author | wenzelm |
Thu, 08 Dec 2005 20:15:50 +0100 | |
changeset 18372 | 2bffdf62fe7f |
parent 18334 | a41ce9c10b73 |
child 18518 | 3b1dfa53e64f |
permissions | -rw-r--r-- |
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(* Title: HOL/Product_Type.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1992 University of Cambridge |
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*) |
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header {* Cartesian products *} |
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theory Product_Type |
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imports Fun |
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uses ("Tools/split_rule.ML") |
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begin |
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subsection {* Unit *} |
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||
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typedef unit = "{True}" |
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proof |
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show "True : ?unit" by blast |
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qed |
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||
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constdefs |
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Unity :: unit ("'(')") |
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"() == Abs_unit True" |
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lemma unit_eq: "u = ()" |
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by (induct u) (simp add: unit_def Unity_def) |
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text {* |
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Simplification procedure for @{thm [source] unit_eq}. Cannot use |
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this rule directly --- it loops! |
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*} |
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ML_setup {* |
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val unit_eq_proc = |
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let val unit_meta_eq = mk_meta_eq (thm "unit_eq") in |
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Simplifier.simproc (Theory.sign_of (the_context ())) "unit_eq" ["x::unit"] |
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(fn _ => fn _ => fn t => if HOLogic.is_unit t then NONE else SOME unit_meta_eq) |
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end; |
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Addsimprocs [unit_eq_proc]; |
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*} |
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lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()" |
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by simp |
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lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P" |
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by (rule triv_forall_equality) |
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lemma unit_induct [induct type: unit]: "P () ==> P x" |
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by simp |
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text {* |
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This rewrite counters the effect of @{text unit_eq_proc} on @{term |
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[source] "%u::unit. f u"}, replacing it by @{term [source] |
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f} rather than by @{term [source] "%u. f ()"}. |
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*} |
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lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f" |
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by (rule ext) simp |
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subsection {* Pairs *} |
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subsubsection {* Type definition *} |
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constdefs |
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Pair_Rep :: "['a, 'b] => ['a, 'b] => bool" |
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"Pair_Rep == (%a b. %x y. x=a & y=b)" |
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global |
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typedef (Prod) |
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('a, 'b) "*" (infixr 20) |
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= "{f. EX a b. f = Pair_Rep (a::'a) (b::'b)}" |
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proof |
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fix a b show "Pair_Rep a b : ?Prod" |
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by blast |
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qed |
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syntax (xsymbols) |
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"*" :: "[type, type] => type" ("(_ \<times>/ _)" [21, 20] 20) |
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syntax (HTML output) |
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"*" :: "[type, type] => type" ("(_ \<times>/ _)" [21, 20] 20) |
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local |
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subsubsection {* Abstract constants and syntax *} |
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global |
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consts |
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fst :: "'a * 'b => 'a" |
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snd :: "'a * 'b => 'b" |
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split :: "[['a, 'b] => 'c, 'a * 'b] => 'c" |
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curry :: "['a * 'b => 'c, 'a, 'b] => 'c" |
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prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd" |
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Pair :: "['a, 'b] => 'a * 'b" |
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Sigma :: "['a set, 'a => 'b set] => ('a * 'b) set" |
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local |
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text {* |
