author | bulwahn |
Fri, 27 Jan 2012 10:31:30 +0100 | |
changeset 46343 | 6d9535e52915 |
parent 46128 | 53e7cc599f58 |
child 46556 | 2848e36e0348 |
permissions | -rw-r--r-- |
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(* Title: HOL/Product_Type.thy |
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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 Typedef Inductive Fun |
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uses |
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("Tools/split_rule.ML") |
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("Tools/inductive_set.ML") |
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begin |
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|
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subsection {* @{typ bool} is a datatype *} |
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rep_datatype True False by (auto intro: bool_induct) |
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declare case_split [cases type: bool] |
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-- "prefer plain propositional version" |
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|
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lemma |
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shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P" |
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and [code]: "HOL.equal True P \<longleftrightarrow> P" |
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and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P" |
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and [code]: "HOL.equal P True \<longleftrightarrow> P" |
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and [code nbe]: "HOL.equal P P \<longleftrightarrow> True" |
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by (simp_all add: equal) |
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|
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lemma If_case_cert: |
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assumes "CASE \<equiv> (\<lambda>b. If b f g)" |
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shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)" |
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using assms by simp_all |
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|
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setup {* |
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Code.add_case @{thm If_case_cert} |
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*} |
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code_const "HOL.equal \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool" |
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(Haskell infix 4 "==") |
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code_instance bool :: equal |
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(Haskell -) |
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subsection {* The @{text unit} type *} |
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typedef (open) unit = "{True}" |
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by auto |
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definition Unity :: unit ("'(')") |
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where "() = Abs_unit True" |
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|
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lemma unit_eq [no_atp]: "u = ()" |
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by (induct u) (simp add: 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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simproc_setup unit_eq ("x::unit") = {* |
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fn _ => fn _ => fn ct => |
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if HOLogic.is_unit (term_of ct) then NONE |
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else SOME (mk_meta_eq @{thm unit_eq}) |
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*} |
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||
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rep_datatype "()" by simp |
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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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text {* |
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This rewrite counters the effect of simproc @{text unit_eq} 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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||
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lemma unit_abs_eta_conv [simp, no_atp]: "(%u::unit. f ()) = f" |
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by (rule ext) simp |
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lemma UNIV_unit [no_atp]: |
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"UNIV = {()}" by auto |
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instantiation unit :: default |
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begin |
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definition "default = ()" |
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instance .. |
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end |
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lemma [code]: |
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"HOL.equal (u\<Colon>unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+ |
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code_type unit |
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(SML "unit") |
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(OCaml "unit") |
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(Haskell "()") |
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(Scala "Unit") |
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105 |
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code_const Unity |
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(SML "()") |
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(OCaml "()") |
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(Haskell "()") |
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(Scala "()") |
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code_instance unit :: equal |
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(Haskell -) |
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code_const "HOL.equal \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool" |
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(Haskell infix 4 "==") |
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code_reserved SML |
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unit |
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code_reserved OCaml |
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unit |
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code_reserved Scala |
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Unit |
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||
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subsection {* The product type *} |
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subsubsection {* Type definition *} |
131 |
||
132 |
definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where |
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"Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)" |
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definition "prod = {f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}" |
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||
137 |
typedef (open) ('a, 'b) prod (infixr "*" 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set" |
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unfolding prod_def by auto |
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10213 | 139 |
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type_notation (xsymbols) |
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141 |
"prod" ("(_ \<times>/ _)" [21, 20] 20) |
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type_notation (HTML output) |
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143 |
"prod" ("(_ \<times>/ _)" [21, 20] 20) |
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|
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definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where |
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146 |
