author | paulson <lp15@cam.ac.uk> |
Thu, 21 Mar 2019 14:18:22 +0000 | |
changeset 69939 | 812ce526da33 |
parent 69922 | 4a9167f377b0 |
child 69994 | cf7150ab1075 |
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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section \<open>Cartesian products\<close> |
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theory Product_Type |
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imports Typedef Inductive Fun |
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keywords "inductive_set" "coinductive_set" :: thy_defn |
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begin |
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subsection \<open>\<^typ>\<open>bool\<close> is a datatype\<close> |
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free_constructors (discs_sels) case_bool for True | False |
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by auto |
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text \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close> |
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setup \<open>Sign.mandatory_path "old"\<close> |
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old_rep_datatype True False by (auto intro: bool_induct) |
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setup \<open>Sign.parent_path\<close> |
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text \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close> |
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setup \<open>Sign.mandatory_path "bool"\<close> |
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lemmas induct = old.bool.induct |
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lemmas inducts = old.bool.inducts |
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lemmas rec = old.bool.rec |
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lemmas simps = bool.distinct bool.case bool.rec |
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setup \<open>Sign.parent_path\<close> |
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declare case_split [cases type: bool] |
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\<comment> \<open>prefer plain propositional version\<close> |
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lemma [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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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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setup \<open>Code.declare_case_global @{thm If_case_cert}\<close> |
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code_printing |
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constant "HOL.equal :: bool \<Rightarrow> bool \<Rightarrow> bool" \<rightharpoonup> (Haskell) infix 4 "==" |
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| class_instance "bool" :: "equal" \<rightharpoonup> (Haskell) - |
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subsection \<open>The \<open>unit\<close> type\<close> |
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typedef 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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lemma unit_eq [no_atp]: "u = ()" |
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by (induct u) (simp add: Unity_def) |
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text \<open> |
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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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\<close> |
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simproc_setup unit_eq ("x::unit") = \<open> |
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fn _ => fn _ => fn ct => |
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if HOLogic.is_unit (Thm.term_of ct) then NONE |
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else SOME (mk_meta_eq @{thm unit_eq}) |
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\<close> |
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free_constructors case_unit for "()" |
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by auto |
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text \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close> |
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setup \<open>Sign.mandatory_path "old"\<close> |
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old_rep_datatype "()" by simp |
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setup \<open>Sign.parent_path\<close> |
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text \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close> |
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setup \<open>Sign.mandatory_path "unit"\<close> |
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lemmas induct = old.unit.induct |
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lemmas inducts = old.unit.inducts |
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lemmas rec = old.unit.rec |
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lemmas simps = unit.case unit.rec |
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setup \<open>Sign.parent_path\<close> |
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lemma unit_all_eq1: "(\<And>x::unit. PROP P x) \<equiv> PROP P ()" |
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by simp |
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lemma unit_all_eq2: "(\<And>x::unit. PROP P) \<equiv> PROP P" |
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by (rule triv_forall_equality) |
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text \<open> |
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This rewrite counters the effect of simproc \<open>unit_eq\<close> on @{term |
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[source] "\<lambda>u::unit. f u"}, replacing it by @{term [source] |
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f} rather than by @{term [source] "\<lambda>u. f ()"}. |
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\<close> |
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lemma unit_abs_eta_conv [simp]: "(\<lambda>u::unit. f ()) = f" |
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by (rule ext) simp |
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lemma UNIV_unit: "UNIV = {()}" |
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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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instantiation unit :: "{complete_boolean_algebra,complete_linorder,wellorder}" |
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begin |
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definition less_eq_unit :: "unit \<Rightarrow> unit \<Rightarrow> bool" |
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where "(_::unit) \<le> _ \<longleftrightarrow> True" |
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lemma less_eq_unit [iff]: "u \<le> v" for u v :: unit |
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by (simp add: less_eq_unit_def) |
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definition less_unit :: "unit \<Rightarrow> unit \<Rightarrow> bool" |
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where "(_::unit) < _ \<longleftrightarrow> False" |
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lemma less_unit [iff]: "\<not> u < v" for u v :: unit |
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by (simp_all add: less_eq_unit_def less_unit_def) |
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definition bot_unit :: unit |
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where [code_unfold]: "\<bottom> = ()" |
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definition top_unit :: unit |
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where [code_unfold]: "\<top> = ()" |
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definition inf_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit" |
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where [simp]: "_ \<sqinter> _ = ()" |
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definition sup_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit" |
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where [simp]: "_ \<squnion> _ = ()" |
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definition Inf_unit :: "unit set \<Rightarrow> unit" |
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where [simp]: "\<Sqinter>_ = ()" |
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definition Sup_unit :: "unit set \<Rightarrow> unit" |
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where [simp]: "\<Squnion>_ = ()" |
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definition uminus_unit :: "unit \<Rightarrow> unit" |
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where [simp]: "- _ = ()" |
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declare less_eq_unit_def [abs_def, code_unfold] |
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less_unit_def [abs_def, code_unfold] |
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inf_unit_def [abs_def, code_unfold] |
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sup_unit_def [abs_def, code_unfold] |
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Inf_unit_def [abs_def, code_unfold] |
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Sup_unit_def [abs_def, code_unfold] |
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uminus_unit_def [abs_def, code_unfold] |
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instance |
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by intro_classes auto |
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end |
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lemma [code]: "HOL.equal u v \<longleftrightarrow> True" for u v :: unit |
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unfolding equal unit_eq [of u] unit_eq [of v] by rule+ |
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code_printing |
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type_constructor unit \<rightharpoonup> |
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(SML) "unit" |
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and (OCaml) "unit" |
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and (Haskell) "()" |
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and (Scala) "Unit" |
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188 |
| constant Unity \<rightharpoonup> |
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|
189 |
(SML) "()" |
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|
190 |
and (OCaml) "()" |
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|
191 |
and (Haskell) "()" |
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|
192 |
and (Scala) "()" |
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|
193 |
| class_instance unit :: equal \<rightharpoonup> |
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|
194 |
(Haskell) - |
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|
195 |
| constant "HOL.equal :: unit \<Rightarrow> unit \<Rightarrow> bool" \<rightharpoonup> |
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|
196 |
(Haskell) infix 4 "==" |
26358
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Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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|
197 |
|
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|
198 |