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Patterns -- extends pre-defined type @{typ pttrn} used in |
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abstractions. |
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*} |
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nonterminals |
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tuple_args patterns |
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syntax |
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"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))") |
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"_tuple_arg" :: "'a => tuple_args" ("_") |
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"_tuple_args" :: "'a => tuple_args => tuple_args" ("_,/ _") |
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"_pattern" :: "[pttrn, patterns] => pttrn" ("'(_,/ _')") |
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"" :: "pttrn => patterns" ("_") |
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"_patterns" :: "[pttrn, patterns] => patterns" ("_,/ _") |
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"@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" 10) |
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"@Times" ::"['a set, 'a => 'b set] => ('a * 'b) set" (infixr "<*>" 80) |
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translations |
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"(x, y)" == "Pair x y" |
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"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" |
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"%(x,y,zs).b" == "split(%x (y,zs).b)" |
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"%(x,y).b" == "split(%x y. b)" |
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"_abs (Pair x y) t" => "%(x,y).t" |
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(* The last rule accommodates tuples in `case C ... (x,y) ... => ...' |
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The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *) |
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"SIGMA x:A. B" => "Sigma A (%x. B)" |
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"A <*> B" => "Sigma A (%_. B)" |
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(* reconstructs pattern from (nested) splits, avoiding eta-contraction of body*) |
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(* works best with enclosing "let", if "let" does not avoid eta-contraction *) |
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print_translation {* |
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let fun split_tr' [Abs (x,T,t as (Abs abs))] = |
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(* split (%x y. t) => %(x,y) t *) |
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let val (y,t') = atomic_abs_tr' abs; |
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val (x',t'') = atomic_abs_tr' (x,T,t'); |
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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end |
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| split_tr' [Abs (x,T,(s as Const ("split",_)$t))] = |
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(* split (%x. (split (%y z. t))) => %(x,y,z). t *) |
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let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t]; |
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val (x',t'') = atomic_abs_tr' (x,T,t'); |
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in Syntax.const "_abs"$ |
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(Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end |
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| split_tr' [Const ("split",_)$t] = |
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(* split (split (%x y z. t)) => %((x,y),z). t *) |
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split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *) |
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| split_tr' [Const ("_abs",_)$x_y$(Abs abs)] = |
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(* split (%pttrn z. t) => %(pttrn,z). t *) |
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let val (z,t) = atomic_abs_tr' abs; |
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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end |
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| split_tr' _ = raise Match; |
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in [("split", split_tr')] |
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end |
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*} |
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(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) |
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typed_print_translation {* |
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let |
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fun split_guess_names_tr' _ T [Abs (x,_,Abs _)] = raise Match |
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| split_guess_names_tr' _ T [Abs (x,xT,t)] = |
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(case (head_of t) of |
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Const ("split",_) => raise Match |
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| _ => let |
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val (_::yT::_) = binder_types (domain_type T) handle Bind => raise Match; |
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val (y,t') = atomic_abs_tr' ("y",yT,(incr_boundvars 1 t)$Bound 0); |
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val (x',t'') = atomic_abs_tr' (x,xT,t'); |
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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end) |
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| split_guess_names_tr' _ T [t] = |
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(case (head_of t) of |
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Const ("split",_) => raise Match |
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| _ => let |
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val (xT::yT::_) = binder_types (domain_type T) handle Bind => raise Match; |
