author | blanchet |
Fri, 21 Feb 2014 00:09:56 +0100 | |
changeset 55642 | 63beb38e9258 |
parent 55469 | b19dd108f0c2 |
child 55932 | 68c5104d2204 |
permissions | -rw-r--r-- |
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(* Title: HOL/Product_Type.thy |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1992 University of Cambridge |
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*) |
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header {* Cartesian products *} |
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theory Product_Type |
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imports Typedef Inductive Fun |
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keywords "inductive_set" "coinductive_set" :: thy_decl |
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begin |
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subsection {* @{typ bool} is a datatype *} |
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free_constructors case_bool for =: True | False |
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by auto |
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text {* Avoid name clashes by prefixing the output of @{text rep_datatype} with @{text old}. *} |
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||
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setup {* Sign.mandatory_path "old" *} |
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rep_datatype True False by (auto intro: bool_induct) |
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setup {* Sign.parent_path *} |
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text {* But erase the prefix for properties that are not generated by @{text free_constructors}. *} |
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setup {* Sign.mandatory_path "bool" *} |
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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 {* Sign.parent_path *} |
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declare case_split [cases type: bool] |
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-- "prefer plain propositional version" |
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lemma |
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shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P" |
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and [code]: "HOL.equal True P \<longleftrightarrow> P" |
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and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P" |
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and [code]: "HOL.equal P True \<longleftrightarrow> P" |
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and [code nbe]: "HOL.equal P P \<longleftrightarrow> True" |
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by (simp_all add: equal) |
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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 {* |
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Code.add_case @{thm If_case_cert} |
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*} |
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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 {* The @{text unit} type *} |
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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 {* |
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Simplification procedure for @{thm [source] unit_eq}. Cannot use |
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this rule directly --- it loops! |
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*} |
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simproc_setup unit_eq ("x::unit") = {* |
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fn _ => fn _ => fn ct => |
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if HOLogic.is_unit (term_of ct) then NONE |
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else SOME (mk_meta_eq @{thm unit_eq}) |
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*} |
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free_constructors case_unit for "()" |
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by auto |
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text {* Avoid name clashes by prefixing the output of @{text rep_datatype} with @{text old}. *} |
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setup {* Sign.mandatory_path "old" *} |
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rep_datatype "()" by simp |
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setup {* Sign.parent_path *} |
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text {* But erase the prefix for properties that are not generated by @{text free_constructors}. *} |
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setup {* Sign.mandatory_path "unit" *} |
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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 {* Sign.parent_path *} |
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lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()" |
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by simp |
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lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P" |
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by (rule triv_forall_equality) |
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text {* |
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This rewrite counters the effect of simproc @{text unit_eq} on @{term |
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[source] "%u::unit. f u"}, replacing it by @{term [source] |
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f} rather than by @{term [source] "%u. f ()"}. |
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*} |
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lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f" |
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by (rule ext) simp |
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lemma UNIV_unit: |
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"UNIV = {()}" by auto |
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instantiation unit :: default |
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begin |
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definition "default = ()" |
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instance .. |
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end |
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lemma [code]: |
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"HOL.equal (u\<Colon>unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+ |
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code_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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| constant Unity \<rightharpoonup> |
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(SML) "()" |
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and (OCaml) "()" |
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and (Haskell) "()" |
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and (Scala) "()" |
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| class_instance unit :: equal \<rightharpoonup> |
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(Haskell) - |
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| constant "HOL.equal :: unit \<Rightarrow> unit \<Rightarrow> bool" \<rightharpoonup> |
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(Haskell) infix 4 "==" |
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code_reserved SML |
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unit |
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code_reserved OCaml |
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unit |
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157 |
|
34886 | 158 |
code_reserved Scala |
159 |
Unit |
|
160 |
||
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161 |
|
37166 | 162 |
subsection {* The product type *} |
10213 | 163 |
|
37166 | 164 |
subsubsection {* Type definition *} |
165 |
||
166 |
definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where |
|
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"Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)" |
10213 | 168 |
|
45696 | 169 |
definition "prod = {f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}" |
170 |
||
49834 | 171 |
typedef ('a, 'b) prod (infixr "*" 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set" |
45696 | 172 |
unfolding prod_def by auto |
10213 | 173 |
|
35427 | 174 |
type_notation (xsymbols) |
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175 |
"prod" ("(_ \<times>/ _)" [21, 20] 20) |
35427 | 176 |
type_notation (HTML output) |
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177 |
"prod" ("(_ \<times>/ _)" [21, 20] 20) |
10213 | 178 |
|
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definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where |
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"Pair a b = Abs_prod (Pair_Rep a b)" |
37166 | 181 |
|
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182 |
lemma prod_cases: "(\<And>a b. P (Pair a b)) \<Longrightarrow> P p" |
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by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def) |
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184 |
|
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185 |
free_constructors case_prod for Pair fst snd |
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186 |
proof - |
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187 |
fix P :: bool and p :: "'a \<times> 'b" |
