--- a/src/HOL/Library/Coinductive_List.thy Tue Feb 27 00:32:52 2007 +0100
+++ b/src/HOL/Library/Coinductive_List.thy Tue Feb 27 00:33:49 2007 +0100
@@ -280,7 +280,7 @@
qed
-subsection {* Equality as greatest fixed-point; the bisimulation principle. *}
+subsection {* Equality as greatest fixed-point -- the bisimulation principle *}
consts
EqLList :: "('a Datatype.item \<times> 'a Datatype.item) set \<Rightarrow>
--- a/src/HOL/Library/Continuity.thy Tue Feb 27 00:32:52 2007 +0100
+++ b/src/HOL/Library/Continuity.thy Tue Feb 27 00:33:49 2007 +0100
@@ -9,20 +9,22 @@
imports Main
begin
-subsection{*Continuity for complete lattices*}
+subsection {* Continuity for complete lattices *}
-constdefs
- chain :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool"
-"chain M == !i. M i \<le> M(Suc i)"
- continuous :: "('a::order \<Rightarrow> 'a::order) \<Rightarrow> bool"
-"continuous F == !M. chain M \<longrightarrow> F(SUP i. M i) = (SUP i. F(M i))"
+definition
+ chain :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
+ "chain M \<longleftrightarrow> (\<forall>i. M i \<le> M (Suc i))"
+
+definition
+ continuous :: "('a::order \<Rightarrow> 'a::order) \<Rightarrow> bool" where
+ "continuous F \<longleftrightarrow> (\<forall>M. chain M \<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))"
abbreviation
bot :: "'a::order" where
- "bot == Sup {}"
+ "bot \<equiv> Sup {}"
lemma SUP_nat_conv:
- "(SUP n::nat. M n::'a::comp_lat) = join (M 0) (SUP n. M(Suc n))"
+ "(SUP n::nat. M n::'a::comp_lat) = join (M 0) (SUP n. M(Suc n))"
apply(rule order_antisym)
apply(rule SUP_leI)
apply(case_tac n)
--- a/src/HOL/Library/GCD.thy Tue Feb 27 00:32:52 2007 +0100
+++ b/src/HOL/Library/GCD.thy Tue Feb 27 00:33:49 2007 +0100
@@ -180,10 +180,10 @@
lemma gcd_add2 [simp]: "gcd (m, m + n) = gcd (m, n)"
proof -
- have "gcd (m, m + n) = gcd (m + n, m)" by (rule gcd_commute)
+ have "gcd (m, m + n) = gcd (m + n, m)" by (rule gcd_commute)
also have "... = gcd (n + m, m)" by (simp add: add_commute)
also have "... = gcd (n, m)" by simp
- also have "... = gcd (m, n)" by (rule gcd_commute)
+ also have "... = gcd (m, n)" by (rule gcd_commute)
finally show ?thesis .
qed
@@ -195,110 +195,124 @@
lemma gcd_add_mult: "gcd (m, k * m + n) = gcd (m, n)"
by (induct k) (simp_all add: add_assoc)
- (* Division by gcd yields rrelatively primes *)
+text {*
+ \medskip Division by gcd yields rrelatively primes.
+*}
lemma div_gcd_relprime:
- assumes nz:"a\<noteq>0 \<or> b\<noteq>0"
+ assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
shows "gcd (a div gcd(a,b), b div gcd(a,b)) = 1"
-proof-
- let ?g = "gcd (a,b)"
+proof -
+ let ?g = "gcd (a, b)"
let ?a' = "a div ?g"
let ?b' = "b div ?g"
let ?g' = "gcd (?a', ?b')"
have dvdg: "?g dvd a" "?g dvd b" by simp_all
have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
- from dvdg dvdg' obtain ka kb ka' kb' where
- kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
+ from dvdg dvdg' obtain ka kb ka' kb' where
+ kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
unfolding dvd_def by blast
- hence "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all
- hence dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
- by (auto simp add: dvd_mult_div_cancel[OF dvdg(1)]
- dvd_mult_div_cancel[OF dvdg(2)] dvd_def)
+ then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'" by simp_all
+ then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
+ by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
+ dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
have "?g \<noteq> 0" using nz by (simp add: gcd_zero)
- hence gp: "?g > 0" by simp
- from gcd_greatest[OF dvdgg'] have "?g * ?g' dvd ?g" .
- with dvd_mult_cancel1[OF gp] show "?g' = 1" by simp
+ then have gp: "?g > 0" by simp
+ from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
+ with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
qed
- (* gcd on integers *)
-constdefs igcd:: "int \<Rightarrow> int \<Rightarrow> int"
- "igcd i j \<equiv> int (gcd (nat (abs i),nat (abs j)))"
-lemma igcd_dvd1[simp]:"igcd i j dvd i"
+
+text {*
+ \medskip Gcd on integers.
