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(* Title: HOL/Extraction/Higman.thy
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ID: $Id$
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Author: Stefan Berghofer, TU Muenchen
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Monika Seisenberger, LMU Muenchen
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*)
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header {* Higman's lemma *}
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haftmann@16417
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theory Higman imports Main begin
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text {*
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Formalization by Stefan Berghofer and Monika Seisenberger,
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based on Coquand and Fridlender \cite{Coquand93}.
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*}
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datatype letter = A | B
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consts
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emb :: "(letter list \<times> letter list) set"
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inductive emb
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intros
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emb0 [Pure.intro]: "([], bs) \<in> emb"
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emb1 [Pure.intro]: "(as, bs) \<in> emb \<Longrightarrow> (as, b # bs) \<in> emb"
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emb2 [Pure.intro]: "(as, bs) \<in> emb \<Longrightarrow> (a # as, a # bs) \<in> emb"
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consts
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L :: "letter list \<Rightarrow> letter list list set"
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inductive "L v"
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intros
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L0 [Pure.intro]: "(w, v) \<in> emb \<Longrightarrow> w # ws \<in> L v"
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L1 [Pure.intro]: "ws \<in> L v \<Longrightarrow> w # ws \<in> L v"
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consts
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good :: "letter list list set"
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inductive good
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intros
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good0 [Pure.intro]: "ws \<in> L w \<Longrightarrow> w # ws \<in> good"
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good1 [Pure.intro]: "ws \<in> good \<Longrightarrow> w # ws \<in> good"
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consts
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R :: "letter \<Rightarrow> (letter list list \<times> letter list list) set"
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inductive "R a"
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intros
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R0 [Pure.intro]: "([], []) \<in> R a"
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R1 [Pure.intro]: "(vs, ws) \<in> R a \<Longrightarrow> (w # vs, (a # w) # ws) \<in> R a"
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consts
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T :: "letter \<Rightarrow> (letter list list \<times> letter list list) set"
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inductive "T a"
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intros
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T0 [Pure.intro]: "a \<noteq> b \<Longrightarrow> (ws, zs) \<in> R b \<Longrightarrow> (w # zs, (a # w) # zs) \<in> T a"
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T1 [Pure.intro]: "(ws, zs) \<in> T a \<Longrightarrow> (w # ws, (a # w) # zs) \<in> T a"
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T2 [Pure.intro]: "a \<noteq> b \<Longrightarrow> (ws, zs) \<in> T a \<Longrightarrow> (ws, (b # w) # zs) \<in> T a"
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consts
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bar :: "letter list list set"
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inductive bar
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intros
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bar1 [Pure.intro]: "ws \<in> good \<Longrightarrow> ws \<in> bar"
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bar2 [Pure.intro]: "(\<And>w. w # ws \<in> bar) \<Longrightarrow> ws \<in> bar"
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theorem prop1: "([] # ws) \<in> bar" by rules
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theorem lemma1: "ws \<in> L as \<Longrightarrow> ws \<in> L (a # as)"
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by (erule L.induct, rules+)
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lemma lemma2': "(vs, ws) \<in> R a \<Longrightarrow> vs \<in> L as \<Longrightarrow> ws \<in> L (a # as)"
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apply (induct set: R)
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apply (erule L.elims)
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apply simp+
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apply (erule L.elims)
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apply simp_all
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apply (rule L0)
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apply (erule emb2)
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apply (erule L1)
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done
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lemma lemma2: "(vs, ws) \<in> R a \<Longrightarrow> vs \<in> good \<Longrightarrow> ws \<in> good"
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apply (induct set: R)
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apply rules
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apply (erule good.elims)
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apply simp_all
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apply (rule good0)
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apply (erule lemma2')
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apply assumption
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apply (erule good1)
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done
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lemma lemma3': "(vs, ws) \<in> T a \<Longrightarrow> vs \<in> L as \<Longrightarrow> ws \<in> L (a # as)"
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apply (induct set: T)
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apply (erule L.elims)
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apply simp_all
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apply (rule L0)
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apply (erule emb2)
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apply (rule L1)
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apply (erule lemma1)
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apply (erule L.elims)
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apply simp_all
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apply rules+
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done
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lemma lemma3: "(ws, zs) \<in> T a \<Longrightarrow> ws \<in> good \<Longrightarrow> zs \<in> good"
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apply (induct set: T)
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apply (erule good.elims)
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apply simp_all
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apply (rule good0)
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apply (erule lemma1)
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apply (erule good1)
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apply (erule good.elims)
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apply simp_all
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apply (rule good0)
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apply (erule lemma3')
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apply rules+
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done
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lemma lemma4: "(ws, zs) \<in> R a \<Longrightarrow> ws \<noteq> [] \<Longrightarrow> (ws, zs) \<in> T a"
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apply (induct set: R)
