--- a/CONTRIBUTORS Wed Apr 22 11:00:25 2009 -0700
+++ b/CONTRIBUTORS Mon Apr 27 07:26:17 2009 -0700
@@ -7,6 +7,10 @@
Contributions to this Isabelle version
--------------------------------------
+
+Contributions to Isabelle2009
+-----------------------------
+
* March 2009: Robert Himmelmann, TUM and Amine Chaieb, University of
Cambridge
Elementary topology in Euclidean space.
--- a/NEWS Wed Apr 22 11:00:25 2009 -0700
+++ b/NEWS Mon Apr 27 07:26:17 2009 -0700
@@ -4,6 +4,26 @@
New in this Isabelle version
----------------------------
+*** Pure ***
+
+* On instantiation of classes, remaining undefined class parameters are
+formally declared. INCOMPATIBILITY.
+
+
+*** HOL ***
+
+* Class semiring_div requires superclass no_zero_divisors and proof of div_mult_mult1;
+theorems div_mult_mult1, div_mult_mult2, div_mult_mult1_if, div_mult_mult1 and
+div_mult_mult2 have been generalized to class semiring_div, subsuming former
+theorems zdiv_zmult_zmult1, zdiv_zmult_zmult1_if, zdiv_zmult_zmult1 and zdiv_zmult_zmult2.
+div_mult_mult1 is now [simp] by default. INCOMPATIBILITY.
+
+* Power operations on relations and functions are now one dedicate constant compow with
+infix syntax "^^". Power operations on multiplicative monoids retains syntax "^"
+and is now defined generic in class power. INCOMPATIBILITY.
+
+* ML antiquotation @{code_datatype} inserts definition of a datatype generated
+by the code generator; see Predicate.thy for an example.
New in Isabelle2009 (April 2009)
@@ -187,7 +207,7 @@
* Keyword 'code_exception' now named 'code_abort'. INCOMPATIBILITY.
-* Unified theorem tables for both code code generators. Thus [code
+* Unified theorem tables for both code generators. Thus [code
func] has disappeared and only [code] remains. INCOMPATIBILITY.
* Command 'find_consts' searches for constants based on type and name
--- a/contrib/SystemOnTPTP/remote Wed Apr 22 11:00:25 2009 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,120 +0,0 @@
-#!/usr/bin/env perl
-#
-# Wrapper for custom remote provers on SystemOnTPTP
-# Author: Fabian Immler, TU Muenchen
-#
-
-use warnings;
-use strict;
-use Getopt::Std;
-use HTTP::Request::Common;
-use LWP;
-
-my $SystemOnTPTPFormReplyURL = "http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTPFormReply";
-
-# default parameters
-my %URLParameters = (
- "NoHTML" => 1,
- "QuietFlag" => "-q01",
- "X2TPTP" => "-S",
- "SubmitButton" => "RunSelectedSystems",
- "ProblemSource" => "UPLOAD",
- );
-
-#----Get format and transform options if specified
-my %Options;
-getopts("hws:t:c:",\%Options);
-
-#----Usage
-sub usage() {
- print("Usage: remote [<options>] <File name>\n");
- print(" <options> are ...\n");
- print(" -h - print this help\n");
- print(" -w - list available ATP systems\n");
- print(" -s<system> - specified system to use\n");
- print(" -t<timelimit> - CPU time limit for system\n");
- print(" -c<command> - custom command for system\n");
- print(" <File name> - TPTP problem file\n");
- exit(0);
-}
-if (exists($Options{'h'})) {
- usage();
-}
-#----What systems flag
-if (exists($Options{'w'})) {
- $URLParameters{"SubmitButton"} = "ListSystems";
- delete($URLParameters{"ProblemSource"});
-}
-#----Selected system
-my $System;
-if (exists($Options{'s'})) {
- $System = $Options{'s'};
-} else {
- # use Vampire as default
- $System = "Vampire---9.0";
-}
-$URLParameters{"System___$System"} = $System;
-
-#----Time limit
-if (exists($Options{'t'})) {
- $URLParameters{"TimeLimit___$System"} = $Options{'t'};
-}
-#----Custom command
-if (exists($Options{'c'})) {
- $URLParameters{"Command___$System"} = $Options{'c'};
-}
-
-#----Get single file name
-if (exists($URLParameters{"ProblemSource"})) {
- if (scalar(@ARGV) >= 1) {
- $URLParameters{"UPLOADProblem"} = [shift(@ARGV)];
- } else {
- print("Missing problem file\n");
- usage();
- die;
- }
-}
-
-# Query Server
-my $Agent = LWP::UserAgent->new;
-if (exists($Options{'t'})) {
- # give server more time to respond
- $Agent->timeout($Options{'t'} + 10);
-}
-my $Request = POST($SystemOnTPTPFormReplyURL,
- Content_Type => 'form-data',Content => \%URLParameters);
-my $Response = $Agent->request($Request);
-
-#catch errors / failure
-if(! $Response->is_success){
- print "HTTP-Error: " . $Response->message . "\n";
- exit(-1);
-} elsif (exists($Options{'w'})) {
- print $Response->content;
- exit (0);
-} elsif ($Response->content =~ /WARNING: (\S*) does not exist/) {
- print "Specified System $1 does not exist\n";
- exit(-1);
-} elsif ($Response->content =~ /%\s*Result\s*:\s*Unsatisfiable.*\n%\s*Output\s*:\s*(CNF)?Refutation.*\n%/) {
- my @lines = split( /\n/, $Response->content);
- my $extract = "";
- foreach my $line (@lines){
- #ignore comments
- if ($line !~ /^%/ && !($line eq "")) {
- $extract .= "$line";
- }
- }
- # insert newlines after ').'
- $extract =~ s/\s//g;
- $extract =~ s/\)\.cnf/\)\.\ncnf/g;
-
- # orientation for res_reconstruct.ML
- print "# SZS output start CNFRefutation.\n";
- print "$extract\n";
- print "# SZS output end CNFRefutation.\n";
- exit(0);
-} else {
- print "Remote-script could not extract proof:\n".$Response->content;
- exit(-1);
-}
-
--- a/doc-src/Codegen/Thy/Program.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/doc-src/Codegen/Thy/Program.thy Mon Apr 27 07:26:17 2009 -0700
@@ -323,7 +323,7 @@
*}
-subsection {* Equality and wellsortedness *}
+subsection {* Equality *}
text {*
Surely you have already noticed how equality is treated
@@ -358,60 +358,7 @@
manually like any other type class.
Though this @{text eq} class is designed to get rarely in
- the way, a subtlety
- enters the stage when definitions of overloaded constants
- are dependent on operational equality. For example, let
- us define a lexicographic ordering on tuples
- (also see theory @{theory Product_ord}):
-*}
-
-instantiation %quote "*" :: (order, order) order
-begin
-
-definition %quote [code del]:
- "x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x \<le> snd y"
-
-definition %quote [code del]:
- "x < y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x < snd y"
-
-instance %quote proof
-qed (auto simp: less_eq_prod_def less_prod_def intro: order_less_trans)
-
-end %quote
-
-lemma %quote order_prod [code]:
- "(x1 \<Colon> 'a\<Colon>order, y1 \<Colon> 'b\<Colon>order) < (x2, y2) \<longleftrightarrow>
- x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
- "(x1 \<Colon> 'a\<Colon>order, y1 \<Colon> 'b\<Colon>order) \<le> (x2, y2) \<longleftrightarrow>
- x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
- by (simp_all add: less_prod_def less_eq_prod_def)
-
-text {*
- \noindent Then code generation will fail. Why? The definition
- of @{term "op \<le>"} depends on equality on both arguments,
- which are polymorphic and impose an additional @{class eq}
- class constraint, which the preprocessor does not propagate
- (for technical reasons).
-
- The solution is to add @{class eq} explicitly to the first sort arguments in the
- code theorems:
-*}
-
-lemma %quote order_prod_code [code]:
- "(x1 \<Colon> 'a\<Colon>{order, eq}, y1 \<Colon> 'b\<Colon>order) < (x2, y2) \<longleftrightarrow>
- x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
- "(x1 \<Colon> 'a\<Colon>{order, eq}, y1 \<Colon> 'b\<Colon>order) \<le> (x2, y2) \<longleftrightarrow>
- x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
- by (simp_all add: less_prod_def less_eq_prod_def)
-
-text {*
- \noindent Then code generation succeeds:
-*}
-
-text %quote {*@{code_stmts "op \<le> \<Colon> _ \<times> _ \<Rightarrow> _ \<times> _ \<Rightarrow> bool" (SML)}*}
-
-text {*
- In some cases, the automatically derived code equations
+ the way, in some cases the automatically derived code equations
for equality on a particular type may not be appropriate.
As example, watch the following datatype representing
monomorphic parametric types (where type constructors
--- a/doc-src/Codegen/Thy/document/Program.tex Wed Apr 22 11:00:25 2009 -0700
+++ b/doc-src/Codegen/Thy/document/Program.tex Mon Apr 27 07:26:17 2009 -0700
@@ -714,7 +714,7 @@
\end{isamarkuptext}%
\isamarkuptrue%
%
-\isamarkupsubsection{Equality and wellsortedness%
+\isamarkupsubsection{Equality%
}
\isamarkuptrue%
%
@@ -801,141 +801,7 @@
manually like any other type class.
Though this \isa{eq} class is designed to get rarely in
- the way, a subtlety
- enters the stage when definitions of overloaded constants
- are dependent on operational equality. For example, let
- us define a lexicographic ordering on tuples
- (also see theory \hyperlink{theory.Product-ord}{\mbox{\isa{Product{\isacharunderscore}ord}}}):%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{instantiation}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharasterisk}{\isachardoublequoteclose}\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}order{\isacharcomma}\ order{\isacharparenright}\ order\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isacharbrackleft}code\ del{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}x\ {\isasymle}\ y\ {\isasymlongleftrightarrow}\ fst\ x\ {\isacharless}\ fst\ y\ {\isasymor}\ fst\ x\ {\isacharequal}\ fst\ y\ {\isasymand}\ snd\ x\ {\isasymle}\ snd\ y{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ {\isacharbrackleft}code\ del{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}x\ {\isacharless}\ y\ {\isasymlongleftrightarrow}\ fst\ x\ {\isacharless}\ fst\ y\ {\isasymor}\ fst\ x\ {\isacharequal}\ fst\ y\ {\isasymand}\ snd\ x\ {\isacharless}\ snd\ y{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{instance}\isamarkupfalse%
-\ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-\ {\isacharparenleft}auto\ simp{\isacharcolon}\ less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def\ less{\isacharunderscore}prod{\isacharunderscore}def\ intro{\isacharcolon}\ order{\isacharunderscore}less{\isacharunderscore}trans{\isacharparenright}\isanewline
-\isanewline
-\isacommand{end}\isamarkupfalse%
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ order{\isacharunderscore}prod\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}order{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}order{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isasymle}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ less{\isacharunderscore}prod{\isacharunderscore}def\ less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Then code generation will fail. Why? The definition
- of \isa{op\ {\isasymle}} depends on equality on both arguments,
- which are polymorphic and impose an additional \isa{eq}
- class constraint, which the preprocessor does not propagate
- (for technical reasons).
-
- The solution is to add \isa{eq} explicitly to the first sort arguments in the
- code theorems:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-\isacommand{lemma}\isamarkupfalse%
-\ order{\isacharunderscore}prod{\isacharunderscore}code\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}order{\isacharcomma}\ eq{\isacharbraceright}{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isacharless}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}x{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}a{\isasymColon}{\isacharbraceleft}order{\isacharcomma}\ eq{\isacharbraceright}{\isacharcomma}\ y{\isadigit{1}}\ {\isasymColon}\ {\isacharprime}b{\isasymColon}order{\isacharparenright}\ {\isasymle}\ {\isacharparenleft}x{\isadigit{2}}{\isacharcomma}\ y{\isadigit{2}}{\isacharparenright}\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ x{\isadigit{1}}\ {\isacharless}\ x{\isadigit{2}}\ {\isasymor}\ x{\isadigit{1}}\ {\isacharequal}\ x{\isadigit{2}}\ {\isasymand}\ y{\isadigit{1}}\ {\isasymle}\ y{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ less{\isacharunderscore}prod{\isacharunderscore}def\ less{\isacharunderscore}eq{\isacharunderscore}prod{\isacharunderscore}def{\isacharparenright}%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-\noindent Then code generation succeeds:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\isatagquote
-%
-\begin{isamarkuptext}%
-\isatypewriter%
-\noindent%
-\hspace*{0pt}structure Example = \\
-\hspace*{0pt}struct\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a eq = {\char123}eq :~'a -> 'a -> bool{\char125};\\
-\hspace*{0pt}fun eq (A{\char95}:'a eq) = {\char35}eq A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a ord = {\char123}less{\char95}eq :~'a -> 'a -> bool,~less :~'a -> 'a -> bool{\char125};\\
-\hspace*{0pt}fun less{\char95}eq (A{\char95}:'a ord) = {\char35}less{\char95}eq A{\char95};\\
-\hspace*{0pt}fun less (A{\char95}:'a ord) = {\char35}less A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun eqop A{\char95}~a b = eq A{\char95}~a b;\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a preorder = {\char123}Orderings{\char95}{\char95}ord{\char95}preorder :~'a ord{\char125};\\
-\hspace*{0pt}fun ord{\char95}preorder (A{\char95}:'a preorder) = {\char35}Orderings{\char95}{\char95}ord{\char95}preorder A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}type 'a order = {\char123}Orderings{\char95}{\char95}preorder{\char95}order :~'a preorder{\char125};\\
-\hspace*{0pt}fun preorder{\char95}order (A{\char95}:'a order) = {\char35}Orderings{\char95}{\char95}preorder{\char95}order A{\char95};\\
-\hspace*{0pt}\\
-\hspace*{0pt}fun less{\char95}eqa (A1{\char95},~A2{\char95}) B{\char95}~(x1,~y1) (x2,~y2) =\\
-\hspace*{0pt} ~less ((ord{\char95}preorder o preorder{\char95}order) A2{\char95}) x1 x2 orelse\\
-\hspace*{0pt} ~~~eqop A1{\char95}~x1 x2 andalso\\
-\hspace*{0pt} ~~~~~less{\char95}eq ((ord{\char95}preorder o preorder{\char95}order) B{\char95}) y1 y2\\
-\hspace*{0pt} ~| less{\char95}eqa (A1{\char95},~A2{\char95}) B{\char95}~(x1,~y1) (x2,~y2) =\\
-\hspace*{0pt} ~~~less ((ord{\char95}preorder o preorder{\char95}order) A2{\char95}) x1 x2 orelse\\
-\hspace*{0pt} ~~~~~eqop A1{\char95}~x1 x2 andalso\\
-\hspace*{0pt} ~~~~~~~less{\char95}eq ((ord{\char95}preorder o preorder{\char95}order) B{\char95}) y1 y2;\\
-\hspace*{0pt}\\
-\hspace*{0pt}end;~(*struct Example*)%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\endisatagquote
-{\isafoldquote}%
-%
-\isadelimquote
-%
-\endisadelimquote
-%
-\begin{isamarkuptext}%
-In some cases, the automatically derived code equations
+ the way, in some cases the automatically derived code equations
for equality on a particular type may not be appropriate.
As example, watch the following datatype representing
monomorphic parametric types (where type constructors
--- a/doc-src/Main/Docs/Main_Doc.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/doc-src/Main/Docs/Main_Doc.thy Mon Apr 27 07:26:17 2009 -0700
@@ -268,6 +268,7 @@
@{const Transitive_Closure.rtrancl} & @{term_type_only Transitive_Closure.rtrancl "('a*'a)set\<Rightarrow>('a*'a)set"}\\
@{const Transitive_Closure.trancl} & @{term_type_only Transitive_Closure.trancl "('a*'a)set\<Rightarrow>('a*'a)set"}\\
@{const Transitive_Closure.reflcl} & @{term_type_only Transitive_Closure.reflcl "('a*'a)set\<Rightarrow>('a*'a)set"}\\
+@{const compower} & @{term_type_only "op ^^ :: ('a*'a)set\<Rightarrow>nat\<Rightarrow>('a*'a)set" "('a*'a)set\<Rightarrow>nat\<Rightarrow>('a*'a)set"}\\
\end{tabular}
\subsubsection*{Syntax}
@@ -318,7 +319,6 @@
@{term "op + :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
@{term "op - :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
@{term "op * :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
-@{term "op ^ :: nat \<Rightarrow> nat \<Rightarrow> nat"} &
@{term "op div :: nat \<Rightarrow> nat \<Rightarrow> nat"}&
@{term "op mod :: nat \<Rightarrow> nat \<Rightarrow> nat"}&
@{term "op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"}\\
@@ -331,7 +331,9 @@
\end{tabular}
\begin{tabular}{@ {} l @ {~::~} l @ {}}
-@{const Nat.of_nat} & @{typeof Nat.of_nat}
+@{const Nat.of_nat} & @{typeof Nat.of_nat}\\
+@{term "op ^^ :: ('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"} &
+ @{term_type_only "op ^^ :: ('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"}
\end{tabular}
\section{Int}
@@ -450,14 +452,6 @@
\end{tabular}
-\section{Iterated Functions and Relations}
-
-Theory: @{theory Relation_Power}
-
-Iterated functions \ @{term[source]"(f::'a\<Rightarrow>'a) ^ n"} \
-and relations \ @{term[source]"(r::('a\<times>'a)set) ^ n"}.
-
-
\section{Option}
@{datatype option}
--- a/doc-src/Main/Docs/document/Main_Doc.tex Wed Apr 22 11:00:25 2009 -0700
+++ b/doc-src/Main/Docs/document/Main_Doc.tex Mon Apr 27 07:26:17 2009 -0700
@@ -279,6 +279,7 @@
\isa{rtrancl} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\
\isa{trancl} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\
\isa{reflcl} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\
+\isa{op\ {\isacharcircum}{\isacharcircum}} & \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}\\
\end{tabular}
\subsubsection*{Syntax}
@@ -328,7 +329,6 @@
\isa{op\ {\isacharplus}} &
\isa{op\ {\isacharminus}} &
\isa{op\ {\isacharasterisk}} &
-\isa{op\ {\isacharcircum}} &
\isa{op\ div}&
\isa{op\ mod}&
\isa{op\ dvd}\\
@@ -341,7 +341,9 @@
\end{tabular}
\begin{tabular}{@ {} l @ {~::~} l @ {}}
-\isa{of{\isacharunderscore}nat} & \isa{nat\ {\isasymRightarrow}\ {\isacharprime}a}
+\isa{of{\isacharunderscore}nat} & \isa{nat\ {\isasymRightarrow}\ {\isacharprime}a}\\
+\isa{op\ {\isacharcircum}{\isacharcircum}} &
+ \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isacharparenright}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a}
\end{tabular}
\section{Int}
@@ -460,14 +462,6 @@
\end{tabular}
-\section{Iterated Functions and Relations}
-
-Theory: \isa{Relation{\isacharunderscore}Power}
-
-Iterated functions \ \isa{{\isachardoublequote}{\isacharparenleft}f{\isacharcolon}{\isacharcolon}{\isacharprime}a{\isasymRightarrow}{\isacharprime}a{\isacharparenright}\ {\isacharcircum}\ n{\isachardoublequote}} \
-and relations \ \isa{{\isachardoublequote}{\isacharparenleft}r{\isacharcolon}{\isacharcolon}{\isacharparenleft}{\isacharprime}a{\isasymtimes}{\isacharprime}a{\isacharparenright}set{\isacharparenright}\ {\isacharcircum}\ n{\isachardoublequote}}.
-
-
\section{Option}
\isa{\isacommand{datatype}\ {\isacharprime}a\ option\ {\isacharequal}\ None\ {\isacharbar}\ Some\ {\isacharprime}a}
--- a/doc-src/TutorialI/tutorial.tex Wed Apr 22 11:00:25 2009 -0700
+++ b/doc-src/TutorialI/tutorial.tex Mon Apr 27 07:26:17 2009 -0700
@@ -39,10 +39,11 @@
%University of Cambridge\\
%Computer Laboratory
}
+\pagenumbering{roman}
\maketitle
+\newpage
-\pagenumbering{roman}
-\setcounter{page}{5}
+%\setcounter{page}{5}
%\vspace*{\fill}
%\begin{center}
%\LARGE In memoriam \\[1ex]
@@ -52,6 +53,7 @@
%\vspace*{\fill}
%\vspace*{\fill}
%\newpage
+
\include{preface}
\tableofcontents
--- a/doc-src/more_antiquote.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/doc-src/more_antiquote.ML Mon Apr 27 07:26:17 2009 -0700
@@ -88,7 +88,7 @@
let
val thy = ProofContext.theory_of ctxt;
val const = Code_Unit.check_const thy raw_const;
- val (_, funcgr) = Code_Wellsorted.make thy [const];
+ val (_, funcgr) = Code_Wellsorted.obtain thy [const] [];
fun holize thm = @{thm meta_eq_to_obj_eq} OF [thm];
val thms = Code_Wellsorted.eqns funcgr const
|> map_filter (fn (thm, linear) => if linear then SOME thm else NONE)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/lib/scripts/SystemOnTPTP Mon Apr 27 07:26:17 2009 -0700
@@ -0,0 +1,120 @@
+#!/usr/bin/env perl
+#
+# Wrapper for custom remote provers on SystemOnTPTP
+# Author: Fabian Immler, TU Muenchen
+#
+
+use warnings;
+use strict;
+use Getopt::Std;
+use HTTP::Request::Common;
+use LWP;
+
+my $SystemOnTPTPFormReplyURL = "http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTPFormReply";
+
+# default parameters
+my %URLParameters = (
+ "NoHTML" => 1,
+ "QuietFlag" => "-q01",
+ "X2TPTP" => "-S",
+ "SubmitButton" => "RunSelectedSystems",
+ "ProblemSource" => "UPLOAD",
+ );
+
+#----Get format and transform options if specified
+my %Options;
+getopts("hws:t:c:",\%Options);
+
+#----Usage
+sub usage() {
+ print("Usage: remote [<options>] <File name>\n");
+ print(" <options> are ...\n");
+ print(" -h - print this help\n");
+ print(" -w - list available ATP systems\n");
+ print(" -s<system> - specified system to use\n");
+ print(" -t<timelimit> - CPU time limit for system\n");
+ print(" -c<command> - custom command for system\n");
+ print(" <File name> - TPTP problem file\n");
+ exit(0);
+}
+if (exists($Options{'h'})) {
+ usage();
+}
+#----What systems flag
+if (exists($Options{'w'})) {
+ $URLParameters{"SubmitButton"} = "ListSystems";
+ delete($URLParameters{"ProblemSource"});
+}
+#----Selected system
+my $System;
+if (exists($Options{'s'})) {
+ $System = $Options{'s'};
+} else {
+ # use Vampire as default
+ $System = "Vampire---9.0";
+}
+$URLParameters{"System___$System"} = $System;
+
+#----Time limit
+if (exists($Options{'t'})) {
+ $URLParameters{"TimeLimit___$System"} = $Options{'t'};
+}
+#----Custom command
+if (exists($Options{'c'})) {
+ $URLParameters{"Command___$System"} = $Options{'c'};
+}
+
+#----Get single file name
+if (exists($URLParameters{"ProblemSource"})) {
+ if (scalar(@ARGV) >= 1) {
+ $URLParameters{"UPLOADProblem"} = [shift(@ARGV)];
+ } else {
+ print("Missing problem file\n");
+ usage();
+ die;
+ }
+}
+
+# Query Server
+my $Agent = LWP::UserAgent->new;
+if (exists($Options{'t'})) {
+ # give server more time to respond
+ $Agent->timeout($Options{'t'} + 10);
+}
+my $Request = POST($SystemOnTPTPFormReplyURL,
+ Content_Type => 'form-data',Content => \%URLParameters);
+my $Response = $Agent->request($Request);
+
+#catch errors / failure
+if(! $Response->is_success){
+ print "HTTP-Error: " . $Response->message . "\n";
+ exit(-1);
+} elsif (exists($Options{'w'})) {
+ print $Response->content;
+ exit (0);
+} elsif ($Response->content =~ /WARNING: (\S*) does not exist/) {
+ print "Specified System $1 does not exist\n";
+ exit(-1);
+} elsif ($Response->content =~ /%\s*Result\s*:\s*Unsatisfiable.*\n%\s*Output\s*:\s*(CNF)?Refutation.*\n%/) {
+ my @lines = split( /\n/, $Response->content);
+ my $extract = "";
+ foreach my $line (@lines){
+ #ignore comments
+ if ($line !~ /^%/ && !($line eq "")) {
+ $extract .= "$line";
+ }
+ }
+ # insert newlines after ').'
+ $extract =~ s/\s//g;
+ $extract =~ s/\)\.cnf/\)\.\ncnf/g;
+
+ # orientation for res_reconstruct.ML
+ print "# SZS output start CNFRefutation.\n";
+ print "$extract\n";
+ print "# SZS output end CNFRefutation.\n";
+ exit(0);
+} else {
+ print "Remote-script could not extract proof:\n".$Response->content;
+ exit(-1);
+}
+
--- a/src/HOL/Algebra/abstract/Ring2.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Algebra/abstract/Ring2.thy Mon Apr 27 07:26:17 2009 -0700
@@ -12,7 +12,7 @@
subsection {* Ring axioms *}
-class ring = zero + one + plus + minus + uminus + times + inverse + power + Ring_and_Field.dvd +
+class ring = zero + one + plus + minus + uminus + times + inverse + power + dvd +
assumes a_assoc: "(a + b) + c = a + (b + c)"
and l_zero: "0 + a = a"
and l_neg: "(-a) + a = 0"
@@ -28,8 +28,6 @@
assumes minus_def: "a - b = a + (-b)"
and inverse_def: "inverse a = (if a dvd 1 then THE x. a*x = 1 else 0)"
and divide_def: "a / b = a * inverse b"
- and power_0 [simp]: "a ^ 0 = 1"
- and power_Suc [simp]: "a ^ Suc n = a ^ n * a"
begin
definition assoc :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "assoc" 50) where
--- a/src/HOL/Algebra/poly/LongDiv.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Algebra/poly/LongDiv.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1,6 +1,5 @@
(*
Experimental theory: long division of polynomials
- $Id$
Author: Clemens Ballarin, started 23 June 1999
*)
@@ -133,9 +132,9 @@
delsimprocs [ring_simproc]) 1 *})
apply (tactic {* asm_simp_tac (@{simpset} delsimprocs [ring_simproc]) 1 *})
apply (tactic {* simp_tac (@{simpset} addsimps [thm "minus_def", thm "smult_r_distr",
- thm "smult_r_minus", thm "monom_mult_smult", thm "smult_assoc1", thm "smult_assoc2"]
+ thm "smult_r_minus", thm "monom_mult_smult", thm "smult_assoc2"]
delsimprocs [ring_simproc]) 1 *})
- apply simp
+ apply (simp add: smult_assoc1 [symmetric])
done
ML {*
--- a/src/HOL/Algebra/poly/UnivPoly2.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Algebra/poly/UnivPoly2.thy Mon Apr 27 07:26:17 2009 -0700
@@ -155,16 +155,6 @@
end
-instantiation up :: ("{times, one, comm_monoid_add}") power
-begin
-
-primrec power_up where
- "(a \<Colon> 'a up) ^ 0 = 1"
- | "(a \<Colon> 'a up) ^ Suc n = a ^ n * a"
-
-instance ..
-
-end
subsection {* Effect of operations on coefficients *}
@@ -328,8 +318,9 @@
qed
show "(p + q) * r = p * r + q * r"
by (rule up_eqI) simp
- show "p * q = q * p"
+ show "\<And>q. p * q = q * p"
proof (rule up_eqI)
+ fix q
fix n
{
fix k
@@ -354,11 +345,11 @@
by (simp add: up_inverse_def)
show "p / q = p * inverse q"
by (simp add: up_divide_def)
- fix n
- show "p ^ 0 = 1" by simp
- show "p ^ Suc n = p ^ n * p" by simp
qed
+instance up :: (ring) recpower proof
+qed simp_all
+
(* Further properties of monom *)
lemma monom_zero [simp]:
--- a/src/HOL/Bali/Trans.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Bali/Trans.thy Mon Apr 27 07:26:17 2009 -0700
@@ -359,7 +359,7 @@
abbreviation
stepn:: "[prog, term \<times> state,nat,term \<times> state] \<Rightarrow> bool" ("_\<turnstile>_ \<mapsto>_ _"[61,82,82] 81)
- where "G\<turnstile>p \<mapsto>n p' \<equiv> (p,p') \<in> {(x, y). step G x y}^n"
+ where "G\<turnstile>p \<mapsto>n p' \<equiv> (p,p') \<in> {(x, y). step G x y}^^n"
abbreviation
steptr:: "[prog,term \<times> state,term \<times> state] \<Rightarrow> bool" ("_\<turnstile>_ \<mapsto>* _"[61,82,82] 81)
@@ -370,25 +370,6 @@
Smallstep zu Bigstep, nur wenn nicht die Ausdrücke Callee, FinA ,\<dots>
*)
-lemma rtrancl_imp_rel_pow: "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R^n"
-proof -
- assume "p \<in> R\<^sup>*"
- moreover obtain x y where p: "p = (x,y)" by (cases p)
- ultimately have "(x,y) \<in> R\<^sup>*" by hypsubst
- hence "\<exists>n. (x,y) \<in> R^n"
- proof induct
- fix a have "(a,a) \<in> R^0" by simp
- thus "\<exists>n. (a,a) \<in> R ^ n" ..
- next
- fix a b c assume "\<exists>n. (a,b) \<in> R ^ n"
- then obtain n where "(a,b) \<in> R^n" ..
- moreover assume "(b,c) \<in> R"
- ultimately have "(a,c) \<in> R^(Suc n)" by auto
- thus "\<exists>n. (a,c) \<in> R^n" ..
- qed
- with p show ?thesis by hypsubst
-qed
-
(*
lemma imp_eval_trans:
assumes eval: "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v,s1)"
--- a/src/HOL/Code_Eval.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Code_Eval.thy Mon Apr 27 07:26:17 2009 -0700
@@ -161,6 +161,7 @@
signature EVAL =
sig
val mk_term: ((string * typ) -> term) -> (typ -> term) -> term -> term
+ val mk_term_of: typ -> term -> term
val eval_ref: (unit -> term) option ref
val eval_term: theory -> term -> term
end;
@@ -175,8 +176,7 @@
fun eval_term thy t =
t
|> Eval.mk_term_of (fastype_of t)
- |> (fn t => Code_ML.eval_term ("Eval.eval_ref", eval_ref) thy t [])
- |> Code.postprocess_term thy;
+ |> (fn t => Code_ML.eval NONE ("Eval.eval_ref", eval_ref) I thy t []);
end;
*}
--- a/src/HOL/Code_Setup.thy Wed Apr 22 11:00:25 2009 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,253 +0,0 @@
-(* Title: HOL/Code_Setup.thy
- ID: $Id$
- Author: Florian Haftmann
-*)
-
-header {* Setup of code generators and related tools *}
-
-theory Code_Setup
-imports HOL
-begin
-
-subsection {* Generic code generator foundation *}
-
-text {* Datatypes *}
-
-code_datatype True False
-
-code_datatype "TYPE('a\<Colon>{})"
-
-code_datatype Trueprop "prop"
-
-text {* Code equations *}
-
-lemma [code]:
- shows "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
- and "(False \<Longrightarrow> Q) \<equiv> Trueprop True"
- and "(PROP P \<Longrightarrow> True) \<equiv> Trueprop True"
- and "(Q \<Longrightarrow> False) \<equiv> Trueprop (\<not> Q)" by (auto intro!: equal_intr_rule)
-
-lemma [code]:
- shows "False \<and> x \<longleftrightarrow> False"
- and "True \<and> x \<longleftrightarrow> x"
- and "x \<and> False \<longleftrightarrow> False"
- and "x \<and> True \<longleftrightarrow> x" by simp_all
-
-lemma [code]:
- shows "False \<or> x \<longleftrightarrow> x"
- and "True \<or> x \<longleftrightarrow> True"
- and "x \<or> False \<longleftrightarrow> x"
- and "x \<or> True \<longleftrightarrow> True" by simp_all
-
-lemma [code]:
- shows "\<not> True \<longleftrightarrow> False"
- and "\<not> False \<longleftrightarrow> True" by (rule HOL.simp_thms)+
-
-lemmas [code] = Let_def if_True if_False
-
-lemmas [code, code unfold, symmetric, code post] = imp_conv_disj
-
-text {* Equality *}
-
-context eq
-begin
-
-lemma equals_eq [code inline, code]: "op = \<equiv> eq"
- by (rule eq_reflection) (rule ext, rule ext, rule sym, rule eq_equals)
-
-declare eq [code unfold, code inline del]
-
-declare equals_eq [symmetric, code post]
-
-end
-
-declare simp_thms(6) [code nbe]
-
-hide (open) const eq
-hide const eq
-
-setup {*
- Code_Unit.add_const_alias @{thm equals_eq}
-*}
-
-text {* Cases *}
-
-lemma Let_case_cert:
- assumes "CASE \<equiv> (\<lambda>x. Let x f)"
- shows "CASE x \<equiv> f x"
- using assms by simp_all
-
-lemma If_case_cert:
- assumes "CASE \<equiv> (\<lambda>b. If b f g)"
- shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)"
- using assms by simp_all
-
-setup {*
- Code.add_case @{thm Let_case_cert}
- #> Code.add_case @{thm If_case_cert}
- #> Code.add_undefined @{const_name undefined}
-*}
-
-code_abort undefined
-
-
-subsection {* Generic code generator preprocessor *}
-
-setup {*
- Code.map_pre (K HOL_basic_ss)
- #> Code.map_post (K HOL_basic_ss)
-*}
-
-
-subsection {* Generic code generator target languages *}
-
-text {* type bool *}
-
-code_type bool
- (SML "bool")
- (OCaml "bool")
- (Haskell "Bool")
-
-code_const True and False and Not and "op &" and "op |" and If
- (SML "true" and "false" and "not"
- and infixl 1 "andalso" and infixl 0 "orelse"
- and "!(if (_)/ then (_)/ else (_))")
- (OCaml "true" and "false" and "not"
- and infixl 4 "&&" and infixl 2 "||"
- and "!(if (_)/ then (_)/ else (_))")
- (Haskell "True" and "False" and "not"
- and infixl 3 "&&" and infixl 2 "||"
- and "!(if (_)/ then (_)/ else (_))")
-
-code_reserved SML
- bool true false not
-
-code_reserved OCaml
- bool not
-
-text {* using built-in Haskell equality *}
-
-code_class eq
- (Haskell "Eq")
-
-code_const "eq_class.eq"
- (Haskell infixl 4 "==")
-
-code_const "op ="
- (Haskell infixl 4 "==")
-
-text {* undefined *}
-
-code_const undefined
- (SML "!(raise/ Fail/ \"undefined\")")
- (OCaml "failwith/ \"undefined\"")
- (Haskell "error/ \"undefined\"")
-
-
-subsection {* SML code generator setup *}
-
-types_code
- "bool" ("bool")
-attach (term_of) {*
-fun term_of_bool b = if b then HOLogic.true_const else HOLogic.false_const;
-*}
-attach (test) {*
-fun gen_bool i =
- let val b = one_of [false, true]
- in (b, fn () => term_of_bool b) end;
-*}
- "prop" ("bool")
-attach (term_of) {*
-fun term_of_prop b =
- HOLogic.mk_Trueprop (if b then HOLogic.true_const else HOLogic.false_const);
-*}
-
-consts_code
- "Trueprop" ("(_)")
- "True" ("true")
- "False" ("false")
- "Not" ("Bool.not")
- "op |" ("(_ orelse/ _)")
- "op &" ("(_ andalso/ _)")
- "If" ("(if _/ then _/ else _)")
-
-setup {*
-let
-
-fun eq_codegen thy defs dep thyname b t gr =
- (case strip_comb t of
- (Const ("op =", Type (_, [Type ("fun", _), _])), _) => NONE
- | (Const ("op =", _), [t, u]) =>
- let
- val (pt, gr') = Codegen.invoke_codegen thy defs dep thyname false t gr;
- val (pu, gr'') = Codegen.invoke_codegen thy defs dep thyname false u gr';
- val (_, gr''') = Codegen.invoke_tycodegen thy defs dep thyname false HOLogic.boolT gr'';
- in
- SOME (Codegen.parens
- (Pretty.block [pt, Codegen.str " =", Pretty.brk 1, pu]), gr''')
- end
- | (t as Const ("op =", _), ts) => SOME (Codegen.invoke_codegen
- thy defs dep thyname b (Codegen.eta_expand t ts 2) gr)
- | _ => NONE);
-
-in
- Codegen.add_codegen "eq_codegen" eq_codegen
-end
-*}
-
-
-subsection {* Evaluation and normalization by evaluation *}
-
-setup {*
- Value.add_evaluator ("SML", Codegen.eval_term o ProofContext.theory_of)
-*}
-
-ML {*
-structure Eval_Method =
-struct
-
-val eval_ref : (unit -> bool) option ref = ref NONE;
-
-end;
-*}
-
-oracle eval_oracle = {* fn ct =>
- let
- val thy = Thm.theory_of_cterm ct;
- val t = Thm.term_of ct;
- val dummy = @{cprop True};
- in case try HOLogic.dest_Trueprop t
- of SOME t' => if Code_ML.eval_term
- ("Eval_Method.eval_ref", Eval_Method.eval_ref) thy t' []
- then Thm.capply (Thm.capply @{cterm "op \<equiv> \<Colon> prop \<Rightarrow> prop \<Rightarrow> prop"} ct) dummy
- else dummy
- | NONE => dummy
- end
-*}
-
-ML {*
-fun gen_eval_method conv ctxt = SIMPLE_METHOD'
- (CONVERSION (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt)
- THEN' rtac TrueI)
-*}
-
-method_setup eval = {* Scan.succeed (gen_eval_method eval_oracle) *}
- "solve goal by evaluation"
-
-method_setup evaluation = {* Scan.succeed (gen_eval_method Codegen.evaluation_conv) *}
- "solve goal by evaluation"
-
-method_setup normalization = {*
- Scan.succeed (K (SIMPLE_METHOD' (CONVERSION Nbe.norm_conv THEN' (fn k => TRY (rtac TrueI k)))))
-*} "solve goal by normalization"
-
-
-subsection {* Quickcheck *}
-
-setup {*
- Quickcheck.add_generator ("SML", Codegen.test_term)
-*}
-
-quickcheck_params [size = 5, iterations = 50]
-
-end
--- a/src/HOL/Complex.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Complex.thy Mon Apr 27 07:26:17 2009 -0700
@@ -159,19 +159,7 @@
subsection {* Exponentiation *}
-instantiation complex :: recpower
-begin
-
-primrec power_complex where
- "z ^ 0 = (1\<Colon>complex)"
-| "z ^ Suc n = (z\<Colon>complex) * z ^ n"
-
-instance proof
-qed simp_all
-
-declare power_complex.simps [simp del]
-
-end
+instance complex :: recpower ..
subsection {* Numerals and Arithmetic *}
--- a/src/HOL/Decision_Procs/Approximation.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Decision_Procs/Approximation.thy Mon Apr 27 07:26:17 2009 -0700
@@ -23,8 +23,8 @@
qed
lemma horner_schema: fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat"
- assumes f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
- shows "horner F G n ((F^j') s) (f j') x = (\<Sum> j = 0..< n. -1^j * (1 / real (f (j' + j))) * x^j)"
+ assumes f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
+ shows "horner F G n ((F ^^ j') s) (f j') x = (\<Sum> j = 0..< n. -1 ^ j * (1 / real (f (j' + j))) * x ^ j)"
proof (induct n arbitrary: i k j')
case (Suc n)
@@ -33,13 +33,13 @@
qed auto
lemma horner_bounds':
- assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
+ assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
and lb_0: "\<And> i k x. lb 0 i k x = 0"
and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)"
and ub_0: "\<And> i k x. ub 0 i k x = 0"
and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)"
- shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> horner F G n ((F^j') s) (f j') (Ifloat x) \<and>
- horner F G n ((F^j') s) (f j') (Ifloat x) \<le> Ifloat (ub n ((F^j') s) (f j') x)"
+ shows "Ifloat (lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') (Ifloat x) \<and>
+ horner F G n ((F ^^ j') s) (f j') (Ifloat x) \<le> Ifloat (ub n ((F ^^ j') s) (f j') x)"
(is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'")
proof (induct n arbitrary: j')
case 0 thus ?case unfolding lb_0 ub_0 horner.simps by auto
@@ -49,15 +49,15 @@
proof (rule add_mono)
show "Ifloat (lapprox_rat prec 1 (int (f j'))) \<le> 1 / real (f j')" using lapprox_rat[of prec 1 "int (f j')"] by auto
from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct2] `0 \<le> Ifloat x`
- show "- Ifloat (x * ub n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) x) \<le> - (Ifloat x * horner F G n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) (Ifloat x))"
+ show "- Ifloat (x * ub n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) x) \<le> - (Ifloat x * horner F G n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) (Ifloat x))"
unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono)
qed
moreover have "?horner (Suc n) j' \<le> ?ub (Suc n) j'" unfolding ub_Suc ub_Suc horner.simps Ifloat_sub diff_def
proof (rule add_mono)
show "1 / real (f j') \<le> Ifloat (rapprox_rat prec 1 (int (f j')))" using rapprox_rat[of 1 "int (f j')" prec] by auto
from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct1] `0 \<le> Ifloat x`
- show "- (Ifloat x * horner F G n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) (Ifloat x)) \<le>
- - Ifloat (x * lb n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) x)"
+ show "- (Ifloat x * horner F G n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) (Ifloat x)) \<le>
+ - Ifloat (x * lb n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) x)"
unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono)
qed
ultimately show ?case by blast
@@ -73,13 +73,13 @@
*}
lemma horner_bounds: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
- assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
+ assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
and lb_0: "\<And> i k x. lb 0 i k x = 0"
and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)"
and ub_0: "\<And> i k x. ub 0 i k x = 0"
and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)"
- shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> (\<Sum>j=0..<n. -1^j * (1 / real (f (j' + j))) * (Ifloat x)^j)" (is "?lb") and
- "(\<Sum>j=0..<n. -1^j * (1 / real (f (j' + j))) * (Ifloat x)^j) \<le> Ifloat (ub n ((F^j') s) (f j') x)" (is "?ub")
+ shows "Ifloat (lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. -1 ^ j * (1 / real (f (j' + j))) * Ifloat x ^ j)" (is "?lb") and
+ "(\<Sum>j=0..<n. -1 ^ j * (1 / real (f (j' + j))) * (Ifloat x ^ j)) \<le> Ifloat (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
proof -
have "?lb \<and> ?ub"
using horner_bounds'[where lb=lb, OF `0 \<le> Ifloat x` f_Suc lb_0 lb_Suc ub_0 ub_Suc]
@@ -88,29 +88,29 @@
qed
lemma horner_bounds_nonpos: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
- assumes "Ifloat x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
+ assumes "Ifloat x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
and lb_0: "\<And> i k x. lb 0 i k x = 0"
and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) + x * (ub n (F i) (G i k) x)"
and ub_0: "\<And> i k x. ub 0 i k x = 0"
and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) + x * (lb n (F i) (G i k) x)"
- shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j)" (is "?lb") and
- "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j) \<le> Ifloat (ub n ((F^j') s) (f j') x)" (is "?ub")
+ shows "Ifloat (lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / real (f (j' + j))) * Ifloat x ^ j)" (is "?lb") and
+ "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x ^ j)) \<le> Ifloat (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
proof -
{ fix x y z :: float have "x - y * z = x + - y * z"
- by (cases x, cases y, cases z, simp add: plus_float.simps minus_float.simps uminus_float.simps times_float.simps algebra_simps)
+ by (cases x, cases y, cases z, simp add: plus_float.simps minus_float_def uminus_float.simps times_float.simps algebra_simps)
} note diff_mult_minus = this
{ fix x :: float have "- (- x) = x" by (cases x, auto simp add: uminus_float.simps) } note minus_minus = this
have move_minus: "Ifloat (-x) = -1 * Ifloat x" by auto
- have sum_eq: "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j) =
+ have sum_eq: "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * Ifloat x ^ j) =
(\<Sum>j = 0..<n. -1 ^ j * (1 / real (f (j' + j))) * Ifloat (- x) ^ j)"
proof (rule setsum_cong, simp)
fix j assume "j \<in> {0 ..< n}"
show "1 / real (f (j' + j)) * Ifloat x ^ j = -1 ^ j * (1 / real (f (j' + j))) * Ifloat (- x) ^ j"
unfolding move_minus power_mult_distrib real_mult_assoc[symmetric]
- unfolding real_mult_commute unfolding real_mult_assoc[of "-1^j", symmetric] power_mult_distrib[symmetric]
+ unfolding real_mult_commute unfolding real_mult_assoc[of "-1 ^ j", symmetric] power_mult_distrib[symmetric]
by auto
qed
@@ -160,21 +160,21 @@
else (0, (max (-l) u) ^ n))"
lemma float_power_bnds: assumes "(l1, u1) = float_power_bnds n l u" and "x \<in> {Ifloat l .. Ifloat u}"
- shows "x^n \<in> {Ifloat l1..Ifloat u1}"
+ shows "x ^ n \<in> {Ifloat l1..Ifloat u1}"
proof (cases "even n")
case True
show ?thesis
proof (cases "0 < l")
case True hence "odd n \<or> 0 < l" and "0 \<le> Ifloat l" unfolding less_float_def by auto
have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \<or> 0 < l`] by auto
- have "Ifloat l^n \<le> x^n" and "x^n \<le> Ifloat u^n " using `0 \<le> Ifloat l` and assms unfolding atLeastAtMost_iff using power_mono[of "Ifloat l" x] power_mono[of x "Ifloat u"] by auto
+ have "Ifloat l ^ n \<le> x ^ n" and "x ^ n \<le> Ifloat u ^ n " using `0 \<le> Ifloat l` and assms unfolding atLeastAtMost_iff using power_mono[of "Ifloat l" x] power_mono[of x "Ifloat u"] by auto
thus ?thesis using assms `0 < l` unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto
next
case False hence P: "\<not> (odd n \<or> 0 < l)" using `even n` by auto
show ?thesis
proof (cases "u < 0")
case True hence "0 \<le> - Ifloat u" and "- Ifloat u \<le> - x" and "0 \<le> - x" and "-x \<le> - Ifloat l" using assms unfolding less_float_def by auto
- hence "Ifloat u^n \<le> x^n" and "x^n \<le> Ifloat l^n" using power_mono[of "-x" "-Ifloat l" n] power_mono[of "-Ifloat u" "-x" n]
+ hence "Ifloat u ^ n \<le> x ^ n" and "x ^ n \<le> Ifloat l ^ n" using power_mono[of "-x" "-Ifloat l" n] power_mono[of "-Ifloat u" "-x" n]
unfolding power_minus_even[OF `even n`] by auto
moreover have u1: "u1 = l ^ n" and l1: "l1 = u ^ n" using assms unfolding float_power_bnds_def if_not_P[OF P] if_P[OF True] by auto
ultimately show ?thesis using float_power by auto
@@ -194,11 +194,11 @@
next
case False hence "odd n \<or> 0 < l" by auto
have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \<or> 0 < l`] by auto
- have "Ifloat l^n \<le> x^n" and "x^n \<le> Ifloat u^n " using assms unfolding atLeastAtMost_iff using power_mono_odd[OF False] by auto
+ have "Ifloat l ^ n \<le> x ^ n" and "x ^ n \<le> Ifloat u ^ n " using assms unfolding atLeastAtMost_iff using power_mono_odd[OF False] by auto
thus ?thesis unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto
qed
-lemma bnds_power: "\<forall> x l u. (l1, u1) = float_power_bnds n l u \<and> x \<in> {Ifloat l .. Ifloat u} \<longrightarrow> Ifloat l1 \<le> x^n \<and> x^n \<le> Ifloat u1"
+lemma bnds_power: "\<forall> x l u. (l1, u1) = float_power_bnds n l u \<and> x \<in> {Ifloat l .. Ifloat u} \<longrightarrow> Ifloat l1 \<le> x ^ n \<and> x ^ n \<le> Ifloat u1"
using float_power_bnds by auto
section "Square root"
@@ -794,8 +794,8 @@
let "?f n" = "fact (2 * n)"
{ fix n
- have F: "\<And>m. ((\<lambda>i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
- have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^ n) 1 * (((\<lambda>i. i + 2) ^ n) 1 + 1)"
+ have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
+ have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)"
unfolding F by auto } note f_eq = this
from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
@@ -811,7 +811,7 @@
have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0
using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto
- { fix x n have "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x^(2 * i))
+ { fix x n have "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x ^ (2 * i))
= (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum")
proof -
have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
@@ -905,8 +905,8 @@
let "?f n" = "fact (2 * n + 1)"
{ fix n
- have F: "\<And>m. ((\<lambda>i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
- have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^ n) 2 * (((\<lambda>i. i + 2) ^ n) 2 + 1)"
+ have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
+ have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)"
unfolding F by auto } note f_eq = this
from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
@@ -1382,8 +1382,8 @@
shows "exp (Ifloat x) \<in> { Ifloat (lb_exp_horner prec (get_even n) 1 1 x) .. Ifloat (ub_exp_horner prec (get_odd n) 1 1 x) }"
proof -
{ fix n
- have F: "\<And> m. ((\<lambda>i. i + 1) ^ n) m = n + m" by (induct n, auto)
- have "fact (Suc n) = fact n * ((\<lambda>i. i + 1) ^ n) 1" unfolding F by auto } note f_eq = this
+ have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m" by (induct n, auto)
+ have "fact (Suc n) = fact n * ((\<lambda>i. i + 1) ^^ n) 1" unfolding F by auto } note f_eq = this
note bounds = horner_bounds_nonpos[where f="fact" and lb="lb_exp_horner prec" and ub="ub_exp_horner prec" and j'=0 and s=1,
OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]
@@ -1462,7 +1462,8 @@
finally have "0 < Ifloat ((?horner x) ^ num)" .