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188 |
show "(\<And>x1 x2. p = Pair x1 x2 \<Longrightarrow> P) \<Longrightarrow> P" |
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189 |
by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def) |
37166 | 190 |
next |
191 |
fix a c :: 'a and b d :: 'b |
|
192 |
have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d" |
|
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193 |
by (auto simp add: Pair_Rep_def fun_eq_iff) |
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194 |
moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod" |
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|
195 |
by (auto simp add: prod_def) |
37166 | 196 |
ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d" |
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|
197 |
by (simp add: Pair_def Abs_prod_inject) |
37166 | 198 |
qed |
199 |
||
55442 | 200 |
text {* Avoid name clashes by prefixing the output of @{text rep_datatype} with @{text old}. *} |
201 |
||
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202 |
setup {* Sign.mandatory_path "old" *} |
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203 |
|
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|
204 |
rep_datatype Pair |
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avoid duplicate 'case' definitions by first looking up 'Ctr_Sugar'
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|
205 |
by (erule prod_cases) (rule prod.inject) |
55393
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206 |
|
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|
207 |
setup {* Sign.parent_path *} |
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c6161bee8486
adapt Nitpick to "prod_case" and "*" -> "sum" renaming;
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parents:
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diff
changeset
|
208 |
|
55468
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
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|
209 |
text {* But erase the prefix for properties that are not generated by @{text free_constructors}. *} |
55442 | 210 |
|
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211 |
setup {* Sign.mandatory_path "prod" *} |
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212 |
|
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213 |
declare |
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214 |
old.prod.inject[iff del] |
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215 |
|
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|
216 |
lemmas induct = old.prod.induct |
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|
217 |
lemmas inducts = old.prod.inducts |
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218 |
lemmas rec = old.prod.rec |
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219 |
lemmas simps = prod.inject prod.case prod.rec |
55393
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220 |
|
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221 |
setup {* Sign.parent_path *} |
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222 |
|
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|
223 |
declare prod.case [nitpick_simp del] |
40929
7ff03a5e044f
theorem names generated by the (rep_)datatype command now have mandatory qualifiers
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parents:
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|
224 |
declare prod.weak_case_cong [cong del] |
37411
c88c44156083
removed simplifier congruence rule of "prod_case"
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|
225 |
|
37166 | 226 |
|
227 |
subsubsection {* Tuple syntax *} |
|
228 |
||
37591 | 229 |
abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where |
55414
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230 |
"split \<equiv> case_prod" |
19535 | 231 |
|
11777 | 232 |
text {* |
233 |
Patterns -- extends pre-defined type @{typ pttrn} used in |
|
234 |
abstractions. |
|
235 |
*} |
|
10213 | 236 |
|
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replaced command 'nonterminals' by slightly modernized version 'nonterminal';
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|
237 |
nonterminal tuple_args and patterns |
10213 | 238 |
|
239 |
syntax |
|
240 |
"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))") |
|
241 |
"_tuple_arg" :: "'a => tuple_args" ("_") |
|
242 |
"_tuple_args" :: "'a => tuple_args => tuple_args" ("_,/ _") |
|
11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
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diff
changeset
|
243 |
"_pattern" :: "[pttrn, patterns] => pttrn" ("'(_,/ _')") |
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
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|
244 |
"" :: "pttrn => patterns" ("_") |
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diff
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|
245 |
"_patterns" :: "[pttrn, patterns] => patterns" ("_,/ _") |
10213 | 246 |
|
247 |
translations |
|
35115 | 248 |
"(x, y)" == "CONST Pair x y" |
51392
635562bc14ef
extended set comprehension notation with {pttrn : A . P}
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diff
changeset
|
249 |
"_pattern x y" => "CONST Pair x y" |
635562bc14ef
extended set comprehension notation with {pttrn : A . P}
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diff
changeset
|
250 |
"_patterns x y" => "CONST Pair x y" |
10213 | 251 |
"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" |
55414
eab03e9cee8a
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parents:
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diff
changeset
|
252 |
"%(x, y, zs). b" == "CONST case_prod (%x (y, zs). b)" |
eab03e9cee8a
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parents:
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diff
changeset
|
253 |
"%(x, y). b" == "CONST case_prod (%x y. b)" |
35115 | 254 |
"_abs (CONST Pair x y) t" => "%(x, y). t" |
37166 | 255 |
-- {* The last rule accommodates tuples in `case C ... (x,y) ... => ...' |
256 |
The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *} |
|
10213 | 257 |
|
35115 | 258 |
(*reconstruct pattern from (nested) splits, avoiding eta-contraction of body; |
259 |
works best with enclosing "let", if "let" does not avoid eta-contraction*) |
|
14359 | 260 |
print_translation {* |
52143 | 261 |
let |
262 |
fun split_tr' [Abs (x, T, t as (Abs abs))] = |
|
263 |
(* split (%x y. t) => %(x,y) t *) |
|
264 |
let |
|
265 |
val (y, t') = Syntax_Trans.atomic_abs_tr' abs; |
|
266 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); |
|
267 |
in |
|
268 |
Syntax.const @{syntax_const "_abs"} $ |
|
269 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' |
|
270 |
end |
|
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|
271 |
| split_tr' [Abs (x, T, (s as Const (@{const_syntax case_prod}, _) $ t))] = |
52143 | 272 |
(* split (%x. (split (%y z. t))) => %(x,y,z). t *) |
273 |
let |
|
274 |
val Const (@{syntax_const "_abs"}, _) $ |
|
275 |
(Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t]; |
|
276 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t'); |
|
277 |
in |
|
278 |
Syntax.const @{syntax_const "_abs"} $ |
|
279 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ |
|
280 |
(Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t'' |
|
281 |
end |
|
55414
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renamed '{prod,sum,bool,unit}_case' to 'case_...'
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parents:
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diff
changeset
|
282 |
| split_tr' [Const (@{const_syntax case_prod}, _) $ t] = |
52143 | 283 |
(* split (split (%x y z. t)) => %((x, y), z). t *) |
284 |
split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *) |
|
285 |
| split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] = |
|
286 |
(* split (%pttrn z. t) => %(pttrn,z). t *) |
|
287 |
let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in |
|
288 |
Syntax.const @{syntax_const "_abs"} $ |
|
289 |
(Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t |
|
290 |
end |
|
291 |
| split_tr' _ = raise Match; |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
292 |
in [(@{const_syntax case_prod}, K split_tr')] end |
14359 | 293 |
*} |
294 |
||
15422
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
295 |
(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) |
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
296 |
typed_print_translation {* |
52143 | 297 |
let |
298 |
fun split_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match |
|
299 |
| split_guess_names_tr' T [Abs (x, xT, t)] = |
|
300 |
(case (head_of t) of |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
301 |
Const (@{const_syntax case_prod}, _) => raise Match |
52143 | 302 |
| _ => |
303 |
let |
|
304 |
val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; |
|
305 |
val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0); |
|
306 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t'); |
|
307 |
in |
|
308 |
Syntax.const @{syntax_const "_abs"} $ |
|
309 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' |
|
310 |
end) |
|
311 |
| split_guess_names_tr' T [t] = |
|
312 |
(case head_of t of |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
313 |
Const (@{const_syntax case_prod}, _) => raise Match |
52143 | 314 |
| _ => |
315 |
let |
|
316 |
val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match; |
|
317 |
val (y, t') = |
|
318 |
Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0); |
|
319 |
val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t'); |
|
320 |
in |
|
321 |
Syntax.const @{syntax_const "_abs"} $ |