code_reserved SML |
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Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
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|
199 |
unit |
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|
200 |
|
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|
201 |
code_reserved OCaml |
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|
202 |
unit |
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|
203 |
|
34886 | 204 |
code_reserved Scala |
205 |
Unit |
|
206 |
||
26358
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|
207 |
|
60758 | 208 |
subsection \<open>The product type\<close> |
10213 | 209 |
|
60758 | 210 |
subsubsection \<open>Type definition\<close> |
37166 | 211 |
|
63400 | 212 |
definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" |
213 |
where "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)" |
|
10213 | 214 |
|
61076 | 215 |
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}" |
45696 | 216 |
|
61955
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|
217 |
typedef ('a, 'b) prod ("(_ \<times>/ _)" [21, 20] 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set" |
45696 | 218 |
unfolding prod_def by auto |
10213 | 219 |
|
61955
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|
220 |
type_notation (ASCII) |
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|
221 |
prod (infixr "*" 20) |
10213 | 222 |
|
63400 | 223 |
definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" |
224 |
where "Pair a b = Abs_prod (Pair_Rep a b)" |
|
37166 | 225 |
|
55393
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|
226 |
lemma prod_cases: "(\<And>a b. P (Pair a b)) \<Longrightarrow> P p" |
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|
227 |
by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def) |
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|
228 |
|
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|
229 |
free_constructors case_prod for Pair fst snd |
55393
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|
230 |
proof - |
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|
231 |
fix P :: bool and p :: "'a \<times> 'b" |
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|
232 |
show "(\<And>x1 x2. p = Pair x1 x2 \<Longrightarrow> P) \<Longrightarrow> P" |
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|
233 |
by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def) |
37166 | 234 |
next |
235 |
fix a c :: 'a and b d :: 'b |
|
236 |
have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
237 |
by (auto simp add: Pair_Rep_def fun_eq_iff) |
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
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parents:
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diff
changeset
|
238 |
moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod" |
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
haftmann
parents:
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diff
changeset
|
239 |
by (auto simp add: prod_def) |
37166 | 240 |
ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d" |
37389
09467cdfa198
qualified type "*"; qualified constants Pair, fst, snd, split
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parents:
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diff
changeset
|
241 |
by (simp add: Pair_def Abs_prod_inject) |
37166 | 242 |
qed |
243 |
||
61799 | 244 |
text \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close> |
55442 | 245 |
|
60758 | 246 |
setup \<open>Sign.mandatory_path "old"\<close> |
55393
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|
247 |
|
58306
117ba6cbe414
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
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parents:
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diff
changeset
|
248 |
old_rep_datatype Pair |
63400 | 249 |
by (erule prod_cases) (rule prod.inject) |
55393
ce5cebfaedda
se 'wrap_free_constructors' to register 'sum' , 'prod', 'unit', 'bool' with their discriminators/selectors
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parents:
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changeset
|
250 |
|
60758 | 251 |
setup \<open>Sign.parent_path\<close> |
37704
c6161bee8486
adapt Nitpick to "prod_case" and "*" -> "sum" renaming;
blanchet
parents:
37678
diff
changeset
|
252 |
|
61799 | 253 |
text \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close> |
55442 | 254 |
|
60758 | 255 |
setup \<open>Sign.mandatory_path "prod"\<close> |
55393
ce5cebfaedda
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parents:
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changeset
|
256 |
|
61424
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changeset
|
257 |
declare old.prod.inject [iff del] |
55393
ce5cebfaedda
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parents:
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changeset
|
258 |
|
ce5cebfaedda
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parents:
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changeset
|
259 |
lemmas induct = old.prod.induct |
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parents:
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|
260 |
lemmas inducts = old.prod.inducts |
55642
63beb38e9258
adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
blanchet
parents:
55469
diff
changeset
|
261 |
lemmas rec = old.prod.rec |
63beb38e9258
adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
blanchet
parents:
55469
diff
changeset
|
262 |
lemmas simps = prod.inject prod.case prod.rec |
55393
ce5cebfaedda
se 'wrap_free_constructors' to register 'sum' , 'prod', 'unit', 'bool' with their discriminators/selectors
blanchet
parents:
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diff
changeset
|
263 |
|
60758 | 264 |
setup \<open>Sign.parent_path\<close> |
55393
ce5cebfaedda
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blanchet
parents:
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diff
changeset
|
265 |
|
ce5cebfaedda
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blanchet
parents:
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diff
changeset
|
266 |
declare prod.case [nitpick_simp del] |
63566 | 267 |
declare old.prod.case_cong_weak [cong del] |
61424
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changeset
|
268 |
declare prod.case_eq_if [mono] |
c3658c18b7bc
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diff
changeset
|
269 |
declare prod.split [no_atp] |
c3658c18b7bc
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diff
changeset
|
270 |
declare prod.split_asm [no_atp] |
c3658c18b7bc
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diff
changeset
|
271 |
|
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changeset
|
272 |
text \<open> |
61799 | 273 |
@{thm [source] prod.split} could be declared as \<open>[split]\<close> |
61424
c3658c18b7bc
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haftmann
parents:
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diff
changeset
|
274 |
done after the Splitter has been speeded up significantly; |
c3658c18b7bc
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haftmann
parents:
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diff
changeset
|
275 |
precompute the constants involved and don't do anything unless the |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
276 |
current goal contains one of those constants. |
c3658c18b7bc
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diff
changeset
|
277 |
\<close> |
37411
c88c44156083
removed simplifier congruence rule of "prod_case"
haftmann
parents:
37389
diff
changeset
|
278 |
|
37166 | 279 |
|
60758 | 280 |
subsubsection \<open>Tuple syntax\<close> |
37166 | 281 |
|
60758 | 282 |
text \<open> |
69593 | 283 |
Patterns -- extends pre-defined type \<^typ>\<open>pttrn\<close> used in |
11777 | 284 |
abstractions. |
60758 | 285 |
\<close> |
10213 | 286 |
|
41229
d797baa3d57c
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents:
40968
diff
changeset
|
287 |
nonterminal tuple_args and patterns |
10213 | 288 |
syntax |
63400 | 289 |
"_tuple" :: "'a \<Rightarrow> tuple_args \<Rightarrow> 'a \<times> 'b" ("(1'(_,/ _'))") |
290 |
"_tuple_arg" :: "'a \<Rightarrow> tuple_args" ("_") |
|
291 |
"_tuple_args" :: "'a \<Rightarrow> tuple_args \<Rightarrow> tuple_args" ("_,/ _") |
|
292 |
"_pattern" :: "pttrn \<Rightarrow> patterns \<Rightarrow> pttrn" ("'(_,/ _')") |
|
293 |
"" :: "pttrn \<Rightarrow> patterns" ("_") |
|
294 |
"_patterns" :: "pttrn \<Rightarrow> patterns \<Rightarrow> patterns" ("_,/ _") |
|
63237 | 295 |
"_unit" :: pttrn ("'(')") |
10213 | 296 |
translations |
61124 | 297 |
"(x, y)" \<rightleftharpoons> "CONST Pair x y" |
298 |
"_pattern x y" \<rightleftharpoons> "CONST Pair x y" |
|
299 |
"_patterns x y" \<rightleftharpoons> "CONST Pair x y" |
|
300 |
"_tuple x (_tuple_args y z)" \<rightleftharpoons> "_tuple x (_tuple_arg (_tuple y z))" |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
301 |
"\<lambda>(x, y, zs). b" \<rightleftharpoons> "CONST case_prod (\<lambda>x (y, zs). b)" |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
302 |
"\<lambda>(x, y). b" \<rightleftharpoons> "CONST case_prod (\<lambda>x y. b)" |
61124 | 303 |
"_abs (CONST Pair x y) t" \<rightharpoonup> "\<lambda>(x, y). t" |
61799 | 304 |
\<comment> \<open>This rule accommodates tuples in \<open>case C \<dots> (x, y) \<dots> \<Rightarrow> \<dots>\<close>: |
305 |
The \<open>(x, y)\<close> is parsed as \<open>Pair x y\<close> because it is \<open>logic\<close>, |
|
306 |
not \<open>pttrn\<close>.\<close> |
|
63237 | 307 |
"\<lambda>(). b" \<rightleftharpoons> "CONST case_unit b" |
308 |
"_abs (CONST Unity) t" \<rightharpoonup> "\<lambda>(). t" |
|
10213 | 309 |
|
69593 | 310 |
text \<open>print \<^term>\<open>case_prod f\<close> as \<^term>\<open>\<lambda>(x, y). f x y\<close> and |
311 |
\<^term>\<open>case_prod (\<lambda>x. f x)\<close> as \<^term>\<open>\<lambda>(x, y). f x y\<close>\<close> |
|
61226
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
312 |
|
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
313 |
typed_print_translation \<open> |
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
314 |
let |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
315 |
fun case_prod_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
316 |
| case_prod_guess_names_tr' T [Abs (x, xT, t)] = |
61226
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
317 |
(case (head_of t) of |
69593 | 318 |
Const (\<^const_syntax>\<open>case_prod\<close>, _) => raise Match |
61226
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
319 |
| _ => |
63575 | 320 |
let |
61226
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
321 |
val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; |
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
322 |
val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0); |
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
323 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t'); |
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
324 |
in |
69593 | 325 |
Syntax.const \<^syntax_const>\<open>_abs\<close> $ |
326 |
(Syntax.const \<^syntax_const>\<open>_pattern\<close> $ x' $ y) $ t'' |
|
61226
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
327 |
end) |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
328 |
| case_prod_guess_names_tr' T [t] = |
61226
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
329 |