+*}
+
+definition
+ igcd :: "int \<Rightarrow> int \<Rightarrow> int" where
+ "igcd i j = int (gcd (nat (abs i), nat (abs j)))"
+
+lemma igcd_dvd1 [simp]: "igcd i j dvd i"
by (simp add: igcd_def int_dvd_iff)
-lemma igcd_dvd2[simp]:"igcd i j dvd j"
-by (simp add: igcd_def int_dvd_iff)
+lemma igcd_dvd2 [simp]: "igcd i j dvd j"
+ by (simp add: igcd_def int_dvd_iff)
lemma igcd_pos: "igcd i j \<ge> 0"
-by (simp add: igcd_def)
-lemma igcd0[simp]: "(igcd i j = 0) = (i = 0 \<and> j = 0)"
-by (simp add: igcd_def gcd_zero) arith
+ by (simp add: igcd_def)
+
+lemma igcd0 [simp]: "(igcd i j = 0) = (i = 0 \<and> j = 0)"
+ by (simp add: igcd_def gcd_zero) arith
lemma igcd_commute: "igcd i j = igcd j i"
unfolding igcd_def by (simp add: gcd_commute)
-lemma igcd_neg1[simp]: "igcd (- i) j = igcd i j"
+
+lemma igcd_neg1 [simp]: "igcd (- i) j = igcd i j"
unfolding igcd_def by simp
-lemma igcd_neg2[simp]: "igcd i (- j) = igcd i j"
+
+lemma igcd_neg2 [simp]: "igcd i (- j) = igcd i j"
unfolding igcd_def by simp
+
lemma zrelprime_dvd_mult: "igcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k"
unfolding igcd_def
-proof-
- assume H: "int (gcd (nat \<bar>i\<bar>, nat \<bar>j\<bar>)) = 1" "i dvd k * j"
- hence g: "gcd (nat \<bar>i\<bar>, nat \<bar>j\<bar>) = 1" by simp
- from H(2) obtain h where h:"k*j = i*h" unfolding dvd_def by blast
+proof -
+ assume "int (gcd (nat \<bar>i\<bar>, nat \<bar>j\<bar>)) = 1" "i dvd k * j"
+ then have g: "gcd (nat \<bar>i\<bar>, nat \<bar>j\<bar>) = 1" by simp
+ from `i dvd k * j` obtain h where h: "k*j = i*h" unfolding dvd_def by blast
have th: "nat \<bar>i\<bar> dvd nat \<bar>k\<bar> * nat \<bar>j\<bar>"
- unfolding dvd_def
- by (rule_tac x= "nat \<bar>h\<bar>" in exI,simp add: h nat_abs_mult_distrib[symmetric])
- from relprime_dvd_mult[OF g th] obtain h' where h': "nat \<bar>k\<bar> = nat \<bar>i\<bar> * h'"
+ unfolding dvd_def
+ by (rule_tac x= "nat \<bar>h\<bar>" in exI, simp add: h nat_abs_mult_distrib [symmetric])
+ from relprime_dvd_mult [OF g th] obtain h' where h': "nat \<bar>k\<bar> = nat \<bar>i\<bar> * h'"
unfolding dvd_def by blast
from h' have "int (nat \<bar>k\<bar>) = int (nat \<bar>i\<bar> * h')" by simp
- hence "\<bar>k\<bar> = \<bar>i\<bar> * int h'" by (simp add: int_mult)
+ then have "\<bar>k\<bar> = \<bar>i\<bar> * int h'" by (simp add: int_mult)
then show ?thesis
- apply (subst zdvd_abs1[symmetric])
- apply (subst zdvd_abs2[symmetric])
+ apply (subst zdvd_abs1 [symmetric])
+ apply (subst zdvd_abs2 [symmetric])
apply (unfold dvd_def)
- apply (rule_tac x="int h'" in exI, simp)
+ apply (rule_tac x = "int h'" in exI, simp)
done
qed
lemma int_nat_abs: "int (nat (abs x)) = abs x" by arith
-lemma igcd_greatest: assumes km:"k dvd m" and kn:"k dvd n" shows "k dvd igcd m n"
-proof-
+
+lemma igcd_greatest:
+ assumes "k dvd m" and "k dvd n"
+ shows "k dvd igcd m n"
+proof -
let ?k' = "nat \<bar>k\<bar>"
let ?m' = "nat \<bar>m\<bar>"
let ?n' = "nat \<bar>n\<bar>"
- from km kn have dvd': "?k' dvd ?m'" "?k' dvd ?n'"
+ from `k dvd m` and `k dvd n` have dvd': "?k' dvd ?m'" "?k' dvd ?n'"
unfolding zdvd_int by (simp_all only: int_nat_abs zdvd_abs1 zdvd_abs2)
- from gcd_greatest[OF dvd'] have "int (nat \<bar>k\<bar>) dvd igcd m n"
+ from gcd_greatest [OF dvd'] have "int (nat \<bar>k\<bar>) dvd igcd m n"
unfolding igcd_def by (simp only: zdvd_int)
- hence "\<bar>k\<bar> dvd igcd m n" by (simp only: int_nat_abs)
- thus "k dvd igcd m n" by (simp add: zdvd_abs1)
+ then have "\<bar>k\<bar> dvd igcd m n" by (simp only: int_nat_abs)
+ then show "k dvd igcd m n" by (simp add: zdvd_abs1)
qed
lemma div_igcd_relprime:
- assumes nz:"a\<noteq>0 \<or> b\<noteq>0"
+ assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
shows "igcd (a div (igcd a b)) (b div (igcd a b)) = 1"
-proof-
+proof -
from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 0" by simp
- have th1: "(1::int) = int 1" by simp
let ?g = "igcd a b"
let ?a' = "a div ?g"
let ?b' = "b div ?g"
let ?g' = "igcd ?a' ?b'"
have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: igcd_dvd1 igcd_dvd2)
have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: igcd_dvd1 igcd_dvd2)
- from dvdg dvdg' obtain ka kb ka' kb' where
- kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
+ from dvdg dvdg' obtain ka kb ka' kb' where
+ kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
unfolding dvd_def by blast
- hence "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all
- hence dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
- by (auto simp add: zdvd_mult_div_cancel[OF dvdg(1)]
- zdvd_mult_div_cancel[OF dvdg(2)] dvd_def)
+ then have "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all
+ then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
+ by (auto simp add: zdvd_mult_div_cancel [OF dvdg(1)]
+ zdvd_mult_div_cancel [OF dvdg(2)] dvd_def)
have "?g \<noteq> 0" using nz by simp
- hence gp: "?g \<noteq> 0" using igcd_pos[where i="a" and j="b"] by arith
- from igcd_greatest[OF dvdgg'] have "?g * ?g' dvd ?g" .