|
322 |
(Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t'' |
|
323 |
end) |
|
324 |
| split_guess_names_tr' _ _ = raise Match; |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
325 |
in [(@{const_syntax case_prod}, K split_guess_names_tr')] end |
15422
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
326 |
*} |
cbdddc0efadf
added print translation for split: split f --> %(x,y). f x y
schirmer
parents:
15404
diff
changeset
|
327 |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
328 |
(* Force eta-contraction for terms of the form "Q A (%p. case_prod P p)" |
42059
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
41792
diff
changeset
|
329 |
where Q is some bounded quantifier or set operator. |
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
41792
diff
changeset
|
330 |
Reason: the above prints as "Q p : A. case p of (x,y) \<Rightarrow> P x y" |
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
41792
diff
changeset
|
331 |
whereas we want "Q (x,y):A. P x y". |
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
41792
diff
changeset
|
332 |
Otherwise prevent eta-contraction. |
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
41792
diff
changeset
|
333 |
*) |
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
41792
diff
changeset
|
334 |
print_translation {* |
52143 | 335 |
let |
336 |
fun contract Q tr ctxt ts = |
|
337 |
(case ts of |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
338 |
[A, Abs (_, _, (s as Const (@{const_syntax case_prod},_) $ t) $ Bound 0)] => |
52143 | 339 |
if Term.is_dependent t then tr ctxt ts |
340 |
else Syntax.const Q $ A $ s |
|
341 |
| _ => tr ctxt ts); |
|
342 |
in |
|
42284 | 343 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"}, |
344 |
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"}, |
|
345 |
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}, |
|
346 |
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}] |
|
52143 | 347 |
|> map (fn (Q, tr) => (Q, contract Q tr)) |
348 |
end |
|
42059
83f3dc509068
fixed a printing problem for bounded quantifiers and bounded set operators in the case of tuples
nipkow
parents:
41792
diff
changeset
|
349 |
*} |
10213 | 350 |
|
37166 | 351 |
subsubsection {* Code generator setup *} |
352 |
||
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|
353 |
code_printing |
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|
354 |
type_constructor prod \<rightharpoonup> |
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parents:
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diff
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|
355 |
(SML) infix 2 "*" |
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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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|
356 |
and (OCaml) infix 2 "*" |
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|
357 |
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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diff
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|
358 |
and (Scala) "((_),/ (_))" |
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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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|
359 |
| constant Pair \<rightharpoonup> |
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parents:
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diff
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|
360 |
(SML) "!((_),/ (_))" |
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parents:
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|
361 |
and (OCaml) "!((_),/ (_))" |
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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
|
362 |
and (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
|
363 |
and (Scala) "!((_),/ (_))" |
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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
|
364 |
| class_instance prod :: equal \<rightharpoonup> |
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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
|
365 |
(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
|
366 |
| constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup> |
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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
|
367 |
(Haskell) infix 4 "==" |
37166 | 368 |
|
369 |
||
370 |
subsubsection {* Fundamental operations and properties *} |
|
11838 | 371 |
|
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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)
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parents:
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diff
changeset
|
372 |
lemma Pair_inject: |
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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)
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parents:
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diff
changeset
|
373 |
assumes "(a, b) = (a', b')" |
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)
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parents:
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diff
changeset
|
374 |
and "a = a' ==> b = b' ==> R" |
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:
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diff
changeset
|
375 |
shows R |
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:
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diff
changeset
|
376 |
using assms by simp |
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:
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diff
changeset
|
377 |
|
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Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
378 |
lemma surj_pair [simp]: "EX x y. p = (x, y)" |
37166 | 379 |
by (cases p) simp |
10213 | 380 |
|
52435
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haftmann
parents:
52143
diff
changeset
|
381 |
code_printing |
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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
|
382 |
constant fst \<rightharpoonup> (Haskell) "fst" |
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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
|
383 |
| constant snd \<rightharpoonup> (Haskell) "snd" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
384 |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
385 |
lemma case_prod_unfold [nitpick_unfold]: "case_prod = (%c p. c (fst p) (snd p))" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
386 |
by (simp add: fun_eq_iff split: prod.split) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
387 |
|
11838 | 388 |
lemma fst_eqD: "fst (x, y) = a ==> x = a" |
389 |
by simp |
|
390 |
||
391 |
lemma snd_eqD: "snd (x, y) = a ==> y = a" |
|
392 |
by simp |
|
393 |
||
55393
ce5cebfaedda
se 'wrap_free_constructors' to register 'sum' , 'prod', 'unit', 'bool' with their discriminators/selectors
blanchet
parents:
54630
diff
changeset
|
394 |
lemmas surjective_pairing = prod.collapse [symmetric] |
11838 | 395 |
|
44066
d74182c93f04
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents:
43866
diff
changeset
|
396 |
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t" |
37166 | 397 |
by (cases s, cases t) simp |
398 |
||
399 |
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
|
400 |
by (simp add: prod_eq_iff) |
37166 | 401 |
|
402 |
lemma split_conv [simp, code]: "split f (a, b) = f a b" |
|
55642
63beb38e9258
adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
blanchet
parents:
55469
diff
changeset
|
403 |
by (fact prod.case) |
37166 | 404 |
|
405 |
lemma splitI: "f a b \<Longrightarrow> split f (a, b)" |
|
406 |
by (rule split_conv [THEN iffD2]) |
|
407 |
||
408 |
lemma splitD: "split f (a, b) \<Longrightarrow> f a b" |
|
409 |
by (rule split_conv [THEN iffD1]) |
|
410 |
||
411 |
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
412 |
by (simp add: fun_eq_iff split: prod.split) |
37166 | 413 |
|
414 |
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f" |
|
415 |
-- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *} |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39272
diff
changeset
|
416 |
by (simp add: fun_eq_iff split: prod.split) |
37166 | 417 |
|
418 |
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)" |
|
419 |
by (cases x) simp |
|
420 |
||
421 |
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p" |
|
422 |
by (cases p) simp |
|
423 |
||
424 |
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
425 |
by (simp add: case_prod_unfold) |
37166 | 426 |
|
427 |
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q" |
|
428 |
-- {* Prevents simplification of @{term c}: much faster *} |
|
40929
7ff03a5e044f
theorem names generated by the (rep_)datatype command now have mandatory qualifiers
huffman
parents:
40702
diff
changeset
|
429 |
by (fact prod.weak_case_cong) |
37166 | 430 |
|
431 |
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" |
|
432 |
by (simp add: split_eta) |
|
433 |
||
47740
a8989fe9a3a5
added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents:
46950
diff
changeset
|
434 |
lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" |
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
435 |
proof |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
436 |
fix a b |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
437 |
assume "!!x. PROP P x" |
19535 | 438 |
then show "PROP P (a, b)" . |
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
439 |
next |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
440 |
fix x |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