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val (y,t') = |
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atomic_abs_tr' ("y",yT,(incr_boundvars 2 t)$Bound 1$Bound 0); |
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val (x',t'') = atomic_abs_tr' ("x",xT,t'); |
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in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end) |
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| split_guess_names_tr' _ _ _ = raise Match; |
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in [("split", split_guess_names_tr')] |
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end |
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*} |
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text{*Deleted x-symbol and html support using @{text"\<Sigma>"} (Sigma) because of the danger of confusion with Sum.*} |
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syntax (xsymbols) |
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"@Times" :: "['a set, 'a => 'b set] => ('a * 'b) set" ("_ \<times> _" [81, 80] 80) |
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syntax (HTML output) |
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"@Times" :: "['a set, 'a => 'b set] => ('a * 'b) set" ("_ \<times> _" [81, 80] 80) |
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print_translation {* [("Sigma", dependent_tr' ("@Sigma", "@Times"))] *} |
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||
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subsubsection {* Definitions *} |
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defs |
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Pair_def: "Pair a b == Abs_Prod(Pair_Rep a b)" |
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fst_def: "fst p == THE a. EX b. p = (a, b)" |
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snd_def: "snd p == THE b. EX a. p = (a, b)" |
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split_def: "split == (%c p. c (fst p) (snd p))" |
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curry_def: "curry == (%c x y. c (x,y))" |
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prod_fun_def: "prod_fun f g == split(%x y.(f(x), g(y)))" |
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Sigma_def: "Sigma A B == UN x:A. UN y:B(x). {(x, y)}" |
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209 |
||
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subsubsection {* Lemmas and proof tool setup *} |
11838 | 211 |
|
212 |
lemma ProdI: "Pair_Rep a b : Prod" |
|
213 |
by (unfold Prod_def) blast |
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214 |
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215 |
lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' ==> a = a' & b = b'" |
|
216 |
apply (unfold Pair_Rep_def) |
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apply (drule fun_cong [THEN fun_cong], blast) |
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done |
10213 | 219 |
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lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod" |
221 |
apply (rule inj_on_inverseI) |
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222 |
apply (erule Abs_Prod_inverse) |
|
223 |
done |
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224 |
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225 |
lemma Pair_inject: |
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18372 | 226 |
assumes "(a, b) = (a', b')" |
227 |
and "a = a' ==> b = b' ==> R" |
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228 |
shows R |
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229 |
apply (insert prems [unfolded Pair_def]) |
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apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE]) |
|
231 |
apply (assumption | rule ProdI)+ |
|
232 |
done |
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10213 | 233 |
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11838 | 234 |
lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')" |
235 |
by (blast elim!: Pair_inject) |
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236 |
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237 |
lemma fst_conv [simp]: "fst (a, b) = a" |
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238 |
by (unfold fst_def) blast |
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239 |
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240 |
lemma snd_conv [simp]: "snd (a, b) = b" |
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241 |
by (unfold snd_def) blast |
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lemma fst_eqD: "fst (x, y) = a ==> x = a" |
244 |
by simp |
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245 |
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246 |
lemma snd_eqD: "snd (x, y) = a ==> y = a" |
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247 |
by simp |
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248 |
||
249 |
lemma PairE_lemma: "EX x y. p = (x, y)" |
|
250 |
apply (unfold Pair_def) |
|
251 |
apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE]) |
|
252 |
apply (erule exE, erule exE, rule exI, rule exI) |
|
253 |
apply (rule Rep_Prod_inverse [symmetric, THEN trans]) |
|
254 |
apply (erule arg_cong) |
|
255 |
done |
|
11032 | 256 |
|
11838 | 257 |
lemma PairE [cases type: *]: "(!!x y. p = (x, y) ==> Q) ==> Q" |
258 |
by (insert PairE_lemma [of p]) blast |
|
259 |
||
16121 | 260 |
ML {* |
11838 | 261 |
local val PairE = thm "PairE" in |
262 |
fun pair_tac s = |
|
263 |
EVERY' [res_inst_tac [("p", s)] PairE, hyp_subst_tac, K prune_params_tac]; |
|
264 |
end; |
|
265 |
*} |
|
11032 | 266 |
|
11838 | 267 |
lemma surjective_pairing: "p = (fst p, snd p)" |
268 |
-- {* Do not add as rewrite rule: invalidates some proofs in IMP *} |
|
269 |
by (cases p) simp |
|
270 |
||
17085 | 271 |
lemmas pair_collapse = surjective_pairing [symmetric] |
272 |
declare pair_collapse [simp] |
|
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changeset
|
273 |
|
11838 | 274 |
lemma surj_pair [simp]: "EX x y. z = (x, y)" |
275 |
apply (rule exI) |
|
276 |
apply (rule exI) |
|
277 |
apply (rule surjective_pairing) |
|
278 |
done |
|
279 |
||
280 |
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" |
|
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|