(case head_of t of |
69593 | 330 |
Const (\<^const_syntax>\<open>case_prod\<close>, _) => raise Match |
61226
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
331 |
| _ => |
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
332 |
let |
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
333 |
val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; |
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
334 |
val (y, t') = |
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
335 |
Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0); |
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
336 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t'); |
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
337 |
in |
69593 | 338 |
Syntax.const \<^syntax_const>\<open>_abs\<close> $ |
339 |
(Syntax.const \<^syntax_const>\<open>_pattern\<close> $ x' $ y) $ t'' |
|
61226
af7bed1360f3
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
340 |
end) |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
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|
341 |
| case_prod_guess_names_tr' _ _ = raise Match; |
69593 | 342 |
in [(\<^const_syntax>\<open>case_prod\<close>, K case_prod_guess_names_tr')] end |
61226
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effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
haftmann
parents:
61144
diff
changeset
|
343 |
\<close> |
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parents:
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diff
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|
344 |
|
69593 | 345 |
text \<open>Reconstruct pattern from (nested) \<^const>\<open>case_prod\<close>s, |
61425
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parents:
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diff
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|
346 |
avoiding eta-contraction of body; required for enclosing "let", |
63400 | 347 |
if "let" does not avoid eta-contraction, which has been observed to occur.\<close> |
61425
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parents:
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|
348 |
|
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|
349 |
print_translation \<open> |
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|
350 |
let |
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parents:
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|
351 |
fun case_prod_tr' [Abs (x, T, t as (Abs abs))] = |
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parents:
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|
352 |
(* case_prod (\<lambda>x y. t) \<Rightarrow> \<lambda>(x, y) t *) |
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parents:
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|
353 |
let |
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parents:
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|
354 |
val (y, t') = Syntax_Trans.atomic_abs_tr' abs; |
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parents:
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changeset
|
355 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); |
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parents:
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diff
changeset
|
356 |
in |
69593 | 357 |
Syntax.const \<^syntax_const>\<open>_abs\<close> $ |
358 |
(Syntax.const \<^syntax_const>\<open>_pattern\<close> $ x' $ y) $ t'' |
|
61425
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parents:
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changeset
|
359 |
end |
69593 | 360 |
| case_prod_tr' [Abs (x, T, (s as Const (\<^const_syntax>\<open>case_prod\<close>, _) $ t))] = |
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|
361 |
(* case_prod (\<lambda>x. (case_prod (\<lambda>y z. t))) \<Rightarrow> \<lambda>(x, y, z). t *) |
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parents:
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|
362 |
let |
69593 | 363 |
val Const (\<^syntax_const>\<open>_abs\<close>, _) $ |
364 |
(Const (\<^syntax_const>\<open>_pattern\<close>, _) $ y $ z) $ t' = |
|
61425
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parents:
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changeset
|
365 |
case_prod_tr' [t]; |
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parents:
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changeset
|
366 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); |
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parents:
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diff
changeset
|
367 |
in |
69593 | 368 |
Syntax.const \<^syntax_const>\<open>_abs\<close> $ |
369 |
(Syntax.const \<^syntax_const>\<open>_pattern\<close> $ x' $ |
|
370 |
(Syntax.const \<^syntax_const>\<open>_patterns\<close> $ y $ z)) $ t'' |
|
61425
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parents:
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changeset
|
371 |
end |
69593 | 372 |
| case_prod_tr' [Const (\<^const_syntax>\<open>case_prod\<close>, _) $ t] = |
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parents:
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changeset
|
373 |
(* case_prod (case_prod (\<lambda>x y z. t)) \<Rightarrow> \<lambda>((x, y), z). t *) |
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|
374 |
case_prod_tr' [(case_prod_tr' [t])] |
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parents:
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changeset
|
375 |
(* inner case_prod_tr' creates next pattern *) |
69593 | 376 |
| case_prod_tr' [Const (\<^syntax_const>\<open>_abs\<close>, _) $ x_y $ Abs abs] = |
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parents:
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changeset
|
377 |
(* case_prod (\<lambda>pttrn z. t) \<Rightarrow> \<lambda>(pttrn, z). t *) |
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parents:
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changeset
|
378 |
let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in |
69593 | 379 |
Syntax.const \<^syntax_const>\<open>_abs\<close> $ |
380 |
(Syntax.const \<^syntax_const>\<open>_pattern\<close> $ x_y $ z) $ t |
|
61425
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parents:
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diff
changeset
|
381 |
end |
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parents:
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diff
changeset
|
382 |
| case_prod_tr' _ = raise Match; |
69593 | 383 |
in [(\<^const_syntax>\<open>case_prod\<close>, K case_prod_tr')] end |
61425
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parents:
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changeset
|
384 |
\<close> |
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parents:
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changeset
|
385 |
|
10213 | 386 |
|
60758 | 387 |
subsubsection \<open>Code generator setup\<close> |
37166 | 388 |
|
52435
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haftmann
parents:
52143
diff
changeset
|
389 |
code_printing |
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haftmann
parents:
52143
diff
changeset
|
390 |
type_constructor prod \<rightharpoonup> |
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haftmann
parents:
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diff
changeset
|
391 |
(SML) infix 2 "*" |
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migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
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diff
changeset
|
392 |
and (OCaml) infix 2 "*" |
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migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
52143
diff
changeset
|
393 |
and (Haskell) "!((_),/ (_))" |
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migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
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parents:
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diff
changeset
|
394 |
and (Scala) "((_),/ (_))" |
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haftmann
parents:
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diff
changeset
|
395 |
| constant Pair \<rightharpoonup> |
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haftmann
parents:
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diff
changeset
|
396 |
(SML) "!((_),/ (_))" |
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parents:
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diff
changeset
|
397 |
and (OCaml) "!((_),/ (_))" |
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migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
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diff
changeset
|
398 |
and (Haskell) "!((_),/ (_))" |
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haftmann
parents:
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diff
changeset
|
399 |
and (Scala) "!((_),/ (_))" |
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haftmann
parents:
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diff
changeset
|
400 |
| class_instance prod :: equal \<rightharpoonup> |
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haftmann
parents:
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diff
changeset
|
401 |
(Haskell) - |
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migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
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diff
changeset
|
402 |
| constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup> |
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haftmann
parents:
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diff
changeset
|
403 |
(Haskell) infix 4 "==" |
61424
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haftmann
parents:
61422
diff
changeset
|
404 |
| constant fst \<rightharpoonup> (Haskell) "fst" |
c3658c18b7bc
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haftmann
parents:
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diff
changeset
|
405 |
| constant snd \<rightharpoonup> (Haskell) "snd" |
37166 | 406 |
|
407 |
||
60758 | 408 |
subsubsection \<open>Fundamental operations and properties\<close> |
11838 | 409 |
|
63400 | 410 |
lemma Pair_inject: "(a, b) = (a', b') \<Longrightarrow> (a = a' \<Longrightarrow> b = b' \<Longrightarrow> R) \<Longrightarrow> R" |
411 |
by simp |
|
49897
cc69be3c8f87
moving Pair_inject from legacy and duplicate section to general section, as Pair_inject was considered a duplicate in e8400e31528a by mistake (cf. communication on dev mailing list)
bulwahn
parents:
49834
diff
changeset
|
412 |
|
63400 | 413 |
lemma surj_pair [simp]: "\<exists>x y. p = (x, y)" |
37166 | 414 |
by (cases p) simp |
10213 | 415 |
|
63400 | 416 |
lemma fst_eqD: "fst (x, y) = a \<Longrightarrow> x = a" |
11838 | 417 |
by simp |
418 |
||
63400 | 419 |
lemma snd_eqD: "snd (x, y) = a \<Longrightarrow> y = a" |
11838 | 420 |
by simp |
421 |
||
61424
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haftmann
parents:
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diff
changeset
|
422 |
lemma case_prod_unfold [nitpick_unfold]: "case_prod = (\<lambda>c p. c (fst p) (snd p))" |
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parents:
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diff
changeset
|
423 |
by (simp add: fun_eq_iff split: prod.split) |
c3658c18b7bc
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parents:
