--- /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