281 |
proof |
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changeset
|
282 |
fix a b |
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|
283 |
assume "!!x. PROP P x" |
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|
284 |
thus "PROP P (a, b)" . |
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proper proof of split_paired_all (presently unused);
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diff
changeset
|
285 |
next |
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proper proof of split_paired_all (presently unused);
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diff
changeset
|
286 |
fix x |
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proper proof of split_paired_all (presently unused);
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changeset
|
287 |
assume "!!a b. PROP P (a, b)" |
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|
288 |
hence "PROP P (fst x, snd x)" . |
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proper proof of split_paired_all (presently unused);
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changeset
|
289 |
thus "PROP P x" by simp |
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proper proof of split_paired_all (presently unused);
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changeset
|
290 |
qed |
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proper proof of split_paired_all (presently unused);
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changeset
|
291 |
|
11838 | 292 |
lemmas split_tupled_all = split_paired_all unit_all_eq2 |
293 |
||
294 |
text {* |
|
295 |
The rule @{thm [source] split_paired_all} does not work with the |
|
296 |
Simplifier because it also affects premises in congrence rules, |
|
297 |
where this can lead to premises of the form @{text "!!a b. ... = |
|
298 |
?P(a, b)"} which cannot be solved by reflexivity. |
|
299 |
*} |
|
300 |
||
16121 | 301 |
ML_setup {* |
11838 | 302 |
(* replace parameters of product type by individual component parameters *) |
303 |
val safe_full_simp_tac = generic_simp_tac true (true, false, false); |
|
304 |
local (* filtering with exists_paired_all is an essential optimization *) |
|
16121 | 305 |
fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) = |
11838 | 306 |
can HOLogic.dest_prodT T orelse exists_paired_all t |
307 |
| exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u |
|
308 |
| exists_paired_all (Abs (_, _, t)) = exists_paired_all t |
|
309 |
| exists_paired_all _ = false; |
|
310 |
val ss = HOL_basic_ss |
|
16121 | 311 |
addsimps [thm "split_paired_all", thm "unit_all_eq2", thm "unit_abs_eta_conv"] |
11838 | 312 |
addsimprocs [unit_eq_proc]; |
313 |
in |
|
314 |
val split_all_tac = SUBGOAL (fn (t, i) => |
|
315 |
if exists_paired_all t then safe_full_simp_tac ss i else no_tac); |
|
316 |
val unsafe_split_all_tac = SUBGOAL (fn (t, i) => |
|
317 |
if exists_paired_all t then full_simp_tac ss i else no_tac); |
|
318 |
fun split_all th = |
|
319 |
if exists_paired_all (#prop (Thm.rep_thm th)) then full_simplify ss th else th; |
|
320 |
end; |
|
321 |
||
17875 | 322 |
change_claset (fn cs => cs addSbefore ("split_all_tac", split_all_tac)); |
16121 | 323 |
*} |
11838 | 324 |
|
325 |
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))" |
|
326 |
-- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} |
|
327 |
by fast |
|
328 |
||
14189 | 329 |
lemma curry_split [simp]: "curry (split f) = f" |
330 |
by (simp add: curry_def split_def) |
|
331 |
||
332 |
lemma split_curry [simp]: "split (curry f) = f" |
|
333 |
by (simp add: curry_def split_def) |
|
334 |
||
335 |
lemma curryI [intro!]: "f (a,b) ==> curry f a b" |
|
336 |
by (simp add: curry_def) |
|
337 |
||
14190
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Fixed blunder in the setup of the classical reasoner wrt. the constant
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parents:
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diff
changeset
|
338 |
lemma curryD [dest!]: "curry f a b ==> f (a,b)" |
14189 | 339 |
by (simp add: curry_def) |
340 |
||
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Fixed blunder in the setup of the classical reasoner wrt. the constant
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parents:
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diff
changeset
|
341 |
lemma curryE: "[| curry f a b ; f (a,b) ==> Q |] ==> Q" |
14189 | 342 |
by (simp add: curry_def) |
343 |
||
344 |
lemma curry_conv [simp]: "curry f a b = f (a,b)" |
|
345 |
by (simp add: curry_def) |
|
346 |
||
11838 | 347 |
lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x" |
348 |
by fast |
|
349 |
||
350 |
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))" |
|
351 |
by fast |
|
352 |
||
353 |
lemma split_conv [simp]: "split c (a, b) = c a b" |
|
354 |
by (simp add: split_def) |
|
355 |
||
356 |
lemmas split = split_conv -- {* for backwards compatibility *} |
|
357 |
||
358 |
lemmas splitI = split_conv [THEN iffD2, standard] |
|
359 |
lemmas splitD = split_conv [THEN iffD1, standard] |
|
360 |
||
361 |
lemma split_Pair_apply: "split (%x y. f (x, y)) = f" |
|
362 |
-- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *} |
|
363 |
apply (rule ext) |
|
14208 | 364 |
apply (tactic {* pair_tac "x" 1 *}, simp) |
11838 | 365 |
done |
366 |
||
367 |
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))" |
|
368 |
-- {* Can't be added to simpset: loops! *} |
|
369 |
by (simp add: split_Pair_apply) |
|
370 |
||
371 |
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" |
|
372 |
by (simp add: split_def) |
|
373 |
||
374 |
lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)" |
|
14208 | 375 |
by (simp only: split_tupled_all, simp) |
11838 | 376 |
|
377 |
lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q" |
|
378 |
by (simp add: Pair_fst_snd_eq) |
|
379 |
||
380 |
lemma split_weak_cong: "p = q ==> split c p = split c q" |
|
381 |
-- {* Prevents simplification of @{term c}: much faster *} |
|
382 |
by (erule arg_cong) |
|
383 |
||
384 |
lemma split_eta: "(%(x, y). f (x, y)) = f" |
|
385 |
apply (rule ext) |
|
386 |
apply (simp only: split_tupled_all) |
|
387 |
apply (rule split_conv) |
|
388 |
done |
|
389 |
||
390 |
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" |
|
391 |
by (simp add: split_eta) |
|
392 |
||
393 |
