--- a/Closures2.thy Thu Sep 01 20:26:30 2011 +0000
+++ b/Closures2.thy Thu Sep 01 23:18:34 2011 +0000
@@ -1,24 +1,50 @@
theory Closure2
imports
Closures
+ Higman
(* "~~/src/HOL/Proofs/Extraction/Higman" *)
begin
+notation
+ emb ("_ \<preceq> _")
+
+declare emb0 [intro]
+declare emb1 [intro]
+declare emb2 [intro]
+
+lemma letter_UNIV:
+ shows "UNIV = {A, B::letter}"
+apply(auto)
+apply(case_tac x)
+apply(auto)
+done
+
+instance letter :: finite
+apply(default)
+apply(simp add: letter_UNIV)
+done
+
+hide_const A
+hide_const B
+
+(*
inductive
emb :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<preceq> _")
where
emb0 [intro]: "emb [] y"
| emb1 [intro]: "emb x y \<Longrightarrow> emb x (c # y)"
| emb2 [intro]: "emb x y \<Longrightarrow> emb (c # x) (c # y)"
+*)
lemma emb_refl [intro]:
shows "x \<preceq> x"
-by (induct x) (auto intro: emb.intros)
+by (induct x) (auto)
lemma emb_right [intro]:
assumes a: "x \<preceq> y"
shows "x \<preceq> y @ y'"
-using a by (induct arbitrary: y') (auto)
+using a
+by (induct arbitrary: y') (auto)
lemma emb_left [intro]:
assumes a: "x \<preceq> y"
@@ -160,15 +186,16 @@
| "UP (Star r) = Star Allreg"
lemma lang_UP:
+ fixes r::"letter rexp"
shows "lang (UP r) = SUPSEQ (lang r)"
by (induct r) (simp_all)
lemma regular_SUPSEQ:
- fixes A::"'a::finite lang"
+ fixes A::"letter lang"
assumes "regular A"
shows "regular (SUPSEQ A)"
proof -
- from assms obtain r::"'a::finite rexp" where "lang r = A" by auto
+ from assms obtain r::"letter rexp" where "lang r = A" by auto
then have "lang (UP r) = SUPSEQ A" by (simp add: lang_UP)
then show "regular (SUPSEQ A)" by auto
qed
@@ -178,6 +205,7 @@
unfolding SUPSEQ_def by auto
lemma w3:
+ fixes T A::"letter lang"
assumes eq: "T = - (SUBSEQ A)"
shows "T = SUPSEQ T"
apply(rule)
@@ -209,16 +237,34 @@
by (rule w3) (simp)
definition
- "minimal_in x L \<equiv> \<forall>y \<in> L. y \<preceq> x \<longrightarrow> y = x"
+ minimal_in :: "letter list \<Rightarrow> letter lang \<Rightarrow> bool"
+where
+ "minimal_in x A \<equiv> \<forall>y \<in> A. y \<preceq> x \<longrightarrow> y = x"
lemma minimal_in2:
- shows "minimal_in x L = (\<forall>y \<in> L. y \<preceq> x \<longrightarrow> x \<preceq> y)"
+ shows "minimal_in x A = (\<forall>y \<in> A. y \<preceq> x \<longrightarrow> x \<preceq> y)"
by (auto simp add: minimal_in_def intro: emb_antisym)
lemma higman:
assumes "\<forall>x \<in> A. \<forall>y \<in> A. x \<noteq> y \<longrightarrow> \<not>(x \<preceq> y) \<and> \<not>(y \<preceq> x)"
shows "finite A"
-sorry
+apply(rule ccontr)
+apply(simp add: infinite_iff_countable_subset)
+apply(auto)
+apply(insert higman_idx)
+apply(drule_tac x="f" in meta_spec)
+apply(auto)
+using assms
+apply -
+apply(drule_tac x="f i" in bspec)
+apply(auto)[1]
+apply(drule_tac x="f j" in bspec)
+apply(auto)[1]
+apply(drule mp)
+apply(simp add: inj_on_def)
+apply(auto)[1]
+apply(auto)[1]
+done
lemma minimal:
assumes "minimal_in x A" "minimal_in y A"
@@ -266,7 +312,7 @@
qed
lemma closure_SUPSEQ:
- fixes A::"'a::finite lang"
+ fixes A::"letter lang"
shows "regular (SUPSEQ A)"
proof -
obtain M where a: "finite M" and b: "SUPSEQ A = SUPSEQ M"
@@ -277,7 +323,7 @@
qed
lemma closure_SUBSEQ:
- fixes A::"'a::finite lang"
+ fixes A::"letter lang"
shows "regular (SUBSEQ A)"
proof -
have "regular (SUPSEQ (- SUBSEQ A))" by (rule closure_SUPSEQ)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Higman.thy Thu Sep 01 23:18:34 2011 +0000
@@ -0,0 +1,314 @@
+(* Title: HOL/Proofs/Extraction/Higman.thy
+ Author: Stefan Berghofer, TU Muenchen
+ Author: Monika Seisenberger, LMU Muenchen
+*)
+
+header {* Higman's lemma *}
+
+theory Higman
+imports Main "~~/src/HOL/Library/State_Monad" Random
+begin
+
+text {*
+ Formalization by Stefan Berghofer and Monika Seisenberger,
+ based on Coquand and Fridlender \cite{Coquand93}.