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apply rules
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apply (case_tac vs)
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apply (erule R.elims)
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apply simp
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apply (case_tac a)
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apply (rule_tac b=B in T0)
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apply simp
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apply (rule R0)
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apply (rule_tac b=A in T0)
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apply simp
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apply (rule R0)
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apply simp
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apply (rule T1)
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apply simp
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done
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lemma letter_neq: "(a::letter) \<noteq> b \<Longrightarrow> c \<noteq> a \<Longrightarrow> c = b"
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apply (case_tac a)
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apply (case_tac b)
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apply (case_tac c, simp, simp)
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apply (case_tac c, simp, simp)
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apply (case_tac b)
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apply (case_tac c, simp, simp)
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apply (case_tac c, simp, simp)
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done
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lemma letter_eq_dec: "(a::letter) = b \<or> a \<noteq> b"
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apply (case_tac a)
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apply (case_tac b)
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apply simp
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apply simp
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apply (case_tac b)
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apply simp
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apply simp
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done
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theorem prop2:
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assumes ab: "a \<noteq> b" and bar: "xs \<in> bar"
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shows "\<And>ys zs. ys \<in> bar \<Longrightarrow> (xs, zs) \<in> T a \<Longrightarrow> (ys, zs) \<in> T b \<Longrightarrow> zs \<in> bar" using bar
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proof induct
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fix xs zs assume "xs \<in> good" and "(xs, zs) \<in> T a"
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show "zs \<in> bar" by (rule bar1) (rule lemma3)
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next
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fix xs ys
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assume I: "\<And>w ys zs. ys \<in> bar \<Longrightarrow> (w # xs, zs) \<in> T a \<Longrightarrow> (ys, zs) \<in> T b \<Longrightarrow> zs \<in> bar"
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assume "ys \<in> bar"
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thus "\<And>zs. (xs, zs) \<in> T a \<Longrightarrow> (ys, zs) \<in> T b \<Longrightarrow> zs \<in> bar"
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proof induct
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fix ys zs assume "ys \<in> good" and "(ys, zs) \<in> T b"
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show "zs \<in> bar" by (rule bar1) (rule lemma3)
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next
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fix ys zs assume I': "\<And>w zs. (xs, zs) \<in> T a \<Longrightarrow> (w # ys, zs) \<in> T b \<Longrightarrow> zs \<in> bar"
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and ys: "\<And>w. w # ys \<in> bar" and Ta: "(xs, zs) \<in> T a" and Tb: "(ys, zs) \<in> T b"
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show "zs \<in> bar"
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proof (rule bar2)
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fix w
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show "w # zs \<in> bar"
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proof (cases w)
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case Nil
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thus ?thesis by simp (rule prop1)
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next
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case (Cons c cs)
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from letter_eq_dec show ?thesis
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proof
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assume ca: "c = a"
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from ab have "(a # cs) # zs \<in> bar" by (rules intro: I ys Ta Tb)
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thus ?thesis by (simp add: Cons ca)
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next
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assume "c \<noteq> a"
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with ab have cb: "c = b" by (rule letter_neq)
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from ab have "(b # cs) # zs \<in> bar" by (rules intro: I' Ta Tb)
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thus ?thesis by (simp add: Cons cb)
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qed
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qed
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qed
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qed
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qed
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theorem prop3:
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assumes bar: "xs \<in> bar"
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shows "\<And>zs. xs \<noteq> [] \<Longrightarrow> (xs, zs) \<in> R a \<Longrightarrow> zs \<in> bar" using bar
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proof induct
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fix xs zs
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assume "xs \<in> good" and "(xs, zs) \<in> R a"
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show "zs \<in> bar" by (rule bar1) (rule lemma2)
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next
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fix xs zs
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assume I: "\<And>w zs. w # xs \<noteq> [] \<Longrightarrow> (w # xs, zs) \<in> R a \<Longrightarrow> zs \<in> bar"
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and xsb: "\<And>w. w # xs \<in> bar" and xsn: "xs \<noteq> []" and R: "(xs, zs) \<in> R a"
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show "zs \<in> bar"
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proof (rule bar2)
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fix w
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show "w # zs \<in> bar"
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proof (induct w)
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case Nil
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show ?case by (rule prop1)
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next
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case (Cons c cs)
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from letter_eq_dec show ?case
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proof
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assume "c = a"
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thus ?thesis by (rules intro: I [simplified] R)
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next
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from R xsn have T: "(xs, zs) \<in> T a" by (rule lemma4)
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assume "c \<noteq> a"
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thus ?thesis by (rules intro: prop2 Cons xsb xsn R T)
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qed
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qed
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qed
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qed
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theorem higman: "[] \<in> bar"