}
ultimately show ?thesis
- unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def by (cases "floor_fl x", cases "x < - 1", auto simp add: le_float_def less_float_def normfloat)
+ unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def
+ by (cases "floor_fl x", cases "x < - 1", auto simp add: float_power le_float_def less_float_def)
qed
lemma exp_boundaries': assumes "x \<le> 0"
@@ -1631,10 +1632,10 @@
lemma ln_bounds:
assumes "0 \<le> x" and "x < 1"
- shows "(\<Sum>i=0..<2*n. -1^i * (1 / real (i + 1)) * x^(Suc i)) \<le> ln (x + 1)" (is "?lb")
- and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. -1^i * (1 / real (i + 1)) * x^(Suc i))" (is "?ub")
+ shows "(\<Sum>i=0..<2*n. -1^i * (1 / real (i + 1)) * x ^ (Suc i)) \<le> ln (x + 1)" (is "?lb")
+ and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. -1^i * (1 / real (i + 1)) * x ^ (Suc i))" (is "?ub")
proof -
- let "?a n" = "(1/real (n +1)) * x^(Suc n)"
+ let "?a n" = "(1/real (n +1)) * x ^ (Suc n)"
have ln_eq: "(\<Sum> i. -1^i * ?a i) = ln (x + 1)"
using ln_series[of "x + 1"] `0 \<le> x` `x < 1` by auto
@@ -2479,7 +2480,7 @@
fun lift_var (Free (varname, _)) = (case AList.lookup (op =) bound_eqs varname of
SOME bound => bound
| NONE => raise TERM ("No bound equations found for " ^ varname, []))
- | lift_var t = raise TERM ("Can not convert expression " ^
+ | lift_var t = raise TERM ("Can not convert expression " ^
(Syntax.string_of_term ctxt t), [t])
val _ $ vs = HOLogic.dest_Trueprop (Logic.strip_imp_concl goal')
--- a/src/HOL/Decision_Procs/cooper_tac.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Decision_Procs/cooper_tac.ML Mon Apr 27 07:26:17 2009 -0700
@@ -76,7 +76,7 @@
@{thm "div_by_1"}, @{thm "mod_by_1"}, @{thm "div_1"}, @{thm "mod_1"},
Suc_plus1]
addsimps @{thms add_ac}
- addsimprocs [cancel_div_mod_proc]
+ addsimprocs [cancel_div_mod_nat_proc, cancel_div_mod_int_proc]
val simpset0 = HOL_basic_ss
addsimps [mod_div_equality', Suc_plus1]
addsimps comp_arith
--- a/src/HOL/Decision_Procs/mir_tac.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Decision_Procs/mir_tac.ML Mon Apr 27 07:26:17 2009 -0700
@@ -99,7 +99,7 @@
@{thm "div_by_1"}, @{thm "mod_by_1"}, @{thm "div_1"}, @{thm "mod_1"},
@{thm "Suc_plus1"}]
addsimps @{thms add_ac}
- addsimprocs [cancel_div_mod_proc]
+ addsimprocs [cancel_div_mod_nat_proc, cancel_div_mod_int_proc]
val simpset0 = HOL_basic_ss
addsimps [mod_div_equality', Suc_plus1]
addsimps comp_ths
--- a/src/HOL/Divides.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Divides.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1,5 +1,4 @@
(* Title: HOL/Divides.thy
- ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
*)
@@ -20,11 +19,12 @@
subsection {* Abstract division in commutative semirings. *}
-class semiring_div = comm_semiring_1_cancel + div +
+class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
assumes mod_div_equality: "a div b * b + a mod b = a"
and div_by_0 [simp]: "a div 0 = 0"
and div_0 [simp]: "0 div a = 0"
and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
+ and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
begin
text {* @{const div} and @{const mod} *}
@@ -38,16 +38,16 @@
by (simp only: add_ac)
lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
-by (simp add: mod_div_equality)
+ by (simp add: mod_div_equality)
lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
-by (simp add: mod_div_equality2)
+ by (simp add: mod_div_equality2)
lemma mod_by_0 [simp]: "a mod 0 = a"
-using mod_div_equality [of a zero] by simp
+ using mod_div_equality [of a zero] by simp
lemma mod_0 [simp]: "0 mod a = 0"
-using mod_div_equality [of zero a] div_0 by simp
+ using mod_div_equality [of zero a] div_0 by simp
lemma div_mult_self2 [simp]:
assumes "b \<noteq> 0"
@@ -72,7 +72,7 @@
qed
lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
-by (simp add: mult_commute [of b])
+ by (simp add: mult_commute [of b])
lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
using div_mult_self2 [of b 0 a] by simp
@@ -238,9 +238,9 @@
by (simp only: mod_add_eq [symmetric])
qed
-lemma div_add[simp]: "z dvd x \<Longrightarrow> z dvd y
+lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y
\<Longrightarrow> (x + y) div z = x div z + y div z"
-by(cases "z=0", simp, unfold dvd_def, auto simp add: algebra_simps)
+by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)
text {* Multiplication respects modular equivalence. *}
@@ -297,21 +297,41 @@
finally show ?thesis .
qed
-end
-
-lemma div_mult_div_if_dvd: "(y::'a::{semiring_div,no_zero_divisors}) dvd x \<Longrightarrow>
- z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
-unfolding dvd_def
- apply clarify
- apply (case_tac "y = 0")
- apply simp
- apply (case_tac "z = 0")
- apply simp
- apply (simp add: algebra_simps)
+lemma div_mult_div_if_dvd:
+ "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
+ apply (cases "y = 0", simp)
+ apply (cases "z = 0", simp)
+ apply (auto elim!: dvdE simp add: algebra_simps)
apply (subst mult_assoc [symmetric])
apply (simp add: no_zero_divisors)
-done
+ done
+
+lemma div_mult_mult2 [simp]:
+ "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
+ by (drule div_mult_mult1) (simp add: mult_commute)
+
+lemma div_mult_mult1_if [simp]:
+ "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
+ by simp_all
+lemma mod_mult_mult1:
+ "(c * a) mod (c * b) = c * (a mod b)"
+proof (cases "c = 0")
+ case True then show ?thesis by simp
+next
+ case False
+ from mod_div_equality
+ have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
+ with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
+ = c * a + c * (a mod b)" by (simp add: algebra_simps)
+ with mod_div_equality show ?thesis by simp
+qed
+
+lemma mod_mult_mult2:
+ "(a * c) mod (b * c) = (a mod b) * c"
+ using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
+
+end
lemma div_power: "(y::'a::{semiring_div,no_zero_divisors,recpower}) dvd x \<Longrightarrow>
(x div y)^n = x^n div y^n"
@@ -398,15 +418,17 @@
@{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
*}
-definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
- "divmod_rel m n q r \<longleftrightarrow> m = q * n + r \<and> (if n > 0 then 0 \<le> r \<and> r < n else q = 0)"
+definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
+ "divmod_rel m n qr \<longleftrightarrow>
+ m = fst qr * n + snd qr \<and>
+ (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"
text {* @{const divmod_rel} is total: *}
lemma divmod_rel_ex:
- obtains q r where "divmod_rel m n q r"
+ obtains q r where "divmod_rel m n (q, r)"
proof (cases "n = 0")
- case True with that show thesis
+ case True with that show thesis
by (auto simp add: divmod_rel_def)
next
case False
@@ -436,13 +458,14 @@
text {* @{const divmod_rel} is injective: *}
-lemma divmod_rel_unique_div:
- assumes "divmod_rel m n q r"
- and "divmod_rel m n q' r'"
- shows "q = q'"
+lemma divmod_rel_unique:
+ assumes "divmod_rel m n qr"
+ and "divmod_rel m n qr'"
+ shows "qr = qr'"
proof (cases "n = 0")
case True with assms show ?thesis
- by (simp add: divmod_rel_def)
+ by (cases qr, cases qr')
+ (simp add: divmod_rel_def)
next
case False
have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
@@ -450,18 +473,11 @@
apply (subst less_iff_Suc_add)
apply (auto simp add: add_mult_distrib)
done
- from `n \<noteq> 0` assms show ?thesis
- by (auto simp add: divmod_rel_def
- intro: order_antisym dest: aux sym)
-qed
-
-lemma divmod_rel_unique_mod:
- assumes "divmod_rel m n q r"
- and "divmod_rel m n q' r'"
- shows "r = r'"
-proof -
- from assms have "q = q'" by (rule divmod_rel_unique_div)
- with assms show ?thesis by (simp add: divmod_rel_def)
+ from `n \<noteq> 0` assms have "fst qr = fst qr'"
+ by (auto simp add: divmod_rel_def intro: order_antisym dest: aux sym)
+ moreover from this assms have "snd qr = snd qr'"
+ by (simp add: divmod_rel_def)
+ ultimately show ?thesis by (cases qr, cases qr') simp
qed
text {*
@@ -473,7 +489,21 @@
begin
definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
- [code del]: "divmod m n = (THE (q, r). divmod_rel m n q r)"
+ [code del]: "divmod m n = (THE qr. divmod_rel m n qr)"
+
+lemma divmod_rel_divmod:
+ "divmod_rel m n (divmod m n)"
+proof -
+ from divmod_rel_ex
+ obtain qr where rel: "divmod_rel m n qr" .
+ then show ?thesis
+ by (auto simp add: divmod_def intro: theI elim: divmod_rel_unique)
+qed
+
+lemma divmod_eq:
+ assumes "divmod_rel m n qr"
+ shows "divmod m n = qr"
+ using assms by (auto intro: divmod_rel_unique divmod_rel_divmod)
definition div_nat where
"m div n = fst (divmod m n)"
@@ -485,30 +515,18 @@
"divmod m n = (m div n, m mod n)"
unfolding div_nat_def mod_nat_def by simp
-lemma divmod_eq:
- assumes "divmod_rel m n q r"
- shows "divmod m n = (q, r)"
- using assms by (auto simp add: divmod_def
- dest: divmod_rel_unique_div divmod_rel_unique_mod)
-
lemma div_eq:
- assumes "divmod_rel m n q r"
+ assumes "divmod_rel m n (q, r)"
shows "m div n = q"
- using assms by (auto dest: divmod_eq simp add: div_nat_def)
+ using assms by (auto dest: divmod_eq simp add: divmod_div_mod)
lemma mod_eq:
- assumes "divmod_rel m n q r"
+ assumes "divmod_rel m n (q, r)"
shows "m mod n = r"
- using assms by (auto dest: divmod_eq simp add: mod_nat_def)
+ using assms by (auto dest: divmod_eq simp add: divmod_div_mod)
-lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)"
-proof -
- from divmod_rel_ex
- obtain q r where rel: "divmod_rel m n q r" .
- moreover with div_eq mod_eq have "m div n = q" and "m mod n = r"
- by simp_all
- ultimately show ?thesis by simp
-qed
+lemma divmod_rel: "divmod_rel m n (m div n, m mod n)"
+ by (simp add: div_nat_def mod_nat_def divmod_rel_divmod)
lemma divmod_zero:
"divmod m 0 = (0, m)"
@@ -531,10 +549,10 @@
assumes "0 < n" and "n \<le> m"
shows "divmod m n = (Suc ((m - n) div n), (m - n) mod n)"
proof -
- from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" .
+ from divmod_rel have divmod_m_n: "divmod_rel m n (m div n, m mod n)" .
with assms have m_div_n: "m div n \<ge> 1"
by (cases "m div n") (auto simp add: divmod_rel_def)
- from assms divmod_m_n have "divmod_rel (m - n) n (m div n - Suc 0) (m mod n)"
+ from assms divmod_m_n have "divmod_rel (m - n) n (m div n - Suc 0, m mod n)"
by (cases "m div n") (auto simp add: divmod_rel_def)
with divmod_eq have "divmod (m - n) n = (m div n - Suc 0, m mod n)" by simp
moreover from divmod_div_mod have "divmod (m - n) n = ((m - n) div n, (m - n) mod n)" .
@@ -569,55 +587,74 @@
shows "m mod n = (m - n) mod n"
using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all
-instance proof
- fix m n :: nat show "m div n * n + m mod n = m"
- using divmod_rel [of m n] by (simp add: divmod_rel_def)
-next
- fix n :: nat show "n div 0 = 0"
- using divmod_zero divmod_div_mod [of n 0] by simp
-next
- fix n :: nat show "0 div n = 0"
- using divmod_rel [of 0 n] by (cases n) (simp_all add: divmod_rel_def)
-next
- fix m n q :: nat assume "n \<noteq> 0" then show "(q + m * n) div n = m + q div n"
- by (induct m) (simp_all add: le_div_geq)
+instance proof -
+ have [simp]: "\<And>n::nat. n div 0 = 0"
+ by (simp add: div_nat_def divmod_zero)
+ have [simp]: "\<And>n::nat. 0 div n = 0"
+ proof -
+ fix n :: nat
+ show "0 div n = 0"
+ by (cases "n = 0") simp_all
+ qed
+ show "OFCLASS(nat, semiring_div_class)" proof
+ fix m n :: nat
+ show "m div n * n + m mod n = m"
+ using divmod_rel [of m n] by (simp add: divmod_rel_def)
+ next
+ fix m n q :: nat
+ assume "n \<noteq> 0"
+ then show "(q + m * n) div n = m + q div n"
+ by (induct m) (simp_all add: le_div_geq)
+ next
+ fix m n q :: nat
+ assume "m \<noteq> 0"
+ then show "(m * n) div (m * q) = n div q"
+ proof (cases "n \<noteq> 0 \<and> q \<noteq> 0")
+ case False then show ?thesis by auto
+ next
+ case True with `m \<noteq> 0`
+ have "m > 0" and "n > 0" and "q > 0" by auto
+ then have "\<And>a b. divmod_rel n q (a, b) \<Longrightarrow> divmod_rel (m * n) (m * q) (a, m * b)"
+ by (auto simp add: divmod_rel_def) (simp_all add: algebra_simps)
+ moreover from divmod_rel have "divmod_rel n q (n div q, n mod q)" .
+ ultimately have "divmod_rel (m * n) (m * q) (n div q, m * (n mod q))" .
+ then show ?thesis by (simp add: div_eq)
+ qed
+ qed simp_all
qed
end
text {* Simproc for cancelling @{const div} and @{const mod} *}
-(*lemmas mod_div_equality_nat = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard]
-lemmas mod_div_equality2_nat = mod_div_equality2 [of "n\<Colon>nat" m, standard*)
+ML {*
+local
+
+structure CancelDivMod = CancelDivModFun(struct
-ML {*
-structure CancelDivModData =
-struct
-
-val div_name = @{const_name div};
-val mod_name = @{const_name mod};
-val mk_binop = HOLogic.mk_binop;
-val mk_sum = Nat_Arith.mk_sum;
-val dest_sum = Nat_Arith.dest_sum;
+ val div_name = @{const_name div};
+ val mod_name = @{const_name mod};
+ val mk_binop = HOLogic.mk_binop;
+ val mk_sum = Nat_Arith.mk_sum;
+ val dest_sum = Nat_Arith.dest_sum;
-(*logic*)
+ val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
-val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
-
-val trans = trans
+ val trans = trans;
-val prove_eq_sums =
- let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
- in Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac simps) end;
+ val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
+ (@{thm monoid_add_class.add_0_left} :: @{thm monoid_add_class.add_0_right} :: @{thms add_ac}))
-end;
+end)
-structure CancelDivMod = CancelDivModFun(CancelDivModData);
+in
-val cancel_div_mod_proc = Simplifier.simproc (the_context ())
+val cancel_div_mod_nat_proc = Simplifier.simproc (the_context ())
"cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
-Addsimprocs[cancel_div_mod_proc];
+val _ = Addsimprocs [cancel_div_mod_nat_proc];
+
+end
*}
text {* code generator setup *}
@@ -658,7 +695,7 @@
fixes m n :: nat
assumes "n > 0"
shows "m mod n < (n::nat)"
- using assms divmod_rel unfolding divmod_rel_def by auto
+ using assms divmod_rel [of m n] unfolding divmod_rel_def by auto
lemma mod_less_eq_dividend [simp]:
fixes m n :: nat
@@ -700,18 +737,19 @@
subsubsection {* Quotient and Remainder *}
lemma divmod_rel_mult1_eq:
- "[| divmod_rel b c q r; c > 0 |]
- ==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)"
+ "divmod_rel b c (q, r) \<Longrightarrow> c > 0
+ \<Longrightarrow> divmod_rel (a * b) c (a * q + a * r div c, a * r mod c)"
by (auto simp add: split_ifs divmod_rel_def algebra_simps)
-lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
+lemma div_mult1_eq:
+ "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
apply (cases "c = 0", simp)
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq])
done
lemma divmod_rel_add1_eq:
- "[| divmod_rel a c aq ar; divmod_rel b c bq br; c > 0 |]
- ==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)"
+ "divmod_rel a c (aq, ar) \<Longrightarrow> divmod_rel b c (bq, br) \<Longrightarrow> c > 0
+ \<Longrightarrow> divmod_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
by (auto simp add: split_ifs divmod_rel_def algebra_simps)
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
@@ -728,8 +766,9 @@
apply (simp add: add_mult_distrib2)
done
-lemma divmod_rel_mult2_eq: "[| divmod_rel a b q r; 0 < b; 0 < c |]
- ==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)"
+lemma divmod_rel_mult2_eq:
+ "divmod_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c
+ \<Longrightarrow> divmod_rel a (b * c) (q div c, b *(q mod c) + r)"
by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
@@ -745,23 +784,6 @@
done
-subsubsection{*Cancellation of Common Factors in Division*}
-
-lemma div_mult_mult_lemma:
- "[| (0::nat) < b; 0 < c |] ==> (c*a) div (c*b) = a div b"
-by (auto simp add: div_mult2_eq)
-
-lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
- apply (cases "b = 0")
- apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
- done
-
-lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
- apply (drule div_mult_mult1)
- apply (auto simp add: mult_commute)
- done
-
-
subsubsection{*Further Facts about Quotient and Remainder*}
lemma div_1 [simp]: "m div Suc 0 = m"
@@ -769,7 +791,7 @@
(* Monotonicity of div in first argument *)
-lemma div_le_mono [rule_format]:
+lemma div_le_mono [rule_format (no_asm)]:
"\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
apply (case_tac "k=0", simp)
apply (induct "n" rule: nat_less_induct, clarify)
@@ -824,12 +846,6 @@
apply (simp_all)
done
-lemma nat_div_eq_0 [simp]: "(n::nat) > 0 ==> ((m div n) = 0) = (m < n)"
-by(auto, subst mod_div_equality [of m n, symmetric], auto)
-
-lemma nat_div_gt_0 [simp]: "(n::nat) > 0 ==> ((m div n) > 0) = (m >= n)"
-by (subst neq0_conv [symmetric], auto)
-
declare div_less_dividend [simp]
text{*A fact for the mutilated chess board*}
@@ -915,16 +931,10 @@
done
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
-by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
-
-lemma nat_dvd_not_less: "(0::nat) < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
-by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
+ by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
- apply (subgoal_tac "m mod n = 0")
- apply (simp add: mult_div_cancel)
- apply (simp only: dvd_eq_mod_eq_0)
- done
+ by (simp add: dvd_eq_mod_eq_0 mult_div_cancel)
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
by (induct n) auto
@@ -1001,9 +1011,11 @@
from A B show ?lhs ..
next
assume P: ?lhs
- then have "divmod_rel m n q (m - n * q)"
+ then have "divmod_rel m n (q, m - n * q)"
unfolding divmod_rel_def by (auto simp add: mult_ac)
- then show ?rhs using divmod_rel by (rule divmod_rel_unique_div)
+ with divmod_rel_unique divmod_rel [of m n]
+ have "(q, m - n * q) = (m div n, m mod n)" by auto
+ then show ?rhs by simp
qed
theorem split_div':
@@ -1155,4 +1167,9 @@
with j show ?thesis by blast
qed
+lemma nat_dvd_not_less:
+ fixes m n :: nat
+ shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
+by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
+
end
--- a/src/HOL/Groebner_Basis.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Groebner_Basis.thy Mon Apr 27 07:26:17 2009 -0700
@@ -5,7 +5,7 @@
header {* Semiring normalization and Groebner Bases *}
theory Groebner_Basis
-imports NatBin
+imports Nat_Numeral
uses
"Tools/Groebner_Basis/misc.ML"
"Tools/Groebner_Basis/normalizer_data.ML"
--- a/src/HOL/HOL.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/HOL.thy Mon Apr 27 07:26:17 2009 -0700
@@ -5,9 +5,10 @@
header {* The basis of Higher-Order Logic *}
theory HOL
-imports Pure
+imports Pure "~~/src/Tools/Code_Generator"
uses
("Tools/hologic.ML")
+ "~~/src/Tools/auto_solve.ML"
"~~/src/Tools/IsaPlanner/zipper.ML"
"~~/src/Tools/IsaPlanner/isand.ML"
"~~/src/Tools/IsaPlanner/rw_tools.ML"
@@ -27,16 +28,6 @@
"~~/src/Tools/atomize_elim.ML"
"~~/src/Tools/induct.ML"
("~~/src/Tools/induct_tacs.ML")
- "~~/src/Tools/value.ML"
- "~~/src/Tools/code/code_name.ML"
- "~~/src/Tools/code/code_funcgr.ML" (*formal dependency*)
- "~~/src/Tools/code/code_wellsorted.ML"
- "~~/src/Tools/code/code_thingol.ML"
- "~~/src/Tools/code/code_printer.ML"
- "~~/src/Tools/code/code_target.ML"
- "~~/src/Tools/code/code_ml.ML"
- "~~/src/Tools/code/code_haskell.ML"
- "~~/src/Tools/nbe.ML"
("Tools/recfun_codegen.ML")
begin
@@ -1674,37 +1665,264 @@
*}
-subsection {* Code generator basics -- see further theory @{text "Code_Setup"} *}
+subsection {* Code generator setup *}
+
+subsubsection {* SML code generator setup *}
+
+use "Tools/recfun_codegen.ML"
+
+setup {*
+ Codegen.setup
+ #> RecfunCodegen.setup
+*}
+
+types_code
+ "bool" ("bool")
+attach (term_of) {*
+fun term_of_bool b = if b then HOLogic.true_const else HOLogic.false_const;
+*}
+attach (test) {*
+fun gen_bool i =
+ let val b = one_of [false, true]
+ in (b, fn () => term_of_bool b) end;
+*}
+ "prop" ("bool")
+attach (term_of) {*
+fun term_of_prop b =
+ HOLogic.mk_Trueprop (if b then HOLogic.true_const else HOLogic.false_const);
+*}
-text {* Equality *}
+consts_code
+ "Trueprop" ("(_)")
+ "True" ("true")
+ "False" ("false")
+ "Not" ("Bool.not")
+ "op |" ("(_ orelse/ _)")
+ "op &" ("(_ andalso/ _)")
+ "If" ("(if _/ then _/ else _)")
+
+setup {*
+let
+
+fun eq_codegen thy defs dep thyname b t gr =
+ (case strip_comb t of
+ (Const ("op =", Type (_, [Type ("fun", _), _])), _) => NONE
+ | (Const ("op =", _), [t, u]) =>
+ let
+ val (pt, gr') = Codegen.invoke_codegen thy defs dep thyname false t gr;
+ val (pu, gr'') = Codegen.invoke_codegen thy defs dep thyname false u gr';
+ val (_, gr''') = Codegen.invoke_tycodegen thy defs dep thyname false HOLogic.boolT gr'';
+ in
+ SOME (Codegen.parens
+ (Pretty.block [pt, Codegen.str " =", Pretty.brk 1, pu]), gr''')
+ end
+ | (t as Const ("op =", _), ts) => SOME (Codegen.invoke_codegen
+ thy defs dep thyname b (Codegen.eta_expand t ts 2) gr)
+ | _ => NONE);
+
+in
+ Codegen.add_codegen "eq_codegen" eq_codegen
+end
+*}
+
+subsubsection {* Equality *}
class eq =
fixes eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
assumes eq_equals: "eq x y \<longleftrightarrow> x = y"
begin
-lemma eq: "eq = (op =)"
+lemma eq [code unfold, code inline del]: "eq = (op =)"
by (rule ext eq_equals)+
lemma eq_refl: "eq x x \<longleftrightarrow> True"
unfolding eq by rule+
+lemma equals_eq [code inline]: "(op =) \<equiv> eq"
+ by (rule eq_reflection) (rule ext, rule ext, rule sym, rule eq_equals)
+
+declare equals_eq [symmetric, code post]
+
end
-text {* Module setup *}
+declare equals_eq [code]
+
+
+subsubsection {* Generic code generator foundation *}
+
+text {* Datatypes *}
+
+code_datatype True False
+
+code_datatype "TYPE('a\<Colon>{})"
+
+code_datatype Trueprop "prop"
+
+text {* Code equations *}
+
+lemma [code]:
+ shows "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
+ and "(False \<Longrightarrow> Q) \<equiv> Trueprop True"
+ and "(PROP P \<Longrightarrow> True) \<equiv> Trueprop True"
+ and "(Q \<Longrightarrow> False) \<equiv> Trueprop (\<not> Q)" by (auto intro!: equal_intr_rule)
+
+lemma [code]:
+ shows "False \<and> x \<longleftrightarrow> False"
+ and "True \<and> x \<longleftrightarrow> x"
+ and "x \<and> False \<longleftrightarrow> False"
+ and "x \<and> True \<longleftrightarrow> x" by simp_all
+
+lemma [code]:
+ shows "False \<or> x \<longleftrightarrow> x"
+ and "True \<or> x \<longleftrightarrow> True"
+ and "x \<or> False \<longleftrightarrow> x"
+ and "x \<or> True \<longleftrightarrow> True" by simp_all
+
+lemma [code]:
+ shows "\<not> True \<longleftrightarrow> False"
+ and "\<not> False \<longleftrightarrow> True" by (rule HOL.simp_thms)+
-use "Tools/recfun_codegen.ML"
+lemmas [code] = Let_def if_True if_False
+
+lemmas [code, code unfold, symmetric, code post] = imp_conv_disj
+
+text {* Equality *}
+
+declare simp_thms(6) [code nbe]
+
+hide (open) const eq
+hide const eq
+
+setup {*
+ Code_Unit.add_const_alias @{thm equals_eq}
+*}
+
+text {* Cases *}
+
+lemma Let_case_cert:
+ assumes "CASE \<equiv> (\<lambda>x. Let x f)"
+ shows "CASE x \<equiv> f x"
+ using assms by simp_all
+
+lemma If_case_cert:
+ assumes "CASE \<equiv> (\<lambda>b. If b f g)"
+ shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)"
+ using assms by simp_all
+
+setup {*
+ Code.add_case @{thm Let_case_cert}
+ #> Code.add_case @{thm If_case_cert}
+ #> Code.add_undefined @{const_name undefined}
+*}
+
+code_abort undefined
+
+subsubsection {* Generic code generator preprocessor *}
setup {*
- Code_ML.setup
- #> Code_Haskell.setup
- #> Nbe.setup
- #> Codegen.setup
- #> RecfunCodegen.setup
+ Code.map_pre (K HOL_basic_ss)
+ #> Code.map_post (K HOL_basic_ss)
*}
+subsubsection {* Generic code generator target languages *}
-subsection {* Nitpick hooks *}
+text {* type bool *}
+
+code_type bool
+ (SML "bool")
+ (OCaml "bool")
+ (Haskell "Bool")
+
+code_const True and False and Not and "op &" and "op |" and If
+ (SML "true" and "false" and "not"
+ and infixl 1 "andalso" and infixl 0 "orelse"
+ and "!(if (_)/ then (_)/ else (_))")
+ (OCaml "true" and "false" and "not"
+ and infixl 4 "&&" and infixl 2 "||"
+ and "!(if (_)/ then (_)/ else (_))")
+ (Haskell "True" and "False" and "not"
+ and infixl 3 "&&" and infixl 2 "||"
+ and "!(if (_)/ then (_)/ else (_))")
+
+code_reserved SML
+ bool true false not
+
+code_reserved OCaml
+ bool not
+
+text {* using built-in Haskell equality *}
+
+code_class eq
+ (Haskell "Eq")
+
+code_const "eq_class.eq"
+ (Haskell infixl 4 "==")
+
+code_const "op ="
+ (Haskell infixl 4 "==")
+
+text {* undefined *}
+
+code_const undefined
+ (SML "!(raise/ Fail/ \"undefined\")")
+ (OCaml "failwith/ \"undefined\"")
+ (Haskell "error/ \"undefined\"")
+
+subsubsection {* Evaluation and normalization by evaluation *}
+
+setup {*
+ Value.add_evaluator ("SML", Codegen.eval_term o ProofContext.theory_of)
+*}
+
+ML {*
+structure Eval_Method =
+struct
+
+val eval_ref : (unit -> bool) option ref = ref NONE;
+
+end;
+*}
+
+oracle eval_oracle = {* fn ct =>
+ let
+ val thy = Thm.theory_of_cterm ct;
+ val t = Thm.term_of ct;
+ val dummy = @{cprop True};
+ in case try HOLogic.dest_Trueprop t
+ of SOME t' => if Code_ML.eval NONE
+ ("Eval_Method.eval_ref", Eval_Method.eval_ref) (K I) thy t' []
+ then Thm.capply (Thm.capply @{cterm "op \<equiv> \<Colon> prop \<Rightarrow> prop \<Rightarrow> prop"} ct) dummy
+ else dummy
+ | NONE => dummy
+ end
+*}
+
+ML {*
+fun gen_eval_method conv ctxt = SIMPLE_METHOD'
+ (CONVERSION (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt)
+ THEN' rtac TrueI)
+*}
+
+method_setup eval = {* Scan.succeed (gen_eval_method eval_oracle) *}
+ "solve goal by evaluation"
+
+method_setup evaluation = {* Scan.succeed (gen_eval_method Codegen.evaluation_conv) *}
+ "solve goal by evaluation"
+
+method_setup normalization = {*
+ Scan.succeed (K (SIMPLE_METHOD' (CONVERSION Nbe.norm_conv THEN' (fn k => TRY (rtac TrueI k)))))
+*} "solve goal by normalization"
+
+subsubsection {* Quickcheck *}
+
+setup {*
+ Quickcheck.add_generator ("SML", Codegen.test_term)
+*}
+
+quickcheck_params [size = 5, iterations = 50]
+
+
+subsection {* Nitpick setup *}
text {* This will be relocated once Nitpick is moved to HOL. *}
@@ -1730,10 +1948,14 @@
val description = "introduction rules for (co)inductive predicates as needed by Nitpick"
)
*}
-setup {* Nitpick_Const_Def_Thms.setup
- #> Nitpick_Const_Simp_Thms.setup
- #> Nitpick_Const_Psimp_Thms.setup
- #> Nitpick_Ind_Intro_Thms.setup *}
+
+setup {*
+ Nitpick_Const_Def_Thms.setup
+ #> Nitpick_Const_Simp_Thms.setup
+ #> Nitpick_Const_Psimp_Thms.setup
+ #> Nitpick_Ind_Intro_Thms.setup
+*}
+
subsection {* Legacy tactics and ML bindings *}
--- a/src/HOL/HoareParallel/OG_Tran.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/HoareParallel/OG_Tran.thy Mon Apr 27 07:26:17 2009 -0700
@@ -74,7 +74,7 @@
abbreviation
ann_transition_n :: "('a ann_com_op \<times> 'a) \<Rightarrow> nat \<Rightarrow> ('a ann_com_op \<times> 'a)
\<Rightarrow> bool" ("_ -_\<rightarrow> _"[81,81] 100) where
- "con_0 -n\<rightarrow> con_1 \<equiv> (con_0, con_1) \<in> ann_transition^n"
+ "con_0 -n\<rightarrow> con_1 \<equiv> (con_0, con_1) \<in> ann_transition ^^ n"
abbreviation
ann_transitions :: "('a ann_com_op \<times> 'a) \<Rightarrow> ('a ann_com_op \<times> 'a) \<Rightarrow> bool"
@@ -84,7 +84,7 @@
abbreviation
transition_n :: "('a com \<times> 'a) \<Rightarrow> nat \<Rightarrow> ('a com \<times> 'a) \<Rightarrow> bool"
("_ -P_\<rightarrow> _"[81,81,81] 100) where
- "con_0 -Pn\<rightarrow> con_1 \<equiv> (con_0, con_1) \<in> transition^n"
+ "con_0 -Pn\<rightarrow> con_1 \<equiv> (con_0, con_1) \<in> transition ^^ n"
subsection {* Definition of Semantics *}
--- a/src/HOL/IMP/Compiler0.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/IMP/Compiler0.thy Mon Apr 27 07:26:17 2009 -0700
@@ -45,7 +45,7 @@
abbreviation
stepan :: "[instr list,state,nat,nat,state,nat] \<Rightarrow> bool"
("_ \<turnstile>/ (3\<langle>_,_\<rangle>/ -(_)\<rightarrow> \<langle>_,_\<rangle>)" [50,0,0,0,0,0] 50) where
- "P \<turnstile> \<langle>s,m\<rangle> -(i)\<rightarrow> \<langle>t,n\<rangle> == ((s,m),t,n) : ((stepa1 P)^i)"
+ "P \<turnstile> \<langle>s,m\<rangle> -(i)\<rightarrow> \<langle>t,n\<rangle> == ((s,m),t,n) : (stepa1 P ^^ i)"
subsection "The compiler"
--- a/src/HOL/IMP/Machines.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/IMP/Machines.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1,7 +1,6 @@
-
-(* $Id$ *)
-
-theory Machines imports Natural begin
+theory Machines
+imports Natural
+begin
lemma rtrancl_eq: "R^* = Id \<union> (R O R^*)"
by (fast intro: rtrancl_into_rtrancl elim: rtranclE)
@@ -11,20 +10,22 @@
lemmas converse_rel_powE = rel_pow_E2
-lemma R_O_Rn_commute: "R O R^n = R^n O R"
+lemma R_O_Rn_commute: "R O R ^^ n = R ^^ n O R"
by (induct n) (simp, simp add: O_assoc [symmetric])
lemma converse_in_rel_pow_eq:
- "((x,z) \<in> R^n) = (n=0 \<and> z=x \<or> (\<exists>m y. n = Suc m \<and> (x,y) \<in> R \<and> (y,z) \<in> R^m))"
+ "((x,z) \<in> R ^^ n) = (n=0 \<and> z=x \<or> (\<exists>m y. n = Suc m \<and> (x,y) \<in> R \<and> (y,z) \<in> R ^^ m))"
apply(rule iffI)
apply(blast elim:converse_rel_powE)
apply (fastsimp simp add:gr0_conv_Suc R_O_Rn_commute)
done
-lemma rel_pow_plus: "R^(m+n) = R^n O R^m"
+lemma rel_pow_plus:
+ "R ^^ (m+n) = R ^^ n O R ^^ m"
by (induct n) (simp, simp add: O_assoc)
-lemma rel_pow_plusI: "\<lbrakk> (x,y) \<in> R^m; (y,z) \<in> R^n \<rbrakk> \<Longrightarrow> (x,z) \<in> R^(m+n)"
+lemma rel_pow_plusI:
+ "\<lbrakk> (x,y) \<in> R ^^ m; (y,z) \<in> R ^^ n \<rbrakk> \<Longrightarrow> (x,z) \<in> R ^^ (m+n)"
by (simp add: rel_pow_plus rel_compI)
subsection "Instructions"
@@ -57,7 +58,7 @@
abbreviation
exec0n :: "[instrs, nat,state, nat, nat,state] \<Rightarrow> bool"
("(_/ \<turnstile> (1\<langle>_,/_\<rangle>)/ -_\<rightarrow> (1\<langle>_,/_\<rangle>))" [50,0,0,0,0] 50) where
- "p \<turnstile> \<langle>i,s\<rangle> -n\<rightarrow> \<langle>j,t\<rangle> == ((i,s),j,t) : (exec01 p)^n"
+ "p \<turnstile> \<langle>i,s\<rangle> -n\<rightarrow> \<langle>j,t\<rangle> == ((i,s),j,t) : (exec01 p)^^n"
subsection "M0 with lists"
@@ -89,7 +90,7 @@
abbreviation
stepan :: "[instrs,instrs,state, nat, instrs,instrs,state] \<Rightarrow> bool"
("((1\<langle>_,/_,/_\<rangle>)/ -_\<rightarrow> (1\<langle>_,/_,/_\<rangle>))" 50) where
- "\<langle>p,q,s\<rangle> -i\<rightarrow> \<langle>p',q',t\<rangle> == ((p,q,s),p',q',t) : (stepa1^i)"
+ "\<langle>p,q,s\<rangle> -i\<rightarrow> \<langle>p',q',t\<rangle> == ((p,q,s),p',q',t) : (stepa1^^i)"
inductive_cases execE: "((i#is,p,s), (is',p',s')) : stepa1"
--- a/src/HOL/IMP/Transition.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/IMP/Transition.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1,5 +1,4 @@
(* Title: HOL/IMP/Transition.thy
- ID: $Id$
Author: Tobias Nipkow & Robert Sandner, TUM
Isar Version: Gerwin Klein, 2001
Copyright 1996 TUM
@@ -69,7 +68,7 @@
abbreviation
evalcn :: "[(com option\<times>state),nat,(com option\<times>state)] \<Rightarrow> bool"
("_ -_\<rightarrow>\<^sub>1 _" [60,60,60] 60) where
- "cs -n\<rightarrow>\<^sub>1 cs' == (cs,cs') \<in> evalc1^n"
+ "cs -n\<rightarrow>\<^sub>1 cs' == (cs,cs') \<in> evalc1^^n"
abbreviation
evalc' :: "[(com option\<times>state),(com option\<times>state)] \<Rightarrow> bool"
@@ -77,28 +76,9 @@
"cs \<longrightarrow>\<^sub>1\<^sup>* cs' == (cs,cs') \<in> evalc1^*"
(*<*)
-(* fixme: move to Relation_Power.thy *)
-lemma rel_pow_Suc_E2 [elim!]:
- "[| (x, z) \<in> R ^ Suc n; !!y. [| (x, y) \<in> R; (y, z) \<in> R ^ n |] ==> P |] ==> P"
- by (blast dest: rel_pow_Suc_D2)
+declare rel_pow_Suc_E2 [elim!]
+(*>*)
-lemma rtrancl_imp_rel_pow: "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R^n"
-proof (induct p)
- fix x y
- assume "(x, y) \<in> R\<^sup>*"
- thus "\<exists>n. (x, y) \<in> R^n"
- proof induct
- fix a have "(a, a) \<in> R^0" by simp
- thus "\<exists>n. (a, a) \<in> R ^ n" ..
- next
- fix a b c assume "\<exists>n. (a, b) \<in> R ^ n"
- then obtain n where "(a, b) \<in> R^n" ..
- moreover assume "(b, c) \<in> R"
- ultimately have "(a, c) \<in> R^(Suc n)" by auto
- thus "\<exists>n. (a, c) \<in> R^n" ..
- qed
-qed
-(*>*)
text {*
As for the big step semantics you can read these rules in a
syntax directed way:
@@ -189,8 +169,8 @@
(*<*)
(* FIXME: relpow.simps don't work *)
lemmas [simp del] = relpow.simps
-lemma rel_pow_0 [simp]: "!!R::('a*'a) set. R^0 = Id" by (simp add: relpow.simps)
-lemma rel_pow_Suc_0 [simp]: "!!R::('a*'a) set. R^(Suc 0) = R" by (simp add: relpow.simps)
+lemma rel_pow_0 [simp]: "!!R::('a*'a) set. R ^^ 0 = Id" by (simp add: relpow.simps)
+lemma rel_pow_Suc_0 [simp]: "!!R::('a*'a) set. R ^^ Suc 0 = R" by (simp add: relpow.simps)
(*>*)
lemma evalc1_None_0 [simp]: "\<langle>s\<rangle> -n\<rightarrow>\<^sub>1 y = (n = 0 \<and> y = \<langle>s\<rangle>)"
--- a/src/HOL/Import/HOL/HOL4Base.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Import/HOL/HOL4Base.thy Mon Apr 27 07:26:17 2009 -0700
@@ -2794,8 +2794,8 @@
by (import numeral numeral_fact)
lemma numeral_funpow: "ALL n::nat.
- ((f::'a::type => 'a::type) ^ n) (x::'a::type) =
- (if n = 0 then x else (f ^ (n - 1)) (f x))"
+ ((f::'a::type => 'a::type) ^^ n) (x::'a::type) =
+ (if n = 0 then x else (f ^^ (n - 1)) (f x))"
by (import numeral numeral_funpow)
;end_setup
--- a/src/HOL/Import/HOL/HOL4Word32.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Import/HOL/HOL4Word32.thy Mon Apr 27 07:26:17 2009 -0700
@@ -434,15 +434,15 @@
by (import word32 EQUIV_QT)
lemma FUNPOW_THM: "ALL (f::'a::type => 'a::type) (n::nat) x::'a::type.
- (f ^ n) (f x) = f ((f ^ n) x)"
+ (f ^^ n) (f x) = f ((f ^^ n) x)"
by (import word32 FUNPOW_THM)
lemma FUNPOW_THM2: "ALL (f::'a::type => 'a::type) (n::nat) x::'a::type.
- (f ^ Suc n) x = f ((f ^ n) x)"
+ (f ^^ Suc n) x = f ((f ^^ n) x)"
by (import word32 FUNPOW_THM2)
lemma FUNPOW_COMP: "ALL (f::'a::type => 'a::type) (m::nat) (n::nat) a::'a::type.
- (f ^ m) ((f ^ n) a) = (f ^ (m + n)) a"
+ (f ^^ m) ((f ^^ n) a) = (f ^^ (m + n)) a"
by (import word32 FUNPOW_COMP)
lemma INw_MODw: "ALL n::nat. INw (MODw n)"
@@ -1170,23 +1170,23 @@
constdefs
word_lsr :: "word32 => nat => word32"
- "word_lsr == %(a::word32) n::nat. (word_lsr1 ^ n) a"
+ "word_lsr == %(a::word32) n::nat. (word_lsr1 ^^ n) a"
-lemma word_lsr: "ALL (a::word32) n::nat. word_lsr a n = (word_lsr1 ^ n) a"
+lemma word_lsr: "ALL (a::word32) n::nat. word_lsr a n = (word_lsr1 ^^ n) a"
by (import word32 word_lsr)
constdefs
word_asr :: "word32 => nat => word32"
- "word_asr == %(a::word32) n::nat. (word_asr1 ^ n) a"
+ "word_asr == %(a::word32) n::nat. (word_asr1 ^^ n) a"
-lemma word_asr: "ALL (a::word32) n::nat. word_asr a n = (word_asr1 ^ n) a"
+lemma word_asr: "ALL (a::word32) n::nat. word_asr a n = (word_asr1 ^^ n) a"
by (import word32 word_asr)
constdefs
word_ror :: "word32 => nat => word32"
- "word_ror == %(a::word32) n::nat. (word_ror1 ^ n) a"
+ "word_ror == %(a::word32) n::nat. (word_ror1 ^^ n) a"
-lemma word_ror: "ALL (a::word32) n::nat. word_ror a n = (word_ror1 ^ n) a"
+lemma word_ror: "ALL (a::word32) n::nat. word_ror a n = (word_ror1 ^^ n) a"
by (import word32 word_ror)
consts
@@ -1583,4 +1583,3 @@
;end_setup
end
-
--- a/src/HOL/Import/HOL/arithmetic.imp Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Import/HOL/arithmetic.imp Mon Apr 27 07:26:17 2009 -0700
@@ -43,7 +43,7 @@
"TWO" > "HOL4Base.arithmetic.TWO"
"TIMES2" > "NatSimprocs.nat_mult_2"
"SUC_SUB1" > "HOL4Base.arithmetic.SUC_SUB1"
- "SUC_ONE_ADD" > "NatBin.Suc_eq_add_numeral_1_left"
+ "SUC_ONE_ADD" > "Nat_Numeral.Suc_eq_add_numeral_1_left"
"SUC_NOT" > "Nat.nat.simps_2"
"SUC_ELIM_THM" > "HOL4Base.arithmetic.SUC_ELIM_THM"
"SUC_ADD_SYM" > "HOL4Base.arithmetic.SUC_ADD_SYM"
@@ -233,7 +233,7 @@
"EVEN_AND_ODD" > "HOL4Base.arithmetic.EVEN_AND_ODD"
"EVEN_ADD" > "HOL4Base.arithmetic.EVEN_ADD"
"EVEN" > "HOL4Base.arithmetic.EVEN"
- "EQ_MULT_LCANCEL" > "NatBin.nat_mult_eq_cancel_disj"
+ "EQ_MULT_LCANCEL" > "Nat_Numeral.nat_mult_eq_cancel_disj"
"EQ_MONO_ADD_EQ" > "Nat.nat_add_right_cancel"
"EQ_LESS_EQ" > "Orderings.order_eq_iff"
"EQ_ADD_RCANCEL" > "Nat.nat_add_right_cancel"
--- a/src/HOL/Import/HOL/real.imp Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Import/HOL/real.imp Mon Apr 27 07:26:17 2009 -0700
@@ -99,7 +99,7 @@
"REAL_POW_INV" > "Power.power_inverse"
"REAL_POW_DIV" > "Power.power_divide"
"REAL_POW_ADD" > "Power.power_add"
- "REAL_POW2_ABS" > "NatBin.power2_abs"
+ "REAL_POW2_ABS" > "Nat_Numeral.power2_abs"
"REAL_POS_NZ" > "HOL4Real.real.REAL_POS_NZ"
"REAL_POS" > "RealDef.real_of_nat_ge_zero"
"REAL_POASQ" > "HOL4Real.real.REAL_POASQ"
@@ -210,7 +210,7 @@
"REAL_LE_RDIV_EQ" > "Ring_and_Field.pos_le_divide_eq"
"REAL_LE_RDIV" > "Ring_and_Field.mult_imp_le_div_pos"
"REAL_LE_RADD" > "OrderedGroup.add_le_cancel_right"
- "REAL_LE_POW2" > "NatBin.zero_compare_simps_12"
+ "REAL_LE_POW2" > "Nat_Numeral.zero_compare_simps_12"
"REAL_LE_NEGTOTAL" > "HOL4Real.real.REAL_LE_NEGTOTAL"
"REAL_LE_NEGR" > "OrderedGroup.le_minus_self_iff"
"REAL_LE_NEGL" > "OrderedGroup.minus_le_self_iff"
@@ -313,7 +313,7 @@
"POW_ONE" > "Power.power_one"
"POW_NZ" > "Power.field_power_not_zero"
"POW_MUL" > "Power.power_mult_distrib"
- "POW_MINUS1" > "NatBin.power_minus1_even"
+ "POW_MINUS1" > "Nat_Numeral.power_minus1_even"
"POW_M1" > "HOL4Real.real.POW_M1"
"POW_LT" > "HOL4Real.real.POW_LT"
"POW_LE" > "Power.power_mono"
@@ -323,7 +323,7 @@
"POW_ABS" > "Power.power_abs"
"POW_2_LT" > "RealPow.two_realpow_gt"
"POW_2_LE1" > "RealPow.two_realpow_ge_one"
- "POW_2" > "NatBin.power2_eq_square"
+ "POW_2" > "Nat_Numeral.power2_eq_square"
"POW_1" > "Power.power_one_right"
"POW_0" > "Power.power_0_Suc"
"ABS_ZERO" > "OrderedGroup.abs_eq_0"
@@ -335,7 +335,7 @@
"ABS_SIGN2" > "HOL4Real.real.ABS_SIGN2"
"ABS_SIGN" > "HOL4Real.real.ABS_SIGN"
"ABS_REFL" > "HOL4Real.real.ABS_REFL"
- "ABS_POW2" > "NatBin.abs_power2"
+ "ABS_POW2" > "Nat_Numeral.abs_power2"
"ABS_POS" > "OrderedGroup.abs_ge_zero"
"ABS_NZ" > "OrderedGroup.zero_less_abs_iff"
"ABS_NEG" > "OrderedGroup.abs_minus_cancel"
--- a/src/HOL/Import/HOL4Compat.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Import/HOL4Compat.thy Mon Apr 27 07:26:17 2009 -0700
@@ -202,19 +202,13 @@
constdefs
FUNPOW :: "('a => 'a) => nat => 'a => 'a"
- "FUNPOW f n == f ^ n"
+ "FUNPOW f n == f ^^ n"
-lemma FUNPOW: "(ALL f x. (f ^ 0) x = x) &
- (ALL f n x. (f ^ Suc n) x = (f ^ n) (f x))"
-proof auto
- fix f n x
- have "ALL x. f ((f ^ n) x) = (f ^ n) (f x)"
- by (induct n,auto)
- thus "f ((f ^ n) x) = (f ^ n) (f x)"
- ..
-qed
+lemma FUNPOW: "(ALL f x. (f ^^ 0) x = x) &
+ (ALL f n x. (f ^^ Suc n) x = (f ^^ n) (f x))"
+ by (simp add: funpow_swap1)
-lemma [hol4rew]: "FUNPOW f n = f ^ n"
+lemma [hol4rew]: "FUNPOW f n = f ^^ n"
by (simp add: FUNPOW_def)
lemma ADD: "(!n. (0::nat) + n = n) & (!m n. Suc m + n = Suc (m + n))"
@@ -224,7 +218,7 @@
by simp
lemma SUB: "(!m. (0::nat) - m = 0) & (!m n. (Suc m) - n = (if m < n then 0 else Suc (m - n)))"
- by (simp, arith)
+ by (simp) arith
lemma MAX_DEF: "max (m::nat) n = (if m < n then n else m)"
by (simp add: max_def)
--- a/src/HOL/Import/HOLLight/hollight.imp Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Import/HOLLight/hollight.imp Mon Apr 27 07:26:17 2009 -0700
@@ -1515,7 +1515,7 @@
"EQ_REFL_T" > "HOL.simp_thms_6"
"EQ_REFL" > "Presburger.fm_modd_pinf"
"EQ_MULT_RCANCEL" > "Nat.mult_cancel2"
- "EQ_MULT_LCANCEL" > "NatBin.nat_mult_eq_cancel_disj"
+ "EQ_MULT_LCANCEL" > "Nat_Numeral.nat_mult_eq_cancel_disj"
"EQ_IMP_LE" > "HOLLight.hollight.EQ_IMP_LE"
"EQ_EXT" > "HOL.meta_eq_to_obj_eq"
"EQ_CLAUSES" > "HOLLight.hollight.EQ_CLAUSES"
--- a/src/HOL/Int.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Int.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1536,7 +1536,7 @@
by (simp add: abs_if)
lemma abs_power_minus_one [simp]:
- "abs(-1 ^ n) = (1::'a::{ordered_idom,number_ring,recpower})"
+ "abs(-1 ^ n) = (1::'a::{ordered_idom,number_ring})"
by (simp add: power_abs)
lemma of_int_number_of_eq [simp]:
@@ -1848,42 +1848,21 @@
subsection {* Integer Powers *}
-instantiation int :: recpower
+context ring_1
begin
-primrec power_int where
- "p ^ 0 = (1\<Colon>int)"
- | "p ^ (Suc n) = (p\<Colon>int) * (p ^ n)"
-
-instance proof
- fix z :: int
- fix n :: nat
- show "z ^ 0 = 1" by simp
- show "z ^ Suc n = z * (z ^ n)" by simp
-qed
-
-declare power_int.simps [simp del]
+lemma of_int_power:
+ "of_int (z ^ n) = of_int z ^ n"
+ by (induct n) simp_all
end
-lemma zpower_zadd_distrib: "x ^ (y + z) = ((x ^ y) * (x ^ z)::int)"
- by (rule Power.power_add)
-
-lemma zpower_zpower: "(x ^ y) ^ z = (x ^ (y * z)::int)"
- by (rule Power.power_mult [symmetric])
-
-lemma zero_less_zpower_abs_iff [simp]:
- "(0 < abs x ^ n) \<longleftrightarrow> (x \<noteq> (0::int) | n = 0)"
- by (induct n) (auto simp add: zero_less_mult_iff)
-
-lemma zero_le_zpower_abs [simp]: "(0::int) \<le> abs x ^ n"
- by (induct n) (auto simp add: zero_le_mult_iff)
-
-lemma of_int_power:
- "of_int (z ^ n) = (of_int z ^ n :: 'a::{recpower, ring_1})"
- by (induct n) simp_all
-
-lemma int_power: "int (m^n) = (int m) ^ n"
+lemma zpower_zpower:
+ "(x ^ y) ^ z = (x ^ (y * z)::int)"
+ by (rule power_mult [symmetric])
+
+lemma int_power:
+ "int (m ^ n) = int m ^ n"
by (rule of_nat_power)
lemmas zpower_int = int_power [symmetric]
@@ -2224,6 +2203,8 @@
subsection {* Legacy theorems *}
+instance int :: recpower ..
+
lemmas zminus_zminus = minus_minus [of "z::int", standard]
lemmas zminus_0 = minus_zero [where 'a=int]
lemmas zminus_zadd_distrib = minus_add_distrib [of "z::int" "w", standard]
@@ -2278,4 +2259,15 @@
lemmas zless_le = less_int_def
lemmas int_eq_of_nat = TrueI
+lemma zpower_zadd_distrib:
+ "x ^ (y + z) = ((x ^ y) * (x ^ z)::int)"
+ by (rule power_add)
+
+lemma zero_less_zpower_abs_iff:
+ "(0 < abs x ^ n) \<longleftrightarrow> (x \<noteq> (0::int) | n = 0)"
+ by (rule zero_less_power_abs_iff)
+
+lemma zero_le_zpower_abs: "(0::int) \<le> abs x ^ n"
+ by (rule zero_le_power_abs)
+
end
--- a/src/HOL/IntDiv.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/IntDiv.thy Mon Apr 27 07:26:17 2009 -0700
@@ -249,33 +249,33 @@
text {* Tool setup *}
ML {*
-local
+local
-structure CancelDivMod = CancelDivModFun(
-struct
- val div_name = @{const_name Divides.div};
- val mod_name = @{const_name Divides.mod};
+structure CancelDivMod = CancelDivModFun(struct
+
+ val div_name = @{const_name div};
+ val mod_name = @{const_name mod};
val mk_binop = HOLogic.mk_binop;
val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;
val dest_sum = Int_Numeral_Simprocs.dest_sum;
- val div_mod_eqs =
- map mk_meta_eq [@{thm zdiv_zmod_equality},
- @{thm zdiv_zmod_equality2}];
+
+ val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];
+
val trans = trans;
- val prove_eq_sums =
- let
- val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac}
- in Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac simps) end;
+
+ val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
+ (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
+
end)
in
-val cancel_zdiv_zmod_proc = Simplifier.simproc (the_context ())
- "cancel_zdiv_zmod" ["(m::int) + n"] (K CancelDivMod.proc)
+val cancel_div_mod_int_proc = Simplifier.simproc (the_context ())
+ "cancel_zdiv_zmod" ["(k::int) + l"] (K CancelDivMod.proc);
-end;
+val _ = Addsimprocs [cancel_div_mod_int_proc];
-Addsimprocs [cancel_zdiv_zmod_proc]
+end
*}
lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
@@ -711,6 +711,26 @@
show "(a + c * b) div b = c + a div b"
unfolding zdiv_zadd1_eq [of a "c * b"] using not0
by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
+next
+ fix a b c :: int
+ assume "a \<noteq> 0"
+ then show "(a * b) div (a * c) = b div c"
+ proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")
+ case False then show ?thesis by auto
+ next
+ case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto
+ with `a \<noteq> 0`
+ have "\<And>q r. divmod_rel b c (q, r) \<Longrightarrow> divmod_rel (a * b) (a * c) (q, a * r)"
+ apply (auto simp add: divmod_rel_def)
+ apply (auto simp add: algebra_simps)
+ apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff)
+ apply (simp_all add: mult_less_cancel_left_disj mult_commute [of _ a])
+ done
+ moreover with `c \<noteq> 0` divmod_rel_div_mod have "divmod_rel b c (b div c, b mod c)" by auto
+ ultimately have "divmod_rel (a * b) (a * c) (b div c, a * (b mod c))" .