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diff
changeset
|
424 |
|
c3658c18b7bc
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haftmann
parents:
61422
diff
changeset
|
425 |
lemma case_prod_conv [simp, code]: "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b" |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
426 |
by (fact prod.case) |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
427 |
|
55393
ce5cebfaedda
se 'wrap_free_constructors' to register 'sum' , 'prod', 'unit', 'bool' with their discriminators/selectors
blanchet
parents:
54630
diff
changeset
|
428 |
lemmas surjective_pairing = prod.collapse [symmetric] |
11838 | 429 |
|
44066
d74182c93f04
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents:
43866
diff
changeset
|
430 |
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t" |
37166 | 431 |
by (cases s, cases t) simp |
432 |
||
433 |
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
|
434 |
by (simp add: prod_eq_iff) |
37166 | 435 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
436 |
lemma case_prodI: "f a b \<Longrightarrow> case (a, b) of (c, d) \<Rightarrow> f c d" |
c3658c18b7bc
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haftmann
parents:
61422
diff
changeset
|
437 |
by (rule prod.case [THEN iffD2]) |
37166 | 438 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
439 |
lemma case_prodD: "(case (a, b) of (c, d) \<Rightarrow> f c d) \<Longrightarrow> f a b" |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
440 |
by (rule prod.case [THEN iffD1]) |
37166 | 441 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
442 |
lemma case_prod_Pair [simp]: "case_prod Pair = 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
|
443 |
by (simp add: fun_eq_iff split: prod.split) |
37166 | 444 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
445 |
lemma case_prod_eta: "(\<lambda>(x, y). f (x, y)) = f" |
69593 | 446 |
\<comment> \<open>Subsumes the old \<open>split_Pair\<close> when \<^term>\<open>f\<close> is the identity function.\<close> |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
447 |
by (simp add: fun_eq_iff split: prod.split) |
37166 | 448 |
|
67575 | 449 |
(* This looks like a sensible simp-rule but appears to do more harm than good: |
450 |
lemma case_prod_const [simp]: "(\<lambda>(_,_). c) = (\<lambda>_. c)" |
|
451 |
by(rule case_prod_eta) |
|
452 |
*) |
|
453 |
||
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
454 |
lemma case_prod_comp: "(case x of (a, b) \<Rightarrow> (f \<circ> g) a b) = f (g (fst x)) (snd x)" |
37166 | 455 |
by (cases x) simp |
456 |
||
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
457 |
lemma The_case_prod: "The (case_prod P) = (THE xy. P (fst xy) (snd xy))" |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
458 |
by (simp add: case_prod_unfold) |
37166 | 459 |
|
63400 | 460 |
lemma cond_case_prod_eta: "(\<And>x y. f x y = g (x, y)) \<Longrightarrow> (\<lambda>(x, y). f x y) = g" |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
461 |
by (simp add: case_prod_eta) |
37166 | 462 |
|
63400 | 463 |
lemma split_paired_all [no_atp]: "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))" |
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
464 |
proof |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
465 |
fix a b |
63400 | 466 |
assume "\<And>x. PROP P x" |
19535 | 467 |
then show "PROP P (a, b)" . |
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
468 |
next |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
469 |
fix x |
63400 | 470 |
assume "\<And>a b. PROP P (a, b)" |
60758 | 471 |
from \<open>PROP P (fst x, snd x)\<close> show "PROP P x" by simp |
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
472 |
qed |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
473 |
|
60758 | 474 |
text \<open> |
11838 | 475 |
The rule @{thm [source] split_paired_all} does not work with the |
476 |
Simplifier because it also affects premises in congrence rules, |
|
63400 | 477 |
where this can lead to premises of the form \<open>\<And>a b. \<dots> = ?P(a, b)\<close> |
478 |
which cannot be solved by reflexivity. |
|
60758 | 479 |
\<close> |
11838 | 480 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
481 |
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
|
482 |
|
60758 | 483 |
ML \<open> |
11838 | 484 |
(* replace parameters of product type by individual component parameters *) |
485 |
local (* filtering with exists_paired_all is an essential optimization *) |
|
69593 | 486 |
fun exists_paired_all (Const (\<^const_name>\<open>Pure.all\<close>, _) $ Abs (_, T, t)) = |
11838 | 487 |
can HOLogic.dest_prodT T orelse exists_paired_all t |
488 |
| exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u |
|
489 |
| exists_paired_all (Abs (_, _, t)) = exists_paired_all t |
|
490 |
| exists_paired_all _ = false; |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
491 |
val ss = |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
492 |
simpset_of |
69593 | 493 |
(put_simpset HOL_basic_ss \<^context> |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
494 |
addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] |
69593 | 495 |
addsimprocs [\<^simproc>\<open>unit_eq\<close>]); |
11838 | 496 |
in |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
497 |
fun split_all_tac ctxt = SUBGOAL (fn (t, i) => |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
498 |
if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac); |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
499 |
|
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
500 |
fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) => |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
501 |
if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac); |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
502 |
|
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
503 |
fun split_all ctxt th = |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
504 |
if exists_paired_all (Thm.prop_of th) |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
505 |
then full_simplify (put_simpset ss ctxt) th else th; |
11838 | 506 |
end; |
60758 | 507 |
\<close> |
11838 | 508 |
|
60758 | 509 |
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac))\<close> |
11838 | 510 |
|
63400 | 511 |
lemma split_paired_All [simp, no_atp]: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>a b. P (a, b))" |
61799 | 512 |
\<comment> \<open>\<open>[iff]\<close> is not a good idea because it makes \<open>blast\<close> loop\<close> |
11838 | 513 |
by fast |
514 |
||
63400 | 515 |
lemma split_paired_Ex [simp, no_atp]: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>a b. P (a, b))" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
516 |
by fast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
517 |
|
47740
a8989fe9a3a5
added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents:
46950
diff
changeset
|
518 |
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))" |
61799 | 519 |
\<comment> \<open>Can't be added to simpset: loops!\<close> |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
520 |
by (simp add: case_prod_eta) |
11838 | 521 |
|
60758 | 522 |
text \<open> |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
523 |
Simplification procedure for @{thm [source] cond_case_prod_eta}. Using |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
524 |
@{thm [source] case_prod_eta} as a rewrite rule is not general enough, |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
525 |
and using @{thm [source] cond_case_prod_eta} directly would render some |
61799 | 526 |
existing proofs very inefficient; similarly for \<open>prod.case_eq_if\<close>. |
60758 | 527 |
\<close> |
11838 | 528 |
|
60758 | 529 |
ML \<open> |
11838 | 530 |
local |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
531 |
val cond_case_prod_eta_ss = |
69593 | 532 |
simpset_of (put_simpset HOL_basic_ss \<^context> addsimps @{thms cond_case_prod_eta}); |
35364 | 533 |
fun Pair_pat k 0 (Bound m) = (m = k) |
69593 | 534 |
| Pair_pat k i (Const (\<^const_name>\<open>Pair\<close>, _) $ Bound m $ t) = |
35364 | 535 |
i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t |
536 |
| Pair_pat _ _ _ = false; |
|
537 |
fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t |
|
538 |
| no_args k i (t $ u) = no_args k i t andalso no_args k i u |
|
539 |
| no_args k i (Bound m) = m < k orelse m > k + i |
|
540 |
| no_args _ _ _ = true; |
|
541 |
fun split_pat tp i (Abs (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE |
|
69593 | 542 |
| split_pat tp i (Const (\<^const_name>\<open>case_prod\<close>, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t |
35364 | 543 |
| split_pat tp i _ = NONE; |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
544 |
fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] [] |
35364 | 545 |
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))) |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
546 |
(K (simp_tac (put_simpset cond_case_prod_eta_ss ctxt) 1))); |
11838 | 547 |
|
35364 | 548 |
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t |
549 |
| beta_term_pat k i (t $ u) = |
|
550 |
Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u) |
|
551 |
| beta_term_pat k i t = no_args k i t; |
|
552 |
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg |
|
553 |
| eta_term_pat _ _ _ = false; |
|
11838 | 554 |
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) |
35364 | 555 |
| subst arg k i (t $ u) = |
556 |
if Pair_pat k i (t $ u) then incr_boundvars k arg |
|
557 |
else (subst arg k i t $ subst arg k i u) |
|
558 |
| subst arg k i t = t; |
|
43595 | 559 |
in |
69593 | 560 |
fun beta_proc ctxt (s as Const (\<^const_name>\<open>case_prod\<close>, _) $ Abs (_, _, t) $ arg) = |
11838 | 561 |
(case split_pat beta_term_pat 1 t of |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
562 |
SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f)) |
15531 | 563 |
| NONE => NONE) |
35364 | 564 |
| beta_proc _ _ = NONE; |
69593 | 565 |
fun eta_proc ctxt (s as Const (\<^const_name>\<open>case_prod\<close>, _) $ Abs (_, _, t)) = |
11838 | 566 |
(case split_pat eta_term_pat 1 t of |
58468 | 567 |
SOME (_, ft) => SOME (metaeq ctxt s (let val f $ _ = ft in f end)) |
15531 | 568 |
| NONE => NONE) |
35364 | 569 |
| eta_proc _ _ = NONE; |
11838 | 570 |
end; |
60758 | 571 |
\<close> |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
572 |
simproc_setup case_prod_beta ("case_prod f z") = |
60758 | 573 |
\<open>fn _ => fn ctxt => fn ct => beta_proc ctxt (Thm.term_of ct)\<close> |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
574 |
simproc_setup case_prod_eta ("case_prod f") = |
60758 | 575 |
\<open>fn _ => fn ctxt => fn ct => eta_proc ctxt (Thm.term_of ct)\<close> |
11838 | 576 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
577 |
lemma case_prod_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))" |
50104 | 578 |
by (auto simp: fun_eq_iff) |
579 |
||
60758 | 580 |
text \<open> |
69593 | 581 |
\<^medskip> \<^const>\<open>case_prod\<close> used as a logical connective or set former. |
11838 | 582 |
|
63400 | 583 |
\<^medskip> These rules are for use with \<open>blast\<close>; could instead |
61799 | 584 |
call \<open>simp\<close> using @{thm [source] prod.split} as rewrite.\<close> |
11838 | 585 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
586 |
lemma case_prodI2: |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
587 |
