author | blanchet |
Tue, 04 Sep 2012 17:23:08 +0200 | |
changeset 49128 | 1a86ef0a0210 |
parent 49074 | d8af889dcbe3 |
child 49220 | a6260b4fb410 |
permissions | -rw-r--r-- |
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(* Title: Codatatype_Examples/ListF.thy |
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Author: Dmitriy Traytel, TU Muenchen |
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Author: Andrei Popescu, TU Muenchen |
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Copyright 2012 |
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Finite lists. |
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*) |
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header {* Finite Lists *} |
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theory ListF |
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imports "../Codatatype" |
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begin |
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data_raw listF: 'list = "unit + 'a \<times> 'list" |
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definition "NilF = listF_fld (Inl ())" |
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definition "Conss a as \<equiv> listF_fld (Inr (a, as))" |
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lemma listF_map_NilF[simp]: "listF_map f NilF = NilF" |
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unfolding listF_map_def listFBNF_map_def NilF_def listF.fld_iter by simp |
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lemma listF_map_Conss[simp]: |
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"listF_map f (Conss x xs) = Conss (f x) (listF_map f xs)" |
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unfolding listF_map_def listFBNF_map_def Conss_def listF.fld_iter by simp |
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lemma listF_set_NilF[simp]: "listF_set NilF = {}" |
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unfolding listF_set_def NilF_def listF.fld_iter listFBNF_set1_def listFBNF_set2_def |
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sum_set_defs listFBNF_map_def collect_def[abs_def] by simp |
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lemma listF_set_Conss[simp]: "listF_set (Conss x xs) = {x} \<union> listF_set xs" |
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unfolding listF_set_def Conss_def listF.fld_iter listFBNF_set1_def listFBNF_set2_def |
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sum_set_defs prod_set_defs listFBNF_map_def collect_def[abs_def] by simp |
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lemma iter_sum_case_NilF: "listF_fld_iter (sum_case f g) NilF = f ()" |
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unfolding NilF_def listF.fld_iter listFBNF_map_def by simp |
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lemma iter_sum_case_Conss: |
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"listF_fld_iter (sum_case f g) (Conss y ys) = g (y, listF_fld_iter (sum_case f g) ys)" |
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unfolding Conss_def listF.fld_iter listFBNF_map_def by simp |
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(* familiar induction principle *) |
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lemma listF_induct: |
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fixes xs :: "'a listF" |
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assumes IB: "P NilF" and IH: "\<And>x xs. P xs \<Longrightarrow> P (Conss x xs)" |
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shows "P xs" |
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proof (rule listF.fld_induct) |
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fix xs :: "unit + 'a \<times> 'a listF" |
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assume raw_IH: "\<And>a. a \<in> listFBNF_set2 xs \<Longrightarrow> P a" |
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show "P (listF_fld xs)" |
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proof (cases xs) |
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case (Inl a) with IB show ?thesis unfolding NilF_def by simp |
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next |
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case (Inr b) |
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then obtain y ys where yys: "listF_fld xs = Conss y ys" |
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unfolding Conss_def listF.fld_inject by (blast intro: prod.exhaust) |
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hence "ys \<in> listFBNF_set2 xs" |
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unfolding listFBNF_set2_def Conss_def listF.fld_inject sum_set_defs prod_set_defs |
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collect_def[abs_def] by simp |
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with raw_IH have "P ys" by blast |
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with IH have "P (Conss y ys)" by blast |
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with yys show ?thesis by simp |
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qed |
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qed |
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rep_datatype NilF Conss |
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by (blast intro: listF_induct) (auto simp add: NilF_def Conss_def listF.fld_inject) |
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definition Singll ("[[_]]") where |
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[simp]: "Singll a \<equiv> Conss a NilF" |
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definition appendd (infixr "@@" 65) where |
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"appendd \<equiv> listF_fld_iter (sum_case (\<lambda> _. id) (\<lambda> (a,f) bs. Conss a (f bs)))" |
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definition "lrev \<equiv> listF_fld_iter (sum_case (\<lambda> _. NilF) (\<lambda> (b,bs). bs @@ [[b]]))" |
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lemma lrev_NilF[simp]: "lrev NilF = NilF" |
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unfolding lrev_def by (simp add: iter_sum_case_NilF) |
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lemma lrev_Conss[simp]: "lrev (Conss y ys) = lrev ys @@ [[y]]" |
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unfolding lrev_def by (simp add: iter_sum_case_Conss) |
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lemma NilF_appendd[simp]: "NilF @@ ys = ys" |
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unfolding appendd_def by (simp add: iter_sum_case_NilF) |
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lemma Conss_append[simp]: "Conss x xs @@ ys = Conss x (xs @@ ys)" |
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unfolding appendd_def by (simp add: iter_sum_case_Conss) |
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lemma appendd_NilF[simp]: "xs @@ NilF = xs" |
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by (rule listF_induct) auto |
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lemma appendd_assoc[simp]: "(xs @@ ys) @@ zs = xs @@ ys @@ zs" |
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by (rule listF_induct) auto |
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lemma lrev_appendd[simp]: "lrev (xs @@ ys) = lrev ys @@ lrev xs" |
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by (rule listF_induct[of _ xs]) auto |
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lemma listF_map_appendd[simp]: |
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"listF_map f (xs @@ ys) = listF_map f xs @@ listF_map f ys" |
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by (rule listF_induct[of _ xs]) auto |