+ moreover from `a \<noteq> 0` `c \<noteq> 0` have "a * c \<noteq> 0" by simp
+ ultimately show ?thesis by (rule divmod_rel_div)
+ qed
qed auto
lemma posDivAlg_div_mod:
@@ -808,52 +828,6 @@
done
-subsection{*Cancellation of Common Factors in div*}
-
-lemma zdiv_zmult_zmult1_aux1:
- "[| (0::int) < b; c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
-by (subst zdiv_zmult2_eq, auto)
-
-lemma zdiv_zmult_zmult1_aux2:
- "[| b < (0::int); c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
-apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
-apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
-done
-
-lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"
-apply (case_tac "b = 0", simp)
-apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
-done
-
-lemma zdiv_zmult_zmult1_if[simp]:
- "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"
-by (simp add:zdiv_zmult_zmult1)
-
-
-subsection{*Distribution of Factors over mod*}
-
-lemma zmod_zmult_zmult1_aux1:
- "[| (0::int) < b; c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
-by (subst zmod_zmult2_eq, auto)
-
-lemma zmod_zmult_zmult1_aux2:
- "[| b < (0::int); c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
-apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
-apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
-done
-
-lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
-apply (case_tac "b = 0", simp)
-apply (case_tac "c = 0", simp)
-apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
-done
-
-lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
-apply (cut_tac c = c in zmod_zmult_zmult1)
-apply (auto simp add: mult_commute)
-done
-
-
subsection {*Splitting Rules for div and mod*}
text{*The proofs of the two lemmas below are essentially identical*}
@@ -937,7 +911,7 @@
right_distrib)
thus ?thesis
by (subst zdiv_zadd1_eq,
- simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
+ simp add: mod_mult_mult1 one_less_a2
div_pos_pos_trivial)
qed
@@ -961,7 +935,7 @@
then number_of v div (number_of w)
else (number_of v + (1::int)) div (number_of w))"
apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)
-apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)
+apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)
done
@@ -977,7 +951,7 @@
apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq
pos_mod_bound)
apply (subst mod_add_eq)
-apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
+apply (simp add: mod_mult_mult2 mod_pos_pos_trivial)
apply (rule mod_pos_pos_trivial)
apply (auto simp add: mod_pos_pos_trivial ring_distribs)
apply (subgoal_tac "0 \<le> b mod a", arith, simp)
@@ -998,7 +972,7 @@
"number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =
(2::int) * (number_of v mod number_of w)"
apply (simp only: number_of_eq numeral_simps)
-apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2
+apply (simp add: mod_mult_mult1 pos_zmod_mult_2
neg_zmod_mult_2 add_ac)
done
@@ -1008,7 +982,7 @@
then 2 * (number_of v mod number_of w) + 1
else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
apply (simp only: number_of_eq numeral_simps)
-apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2
+apply (simp add: mod_mult_mult1 pos_zmod_mult_2
neg_zmod_mult_2 add_ac)
done
@@ -1090,9 +1064,7 @@
done
lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
- apply (simp add: dvd_def)
- apply (auto simp add: zmod_zmult_zmult1)
- done
+ by (auto elim!: dvdE simp add: mod_mult_mult1)
lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
apply (subgoal_tac "k dvd n * (m div n) + m mod n")
@@ -1247,9 +1219,9 @@
lemmas zmod_simps =
mod_add_left_eq [symmetric]
mod_add_right_eq [symmetric]
- IntDiv.zmod_zmult1_eq [symmetric]
- mod_mult_left_eq [symmetric]
- IntDiv.zpower_zmod
+ zmod_zmult1_eq [symmetric]
+ mod_mult_left_eq [symmetric]
+ zpower_zmod
zminus_zmod zdiff_zmod_left zdiff_zmod_right
text {* Distributive laws for function @{text nat}. *}
--- a/src/HOL/IsaMakefile Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/IsaMakefile Mon Apr 27 07:26:17 2009 -0700
@@ -89,7 +89,7 @@
$(SRC)/Tools/IsaPlanner/rw_tools.ML \
$(SRC)/Tools/IsaPlanner/zipper.ML \
$(SRC)/Tools/atomize_elim.ML \
- $(SRC)/Tools/code/code_funcgr.ML \
+ $(SRC)/Tools/auto_solve.ML \
$(SRC)/Tools/code/code_haskell.ML \
$(SRC)/Tools/code/code_ml.ML \
$(SRC)/Tools/code/code_name.ML \
@@ -103,10 +103,11 @@
$(SRC)/Tools/intuitionistic.ML \
$(SRC)/Tools/induct_tacs.ML \
$(SRC)/Tools/nbe.ML \
+ $(SRC)/Tools/quickcheck.ML \
$(SRC)/Tools/project_rule.ML \
$(SRC)/Tools/random_word.ML \
$(SRC)/Tools/value.ML \
- Code_Setup.thy \
+ $(SRC)/Tools/Code_Generator.thy \
HOL.thy \
Tools/hologic.ML \
Tools/recfun_codegen.ML \
@@ -216,10 +217,9 @@
List.thy \
Main.thy \
Map.thy \
- NatBin.thy \
+ Nat_Numeral.thy \
Presburger.thy \
Recdef.thy \
- Relation_Power.thy \
SetInterval.thy \
$(SRC)/Provers/Arith/assoc_fold.ML \
$(SRC)/Provers/Arith/cancel_numeral_factor.ML \
--- a/src/HOL/Library/Code_Index.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Code_Index.thy Mon Apr 27 07:26:17 2009 -0700
@@ -144,7 +144,7 @@
subsection {* Basic arithmetic *}
-instantiation index :: "{minus, ordered_semidom, Divides.div, linorder}"
+instantiation index :: "{minus, ordered_semidom, semiring_div, linorder}"
begin
definition [simp, code del]:
@@ -172,7 +172,7 @@
"n < m \<longleftrightarrow> nat_of n < nat_of m"
instance proof
-qed (auto simp add: left_distrib)
+qed (auto simp add: index left_distrib div_mult_self1)
end
--- a/src/HOL/Library/Coinductive_List.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Coinductive_List.thy Mon Apr 27 07:26:17 2009 -0700
@@ -786,7 +786,7 @@
lemma funpow_lmap:
fixes f :: "'a \<Rightarrow> 'a"
- shows "(lmap f ^ n) (LCons b l) = LCons ((f ^ n) b) ((lmap f ^ n) l)"
+ shows "(lmap f ^^ n) (LCons b l) = LCons ((f ^^ n) b) ((lmap f ^^ n) l)"
by (induct n) simp_all
@@ -796,35 +796,35 @@
proof
fix x
have "(h x, iterates f x) \<in>
- {((lmap f ^ n) (h u), (lmap f ^ n) (iterates f u)) | u n. True}"
+ {((lmap f ^^ n) (h u), (lmap f ^^ n) (iterates f u)) | u n. True}"
proof -
- have "(h x, iterates f x) = ((lmap f ^ 0) (h x), (lmap f ^ 0) (iterates f x))"
+ have "(h x, iterates f x) = ((lmap f ^^ 0) (h x), (lmap f ^^ 0) (iterates f x))"
by simp
then show ?thesis by blast
qed
then show "h x = iterates f x"
proof (coinduct rule: llist_equalityI)
case (Eqllist q)
- then obtain u n where "q = ((lmap f ^ n) (h u), (lmap f ^ n) (iterates f u))"
+ then obtain u n where "q = ((lmap f ^^ n) (h u), (lmap f ^^ n) (iterates f u))"
(is "_ = (?q1, ?q2)")
by auto
- also have "?q1 = LCons ((f ^ n) u) ((lmap f ^ Suc n) (h u))"
+ also have "?q1 = LCons ((f ^^ n) u) ((lmap f ^^ Suc n) (h u))"
proof -
- have "?q1 = (lmap f ^ n) (LCons u (lmap f (h u)))"
+ have "?q1 = (lmap f ^^ n) (LCons u (lmap f (h u)))"
by (subst h) rule
- also have "\<dots> = LCons ((f ^ n) u) ((lmap f ^ n) (lmap f (h u)))"
+ also have "\<dots> = LCons ((f ^^ n) u) ((lmap f ^^ n) (lmap f (h u)))"
by (rule funpow_lmap)
- also have "(lmap f ^ n) (lmap f (h u)) = (lmap f ^ Suc n) (h u)"
+ also have "(lmap f ^^ n) (lmap f (h u)) = (lmap f ^^ Suc n) (h u)"
by (simp add: funpow_swap1)
finally show ?thesis .
qed
- also have "?q2 = LCons ((f ^ n) u) ((lmap f ^ Suc n) (iterates f u))"
+ also have "?q2 = LCons ((f ^^ n) u) ((lmap f ^^ Suc n) (iterates f u))"
proof -
- have "?q2 = (lmap f ^ n) (LCons u (iterates f (f u)))"
+ have "?q2 = (lmap f ^^ n) (LCons u (iterates f (f u)))"
by (subst iterates) rule
- also have "\<dots> = LCons ((f ^ n) u) ((lmap f ^ n) (iterates f (f u)))"
+ also have "\<dots> = LCons ((f ^^ n) u) ((lmap f ^^ n) (iterates f (f u)))"
by (rule funpow_lmap)
- also have "(lmap f ^ n) (iterates f (f u)) = (lmap f ^ Suc n) (iterates f u)"
+ also have "(lmap f ^^ n) (iterates f (f u)) = (lmap f ^^ Suc n) (iterates f u)"
by (simp add: lmap_iterates funpow_swap1)
finally show ?thesis .
qed
--- a/src/HOL/Library/Continuity.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Continuity.thy Mon Apr 27 07:26:17 2009 -0700
@@ -5,7 +5,7 @@
header {* Continuity and iterations (of set transformers) *}
theory Continuity
-imports Relation_Power Main
+imports Transitive_Closure Main
begin
subsection {* Continuity for complete lattices *}
@@ -48,25 +48,25 @@
qed
lemma continuous_lfp:
- assumes "continuous F" shows "lfp F = (SUP i. (F^i) bot)"
+ assumes "continuous F" shows "lfp F = (SUP i. (F ^^ i) bot)"
proof -
note mono = continuous_mono[OF `continuous F`]
- { fix i have "(F^i) bot \<le> lfp F"
+ { fix i have "(F ^^ i) bot \<le> lfp F"
proof (induct i)
- show "(F^0) bot \<le> lfp F" by simp
+ show "(F ^^ 0) bot \<le> lfp F" by simp
next
case (Suc i)
- have "(F^(Suc i)) bot = F((F^i) bot)" by simp
+ have "(F ^^ Suc i) bot = F((F ^^ i) bot)" by simp
also have "\<dots> \<le> F(lfp F)" by(rule monoD[OF mono Suc])
also have "\<dots> = lfp F" by(simp add:lfp_unfold[OF mono, symmetric])
finally show ?case .
qed }
- hence "(SUP i. (F^i) bot) \<le> lfp F" by (blast intro!:SUP_leI)
- moreover have "lfp F \<le> (SUP i. (F^i) bot)" (is "_ \<le> ?U")
+ hence "(SUP i. (F ^^ i) bot) \<le> lfp F" by (blast intro!:SUP_leI)
+ moreover have "lfp F \<le> (SUP i. (F ^^ i) bot)" (is "_ \<le> ?U")
proof (rule lfp_lowerbound)
- have "chain(%i. (F^i) bot)"
+ have "chain(%i. (F ^^ i) bot)"
proof -
- { fix i have "(F^i) bot \<le> (F^(Suc i)) bot"
+ { fix i have "(F ^^ i) bot \<le> (F ^^ (Suc i)) bot"
proof (induct i)
case 0 show ?case by simp
next
@@ -74,7 +74,7 @@
qed }
thus ?thesis by(auto simp add:chain_def)
qed
- hence "F ?U = (SUP i. (F^(i+1)) bot)" using `continuous F` by (simp add:continuous_def)
+ hence "F ?U = (SUP i. (F ^^ (i+1)) bot)" using `continuous F` by (simp add:continuous_def)
also have "\<dots> \<le> ?U" by(fast intro:SUP_leI le_SUPI)
finally show "F ?U \<le> ?U" .
qed
@@ -193,7 +193,7 @@
definition
up_iterate :: "('a set => 'a set) => nat => 'a set" where
- "up_iterate f n = (f^n) {}"
+ "up_iterate f n = (f ^^ n) {}"
lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"
by (simp add: up_iterate_def)
@@ -245,7 +245,7 @@
definition
down_iterate :: "('a set => 'a set) => nat => 'a set" where
- "down_iterate f n = (f^n) UNIV"
+ "down_iterate f n = (f ^^ n) UNIV"
lemma down_iterate_0 [simp]: "down_iterate f 0 = UNIV"
by (simp add: down_iterate_def)
--- a/src/HOL/Library/Euclidean_Space.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Euclidean_Space.thy Mon Apr 27 07:26:17 2009 -0700
@@ -253,12 +253,7 @@
"vector_power x 0 = 1"
| "vector_power x (Suc n) = x * vector_power x n"
-instantiation "^" :: (recpower,type) recpower
-begin
- definition vec_power_def: "op ^ \<equiv> vector_power"
- instance
- apply (intro_classes) by (simp_all add: vec_power_def)
-end
+instance "^" :: (recpower,type) recpower ..
instance "^" :: (semiring,type) semiring
apply (intro_classes) by (vector ring_simps)+
--- a/src/HOL/Library/Eval_Witness.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Eval_Witness.thy Mon Apr 27 07:26:17 2009 -0700
@@ -68,7 +68,7 @@
| dest_exs _ _ = sys_error "dest_exs";
val t = dest_exs (length ws) (HOLogic.dest_Trueprop goal);
in
- if Code_ML.eval_term ("Eval_Witness_Method.eval_ref", Eval_Witness_Method.eval_ref) thy t ws
+ if Code_ML.eval NONE ("Eval_Witness_Method.eval_ref", Eval_Witness_Method.eval_ref) (K I) thy t ws
then Thm.cterm_of thy goal
else @{cprop True} (*dummy*)
end
--- a/src/HOL/Library/Float.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Float.thy Mon Apr 27 07:26:17 2009 -0700
@@ -15,8 +15,8 @@
datatype float = Float int int
-fun Ifloat :: "float \<Rightarrow> real" where
-"Ifloat (Float a b) = real a * pow2 b"
+primrec Ifloat :: "float \<Rightarrow> real" where
+ "Ifloat (Float a b) = real a * pow2 b"
instantiation float :: zero begin
definition zero_float where "0 = Float 0 0"
@@ -33,11 +33,11 @@
instance ..
end
-fun mantissa :: "float \<Rightarrow> int" where
-"mantissa (Float a b) = a"
+primrec mantissa :: "float \<Rightarrow> int" where
+ "mantissa (Float a b) = a"
-fun scale :: "float \<Rightarrow> int" where
-"scale (Float a b) = b"
+primrec scale :: "float \<Rightarrow> int" where
+ "scale (Float a b) = b"
lemma Ifloat_neg_exp: "e < 0 \<Longrightarrow> Ifloat (Float m e) = real m * inverse (2^nat (-e))" by auto
lemma Ifloat_nge0_exp: "\<not> 0 \<le> e \<Longrightarrow> Ifloat (Float m e) = real m * inverse (2^nat (-e))" by auto
@@ -320,12 +320,12 @@
end
instantiation float :: uminus begin
-fun uminus_float where [simp del]: "uminus_float (Float m e) = Float (-m) e"
+primrec uminus_float where [simp del]: "uminus_float (Float m e) = Float (-m) e"
instance ..
end
instantiation float :: minus begin
-fun minus_float where [simp del]: "(z::float) - w = z + (- w)"
+definition minus_float where [simp del]: "(z::float) - w = z + (- w)"
instance ..
end
@@ -334,11 +334,11 @@
instance ..
end
-fun float_pprt :: "float \<Rightarrow> float" where
-"float_pprt (Float a e) = (if 0 <= a then (Float a e) else 0)"
+primrec float_pprt :: "float \<Rightarrow> float" where
+ "float_pprt (Float a e) = (if 0 <= a then (Float a e) else 0)"
-fun float_nprt :: "float \<Rightarrow> float" where
-"float_nprt (Float a e) = (if 0 <= a then 0 else (Float a e))"
+primrec float_nprt :: "float \<Rightarrow> float" where
+ "float_nprt (Float a e) = (if 0 <= a then 0 else (Float a e))"
instantiation float :: ord begin
definition le_float_def: "z \<le> w \<equiv> Ifloat z \<le> Ifloat w"
@@ -354,7 +354,7 @@
by (cases a, simp add: uminus_float.simps)
lemma Ifloat_sub[simp]: "Ifloat (a - b) = Ifloat a - Ifloat b"
- by (cases a, cases b, simp add: minus_float.simps)
+ by (cases a, cases b, simp add: minus_float_def)
lemma Ifloat_mult[simp]: "Ifloat (a*b) = Ifloat a * Ifloat b"
by (cases a, cases b, simp add: times_float.simps pow2_add)
@@ -443,37 +443,10 @@
lemma Ifloat_min: "Ifloat (min x y) = min (Ifloat x) (Ifloat y)" unfolding min_def le_float_def by auto
lemma Ifloat_max: "Ifloat (max a b) = max (Ifloat a) (Ifloat b)" unfolding max_def le_float_def by auto
-instantiation float :: power begin
-fun power_float where [simp del]: "(Float m e) ^ n = Float (m ^ n) (e * int n)"
-instance ..
-end
-
-instance float :: recpower
-proof (intro_classes)
- fix a :: float show "a ^ 0 = 1" by (cases a, auto simp add: power_float.simps one_float_def)
-next
- fix a :: float and n :: nat show "a ^ (Suc n) = a * a ^ n"
- by (cases a, auto simp add: power_float.simps times_float.simps algebra_simps)
-qed
+instance float :: recpower ..
-lemma float_power: "Ifloat (x ^ n) = (Ifloat x) ^ n"
-proof (cases x)
- case (Float m e)
-
- have "pow2 e ^ n = pow2 (e * int n)"
- proof (cases "e >= 0")
- case True hence e_nat: "e = int (nat e)" by auto
- hence "pow2 e ^ n = (2 ^ nat e) ^ n" using pow2_int[of "nat e"] by auto
- thus ?thesis unfolding power_mult[symmetric] unfolding pow2_int[symmetric] int_mult e_nat[symmetric] .
- next
- case False hence e_minus: "-e = int (nat (-e))" by auto
- hence "pow2 (-e) ^ n = (2 ^ nat (-e)) ^ n" using pow2_int[of "nat (-e)"] by auto
- hence "pow2 (-e) ^ n = pow2 ((-e) * int n)" unfolding power_mult[symmetric] unfolding pow2_int[symmetric] int_mult e_minus[symmetric] zmult_zminus .
- thus ?thesis unfolding pow2_neg[of "-e"] pow2_neg[of "-e * int n"] unfolding zmult_zminus zminus_zminus nonzero_power_inverse[OF pow2_neq_zero, symmetric]
- using nonzero_inverse_eq_imp_eq[OF _ pow2_neq_zero pow2_neq_zero] by auto
- qed
- thus ?thesis by (auto simp add: Float power_mult_distrib Ifloat.simps power_float.simps)
-qed
+lemma float_power: "Ifloat (x ^ n) = Ifloat x ^ n"
+ by (induct n) simp_all
lemma zero_le_pow2[simp]: "0 \<le> pow2 s"
apply (subgoal_tac "0 < pow2 s")
@@ -1182,12 +1155,12 @@
unfolding x_eq y_eq float_divr.simps Let_def le_float_def Ifloat_0 Ifloat_mult by (auto intro!: mult_nonneg_nonpos)
qed
-fun round_down :: "nat \<Rightarrow> float \<Rightarrow> float" where
+primrec round_down :: "nat \<Rightarrow> float \<Rightarrow> float" where
"round_down prec (Float m e) = (let d = bitlen m - int prec in
if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)
else Float m e)"
-fun round_up :: "nat \<Rightarrow> float \<Rightarrow> float" where
+primrec round_up :: "nat \<Rightarrow> float \<Rightarrow> float" where
"round_up prec (Float m e) = (let d = bitlen m - int prec in
if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P in Float (n + (if r = 0 then 0 else 1)) (e + d)
else Float m e)"
@@ -1314,8 +1287,8 @@
finally show ?thesis .
qed
-fun float_abs :: "float \<Rightarrow> float" where
-"float_abs (Float m e) = Float \<bar>m\<bar> e"
+primrec float_abs :: "float \<Rightarrow> float" where
+ "float_abs (Float m e) = Float \<bar>m\<bar> e"
instantiation float :: abs begin
definition abs_float_def: "\<bar>x\<bar> = float_abs x"
@@ -1329,8 +1302,8 @@
thus ?thesis unfolding Float abs_float_def float_abs.simps Ifloat.simps by auto
qed
-fun floor_fl :: "float \<Rightarrow> float" where
-"floor_fl (Float m e) = (if 0 \<le> e then Float m e
+primrec floor_fl :: "float \<Rightarrow> float" where
+ "floor_fl (Float m e) = (if 0 \<le> e then Float m e
else Float (m div (2 ^ (nat (-e)))) 0)"
lemma floor_fl: "Ifloat (floor_fl x) \<le> Ifloat x"
@@ -1358,8 +1331,8 @@
declare floor_fl.simps[simp del]
-fun ceiling_fl :: "float \<Rightarrow> float" where
-"ceiling_fl (Float m e) = (if 0 \<le> e then Float m e
+primrec ceiling_fl :: "float \<Rightarrow> float" where
+ "ceiling_fl (Float m e) = (if 0 \<le> e then Float m e
else Float (m div (2 ^ (nat (-e))) + 1) 0)"
lemma ceiling_fl: "Ifloat x \<le> Ifloat (ceiling_fl x)"
--- a/src/HOL/Library/Formal_Power_Series.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Formal_Power_Series.thy Mon Apr 27 07:26:17 2009 -0700
@@ -680,30 +680,14 @@
subsection {* Powers*}
-instantiation fps :: (semiring_1) power
-begin
-
-fun fps_pow :: "nat \<Rightarrow> 'a fps \<Rightarrow> 'a fps" where
- "fps_pow 0 f = 1"
-| "fps_pow (Suc n) f = f * fps_pow n f"
-
-definition fps_power_def: "power (f::'a fps) n = fps_pow n f"
-instance ..
-end
-
-instantiation fps :: (comm_ring_1) recpower
-begin
-instance
- apply (intro_classes)
- by (simp_all add: fps_power_def)
-end
+instance fps :: (semiring_1) recpower ..
lemma fps_power_zeroth_eq_one: "a$0 =1 \<Longrightarrow> a^n $ 0 = (1::'a::semiring_1)"
- by (induct n, auto simp add: fps_power_def expand_fps_eq fps_mult_nth)
+ by (induct n, auto simp add: expand_fps_eq fps_mult_nth)
lemma fps_power_first_eq: "(a:: 'a::comm_ring_1 fps)$0 =1 \<Longrightarrow> a^n $ 1 = of_nat n * a$1"
proof(induct n)
- case 0 thus ?case by (simp add: fps_power_def)
+ case 0 thus ?case by simp
next
case (Suc n)
note h = Suc.hyps[OF `a$0 = 1`]
@@ -712,13 +696,13 @@
qed
lemma startsby_one_power:"a $ 0 = (1::'a::comm_ring_1) \<Longrightarrow> a^n $ 0 = 1"
- by (induct n, auto simp add: fps_power_def fps_mult_nth)
+ by (induct n, auto simp add: fps_mult_nth)
lemma startsby_zero_power:"a $0 = (0::'a::comm_ring_1) \<Longrightarrow> n > 0 \<Longrightarrow> a^n $0 = 0"
- by (induct n, auto simp add: fps_power_def fps_mult_nth)
+ by (induct n, auto simp add: fps_mult_nth)
lemma startsby_power:"a $0 = (v::'a::{comm_ring_1, recpower}) \<Longrightarrow> a^n $0 = v^n"
- by (induct n, auto simp add: fps_power_def fps_mult_nth power_Suc)
+ by (induct n, auto simp add: fps_mult_nth power_Suc)
lemma startsby_zero_power_iff[simp]:
"a^n $0 = (0::'a::{idom, recpower}) \<longleftrightarrow> (n \<noteq> 0 \<and> a$0 = 0)"
@@ -901,7 +885,7 @@
lemma X_power_iff: "X^k = Abs_fps (\<lambda>n. if n = k then (1::'a::comm_ring_1) else 0)"
proof(induct k)
- case 0 thus ?case by (simp add: X_def fps_power_def fps_eq_iff)
+ case 0 thus ?case by (simp add: X_def fps_eq_iff)
next
case (Suc k)
{fix m
@@ -979,7 +963,7 @@
(* {a_{n+k}}_0^infty Corresponds to (f - setsum (\<lambda>i. a_i * x^i))/x^h, for h>0*)
lemma fps_power_mult_eq_shift:
- "X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) = Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. k}" (is "?lhs = ?rhs")
+ "X^Suc k * Abs_fps (\<lambda>n. a (n + Suc k)) = Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: comm_ring_1) * X^i) {0 .. k}" (is "?lhs = ?rhs")
proof-
{fix n:: nat
have "?lhs $ n = (if n < Suc k then 0 else a n)"
@@ -990,7 +974,7 @@
next
case (Suc k)
note th = Suc.hyps[symmetric]
- have "(Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. Suc k})$n = (Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a:: field) * X^i) {0 .. k} - fps_const (a (Suc k)) * X^ Suc k) $ n" by (simp add: ring_simps)
+ have "(Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. Suc k})$n = (Abs_fps a - setsum (\<lambda>i. fps_const (a i :: 'a) * X^i) {0 .. k} - fps_const (a (Suc k)) * X^ Suc k) $ n" by (simp add: ring_simps)
also have "\<dots> = (if n < Suc k then 0 else a n) - (fps_const (a (Suc k)) * X^ Suc k)$n"
using th
unfolding fps_sub_nth by simp
@@ -1022,13 +1006,16 @@
lemma XD_linear[simp]: "XD (fps_const c * a + fps_const d * b) = fps_const c * XD a + fps_const d * XD (b :: ('a::comm_ring_1) fps)"
by simp
-lemma XDN_linear: "(XD^n) (fps_const c * a + fps_const d * b) = fps_const c * (XD^n) a + fps_const d * (XD^n) (b :: ('a::comm_ring_1) fps)"
+lemma XDN_linear:
+ "(XD ^^ n) (fps_const c * a + fps_const d * b) = fps_const c * (XD ^^ n) a + fps_const d * (XD ^^ n) (b :: ('a::comm_ring_1) fps)"
by (induct n, simp_all)
lemma fps_mult_X_deriv_shift: "X* fps_deriv a = Abs_fps (\<lambda>n. of_nat n* a$n)" by (simp add: fps_eq_iff)
-lemma fps_mult_XD_shift: "(XD ^k) (a:: ('a::{comm_ring_1, recpower, ring_char_0}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
-by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def)
+
+lemma fps_mult_XD_shift:
+ "(XD ^^ k) (a:: ('a::{comm_ring_1, recpower}) fps) = Abs_fps (\<lambda>n. (of_nat n ^ k) * a$n)"
+ by (induct k arbitrary: a) (simp_all add: power_Suc XD_def fps_eq_iff ring_simps del: One_nat_def)
subsubsection{* Rule 3 is trivial and is given by @{text fps_times_def}*}
subsubsection{* Rule 5 --- summation and "division" by (1 - X)*}
@@ -1901,19 +1888,16 @@
done
lemma fps_compose_1[simp]: "1 oo a = 1"
- by (simp add: fps_eq_iff fps_compose_nth fps_power_def mult_delta_left setsum_delta)
+ by (simp add: fps_eq_iff fps_compose_nth mult_delta_left setsum_delta)
lemma fps_compose_0[simp]: "0 oo a = 0"
by (simp add: fps_eq_iff fps_compose_nth)
-lemma fps_pow_0: "fps_pow n 0 = (if n = 0 then 1 else 0)"
- by (induct n, simp_all)
-
lemma fps_compose_0_right[simp]: "a oo 0 = fps_const (a$0)"
- by (auto simp add: fps_eq_iff fps_compose_nth fps_power_def fps_pow_0 setsum_0')
+ by (auto simp add: fps_eq_iff fps_compose_nth power_0_left setsum_0')
lemma fps_compose_add_distrib: "(a + b) oo c = (a oo c) + (b oo c)"
- by (simp add: fps_eq_iff fps_compose_nth ring_simps setsum_addf)
+ by (simp add: fps_eq_iff fps_compose_nth ring_simps setsum_addf)
lemma fps_compose_setsum_distrib: "(setsum f S) oo a = setsum (\<lambda>i. f i oo a) S"
proof-
@@ -2341,11 +2325,11 @@
proof-
have "fps_deriv ?lhs = 0"
apply (simp add: fps_deriv_power fps_sin_deriv fps_cos_deriv power_Suc)
- by (simp add: fps_power_def ring_simps fps_const_neg[symmetric] del: fps_const_neg)
+ by (simp add: ring_simps fps_const_neg[symmetric] del: fps_const_neg)
then have "?lhs = fps_const (?lhs $ 0)"
unfolding fps_deriv_eq_0_iff .
also have "\<dots> = 1"
- by (auto simp add: fps_eq_iff fps_power_def numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def)
+ by (auto simp add: fps_eq_iff numeral_2_eq_2 fps_mult_nth fps_cos_def fps_sin_def)
finally show ?thesis .
qed
--- a/src/HOL/Library/Numeral_Type.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Numeral_Type.thy Mon Apr 27 07:26:17 2009 -0700
@@ -154,8 +154,8 @@
locale mod_type =
fixes n :: int
- and Rep :: "'a::{zero,one,plus,times,uminus,minus,power} \<Rightarrow> int"
- and Abs :: "int \<Rightarrow> 'a::{zero,one,plus,times,uminus,minus,power}"
+ and Rep :: "'a::{zero,one,plus,times,uminus,minus} \<Rightarrow> int"
+ and Abs :: "int \<Rightarrow> 'a::{zero,one,plus,times,uminus,minus}"
assumes type: "type_definition Rep Abs {0..<n}"
and size1: "1 < n"
and zero_def: "0 = Abs 0"
@@ -164,14 +164,13 @@
and mult_def: "x * y = Abs ((Rep x * Rep y) mod n)"
and diff_def: "x - y = Abs ((Rep x - Rep y) mod n)"
and minus_def: "- x = Abs ((- Rep x) mod n)"
- and power_def: "x ^ k = Abs (Rep x ^ k mod n)"
begin
lemma size0: "0 < n"
by (cut_tac size1, simp)
lemmas definitions =
- zero_def one_def add_def mult_def minus_def diff_def power_def
+ zero_def one_def add_def mult_def minus_def diff_def
lemma Rep_less_n: "Rep x < n"
by (rule type_definition.Rep [OF type, simplified, THEN conjunct2])
@@ -227,8 +226,8 @@
locale mod_ring = mod_type +
constrains n :: int
- and Rep :: "'a::{number_ring,power} \<Rightarrow> int"
- and Abs :: "int \<Rightarrow> 'a::{number_ring,power}"
+ and Rep :: "'a::{number_ring} \<Rightarrow> int"
+ and Abs :: "int \<Rightarrow> 'a::{number_ring}"
begin
lemma of_nat_eq: "of_nat k = Abs (int k mod n)"
@@ -272,7 +271,7 @@
@{typ num1}, since 0 and 1 are not distinct.
*}
-instantiation num1 :: "{comm_ring,comm_monoid_mult,number,recpower}"
+instantiation num1 :: "{comm_ring,comm_monoid_mult,number}"
begin
lemma num1_eq_iff: "(x::num1) = (y::num1) \<longleftrightarrow> True"
@@ -284,7 +283,7 @@
end
instantiation
- bit0 and bit1 :: (finite) "{zero,one,plus,times,uminus,minus,power}"
+ bit0 and bit1 :: (finite) "{zero,one,plus,times,uminus,minus}"
begin
definition Abs_bit0' :: "int \<Rightarrow> 'a bit0" where
@@ -299,7 +298,6 @@
definition "x * y = Abs_bit0' (Rep_bit0 x * Rep_bit0 y)"
definition "x - y = Abs_bit0' (Rep_bit0 x - Rep_bit0 y)"
definition "- x = Abs_bit0' (- Rep_bit0 x)"
-definition "x ^ k = Abs_bit0' (Rep_bit0 x ^ k)"
definition "0 = Abs_bit1 0"
definition "1 = Abs_bit1 1"
@@ -307,7 +305,6 @@
definition "x * y = Abs_bit1' (Rep_bit1 x * Rep_bit1 y)"
definition "x - y = Abs_bit1' (Rep_bit1 x - Rep_bit1 y)"
definition "- x = Abs_bit1' (- Rep_bit1 x)"
-definition "x ^ k = Abs_bit1' (Rep_bit1 x ^ k)"
instance ..
@@ -326,7 +323,6 @@
apply (rule times_bit0_def [unfolded Abs_bit0'_def])
apply (rule minus_bit0_def [unfolded Abs_bit0'_def])
apply (rule uminus_bit0_def [unfolded Abs_bit0'_def])
-apply (rule power_bit0_def [unfolded Abs_bit0'_def])
done
interpretation bit1:
@@ -342,7 +338,6 @@
apply (rule times_bit1_def [unfolded Abs_bit1'_def])
apply (rule minus_bit1_def [unfolded Abs_bit1'_def])
apply (rule uminus_bit1_def [unfolded Abs_bit1'_def])
-apply (rule power_bit1_def [unfolded Abs_bit1'_def])
done
instance bit0 :: (finite) "{comm_ring_1,recpower}"
@@ -386,9 +381,6 @@
lemmas bit0_iszero_number_of [simp] = bit0.iszero_number_of
lemmas bit1_iszero_number_of [simp] = bit1.iszero_number_of
-declare power_Suc [where ?'a="'a::finite bit0", standard, simp]
-declare power_Suc [where ?'a="'a::finite bit1", standard, simp]
-
subsection {* Syntax *}
--- a/src/HOL/Library/Polynomial.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Polynomial.thy Mon Apr 27 07:26:17 2009 -0700
@@ -632,19 +632,7 @@
shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"
by (safe elim!: dvd_smult dvd_smult_cancel)
-instantiation poly :: (comm_semiring_1) recpower
-begin
-
-primrec power_poly where
- "(p::'a poly) ^ 0 = 1"
-| "(p::'a poly) ^ (Suc n) = p * p ^ n"
-
-instance
- by default simp_all
-
-declare power_poly.simps [simp del]
-
-end
+instance poly :: (comm_semiring_1) recpower ..
lemma degree_power_le: "degree (p ^ n) \<le> degree p * n"
by (induct n, simp, auto intro: order_trans degree_mult_le)
@@ -987,6 +975,30 @@
by (simp add: pdivmod_rel_def left_distrib)
thus "(x + z * y) div y = z + x div y"
by (rule div_poly_eq)
+next
+ fix x y z :: "'a poly"
+ assume "x \<noteq> 0"
+ show "(x * y) div (x * z) = y div z"
+ proof (cases "y \<noteq> 0 \<and> z \<noteq> 0")
+ have "\<And>x::'a poly. pdivmod_rel x 0 0 x"
+ by (rule pdivmod_rel_by_0)
+ then have [simp]: "\<And>x::'a poly. x div 0 = 0"
+ by (rule div_poly_eq)
+ have "\<And>x::'a poly. pdivmod_rel 0 x 0 0"
+ by (rule pdivmod_rel_0)
+ then have [simp]: "\<And>x::'a poly. 0 div x = 0"
+ by (rule div_poly_eq)
+ case False then show ?thesis by auto
+ next
+ case True then have "y \<noteq> 0" and "z \<noteq> 0" by auto
+ with `x \<noteq> 0`
+ have "\<And>q r. pdivmod_rel y z q r \<Longrightarrow> pdivmod_rel (x * y) (x * z) q (x * r)"
+ by (auto simp add: pdivmod_rel_def algebra_simps)
+ (rule classical, simp add: degree_mult_eq)
+ moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" .
+ ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" .
+ then show ?thesis by (simp add: div_poly_eq)
+ qed
qed
end
--- a/src/HOL/Library/Quickcheck.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Quickcheck.thy Mon Apr 27 07:26:17 2009 -0700
@@ -47,6 +47,8 @@
val eval_ref : (unit -> int -> int * int -> term list option * (int * int)) option ref = ref NONE;
+val target = "Quickcheck";
+
fun mk_generator_expr thy prop tys =
let
val bound_max = length tys - 1;
@@ -72,14 +74,75 @@
let
val tys = (map snd o fst o strip_abs) t;
val t' = mk_generator_expr thy t tys;
- val f = Code_ML.eval_term ("Quickcheck.eval_ref", eval_ref) thy t' [];
- in f #> Random_Engine.run #> (Option.map o map) (Code.postprocess_term thy) end;
+ val f = Code_ML.eval (SOME target) ("Quickcheck.eval_ref", eval_ref)
+ (fn proc => fn g => fn s => g s #>> (Option.map o map) proc) thy t' [];
+ in f #> Random_Engine.run end;
end
*}
setup {*
- Quickcheck.add_generator ("code", Quickcheck.compile_generator_expr o ProofContext.theory_of)
+ Code_Target.extend_target (Quickcheck.target, (Code_ML.target_Eval, K I))
+ #> Quickcheck.add_generator ("code", Quickcheck.compile_generator_expr o ProofContext.theory_of)
*}
+
+subsection {* Type @{typ "'a \<Rightarrow> 'b"} *}
+
+ML {*
+structure Random_Engine =
+struct
+
+open Random_Engine;
+
+fun random_fun (T1 : typ) (T2 : typ) (eq : 'a -> 'a -> bool) (term_of : 'a -> term)
+ (random : Random_Engine.seed -> ('b * (unit -> term)) * Random_Engine.seed)
+ (random_split : Random_Engine.seed -> Random_Engine.seed * Random_Engine.seed)
+ (seed : Random_Engine.seed) =
+ let
+ val (seed', seed'') = random_split seed;
+ val state = ref (seed', [], Const (@{const_name undefined}, T1 --> T2));
+ val fun_upd = Const (@{const_name fun_upd},
+ (T1 --> T2) --> T1 --> T2 --> T1 --> T2);
+ fun random_fun' x =
+ let
+ val (seed, fun_map, f_t) = ! state;
+ in case AList.lookup (uncurry eq) fun_map x
+ of SOME y => y
+ | NONE => let
+ val t1 = term_of x;
+ val ((y, t2), seed') = random seed;
+ val fun_map' = (x, y) :: fun_map;
+ val f_t' = fun_upd $ f_t $ t1 $ t2 ();
+ val _ = state := (seed', fun_map', f_t');
+ in y end
+ end;
+ fun term_fun' () = #3 (! state);
+ in ((random_fun', term_fun'), seed'') end;
+
end
+*}
+
+axiomatization
+ random_fun_aux :: "typerep \<Rightarrow> typerep \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> term)
+ \<Rightarrow> (seed \<Rightarrow> ('b \<times> (unit \<Rightarrow> term)) \<times> seed) \<Rightarrow> (seed \<Rightarrow> seed \<times> seed)
+ \<Rightarrow> seed \<Rightarrow> (('a \<Rightarrow> 'b) \<times> (unit \<Rightarrow> term)) \<times> seed"
+
+code_const random_fun_aux (Quickcheck "Random'_Engine.random'_fun")
+ -- {* With enough criminal energy this can be abused to derive @{prop False};
+ for this reason we use a distinguished target @{text Quickcheck}
+ not spoiling the regular trusted code generation *}
+
+instantiation "fun" :: ("{eq, term_of}", "{type, random}") random
+begin
+
+definition random_fun :: "index \<Rightarrow> seed \<Rightarrow> (('a \<Rightarrow> 'b) \<times> (unit \<Rightarrow> term)) \<times> seed" where
+ "random n = random_fun_aux TYPEREP('a) TYPEREP('b) (op =) Code_Eval.term_of (random n) split_seed"
+
+instance ..
+
+end
+
+code_reserved Quickcheck Random_Engine
+
+end
--- a/src/HOL/Library/Topology_Euclidean_Space.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Topology_Euclidean_Space.thy Mon Apr 27 07:26:17 2009 -0700
@@ -5441,7 +5441,7 @@
have "1 - c > 0" using c by auto
from s(2) obtain z0 where "z0 \<in> s" by auto
- def z \<equiv> "\<lambda> n::nat. fun_pow n f z0"
+ def z \<equiv> "\<lambda>n. (f ^^ n) z0"
{ fix n::nat
have "z n \<in> s" unfolding z_def
proof(induct n) case 0 thus ?case using `z0 \<in>s` by auto
@@ -5580,7 +5580,7 @@
using dist[THEN bspec[where x=x], THEN bspec[where x=y]] by auto } note dist' = this
def y \<equiv> "g x"
have [simp]:"y\<in>s" unfolding y_def using gs[unfolded image_subset_iff] and `x\<in>s` by blast
- def f \<equiv> "\<lambda> n. fun_pow n g"
+ def f \<equiv> "\<lambda>n. g ^^ n"
have [simp]:"\<And>n z. g (f n z) = f (Suc n) z" unfolding f_def by auto
have [simp]:"\<And>z. f 0 z = z" unfolding f_def by auto
{ fix n::nat and z assume "z\<in>s"
--- a/src/HOL/Library/Word.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/Word.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1,5 +1,4 @@
(* Title: HOL/Library/Word.thy
- ID: $Id$
Author: Sebastian Skalberg (TU Muenchen)
*)
@@ -40,10 +39,8 @@
Zero ("\<zero>")
| One ("\<one>")
-primrec
- bitval :: "bit => nat"
-where
- "bitval \<zero> = 0"
+primrec bitval :: "bit => nat" where
+ "bitval \<zero> = 0"
| "bitval \<one> = 1"
consts
@@ -1531,7 +1528,7 @@
show ?thesis
apply simp
apply (subst power_Suc [symmetric])
- apply (simp del: power_int.simps)
+ apply simp
done
qed
finally show ?thesis .
--- a/src/HOL/Library/reflection.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Library/reflection.ML Mon Apr 27 07:26:17 2009 -0700
@@ -314,5 +314,6 @@
in (rtac th i THEN TRY(rtac TrueI i)) st end);
fun reflection_tac ctxt = gen_reflection_tac ctxt Codegen.evaluation_conv;
+ (*FIXME why Codegen.evaluation_conv? very specific...*)
end
--- a/src/HOL/List.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/List.thy Mon Apr 27 07:26:17 2009 -0700
@@ -5,7 +5,7 @@
header {* The datatype of finite lists *}
theory List
-imports Plain Relation_Power Presburger Recdef ATP_Linkup
+imports Plain Presburger Recdef ATP_Linkup
uses "Tools/string_syntax.ML"
begin
@@ -198,7 +198,7 @@
definition
rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
- "rotate n = rotate1 ^ n"
+ "rotate n = rotate1 ^^ n"
definition
list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
--- a/src/HOL/Map.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Map.thy Mon Apr 27 07:26:17 2009 -0700
@@ -11,7 +11,7 @@
imports List
begin
-types ('a,'b) "~=>" = "'a => 'b option" (infixr 0)
+types ('a,'b) "~=>" = "'a => 'b option" (infixr "~=>" 0)
translations (type) "a ~=> b " <= (type) "a => b option"
syntax (xsymbols)
--- a/src/HOL/NSA/HyperDef.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/NSA/HyperDef.thy Mon Apr 27 07:26:17 2009 -0700
@@ -459,7 +459,7 @@
by transfer (rule power_add)
lemma hyperpow_one [simp]:
- "\<And>r. (r::'a::recpower star) pow (1::hypnat) = r"
+ "\<And>r. (r::'a::monoid_mult star) pow (1::hypnat) = r"
by transfer (rule power_one_right)
lemma hyperpow_two:
--- a/src/HOL/NSA/StarDef.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/NSA/StarDef.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1,5 +1,4 @@
(* Title : HOL/Hyperreal/StarDef.thy
- ID : $Id$
Author : Jacques D. Fleuriot and Brian Huffman
*)
@@ -546,16 +545,6 @@
end
-instantiation star :: (power) power
-begin
-
-definition
- star_power_def: "(op ^) \<equiv> \<lambda>x n. ( *f* (\<lambda>x. x ^ n)) x"
-
-instance ..
-
-end
-
instantiation star :: (ord) ord
begin
@@ -574,7 +563,7 @@
star_add_def star_diff_def star_minus_def
star_mult_def star_divide_def star_inverse_def
star_le_def star_less_def star_abs_def star_sgn_def
- star_div_def star_mod_def star_power_def
+ star_div_def star_mod_def
text {* Class operations preserve standard elements *}
@@ -614,15 +603,11 @@
lemma Standard_mod: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x mod y \<in> Standard"
by (simp add: star_mod_def)
-lemma Standard_power: "x \<in> Standard \<Longrightarrow> x ^ n \<in> Standard"
-by (simp add: star_power_def)
-
lemmas Standard_simps [simp] =
Standard_zero Standard_one Standard_number_of
Standard_add Standard_diff Standard_minus
Standard_mult Standard_divide Standard_inverse
Standard_abs Standard_div Standard_mod
- Standard_power
text {* @{term star_of} preserves class operations *}
@@ -650,9 +635,6 @@
lemma star_of_mod: "star_of (x mod y) = star_of x mod star_of y"
by transfer (rule refl)
-lemma star_of_power: "star_of (x ^ n) = star_of x ^ n"
-by transfer (rule refl)
-
lemma star_of_abs: "star_of (abs x) = abs (star_of x)"
by transfer (rule refl)
@@ -717,8 +699,7 @@
lemmas star_of_simps [simp] =
star_of_add star_of_diff star_of_minus
star_of_mult star_of_divide star_of_inverse
- star_of_div star_of_mod
- star_of_power star_of_abs
+ star_of_div star_of_mod star_of_abs
star_of_zero star_of_one star_of_number_of
star_of_less star_of_le star_of_eq
star_of_0_less star_of_0_le star_of_0_eq
@@ -970,25 +951,35 @@
instance star :: (ordered_idom) ordered_idom ..
instance star :: (ordered_field) ordered_field ..
-subsection {* Power classes *}
+
+subsection {* Power *}
+
+instance star :: (recpower) recpower ..
-text {*
- Proving the class axiom @{thm [source] power_Suc} for type
- @{typ "'a star"} is a little tricky, because it quantifies
- over values of type @{typ nat}. The transfer principle does
- not handle quantification over non-star types in general,
- but we can work around this by fixing an arbitrary @{typ nat}
- value, and then applying the transfer principle.
-*}
+lemma star_power_def [transfer_unfold]:
+ "(op ^) \<equiv> \<lambda>x n. ( *f* (\<lambda>x. x ^ n)) x"
+proof (rule eq_reflection, rule ext, rule ext)
+ fix n :: nat
+ show "\<And>x::'a star. x ^ n = ( *f* (\<lambda>x. x ^ n)) x"
+ proof (induct n)
+ case 0
+ have "\<And>x::'a star. ( *f* (\<lambda>x. 1)) x = 1"
+ by transfer simp
+ then show ?case by simp
+ next
+ case (Suc n)
+ have "\<And>x::'a star. x * ( *f* (\<lambda>x\<Colon>'a. x ^ n)) x = ( *f* (\<lambda>x\<Colon>'a. x * x ^ n)) x"
+ by transfer simp
+ with Suc show ?case by simp
+ qed
+qed
-instance star :: (recpower) recpower
-proof
- show "\<And>a::'a star. a ^ 0 = 1"
- by transfer (rule power_0)
-next
- fix n show "\<And>a::'a star. a ^ Suc n = a * a ^ n"
- by transfer (rule power_Suc)
-qed
+lemma Standard_power [simp]: "x \<in> Standard \<Longrightarrow> x ^ n \<in> Standard"
+ by (simp add: star_power_def)
+
+lemma star_of_power [simp]: "star_of (x ^ n) = star_of x ^ n"
+ by transfer (rule refl)
+
subsection {* Number classes *}
--- a/src/HOL/Nat.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Nat.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1164,6 +1164,64 @@
end
+subsection {* Natural operation of natural numbers on functions *}
+
+text {*
+ We use the same logical constant for the power operations on
+ functions and relations, in order to share the same syntax.
+*}
+
+consts compow :: "nat \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
+
+abbreviation compower :: "('a \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'b" (infixr "^^" 80) where
+ "f ^^ n \<equiv> compow n f"
+
+notation (latex output)
+ compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
+
+notation (HTML output)
+ compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
+
+text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
+
+overloading
+ funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
+begin
+
+primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
+ "funpow 0 f = id"
+ | "funpow (Suc n) f = f o funpow n f"
+
+end
+
+text {* for code generation *}
+
+definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
+ funpow_code_def [code post]: "funpow = compow"
+
+lemmas [code inline] = funpow_code_def [symmetric]
+
+lemma [code]:
+ "funpow 0 f = id"
+ "funpow (Suc n) f = f o funpow n f"
+ unfolding funpow_code_def by simp_all
+
+hide (open) const funpow
+
+lemma funpow_add:
+ "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
+ by (induct m) simp_all
+
+lemma funpow_swap1:
+ "f ((f ^^ n) x) = (f ^^ n) (f x)"
+proof -
+ have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
+ also have "\<dots> = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
+ also have "\<dots> = (f ^^ n) (f x)" by simp
+ finally show ?thesis .
+qed
+
+
subsection {* Embedding of the Naturals into any
@{text semiring_1}: @{term of_nat} *}
@@ -1189,7 +1247,7 @@
"of_nat_aux inc 0 i = i"
| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
-lemma of_nat_code [code, code unfold, code inline del]:
+lemma of_nat_code:
"of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
proof (induct n)
case 0 then show ?case by simp
@@ -1201,9 +1259,11 @@
by simp
with Suc show ?case by (simp add: add_commute)
qed
-
+
end
+declare of_nat_code [code, code unfold, code inline del]
+
text{*Class for unital semirings with characteristic zero.
Includes non-ordered rings like the complex numbers.*}
--- a/src/HOL/NatBin.thy Wed Apr 22 11:00:25 2009 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,975 +0,0 @@
-(* Title: HOL/NatBin.thy
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1999 University of Cambridge
-*)
-
-header {* Binary arithmetic for the natural numbers *}
-
-theory NatBin
-imports IntDiv
-uses ("Tools/nat_simprocs.ML")
-begin
-
-text {*
- Arithmetic for naturals is reduced to that for the non-negative integers.
-*}
-
-instantiation nat :: number
-begin
-
-definition
- nat_number_of_def [code inline, code del]: "number_of v = nat (number_of v)"
-
-instance ..
-
-end
-
-lemma [code post]:
- "nat (number_of v) = number_of v"
- unfolding nat_number_of_def ..