text {* |
|
394 |
Simplification procedure for @{thm [source] cond_split_eta}. Using |
|
395 |
@{thm [source] split_eta} as a rewrite rule is not general enough, |
|
396 |
and using @{thm [source] cond_split_eta} directly would render some |
|
397 |
existing proofs very inefficient; similarly for @{text |
|
398 |
split_beta}. *} |
|
399 |
||
400 |
ML_setup {* |
|
401 |
||
402 |
local |
|
18328 | 403 |
val cond_split_eta_ss = HOL_basic_ss addsimps [thm "cond_split_eta"] |
11838 | 404 |
fun Pair_pat k 0 (Bound m) = (m = k) |
405 |
| Pair_pat k i (Const ("Pair", _) $ Bound m $ t) = i > 0 andalso |
|
406 |
m = k+i andalso Pair_pat k (i-1) t |
|
407 |
| Pair_pat _ _ _ = false; |
|
408 |
fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t |
|
409 |
| no_args k i (t $ u) = no_args k i t andalso no_args k i u |
|
410 |
| no_args k i (Bound m) = m < k orelse m > k+i |
|
411 |
| no_args _ _ _ = true; |
|
15531 | 412 |
fun split_pat tp i (Abs (_,_,t)) = if tp 0 i t then SOME (i,t) else NONE |
11838 | 413 |
| split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t |
15531 | 414 |
| split_pat tp i _ = NONE; |
17956 | 415 |
fun metaeq thy ss lhs rhs = mk_meta_eq (Goal.prove thy [] [] |
13480
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents:
13462
diff
changeset
|
416 |
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs))) |
18328 | 417 |
(K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1))); |
11838 | 418 |
|
419 |
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t |
|
420 |
| beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse |
|
421 |
(beta_term_pat k i t andalso beta_term_pat k i u) |
|
422 |
| beta_term_pat k i t = no_args k i t; |
|
423 |
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg |
|
424 |
| eta_term_pat _ _ _ = false; |
|
425 |
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) |
|
426 |
| subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg |
|
427 |
else (subst arg k i t $ subst arg k i u) |
|
428 |
| subst arg k i t = t; |
|
17002 | 429 |
fun beta_proc thy ss (s as Const ("split", _) $ Abs (_, _, t) $ arg) = |
11838 | 430 |
(case split_pat beta_term_pat 1 t of |
17002 | 431 |
SOME (i,f) => SOME (metaeq thy ss s (subst arg 0 i f)) |
15531 | 432 |
| NONE => NONE) |
433 |
| beta_proc _ _ _ = NONE; |
|
17002 | 434 |
fun eta_proc thy ss (s as Const ("split", _) $ Abs (_, _, t)) = |
11838 | 435 |
(case split_pat eta_term_pat 1 t of |
17002 | 436 |
SOME (_,ft) => SOME (metaeq thy ss s (let val (f $ arg) = ft in f end)) |
15531 | 437 |
| NONE => NONE) |
438 |
| eta_proc _ _ _ = NONE; |
|
11838 | 439 |
in |
13462 | 440 |
val split_beta_proc = Simplifier.simproc (Theory.sign_of (the_context ())) |
441 |
"split_beta" ["split f z"] beta_proc; |
|
442 |
val split_eta_proc = Simplifier.simproc (Theory.sign_of (the_context ())) |
|
443 |
"split_eta" ["split f"] eta_proc; |
|
11838 | 444 |
end; |
445 |
||
446 |
Addsimprocs [split_beta_proc, split_eta_proc]; |
|
447 |
*} |
|
448 |
||
449 |
lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)" |
|
450 |
by (subst surjective_pairing, rule split_conv) |
|
451 |
||
452 |
lemma split_split: "R (split c p) = (ALL x y. p = (x, y) --> R (c x y))" |
|
453 |
-- {* For use with @{text split} and the Simplifier. *} |
|
15481 | 454 |
by (insert surj_pair [of p], clarify, simp) |
11838 | 455 |
|
456 |
text {* |
|
457 |
@{thm [source] split_split} could be declared as @{text "[split]"} |
|
458 |
done after the Splitter has been speeded up significantly; |
|
459 |
precompute the constants involved and don't do anything unless the |
|
460 |
current goal contains one of those constants. |
|
461 |
*} |
|
462 |
||
463 |
lemma split_split_asm: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" |
|
14208 | 464 |
by (subst split_split, simp) |
11838 | 465 |
|
466 |
||
467 |
text {* |
|
468 |
\medskip @{term split} used as a logical connective or set former. |
|
469 |
||
470 |
\medskip These rules are for use with @{text blast}; could instead |
|
471 |
call @{text simp} using @{thm [source] split} as rewrite. *} |
|
472 |
||
473 |
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p" |
|
474 |
apply (simp only: split_tupled_all) |
|
475 |
apply (simp (no_asm_simp)) |
|
476 |
done |
|
477 |
||
478 |
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x" |
|
479 |
apply (simp only: split_tupled_all) |
|
480 |
apply (simp (no_asm_simp)) |
|
481 |
done |
|
482 |
||
483 |
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" |
|
484 |
by (induct p) (auto simp add: split_def) |
|
485 |
||
486 |
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" |
|
487 |
by (induct p) (auto simp add: split_def) |
|
488 |
||
489 |
lemma splitE2: |
|
490 |
"[| Q (split P z); !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R" |
|
491 |
proof - |
|
492 |
assume q: "Q (split P z)" |
|
493 |
assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R" |
|
494 |
show R |
|
495 |
apply (rule r surjective_pairing)+ |
|
496 |
apply (rule split_beta [THEN subst], rule q) |
|
497 |
done |
|
498 |
qed |
|
499 |
||
500 |
lemma splitD': "split R (a,b) c ==> R a b c" |
|
501 |
by simp |
|
502 |
||
503 |
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" |
|
504 |
by simp |
|
505 |
||
506 |
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p" |
|
14208 | 507 |
by (simp only: split_tupled_all, simp) |
11838 | 508 |
|
18372 | 509 |
lemma mem_splitE: |
510 |
assumes major: "z: split c p" |
|
511 |
and cases: "!!x y. [| p = (x,y); z: c x y |] ==> Q" |
|
512 |
shows Q |
|
513 |
by (rule major [unfolded split_def] cases surjective_pairing)+ |
|
11838 | 514 |
|
515 |
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] |
|
516 |
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] |
|
517 |
||
16121 | 518 |
ML_setup {* |
11838 | 519 |
local (* filtering with exists_p_split is an essential optimization *) |
16121 | 520 |
fun exists_p_split (Const ("split",_) $ _ $ (Const ("Pair",_)$_$_)) = true |
11838 | 521 |
| exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u |
522 |
| exists_p_split (Abs (_, _, t)) = exists_p_split t |
|
523 |
| exists_p_split _ = false; |
|
16121 | 524 |
val ss = HOL_basic_ss addsimps [thm "split_conv"]; |
11838 | 525 |
in |
526 |
val split_conv_tac = SUBGOAL (fn (t, i) => |
|
527 |
if exists_p_split t then safe_full_simp_tac ss i else no_tac); |
|
528 |