+*}
+
+datatype letter = A | B
+
+inductive emb :: "letter list \<Rightarrow> letter list \<Rightarrow> bool"
+where
+ emb0 [Pure.intro]: "emb [] bs"
+ | emb1 [Pure.intro]: "emb as bs \<Longrightarrow> emb as (b # bs)"
+ | emb2 [Pure.intro]: "emb as bs \<Longrightarrow> emb (a # as) (a # bs)"
+
+inductive L :: "letter list \<Rightarrow> letter list list \<Rightarrow> bool"
+ for v :: "letter list"
+where
+ L0 [Pure.intro]: "emb w v \<Longrightarrow> L v (w # ws)"
+ | L1 [Pure.intro]: "L v ws \<Longrightarrow> L v (w # ws)"
+
+inductive good :: "letter list list \<Rightarrow> bool"
+where
+ good0 [Pure.intro]: "L w ws \<Longrightarrow> good (w # ws)"
+ | good1 [Pure.intro]: "good ws \<Longrightarrow> good (w # ws)"
+
+inductive R :: "letter \<Rightarrow> letter list list \<Rightarrow> letter list list \<Rightarrow> bool"
+ for a :: letter
+where
+ R0 [Pure.intro]: "R a [] []"
+ | R1 [Pure.intro]: "R a vs ws \<Longrightarrow> R a (w # vs) ((a # w) # ws)"
+
+inductive T :: "letter \<Rightarrow> letter list list \<Rightarrow> letter list list \<Rightarrow> bool"
+ for a :: letter
+where
+ T0 [Pure.intro]: "a \<noteq> b \<Longrightarrow> R b ws zs \<Longrightarrow> T a (w # zs) ((a # w) # zs)"
+ | T1 [Pure.intro]: "T a ws zs \<Longrightarrow> T a (w # ws) ((a # w) # zs)"
+ | T2 [Pure.intro]: "a \<noteq> b \<Longrightarrow> T a ws zs \<Longrightarrow> T a ws ((b # w) # zs)"
+
+inductive bar :: "letter list list \<Rightarrow> bool"
+where
+ bar1 [Pure.intro]: "good ws \<Longrightarrow> bar ws"
+ | bar2 [Pure.intro]: "(\<And>w. bar (w # ws)) \<Longrightarrow> bar ws"
+
+theorem prop1: "bar ([] # ws)" by iprover
+
+theorem lemma1: "L as ws \<Longrightarrow> L (a # as) ws"
+ by (erule L.induct, iprover+)
+
+lemma lemma2': "R a vs ws \<Longrightarrow> L as vs \<Longrightarrow> L (a # as) ws"
+ apply (induct set: R)
+ apply (erule L.cases)
+ apply simp+
+ apply (erule L.cases)
+ apply simp_all
+ apply (rule L0)
+ apply (erule emb2)
+ apply (erule L1)
+ done
+
+lemma lemma2: "R a vs ws \<Longrightarrow> good vs \<Longrightarrow> good ws"
+ apply (induct set: R)
+ apply iprover
+ apply (erule good.cases)
+ apply simp_all
+ apply (rule good0)
+ apply (erule lemma2')
+ apply assumption
+ apply (erule good1)
+ done
+
+lemma lemma3': "T a vs ws \<Longrightarrow> L as vs \<Longrightarrow> L (a # as) ws"
+ apply (induct set: T)
+ apply (erule L.cases)
+ apply simp_all
+ apply (rule L0)
+ apply (erule emb2)
+ apply (rule L1)
+ apply (erule lemma1)
+ apply (erule L.cases)
+ apply simp_all
+ apply iprover+
+ done
+
+lemma lemma3: "T a ws zs \<Longrightarrow> good ws \<Longrightarrow> good zs"
+ apply (induct set: T)
+ apply (erule good.cases)
+ apply simp_all
+ apply (rule good0)
+ apply (erule lemma1)
+ apply (erule good1)
+ apply (erule good.cases)
+ apply simp_all
+ apply (rule good0)
