply (induct_tac P x rule: accp_induct)
- apply blast
- apply blast
- done
-
-theorem accp_wfPD: "wfP r ==> accp r x"
- apply (erule wfP_induct_rule)
- apply (rule accp.accI)
- apply blast
- done
-
-theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
- apply (blast intro: accp_wfPI dest: accp_wfPD)
- done
-
-
-text {* Smaller relations have bigger accessible parts: *}
-
-lemma accp_subset:
- assumes sub: "R1 \<le> R2"
- shows "accp R2 \<le> accp R1"
-proof
- fix x assume "accp R2 x"
- then show "accp R1 x"
- proof (induct x)
- fix x
- assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
- with sub show "accp R1 x"
- by (blast intro: accp.accI)
- qed
-qed
-
-
-text {* This is a generalized induction theorem that works on
- subsets of the accessible part. *}
-
-lemma accp_subset_induct:
- assumes subset: "D \<le> accp R"
- and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
- and "D x"
- and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
- shows "P x"
-proof -
- from subset and `D x`
- have "accp R x" ..
- then show "P x" using `D x`
- proof (induct x)
- fix x
- assume "D x"
- and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
- with dcl and istep show "P x" by blast
- qed
-qed
-
-
-text {* Set versions of the above theorems *}
-
-lemmas acc_induct = accp_induct [to_set]
-
-lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
-
-lemmas acc_downward = accp_downward [to_set]
-
-lemmas not_acc_down = not_accp_down [to_set]
-
-lemmas acc_downwards_aux = accp_downwards_aux [to_set]
-
-lemmas acc_downwards = accp_downwards [to_set]
-
-lemmas acc_wfI = accp_wfPI [to_set]
-
-lemmas acc_wfD = accp_wfPD [to_set]
-
-lemmas wf_acc_iff = wfP_accp_iff [to_set]
-
-lemmas acc_subset = accp_subset [to_set]
-
-lemmas acc_subset_induct = accp_subset_induct [to_set]
-
-end
--- a/src/HOL/Datatype.thy Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/Datatype.thy Fri Apr 25 15:30:33 2008 +0200
@@ -9,7 +9,7 @@
header {* Analogues of the Cartesian Product and Disjoint Sum for Datatypes *}
theory Datatype
-imports Finite_Set
+imports Finite_Set Wellfounded
begin
lemma size_bool [code func]:
--- a/src/HOL/Divides.thy Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/Divides.thy Fri Apr 25 15:30:33 2008 +0200
@@ -7,7 +7,7 @@
header {* The division operators div, mod and the divides relation dvd *}
theory Divides
-imports Nat Power Product_Type Wellfounded_Recursion
+imports Nat Power Product_Type
uses "~~/src/Provers/Arith/cancel_div_mod.ML"
begin
--- a/src/HOL/Finite_Set.thy Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/Finite_Set.thy Fri Apr 25 15:30:33 2008 +0200
@@ -7,7 +7,7 @@
header {* Finite sets *}
theory Finite_Set
-imports Divides
+imports Divides Transitive_Closure
begin
subsection {* Definition and basic properties *}
@@ -2639,6 +2639,35 @@
by (simp add: Max fold1_antimono [folded dual_max])
qed
+lemma finite_linorder_induct[consumes 1, case_names empty insert]:
+ "finite A \<Longrightarrow> P {} \<Longrightarrow>
+ (!!A b. finite A \<Longrightarrow> ALL a:A. a < b \<Longrightarrow> P A \<Longrightarrow> P(insert b A))
+ \<Longrightarrow> P A"
+proof (induct A rule: measure_induct_rule[where f=card])
+ fix A :: "'a set"
+ assume IH: "!! B. card B < card A \<Longrightarrow> finite B \<Longrightarrow> P {} \<Longrightarrow>
+ (!!A b. finite A \<Longrightarrow> (\<forall>a\<in>A. a<b) \<Longrightarrow> P A \<Longrightarrow> P (insert b A))
+ \<Longrightarrow> P B"
+ and "finite A" and "P {}"
+ and step: "!!A b. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)"
+ show "P A"
+ proof cases
+ assume "A = {}" thus "P A" using `P {}` by simp
+ next
+ let ?B = "A - {Max A}" let ?A = "insert (Max A) ?B"
+ assume "A \<noteq> {}"
+ with `finite A` have "Max A : A" by auto
+ hence A: "?A = A" using insert_Diff_single insert_absorb by auto
+ note card_Diff1_less[OF `finite A` `Max A : A`]
+ moreover have "finite ?B" using `finite A` by simp
+ ultimately have "P ?B" using `P {}` step IH by blast
+ moreover have "\<forall>a\<in>?B. a < Max A"
+ using Max_ge[OF `finite A` `A \<noteq> {}`] by fastsimp
+ ultimately show "P A"
+ using A insert_Diff_single step[OF `finite ?B`] by fastsimp
+ qed
+qed
+
end
context ordered_ab_semigroup_add
--- a/src/HOL/FunDef.thy Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/FunDef.thy Fri Apr 25 15:30:33 2008 +0200
@@ -6,7 +6,7 @@
header {* General recursive function definitions *}
theory FunDef
-imports Accessible_Part
+imports Wellfounded
uses
("Tools/function_package/fundef_lib.ML")
("Tools/function_package/fundef_common.ML")
@@ -19,6 +19,8 @@
("Tools/function_package/fundef_package.ML")
("Tools/function_package/auto_term.ML")
("Tools/function_package/induction_scheme.ML")
+ ("Tools/function_package/lexicographic_order.ML")
+ ("Tools/function_package/fundef_datatype.ML")
begin
text {* Definitions with default value. *}
@@ -106,10 +108,14 @@
use "Tools/function_package/auto_term.ML"
use "Tools/function_package/fundef_package.ML"
use "Tools/function_package/induction_scheme.ML"
+use "Tools/function_package/lexicographic_order.ML"
+use "Tools/function_package/fundef_datatype.ML"
setup {*
FundefPackage.setup
#> InductionScheme.setup
+ #> LexicographicOrder.setup
+ #> FundefDatatype.setup
*}
lemma let_cong [fundef_cong]:
--- a/src/HOL/Hilbert_Choice.thy Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/Hilbert_Choice.thy Fri Apr 25 15:30:33 2008 +0200
@@ -7,7 +7,7 @@
header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
theory Hilbert_Choice
-imports Nat Wellfounded_Recursion
+imports Nat Wellfounded
uses ("Tools/meson.ML") ("Tools/specification_package.ML")
begin
--- a/src/HOL/Int.thy Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/Int.thy Fri Apr 25 15:30:33 2008 +0200
@@ -9,7 +9,7 @@
header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *}
theory Int
-imports Equiv_Relations Nat Wellfounded_Relations
+imports Equiv_Relations Nat Wellfounded
uses
("Tools/numeral.ML")
("Tools/numeral_syntax.ML")
--- a/src/HOL/IsaMakefile Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/IsaMakefile Fri Apr 25 15:30:33 2008 +0200
@@ -92,7 +92,7 @@
$(SRC)/Tools/code/code_package.ML $(SRC)/Tools/code/code_target.ML \
$(SRC)/Tools/code/code_thingol.ML $(SRC)/Tools/nbe.ML $(SRC)/Tools/atomize_elim.ML \
$(SRC)/Tools/random_word.ML $(SRC)/Tools/rat.ML Tools/TFL/casesplit.ML ATP_Linkup.thy \
- Accessible_Part.thy Arith_Tools.thy Code_Setup.thy Datatype.thy \
+ Arith_Tools.thy Code_Setup.thy Datatype.thy \
Divides.thy Equiv_Relations.thy Extraction.thy \
Finite_Set.thy Fun.thy FunDef.thy HOL.thy \
Hilbert_Choice.thy Inductive.thy Int.thy IntDiv.thy \
@@ -142,8 +142,8 @@
Tools/sat_funcs.ML Tools/sat_solver.ML Tools/specification_package.ML \
Tools/split_rule.ML Tools/string_syntax.ML Tools/typecopy_package.ML \
Tools/typedef_codegen.ML Tools/typedef_package.ML \
- Transitive_Closure.thy Typedef.thy Wellfounded_Recursion.thy \
- Wellfounded_Relations.thy arith_data.ML document/root.tex hologic.ML \
+ Transitive_Closure.thy Typedef.thy Wellfounded.thy \
+ arith_data.ML document/root.tex hologic.ML \
int_arith1.ML int_factor_simprocs.ML nat_simprocs.ML simpdata.ML
@$(ISATOOL) usedir $(HOL_USEDIR_OPTIONS) -b -g true $(OUT)/Pure HOL
--- a/src/HOL/Lambda/ROOT.ML Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/Lambda/ROOT.ML Fri Apr 25 15:30:33 2008 +0200
@@ -7,7 +7,7 @@
Syntax.ambiguity_level := 100;
Proofterm.proofs := 2;
-no_document use_thys ["Accessible_Part", "Code_Integer"];
+no_document use_thys ["Code_Integer"];
use_thys ["Eta", "StrongNorm", "Standardization"];
if HOL_proofs < 2 then
warning "HOL proof terms required for running theory WeakNorm"
--- a/src/HOL/Nat.thy Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/Nat.thy Fri Apr 25 15:30:33 2008 +0200
@@ -734,7 +734,55 @@
by simp
-subsubsection {* Additional theorems about "less than" *}
+subsubsection {* Additional theorems about @{term "op \<le>"} *}
+
+text {* Complete induction, aka course-of-values induction *}
+
+lemma less_induct [case_names less]:
+ fixes P :: "nat \<Rightarrow> bool"
+ assumes step: "\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x"
+ shows "P a"
+proof -
+ have "\<And>z. z\<le>a \<Longrightarrow> P z"
+ proof (induct a)
+ case (0 z)
+ have "P 0" by (rule step) auto
+ thus ?case using 0 by auto
+ next
+ case (Suc x z)
+ then have "z \<le> x \<or> z = Suc x" by (simp add: le_Suc_eq)
+ thus ?case
+ proof
+ assume "z \<le> x" thus "P z" by (rule Suc(1))
+ next
+ assume z: "z = Suc x"
+ show "P z"
+ by (rule step) (rule Suc(1), simp add: z le_simps)
+ qed
+ qed
+ thus ?thesis by auto
+qed
+
+lemma nat_less_induct:
+ assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
+ using assms less_induct by blast
+
+lemma measure_induct_rule [case_names less]:
+ fixes f :: "'a \<Rightarrow> nat"
+ assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
+ shows "P a"
+by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
+
+text {* old style induction rules: *}
+lemma measure_induct:
+ fixes f :: "'a \<Rightarrow> nat"
+ shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
+ by (rule measure_induct_rule [of f P a]) iprover
+
+lemma full_nat_induct:
+ assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
+ shows "P n"
+ by (rule less_induct) (auto intro: step simp:le_simps)
text{*An induction rule for estabilishing binary relations*}
lemma less_Suc_induct:
@@ -755,6 +803,73 @@
thus "P i j" by (simp add: j)
qed
+lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
+ apply (rule nat_less_induct)
+ apply (case_tac n)
+ apply (case_tac [2] nat)
+ apply (blast intro: less_trans)+
+ done
+
+text {* The method of infinite descent, frequently used in number theory.