end; |
|
529 |
(* This prevents applications of splitE for already splitted arguments leading |
|
530 |
to quite time-consuming computations (in particular for nested tuples) *) |
|
17875 | 531 |
change_claset (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac)); |
16121 | 532 |
*} |
11838 | 533 |
|
534 |
lemma split_eta_SetCompr [simp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" |
|
18372 | 535 |
by (rule ext) fast |
11838 | 536 |
|
537 |
lemma split_eta_SetCompr2 [simp]: "(%u. EX x y. u = (x, y) & P x y) = split P" |
|
18372 | 538 |
by (rule ext) fast |
11838 | 539 |
|
540 |
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" |
|
541 |
-- {* Allows simplifications of nested splits in case of independent predicates. *} |
|
18372 | 542 |
by (rule ext) blast |
11838 | 543 |
|
14337
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
544 |
(* Do NOT make this a simp rule as it |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
545 |
a) only helps in special situations |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
546 |
b) can lead to nontermination in the presence of split_def |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
547 |
*) |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
548 |
lemma split_comp_eq: |
14101 | 549 |
"(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" |
18372 | 550 |
by (rule ext) auto |
14101 | 551 |
|
11838 | 552 |
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" |
553 |
by blast |
|
554 |
||
555 |
(* |
|
556 |
the following would be slightly more general, |
|
557 |
but cannot be used as rewrite rule: |
|
558 |
### Cannot add premise as rewrite rule because it contains (type) unknowns: |
|
559 |
### ?y = .x |
|
560 |
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)" |
|
14208 | 561 |
by (rtac some_equality 1) |
562 |
by ( Simp_tac 1) |
|
563 |
by (split_all_tac 1) |
|
564 |
by (Asm_full_simp_tac 1) |
|
11838 | 565 |
qed "The_split_eq"; |
566 |
*) |
|
567 |
||
568 |
lemma injective_fst_snd: "!!x y. [|fst x = fst y; snd x = snd y|] ==> x = y" |
|
569 |
by auto |
|
570 |
||
571 |
||
572 |
text {* |
|
573 |
\bigskip @{term prod_fun} --- action of the product functor upon |
|
574 |
functions. |
|
575 |
*} |
|
576 |
||
577 |
lemma prod_fun [simp]: "prod_fun f g (a, b) = (f a, g b)" |
|
578 |
by (simp add: prod_fun_def) |
|
579 |
||
580 |
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)" |
|
581 |
apply (rule ext) |
|
14208 | 582 |
apply (tactic {* pair_tac "x" 1 *}, simp) |
11838 | 583 |
done |
584 |
||
585 |
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)" |
|
586 |
apply (rule ext) |
|
14208 | 587 |
apply (tactic {* pair_tac "z" 1 *}, simp) |
11838 | 588 |
done |
589 |
||
590 |
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r" |
|
591 |
apply (rule image_eqI) |
|
14208 | 592 |
apply (rule prod_fun [symmetric], assumption) |
11838 | 593 |
done |
594 |
||
595 |
lemma prod_fun_imageE [elim!]: |
|
18372 | 596 |
assumes major: "c: (prod_fun f g)`r" |
597 |
and cases: "!!x y. [| c=(f(x),g(y)); (x,y):r |] ==> P" |
|
598 |
shows P |
|
599 |
apply (rule major [THEN imageE]) |
|
600 |
apply (rule_tac p = x in PairE) |
|
601 |
apply (rule cases) |
|
602 |
apply (blast intro: prod_fun) |
|
603 |
apply blast |
|
604 |
done |
|
11838 | 605 |
|
606 |
||
14101 | 607 |
constdefs |
608 |
upd_fst :: "('a => 'c) => 'a * 'b => 'c * 'b" |
|
609 |
"upd_fst f == prod_fun f id" |
|
610 |
||
611 |
upd_snd :: "('b => 'c) => 'a * 'b => 'a * 'c" |
|
612 |
"upd_snd f == prod_fun id f" |
|
613 |
||
614 |
lemma upd_fst_conv [simp]: "upd_fst f (x,y) = (f x,y)" |
|
18372 | 615 |
by (simp add: upd_fst_def) |
14101 | 616 |
|
617 |
lemma upd_snd_conv [simp]: "upd_snd f (x,y) = (x,f y)" |
|
18372 | 618 |
by (simp add: upd_snd_def) |
14101 | 619 |
|
11838 | 620 |
text {* |
621 |
\bigskip Disjoint union of a family of sets -- Sigma. |
|
622 |
*} |
|
623 |
||
624 |
lemma SigmaI [intro!]: "[| a:A; b:B(a) |] ==> (a,b) : Sigma A B" |
|
625 |
by (unfold Sigma_def) blast |
|
626 |
||
14952
47455995693d
removal of x-symbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset
|
627 |
lemma SigmaE [elim!]: |
11838 | 628 |
"[| c: Sigma A B; |
629 |
!!x y.[| x:A; y:B(x); c=(x,y) |] ==> P |
|
630 |
|] ==> P" |
|
631 |
-- {* The general elimination rule. *} |
|
632 |
by (unfold Sigma_def) blast |
|
633 |
||
634 |
text {* |
|
15422
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
635 |
Elimination of @{term "(a, b) : A \<times> B"} -- introduces no |
11838 | 636 |
eigenvariables. |
637 |
*} |
|
638 |
||
639 |
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A" |
|
18372 | 640 |
by blast |
11838 | 641 |
|
642 |
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a" |
|
18372 | 643 |
by blast |
11838 | 644 |
|
645 |
lemma SigmaE2: |
|
646 |
"[| (a, b) : Sigma A B; |
|
647 |
[| a:A; b:B(a) |] ==> P |
|
648 |
|] ==> P" |
|
14952
47455995693d
removal of x-symbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset
|
649 |
by blast |
11838 | 650 |
|
14952
47455995693d
removal of x-symbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset
|
651 |
lemma Sigma_cong: |
15422
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
652 |
"\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk> |
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
653 |
\<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)" |
18372 | 654 |
by auto |
11838 | 655 |
|
656 |
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D" |
|
657 |
by blast |
|
658 |
||
659 |
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" |
|
660 |
by blast |
|
661 |
||
662 |
lemma Sigma_empty2 [simp]: "A <*> {} = {}" |
|
663 |
by blast |
|
664 |
||
665 |
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" |
|
666 |
by auto |
|
667 |
||
668 |
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)" |
|
669 |
by auto |
|
670 |
||
671 |
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV" |
|
672 |
by auto |
|
673 |
||
674 |
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))" |
|
675 |
by blast |
|
676 |
||
677 |
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)" |
|
678 |
by blast |
|
679 |
||
680 |
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)" |
|
681 |
by (blast elim: equalityE) |
|
682 |
||
683 |
lemma SetCompr_Sigma_eq: |
|
684 |
"Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))" |