+ apply (erule lemma3')
+ apply iprover+
+ done
+
+lemma lemma4: "R a ws zs \<Longrightarrow> ws \<noteq> [] \<Longrightarrow> T a ws zs"
+ apply (induct set: R)
+ apply iprover
+ apply (case_tac vs)
+ apply (erule R.cases)
+ apply simp
+ apply (case_tac a)
+ apply (rule_tac b=B in T0)
+ apply simp
+ apply (rule R0)
+ apply (rule_tac b=A in T0)
+ apply simp
+ apply (rule R0)
+ apply simp
+ apply (rule T1)
+ apply simp
+ done
+
+lemma letter_neq: "(a::letter) \<noteq> b \<Longrightarrow> c \<noteq> a \<Longrightarrow> c = b"
+ apply (case_tac a)
+ apply (case_tac b)
+ apply (case_tac c, simp, simp)
+ apply (case_tac c, simp, simp)
+ apply (case_tac b)
+ apply (case_tac c, simp, simp)
+ apply (case_tac c, simp, simp)
+ done
+
+lemma letter_eq_dec: "(a::letter) = b \<or> a \<noteq> b"
+ apply (case_tac a)
+ apply (case_tac b)
+ apply simp
+ apply simp
+ apply (case_tac b)
+ apply simp
+ apply simp
+ done
+
+theorem prop2:
+ assumes ab: "a \<noteq> b" and bar: "bar xs"
+ shows "\<And>ys zs. bar ys \<Longrightarrow> T a xs zs \<Longrightarrow> T b ys zs \<Longrightarrow> bar zs" using bar
+proof induct
+ fix xs zs assume "T a xs zs" and "good xs"
+ hence "good zs" by (rule lemma3)
+ then show "bar zs" by (rule bar1)
+next
+ fix xs ys
+ assume I: "\<And>w ys zs. bar ys \<Longrightarrow> T a (w # xs) zs \<Longrightarrow> T b ys zs \<Longrightarrow> bar zs"
+ assume "bar ys"
+ thus "\<And>zs. T a xs zs \<Longrightarrow> T b ys zs \<Longrightarrow> bar zs"
+ proof induct
+ fix ys zs assume "T b ys zs" and "good ys"
+ then have "good zs" by (rule lemma3)
+ then show "bar zs" by (rule bar1)
+ next
+ fix ys zs assume I': "\<And>w zs. T a xs zs \<Longrightarrow> T b (w # ys) zs \<Longrightarrow> bar zs"
+ and ys: "\<And>w. bar (w # ys)" and Ta: "T a xs zs" and Tb: "T b ys zs"
+ show "bar zs"
+ proof (rule bar2)
+ fix w
+ show "bar (w # zs)"
+ proof (cases w)
+ case Nil
+ thus ?thesis by simp (rule prop1)
+ next
+ case (Cons c cs)
+ from letter_eq_dec show ?thesis
+ proof
+ assume ca: "c = a"
+ from ab have "bar ((a # cs) # zs)" by (iprover intro: I ys Ta Tb)
+ thus ?thesis by (simp add: Cons ca)
+ next
+ assume "c \<noteq> a"
+ with ab have cb: "c = b" by (rule letter_neq)
+ from ab have "bar ((b # cs) # zs)" by (iprover intro: I' Ta Tb)
+ thus ?thesis by (simp add: Cons cb)
+ qed
+ qed
+ qed
+ qed
+qed
+
+theorem prop3:
+ assumes bar: "bar xs"
+ shows "\<And>zs. xs \<noteq> [] \<Longrightarrow> R a xs zs \<Longrightarrow> bar zs" using bar
+proof induct
+ fix xs zs
+ assume "R a xs zs" and "good xs"
+ then have "good zs" by (rule lemma2)
+ then show "bar zs" by (rule bar1)
+next
+ fix xs zs
+ assume I: "\<And>w zs. w # xs \<noteq> [] \<Longrightarrow> R a (w # xs) zs \<Longrightarrow> bar zs"
+ and xsb: "\<And>w. bar (w # xs)" and xsn: "xs \<noteq> []" and R: "R a xs zs"