"Pair a b = Abs_prod (Pair_Rep a b)" |
37166 | 147 |
|
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148 |
rep_datatype Pair proof - |
37166 | 149 |
fix P :: "'a \<times> 'b \<Rightarrow> bool" and p |
150 |
assume "\<And>a b. P (Pair a b)" |
|
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then show "P p" by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def) |
37166 | 152 |
next |
153 |
fix a c :: 'a and b d :: 'b |
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154 |
have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d" |
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by (auto simp add: Pair_Rep_def fun_eq_iff) |
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156 |
moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod" |
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157 |
by (auto simp add: prod_def) |
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ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d" |
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159 |
by (simp add: Pair_def Abs_prod_inject) |
37166 | 160 |
qed |
161 |
||
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162 |
declare prod.simps(2) [nitpick_simp del] |
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163 |
|
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164 |
declare prod.weak_case_cong [cong del] |
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165 |
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37166 | 166 |
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167 |
subsubsection {* Tuple syntax *} |
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168 |
||
37591 | 169 |
abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where |
170 |
"split \<equiv> prod_case" |
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19535 | 171 |
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11777 | 172 |
text {* |
173 |
Patterns -- extends pre-defined type @{typ pttrn} used in |
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174 |
abstractions. |
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175 |
*} |
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10213 | 176 |
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177 |
nonterminal tuple_args and patterns |
10213 | 178 |
|
179 |
syntax |
|
180 |
"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))") |
|
181 |
"_tuple_arg" :: "'a => tuple_args" ("_") |
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182 |
"_tuple_args" :: "'a => tuple_args => tuple_args" ("_,/ _") |
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183 |
"_pattern" :: "[pttrn, patterns] => pttrn" ("'(_,/ _')") |
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184 |
"" :: "pttrn => patterns" ("_") |
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185 |
"_patterns" :: "[pttrn, patterns] => patterns" ("_,/ _") |
10213 | 186 |
|
187 |
translations |
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35115 | 188 |
"(x, y)" == "CONST Pair x y" |
10213 | 189 |
"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" |
37591 | 190 |
"%(x, y, zs). b" == "CONST prod_case (%x (y, zs). b)" |
191 |
"%(x, y). b" == "CONST prod_case (%x y. b)" |
|
35115 | 192 |
"_abs (CONST Pair x y) t" => "%(x, y). t" |
37166 | 193 |
-- {* The last rule accommodates tuples in `case C ... (x,y) ... => ...' |
194 |
The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *} |
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10213 | 195 |
|
35115 | 196 |
(*reconstruct pattern from (nested) splits, avoiding eta-contraction of body; |
197 |
works best with enclosing "let", if "let" does not avoid eta-contraction*) |
|
14359 | 198 |
print_translation {* |
35115 | 199 |
let |
200 |
fun split_tr' [Abs (x, T, t as (Abs abs))] = |
|
201 |
(* split (%x y. t) => %(x,y) t *) |
|
202 |
let |
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42284 | 203 |
val (y, t') = Syntax_Trans.atomic_abs_tr' abs; |
204 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); |
|
35115 | 205 |
in |
206 |
Syntax.const @{syntax_const "_abs"} $ |
|
207 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' |
|
208 |
end |
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37591 | 209 |
| split_tr' [Abs (x, T, (s as Const (@{const_syntax prod_case}, _) $ t))] = |
35115 | 210 |
(* split (%x. (split (%y z. t))) => %(x,y,z). t *) |
211 |
let |
|
212 |
val Const (@{syntax_const "_abs"}, _) $ |
|
213 |
(Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t]; |
|
42284 | 214 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); |
35115 | 215 |
in |
216 |
Syntax.const @{syntax_const "_abs"} $ |
|
217 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ |
|
218 |
(Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t'' |
|
219 |
end |
|
37591 | 220 |
| split_tr' [Const (@{const_syntax prod_case}, _) $ t] = |
35115 | 221 |
(* split (split (%x y z. t)) => %((x, y), z). t *) |
222 |
split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *) |
|
223 |
| split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] = |
|
224 |
(* split (%pttrn z. t) => %(pttrn,z). t *) |
|
42284 | 225 |
let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in |
35115 | 226 |
Syntax.const @{syntax_const "_abs"} $ |
227 |
(Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t |
|
228 |
end |
|
229 |
| split_tr' _ = raise Match; |
|
37591 | 230 |
in [(@{const_syntax prod_case}, split_tr')] end |
14359 | 231 |
*} |
232 |
||
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|
233 |
(* 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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|
234 |
typed_print_translation {* |
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changeset
|
235 |
let |
42247
12fe41a92cd5
typed_print_translation: discontinued show_sorts argument;
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parents:
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|
236 |
fun split_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match |
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|
237 |
| split_guess_names_tr' T [Abs (x, xT, t)] = |
15422
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|
238 |
(case (head_of t) of |
37591 | 239 |
Const (@{const_syntax prod_case}, _) => raise Match |
35115 | 240 |
| _ => |
241 |
let |
|
242 |
val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; |
|
42284 | 243 |
val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0); |
244 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t'); |
|
35115 | 245 |
in |
246 |
Syntax.const @{syntax_const "_abs"} $ |
|
247 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' |
|
248 |
end) |
|
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|
249 |
| split_guess_names_tr' T [t] = |
35115 | 250 |
(case head_of t of |
37591 | 251 |
Const (@{const_syntax prod_case}, _) => raise Match |
35115 | 252 |
| _ => |
253 |
let |
|
254 |
val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; |
|
42284 | 255 |
val (y, t') = |
256 |
Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0); |
|
257 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t'); |
|
35115 | 258 |
in |
259 |
Syntax.const @{syntax_const "_abs"} $ |
|
260 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' |
|
261 |
end) |
|
42247
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parents:
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|
262 |
| split_guess_names_tr' _ _ = raise Match; |
37591 | 263 |
in [(@{const_syntax prod_case}, split_guess_names_tr')] end |
15422
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
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diff
changeset
|
264 |
*} |
cbdddc0efadf
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schirmer
parents:
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diff
changeset
|
265 |
|
42059
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
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|
266 |
(* Force eta-contraction for terms of the form "Q A (%p. prod_case P p)" |
83f3dc509068
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|
267 |
where Q is some bounded quantifier or set operator. |
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|
268 |
Reason: the above prints as "Q p : A. case p of (x,y) \<Rightarrow> P x y" |
83f3dc509068
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|
269 |
whereas we want "Q (x,y):A. P x y". |
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