-
-abbreviation (xsymbols)
- power2 :: "'a::power => 'a" ("(_\<twosuperior>)" [1000] 999) where
- "x\<twosuperior> == x^2"
-
-notation (latex output)
- power2 ("(_\<twosuperior>)" [1000] 999)
-
-notation (HTML output)
- power2 ("(_\<twosuperior>)" [1000] 999)
-
-
-subsection {* Predicate for negative binary numbers *}
-
-definition neg :: "int \<Rightarrow> bool" where
- "neg Z \<longleftrightarrow> Z < 0"
-
-lemma not_neg_int [simp]: "~ neg (of_nat n)"
-by (simp add: neg_def)
-
-lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
-by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
-
-lemmas neg_eq_less_0 = neg_def
-
-lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
-by (simp add: neg_def linorder_not_less)
-
-text{*To simplify inequalities when Numeral1 can get simplified to 1*}
-
-lemma not_neg_0: "~ neg 0"
-by (simp add: One_int_def neg_def)
-
-lemma not_neg_1: "~ neg 1"
-by (simp add: neg_def linorder_not_less zero_le_one)
-
-lemma neg_nat: "neg z ==> nat z = 0"
-by (simp add: neg_def order_less_imp_le)
-
-lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
-by (simp add: linorder_not_less neg_def)
-
-text {*
- If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
- @{term Numeral0} IS @{term "number_of Pls"}
-*}
-
-lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
- by (simp add: neg_def)
-
-lemma neg_number_of_Min: "neg (number_of Int.Min)"
- by (simp add: neg_def)
-
-lemma neg_number_of_Bit0:
- "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
- by (simp add: neg_def)
-
-lemma neg_number_of_Bit1:
- "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
- by (simp add: neg_def)
-
-lemmas neg_simps [simp] =
- not_neg_0 not_neg_1
- not_neg_number_of_Pls neg_number_of_Min
- neg_number_of_Bit0 neg_number_of_Bit1
-
-
-subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
-
-declare nat_0 [simp] nat_1 [simp]
-
-lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
-by (simp add: nat_number_of_def)
-
-lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
-by (simp add: nat_number_of_def)
-
-lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
-by (simp add: nat_1 nat_number_of_def)
-
-lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
-by (simp add: nat_numeral_1_eq_1)
-
-lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
-apply (unfold nat_number_of_def)
-apply (rule nat_2)
-done
-
-
-subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
-
-lemma int_nat_number_of [simp]:
- "int (number_of v) =
- (if neg (number_of v :: int) then 0
- else (number_of v :: int))"
- unfolding nat_number_of_def number_of_is_id neg_def
- by simp
-
-
-subsubsection{*Successor *}
-
-lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
-apply (rule sym)
-apply (simp add: nat_eq_iff int_Suc)
-done
-
-lemma Suc_nat_number_of_add:
- "Suc (number_of v + n) =
- (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)"
- unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
- by (simp add: Suc_nat_eq_nat_zadd1 add_ac)
-
-lemma Suc_nat_number_of [simp]:
- "Suc (number_of v) =
- (if neg (number_of v :: int) then 1 else number_of (Int.succ v))"
-apply (cut_tac n = 0 in Suc_nat_number_of_add)
-apply (simp cong del: if_weak_cong)
-done
-
-
-subsubsection{*Addition *}
-
-lemma add_nat_number_of [simp]:
- "(number_of v :: nat) + number_of v' =
- (if v < Int.Pls then number_of v'
- else if v' < Int.Pls then number_of v
- else number_of (v + v'))"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by (simp add: nat_add_distrib)
-
-lemma nat_number_of_add_1 [simp]:
- "number_of v + (1::nat) =
- (if v < Int.Pls then 1 else number_of (Int.succ v))"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by (simp add: nat_add_distrib)
-
-lemma nat_1_add_number_of [simp]:
- "(1::nat) + number_of v =
- (if v < Int.Pls then 1 else number_of (Int.succ v))"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by (simp add: nat_add_distrib)
-
-lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
- by (rule int_int_eq [THEN iffD1]) simp
-
-
-subsubsection{*Subtraction *}
-
-lemma diff_nat_eq_if:
- "nat z - nat z' =
- (if neg z' then nat z
- else let d = z-z' in
- if neg d then 0 else nat d)"
-by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
-
-
-lemma diff_nat_number_of [simp]:
- "(number_of v :: nat) - number_of v' =
- (if v' < Int.Pls then number_of v
- else let d = number_of (v + uminus v') in
- if neg d then 0 else nat d)"
- unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
- by auto
-
-lemma nat_number_of_diff_1 [simp]:
- "number_of v - (1::nat) =
- (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by auto
-
-
-subsubsection{*Multiplication *}
-
-lemma mult_nat_number_of [simp]:
- "(number_of v :: nat) * number_of v' =
- (if v < Int.Pls then 0 else number_of (v * v'))"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by (simp add: nat_mult_distrib)
-
-
-subsubsection{*Quotient *}
-
-lemma div_nat_number_of [simp]:
- "(number_of v :: nat) div number_of v' =
- (if neg (number_of v :: int) then 0
- else nat (number_of v div number_of v'))"
- unfolding nat_number_of_def number_of_is_id neg_def
- by (simp add: nat_div_distrib)
-
-lemma one_div_nat_number_of [simp]:
- "Suc 0 div number_of v' = nat (1 div number_of v')"
-by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
-
-
-subsubsection{*Remainder *}
-
-lemma mod_nat_number_of [simp]:
- "(number_of v :: nat) mod number_of v' =
- (if neg (number_of v :: int) then 0
- else if neg (number_of v' :: int) then number_of v
- else nat (number_of v mod number_of v'))"
- unfolding nat_number_of_def number_of_is_id neg_def
- by (simp add: nat_mod_distrib)
-
-lemma one_mod_nat_number_of [simp]:
- "Suc 0 mod number_of v' =
- (if neg (number_of v' :: int) then Suc 0
- else nat (1 mod number_of v'))"
-by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
-
-
-subsubsection{* Divisibility *}
-
-lemmas dvd_eq_mod_eq_0_number_of =
- dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
-
-declare dvd_eq_mod_eq_0_number_of [simp]
-
-ML
-{*
-val nat_number_of_def = thm"nat_number_of_def";
-
-val nat_number_of = thm"nat_number_of";
-val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
-val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
-val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
-val numeral_2_eq_2 = thm"numeral_2_eq_2";
-val nat_div_distrib = thm"nat_div_distrib";
-val nat_mod_distrib = thm"nat_mod_distrib";
-val int_nat_number_of = thm"int_nat_number_of";
-val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
-val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
-val Suc_nat_number_of = thm"Suc_nat_number_of";
-val add_nat_number_of = thm"add_nat_number_of";
-val diff_nat_eq_if = thm"diff_nat_eq_if";
-val diff_nat_number_of = thm"diff_nat_number_of";
-val mult_nat_number_of = thm"mult_nat_number_of";
-val div_nat_number_of = thm"div_nat_number_of";
-val mod_nat_number_of = thm"mod_nat_number_of";
-*}
-
-
-subsection{*Comparisons*}
-
-subsubsection{*Equals (=) *}
-
-lemma eq_nat_nat_iff:
- "[| (0::int) <= z; 0 <= z' |] ==> (nat z = nat z') = (z=z')"
-by (auto elim!: nonneg_eq_int)
-
-lemma eq_nat_number_of [simp]:
- "((number_of v :: nat) = number_of v') =
- (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
- else if neg (number_of v' :: int) then (number_of v :: int) = 0
- else v = v')"
- unfolding nat_number_of_def number_of_is_id neg_def
- by auto
-
-
-subsubsection{*Less-than (<) *}
-
-lemma less_nat_number_of [simp]:
- "(number_of v :: nat) < number_of v' \<longleftrightarrow>
- (if v < v' then Int.Pls < v' else False)"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by auto
-
-
-subsubsection{*Less-than-or-equal *}
-
-lemma le_nat_number_of [simp]:
- "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
- (if v \<le> v' then True else v \<le> Int.Pls)"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by auto
-
-(*Maps #n to n for n = 0, 1, 2*)
-lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
-
-
-subsection{*Powers with Numeric Exponents*}
-
-text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
-We cannot prove general results about the numeral @{term "-1"}, so we have to
-use @{term "- 1"} instead.*}
-
-lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a"
- by (simp add: numeral_2_eq_2 Power.power_Suc)
-
-lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"
- by (simp add: power2_eq_square)
-
-lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"
- by (simp add: power2_eq_square)
-
-lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
- apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
- apply (erule ssubst)
- apply (simp add: power_Suc mult_ac)
- apply (unfold nat_number_of_def)
- apply (subst nat_eq_iff)
- apply simp
-done
-
-text{*Squares of literal numerals will be evaluated.*}
-lemmas power2_eq_square_number_of =
- power2_eq_square [of "number_of w", standard]
-declare power2_eq_square_number_of [simp]
-
-
-lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
- by (simp add: power2_eq_square)
-
-lemma zero_less_power2[simp]:
- "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
- by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
-
-lemma power2_less_0[simp]:
- fixes a :: "'a::{ordered_idom,recpower}"
- shows "~ (a\<twosuperior> < 0)"
-by (force simp add: power2_eq_square mult_less_0_iff)
-
-lemma zero_eq_power2[simp]:
- "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
- by (force simp add: power2_eq_square mult_eq_0_iff)
-
-lemma abs_power2[simp]:
- "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
- by (simp add: power2_eq_square abs_mult abs_mult_self)
-
-lemma power2_abs[simp]:
- "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
- by (simp add: power2_eq_square abs_mult_self)
-
-lemma power2_minus[simp]:
- "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
- by (simp add: power2_eq_square)
-
-lemma power2_le_imp_le:
- fixes x y :: "'a::{ordered_semidom,recpower}"
- shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
-unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
-
-lemma power2_less_imp_less:
- fixes x y :: "'a::{ordered_semidom,recpower}"
- shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
-by (rule power_less_imp_less_base)
-
-lemma power2_eq_imp_eq:
- fixes x y :: "'a::{ordered_semidom,recpower}"
- shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
-unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
-
-lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
-proof (induct n)
- case 0 show ?case by simp
-next
- case (Suc n) then show ?case by (simp add: power_Suc power_add)
-qed
-
-lemma power_minus1_odd: "(- 1) ^ Suc(2*n) = -(1::'a::{comm_ring_1,recpower})"
- by (simp add: power_Suc)
-
-lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
-by (subst mult_commute) (simp add: power_mult)
-
-lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
-by (simp add: power_even_eq)
-
-lemma power_minus_even [simp]:
- "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
-by (simp add: power_minus1_even power_minus [of a])
-
-lemma zero_le_even_power'[simp]:
- "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
-proof (induct "n")
- case 0
- show ?case by (simp add: zero_le_one)
-next
- case (Suc n)
- have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
- by (simp add: mult_ac power_add power2_eq_square)
- thus ?case
- by (simp add: prems zero_le_mult_iff)
-qed
-
-lemma odd_power_less_zero:
- "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
-proof (induct "n")
- case 0
- then show ?case by simp
-next
- case (Suc n)
- have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
- by (simp add: mult_ac power_add power2_eq_square)
- thus ?case
- by (simp del: power_Suc add: prems mult_less_0_iff mult_neg_neg)
-qed
-
-lemma odd_0_le_power_imp_0_le:
- "0 \<le> a ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
-apply (insert odd_power_less_zero [of a n])
-apply (force simp add: linorder_not_less [symmetric])
-done
-
-text{*Simprules for comparisons where common factors can be cancelled.*}
-lemmas zero_compare_simps =
- add_strict_increasing add_strict_increasing2 add_increasing
- zero_le_mult_iff zero_le_divide_iff
- zero_less_mult_iff zero_less_divide_iff
- mult_le_0_iff divide_le_0_iff
- mult_less_0_iff divide_less_0_iff
- zero_le_power2 power2_less_0
-
-subsubsection{*Nat *}
-
-lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
-by (simp add: numerals)
-
-(*Expresses a natural number constant as the Suc of another one.
- NOT suitable for rewriting because n recurs in the condition.*)
-lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
-
-subsubsection{*Arith *}
-
-lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
-by (simp add: numerals)
-
-lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
-by (simp add: numerals)
-
-(* These two can be useful when m = number_of... *)
-
-lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
- unfolding One_nat_def by (cases m) simp_all
-
-lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
- unfolding One_nat_def by (cases m) simp_all
-
-lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
- unfolding One_nat_def by (cases m) simp_all
-
-
-subsection{*Comparisons involving (0::nat) *}
-
-text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
-
-lemma eq_number_of_0 [simp]:
- "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by auto
-
-lemma eq_0_number_of [simp]:
- "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
-by (rule trans [OF eq_sym_conv eq_number_of_0])
-
-lemma less_0_number_of [simp]:
- "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
- unfolding nat_number_of_def number_of_is_id numeral_simps
- by simp
-
-lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
-by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
-
-
-
-subsection{*Comparisons involving @{term Suc} *}
-
-lemma eq_number_of_Suc [simp]:
- "(number_of v = Suc n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then False else nat pv = n)"
-apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
- number_of_pred nat_number_of_def
- split add: split_if)
-apply (rule_tac x = "number_of v" in spec)
-apply (auto simp add: nat_eq_iff)
-done
-
-lemma Suc_eq_number_of [simp]:
- "(Suc n = number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then False else nat pv = n)"
-by (rule trans [OF eq_sym_conv eq_number_of_Suc])
-
-lemma less_number_of_Suc [simp]:
- "(number_of v < Suc n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then True else nat pv < n)"
-apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
- number_of_pred nat_number_of_def
- split add: split_if)
-apply (rule_tac x = "number_of v" in spec)
-apply (auto simp add: nat_less_iff)
-done
-
-lemma less_Suc_number_of [simp]:
- "(Suc n < number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then False else n < nat pv)"
-apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
- number_of_pred nat_number_of_def
- split add: split_if)
-apply (rule_tac x = "number_of v" in spec)
-apply (auto simp add: zless_nat_eq_int_zless)
-done
-
-lemma le_number_of_Suc [simp]:
- "(number_of v <= Suc n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then True else nat pv <= n)"
-by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
-
-lemma le_Suc_number_of [simp]:
- "(Suc n <= number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then False else n <= nat pv)"
-by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
-
-
-lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
-by auto
-
-
-
-subsection{*Max and Min Combined with @{term Suc} *}
-
-lemma max_number_of_Suc [simp]:
- "max (Suc n) (number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then Suc n else Suc(max n (nat pv)))"
-apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
- split add: split_if nat.split)
-apply (rule_tac x = "number_of v" in spec)
-apply auto
-done
-
-lemma max_Suc_number_of [simp]:
- "max (number_of v) (Suc n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then Suc n else Suc(max (nat pv) n))"
-apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
- split add: split_if nat.split)
-apply (rule_tac x = "number_of v" in spec)
-apply auto
-done
-
-lemma min_number_of_Suc [simp]:
- "min (Suc n) (number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then 0 else Suc(min n (nat pv)))"
-apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
- split add: split_if nat.split)
-apply (rule_tac x = "number_of v" in spec)
-apply auto
-done
-
-lemma min_Suc_number_of [simp]:
- "min (number_of v) (Suc n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then 0 else Suc(min (nat pv) n))"
-apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
- split add: split_if nat.split)
-apply (rule_tac x = "number_of v" in spec)
-apply auto
-done
-
-subsection{*Literal arithmetic involving powers*}
-
-lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
-apply (induct "n")
-apply (simp_all (no_asm_simp) add: nat_mult_distrib)
-done
-
-lemma power_nat_number_of:
- "(number_of v :: nat) ^ n =
- (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
-by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
- split add: split_if cong: imp_cong)
-
-
-lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
-declare power_nat_number_of_number_of [simp]
-
-
-
-text{*For arbitrary rings*}
-
-lemma power_number_of_even:
- fixes z :: "'a::{number_ring,recpower}"
- shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
-unfolding Let_def nat_number_of_def number_of_Bit0
-apply (rule_tac x = "number_of w" in spec, clarify)
-apply (case_tac " (0::int) <= x")
-apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
-done
-
-lemma power_number_of_odd:
- fixes z :: "'a::{number_ring,recpower}"
- shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
- then (let w = z ^ (number_of w) in z * w * w) else 1)"
-unfolding Let_def nat_number_of_def number_of_Bit1
-apply (rule_tac x = "number_of w" in spec, auto)
-apply (simp only: nat_add_distrib nat_mult_distrib)
-apply simp
-apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc)
-done
-
-lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
-lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
-
-lemmas power_number_of_even_number_of [simp] =
- power_number_of_even [of "number_of v", standard]
-
-lemmas power_number_of_odd_number_of [simp] =
- power_number_of_odd [of "number_of v", standard]
-
-
-
-ML
-{*
-val numeral_ss = @{simpset} addsimps @{thms numerals};
-
-val nat_bin_arith_setup =
- Lin_Arith.map_data
- (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
- {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
- inj_thms = inj_thms,
- lessD = lessD, neqE = neqE,
- simpset = simpset addsimps @{thms neg_simps} @
- [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}]})
-*}
-
-declaration {* K nat_bin_arith_setup *}
-
-(* Enable arith to deal with div/mod k where k is a numeral: *)
-declare split_div[of _ _ "number_of k", standard, arith_split]
-declare split_mod[of _ _ "number_of k", standard, arith_split]
-
-lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
- by (simp add: number_of_Pls nat_number_of_def)
-
-lemma nat_number_of_Min: "number_of Int.Min = (0::nat)"
- apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
- done
-
-lemma nat_number_of_Bit0:
- "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
- unfolding nat_number_of_def number_of_is_id numeral_simps Let_def
- by auto
-
-lemma nat_number_of_Bit1:
- "number_of (Int.Bit1 w) =
- (if neg (number_of w :: int) then 0
- else let n = number_of w in Suc (n + n))"
- unfolding nat_number_of_def number_of_is_id numeral_simps neg_def Let_def
- by auto
-
-lemmas nat_number =
- nat_number_of_Pls nat_number_of_Min
- nat_number_of_Bit0 nat_number_of_Bit1
-
-lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
- by (simp add: Let_def)
-
-lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
-by (simp add: power_mult power_Suc);
-
-lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
-by (simp add: power_mult power_Suc);
-
-
-subsection{*Literal arithmetic and @{term of_nat}*}
-
-lemma of_nat_double:
- "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
-by (simp only: mult_2 nat_add_distrib of_nat_add)
-
-lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
-by (simp only: nat_number_of_def)
-
-lemma of_nat_number_of_lemma:
- "of_nat (number_of v :: nat) =
- (if 0 \<le> (number_of v :: int)
- then (number_of v :: 'a :: number_ring)
- else 0)"
-by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
-
-lemma of_nat_number_of_eq [simp]:
- "of_nat (number_of v :: nat) =
- (if neg (number_of v :: int) then 0
- else (number_of v :: 'a :: number_ring))"
-by (simp only: of_nat_number_of_lemma neg_def, simp)
-
-
-subsection {*Lemmas for the Combination and Cancellation Simprocs*}
-
-lemma nat_number_of_add_left:
- "number_of v + (number_of v' + (k::nat)) =
- (if neg (number_of v :: int) then number_of v' + k
- else if neg (number_of v' :: int) then number_of v + k
- else number_of (v + v') + k)"
- unfolding nat_number_of_def number_of_is_id neg_def
- by auto
-
-lemma nat_number_of_mult_left:
- "number_of v * (number_of v' * (k::nat)) =
- (if v < Int.Pls then 0
- else number_of (v * v') * k)"
-by simp
-
-
-subsubsection{*For @{text combine_numerals}*}
-
-lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
-by (simp add: add_mult_distrib)
-
-
-subsubsection{*For @{text cancel_numerals}*}
-
-lemma nat_diff_add_eq1:
- "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
-by (simp split add: nat_diff_split add: add_mult_distrib)
-
-lemma nat_diff_add_eq2:
- "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
-by (simp split add: nat_diff_split add: add_mult_distrib)
-
-lemma nat_eq_add_iff1:
- "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_eq_add_iff2:
- "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_less_add_iff1:
- "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_less_add_iff2:
- "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_le_add_iff1:
- "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_le_add_iff2:
- "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-
-subsubsection{*For @{text cancel_numeral_factors} *}
-
-lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
-by auto
-
-lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
-by auto
-
-lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
-by auto
-
-lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
-by auto
-
-lemma nat_mult_dvd_cancel_disj[simp]:
- "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
-by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
-
-lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
-by(auto)
-
-
-subsubsection{*For @{text cancel_factor} *}
-
-lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
-by auto
-
-lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
-by auto
-
-lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
-by auto
-
-lemma nat_mult_div_cancel_disj[simp]:
- "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
-by (simp add: nat_mult_div_cancel1)
-
-
-subsection {* Simprocs for the Naturals *}
-
-use "Tools/nat_simprocs.ML"
-declaration {* K nat_simprocs_setup *}
-
-subsubsection{*For simplifying @{term "Suc m - K"} and @{term "K - Suc m"}*}
-
-text{*Where K above is a literal*}
-
-lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
-by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
-
-text {*Now just instantiating @{text n} to @{text "number_of v"} does
- the right simplification, but with some redundant inequality
- tests.*}
-lemma neg_number_of_pred_iff_0:
- "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
-apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
-apply (simp only: less_Suc_eq_le le_0_eq)
-apply (subst less_number_of_Suc, simp)
-done
-
-text{*No longer required as a simprule because of the @{text inverse_fold}
- simproc*}
-lemma Suc_diff_number_of:
- "Int.Pls < v ==>
- Suc m - (number_of v) = m - (number_of (Int.pred v))"
-apply (subst Suc_diff_eq_diff_pred)
-apply simp
-apply (simp del: nat_numeral_1_eq_1)
-apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
- neg_number_of_pred_iff_0)
-done
-
-lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
-by (simp add: numerals split add: nat_diff_split)
-
-
-subsubsection{*For @{term nat_case} and @{term nat_rec}*}
-
-lemma nat_case_number_of [simp]:
- "nat_case a f (number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then a else f (nat pv))"
-by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
-
-lemma nat_case_add_eq_if [simp]:
- "nat_case a f ((number_of v) + n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then nat_case a f n else f (nat pv + n))"
-apply (subst add_eq_if)
-apply (simp split add: nat.split
- del: nat_numeral_1_eq_1
- add: nat_numeral_1_eq_1 [symmetric]
- numeral_1_eq_Suc_0 [symmetric]
- neg_number_of_pred_iff_0)
-done
-
-lemma nat_rec_number_of [simp]:
- "nat_rec a f (number_of v) =
- (let pv = number_of (Int.pred v) in
- if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
-apply (case_tac " (number_of v) ::nat")
-apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
-apply (simp split add: split_if_asm)
-done
-
-lemma nat_rec_add_eq_if [simp]:
- "nat_rec a f (number_of v + n) =
- (let pv = number_of (Int.pred v) in
- if neg pv then nat_rec a f n
- else f (nat pv + n) (nat_rec a f (nat pv + n)))"
-apply (subst add_eq_if)
-apply (simp split add: nat.split
- del: nat_numeral_1_eq_1
- add: nat_numeral_1_eq_1 [symmetric]
- numeral_1_eq_Suc_0 [symmetric]
- neg_number_of_pred_iff_0)
-done
-
-
-subsubsection{*Various Other Lemmas*}
-
-text {*Evens and Odds, for Mutilated Chess Board*}
-
-text{*Lemmas for specialist use, NOT as default simprules*}
-lemma nat_mult_2: "2 * z = (z+z::nat)"
-proof -
- have "2*z = (1 + 1)*z" by simp
- also have "... = z+z" by (simp add: left_distrib)
- finally show ?thesis .
-qed
-
-lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
-by (subst mult_commute, rule nat_mult_2)
-
-text{*Case analysis on @{term "n<2"}*}
-lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
-by arith
-
-lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
-by arith
-
-lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
-by (simp add: nat_mult_2 [symmetric])
-
-lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
-apply (subgoal_tac "m mod 2 < 2")
-apply (erule less_2_cases [THEN disjE])
-apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
-done
-
-lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
-apply (subgoal_tac "m mod 2 < 2")
-apply (force simp del: mod_less_divisor, simp)
-done
-
-text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
-
-lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
-by simp
-
-lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
-by simp
-
-text{*Can be used to eliminate long strings of Sucs, but not by default*}
-lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
-by simp
-
-
-text{*These lemmas collapse some needless occurrences of Suc:
- at least three Sucs, since two and fewer are rewritten back to Suc again!
- We already have some rules to simplify operands smaller than 3.*}
-
-lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
-by (simp add: Suc3_eq_add_3)
-
-lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
-by (simp add: Suc3_eq_add_3)
-
-lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
-by (simp add: Suc3_eq_add_3)
-
-lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
-by (simp add: Suc3_eq_add_3)
-
-lemmas Suc_div_eq_add3_div_number_of =
- Suc_div_eq_add3_div [of _ "number_of v", standard]
-declare Suc_div_eq_add3_div_number_of [simp]
-
-lemmas Suc_mod_eq_add3_mod_number_of =
- Suc_mod_eq_add3_mod [of _ "number_of v", standard]
-declare Suc_mod_eq_add3_mod_number_of [simp]
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Nat_Numeral.thy Mon Apr 27 07:26:17 2009 -0700
@@ -0,0 +1,980 @@
+(* Title: HOL/Nat_Numeral.thy
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1999 University of Cambridge
+*)
+
+header {* Binary numerals for the natural numbers *}
+
+theory Nat_Numeral
+imports IntDiv
+uses ("Tools/nat_simprocs.ML")
+begin
+
+text {*
+ Arithmetic for naturals is reduced to that for the non-negative integers.
+*}
+
+instantiation nat :: number
+begin
+
+definition
+ nat_number_of_def [code inline, code del]: "number_of v = nat (number_of v)"
+
+instance ..
+
+end
+
+lemma [code post]:
+ "nat (number_of v) = number_of v"
+ unfolding nat_number_of_def ..
+
+context recpower
+begin
+
+abbreviation (xsymbols)
+ power2 :: "'a \<Rightarrow> 'a" ("(_\<twosuperior>)" [1000] 999) where
+ "x\<twosuperior> \<equiv> x ^ 2"
+
+notation (latex output)
+ power2 ("(_\<twosuperior>)" [1000] 999)
+
+notation (HTML output)
+ power2 ("(_\<twosuperior>)" [1000] 999)
+
+end
+
+
+subsection {* Predicate for negative binary numbers *}
+
+definition neg :: "int \<Rightarrow> bool" where
+ "neg Z \<longleftrightarrow> Z < 0"
+
+lemma not_neg_int [simp]: "~ neg (of_nat n)"
+by (simp add: neg_def)
+
+lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
+by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
+
+lemmas neg_eq_less_0 = neg_def
+
+lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
+by (simp add: neg_def linorder_not_less)
+
+text{*To simplify inequalities when Numeral1 can get simplified to 1*}
+
+lemma not_neg_0: "~ neg 0"
+by (simp add: One_int_def neg_def)
+
+lemma not_neg_1: "~ neg 1"
+by (simp add: neg_def linorder_not_less zero_le_one)
+
+lemma neg_nat: "neg z ==> nat z = 0"
+by (simp add: neg_def order_less_imp_le)
+
+lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
+by (simp add: linorder_not_less neg_def)
+
+text {*
+ If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
+ @{term Numeral0} IS @{term "number_of Pls"}
+*}
+
+lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
+ by (simp add: neg_def)
+
+lemma neg_number_of_Min: "neg (number_of Int.Min)"
+ by (simp add: neg_def)
+
+lemma neg_number_of_Bit0:
+ "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
+ by (simp add: neg_def)
+
+lemma neg_number_of_Bit1:
+ "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
+ by (simp add: neg_def)
+
+lemmas neg_simps [simp] =
+ not_neg_0 not_neg_1
+ not_neg_number_of_Pls neg_number_of_Min
+ neg_number_of_Bit0 neg_number_of_Bit1
+
+
+subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}
+
+declare nat_0 [simp] nat_1 [simp]
+
+lemma nat_number_of [simp]: "nat (number_of w) = number_of w"
+by (simp add: nat_number_of_def)
+
+lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"
+by (simp add: nat_number_of_def)
+
+lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
+by (simp add: nat_1 nat_number_of_def)
+
+lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
+by (simp add: nat_numeral_1_eq_1)
+
+lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
+apply (unfold nat_number_of_def)
+apply (rule nat_2)
+done
+
+
+subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}
+
+lemma int_nat_number_of [simp]:
+ "int (number_of v) =
+ (if neg (number_of v :: int) then 0
+ else (number_of v :: int))"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by simp
+
+
+subsubsection{*Successor *}
+
+lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
+apply (rule sym)
+apply (simp add: nat_eq_iff int_Suc)
+done
+
+lemma Suc_nat_number_of_add:
+ "Suc (number_of v + n) =
+ (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)"
+ unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
+ by (simp add: Suc_nat_eq_nat_zadd1 add_ac)
+
+lemma Suc_nat_number_of [simp]:
+ "Suc (number_of v) =
+ (if neg (number_of v :: int) then 1 else number_of (Int.succ v))"
+apply (cut_tac n = 0 in Suc_nat_number_of_add)
+apply (simp cong del: if_weak_cong)
+done
+
+
+subsubsection{*Addition *}
+
+lemma add_nat_number_of [simp]:
+ "(number_of v :: nat) + number_of v' =
+ (if v < Int.Pls then number_of v'
+ else if v' < Int.Pls then number_of v
+ else number_of (v + v'))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by (simp add: nat_add_distrib)
+
+lemma nat_number_of_add_1 [simp]:
+ "number_of v + (1::nat) =
+ (if v < Int.Pls then 1 else number_of (Int.succ v))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by (simp add: nat_add_distrib)
+
+lemma nat_1_add_number_of [simp]:
+ "(1::nat) + number_of v =
+ (if v < Int.Pls then 1 else number_of (Int.succ v))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by (simp add: nat_add_distrib)
+
+lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
+ by (rule int_int_eq [THEN iffD1]) simp
+
+
+subsubsection{*Subtraction *}
+
+lemma diff_nat_eq_if:
+ "nat z - nat z' =
+ (if neg z' then nat z
+ else let d = z-z' in
+ if neg d then 0 else nat d)"
+by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)
+
+
+lemma diff_nat_number_of [simp]:
+ "(number_of v :: nat) - number_of v' =
+ (if v' < Int.Pls then number_of v
+ else let d = number_of (v + uminus v') in
+ if neg d then 0 else nat d)"
+ unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
+ by auto
+
+lemma nat_number_of_diff_1 [simp]:
+ "number_of v - (1::nat) =
+ (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by auto
+
+
+subsubsection{*Multiplication *}
+
+lemma mult_nat_number_of [simp]:
+ "(number_of v :: nat) * number_of v' =
+ (if v < Int.Pls then 0 else number_of (v * v'))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by (simp add: nat_mult_distrib)
+
+
+subsubsection{*Quotient *}
+
+lemma div_nat_number_of [simp]:
+ "(number_of v :: nat) div number_of v' =
+ (if neg (number_of v :: int) then 0
+ else nat (number_of v div number_of v'))"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by (simp add: nat_div_distrib)
+
+lemma one_div_nat_number_of [simp]:
+ "Suc 0 div number_of v' = nat (1 div number_of v')"
+by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
+
+
+subsubsection{*Remainder *}
+
+lemma mod_nat_number_of [simp]:
+ "(number_of v :: nat) mod number_of v' =
+ (if neg (number_of v :: int) then 0
+ else if neg (number_of v' :: int) then number_of v
+ else nat (number_of v mod number_of v'))"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by (simp add: nat_mod_distrib)
+
+lemma one_mod_nat_number_of [simp]:
+ "Suc 0 mod number_of v' =
+ (if neg (number_of v' :: int) then Suc 0
+ else nat (1 mod number_of v'))"
+by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])
+
+
+subsubsection{* Divisibility *}
+
+lemmas dvd_eq_mod_eq_0_number_of =
+ dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
+
+declare dvd_eq_mod_eq_0_number_of [simp]
+
+ML
+{*
+val nat_number_of_def = thm"nat_number_of_def";
+
+val nat_number_of = thm"nat_number_of";
+val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";
+val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";
+val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";
+val numeral_2_eq_2 = thm"numeral_2_eq_2";
+val nat_div_distrib = thm"nat_div_distrib";
+val nat_mod_distrib = thm"nat_mod_distrib";
+val int_nat_number_of = thm"int_nat_number_of";
+val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";
+val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";
+val Suc_nat_number_of = thm"Suc_nat_number_of";
+val add_nat_number_of = thm"add_nat_number_of";
+val diff_nat_eq_if = thm"diff_nat_eq_if";
+val diff_nat_number_of = thm"diff_nat_number_of";
+val mult_nat_number_of = thm"mult_nat_number_of";
+val div_nat_number_of = thm"div_nat_number_of";
+val mod_nat_number_of = thm"mod_nat_number_of";
+*}
+
+
+subsection{*Comparisons*}
+
+subsubsection{*Equals (=) *}
+
+lemma eq_nat_nat_iff:
+ "[| (0::int) <= z; 0 <= z' |] ==> (nat z = nat z') = (z=z')"
+by (auto elim!: nonneg_eq_int)
+
+lemma eq_nat_number_of [simp]:
+ "((number_of v :: nat) = number_of v') =
+ (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
+ else if neg (number_of v' :: int) then (number_of v :: int) = 0
+ else v = v')"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by auto
+
+
+subsubsection{*Less-than (<) *}
+
+lemma less_nat_number_of [simp]:
+ "(number_of v :: nat) < number_of v' \<longleftrightarrow>
+ (if v < v' then Int.Pls < v' else False)"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by auto
+
+
+subsubsection{*Less-than-or-equal *}
+
+lemma le_nat_number_of [simp]:
+ "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
+ (if v \<le> v' then True else v \<le> Int.Pls)"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by auto
+
+(*Maps #n to n for n = 0, 1, 2*)
+lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2
+
+
+subsection{*Powers with Numeric Exponents*}
+
+text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.
+We cannot prove general results about the numeral @{term "-1"}, so we have to
+use @{term "- 1"} instead.*}
+
+lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a"
+ by (simp add: numeral_2_eq_2 Power.power_Suc)
+
+lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"
+ by (simp add: power2_eq_square)
+
+lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"
+ by (simp add: power2_eq_square)
+
+lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"
+ apply (subgoal_tac "3 = Suc (Suc (Suc 0))")
+ apply (erule ssubst)
+ apply (simp add: power_Suc mult_ac)
+ apply (unfold nat_number_of_def)
+ apply (subst nat_eq_iff)
+ apply simp
+done
+
+text{*Squares of literal numerals will be evaluated.*}
+lemmas power2_eq_square_number_of =
+ power2_eq_square [of "number_of w", standard]
+declare power2_eq_square_number_of [simp]
+
+
+lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})"
+ by (simp add: power2_eq_square)
+
+lemma zero_less_power2[simp]:
+ "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))"
+ by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
+
+lemma power2_less_0[simp]:
+ fixes a :: "'a::{ordered_idom,recpower}"
+ shows "~ (a\<twosuperior> < 0)"
+by (force simp add: power2_eq_square mult_less_0_iff)
+
+lemma zero_eq_power2[simp]:
+ "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))"
+ by (force simp add: power2_eq_square mult_eq_0_iff)
+
+lemma abs_power2[simp]:
+ "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})"
+ by (simp add: power2_eq_square abs_mult abs_mult_self)
+
+lemma power2_abs[simp]:
+ "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})"
+ by (simp add: power2_eq_square abs_mult_self)
+
+lemma power2_minus[simp]:
+ "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})"
+ by (simp add: power2_eq_square)
+
+lemma power2_le_imp_le:
+ fixes x y :: "'a::{ordered_semidom,recpower}"
+ shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"
+unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
+
+lemma power2_less_imp_less:
+ fixes x y :: "'a::{ordered_semidom,recpower}"
+ shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"
+by (rule power_less_imp_less_base)
+
+lemma power2_eq_imp_eq:
+ fixes x y :: "'a::{ordered_semidom,recpower}"
+ shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"
+unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)
+
+lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"
+proof (induct n)
+ case 0 show ?case by simp
+next
+ case (Suc n) then show ?case by (simp add: power_Suc power_add)
+qed
+
+lemma power_minus1_odd: "(- 1) ^ Suc(2*n) = -(1::'a::{comm_ring_1,recpower})"
+ by (simp add: power_Suc)
+
+lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2"
+ by (subst mult_commute) (simp add: power_mult)
+
+lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"
+ by (simp add: power_even_eq)
+
+lemma power_minus_even [simp]:
+ "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)"
+ by (simp add: power_minus [of a])
+
+lemma zero_le_even_power'[simp]:
+ "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)"
+proof (induct "n")
+ case 0
+ show ?case by (simp add: zero_le_one)
+next
+ case (Suc n)
+ have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
+ by (simp add: mult_ac power_add power2_eq_square)
+ thus ?case
+ by (simp add: prems zero_le_mult_iff)
+qed
+
+lemma odd_power_less_zero:
+ "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0"
+proof (induct "n")
+ case 0
+ then show ?case by simp
+next
+ case (Suc n)
+ have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
+ by (simp add: mult_ac power_add power2_eq_square)
+ thus ?case
+ by (simp del: power_Suc add: prems mult_less_0_iff mult_neg_neg)
+qed
+
+lemma odd_0_le_power_imp_0_le:
+ "0 \<le> a ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})"
+apply (insert odd_power_less_zero [of a n])
+apply (force simp add: linorder_not_less [symmetric])
+done
+
+text{*Simprules for comparisons where common factors can be cancelled.*}
+lemmas zero_compare_simps =
+ add_strict_increasing add_strict_increasing2 add_increasing
+ zero_le_mult_iff zero_le_divide_iff
+ zero_less_mult_iff zero_less_divide_iff
+ mult_le_0_iff divide_le_0_iff
+ mult_less_0_iff divide_less_0_iff
+ zero_le_power2 power2_less_0
+
+subsubsection{*Nat *}
+
+lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
+by (simp add: numerals)
+
+(*Expresses a natural number constant as the Suc of another one.
+ NOT suitable for rewriting because n recurs in the condition.*)
+lemmas expand_Suc = Suc_pred' [of "number_of v", standard]
+
+subsubsection{*Arith *}
+
+lemma Suc_eq_add_numeral_1: "Suc n = n + 1"
+by (simp add: numerals)
+
+lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"
+by (simp add: numerals)
+
+(* These two can be useful when m = number_of... *)
+
+lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
+ unfolding One_nat_def by (cases m) simp_all
+
+lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
+ unfolding One_nat_def by (cases m) simp_all
+
+lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
+ unfolding One_nat_def by (cases m) simp_all
+
+
+subsection{*Comparisons involving (0::nat) *}
+
+text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}
+
+lemma eq_number_of_0 [simp]:
+ "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by auto
+
+lemma eq_0_number_of [simp]:
+ "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
+by (rule trans [OF eq_sym_conv eq_number_of_0])
+
+lemma less_0_number_of [simp]:
+ "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
+ unfolding nat_number_of_def number_of_is_id numeral_simps
+ by simp
+
+lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"
+by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
+
+
+
+subsection{*Comparisons involving @{term Suc} *}
+
+lemma eq_number_of_Suc [simp]:
+ "(number_of v = Suc n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then False else nat pv = n)"
+apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
+ number_of_pred nat_number_of_def
+ split add: split_if)
+apply (rule_tac x = "number_of v" in spec)
+apply (auto simp add: nat_eq_iff)
+done
+
+lemma Suc_eq_number_of [simp]:
+ "(Suc n = number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then False else nat pv = n)"
+by (rule trans [OF eq_sym_conv eq_number_of_Suc])
+
+lemma less_number_of_Suc [simp]:
+ "(number_of v < Suc n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then True else nat pv < n)"
+apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
+ number_of_pred nat_number_of_def
+ split add: split_if)
+apply (rule_tac x = "number_of v" in spec)
+apply (auto simp add: nat_less_iff)
+done
+
+lemma less_Suc_number_of [simp]:
+ "(Suc n < number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then False else n < nat pv)"
+apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less
+ number_of_pred nat_number_of_def
+ split add: split_if)
+apply (rule_tac x = "number_of v" in spec)
+apply (auto simp add: zless_nat_eq_int_zless)
+done
+
+lemma le_number_of_Suc [simp]:
+ "(number_of v <= Suc n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then True else nat pv <= n)"
+by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])
+
+lemma le_Suc_number_of [simp]:
+ "(Suc n <= number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then False else n <= nat pv)"
+by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])
+
+
+lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min"
+by auto
+
+
+
+subsection{*Max and Min Combined with @{term Suc} *}
+
+lemma max_number_of_Suc [simp]:
+ "max (Suc n) (number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then Suc n else Suc(max n (nat pv)))"
+apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
+ split add: split_if nat.split)
+apply (rule_tac x = "number_of v" in spec)
+apply auto
+done
+
+lemma max_Suc_number_of [simp]:
+ "max (number_of v) (Suc n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then Suc n else Suc(max (nat pv) n))"
+apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
+ split add: split_if nat.split)
+apply (rule_tac x = "number_of v" in spec)
+apply auto
+done
+
+lemma min_number_of_Suc [simp]:
+ "min (Suc n) (number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then 0 else Suc(min n (nat pv)))"
+apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
+ split add: split_if nat.split)
+apply (rule_tac x = "number_of v" in spec)
+apply auto
+done
+
+lemma min_Suc_number_of [simp]:
+ "min (number_of v) (Suc n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then 0 else Suc(min (nat pv) n))"
+apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def
+ split add: split_if nat.split)
+apply (rule_tac x = "number_of v" in spec)
+apply auto
+done
+
+subsection{*Literal arithmetic involving powers*}
+
+lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"
+apply (induct "n")
+apply (simp_all (no_asm_simp) add: nat_mult_distrib)
+done
+
+lemma power_nat_number_of:
+ "(number_of v :: nat) ^ n =
+ (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"
+by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq
+ split add: split_if cong: imp_cong)
+
+
+lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]
+declare power_nat_number_of_number_of [simp]
+
+
+
+text{*For arbitrary rings*}
+
+lemma power_number_of_even:
+ fixes z :: "'a::{number_ring,recpower}"
+ shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
+unfolding Let_def nat_number_of_def number_of_Bit0
+apply (rule_tac x = "number_of w" in spec, clarify)
+apply (case_tac " (0::int) <= x")
+apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)
+done
+
+lemma power_number_of_odd:
+ fixes z :: "'a::{number_ring,recpower}"
+ shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
+ then (let w = z ^ (number_of w) in z * w * w) else 1)"
+unfolding Let_def nat_number_of_def number_of_Bit1
+apply (rule_tac x = "number_of w" in spec, auto)
+apply (simp only: nat_add_distrib nat_mult_distrib)
+apply simp
+apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc)
+done
+
+lemmas zpower_number_of_even = power_number_of_even [where 'a=int]
+lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]
+
+lemmas power_number_of_even_number_of [simp] =
+ power_number_of_even [of "number_of v", standard]
+
+lemmas power_number_of_odd_number_of [simp] =
+ power_number_of_odd [of "number_of v", standard]
+
+
+
+ML
+{*
+val numeral_ss = @{simpset} addsimps @{thms numerals};
+
+val nat_bin_arith_setup =
+ Lin_Arith.map_data
+ (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
+ {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,
+ inj_thms = inj_thms,
+ lessD = lessD, neqE = neqE,
+ simpset = simpset addsimps @{thms neg_simps} @
+ [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}]})
+*}
+
+declaration {* K nat_bin_arith_setup *}
+
+(* Enable arith to deal with div/mod k where k is a numeral: *)
+declare split_div[of _ _ "number_of k", standard, arith_split]
+declare split_mod[of _ _ "number_of k", standard, arith_split]
+
+lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
+ by (simp add: number_of_Pls nat_number_of_def)
+
+lemma nat_number_of_Min: "number_of Int.Min = (0::nat)"
+ apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
+ done
+
+lemma nat_number_of_Bit0:
+ "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
+ unfolding nat_number_of_def number_of_is_id numeral_simps Let_def
+ by auto
+
+lemma nat_number_of_Bit1:
+ "number_of (Int.Bit1 w) =
+ (if neg (number_of w :: int) then 0
+ else let n = number_of w in Suc (n + n))"
+ unfolding nat_number_of_def number_of_is_id numeral_simps neg_def Let_def
+ by auto
+
+lemmas nat_number =
+ nat_number_of_Pls nat_number_of_Min
+ nat_number_of_Bit0 nat_number_of_Bit1
+
+lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
+ by (simp add: Let_def)
+
+lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})"
+by (simp add: power_mult power_Suc);
+
+lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})"
+by (simp add: power_mult power_Suc);
+
+
+subsection{*Literal arithmetic and @{term of_nat}*}
+
+lemma of_nat_double:
+ "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"
+by (simp only: mult_2 nat_add_distrib of_nat_add)
+
+lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"
+by (simp only: nat_number_of_def)
+
+lemma of_nat_number_of_lemma:
+ "of_nat (number_of v :: nat) =
+ (if 0 \<le> (number_of v :: int)
+ then (number_of v :: 'a :: number_ring)
+ else 0)"
+by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);
+
+lemma of_nat_number_of_eq [simp]:
+ "of_nat (number_of v :: nat) =
+ (if neg (number_of v :: int) then 0
+ else (number_of v :: 'a :: number_ring))"
+by (simp only: of_nat_number_of_lemma neg_def, simp)
+
+
+subsection {*Lemmas for the Combination and Cancellation Simprocs*}
+
+lemma nat_number_of_add_left:
+ "number_of v + (number_of v' + (k::nat)) =
+ (if neg (number_of v :: int) then number_of v' + k
+ else if neg (number_of v' :: int) then number_of v + k
+ else number_of (v + v') + k)"
+ unfolding nat_number_of_def number_of_is_id neg_def
+ by auto
+
+lemma nat_number_of_mult_left:
+ "number_of v * (number_of v' * (k::nat)) =
+ (if v < Int.Pls then 0
+ else number_of (v * v') * k)"
+by simp
+
+
+subsubsection{*For @{text combine_numerals}*}
+
+lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
+by (simp add: add_mult_distrib)
+
+
+subsubsection{*For @{text cancel_numerals}*}
+
+lemma nat_diff_add_eq1:
+ "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
+by (simp split add: nat_diff_split add: add_mult_distrib)
+
+lemma nat_diff_add_eq2:
+ "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
+by (simp split add: nat_diff_split add: add_mult_distrib)
+
+lemma nat_eq_add_iff1:
+ "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+lemma nat_eq_add_iff2:
+ "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+lemma nat_less_add_iff1:
+ "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+lemma nat_less_add_iff2:
+ "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+lemma nat_le_add_iff1:
+ "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+lemma nat_le_add_iff2:
+ "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
+by (auto split add: nat_diff_split simp add: add_mult_distrib)
+
+
+subsubsection{*For @{text cancel_numeral_factors} *}
+
+lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
+by auto
+
+lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
+by auto
+
+lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
+by auto
+
+lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
+by auto
+
+lemma nat_mult_dvd_cancel_disj[simp]:
+ "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
+by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
+
+lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
+by(auto)
+
+
+subsubsection{*For @{text cancel_factor} *}
+
+lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
+by auto
+
+lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
+by auto
+
+lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
+by auto
+
+lemma nat_mult_div_cancel_disj[simp]:
+ "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
+by (simp add: nat_mult_div_cancel1)
+
+
+subsection {* Simprocs for the Naturals *}
+
+use "Tools/nat_simprocs.ML"
+declaration {* K nat_simprocs_setup *}
+
+subsubsection{*For simplifying @{term "Suc m - K"} and @{term "K - Suc m"}*}
+
+text{*Where K above is a literal*}
+
+lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
+by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split)
+
+text {*Now just instantiating @{text n} to @{text "number_of v"} does
+ the right simplification, but with some redundant inequality
+ tests.*}
+lemma neg_number_of_pred_iff_0:
+ "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
+apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
+apply (simp only: less_Suc_eq_le le_0_eq)
+apply (subst less_number_of_Suc, simp)
+done
+
+text{*No longer required as a simprule because of the @{text inverse_fold}
+ simproc*}
+lemma Suc_diff_number_of:
+ "Int.Pls < v ==>
+ Suc m - (number_of v) = m - (number_of (Int.pred v))"
+apply (subst Suc_diff_eq_diff_pred)
+apply simp
+apply (simp del: nat_numeral_1_eq_1)
+apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
+ neg_number_of_pred_iff_0)
+done
+
+lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
+by (simp add: numerals split add: nat_diff_split)
+
+
+subsubsection{*For @{term nat_case} and @{term nat_rec}*}
+
+lemma nat_case_number_of [simp]:
+ "nat_case a f (number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then a else f (nat pv))"
+by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
+
+lemma nat_case_add_eq_if [simp]:
+ "nat_case a f ((number_of v) + n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then nat_case a f n else f (nat pv + n))"
+apply (subst add_eq_if)
+apply (simp split add: nat.split
+ del: nat_numeral_1_eq_1
+ add: nat_numeral_1_eq_1 [symmetric]
+ numeral_1_eq_Suc_0 [symmetric]
+ neg_number_of_pred_iff_0)
+done
+
+lemma nat_rec_number_of [simp]:
+ "nat_rec a f (number_of v) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
+apply (case_tac " (number_of v) ::nat")
+apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
+apply (simp split add: split_if_asm)
+done
+
+lemma nat_rec_add_eq_if [simp]:
+ "nat_rec a f (number_of v + n) =
+ (let pv = number_of (Int.pred v) in
+ if neg pv then nat_rec a f n
+ else f (nat pv + n) (nat_rec a f (nat pv + n)))"
+apply (subst add_eq_if)
+apply (simp split add: nat.split
+ del: nat_numeral_1_eq_1
+ add: nat_numeral_1_eq_1 [symmetric]
+ numeral_1_eq_Suc_0 [symmetric]
+ neg_number_of_pred_iff_0)
+done
+
+
+subsubsection{*Various Other Lemmas*}
+
+text {*Evens and Odds, for Mutilated Chess Board*}
+
+text{*Lemmas for specialist use, NOT as default simprules*}
+lemma nat_mult_2: "2 * z = (z+z::nat)"
+proof -
+ have "2*z = (1 + 1)*z" by simp
+ also have "... = z+z" by (simp add: left_distrib)
+ finally show ?thesis .
+qed
+
+lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
+by (subst mult_commute, rule nat_mult_2)
+
+text{*Case analysis on @{term "n<2"}*}
+lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
+by arith
+
+lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)"
+by arith
+
+lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
+by (simp add: nat_mult_2 [symmetric])
+
+lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
+apply (subgoal_tac "m mod 2 < 2")
+apply (erule less_2_cases [THEN disjE])
+apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
+done
+
+lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
+apply (subgoal_tac "m mod 2 < 2")
+apply (force simp del: mod_less_divisor, simp)
+done
+
+text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
+
+lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
+by simp
+
+lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
+by simp
+
+text{*Can be used to eliminate long strings of Sucs, but not by default*}
+lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
+by simp
+
+
+text{*These lemmas collapse some needless occurrences of Suc:
+ at least three Sucs, since two and fewer are rewritten back to Suc again!