441 |
assume "!!a b. PROP P (a, b)" |
19535 | 442 |
from `PROP P (fst x, snd x)` show "PROP P x" by simp |
11820
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
443 |
qed |
015a82d4ee96
proper proof of split_paired_all (presently unused);
wenzelm
parents:
11777
diff
changeset
|
444 |
|
50104 | 445 |
lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))" |
446 |
by (cases x) simp |
|
447 |
||
11838 | 448 |
text {* |
449 |
The rule @{thm [source] split_paired_all} does not work with the |
|
450 |
Simplifier because it also affects premises in congrence rules, |
|
451 |
where this can lead to premises of the form @{text "!!a b. ... = |
|
452 |
?P(a, b)"} which cannot be solved by reflexivity. |
|
453 |
*} |
|
454 |
||
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
455 |
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
|
456 |
|
26480 | 457 |
ML {* |
11838 | 458 |
(* replace parameters of product type by individual component parameters *) |
459 |
local (* filtering with exists_paired_all is an essential optimization *) |
|
16121 | 460 |
fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) = |
11838 | 461 |
can HOLogic.dest_prodT T orelse exists_paired_all t |
462 |
| exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u |
|
463 |
| exists_paired_all (Abs (_, _, t)) = exists_paired_all t |
|
464 |
| exists_paired_all _ = false; |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
465 |
val ss = |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
466 |
simpset_of |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
467 |
(put_simpset HOL_basic_ss @{context} |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
468 |
addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}] |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
469 |
addsimprocs [@{simproc unit_eq}]); |
11838 | 470 |
in |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
471 |
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
|
472 |
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
|
473 |
|
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
474 |
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
|
475 |
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
|
476 |
|
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
477 |
fun split_all ctxt th = |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
478 |
if exists_paired_all (Thm.prop_of th) |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
479 |
then full_simplify (put_simpset ss ctxt) th else th; |
11838 | 480 |
end; |
26340 | 481 |
*} |
11838 | 482 |
|
51703
f2e92fc0c8aa
modifiers for classical wrappers operate on Proof.context instead of claset;
wenzelm
parents:
51392
diff
changeset
|
483 |
setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac)) *} |
11838 | 484 |
|
47740
a8989fe9a3a5
added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents:
46950
diff
changeset
|
485 |
lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))" |
11838 | 486 |
-- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} |
487 |
by fast |
|
488 |
||
47740
a8989fe9a3a5
added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents:
46950
diff
changeset
|
489 |
lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
490 |
by fast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
491 |
|
47740
a8989fe9a3a5
added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents:
46950
diff
changeset
|
492 |
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))" |
11838 | 493 |
-- {* Can't be added to simpset: loops! *} |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
494 |
by (simp add: split_eta) |
11838 | 495 |
|
496 |
text {* |
|
497 |
Simplification procedure for @{thm [source] cond_split_eta}. Using |
|
498 |
@{thm [source] split_eta} as a rewrite rule is not general enough, |
|
499 |
and using @{thm [source] cond_split_eta} directly would render some |
|
500 |
existing proofs very inefficient; similarly for @{text |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
501 |
split_beta}. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
502 |
*} |
11838 | 503 |
|
26480 | 504 |
ML {* |
11838 | 505 |
local |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
506 |
val cond_split_eta_ss = |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
507 |
simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_split_eta}); |
35364 | 508 |
fun Pair_pat k 0 (Bound m) = (m = k) |
509 |
| Pair_pat k i (Const (@{const_name Pair}, _) $ Bound m $ t) = |
|
510 |
i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t |
|
511 |
| Pair_pat _ _ _ = false; |
|
512 |
fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t |
|
513 |
| no_args k i (t $ u) = no_args k i t andalso no_args k i u |
|
514 |
| no_args k i (Bound m) = m < k orelse m > k + i |
|
515 |
| no_args _ _ _ = true; |
|
516 |
fun split_pat tp i (Abs (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
517 |
| split_pat tp i (Const (@{const_name case_prod}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t |
35364 | 518 |
| split_pat tp i _ = NONE; |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
519 |
fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] [] |
35364 | 520 |
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))) |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
521 |
(K (simp_tac (put_simpset cond_split_eta_ss ctxt) 1))); |
11838 | 522 |
|
35364 | 523 |
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t |
524 |
| beta_term_pat k i (t $ u) = |
|
525 |
Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u) |
|
526 |
| beta_term_pat k i t = no_args k i t; |
|
527 |
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg |
|
528 |
| eta_term_pat _ _ _ = false; |
|
11838 | 529 |
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) |
35364 | 530 |
| subst arg k i (t $ u) = |
531 |
if Pair_pat k i (t $ u) then incr_boundvars k arg |
|
532 |
else (subst arg k i t $ subst arg k i u) |
|
533 |
| subst arg k i t = t; |
|
43595 | 534 |
in |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
535 |
fun beta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t) $ arg) = |
11838 | 536 |
(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
|
537 |
SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f)) |
15531 | 538 |
| NONE => NONE) |
35364 | 539 |
| beta_proc _ _ = NONE; |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
540 |
fun eta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t)) = |
11838 | 541 |
(case split_pat eta_term_pat 1 t of |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
542 |
SOME (_, ft) => SOME (metaeq ctxt s (let val (f $ arg) = ft in f end)) |
15531 | 543 |
| NONE => NONE) |
35364 | 544 |
| eta_proc _ _ = NONE; |
11838 | 545 |
end; |
546 |
*} |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
547 |
simproc_setup split_beta ("split f z") = {* fn _ => fn ctxt => fn ct => beta_proc ctxt (term_of ct) *} |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
548 |
simproc_setup split_eta ("split f") = {* fn _ => fn ctxt => fn ct => eta_proc ctxt (term_of ct) *} |
11838 | 549 |
|
26798
a9134a089106
split_beta is now declared as monotonicity rule, to allow bounded
berghofe
parents:
26588
diff
changeset
|
550 |
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)" |
11838 | 551 |
by (subst surjective_pairing, rule split_conv) |
552 |
||
50104 | 553 |
lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))" |
554 |
by (auto simp: fun_eq_iff) |
|
555 |
||
556 |
||
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
557 |
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))" |
11838 | 558 |
-- {* For use with @{text split} and the Simplifier. *} |
15481 | 559 |
by (insert surj_pair [of p], clarify, simp) |
11838 | 560 |
|
561 |
text {* |
|
562 |
@{thm [source] split_split} could be declared as @{text "[split]"} |
|
563 |
done after the Splitter has been speeded up significantly; |
|
564 |
precompute the constants involved and don't do anything unless the |
|
565 |
current goal contains one of those constants. |
|
566 |
*} |
|
567 |
||
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35427
diff
changeset
|
568 |
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" |
14208 | 569 |
by (subst split_split, simp) |
11838 | 570 |
|
571 |
text {* |
|
572 |
\medskip @{term split} used as a logical connective or set former. |
|
573 |
||
574 |
\medskip These rules are for use with @{text blast}; could instead |
|
40929
7ff03a5e044f
theorem names generated by the (rep_)datatype command now have mandatory qualifiers
huffman
parents:
40702
diff
changeset
|
575 |
call @{text simp} using @{thm [source] prod.split} as rewrite. *} |
11838 | 576 |
|
577 |
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p" |
|
578 |
apply (simp only: split_tupled_all) |
|
579 |
apply (simp (no_asm_simp)) |
|
580 |
done |
|
581 |
||
582 |
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x" |
|
583 |
apply (simp only: split_tupled_all) |
|
584 |
apply (simp (no_asm_simp)) |
|
585 |
done |
|
586 |
||
587 |
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" |
|
37591 | 588 |
by (induct p) auto |
11838 | 589 |
|
590 |
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" |
|
37591 | 591 |
by (induct p) auto |
11838 | 592 |
|
593 |
lemma splitE2: |
|
594 |
"[| Q (split P z); !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R" |
|
595 |
proof - |
|
596 |
assume q: "Q (split P z)" |
|
597 |
assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R" |
|
598 |
show R |
|
599 |
apply (rule r surjective_pairing)+ |
|
600 |
apply (rule split_beta [THEN subst], rule q) |
|
601 |
done |
|
602 |
qed |
|
603 |
||
604 |
lemma splitD': "split R (a,b) c ==> R a b c" |
|
605 |
by simp |
|
