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(* Title: HOL/Proofs/Extraction/Higman.thy
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Author: Stefan Berghofer, TU Muenchen
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Author: Monika Seisenberger, LMU Muenchen
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*)
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header {* Higman's lemma *}
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theory Higman
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imports Main "~~/src/HOL/Library/State_Monad" Random
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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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inductive emb :: "letter list \<Rightarrow> letter list \<Rightarrow> bool"
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where
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emb0 [Pure.intro]: "emb [] bs"
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| emb1 [Pure.intro]: "emb as bs \<Longrightarrow> emb as (b # bs)"
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| emb2 [Pure.intro]: "emb as bs \<Longrightarrow> emb (a # as) (a # bs)"
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inductive L :: "letter list \<Rightarrow> letter list list \<Rightarrow> bool"
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for v :: "letter list"
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where
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L0 [Pure.intro]: "emb w v \<Longrightarrow> L v (w # ws)"
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| L1 [Pure.intro]: "L v ws \<Longrightarrow> L v (w # ws)"
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inductive good :: "letter list list \<Rightarrow> bool"
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where
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good0 [Pure.intro]: "L w ws \<Longrightarrow> good (w # ws)"
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| good1 [Pure.intro]: "good ws \<Longrightarrow> good (w # ws)"
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inductive R :: "letter \<Rightarrow> letter list list \<Rightarrow> letter list list \<Rightarrow> bool"
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for a :: letter
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where
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R0 [Pure.intro]: "R a [] []"
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| R1 [Pure.intro]: "R a vs ws \<Longrightarrow> R a (w # vs) ((a # w) # ws)"
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inductive T :: "letter \<Rightarrow> letter list list \<Rightarrow> letter list list \<Rightarrow> bool"
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for a :: letter
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where
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T0 [Pure.intro]: "a \<noteq> b \<Longrightarrow> R b ws zs \<Longrightarrow> T a (w # zs) ((a # w) # zs)"
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| T1 [Pure.intro]: "T a ws zs \<Longrightarrow> T a (w # ws) ((a # w) # zs)"
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| T2 [Pure.intro]: "a \<noteq> b \<Longrightarrow> T a ws zs \<Longrightarrow> T a ws ((b # w) # zs)"
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inductive bar :: "letter list list \<Rightarrow> bool"
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where
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bar1 [Pure.intro]: "good ws \<Longrightarrow> bar ws"
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| bar2 [Pure.intro]: "(\<And>w. bar (w # ws)) \<Longrightarrow> bar ws"
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theorem prop1: "bar ([] # ws)" by iprover
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theorem lemma1: "L as ws \<Longrightarrow> L (a # as) ws"
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by (erule L.induct, iprover+)
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lemma lemma2': "R a vs ws \<Longrightarrow> L as vs \<Longrightarrow> L (a # as) ws"
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apply (induct set: R)
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apply (erule L.cases)
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apply simp+
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apply (erule L.cases)
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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: "R a vs ws \<Longrightarrow> good vs \<Longrightarrow> good ws"
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apply (induct set: R)
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apply iprover
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apply (erule good.cases)
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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': "T a vs ws \<Longrightarrow> L as vs \<Longrightarrow> L (a # as) ws"
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apply (induct set: T)
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apply (erule L.cases)
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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.cases)
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apply simp_all
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apply iprover+
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done
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lemma lemma3: "T a ws zs \<Longrightarrow> good ws \<Longrightarrow> good zs"
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apply (induct set: T)
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apply (erule good.cases)
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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.cases)
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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 iprover+
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done
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lemma lemma4: "R a ws zs \<Longrightarrow> ws \<noteq> [] \<Longrightarrow> T a ws zs"
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apply (induct set: R)
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apply iprover
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apply (case_tac vs)
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apply (erule R.cases)
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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: "bar xs"
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shows "\<And>ys zs. bar ys \<Longrightarrow> T a xs zs \<Longrightarrow> T b ys zs \<Longrightarrow> bar zs" using bar
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proof induct
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fix xs zs assume "T a xs zs" and "good xs"
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hence "good zs" by (rule lemma3)
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then show "bar zs" by (rule bar1)
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next
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fix xs ys
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assume I: "\<And>w ys zs. bar ys \<Longrightarrow> T a (w # xs) zs \<Longrightarrow> T b ys zs \<Longrightarrow> bar zs"
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assume "bar ys"
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thus "\<And>zs. T a xs zs \<Longrightarrow> T b ys zs \<Longrightarrow> bar zs"
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proof induct
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fix ys zs assume "T b ys zs" and "good ys"
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then have "good zs" by (rule lemma3)
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then show "bar zs" by (rule bar1)
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next
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fix ys zs assume I': "\<And>w zs. T a xs zs \<Longrightarrow> T b (w # ys) zs \<Longrightarrow> bar zs"
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and ys: "\<And>w. bar (w # ys)" and Ta: "T a xs zs" and Tb: "T b ys zs"
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show "bar zs"
