more on paper; modified schs functions; it is still compatible with the old definition
(* Title: HOL/Proofs/Extraction/Higman.thy Author: Stefan Berghofer, TU Muenchen Author: Monika Seisenberger, LMU Muenchen*)header {* Higman's lemma *}theory Higmanimports Main "~~/src/HOL/Library/State_Monad" Randombegintext {* Formalization by Stefan Berghofer and Monika Seisenberger, based on Coquand and Fridlender \cite{Coquand93}.*}datatype letter = A | Binductive 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 :: letterwhere 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 :: letterwhere 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 iprovertheorem 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) donelemma 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) donelemma 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+ donelemma 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+ donelemma 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 donelemma 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) donelemma 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 donetheorem 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 barproof 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 qedqedtheorem prop3: assumes bar: "bar xs" shows "\<And>zs. xs \<noteq> [] \<Longrightarrow> R a xs zs \<Longrightarrow> bar zs" using barproof 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 qedqedtheorem 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) qedqedprimrec 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 Lproof 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 iprovernext 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 iproverqedtheorem good_idx: assumes good: "good ws" shows "is_prefix ws f \<Longrightarrow> \<exists>i j. emb (f i) (f j) \<and> i < j" using goodproof 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 simpqedtheorem bar_idx: assumes bar: "bar ws" shows "is_prefix ws f \<Longrightarrow> \<exists>i j. emb (f i) (f j) \<and> i < j" using barproof 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)qedtext {*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 simpqedtext {*Weak version: only yield sequence containing wordsthat 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 barproof induct case bar1 thus ?case by iprovernext case (bar2 ws) from bar2.prems have "is_prefix (f (length ws) # ws) f" by simp thus ?case by (iprover intro: bar2)qedtheorem good_prefix: "\<exists>vs. is_prefix vs f \<and> good vs" using higman by (rule good_prefix_lemma) simp+end