"\<And>p. (\<And>a b. p = (a, b) \<Longrightarrow> c a b) \<Longrightarrow> case p of (a, b) \<Rightarrow> c a b" |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
588 |
by (simp add: split_tupled_all) |
11838 | 589 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
590 |
lemma case_prodI2': |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
591 |
"\<And>p. (\<And>a b. (a, b) = p \<Longrightarrow> c a b x) \<Longrightarrow> (case p of (a, b) \<Rightarrow> c a b) x" |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
592 |
by (simp add: split_tupled_all) |
11838 | 593 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
594 |
lemma case_prodE [elim!]: |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
595 |
"(case p of (a, b) \<Rightarrow> c a b) \<Longrightarrow> (\<And>x y. p = (x, y) \<Longrightarrow> c x y \<Longrightarrow> Q) \<Longrightarrow> Q" |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
596 |
by (induct p) simp |
11838 | 597 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
598 |
lemma case_prodE' [elim!]: |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
599 |
"(case p of (a, b) \<Rightarrow> c a b) z \<Longrightarrow> (\<And>x y. p = (x, y) \<Longrightarrow> c x y z \<Longrightarrow> Q) \<Longrightarrow> Q" |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
600 |
by (induct p) simp |
11838 | 601 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
602 |
lemma case_prodE2: |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
603 |
assumes q: "Q (case z of (a, b) \<Rightarrow> P a b)" |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
604 |
and r: "\<And>x y. z = (x, y) \<Longrightarrow> Q (P x y) \<Longrightarrow> R" |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
605 |
shows R |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
606 |
proof (rule r) |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
607 |
show "z = (fst z, snd z)" by simp |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
608 |
then show "Q (P (fst z) (snd z))" |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
609 |
using q by (simp add: case_prod_unfold) |
11838 | 610 |
qed |
611 |
||
63400 | 612 |
lemma case_prodD': "(case (a, b) of (c, d) \<Rightarrow> R c d) c \<Longrightarrow> R a b c" |
11838 | 613 |
by simp |
614 |
||
63400 | 615 |
lemma mem_case_prodI: "z \<in> c a b \<Longrightarrow> z \<in> (case (a, b) of (d, e) \<Rightarrow> c d e)" |
11838 | 616 |
by simp |
617 |
||
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
618 |
lemma mem_case_prodI2 [intro!]: |
61127 | 619 |
"\<And>p. (\<And>a b. p = (a, b) \<Longrightarrow> z \<in> c a b) \<Longrightarrow> z \<in> (case p of (a, b) \<Rightarrow> c a b)" |
620 |
by (simp only: split_tupled_all) simp |
|
11838 | 621 |
|
61799 | 622 |
declare mem_case_prodI [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close> |
623 |
declare case_prodI2' [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close> |
|
624 |
declare case_prodI2 [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close> |
|
625 |
declare case_prodI [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close> |
|
63575 | 626 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
627 |
lemma mem_case_prodE [elim!]: |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
628 |
assumes "z \<in> case_prod c p" |
58468 | 629 |
obtains x y where "p = (x, y)" and "z \<in> c x y" |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
630 |
using assms by (rule case_prodE2) |
11838 | 631 |
|
60758 | 632 |
ML \<open> |
11838 | 633 |
local (* filtering with exists_p_split is an essential optimization *) |
69593 | 634 |
fun exists_p_split (Const (\<^const_name>\<open>case_prod\<close>,_) $ _ $ (Const (\<^const_name>\<open>Pair\<close>,_)$_$_)) = true |
11838 | 635 |
| exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u |
636 |
| exists_p_split (Abs (_, _, t)) = exists_p_split t |
|
637 |
| exists_p_split _ = false; |
|
638 |
in |
|
63575 | 639 |
fun split_conv_tac ctxt = SUBGOAL (fn (t, i) => |
640 |
if exists_p_split t |
|
641 |
then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms case_prod_conv}) i |
|
642 |
else no_tac); |
|
11838 | 643 |
end; |
60758 | 644 |
\<close> |
26340 | 645 |
|
11838 | 646 |
(* This prevents applications of splitE for already splitted arguments leading |
647 |
to quite time-consuming computations (in particular for nested tuples) *) |
|
60758 | 648 |
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac))\<close> |
11838 | 649 |
|
63400 | 650 |
lemma split_eta_SetCompr [simp, no_atp]: "(\<lambda>u. \<exists>x y. u = (x, y) \<and> P (x, y)) = P" |
18372 | 651 |
by (rule ext) fast |
11838 | 652 |
|
63400 | 653 |
lemma split_eta_SetCompr2 [simp, no_atp]: "(\<lambda>u. \<exists>x y. u = (x, y) \<and> P x y) = case_prod P" |
18372 | 654 |
by (rule ext) fast |
11838 | 655 |
|
63400 | 656 |
lemma split_part [simp]: "(\<lambda>(a,b). P \<and> Q a b) = (\<lambda>ab. P \<and> case_prod Q ab)" |
61799 | 657 |
\<comment> \<open>Allows simplifications of nested splits in case of independent predicates.\<close> |
18372 | 658 |
by (rule ext) blast |
11838 | 659 |
|
14337
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
660 |
(* 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
|
661 |
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
|
662 |
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
|
663 |
*) |
63575 | 664 |
lemma split_comp_eq: |
63400 | 665 |
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" |
666 |
and g :: "'d \<Rightarrow> 'a" |
|
667 |
shows "(\<lambda>u. f (g (fst u)) (snd u)) = case_prod (\<lambda>x. f (g x))" |
|
18372 | 668 |
by (rule ext) auto |
14101 | 669 |
|
63400 | 670 |
lemma pair_imageI [intro]: "(a, b) \<in> A \<Longrightarrow> f a b \<in> (\<lambda>(a, b). f a b) ` A" |
671 |
by (rule image_eqI [where x = "(a, b)"]) auto |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
672 |
|
68457 | 673 |
lemma Collect_const_case_prod[simp]: "{(a,b). P} = (if P then UNIV else {})" |
674 |
by auto |
|
675 |
||
63400 | 676 |
lemma The_split_eq [simp]: "(THE (x',y'). x = x' \<and> y = y') = (x, y)" |
11838 | 677 |
by blast |
678 |
||
679 |
(* |
|
680 |
the following would be slightly more general, |
|
681 |
but cannot be used as rewrite rule: |
|
682 |
### Cannot add premise as rewrite rule because it contains (type) unknowns: |
|
683 |
### ?y = .x |
|
684 |
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)" |
|
14208 | 685 |
by (rtac some_equality 1) |
686 |
by ( Simp_tac 1) |
|
687 |
by (split_all_tac 1) |
|
688 |
by (Asm_full_simp_tac 1) |
|
11838 | 689 |
qed "The_split_eq"; |
690 |
*) |
|
691 |
||
63400 | 692 |
lemma case_prod_beta: "case_prod f p = f (fst p) (snd p)" |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
693 |
by (fact prod.case_eq_if) |
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset
|
694 |
|
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset
|
695 |
lemma prod_cases3 [cases type]: |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
696 |
obtains (fields) a b c where "y = (a, b, c)" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
697 |
by (cases y, case_tac b) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
698 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
699 |
lemma prod_induct3 [case_names fields, induct type]: |
63400 | 700 |
"(\<And>a b c. P (a, b, c)) \<Longrightarrow> P x" |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
701 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
702 |
|
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset
|
703 |
lemma prod_cases4 [cases type]: |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
704 |
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
|
705 |
by (cases y, case_tac c) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
706 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
707 |
lemma prod_induct4 [case_names fields, induct type]: |
63400 | 708 |
"(\<And>a b c d. P (a, b, c, d)) \<Longrightarrow> P x" |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
709 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
710 |
|
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset
|
711 |
lemma prod_cases5 [cases type]: |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
712 |
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
|
713 |
by (cases y, case_tac d) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
714 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
715 |
lemma prod_induct5 [case_names fields, induct type]: |
63400 | 716 |
"(\<And>a b c d e. P (a, b, c, d, e)) \<Longrightarrow> P x" |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
717 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
718 |
|
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset
|
719 |
lemma prod_cases6 [cases type]: |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
720 |
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
|
721 |
by (cases y, case_tac e) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
722 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
723 |
lemma prod_induct6 [case_names fields, induct type]: |
63400 | 724 |
"(\<And>a b c d e f. P (a, b, c, d, e, f)) \<Longrightarrow> P x" |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
725 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
726 |
|
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset
|
727 |
lemma prod_cases7 [cases type]: |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
728 |
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
|
729 |
by (cases y, case_tac f) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
730 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
731 |
lemma prod_induct7 [case_names fields, induct type]: |
63400 | 732 |
"(\<And>a b c d e f g. P (a, b, c, d, e, f, g)) \<Longrightarrow> P x" |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
733 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
734 |
|
63400 | 735 |
definition internal_case_prod :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" |
736 |
where "internal_case_prod \<equiv> case_prod" |
|
37166 | 737 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
738 |
lemma internal_case_prod_conv: "internal_case_prod c (a, b) = c a b" |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
739 |
by (simp only: internal_case_prod_def case_prod_conv) |
37166 | 740 |
|
69605 | 741 |
ML_file \<open>Tools/split_rule.ML\<close> |
37166 | 742 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
743 |
hide_const internal_case_prod |
37166 | 744 |
|
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
745 |
|
60758 | 746 |
subsubsection \<open>Derived operations\<close> |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
747 |
|
63400 | 748 |
definition curry :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" |
749 |
where "curry = (\<lambda>c x y. c (x, y))" |
|
37166 | 750 |
|
751 |
lemma curry_conv [simp, code]: "curry f a b = f (a, b)" |
|
752 |
by (simp add: curry_def) |
|
753 |
||
754 |
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b" |
|
755 |
by (simp add: curry_def) |
|
756 |
||
757 |
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)" |
|
758 |
by (simp add: curry_def) |
|
759 |
||
760 |
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q" |
|
761 |
by (simp add: curry_def) |
|
762 |
||
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
763 |
lemma curry_case_prod [simp]: "curry (case_prod f) = f" |
61032
b57df8eecad6
standardized some occurences of ancient "split" alias
haftmann
parents:
60758
diff
changeset
|