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changeset
|
270 |
Otherwise prevent eta-contraction. |
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|
271 |
*) |
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|
272 |
print_translation {* |
83f3dc509068
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nipkow
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|
273 |
let |
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
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diff
changeset
|
274 |
fun contract Q f ts = |
83f3dc509068
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nipkow
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diff
changeset
|
275 |
case ts of |
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
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changeset
|
276 |
[A, Abs(_, _, (s as Const (@{const_syntax prod_case},_) $ t) $ Bound 0)] |
42083
e1209fc7ecdc
added Term.is_open and Term.is_dependent convenience, to cover common situations of loose bounds;
wenzelm
parents:
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diff
changeset
|
277 |
=> if Term.is_dependent t then f ts else Syntax.const Q $ A $ s |
42059
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
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diff
changeset
|
278 |
| _ => f ts; |
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
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diff
changeset
|
279 |
fun contract2 (Q,f) = (Q, contract Q f); |
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
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parents:
41792
diff
changeset
|
280 |
val pairs = |
42284 | 281 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"}, |
282 |
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"}, |
|
283 |
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}, |
|
284 |
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}] |
|
42059
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
41792
diff
changeset
|
285 |
in map contract2 pairs end |
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
41792
diff
changeset
|
286 |
*} |
10213 | 287 |
|
37166 | 288 |
subsubsection {* Code generator setup *} |
289 |
||
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
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diff
changeset
|
290 |
code_type prod |
37166 | 291 |
(SML infix 2 "*") |
292 |
(OCaml infix 2 "*") |
|
293 |
(Haskell "!((_),/ (_))") |
|
294 |
(Scala "((_),/ (_))") |
|
295 |
||
296 |
code_const Pair |
|
297 |
(SML "!((_),/ (_))") |
|
298 |
(OCaml "!((_),/ (_))") |
|
299 |
(Haskell "!((_),/ (_))") |
|
300 |
(Scala "!((_),/ (_))") |
|
301 |
||
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
302 |
code_instance prod :: equal |
37166 | 303 |
(Haskell -) |
304 |
||
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38715
diff
changeset
|
305 |
code_const "HOL.equal \<Colon> 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" |
39272 | 306 |
(Haskell infix 4 "==") |
37166 | 307 |
|
308 |
||
309 |
subsubsection {* Fundamental operations and properties *} |
|
11838 | 310 |
|
26358
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Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
311 |
lemma surj_pair [simp]: "EX x y. p = (x, y)" |
37166 | 312 |
by (cases p) simp |
10213 | 313 |
|
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
314 |
definition fst :: "'a \<times> 'b \<Rightarrow> 'a" where |
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
315 |
"fst p = (case p of (a, b) \<Rightarrow> a)" |
11838 | 316 |
|
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
317 |
definition snd :: "'a \<times> 'b \<Rightarrow> 'b" where |
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
37387
diff
changeset
|
318 |
"snd p = (case p of (a, b) \<Rightarrow> b)" |
11838 | 319 |
|
22886 | 320 |
lemma fst_conv [simp, code]: "fst (a, b) = a" |
37166 | 321 |
unfolding fst_def by simp |
11838 | 322 |
|
22886 | 323 |
lemma snd_conv [simp, code]: "snd (a, b) = b" |
37166 | 324 |
unfolding snd_def by simp |
11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset
|
325 |
|
37166 | 326 |
code_const fst and snd |
327 |
(Haskell "fst" and "snd") |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
328 |
|
41792
ff3cb0c418b7
renamed "nitpick\_def" to "nitpick_unfold" to reflect its new semantics
blanchet
parents:
41505
diff
changeset
|
329 |
lemma prod_case_unfold [nitpick_unfold]: "prod_case = (%c p. c (fst p) (snd p))" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
330 |
by (simp add: fun_eq_iff split: prod.split) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
331 |
|
11838 | 332 |
lemma fst_eqD: "fst (x, y) = a ==> x = a" |
333 |
by simp |
|
334 |
||
335 |
lemma snd_eqD: "snd (x, y) = a ==> y = a" |
|
336 |
by simp |
|
337 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
338 |
lemma pair_collapse [simp]: "(fst p, snd p) = p" |
11838 | 339 |
by (cases p) simp |
340 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
341 |
lemmas surjective_pairing = pair_collapse [symmetric] |
11838 | 342 |
|
44066
d74182c93f04
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents:
43866
diff
changeset
|
343 |
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t" |
37166 | 344 |
by (cases s, cases t) simp |
345 |
||
346 |
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q" |
|
44066
d74182c93f04
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents:
43866
diff
changeset
|
347 |
by (simp add: prod_eq_iff) |
37166 | 348 |
|
349 |
lemma split_conv [simp, code]: "split f (a, b) = f a b" |
|
37591 | 350 |
by (fact prod.cases) |
37166 | 351 |
|
352 |
lemma splitI: "f a b \<Longrightarrow> split f (a, b)" |
|
353 |
by (rule split_conv [THEN iffD2]) |
|
354 |
||
355 |
lemma splitD: "split f (a, b) \<Longrightarrow> f a b" |
|
356 |
by (rule split_conv [THEN iffD1]) |
|
357 |
||
358 |
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
359 |
by (simp add: fun_eq_iff split: prod.split) |
37166 | 360 |
|
361 |
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f" |
|
362 |
-- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *} |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
363 |
by (simp add: fun_eq_iff split: prod.split) |
37166 | 364 |
|
365 |
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)" |
|
366 |
by (cases x) simp |
|
367 |
||
368 |
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p" |
|
369 |
by (cases p) simp |
|
370 |
||
371 |
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" |
|
37591 | 372 |
by (simp add: prod_case_unfold) |
37166 | 373 |
|
374 |
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q" |
|
375 |
-- {* Prevents simplification of @{term c}: much faster *} |
|
40929
7ff03a5e044f
theorem names generated by the (rep_)datatype command now have mandatory qualifiers
huffman
parents:
40702
diff
changeset
|
376 |
by (fact prod.weak_case_cong) |
37166 | 377 |
|
378 |
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" |
|
379 |
by (simp add: split_eta) |
|
380 |
||
11838 | 381 |
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" |
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
382 |
proof |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
383 |
fix a b |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
384 |
assume "!!x. PROP P x" |
19535 | 385 |
then show "PROP P (a, b)" . |
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
386 |
next |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
387 |
fix x |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
388 |
assume "!!a b. PROP P (a, b)" |
19535 | 389 |
from `PROP P (fst x, snd x)` show "PROP P x" by simp |
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
390 |
qed |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
391 |
|
11838 | 392 |
text {* |
393 |
The rule @{thm [source] split_paired_all} does not work with the |
|
394 |
Simplifier because it also affects premises in congrence rules, |
|
395 |
where this can lead to premises of the form @{text "!!a b. ... = |
|
396 |
?P(a, b)"} which cannot be solved by reflexivity. |
|
397 |
*} |
|
398 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
399 |
lemmas split_tupled_all = split_paired_all unit_all_eq2 |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
400 |
|
26480 | 401 |
ML {* |
11838 | 402 |
(* replace parameters of product type by individual component parameters *) |
403 |
val safe_full_simp_tac = generic_simp_tac true (true, false, false); |