+Provided by Roelof Oosterhuis.
+$P(n)$ is true for all $n\in\mathbb{N}$ if
+\begin{itemize}
+ \item case ``0'': given $n=0$ prove $P(n)$,
+ \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
+ a smaller integer $m$ such that $\neg P(m)$.
+\end{itemize} *}
+
+text{* A compact version without explicit base case: *}
+lemma infinite_descent:
+ "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m \<rbrakk> \<Longrightarrow> P n"
+by (induct n rule: less_induct, auto)
+
+lemma infinite_descent0[case_names 0 smaller]:
+ "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
+by (rule infinite_descent) (case_tac "n>0", auto)
+
+text {*
+Infinite descent using a mapping to $\mathbb{N}$:
+$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
+\begin{itemize}
+\item case ``0'': given $V(x)=0$ prove $P(x)$,
+\item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$.
+\end{itemize}
+NB: the proof also shows how to use the previous lemma. *}
+
+corollary infinite_descent0_measure [case_names 0 smaller]:
+ assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
+ and A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
+ shows "P x"
+proof -
+ obtain n where "n = V x" by auto
+ moreover have "\<And>x. V x = n \<Longrightarrow> P x"
+ proof (induct n rule: infinite_descent0)
+ case 0 -- "i.e. $V(x) = 0$"
+ with A0 show "P x" by auto
+ next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
+ case (smaller n)
+ then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
+ with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
+ with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
+ then show ?case by auto
+ qed
+ ultimately show "P x" by auto
+qed
+
+text{* Again, without explicit base case: *}
+lemma infinite_descent_measure:
+assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
+proof -
+ from assms obtain n where "n = V x" by auto
+ moreover have "!!x. V x = n \<Longrightarrow> P x"
+ proof (induct n rule: infinite_descent, auto)
+ fix x assume "\<not> P x"
+ with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
+ qed
+ ultimately show "P x" by auto
+qed
+
text {* A [clumsy] way of lifting @{text "<"}
monotonicity to @{text "\<le>"} monotonicity *}
lemma less_mono_imp_le_mono:
@@ -809,7 +924,7 @@
done
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
-by (simp add: add_commute not_add_less1)
+by (simp add: add_commute)
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
apply (rule order_trans [of _ "m+k"])
@@ -841,7 +956,7 @@
by (simp add: add_diff_inverse linorder_not_less)
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
-by (simp add: le_add_diff_inverse add_commute)
+by (simp add: add_commute)
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
by (induct m n rule: diff_induct) simp_all
@@ -1328,6 +1443,6 @@
subsection {* size of a datatype value *}
class size = type +
- fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded_Recursion} *}
+ fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}
end
--- a/src/HOL/Recdef.thy Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/Recdef.thy Fri Apr 25 15:30:33 2008 +0200
@@ -6,7 +6,7 @@
header {* TFL: recursive function definitions *}
theory Recdef
-imports Wellfounded_Relations FunDef
+imports FunDef
uses
("Tools/TFL/casesplit.ML")
("Tools/TFL/utils.ML")
@@ -20,6 +20,30 @@
("Tools/recdef_package.ML")
begin
+text{** This form avoids giant explosions in proofs. NOTE USE OF ==*}
+lemma def_wfrec: "[| f==wfrec r H; wf(r) |] ==> f(a) = H (cut f r a) a"
+apply auto
+apply (blast intro: wfrec)
+done
+
+
+lemma tfl_wf_induct: "ALL R. wf R -->
+ (ALL P. (ALL x. (ALL y. (y,x):R --> P y) --> P x) --> (ALL x. P x))"
+apply clarify
+apply (rule_tac r = R and P = P and a = x in wf_induct, assumption, blast)
+done
+
+lemma tfl_cut_apply: "ALL f R. (x,a):R --> (cut f R a)(x) = f(x)"
+apply clarify
+apply (rule cut_apply, assumption)
+done
+
+lemma tfl_wfrec:
+ "ALL M R f. (f=wfrec R M) --> wf R --> (ALL x. f x = M (cut f R x) x)"
+apply clarify
+apply (erule wfrec)
+done
+
lemma tfl_eq_True: "(x = True) --> x"
by blast
--- a/src/HOL/Tools/function_package/fundef_common.ML Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/Tools/function_package/fundef_common.ML Fri Apr 25 15:30:33 2008 +0200
@@ -17,7 +17,7 @@
fun PROFILE msg = if !profile then timeap_msg msg else I
-val acc_const_name = "Accessible_Part.accp"
+val acc_const_name = @{const_name "accp"}
fun mk_acc domT R =
Const (acc_const_name, (domT --> domT --> HOLogic.boolT) --> domT --> HOLogic.boolT) $ R
--- a/src/HOL/Tools/function_package/fundef_core.ML Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/Tools/function_package/fundef_core.ML Fri Apr 25 15:30:33 2008 +0200
@@ -95,17 +95,17 @@
(* Theory dependencies. *)
val Pair_inject = @{thm Product_Type.Pair_inject};
-val acc_induct_rule = @{thm Accessible_Part.accp_induct_rule};
+val acc_induct_rule = @{thm accp_induct_rule};
val ex1_implies_ex = @{thm FunDef.fundef_ex1_existence};
val ex1_implies_un = @{thm FunDef.fundef_ex1_uniqueness};
val ex1_implies_iff = @{thm FunDef.fundef_ex1_iff};
-val acc_downward = @{thm Accessible_Part.accp_downward};
-val accI = @{thm Accessible_Part.accp.accI};
+val acc_downward = @{thm accp_downward};
+val accI = @{thm accp.accI};
val case_split = @{thm HOL.case_split_thm};
val fundef_default_value = @{thm FunDef.fundef_default_value};
-val not_acc_down = @{thm Accessible_Part.not_accp_down};
+val not_acc_down = @{thm not_accp_down};
@@ -575,7 +575,7 @@
(** Induction rule **)
-val acc_subset_induct = @{thm Orderings.predicate1I} RS @{thm Accessible_Part.accp_subset_induct}
+val acc_subset_induct = @{thm Orderings.predicate1I} RS @{thm accp_subset_induct}
fun binder_conv cv ctxt = Conv.arg_conv (Conv.abs_conv (K cv) ctxt);
@@ -710,7 +710,7 @@
(** Termination rule **)
-val wf_induct_rule = @{thm Wellfounded_Recursion.wfP_induct_rule};
+val wf_induct_rule = @{thm Wellfounded.wfP_induct_rule};
val wf_in_rel = @{thm FunDef.wf_in_rel};
val in_rel_def = @{thm FunDef.in_rel_def};
@@ -770,7 +770,7 @@
val Rrel = Free ("R", HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT)))
val inrel_R = Const ("FunDef.in_rel", HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT)) --> fastype_of R) $ Rrel
- val wfR' = cterm_of thy (HOLogic.mk_Trueprop (Const ("Wellfounded_Recursion.wfP", (domT --> domT --> boolT) --> boolT) $ R')) (* "wf R'" *)
+ val wfR' = cterm_of thy (HOLogic.mk_Trueprop (Const (@{const_name "Wellfounded.wfP"}, (domT --> domT --> boolT) --> boolT) $ R')) (* "wf R'" *)
(* Inductive Hypothesis: !!z. (z,x):R' ==> z : acc R *)
val ihyp = all domT $ Abs ("z", domT,
--- a/src/HOL/Tools/function_package/lexicographic_order.ML Thu Apr 24 16:53:04 2008 +0200
+++ b/src/HOL/Tools/function_package/lexicographic_order.ML Fri Apr 25 15:30:33 2008 +0200
@@ -50,7 +50,7 @@
val mlexT = (domT --> HOLogic.natT) --> relT --> relT
fun mk_ms [] = Const (@{const_name "{}"}, relT)
| mk_ms (f::fs) =
- Const (@{const_name "Wellfounded_Relations.mlex_prod"}, mlexT) $ f $ mk_ms fs
+ Const (@{const_name "mlex_prod"}, mlexT) $ f $ mk_ms fs
in
mk_ms mfuns
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Wellfounded.thy Fri Apr 25 15:30:33 2008 +0200
@@ -0,0 +1,919 @@
+(* ID: $Id$
+ Author: Tobias Nipkow
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Author: Konrad Slind, Alexander Krauss
+ Copyright 1992-2008 University of Cambridge and TU Muenchen
+*)
+
+header {*Well-founded Recursion*}
+
+theory Wellfounded
+imports Finite_Set Nat
+uses ("Tools/function_package/size.ML")
+begin
+
+inductive
+ wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"
+ for R :: "('a * 'a) set"
+ and F :: "('a => 'b) => 'a => 'b"
+where
+ wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>
+ wfrec_rel R F x (F g x)"
+
+constdefs
+ wf :: "('a * 'a)set => bool"
+ "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
+
+ wfP :: "('a => 'a => bool) => bool"
+ "wfP r == wf {(x, y). r x y}"
+
+ acyclic :: "('a*'a)set => bool"
+ "acyclic r == !x. (x,x) ~: r^+"
+
+ cut :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b"
+ "cut f r x == (%y. if (y,x):r then f y else arbitrary)"
+
+ adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool"
+ "adm_wf R F == ALL f g x.