606 |
||
607 |
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" |
|
608 |
by simp |
|
609 |
||
610 |
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p" |
|
14208 | 611 |
by (simp only: split_tupled_all, simp) |
11838 | 612 |
|
18372 | 613 |
lemma mem_splitE: |
37166 | 614 |
assumes major: "z \<in> split c p" |
615 |
and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q" |
|
18372 | 616 |
shows Q |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
617 |
by (rule major [unfolded case_prod_unfold] cases surjective_pairing)+ |
11838 | 618 |
|
619 |
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] |
|
620 |
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] |
|
621 |
||
26340 | 622 |
ML {* |
11838 | 623 |
local (* filtering with exists_p_split is an essential optimization *) |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
624 |
fun exists_p_split (Const (@{const_name case_prod},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true |
11838 | 625 |
| exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u |
626 |
| exists_p_split (Abs (_, _, t)) = exists_p_split t |
|
627 |
| exists_p_split _ = false; |
|
628 |
in |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
629 |
fun split_conv_tac ctxt = SUBGOAL (fn (t, i) => |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
630 |
if exists_p_split t |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
631 |
then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms split_conv}) i |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
632 |
else no_tac); |
11838 | 633 |
end; |
26340 | 634 |
*} |
635 |
||
11838 | 636 |
(* This prevents applications of splitE for already splitted arguments leading |
637 |
to quite time-consuming computations (in particular for nested tuples) *) |
|
51703
f2e92fc0c8aa
modifiers for classical wrappers operate on Proof.context instead of claset;
wenzelm
parents:
51392
diff
changeset
|
638 |
setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac)) *} |
11838 | 639 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
640 |
lemma split_eta_SetCompr [simp, no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" |
18372 | 641 |
by (rule ext) fast |
11838 | 642 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
52435
diff
changeset
|
643 |
lemma split_eta_SetCompr2 [simp, no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P" |
18372 | 644 |
by (rule ext) fast |
11838 | 645 |
|
646 |
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" |
|
647 |
-- {* Allows simplifications of nested splits in case of independent predicates. *} |
|
18372 | 648 |
by (rule ext) blast |
11838 | 649 |
|
14337
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
650 |
(* 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
|
651 |
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
|
652 |
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
|
653 |
*) |
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset
|
654 |
lemma split_comp_eq: |
20415 | 655 |
fixes f :: "'a => 'b => 'c" and g :: "'d => 'a" |
656 |
shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" |
|
18372 | 657 |
by (rule ext) auto |
14101 | 658 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
659 |
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
660 |
apply (rule_tac x = "(a, b)" in image_eqI) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
661 |
apply auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
662 |
done |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
663 |
|
11838 | 664 |
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" |
665 |
by blast |
|
666 |
||
667 |
(* |
|
668 |
the following would be slightly more general, |
|
669 |
but cannot be used as rewrite rule: |
|
670 |
### Cannot add premise as rewrite rule because it contains (type) unknowns: |
|
671 |
### ?y = .x |
|
672 |
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)" |
|
14208 | 673 |
by (rtac some_equality 1) |
674 |
by ( Simp_tac 1) |
|
675 |
by (split_all_tac 1) |
|
676 |
by (Asm_full_simp_tac 1) |
|
11838 | 677 |
qed "The_split_eq"; |
678 |
*) |
|
679 |
||
680 |
text {* |
|
681 |
Setup of internal @{text split_rule}. |
|
682 |
*} |
|
683 |
||
55642
63beb38e9258
adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
blanchet
parents:
55469
diff
changeset
|
684 |
lemmas case_prodI = prod.case [THEN iffD2] |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
685 |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
686 |
lemma case_prodI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> case_prod c p" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
687 |
by (fact splitI2) |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
688 |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
689 |
lemma case_prodI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> case_prod c p x" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
690 |
by (fact splitI2') |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
691 |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
692 |
lemma case_prodE: "case_prod c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
693 |
by (fact splitE) |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
694 |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
695 |
lemma case_prodE': "case_prod c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
696 |
by (fact splitE') |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
697 |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
698 |
declare case_prodI [intro!] |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
699 |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
700 |
lemma case_prod_beta: |
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
701 |
"case_prod f p = f (fst p) (snd p)" |
37591 | 702 |
by (fact split_beta) |
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
25885
diff
changeset
|
703 |
|
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset
|
704 |
lemma prod_cases3 [cases type]: |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
705 |
obtains (fields) a b c where "y = (a, b, c)" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
706 |
by (cases y, case_tac b) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
707 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
708 |
lemma prod_induct3 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
709 |
"(!!a b c. P (a, b, c)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
710 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
711 |
|
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset
|
712 |
lemma prod_cases4 [cases type]: |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
713 |
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
|
714 |
by (cases y, case_tac c) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
715 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
716 |
lemma prod_induct4 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
717 |
"(!!a b c d. P (a, b, c, d)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
718 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
719 |
|
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset
|
720 |
lemma prod_cases5 [cases type]: |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
721 |
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
|
722 |
by (cases y, case_tac d) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
723 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
724 |
lemma prod_induct5 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
725 |
"(!!a b c d e. P (a, b, c, d, e)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
726 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
727 |
|
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset
|
728 |
lemma prod_cases6 [cases type]: |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
729 |
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
|
730 |
by (cases y, case_tac e) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
731 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
732 |
lemma prod_induct6 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
733 |
"(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
734 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
735 |
|
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
55414
diff
changeset
|
736 |
lemma prod_cases7 [cases type]: |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
737 |
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
|
738 |
by (cases y, case_tac f) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
739 |
|
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
740 |
lemma prod_induct7 [case_names fields, induct type]: |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
741 |
"(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x" |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
742 |
by (cases x) blast |
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
743 |
|
37166 | 744 |
lemma split_def: |
745 |
"split = (\<lambda>c p. c (fst p) (snd p))" |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
746 |
by (fact case_prod_unfold) |
37166 | 747 |
|
748 |
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where |
|
749 |
"internal_split == split" |
|
750 |
||
751 |
lemma internal_split_conv: "internal_split c (a, b) = c a b" |
|
752 |
by (simp only: internal_split_def split_conv) |
|
753 |
||
48891 | 754 |
ML_file "Tools/split_rule.ML" |
37166 | 755 |
setup Split_Rule.setup |
756 |
||
757 |
hide_const internal_split |
|
758 |
||
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
759 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
760 |
subsubsection {* Derived operations *} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