|
404 |
local (* filtering with exists_paired_all is an essential optimization *) |
|
16121 | 405 |
fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) = |
11838 | 406 |
can HOLogic.dest_prodT T orelse exists_paired_all t |
407 |
| exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u |
|
408 |
| exists_paired_all (Abs (_, _, t)) = exists_paired_all t |
|
409 |
| exists_paired_all _ = false; |
|
410 |
val ss = HOL_basic_ss |
|
26340 | 411 |
addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] |
43594 | 412 |
addsimprocs [@{simproc unit_eq}]; |
11838 | 413 |
in |
414 |
val split_all_tac = SUBGOAL (fn (t, i) => |
|
415 |
if exists_paired_all t then safe_full_simp_tac ss i else no_tac); |
|
416 |
val unsafe_split_all_tac = SUBGOAL (fn (t, i) => |
|
417 |
if exists_paired_all t then full_simp_tac ss i else no_tac); |
|
418 |
fun split_all th = |
|
26340 | 419 |
if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th; |
11838 | 420 |
end; |
26340 | 421 |
*} |
11838 | 422 |
|
26340 | 423 |
declaration {* fn _ => |
424 |
Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac)) |
|
16121 | 425 |
*} |
11838 | 426 |
|
427 |
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))" |
|
428 |
-- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} |
|
429 |
by fast |
|
430 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
431 |
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
432 |
by fast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
433 |
|
11838 | 434 |
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))" |
435 |
-- {* Can't be added to simpset: loops! *} |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
436 |
by (simp add: split_eta) |
11838 | 437 |
|
438 |
text {* |
|
439 |
Simplification procedure for @{thm [source] cond_split_eta}. Using |
|
440 |
@{thm [source] split_eta} as a rewrite rule is not general enough, |
|
441 |
and using @{thm [source] cond_split_eta} directly would render some |
|
442 |
existing proofs very inefficient; similarly for @{text |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
443 |
split_beta}. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
444 |
*} |
11838 | 445 |
|
26480 | 446 |
ML {* |
11838 | 447 |
local |
35364 | 448 |
val cond_split_eta_ss = HOL_basic_ss addsimps @{thms cond_split_eta}; |
449 |
fun Pair_pat k 0 (Bound m) = (m = k) |
|
450 |
| Pair_pat k i (Const (@{const_name Pair}, _) $ Bound m $ t) = |
|
451 |
i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t |
|
452 |
| Pair_pat _ _ _ = false; |
|
453 |
fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t |
|
454 |
| no_args k i (t $ u) = no_args k i t andalso no_args k i u |
|
455 |
| no_args k i (Bound m) = m < k orelse m > k + i |
|
456 |
| no_args _ _ _ = true; |
|
457 |
fun split_pat tp i (Abs (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE |
|
37591 | 458 |
| split_pat tp i (Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t |
35364 | 459 |
| split_pat tp i _ = NONE; |
20044
92cc2f4c7335
simprocs: no theory argument -- use simpset context instead;
wenzelm
parents:
19656
diff
changeset
|
460 |
fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] [] |
35364 | 461 |
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))) |
18328 | 462 |
(K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1))); |
11838 | 463 |
|
35364 | 464 |
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t |
465 |
| beta_term_pat k i (t $ u) = |
|
466 |
Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u) |
|
467 |
| beta_term_pat k i t = no_args k i t; |
|
468 |
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg |
|
469 |
| eta_term_pat _ _ _ = false; |
|
11838 | 470 |
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) |
35364 | 471 |
| subst arg k i (t $ u) = |
472 |
if Pair_pat k i (t $ u) then incr_boundvars k arg |
|
473 |
else (subst arg k i t $ subst arg k i u) |
|
474 |
| subst arg k i t = t; |
|
43595 | 475 |
in |
37591 | 476 |
fun beta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t) $ arg) = |
11838 | 477 |
(case split_pat beta_term_pat 1 t of |
35364 | 478 |
SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f)) |
15531 | 479 |
| NONE => NONE) |
35364 | 480 |
| beta_proc _ _ = NONE; |
37591 | 481 |
fun eta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = |
11838 | 482 |
(case split_pat eta_term_pat 1 t of |
35364 | 483 |
SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end)) |
15531 | 484 |
| NONE => NONE) |
35364 | 485 |
| eta_proc _ _ = NONE; |
11838 | 486 |
end; |
487 |
*} |
|
43595 | 488 |
simproc_setup split_beta ("split f z") = {* fn _ => fn ss => fn ct => beta_proc ss (term_of ct) *} |
489 |
simproc_setup split_eta ("split f") = {* fn _ => fn ss => fn ct => eta_proc ss (term_of ct) *} |
|
11838 | 490 |
|
26798
a9134a089106
split_beta is now declared as monotonicity rule, to allow bounded
berghofe
parents:
26588
diff
changeset
|
491 |
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)" |
11838 | 492 |
by (subst surjective_pairing, rule split_conv) |
493 |
||
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
494 |
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))" |
11838 | 495 |
-- {* For use with @{text split} and the Simplifier. *} |
15481 | 496 |
by (insert surj_pair [of p], clarify, simp) |
11838 | 497 |
|
498 |
text {* |
|
499 |
@{thm [source] split_split} could be declared as @{text "[split]"} |
|
500 |
done after the Splitter has been speeded up significantly; |
|
501 |
precompute the constants involved and don't do anything unless the |
|
502 |
current goal contains one of those constants. |
|
503 |
*} |
|
504 |
||
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
505 |
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" |
14208 | 506 |
by (subst split_split, simp) |
11838 | 507 |
|
508 |
text {* |
|
509 |
\medskip @{term split} used as a logical connective or set former. |
|
510 |
||
511 |
\medskip These rules are for use with @{text blast}; could instead |
|
40929
7ff03a5e044f
theorem names generated by the (rep_)datatype command now have mandatory qualifiers
huffman
parents:
40702
diff
changeset
|
512 |
call @{text simp} using @{thm [source] prod.split} as rewrite. *} |
11838 | 513 |
|
514 |
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p" |
|
515 |
apply (simp only: split_tupled_all) |
|
516 |
apply (simp (no_asm_simp)) |
|
517 |
done |
|
518 |
||
519 |
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x" |
|
520 |
apply (simp only: split_tupled_all) |
|
521 |
apply (simp (no_asm_simp)) |
|
522 |
done |
|
523 |
||
524 |
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" |
|
37591 | 525 |
by (induct p) auto |
11838 | 526 |
|
527 |
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" |
|
37591 | 528 |
by (induct p) auto |
11838 | 529 |
|
530 |
lemma splitE2: |
|
531 |
"[| Q (split P z); !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R" |
|
532 |
proof - |
|
533 |
assume q: "Q (split P z)" |
|
534 |
assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R" |
|
535 |
show R |
|
536 |
apply (rule r surjective_pairing)+ |
|
537 |
apply (rule split_beta [THEN subst], rule q) |
|
538 |
done |
|
539 |
qed |
|
540 |
||
541 |
lemma splitD': "split R (a,b) c ==> R a b c" |
|
542 |
by simp |
|
543 |
||
544 |
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" |
|
545 |
by simp |
|
546 |
||
547 |
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p" |
|
14208 | 548 |
by (simp only: split_tupled_all, simp) |
11838 | 549 |
|
18372 | 550 |
lemma mem_splitE: |
37166 | 551 |
assumes major: "z \<in> split c p" |
552 |
and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q" |
|
18372 | 553 |
shows Q |
37591 | 554 |
by (rule major [unfolded prod_case_unfold] cases surjective_pairing)+ |
11838 | 555 |
|
556 |
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] |
|
557 |
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] |
|
558 |
||
26340 | 559 |
ML {* |
11838 | 560 |
local (* filtering with exists_p_split is an essential optimization *) |
37591 | 561 |
fun exists_p_split (Const (@{const_name prod_case},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true |
11838 | 562 |
| exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u |
563 |
| exists_p_split (Abs (_, _, t)) = exists_p_split t |
|
564 |
| exists_p_split _ = false; |
|
35364 | 565 |
val ss = HOL_basic_ss addsimps @{thms split_conv}; |
11838 | 566 |
in |
567 |
val split_conv_tac = SUBGOAL (fn (t, i) => |
|
568 |