+ (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
+
+ wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"
+ [code func del]: "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
+
+abbreviation acyclicP :: "('a => 'a => bool) => bool" where
+ "acyclicP r == acyclic {(x, y). r x y}"
+
+class wellorder = linorder +
+ assumes wf: "wf {(x, y). x < y}"
+
+
+lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
+ by (simp add: wfP_def)
+
+lemma wfUNIVI:
+ "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
+ unfolding wf_def by blast
+
+lemmas wfPUNIVI = wfUNIVI [to_pred]
+
+text{*Restriction to domain @{term A} and range @{term B}. If @{term r} is
+ well-founded over their intersection, then @{term "wf r"}*}
+lemma wfI:
+ "[| r \<subseteq> A <*> B;
+ !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x; x : A; x : B |] ==> P x |]
+ ==> wf r"
+ unfolding wf_def by blast
+
+lemma wf_induct:
+ "[| wf(r);
+ !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)
+ |] ==> P(a)"
+ unfolding wf_def by blast
+
+lemmas wfP_induct = wf_induct [to_pred]
+
+lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
+
+lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
+
+lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"
+ by (induct a arbitrary: x set: wf) blast
+
+(* [| wf r; ~Z ==> (a,x) : r; (x,a) ~: r ==> Z |] ==> Z *)
+lemmas wf_asym = wf_not_sym [elim_format]
+
+lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"
+ by (blast elim: wf_asym)
+
+(* [| wf r; (a,a) ~: r ==> PROP W |] ==> PROP W *)
+lemmas wf_irrefl = wf_not_refl [elim_format]
+
+text{*transitive closure of a well-founded relation is well-founded! *}
+lemma wf_trancl:
+ assumes "wf r"
+ shows "wf (r^+)"
+proof -
+ {
+ fix P and x
+ assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"
+ have "P x"
+ proof (rule induct_step)
+ fix y assume "(y, x) : r^+"
+ with `wf r` show "P y"
+ proof (induct x arbitrary: y)
+ case (less x)
+ note hyp = `\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'`
+ from `(y, x) : r^+` show "P y"
+ proof cases
+ case base
+ show "P y"
+ proof (rule induct_step)
+ fix y' assume "(y', y) : r^+"
+ with `(y, x) : r` show "P y'" by (rule hyp [of y y'])
+ qed
+ next
+ case step
+ then obtain x' where "(x', x) : r" and "(y, x') : r^+" by simp
+ then show "P y" by (rule hyp [of x' y])
+ qed
+ qed
+ qed
+ } then show ?thesis unfolding wf_def by blast
+qed
+
+lemmas wfP_trancl = wf_trancl [to_pred]
+
+lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
+ apply (subst trancl_converse [symmetric])
+ apply (erule wf_trancl)
+ done
+
+
+subsubsection {* Other simple well-foundedness results *}
+
+text{*Minimal-element characterization of well-foundedness*}
+lemma wf_eq_minimal: "wf r = (\<forall>Q x. x\<in>Q --> (\<exists>z\<in>Q. \<forall>y. (y,z)\<in>r --> y\<notin>Q))"
+proof (intro iffI strip)
+ fix Q :: "'a set" and x
+ assume "wf r" and "x \<in> Q"
+ then show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
+ unfolding wf_def
+ by (blast dest: spec [of _ "%x. x\<in>Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y,z) \<in> r \<longrightarrow> y\<notin>Q)"])
+next
+ assume 1: "\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q)"
+ show "wf r"
+ proof (rule wfUNIVI)
+ fix P :: "'a \<Rightarrow> bool" and x
+ assume 2: "\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x"
+ let ?Q = "{x. \<not> P x}"
+ have "x \<in> ?Q \<longrightarrow> (\<exists>z \<in> ?Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> ?Q)"
+ by (rule 1 [THEN spec, THEN spec])
+ then have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> (\<forall>y. (y, z) \<in> r \<longrightarrow> P y))" by simp
+ with 2 have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> P z)" by fast
+ then show "P x" by simp
+ qed
+qed
+
+lemma wfE_min:
+ assumes "wf R" "x \<in> Q"
+ obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
+ using assms unfolding wf_eq_minimal by blast
+
+lemma wfI_min:
+ "(\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q)
+ \<Longrightarrow> wf R"
+ unfolding wf_eq_minimal by blast
+
+lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
+
+text {* Well-foundedness of subsets *}
+lemma wf_subset: "[| wf(r); p<=r |] ==> wf(p)"
+ apply (simp (no_asm_use) add: wf_eq_minimal)
+ apply fast
+ done
+
+lemmas wfP_subset = wf_subset [to_pred]
+
+text {* Well-foundedness of the empty relation *}
+lemma wf_empty [iff]: "wf({})"
+ by (simp add: wf_def)
+
+lemmas wfP_empty [iff] =
+ wf_empty [to_pred bot_empty_eq2, simplified bot_fun_eq bot_bool_eq]
+
+lemma wf_Int1: "wf r ==> wf (r Int r')"
+ apply (erule wf_subset)
+ apply (rule Int_lower1)
+ done
+
+lemma wf_Int2: "wf r ==> wf (r' Int r)"
+ apply (erule wf_subset)
+ apply (rule Int_lower2)
+ done
+
+text{*Well-foundedness of insert*}
+lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
+apply (rule iffI)
+ apply (blast elim: wf_trancl [THEN wf_irrefl]
+ intro: rtrancl_into_trancl1 wf_subset
+ rtrancl_mono [THEN [2] rev_subsetD])
+apply (simp add: wf_eq_minimal, safe)
+apply (rule allE, assumption, erule impE, blast)
+apply (erule bexE)
+apply (rename_tac "a", case_tac "a = x")
+ prefer 2
+apply blast
+apply (case_tac "y:Q")
+ prefer 2 apply blast
+apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
+ apply assumption
+apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl)
+ --{*essential for speed*}
+txt{*Blast with new substOccur fails*}
+apply (fast intro: converse_rtrancl_into_rtrancl)
+done
+
+text{*Well-foundedness of image*}
+lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun f f ` r)"
+apply (simp only: wf_eq_minimal, clarify)
+apply (case_tac "EX p. f p : Q")
+apply (erule_tac x = "{p. f p : Q}" in allE)
+apply (fast dest: inj_onD, blast)
+done
+
+
+subsubsection {* Well-Foundedness Results for Unions *}
+
+lemma wf_union_compatible:
+ assumes "wf R" "wf S"
+ assumes "S O R \<subseteq> R"
+ shows "wf (R \<union> S)"
+proof (rule wfI_min)
+ fix x :: 'a and Q
+ let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
+ assume "x \<in> Q"
+ obtain a where "a \<in> ?Q'"
+ by (rule wfE_min [OF `wf R` `x \<in> Q`]) blast
+ with `wf S`
+ obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'" by (erule wfE_min)
+ {
+ fix y assume "(y, z) \<in> S"
+ then have "y \<notin> ?Q'" by (rule zmin)
+
+ have "y \<notin> Q"
+ proof
+ assume "y \<in> Q"
+ with `y \<notin> ?Q'`
+ obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
+ from `(w, y) \<in> R` `(y, z) \<in> S` have "(w, z) \<in> S O R" by (rule rel_compI)
+ with `S O R \<subseteq> R` have "(w, z) \<in> R" ..
+ with `z \<in> ?Q'` have "w \<notin> Q" by blast
+ with `w \<in> Q` show False by contradiction
+ qed
+ }
+ with `z \<in> ?Q'` show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
+qed
+
+
+text {* Well-foundedness of indexed union with disjoint domains and ranges *}
+
+lemma wf_UN: "[| ALL i:I. wf(r i);
+ ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}
+ |] ==> wf(UN i:I. r i)"
+apply (simp only: wf_eq_minimal, clarify)
+apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
+ prefer 2
+ apply force
+apply clarify
+apply (drule bspec, assumption)
+apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
+apply (blast elim!: allE)
+done
+
+lemmas wfP_SUP = wf_UN [where I=UNIV and r="\<lambda>i. {(x, y). r i x y}",
+ to_pred SUP_UN_eq2 bot_empty_eq, simplified, standard]
+
+lemma wf_Union:
+ "[| ALL r:R. wf r;
+ ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}
+ |] ==> wf(Union R)"
+apply (simp add: Union_def)
+apply (blast intro: wf_UN)
+done
+
+(*Intuition: we find an (R u S)-min element of a nonempty subset A
+ by case distinction.
+ 1. There is a step a -R-> b with a,b : A.
+ Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
+ By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
+ subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
+ have an S-successor and is thus S-min in A as well.
+ 2. There is no such step.
+ Pick an S-min element of A. In this case it must be an R-min
+ element of A as well.