761 |
|
37387
3581483cca6c
qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents:
37278
diff
changeset
|
762 |
definition curry :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where |
3581483cca6c
qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents:
37278
diff
changeset
|
763 |
"curry = (\<lambda>c x y. c (x, y))" |
37166 | 764 |
|
765 |
lemma curry_conv [simp, code]: "curry f a b = f (a, b)" |
|
766 |
by (simp add: curry_def) |
|
767 |
||
768 |
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b" |
|
769 |
by (simp add: curry_def) |
|
770 |
||
771 |
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)" |
|
772 |
by (simp add: curry_def) |
|
773 |
||
774 |
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q" |
|
775 |
by (simp add: curry_def) |
|
776 |
||
777 |
lemma curry_split [simp]: "curry (split f) = f" |
|
778 |
by (simp add: curry_def split_def) |
|
779 |
||
780 |
lemma split_curry [simp]: "split (curry f) = f" |
|
781 |
by (simp add: curry_def split_def) |
|
782 |
||
54630
9061af4d5ebc
restrict admissibility to non-empty chains to allow more syntax-directed proof rules
Andreas Lochbihler
parents:
54147
diff
changeset
|
783 |
lemma curry_K: "curry (\<lambda>x. c) = (\<lambda>x y. c)" |
9061af4d5ebc
restrict admissibility to non-empty chains to allow more syntax-directed proof rules
Andreas Lochbihler
parents:
54147
diff
changeset
|
784 |
by(simp add: fun_eq_iff) |
9061af4d5ebc
restrict admissibility to non-empty chains to allow more syntax-directed proof rules
Andreas Lochbihler
parents:
54147
diff
changeset
|
785 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
786 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
787 |
The composition-uncurry combinator. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
788 |
*} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
789 |
|
37751 | 790 |
notation fcomp (infixl "\<circ>>" 60) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
791 |
|
37751 | 792 |
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
793 |
"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
|
794 |
|
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
795 |
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
|
796 |
by (simp add: fun_eq_iff scomp_def case_prod_unfold) |
37678
0040bafffdef
"prod" and "sum" replace "*" and "+" respectively
haftmann
parents:
37591
diff
changeset
|
797 |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
798 |
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = case_prod g (f x)" |
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55403
diff
changeset
|
799 |
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
|
800 |
|
37751 | 801 |
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x" |
44921 | 802 |
by (simp add: fun_eq_iff) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
803 |
|
37751 | 804 |
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x" |
44921 | 805 |
by (simp add: fun_eq_iff) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
806 |
|
37751 | 807 |
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
|
808 |
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
|
809 |
|
37751 | 810 |
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
|
811 |
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
|
812 |
|
37751 | 813 |
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)" |
44921 | 814 |
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
|
815 |
|
52435
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
52143
diff
changeset
|
816 |
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
|
817 |
constant scomp \<rightharpoonup> (Eval) infixl 3 "#->" |
31202
52d332f8f909
pretty printing of functional combinators for evaluation code
haftmann
parents:
30924
diff
changeset
|
818 |
|
37751 | 819 |
no_notation fcomp (infixl "\<circ>>" 60) |
820 |
no_notation scomp (infixl "\<circ>\<rightarrow>" 60) |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
821 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
822 |
text {* |
40607 | 823 |
@{term map_pair} --- action of the product functor upon |
36664
6302f9ad7047
repaired comments where SOMEthing went utterly wrong (cf. 2b04504fcb69)
krauss
parents:
36622
diff
changeset
|
824 |
functions. |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
825 |
*} |
21195 | 826 |
|
40607 | 827 |
definition map_pair :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where |
828 |
"map_pair f g = (\<lambda>(x, y). (f x, g y))" |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
829 |
|
40607 | 830 |
lemma map_pair_simp [simp, code]: |
831 |
"map_pair f g (a, b) = (f a, g b)" |
|
832 |
by (simp add: map_pair_def) |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
833 |
|
55467
a5c9002bc54d
renamed 'enriched_type' to more informative 'functor' (following the renaming of enriched type constructors to bounded natural functors)
blanchet
parents:
55442
diff
changeset
|
834 |
functor map_pair: map_pair |
44921 | 835 |
by (auto simp add: split_paired_all) |
37278 | 836 |
|
40607 | 837 |
lemma fst_map_pair [simp]: |
838 |
"fst (map_pair f g x) = f (fst x)" |
|
839 |
by (cases x) simp_all |
|
37278 | 840 |
|
40607 | 841 |
lemma snd_prod_fun [simp]: |
842 |
"snd (map_pair f g x) = g (snd x)" |
|
843 |
by (cases x) simp_all |
|
37278 | 844 |
|
40607 | 845 |
lemma fst_comp_map_pair [simp]: |
846 |
"fst \<circ> map_pair f g = f \<circ> fst" |
|
847 |
by (rule ext) simp_all |
|
37278 | 848 |
|
40607 | 849 |
lemma snd_comp_map_pair [simp]: |
850 |
"snd \<circ> map_pair f g = g \<circ> snd" |
|
851 |
by (rule ext) simp_all |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
852 |
|
40607 | 853 |
lemma map_pair_compose: |
854 |
"map_pair (f1 o f2) (g1 o g2) = (map_pair f1 g1 o map_pair f2 g2)" |
|
855 |
by (rule ext) (simp add: map_pair.compositionality comp_def) |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
856 |
|
40607 | 857 |
lemma map_pair_ident [simp]: |
858 |
"map_pair (%x. x) (%y. y) = (%z. z)" |
|
859 |
by (rule ext) (simp add: map_pair.identity) |
|
860 |
||
861 |
lemma map_pair_imageI [intro]: |
|
862 |
"(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_pair f g ` R" |
|
863 |
by (rule image_eqI) simp_all |
|
21195 | 864 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
865 |
lemma prod_fun_imageE [elim!]: |
40607 | 866 |
assumes major: "c \<in> map_pair f g ` R" |
867 |
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
|
868 |
shows P |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
869 |
apply (rule major [THEN imageE]) |
37166 | 870 |
apply (case_tac x) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
871 |
apply (rule cases) |
40607 | 872 |
apply simp_all |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
873 |
done |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
874 |
|
37166 | 875 |
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where |
40607 | 876 |
"apfst f = map_pair f id" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
877 |
|
37166 | 878 |
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where |
40607 | 879 |
"apsnd f = map_pair id f" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
880 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
881 |
lemma apfst_conv [simp, code]: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
882 |
"apfst f (x, y) = (f x, y)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
883 |
by (simp add: apfst_def) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
884 |
|
33638
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
885 |
lemma apsnd_conv [simp, code]: |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
886 |
"apsnd f (x, y) = (x, f y)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
887 |
by (simp add: apsnd_def) |
21195 | 888 |
|
33594 | 889 |
lemma fst_apfst [simp]: |
890 |
"fst (apfst f x) = f (fst x)" |
|
891 |
by (cases x) simp |
|
892 |
||
51173 | 893 |
lemma fst_comp_apfst [simp]: |
894 |
"fst \<circ> apfst f = f \<circ> fst" |
|
895 |
by (simp add: fun_eq_iff) |
|
896 |
||
33594 | 897 |
lemma fst_apsnd [simp]: |
898 |
"fst (apsnd f x) = fst x" |
|
899 |
by (cases x) simp |
|
900 |
||
51173 | 901 |
lemma fst_comp_apsnd [simp]: |
902 |
"fst \<circ> apsnd f = fst" |
|
903 |
by (simp add: fun_eq_iff) |
|
904 |
||
33594 | 905 |
lemma snd_apfst [simp]: |
906 |
"snd (apfst f x) = snd x" |
|
907 |
by (cases x) simp |
|
908 |
||
51173 | 909 |
lemma snd_comp_apfst [simp]: |
910 |
"snd \<circ> apfst f = snd" |
|
911 |
by (simp add: fun_eq_iff) |
|
912 |
||
33594 | 913 |
lemma snd_apsnd [simp]: |
914 |
"snd (apsnd f x) = f (snd x)" |
|
915 |
by (cases x) simp |
|
916 |
||
51173 | 917 |
lemma snd_comp_apsnd [simp]: |
918 |
"snd \<circ> apsnd f = f \<circ> snd" |
|
919 |
by (simp add: fun_eq_iff) |
|
920 |
||
33594 | 921 |
lemma apfst_compose: |
922 |
"apfst f (apfst g x) = apfst (f \<circ> g) x" |
|
923 |
by (cases x) simp |
|
924 |
||
925 |
lemma apsnd_compose: |
|
926 |
"apsnd f (apsnd g x) = apsnd (f \<circ> g) x" |
|
927 |
by (cases x) simp |
|
928 |
||
929 |
lemma apfst_apsnd [simp]: |
|
930 |
"apfst f (apsnd g x) = (f (fst x), g (snd x))" |
|
931 |
by (cases x) simp |
|
932 |
||
933 |
lemma apsnd_apfst [simp]: |
|
934 |
"apsnd f (apfst g x) = (g (fst x), f (snd x))" |
|
935 |
by (cases x) simp |
|
936 |
||
937 |
lemma apfst_id [simp] : |
|
938 |
"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
|
939 |
by (simp add: fun_eq_iff) |
33594 | 940 |
|
941 |
lemma apsnd_id [simp] : |
|
942 |
"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
|
943 |
by (simp add: fun_eq_iff) |
33594 | 944 |
|
945 |
lemma apfst_eq_conv [simp]: |
|
946 |
"apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)" |
|
947 |
by (cases x) simp |