+ show "bar zs"
+ proof (rule bar2)
+ fix w
+ show "bar (w # zs)"
+ proof (induct w)
+ case Nil
+ show ?case by (rule prop1)
+ next
+ case (Cons c cs)
+ from letter_eq_dec show ?case
+ proof
+ assume "c = a"
+ thus ?thesis by (iprover intro: I [simplified] R)
+ next
+ from R xsn have T: "T a xs zs" by (rule lemma4)
+ assume "c \<noteq> a"
+ thus ?thesis by (iprover intro: prop2 Cons xsb xsn R T)
+ qed
+ qed
+ qed
+qed
+
+theorem higman: "bar []"
+proof (rule bar2)
+ fix w
+ show "bar [w]"
+ proof (induct w)
+ show "bar [[]]" by (rule prop1)
+ next
+ fix c cs assume "bar [cs]"
+ thus "bar [c # cs]" by (rule prop3) (simp, iprover)
+ qed
+qed
+
+primrec
+ is_prefix :: "'a list \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"
+where
+ "is_prefix [] f = True"
+ | "is_prefix (x # xs) f = (x = f (length xs) \<and> is_prefix xs f)"
+
+theorem L_idx:
+ assumes L: "L w ws"
+ shows "is_prefix ws f \<Longrightarrow> \<exists>i. emb (f i) w \<and> i < length ws" using L
+proof induct
+ case (L0 v ws)
+ hence "emb (f (length ws)) w" by simp
+ moreover have "length ws < length (v # ws)" by simp
+ ultimately show ?case by iprover
+next
+ case (L1 ws v)
+ then obtain i where emb: "emb (f i) w" and "i < length ws"
+ by simp iprover
+ hence "i < length (v # ws)" by simp
+ with emb show ?case by iprover
+qed
+
+theorem good_idx:
+ assumes good: "good ws"
+ shows "is_prefix ws f \<Longrightarrow> \<exists>i j. emb (f i) (f j) \<and> i < j" using good
+proof induct
+ case (good0 w ws)
+ hence "w = f (length ws)" and "is_prefix ws f" by simp_all
+ with good0 show ?case by (iprover dest: L_idx)
+next
+ case (good1 ws w)
+ thus ?case by simp
+qed
+
+theorem bar_idx:
+ assumes bar: "bar ws"
+ shows "is_prefix ws f \<Longrightarrow> \<exists>i j. emb (f i) (f j) \<and> i < j" using bar
+proof induct
+ case (bar1 ws)
+ thus ?case by (rule good_idx)
+next
+ case (bar2 ws)
+ hence "is_prefix (f (length ws) # ws) f" by simp
+ thus ?case by (rule bar2)
+qed
+
+text {*
+Strong version: yields indices of words that can be embedded into each other.
+*}
+
+theorem higman_idx: "\<exists>(i::nat) j. emb (f i) (f j) \<and> i < j"
+proof (rule bar_idx)
+ show "bar []" by (rule higman)
+ show "is_prefix [] f" by simp
+qed
+
+text {*
+Weak version: only yield sequence containing words
+that can be embedded into each other.
+*}
+
+theorem good_prefix_lemma:
+ assumes bar: "bar ws"
+ shows "is_prefix ws f \<Longrightarrow> \<exists>vs. is_prefix vs f \<and> good vs" using bar
+proof induct
+ case bar1
+ thus ?case by iprover
+next
+ case (bar2 ws)
+ from bar2.prems have "is_prefix (f (length ws) # ws) f" by simp
+ thus ?case by (iprover intro: bar2)
+qed
+
+theorem good_prefix: "\<exists>vs. is_prefix vs f \<and> good vs"
+ using higman
+ by (rule good_prefix_lemma) simp+
+
+
+end