+
+*)
+lemma wf_Un:
+ "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
+ using wf_union_compatible[of s r]
+ by (auto simp: Un_ac)
+
+lemma wf_union_merge:
+ "wf (R \<union> S) = wf (R O R \<union> R O S \<union> S)" (is "wf ?A = wf ?B")
+proof
+ assume "wf ?A"
+ with wf_trancl have wfT: "wf (?A^+)" .
+ moreover have "?B \<subseteq> ?A^+"
+ by (subst trancl_unfold, subst trancl_unfold) blast
+ ultimately show "wf ?B" by (rule wf_subset)
+next
+ assume "wf ?B"
+
+ show "wf ?A"
+ proof (rule wfI_min)
+ fix Q :: "'a set" and x
+ assume "x \<in> Q"
+
+ with `wf ?B`
+ obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q"
+ by (erule wfE_min)
+ then have A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
+ and A2: "\<And>y. (y, z) \<in> R O S \<Longrightarrow> y \<notin> Q"
+ and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
+ by auto
+
+ show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
+ proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
+ case True
+ with `z \<in> Q` A3 show ?thesis by blast
+ next
+ case False
+ then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
+
+ have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
+ proof (intro allI impI)
+ fix y assume "(y, z') \<in> ?A"
+ then show "y \<notin> Q"
+ proof
+ assume "(y, z') \<in> R"
+ then have "(y, z) \<in> R O R" using `(z', z) \<in> R` ..
+ with A1 show "y \<notin> Q" .
+ next
+ assume "(y, z') \<in> S"
+ then have "(y, z) \<in> R O S" using `(z', z) \<in> R` ..
+ with A2 show "y \<notin> Q" .
+ qed
+ qed
+ with `z' \<in> Q` show ?thesis ..
+ qed
+ qed
+qed
+
+lemma wf_comp_self: "wf R = wf (R O R)" -- {* special case *}
+ by (rule wf_union_merge [where S = "{}", simplified])
+
+
+subsubsection {* acyclic *}
+
+lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
+ by (simp add: acyclic_def)
+
+lemma wf_acyclic: "wf r ==> acyclic r"
+apply (simp add: acyclic_def)
+apply (blast elim: wf_trancl [THEN wf_irrefl])
+done
+
+lemmas wfP_acyclicP = wf_acyclic [to_pred]
+
+lemma acyclic_insert [iff]:
+ "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
+apply (simp add: acyclic_def trancl_insert)
+apply (blast intro: rtrancl_trans)
+done
+
+lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
+by (simp add: acyclic_def trancl_converse)
+
+lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
+
+lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
+apply (simp add: acyclic_def antisym_def)
+apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
+done
+
+(* Other direction:
+acyclic = no loops
+antisym = only self loops
+Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
+==> antisym( r^* ) = acyclic(r - Id)";
+*)
+
+lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
+apply (simp add: acyclic_def)
+apply (blast intro: trancl_mono)
+done
+
+text{* Wellfoundedness of finite acyclic relations*}
+
+lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
+apply (erule finite_induct, blast)
+apply (simp (no_asm_simp) only: split_tupled_all)
+apply simp
+done
+
+lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
+apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
+apply (erule acyclic_converse [THEN iffD2])
+done
+
+lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
+by (blast intro: finite_acyclic_wf wf_acyclic)
+
+
+subsection{*Well-Founded Recursion*}
+
+text{*cut*}
+
+lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
+by (simp add: expand_fun_eq cut_def)
+
+lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
+by (simp add: cut_def)
+
+text{*Inductive characterization of wfrec combinator; for details see:
+John Harrison, "Inductive definitions: automation and application"*}
+
+lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y"
+apply (simp add: adm_wf_def)
+apply (erule_tac a=x in wf_induct)
+apply (rule ex1I)
+apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI)
+apply (fast dest!: theI')
+apply (erule wfrec_rel.cases, simp)
+apply (erule allE, erule allE, erule allE, erule mp)
+apply (fast intro: the_equality [symmetric])
+done
+
+lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
+apply (simp add: adm_wf_def)
+apply (intro strip)
+apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
+apply (rule refl)
+done
+
+lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
+apply (simp add: wfrec_def)
+apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
+apply (rule wfrec_rel.wfrecI)
+apply (intro strip)
+apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
+done
+
+subsection {* Code generator setup *}
+
+consts_code
+ "wfrec" ("\<module>wfrec?")
+attach {*
+fun wfrec f x = f (wfrec f) x;
+*}
+
+
+subsection {*LEAST and wellorderings*}
+
+text{* See also @{text wf_linord_ex_has_least} and its consequences in
+ @{text Wellfounded_Relations.ML}*}
+
+lemma wellorder_Least_lemma [rule_format]:
+ "P (k::'a::wellorder) --> P (LEAST x. P(x)) & (LEAST x. P(x)) <= k"
+apply (rule_tac a = k in wf [THEN wf_induct])
+apply (rule impI)
+apply (rule classical)
+apply (rule_tac s = x in Least_equality [THEN ssubst], auto)
+apply (auto simp add: linorder_not_less [symmetric])
+done
+
+lemmas LeastI = wellorder_Least_lemma [THEN conjunct1, standard]
+lemmas Least_le = wellorder_Least_lemma [THEN conjunct2, standard]
+
+-- "The following 3 lemmas are due to Brian Huffman"
+lemma LeastI_ex: "EX x::'a::wellorder. P x ==> P (Least P)"
+apply (erule exE)
+apply (erule LeastI)
+done
+
+lemma LeastI2:
+ "[| P (a::'a::wellorder); !!x. P x ==> Q x |] ==> Q (Least P)"
+by (blast intro: LeastI)
+
+lemma LeastI2_ex:
+ "[| EX a::'a::wellorder. P a; !!x. P x ==> Q x |] ==> Q (Least P)"
+by (blast intro: LeastI_ex)
+
+lemma not_less_Least: "[| k < (LEAST x. P x) |] ==> ~P (k::'a::wellorder)"
+apply (simp (no_asm_use) add: linorder_not_le [symmetric])
+apply (erule contrapos_nn)
+apply (erule Least_le)
+done
+
+subsection {* @{typ nat} is well-founded *}
+
+lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
+proof (rule ext, rule ext, rule iffI)
+ fix n m :: nat
+ assume "m < n"
+ then show "(\<lambda>m n. n = Suc m)^++ m n"
+ proof (induct n)
+ case 0 then show ?case by auto
+ next
+ case (Suc n) then show ?case
+ by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
+ qed
+next
+ fix n m :: nat
+ assume "(\<lambda>m n. n = Suc m)^++ m n"
+ then show "m < n"
+ by (induct n)
+ (simp_all add: less_Suc_eq_le reflexive le_less)
+qed
+
+definition
+ pred_nat :: "(nat * nat) set" where
+ "pred_nat = {(m, n). n = Suc m}"
+
+definition
+ less_than :: "(nat * nat) set" where
+ "less_than = pred_nat^+"
+
+lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
+ unfolding less_nat_rel pred_nat_def trancl_def by simp
+
+lemma pred_nat_trancl_eq_le:
+ "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
+ unfolding less_eq rtrancl_eq_or_trancl by auto
+
+lemma wf_pred_nat: "wf pred_nat"
+ apply (unfold wf_def pred_nat_def, clarify)
+ apply (induct_tac x, blast+)
+ done
+
+lemma wf_less_than [iff]: "wf less_than"
+ by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
+
+lemma trans_less_than [iff]: "trans less_than"
+ by (simp add: less_than_def trans_trancl)
+
+lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
+ by (simp add: less_than_def less_eq)
+
+lemma wf_less: "wf {(x, y::nat). x < y}"
+ using wf_less_than by (simp add: less_than_def less_eq [symmetric])
+
+text {* Type @{typ nat} is a wellfounded order *}
+
+instance nat :: wellorder
+ by intro_classes
+ (assumption |
+ rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+
+
+text {* @{text LEAST} theorems for type @{typ nat}*}
+
+lemma Least_Suc:
+ "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
+ apply (case_tac "n", auto)
+ apply (frule LeastI)
+ apply (drule_tac P = "%x. P (Suc x) " in LeastI)
+ apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
+ apply (erule_tac [2] Least_le)
+ apply (case_tac "LEAST x. P x", auto)
+ apply (drule_tac P = "%x. P (Suc x) " in Least_le)
+ apply (blast intro: order_antisym)
+ done
+
+lemma Least_Suc2:
+ "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
+ apply (erule (1) Least_Suc [THEN ssubst])
+ apply simp
+ done
+
+lemma ex_least_nat_le: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not>P i) & P(k)"
+ apply (cases n)
+ apply blast
+ apply (rule_tac x="LEAST k. P(k)" in exI)
+ apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
+ done
+
+lemma ex_least_nat_less: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not>P i) & P(k+1)"
+ apply (cases n)
+ apply blast
+ apply (frule (1) ex_least_nat_le)
+ apply (erule exE)
+ apply (case_tac k)
+ apply simp
+ apply (rename_tac k1)
+ apply (rule_tac x=k1 in exI)
+ apply fastsimp
+ done
+
+
+subsection {* Accessible Part *}
+
+text {*
+ Inductive definition of the accessible part @{term "acc r"} of a
+ relation; see also \cite{paulin-tlca}.