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proof (rule bar2)
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fix w
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show "[w] \<in> bar"
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proof (induct w)
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show "[[]] \<in> bar" by (rule prop1)
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next
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fix c cs assume "[cs] \<in> bar"
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thus "[c # cs] \<in> bar" by (rule prop3) (simp, rules)
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qed
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qed
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consts
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is_prefix :: "'a list \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"
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primrec
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"is_prefix [] f = True"
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"is_prefix (x # xs) f = (x = f (length xs) \<and> is_prefix xs f)"
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theorem good_prefix_lemma:
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assumes bar: "ws \<in> bar"
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shows "is_prefix ws f \<Longrightarrow> \<exists>vs. is_prefix vs f \<and> vs \<in> good" using bar
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proof induct
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case bar1
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thus ?case by rules
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next
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case (bar2 ws)
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have "is_prefix (f (length ws) # ws) f" by simp
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thus ?case by (rules intro: bar2)
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qed
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theorem good_prefix: "\<exists>vs. is_prefix vs f \<and> vs \<in> good"
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using higman
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by (rule good_prefix_lemma) simp+
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subsection {* Extracting the program *}
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declare bar.induct [ind_realizer]
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extract good_prefix
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text {*
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Program extracted from the proof of @{text good_prefix}:
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@{thm [display] good_prefix_def [no_vars]}
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Corresponding correctness theorem:
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@{thm [display] good_prefix_correctness [no_vars]}
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berghofe@13405
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281 |
Program extracted from the proof of @{text good_prefix_lemma}:
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berghofe@13405
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@{thm [display] good_prefix_lemma_def [no_vars]}
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berghofe@13405
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283 |
Program extracted from the proof of @{text higman}:
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@{thm [display] higman_def [no_vars]}
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berghofe@13405
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285 |
Program extracted from the proof of @{text prop1}:
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@{thm [display] prop1_def [no_vars]}
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berghofe@13405
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Program extracted from the proof of @{text prop2}:
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berghofe@13405
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@{thm [display] prop2_def [no_vars]}
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berghofe@13405
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Program extracted from the proof of @{text prop3}:
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@{thm [display] prop3_def [no_vars]}
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*}
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292 |
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code_module Higman
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contains
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test = good_prefix
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296 |
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ML {*
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local open Higman in
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299 |
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val a = 16807.0;
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|
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val m = 2147483647.0;
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|
302 |
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fun nextRand seed =
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let val t = a*seed
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|
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in t - m * real (Real.floor(t/m)) end;
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|
306 |
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|
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fun mk_word seed l =
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let
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val r = nextRand seed;
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val i = Real.round (r / m * 10.0);
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|
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in if i > 7 andalso l > 2 then (r, []) else
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apsnd (cons (if i mod 2 = 0 then A else B)) (mk_word r (l+1))
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|
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end;
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|
314 |
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|
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fun f s id_0 = mk_word s 0
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|
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| f s (Suc n) = f (fst (mk_word s 0)) n;
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|
317 |
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|
318 |
val g1 = snd o (f 20000.0);
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|
319 |
|
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|
320 |
val g2 = snd o (f 50000.0);
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|
321 |
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|
322 |
fun f1 id_0 = [A,A]
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|
323 |
| f1 (Suc id_0) = [B]
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|
324 |
| f1 (Suc (Suc id_0)) = [A,B]
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|
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| f1 _ = [];
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|
326 |
|
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|
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fun f2 id_0 = [A,A]
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|
328 |
| f2 (Suc id_0) = [B]
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berghofe@17145
|
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| f2 (Suc (Suc id_0)) = [B,A]
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|
330 |
| f2 _ = [];
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|
331 |
|
berghofe@13405
|
332 |
val xs1 = test g1;
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berghofe@13405
|
333 |
val xs2 = test g2;
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berghofe@13405
|
334 |
val xs3 = test f1;
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berghofe@13405
|
335 |
val xs4 = test f2;
|
berghofe@17145
|
336 |
|
berghofe@17145
|
337 |
end;
|
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|
338 |
*}
|
berghofe@13405
|
339 |
|
berghofe@13405
|
340 |
end
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