Renamed inductive2 to inductive.
(* Title: HOL/Extraction/Higman.thy
ID: $Id$
Author: Stefan Berghofer, TU Muenchen
Monika Seisenberger, LMU Muenchen
*)
header {* Higman's lemma *}
theory Higman imports Main 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
consts
is_prefix :: "'a list \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"
primrec
"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+
subsection {* Extracting the program *}
declare R.induct [ind_realizer]
declare T.induct [ind_realizer]
declare L.induct [ind_realizer]
declare good.induct [ind_realizer]
declare bar.induct [ind_realizer]
extract higman_idx
text {*
Program extracted from the proof of @{text higman_idx}:
@{thm [display] higman_idx_def [no_vars]}
Corresponding correctness theorem:
@{thm [display] higman_idx_correctness [no_vars]}
Program extracted from the proof of @{text higman}:
@{thm [display] higman_def [no_vars]}
Program extracted from the proof of @{text prop1}:
@{thm [display] prop1_def [no_vars]}
Program extracted from the proof of @{text prop2}:
@{thm [display] prop2_def [no_vars]}
Program extracted from the proof of @{text prop3}:
@{thm [display] prop3_def [no_vars]}
*}
consts_code
"arbitrary :: LT" ("({* L0 [] [] *})")
"arbitrary :: TT" ("({* T0 A [] [] [] R0 *})")
code_module Higman
contains
test = higman_idx
ML {*
local open Higman in
val a = 16807.0;
val m = 2147483647.0;
fun nextRand seed =
let val t = a*seed
in t - m * real (Real.floor(t/m)) end;
fun mk_word seed l =
let
val r = nextRand seed;
val i = Real.round (r / m * 10.0);
in if i > 7 andalso l > 2 then (r, []) else
apsnd (cons (if i mod 2 = 0 then A else B)) (mk_word r (l+1))
end;
fun f s zero = mk_word s 0
| f s (Suc n) = f (fst (mk_word s 0)) n;
val g1 = snd o (f 20000.0);
val g2 = snd o (f 50000.0);
fun f1 zero = [A,A]
| f1 (Suc zero) = [B]
| f1 (Suc (Suc zero)) = [A,B]
| f1 _ = [];
fun f2 zero = [A,A]
| f2 (Suc zero) = [B]
| f2 (Suc (Suc zero)) = [B,A]
| f2 _ = [];
val (i1, j1) = test g1;
val (v1, w1) = (g1 i1, g1 j1);
val (i2, j2) = test g2;
val (v2, w2) = (g2 i2, g2 j2);
val (i3, j3) = test f1;
val (v3, w3) = (f1 i3, f1 j3);
val (i4, j4) = test f2;
val (v4, w4) = (f2 i4, f2 j4);
end;
*}
definition
arbitrary_LT :: "LT" where
[symmetric, code inline]: "arbitrary_LT = arbitrary"
definition
arbitrary_TT :: "TT" where
[symmetric, code inline]: "arbitrary_TT = arbitrary"
definition
"arbitrary_LT' = L0 [] []"
definition
"arbitrary_TT' = T0 A [] [] [] R0"
code_axioms
arbitrary_LT \<equiv> arbitrary_LT'
arbitrary_TT \<equiv> arbitrary_TT'
code_gen higman_idx in SML
ML {*
local
open ROOT.Higman
open ROOT.Nat
in
val a = 16807.0;
val m = 2147483647.0;
fun nextRand seed =
let val t = a*seed
in t - m * real (Real.floor(t/m)) end;
fun mk_word seed l =
let
val r = nextRand seed;
val i = Real.round (r / m * 10.0);
in if i > 7 andalso l > 2 then (r, []) else
apsnd (cons (if i mod 2 = 0 then A else B)) (mk_word r (l+1))
end;
fun f s Zero_nat = mk_word s 0
| f s (Suc n) = f (fst (mk_word s 0)) n;
val g1 = snd o (f 20000.0);
val g2 = snd o (f 50000.0);
fun f1 Zero_nat = [A,A]
| f1 (Suc Zero_nat) = [B]
| f1 (Suc (Suc Zero_nat)) = [A,B]
| f1 _ = [];
fun f2 Zero_nat = [A,A]
| f2 (Suc Zero_nat) = [B]
| f2 (Suc (Suc Zero_nat)) = [B,A]
| f2 _ = [];
val (i1, j1) = higman_idx g1;
val (v1, w1) = (g1 i1, g1 j1);
val (i2, j2) = higman_idx g2;
val (v2, w2) = (g2 i2, g2 j2);
val (i3, j3) = higman_idx f1;
val (v3, w3) = (f1 i3, f1 j3);
val (i4, j4) = higman_idx f2;
val (v4, w4) = (f2 i4, f2 j4);
end;
*}
end