src/HOL/Datatype_Examples/Derivation_Trees/DTree.thy
author wenzelm
Sun Nov 02 18:21:45 2014 +0100 (2014-11-02)
changeset 58889 5b7a9633cfa8
parent 58309 a09ec6daaa19
child 63167 0909deb8059b
permissions -rw-r--r--
modernized header uniformly as section;
     1 (*  Title:      HOL/Datatype_Examples/Derivation_Trees/DTree.thy
     2     Author:     Andrei Popescu, TU Muenchen
     3     Copyright   2012
     4 
     5 Derivation trees with nonterminal internal nodes and terminal leaves.
     6 *)
     7 
     8 section {* Trees with Nonterminal Internal Nodes and Terminal Leaves *}
     9 
    10 theory DTree
    11 imports Prelim
    12 begin
    13 
    14 typedecl N
    15 typedecl T
    16 
    17 codatatype dtree = NNode (root: N) (ccont: "(T + dtree) fset")
    18 
    19 subsection{* Transporting the Characteristic Lemmas from @{text "fset"} to @{text "set"} *}
    20 
    21 definition "Node n as \<equiv> NNode n (the_inv fset as)"
    22 definition "cont \<equiv> fset o ccont"
    23 definition "unfold rt ct \<equiv> corec_dtree rt (the_inv fset o image (map_sum id Inr) o ct)"
    24 definition "corec rt ct \<equiv> corec_dtree rt (the_inv fset o ct)"
    25 
    26 lemma finite_cont[simp]: "finite (cont tr)"
    27   unfolding cont_def comp_apply by (cases tr, clarsimp)
    28 
    29 lemma Node_root_cont[simp]:
    30   "Node (root tr) (cont tr) = tr"
    31   unfolding Node_def cont_def comp_apply
    32   apply (rule trans[OF _ dtree.collapse])
    33   apply (rule arg_cong2[OF refl the_inv_into_f_f[unfolded inj_on_def]])
    34   apply (simp_all add: fset_inject)
    35   done
    36 
    37 lemma dtree_simps[simp]:
    38 assumes "finite as" and "finite as'"
    39 shows "Node n as = Node n' as' \<longleftrightarrow> n = n' \<and> as = as'"
    40 using assms dtree.inject unfolding Node_def
    41 by (metis fset_to_fset)
    42 
    43 lemma dtree_cases[elim, case_names Node Choice]:
    44 assumes Node: "\<And> n as. \<lbrakk>finite as; tr = Node n as\<rbrakk> \<Longrightarrow> phi"
    45 shows phi
    46 apply(cases rule: dtree.exhaust[of tr])
    47 using Node unfolding Node_def
    48 by (metis Node Node_root_cont finite_cont)
    49 
    50 lemma dtree_sel_ctr[simp]:
    51 "root (Node n as) = n"
    52 "finite as \<Longrightarrow> cont (Node n as) = as"
    53 unfolding Node_def cont_def by auto
    54 
    55 lemmas root_Node = dtree_sel_ctr(1)
    56 lemmas cont_Node = dtree_sel_ctr(2)
    57 
    58 lemma dtree_cong:
    59 assumes "root tr = root tr'" and "cont tr = cont tr'"
    60 shows "tr = tr'"
    61 by (metis Node_root_cont assms)
    62 
    63 lemma rel_set_cont:
    64 "rel_set \<chi> (cont tr1) (cont tr2) = rel_fset \<chi> (ccont tr1) (ccont tr2)"
    65 unfolding cont_def comp_def rel_fset_fset ..
    66 
    67 lemma dtree_coinduct[elim, consumes 1, case_names Lift, induct pred: "HOL.eq"]:
    68 assumes phi: "\<phi> tr1 tr2" and
    69 Lift: "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow>
    70                   root tr1 = root tr2 \<and> rel_set (rel_sum op = \<phi>) (cont tr1) (cont tr2)"
    71 shows "tr1 = tr2"
    72 using phi apply(elim dtree.coinduct)
    73 apply(rule Lift[unfolded rel_set_cont]) .
    74 
    75 lemma unfold:
    76 "root (unfold rt ct b) = rt b"
    77 "finite (ct b) \<Longrightarrow> cont (unfold rt ct b) = image (id \<oplus> unfold rt ct) (ct b)"
    78 using dtree.corec_sel[of rt "the_inv fset o image (map_sum id Inr) o ct" b] unfolding unfold_def
    79 apply blast
    80 unfolding cont_def comp_def
    81 by (simp add: case_sum_o_inj map_sum.compositionality image_image)
    82 
    83 lemma corec:
    84 "root (corec rt ct b) = rt b"
    85 "finite (ct b) \<Longrightarrow> cont (corec rt ct b) = image (id \<oplus> ([[id, corec rt ct]])) (ct b)"
    86 using dtree.corec_sel[of rt "the_inv fset \<circ> ct" b] unfolding corec_def
    87 unfolding cont_def comp_def id_def
    88 by simp_all
    89 
    90 end