764 |
by (simp add: curry_def case_prod_unfold) |
37166 | 765 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
766 |
lemma case_prod_curry [simp]: "case_prod (curry f) = f" |
61032
b57df8eecad6
standardized some occurences of ancient "split" alias
haftmann
parents:
60758
diff
changeset
|
767 |
by (simp add: curry_def case_prod_unfold) |
37166 | 768 |
|
54630
9061af4d5ebc
restrict admissibility to non-empty chains to allow more syntax-directed proof rules
Andreas Lochbihler
parents:
54147
diff
changeset
|
769 |
lemma curry_K: "curry (\<lambda>x. c) = (\<lambda>x y. c)" |
63400 | 770 |
by (simp add: fun_eq_iff) |
54630
9061af4d5ebc
restrict admissibility to non-empty chains to allow more syntax-directed proof rules
Andreas Lochbihler
parents:
54147
diff
changeset
|
771 |
|
63400 | 772 |
text \<open>The composition-uncurry combinator.\<close> |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
773 |
|
37751 | 774 |
notation fcomp (infixl "\<circ>>" 60) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
775 |
|
63400 | 776 |
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) |
777 |
where "f \<circ>\<rightarrow> g = (\<lambda>x. case_prod g (f x))" |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
778 |
|
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
779 |
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))" |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
780 |
by (simp add: fun_eq_iff scomp_def case_prod_unfold) |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
781 |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
782 |
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = case_prod g (f x)" |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
783 |
by (simp add: scomp_unfold case_prod_unfold) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
784 |
|
37751 | 785 |
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x" |
44921 | 786 |
by (simp add: fun_eq_iff) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
787 |
|
37751 | 788 |
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x" |
44921 | 789 |
by (simp add: fun_eq_iff) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
790 |
|
37751 | 791 |
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
|
792 |
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
|
793 |
|
37751 | 794 |
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
|
795 |
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
|
796 |
|
37751 | 797 |
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)" |
44921 | 798 |
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
|
799 |
|
52435
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
52143
diff
changeset
|
800 |
code_printing |
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
52143
diff
changeset
|
801 |
constant scomp \<rightharpoonup> (Eval) infixl 3 "#->" |
31202
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset
|
802 |
|
37751 | 803 |
no_notation fcomp (infixl "\<circ>>" 60) |
804 |
no_notation scomp (infixl "\<circ>\<rightarrow>" 60) |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
805 |
|
60758 | 806 |
text \<open> |
69593 | 807 |
\<^term>\<open>map_prod\<close> --- action of the product functor upon functions. |
60758 | 808 |
\<close> |
21195 | 809 |
|
63400 | 810 |
definition map_prod :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" |
811 |
where "map_prod 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
|
812 |
|
63400 | 813 |
lemma map_prod_simp [simp, code]: "map_prod f g (a, b) = (f a, g b)" |
55932 | 814 |
by (simp add: map_prod_def) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
815 |
|
55932 | 816 |
functor map_prod: map_prod |
44921 | 817 |
by (auto simp add: split_paired_all) |
37278 | 818 |
|
63400 | 819 |
lemma fst_map_prod [simp]: "fst (map_prod f g x) = f (fst x)" |
40607 | 820 |
by (cases x) simp_all |
37278 | 821 |
|
63400 | 822 |
lemma snd_map_prod [simp]: "snd (map_prod f g x) = g (snd x)" |
40607 | 823 |
by (cases x) simp_all |
37278 | 824 |
|
63400 | 825 |
lemma fst_comp_map_prod [simp]: "fst \<circ> map_prod f g = f \<circ> fst" |
40607 | 826 |
by (rule ext) simp_all |
37278 | 827 |
|
63400 | 828 |
lemma snd_comp_map_prod [simp]: "snd \<circ> map_prod f g = g \<circ> snd" |
40607 | 829 |
by (rule ext) simp_all |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
830 |
|
63400 | 831 |
lemma map_prod_compose: "map_prod (f1 \<circ> f2) (g1 \<circ> g2) = (map_prod f1 g1 \<circ> map_prod f2 g2)" |
55932 | 832 |
by (rule ext) (simp add: map_prod.compositionality comp_def) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
833 |
|
63400 | 834 |
lemma map_prod_ident [simp]: "map_prod (\<lambda>x. x) (\<lambda>y. y) = (\<lambda>z. z)" |
55932 | 835 |
by (rule ext) (simp add: map_prod.identity) |
40607 | 836 |
|
63400 | 837 |
lemma map_prod_imageI [intro]: "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_prod f g ` R" |
40607 | 838 |
by (rule image_eqI) simp_all |
21195 | 839 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
840 |
lemma prod_fun_imageE [elim!]: |
55932 | 841 |
assumes major: "c \<in> map_prod f g ` R" |
40607 | 842 |
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
|
843 |
shows P |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
844 |
apply (rule major [THEN imageE]) |
37166 | 845 |
apply (case_tac x) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
846 |
apply (rule cases) |
63575 | 847 |
apply simp_all |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
848 |
done |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
849 |
|
63400 | 850 |
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" |
851 |
where "apfst f = map_prod f id" |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
852 |
|
63400 | 853 |
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" |
854 |
where "apsnd f = map_prod id f" |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
855 |
|
63575 | 856 |
lemma apfst_conv [simp, code]: "apfst f (x, y) = (f x, y)" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
857 |
by (simp add: apfst_def) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
858 |
|
63575 | 859 |
lemma apsnd_conv [simp, code]: "apsnd f (x, y) = (x, f y)" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
860 |
by (simp add: apsnd_def) |
21195 | 861 |
|
63400 | 862 |
lemma fst_apfst [simp]: "fst (apfst f x) = f (fst x)" |
33594 | 863 |
by (cases x) simp |
864 |
||
63400 | 865 |
lemma fst_comp_apfst [simp]: "fst \<circ> apfst f = f \<circ> fst" |
51173 | 866 |
by (simp add: fun_eq_iff) |
867 |
||
63400 | 868 |
lemma fst_apsnd [simp]: "fst (apsnd f x) = fst x" |
33594 | 869 |
by (cases x) simp |
870 |
||
63400 | 871 |
lemma fst_comp_apsnd [simp]: "fst \<circ> apsnd f = fst" |
51173 | 872 |
by (simp add: fun_eq_iff) |
873 |
||
63400 | 874 |
lemma snd_apfst [simp]: "snd (apfst f x) = snd x" |
33594 | 875 |
by (cases x) simp |
876 |
||
63400 | 877 |
lemma snd_comp_apfst [simp]: "snd \<circ> apfst f = snd" |
51173 | 878 |
by (simp add: fun_eq_iff) |
879 |
||
63400 | 880 |
lemma snd_apsnd [simp]: "snd (apsnd f x) = f (snd x)" |
33594 | 881 |
by (cases x) simp |
882 |
||
63400 | 883 |
lemma snd_comp_apsnd [simp]: "snd \<circ> apsnd f = f \<circ> snd" |
51173 | 884 |
by (simp add: fun_eq_iff) |
885 |
||
63400 | 886 |
lemma apfst_compose: "apfst f (apfst g x) = apfst (f \<circ> g) x" |
33594 | 887 |
by (cases x) simp |
888 |
||
63400 | 889 |
lemma apsnd_compose: "apsnd f (apsnd g x) = apsnd (f \<circ> g) x" |
33594 | 890 |
by (cases x) simp |
891 |
||
63400 | 892 |
lemma apfst_apsnd [simp]: "apfst f (apsnd g x) = (f (fst x), g (snd x))" |
33594 | 893 |
by (cases x) simp |
894 |
||
63400 | 895 |
lemma apsnd_apfst [simp]: "apsnd f (apfst g x) = (g (fst x), f (snd x))" |
33594 | 896 |
by (cases x) simp |
897 |
||
63400 | 898 |
lemma apfst_id [simp]: "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
|
899 |
by (simp add: fun_eq_iff) |
33594 | 900 |
|
63400 | 901 |
lemma apsnd_id [simp]: "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
|
902 |
by (simp add: fun_eq_iff) |
33594 | 903 |
|
63400 | 904 |
lemma apfst_eq_conv [simp]: "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)" |
33594 | 905 |
by (cases x) simp |
906 |
||
63400 | 907 |
lemma apsnd_eq_conv [simp]: "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)" |
33594 | 908 |
by (cases x) simp |
909 |
||
63400 | 910 |
lemma apsnd_apfst_commute: "apsnd f (apfst g p) = apfst g (apsnd f p)" |
33638
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
911 |
by simp |
21195 | 912 |
|
56626 | 913 |
context |
914 |
begin |
|
915 |
||
60758 | 916 |
local_setup \<open>Local_Theory.map_background_naming (Name_Space.mandatory_path "prod")\<close> |
56626 | 917 |
|
56545 | 918 |
definition swap :: "'a \<times> 'b \<Rightarrow> 'b \<times> 'a" |
63400 | 919 |
where "swap p = (snd p, fst p)" |
56545 | 920 |
|
56626 | 921 |
end |
922 |
||
63400 | 923 |
lemma swap_simp [simp]: "prod.swap (x, y) = (y, x)" |
56626 | 924 |
by (simp add: prod.swap_def) |
56545 | 925 |
|
63400 | 926 |
lemma swap_swap [simp]: "prod.swap (prod.swap p) = p" |
58195 | 927 |
by (cases p) simp |
928 |
||
63400 | 929 |
lemma swap_comp_swap [simp]: "prod.swap \<circ> prod.swap = id" |
58195 | 930 |
by (simp add: fun_eq_iff) |
931 |
||
63400 | 932 |
lemma pair_in_swap_image [simp]: "(y, x) \<in> prod.swap ` A \<longleftrightarrow> (x, y) \<in> A" |
56545 | 933 |
by (auto intro!: image_eqI) |
934 |
||
63400 | 935 |
lemma inj_swap [simp]: "inj_on prod.swap A" |
56626 | 936 |
by (rule inj_onI) auto |
937 |
||
63400 | 938 |
lemma swap_inj_on: "inj_on (\<lambda>(i, j). (j, i)) A" |
56626 | 939 |
by (rule inj_onI) auto |
56545 | 940 |
|
63400 | 941 |
lemma surj_swap [simp]: "surj prod.swap" |
58195 | 942 |
by (rule surjI [of _ prod.swap]) simp |
943 |
||
63400 | 944 |
lemma bij_swap [simp]: "bij prod.swap" |
58195 | 945 |
by (simp add: bij_def) |
946 |
||
63400 | 947 |
lemma case_swap [simp]: "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)" |
56545 | 948 |
by (cases p) simp |
949 |
||
62139 | 950 |
lemma fst_swap [simp]: "fst (prod.swap x) = snd x" |
63400 | 951 |
by (cases x) simp |
62139 | 952 |
|
953 |
lemma snd_swap [simp]: "snd (prod.swap x) = fst x" |
|
63400 | 954 |
by (cases x) simp |
62139 | 955 |
|
63400 | 956 |
text \<open>Disjoint union of a family of sets -- Sigma.\<close> |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
957 |
|
63400 | 958 |
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" |
959 |
where "Sigma A B \<equiv> \<Union>x\<in>A. \<Union>y\<in>B x. {Pair x y}" |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
960 |
|
63400 | 961 |
abbreviation Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" (infixr "\<times>" 80) |
962 |
where "A \<times> B \<equiv> Sigma A (\<lambda>_. B)" |
|
15394 | 963 |
|
45662
4f7c05990420
Hide Product_Type.Times - too precious an identifier
nipkow
parents:
45607
diff
changeset
|
964 |
hide_const (open) Times |
4f7c05990420
Hide Product_Type.Times - too precious an identifier
nipkow
parents:
45607
diff
changeset
|
965 |
|
68467
44ffc5b9cd76
fixing overloading problems involving vector cross products
paulson <lp15@cam.ac.uk>
parents:
68457
diff
changeset
|
966 |
bundle no_Set_Product_syntax begin |
44ffc5b9cd76
fixing overloading problems involving vector cross products
paulson <lp15@cam.ac.uk>
parents:
68457
diff
changeset
|
967 |
no_notation Product_Type.Times (infixr "\<times>" 80) |
44ffc5b9cd76
fixing overloading problems involving vector cross products
paulson <lp15@cam.ac.uk>
parents:
68457
diff
changeset
|
968 |