if exists_p_split t then safe_full_simp_tac ss i else no_tac); |
|
569 |
end; |
|
26340 | 570 |
*} |
571 |
||
11838 | 572 |
(* This prevents applications of splitE for already splitted arguments leading |
573 |
to quite time-consuming computations (in particular for nested tuples) *) |
|
26340 | 574 |
declaration {* fn _ => |
575 |
Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac)) |
|
16121 | 576 |
*} |
11838 | 577 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
578 |
lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" |
18372 | 579 |
by (rule ext) fast |
11838 | 580 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
581 |
lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P" |
18372 | 582 |
by (rule ext) fast |
11838 | 583 |
|
584 |
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" |
|
585 |
-- {* Allows simplifications of nested splits in case of independent predicates. *} |
|
18372 | 586 |
by (rule ext) blast |
11838 | 587 |
|
14337
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
588 |
(* 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
|
589 |
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
|
590 |
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
|
591 |
*) |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
592 |
lemma split_comp_eq: |
20415 | 593 |
fixes f :: "'a => 'b => 'c" and g :: "'d => 'a" |
594 |
shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" |
|
18372 | 595 |
by (rule ext) auto |
14101 | 596 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
597 |
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
598 |
apply (rule_tac x = "(a, b)" in image_eqI) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
599 |
apply auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
600 |
done |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
601 |
|
11838 | 602 |
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" |
603 |
by blast |
|
604 |
||
605 |
(* |
|
606 |
the following would be slightly more general, |
|
607 |
but cannot be used as rewrite rule: |
|
608 |
### Cannot add premise as rewrite rule because it contains (type) unknowns: |
|
609 |
### ?y = .x |
|
610 |
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)" |
|
14208 | 611 |
by (rtac some_equality 1) |
612 |
by ( Simp_tac 1) |
|
613 |
by (split_all_tac 1) |
|
614 |
by (Asm_full_simp_tac 1) |
|
11838 | 615 |
qed "The_split_eq"; |
616 |
*) |
|
617 |
||
618 |
text {* |
|
619 |
Setup of internal @{text split_rule}. |
|
620 |
*} |
|
621 |
||
45607 | 622 |
lemmas prod_caseI = prod.cases [THEN iffD2] |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
623 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
624 |
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
625 |
by (fact splitI2) |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
626 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
627 |
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
628 |
by (fact splitI2') |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
629 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
630 |
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
631 |
by (fact splitE) |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
632 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
633 |
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
634 |
by (fact splitE') |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
635 |
|
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
636 |
declare prod_caseI [intro!] |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
637 |
|
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset
|
638 |
lemma prod_case_beta: |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset
|
639 |
"prod_case f p = f (fst p) (snd p)" |
37591 | 640 |
by (fact split_beta) |
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset
|
641 |
|
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
642 |
lemma prod_cases3 [cases type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
643 |
obtains (fields) a b c where "y = (a, b, c)" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
644 |
by (cases y, case_tac b) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
645 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
646 |
lemma prod_induct3 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
647 |
"(!!a b c. P (a, b, c)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
648 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
649 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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diff
changeset
|
650 |
lemma prod_cases4 [cases type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
651 |
obtains (fields) a b c d where "y = (a, b, c, d)" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
652 |
by (cases y, case_tac c) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
653 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
654 |
lemma prod_induct4 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
655 |
"(!!a b c d. P (a, b, c, d)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
656 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
657 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
658 |
lemma prod_cases5 [cases type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
659 |
obtains (fields) a b c d e where "y = (a, b, c, d, e)" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
660 |
by (cases y, case_tac d) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
661 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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changeset
|
662 |
lemma prod_induct5 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
663 |
"(!!a b c d e. P (a, b, c, d, e)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
664 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
665 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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changeset
|
666 |
lemma prod_cases6 [cases type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
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parents:
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changeset
|
667 |
obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
668 |
by (cases y, case_tac e) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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diff
changeset
|
669 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
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diff
changeset
|
670 |
lemma prod_induct6 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
671 |
"(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
672 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
673 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
674 |
lemma prod_cases7 [cases type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
675 |
obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
676 |
by (cases y, case_tac f) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
677 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
678 |
lemma prod_induct7 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
679 |
"(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
680 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
681 |
|
37166 | 682 |
lemma split_def: |
683 |
"split = (\<lambda>c p. c (fst p) (snd p))" |
|
37591 | 684 |
by (fact prod_case_unfold) |
37166 | 685 |
|
686 |
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where |
|
687 |
"internal_split == split" |
|
688 |
||
689 |
lemma internal_split_conv: "internal_split c (a, b) = c a b" |
|
690 |
by (simp only: internal_split_def split_conv) |
|
691 |
||
692 |
use "Tools/split_rule.ML" |
|
693 |
setup Split_Rule.setup |
|
694 |
||
695 |
hide_const internal_split |
|
696 |
||
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
697 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
698 |
subsubsection {* Derived operations *} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
699 |
|
37387
3581483cca6c
qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents:
37278
diff
changeset
|
700 |
definition curry :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where |
3581483cca6c
qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents:
37278
diff
changeset
|
701 |
"curry = (\<lambda>c x y. c (x, y))" |
37166 | 702 |
|
703 |
lemma curry_conv [simp, code]: "curry f a b = f (a, b)" |
|
704 |
by (simp add: curry_def) |