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proof (rule bar2)
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fix w
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show "bar (w # zs)"
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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 "bar ((a # cs) # zs)" by (iprover 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 "bar ((b # cs) # zs)" by (iprover 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: "bar xs"
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shows "\<And>zs. xs \<noteq> [] \<Longrightarrow> R a xs zs \<Longrightarrow> bar zs" using bar
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proof induct
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fix xs zs
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assume "R a xs zs" and "good xs"
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then have "good zs" by (rule lemma2)
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then show "bar zs" by (rule bar1)
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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> R a (w # xs) zs \<Longrightarrow> bar zs"
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and xsb: "\<And>w. bar (w # xs)" and xsn: "xs \<noteq> []" and R: "R a xs zs"
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show "bar zs"
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proof (rule bar2)
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fix w
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show "bar (w # zs)"
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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 (iprover intro: I [simplified] R)
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next
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from R xsn have T: "T a xs zs" by (rule lemma4)
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assume "c \<noteq> a"
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thus ?thesis by (iprover 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: "bar []"
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proof (rule bar2)
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fix w
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show "bar [w]"
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proof (induct w)
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show "bar [[]]" by (rule prop1)
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next
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fix c cs assume "bar [cs]"
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thus "bar [c # cs]" by (rule prop3) (simp, iprover)
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qed
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qed
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primrec
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is_prefix :: "'a list \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"
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where
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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 L_idx:
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assumes L: "L w ws"
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shows "is_prefix ws f \<Longrightarrow> \<exists>i. emb (f i) w \<and> i < length ws" using L
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proof induct
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case (L0 v ws)
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hence "emb (f (length ws)) w" by simp
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moreover have "length ws < length (v # ws)" by simp
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ultimately show ?case by iprover
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next
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case (L1 ws v)
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then obtain i where emb: "emb (f i) w" and "i < length ws"
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by simp iprover
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hence "i < length (v # ws)" by simp
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with emb show ?case by iprover
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qed
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theorem good_idx:
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assumes good: "good ws"
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shows "is_prefix ws f \<Longrightarrow> \<exists>i j. emb (f i) (f j) \<and> i < j" using good
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proof induct
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case (good0 w ws)
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hence "w = f (length ws)" and "is_prefix ws f" by simp_all
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with good0 show ?case by (iprover dest: L_idx)
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next
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case (good1 ws w)
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thus ?case by simp
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qed
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theorem bar_idx:
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assumes bar: "bar ws"
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shows "is_prefix ws f \<Longrightarrow> \<exists>i j. emb (f i) (f j) \<and> i < j" using bar
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proof induct
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case (bar1 ws)
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thus ?case by (rule good_idx)
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next
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case (bar2 ws)
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hence "is_prefix (f (length ws) # ws) f" by simp
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thus ?case by (rule bar2)
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qed
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text {*
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Strong version: yields indices of words that can be embedded into each other.
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*}
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theorem higman_idx: "\<exists>(i::nat) j. emb (f i) (f j) \<and> i < j"
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proof (rule bar_idx)
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show "bar []" by (rule higman)
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show "is_prefix [] f" by simp
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qed
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text {*
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Weak version: only yield sequence containing words
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that can be embedded into each other.
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*}
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theorem good_prefix_lemma:
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assumes bar: "bar ws"
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shows "is_prefix ws f \<Longrightarrow> \<exists>vs. is_prefix vs f \<and> good vs" using bar
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proof induct
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case bar1
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thus ?case by iprover
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next
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case (bar2 ws)
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from bar2.prems have "is_prefix (f (length ws) # ws) f" by simp
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thus ?case by (iprover intro: bar2)
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qed
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theorem good_prefix: "\<exists>vs. is_prefix vs f \<and> good vs"
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using higman
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by (rule good_prefix_lemma) simp+
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end
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