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lemma lrev_listF_map[simp]: "lrev (listF_map f xs) = listF_map f (lrev xs)" |
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by (rule listF_induct[of _ xs]) auto |
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added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
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105 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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106 |
lemma lrev_lrev[simp]: "lrev (lrev as) = as" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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107 |
by (rule listF_induct) auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
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|
108 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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109 |
fun lengthh where |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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110 |
"lengthh NilF = 0" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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111 |
| "lengthh (Conss x xs) = Suc (lengthh xs)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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|
112 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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113 |
fun nthh where |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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114 |
"nthh (Conss x xs) 0 = x" |
7f79f94a432c
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parents:
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115 |
| "nthh (Conss x xs) (Suc n) = nthh xs n" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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116 |
| "nthh xs i = undefined" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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|
117 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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118 |
lemma lengthh_listF_map[simp]: "lengthh (listF_map f xs) = lengthh xs" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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119 |
by (rule listF_induct[of _ xs]) auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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|
120 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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|
121 |
lemma nthh_listF_map[simp]: |
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parents:
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122 |
"i < lengthh xs \<Longrightarrow> nthh (listF_map f xs) i = f (nthh xs i)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
changeset
|
123 |
by (induct rule: nthh.induct) auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
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|
124 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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125 |
lemma nthh_listF_set[simp]: "i < lengthh xs \<Longrightarrow> nthh xs i \<in> listF_set xs" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
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|
126 |
by (induct rule: nthh.induct) auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
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|
127 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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|
128 |
lemma NilF_iff[iff]: "(lengthh xs = 0) = (xs = NilF)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
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|
129 |
by (induct xs) auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
changeset
|
130 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
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|
131 |
lemma Conss_iff[iff]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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|
132 |
"(lengthh xs = Suc n) = (\<exists>y ys. xs = Conss y ys \<and> lengthh ys = n)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
changeset
|
133 |
by (induct xs) auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
changeset
|
134 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
diff
changeset
|
135 |
lemma Conss_iff'[iff]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
136 |
"(Suc n = lengthh xs) = (\<exists>y ys. xs = Conss y ys \<and> lengthh ys = n)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
137 |
by (induct xs) (simp, simp, blast) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
138 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
139 |
lemma listF_induct2: "\<lbrakk>lengthh xs = lengthh ys; P NilF NilF; |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
140 |
\<And>x xs y ys. P xs ys \<Longrightarrow> P (Conss x xs) (Conss y ys)\<rbrakk> \<Longrightarrow> P xs ys" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
141 |
by (induct xs arbitrary: ys rule: listF_induct) auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
142 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
143 |
fun zipp where |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
144 |
"zipp NilF NilF = NilF" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
145 |
| "zipp (Conss x xs) (Conss y ys) = Conss (x, y) (zipp xs ys)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
146 |
| "zipp xs ys = undefined" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
147 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
148 |
lemma listF_map_fst_zip[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
149 |
"lengthh xs = lengthh ys \<Longrightarrow> listF_map fst (zipp xs ys) = xs" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
150 |
by (erule listF_induct2) auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
151 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
152 |
lemma listF_map_snd_zip[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
153 |
"lengthh xs = lengthh ys \<Longrightarrow> listF_map snd (zipp xs ys) = ys" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
154 |
by (erule listF_induct2) auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
155 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
156 |
lemma lengthh_zip[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
157 |
"lengthh xs = lengthh ys \<Longrightarrow> lengthh (zipp xs ys) = lengthh xs" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
158 |
by (erule listF_induct2) auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
159 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
160 |
lemma nthh_zip[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
161 |
assumes *: "lengthh xs = lengthh ys" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
162 |
shows "i < lengthh xs \<Longrightarrow> nthh (zipp xs ys) i = (nthh xs i, nthh ys i)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
163 |
proof (induct arbitrary: i rule: listF_induct2[OF *]) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
164 |
case (2 x xs y ys) thus ?case by (induct i) auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
165 |
qed simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
166 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
167 |
lemma list_set_nthh[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
168 |
"(x \<in> listF_set xs) \<Longrightarrow> (\<exists>i < lengthh xs. nthh xs i = x)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
169 |
by (induct xs) (auto, induct rule: nthh.induct, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
170 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
171 |
end |