+*}
+
+inductive_set
+ acc :: "('a * 'a) set => 'a set"
+ for r :: "('a * 'a) set"
+ where
+ accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
+
+abbreviation
+ termip :: "('a => 'a => bool) => 'a => bool" where
+ "termip r == accp (r\<inverse>\<inverse>)"
+
+abbreviation
+ termi :: "('a * 'a) set => 'a set" where
+ "termi r == acc (r\<inverse>)"
+
+lemmas accpI = accp.accI
+
+text {* Induction rules *}
+
+theorem accp_induct:
+ assumes major: "accp r a"
+ assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"
+ shows "P a"
+ apply (rule major [THEN accp.induct])
+ apply (rule hyp)
+ apply (rule accp.accI)
+ apply fast
+ apply fast
+ done
+
+theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]
+
+theorem accp_downward: "accp r b ==> r a b ==> accp r a"
+ apply (erule accp.cases)
+ apply fast
+ done
+
+lemma not_accp_down:
+ assumes na: "\<not> accp R x"
+ obtains z where "R z x" and "\<not> accp R z"
+proof -
+ assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
+
+ show thesis
+ proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
+ case True
+ hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto
+ hence "accp R x"
+ by (rule accp.accI)
+ with na show thesis ..
+ next
+ case False then obtain z where "R z x" and "\<not> accp R z"
+ by auto
+ with a show thesis .
+ qed
+qed
+
+lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"
+ apply (erule rtranclp_induct)
+ apply blast
+ apply (blast dest: accp_downward)
+ done
+
+theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"
+ apply (blast dest: accp_downwards_aux)
+ done
+
+theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
+ apply (rule wfPUNIVI)
+ apply (induct_tac P x rule: accp_induct)
+ apply blast
+ apply blast
+ done
+
+theorem accp_wfPD: "wfP r ==> accp r x"
+ apply (erule wfP_induct_rule)
+ apply (rule accp.accI)
+ apply blast
+ done
+
+theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
+ apply (blast intro: accp_wfPI dest: accp_wfPD)
+ done
+
+
+text {* Smaller relations have bigger accessible parts: *}
+
+lemma accp_subset:
+ assumes sub: "R1 \<le> R2"
+ shows "accp R2 \<le> accp R1"
+proof
+ fix x assume "accp R2 x"
+ then show "accp R1 x"
+ proof (induct x)
+ fix x
+ assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
+ with sub show "accp R1 x"
+ by (blast intro: accp.accI)
+ qed
+qed
+
+
+text {* This is a generalized induction theorem that works on
+ subsets of the accessible part. *}
+
+lemma accp_subset_induct:
+ assumes subset: "D \<le> accp R"
+ and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
+ and "D x"
+ and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
+ shows "P x"
+proof -
+ from subset and `D x`
+ have "accp R x" ..
+ then show "P x" using `D x`
+ proof (induct x)
+ fix x
+ assume "D x"
+ and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
+ with dcl and istep show "P x" by blast
+ qed
+qed
+
+
+text {* Set versions of the above theorems *}
+
+lemmas acc_induct = accp_induct [to_set]
+
+lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
+
+lemmas acc_downward = accp_downward [to_set]
+
+lemmas not_acc_down = not_accp_down [to_set]
+
+lemmas acc_downwards_aux = accp_downwards_aux [to_set]
+
+lemmas acc_downwards = accp_downwards [to_set]
+
+lemmas acc_wfI = accp_wfPI [to_set]
+
+lemmas acc_wfD = accp_wfPD [to_set]
+
+lemmas wf_acc_iff = wfP_accp_iff [to_set]
+
+lemmas acc_subset = accp_subset [to_set]
+
+lemmas acc_subset_induct = accp_subset_induct [to_set]
+
+
+subsection {* Tools for building wellfounded relations *}
+
+text {* Inverse Image *}
+
+lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
+apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
+apply clarify
+apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
+prefer 2 apply (blast del: allE)
+apply (erule allE)
+apply (erule (1) notE impE)
+apply blast
+done
+
+lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
+ by (auto simp:inv_image_def)
+
+text {* Measure functions into @{typ nat} *}
+
+definition measure :: "('a => nat) => ('a * 'a)set"
+where "measure == inv_image less_than"
+
+lemma in_measure[simp]: "((x,y) : measure f) = (f x < f y)"
+ by (simp add:measure_def)
+
+lemma wf_measure [iff]: "wf (measure f)"
+apply (unfold measure_def)
+apply (rule wf_less_than [THEN wf_inv_image])
+done
+
+text{* Lexicographic combinations *}
+
+definition
+ lex_prod :: "[('a*'a)set, ('b*'b)set] => (('a*'b)*('a*'b))set"
+ (infixr "<*lex*>" 80)
+where
+ "ra <*lex*> rb == {((a,b),(a',b')). (a,a') : ra | a=a' & (b,b') : rb}"
+
+lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
+apply (unfold wf_def lex_prod_def)
+apply (rule allI, rule impI)
+apply (simp (no_asm_use) only: split_paired_All)
+apply (drule spec, erule mp)
+apply (rule allI, rule impI)
+apply (drule spec, erule mp, blast)
+done
+
+lemma in_lex_prod[simp]:
+ "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
+ by (auto simp:lex_prod_def)
+
+text{* @{term "op <*lex*>"} preserves transitivity *}
+
+lemma trans_lex_prod [intro!]:
+ "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
+by (unfold trans_def lex_prod_def, blast)
+
+text {* lexicographic combinations with measure functions *}
+
+definition
+ mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
+where
+ "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
+
+lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
+unfolding mlex_prod_def
+by auto
+
+lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
+unfolding mlex_prod_def by simp
+
+lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
+unfolding mlex_prod_def by auto
+
+text {* proper subset relation on finite sets *}
+
+definition finite_psubset :: "('a set * 'a set) set"
+where "finite_psubset == {(A,B). A < B & finite B}"
+
+lemma wf_finite_psubset: "wf(finite_psubset)"
+apply (unfold finite_psubset_def)
+apply (rule wf_measure [THEN wf_subset])
+apply (simp add: measure_def inv_image_def less_than_def less_eq)
+apply (fast elim!: psubset_card_mono)
+done
+
+lemma trans_finite_psubset: "trans finite_psubset"
+by (simp add: finite_psubset_def psubset_def trans_def, blast)
+
+
+
+
+text {*Wellfoundedness of @{text same_fst}*}
+
+definition
+ same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"
+where
+ "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"
+ --{*For @{text rec_def} declarations where the first n parameters
+ stay unchanged in the recursive call.
+ See @{text "Library/While_Combinator.thy"} for an application.*}
+
+lemma same_fstI [intro!]:
+ "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"
+by (simp add: same_fst_def)
+
+lemma wf_same_fst:
+ assumes prem: "(!!x. P x ==> wf(R x))"
+ shows "wf(same_fst P R)"
+apply (simp cong del: imp_cong add: wf_def same_fst_def)
+apply (intro strip)
+apply (rename_tac a b)
+apply (case_tac "wf (R a)")
+ apply (erule_tac a = b in wf_induct, blast)
+apply (blast intro: prem)
+done
+
+
+subsection{*Weakly decreasing sequences (w.r.t. some well-founded order)
+ stabilize.*}
+
+text{*This material does not appear to be used any longer.*}
+
+lemma lemma1: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"
+apply (induct_tac "k", simp_all)
+apply (blast intro: rtrancl_trans)
+done
+
+lemma lemma2: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]
+ ==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"
+apply (erule wf_induct, clarify)
+apply (case_tac "EX j. (f (m+j), f m) : r^+")
+ apply clarify
+ apply (subgoal_tac "EX i. ALL k. f ((m+j) +i+k) = f ( (m+j) +i) ")
+ apply clarify
+ apply (rule_tac x = "j+i" in exI)
+ apply (simp add: add_ac, blast)
+apply (rule_tac x = 0 in exI, clarsimp)
+apply (drule_tac i = m and k = k in lemma1)
+apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
+done
+
+lemma wf_weak_decr_stable: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]
+ ==> EX i. ALL k. f (i+k) = f i"
+apply (drule_tac x = 0 in lemma2 [THEN spec], auto)
+done
+
+(* special case of the theorem above: <= *)
+lemma weak_decr_stable:
+ "ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"
+apply (rule_tac r = pred_nat in wf_weak_decr_stable)
+apply (simp add: pred_nat_trancl_eq_le)
+apply (intro wf_trancl wf_pred_nat)
+done
+
+
+subsection {* size of a datatype value *}
+
+use "Tools/function_package/size.ML"
+
+setup Size.setup
+
+lemma nat_size [simp, code func]: "size (n\<Colon>nat) = n"
+ by (induct n) simp_all
+
+
+end
--- a/src/HOL/Wellfounded_Recursion.thy Thu Apr 24 16:53:04 2008 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,693 +0,0 @@
-(* ID: $Id$
- Author: Tobias Nipkow
- Copyright 1992 University of Cambridge
-*)
-
-header {*Well-founded Recursion*}
-
-theory Wellfounded_Recursion
-imports Transitive_Closure Nat
-uses ("Tools/function_package/size.ML")
-begin
-
-inductive
- wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"
- for R :: "('a * 'a) set"
- and F :: "('a => 'b) => 'a => 'b"
-where
- wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>
- wfrec_rel R F x (F g x)"
-
-constdefs
- wf :: "('a * 'a)set => bool"
- "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
-
- wfP :: "('a => 'a => bool) => bool"
- "wfP r == wf {(x, y). r x y}"
-
- acyclic :: "('a*'a)set => bool"
- "acyclic r == !x. (x,x) ~: r^+"
-
- cut :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b"
- "cut f r x == (%y. if (y,x):r then f y else arbitrary)"
-
- adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool"
- "adm_wf R F == ALL f g x.