|
705 |
||
706 |
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b" |
|
707 |
by (simp add: curry_def) |
|
708 |
||
709 |
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)" |
|
710 |
by (simp add: curry_def) |
|
711 |
||
712 |
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q" |
|
713 |
by (simp add: curry_def) |
|
714 |
||
715 |
lemma curry_split [simp]: "curry (split f) = f" |
|
716 |
by (simp add: curry_def split_def) |
|
717 |
||
718 |
lemma split_curry [simp]: "split (curry f) = f" |
|
719 |
by (simp add: curry_def split_def) |
|
720 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
721 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
722 |
The composition-uncurry combinator. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
723 |
*} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
724 |
|
37751 | 725 |
notation fcomp (infixl "\<circ>>" 60) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
726 |
|
37751 | 727 |
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where |
728 |
"f \<circ>\<rightarrow> g = (\<lambda>x. prod_case g (f x))" |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
729 |
|
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
730 |
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
731 |
by (simp add: fun_eq_iff scomp_def prod_case_unfold) |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
732 |
|
37751 | 733 |
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = prod_case g (f x)" |
734 |
by (simp add: scomp_unfold prod_case_unfold) |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
735 |
|
37751 | 736 |
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x" |
44921 | 737 |
by (simp add: fun_eq_iff) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
738 |
|
37751 | 739 |
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x" |
44921 | 740 |
by (simp add: fun_eq_iff) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
741 |
|
37751 | 742 |
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
743 |
by (simp add: fun_eq_iff scomp_unfold) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
744 |
|
37751 | 745 |
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
746 |
by (simp add: fun_eq_iff scomp_unfold fcomp_def) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
747 |
|
37751 | 748 |
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)" |
44921 | 749 |
by (simp add: fun_eq_iff scomp_unfold) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
750 |
|
31202
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset
|
751 |
code_const scomp |
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset
|
752 |
(Eval infixl 3 "#->") |
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset
|
753 |
|
37751 | 754 |
no_notation fcomp (infixl "\<circ>>" 60) |
755 |
no_notation scomp (infixl "\<circ>\<rightarrow>" 60) |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
756 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
757 |
text {* |
40607 | 758 |
@{term map_pair} --- action of the product functor upon |
36664
6302f9ad7047
repaired comments where SOMEthing went utterly wrong (cf. 2b04504fcb69)
krauss
parents:
36622
diff
changeset
|
759 |
functions. |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
760 |
*} |
21195 | 761 |
|
40607 | 762 |
definition map_pair :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where |
763 |
"map_pair f g = (\<lambda>(x, y). (f x, g y))" |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
764 |
|
40607 | 765 |
lemma map_pair_simp [simp, code]: |
766 |
"map_pair f g (a, b) = (f a, g b)" |
|
767 |
by (simp add: map_pair_def) |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
768 |
|
41505
6d19301074cf
"enriched_type" replaces less specific "type_lifting"
haftmann
parents:
41372
diff
changeset
|
769 |
enriched_type map_pair: map_pair |
44921 | 770 |
by (auto simp add: split_paired_all) |
37278 | 771 |
|
40607 | 772 |
lemma fst_map_pair [simp]: |
773 |
"fst (map_pair f g x) = f (fst x)" |
|
774 |
by (cases x) simp_all |
|
37278 | 775 |
|
40607 | 776 |
lemma snd_prod_fun [simp]: |
777 |
"snd (map_pair f g x) = g (snd x)" |
|
778 |
by (cases x) simp_all |
|
37278 | 779 |
|
40607 | 780 |
lemma fst_comp_map_pair [simp]: |
781 |
"fst \<circ> map_pair f g = f \<circ> fst" |
|
782 |
by (rule ext) simp_all |
|
37278 | 783 |
|
40607 | 784 |
lemma snd_comp_map_pair [simp]: |
785 |
"snd \<circ> map_pair f g = g \<circ> snd" |
|
786 |
by (rule ext) simp_all |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
787 |
|
40607 | 788 |
lemma map_pair_compose: |
789 |
"map_pair (f1 o f2) (g1 o g2) = (map_pair f1 g1 o map_pair f2 g2)" |
|
790 |
by (rule ext) (simp add: map_pair.compositionality comp_def) |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
791 |
|
40607 | 792 |
lemma map_pair_ident [simp]: |
793 |
"map_pair (%x. x) (%y. y) = (%z. z)" |
|
794 |
by (rule ext) (simp add: map_pair.identity) |
|
795 |
||
796 |
lemma map_pair_imageI [intro]: |
|
797 |
"(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_pair f g ` R" |
|
798 |
by (rule image_eqI) simp_all |
|
21195 | 799 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
800 |
lemma prod_fun_imageE [elim!]: |
40607 | 801 |
assumes major: "c \<in> map_pair f g ` R" |
802 |
and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P" |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
803 |
shows P |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
804 |
apply (rule major [THEN imageE]) |
37166 | 805 |
apply (case_tac x) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
806 |
apply (rule cases) |
40607 | 807 |
apply simp_all |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
808 |
done |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
809 |
|
37166 | 810 |
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where |
40607 | 811 |
"apfst f = map_pair f id" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
812 |
|
37166 | 813 |
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where |
40607 | 814 |
"apsnd f = map_pair id f" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
815 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
816 |
lemma apfst_conv [simp, code]: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
817 |
"apfst f (x, y) = (f x, y)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
818 |
by (simp add: apfst_def) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
819 |
|
33638
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
820 |
lemma apsnd_conv [simp, code]: |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
821 |
"apsnd f (x, y) = (x, f y)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
822 |
by (simp add: apsnd_def) |
21195 | 823 |
|
33594 | 824 |
lemma fst_apfst [simp]: |
825 |
"fst (apfst f x) = f (fst x)" |
|
826 |
by (cases x) simp |
|
827 |
||
828 |
lemma fst_apsnd [simp]: |
|
829 |
"fst (apsnd f x) = fst x" |
|
830 |
by (cases x) simp |
|
831 |
||
832 |
lemma snd_apfst [simp]: |
|
833 |
"snd (apfst f x) = snd x" |
|
834 |
by (cases x) simp |
|
835 |
||
836 |
lemma snd_apsnd [simp]: |
|
837 |
"snd (apsnd f x) = f (snd x)" |
|
838 |
by (cases x) simp |
|
839 |
||
840 |
lemma apfst_compose: |
|
841 |
"apfst f (apfst g x) = apfst (f \<circ> g) x" |
|
842 |
by (cases x) simp |
|
843 |
||
844 |
lemma apsnd_compose: |
|
845 |
"apsnd f (apsnd g x) = apsnd (f \<circ> g) x" |
|
846 |
by (cases x) simp |
|
847 |
||
848 |
lemma apfst_apsnd [simp]: |
|
849 |
"apfst f (apsnd g x) = (f (fst x), g (snd x))" |
|
850 |
by (cases x) simp |
|
851 |
||
852 |
lemma apsnd_apfst [simp]: |
|
853 |
"apsnd f (apfst g x) = (g (fst x), f (snd x))" |
|
854 |
by (cases x) simp |
|
855 |
||
856 |
lemma apfst_id [simp] : |
|
857 |
"apfst id = id" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
858 |
by (simp add: fun_eq_iff) |
33594 | 859 |
|
860 |
lemma apsnd_id [simp] : |
|
861 |
"apsnd id = id" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
862 |
by (simp add: fun_eq_iff) |
33594 | 863 |
|
864 |
lemma apfst_eq_conv [simp]: |
|
865 |
"apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)" |
|
866 |
by (cases x) simp |
|
867 |
||
868 |
lemma apsnd_eq_conv [simp]: |
|
869 |
"apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)" |
|
870 |
by (cases x) simp |
|
871 |
||
33638
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
872 |
lemma apsnd_apfst_commute: |
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
873 |
"apsnd f (apfst g p) = apfst g (apsnd f p)" |