|
685 |
by blast |
|
686 |
||
687 |
text {* |
|
688 |
\bigskip Complex rules for Sigma. |
|
689 |
*} |
|
690 |
||
691 |
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q" |
|
692 |
by blast |
|
693 |
||
694 |
lemma UN_Times_distrib: |
|
695 |
"(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)" |
|
696 |
-- {* Suggested by Pierre Chartier *} |
|
697 |
by blast |
|
698 |
||
699 |
lemma split_paired_Ball_Sigma [simp]: |
|
700 |
"(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))" |
|
701 |
by blast |
|
702 |
||
703 |
lemma split_paired_Bex_Sigma [simp]: |
|
704 |
"(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))" |
|
705 |
by blast |
|
706 |
||
707 |
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))" |
|
708 |
by blast |
|
709 |
||
710 |
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))" |
|
711 |
by blast |
|
712 |
||
713 |
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))" |
|
714 |
by blast |
|
715 |
||
716 |
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))" |
|
717 |
by blast |
|
718 |
||
719 |
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))" |
|
720 |
by blast |
|
721 |
||
722 |
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))" |
|
723 |
by blast |
|
724 |
||
725 |
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)" |
|
726 |
by blast |
|
727 |
||
728 |
text {* |
|
729 |
Non-dependent versions are needed to avoid the need for higher-order |
|
730 |
matching, especially when the rules are re-oriented. |
|
731 |
*} |
|
732 |
||
733 |
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" |
|
734 |
by blast |
|
735 |
||
736 |
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" |
|
737 |
by blast |
|
738 |
||
739 |
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)" |
|
740 |
by blast |
|
741 |
||
742 |
||
11493 | 743 |
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A" |
11777 | 744 |
apply (rule_tac x = "(a, b)" in image_eqI) |
745 |
apply auto |
|
746 |
done |
|
747 |
||
11493 | 748 |
|
11838 | 749 |
text {* |
750 |
Setup of internal @{text split_rule}. |
|
751 |
*} |
|
752 |
||
11032 | 753 |
constdefs |
11425 | 754 |
internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c" |
11032 | 755 |
"internal_split == split" |
756 |
||
757 |
lemma internal_split_conv: "internal_split c (a, b) = c a b" |
|
758 |
by (simp only: internal_split_def split_conv) |
|
759 |
||
760 |
hide const internal_split |
|
761 |
||
11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset
|
762 |
use "Tools/split_rule.ML" |
11032 | 763 |
setup SplitRule.setup |
10213 | 764 |
|
15394 | 765 |
|
766 |
subsection {* Code generator setup *} |
|
767 |
||
768 |
types_code |
|
769 |
"*" ("(_ */ _)") |
|
16770
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
770 |
attach (term_of) {* |
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
771 |
fun term_of_id_42 f T g U (x, y) = HOLogic.pair_const T U $ f x $ g y; |
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
772 |
*} |
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
773 |
attach (test) {* |
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
774 |
fun gen_id_42 aG bG i = (aG i, bG i); |
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
775 |
*} |
15394 | 776 |
|
777 |
consts_code |
|
778 |
"Pair" ("(_,/ _)") |
|
779 |
"fst" ("fst") |
|
780 |
"snd" ("snd") |
|
781 |
||
782 |
ML {* |
|
18013 | 783 |
|
784 |
signature PRODUCT_TYPE_CODEGEN = |
|
785 |
sig |
|
786 |
val strip_abs: int -> term -> term list * term; |
|
787 |
end; |
|
788 |
||
789 |
structure ProductTypeCodegen : PRODUCT_TYPE_CODEGEN = |
|
790 |
struct |
|
15394 | 791 |
|
792 |
fun strip_abs 0 t = ([], t) |
|
793 |
| strip_abs i (Abs (s, T, t)) = |
|
18013 | 794 |
let |
795 |
val s' = Codegen.new_name t s; |
|
796 |
val v = Free (s', T) |
|
797 |
in apfst (cons v) (strip_abs (i-1) (subst_bound (v, t))) end |
|
15394 | 798 |
| strip_abs i (u as Const ("split", _) $ t) = (case strip_abs (i+1) t of |
799 |
(v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u) |
|
800 |
| _ => ([], u)) |
|
801 |
| strip_abs i t = ([], t); |
|
802 |
||
18013 | 803 |
end; |
804 |
||
805 |
local |
|
806 |
||
807 |
open ProductTypeCodegen; |
|
808 |
||
16634 | 809 |
fun let_codegen thy defs gr dep thyname brack t = (case strip_comb t of |
810 |
(t1 as Const ("Let", _), t2 :: t3 :: ts) => |
|
15394 | 811 |
let |
812 |
fun dest_let (l as Const ("Let", _) $ t $ u) = |
|
813 |
(case strip_abs 1 u of |
|
814 |
([p], u') => apfst (cons (p, t)) (dest_let u') |
|
815 |
| _ => ([], l)) |
|
816 |
| dest_let t = ([], t); |
|
817 |
fun mk_code (gr, (l, r)) = |
|
818 |
let |
|
16634 | 819 |
val (gr1, pl) = Codegen.invoke_codegen thy defs dep thyname false (gr, l); |
820 |
val (gr2, pr) = Codegen.invoke_codegen thy defs dep thyname false (gr1, r); |
|
15394 | 821 |
in (gr2, (pl, pr)) end |
16634 | 822 |
in case dest_let (t1 $ t2 $ t3) of |
15531 | 823 |
([], _) => NONE |
15394 | 824 |
| (ps, u) => |
825 |
let |
|
826 |
val (gr1, qs) = foldl_map mk_code (gr, ps); |
|
16634 | 827 |
val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u); |
828 |
val (gr3, pargs) = foldl_map |
|
17021
1c361a3de73d
Fixed bug in code generator for let and split leading to ill-formed code.
berghofe
parents:
17002
diff
changeset
|
829 |
(Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts) |
15394 | 830 |
in |
16634 | 831 |
SOME (gr3, Codegen.mk_app brack |
832 |
(Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, List.concat |
|
833 |
(separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) => |
|
834 |
[Pretty.block [Pretty.str "val ", pl, Pretty.str " =", |
|
835 |
Pretty.brk 1, pr]]) qs))), |
|
836 |
Pretty.brk 1, Pretty.str "in ", pu, |
|
837 |
Pretty.brk 1, Pretty.str "end"])) pargs) |
|
15394 | 838 |
end |
839 |
end |
|
16634 | 840 |
| _ => NONE); |
15394 | 841 |
|
16634 | 842 |
fun split_codegen thy defs gr dep thyname brack t = (case strip_comb t of |
843 |
(t1 as Const ("split", _), t2 :: ts) => |
|
844 |
(case strip_abs 1 (t1 $ t2) of |
|
845 |
([p], u) => |
|
846 |
let |
|
847 |
val (gr1, q) = Codegen.invoke_codegen thy defs dep thyname false (gr, p); |
|
848 |
val (gr2, pu) = Codegen.invoke_codegen thy defs dep thyname false (gr1, u); |
|
849 |
val (gr3, pargs) = foldl_map |
|
17021
1c361a3de73d
Fixed bug in code generator for let and split leading to ill-formed code.