- (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
-
- wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"
- [code func del]: "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
-
-abbreviation acyclicP :: "('a => 'a => bool) => bool" where
- "acyclicP r == acyclic {(x, y). r x y}"
-
-class wellorder = linorder +
- assumes wf: "wf {(x, y). x < y}"
-
-
-lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
- by (simp add: wfP_def)
-
-lemma wfUNIVI:
- "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
- unfolding wf_def by blast
-
-lemmas wfPUNIVI = wfUNIVI [to_pred]
-
-text{*Restriction to domain @{term A} and range @{term B}. If @{term r} is
- well-founded over their intersection, then @{term "wf r"}*}
-lemma wfI:
- "[| r \<subseteq> A <*> B;
- !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x; x : A; x : B |] ==> P x |]
- ==> wf r"
- unfolding wf_def by blast
-
-lemma wf_induct:
- "[| wf(r);
- !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)
- |] ==> P(a)"
- unfolding wf_def by blast
-
-lemmas wfP_induct = wf_induct [to_pred]
-
-lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
-
-lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
-
-lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"
- by (induct a arbitrary: x set: wf) blast
-
-(* [| wf r; ~Z ==> (a,x) : r; (x,a) ~: r ==> Z |] ==> Z *)
-lemmas wf_asym = wf_not_sym [elim_format]
-
-lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"
- by (blast elim: wf_asym)
-
-(* [| wf r; (a,a) ~: r ==> PROP W |] ==> PROP W *)
-lemmas wf_irrefl = wf_not_refl [elim_format]
-
-text{*transitive closure of a well-founded relation is well-founded! *}
-lemma wf_trancl:
- assumes "wf r"
- shows "wf (r^+)"
-proof -
- {
- fix P and x
- assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"
- have "P x"
- proof (rule induct_step)
- fix y assume "(y, x) : r^+"
- with `wf r` show "P y"
- proof (induct x arbitrary: y)
- case (less x)
- note hyp = `\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'`
- from `(y, x) : r^+` show "P y"
- proof cases
- case base
- show "P y"
- proof (rule induct_step)
- fix y' assume "(y', y) : r^+"
- with `(y, x) : r` show "P y'" by (rule hyp [of y y'])
- qed
- next
- case step
- then obtain x' where "(x', x) : r" and "(y, x') : r^+" by simp
- then show "P y" by (rule hyp [of x' y])
- qed
- qed
- qed
- } then show ?thesis unfolding wf_def by blast
-qed
-
-lemmas wfP_trancl = wf_trancl [to_pred]
-
-lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
- apply (subst trancl_converse [symmetric])
- apply (erule wf_trancl)
- done
-
-
-subsubsection {* Other simple well-foundedness results *}
-
-text{*Minimal-element characterization of well-foundedness*}
-lemma wf_eq_minimal: "wf r = (\<forall>Q x. x\<in>Q --> (\<exists>z\<in>Q. \<forall>y. (y,z)\<in>r --> y\<notin>Q))"
-proof (intro iffI strip)
- fix Q :: "'a set" and x
- assume "wf r" and "x \<in> Q"
- then show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
- unfolding wf_def
- by (blast dest: spec [of _ "%x. x\<in>Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y,z) \<in> r \<longrightarrow> y\<notin>Q)"])
-next
- assume 1: "\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q)"
- show "wf r"
- proof (rule wfUNIVI)
- fix P :: "'a \<Rightarrow> bool" and x
- assume 2: "\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x"
- let ?Q = "{x. \<not> P x}"
- have "x \<in> ?Q \<longrightarrow> (\<exists>z \<in> ?Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> ?Q)"
- by (rule 1 [THEN spec, THEN spec])
- then have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> (\<forall>y. (y, z) \<in> r \<longrightarrow> P y))" by simp
- with 2 have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> P z)" by fast
- then show "P x" by simp
- qed
-qed
-
-lemma wfE_min:
- assumes "wf R" "x \<in> Q"
- obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
- using assms unfolding wf_eq_minimal by blast
-
-lemma wfI_min:
- "(\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q)
- \<Longrightarrow> wf R"
- unfolding wf_eq_minimal by blast
-
-lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
-
-text {* Well-foundedness of subsets *}
-lemma wf_subset: "[| wf(r); p<=r |] ==> wf(p)"
- apply (simp (no_asm_use) add: wf_eq_minimal)
- apply fast
- done
-
-lemmas wfP_subset = wf_subset [to_pred]
-
-text {* Well-foundedness of the empty relation *}
-lemma wf_empty [iff]: "wf({})"
- by (simp add: wf_def)
-
-lemmas wfP_empty [iff] =
- wf_empty [to_pred bot_empty_eq2, simplified bot_fun_eq bot_bool_eq]
-
-lemma wf_Int1: "wf r ==> wf (r Int r')"
- apply (erule wf_subset)
- apply (rule Int_lower1)
- done
-
-lemma wf_Int2: "wf r ==> wf (r' Int r)"
- apply (erule wf_subset)
- apply (rule Int_lower2)
- done
-
-text{*Well-foundedness of insert*}
-lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
-apply (rule iffI)
- apply (blast elim: wf_trancl [THEN wf_irrefl]
- intro: rtrancl_into_trancl1 wf_subset
- rtrancl_mono [THEN [2] rev_subsetD])
-apply (simp add: wf_eq_minimal, safe)
-apply (rule allE, assumption, erule impE, blast)
-apply (erule bexE)
-apply (rename_tac "a", case_tac "a = x")
- prefer 2
-apply blast
-apply (case_tac "y:Q")
- prefer 2 apply blast
-apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
- apply assumption
-apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl)
- --{*essential for speed*}
-txt{*Blast with new substOccur fails*}
-apply (fast intro: converse_rtrancl_into_rtrancl)
-done
-
-text{*Well-foundedness of image*}
-lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun f f ` r)"
-apply (simp only: wf_eq_minimal, clarify)
-apply (case_tac "EX p. f p : Q")
-apply (erule_tac x = "{p. f p : Q}" in allE)
-apply (fast dest: inj_onD, blast)
-done
-
-
-subsubsection {* Well-Foundedness Results for Unions *}
-
-lemma wf_union_compatible:
- assumes "wf R" "wf S"
- assumes "S O R \<subseteq> R"
- shows "wf (R \<union> S)"
-proof (rule wfI_min)
- fix x :: 'a and Q
- let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
- assume "x \<in> Q"
- obtain a where "a \<in> ?Q'"
- by (rule wfE_min [OF `wf R` `x \<in> Q`]) blast
- with `wf S`
- obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'" by (erule wfE_min)
- {
- fix y assume "(y, z) \<in> S"
- then have "y \<notin> ?Q'" by (rule zmin)
-
- have "y \<notin> Q"
- proof
- assume "y \<in> Q"
- with `y \<notin> ?Q'`
- obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
- from `(w, y) \<in> R` `(y, z) \<in> S` have "(w, z) \<in> S O R" by (rule rel_compI)
- with `S O R \<subseteq> R` have "(w, z) \<in> R" ..
- with `z \<in> ?Q'` have "w \<notin> Q" by blast
- with `w \<in> Q` show False by contradiction
- qed
- }
- with `z \<in> ?Q'` show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
-qed
-
-
-text {* Well-foundedness of indexed union with disjoint domains and ranges *}
-
-lemma wf_UN: "[| ALL i:I. wf(r i);
- ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}
- |] ==> wf(UN i:I. r i)"
-apply (simp only: wf_eq_minimal, clarify)
-apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
- prefer 2
- apply force
-apply clarify
-apply (drule bspec, assumption)
-apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
-apply (blast elim!: allE)
-done
-
-lemmas wfP_SUP = wf_UN [where I=UNIV and r="\<lambda>i. {(x, y). r i x y}",
- to_pred SUP_UN_eq2 bot_empty_eq, simplified, standard]
-
-lemma wf_Union:
- "[| ALL r:R. wf r;
- ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}
- |] ==> wf(Union R)"
-apply (simp add: Union_def)
-apply (blast intro: wf_UN)
-done
-
-(*Intuition: we find an (R u S)-min element of a nonempty subset A
- by case distinction.
- 1. There is a step a -R-> b with a,b : A.
- Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
- By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
- subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
- have an S-successor and is thus S-min in A as well.
- 2. There is no such step.
- Pick an S-min element of A. In this case it must be an R-min
- element of A as well.
-
-*)
-lemma wf_Un:
- "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
- using wf_union_compatible[of s r]
- by (auto simp: Un_ac)
-
-lemma wf_union_merge:
- "wf (R \<union> S) = wf (R O R \<union> R O S \<union> S)" (is "wf ?A = wf ?B")
-proof
- assume "wf ?A"
- with wf_trancl have wfT: "wf (?A^+)" .
- moreover have "?B \<subseteq> ?A^+"
- by (subst trancl_unfold, subst trancl_unfold) blast
- ultimately show "wf ?B" by (rule wf_subset)
-next
- assume "wf ?B"
-
- show "wf ?A"
- proof (rule wfI_min)
- fix Q :: "'a set" and x
- assume "x \<in> Q"
-
- with `wf ?B`
- obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q"
- by (erule wfE_min)
- then have A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
- and A2: "\<And>y. (y, z) \<in> R O S \<Longrightarrow> y \<notin> Q"
- and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
- by auto
-
- show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
- proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
- case True
- with `z \<in> Q` A3 show ?thesis by blast
- next
- case False
- then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
-
- have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
- proof (intro allI impI)
- fix y assume "(y, z') \<in> ?A"
- then show "y \<notin> Q"
- proof
- assume "(y, z') \<in> R"
- then have "(y, z) \<in> R O R" using `(z', z) \<in> R` ..
- with A1 show "y \<notin> Q" .
- next
- assume "(y, z') \<in> S"
- then have "(y, z) \<in> R O S" using `(z', z) \<in> R` ..
- with A2 show "y \<notin> Q" .
- qed
- qed
- with `z' \<in> Q` show ?thesis ..