+ We already have some rules to simplify operands smaller than 3.*}
+
+lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
+by (simp add: Suc3_eq_add_3)
+
+lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
+by (simp add: Suc3_eq_add_3)
+
+lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
+by (simp add: Suc3_eq_add_3)
+
+lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
+by (simp add: Suc3_eq_add_3)
+
+lemmas Suc_div_eq_add3_div_number_of =
+ Suc_div_eq_add3_div [of _ "number_of v", standard]
+declare Suc_div_eq_add3_div_number_of [simp]
+
+lemmas Suc_mod_eq_add3_mod_number_of =
+ Suc_mod_eq_add3_mod [of _ "number_of v", standard]
+declare Suc_mod_eq_add3_mod_number_of [simp]
+
+end
\ No newline at end of file
--- a/src/HOL/Nominal/Examples/Fsub.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Nominal/Examples/Fsub.thy Mon Apr 27 07:26:17 2009 -0700
@@ -245,7 +245,7 @@
apply (simp add: dj_perm_forget[OF dj_tyvrs_vrs])
done
-lemma ty_vrs_fresh[fresh]:
+lemma ty_vrs_fresh:
fixes x::"vrs"
and T::"ty"
shows "x \<sharp> T"
@@ -422,7 +422,7 @@
by (nominal_induct T avoiding: X T' rule: ty.strong_induct)
(perm_simp add: fresh_left)+
-lemma type_subst_fresh[fresh]:
+lemma type_subst_fresh:
fixes X::"tyvrs"
assumes "X \<sharp> T" and "X \<sharp> P"
shows "X \<sharp> T[Y \<mapsto> P]\<^sub>\<tau>"
@@ -430,7 +430,7 @@
by (nominal_induct T avoiding: X Y P rule:ty.strong_induct)
(auto simp add: abs_fresh)
-lemma fresh_type_subst_fresh[fresh]:
+lemma fresh_type_subst_fresh:
assumes "X\<sharp>T'"
shows "X\<sharp>T[X \<mapsto> T']\<^sub>\<tau>"
using assms
@@ -458,18 +458,19 @@
| "(VarB X U)[Y \<mapsto> T]\<^sub>b = VarB X (U[Y \<mapsto> T]\<^sub>\<tau>)"
by auto
-lemma binding_subst_fresh[fresh]:
+lemma binding_subst_fresh:
fixes X::"tyvrs"
assumes "X \<sharp> a"
and "X \<sharp> P"
shows "X \<sharp> a[Y \<mapsto> P]\<^sub>b"
using assms
-by (nominal_induct a rule:binding.strong_induct)
- (auto simp add: freshs)
+by (nominal_induct a rule: binding.strong_induct)
+ (auto simp add: type_subst_fresh)
-lemma binding_subst_identity: "X \<sharp> B \<Longrightarrow> B[X \<mapsto> U]\<^sub>b = B"
- by (induct B rule: binding.induct)
- (simp_all add: fresh_atm type_subst_identity)
+lemma binding_subst_identity:
+ shows "X \<sharp> B \<Longrightarrow> B[X \<mapsto> U]\<^sub>b = B"
+by (induct B rule: binding.induct)
+ (simp_all add: fresh_atm type_subst_identity)
consts
subst_tyc :: "env \<Rightarrow> tyvrs \<Rightarrow> ty \<Rightarrow> env" ("_[_ \<mapsto> _]\<^sub>e" [100,100,100] 100)
@@ -478,14 +479,14 @@
"([])[Y \<mapsto> T]\<^sub>e= []"
"(B#\<Gamma>)[Y \<mapsto> T]\<^sub>e = (B[Y \<mapsto> T]\<^sub>b)#(\<Gamma>[Y \<mapsto> T]\<^sub>e)"
-lemma ctxt_subst_fresh'[fresh]:
+lemma ctxt_subst_fresh':
fixes X::"tyvrs"
assumes "X \<sharp> \<Gamma>"
and "X \<sharp> P"
shows "X \<sharp> \<Gamma>[Y \<mapsto> P]\<^sub>e"
using assms
by (induct \<Gamma>)
- (auto simp add: fresh_list_cons freshs)
+ (auto simp add: fresh_list_cons binding_subst_fresh)
lemma ctxt_subst_mem_TVarB: "TVarB X T \<in> set \<Gamma> \<Longrightarrow> TVarB X (T[Y \<mapsto> U]\<^sub>\<tau>) \<in> set (\<Gamma>[Y \<mapsto> U]\<^sub>e)"
by (induct \<Gamma>) auto
@@ -1188,8 +1189,8 @@
using assms by (induct, auto)
nominal_inductive typing
- by (auto dest!: typing_ok intro: closed_in_fresh fresh_domain
- simp: abs_fresh fresh_prod fresh_atm freshs valid_ty_domain_fresh fresh_trm_domain)
+by (auto dest!: typing_ok intro: closed_in_fresh fresh_domain type_subst_fresh
+ simp: abs_fresh fresh_type_subst_fresh ty_vrs_fresh valid_ty_domain_fresh fresh_trm_domain)
lemma ok_imp_VarB_closed_in:
assumes ok: "\<turnstile> \<Gamma> ok"
--- a/src/HOL/Nominal/Nominal.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Nominal/Nominal.thy Mon Apr 27 07:26:17 2009 -0700
@@ -18,25 +18,98 @@
types
'x prm = "('x \<times> 'x) list"
-(* polymorphic operations for permutation and swapping *)
+(* polymorphic constants for permutation and swapping *)
consts
perm :: "'x prm \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<bullet>" 80)
swap :: "('x \<times> 'x) \<Rightarrow> 'x \<Rightarrow> 'x"
+(* a "private" copy of the option type used in the abstraction function *)
+datatype 'a noption = nSome 'a | nNone
+
+(* a "private" copy of the product type used in the nominal induct method *)
+datatype ('a,'b) nprod = nPair 'a 'b
+
(* an auxiliary constant for the decision procedure involving *)
-(* permutations (to avoid loops when using perm-composition) *)
+(* permutations (to avoid loops when using perm-compositions) *)
constdefs
"perm_aux pi x \<equiv> pi\<bullet>x"
-(* permutation on functions *)
-defs (unchecked overloaded)
- perm_fun_def: "pi\<bullet>(f::'a\<Rightarrow>'b) \<equiv> (\<lambda>x. pi\<bullet>f((rev pi)\<bullet>x))"
-
-(* permutation on bools *)
-primrec (unchecked perm_bool)
- true_eqvt: "pi\<bullet>True = True"
- false_eqvt: "pi\<bullet>False = False"
-
+(* overloaded permutation operations *)
+overloading
+ perm_fun \<equiv> "perm :: 'x prm \<Rightarrow> ('a\<Rightarrow>'b) \<Rightarrow> ('a\<Rightarrow>'b)" (unchecked)
+ perm_bool \<equiv> "perm :: 'x prm \<Rightarrow> bool \<Rightarrow> bool" (unchecked)
+ perm_unit \<equiv> "perm :: 'x prm \<Rightarrow> unit \<Rightarrow> unit" (unchecked)
+ perm_prod \<equiv> "perm :: 'x prm \<Rightarrow> ('a\<times>'b) \<Rightarrow> ('a\<times>'b)" (unchecked)
+ perm_list \<equiv> "perm :: 'x prm \<Rightarrow> 'a list \<Rightarrow> 'a list" (unchecked)
+ perm_option \<equiv> "perm :: 'x prm \<Rightarrow> 'a option \<Rightarrow> 'a option" (unchecked)
+ perm_char \<equiv> "perm :: 'x prm \<Rightarrow> char \<Rightarrow> char" (unchecked)
+ perm_nat \<equiv> "perm :: 'x prm \<Rightarrow> nat \<Rightarrow> nat" (unchecked)
+ perm_int \<equiv> "perm :: 'x prm \<Rightarrow> int \<Rightarrow> int" (unchecked)
+
+ perm_noption \<equiv> "perm :: 'x prm \<Rightarrow> 'a noption \<Rightarrow> 'a noption" (unchecked)
+ perm_nprod \<equiv> "perm :: 'x prm \<Rightarrow> ('a, 'b) nprod \<Rightarrow> ('a, 'b) nprod" (unchecked)
+begin
+
+definition
+ perm_fun_def: "perm_fun pi (f::'a\<Rightarrow>'b) \<equiv> (\<lambda>x. pi\<bullet>f((rev pi)\<bullet>x))"
+
+fun
+ perm_bool :: "'x prm \<Rightarrow> bool \<Rightarrow> bool"
+where
+ true_eqvt: "perm_bool pi True = True"
+| false_eqvt: "perm_bool pi False = False"
+
+fun
+ perm_unit :: "'x prm \<Rightarrow> unit \<Rightarrow> unit"
+where
+ "perm_unit pi () = ()"
+
+fun
+ perm_prod :: "'x prm \<Rightarrow> ('a\<times>'b) \<Rightarrow> ('a\<times>'b)"
+where
+ "perm_prod pi (x,y) = (pi\<bullet>x,pi\<bullet>y)"
+
+fun
+ perm_list :: "'x prm \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+ nil_eqvt: "perm_list pi [] = []"
+| cons_eqvt: "perm_list pi (x#xs) = (pi\<bullet>x)#(pi\<bullet>xs)"
+
+fun
+ perm_option :: "'x prm \<Rightarrow> 'a option \<Rightarrow> 'a option"
+where
+ some_eqvt: "perm_option pi (Some x) = Some (pi\<bullet>x)"
+| none_eqvt: "perm_option pi None = None"
+
+definition
+ perm_char :: "'x prm \<Rightarrow> char \<Rightarrow> char"
+where
+ perm_char_def: "perm_char pi c \<equiv> c"
+
+definition
+ perm_nat :: "'x prm \<Rightarrow> nat \<Rightarrow> nat"
+where
+ perm_nat_def: "perm_nat pi i \<equiv> i"
+
+definition
+ perm_int :: "'x prm \<Rightarrow> int \<Rightarrow> int"
+where
+ perm_int_def: "perm_int pi i \<equiv> i"
+
+fun
+ perm_noption :: "'x prm \<Rightarrow> 'a noption \<Rightarrow> 'a noption"
+where
+ nsome_eqvt: "perm_noption pi (nSome x) = nSome (pi\<bullet>x)"
+| nnone_eqvt: "perm_noption pi nNone = nNone"
+
+fun
+ perm_nprod :: "'x prm \<Rightarrow> ('a, 'b) nprod \<Rightarrow> ('a, 'b) nprod"
+where
+ "perm_nprod pi (nPair x y) = nPair (pi\<bullet>x) (pi\<bullet>y)"
+end
+
+
+(* permutations on booleans *)
lemma perm_bool:
shows "pi\<bullet>(b::bool) = b"
by (cases b) auto
@@ -54,8 +127,7 @@
lemma if_eqvt:
fixes pi::"'a prm"
shows "pi\<bullet>(if b then c1 else c2) = (if (pi\<bullet>b) then (pi\<bullet>c1) else (pi\<bullet>c2))"
-apply(simp add: perm_fun_def)
-done
+ by (simp add: perm_fun_def)
lemma imp_eqvt:
shows "pi\<bullet>(A\<longrightarrow>B) = ((pi\<bullet>A)\<longrightarrow>(pi\<bullet>B))"
@@ -82,13 +154,7 @@
shows "(pi\<bullet>(X\<union>Y)) = (pi\<bullet>X) \<union> (pi\<bullet>Y)"
by (simp add: perm_fun_def perm_bool Un_iff [unfolded mem_def] expand_fun_eq)
-(* permutation on units and products *)
-primrec (unchecked perm_unit)
- "pi\<bullet>() = ()"
-
-primrec (unchecked perm_prod)
- "pi\<bullet>(x,y) = (pi\<bullet>x,pi\<bullet>y)"
-
+(* permutations on products *)
lemma fst_eqvt:
"pi\<bullet>(fst x) = fst (pi\<bullet>x)"
by (cases x) simp
@@ -98,10 +164,6 @@
by (cases x) simp
(* permutation on lists *)
-primrec (unchecked perm_list)
- nil_eqvt: "pi\<bullet>[] = []"
- cons_eqvt: "pi\<bullet>(x#xs) = (pi\<bullet>x)#(pi\<bullet>xs)"
-
lemma append_eqvt:
fixes pi :: "'x prm"
and l1 :: "'a list"
@@ -115,41 +177,12 @@
shows "pi\<bullet>(rev l) = rev (pi\<bullet>l)"
by (induct l) (simp_all add: append_eqvt)
-(* permutation on options *)
-
-primrec (unchecked perm_option)
- some_eqvt: "pi\<bullet>Some(x) = Some(pi\<bullet>x)"
- none_eqvt: "pi\<bullet>None = None"
-
-(* a "private" copy of the option type used in the abstraction function *)
-datatype 'a noption = nSome 'a | nNone
-
-primrec (unchecked perm_noption)
- nSome_eqvt: "pi\<bullet>nSome(x) = nSome(pi\<bullet>x)"
- nNone_eqvt: "pi\<bullet>nNone = nNone"
-
-(* a "private" copy of the product type used in the nominal induct method *)
-datatype ('a,'b) nprod = nPair 'a 'b
-
-primrec (unchecked perm_nprod)
- perm_nProd_def: "pi\<bullet>(nPair x1 x2) = nPair (pi\<bullet>x1) (pi\<bullet>x2)"
-
-(* permutation on characters (used in strings) *)
-defs (unchecked overloaded)
- perm_char_def: "pi\<bullet>(c::char) \<equiv> c"
-
+(* permutation on characters and strings *)
lemma perm_string:
fixes s::"string"
shows "pi\<bullet>s = s"
-by (induct s)(auto simp add: perm_char_def)
-
-(* permutation on ints *)
-defs (unchecked overloaded)
- perm_int_def: "pi\<bullet>(i::int) \<equiv> i"
-
-(* permutation on nats *)
-defs (unchecked overloaded)
- perm_nat_def: "pi\<bullet>(i::nat) \<equiv> i"
+ by (induct s)(auto simp add: perm_char_def)
+
section {* permutation equality *}
(*==============================*)
@@ -170,11 +203,12 @@
supports :: "'x set \<Rightarrow> 'a \<Rightarrow> bool" (infixl "supports" 80)
"S supports x \<equiv> \<forall>a b. (a\<notin>S \<and> b\<notin>S \<longrightarrow> [(a,b)]\<bullet>x=x)"
+(* lemmas about supp *)
lemma supp_fresh_iff:
fixes x :: "'a"
shows "(supp x) = {a::'x. \<not>a\<sharp>x}"
-apply(simp add: fresh_def)
-done
+ by (simp add: fresh_def)
+
lemma supp_unit:
shows "supp () = {}"
@@ -205,14 +239,13 @@
fixes x :: "'a"
and xs :: "'a list"
shows "supp (x#xs) = (supp x)\<union>(supp xs)"
-apply(auto simp add: supp_def Collect_imp_eq Collect_neg_eq)
-done
+ by (auto simp add: supp_def Collect_imp_eq Collect_neg_eq)
lemma supp_list_append:
fixes xs :: "'a list"
and ys :: "'a list"
shows "supp (xs@ys) = (supp xs)\<union>(supp ys)"
- by (induct xs, auto simp add: supp_list_nil supp_list_cons)
+ by (induct xs) (auto simp add: supp_list_nil supp_list_cons)
lemma supp_list_rev:
fixes xs :: "'a list"
@@ -221,47 +254,40 @@
lemma supp_bool:
fixes x :: "bool"
- shows "supp (x) = {}"
- apply(case_tac "x")
- apply(simp_all add: supp_def)
-done
+ shows "supp x = {}"
+ by (cases "x") (simp_all add: supp_def)
lemma supp_some:
fixes x :: "'a"
shows "supp (Some x) = (supp x)"
- apply(simp add: supp_def)
- done
+ by (simp add: supp_def)
lemma supp_none:
fixes x :: "'a"
shows "supp (None) = {}"
- apply(simp add: supp_def)
- done
+ by (simp add: supp_def)
lemma supp_int:
fixes i::"int"
shows "supp (i) = {}"
- apply(simp add: supp_def perm_int_def)
- done
+ by (simp add: supp_def perm_int_def)
lemma supp_nat:
fixes n::"nat"
- shows "supp (n) = {}"
- apply(simp add: supp_def perm_nat_def)
- done
+ shows "(supp n) = {}"
+ by (simp add: supp_def perm_nat_def)
lemma supp_char:
fixes c::"char"
- shows "supp (c) = {}"
- apply(simp add: supp_def perm_char_def)
- done
+ shows "(supp c) = {}"
+ by (simp add: supp_def perm_char_def)
lemma supp_string:
fixes s::"string"
- shows "supp (s) = {}"
-apply(simp add: supp_def perm_string)
-done
-
+ shows "(supp s) = {}"
+ by (simp add: supp_def perm_string)
+
+(* lemmas about freshness *)
lemma fresh_set_empty:
shows "a\<sharp>{}"
by (simp add: fresh_def supp_set_empty)
@@ -344,7 +370,6 @@
by (simp add: fresh_def supp_bool)
text {* Normalization of freshness results; cf.\ @{text nominal_induct} *}
-
lemma fresh_unit_elim:
shows "(a\<sharp>() \<Longrightarrow> PROP C) \<equiv> PROP C"
by (simp add: fresh_def supp_unit)
@@ -371,63 +396,6 @@
Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))
*}
-section {* generalisation of freshness to lists and sets of atoms *}
-(*================================================================*)
-
-consts
- fresh_star :: "'b \<Rightarrow> 'a \<Rightarrow> bool" ("_ \<sharp>* _" [100,100] 100)
-
-defs (overloaded)
- fresh_star_set: "xs\<sharp>*c \<equiv> \<forall>x\<in>xs. x\<sharp>c"
-
-defs (overloaded)
- fresh_star_list: "xs\<sharp>*c \<equiv> \<forall>x\<in>set xs. x\<sharp>c"
-
-lemmas fresh_star_def = fresh_star_list fresh_star_set
-
-lemma fresh_star_prod_set:
- fixes xs::"'a set"
- shows "xs\<sharp>*(a,b) = (xs\<sharp>*a \<and> xs\<sharp>*b)"
-by (auto simp add: fresh_star_def fresh_prod)
-
-lemma fresh_star_prod_list:
- fixes xs::"'a list"
- shows "xs\<sharp>*(a,b) = (xs\<sharp>*a \<and> xs\<sharp>*b)"
-by (auto simp add: fresh_star_def fresh_prod)
-
-lemmas fresh_star_prod = fresh_star_prod_list fresh_star_prod_set
-
-lemma fresh_star_set_eq: "set xs \<sharp>* c = xs \<sharp>* c"
- by (simp add: fresh_star_def)
-
-lemma fresh_star_Un_elim:
- "((S \<union> T) \<sharp>* c \<Longrightarrow> PROP C) \<equiv> (S \<sharp>* c \<Longrightarrow> T \<sharp>* c \<Longrightarrow> PROP C)"
- apply rule
- apply (simp_all add: fresh_star_def)
- apply (erule meta_mp)
- apply blast
- done
-
-lemma fresh_star_insert_elim:
- "(insert x S \<sharp>* c \<Longrightarrow> PROP C) \<equiv> (x \<sharp> c \<Longrightarrow> S \<sharp>* c \<Longrightarrow> PROP C)"
- by rule (simp_all add: fresh_star_def)
-
-lemma fresh_star_empty_elim:
- "({} \<sharp>* c \<Longrightarrow> PROP C) \<equiv> PROP C"
- by (simp add: fresh_star_def)
-
-text {* Normalization of freshness results; cf.\ @{text nominal_induct} *}
-
-lemma fresh_star_unit_elim:
- shows "((a::'a set)\<sharp>*() \<Longrightarrow> PROP C) \<equiv> PROP C"
- and "((b::'a list)\<sharp>*() \<Longrightarrow> PROP C) \<equiv> PROP C"
- by (simp_all add: fresh_star_def fresh_def supp_unit)
-
-lemma fresh_star_prod_elim:
- shows "((a::'a set)\<sharp>*(x,y) \<Longrightarrow> PROP C) \<equiv> (a\<sharp>*x \<Longrightarrow> a\<sharp>*y \<Longrightarrow> PROP C)"
- and "((b::'a list)\<sharp>*(x,y) \<Longrightarrow> PROP C) \<equiv> (b\<sharp>*x \<Longrightarrow> b\<sharp>*y \<Longrightarrow> PROP C)"
- by (rule, simp_all add: fresh_star_prod)+
-
section {* Abstract Properties for Permutations and Atoms *}
(*=========================================================*)
@@ -487,7 +455,7 @@
shows "swap (a,b) c = (if a=c then b else (if b=c then a else c))"
using a by (simp only: at_def)
-(* rules to calculate simple premutations *)
+(* rules to calculate simple permutations *)
lemmas at_calc = at2 at1 at3
lemma at_swap_simps:
@@ -682,7 +650,6 @@
shows "pi1 \<triangleq> pi2 \<Longrightarrow> (rev pi1) \<triangleq> (rev pi2)"
by (simp add: at_prm_rev_eq[OF at])
-
lemma at_ds1:
fixes a :: "'x"
assumes at: "at TYPE('x)"
@@ -838,15 +805,18 @@
by (auto intro: ex_in_inf[OF at, OF fs] simp add: fresh_def)
lemma at_finite_select:
- shows "at (TYPE('a)) \<Longrightarrow> finite (S::'a set) \<Longrightarrow> \<exists>x. x \<notin> S"
- apply (drule Diff_infinite_finite)
- apply (simp add: at_def)
- apply blast
- apply (subgoal_tac "UNIV - S \<noteq> {}")
- apply (simp only: ex_in_conv [symmetric])
- apply blast
- apply (rule notI)
- apply simp
+ fixes S::"'a set"
+ assumes a: "at TYPE('a)"
+ and b: "finite S"
+ shows "\<exists>x. x \<notin> S"
+ using a b
+ apply(drule_tac S="UNIV::'a set" in Diff_infinite_finite)
+ apply(simp add: at_def)
+ apply(subgoal_tac "UNIV - S \<noteq> {}")
+ apply(simp only: ex_in_conv [symmetric])
+ apply(blast)
+ apply(rule notI)
+ apply(simp)
done
lemma at_different:
@@ -1222,8 +1192,8 @@
assumes pt: "pt TYPE('a) TYPE('x)"
and at: "at TYPE('x)"
shows "pi\<bullet>(x=y) = (pi\<bullet>x = pi\<bullet>y)"
-using assms
-by (auto simp add: pt_bij perm_bool)
+ using pt at
+ by (auto simp add: pt_bij perm_bool)
lemma pt_bij3:
fixes pi :: "'x prm"
@@ -1231,7 +1201,7 @@
and y :: "'a"
assumes a: "x=y"
shows "(pi\<bullet>x = pi\<bullet>y)"
-using a by simp
+ using a by simp
lemma pt_bij4:
fixes pi :: "'x prm"
@@ -1241,7 +1211,7 @@
and at: "at TYPE('x)"
and a: "pi\<bullet>x = pi\<bullet>y"
shows "x = y"
-using a by (simp add: pt_bij[OF pt, OF at])
+ using a by (simp add: pt_bij[OF pt, OF at])
lemma pt_swap_bij:
fixes a :: "'x"
@@ -1574,35 +1544,6 @@
apply(simp add: pt_rev_pi[OF ptb, OF at])
done
-lemma pt_fresh_star_bij_ineq:
- fixes pi :: "'x prm"
- and x :: "'a"
- and a :: "'y set"
- and b :: "'y list"
- assumes pta: "pt TYPE('a) TYPE('x)"
- and ptb: "pt TYPE('y) TYPE('x)"
- and at: "at TYPE('x)"
- and cp: "cp TYPE('a) TYPE('x) TYPE('y)"
- shows "(pi\<bullet>a)\<sharp>*(pi\<bullet>x) = a\<sharp>*x"
- and "(pi\<bullet>b)\<sharp>*(pi\<bullet>x) = b\<sharp>*x"
-apply(unfold fresh_star_def)
-apply(auto)
-apply(drule_tac x="pi\<bullet>xa" in bspec)
-apply(rule pt_set_bij2[OF ptb, OF at])
-apply(assumption)
-apply(simp add: fresh_star_def pt_fresh_bij_ineq[OF pta, OF ptb, OF at, OF cp])
-apply(drule_tac x="(rev pi)\<bullet>xa" in bspec)
-apply(simp add: pt_set_bij1[OF ptb, OF at])
-apply(simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp])
-apply(drule_tac x="pi\<bullet>xa" in bspec)
-apply(simp add: pt_set_bij1[OF ptb, OF at])
-apply(simp add: pt_set_eqvt [OF ptb at] pt_rev_pi[OF pt_list_inst[OF ptb], OF at])
-apply(simp add: pt_fresh_bij_ineq[OF pta, OF ptb, OF at, OF cp])
-apply(drule_tac x="(rev pi)\<bullet>xa" in bspec)
-apply(simp add: pt_set_bij1[OF ptb, OF at] pt_set_eqvt [OF ptb at])
-apply(simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp])
-done
-
lemma pt_fresh_left:
fixes pi :: "'x prm"
and x :: "'a"
@@ -1651,56 +1592,6 @@
apply(rule at)
done
-lemma pt_fresh_star_bij:
- fixes pi :: "'x prm"
- and x :: "'a"
- and a :: "'x set"
- and b :: "'x list"
- assumes pt: "pt TYPE('a) TYPE('x)"
- and at: "at TYPE('x)"
- shows "(pi\<bullet>a)\<sharp>*(pi\<bullet>x) = a\<sharp>*x"
- and "(pi\<bullet>b)\<sharp>*(pi\<bullet>x) = b\<sharp>*x"
-apply(rule pt_fresh_star_bij_ineq(1))
-apply(rule pt)
-apply(rule at_pt_inst)
-apply(rule at)+
-apply(rule cp_pt_inst)
-apply(rule pt)
-apply(rule at)
-apply(rule pt_fresh_star_bij_ineq(2))
-apply(rule pt)
-apply(rule at_pt_inst)
-apply(rule at)+
-apply(rule cp_pt_inst)
-apply(rule pt)
-apply(rule at)
-done
-
-lemma pt_fresh_star_eqvt:
- fixes pi :: "'x prm"
- and x :: "'a"
- and a :: "'x set"
- and b :: "'x list"
- assumes pt: "pt TYPE('a) TYPE('x)"
- and at: "at TYPE('x)"
- shows "pi\<bullet>(a\<sharp>*x) = (pi\<bullet>a)\<sharp>*(pi\<bullet>x)"
- and "pi\<bullet>(b\<sharp>*x) = (pi\<bullet>b)\<sharp>*(pi\<bullet>x)"
- by (simp_all add: perm_bool pt_fresh_star_bij[OF pt, OF at])
-
-lemma pt_fresh_star_eqvt_ineq:
- fixes pi::"'x prm"
- and a::"'y set"
- and b::"'y list"
- and x::"'a"
- assumes pta: "pt TYPE('a) TYPE('x)"
- and ptb: "pt TYPE('y) TYPE('x)"
- and at: "at TYPE('x)"
- and cp: "cp TYPE('a) TYPE('x) TYPE('y)"
- and dj: "disjoint TYPE('y) TYPE('x)"
- shows "pi\<bullet>(a\<sharp>*x) = (pi\<bullet>a)\<sharp>*(pi\<bullet>x)"
- and "pi\<bullet>(b\<sharp>*x) = (pi\<bullet>b)\<sharp>*(pi\<bullet>x)"
- by (simp_all add: pt_fresh_star_bij_ineq[OF pta, OF ptb, OF at, OF cp] dj_perm_forget[OF dj] perm_bool)
-
lemma pt_fresh_bij1:
fixes pi :: "'x prm"
and x :: "'a"
@@ -1753,7 +1644,6 @@
(* the next two lemmas are needed in the proof *)
(* of the structural induction principle *)
-
lemma pt_fresh_aux:
fixes a::"'x"
and b::"'x"
@@ -1857,27 +1747,6 @@
thus ?thesis using eq3 by simp
qed
-lemma pt_freshs_freshs:
- assumes pt: "pt TYPE('a) TYPE('x)"
- and at: "at TYPE ('x)"
- and pi: "set (pi::'x prm) \<subseteq> Xs \<times> Ys"
- and Xs: "Xs \<sharp>* (x::'a)"
- and Ys: "Ys \<sharp>* x"
- shows "pi \<bullet> x = x"
- using pi
-proof (induct pi)
- case Nil
- show ?case by (simp add: pt1 [OF pt])
-next
- case (Cons p pi)
- obtain a b where p: "p = (a, b)" by (cases p)
- with Cons Xs Ys have "a \<sharp> x" "b \<sharp> x"
- by (simp_all add: fresh_star_def)
- with Cons p show ?case
- by (simp add: pt_fresh_fresh [OF pt at]
- pt2 [OF pt, of "[(a, b)]" pi, simplified])
-qed
-
lemma pt_pi_fresh_fresh:
fixes x :: "'a"
and pi :: "'x prm"
@@ -1943,8 +1812,7 @@
thus ?thesis by (simp add: pt2[OF pt])
qed
-section {* equivaraince for some connectives *}
-
+section {* equivariance for some connectives *}
lemma pt_all_eqvt:
fixes pi :: "'x prm"
and x :: "'a"
@@ -1990,8 +1858,6 @@
apply(rule theI'[OF unique])
done
-
-
section {* facts about supports *}
(*==============================*)
@@ -2160,6 +2026,7 @@
shows "(x \<sharp> X) = (x \<notin> X)"
by (simp add: at_fin_set_supp fresh_def at fs)
+
section {* Permutations acting on Functions *}
(*==========================================*)
@@ -2540,9 +2407,8 @@
and a1: "a\<sharp>x"
and a2: "a\<sharp>X"
shows "a\<sharp>(insert x X)"
-using a1 a2
-apply(simp add: fresh_fin_insert[OF pt, OF at, OF fs, OF f])
-done
+ using a1 a2
+ by (simp add: fresh_fin_insert[OF pt, OF at, OF fs, OF f])
lemma pt_list_set_supp:
fixes xs :: "'a list"
@@ -2571,14 +2437,191 @@
shows "a\<sharp>(set xs) = a\<sharp>xs"
by (simp add: fresh_def pt_list_set_supp[OF pt, OF at, OF fs])
+
+section {* generalisation of freshness to lists and sets of atoms *}
+(*================================================================*)
+
+consts
+ fresh_star :: "'b \<Rightarrow> 'a \<Rightarrow> bool" ("_ \<sharp>* _" [100,100] 100)
+
+defs (overloaded)
+ fresh_star_set: "xs\<sharp>*c \<equiv> \<forall>x\<in>xs. x\<sharp>c"
+
+defs (overloaded)
+ fresh_star_list: "xs\<sharp>*c \<equiv> \<forall>x\<in>set xs. x\<sharp>c"
+
+lemmas fresh_star_def = fresh_star_list fresh_star_set
+
+lemma fresh_star_prod_set:
+ fixes xs::"'a set"
+ shows "xs\<sharp>*(a,b) = (xs\<sharp>*a \<and> xs\<sharp>*b)"
+by (auto simp add: fresh_star_def fresh_prod)
+
+lemma fresh_star_prod_list:
+ fixes xs::"'a list"
+ shows "xs\<sharp>*(a,b) = (xs\<sharp>*a \<and> xs\<sharp>*b)"
+ by (auto simp add: fresh_star_def fresh_prod)
+
+lemmas fresh_star_prod = fresh_star_prod_list fresh_star_prod_set
+
+lemma fresh_star_set_eq: "set xs \<sharp>* c = xs \<sharp>* c"
+ by (simp add: fresh_star_def)
+
+lemma fresh_star_Un_elim:
+ "((S \<union> T) \<sharp>* c \<Longrightarrow> PROP C) \<equiv> (S \<sharp>* c \<Longrightarrow> T \<sharp>* c \<Longrightarrow> PROP C)"
+ apply rule
+ apply (simp_all add: fresh_star_def)
+ apply (erule meta_mp)
+ apply blast
+ done
+
+lemma fresh_star_insert_elim:
+ "(insert x S \<sharp>* c \<Longrightarrow> PROP C) \<equiv> (x \<sharp> c \<Longrightarrow> S \<sharp>* c \<Longrightarrow> PROP C)"
+ by rule (simp_all add: fresh_star_def)
+
+lemma fresh_star_empty_elim:
+ "({} \<sharp>* c \<Longrightarrow> PROP C) \<equiv> PROP C"
+ by (simp add: fresh_star_def)
+
+text {* Normalization of freshness results; see \ @{text nominal_induct} *}
+
+lemma fresh_star_unit_elim:
+ shows "((a::'a set)\<sharp>*() \<Longrightarrow> PROP C) \<equiv> PROP C"
+ and "((b::'a list)\<sharp>*() \<Longrightarrow> PROP C) \<equiv> PROP C"
+ by (simp_all add: fresh_star_def fresh_def supp_unit)
+
+lemma fresh_star_prod_elim:
+ shows "((a::'a set)\<sharp>*(x,y) \<Longrightarrow> PROP C) \<equiv> (a\<sharp>*x \<Longrightarrow> a\<sharp>*y \<Longrightarrow> PROP C)"
+ and "((b::'a list)\<sharp>*(x,y) \<Longrightarrow> PROP C) \<equiv> (b\<sharp>*x \<Longrightarrow> b\<sharp>*y \<Longrightarrow> PROP C)"
+ by (rule, simp_all add: fresh_star_prod)+
+
+
+lemma pt_fresh_star_bij_ineq:
+ fixes pi :: "'x prm"
+ and x :: "'a"
+ and a :: "'y set"
+ and b :: "'y list"
+ assumes pta: "pt TYPE('a) TYPE('x)"
+ and ptb: "pt TYPE('y) TYPE('x)"
+ and at: "at TYPE('x)"
+ and cp: "cp TYPE('a) TYPE('x) TYPE('y)"
+ shows "(pi\<bullet>a)\<sharp>*(pi\<bullet>x) = a\<sharp>*x"
+ and "(pi\<bullet>b)\<sharp>*(pi\<bullet>x) = b\<sharp>*x"
+apply(unfold fresh_star_def)
+apply(auto)
+apply(drule_tac x="pi\<bullet>xa" in bspec)
+apply(erule pt_set_bij2[OF ptb, OF at])
+apply(simp add: fresh_star_def pt_fresh_bij_ineq[OF pta, OF ptb, OF at, OF cp])
+apply(drule_tac x="(rev pi)\<bullet>xa" in bspec)
+apply(simp add: pt_set_bij1[OF ptb, OF at])
+apply(simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp])
+apply(drule_tac x="pi\<bullet>xa" in bspec)
+apply(simp add: pt_set_bij1[OF ptb, OF at])
+apply(simp add: pt_set_eqvt [OF ptb at] pt_rev_pi[OF pt_list_inst[OF ptb], OF at])
+apply(simp add: pt_fresh_bij_ineq[OF pta, OF ptb, OF at, OF cp])
+apply(drule_tac x="(rev pi)\<bullet>xa" in bspec)
+apply(simp add: pt_set_bij1[OF ptb, OF at] pt_set_eqvt [OF ptb at])
+apply(simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp])
+done
+
+lemma pt_fresh_star_bij:
+ fixes pi :: "'x prm"
+ and x :: "'a"
+ and a :: "'x set"
+ and b :: "'x list"
+ assumes pt: "pt TYPE('a) TYPE('x)"
+ and at: "at TYPE('x)"
+ shows "(pi\<bullet>a)\<sharp>*(pi\<bullet>x) = a\<sharp>*x"
+ and "(pi\<bullet>b)\<sharp>*(pi\<bullet>x) = b\<sharp>*x"
+apply(rule pt_fresh_star_bij_ineq(1))
+apply(rule pt)
+apply(rule at_pt_inst)
+apply(rule at)+
+apply(rule cp_pt_inst)
+apply(rule pt)
+apply(rule at)
+apply(rule pt_fresh_star_bij_ineq(2))
+apply(rule pt)
+apply(rule at_pt_inst)
+apply(rule at)+
+apply(rule cp_pt_inst)
+apply(rule pt)
+apply(rule at)
+done
+
+lemma pt_fresh_star_eqvt:
+ fixes pi :: "'x prm"
+ and x :: "'a"
+ and a :: "'x set"
+ and b :: "'x list"
+ assumes pt: "pt TYPE('a) TYPE('x)"
+ and at: "at TYPE('x)"
+ shows "pi\<bullet>(a\<sharp>*x) = (pi\<bullet>a)\<sharp>*(pi\<bullet>x)"
+ and "pi\<bullet>(b\<sharp>*x) = (pi\<bullet>b)\<sharp>*(pi\<bullet>x)"
+ by (simp_all add: perm_bool pt_fresh_star_bij[OF pt, OF at])
+
+lemma pt_fresh_star_eqvt_ineq:
+ fixes pi::"'x prm"
+ and a::"'y set"
+ and b::"'y list"
+ and x::"'a"
+ assumes pta: "pt TYPE('a) TYPE('x)"
+ and ptb: "pt TYPE('y) TYPE('x)"
+ and at: "at TYPE('x)"
+ and cp: "cp TYPE('a) TYPE('x) TYPE('y)"
+ and dj: "disjoint TYPE('y) TYPE('x)"
+ shows "pi\<bullet>(a\<sharp>*x) = (pi\<bullet>a)\<sharp>*(pi\<bullet>x)"
+ and "pi\<bullet>(b\<sharp>*x) = (pi\<bullet>b)\<sharp>*(pi\<bullet>x)"
+ by (simp_all add: pt_fresh_star_bij_ineq[OF pta, OF ptb, OF at, OF cp] dj_perm_forget[OF dj] perm_bool)
+
+lemma pt_freshs_freshs:
+ assumes pt: "pt TYPE('a) TYPE('x)"
+ and at: "at TYPE ('x)"
+ and pi: "set (pi::'x prm) \<subseteq> Xs \<times> Ys"
+ and Xs: "Xs \<sharp>* (x::'a)"
+ and Ys: "Ys \<sharp>* x"
+ shows "pi\<bullet>x = x"
+ using pi
+proof (induct pi)
+ case Nil
+ show ?case by (simp add: pt1 [OF pt])
+next
+ case (Cons p pi)
+ obtain a b where p: "p = (a, b)" by (cases p)
+ with Cons Xs Ys have "a \<sharp> x" "b \<sharp> x"
+ by (simp_all add: fresh_star_def)
+ with Cons p show ?case
+ by (simp add: pt_fresh_fresh [OF pt at]
+ pt2 [OF pt, of "[(a, b)]" pi, simplified])
+qed
+
+lemma pt_fresh_star_pi:
+ fixes x::"'a"
+ and pi::"'x prm"
+ assumes pt: "pt TYPE('a) TYPE('x)"
+ and at: "at TYPE('x)"
+ and a: "((supp x)::'x set)\<sharp>* pi"
+ shows "pi\<bullet>x = x"
+using a
+apply(induct pi)
+apply(auto simp add: fresh_star_def fresh_list_cons fresh_prod pt1[OF pt])
+apply(subgoal_tac "((a,b)#pi)\<bullet>x = ([(a,b)]@pi)\<bullet>x")
+apply(simp only: pt2[OF pt])
+apply(rule pt_fresh_fresh[OF pt at])
+apply(simp add: fresh_def at_supp[OF at])
+apply(blast)
+apply(simp add: fresh_def at_supp[OF at])
+apply(blast)
+apply(simp add: pt2[OF pt])
+done
+
section {* Infrastructure lemmas for strong rule inductions *}
(*==========================================================*)
-
text {*
For every set of atoms, there is another set of atoms
avoiding a finitely supported c and there is a permutation
- which make 'translates' between both sets.
+ which 'translates' between both sets.
*}
lemma at_set_avoiding_aux:
fixes Xs::"'a set"
@@ -3365,7 +3408,6 @@
syntax ABS :: "type \<Rightarrow> type \<Rightarrow> type" ("\<guillemotleft>_\<guillemotright>_" [1000,1000] 1000)
-
section {* lemmas for deciding permutation equations *}
(*===================================================*)
@@ -3526,8 +3568,8 @@
shows "pi\<bullet>(x div y) = (pi\<bullet>x) div (pi\<bullet>y)"
by (simp add:perm_int_def)
-(*******************************************************************)
-(* Setup of the theorem attributes eqvt, eqvt_force, fresh and bij *)
+(*******************************************************)
+(* Setup of the theorem attributes eqvt and eqvt_force *)
use "nominal_thmdecls.ML"
setup "NominalThmDecls.setup"
--- a/src/HOL/Nominal/nominal_thmdecls.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Nominal/nominal_thmdecls.ML Mon Apr 27 07:26:17 2009 -0700
@@ -1,12 +1,12 @@
(* Authors: Julien Narboux and Christian Urban
This file introduces the infrastructure for the lemma
- declaration "eqvts" "bijs" and "freshs".
+ collection "eqvts".
- By attaching [eqvt] [bij] or [fresh] to a lemma, the lemma gets stored
- in a data-slot in the context. Possible modifiers
- are [attribute add] and [attribute del] for adding and deleting,
- respectively the lemma from the data-slot.
+ By attaching [eqvt] or [eqvt_force] to a lemma, it will get
+ stored in a data-slot in the context. Possible modifiers
+ are [... add] and [... del] for adding and deleting,
+ respectively, the lemma from the data-slot.
*)
signature NOMINAL_THMDECLS =
@@ -17,9 +17,6 @@
val eqvt_force_del: attribute
val setup: theory -> theory
val get_eqvt_thms: Proof.context -> thm list
- val get_fresh_thms: Proof.context -> thm list
- val get_bij_thms: Proof.context -> thm list
-
val NOMINAL_EQVT_DEBUG : bool ref
end;
@@ -29,13 +26,11 @@
structure Data = GenericDataFun
(
- type T = {eqvts:thm list, freshs:thm list, bijs:thm list};
- val empty = ({eqvts=[], freshs=[], bijs=[]}:T);
- val extend = I;
- fun merge _ (r1:T,r2:T) = {eqvts = Thm.merge_thms (#eqvts r1, #eqvts r2),
- freshs = Thm.merge_thms (#freshs r1, #freshs r2),
- bijs = Thm.merge_thms (#bijs r1, #bijs r2)}
-);
+ type T = thm list
+ val empty = []:T
+ val extend = I
+ fun merge _ (r1:T, r2:T) = Thm.merge_thms (r1, r2)
+)
(* Exception for when a theorem does not conform with form of an equivariance lemma. *)
(* There are two forms: one is an implication (for relations) and the other is an *)
@@ -46,72 +41,68 @@
(* the implicational case it is also checked that the variables and permutation fit *)
(* together, i.e. are of the right "pt_class", so that a stronger version of the *)
(* equality-lemma can be derived. *)
-exception EQVT_FORM of string;
+exception EQVT_FORM of string
-val get_eqvt_thms = Context.Proof #> Data.get #> #eqvts;
-val get_fresh_thms = Context.Proof #> Data.get #> #freshs;
-val get_bij_thms = Context.Proof #> Data.get #> #bijs;
+val NOMINAL_EQVT_DEBUG = ref false
-(* FIXME: should be a function in a library *)
-fun mk_permT T = HOLogic.listT (HOLogic.mk_prodT (T, T));
-
-val NOMINAL_EQVT_DEBUG = ref false;
-
-fun tactic (msg,tac) =
- if !NOMINAL_EQVT_DEBUG
- then tac THEN print_tac ("after "^msg)
- else tac
+fun tactic (msg, tac) =
+ if !NOMINAL_EQVT_DEBUG
+ then tac THEN' (K (print_tac ("after " ^ msg)))
+ else tac
-fun tactic_eqvt ctx orig_thm pi pi' =
- let
- val mypi = Thm.cterm_of ctx pi
- val T = fastype_of pi'
- val mypifree = Thm.cterm_of ctx (Const ("List.rev", T --> T) $ pi')
- val perm_pi_simp = PureThy.get_thms ctx "perm_pi_simp"
- in
- EVERY [tactic ("iffI applied",rtac iffI 1),
- tactic ("remove pi with perm_boolE", (dtac @{thm perm_boolE} 1)),
- tactic ("solve with orig_thm", (etac orig_thm 1)),
- tactic ("applies orig_thm instantiated with rev pi",
- dtac (Drule.cterm_instantiate [(mypi,mypifree)] orig_thm) 1),
- tactic ("getting rid of the pi on the right",
- (rtac @{thm perm_boolI} 1)),
- tactic ("getting rid of all remaining perms",
- full_simp_tac (HOL_basic_ss addsimps perm_pi_simp) 1)]
- end;
+fun prove_eqvt_tac ctxt orig_thm pi pi' =
+let
+ val mypi = Thm.cterm_of ctxt pi
+ val T = fastype_of pi'
+ val mypifree = Thm.cterm_of ctxt (Const (@{const_name "rev"}, T --> T) $ pi')
+ val perm_pi_simp = PureThy.get_thms ctxt "perm_pi_simp"
+in
+ EVERY1 [tactic ("iffI applied", rtac @{thm iffI}),
+ tactic ("remove pi with perm_boolE", dtac @{thm perm_boolE}),
+ tactic ("solve with orig_thm", etac orig_thm),
+ tactic ("applies orig_thm instantiated with rev pi",
+ dtac (Drule.cterm_instantiate [(mypi,mypifree)] orig_thm)),
+ tactic ("getting rid of the pi on the right", rtac @{thm perm_boolI}),
+ tactic ("getting rid of all remaining perms",
+ full_simp_tac (HOL_basic_ss addsimps perm_pi_simp))]
+end;
fun get_derived_thm ctxt hyp concl orig_thm pi typi =
let
val thy = ProofContext.theory_of ctxt;
val pi' = Var (pi, typi);
- val lhs = Const ("Nominal.perm", typi --> HOLogic.boolT --> HOLogic.boolT) $ pi' $ hyp;
+ val lhs = Const (@{const_name "perm"}, typi --> HOLogic.boolT --> HOLogic.boolT) $ pi' $ hyp;
val ([goal_term, pi''], ctxt') = Variable.import_terms false
[HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, concl)), pi'] ctxt
val _ = Display.print_cterm (cterm_of thy goal_term)
in
Goal.prove ctxt' [] [] goal_term
- (fn _ => tactic_eqvt thy orig_thm pi' pi'') |>
+ (fn _ => prove_eqvt_tac thy orig_thm pi' pi'') |>
singleton (ProofContext.export ctxt' ctxt)
end
-(* replaces every variable x in t with pi o x *)
-fun apply_pi trm (pi,typi) =
- let
- fun only_vars t =
- (case t of
- Var (n,ty) => (Const ("Nominal.perm",typi --> ty --> ty) $ (Var (pi,typi)) $ (Var (n,ty)))
- | _ => t)
+(* replaces in t every variable, say x, with pi o x *)
+fun apply_pi trm (pi, typi) =
+let
+ fun replace n ty =
+ let
+ val c = Const (@{const_name "perm"}, typi --> ty --> ty)
+ val v1 = Var (pi, typi)
+ val v2 = Var (n, ty)
in
- map_aterms only_vars trm
- end;
+ c $ v1 $ v2
+ end
+in
+ map_aterms (fn Var (n, ty) => replace n ty | t => t) trm
+end
(* returns *the* pi which is in front of all variables, provided there *)
(* exists such a pi; otherwise raises EQVT_FORM *)
fun get_pi t thy =
let fun get_pi_aux s =
(case s of
- (Const ("Nominal.perm",typrm) $
- (Var (pi,typi as Type("List.list",[Type ("*",[Type (tyatm,[]),_])]))) $
+ (Const (@{const_name "perm"} ,typrm) $
+ (Var (pi,typi as Type(@{type_name "list"}, [Type ("*", [Type (tyatm,[]),_])]))) $
(Var (n,ty))) =>
let
(* FIXME: this should be an operation the library *)
@@ -130,7 +121,7 @@
(* to ensure that all pi's must have been the same, i.e. distinct returns *)
(* a singleton-list *)
(case (distinct (op =) (get_pi_aux t)) of
- [(pi,typi)] => (pi,typi)
+ [(pi,typi)] => (pi, typi)
| _ => raise EQVT_FORM "All permutation should be the same")
end;
@@ -155,8 +146,8 @@
else raise EQVT_FORM "Type Implication"
end
(* case: eqvt-lemma is of the equational form *)
- | (Const ("Trueprop", _) $ (Const ("op =", _) $
- (Const ("Nominal.perm",typrm) $ Var (pi,typi) $ lhs) $ rhs)) =>
+ | (Const (@{const_name "Trueprop"}, _) $ (Const (@{const_name "op ="}, _) $
+ (Const (@{const_name "perm"},typrm) $ Var (pi,typi) $ lhs) $ rhs)) =>
(if (apply_pi lhs (pi,typi)) = rhs
then [orig_thm]
else raise EQVT_FORM "Type Equality")
@@ -165,38 +156,24 @@
fold (fn thm => Data.map (flag thm)) thms_to_be_added context
end
handle EQVT_FORM s =>
- error (Display.string_of_thm orig_thm ^ " does not comply with the form of an equivariance lemma ("^s^").")
-
-(* in cases of bij- and freshness, we just add the lemmas to the *)
-(* data-slot *)
-
-fun eqvt_map f (r:Data.T) = {eqvts = f (#eqvts r), freshs = #freshs r, bijs = #bijs r};
-fun fresh_map f (r:Data.T) = {eqvts = #eqvts r, freshs = f (#freshs r), bijs = #bijs r};
-fun bij_map f (r:Data.T) = {eqvts = #eqvts r, freshs = #freshs r, bijs = f (#bijs r)};
-
-val eqvt_add = Thm.declaration_attribute (eqvt_add_del_aux (eqvt_map o Thm.add_thm));
-val eqvt_del = Thm.declaration_attribute (eqvt_add_del_aux (eqvt_map o Thm.del_thm));
-
-val eqvt_force_add = Thm.declaration_attribute (Data.map o eqvt_map o Thm.add_thm);
-val eqvt_force_del = Thm.declaration_attribute (Data.map o eqvt_map o Thm.del_thm);
-val bij_add = Thm.declaration_attribute (Data.map o bij_map o Thm.add_thm);
-val bij_del = Thm.declaration_attribute (Data.map o bij_map o Thm.del_thm);
-val fresh_add = Thm.declaration_attribute (Data.map o fresh_map o Thm.add_thm);
-val fresh_del = Thm.declaration_attribute (Data.map o fresh_map o Thm.del_thm);
+ error (Display.string_of_thm orig_thm ^
+ " does not comply with the form of an equivariance lemma (" ^ s ^").")
+val eqvt_add = Thm.declaration_attribute (eqvt_add_del_aux (Thm.add_thm));
+val eqvt_del = Thm.declaration_attribute (eqvt_add_del_aux (Thm.del_thm));
+
+val eqvt_force_add = Thm.declaration_attribute (Data.map o Thm.add_thm);
+val eqvt_force_del = Thm.declaration_attribute (Data.map o Thm.del_thm);
+
+val get_eqvt_thms = Context.Proof #> Data.get;
val setup =
- Attrib.setup @{binding eqvt} (Attrib.add_del eqvt_add eqvt_del)
- "equivariance theorem declaration" #>
- Attrib.setup @{binding eqvt_force} (Attrib.add_del eqvt_force_add eqvt_force_del)
- "equivariance theorem declaration (without checking the form of the lemma)" #>
- Attrib.setup @{binding fresh} (Attrib.add_del fresh_add fresh_del)
- "freshness theorem declaration" #>
- Attrib.setup @{binding "bij"} (Attrib.add_del bij_add bij_del)
- "bijection theorem declaration" #>
- PureThy.add_thms_dynamic (Binding.name "eqvts", #eqvts o Data.get) #>
- PureThy.add_thms_dynamic (Binding.name "freshs", #freshs o Data.get) #>
- PureThy.add_thms_dynamic (Binding.name "bijs", #bijs o Data.get);
+ Attrib.setup @{binding eqvt} (Attrib.add_del eqvt_add eqvt_del)
+ "equivariance theorem declaration"
+ #> Attrib.setup @{binding eqvt_force} (Attrib.add_del eqvt_force_add eqvt_force_del)
+ "equivariance theorem declaration (without checking the form of the lemma)"
+ #> PureThy.add_thms_dynamic (Binding.name "eqvts", Data.get)
+
end;
--- a/src/HOL/Orderings.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Orderings.thy Mon Apr 27 07:26:17 2009 -0700
@@ -5,7 +5,7 @@
header {* Abstract orderings *}
theory Orderings
-imports Code_Setup
+imports HOL
uses "~~/src/Provers/order.ML"
begin
--- a/src/HOL/Power.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Power.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1,95 +1,179 @@
(* Title: HOL/Power.thy
- ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1997 University of Cambridge
-
*)
-header{*Exponentiation*}
+header {* Exponentiation *}
theory Power
imports Nat
begin
-class power =
- fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
+subsection {* Powers for Arbitrary Monoids *}
+
+class power = one + times
+begin
-subsection{*Powers for Arbitrary Monoids*}
+primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
+ power_0: "a ^ 0 = 1"
+ | power_Suc: "a ^ Suc n = a * a ^ n"
+
+notation (latex output)
+ power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
-class recpower = monoid_mult + power +
- assumes power_0 [simp]: "a ^ 0 = 1"
- assumes power_Suc [simp]: "a ^ Suc n = a * (a ^ n)"
+notation (HTML output)
+ power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
+
+end
+
+context monoid_mult
+begin
-lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
+subclass power ..
+
+lemma power_one [simp]:
+ "1 ^ n = 1"
+ by (induct n) simp_all
+
+lemma power_one_right [simp]:
+ "a ^ 1 = a"
by simp
-text{*It looks plausible as a simprule, but its effect can be strange.*}
-lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"
- by (induct n) simp_all
-
-lemma power_one [simp]: "1^n = (1::'a::recpower)"
- by (induct n) simp_all
-
-lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
- unfolding One_nat_def by simp
-
-lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
+lemma power_commutes:
+ "a ^ n * a = a * a ^ n"
by (induct n) (simp_all add: mult_assoc)
-lemma power_Suc2: "(a::'a::recpower) ^ Suc n = a ^ n * a"
+lemma power_Suc2:
+ "a ^ Suc n = a ^ n * a"
by (simp add: power_commutes)
-lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
- by (induct m) (simp_all add: mult_ac)
+lemma power_add:
+ "a ^ (m + n) = a ^ m * a ^ n"
+ by (induct m) (simp_all add: algebra_simps)
-lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
+lemma power_mult:
+ "a ^ (m * n) = (a ^ m) ^ n"
by (induct n) (simp_all add: power_add)
-lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
+end
+
+context comm_monoid_mult
+begin
+
+lemma power_mult_distrib:
+ "(a * b) ^ n = (a ^ n) * (b ^ n)"
by (induct n) (simp_all add: mult_ac)
-lemma zero_less_power[simp]:
- "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
-by (induct n) (simp_all add: mult_pos_pos)
+end
+
+context semiring_1
+begin
+
+lemma of_nat_power:
+ "of_nat (m ^ n) = of_nat m ^ n"
+ by (induct n) (simp_all add: of_nat_mult)
+
+end
+
+context comm_semiring_1
+begin
+
+text {* The divides relation *}
+
+lemma le_imp_power_dvd:
+ assumes "m \<le> n" shows "a ^ m dvd a ^ n"
+proof
+ have "a ^ n = a ^ (m + (n - m))"
+ using `m \<le> n` by simp
+ also have "\<dots> = a ^ m * a ^ (n - m)"
+ by (rule power_add)
+ finally show "a ^ n = a ^ m * a ^ (n - m)" .