- with zdvd_mult_cancel1[OF gp] have "\<bar>?g'\<bar> = 1" by simp
+ then have gp: "?g \<noteq> 0" using igcd_pos[where i="a" and j="b"] by arith
+ from igcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
+ with zdvd_mult_cancel1 [OF gp] have "\<bar>?g'\<bar> = 1" by simp
with igcd_pos show "?g' = 1" by simp
qed
--- a/src/HOL/Library/Library/document/root.bib Tue Feb 27 00:32:52 2007 +0100
+++ b/src/HOL/Library/Library/document/root.bib Tue Feb 27 00:33:49 2007 +0100
@@ -1,5 +1,12 @@
- @InProceedings{Avigad-Donnelly,
+@Unpublished{Abrial-Laffitte,
+ author = {Abrial and Laffitte},
+ title = {Towards the Mechanization of the Proofs of
+ Some Classical Theorems of Set Theory},
+ note = {Unpublished}
+}
+
+@InProceedings{Avigad-Donnelly,
author = {Jeremy Avigad and Kevin Donnelly},
title = {Formalizing {O} notation in {Isabelle/HOL}},
booktitle = {Automated Reasoning: second international conference, IJCAR 2004},
@@ -9,13 +16,6 @@
publisher = {Springer}
}
-@Unpublished{Abrial-Laffitte,
- author = {Abrial and Laffitte},
- title = {Towards the Mechanization of the Proofs of
- Some Classical Theorems of Set Theory},
- note = {Unpublished}
-}
-
@Book{Oberschelp:1993,
author = {Arnold Oberschelp},
title = {Rekursionstheorie},
@@ -23,6 +23,21 @@
year = 1993
}
+@InProceedings{Podelski-Rybalchenko,
+ author = {Andreas Podelski and Andrey Rybalchenko},
+ title = {Transition Invariants},
+ booktitle = {19th Annual IEEE Symposium on Logic in Computer Science (LICS'04)},
+ pages = {32--41},
+ year = 2004
+}
+
+@Book{davenport92,
+ author = {H. Davenport},
+ title = {The Higher Arithmetic},
+ publisher = {Cambridge University Press},
+ year = 1992
+}
+
@InProceedings{paulin-tlca,
author = {Christine Paulin-Mohring},
title = {Inductive Definitions in the System {Coq}: Rules and
--- a/src/HOL/Library/Ramsey.thy Tue Feb 27 00:32:52 2007 +0100
+++ b/src/HOL/Library/Ramsey.thy Tue Feb 27 00:33:49 2007 +0100
@@ -235,10 +235,9 @@
subsection {*Disjunctive Well-Foundedness*}
-text{*An application of Ramsey's theorem to program termination. See
-
-Andreas Podelski and Andrey Rybalchenko, Transition Invariants, 19th Annual
-IEEE Symposium on Logic in Computer Science (LICS'04), pages 32--41 (2004).
+text {*
+ An application of Ramsey's theorem to program termination. See
+ \cite{Podelski-Rybalchenko}.
*}
definition
--- a/src/HOL/document/root.bib Tue Feb 27 00:32:52 2007 +0100
+++ b/src/HOL/document/root.bib Tue Feb 27 00:33:49 2007 +0100
@@ -1,9 +1,3 @@
-@Book{davenport92,
- author = {H. Davenport},
- title = {The Higher Arithmetic},
- publisher = {Cambridge University Press},
- year = 1992
-}
@book{Birkhoff79,author={Garret Birkhoff},title={Lattice Theory},
publisher={American Mathematical Society},year=1979}
\ No newline at end of file