- qed
- qed
-qed
-
-lemma wf_comp_self: "wf R = wf (R O R)" -- {* special case *}
- by (rule wf_union_merge [where S = "{}", simplified])
-
-
-subsubsection {* acyclic *}
-
-lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
- by (simp add: acyclic_def)
-
-lemma wf_acyclic: "wf r ==> acyclic r"
-apply (simp add: acyclic_def)
-apply (blast elim: wf_trancl [THEN wf_irrefl])
-done
-
-lemmas wfP_acyclicP = wf_acyclic [to_pred]
-
-lemma acyclic_insert [iff]:
- "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
-apply (simp add: acyclic_def trancl_insert)
-apply (blast intro: rtrancl_trans)
-done
-
-lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
-by (simp add: acyclic_def trancl_converse)
-
-lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
-
-lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
-apply (simp add: acyclic_def antisym_def)
-apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
-done
-
-(* Other direction:
-acyclic = no loops
-antisym = only self loops
-Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
-==> antisym( r^* ) = acyclic(r - Id)";
-*)
-
-lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
-apply (simp add: acyclic_def)
-apply (blast intro: trancl_mono)
-done
-
-
-subsection{*Well-Founded Recursion*}
-
-text{*cut*}
-
-lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
-by (simp add: expand_fun_eq cut_def)
-
-lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
-by (simp add: cut_def)
-
-text{*Inductive characterization of wfrec combinator; for details see:
-John Harrison, "Inductive definitions: automation and application"*}
-
-lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y"
-apply (simp add: adm_wf_def)
-apply (erule_tac a=x in wf_induct)
-apply (rule ex1I)
-apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI)
-apply (fast dest!: theI')
-apply (erule wfrec_rel.cases, simp)
-apply (erule allE, erule allE, erule allE, erule mp)
-apply (fast intro: the_equality [symmetric])
-done
-
-lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
-apply (simp add: adm_wf_def)
-apply (intro strip)
-apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
-apply (rule refl)
-done
-
-lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
-apply (simp add: wfrec_def)
-apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
-apply (rule wfrec_rel.wfrecI)
-apply (intro strip)
-apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
-done
-
-
-text{** This form avoids giant explosions in proofs. NOTE USE OF ==*}
-lemma def_wfrec: "[| f==wfrec r H; wf(r) |] ==> f(a) = H (cut f r a) a"
-apply auto
-apply (blast intro: wfrec)
-done
-
-
-subsection {* Code generator setup *}
-
-consts_code
- "wfrec" ("\<module>wfrec?")
-attach {*
-fun wfrec f x = f (wfrec f) x;
-*}
-
-
-subsection{*Variants for TFL: the Recdef Package*}
-
-lemma tfl_wf_induct: "ALL R. wf R -->
- (ALL P. (ALL x. (ALL y. (y,x):R --> P y) --> P x) --> (ALL x. P x))"
-apply clarify
-apply (rule_tac r = R and P = P and a = x in wf_induct, assumption, blast)
-done
-
-lemma tfl_cut_apply: "ALL f R. (x,a):R --> (cut f R a)(x) = f(x)"
-apply clarify
-apply (rule cut_apply, assumption)
-done
-
-lemma tfl_wfrec:
- "ALL M R f. (f=wfrec R M) --> wf R --> (ALL x. f x = M (cut f R x) x)"
-apply clarify
-apply (erule wfrec)
-done
-
-subsection {*LEAST and wellorderings*}
-
-text{* See also @{text wf_linord_ex_has_least} and its consequences in
- @{text Wellfounded_Relations.ML}*}
-
-lemma wellorder_Least_lemma [rule_format]:
- "P (k::'a::wellorder) --> P (LEAST x. P(x)) & (LEAST x. P(x)) <= k"
-apply (rule_tac a = k in wf [THEN wf_induct])
-apply (rule impI)
-apply (rule classical)
-apply (rule_tac s = x in Least_equality [THEN ssubst], auto)
-apply (auto simp add: linorder_not_less [symmetric])
-done
-
-lemmas LeastI = wellorder_Least_lemma [THEN conjunct1, standard]
-lemmas Least_le = wellorder_Least_lemma [THEN conjunct2, standard]
-
--- "The following 3 lemmas are due to Brian Huffman"
-lemma LeastI_ex: "EX x::'a::wellorder. P x ==> P (Least P)"
-apply (erule exE)
-apply (erule LeastI)
-done
-
-lemma LeastI2:
- "[| P (a::'a::wellorder); !!x. P x ==> Q x |] ==> Q (Least P)"
-by (blast intro: LeastI)
-
-lemma LeastI2_ex:
- "[| EX a::'a::wellorder. P a; !!x. P x ==> Q x |] ==> Q (Least P)"
-by (blast intro: LeastI_ex)
-
-lemma not_less_Least: "[| k < (LEAST x. P x) |] ==> ~P (k::'a::wellorder)"
-apply (simp (no_asm_use) add: linorder_not_le [symmetric])
-apply (erule contrapos_nn)
-apply (erule Least_le)
-done
-
-subsection {* @{typ nat} is well-founded *}
-
-lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
-proof (rule ext, rule ext, rule iffI)
- fix n m :: nat
- assume "m < n"
- then show "(\<lambda>m n. n = Suc m)^++ m n"
- proof (induct n)
- case 0 then show ?case by auto
- next
- case (Suc n) then show ?case
- by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
- qed
-next
- fix n m :: nat
- assume "(\<lambda>m n. n = Suc m)^++ m n"
- then show "m < n"
- by (induct n)
- (simp_all add: less_Suc_eq_le reflexive le_less)
-qed
-
-definition
- pred_nat :: "(nat * nat) set" where
- "pred_nat = {(m, n). n = Suc m}"
-
-definition
- less_than :: "(nat * nat) set" where
- "less_than = pred_nat^+"
-
-lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
- unfolding less_nat_rel pred_nat_def trancl_def by simp
-
-lemma pred_nat_trancl_eq_le:
- "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
- unfolding less_eq rtrancl_eq_or_trancl by auto
-
-lemma wf_pred_nat: "wf pred_nat"
- apply (unfold wf_def pred_nat_def, clarify)
- apply (induct_tac x, blast+)
- done
-
-lemma wf_less_than [iff]: "wf less_than"
- by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
-
-lemma trans_less_than [iff]: "trans less_than"
- by (simp add: less_than_def trans_trancl)
-
-lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
- by (simp add: less_than_def less_eq)
-
-lemma wf_less: "wf {(x, y::nat). x < y}"
- using wf_less_than by (simp add: less_than_def less_eq [symmetric])
-
-text {* Complete induction, aka course-of-values induction *}
-lemma nat_less_induct:
- assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
- apply (induct n rule: wf_induct [OF wf_pred_nat [THEN wf_trancl]])
- apply (rule assms)
- apply (unfold less_eq [symmetric], assumption)
- done
-
-lemmas less_induct = nat_less_induct [rule_format, case_names less]
-
-text {* Type @{typ nat} is a wellfounded order *}
-
-instance nat :: wellorder
- by intro_classes
- (assumption |
- rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+
-
-lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
- apply (rule nat_less_induct)
- apply (case_tac n)
- apply (case_tac [2] nat)
- apply (blast intro: less_trans)+
- done
-
-text {* The method of infinite descent, frequently used in number theory.
-Provided by Roelof Oosterhuis.
-$P(n)$ is true for all $n\in\mathbb{N}$ if
-\begin{itemize}
- \item case ``0'': given $n=0$ prove $P(n)$,
- \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
- a smaller integer $m$ such that $\neg P(m)$.
-\end{itemize} *}
-
-lemma infinite_descent0[case_names 0 smaller]:
- "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
-by (induct n rule: less_induct, case_tac "n>0", auto)
-
-text{* A compact version without explicit base case: *}
-lemma infinite_descent:
- "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m \<rbrakk> \<Longrightarrow> P n"
-by (induct n rule: less_induct, auto)
-
-text {*
-Infinite descent using a mapping to $\mathbb{N}$:
-$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
-\begin{itemize}
-\item case ``0'': given $V(x)=0$ prove $P(x)$,
-\item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$.