+qed
+
+lemma power_le_dvd:
+ "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
+ by (rule dvd_trans [OF le_imp_power_dvd])
+
+lemma dvd_power_same:
+ "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
+ by (induct n) (auto simp add: mult_dvd_mono)
+
+lemma dvd_power_le:
+ "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
+ by (rule power_le_dvd [OF dvd_power_same])
-lemma zero_le_power[simp]:
- "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
-by (induct n) (simp_all add: mult_nonneg_nonneg)
+lemma dvd_power [simp]:
+ assumes "n > (0::nat) \<or> x = 1"
+ shows "x dvd (x ^ n)"
+using assms proof
+ assume "0 < n"
+ then have "x ^ n = x ^ Suc (n - 1)" by simp
+ then show "x dvd (x ^ n)" by simp
+next
+ assume "x = 1"
+ then show "x dvd (x ^ n)" by simp
+qed
+
+end
+
+context ring_1
+begin
+
+lemma power_minus:
+ "(- a) ^ n = (- 1) ^ n * a ^ n"
+proof (induct n)
+ case 0 show ?case by simp
+next
+ case (Suc n) then show ?case
+ by (simp del: power_Suc add: power_Suc2 mult_assoc)
+qed
+
+end
+
+context ordered_semidom
+begin
+
+lemma zero_less_power [simp]:
+ "0 < a \<Longrightarrow> 0 < a ^ n"
+ by (induct n) (simp_all add: mult_pos_pos)
+
+lemma zero_le_power [simp]:
+ "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
+ by (induct n) (simp_all add: mult_nonneg_nonneg)
lemma one_le_power[simp]:
- "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
-apply (induct "n")
-apply simp_all
-apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
-apply (simp_all add: order_trans [OF zero_le_one])
-done
-
-lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
- by (simp add: order_trans [OF zero_le_one order_less_imp_le])
+ "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
+ apply (induct n)
+ apply simp_all
+ apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
+ apply (simp_all add: order_trans [OF zero_le_one])
+ done
lemma power_gt1_lemma:
- assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
- shows "1 < a * a^n"
+ assumes gt1: "1 < a"
+ shows "1 < a * a ^ n"
proof -
- have "1*1 < a*1" using gt1 by simp
- also have "\<dots> \<le> a * a^n" using gt1
- by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le
+ from gt1 have "0 \<le> a"
+ by (fact order_trans [OF zero_le_one less_imp_le])
+ have "1 * 1 < a * 1" using gt1 by simp
+ also have "\<dots> \<le> a * a ^ n" using gt1
+ by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
zero_le_one order_refl)
finally show ?thesis by simp
qed
-lemma one_less_power[simp]:
- "\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n"
-by (cases n, simp_all add: power_gt1_lemma)
+lemma power_gt1:
+ "1 < a \<Longrightarrow> 1 < a ^ Suc n"
+ by (simp add: power_gt1_lemma)
-lemma power_gt1:
- "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
-by (simp add: power_gt1_lemma)
+lemma one_less_power [simp]:
+ "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
+ by (cases n) (simp_all add: power_gt1_lemma)
lemma power_le_imp_le_exp:
- assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
- shows "!!n. a^m \<le> a^n ==> m \<le> n"
-proof (induct m)
+ assumes gt1: "1 < a"
+ shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
+proof (induct m arbitrary: n)
case 0
show ?case by simp
next
@@ -97,212 +181,128 @@
show ?case
proof (cases n)
case 0
- from prems have "a * a^m \<le> 1" by simp
+ with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
with gt1 show ?thesis
by (force simp only: power_gt1_lemma
- linorder_not_less [symmetric])
+ not_less [symmetric])
next
case (Suc n)
- from prems show ?thesis
+ with Suc.prems Suc.hyps show ?thesis
by (force dest: mult_left_le_imp_le
- simp add: order_less_trans [OF zero_less_one gt1])
+ simp add: less_trans [OF zero_less_one gt1])
qed
qed
text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
lemma power_inject_exp [simp]:
- "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"
+ "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
by (force simp add: order_antisym power_le_imp_le_exp)
text{*Can relax the first premise to @{term "0<a"} in the case of the
natural numbers.*}
lemma power_less_imp_less_exp:
- "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
-by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
- power_le_imp_le_exp)
-
+ "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
+ by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
+ power_le_imp_le_exp)
lemma power_mono:
- "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
-apply (induct "n")
-apply simp_all
-apply (auto intro: mult_mono order_trans [of 0 a b])
-done
+ "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
+ by (induct n)
+ (auto intro: mult_mono order_trans [of 0 a b])
lemma power_strict_mono [rule_format]:
- "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
- ==> 0 < n --> a^n < b^n"
-apply (induct "n")
-apply (auto simp add: mult_strict_mono order_le_less_trans [of 0 a b])
-done
-
-lemma power_eq_0_iff [simp]:
- "(a^n = 0) \<longleftrightarrow>
- (a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,recpower}) & n\<noteq>0)"
-apply (induct "n")
-apply (auto simp add: no_zero_divisors)
-done
-
-
-lemma field_power_not_zero:
- "a \<noteq> (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \<noteq> 0"
-by force
-
-lemma nonzero_power_inverse:
- fixes a :: "'a::{division_ring,recpower}"
- shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n"
-apply (induct "n")
-apply (auto simp add: nonzero_inverse_mult_distrib power_commutes)
-done (* TODO: reorient or rename to nonzero_inverse_power *)
-
-text{*Perhaps these should be simprules.*}
-lemma power_inverse:
- fixes a :: "'a::{division_ring,division_by_zero,recpower}"
- shows "inverse (a ^ n) = (inverse a) ^ n"
-apply (cases "a = 0")
-apply (simp add: power_0_left)
-apply (simp add: nonzero_power_inverse)
-done (* TODO: reorient or rename to inverse_power *)
-
-lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n =
- (1 / a)^n"
-apply (simp add: divide_inverse)
-apply (rule power_inverse)
-done
-
-lemma nonzero_power_divide:
- "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
-by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
-
-lemma power_divide:
- "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
-apply (case_tac "b=0", simp add: power_0_left)
-apply (rule nonzero_power_divide)
-apply assumption
-done
-
-lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
-apply (induct "n")
-apply (auto simp add: abs_mult)
-done
-
-lemma abs_power_minus [simp]:
- fixes a:: "'a::{ordered_idom,recpower}" shows "abs((-a) ^ n) = abs(a ^ n)"
- by (simp add: abs_minus_cancel power_abs)
-
-lemma zero_less_power_abs_iff [simp,noatp]:
- "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
-proof (induct "n")
- case 0
- show ?case by simp
-next
- case (Suc n)
- show ?case by (auto simp add: prems zero_less_mult_iff)
-qed
-
-lemma zero_le_power_abs [simp]:
- "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
-by (rule zero_le_power [OF abs_ge_zero])
-
-lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{ring_1,recpower}) ^ n"
-proof (induct n)
- case 0 show ?case by simp
-next
- case (Suc n) then show ?case
- by (simp del: power_Suc add: power_Suc2 mult_assoc)
-qed
+ "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
+ by (induct n)
+ (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
text{*Lemma for @{text power_strict_decreasing}*}
lemma power_Suc_less:
- "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
- ==> a * a^n < a^n"
-apply (induct n)
-apply (auto simp add: mult_strict_left_mono)
-done
+ "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
+ by (induct n)
+ (auto simp add: mult_strict_left_mono)
-lemma power_strict_decreasing:
- "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
- ==> a^N < a^n"
-apply (erule rev_mp)
-apply (induct "N")
-apply (auto simp add: power_Suc_less less_Suc_eq)
-apply (rename_tac m)
-apply (subgoal_tac "a * a^m < 1 * a^n", simp)
-apply (rule mult_strict_mono)
-apply (auto simp add: order_less_imp_le)
-done
+lemma power_strict_decreasing [rule_format]:
+ "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
+proof (induct N)
+ case 0 then show ?case by simp
+next
+ case (Suc N) then show ?case
+ apply (auto simp add: power_Suc_less less_Suc_eq)
+ apply (subgoal_tac "a * a^N < 1 * a^n")
+ apply simp
+ apply (rule mult_strict_mono) apply auto
+ done
+qed
text{*Proof resembles that of @{text power_strict_decreasing}*}
-lemma power_decreasing:
- "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
- ==> a^N \<le> a^n"
-apply (erule rev_mp)
-apply (induct "N")
-apply (auto simp add: le_Suc_eq)
-apply (rename_tac m)
-apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
-apply (rule mult_mono)
-apply auto
-done
+lemma power_decreasing [rule_format]:
+ "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
+proof (induct N)
+ case 0 then show ?case by simp
+next
+ case (Suc N) then show ?case
+ apply (auto simp add: le_Suc_eq)
+ apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
+ apply (rule mult_mono) apply auto
+ done
+qed
lemma power_Suc_less_one:
- "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
-apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
-done
+ "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
+ using power_strict_decreasing [of 0 "Suc n" a] by simp
text{*Proof again resembles that of @{text power_strict_decreasing}*}
-lemma power_increasing:
- "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
-apply (erule rev_mp)
-apply (induct "N")
-apply (auto simp add: le_Suc_eq)
-apply (rename_tac m)
-apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
-apply (rule mult_mono)
-apply (auto simp add: order_trans [OF zero_le_one])
-done
+lemma power_increasing [rule_format]:
+ "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
+proof (induct N)
+ case 0 then show ?case by simp
+next
+ case (Suc N) then show ?case
+ apply (auto simp add: le_Suc_eq)
+ apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
+ apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
+ done
+qed
text{*Lemma for @{text power_strict_increasing}*}
lemma power_less_power_Suc:
- "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
-apply (induct n)
-apply (auto simp add: mult_strict_left_mono order_less_trans [OF zero_less_one])
-done
+ "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
+ by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
-lemma power_strict_increasing:
- "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
-apply (erule rev_mp)
-apply (induct "N")
-apply (auto simp add: power_less_power_Suc less_Suc_eq)
-apply (rename_tac m)
-apply (subgoal_tac "1 * a^n < a * a^m", simp)
-apply (rule mult_strict_mono)
-apply (auto simp add: order_less_trans [OF zero_less_one] order_less_imp_le)
-done
+lemma power_strict_increasing [rule_format]:
+ "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
+proof (induct N)
+ case 0 then show ?case by simp
+next
+ case (Suc N) then show ?case
+ apply (auto simp add: power_less_power_Suc less_Suc_eq)
+ apply (subgoal_tac "1 * a^n < a * a^N", simp)
+ apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
+ done
+qed
lemma power_increasing_iff [simp]:
- "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
-by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le)
+ "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
+ by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
lemma power_strict_increasing_iff [simp]:
- "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
+ "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
by (blast intro: power_less_imp_less_exp power_strict_increasing)
lemma power_le_imp_le_base:
-assumes le: "a ^ Suc n \<le> b ^ Suc n"
- and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"
-shows "a \<le> b"
+ assumes le: "a ^ Suc n \<le> b ^ Suc n"
+ and ynonneg: "0 \<le> b"
+ shows "a \<le> b"
proof (rule ccontr)
assume "~ a \<le> b"
then have "b < a" by (simp only: linorder_not_le)
then have "b ^ Suc n < a ^ Suc n"
by (simp only: prems power_strict_mono)
- from le and this show "False"
+ from le and this show False
by (simp add: linorder_not_less [symmetric])
qed
lemma power_less_imp_less_base:
- fixes a b :: "'a::{ordered_semidom,recpower}"
assumes less: "a ^ n < b ^ n"
assumes nonneg: "0 \<le> b"
shows "a < b"
@@ -310,98 +310,144 @@
assume "~ a < b"
hence "b \<le> a" by (simp only: linorder_not_less)
hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
- thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less)
+ thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
qed
lemma power_inject_base:
- "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
- ==> a = (b::'a::{ordered_semidom,recpower})"
-by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
+ "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
+by (blast intro: power_le_imp_le_base antisym eq_refl sym)
lemma power_eq_imp_eq_base:
- fixes a b :: "'a::{ordered_semidom,recpower}"
- shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"
-by (cases n, simp_all del: power_Suc, rule power_inject_base)
+ "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
+ by (cases n) (simp_all del: power_Suc, rule power_inject_base)
-text {* The divides relation *}
+end
+
+context ordered_idom
+begin
-lemma le_imp_power_dvd:
- fixes a :: "'a::{comm_semiring_1,recpower}"
- assumes "m \<le> n" shows "a^m dvd a^n"
-proof
- have "a^n = a^(m + (n - m))"
- using `m \<le> n` by simp
- also have "\<dots> = a^m * a^(n - m)"
- by (rule power_add)
- finally show "a^n = a^m * a^(n - m)" .
+lemma power_abs:
+ "abs (a ^ n) = abs a ^ n"
+ by (induct n) (auto simp add: abs_mult)
+
+lemma abs_power_minus [simp]:
+ "abs ((-a) ^ n) = abs (a ^ n)"
+ by (simp add: abs_minus_cancel power_abs)
+
+lemma zero_less_power_abs_iff [simp, noatp]:
+ "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
+proof (induct n)
+ case 0 show ?case by simp
+next
+ case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
qed
-lemma power_le_dvd:
- fixes a b :: "'a::{comm_semiring_1,recpower}"
- shows "a^n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a^m dvd b"
- by (rule dvd_trans [OF le_imp_power_dvd])
-
-
-lemma dvd_power_same:
- "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> x^n dvd y^n"
-by (induct n) (auto simp add: mult_dvd_mono)
-
-lemma dvd_power_le:
- "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> m >= n \<Longrightarrow> x^n dvd y^m"
-by(rule power_le_dvd[OF dvd_power_same])
+lemma zero_le_power_abs [simp]:
+ "0 \<le> abs a ^ n"
+ by (rule zero_le_power [OF abs_ge_zero])
-lemma dvd_power [simp]:
- "n > 0 | (x::'a::{comm_semiring_1,recpower}) = 1 \<Longrightarrow> x dvd x^n"
-apply (erule disjE)
- apply (subgoal_tac "x ^ n = x^(Suc (n - 1))")
- apply (erule ssubst)
- apply (subst power_Suc)
- apply auto
-done
+end
-
-subsection{*Exponentiation for the Natural Numbers*}
-
-instantiation nat :: recpower
+context ring_1_no_zero_divisors
begin
-primrec power_nat where
- "p ^ 0 = (1\<Colon>nat)"
- | "p ^ (Suc n) = (p\<Colon>nat) * (p ^ n)"
+lemma field_power_not_zero:
+ "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
+ by (induct n) auto
+
+end
+
+context division_ring
+begin
-instance proof
- fix z n :: nat
- show "z^0 = 1" by simp
- show "z^(Suc n) = z * (z^n)" by simp
-qed
+text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
+lemma nonzero_power_inverse:
+ "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
+ by (induct n)
+ (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
-declare power_nat.simps [simp del]
+end
+
+context field
+begin
+
+lemma nonzero_power_divide:
+ "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
+ by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
end
-lemma of_nat_power:
- "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
-by (induct n, simp_all add: of_nat_mult)
+lemma power_0_Suc [simp]:
+ "(0::'a::{power, semiring_0}) ^ Suc n = 0"
+ by simp
+
+text{*It looks plausible as a simprule, but its effect can be strange.*}
+lemma power_0_left:
+ "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
+ by (induct n) simp_all
+
+lemma power_eq_0_iff [simp]:
+ "a ^ n = 0 \<longleftrightarrow>
+ a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
+ by (induct n)
+ (auto simp add: no_zero_divisors elim: contrapos_pp)
+
+lemma power_diff:
+ fixes a :: "'a::field"
+ assumes nz: "a \<noteq> 0"
+ shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
+ by (induct m n rule: diff_induct) (simp_all add: nz)
-lemma nat_one_le_power [simp]: "Suc 0 \<le> i ==> Suc 0 \<le> i^n"
-by (rule one_le_power [of i n, unfolded One_nat_def])
+text{*Perhaps these should be simprules.*}
+lemma power_inverse:
+ fixes a :: "'a::{division_ring,division_by_zero,power}"
+ shows "inverse (a ^ n) = (inverse a) ^ n"
+apply (cases "a = 0")
+apply (simp add: power_0_left)
+apply (simp add: nonzero_power_inverse)
+done (* TODO: reorient or rename to inverse_power *)
+
+lemma power_one_over:
+ "1 / (a::'a::{field,division_by_zero, power}) ^ n = (1 / a) ^ n"
+ by (simp add: divide_inverse) (rule power_inverse)
-lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
-by (induct "n", auto)
+lemma power_divide:
+ "(a / b) ^ n = (a::'a::{field,division_by_zero}) ^ n / b ^ n"
+apply (cases "b = 0")
+apply (simp add: power_0_left)
+apply (rule nonzero_power_divide)
+apply assumption
+done
+
+class recpower = monoid_mult
+
+
+subsection {* Exponentiation for the Natural Numbers *}
+
+instance nat :: recpower ..
+
+lemma nat_one_le_power [simp]:
+ "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
+ by (rule one_le_power [of i n, unfolded One_nat_def])
+
+lemma nat_zero_less_power_iff [simp]:
+ "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
+ by (induct n) auto
lemma nat_power_eq_Suc_0_iff [simp]:
- "((x::nat)^m = Suc 0) = (m = 0 | x = Suc 0)"
-by (induct_tac m, auto)
+ "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
+ by (induct m) auto
-lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0"
-by simp
+lemma power_Suc_0 [simp]:
+ "Suc 0 ^ n = Suc 0"
+ by simp
text{*Valid for the naturals, but what if @{text"0<i<1"}?
Premises cannot be weakened: consider the case where @{term "i=0"},
@{term "m=1"} and @{term "n=0"}.*}
lemma nat_power_less_imp_less:
assumes nonneg: "0 < (i\<Colon>nat)"
- assumes less: "i^m < i^n"
+ assumes less: "i ^ m < i ^ n"
shows "m < n"
proof (cases "i = 1")
case True with less power_one [where 'a = nat] show ?thesis by simp
@@ -410,10 +456,4 @@
from power_strict_increasing_iff [OF this] less show ?thesis ..
qed
-lemma power_diff:
- assumes nz: "a ~= 0"
- shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"
- by (induct m n rule: diff_induct)
- (simp_all add: nonzero_mult_divide_cancel_left nz)
-
end
--- a/src/HOL/Predicate.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Predicate.thy Mon Apr 27 07:26:17 2009 -0700
@@ -622,6 +622,51 @@
"pred_rec f P = f (eval P)"
by (cases P) simp
+text {* for evaluation of predicate enumerations *}
+
+ML {*
+signature PREDICATE =
+sig
+ datatype 'a pred = Seq of (unit -> 'a seq)
+ and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
+ val yield: 'a pred -> ('a * 'a pred) option
+ val yieldn: int -> 'a pred -> 'a list * 'a pred
+end;
+
+structure Predicate : PREDICATE =
+struct
+
+@{code_datatype pred = Seq};
+@{code_datatype seq = Empty | Insert | Join};
+
+fun yield (Seq f) = next (f ())
+and next @{code Empty} = NONE
+ | next (@{code Insert} (x, P)) = SOME (x, P)
+ | next (@{code Join} (P, xq)) = (case yield P
+ of NONE => next xq
+ | SOME (x, Q) => SOME (x, @{code Seq} (fn _ => @{code Join} (Q, xq))))
+
+fun anamorph f k x = (if k = 0 then ([], x)
+ else case f x
+ of NONE => ([], x)
+ | SOME (v, y) => let
+ val (vs, z) = anamorph f (k - 1) y
+ in (v :: vs, z) end)
+
+fun yieldn P = anamorph yield P;
+
+end;
+*}
+
+code_reserved Eval Predicate
+
+code_type pred and seq
+ (Eval "_/ Predicate.pred" and "_/ Predicate.seq")
+
+code_const Seq and Empty and Insert and Join
+ (Eval "Predicate.Seq" and "Predicate.Empty" and "Predicate.Insert/ (_,/ _)" and "Predicate.Join/ (_,/ _)")
+
+
no_notation
inf (infixl "\<sqinter>" 70) and
sup (infixl "\<squnion>" 65) and
--- a/src/HOL/Product_Type.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Product_Type.thy Mon Apr 27 07:26:17 2009 -0700
@@ -84,6 +84,14 @@
lemma unit_abs_eta_conv [simp,noatp]: "(%u::unit. f ()) = f"
by (rule ext) simp
+instantiation unit :: default
+begin
+
+definition "default = ()"
+
+instance ..
+
+end
text {* code generator setup *}
--- a/src/HOL/Rational.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Rational.thy Mon Apr 27 07:26:17 2009 -0700
@@ -156,11 +156,6 @@
then show ?thesis by (simp add: mult_rat [symmetric])
qed
-primrec power_rat
-where
- "q ^ 0 = (1\<Colon>rat)"
-| "q ^ Suc n = (q\<Colon>rat) * (q ^ n)"
-
instance proof
fix q r s :: rat show "(q * r) * s = q * (r * s)"
by (cases q, cases r, cases s) (simp add: eq_rat)
@@ -193,15 +188,8 @@
next
fix q :: rat show "q * 1 = q"
by (cases q) (simp add: One_rat_def eq_rat)
-next
- fix q :: rat
- fix n :: nat
- show "q ^ 0 = 1" by simp
- show "q ^ (Suc n) = q * (q ^ n)" by simp
qed
-declare power_rat.simps [simp del]
-
end
lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
@@ -222,7 +210,8 @@
definition
rat_number_of_def [code del]: "number_of w = Fract w 1"
-instance by intro_classes (simp add: rat_number_of_def of_int_rat)
+instance proof
+qed (simp add: rat_number_of_def of_int_rat)
end
--- a/src/HOL/RealPow.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/RealPow.thy Mon Apr 27 07:26:17 2009 -0700
@@ -12,24 +12,7 @@
declare abs_mult_self [simp]
-instantiation real :: recpower
-begin
-
-primrec power_real where
- "r ^ 0 = (1\<Colon>real)"
-| "r ^ Suc n = (r\<Colon>real) * r ^ n"
-
-instance proof
- fix z :: real
- fix n :: nat
- show "z^0 = 1" by simp
- show "z^(Suc n) = z * (z^n)" by simp
-qed
-
-declare power_real.simps [simp del]
-
-end
-
+instance real :: recpower ..
lemma two_realpow_ge_one [simp]: "(1::real) \<le> 2 ^ n"
by simp
@@ -47,7 +30,6 @@
lemma realpow_minus_mult [rule_format]:
"0 < n --> (x::real) ^ (n - 1) * x = x ^ n"
-unfolding One_nat_def
apply (simp split add: nat_diff_split)
done
--- a/src/HOL/Relation_Power.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Relation_Power.thy Mon Apr 27 07:26:17 2009 -0700
@@ -9,132 +9,124 @@
imports Power Transitive_Closure Plain
begin
-instance
- "fun" :: (type, type) power ..
- --{* only type @{typ "'a => 'a"} should be in class @{text power}!*}
+consts funpower :: "('a \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'b" (infixr "^^" 80)
overloading
- relpow \<equiv> "power \<Colon> ('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a \<times> 'a) set" (unchecked)
+ relpow \<equiv> "funpower \<Colon> ('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a \<times> 'a) set"
begin
-text {* @{text "R ^ n = R O ... O R"}, the n-fold composition of @{text R} *}
+text {* @{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R} *}
primrec relpow where
- "(R \<Colon> ('a \<times> 'a) set) ^ 0 = Id"
- | "(R \<Colon> ('a \<times> 'a) set) ^ Suc n = R O (R ^ n)"
+ "(R \<Colon> ('a \<times> 'a) set) ^^ 0 = Id"
+ | "(R \<Colon> ('a \<times> 'a) set) ^^ Suc n = R O (R ^^ n)"
end
overloading
- funpow \<equiv> "power \<Colon> ('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" (unchecked)
+ funpow \<equiv> "funpower \<Colon> ('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"
begin
-text {* @{text "f ^ n = f o ... o f"}, the n-fold composition of @{text f} *}
+text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
primrec funpow where
- "(f \<Colon> 'a \<Rightarrow> 'a) ^ 0 = id"
- | "(f \<Colon> 'a \<Rightarrow> 'a) ^ Suc n = f o (f ^ n)"
+ "(f \<Colon> 'a \<Rightarrow> 'a) ^^ 0 = id"
+ | "(f \<Colon> 'a \<Rightarrow> 'a) ^^ Suc n = f o (f ^^ n)"
end
-text{*WARNING: due to the limits of Isabelle's type classes, exponentiation on
-functions and relations has too general a domain, namely @{typ "('a * 'b)set"}
-and @{typ "'a => 'b"}. Explicit type constraints may therefore be necessary.
-For example, @{term "range(f^n) = A"} and @{term "Range(R^n) = B"} need
-constraints.*}
-
-text {*
- Circumvent this problem for code generation:
-*}
-
-primrec
- fun_pow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
-where
- "fun_pow 0 f = id"
+primrec fun_pow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
+ "fun_pow 0 f = id"
| "fun_pow (Suc n) f = f o fun_pow n f"
-lemma funpow_fun_pow [code unfold]: "f ^ n = fun_pow n f"
+lemma funpow_fun_pow [code unfold]:
+ "f ^^ n = fun_pow n f"
unfolding funpow_def fun_pow_def ..
-lemma funpow_add: "f ^ (m+n) = f^m o f^n"
+lemma funpow_add:
+ "f ^^ (m + n) = f ^^ m o f ^^ n"
by (induct m) simp_all
-lemma funpow_swap1: "f((f^n) x) = (f^n)(f x)"
+lemma funpow_swap1:
+ "f ((f ^^ n) x) = (f ^^ n) (f x)"
proof -
- have "f((f^n) x) = (f^(n+1)) x" unfolding One_nat_def by simp
- also have "\<dots> = (f^n o f^1) x" by (simp only: funpow_add)
- also have "\<dots> = (f^n)(f x)" unfolding One_nat_def by simp
+ have "f ((f ^^ n) x) = (f ^^ (n+1)) x" unfolding One_nat_def by simp
+ also have "\<dots> = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
+ also have "\<dots> = (f ^^ n) (f x)" unfolding One_nat_def by simp
finally show ?thesis .
qed
lemma rel_pow_1 [simp]:
- fixes R :: "('a*'a)set"
- shows "R^1 = R"
- unfolding One_nat_def by simp
-
-lemma rel_pow_0_I: "(x,x) : R^0"
+ fixes R :: "('a * 'a) set"
+ shows "R ^^ 1 = R"
by simp
-lemma rel_pow_Suc_I: "[| (x,y) : R^n; (y,z):R |] ==> (x,z):R^(Suc n)"
+lemma rel_pow_0_I:
+ "(x, x) \<in> R ^^ 0"
+ by simp
+
+lemma rel_pow_Suc_I:
+ "(x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
by auto
lemma rel_pow_Suc_I2:
- "(x, y) : R \<Longrightarrow> (y, z) : R^n \<Longrightarrow> (x,z) : R^(Suc n)"
- apply (induct n arbitrary: z)
- apply simp
- apply fastsimp
- done
+ "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
+ by (induct n arbitrary: z) (simp, fastsimp)
-lemma rel_pow_0_E: "[| (x,y) : R^0; x=y ==> P |] ==> P"
+lemma rel_pow_0_E:
+ "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
by simp
lemma rel_pow_Suc_E:
- "[| (x,z) : R^(Suc n); !!y. [| (x,y) : R^n; (y,z) : R |] ==> P |] ==> P"
+ "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
by auto
lemma rel_pow_E:
- "[| (x,z) : R^n; [| n=0; x = z |] ==> P;
- !!y m. [| n = Suc m; (x,y) : R^m; (y,z) : R |] ==> P
- |] ==> P"
+ "(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
+ \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
+ \<Longrightarrow> P"
by (cases n) auto
lemma rel_pow_Suc_D2:
- "(x, z) : R^(Suc n) \<Longrightarrow> (\<exists>y. (x,y) : R & (y,z) : R^n)"
+ "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
apply (induct n arbitrary: x z)
apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)
apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)
done
lemma rel_pow_Suc_D2':
- "\<forall>x y z. (x,y) : R^n & (y,z) : R --> (\<exists>w. (x,w) : R & (w,z) : R^n)"
+ "\<forall>x y z. (x, y) \<in> R ^^ n \<and> (y, z) \<in> R \<longrightarrow> (\<exists>w. (x, w) \<in> R \<and> (w, z) \<in> R ^^ n)"
by (induct n) (simp_all, blast)
lemma rel_pow_E2:
- "[| (x,z) : R^n; [| n=0; x = z |] ==> P;
- !!y m. [| n = Suc m; (x,y) : R; (y,z) : R^m |] ==> P
- |] ==> P"
- apply (case_tac n, simp)
+ "(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
+ \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
+ \<Longrightarrow> P"
+ apply (cases n, simp)
apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast)
done
-lemma rtrancl_imp_UN_rel_pow: "!!p. p:R^* ==> p : (UN n. R^n)"
- apply (simp only: split_tupled_all)
+lemma rtrancl_imp_UN_rel_pow:
+ "p \<in> R^* \<Longrightarrow> p \<in> (\<Union>n. R ^^ n)"
+ apply (cases p) apply (simp only:)
apply (erule rtrancl_induct)
apply (blast intro: rel_pow_0_I rel_pow_Suc_I)+
done
-lemma rel_pow_imp_rtrancl: "!!p. p:R^n ==> p:R^*"
- apply (simp only: split_tupled_all)
- apply (induct n)
+lemma rel_pow_imp_rtrancl:
+ "p \<in> R ^^ n \<Longrightarrow> p \<in> R^*"
+ apply (induct n arbitrary: p)
+ apply (simp_all only: split_tupled_all)
apply (blast intro: rtrancl_refl elim: rel_pow_0_E)
apply (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)
done
-lemma rtrancl_is_UN_rel_pow: "R^* = (UN n. R^n)"
+lemma rtrancl_is_UN_rel_pow:
+ "R^* = (UN n. R ^^ n)"
by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)
lemma trancl_power:
- "x \<in> r^+ = (\<exists>n > 0. x \<in> r^n)"
+ "x \<in> r^+ = (\<exists>n > 0. x \<in> r ^^ n)"
apply (cases x)
apply simp
apply (rule iffI)
@@ -151,30 +143,12 @@
done
lemma single_valued_rel_pow:
- "!!r::('a * 'a)set. single_valued r ==> single_valued (r^n)"
+ fixes R :: "('a * 'a) set"
+ shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
+ apply (induct n arbitrary: R)
+ apply simp_all
apply (rule single_valuedI)
- apply (induct n)
- apply simp
apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
done
-ML
-{*
-val funpow_add = thm "funpow_add";
-val rel_pow_1 = thm "rel_pow_1";
-val rel_pow_0_I = thm "rel_pow_0_I";
-val rel_pow_Suc_I = thm "rel_pow_Suc_I";
-val rel_pow_Suc_I2 = thm "rel_pow_Suc_I2";
-val rel_pow_0_E = thm "rel_pow_0_E";
-val rel_pow_Suc_E = thm "rel_pow_Suc_E";
-val rel_pow_E = thm "rel_pow_E";
-val rel_pow_Suc_D2 = thm "rel_pow_Suc_D2";
-val rel_pow_Suc_D2 = thm "rel_pow_Suc_D2";
-val rel_pow_E2 = thm "rel_pow_E2";
-val rtrancl_imp_UN_rel_pow = thm "rtrancl_imp_UN_rel_pow";
-val rel_pow_imp_rtrancl = thm "rel_pow_imp_rtrancl";
-val rtrancl_is_UN_rel_pow = thm "rtrancl_is_UN_rel_pow";
-val single_valued_rel_pow = thm "single_valued_rel_pow";
-*}
-
end
--- a/src/HOL/Ring_and_Field.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Ring_and_Field.thy Mon Apr 27 07:26:17 2009 -0700
@@ -2226,15 +2226,21 @@
qed
qed
-instance ordered_idom \<subseteq> pordered_ring_abs
-by default (auto simp add: abs_if not_less
- equal_neg_zero neg_equal_zero mult_less_0_iff)
-
-lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)"
-by (simp add: abs_eq_mult linorder_linear)
-
-lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
-by (simp add: abs_if)
+context ordered_idom
+begin
+
+subclass pordered_ring_abs proof
+qed (auto simp add: abs_if not_less equal_neg_zero neg_equal_zero mult_less_0_iff)
+
+lemma abs_mult:
+ "abs (a * b) = abs a * abs b"
+ by (rule abs_eq_mult) auto
+
+lemma abs_mult_self:
+ "abs a * abs a = a * a"
+ by (simp add: abs_if)
+
+end
lemma nonzero_abs_inverse:
"a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
--- a/src/HOL/SizeChange/Graphs.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/SizeChange/Graphs.thy Mon Apr 27 07:26:17 2009 -0700
@@ -228,18 +228,8 @@
qed
qed
-instantiation graph :: (type, monoid_mult) "{semiring_1, idem_add, recpower, star}"
-begin
-
-primrec power_graph :: "('a\<Colon>type, 'b\<Colon>monoid_mult) graph \<Rightarrow> nat => ('a, 'b) graph"
-where
- "(A \<Colon> ('a, 'b) graph) ^ 0 = 1"
-| "(A \<Colon> ('a, 'b) graph) ^ Suc n = A * (A ^ n)"
-
-definition
- graph_star_def: "star (G \<Colon> ('a, 'b) graph) = (SUP n. G ^ n)"
-
-instance proof
+instance graph :: (type, monoid_mult) "{semiring_1, idem_add, recpower}"
+proof
fix a b c :: "('a, 'b) graph"
show "1 * a = a"
@@ -258,10 +248,16 @@
show "a + a = a" unfolding graph_plus_def by simp
- show "a ^ 0 = 1" "\<And>n. a ^ (Suc n) = a * a ^ n"
- by simp_all
qed
+instantiation graph :: (type, monoid_mult) star
+begin
+
+definition
+ graph_star_def: "star (G \<Colon> ('a, 'b) graph) = (SUP n. G ^ n)"
+
+instance ..
+
end
lemma graph_leqI:
@@ -351,7 +347,7 @@
lemma in_tcl:
"has_edge (tcl G) a x b = (\<exists>n>0. has_edge (G ^ n) a x b)"
- apply (auto simp: tcl_is_SUP in_SUP simp del: power_graph.simps power_Suc)
+ apply (auto simp: tcl_is_SUP in_SUP simp del: power.simps power_Suc)
apply (rule_tac x = "n - 1" in exI, auto)
done
--- a/src/HOL/SizeChange/Interpretation.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/SizeChange/Interpretation.thy Mon Apr 27 07:26:17 2009 -0700
@@ -35,7 +35,7 @@
and nia: "\<And>x. \<not>accp R x \<Longrightarrow> \<not>accp R (f x)"
by blast
- let ?s = "\<lambda>i. (f ^ i) x"
+ let ?s = "\<lambda>i. (f ^^ i) x"
{
fix i
--- a/src/HOL/SizeChange/Size_Change_Termination.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/SizeChange/Size_Change_Termination.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1,5 +1,4 @@
(* Title: HOL/Library/Size_Change_Termination.thy
- ID: $Id$
Author: Alexander Krauss, TU Muenchen
*)
--- a/src/HOL/Tools/Qelim/presburger.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Tools/Qelim/presburger.ML Mon Apr 27 07:26:17 2009 -0700
@@ -131,7 +131,7 @@
@{thm "div_0"}, @{thm "mod_0"}, @{thm "div_by_1"}, @{thm "mod_by_1"}, @{thm "div_1"},
@{thm "mod_1"}, @{thm "Suc_plus1"}]
@ @{thms add_ac}
- addsimprocs [cancel_div_mod_proc]
+ addsimprocs [cancel_div_mod_nat_proc, cancel_div_mod_int_proc]
val splits_ss = comp_ss addsimps [@{thm "mod_div_equality'"}] addsplits
[@{thm "split_zdiv"}, @{thm "split_zmod"}, @{thm "split_div'"},
@{thm "split_min"}, @{thm "split_max"}, @{thm "abs_split"}]
--- a/src/HOL/Tools/atp_manager.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Tools/atp_manager.ML Mon Apr 27 07:26:17 2009 -0700
@@ -51,15 +51,17 @@
fun set_timeout time = CRITICAL (fn () => timeout := time);
val _ =
- ProofGeneralPgip.add_preference "Proof"
+ ProofGeneralPgip.add_preference Preferences.category_proof
(Preferences.string_pref atps
"ATP: provers" "Default automatic provers (separated by whitespace)");
-val _ = ProofGeneralPgip.add_preference "Proof"
+val _ =
+ ProofGeneralPgip.add_preference Preferences.category_proof
(Preferences.int_pref max_atps
"ATP: maximum number" "How many provers may run in parallel");
-val _ = ProofGeneralPgip.add_preference "Proof"
+val _ =
+ ProofGeneralPgip.add_preference Preferences.category_proof
(Preferences.int_pref timeout
"ATP: timeout" "ATPs will be interrupted after this time (in seconds)");
--- a/src/HOL/Tools/atp_wrapper.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Tools/atp_wrapper.ML Mon Apr 27 07:26:17 2009 -0700
@@ -1,5 +1,4 @@
(* Title: HOL/Tools/atp_wrapper.ML
- ID: $Id$
Author: Fabian Immler, TU Muenchen
Wrapper functions for external ATPs.
@@ -179,7 +178,7 @@
fun remote_prover_opts max_new theory_const args timeout =
tptp_prover_opts max_new theory_const
- (Path.explode "$ISABELLE_HOME/contrib/SystemOnTPTP/remote", args ^ " -t " ^ string_of_int timeout)
+ (Path.explode "$ISABELLE_HOME/lib/scripts/SystemOnTPTP", args ^ " -t " ^ string_of_int timeout)
timeout;
val remote_prover = remote_prover_opts 60 false;
--- a/src/HOL/Tools/int_factor_simprocs.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Tools/int_factor_simprocs.ML Mon Apr 27 07:26:17 2009 -0700
@@ -1,5 +1,4 @@
(* Title: HOL/int_factor_simprocs.ML
- ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2000 University of Cambridge
@@ -46,13 +45,13 @@
@{thm mult_zero_right}, @{thm mult_Bit1}, @{thm mult_1_right}];
end
-(*Version for integer division*)
-structure IntDivCancelNumeralFactor = CancelNumeralFactorFun
+(*Version for semiring_div*)
+structure DivCancelNumeralFactor = CancelNumeralFactorFun
(open CancelNumeralFactorCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_binop @{const_name Divides.div}
- val dest_bal = HOLogic.dest_bin @{const_name Divides.div} HOLogic.intT
- val cancel = @{thm zdiv_zmult_zmult1} RS trans
+ val dest_bal = HOLogic.dest_bin @{const_name Divides.div} Term.dummyT
+ val cancel = @{thm div_mult_mult1} RS trans
val neg_exchanges = false
)
@@ -108,8 +107,9 @@
"(l::'a::{ordered_idom,number_ring}) <= m * n"],
K LeCancelNumeralFactor.proc),
("int_div_cancel_numeral_factors",
- ["((l::int) * m) div n", "(l::int) div (m * n)"],
- K IntDivCancelNumeralFactor.proc),
+ ["((l::'a::{semiring_div,number_ring}) * m) div n",
+ "(l::'a::{semiring_div,number_ring}) div (m * n)"],
+ K DivCancelNumeralFactor.proc),
("divide_cancel_numeral_factor",
["((l::'a::{division_by_zero,field,number_ring}) * m) / n",
"(l::'a::{division_by_zero,field,number_ring}) / (m * n)",
@@ -284,24 +284,25 @@
@{thm mult_less_cancel_left_pos} @{thm mult_less_cancel_left_neg}
);
-(*zdiv_zmult_zmult1_if is for integer division (div).*)
-structure IntDivCancelFactor = ExtractCommonTermFun
+(*for semirings with division*)
+structure DivCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_binop @{const_name Divides.div}
- val dest_bal = HOLogic.dest_bin @{const_name Divides.div} HOLogic.intT
- val simp_conv = K (K (SOME @{thm zdiv_zmult_zmult1_if}))
+ val dest_bal = HOLogic.dest_bin @{const_name Divides.div} Term.dummyT
+ val simp_conv = K (K (SOME @{thm div_mult_mult1_if}))
);
-structure IntModCancelFactor = ExtractCommonTermFun
+structure ModCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_binop @{const_name Divides.mod}
val dest_bal = HOLogic.dest_bin @{const_name Divides.mod} HOLogic.intT
- val simp_conv = K (K (SOME @{thm zmod_zmult_zmult1}))
+ val simp_conv = K (K (SOME @{thm mod_mult_mult1}))
);
-structure IntDvdCancelFactor = ExtractCommonTermFun
+(*for idoms*)
+structure DvdCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_binrel @{const_name Ring_and_Field.dvd}
@@ -321,8 +322,8 @@
val cancel_factors =
map Arith_Data.prep_simproc
[("ring_eq_cancel_factor",
- ["(l::'a::{idom}) * m = n",
- "(l::'a::{idom}) = m * n"],
+ ["(l::'a::idom) * m = n",
+ "(l::'a::idom) = m * n"],
K EqCancelFactor.proc),
("ordered_ring_le_cancel_factor",
["(l::'a::ordered_ring) * m <= n",
@@ -333,14 +334,14 @@
"(l::'a::ordered_ring) < m * n"],
K LessCancelFactor.proc),
("int_div_cancel_factor",
- ["((l::int) * m) div n", "(l::int) div (m * n)"],
- K IntDivCancelFactor.proc),
+ ["((l::'a::semiring_div) * m) div n", "(l::'a::semiring_div) div (m * n)"],
+ K DivCancelFactor.proc),
("int_mod_cancel_factor",
- ["((l::int) * m) mod n", "(l::int) mod (m * n)"],
- K IntModCancelFactor.proc),
+ ["((l::'a::semiring_div) * m) mod n", "(l::'a::semiring_div) mod (m * n)"],
+ K ModCancelFactor.proc),
("dvd_cancel_factor",
["((l::'a::idom) * m) dvd n", "(l::'a::idom) dvd (m * n)"],
- K IntDvdCancelFactor.proc),
+ K DvdCancelFactor.proc),
("divide_cancel_factor",
["((l::'a::{division_by_zero,field}) * m) / n",
"(l::'a::{division_by_zero,field}) / (m * n)"],
--- a/src/HOL/Transitive_Closure.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Transitive_Closure.thy Mon Apr 27 07:26:17 2009 -0700
@@ -630,6 +630,140 @@
declare trancl_into_rtrancl [elim]
+subsection {* The power operation on relations *}
+
+text {* @{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R} *}
+
+overloading
+ relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
+begin
+
+primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
+ "relpow 0 R = Id"
+ | "relpow (Suc n) R = R O (R ^^ n)"
+
+end
+
+lemma rel_pow_1 [simp]:
+ fixes R :: "('a \<times> 'a) set"
+ shows "R ^^ 1 = R"
+ by simp
+
+lemma rel_pow_0_I:
+ "(x, x) \<in> R ^^ 0"
+ by simp
+
+lemma rel_pow_Suc_I:
+ "(x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
+ by auto
+
+lemma rel_pow_Suc_I2:
+ "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
+ by (induct n arbitrary: z) (simp, fastsimp)
+
+lemma rel_pow_0_E:
+ "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
+ by simp
+
+lemma rel_pow_Suc_E:
+ "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
+ by auto
+
+lemma rel_pow_E:
+ "(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
+ \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
+ \<Longrightarrow> P"
+ by (cases n) auto
+
+lemma rel_pow_Suc_D2:
+ "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
+ apply (induct n arbitrary: x z)
+ apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)
+ apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)
+ done
+
+lemma rel_pow_Suc_E2:
+ "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P"
+ by (blast dest: rel_pow_Suc_D2)
+
+lemma rel_pow_Suc_D2':
+ "\<forall>x y z. (x, y) \<in> R ^^ n \<and> (y, z) \<in> R \<longrightarrow> (\<exists>w. (x, w) \<in> R \<and> (w, z) \<in> R ^^ n)"
+ by (induct n) (simp_all, blast)
+
+lemma rel_pow_E2:
+ "(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
+ \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
+ \<Longrightarrow> P"
+ apply (cases n, simp)
+ apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast)
+ done
+
+lemma rtrancl_imp_UN_rel_pow:
+ assumes "p \<in> R^*"
+ shows "p \<in> (\<Union>n. R ^^ n)"
+proof (cases p)
+ case (Pair x y)
+ with assms have "(x, y) \<in> R^*" by simp
+ then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
+ case base show ?case by (blast intro: rel_pow_0_I)
+ next
+ case step then show ?case by (blast intro: rel_pow_Suc_I)
+ qed
+ with Pair show ?thesis by simp
+qed
+
+lemma rel_pow_imp_rtrancl:
+ assumes "p \<in> R ^^ n"
+ shows "p \<in> R^*"
+proof (cases p)
+ case (Pair x y)
+ with assms have "(x, y) \<in> R ^^ n" by simp
+ then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)
+ case 0 then show ?case by simp
+ next
+ case Suc then show ?case
+ by (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)
+ qed
+ with Pair show ?thesis by simp
+qed
+
+lemma rtrancl_is_UN_rel_pow:
+ "R^* = (\<Union>n. R ^^ n)"
+ by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)
+
+lemma rtrancl_power:
+ "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
+ by (simp add: rtrancl_is_UN_rel_pow)
+
+lemma trancl_power:
+ "p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
+ apply (cases p)
+ apply simp
+ apply (rule iffI)
+ apply (drule tranclD2)
+ apply (clarsimp simp: rtrancl_is_UN_rel_pow)
+ apply (rule_tac x="Suc n" in exI)
+ apply (clarsimp simp: rel_comp_def)
+ apply fastsimp
+ apply clarsimp
+ apply (case_tac n, simp)
+ apply clarsimp
+ apply (drule rel_pow_imp_rtrancl)
+ apply (drule rtrancl_into_trancl1) apply auto
+ done
+
+lemma rtrancl_imp_rel_pow:
+ "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
+ by (auto dest: rtrancl_imp_UN_rel_pow)
+
+lemma single_valued_rel_pow:
+ fixes R :: "('a * 'a) set"
+ shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
+ apply (induct n arbitrary: R)
+ apply simp_all
+ apply (rule single_valuedI)
+ apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
+ done
subsection {* Setup of transitivity reasoner *}
--- a/src/HOL/UNITY/Comp.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/UNITY/Comp.thy Mon Apr 27 07:26:17 2009 -0700
@@ -15,14 +15,22 @@
header{*Composition: Basic Primitives*}
-theory Comp imports Union begin
+theory Comp
+imports Union
+begin
-instance program :: (type) ord ..
+instantiation program :: (type) ord
+begin
-defs
- component_def: "F \<le> H == \<exists>G. F\<squnion>G = H"
- strict_component_def: "(F < (H::'a program)) == (F \<le> H & F \<noteq> H)"
+definition
+ component_def: "F \<le> H <-> (\<exists>G. F\<squnion>G = H)"
+definition
+ strict_component_def: "F < (H::'a program) <-> (F \<le> H & F \<noteq> H)"
+
+instance ..
+
+end
constdefs
component_of :: "'a program =>'a program=> bool"
@@ -114,7 +122,7 @@
by (auto simp add: stable_def component_constrains)
(*Used in Guar.thy to show that programs are partially ordered*)
-lemmas program_less_le = strict_component_def [THEN meta_eq_to_obj_eq]
+lemmas program_less_le = strict_component_def
subsection{*The preserves property*}
@@ -229,8 +237,7 @@
apply (blast intro: Join_assoc [symmetric])
done
-lemmas strict_component_of_eq =
- strict_component_of_def [THEN meta_eq_to_obj_eq, standard]
+lemmas strict_component_of_eq = strict_component_of_def
(** localize **)
lemma localize_Init_eq [simp]: "Init (localize v F) = Init F"
--- a/src/HOL/UNITY/Transformers.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/UNITY/Transformers.thy Mon Apr 27 07:26:17 2009 -0700
@@ -338,10 +338,10 @@
constdefs
wens_single_finite :: "[('a*'a) set, 'a set, nat] => 'a set"
- "wens_single_finite act B k == \<Union>i \<in> atMost k. ((wp act)^i) B"
+ "wens_single_finite act B k == \<Union>i \<in> atMost k. (wp act ^^ i) B"
wens_single :: "[('a*'a) set, 'a set] => 'a set"
- "wens_single act B == \<Union>i. ((wp act)^i) B"
+ "wens_single act B == \<Union>i. (wp act ^^ i) B"
lemma wens_single_Un_eq:
"single_valued act
--- a/src/HOL/Wellfounded.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Wellfounded.thy Mon Apr 27 07:26:17 2009 -0700
@@ -7,7 +7,7 @@
header {*Well-founded Recursion*}
theory Wellfounded
-imports Finite_Set Transitive_Closure Nat
+imports Finite_Set Transitive_Closure
uses ("Tools/function_package/size.ML")
begin
--- a/src/HOL/Word/BinBoolList.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Word/BinBoolList.thy Mon Apr 27 07:26:17 2009 -0700
@@ -38,7 +38,7 @@
if y then rbl_add ws x else ws)"
lemma butlast_power:
- "(butlast ^ n) bl = take (length bl - n) bl"
+ "(butlast ^^ n) bl = take (length bl - n) bl"
by (induct n) (auto simp: butlast_take)
lemma bin_to_bl_aux_Pls_minus_simp [simp]:
@@ -370,14 +370,14 @@
done
lemma nth_rest_power_bin [rule_format] :
- "ALL n. bin_nth ((bin_rest ^ k) w) n = bin_nth w (n + k)"
+ "ALL n. bin_nth ((bin_rest ^^ k) w) n = bin_nth w (n + k)"
apply (induct k, clarsimp)
apply clarsimp
apply (simp only: bin_nth.Suc [symmetric] add_Suc)
done
lemma take_rest_power_bin:
- "m <= n ==> take m (bin_to_bl n w) = bin_to_bl m ((bin_rest ^ (n - m)) w)"
+ "m <= n ==> take m (bin_to_bl n w) = bin_to_bl m ((bin_rest ^^ (n - m)) w)"
apply (rule nth_equalityI)
apply simp
apply (clarsimp simp add: nth_bin_to_bl nth_rest_power_bin)
--- a/src/HOL/Word/BinGeneral.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Word/BinGeneral.thy Mon Apr 27 07:26:17 2009 -0700
@@ -439,7 +439,7 @@
apply clarsimp
apply (simp add: bin_last_mod bin_rest_div Bit_def
cong: number_of_False_cong)
- apply (clarsimp simp: zmod_zmult_zmult1 [symmetric]
+ apply (clarsimp simp: mod_mult_mult1 [symmetric]
zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2 [THEN sym]]])
apply (rule trans [symmetric, OF _ emep1])
apply auto
@@ -822,8 +822,8 @@
by (induct n) auto
lemma bin_rest_power_trunc [rule_format] :
- "(bin_rest ^ k) (bintrunc n bin) =
- bintrunc (n - k) ((bin_rest ^ k) bin)"
+ "(bin_rest ^^ k) (bintrunc n bin) =
+ bintrunc (n - k) ((bin_rest ^^ k) bin)"
by (induct k) (auto simp: bin_rest_trunc)
lemma bin_rest_trunc_i:
@@ -857,7 +857,7 @@
by (rule ext) auto
lemma rco_lem:
- "f o g o f = g o f ==> f o (g o f) ^ n = g ^ n o f"
+ "f o g o f = g o f ==> f o (g o f) ^^ n = g ^^ n o f"
apply (rule ext)
apply (induct_tac n)
apply (simp_all (no_asm))
@@ -867,7 +867,7 @@
apply simp
done
-lemma rco_alt: "(f o g) ^ n o f = f o (g o f) ^ n"
+lemma rco_alt: "(f o g) ^^ n o f = f o (g o f) ^^ n"
apply (rule ext)
apply (induct n)
apply (simp_all add: o_def)
@@ -891,8 +891,9 @@
subsection {* Miscellaneous lemmas *}
-lemmas funpow_minus_simp =
- trans [OF gen_minus [where f = "power f"] funpow_Suc, standard]
+lemma funpow_minus_simp:
+ "0 < n \<Longrightarrow> f ^^ n = f \<circ> f ^^ (n - 1)"
+ by (cases n) simp_all
lemmas funpow_pred_simp [simp] =
funpow_minus_simp [of "number_of bin", simplified nobm1, standard]
--- a/src/HOL/Word/BinOperations.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Word/BinOperations.thy Mon Apr 27 07:26:17 2009 -0700
@@ -641,7 +641,7 @@
apply (simp add: bin_rest_div zdiv_zmult2_eq)
apply (case_tac b rule: bin_exhaust)
apply simp
- apply (simp add: Bit_def zmod_zmult_zmult1 p1mod22k
+ apply (simp add: Bit_def mod_mult_mult1 p1mod22k
split: bit.split
cong: number_of_False_cong)
done
--- a/src/HOL/Word/Num_Lemmas.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Word/Num_Lemmas.thy Mon Apr 27 07:26:17 2009 -0700
@@ -66,7 +66,7 @@
apply (safe dest!: even_equiv_def [THEN iffD1])
apply (subst pos_zmod_mult_2)
apply arith
- apply (simp add: zmod_zmult_zmult1)
+ apply (simp add: mod_mult_mult1)
done
lemmas eme1p = emep1 [simplified add_commute]
--- a/src/HOL/Word/TdThs.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Word/TdThs.thy Mon Apr 27 07:26:17 2009 -0700
@@ -110,7 +110,7 @@
done
lemma fn_comm_power:
- "fa o tr = tr o fr ==> fa ^ n o tr = tr o fr ^ n"
+ "fa o tr = tr o fr ==> fa ^^ n o tr = tr o fr ^^ n"
apply (rule ext)
apply (induct n)
apply (auto dest: fun_cong)
--- a/src/HOL/Word/WordArith.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Word/WordArith.thy Mon Apr 27 07:26:17 2009 -0700
@@ -778,6 +778,8 @@
apply (simp add: word_mult_1)
done
+instance word :: (len0) recpower ..
+
instance word :: (len0) comm_semiring
by (intro_classes) (simp add : word_left_distrib)
--- a/src/HOL/Word/WordBitwise.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Word/WordBitwise.thy Mon Apr 27 07:26:17 2009 -0700
@@ -443,8 +443,10 @@
lemmas test_bit_2p = refl [THEN test_bit_2p', unfolded word_size]
-lemmas nth_w2p = test_bit_2p [unfolded of_int_number_of_eq
- word_of_int [symmetric] Int.of_int_power]
+lemma nth_w2p:
+ "((2\<Colon>'a\<Colon>len word) ^ n) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a\<Colon>len)"
+ unfolding test_bit_2p [symmetric] word_of_int [symmetric]
+ by (simp add: of_int_power)
lemma uint_2p:
"(0::'a::len word) < 2 ^ n ==> uint (2 ^ n::'a::len word) = 2 ^ n"
--- a/src/HOL/Word/WordDefinition.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Word/WordDefinition.thy Mon Apr 27 07:26:17 2009 -0700
@@ -99,7 +99,7 @@
subsection "Arithmetic operations"
-instantiation word :: (len0) "{number, uminus, minus, plus, one, zero, times, Divides.div, power, ord, bit}"
+instantiation word :: (len0) "{number, uminus, minus, plus, one, zero, times, Divides.div, ord, bit}"
begin
definition
@@ -126,10 +126,6 @@
definition
word_mod_def: "a mod b = word_of_int (uint a mod uint b)"
-primrec power_word where
- "(a\<Colon>'a word) ^ 0 = 1"
- | "(a\<Colon>'a word) ^ Suc n = a * a ^ n"
-
definition
word_number_of_def: "number_of w = word_of_int w"
@@ -157,7 +153,7 @@
instance ..
-end
+end
definition
word_succ :: "'a :: len0 word => 'a word"
@@ -207,10 +203,10 @@
"shiftr1 w = word_of_int (bin_rest (uint w))"
definition
- shiftl_def: "w << n = (shiftl1 ^ n) w"
+ shiftl_def: "w << n = (shiftl1 ^^ n) w"
definition
- shiftr_def: "w >> n = (shiftr1 ^ n) w"
+ shiftr_def: "w >> n = (shiftr1 ^^ n) w"
instance ..