|
948 |
||
949 |
lemma apsnd_eq_conv [simp]: |
|
950 |
"apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)" |
|
951 |
by (cases x) simp |
|
952 |
||
33638
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
953 |
lemma apsnd_apfst_commute: |
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
954 |
"apsnd f (apfst g p) = apfst g (apsnd f p)" |
548a34929e98
Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents:
33594
diff
changeset
|
955 |
by simp |
21195 | 956 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
957 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
958 |
Disjoint union of a family of sets -- Sigma. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
959 |
*} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
960 |
|
45986
c9e50153e5ae
moved various set operations to theory Set (resp. Product_Type)
haftmann
parents:
45696
diff
changeset
|
961 |
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
962 |
Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
963 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
964 |
abbreviation |
45986
c9e50153e5ae
moved various set operations to theory Set (resp. Product_Type)
haftmann
parents:
45696
diff
changeset
|
965 |
Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
966 |
(infixr "<*>" 80) where |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
967 |
"A <*> B == Sigma A (%_. B)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
968 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
969 |
notation (xsymbols) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
970 |
Times (infixr "\<times>" 80) |
15394 | 971 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
972 |
notation (HTML output) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
973 |
Times (infixr "\<times>" 80) |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
974 |
|
45662
4f7c05990420
Hide Product_Type.Times - too precious an identifier
nipkow
parents:
45607
diff
changeset
|
975 |
hide_const (open) Times |
4f7c05990420
Hide Product_Type.Times - too precious an identifier
nipkow
parents:
45607
diff
changeset
|
976 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
977 |
syntax |
35115 | 978 |
"_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10) |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
979 |
translations |
35115 | 980 |
"SIGMA x:A. B" == "CONST Sigma A (%x. B)" |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
981 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
982 |
lemma SigmaI [intro!]: "[| a:A; b:B(a) |] ==> (a,b) : Sigma A B" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
983 |
by (unfold Sigma_def) blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
984 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
985 |
lemma SigmaE [elim!]: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
986 |
"[| c: Sigma A B; |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
987 |
!!x y.[| x:A; y:B(x); c=(x,y) |] ==> P |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
988 |
|] ==> P" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
989 |
-- {* The general elimination rule. *} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
990 |
by (unfold Sigma_def) blast |
20588 | 991 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
992 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
993 |
Elimination of @{term "(a, b) : A \<times> B"} -- introduces no |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
994 |
eigenvariables. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
995 |
*} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
996 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
997 |
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
998 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
999 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1000 |
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1001 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1002 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1003 |
lemma SigmaE2: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1004 |
"[| (a, b) : Sigma A B; |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1005 |
[| a:A; b:B(a) |] ==> P |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1006 |
|] ==> P" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1007 |
by blast |
20588 | 1008 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1009 |
lemma Sigma_cong: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1010 |
"\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk> |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1011 |
\<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1012 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1013 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1014 |
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1015 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1016 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1017 |
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1018 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1019 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1020 |
lemma Sigma_empty2 [simp]: "A <*> {} = {}" |
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 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1023 |
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1024 |
by auto |
21908 | 1025 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1026 |
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1027 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1028 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1029 |
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1030 |
by auto |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1031 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1032 |
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1033 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1034 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1035 |
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1036 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1037 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1038 |
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1039 |
by (blast elim: equalityE) |
20588 | 1040 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1041 |
lemma SetCompr_Sigma_eq: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1042 |
"Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1043 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1044 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1045 |
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1046 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1047 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1048 |
lemma UN_Times_distrib: |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1049 |
"(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1050 |
-- {* Suggested by Pierre Chartier *} |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1051 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1052 |
|
47740
a8989fe9a3a5
added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents:
46950
diff
changeset
|
1053 |
lemma split_paired_Ball_Sigma [simp, no_atp]: |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1054 |
"(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1055 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1056 |
|
47740
a8989fe9a3a5
added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents:
46950
diff
changeset
|
1057 |
lemma split_paired_Bex_Sigma [simp, no_atp]: |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1058 |
"(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1059 |
by blast |
21908 | 1060 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1061 |
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1062 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1063 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1064 |
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1065 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1066 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1067 |
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1068 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1069 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1070 |
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1071 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1072 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1073 |
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1074 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1075 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1076 |
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1077 |
by blast |
21908 | 1078 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1079 |
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)" |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1080 |
by blast |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1081 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1082 |
text {* |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1083 |
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
|
1084 |
matching, especially when the rules are re-oriented. |
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1085 |
*} |
21908 | 1086 |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1087 |
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" |
28719 | 1088 |
by blast |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1089 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1090 |
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" |
28719 | 1091 |
by blast |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1092 |
|