-\end{itemize}
-NB: the proof also shows how to use the previous lemma. *}
-
-corollary infinite_descent0_measure [case_names 0 smaller]:
- assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
- and A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
- shows "P x"
-proof -
- obtain n where "n = V x" by auto
- moreover have "\<And>x. V x = n \<Longrightarrow> P x"
- proof (induct n rule: infinite_descent0)
- case 0 -- "i.e. $V(x) = 0$"
- with A0 show "P x" by auto
- next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
- case (smaller n)
- then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
- with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
- with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
- then show ?case by auto
- qed
- ultimately show "P x" by auto
-qed
-
-text{* Again, without explicit base case: *}
-lemma infinite_descent_measure:
-assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
-proof -
- from assms obtain n where "n = V x" by auto
- moreover have "!!x. V x = n \<Longrightarrow> P x"
- proof (induct n rule: infinite_descent, auto)
- fix x assume "\<not> P x"
- with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
- qed
- ultimately show "P x" by auto
-qed
-
-text {* @{text LEAST} theorems for type @{typ nat}*}
-
-lemma Least_Suc:
- "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
- apply (case_tac "n", auto)
- apply (frule LeastI)
- apply (drule_tac P = "%x. P (Suc x) " in LeastI)
- apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
- apply (erule_tac [2] Least_le)
- apply (case_tac "LEAST x. P x", auto)
- apply (drule_tac P = "%x. P (Suc x) " in Least_le)
- apply (blast intro: order_antisym)
- done
-
-lemma Least_Suc2:
- "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
- apply (erule (1) Least_Suc [THEN ssubst])
- apply simp
- done
-
-lemma ex_least_nat_le: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not>P i) & P(k)"
- apply (cases n)
- apply blast
- apply (rule_tac x="LEAST k. P(k)" in exI)
- apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
- done
-
-lemma ex_least_nat_less: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not>P i) & P(k+1)"
- apply (cases n)
- apply blast
- apply (frule (1) ex_least_nat_le)
- apply (erule exE)
- apply (case_tac k)
- apply simp
- apply (rename_tac k1)
- apply (rule_tac x=k1 in exI)
- apply fastsimp
- done
-
-
-subsection {* size of a datatype value *}
-
-use "Tools/function_package/size.ML"
-
-setup Size.setup
-
-lemma nat_size [simp, code func]: "size (n\<Colon>nat) = n"
- by (induct n) simp_all
-
-end
--- a/src/HOL/Wellfounded_Relations.thy Thu Apr 24 16:53:04 2008 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,269 +0,0 @@
-(* ID: $Id$
- Author: Konrad Slind
- Copyright 1995 TU Munich
-*)
-
-header {*Well-founded Relations*}
-
-theory Wellfounded_Relations
-imports Finite_Set FunDef
-uses
- ("Tools/function_package/lexicographic_order.ML")
- ("Tools/function_package/fundef_datatype.ML")
-begin
-
-text{*Derived WF relations such as inverse image, lexicographic product and
-measure. The simple relational product, in which @{term "(x',y')"} precedes
-@{term "(x,y)"} if @{term "x'<x"} and @{term "y'<y"}, is a subset of the
-lexicographic product, and therefore does not need to be defined separately.*}
-
-constdefs
- measure :: "('a => nat) => ('a * 'a)set"
- "measure == inv_image less_than"
-
- lex_prod :: "[('a*'a)set, ('b*'b)set] => (('a*'b)*('a*'b))set"
- (infixr "<*lex*>" 80)
- "ra <*lex*> rb == {((a,b),(a',b')). (a,a') : ra | a=a' & (b,b') : rb}"
-
- finite_psubset :: "('a set * 'a set) set"
- --{* finite proper subset*}
- "finite_psubset == {(A,B). A < B & finite B}"
-
- same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"
- "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"
- --{*For @{text rec_def} declarations where the first n parameters
- stay unchanged in the recursive call.
- See @{text "Library/While_Combinator.thy"} for an application.*}
-
-
-subsection{*Measure Functions make Wellfounded Relations*}
-
-subsubsection{*`Less than' on the natural numbers*}
-
-lemma full_nat_induct:
- assumes ih: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
- shows "P n"
-apply (rule wf_less_than [THEN wf_induct])
-apply (rule ih, auto)
-done
-
-subsubsection{*The Inverse Image into a Wellfounded Relation is Wellfounded.*}
-
-lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
-apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
-apply clarify
-apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
-prefer 2 apply (blast del: allE)
-apply (erule allE)
-apply (erule (1) notE impE)
-apply blast
-done
-
-lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
- by (auto simp:inv_image_def)
-
-subsubsection{*Finally, All Measures are Wellfounded.*}
-
-lemma in_measure[simp]: "((x,y) : measure f) = (f x < f y)"
- by (simp add:measure_def)
-
-lemma wf_measure [iff]: "wf (measure f)"
-apply (unfold measure_def)
-apply (rule wf_less_than [THEN wf_inv_image])
-done
-
-lemma measure_induct_rule [case_names less]:
- fixes f :: "'a \<Rightarrow> nat"
- assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
- shows "P a"
-proof -
- have "wf (measure f)" ..
- then show ?thesis
- proof induct
- case (less x)
- show ?case
- proof (rule step)
- fix y
- assume "f y < f x"
- hence "(y, x) \<in> measure f" by simp
- thus "P y" by (rule less)
- qed
- qed
-qed
-
-lemma measure_induct:
- fixes f :: "'a \<Rightarrow> nat"
- shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
- by (rule measure_induct_rule [of f P a]) iprover
-
-(* Should go into Finite_Set, but needs measure.
- Maybe move Wf_Rel before Finite_Set and finite_psubset to Finite_set?
-*)
-lemma (in linorder)
- finite_linorder_induct[consumes 1, case_names empty insert]:
- "finite A \<Longrightarrow> P {} \<Longrightarrow>
- (!!A b. finite A \<Longrightarrow> ALL a:A. a < b \<Longrightarrow> P A \<Longrightarrow> P(insert b A))
- \<Longrightarrow> P A"
-proof (induct A rule: measure_induct[where f=card])
- fix A :: "'a set"
- assume IH: "ALL B. card B < card A \<longrightarrow> finite B \<longrightarrow> P {} \<longrightarrow>
- (\<forall>A b. finite A \<longrightarrow> (\<forall>a\<in>A. a<b) \<longrightarrow> P A \<longrightarrow> P (insert b A))
- \<longrightarrow> P B"
- and "finite A" and "P {}"
- and step: "!!A b. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)"
- show "P A"
- proof cases
- assume "A = {}" thus "P A" using `P {}` by simp
- next
- let ?B = "A - {Max A}" let ?A = "insert (Max A) ?B"
- assume "A \<noteq> {}"
- with `finite A` have "Max A : A" by auto
- hence A: "?A = A" using insert_Diff_single insert_absorb by auto
- note card_Diff1_less[OF `finite A` `Max A : A`]
- moreover have "finite ?B" using `finite A` by simp
- ultimately have "P ?B" using `P {}` step IH by blast
- moreover have "\<forall>a\<in>?B. a < Max A"
- using Max_ge[OF `finite A` `A \<noteq> {}`] by fastsimp
- ultimately show "P A"
- using A insert_Diff_single step[OF `finite ?B`] by fastsimp
- qed
-qed
-
-
-subsection{*Other Ways of Constructing Wellfounded Relations*}
-
-text{*Wellfoundedness of lexicographic combinations*}
-lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
-apply (unfold wf_def lex_prod_def)
-apply (rule allI, rule impI)
-apply (simp (no_asm_use) only: split_paired_All)
-apply (drule spec, erule mp)
-apply (rule allI, rule impI)
-apply (drule spec, erule mp, blast)
-done
-
-lemma in_lex_prod[simp]:
- "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
- by (auto simp:lex_prod_def)
-
-text {* lexicographic combinations with measure functions *}
-
-definition
- mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
-where
- "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
-
-lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
-unfolding mlex_prod_def
-by auto
-
-lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
-unfolding mlex_prod_def by simp
-
-lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
-unfolding mlex_prod_def by auto
-
-
-text{*Transitivity of WF combinators.*}
-lemma trans_lex_prod [intro!]:
- "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
-by (unfold trans_def lex_prod_def, blast)
-
-subsubsection{*Wellfoundedness of proper subset on finite sets.*}
-lemma wf_finite_psubset: "wf(finite_psubset)"
-apply (unfold finite_psubset_def)
-apply (rule wf_measure [THEN wf_subset])
-apply (simp add: measure_def inv_image_def less_than_def less_eq)
-apply (fast elim!: psubset_card_mono)
-done
-
-lemma trans_finite_psubset: "trans finite_psubset"
-by (simp add: finite_psubset_def psubset_def trans_def, blast)
-
-
-subsubsection{*Wellfoundedness of finite acyclic relations*}
-
-text{*This proof belongs in this theory because it needs Finite.*}
-
-lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
-apply (erule finite_induct, blast)
-apply (simp (no_asm_simp) only: split_tupled_all)
-apply simp
-done
-
-lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
-apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
-apply (erule acyclic_converse [THEN iffD2])
-done
-
-lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
-by (blast intro: finite_acyclic_wf wf_acyclic)
-
-
-subsubsection{*Wellfoundedness of @{term same_fst}*}
-
-lemma same_fstI [intro!]:
- "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"
-by (simp add: same_fst_def)
-
-lemma wf_same_fst:
- assumes prem: "(!!x. P x ==> wf(R x))"
- shows "wf(same_fst P R)"
-apply (simp cong del: imp_cong add: wf_def same_fst_def)
-apply (intro strip)
-apply (rename_tac a b)
-apply (case_tac "wf (R a)")
- apply (erule_tac a = b in wf_induct, blast)
-apply (blast intro: prem)
-done
-
-
-subsection{*Weakly decreasing sequences (w.r.t. some well-founded order)
- stabilize.*}
-
-text{*This material does not appear to be used any longer.*}
-
-lemma lemma1: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"
-apply (induct_tac "k", simp_all)
-apply (blast intro: rtrancl_trans)
-done
-
-lemma lemma2: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]
- ==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"
-apply (erule wf_induct, clarify)
-apply (case_tac "EX j. (f (m+j), f m) : r^+")
- apply clarify
- apply (subgoal_tac "EX i. ALL k. f ((m+j) +i+k) = f ( (m+j) +i) ")
- apply clarify
- apply (rule_tac x = "j+i" in exI)
- apply (simp add: add_ac, blast)
-apply (rule_tac x = 0 in exI, clarsimp)
-apply (drule_tac i = m and k = k in lemma1)
-apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
-done
-
-lemma wf_weak_decr_stable: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]
- ==> EX i. ALL k. f (i+k) = f i"
-apply (drule_tac x = 0 in lemma2 [THEN spec], auto)
-done
-
-(* special case of the theorem above: <= *)
-lemma weak_decr_stable:
- "ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"
-apply (rule_tac r = pred_nat in wf_weak_decr_stable)
-apply (simp add: pred_nat_trancl_eq_le)
-apply (intro wf_trancl wf_pred_nat)
-done
-
-
-text {*
- Setup of @{text lexicographic_order} method
- and @{text fun} command
-*}
-
-use "Tools/function_package/lexicographic_order.ML"
-use "Tools/function_package/fundef_datatype.ML"
-
-setup "LexicographicOrder.setup #> FundefDatatype.setup"
-
-end