berghofe
parents:
17002
diff
changeset
|
850 |
(Codegen.invoke_codegen thy defs dep thyname true) (gr2, ts) |
16634 | 851 |
in |
852 |
SOME (gr2, Codegen.mk_app brack |
|
853 |
(Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>", |
|
854 |
Pretty.brk 1, pu, Pretty.str ")"]) pargs) |
|
855 |
end |
|
856 |
| _ => NONE) |
|
857 |
| _ => NONE); |
|
15394 | 858 |
|
859 |
in |
|
860 |
||
18220 | 861 |
val prod_codegen_setup = [ |
862 |
Codegen.add_codegen "let_codegen" let_codegen, |
|
863 |
Codegen.add_codegen "split_codegen" split_codegen, |
|
18334 | 864 |
CodegenPackage.add_appgen |
865 |
("let", CodegenPackage.appgen_let strip_abs), |
|
866 |
CodegenPackage.add_appgen |
|
867 |
("split", CodegenPackage.appgen_split strip_abs) |
|
18220 | 868 |
]; |
15394 | 869 |
|
870 |
end; |
|
871 |
*} |
|
872 |
||
873 |
setup prod_codegen_setup |
|
874 |
||
15404 | 875 |
ML |
876 |
{* |
|
877 |
val Collect_split = thm "Collect_split"; |
|
878 |
val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1"; |
|
879 |
val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2"; |
|
880 |
val PairE = thm "PairE"; |
|
881 |
val PairE_lemma = thm "PairE_lemma"; |
|
882 |
val Pair_Rep_inject = thm "Pair_Rep_inject"; |
|
883 |
val Pair_def = thm "Pair_def"; |
|
884 |
val Pair_eq = thm "Pair_eq"; |
|
885 |
val Pair_fst_snd_eq = thm "Pair_fst_snd_eq"; |
|
886 |
val Pair_inject = thm "Pair_inject"; |
|
887 |
val ProdI = thm "ProdI"; |
|
888 |
val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq"; |
|
889 |
val SigmaD1 = thm "SigmaD1"; |
|
890 |
val SigmaD2 = thm "SigmaD2"; |
|
891 |
val SigmaE = thm "SigmaE"; |
|
892 |
val SigmaE2 = thm "SigmaE2"; |
|
893 |
val SigmaI = thm "SigmaI"; |
|
894 |
val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1"; |
|
895 |
val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2"; |
|
896 |
val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1"; |
|
897 |
val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2"; |
|
898 |
val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1"; |
|
899 |
val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2"; |
|
900 |
val Sigma_Union = thm "Sigma_Union"; |
|
901 |
val Sigma_def = thm "Sigma_def"; |
|
902 |
val Sigma_empty1 = thm "Sigma_empty1"; |
|
903 |
val Sigma_empty2 = thm "Sigma_empty2"; |
|
904 |
val Sigma_mono = thm "Sigma_mono"; |
|
905 |
val The_split = thm "The_split"; |
|
906 |
val The_split_eq = thm "The_split_eq"; |
|
907 |
val The_split_eq = thm "The_split_eq"; |
|
908 |
val Times_Diff_distrib1 = thm "Times_Diff_distrib1"; |
|
909 |
val Times_Int_distrib1 = thm "Times_Int_distrib1"; |
|
910 |
val Times_Un_distrib1 = thm "Times_Un_distrib1"; |
|
911 |
val Times_eq_cancel2 = thm "Times_eq_cancel2"; |
|
912 |
val Times_subset_cancel2 = thm "Times_subset_cancel2"; |
|
913 |
val UNIV_Times_UNIV = thm "UNIV_Times_UNIV"; |
|
914 |
val UN_Times_distrib = thm "UN_Times_distrib"; |
|
915 |
val Unity_def = thm "Unity_def"; |
|
916 |
val cond_split_eta = thm "cond_split_eta"; |
|
917 |
val fst_conv = thm "fst_conv"; |
|
918 |
val fst_def = thm "fst_def"; |
|
919 |
val fst_eqD = thm "fst_eqD"; |
|
920 |
val inj_on_Abs_Prod = thm "inj_on_Abs_Prod"; |
|
921 |
val injective_fst_snd = thm "injective_fst_snd"; |
|
922 |
val mem_Sigma_iff = thm "mem_Sigma_iff"; |
|
923 |
val mem_splitE = thm "mem_splitE"; |
|
924 |
val mem_splitI = thm "mem_splitI"; |
|
925 |
val mem_splitI2 = thm "mem_splitI2"; |
|
926 |
val prod_eqI = thm "prod_eqI"; |
|
927 |
val prod_fun = thm "prod_fun"; |
|
928 |
val prod_fun_compose = thm "prod_fun_compose"; |
|
929 |
val prod_fun_def = thm "prod_fun_def"; |
|
930 |
val prod_fun_ident = thm "prod_fun_ident"; |
|
931 |
val prod_fun_imageE = thm "prod_fun_imageE"; |
|
932 |
val prod_fun_imageI = thm "prod_fun_imageI"; |
|
933 |
val prod_induct = thm "prod_induct"; |
|
934 |
val snd_conv = thm "snd_conv"; |
|
935 |
val snd_def = thm "snd_def"; |
|
936 |
val snd_eqD = thm "snd_eqD"; |
|
937 |
val split = thm "split"; |
|
938 |
val splitD = thm "splitD"; |
|
939 |
val splitD' = thm "splitD'"; |
|
940 |
val splitE = thm "splitE"; |
|
941 |
val splitE' = thm "splitE'"; |
|
942 |
val splitE2 = thm "splitE2"; |
|
943 |
val splitI = thm "splitI"; |
|
944 |
val splitI2 = thm "splitI2"; |
|
945 |
val splitI2' = thm "splitI2'"; |
|
946 |
val split_Pair_apply = thm "split_Pair_apply"; |
|
947 |
val split_beta = thm "split_beta"; |
|
948 |
val split_conv = thm "split_conv"; |
|
949 |
val split_def = thm "split_def"; |
|
950 |
val split_eta = thm "split_eta"; |
|
951 |
val split_eta_SetCompr = thm "split_eta_SetCompr"; |
|
952 |
val split_eta_SetCompr2 = thm "split_eta_SetCompr2"; |
|
953 |
val split_paired_All = thm "split_paired_All"; |
|
954 |
val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma"; |
|
955 |
val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma"; |
|
956 |
val split_paired_Ex = thm "split_paired_Ex"; |
|
957 |
val split_paired_The = thm "split_paired_The"; |
|
958 |
val split_paired_all = thm "split_paired_all"; |
|
959 |
val split_part = thm "split_part"; |
|
960 |
val split_split = thm "split_split"; |
|
961 |
val split_split_asm = thm "split_split_asm"; |
|
962 |
val split_tupled_all = thms "split_tupled_all"; |
|
963 |
val split_weak_cong = thm "split_weak_cong"; |
|
964 |
val surj_pair = thm "surj_pair"; |
|
965 |
val surjective_pairing = thm "surjective_pairing"; |
|
966 |
val unit_abs_eta_conv = thm "unit_abs_eta_conv"; |
|
967 |
val unit_all_eq1 = thm "unit_all_eq1"; |
|
968 |
val unit_all_eq2 = thm "unit_all_eq2"; |
|
969 |
val unit_eq = thm "unit_eq"; |
|
970 |
val unit_induct = thm "unit_induct"; |
|
971 |
*} |
|
972 |
||
10213 | 973 |
end |