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1093 |
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)" |
28719 | 1094 |
by blast |
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1095 |
|
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
|
1096 |
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1097 |
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
|
1098 |
|
50104 | 1099 |
lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))" |
1100 |
by auto |
|
1101 |
||
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
|
1102 |
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)" |
44921 | 1103 |
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
|
1104 |
|
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
|
1105 |
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)" |
44921 | 1106 |
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
|
1107 |
|
28719 | 1108 |
lemma insert_times_insert[simp]: |
1109 |
"insert a A \<times> insert b B = |
|
1110 |
insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)" |
|
1111 |
by blast |
|
26358
d6a508c16908
Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents:
26340
diff
changeset
|
1112 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset
|
1113 |
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)" |
47988 | 1114 |
apply auto |
1115 |
apply (case_tac "f x") |
|
1116 |
apply auto |
|
1117 |
done |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33089
diff
changeset
|
1118 |
|
50104 | 1119 |
lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)" |
1120 |
by auto |
|
1121 |
||
35822 | 1122 |
lemma swap_inj_on: |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1123 |
"inj_on (\<lambda>(i, j). (j, i)) A" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1124 |
by (auto intro!: inj_onI) |
35822 | 1125 |
|
1126 |
lemma swap_product: |
|
1127 |
"(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A" |
|
1128 |
by (simp add: split_def image_def) blast |
|
1129 |
||
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
|
1130 |
lemma image_split_eq_Sigma: |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1131 |
"(\<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
|
1132 |
proof (safe intro!: imageI) |
36622
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1133 |
fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b" |
e393a91f86df
Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents:
36176
diff
changeset
|
1134 |
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
|
1135 |
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
|
1136 |
qed simp_all |
35822 | 1137 |
|
46128
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1138 |
definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1139 |
[code_abbrev]: "product A B = A \<times> B" |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1140 |
|
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1141 |
hide_const (open) product |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1142 |
|
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1143 |
lemma member_product: |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1144 |
"x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B" |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1145 |
by (simp add: product_def) |
53e7cc599f58
interaction of set operations for execution and membership predicate
haftmann
parents:
46028
diff
changeset
|
1146 |
|
40607 | 1147 |
text {* The following @{const map_pair} lemmas are due to Joachim Breitner: *} |
1148 |
||
1149 |
lemma map_pair_inj_on: |
|
1150 |
assumes "inj_on f A" and "inj_on g B" |
|
1151 |
shows "inj_on (map_pair f g) (A \<times> B)" |
|
1152 |
proof (rule inj_onI) |
|
1153 |
fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c" |
|
1154 |
assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto |
|
1155 |
assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto |
|
1156 |
assume "map_pair f g x = map_pair f g y" |
|
1157 |
hence "fst (map_pair f g x) = fst (map_pair f g y)" by (auto) |
|
1158 |
hence "f (fst x) = f (fst y)" by (cases x,cases y,auto) |
|
1159 |
with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A` |
|
1160 |
have "fst x = fst y" by (auto dest:dest:inj_onD) |
|
1161 |
moreover from `map_pair f g x = map_pair f g y` |
|
1162 |
have "snd (map_pair f g x) = snd (map_pair f g y)" by (auto) |
|
1163 |
hence "g (snd x) = g (snd y)" by (cases x,cases y,auto) |
|
1164 |
with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B` |
|
1165 |
have "snd x = snd y" by (auto dest:dest:inj_onD) |
|
1166 |
ultimately show "x = y" by(rule prod_eqI) |
|
1167 |
qed |
|
1168 |
||
1169 |
lemma map_pair_surj: |
|
40702 | 1170 |
fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd" |
40607 | 1171 |
assumes "surj f" and "surj g" |
1172 |
shows "surj (map_pair f g)" |
|
1173 |
unfolding surj_def |
|
1174 |
proof |
|
1175 |
fix y :: "'b \<times> 'd" |
|
1176 |
from `surj f` obtain a where "fst y = f a" by (auto elim:surjE) |
|
1177 |
moreover |
|
1178 |
from `surj g` obtain b where "snd y = g b" by (auto elim:surjE) |
|
1179 |
ultimately have "(fst y, snd y) = map_pair f g (a,b)" by auto |
|
1180 |
thus "\<exists>x. y = map_pair f g x" by auto |
|
1181 |
qed |
|
1182 |
||
1183 |
lemma map_pair_surj_on: |
|
1184 |
assumes "f ` A = A'" and "g ` B = B'" |
|
1185 |
shows "map_pair f g ` (A \<times> B) = A' \<times> B'" |
|
1186 |
unfolding image_def |
|
1187 |
proof(rule set_eqI,rule iffI) |
|
1188 |
fix x :: "'a \<times> 'c" |
|
1189 |
assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = map_pair f g x}" |
|
1190 |
then obtain y where "y \<in> A \<times> B" and "x = map_pair f g y" by blast |
|
1191 |
from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto |
|
1192 |
moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto |
|
1193 |
ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto |
|
1194 |
with `x = map_pair f g y` show "x \<in> A' \<times> B'" by (cases y, auto) |
|
1195 |
next |
|
1196 |
fix x :: "'a \<times> 'c" |
|
1197 |
assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto |
|
1198 |
from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto |
|
1199 |
then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE) |
|
1200 |
moreover from `image g B = B'` and `snd x \<in> B'` |
|
1201 |
obtain b where "b \<in> B" and "snd x = g b" by auto |
|
1202 |
ultimately have "(fst x, snd x) = map_pair f g (a,b)" by auto |
|
1203 |
moreover from `a \<in> A` and `b \<in> B` have "(a , b) \<in> A \<times> B" by auto |
|
1204 |
ultimately have "\<exists>y \<in> A \<times> B. x = map_pair f g y" by auto |
|
1205 |
thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_pair f g y}" by auto |
|
1206 |
qed |
|
1207 |
||
21908 | 1208 |
|
49764
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1209 |
subsection {* Simproc for rewriting a set comprehension into a pointfree expression *} |
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1210 |
|
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1211 |
ML_file "Tools/set_comprehension_pointfree.ML" |
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1212 |
|
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1213 |
setup {* |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51703
diff
changeset
|
1214 |
Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs |
49764
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1215 |
[Raw_Simplifier.make_simproc {name = "set comprehension", lhss = [@{cpat "Collect ?P"}], |
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1216 |
proc = K Set_Comprehension_Pointfree.code_simproc, identifier = []}]) |
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1217 |
*} |
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1218 |
|
9979d64b8016
moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents:
48891
diff
changeset
|
1219 |
|
37166 | 1220 |
subsection {* Inductively defined sets *} |
15394 | 1221 |
|
48891 | 1222 |
ML_file "Tools/inductive_set.ML" |
31723
f5cafe803b55
discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents:
31667
diff
changeset
|
1223 |
setup Inductive_Set.setup |
24699
c6674504103f
datatype interpretators for size and datatype_realizer
haftmann
parents:
24286
diff
changeset
|
1224 |
|
37166 | 1225 |
|
1226 |
subsection {* Legacy theorem bindings and duplicates *} |
|
1227 |
||
1228 |
lemma PairE: |
|
1229 |
obtains x y where "p = (x, y)" |
|
1230 |
by (fact prod.exhaust) |
|
1231 |
||
1232 |
lemmas Pair_eq = prod.inject |
|
55393
ce5cebfaedda
se 'wrap_free_constructors' to register 'sum' , 'prod', 'unit', 'bool' with their discriminators/selectors
blanchet
parents:
54630
diff
changeset
|
1233 |
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
|
1234 |
lemmas snd_conv = prod.sel(2) |
ce5cebfaedda
se 'wrap_free_constructors' to register 'sum' , 'prod', 'unit', 'bool' with their discriminators/selectors
blanchet
parents:
54630
diff
changeset
|
1235 |
lemmas pair_collapse = prod.collapse |
ce5cebfaedda
se 'wrap_free_constructors' to register 'sum' , 'prod', 'unit', 'bool' with their discriminators/selectors
blanchet
parents:
54630
diff
changeset
|
1236 |
lemmas split = split_conv |
44066
d74182c93f04
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents:
43866
diff
changeset
|
1237 |
lemmas Pair_fst_snd_eq = prod_eq_iff |
d74182c93f04
rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents:
43866
diff
changeset
|
1238 |
|
45204
5e4a1270c000
hide typedef-generated constants Product_Type.prod and Sum_Type.sum
huffman
parents:
44921
diff
changeset
|
1239 |
hide_const (open) prod |
5e4a1270c000
hide typedef-generated constants Product_Type.prod and Sum_Type.sum
huffman
parents:
44921
diff
changeset
|
1240 |
|
10213 | 1241 |
end |