@@ -245,7 +241,7 @@
"bshiftr1 b w == of_bl (b # butlast (to_bl w))"
sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55)
- "w >>> n == (sshiftr1 ^ n) w"
+ "w >>> n == (sshiftr1 ^^ n) w"
mask :: "nat => 'a::len word"
"mask n == (1 << n) - 1"
@@ -268,7 +264,7 @@
case ys of [] => [] | x # xs => last ys # butlast ys"
rotater :: "nat => 'a list => 'a list"
- "rotater n == rotater1 ^ n"
+ "rotater n == rotater1 ^^ n"
word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word"
"word_rotr n w == of_bl (rotater n (to_bl w))"
@@ -303,7 +299,7 @@
constdefs
-- "Largest representable machine integer."
max_word :: "'a::len word"
- "max_word \<equiv> word_of_int (2^len_of TYPE('a) - 1)"
+ "max_word \<equiv> word_of_int (2 ^ len_of TYPE('a) - 1)"
consts
of_bool :: "bool \<Rightarrow> 'a::len word"
--- a/src/HOL/Word/WordShift.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/Word/WordShift.thy Mon Apr 27 07:26:17 2009 -0700
@@ -361,14 +361,14 @@
lemma shiftr_no':
"w = number_of bin ==>
- (w::'a::len0 word) >> n = number_of ((bin_rest ^ n) (bintrunc (size w) bin))"
+ (w::'a::len0 word) >> n = number_of ((bin_rest ^^ n) (bintrunc (size w) bin))"
apply clarsimp
apply (rule word_eqI)
apply (auto simp: nth_shiftr nth_rest_power_bin nth_bintr word_size)
done
lemma sshiftr_no':
- "w = number_of bin ==> w >>> n = number_of ((bin_rest ^ n)
+ "w = number_of bin ==> w >>> n = number_of ((bin_rest ^^ n)
(sbintrunc (size w - 1) bin))"
apply clarsimp
apply (rule word_eqI)
--- a/src/HOL/base.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/base.ML Mon Apr 27 07:26:17 2009 -0700
@@ -1,2 +1,2 @@
(*side-entry for HOL-Base*)
-use_thy "Code_Setup";
+use_thy "HOL";
--- a/src/HOL/ex/NormalForm.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/ex/NormalForm.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1,7 +1,6 @@
-(* Authors: Klaus Aehlig, Tobias Nipkow
-*)
+(* Authors: Klaus Aehlig, Tobias Nipkow *)
-header {* Test of normalization function *}
+header {* Testing implementation of normalization by evaluation *}
theory NormalForm
imports Main Rational
@@ -19,18 +18,13 @@
datatype n = Z | S n
-consts
- add :: "n \<Rightarrow> n \<Rightarrow> n"
- add2 :: "n \<Rightarrow> n \<Rightarrow> n"
- mul :: "n \<Rightarrow> n \<Rightarrow> n"
- mul2 :: "n \<Rightarrow> n \<Rightarrow> n"
- exp :: "n \<Rightarrow> n \<Rightarrow> n"
-primrec
- "add Z = id"
- "add (S m) = S o add m"
-primrec
- "add2 Z n = n"
- "add2 (S m) n = S(add2 m n)"
+primrec add :: "n \<Rightarrow> n \<Rightarrow> n" where
+ "add Z = id"
+ | "add (S m) = S o add m"
+
+primrec add2 :: "n \<Rightarrow> n \<Rightarrow> n" where
+ "add2 Z n = n"
+ | "add2 (S m) n = S(add2 m n)"
declare add2.simps [code]
lemma [code nbe]: "add2 (add2 n m) k = add2 n (add2 m k)"
@@ -44,15 +38,17 @@
lemma "add2 (add2 (S n) (S m)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization
lemma "add2 (add2 (S n) (add2 (S m) Z)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization
-primrec
- "mul Z = (%n. Z)"
- "mul (S m) = (%n. add (mul m n) n)"
-primrec
- "mul2 Z n = Z"
- "mul2 (S m) n = add2 n (mul2 m n)"
-primrec
- "exp m Z = S Z"
- "exp m (S n) = mul (exp m n) m"
+primrec mul :: "n \<Rightarrow> n \<Rightarrow> n" where
+ "mul Z = (%n. Z)"
+ | "mul (S m) = (%n. add (mul m n) n)"
+
+primrec mul2 :: "n \<Rightarrow> n \<Rightarrow> n" where
+ "mul2 Z n = Z"
+ | "mul2 (S m) n = add2 n (mul2 m n)"
+
+primrec exp :: "n \<Rightarrow> n \<Rightarrow> n" where
+ "exp m Z = S Z"
+ | "exp m (S n) = mul (exp m n) m"
lemma "mul2 (S(S(S(S(S Z))))) (S(S(S Z))) = S(S(S(S(S(S(S(S(S(S(S(S(S(S(S Z))))))))))))))" by normalization
lemma "mul (S(S(S(S(S Z))))) (S(S(S Z))) = S(S(S(S(S(S(S(S(S(S(S(S(S(S(S Z))))))))))))))" by normalization
--- a/src/HOL/ex/Predicate_Compile.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/ex/Predicate_Compile.thy Mon Apr 27 07:26:17 2009 -0700
@@ -1,37 +1,86 @@
theory Predicate_Compile
-imports Complex_Main Code_Index Lattice_Syntax
+imports Complex_Main Lattice_Syntax Code_Eval
uses "predicate_compile.ML"
begin
+text {* Package setup *}
+
setup {* Predicate_Compile.setup *}
-primrec "next" :: "('a Predicate.pred \<Rightarrow> ('a \<times> 'a Predicate.pred) option)
- \<Rightarrow> 'a Predicate.seq \<Rightarrow> ('a \<times> 'a Predicate.pred) option" where
- "next yield Predicate.Empty = None"
- | "next yield (Predicate.Insert x P) = Some (x, P)"
- | "next yield (Predicate.Join P xq) = (case yield P
- of None \<Rightarrow> next yield xq | Some (x, Q) \<Rightarrow> Some (x, Predicate.Seq (\<lambda>_. Predicate.Join Q xq)))"
+
+text {* Experimental code *}
+
+definition pred_map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'b Predicate.pred" where
+ "pred_map f P = Predicate.bind P (Predicate.single o f)"
ML {*
-let
- fun yield (@{code Predicate.Seq} f) = @{code next} yield (f ())
-in
- yield @{code "\<bottom> :: 'a Predicate.pred"} (*replace bottom with sequence to evaluate*)
-end
+structure Predicate =
+struct
+
+open Predicate;
+
+val pred_ref = ref (NONE : (unit -> term Predicate.pred) option);
+
+fun eval_pred thy t =
+ t
+ |> Eval.mk_term_of (fastype_of t)
+ |> (fn t => Code_ML.eval NONE ("Predicate.pred_ref", pred_ref) @{code pred_map} thy t []);
+
+fun eval_pred_elems thy t T length =
+ t |> eval_pred thy |> yieldn length |> fst |> HOLogic.mk_list T;
+
+fun analyze_compr thy t =
+ let
+ val split = case t of (Const (@{const_name Collect}, _) $ t') => t'
+ | _ => error ("Not a set comprehension: " ^ Syntax.string_of_term_global thy t);
+ val (body, Ts, fp) = HOLogic.strip_split split;
+ val (t_pred, args) = strip_comb body;
+ val pred = case t_pred of Const (pred, _) => pred
+ | _ => error ("Not a constant: " ^ Syntax.string_of_term_global thy t_pred);
+ val mode = map is_Bound args; (*FIXME what about higher-order modes?*)
+ val args' = filter_out is_Bound args;
+ val T = HOLogic.mk_tupleT fp Ts;
+ val mk = HOLogic.mk_tuple' fp T;
+ in (((pred, mode), args), (mk, T)) end;
+
+end;
*}
-fun anamorph :: "('b \<Rightarrow> ('a \<times> 'b) option) \<Rightarrow> index \<Rightarrow> 'b \<Rightarrow> 'a list \<times> 'b" where
- "anamorph f k x = (if k = 0 then ([], x)
- else case f x of None \<Rightarrow> ([], x) | Some (v, y) \<Rightarrow> let (vs, z) = anamorph f (k - 1) y in (v # vs, z))"
+
+text {* Example(s) *}
+
+inductive even :: "nat \<Rightarrow> bool" and odd :: "nat \<Rightarrow> bool" where
+ "even 0"
+ | "even n \<Longrightarrow> odd (Suc n)"
+ | "odd n \<Longrightarrow> even (Suc n)"
+
+setup {* pred_compile "even" *}
+thm even_codegen
+
+
+inductive append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
+ append_Nil: "append [] xs xs"
+ | append_Cons: "append xs ys zs \<Longrightarrow> append (x # xs) ys (x # zs)"
-ML {*
-let
- fun yield (@{code Predicate.Seq} f) = @{code next} yield (f ())
- fun yieldn k = @{code anamorph} yield k
-in
- yieldn 0 (*replace with number of elements to retrieve*)
- @{code "\<bottom> :: 'a Predicate.pred"} (*replace bottom with sequence to evaluate*)
-end
-*}
+setup {* pred_compile "append" *}
+thm append_codegen
+
+
+inductive partition :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
+ for f where
+ "partition f [] [] []"
+ | "f x \<Longrightarrow> partition f xs ys zs \<Longrightarrow> partition f (x # xs) (x # ys) zs"
+ | "\<not> f x \<Longrightarrow> partition f xs ys zs \<Longrightarrow> partition f (x # xs) ys (x # zs)"
+
+setup {* pred_compile "partition" *}
+thm partition_codegen
+
+setup {* pred_compile "tranclp" *}
+thm tranclp_codegen
+
+ML_val {* Predicate_Compile.modes_of @{theory} @{const_name partition} *}
+ML_val {* Predicate_Compile.modes_of @{theory} @{const_name tranclp} *}
+
+ML_val {* Predicate.analyze_compr @{theory} @{term "{n. odd n}"} *}
end
\ No newline at end of file
--- a/src/HOL/ex/Quickcheck_Generators.thy Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/ex/Quickcheck_Generators.thy Mon Apr 27 07:26:17 2009 -0700
@@ -6,62 +6,6 @@
imports Quickcheck State_Monad
begin
-subsection {* Type @{typ "'a \<Rightarrow> 'b"} *}
-
-ML {*
-structure Random_Engine =
-struct
-
-open Random_Engine;
-
-fun random_fun (T1 : typ) (T2 : typ) (eq : 'a -> 'a -> bool) (term_of : 'a -> term)
- (random : Random_Engine.seed -> ('b * (unit -> term)) * Random_Engine.seed)
- (random_split : Random_Engine.seed -> Random_Engine.seed * Random_Engine.seed)
- (seed : Random_Engine.seed) =
- let
- val (seed', seed'') = random_split seed;
- val state = ref (seed', [], Const (@{const_name undefined}, T1 --> T2));
- val fun_upd = Const (@{const_name fun_upd},
- (T1 --> T2) --> T1 --> T2 --> T1 --> T2);
- fun random_fun' x =
- let
- val (seed, fun_map, f_t) = ! state;
- in case AList.lookup (uncurry eq) fun_map x
- of SOME y => y
- | NONE => let
- val t1 = term_of x;
- val ((y, t2), seed') = random seed;
- val fun_map' = (x, y) :: fun_map;
- val f_t' = fun_upd $ f_t $ t1 $ t2 ();
- val _ = state := (seed', fun_map', f_t');
- in y end
- end;
- fun term_fun' () = #3 (! state);
- in ((random_fun', term_fun'), seed'') end;
-
-end
-*}
-
-axiomatization
- random_fun_aux :: "typerep \<Rightarrow> typerep \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> term)
- \<Rightarrow> (seed \<Rightarrow> ('b \<times> (unit \<Rightarrow> term)) \<times> seed) \<Rightarrow> (seed \<Rightarrow> seed \<times> seed)
- \<Rightarrow> seed \<Rightarrow> (('a \<Rightarrow> 'b) \<times> (unit \<Rightarrow> term)) \<times> seed"
-
-code_const random_fun_aux (SML "Random'_Engine.random'_fun")
-
-instantiation "fun" :: ("{eq, term_of}", "{type, random}") random
-begin
-
-definition random_fun :: "index \<Rightarrow> seed \<Rightarrow> (('a \<Rightarrow> 'b) \<times> (unit \<Rightarrow> term)) \<times> seed" where
- "random n = random_fun_aux TYPEREP('a) TYPEREP('b) (op =) Code_Eval.term_of (random n) split_seed"
-
-instance ..
-
-end
-
-code_reserved SML Random_Engine
-
-
subsection {* Datatypes *}
definition
--- a/src/HOL/ex/predicate_compile.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/HOL/ex/predicate_compile.ML Mon Apr 27 07:26:17 2009 -0700
@@ -6,38 +6,119 @@
signature PREDICATE_COMPILE =
sig
- val create_def_equation': string -> (int list option list * int list) option -> theory -> theory
+ type mode = int list option list * int list
+ val create_def_equation': string -> mode option -> theory -> theory
val create_def_equation: string -> theory -> theory
- val intro_rule: theory -> string -> (int list option list * int list) -> thm
- val elim_rule: theory -> string -> (int list option list * int list) -> thm
+ val intro_rule: theory -> string -> mode -> thm
+ val elim_rule: theory -> string -> mode -> thm
val strip_intro_concl : term -> int -> (term * (term list * term list))
val code_ind_intros_attrib : attribute
val code_ind_cases_attrib : attribute
+ val print_alternative_rules : theory -> theory
+ val modename_of: theory -> string -> mode -> string
+ val modes_of: theory -> string -> mode list
val setup : theory -> theory
- val print_alternative_rules : theory -> theory
val do_proofs: bool ref
end;
structure Predicate_Compile: PREDICATE_COMPILE =
struct
+(** auxiliary **)
+
+(* debug stuff *)
+
+fun tracing s = (if ! Toplevel.debug then Output.tracing s else ());
+
+fun print_tac s = (if ! Toplevel.debug then Tactical.print_tac s else Seq.single);
+fun debug_tac msg = (fn st => (tracing msg; Seq.single st));
+
+val do_proofs = ref true;
+
+
+(** fundamentals **)
+
+(* syntactic operations *)
+
+fun mk_eq (x, xs) =
+ let fun mk_eqs _ [] = []
+ | mk_eqs a (b::cs) =
+ HOLogic.mk_eq (Free (a, fastype_of b), b) :: mk_eqs a cs
+ in mk_eqs x xs end;
+
+fun mk_tupleT [] = HOLogic.unitT
+ | mk_tupleT Ts = foldr1 HOLogic.mk_prodT Ts;
+
+fun mk_tuple [] = HOLogic.unit
+ | mk_tuple ts = foldr1 HOLogic.mk_prod ts;
+
+fun dest_tuple (Const (@{const_name Product_Type.Unity}, _)) = []
+ | dest_tuple (Const (@{const_name Pair}, _) $ t1 $ t2) = t1 :: (dest_tuple t2)
+ | dest_tuple t = [t]
+
+fun mk_pred_enumT T = Type ("Predicate.pred", [T])
+
+fun dest_pred_enumT (Type ("Predicate.pred", [T])) = T
+ | dest_pred_enumT T = raise TYPE ("dest_pred_enumT", [T], []);
+
+fun mk_Enum f =
+ let val T as Type ("fun", [T', _]) = fastype_of f
+ in
+ Const (@{const_name Predicate.Pred}, T --> mk_pred_enumT T') $ f
+ end;
+
+fun mk_Eval (f, x) =
+ let val T = fastype_of x
+ in
+ Const (@{const_name Predicate.eval}, mk_pred_enumT T --> T --> HOLogic.boolT) $ f $ x
+ end;
+
+fun mk_empty T = Const (@{const_name Orderings.bot}, mk_pred_enumT T);
+
+fun mk_single t =
+ let val T = fastype_of t
+ in Const(@{const_name Predicate.single}, T --> mk_pred_enumT T) $ t end;
+
+fun mk_bind (x, f) =
+ let val T as Type ("fun", [_, U]) = fastype_of f
+ in
+ Const (@{const_name Predicate.bind}, fastype_of x --> T --> U) $ x $ f
+ end;
+
+val mk_sup = HOLogic.mk_binop @{const_name sup};
+
+fun mk_if_predenum cond = Const (@{const_name Predicate.if_pred},
+ HOLogic.boolT --> mk_pred_enumT HOLogic.unitT) $ cond;
+
+fun mk_not_pred t = let val T = mk_pred_enumT HOLogic.unitT
+ in Const (@{const_name Predicate.not_pred}, T --> T) $ t end
+
+
+(* data structures *)
+
+type mode = int list option list * int list;
+
+val mode_ord = prod_ord (list_ord (option_ord (list_ord int_ord))) (list_ord int_ord);
+
structure PredModetab = TableFun(
- type key = (string * (int list option list * int list))
- val ord = prod_ord fast_string_ord (prod_ord
- (list_ord (option_ord (list_ord int_ord))) (list_ord int_ord)))
+ type key = string * mode
+ val ord = prod_ord fast_string_ord mode_ord
+);
+(*FIXME scrap boilerplate*)
+
structure IndCodegenData = TheoryDataFun
(
type T = {names : string PredModetab.table,
- modes : ((int list option list * int list) list) Symtab.table,
+ modes : mode list Symtab.table,
function_defs : Thm.thm Symtab.table,
function_intros : Thm.thm Symtab.table,
function_elims : Thm.thm Symtab.table,
- intro_rules : (Thm.thm list) Symtab.table,
+ intro_rules : Thm.thm list Symtab.table,
elim_rules : Thm.thm Symtab.table,
nparams : int Symtab.table
- };
+ }; (*FIXME: better group tables according to key*)
(* names: map from inductive predicate and mode to function name (string).
modes: map from inductive predicates to modes
function_defs: map from function name to definition
@@ -115,26 +196,12 @@
intro_rules = #intro_rules x, elim_rules = #elim_rules x,
nparams = f (#nparams x)}) thy
-(* Debug stuff and tactics ***********************************************************)
-
-fun tracing s = (if ! Toplevel.debug then Output.tracing s else ());
-fun print_tac s = (if ! Toplevel.debug then Tactical.print_tac s else Seq.single);
-
-fun debug_tac msg = (fn st =>
- (tracing msg; Seq.single st));
-
(* removes first subgoal *)
fun mycheat_tac thy i st =
(Tactic.rtac (SkipProof.make_thm thy (Var (("A", 0), propT))) i) st
-val (do_proofs : bool ref) = ref true;
-
(* Lightweight mode analysis **********************************************)
-(* Hack for message from old code generator *)
-val message = tracing;
-
-
(**************************************************************************)
(* source code from old code generator ************************************)
@@ -153,7 +220,8 @@
| _ => false)
in check end;
-(**** check if a type is an equality type (i.e. doesn't contain fun) ****)
+(**** check if a type is an equality type (i.e. doesn't contain fun)
+ FIXME this is only an approximation ****)
fun is_eqT (Type (s, Ts)) = s <> "fun" andalso forall is_eqT Ts
| is_eqT _ = true;
@@ -165,7 +233,7 @@
| SOME js => enclose "[" "]" (commas (map string_of_int js)))
(iss @ [SOME is]));
-fun print_modes modes = message ("Inferred modes:\n" ^
+fun print_modes modes = tracing ("Inferred modes:\n" ^
cat_lines (map (fn (s, ms) => s ^ ": " ^ commas (map
string_of_mode ms)) modes));
@@ -182,6 +250,7 @@
(get_args' is (i+1) ts)
in get_args' is 1 ts end
+(*FIXME this function should not be named merge... make it local instead*)
fun merge xs [] = xs
| merge [] ys = ys
| merge (x::xs) (y::ys) = if length x >= length y then x::merge xs (y::ys)
@@ -197,7 +266,8 @@
fun cprods xss = foldr (map op :: o cprod) [[]] xss;
-datatype mode = Mode of (int list option list * int list) * int list * mode option list;
+datatype hmode = Mode of mode * int list * hmode option list; (*FIXME don't understand
+ why there is another mode type!?*)
fun modes_of modes t =
let
@@ -285,11 +355,11 @@
in (p, List.filter (fn m => case find_index
(not o check_mode_clause thy param_vs modes m) rs of
~1 => true
- | i => (message ("Clause " ^ string_of_int (i+1) ^ " of " ^
+ | i => (tracing ("Clause " ^ string_of_int (i+1) ^ " of " ^
p ^ " violates mode " ^ string_of_mode m); false)) ms)
end;
-fun fixp f (x : (string * (int list option list * int list) list) list) =
+fun fixp f (x : (string * mode list) list) =
let val y = f x
in if x = y then x else fixp f y end;
@@ -306,66 +376,6 @@
(*****************************************************************************************)
(**** term construction ****)
-fun mk_eq (x, xs) =
- let fun mk_eqs _ [] = []
- | mk_eqs a (b::cs) =
- HOLogic.mk_eq (Free (a, fastype_of b), b) :: mk_eqs a cs
- in mk_eqs x xs end;
-
-fun mk_tuple [] = HOLogic.unit
- | mk_tuple ts = foldr1 HOLogic.mk_prod ts;
-
-fun dest_tuple (Const (@{const_name Product_Type.Unity}, _)) = []
- | dest_tuple (Const (@{const_name Pair}, _) $ t1 $ t2) = t1 :: (dest_tuple t2)
- | dest_tuple t = [t]
-
-fun mk_tupleT [] = HOLogic.unitT
- | mk_tupleT Ts = foldr1 HOLogic.mk_prodT Ts;
-
-fun mk_pred_enumT T = Type ("Predicate.pred", [T])
-
-fun dest_pred_enumT (Type ("Predicate.pred", [T])) = T
- | dest_pred_enumT T = raise TYPE ("dest_pred_enumT", [T], []);
-
-fun mk_single t =
- let val T = fastype_of t
- in Const(@{const_name Predicate.single}, T --> mk_pred_enumT T) $ t end;
-
-fun mk_empty T = Const (@{const_name Orderings.bot}, mk_pred_enumT T);
-
-fun mk_if_predenum cond = Const (@{const_name Predicate.if_pred},
- HOLogic.boolT --> mk_pred_enumT HOLogic.unitT)
- $ cond
-
-fun mk_not_pred t = let val T = mk_pred_enumT HOLogic.unitT
- in Const (@{const_name Predicate.not_pred}, T --> T) $ t end
-
-fun mk_bind (x, f) =
- let val T as Type ("fun", [_, U]) = fastype_of f
- in
- Const (@{const_name Predicate.bind}, fastype_of x --> T --> U) $ x $ f
- end;
-
-fun mk_Enum f =
- let val T as Type ("fun", [T', _]) = fastype_of f
- in
- Const (@{const_name Predicate.Pred}, T --> mk_pred_enumT T') $ f
- end;
-
-fun mk_Eval (f, x) =
- let val T = fastype_of x
- in
- Const (@{const_name Predicate.eval}, mk_pred_enumT T --> T --> HOLogic.boolT) $ f $ x
- end;
-
-fun mk_Eval' f =
- let val T = fastype_of f
- in
- Const (@{const_name Predicate.eval}, T --> dest_pred_enumT T --> HOLogic.boolT) $ f
- end;
-
-val mk_sup = HOLogic.mk_binop @{const_name sup};
-
(* for simple modes (e.g. parameters) only: better call it param_funT *)
(* or even better: remove it and only use funT'_of - some modifications to funT'_of necessary *)
fun funT_of T NONE = T
@@ -424,13 +434,16 @@
(v', mk_empty U')]))
end;
-fun modename thy name mode = let
+fun modename_of thy name mode = let
val v = (PredModetab.lookup (#names (IndCodegenData.get thy)) (name, mode))
- in if (is_some v) then the v
- else error ("fun modename - definition not found: name: " ^ name ^ " mode: " ^ (makestring mode))
+ in if (is_some v) then the v (*FIXME use case here*)
+ else error ("fun modename_of - definition not found: name: " ^ name ^ " mode: " ^ (makestring mode))
end
-(* function can be removed *)
+fun modes_of thy =
+ these o Symtab.lookup ((#modes o IndCodegenData.get) thy);
+
+(*FIXME function can be removed*)
fun mk_funcomp f t =
let
val names = Term.add_free_names t [];
@@ -449,7 +462,7 @@
val f' = case f of
Const (name, T) =>
if AList.defined op = modes name then
- Const (modename thy name (iss, is'), funT'_of (iss, is') T)
+ Const (modename_of thy name (iss, is'), funT'_of (iss, is') T)
else error "compile param: Not an inductive predicate with correct mode"
| Free (name, T) => Free (name, funT_of T (SOME is'))
in list_comb (f', params' @ args') end
@@ -463,7 +476,7 @@
val (Ts, Us) = get_args is
(curry Library.drop (length ms) (fst (strip_type T)))
val params' = map (compile_param thy modes) (ms ~~ params)
- val mode_id = modename thy name mode
+ val mode_id = modename_of thy name mode
in list_comb (Const (mode_id, ((map fastype_of params') @ Ts) --->
mk_pred_enumT (mk_tupleT Us)), params')
end
@@ -556,7 +569,7 @@
val cl_ts =
map (fn cl => compile_clause thy
all_vs param_vs modes mode cl (mk_tuple xs)) cls;
- val mode_id = modename thy s mode
+ val mode_id = modename_of thy s mode
in
HOLogic.mk_Trueprop (HOLogic.mk_eq
(list_comb (Const (mode_id, (Ts1' @ Us1) --->
@@ -591,7 +604,7 @@
fold Term.add_consts intrs [] |> map fst
|> filter_out (member (op =) preds) |> filter (is_ind_pred thy)
-fun print_arities arities = message ("Arities:\n" ^
+fun print_arities arities = tracing ("Arities:\n" ^
cat_lines (map (fn (s, (ks, k)) => s ^ ": " ^
space_implode " -> " (map
(fn NONE => "X" | SOME k' => string_of_int k')
@@ -691,10 +704,10 @@
(* Proving equivalence of term *)
-fun intro_rule thy pred mode = modename thy pred mode
+fun intro_rule thy pred mode = modename_of thy pred mode
|> Symtab.lookup (#function_intros (IndCodegenData.get thy)) |> the
-fun elim_rule thy pred mode = modename thy pred mode
+fun elim_rule thy pred mode = modename_of thy pred mode
|> Symtab.lookup (#function_elims (IndCodegenData.get thy)) |> the
fun pred_intros thy predname = let
@@ -711,7 +724,7 @@
end
fun function_definition thy pred mode =
- modename thy pred mode |> Symtab.lookup (#function_defs (IndCodegenData.get thy)) |> the
+ modename_of thy pred mode |> Symtab.lookup (#function_defs (IndCodegenData.get thy)) |> the
fun is_Type (Type _) = true
| is_Type _ = false
@@ -973,7 +986,7 @@
in nth (#elims (snd ind_result)) index end)
fun prove_one_direction thy all_vs param_vs modes clauses ((pred, T), mode) = let
- val elim_rule = the (Symtab.lookup (#function_elims (IndCodegenData.get thy)) (modename thy pred mode))
+ val elim_rule = the (Symtab.lookup (#function_elims (IndCodegenData.get thy)) (modename_of thy pred mode))
(* val ind_result = InductivePackage.the_inductive (ProofContext.init thy) pred
val index = find_index (fn s => s = pred) (#names (fst ind_result))
val (_, T) = dest_Const (nth (#preds (snd ind_result)) index) *)
@@ -1225,7 +1238,7 @@
(* main function *********************************************************************)
(*************************************************************************************)
-fun create_def_equation' ind_name (mode : (int list option list * int list) option) thy =
+fun create_def_equation' ind_name (mode : mode option) thy =
let
val _ = tracing ("starting create_def_equation' with " ^ ind_name)
val (prednames, preds) =
@@ -1249,6 +1262,7 @@
val _ = tracing ("calling preds: " ^ makestring name_of_calls)
val _ = tracing "starting recursive compilations"
fun rec_call name thy =
+ (*FIXME use member instead of infix mem*)
if not (name mem (Symtab.keys (#modes (IndCodegenData.get thy)))) then
create_def_equation name thy else thy
val thy'' = fold rec_call name_of_calls thy'
--- a/src/Provers/Arith/cancel_div_mod.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Provers/Arith/cancel_div_mod.ML Mon Apr 27 07:26:17 2009 -0700
@@ -69,7 +69,7 @@
fun cancel ss t pq =
let val teqt' = Data.prove_eq_sums ss (t, rearrange t pq)
- in hd(Data.div_mod_eqs RL [teqt' RS transitive_thm]) end;
+ in hd (Data.div_mod_eqs RL [teqt' RS transitive_thm]) end;
fun proc ss t =
let val (divs,mods) = coll_div_mod t ([],[])
--- a/src/Pure/IsaMakefile Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Pure/IsaMakefile Mon Apr 27 07:26:17 2009 -0700
@@ -40,9 +40,8 @@
Pure: $(OUT)/Pure
-$(OUT)/Pure: $(BOOTSTRAP_FILES) ../Tools/auto_solve.ML \
- ../Tools/quickcheck.ML Concurrent/ROOT.ML Concurrent/future.ML \
- Concurrent/mailbox.ML Concurrent/par_list.ML \
+$(OUT)/Pure: $(BOOTSTRAP_FILES) Concurrent/ROOT.ML \
+ Concurrent/future.ML Concurrent/mailbox.ML Concurrent/par_list.ML \
Concurrent/par_list_dummy.ML Concurrent/simple_thread.ML \
Concurrent/synchronized.ML Concurrent/task_queue.ML General/ROOT.ML \
General/alist.ML General/antiquote.ML General/balanced_tree.ML \
--- a/src/Pure/Isar/code.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Pure/Isar/code.ML Mon Apr 27 07:26:17 2009 -0700
@@ -29,8 +29,6 @@
val add_undefined: string -> theory -> theory
val purge_data: theory -> theory
- val coregular_algebra: theory -> Sorts.algebra
- val operational_algebra: theory -> (sort -> sort) * Sorts.algebra
val these_eqns: theory -> string -> (thm * bool) list
val these_raw_eqns: theory -> string -> (thm * bool) list
val get_datatype: theory -> string -> ((string * sort) list * (string * typ list) list)
@@ -469,39 +467,6 @@
fun mk_default_eqn thy = Code_Unit.try_thm (check_linear o Code_Unit.mk_eqn thy);
-(** operational sort algebra and class discipline **)
-
-local
-
-fun arity_constraints thy algebra (class, tyco) =
- let
- val base_constraints = Sorts.mg_domain algebra tyco [class];
- val classparam_constraints = Sorts.complete_sort algebra [class]
- |> maps (map fst o these o try (#params o AxClass.get_info thy))
- |> map_filter (fn c => try (AxClass.param_of_inst thy) (c, tyco))
- |> maps (map fst o get_eqns thy)
- |> map (map (snd o dest_TVar) o Sign.const_typargs thy o Code_Unit.const_typ_eqn);
- val inter_sorts = map2 (curry (Sorts.inter_sort algebra));
- in fold inter_sorts classparam_constraints base_constraints end;
-
-fun retrieve_algebra thy operational =
- Sorts.subalgebra (Syntax.pp_global thy) operational
- (SOME o arity_constraints thy (Sign.classes_of thy))
- (Sign.classes_of thy);
-
-in
-
-fun coregular_algebra thy = retrieve_algebra thy (K true) |> snd;
-fun operational_algebra thy =
- let
- fun add_iff_operational class =
- can (AxClass.get_info thy) class ? cons class;
- val operational_classes = fold add_iff_operational (Sign.all_classes thy) []
- in retrieve_algebra thy (member (op =) operational_classes) end;
-
-end; (*local*)
-
-
(** interfaces and attributes **)
fun delete_force msg key xs =
--- a/src/Pure/ProofGeneral/ROOT.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Pure/ProofGeneral/ROOT.ML Mon Apr 27 07:26:17 2009 -0700
@@ -14,11 +14,7 @@
use "pgip_isabelle.ML";
-(use
- |> setmp Proofterm.proofs 1
- |> setmp quick_and_dirty true
- |> setmp auto_quickcheck true
- |> setmp auto_solve true) "preferences.ML";
+use "preferences.ML";
use "pgip_parser.ML";
--- a/src/Pure/ProofGeneral/preferences.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Pure/ProofGeneral/preferences.ML Mon Apr 27 07:26:17 2009 -0700
@@ -6,6 +6,10 @@
signature PREFERENCES =
sig
+ val category_display: string
+ val category_advanced_display: string
+ val category_tracing: string
+ val category_proof: string
type preference =
{name: string,
descr: string,
@@ -29,6 +33,14 @@
structure Preferences: PREFERENCES =
struct
+(* categories *)
+
+val category_display = "Display";
+val category_advanced_display = "Advanced Display";
+val category_tracing = "Tracing";
+val category_proof = "Proof"
+
+
(* preferences and preference tables *)
type preference =
@@ -66,11 +78,11 @@
(* preferences of Pure *)
-val proof_pref =
+val proof_pref = setmp Proofterm.proofs 1 (fn () =>
let
fun get () = PgipTypes.bool_to_pgstring (! Proofterm.proofs >= 2);
fun set s = Proofterm.proofs := (if PgipTypes.read_pgipbool s then 2 else 1);
- in mkpref get set PgipTypes.Pgipbool "full-proofs" "Record full proof objects internally" end;
+ in mkpref get set PgipTypes.Pgipbool "full-proofs" "Record full proof objects internally" end) ();
val thm_depsN = "thm_deps";
val thm_deps_pref =
@@ -145,24 +157,13 @@
bool_pref Toplevel.debug
"debugging"
"Whether to enable debugging.",
- bool_pref Quickcheck.auto
- "auto-quickcheck"
- "Whether to enable quickcheck automatically.",
- nat_pref Quickcheck.auto_time_limit
- "auto-quickcheck-time-limit"
- "Time limit for automatic quickcheck (in milliseconds).",
- bool_pref AutoSolve.auto
- "auto-solve"
- "Try to solve newly declared lemmas with existing theorems.",
- nat_pref AutoSolve.auto_time_limit
- "auto-solve-time-limit"
- "Time limit for seeking automatic solutions (in milliseconds).",
thm_deps_pref];
val proof_preferences =
- [bool_pref quick_and_dirty
- "quick-and-dirty"
- "Take a few short cuts",
+ [setmp quick_and_dirty true (fn () =>
+ bool_pref quick_and_dirty
+ "quick-and-dirty"
+ "Take a few short cuts") (),
bool_pref Toplevel.skip_proofs
"skip-proofs"
"Skip over proofs (interactive-only)",
@@ -175,10 +176,10 @@
"Check proofs in parallel"];
val pure_preferences =
- [("Display", display_preferences),
- ("Advanced Display", advanced_display_preferences),
- ("Tracing", tracing_preferences),
- ("Proof", proof_preferences)];
+ [(category_display, display_preferences),
+ (category_advanced_display, advanced_display_preferences),
+ (category_tracing, tracing_preferences),
+ (category_proof, proof_preferences)];
(* table of categories and preferences; names must be unique *)
@@ -203,6 +204,6 @@
else
if exists (fn {name, ...} => name = #name pref) prefs
then (warning ("Preference already exists: " ^ quote (#name pref)); (cat, prefs))
- else (cat, pref :: prefs));
+ else (cat, prefs @ [pref]));
end;
--- a/src/Pure/Tools/ROOT.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Pure/Tools/ROOT.ML Mon Apr 27 07:26:17 2009 -0700
@@ -1,16 +1,9 @@
-(* Title: Pure/Tools/ROOT.ML
-
-Miscellaneous tools and packages for Pure Isabelle.
-*)
+(* Miscellaneous tools and packages for Pure Isabelle *)
use "named_thms.ML";
-(*basic XML support*)
use "xml_syntax.ML";
use "find_theorems.ML";
use "find_consts.ML";
-(*quickcheck/autosolve needed here because of pg preferences*)
-use "../../Tools/quickcheck.ML";
-use "../../Tools/auto_solve.ML";
--- a/src/Pure/axclass.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Pure/axclass.ML Mon Apr 27 07:26:17 2009 -0700
@@ -286,74 +286,6 @@
handle TYPE (msg, _, _) => error msg;
-(* primitive rules *)
-
-fun add_classrel th thy =
- let
- fun err () = raise THM ("add_classrel: malformed class relation", 0, [th]);
- val prop = Thm.plain_prop_of (Thm.transfer thy th);
- val rel = Logic.dest_classrel prop handle TERM _ => err ();
- val (c1, c2) = cert_classrel thy rel handle TYPE _ => err ();
- in
- thy
- |> Sign.primitive_classrel (c1, c2)
- |> put_classrel ((c1, c2), Thm.close_derivation (Drule.unconstrainTs th))
- |> perhaps complete_arities
- end;
-
-fun add_arity th thy =
- let
- fun err () = raise THM ("add_arity: malformed type arity", 0, [th]);
- val prop = Thm.plain_prop_of (Thm.transfer thy th);
- val (t, Ss, c) = Logic.dest_arity prop handle TERM _ => err ();
- val _ = map (Sign.certify_sort thy) Ss = Ss orelse err ();
- val _ = case filter_out (fn c => can (get_inst_param thy) (c, t)) (params_of thy c)
- of [] => ()
- | cs => Output.legacy_feature
- ("Missing specifications for overloaded parameters " ^ commas_quote cs)
- val th' = Drule.unconstrainTs th;
- in
- thy
- |> Sign.primitive_arity (t, Ss, [c])
- |> put_arity ((t, Ss, c), th')
- end;
-
-
-(* tactical proofs *)
-
-fun prove_classrel raw_rel tac thy =
- let
- val ctxt = ProofContext.init thy;
- val (c1, c2) = cert_classrel thy raw_rel;
- val th = Goal.prove ctxt [] [] (Logic.mk_classrel (c1, c2)) (K tac) handle ERROR msg =>
- cat_error msg ("The error(s) above occurred while trying to prove class relation " ^
- quote (Syntax.string_of_classrel ctxt [c1, c2]));
- in
- thy
- |> PureThy.add_thms [((Binding.name
- (prefix classrel_prefix (Logic.name_classrel (c1, c2))), th), [])]
- |-> (fn [th'] => add_classrel th')
- end;
-
-fun prove_arity raw_arity tac thy =
- let
- val ctxt = ProofContext.init thy;
- val arity = Sign.cert_arity thy raw_arity;
- val names = map (prefix arity_prefix) (Logic.name_arities arity);
- val props = Logic.mk_arities arity;
- val ths = Goal.prove_multi ctxt [] [] props
- (fn _ => Goal.precise_conjunction_tac (length props) 1 THEN tac) handle ERROR msg =>
- cat_error msg ("The error(s) above occurred while trying to prove type arity " ^
- quote (Syntax.string_of_arity ctxt arity));
- in
- thy
- |> PureThy.add_thms (map (rpair []) (map Binding.name names ~~ ths))
- |-> fold add_arity
- end;
-
-
-(* instance parameters and overloaded definitions *)
-
(* declaration and definition of instances of overloaded constants *)
fun declare_overloaded (c, T) thy =
@@ -398,6 +330,74 @@
end;
+(* primitive rules *)
+
+fun add_classrel th thy =
+ let
+ fun err () = raise THM ("add_classrel: malformed class relation", 0, [th]);
+ val prop = Thm.plain_prop_of (Thm.transfer thy th);
+ val rel = Logic.dest_classrel prop handle TERM _ => err ();
+ val (c1, c2) = cert_classrel thy rel handle TYPE _ => err ();
+ in
+ thy
+ |> Sign.primitive_classrel (c1, c2)
+ |> put_classrel ((c1, c2), Thm.close_derivation (Drule.unconstrainTs th))
+ |> perhaps complete_arities
+ end;
+
+fun add_arity th thy =
+ let
+ fun err () = raise THM ("add_arity: malformed type arity", 0, [th]);
+ val prop = Thm.plain_prop_of (Thm.transfer thy th);
+ val (t, Ss, c) = Logic.dest_arity prop handle TERM _ => err ();
+ val T = Type (t, map TFree (Name.names Name.context Name.aT Ss));
+ val missing_params = Sign.complete_sort thy [c]
+ |> maps (these o Option.map #params o try (get_info thy))
+ |> filter_out (fn (const, _) => can (get_inst_param thy) (const, t))
+ |> (map o apsnd o map_atyps) (K T);
+ val _ = map (Sign.certify_sort thy) Ss = Ss orelse err ();
+ val th' = Drule.unconstrainTs th;
+ in
+ thy
+ |> fold (snd oo declare_overloaded) missing_params
+ |> Sign.primitive_arity (t, Ss, [c])
+ |> put_arity ((t, Ss, c), th')
+ end;
+
+
+(* tactical proofs *)
+
+fun prove_classrel raw_rel tac thy =
+ let
+ val ctxt = ProofContext.init thy;
+ val (c1, c2) = cert_classrel thy raw_rel;
+ val th = Goal.prove ctxt [] [] (Logic.mk_classrel (c1, c2)) (K tac) handle ERROR msg =>
+ cat_error msg ("The error(s) above occurred while trying to prove class relation " ^
+ quote (Syntax.string_of_classrel ctxt [c1, c2]));
+ in
+ thy
+ |> PureThy.add_thms [((Binding.name
+ (prefix classrel_prefix (Logic.name_classrel (c1, c2))), th), [])]
+ |-> (fn [th'] => add_classrel th')
+ end;
+
+fun prove_arity raw_arity tac thy =
+ let
+ val ctxt = ProofContext.init thy;
+ val arity = Sign.cert_arity thy raw_arity;
+ val names = map (prefix arity_prefix) (Logic.name_arities arity);
+ val props = Logic.mk_arities arity;
+ val ths = Goal.prove_multi ctxt [] [] props
+ (fn _ => Goal.precise_conjunction_tac (length props) 1 THEN tac) handle ERROR msg =>
+ cat_error msg ("The error(s) above occurred while trying to prove type arity " ^
+ quote (Syntax.string_of_arity ctxt arity));
+ in
+ thy
+ |> PureThy.add_thms (map (rpair []) (map Binding.name names ~~ ths))
+ |-> fold add_arity
+ end;
+
+
(** class definitions **)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/Code_Generator.thy Mon Apr 27 07:26:17 2009 -0700
@@ -0,0 +1,28 @@
+(* Title: Tools/Code_Generator.thy
+ Author: Florian Haftmann, TU Muenchen
+*)
+
+header {* Loading the code generator modules *}
+
+theory Code_Generator
+imports Pure
+uses
+ "~~/src/Tools/value.ML"
+ "~~/src/Tools/quickcheck.ML"
+ "~~/src/Tools/code/code_name.ML"
+ "~~/src/Tools/code/code_wellsorted.ML"
+ "~~/src/Tools/code/code_thingol.ML"
+ "~~/src/Tools/code/code_printer.ML"
+ "~~/src/Tools/code/code_target.ML"
+ "~~/src/Tools/code/code_ml.ML"
+ "~~/src/Tools/code/code_haskell.ML"
+ "~~/src/Tools/nbe.ML"
+begin
+
+setup {*
+ Code_ML.setup
+ #> Code_Haskell.setup
+ #> Nbe.setup
+*}
+
+end
\ No newline at end of file
--- a/src/Tools/auto_solve.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Tools/auto_solve.ML Mon Apr 27 07:26:17 2009 -0700
@@ -14,18 +14,34 @@
val auto : bool ref
val auto_time_limit : int ref
val limit : int ref
-
- val seek_solution : bool -> Proof.state -> Proof.state
end;
structure AutoSolve : AUTO_SOLVE =
struct
+(* preferences *)
+
val auto = ref false;
val auto_time_limit = ref 2500;
val limit = ref 5;
-fun seek_solution int state =
+val _ =
+ ProofGeneralPgip.add_preference Preferences.category_tracing
+ (setmp auto true (fn () =>
+ Preferences.bool_pref auto
+ "auto-solve"
+ "Try to solve newly declared lemmas with existing theorems.") ());
+
+val _ =
+ ProofGeneralPgip.add_preference Preferences.category_tracing
+ (Preferences.nat_pref auto_time_limit
+ "auto-solve-time-limit"
+ "Time limit for seeking automatic solutions (in milliseconds).");
+
+
+(* hook *)
+
+val _ = Context.>> (Specification.add_theorem_hook (fn int => fn state =>
let
val ctxt = Proof.context_of state;
@@ -76,12 +92,10 @@
if int andalso ! auto andalso not (! Toplevel.quiet)
then go ()
else state
- end;
+ end));
end;
-val _ = Context.>> (Specification.add_theorem_hook AutoSolve.seek_solution);
-
val auto_solve = AutoSolve.auto;
val auto_solve_time_limit = AutoSolve.auto_time_limit;
--- a/src/Tools/code/code_funcgr.ML Wed Apr 22 11:00:25 2009 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,335 +0,0 @@
-(* Title: Tools/code/code_funcgr.ML
- Author: Florian Haftmann, TU Muenchen
-
-Retrieving, normalizing and structuring code equations in graph
-with explicit dependencies.