end |
44ffc5b9cd76
fixing overloading problems involving vector cross products
paulson <lp15@cam.ac.uk>
parents:
68457
diff
changeset
|
969 |
bundle Set_Product_syntax begin |
44ffc5b9cd76
fixing overloading problems involving vector cross products
paulson <lp15@cam.ac.uk>
parents:
68457
diff
changeset
|
970 |
notation Product_Type.Times (infixr "\<times>" 80) |
44ffc5b9cd76
fixing overloading problems involving vector cross products
paulson <lp15@cam.ac.uk>
parents:
68457
diff
changeset
|
971 |
end |
44ffc5b9cd76
fixing overloading problems involving vector cross products
paulson <lp15@cam.ac.uk>
parents:
68457
diff
changeset
|
972 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
973 |
syntax |
63400 | 974 |
"_Sigma" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
975 |
translations |
63400 | 976 |
"SIGMA x:A. B" \<rightleftharpoons> "CONST Sigma A (\<lambda>x. B)" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
977 |
|
63400 | 978 |
lemma SigmaI [intro!]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> (a, b) \<in> Sigma A B" |
979 |
unfolding Sigma_def by blast |
|
980 |
||
981 |
lemma SigmaE [elim!]: "c \<in> Sigma A B \<Longrightarrow> (\<And>x y. x \<in> A \<Longrightarrow> y \<in> B x \<Longrightarrow> c = (x, y) \<Longrightarrow> P) \<Longrightarrow> P" |
|
61799 | 982 |
\<comment> \<open>The general elimination rule.\<close> |
63400 | 983 |
unfolding Sigma_def by blast |
20588 | 984 |
|
60758 | 985 |
text \<open> |
69593 | 986 |
Elimination of \<^term>\<open>(a, b) \<in> A \<times> B\<close> -- introduces no |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
987 |
eigenvariables. |
60758 | 988 |
\<close> |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
989 |
|
63400 | 990 |
lemma SigmaD1: "(a, b) \<in> Sigma A B \<Longrightarrow> a \<in> A" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
991 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
992 |
|
63400 | 993 |
lemma SigmaD2: "(a, b) \<in> Sigma A B \<Longrightarrow> b \<in> B a" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
994 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
995 |
|
63400 | 996 |
lemma SigmaE2: "(a, b) \<in> Sigma A B \<Longrightarrow> (a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> P) \<Longrightarrow> P" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
997 |
by blast |
20588 | 998 |
|
63400 | 999 |
lemma Sigma_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (SIGMA x:A. C x) = (SIGMA x:B. D x)" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1000 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1001 |
|
63400 | 1002 |
lemma Sigma_mono: "A \<subseteq> C \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> D x) \<Longrightarrow> Sigma A B \<subseteq> Sigma C D" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1003 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1004 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1005 |
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1006 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1007 |
|
61943 | 1008 |
lemma Sigma_empty2 [simp]: "A \<times> {} = {}" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1009 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1010 |
|
61943 | 1011 |
lemma UNIV_Times_UNIV [simp]: "UNIV \<times> UNIV = UNIV" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1012 |
by auto |
21908 | 1013 |
|
61943 | 1014 |
lemma Compl_Times_UNIV1 [simp]: "- (UNIV \<times> A) = UNIV \<times> (-A)" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1015 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1016 |
|
61943 | 1017 |
lemma Compl_Times_UNIV2 [simp]: "- (A \<times> UNIV) = (-A) \<times> UNIV" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1018 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1019 |
|
63400 | 1020 |
lemma mem_Sigma_iff [iff]: "(a, b) \<in> Sigma A B \<longleftrightarrow> a \<in> A \<and> b \<in> B a" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1021 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1022 |
|
62101 | 1023 |
lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B" |
1024 |
by (induct x) simp |
|
1025 |
||
59000 | 1026 |
lemma Sigma_empty_iff: "(SIGMA i:I. X i) = {} \<longleftrightarrow> (\<forall>i\<in>I. X i = {})" |
1027 |
by auto |
|
1028 |
||
63400 | 1029 |
lemma Times_subset_cancel2: "x \<in> C \<Longrightarrow> A \<times> C \<subseteq> B \<times> C \<longleftrightarrow> A \<subseteq> B" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1030 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1031 |
|
63400 | 1032 |
lemma Times_eq_cancel2: "x \<in> C \<Longrightarrow> A \<times> C = B \<times> C \<longleftrightarrow> A = B" |
1033 |
by (blast elim: equalityE) |
|
1034 |
||
1035 |
lemma Collect_case_prod_Sigma: "{(x, y). P x \<and> Q x y} = (SIGMA x:Collect P. Collect (Q x))" |
|
1036 |
by blast |
|
1037 |
||
1038 |
lemma Collect_case_prod [simp]: "{(a, b). P a \<and> Q b} = Collect P \<times> Collect Q " |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
1039 |
by (fact Collect_case_prod_Sigma) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1040 |
|
63400 | 1041 |
lemma Collect_case_prodD: "x \<in> Collect (case_prod A) \<Longrightarrow> A (fst x) (snd x)" |
61422 | 1042 |
by auto |
1043 |
||
63400 | 1044 |
lemma Collect_case_prod_mono: "A \<le> B \<Longrightarrow> Collect (case_prod A) \<subseteq> Collect (case_prod B)" |
61422 | 1045 |
by auto (auto elim!: le_funE) |
1046 |
||
63575 | 1047 |
lemma Collect_split_mono_strong: |
61422 | 1048 |
"X = fst ` A \<Longrightarrow> Y = snd ` A \<Longrightarrow> \<forall>a\<in>X. \<forall>b \<in> Y. P a b \<longrightarrow> Q a b |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
1049 |
\<Longrightarrow> A \<subseteq> Collect (case_prod P) \<Longrightarrow> A \<subseteq> Collect (case_prod Q)" |
61422 | 1050 |
by fastforce |
63575 | 1051 |
|
69275 | 1052 |
lemma UN_Times_distrib: "(\<Union>(a, b)\<in>A \<times> B. E a \<times> F b) = \<Union>(E ` A) \<times> \<Union>(F ` B)" |
61799 | 1053 |
\<comment> \<open>Suggested by Pierre Chartier\<close> |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1054 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1055 |
|
63400 | 1056 |
lemma split_paired_Ball_Sigma [simp, no_atp]: "(\<forall>z\<in>Sigma A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>B x. P (x, y))" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1057 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1058 |
|
63400 | 1059 |
lemma split_paired_Bex_Sigma [simp, no_atp]: "(\<exists>z\<in>Sigma A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>y\<in>B x. P (x, y))" |
61127 | 1060 |
by blast |
1061 |
||
63400 | 1062 |
lemma Sigma_Un_distrib1: "Sigma (I \<union> J) C = Sigma I C \<union> Sigma J C" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1063 |
by blast |
21908 | 1064 |
|
63400 | 1065 |
lemma Sigma_Un_distrib2: "(SIGMA i:I. A i \<union> B i) = Sigma I A \<union> Sigma I B" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1066 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1067 |
|
63400 | 1068 |
lemma Sigma_Int_distrib1: "Sigma (I \<inter> J) C = Sigma I C \<inter> Sigma J C" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1069 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1070 |
|
63400 | 1071 |
lemma Sigma_Int_distrib2: "(SIGMA i:I. A i \<inter> B i) = Sigma I A \<inter> Sigma I B" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1072 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1073 |
|
63400 | 1074 |
lemma Sigma_Diff_distrib1: "Sigma (I - J) C = Sigma I C - Sigma J C" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1075 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1076 |
|
63400 | 1077 |
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A i - B i) = Sigma I A - Sigma I B" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1078 |
by blast |
21908 | 1079 |
|
63400 | 1080 |
lemma Sigma_Union: "Sigma (\<Union>X) B = (\<Union>A\<in>X. Sigma A B)" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1081 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1082 |
|
61630 | 1083 |
lemma Pair_vimage_Sigma: "Pair x -` Sigma A f = (if x \<in> A then f x else {})" |
1084 |
by auto |
|
1085 |
||
60758 | 1086 |
text \<open> |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1087 |
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
|
1088 |
matching, especially when the rules are re-oriented. |
60758 | 1089 |
\<close> |
21908 | 1090 |
|
63400 | 1091 |
lemma Times_Un_distrib1: "(A \<union> B) \<times> C = A \<times> C \<union> B \<times> C " |
56545 | 1092 |
by (fact Sigma_Un_distrib1) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1093 |
|
63400 | 1094 |
lemma Times_Int_distrib1: "(A \<inter> B) \<times> C = A \<times> C \<inter> B \<times> C " |
56545 | 1095 |
by (fact Sigma_Int_distrib1) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1096 |
|
63400 | 1097 |
lemma Times_Diff_distrib1: "(A - B) \<times> C = A \<times> C - B \<times> C " |
56545 | 1098 |
by (fact Sigma_Diff_distrib1) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1099 |
|
63400 | 1100 |
lemma Times_empty [simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}" |
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
|
1101 |
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
|
1102 |
|
69144
f13b82281715
new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents:
68467
diff
changeset
|
1103 |
lemma times_subset_iff: "A \<times> C \<subseteq> B \<times> D \<longleftrightarrow> A={} \<or> C={} \<or> A \<subseteq> B \<and> C \<subseteq> D" |
f13b82281715
new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents:
68467
diff
changeset
|
1104 |
by blast |
f13b82281715
new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents:
68467
diff
changeset
|
1105 |
|
63400 | 1106 |
lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> (A = {} \<or> B = {}) \<and> (C = {} \<or> D = {})" |
50104 | 1107 |
by auto |
1108 |
||
63400 | 1109 |
lemma fst_image_times [simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)" |
44921 | 1110 |
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
|
1111 |
|
63400 | 1112 |
lemma snd_image_times [simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)" |
44921 | 1113 |
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
|
1114 |
|
63400 | 1115 |
lemma fst_image_Sigma: "fst ` (Sigma A B) = {x \<in> A. B(x) \<noteq> {}}" |
62379
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62139
diff
changeset
|
1116 |
by force |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62139
diff
changeset
|
1117 |
|
63400 | 1118 |
lemma snd_image_Sigma: "snd ` (Sigma A B) = (\<Union> x \<in> A. B x)" |
62379
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62139
diff
changeset
|
1119 |
by force |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62139
diff
changeset
|
1120 |
|
63400 | 1121 |
lemma vimage_fst: "fst -` A = A \<times> UNIV" |
56545 | 1122 |
by auto |
1123 |
||
63400 | 1124 |
lemma vimage_snd: "snd -` A = UNIV \<times> A" |
56545 | 1125 |
by auto |
1126 |
||
69922
4a9167f377b0
new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents:
69913
diff
changeset
|
1127 |
lemma insert_Times_insert [simp]: |
63400 | 1128 |
"insert a A \<times> insert b B = insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)" |
61127 | 1129 |
by blast |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1130 |
|
63400 | 1131 |
lemma vimage_Times: "f -` (A \<times> B) = (fst \<circ> f) -` A \<inter> (snd \<circ> f) -` B" |