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
874 |
by simp |
21195 | 875 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
876 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
877 |
Disjoint union of a family of sets -- Sigma. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
878 |
*} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
879 |
|
45986
c9e50153e5ae
moved various set operations to theory Set (resp. Product_Type)
haftmann
parents:
45696
diff
changeset
|
880 |
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
881 |
Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
882 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
883 |
abbreviation |
45986
c9e50153e5ae
moved various set operations to theory Set (resp. Product_Type)
haftmann
parents:
45696
diff
changeset
|
884 |
Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
885 |
(infixr "<*>" 80) where |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
886 |
"A <*> B == Sigma A (%_. B)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
887 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
888 |
notation (xsymbols) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
889 |
Times (infixr "\<times>" 80) |
15394 | 890 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
891 |
notation (HTML output) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
892 |
Times (infixr "\<times>" 80) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
893 |
|
45662
4f7c05990420
Hide Product_Type.Times - too precious an identifier
nipkow
parents:
45607
diff
changeset
|
894 |
hide_const (open) Times |
4f7c05990420
Hide Product_Type.Times - too precious an identifier
nipkow
parents:
45607
diff
changeset
|
895 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
896 |
syntax |
35115 | 897 |
"_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
898 |
translations |
35115 | 899 |
"SIGMA x:A. B" == "CONST Sigma A (%x. B)" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
900 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
901 |
lemma SigmaI [intro!]: "[| a:A; b:B(a) |] ==> (a,b) : Sigma A B" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
902 |
by (unfold Sigma_def) blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
903 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
904 |
lemma SigmaE [elim!]: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
905 |
"[| c: Sigma A B; |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
906 |
!!x y.[| x:A; y:B(x); c=(x,y) |] ==> P |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
907 |
|] ==> P" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
908 |
-- {* The general elimination rule. *} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
909 |
by (unfold Sigma_def) blast |
20588 | 910 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
911 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
912 |
Elimination of @{term "(a, b) : A \<times> B"} -- introduces no |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
913 |
eigenvariables. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
914 |
*} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
915 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
916 |
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
917 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
918 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
919 |
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
920 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
921 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
922 |
lemma SigmaE2: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
923 |
"[| (a, b) : Sigma A B; |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
924 |
[| a:A; b:B(a) |] ==> P |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
925 |
|] ==> P" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
926 |
by blast |
20588 | 927 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
928 |
lemma Sigma_cong: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
929 |
"\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk> |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
930 |
\<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
931 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
932 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
933 |
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
934 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
935 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
936 |
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
937 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
938 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
939 |
lemma Sigma_empty2 [simp]: "A <*> {} = {}" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
940 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
941 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
942 |
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
943 |
by auto |
21908 | 944 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
945 |
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
946 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
947 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
948 |
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
949 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
950 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
951 |
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
952 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
953 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
954 |
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
955 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
956 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
957 |
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
958 |
by (blast elim: equalityE) |
20588 | 959 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
960 |
lemma SetCompr_Sigma_eq: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
961 |
"Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
962 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
963 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
964 |
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
965 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
966 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
967 |
lemma UN_Times_distrib: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
968 |
"(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
969 |
-- {* Suggested by Pierre Chartier *} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
970 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
971 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
972 |
lemma split_paired_Ball_Sigma [simp,no_atp]: |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
973 |
"(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
974 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
975 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
976 |
lemma split_paired_Bex_Sigma [simp,no_atp]: |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
977 |
"(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
978 |
by blast |
21908 | 979 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
980 |
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
981 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
982 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
983 |
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
984 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
985 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
986 |
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
987 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
988 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
989 |
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
990 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
991 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
992 |
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
993 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
994 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
995 |
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
996 |
by blast |
21908 | 997 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
998 |
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
999 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1000 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1001 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1002 |