-
-Legacy. To be replaced by Tools/code/code_wellsorted.ML
-*)
-
-signature CODE_WELLSORTED =
-sig
- type T
- val eqns: T -> string -> (thm * bool) list
- val typ: T -> string -> (string * sort) list * typ
- val all: T -> string list
- val pretty: theory -> T -> Pretty.T
- val make: theory -> string list
- -> ((sort -> sort) * Sorts.algebra) * T
- val eval_conv: theory
- -> (term -> term * (((sort -> sort) * Sorts.algebra) -> T -> thm)) -> cterm -> thm
- val eval_term: theory
- -> (term -> term * (((sort -> sort) * Sorts.algebra) -> T -> 'a)) -> term -> 'a
- val timing: bool ref
-end
-
-structure Code_Wellsorted : CODE_WELLSORTED =
-struct
-
-(** the graph type **)
-
-type T = (((string * sort) list * typ) * (thm * bool) list) Graph.T;
-
-fun eqns funcgr =
- these o Option.map snd o try (Graph.get_node funcgr);
-
-fun typ funcgr =
- fst o Graph.get_node funcgr;
-
-fun all funcgr = Graph.keys funcgr;
-
-fun pretty thy funcgr =
- AList.make (snd o Graph.get_node funcgr) (Graph.keys funcgr)
- |> (map o apfst) (Code_Unit.string_of_const thy)
- |> sort (string_ord o pairself fst)
- |> map (fn (s, thms) =>
- (Pretty.block o Pretty.fbreaks) (
- Pretty.str s
- :: map (Display.pretty_thm o fst) thms
- ))
- |> Pretty.chunks;
-
-
-(** generic combinators **)
-
-fun fold_consts f thms =
- thms
- |> maps (op :: o swap o apfst (snd o strip_comb) o Logic.dest_equals o Thm.plain_prop_of)
- |> (fold o fold_aterms) (fn Const c => f c | _ => I);
-
-fun consts_of (const, []) = []
- | consts_of (const, thms as _ :: _) =
- let
- fun the_const (c, _) = if c = const then I else insert (op =) c
- in fold_consts the_const (map fst thms) [] end;
-
-fun insts_of thy algebra tys sorts =
- let
- fun class_relation (x, _) _ = x;
- fun type_constructor tyco xs class =
- (tyco, class) :: (maps o maps) fst xs;
- fun type_variable (TVar (_, sort)) = map (pair []) sort
- | type_variable (TFree (_, sort)) = map (pair []) sort;
- fun of_sort_deriv ty sort =
- Sorts.of_sort_derivation (Syntax.pp_global thy) algebra
- { class_relation = class_relation, type_constructor = type_constructor,
- type_variable = type_variable }
- (ty, sort) handle Sorts.CLASS_ERROR _ => [] (*permissive!*)
- in (flat o flat) (map2 of_sort_deriv tys sorts) end;
-
-fun meets_of thy algebra =
- let
- fun meet_of ty sort tab =
- Sorts.meet_sort algebra (ty, sort) tab
- handle Sorts.CLASS_ERROR _ => tab (*permissive!*);
- in fold2 meet_of end;
-
-
-(** graph algorithm **)
-
-val timing = ref false;
-
-local
-
-fun resort_thms thy algebra typ_of thms =
- let
- val cs = fold_consts (insert (op =)) thms [];
- fun meets (c, ty) = case typ_of c
- of SOME (vs, _) =>
- meets_of thy algebra (Sign.const_typargs thy (c, ty)) (map snd vs)
- | NONE => I;
- val tab = fold meets cs Vartab.empty;
- in map (Code_Unit.inst_thm thy tab) thms end;
-
-fun resort_eqnss thy algebra funcgr =
- let
- val typ_funcgr = try (fst o Graph.get_node funcgr);
- val resort_dep = (apsnd o burrow_fst) (resort_thms thy algebra typ_funcgr);
- fun resort_rec typ_of (c, []) = (true, (c, []))
- | resort_rec typ_of (c, thms as (thm, _) :: _) = if is_some (AxClass.inst_of_param thy c)
- then (true, (c, thms))
- else let
- val (_, (vs, ty)) = Code_Unit.head_eqn thy thm;
- val thms' as (thm', _) :: _ = burrow_fst (resort_thms thy algebra typ_of) thms
- val (_, (vs', ty')) = Code_Unit.head_eqn thy thm'; (*FIXME simplify check*)
- in (Sign.typ_equiv thy (ty, ty'), (c, thms')) end;
- fun resort_recs eqnss =
- let
- fun typ_of c = case these (AList.lookup (op =) eqnss c)
- of (thm, _) :: _ => (SOME o snd o Code_Unit.head_eqn thy) thm
- | [] => NONE;
- val (unchangeds, eqnss') = split_list (map (resort_rec typ_of) eqnss);
- val unchanged = fold (fn x => fn y => x andalso y) unchangeds true;
- in (unchanged, eqnss') end;
- fun resort_rec_until eqnss =
- let
- val (unchanged, eqnss') = resort_recs eqnss;
- in if unchanged then eqnss' else resort_rec_until eqnss' end;
- in map resort_dep #> resort_rec_until end;
-
-fun instances_of thy algebra insts =
- let
- val thy_classes = (#classes o Sorts.rep_algebra o Sign.classes_of) thy;
- fun all_classparams tyco class =
- these (try (#params o AxClass.get_info thy) class)
- |> map_filter (fn (c, _) => try (AxClass.param_of_inst thy) (c, tyco))
- in
- Symtab.empty
- |> fold (fn (tyco, class) =>
- Symtab.map_default (tyco, []) (insert (op =) class)) insts
- |> (fn tab => Symtab.fold (fn (tyco, classes) => append (maps (all_classparams tyco)
- (Graph.all_succs thy_classes classes))) tab [])
- end;
-
-fun instances_of_consts thy algebra funcgr consts =
- let
- fun inst (cexpr as (c, ty)) = insts_of thy algebra
- (Sign.const_typargs thy (c, ty)) ((map snd o fst) (typ funcgr c));
- in
- []
- |> fold (fold (insert (op =)) o inst) consts
- |> instances_of thy algebra
- end;
-
-fun ensure_const' thy algebra funcgr const auxgr =
- if can (Graph.get_node funcgr) const
- then (NONE, auxgr)
- else if can (Graph.get_node auxgr) const
- then (SOME const, auxgr)
- else if is_some (Code.get_datatype_of_constr thy const) then
- auxgr
- |> Graph.new_node (const, [])
- |> pair (SOME const)
- else let
- val thms = Code.these_eqns thy const
- |> burrow_fst (Code_Unit.norm_args thy)
- |> burrow_fst (Code_Unit.norm_varnames thy Code_Name.purify_tvar Code_Name.purify_var);
- val rhs = consts_of (const, thms);
- in
- auxgr
- |> Graph.new_node (const, thms)
- |> fold_map (ensure_const thy algebra funcgr) rhs
- |-> (fn rhs' => fold (fn SOME const' => Graph.add_edge (const, const')
- | NONE => I) rhs')
- |> pair (SOME const)
- end
-and ensure_const thy algebra funcgr const =
- let
- val timeap = if !timing
- then Output.timeap_msg ("time for " ^ Code_Unit.string_of_const thy const)
- else I;
- in timeap (ensure_const' thy algebra funcgr const) end;
-
-fun merge_eqnss thy algebra raw_eqnss funcgr =
- let
- val eqnss = raw_eqnss
- |> resort_eqnss thy algebra funcgr
- |> filter_out (can (Graph.get_node funcgr) o fst);
- fun typ_eqn c [] = Code.default_typscheme thy c
- | typ_eqn c (thms as (thm, _) :: _) = (snd o Code_Unit.head_eqn thy) thm;
- fun add_eqns (const, thms) =
- Graph.new_node (const, (typ_eqn const thms, thms));
- fun add_deps (eqns as (const, thms)) funcgr =
- let
- val deps = consts_of eqns;
- val insts = instances_of_consts thy algebra funcgr
- (fold_consts (insert (op =)) (map fst thms) []);
- in
- funcgr
- |> ensure_consts thy algebra insts
- |> fold (curry Graph.add_edge const) deps
- |> fold (curry Graph.add_edge const) insts
- end;
- in
- funcgr
- |> fold add_eqns eqnss
- |> fold add_deps eqnss
- end
-and ensure_consts thy algebra cs funcgr =
- let
- val auxgr = Graph.empty
- |> fold (snd oo ensure_const thy algebra funcgr) cs;
- in
- funcgr
- |> fold (merge_eqnss thy algebra)
- (map (AList.make (Graph.get_node auxgr))
- (rev (Graph.strong_conn auxgr)))
- end;
-
-in
-
-(** retrieval interfaces **)
-
-val ensure_consts = ensure_consts;
-
-fun proto_eval thy cterm_of evaluator_lift evaluator proto_ct funcgr =
- let
- val ct = cterm_of proto_ct;
- val _ = Sign.no_vars (Syntax.pp_global thy) (Thm.term_of ct);
- val _ = Term.fold_types (Type.no_tvars #> K I) (Thm.term_of ct) ();
- fun consts_of t =
- fold_aterms (fn Const c_ty => cons c_ty | _ => I) t [];
- val algebra = Code.coregular_algebra thy;
- val thm = Code.preprocess_conv thy ct;
- val ct' = Thm.rhs_of thm;
- val t' = Thm.term_of ct';
- val consts = map fst (consts_of t');
- val funcgr' = ensure_consts thy algebra consts funcgr;
- val (t'', evaluator_funcgr) = evaluator t';
- val consts' = consts_of t'';
- val dicts = instances_of_consts thy algebra funcgr' consts';
- val funcgr'' = ensure_consts thy algebra dicts funcgr';
- in (evaluator_lift (evaluator_funcgr (Code.operational_algebra thy)) thm funcgr'', funcgr'') end;
-
-fun proto_eval_conv thy =
- let
- fun evaluator_lift evaluator thm1 funcgr =
- let
- val thm2 = evaluator funcgr;
- val thm3 = Code.postprocess_conv thy (Thm.rhs_of thm2);
- in
- Thm.transitive thm1 (Thm.transitive thm2 thm3) handle THM _ =>
- error ("could not construct evaluation proof:\n"
- ^ (cat_lines o map Display.string_of_thm) [thm1, thm2, thm3])
- end;
- in proto_eval thy I evaluator_lift end;
-
-fun proto_eval_term thy =
- let
- fun evaluator_lift evaluator _ funcgr = evaluator funcgr;
- in proto_eval thy (Thm.cterm_of thy) evaluator_lift end;
-
-end; (*local*)
-
-structure Funcgr = CodeDataFun
-(
- type T = T;
- val empty = Graph.empty;
- fun purge _ cs funcgr =
- Graph.del_nodes ((Graph.all_preds funcgr
- o filter (can (Graph.get_node funcgr))) cs) funcgr;
-);
-
-fun make thy =
- pair (Code.operational_algebra thy)
- o Funcgr.change thy o ensure_consts thy (Code.coregular_algebra thy);
-
-fun eval_conv thy f =
- fst o Funcgr.change_yield thy o proto_eval_conv thy f;
-
-fun eval_term thy f =
- fst o Funcgr.change_yield thy o proto_eval_term thy f;
-
-
-(** diagnostic commands **)
-
-fun code_depgr thy consts =
- let
- val (_, gr) = make thy consts;
- val select = Graph.all_succs gr consts;
- in
- gr
- |> not (null consts) ? Graph.subgraph (member (op =) select)
- |> Graph.map_nodes ((apsnd o map o apfst) (AxClass.overload thy))
- end;
-
-fun code_thms thy = Pretty.writeln o pretty thy o code_depgr thy;
-
-fun code_deps thy consts =
- let
- val gr = code_depgr thy consts;
- fun mk_entry (const, (_, (_, parents))) =
- let
- val name = Code_Unit.string_of_const thy const;
- val nameparents = map (Code_Unit.string_of_const thy) parents;
- in { name = name, ID = name, dir = "", unfold = true,
- path = "", parents = nameparents }
- end;
- val prgr = Graph.fold ((fn x => fn xs => xs @ [x]) o mk_entry) gr [];
- in Present.display_graph prgr end;
-
-local
-
-structure P = OuterParse
-and K = OuterKeyword
-
-fun code_thms_cmd thy = code_thms thy o op @ o Code_Name.read_const_exprs thy;
-fun code_deps_cmd thy = code_deps thy o op @ o Code_Name.read_const_exprs thy;
-
-in
-
-val _ =
- OuterSyntax.improper_command "code_thms" "print system of code equations for code" OuterKeyword.diag
- (Scan.repeat P.term_group
- >> (fn cs => Toplevel.no_timing o Toplevel.unknown_theory
- o Toplevel.keep ((fn thy => code_thms_cmd thy cs) o Toplevel.theory_of)));
-
-val _ =
- OuterSyntax.improper_command "code_deps" "visualize dependencies of code equations for code" OuterKeyword.diag
- (Scan.repeat P.term_group
- >> (fn cs => Toplevel.no_timing o Toplevel.unknown_theory
- o Toplevel.keep ((fn thy => code_deps_cmd thy cs) o Toplevel.theory_of)));
-
-end;
-
-end; (*struct*)
--- a/src/Tools/code/code_ml.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Tools/code/code_ml.ML Mon Apr 27 07:26:17 2009 -0700
@@ -6,8 +6,9 @@
signature CODE_ML =
sig
- val eval_term: string * (unit -> 'a) option ref
- -> theory -> term -> string list -> 'a
+ val eval: string option -> string * (unit -> 'a) option ref
+ -> ((term -> term) -> 'a -> 'a) -> theory -> term -> string list -> 'a
+ val target_Eval: string
val setup: theory -> theory
end;
@@ -22,6 +23,7 @@
val target_SML = "SML";
val target_OCaml = "OCaml";
+val target_Eval = "Eval";
datatype ml_stmt =
MLExc of string * int
@@ -907,36 +909,38 @@
in (deresolver, nodes) end;
fun serialize_ml target compile pr_module pr_stmt raw_module_name labelled_name reserved_names includes raw_module_alias
- _ syntax_tyco syntax_const naming program cs destination =
+ _ syntax_tyco syntax_const naming program stmt_names destination =
let
val is_cons = Code_Thingol.is_cons program;
- val stmt_names = Code_Target.stmt_names_of_destination destination;
- val module_name = if null stmt_names then raw_module_name else SOME "Code";
+ val present_stmt_names = Code_Target.stmt_names_of_destination destination;
+ val is_present = not (null present_stmt_names);
+ val module_name = if is_present then SOME "Code" else raw_module_name;
val (deresolver, nodes) = ml_node_of_program labelled_name module_name
reserved_names raw_module_alias program;
val reserved_names = Code_Printer.make_vars reserved_names;
fun pr_node prefix (Dummy _) =
NONE
- | pr_node prefix (Stmt (_, stmt)) = if null stmt_names orelse
- (not o null o filter (member (op =) stmt_names) o stmt_names_of) stmt then SOME
+ | pr_node prefix (Stmt (_, stmt)) = if is_present andalso
+ (null o filter (member (op =) present_stmt_names) o stmt_names_of) stmt
+ then NONE
+ else SOME
(pr_stmt naming labelled_name syntax_tyco syntax_const reserved_names
(deresolver prefix) is_cons stmt)
- else NONE
| pr_node prefix (Module (module_name, (_, nodes))) =
separate (str "")
((map_filter (pr_node (prefix @ [module_name]) o Graph.get_node nodes)
o rev o flat o Graph.strong_conn) nodes)
- |> (if null stmt_names then pr_module module_name else Pretty.chunks)
+ |> (if is_present then Pretty.chunks else pr_module module_name)
|> SOME;
- val cs' = (map o try)
- (deresolver (if is_some module_name then the_list module_name else [])) cs;
+ val stmt_names' = (map o try)
+ (deresolver (if is_some module_name then the_list module_name else [])) stmt_names;
val p = Pretty.chunks (separate (str "") (map snd includes @ (map_filter
(pr_node [] o Graph.get_node nodes) o rev o flat o Graph.strong_conn) nodes));
in
Code_Target.mk_serialization target
(case compile of SOME compile => SOME (compile o Code_Target.code_of_pretty) | NONE => NONE)
(fn NONE => Code_Target.code_writeln | SOME file => File.write file o Code_Target.code_of_pretty)
- (rpair cs' o Code_Target.code_of_pretty) p destination
+ (rpair stmt_names' o Code_Target.code_of_pretty) p destination
end;
end; (*local*)
@@ -944,20 +948,20 @@
(** ML (system language) code for evaluation and instrumentalization **)
-fun ml_code_of thy = Code_Target.serialize_custom thy (target_SML,
+fun eval_code_of some_target thy = Code_Target.serialize_custom thy (the_default target_Eval some_target,
(fn _ => fn [] => serialize_ml target_SML (SOME (K ())) (K Pretty.chunks) pr_sml_stmt (SOME ""),
literals_sml));
(* evaluation *)
-fun eval eval'' term_of reff thy ct args =
+fun eval some_target reff postproc thy t args =
let
val ctxt = ProofContext.init thy;
- val _ = if null (Term.add_frees (term_of ct) []) then () else error ("Term "
- ^ quote (Syntax.string_of_term_global thy (term_of ct))
+ val _ = if null (Term.add_frees t []) then () else error ("Term "
+ ^ quote (Syntax.string_of_term_global thy t)
^ " to be evaluated contains free variables");
- fun eval' naming program ((vs, ty), t) deps =
+ fun evaluator naming program (((_, (_, ty)), _), t) deps =
let
val _ = if Code_Thingol.contains_dictvar t then
error "Term to be evaluated contains free dictionaries" else ();
@@ -966,13 +970,11 @@
|> Graph.new_node (value_name,
Code_Thingol.Fun (Term.dummy_patternN, (([], ty), [(([], t), (Drule.dummy_thm, true))])))
|> fold (curry Graph.add_edge value_name) deps;
- val (value_code, [SOME value_name']) = ml_code_of thy naming program' [value_name];
+ val (value_code, [SOME value_name']) = eval_code_of some_target thy naming program' [value_name];
val sml_code = "let\n" ^ value_code ^ "\nin " ^ value_name'
^ space_implode " " (map (enclose "(" ")") args) ^ " end";
in ML_Context.evaluate ctxt false reff sml_code end;
- in eval'' thy (rpair eval') ct end;
-
-fun eval_term reff = eval Code_Thingol.eval_term I reff;
+ in Code_Thingol.eval thy I postproc evaluator t end;
(* instrumentalization by antiquotation *)
@@ -981,42 +983,69 @@
structure CodeAntiqData = ProofDataFun
(
- type T = string list * (bool * (string * (string * (string * string) list) lazy));
- fun init _ = ([], (true, ("", Lazy.value ("", []))));
+ type T = (string list * string list) * (bool * (string
+ * (string * ((string * string) list * (string * string) list)) lazy));
+ fun init _ = (([], []), (true, ("", Lazy.value ("", ([], [])))));
);
val is_first_occ = fst o snd o CodeAntiqData.get;
-fun delayed_code thy consts () =
+fun delayed_code thy tycos consts () =
let
val (consts', (naming, program)) = Code_Thingol.consts_program thy consts;
- val (ml_code, consts'') = ml_code_of thy naming program consts';
- val const_tab = map2 (fn const => fn NONE =>
- error ("Constant " ^ (quote o Code_Unit.string_of_const thy) const
- ^ "\nhas a user-defined serialization")
- | SOME const' => (const, const')) consts consts''
- in (ml_code, const_tab) end;
+ val tycos' = map (the o Code_Thingol.lookup_tyco naming) tycos;
+ val (ml_code, target_names) = eval_code_of NONE thy naming program (consts' @ tycos');
+ val (consts'', tycos'') = chop (length consts') target_names;
+ val consts_map = map2 (fn const => fn NONE =>
+ error ("Constant " ^ (quote o Code_Unit.string_of_const thy) const
+ ^ "\nhas a user-defined serialization")
+ | SOME const'' => (const, const'')) consts consts''
+ val tycos_map = map2 (fn tyco => fn NONE =>
+ error ("Type " ^ (quote o Sign.extern_type thy) tyco
+ ^ "\nhas a user-defined serialization")
+ | SOME tyco'' => (tyco, tyco'')) tycos tycos'';
+ in (ml_code, (tycos_map, consts_map)) end;
-fun register_const const ctxt =
+fun register_code new_tycos new_consts ctxt =
let
- val (consts, (_, (struct_name, _))) = CodeAntiqData.get ctxt;
- val consts' = insert (op =) const consts;
+ val ((tycos, consts), (_, (struct_name, _))) = CodeAntiqData.get ctxt;
+ val tycos' = fold (insert (op =)) new_tycos tycos;
+ val consts' = fold (insert (op =)) new_consts consts;
val (struct_name', ctxt') = if struct_name = ""
then ML_Antiquote.variant "Code" ctxt
else (struct_name, ctxt);
- val acc_code = Lazy.lazy (delayed_code (ProofContext.theory_of ctxt) consts');
- in CodeAntiqData.put (consts', (false, (struct_name', acc_code))) ctxt' end;
+ val acc_code = Lazy.lazy (delayed_code (ProofContext.theory_of ctxt) tycos' consts');
+ in CodeAntiqData.put ((tycos', consts'), (false, (struct_name', acc_code))) ctxt' end;
+
+fun register_const const = register_code [] [const];
-fun print_code struct_name is_first const ctxt =
+fun register_datatype tyco constrs = register_code [tyco] constrs;
+
+fun print_const const all_struct_name tycos_map consts_map =
+ (Long_Name.append all_struct_name o the o AList.lookup (op =) consts_map) const;
+
+fun print_datatype tyco constrs all_struct_name tycos_map consts_map =
let
- val (consts, (_, (struct_code_name, acc_code))) = CodeAntiqData.get ctxt;
- val (raw_ml_code, consts_map) = Lazy.force acc_code;
- val const'' = Long_Name.append (Long_Name.append struct_name struct_code_name)
- ((the o AList.lookup (op =) consts_map) const);
+ val upperize = implode o nth_map 0 Symbol.to_ascii_upper o explode;
+ fun check_base name name'' =
+ if upperize (Long_Name.base_name name) = upperize name''
+ then () else error ("Name as printed " ^ quote name''
+ ^ "\ndiffers from logical base name " ^ quote (Long_Name.base_name name) ^ "; sorry.");
+ val tyco'' = (the o AList.lookup (op =) tycos_map) tyco;
+ val constrs'' = map (the o AList.lookup (op =) consts_map) constrs;
+ val _ = check_base tyco tyco'';
+ val _ = map2 check_base constrs constrs'';
+ in "datatype " ^ tyco'' ^ " = datatype " ^ Long_Name.append all_struct_name tyco'' end;
+
+fun print_code struct_name is_first print_it ctxt =
+ let
+ val (_, (_, (struct_code_name, acc_code))) = CodeAntiqData.get ctxt;
+ val (raw_ml_code, (tycos_map, consts_map)) = Lazy.force acc_code;
val ml_code = if is_first then "\nstructure " ^ struct_code_name
^ " =\nstruct\n\n" ^ raw_ml_code ^ "\nend;\n\n"
else "";
- in (ml_code, const'') end;
+ val all_struct_name = Long_Name.append struct_name struct_code_name;
+ in (ml_code, print_it all_struct_name tycos_map consts_map) end;
in
@@ -1025,7 +1054,19 @@
val const = Code_Unit.check_const (ProofContext.theory_of background) raw_const;
val is_first = is_first_occ background;
val background' = register_const const background;
- in (print_code struct_name is_first const, background') end;
+ in (print_code struct_name is_first (print_const const), background') end;
+
+fun ml_code_datatype_antiq (raw_tyco, raw_constrs) {struct_name, background} =
+ let
+ val thy = ProofContext.theory_of background;
+ val tyco = Sign.intern_type thy raw_tyco;
+ val constrs = map (Code_Unit.check_const thy) raw_constrs;
+ val constrs' = (map fst o snd o Code.get_datatype thy) tyco;
+ val _ = if gen_eq_set (op =) (constrs, constrs') then ()
+ else error ("Type " ^ quote tyco ^ ": given constructors diverge from real constructors")
+ val is_first = is_first_occ background;
+ val background' = register_datatype tyco constrs background;
+ in (print_code struct_name is_first (print_datatype tyco constrs), background') end;
end; (*local*)
@@ -1033,6 +1074,10 @@
(** Isar setup **)
val _ = ML_Context.add_antiq "code" (fn _ => Args.term >> ml_code_antiq);
+val _ = ML_Context.add_antiq "code_datatype" (fn _ =>
+ (Args.tyname --| Scan.lift (Args.$$$ "=")
+ -- (Args.term ::: Scan.repeat (Scan.lift (Args.$$$ "|") |-- Args.term)))
+ >> ml_code_datatype_antiq);
fun isar_seri_sml module_name =
Code_Target.parse_args (Scan.succeed ())
@@ -1048,6 +1093,7 @@
val setup =
Code_Target.add_target (target_SML, (isar_seri_sml, literals_sml))
#> Code_Target.add_target (target_OCaml, (isar_seri_ocaml, literals_ocaml))
+ #> Code_Target.extend_target (target_Eval, (target_SML, K I))
#> Code_Target.add_syntax_tyco target_SML "fun" (SOME (2, fn pr_typ => fn fxy => fn [ty1, ty2] =>
brackify_infix (1, R) fxy [
pr_typ (INFX (1, X)) ty1,
--- a/src/Tools/code/code_target.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Tools/code/code_target.ML Mon Apr 27 07:26:17 2009 -0700
@@ -68,7 +68,7 @@
fun compile f = (code_setmp f Compile; ());
fun export f = (code_setmp f Export; ());
fun file p f = (code_setmp f (File p); ());
-fun string cs f = fst (the (code_setmp f (String cs)));
+fun string stmts f = fst (the (code_setmp f (String stmts)));
fun stmt_names_of_destination (String stmts) = stmts
| stmt_names_of_destination _ = [];
--- a/src/Tools/code/code_thingol.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Tools/code/code_thingol.ML Mon Apr 27 07:26:17 2009 -0700
@@ -83,11 +83,11 @@
val consts_program: theory -> string list -> string list * (naming * program)
val cached_program: theory -> naming * program
- val eval_conv: theory
- -> (term -> term * (naming -> program -> typscheme * iterm -> string list -> thm))
+ val eval_conv: theory -> (sort -> sort)
+ -> (naming -> program -> (((string * sort) list * typscheme) * (string * itype) list) * iterm -> string list -> cterm -> thm)
-> cterm -> thm
- val eval_term: theory
- -> (term -> term * (naming -> program -> typscheme * iterm -> string list -> 'a))
+ val eval: theory -> (sort -> sort) -> ((term -> term) -> 'a -> 'a)
+ -> (naming -> program -> (((string * sort) list * typscheme) * (string * itype) list) * iterm -> string list -> 'a)
-> term -> 'a
end;
@@ -459,7 +459,45 @@
(* translation *)
-fun ensure_class thy (algbr as (_, algebra)) funcgr class =
+fun ensure_tyco thy algbr funcgr tyco =
+ let
+ val stmt_datatype =
+ let
+ val (vs, cos) = Code.get_datatype thy tyco;
+ in
+ fold_map (translate_tyvar_sort thy algbr funcgr) vs
+ ##>> fold_map (fn (c, tys) =>
+ ensure_const thy algbr funcgr c
+ ##>> fold_map (translate_typ thy algbr funcgr) tys) cos
+ #>> (fn info => Datatype (tyco, info))
+ end;
+ in ensure_stmt lookup_tyco (declare_tyco thy) stmt_datatype tyco end
+and ensure_const thy algbr funcgr c =
+ let
+ fun stmt_datatypecons tyco =
+ ensure_tyco thy algbr funcgr tyco
+ #>> (fn tyco => Datatypecons (c, tyco));
+ fun stmt_classparam class =
+ ensure_class thy algbr funcgr class
+ #>> (fn class => Classparam (c, class));
+ fun stmt_fun ((vs, ty), raw_thms) =
+ let
+ val thms = if null (Term.add_tfreesT ty []) orelse (null o fst o strip_type) ty
+ then raw_thms
+ else (map o apfst) (Code_Unit.expand_eta thy 1) raw_thms;
+ in
+ fold_map (translate_tyvar_sort thy algbr funcgr) vs
+ ##>> translate_typ thy algbr funcgr ty
+ ##>> fold_map (translate_eq thy algbr funcgr) thms
+ #>> (fn info => Fun (c, info))
+ end;
+ val stmt_const = case Code.get_datatype_of_constr thy c
+ of SOME tyco => stmt_datatypecons tyco
+ | NONE => (case AxClass.class_of_param thy c
+ of SOME class => stmt_classparam class
+ | NONE => stmt_fun (Code_Wellsorted.typ funcgr c, Code_Wellsorted.eqns funcgr c))
+ in ensure_stmt lookup_const (declare_const thy) stmt_const c end
+and ensure_class thy (algbr as (_, algebra)) funcgr class =
let
val superclasses = (Sorts.minimize_sort algebra o Sorts.super_classes algebra) class;
val cs = #params (AxClass.get_info thy class);
@@ -477,65 +515,6 @@
##>> ensure_class thy algbr funcgr superclass
#>> Classrel;
in ensure_stmt lookup_classrel (declare_classrel thy) stmt_classrel (subclass, superclass) end
-and ensure_tyco thy algbr funcgr tyco =
- let
- val stmt_datatype =
- let
- val (vs, cos) = Code.get_datatype thy tyco;
- in
- fold_map (translate_tyvar_sort thy algbr funcgr) vs
- ##>> fold_map (fn (c, tys) =>
- ensure_const thy algbr funcgr c
- ##>> fold_map (translate_typ thy algbr funcgr) tys) cos
- #>> (fn info => Datatype (tyco, info))
- end;
- in ensure_stmt lookup_tyco (declare_tyco thy) stmt_datatype tyco end
-and translate_tyvar_sort thy (algbr as (proj_sort, _)) funcgr (v, sort) =
- fold_map (ensure_class thy algbr funcgr) (proj_sort sort)
- #>> (fn sort => (unprefix "'" v, sort))
-and translate_typ thy algbr funcgr (TFree (v, _)) =
- pair (ITyVar (unprefix "'" v))
- | translate_typ thy algbr funcgr (Type (tyco, tys)) =
- ensure_tyco thy algbr funcgr tyco
- ##>> fold_map (translate_typ thy algbr funcgr) tys
- #>> (fn (tyco, tys) => tyco `%% tys)
-and translate_dicts thy (algbr as (proj_sort, algebra)) funcgr thm (ty, sort) =
- let
- val pp = Syntax.pp_global thy;
- datatype typarg =
- Global of (class * string) * typarg list list
- | Local of (class * class) list * (string * (int * sort));
- fun class_relation (Global ((_, tyco), yss), _) class =
- Global ((class, tyco), yss)
- | class_relation (Local (classrels, v), subclass) superclass =
- Local ((subclass, superclass) :: classrels, v);
- fun type_constructor tyco yss class =
- Global ((class, tyco), (map o map) fst yss);
- fun type_variable (TFree (v, sort)) =
- let
- val sort' = proj_sort sort;
- in map_index (fn (n, class) => (Local ([], (v, (n, sort'))), class)) sort' end;
- val typargs = Sorts.of_sort_derivation pp algebra
- {class_relation = class_relation, type_constructor = type_constructor,
- type_variable = type_variable} (ty, proj_sort sort)
- handle Sorts.CLASS_ERROR e => not_wellsorted thy thm ty sort e;
- fun mk_dict (Global (inst, yss)) =
- ensure_inst thy algbr funcgr inst
- ##>> (fold_map o fold_map) mk_dict yss
- #>> (fn (inst, dss) => DictConst (inst, dss))
- | mk_dict (Local (classrels, (v, (k, sort)))) =
- fold_map (ensure_classrel thy algbr funcgr) classrels
- #>> (fn classrels => DictVar (classrels, (unprefix "'" v, (k, length sort))))
- in fold_map mk_dict typargs end
-and translate_eq thy algbr funcgr (thm, linear) =
- let
- val (args, rhs) = (apfst (snd o strip_comb) o Logic.dest_equals
- o Logic.unvarify o prop_of) thm;
- in
- fold_map (translate_term thy algbr funcgr (SOME thm)) args
- ##>> translate_term thy algbr funcgr (SOME thm) rhs
- #>> rpair (thm, linear)
- end
and ensure_inst thy (algbr as (_, algebra)) funcgr (class, tyco) =
let
val superclasses = (Sorts.minimize_sort algebra o Sorts.super_classes algebra) class;
@@ -572,31 +551,12 @@
#>> (fn ((((class, tyco), arity), superarities), classparams) =>
Classinst ((class, (tyco, arity)), (superarities, classparams)));
in ensure_stmt lookup_instance (declare_instance thy) stmt_inst (class, tyco) end
-and ensure_const thy algbr funcgr c =
- let
- fun stmt_datatypecons tyco =
+and translate_typ thy algbr funcgr (TFree (v, _)) =
+ pair (ITyVar (unprefix "'" v))
+ | translate_typ thy algbr funcgr (Type (tyco, tys)) =
ensure_tyco thy algbr funcgr tyco
- #>> (fn tyco => Datatypecons (c, tyco));
- fun stmt_classparam class =
- ensure_class thy algbr funcgr class
- #>> (fn class => Classparam (c, class));
- fun stmt_fun ((vs, ty), raw_thms) =
- let
- val thms = if null (Term.add_tfreesT ty []) orelse (null o fst o strip_type) ty
- then raw_thms
- else (map o apfst) (Code_Unit.expand_eta thy 1) raw_thms;
- in
- fold_map (translate_tyvar_sort thy algbr funcgr) vs
- ##>> translate_typ thy algbr funcgr ty
- ##>> fold_map (translate_eq thy algbr funcgr) thms
- #>> (fn info => Fun (c, info))
- end;
- val stmt_const = case Code.get_datatype_of_constr thy c
- of SOME tyco => stmt_datatypecons tyco
- | NONE => (case AxClass.class_of_param thy c
- of SOME class => stmt_classparam class
- | NONE => stmt_fun (Code_Wellsorted.typ funcgr c, Code_Wellsorted.eqns funcgr c))
- in ensure_stmt lookup_const (declare_const thy) stmt_const c end
+ ##>> fold_map (translate_typ thy algbr funcgr) tys
+ #>> (fn (tyco, tys) => tyco `%% tys)
and translate_term thy algbr funcgr thm (Const (c, ty)) =
translate_app thy algbr funcgr thm ((c, ty), [])
| translate_term thy algbr funcgr thm (Free (v, _)) =
@@ -617,6 +577,15 @@
translate_term thy algbr funcgr thm t'
##>> fold_map (translate_term thy algbr funcgr thm) ts
#>> (fn (t, ts) => t `$$ ts)
+and translate_eq thy algbr funcgr (thm, linear) =
+ let
+ val (args, rhs) = (apfst (snd o strip_comb) o Logic.dest_equals
+ o Logic.unvarify o prop_of) thm;
+ in
+ fold_map (translate_term thy algbr funcgr (SOME thm)) args
+ ##>> translate_term thy algbr funcgr (SOME thm) rhs
+ #>> rpair (thm, linear)
+ end
and translate_const thy algbr funcgr thm (c, ty) =
let
val tys = Sign.const_typargs thy (c, ty);
@@ -695,7 +664,38 @@
and translate_app thy algbr funcgr thm (c_ty_ts as ((c, _), _)) =
case Code.get_case_scheme thy c
of SOME case_scheme => translate_app_case thy algbr funcgr thm case_scheme c_ty_ts
- | NONE => translate_app_const thy algbr funcgr thm c_ty_ts;
+ | NONE => translate_app_const thy algbr funcgr thm c_ty_ts
+and translate_tyvar_sort thy (algbr as (proj_sort, _)) funcgr (v, sort) =
+ fold_map (ensure_class thy algbr funcgr) (proj_sort sort)
+ #>> (fn sort => (unprefix "'" v, sort))
+and translate_dicts thy (algbr as (proj_sort, algebra)) funcgr thm (ty, sort) =
+ let
+ val pp = Syntax.pp_global thy;
+ datatype typarg =
+ Global of (class * string) * typarg list list
+ | Local of (class * class) list * (string * (int * sort));
+ fun class_relation (Global ((_, tyco), yss), _) class =
+ Global ((class, tyco), yss)
+ | class_relation (Local (classrels, v), subclass) superclass =
+ Local ((subclass, superclass) :: classrels, v);
+ fun type_constructor tyco yss class =
+ Global ((class, tyco), (map o map) fst yss);
+ fun type_variable (TFree (v, sort)) =
+ let
+ val sort' = proj_sort sort;
+ in map_index (fn (n, class) => (Local ([], (v, (n, sort'))), class)) sort' end;
+ val typargs = Sorts.of_sort_derivation pp algebra
+ {class_relation = class_relation, type_constructor = type_constructor,
+ type_variable = type_variable} (ty, proj_sort sort)
+ handle Sorts.CLASS_ERROR e => not_wellsorted thy thm ty sort e;
+ fun mk_dict (Global (inst, yss)) =
+ ensure_inst thy algbr funcgr inst
+ ##>> (fold_map o fold_map) mk_dict yss
+ #>> (fn (inst, dss) => DictConst (inst, dss))
+ | mk_dict (Local (classrels, (v, (k, sort)))) =
+ fold_map (ensure_classrel thy algbr funcgr) classrels
+ #>> (fn classrels => DictVar (classrels, (unprefix "'" v, (k, length sort))))
+ in fold_map mk_dict typargs end;
(* store *)
@@ -733,14 +733,14 @@
fun generate_consts thy algebra funcgr =
fold_map (ensure_const thy algebra funcgr);
in
- invoke_generation thy (Code_Wellsorted.make thy cs) generate_consts cs
+ invoke_generation thy (Code_Wellsorted.obtain thy cs []) generate_consts cs
|-> project_consts
end;
(* value evaluation *)
-fun ensure_value thy algbr funcgr t =
+fun ensure_value thy algbr funcgr (fs, t) =
let
val ty = fastype_of t;
val vs = fold_term_types (K (fold_atyps (insert (eq_fst op =)
@@ -751,33 +751,91 @@
##>> translate_term thy algbr funcgr NONE t
#>> (fn ((vs, ty), t) => Fun
(Term.dummy_patternN, ((vs, ty), [(([], t), (Drule.dummy_thm, true))])));
- fun term_value (dep, (naming, program1)) =
+ fun term_value fs (dep, (naming, program1)) =
let
- val Fun (_, ((vs, ty), [(([], t), _)])) =
+ val Fun (_, (vs_ty, [(([], t), _)])) =
Graph.get_node program1 Term.dummy_patternN;
val deps = Graph.imm_succs program1 Term.dummy_patternN;
val program2 = Graph.del_nodes [Term.dummy_patternN] program1;
val deps_all = Graph.all_succs program2 deps;
val program3 = Graph.subgraph (member (op =) deps_all) program2;
- in (((naming, program3), (((vs, ty), t), deps)), (dep, (naming, program2))) end;
+ in (((naming, program3), (((vs_ty, fs), t), deps)), (dep, (naming, program2))) end;
in
ensure_stmt ((K o K) NONE) pair stmt_value Term.dummy_patternN
#> snd
- #> term_value
+ #> fold_map (fn (v, ty) => translate_typ thy algbr funcgr ty
+ #-> (fn ty' => pair (v, ty'))) fs
+ #-> (fn fs' => term_value fs')
+ end;
+
+fun base_evaluator thy evaluator algebra funcgr vs t =
+ let
+ val fs = Term.add_frees t [];
+ val (((naming, program), ((((vs', ty'), fs'), t'), deps)), _) =
+ invoke_generation thy (algebra, funcgr) ensure_value (fs, t);
+ val vs'' = map (fn (v, _) => (v, (the o AList.lookup (op =) vs o prefix "'") v)) vs';
+ in evaluator naming program (((vs'', (vs', ty')), fs'), t') deps end;
+
+fun eval_conv thy prep_sort = Code_Wellsorted.eval_conv thy prep_sort o base_evaluator thy;
+fun eval thy prep_sort postproc = Code_Wellsorted.eval thy prep_sort postproc o base_evaluator thy;
+
+
+(** diagnostic commands **)
+
+fun code_depgr thy consts =
+ let
+ val (_, eqngr) = Code_Wellsorted.obtain thy consts [];
+ val select = Graph.all_succs eqngr consts;
+ in
+ eqngr
+ |> not (null consts) ? Graph.subgraph (member (op =) select)
+ |> Graph.map_nodes ((apsnd o map o apfst) (AxClass.overload thy))
end;
-fun eval thy evaluator t =
+fun code_thms thy = Pretty.writeln o Code_Wellsorted.pretty thy o code_depgr thy;
+
+fun code_deps thy consts =
let
- val (t', evaluator'') = evaluator t;
- fun evaluator' algebra funcgr =
- let
- val (((naming, program), (vs_ty_t, deps)), _) =
- invoke_generation thy (algebra, funcgr) ensure_value t';
- in evaluator'' naming program vs_ty_t deps end;
- in (t', evaluator') end
+ val eqngr = code_depgr thy consts;
+ val constss = Graph.strong_conn eqngr;
+ val mapping = Symtab.empty |> fold (fn consts => fold (fn const =>
+ Symtab.update (const, consts)) consts) constss;
+ fun succs consts = consts
+ |> maps (Graph.imm_succs eqngr)
+ |> subtract (op =) consts
+ |> map (the o Symtab.lookup mapping)
+ |> distinct (op =);
+ val conn = [] |> fold (fn consts => cons (consts, succs consts)) constss;
+ fun namify consts = map (Code_Unit.string_of_const thy) consts
+ |> commas;
+ val prgr = map (fn (consts, constss) =>
+ { name = namify consts, ID = namify consts, dir = "", unfold = true,
+ path = "", parents = map namify constss }) conn;
+ in Present.display_graph prgr end;
+
+local
-fun eval_conv thy = Code_Wellsorted.eval_conv thy o eval thy;
-fun eval_term thy = Code_Wellsorted.eval_term thy o eval thy;
+structure P = OuterParse
+and K = OuterKeyword
+
+fun code_thms_cmd thy = code_thms thy o op @ o Code_Name.read_const_exprs thy;
+fun code_deps_cmd thy = code_deps thy o op @ o Code_Name.read_const_exprs thy;
+
+in
+
+val _ =
+ OuterSyntax.improper_command "code_thms" "print system of code equations for code" OuterKeyword.diag
+ (Scan.repeat P.term_group
+ >> (fn cs => Toplevel.no_timing o Toplevel.unknown_theory
+ o Toplevel.keep ((fn thy => code_thms_cmd thy cs) o Toplevel.theory_of)));
+
+val _ =
+ OuterSyntax.improper_command "code_deps" "visualize dependencies of code equations for code" OuterKeyword.diag
+ (Scan.repeat P.term_group
+ >> (fn cs => Toplevel.no_timing o Toplevel.unknown_theory
+ o Toplevel.keep ((fn thy => code_deps_cmd thy cs) o Toplevel.theory_of)));
+
+end;
end; (*struct*)
--- a/src/Tools/code/code_wellsorted.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Tools/code/code_wellsorted.ML Mon Apr 27 07:26:17 2009 -0700
@@ -7,25 +7,26 @@
signature CODE_WELLSORTED =
sig
- type T
- val eqns: T -> string -> (thm * bool) list
- val typ: T -> string -> (string * sort) list * typ
- val all: T -> string list
- val pretty: theory -> T -> Pretty.T
- val make: theory -> string list
- -> ((sort -> sort) * Sorts.algebra) * T
- val eval_conv: theory
- -> (term -> term * (((sort -> sort) * Sorts.algebra) -> T -> thm)) -> cterm -> thm
- val eval_term: theory
- -> (term -> term * (((sort -> sort) * Sorts.algebra) -> T -> 'a)) -> term -> 'a
+ type code_algebra
+ type code_graph
+ val eqns: code_graph -> string -> (thm * bool) list
+ val typ: code_graph -> string -> (string * sort) list * typ
+ val all: code_graph -> string list
+ val pretty: theory -> code_graph -> Pretty.T
+ val obtain: theory -> string list -> term list -> code_algebra * code_graph
+ val eval_conv: theory -> (sort -> sort)
+ -> (code_algebra -> code_graph -> (string * sort) list -> term -> cterm -> thm) -> cterm -> thm
+ val eval: theory -> (sort -> sort) -> ((term -> term) -> 'a -> 'a)
+ -> (code_algebra -> code_graph -> (string * sort) list -> term -> 'a) -> term -> 'a
end
structure Code_Wellsorted : CODE_WELLSORTED =
struct
-(** the equation graph type **)
+(** the algebra and code equation graph types **)
-type T = (((string * sort) list * typ) * (thm * bool) list) Graph.T;
+type code_algebra = (sort -> sort) * Sorts.algebra;
+type code_graph = (((string * sort) list * typ) * (thm * bool) list) Graph.T;
fun eqns eqngr = these o Option.map snd o try (Graph.get_node eqngr);
fun typ eqngr = fst o Graph.get_node eqngr;
@@ -47,8 +48,10 @@
(* auxiliary *)
+fun is_proper_class thy = can (AxClass.get_info thy);
+
fun complete_proper_sort thy =
- Sign.complete_sort thy #> filter (can (AxClass.get_info thy));
+ Sign.complete_sort thy #> filter (is_proper_class thy);
fun inst_params thy tyco =
map (fn (c, _) => AxClass.param_of_inst thy (c, tyco))
@@ -232,8 +235,7 @@
((class, tyco), map (fn k => (snd o Vargraph.get_node vardeps) (Inst (class, tyco), k))
(0 upto Sign.arity_number thy tyco - 1));
-fun add_eqs thy (proj_sort, algebra) vardeps
- (c, (proto_lhs, proto_eqns)) (rhss, eqngr) =
+fun add_eqs thy vardeps (c, (proto_lhs, proto_eqns)) (rhss, eqngr) =
if can (Graph.get_node eqngr) c then (rhss, eqngr)
else let
val lhs = map_index (fn (k, (v, _)) =>
@@ -246,72 +248,30 @@
val eqngr' = Graph.new_node (c, (tyscm, eqns)) eqngr;
in (map (pair c) rhss' @ rhss, eqngr') end;
-fun extend_arities_eqngr thy cs cs_rhss (arities, eqngr) =
+fun extend_arities_eqngr thy cs ts (arities, eqngr) =
let
- val cs_rhss' = (map o apsnd o map) (styp_of NONE) cs_rhss;
+ val cs_rhss = (fold o fold_aterms) (fn Const (c_ty as (c, _)) =>
+ insert (op =) (c, (map (styp_of NONE) o Sign.const_typargs thy) c_ty) | _ => I) ts [];
val (vardeps, (eqntab, insts)) = empty_vardeps_data
|> fold (assert_fun thy arities eqngr) cs
- |> fold (assert_rhs thy arities eqngr) cs_rhss';
+ |> fold (assert_rhs thy arities eqngr) cs_rhss;
val arities' = fold (add_arity thy vardeps) insts arities;
val pp = Syntax.pp_global thy;
- val is_proper_class = can (AxClass.get_info thy);
- val (proj_sort, algebra) = Sorts.subalgebra pp is_proper_class
+ val algebra = Sorts.subalgebra pp (is_proper_class thy)
(AList.lookup (op =) arities') (Sign.classes_of thy);
- val (rhss, eqngr') = Symtab.fold
- (add_eqs thy (proj_sort, algebra) vardeps) eqntab ([], eqngr);
- fun deps_of (c, rhs) = c ::
- maps (dicts_of thy (proj_sort, algebra))
- (rhs ~~ (map snd o fst o fst o Graph.get_node eqngr') c);
+ val (rhss, eqngr') = Symtab.fold (add_eqs thy vardeps) eqntab ([], eqngr);
+ fun deps_of (c, rhs) = c :: maps (dicts_of thy algebra)
+ (rhs ~~ (map snd o fst o fst o Graph.get_node eqngr') c);
val eqngr'' = fold (fn (c, rhs) => fold
(curry Graph.add_edge c) (deps_of rhs)) rhss eqngr';
- in ((proj_sort, algebra), (arities', eqngr'')) end;
+ in (algebra, (arities', eqngr'')) end;
-(** retrieval interfaces **)
-
-fun proto_eval thy cterm_of evaluator_lift evaluator proto_ct arities_eqngr =
- let
- val ct = cterm_of proto_ct;
- val _ = Sign.no_vars (Syntax.pp_global thy) (Thm.term_of ct);
- val _ = Term.fold_types (Type.no_tvars #> K I) (Thm.term_of ct) ();
- fun consts_of t =
- fold_aterms (fn Const c_ty => cons c_ty | _ => I) t [];
- val thm = Code.preprocess_conv thy ct;
- val ct' = Thm.rhs_of thm;
- val t' = Thm.term_of ct';
- val (t'', evaluator_eqngr) = evaluator t';
- val consts = map fst (consts_of t');
- val consts' = consts_of t'';
- val const_matches' = fold (fn (c, ty) =>
- insert (op =) (c, Sign.const_typargs thy (c, ty))) consts' [];
- val (algebra', arities_eqngr') =
- extend_arities_eqngr thy consts const_matches' arities_eqngr;
- in
- (evaluator_lift (evaluator_eqngr algebra') thm (snd arities_eqngr'),
- arities_eqngr')
- end;
-
-fun proto_eval_conv thy =
- let
- fun evaluator_lift evaluator thm1 eqngr =
- let
- val thm2 = evaluator eqngr;
- val thm3 = Code.postprocess_conv thy (Thm.rhs_of thm2);
- in
- Thm.transitive thm1 (Thm.transitive thm2 thm3) handle THM _ =>
- error ("could not construct evaluation proof:\n"
- ^ (cat_lines o map Display.string_of_thm) [thm1, thm2, thm3])
- end;
- in proto_eval thy I evaluator_lift end;
-
-fun proto_eval_term thy =
- let
- fun evaluator_lift evaluator _ eqngr = evaluator eqngr;
- in proto_eval thy (Thm.cterm_of thy) evaluator_lift end;
+(** store **)
structure Wellsorted = CodeDataFun
(
- type T = ((string * class) * sort list) list * T;
+ type T = ((string * class) * sort list) list * code_graph;
val empty = ([], Graph.empty);
fun purge thy cs (arities, eqngr) =
let
@@ -327,71 +287,52 @@
in (arities', eqngr') end;
);
-fun make thy cs = apsnd snd
- (Wellsorted.change_yield thy (extend_arities_eqngr thy cs []));
-fun eval_conv thy f =
- fst o Wellsorted.change_yield thy o proto_eval_conv thy f;
+(** retrieval interfaces **)
-fun eval_term thy f =
- fst o Wellsorted.change_yield thy o proto_eval_term thy f;
-
-
-(** diagnostic commands **)
+fun obtain thy cs ts = apsnd snd
+ (Wellsorted.change_yield thy (extend_arities_eqngr thy cs ts));
-fun code_depgr thy consts =
- let
- val (_, eqngr) = make thy consts;
- val select = Graph.all_succs eqngr consts;
- in
- eqngr
- |> not (null consts) ? Graph.subgraph (member (op =) select)
- |> Graph.map_nodes ((apsnd o map o apfst) (AxClass.overload thy))
- end;
+fun prepare_sorts prep_sort (Const (c, ty)) = Const (c, map_type_tfree
+ (fn (v, sort) => TFree (v, prep_sort sort)) ty)
+ | prepare_sorts prep_sort (t1 $ t2) =
+ prepare_sorts prep_sort t1 $ prepare_sorts prep_sort t2
+ | prepare_sorts prep_sort (Abs (v, ty, t)) =
+ Abs (v, Type.strip_sorts ty, prepare_sorts prep_sort t)
+ | prepare_sorts _ (Term.Free (v, ty)) = Term.Free (v, Type.strip_sorts ty)
+ | prepare_sorts _ (t as Bound _) = t;
-fun code_thms thy = Pretty.writeln o pretty thy o code_depgr thy;
-
-fun code_deps thy consts =
+fun gen_eval thy cterm_of conclude_evaluation prep_sort evaluator proto_ct =
let
- val eqngr = code_depgr thy consts;
- val constss = Graph.strong_conn eqngr;
- val mapping = Symtab.empty |> fold (fn consts => fold (fn const =>
- Symtab.update (const, consts)) consts) constss;
- fun succs consts = consts
- |> maps (Graph.imm_succs eqngr)
- |> subtract (op =) consts
- |> map (the o Symtab.lookup mapping)
- |> distinct (op =);
- val conn = [] |> fold (fn consts => cons (consts, succs consts)) constss;
- fun namify consts = map (Code_Unit.string_of_const thy) consts
- |> commas;
- val prgr = map (fn (consts, constss) =>
- { name = namify consts, ID = namify consts, dir = "", unfold = true,
- path = "", parents = map namify constss }) conn;
- in Present.display_graph prgr end;
-
-local
+ val ct = cterm_of proto_ct;
+ val _ = (Term.map_types Type.no_tvars o Sign.no_vars (Syntax.pp_global thy))
+ (Thm.term_of ct);
+ val thm = Code.preprocess_conv thy ct;
+ val ct' = Thm.rhs_of thm;
+ val t' = Thm.term_of ct';
+ val vs = Term.add_tfrees t' [];
+ val consts = fold_aterms
+ (fn Const (c, _) => insert (op =) c | _ => I) t' [];
+ val t'' = prepare_sorts prep_sort t';
+ val (algebra', eqngr') = obtain thy consts [t''];
+ in conclude_evaluation (evaluator algebra' eqngr' vs t'' ct') thm end;
-structure P = OuterParse
-and K = OuterKeyword
-
-fun code_thms_cmd thy = code_thms thy o op @ o Code_Name.read_const_exprs thy;
-fun code_deps_cmd thy = code_deps thy o op @ o Code_Name.read_const_exprs thy;
-
-in
+fun simple_evaluator evaluator algebra eqngr vs t ct =
+ evaluator algebra eqngr vs t;
-val _ =
- OuterSyntax.improper_command "code_thms" "print system of code equations for code" OuterKeyword.diag
- (Scan.repeat P.term_group
- >> (fn cs => Toplevel.no_timing o Toplevel.unknown_theory
- o Toplevel.keep ((fn thy => code_thms_cmd thy cs) o Toplevel.theory_of)));
+fun eval_conv thy =
+ let
+ fun conclude_evaluation thm2 thm1 =
+ let
+ val thm3 = Code.postprocess_conv thy (Thm.rhs_of thm2);
+ in
+ Thm.transitive thm1 (Thm.transitive thm2 thm3) handle THM _ =>
+ error ("could not construct evaluation proof:\n"
+ ^ (cat_lines o map Display.string_of_thm) [thm1, thm2, thm3])
+ end;
+ in gen_eval thy I conclude_evaluation end;
-val _ =
- OuterSyntax.improper_command "code_deps" "visualize dependencies of code equations for code" OuterKeyword.diag
- (Scan.repeat P.term_group
- >> (fn cs => Toplevel.no_timing o Toplevel.unknown_theory
- o Toplevel.keep ((fn thy => code_deps_cmd thy cs) o Toplevel.theory_of)));
-
-end;
+fun eval thy prep_sort postproc evaluator = gen_eval thy (Thm.cterm_of thy)
+ (K o postproc (Code.postprocess_term thy)) prep_sort (simple_evaluator evaluator);
end; (*struct*)
--- a/src/Tools/nbe.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Tools/nbe.ML Mon Apr 27 07:26:17 2009 -0700
@@ -7,7 +7,7 @@
signature NBE =
sig
val norm_conv: cterm -> thm
- val norm_term: theory -> term -> term
+ val norm: theory -> term -> term
datatype Univ =
Const of int * Univ list (*named (uninterpreted) constants*)
@@ -350,7 +350,7 @@
(* term evaluation *)
-fun eval_term ctxt gr deps ((vs, ty) : typscheme, t) =
+fun eval_term ctxt gr deps (vs : (string * sort) list, t) =
let
val frees = Code_Thingol.fold_unbound_varnames (insert (op =)) t []
val frees' = map (fn v => Free (v, [])) frees;
@@ -364,6 +364,15 @@
(* reification *)
+fun typ_of_itype program vs (ityco `%% itys) =
+ let
+ val Code_Thingol.Datatype (tyco, _) = Graph.get_node program ityco;
+ in Type (tyco, map (typ_of_itype program vs) itys) end
+ | typ_of_itype program vs (ITyVar v) =
+ let
+ val sort = (the o AList.lookup (op =) vs) v;
+ in TFree ("'" ^ v, sort) end;
+
fun term_of_univ thy program idx_tab t =
let
fun take_until f [] = []
@@ -418,41 +427,40 @@
(* compilation, evaluation and reification *)
-fun compile_eval thy naming program vs_ty_t deps =
+fun compile_eval thy naming program vs_t deps =
let
val ctxt = ProofContext.init thy;
val (_, (gr, (_, idx_tab))) =
Nbe_Functions.change thy (ensure_stmts ctxt naming program o snd);
in
- vs_ty_t
+ vs_t
|> eval_term ctxt gr deps
|> term_of_univ thy program idx_tab
end;
(* evaluation with type reconstruction *)
-fun eval thy t naming program vs_ty_t deps =
+fun normalize thy naming program (((vs0, (vs, ty)), fs), t) deps =
let
fun subst_const f = map_aterms (fn t as Term.Const (c, ty) => Term.Const (f c, ty)
| t => t);
- val subst_triv_consts = subst_const (Code_Unit.resubst_alias thy);
- val ty = type_of t;
- val type_free = AList.lookup (op =)
- (map (fn (s, T) => (s, Term.Free (s, T))) (Term.add_frees t []));
- val type_frees = Term.map_aterms
- (fn (t as Term.Free (s, _)) => the_default t (type_free s) | t => t);
+ val resubst_triv_consts = subst_const (Code_Unit.resubst_alias thy);
+ val ty' = typ_of_itype program vs0 ty;
+ val fs' = (map o apsnd) (typ_of_itype program vs0) fs;
+ val type_frees = Term.map_aterms (fn (t as Term.Free (s, _)) =>
+ Term.Free (s, (the o AList.lookup (op =) fs') s) | t => t);
fun type_infer t =
singleton (TypeInfer.infer_types (Syntax.pp_global thy) (Sign.tsig_of thy) I
(try (Type.strip_sorts o Sign.the_const_type thy)) (K NONE) Name.context 0)
- (TypeInfer.constrain ty t);
+ (TypeInfer.constrain ty' t);
fun check_tvars t = if null (Term.add_tvars t []) then t else
error ("Illegal schematic type variables in normalized term: "
^ setmp show_types true (Syntax.string_of_term_global thy) t);
val string_of_term = setmp show_types true (Syntax.string_of_term_global thy);
in
- compile_eval thy naming program vs_ty_t deps
+ compile_eval thy naming program (vs, t) deps
|> tracing (fn t => "Normalized:\n" ^ string_of_term t)
- |> subst_triv_consts
+ |> resubst_triv_consts
|> type_frees
|> tracing (fn t => "Vars typed:\n" ^ string_of_term t)
|> type_infer
@@ -463,39 +471,36 @@
(* evaluation oracle *)
-val (_, norm_oracle) = Context.>>> (Context.map_theory_result
- (Thm.add_oracle (Binding.name "norm", fn (thy, t, naming, program, vs_ty_t, deps) =>
- Thm.cterm_of thy (Logic.mk_equals (t, eval thy t naming program vs_ty_t deps)))));
+fun add_triv_classes thy = curry (Sorts.inter_sort (Sign.classes_of thy))
+ (Code_Unit.triv_classes thy);
-fun add_triv_classes thy =
+fun mk_equals thy lhs raw_rhs =
let
- val inters = curry (Sorts.inter_sort (Sign.classes_of thy))
- (Code_Unit.triv_classes thy);
- fun map_sorts f = (map_types o map_atyps)
- (fn TVar (v, sort) => TVar (v, f sort)
- | TFree (v, sort) => TFree (v, f sort));
- in map_sorts inters end;
+ val ty = Thm.typ_of (Thm.ctyp_of_term lhs);
+ val eq = Thm.cterm_of thy (Term.Const ("==", ty --> ty --> propT));
+ val rhs = Thm.cterm_of thy raw_rhs;
+ in Thm.mk_binop eq lhs rhs end;
+
+val (_, raw_norm_oracle) = Context.>>> (Context.map_theory_result
+ (Thm.add_oracle (Binding.name "norm", fn (thy, naming, program, vsp_ty_fs_t, deps, ct) =>
+ mk_equals thy ct (normalize thy naming program vsp_ty_fs_t deps))));
+
+fun norm_oracle thy naming program vsp_ty_fs_t deps ct =
+ raw_norm_oracle (thy, naming, program, vsp_ty_fs_t, deps, ct);
fun norm_conv ct =
let
val thy = Thm.theory_of_cterm ct;
- fun evaluator' t naming program vs_ty_t deps =
- norm_oracle (thy, t, naming, program, vs_ty_t, deps);
- fun evaluator t = (add_triv_classes thy t, evaluator' t);
- in Code_Thingol.eval_conv thy evaluator ct end;
+ in Code_Thingol.eval_conv thy (add_triv_classes thy) (norm_oracle thy) ct end;
-fun norm_term thy t =
- let
- fun evaluator' t naming program vs_ty_t deps = eval thy t naming program vs_ty_t deps;
- fun evaluator t = (add_triv_classes thy t, evaluator' t);
- in (Code.postprocess_term thy o Code_Thingol.eval_term thy evaluator) t end;
+fun norm thy = Code_Thingol.eval thy (add_triv_classes thy) I (normalize thy);
(* evaluation command *)
fun norm_print_term ctxt modes t =
let
val thy = ProofContext.theory_of ctxt;
- val t' = norm_term thy t;
+ val t' = norm thy t;
val ty' = Term.type_of t';
val ctxt' = Variable.auto_fixes t ctxt;
val p = PrintMode.with_modes modes (fn () =>
@@ -510,8 +515,7 @@
let val ctxt = Toplevel.context_of state
in norm_print_term ctxt modes (Syntax.read_term ctxt s) end;
-val setup =
- Value.add_evaluator ("nbe", norm_term o ProofContext.theory_of);
+val setup = Value.add_evaluator ("nbe", norm o ProofContext.theory_of);
local structure P = OuterParse and K = OuterKeyword in
--- a/src/Tools/quickcheck.ML Wed Apr 22 11:00:25 2009 -0700
+++ b/src/Tools/quickcheck.ML Mon Apr 27 07:26:17 2009 -0700
@@ -6,28 +6,48 @@
signature QUICKCHECK =
sig
- val test_term: Proof.context -> bool -> string option -> int -> int -> term -> (string * term) list option;
- val add_generator: string * (Proof.context -> term -> int -> term list option) -> theory -> theory
val auto: bool ref
val auto_time_limit: int ref
+ val test_term: Proof.context -> bool -> string option -> int -> int -> term ->
+ (string * term) list option
+ val add_generator: string * (Proof.context -> term -> int -> term list option) -> theory -> theory
end;
structure Quickcheck : QUICKCHECK =
struct
+(* preferences *)
+
+val auto = ref false;
+val auto_time_limit = ref 2500;
+
+val _ =
+ ProofGeneralPgip.add_preference Preferences.category_tracing
+ (setmp auto true (fn () =>
+ Preferences.bool_pref auto
+ "auto-quickcheck"
+ "Whether to enable quickcheck automatically.") ());
+
+val _ =
+ ProofGeneralPgip.add_preference Preferences.category_tracing
+ (Preferences.nat_pref auto_time_limit
+ "auto-quickcheck-time-limit"
+ "Time limit for automatic quickcheck (in milliseconds).");
+
+
(* quickcheck configuration -- default parameters, test generators *)
datatype test_params = Test_Params of
{ size: int, iterations: int, default_type: typ option };
-fun dest_test_params (Test_Params { size, iterations, default_type}) =
+fun dest_test_params (Test_Params { size, iterations, default_type }) =
((size, iterations), default_type);
fun mk_test_params ((size, iterations), default_type) =
Test_Params { size = size, iterations = iterations, default_type = default_type };
fun map_test_params f (Test_Params { size, iterations, default_type}) =
mk_test_params (f ((size, iterations), default_type));
-fun merge_test_params (Test_Params {size = size1, iterations = iterations1, default_type = default_type1},
- Test_Params {size = size2, iterations = iterations2, default_type = default_type2}) =
+fun merge_test_params (Test_Params { size = size1, iterations = iterations1, default_type = default_type1 },
+ Test_Params { size = size2, iterations = iterations2, default_type = default_type2 }) =
mk_test_params ((Int.max (size1, size2), Int.max (iterations1, iterations2)),
case default_type1 of NONE => default_type2 | _ => default_type1);
@@ -138,10 +158,7 @@
(* automatic testing *)
-val auto = ref false;
-val auto_time_limit = ref 5000;
-
-fun test_goal_auto int state =
+val _ = Context.>> (Specification.add_theorem_hook (fn int => fn state =>
let
val ctxt = Proof.context_of state;
val assms = map term_of (Assumption.all_assms_of ctxt);
@@ -162,12 +179,10 @@
if int andalso !auto andalso not (!Toplevel.quiet)
then test ()
else state
- end;
-
-val _ = Context.>> (Specification.add_theorem_hook test_goal_auto);
+ end));
-(* Isar interfaces *)
+(* Isar commands *)
fun read_nat s = case (Library.read_int o Symbol.explode) s
of (k, []) => if k >= 0 then k