61127 | 1132 |
proof (rule set_eqI) |
63400 | 1133 |
show "x \<in> f -` (A \<times> B) \<longleftrightarrow> x \<in> (fst \<circ> f) -` A \<inter> (snd \<circ> f) -` B" for x |
61127 | 1134 |
by (cases "f x") (auto split: prod.split) |
1135 |
qed |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset
|
1136 |
|
69939
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69922
diff
changeset
|
1137 |
lemma Times_Int_Times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)" |
50104 | 1138 |
by auto |
1139 |
||
63400 | 1140 |
lemma product_swap: "prod.swap ` (A \<times> B) = B \<times> A" |
56626 | 1141 |
by (auto simp add: set_eq_iff) |
35822 | 1142 |
|
63400 | 1143 |
lemma swap_product: "(\<lambda>(i, j). (j, i)) ` (A \<times> B) = B \<times> A" |
56626 | 1144 |
by (auto simp add: set_eq_iff) |
35822 | 1145 |
|
63400 | 1146 |
lemma image_split_eq_Sigma: "(\<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
|
1147 |
proof (safe intro!: imageI) |
63400 | 1148 |
fix a b |
1149 |
assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b" |
|
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
|
1150 |
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
|
1151 |
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
|
1152 |
qed simp_all |
35822 | 1153 |
|
63007
aa894a49f77d
new theorems about convex hulls, etc.; also, renamed some theorems
paulson <lp15@cam.ac.uk>
parents:
62913
diff
changeset
|
1154 |
lemma subset_fst_snd: "A \<subseteq> (fst ` A \<times> snd ` A)" |
aa894a49f77d
new theorems about convex hulls, etc.; also, renamed some theorems
paulson <lp15@cam.ac.uk>
parents:
62913
diff
changeset
|
1155 |
by force |
aa894a49f77d
new theorems about convex hulls, etc.; also, renamed some theorems
paulson <lp15@cam.ac.uk>
parents:
62913
diff
changeset
|
1156 |
|
60057 | 1157 |
lemma inj_on_apfst [simp]: "inj_on (apfst f) (A \<times> UNIV) \<longleftrightarrow> inj_on f A" |
63400 | 1158 |
by (auto simp add: inj_on_def) |
60057 | 1159 |
|
1160 |
lemma inj_apfst [simp]: "inj (apfst f) \<longleftrightarrow> inj f" |
|
63400 | 1161 |
using inj_on_apfst[of f UNIV] by simp |
60057 | 1162 |
|
1163 |
lemma inj_on_apsnd [simp]: "inj_on (apsnd f) (UNIV \<times> A) \<longleftrightarrow> inj_on f A" |
|
63400 | 1164 |
by (auto simp add: inj_on_def) |
60057 | 1165 |
|
1166 |
lemma inj_apsnd [simp]: "inj (apsnd f) \<longleftrightarrow> inj f" |
|
63400 | 1167 |
using inj_on_apsnd[of f UNIV] by simp |
60057 | 1168 |
|
61127 | 1169 |
context |
1170 |
begin |
|
1171 |
||
63575 | 1172 |
qualified definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" |
1173 |
where [code_abbrev]: "product A B = A \<times> B" |
|
46128
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1174 |
|
63400 | 1175 |
lemma member_product: "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B" |
1176 |
by (simp add: product_def) |
|
46128
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1177 |
|
61127 | 1178 |
end |
63575 | 1179 |
|
69593 | 1180 |
text \<open>The following \<^const>\<open>map_prod\<close> lemmas are due to Joachim Breitner:\<close> |
40607 | 1181 |
|
55932 | 1182 |
lemma map_prod_inj_on: |
63575 | 1183 |
assumes "inj_on f A" |
1184 |
and "inj_on g B" |
|
55932 | 1185 |
shows "inj_on (map_prod f g) (A \<times> B)" |
40607 | 1186 |
proof (rule inj_onI) |
63400 | 1187 |
fix x :: "'a \<times> 'c" |
1188 |
fix y :: "'a \<times> 'c" |
|
1189 |
assume "x \<in> A \<times> B" |
|
1190 |
then have "fst x \<in> A" and "snd x \<in> B" by auto |
|
1191 |
assume "y \<in> A \<times> B" |
|
1192 |
then have "fst y \<in> A" and "snd y \<in> B" by auto |
|
55932 | 1193 |
assume "map_prod f g x = map_prod f g y" |
63400 | 1194 |
then have "fst (map_prod f g x) = fst (map_prod f g y)" by auto |
1195 |
then have "f (fst x) = f (fst y)" by (cases x, cases y) auto |
|
1196 |
with \<open>inj_on f A\<close> and \<open>fst x \<in> A\<close> and \<open>fst y \<in> A\<close> have "fst x = fst y" |
|
1197 |
by (auto dest: inj_onD) |
|
60758 | 1198 |
moreover from \<open>map_prod f g x = map_prod f g y\<close> |
63400 | 1199 |
have "snd (map_prod f g x) = snd (map_prod f g y)" by auto |
1200 |
then have "g (snd x) = g (snd y)" by (cases x, cases y) auto |
|
1201 |
with \<open>inj_on g B\<close> and \<open>snd x \<in> B\<close> and \<open>snd y \<in> B\<close> have "snd x = snd y" |
|
1202 |
by (auto dest: inj_onD) |
|
1203 |
ultimately show "x = y" by (rule prod_eqI) |
|
40607 | 1204 |
qed |
1205 |
||
55932 | 1206 |
lemma map_prod_surj: |
63400 | 1207 |
fixes f :: "'a \<Rightarrow> 'b" |
1208 |
and g :: "'c \<Rightarrow> 'd" |
|
40607 | 1209 |
assumes "surj f" and "surj g" |
55932 | 1210 |
shows "surj (map_prod f g)" |
63400 | 1211 |
unfolding surj_def |
40607 | 1212 |
proof |
1213 |
fix y :: "'b \<times> 'd" |
|
63400 | 1214 |
from \<open>surj f\<close> obtain a where "fst y = f a" |
1215 |
by (auto elim: surjE) |
|
40607 | 1216 |
moreover |
63400 | 1217 |
from \<open>surj g\<close> obtain b where "snd y = g b" |
1218 |
by (auto elim: surjE) |
|
1219 |
ultimately have "(fst y, snd y) = map_prod f g (a,b)" |
|
1220 |
by auto |
|
1221 |
then show "\<exists>x. y = map_prod f g x" |
|
1222 |
by auto |
|
40607 | 1223 |
qed |
1224 |
||
55932 | 1225 |
lemma map_prod_surj_on: |
40607 | 1226 |
assumes "f ` A = A'" and "g ` B = B'" |
55932 | 1227 |
shows "map_prod f g ` (A \<times> B) = A' \<times> B'" |
63400 | 1228 |
unfolding image_def |
1229 |
proof (rule set_eqI, rule iffI) |
|
40607 | 1230 |
fix x :: "'a \<times> 'c" |
61076 | 1231 |
assume "x \<in> {y::'a \<times> 'c. \<exists>x::'b \<times> 'd\<in>A \<times> B. y = map_prod f g x}" |
63400 | 1232 |
then obtain y where "y \<in> A \<times> B" and "x = map_prod f g y" |
1233 |
by blast |
|
1234 |
from \<open>image f A = A'\<close> and \<open>y \<in> A \<times> B\<close> have "f (fst y) \<in> A'" |
|
1235 |
by auto |
|
1236 |
moreover from \<open>image g B = B'\<close> and \<open>y \<in> A \<times> B\<close> have "g (snd y) \<in> B'" |
|
1237 |
by auto |
|
1238 |
ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" |
|
1239 |
by auto |
|
1240 |
with \<open>x = map_prod f g y\<close> show "x \<in> A' \<times> B'" |
|
1241 |
by (cases y) auto |
|
40607 | 1242 |
next |
1243 |
fix x :: "'a \<times> 'c" |
|
63400 | 1244 |
assume "x \<in> A' \<times> B'" |
1245 |
then have "fst x \<in> A'" and "snd x \<in> B'" |
|
1246 |
by auto |
|
1247 |
from \<open>image f A = A'\<close> and \<open>fst x \<in> A'\<close> have "fst x \<in> image f A" |
|
1248 |
by auto |
|
1249 |
then obtain a where "a \<in> A" and "fst x = f a" |
|
1250 |
by (rule imageE) |
|
1251 |
moreover from \<open>image g B = B'\<close> and \<open>snd x \<in> B'\<close> obtain b where "b \<in> B" and "snd x = g b" |
|
1252 |
by auto |
|
1253 |
ultimately have "(fst x, snd x) = map_prod f g (a, b)" |
|
1254 |
by auto |
|
1255 |
moreover from \<open>a \<in> A\<close> and \<open>b \<in> B\<close> have "(a , b) \<in> A \<times> B" |
|
1256 |
by auto |
|
1257 |
ultimately have "\<exists>y \<in> A \<times> B. x = map_prod f g y" |
|
1258 |
by auto |
|
1259 |
then show "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_prod f g y}" |
|
1260 |
by auto |
|
40607 | 1261 |
qed |
1262 |
||
21908 | 1263 |
|
60758 | 1264 |
subsection \<open>Simproc for rewriting a set comprehension into a pointfree expression\<close> |
49764
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1265 |
|
69605 | 1266 |
ML_file \<open>Tools/set_comprehension_pointfree.ML\<close> |
49764
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1267 |
|
60758 | 1268 |
setup \<open> |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
1269 |
Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs |
69593 | 1270 |
[Simplifier.make_simproc \<^context> "set comprehension" |
1271 |
{lhss = [\<^term>\<open>Collect P\<close>], |
|
62913 | 1272 |
proc = K Set_Comprehension_Pointfree.code_simproc}]) |
60758 | 1273 |
\<close> |
49764
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1274 |
|
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1275 |
|
60758 | 1276 |
subsection \<open>Inductively defined sets\<close> |
15394 | 1277 |
|
56512 | 1278 |
(* simplify {(x1, ..., xn). (x1, ..., xn) : S} to S *) |
60758 | 1279 |
simproc_setup Collect_mem ("Collect t") = \<open> |
56512 | 1280 |
fn _ => fn ctxt => fn ct => |
59582 | 1281 |
(case Thm.term_of ct of |
69593 | 1282 |
S as Const (\<^const_name>\<open>Collect\<close>, Type (\<^type_name>\<open>fun\<close>, [_, T])) $ t => |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
1283 |
let val (u, _, ps) = HOLogic.strip_ptupleabs t in |
56512 | 1284 |
(case u of |
69593 | 1285 |
(c as Const (\<^const_name>\<open>Set.member\<close>, _)) $ q $ S' => |
56512 | 1286 |
(case try (HOLogic.strip_ptuple ps) q of |
1287 |
NONE => NONE |
|
1288 |
| SOME ts => |
|
1289 |
if not (Term.is_open S') andalso |
|
1290 |
ts = map Bound (length ps downto 0) |
|
1291 |
then |
|
1292 |
let val simp = |
|
1293 |
full_simp_tac (put_simpset HOL_basic_ss ctxt |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
1294 |
addsimps [@{thm split_paired_all}, @{thm case_prod_conv}]) 1 |
56512 | 1295 |
in |
1296 |
SOME (Goal.prove ctxt [] [] |
|
69593 | 1297 |
(Const (\<^const_name>\<open>Pure.eq\<close>, T --> T --> propT) $ S $ S') |
56512 | 1298 |
(K (EVERY |
59498
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents:
59000
diff
changeset
|
1299 |
[resolve_tac ctxt [eq_reflection] 1, |
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents:
59000
diff
changeset
|
1300 |
resolve_tac ctxt @{thms subset_antisym} 1, |
63399 | 1301 |
resolve_tac ctxt @{thms subsetI} 1, |
1302 |
dresolve_tac ctxt @{thms CollectD} 1, simp, |
|
1303 |
resolve_tac ctxt @{thms subsetI} 1, |
|
1304 |
resolve_tac ctxt @{thms CollectI} 1, simp]))) |
|
56512 | 1305 |
end |
1306 |
else NONE) |
|
1307 |
| _ => NONE) |
|
1308 |
end |
|
1309 |
| _ => NONE) |
|
60758 | 1310 |
\<close> |
58389 | 1311 |
|
69605 | 1312 |
ML_file \<open>Tools/inductive_set.ML\<close> |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
1313 |
|
37166 | 1314 |
|
60758 | 1315 |
subsection \<open>Legacy theorem bindings and duplicates\<close> |
37166 | 1316 |
|
55393
ce5cebfaedda
se 'wrap_free_constructors' to register 'sum' , 'prod', 'unit', 'bool' with their discriminators/selectors
blanchet
parents:
54630
diff
changeset
|
1317 |
lemmas fst_conv = prod.sel(1) |
ce5cebfaedda
se 'wrap_free_constructors' to register 'sum' , 'prod', 'unit', 'bool' with their discriminators/selectors
blanchet
parents:
54630
diff
changeset
|
1318 |
lemmas snd_conv = prod.sel(2) |
61032
b57df8eecad6
standardized some occurences of ancient "split" alias
haftmann
parents:
60758
diff
changeset
|
1319 |
lemmas split_def = case_prod_unfold |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
1320 |
lemmas split_beta' = case_prod_beta' |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
1321 |
lemmas split_beta = prod.case_eq_if |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
1322 |
lemmas split_conv = case_prod_conv |
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61422
diff
changeset
|
1323 |
lemmas split = case_prod_conv |
44066
d74182c93f04
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents:
43866
diff
changeset
|
1324 |
|
45204
5e4a1270c000
hide typedef-generated constants Product_Type.prod and Sum_Type.sum
huffman
parents:
44921
diff
changeset
|
1325 |
hide_const (open) prod |
5e4a1270c000
hide typedef-generated constants Product_Type.prod and Sum_Type.sum
huffman
parents:
44921
diff
changeset
|
1326 |
|
10213 | 1327 |
end |