Non-dependent versions are needed to avoid the need for higher-order |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1003 |
matching, especially when the rules are re-oriented. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1004 |
*} |
21908 | 1005 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1006 |
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" |
28719 | 1007 |
by blast |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1008 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1009 |
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" |
28719 | 1010 |
by blast |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1011 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1012 |
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)" |
28719 | 1013 |
by blast |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1014 |
|
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1015 |
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1016 |
by auto |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1017 |
|
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1018 |
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)" |
44921 | 1019 |
by force |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1020 |
|
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1021 |
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)" |
44921 | 1022 |
by force |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1023 |
|
28719 | 1024 |
lemma insert_times_insert[simp]: |
1025 |
"insert a A \<times> insert b B = |
|
1026 |
insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)" |
|
1027 |
by blast |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1028 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset
|
1029 |
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)" |
37166 | 1030 |
by (auto, case_tac "f x", auto) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset
|
1031 |
|
35822 | 1032 |
lemma swap_inj_on: |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1033 |
"inj_on (\<lambda>(i, j). (j, i)) A" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1034 |
by (auto intro!: inj_onI) |
35822 | 1035 |
|
1036 |
lemma swap_product: |
|
1037 |
"(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A" |
|
1038 |
by (simp add: split_def image_def) blast |
|
1039 |
||
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1040 |
lemma image_split_eq_Sigma: |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1041 |
"(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))" |
46128
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1042 |
proof (safe intro!: imageI) |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1043 |
fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1044 |
show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1045 |
using * eq[symmetric] by auto |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1046 |
qed simp_all |
35822 | 1047 |
|
46128
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1048 |
definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1049 |
[code_abbrev]: "product A B = A \<times> B" |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1050 |
|
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1051 |
hide_const (open) product |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1052 |
|
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1053 |
lemma member_product: |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1054 |
"x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B" |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1055 |
by (simp add: product_def) |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1056 |
|
40607 | 1057 |
text {* The following @{const map_pair} lemmas are due to Joachim Breitner: *} |
1058 |
||
1059 |
lemma map_pair_inj_on: |
|
1060 |
assumes "inj_on f A" and "inj_on g B" |
|
1061 |
shows "inj_on (map_pair f g) (A \<times> B)" |
|
1062 |
proof (rule inj_onI) |
|
1063 |
fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c" |
|
1064 |
assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto |
|
1065 |
assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto |
|
1066 |
assume "map_pair f g x = map_pair f g y" |
|
1067 |
hence "fst (map_pair f g x) = fst (map_pair f g y)" by (auto) |
|
1068 |
hence "f (fst x) = f (fst y)" by (cases x,cases y,auto) |
|
1069 |
with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A` |
|
1070 |
have "fst x = fst y" by (auto dest:dest:inj_onD) |
|
1071 |
moreover from `map_pair f g x = map_pair f g y` |
|
1072 |
have "snd (map_pair f g x) = snd (map_pair f g y)" by (auto) |
|
1073 |
hence "g (snd x) = g (snd y)" by (cases x,cases y,auto) |
|
1074 |
with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B` |
|
1075 |
have "snd x = snd y" by (auto dest:dest:inj_onD) |
|
1076 |
ultimately show "x = y" by(rule prod_eqI) |
|
1077 |
qed |
|
1078 |
||
1079 |
lemma map_pair_surj: |
|
40702 | 1080 |
fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd" |
40607 | 1081 |
assumes "surj f" and "surj g" |
1082 |
shows "surj (map_pair f g)" |
|
1083 |
unfolding surj_def |
|
1084 |
proof |
|
1085 |
fix y :: "'b \<times> 'd" |
|
1086 |
from `surj f` obtain a where "fst y = f a" by (auto elim:surjE) |
|
1087 |
moreover |
|
1088 |
from `surj g` obtain b where "snd y = g b" by (auto elim:surjE) |
|
1089 |
ultimately have "(fst y, snd y) = map_pair f g (a,b)" by auto |
|
1090 |
thus "\<exists>x. y = map_pair f g x" by auto |
|
1091 |
qed |
|
1092 |
||
1093 |
lemma map_pair_surj_on: |
|
1094 |
assumes "f ` A = A'" and "g ` B = B'" |
|
1095 |
shows "map_pair f g ` (A \<times> B) = A' \<times> B'" |
|
1096 |
unfolding image_def |
|
1097 |
proof(rule set_eqI,rule iffI) |
|
1098 |
fix x :: "'a \<times> 'c" |
|
1099 |
assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = map_pair f g x}" |
|
1100 |
then obtain y where "y \<in> A \<times> B" and "x = map_pair f g y" by blast |
|
1101 |
from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto |
|
1102 |
moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto |
|
1103 |
ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto |
|
1104 |
with `x = map_pair f g y` show "x \<in> A' \<times> B'" by (cases y, auto) |
|
1105 |
next |
|
1106 |
fix x :: "'a \<times> 'c" |
|
1107 |
assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto |
|
1108 |
from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto |
|
1109 |
then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE) |
|
1110 |
moreover from `image g B = B'` and `snd x \<in> B'` |
|
1111 |
obtain b where "b \<in> B" and "snd x = g b" by auto |
|
1112 |
ultimately have "(fst x, snd x) = map_pair f g (a,b)" by auto |
|
1113 |
moreover from `a \<in> A` and `b \<in> B` have "(a , b) \<in> A \<times> B" by auto |
|
1114 |
ultimately have "\<exists>y \<in> A \<times> B. x = map_pair f g y" by auto |
|
1115 |
thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_pair f g y}" by auto |
|
1116 |
qed |
|
1117 |
||
21908 | 1118 |
|
37166 | 1119 |
subsection {* Inductively defined sets *} |
15394 | 1120 |
|
31723
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31667
diff
changeset
|
1121 |
use "Tools/inductive_set.ML" |
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31667
diff
changeset
|
1122 |
setup Inductive_Set.setup |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
1123 |
|
37166 | 1124 |
|
1125 |
subsection {* Legacy theorem bindings and duplicates *} |
|
1126 |
||
1127 |
lemma PairE: |
|
1128 |
obtains x y where "p = (x, y)" |
|
1129 |
by (fact prod.exhaust) |
|
1130 |
||
1131 |
lemma Pair_inject: |
|
1132 |
assumes "(a, b) = (a', b')" |
|
1133 |
and "a = a' ==> b = b' ==> R" |
|
1134 |
shows R |
|
1135 |
using assms by simp |
|
1136 |
||
1137 |
lemmas Pair_eq = prod.inject |
|
1138 |
||
1139 |
lemmas split = split_conv -- {* for backwards compatibility *} |
|
1140 |
||
44066
d74182c93f04
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents:
43866
diff
changeset
|
1141 |
lemmas Pair_fst_snd_eq = prod_eq_iff |
d74182c93f04
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents:
43866
diff
changeset
|
1142 |
|
45204
5e4a1270c000
hide typedef-generated constants Product_Type.prod and Sum_Type.sum
huffman
parents:
44921
diff
changeset
|
1143 |
hide_const (open) prod |
5e4a1270c000
hide typedef-generated constants Product_Type.prod and Sum_Type.sum
huffman
parents:
44921
diff
changeset
|
1144 |
|
10213 | 1145 |
end |