header{*Examples of Reasoning in ZF Set Theory*}

 theory ZF_examples imports Main_ZFC begin

 subsection {* Binary Trees *}

 consts

   bt :: "i => i"

 datatype "bt(A)" =

   Lf | Br ("a \<in> A", "t1 \<in> bt(A)", "t2 \<in> bt(A)")

 declare bt.intros [simp]

 text{*Induction via tactic emulation*}

 lemma Br_neq_left [rule_format]: "l \<in> bt(A) ==> \<forall>x r. Br(x, l, r) \<noteq> l"

   --{* @{subgoals[display,indent=0,margin=65]} *}

   apply (induct_tac l)

   --{* @{subgoals[display,indent=0,margin=65]} *}

   apply auto

   done

 (*

   apply (Inductive.case_tac l)

   apply (tactic {*exhaust_tac "l" 1*})

 *)

 text{*The new induction method, which I don't understand*}

 lemma Br_neq_left': "l \<in> bt(A) ==> (!!x r. Br(x, l, r) \<noteq> l)"

   --{* @{subgoals[display,indent=0,margin=65]} *}

   apply (induct set: bt)

   --{* @{subgoals[display,indent=0,margin=65]} *}

   apply auto

   done

 lemma Br_iff: "Br(a,l,r) = Br(a',l',r') <-> a=a' & l=l' & r=r'"

   -- "Proving a freeness theorem."

   by (blast elim!: bt.free_elims)

 inductive_cases Br_in_bt: "Br(a,l,r) \<in> bt(A)"

   -- "An elimination rule, for type-checking."

 text {*

 @{thm[display] Br_in_bt[no_vars]}

 *};

 subsection{*Primitive recursion*}

 consts  n_nodes :: "i => i"

 primrec

   "n_nodes(Lf) = 0"

   "n_nodes(Br(a,l,r)) = succ(n_nodes(l) #+ n_nodes(r))"

 lemma n_nodes_type [simp]: "t \<in> bt(A) ==> n_nodes(t) \<in> nat"

   by (induct_tac t, auto) 

 consts  n_nodes_aux :: "i => i"

 primrec

   "n_nodes_aux(Lf) = (\<lambda>k \<in> nat. k)"

   "n_nodes_aux(Br(a,l,r)) = 

       (\<lambda>k \<in> nat. n_nodes_aux(r) `  (n_nodes_aux(l) ` succ(k)))"

 lemma n_nodes_aux_eq [rule_format]:

      "t \<in> bt(A) ==> \<forall>k \<in> nat. n_nodes_aux(t)`k = n_nodes(t) #+ k"

   by (induct_tac t, simp_all) 

 definition n_nodes_tail :: "i => i" where

    "n_nodes_tail(t) == n_nodes_aux(t) ` 0"

 lemma "t \<in> bt(A) ==> n_nodes_tail(t) = n_nodes(t)"

  by (simp add: n_nodes_tail_def n_nodes_aux_eq) 

 subsection {*Inductive definitions*}

 consts  Fin       :: "i=>i"

 inductive

   domains   "Fin(A)" \<subseteq> "Pow(A)"

   intros

     emptyI:  "0 \<in> Fin(A)"

     consI:   "[| a \<in> A;  b \<in> Fin(A) |] ==> cons(a,b) \<in> Fin(A)"

   type_intros  empty_subsetI cons_subsetI PowI

   type_elims   PowD [THEN revcut_rl]

 consts  acc :: "i => i"

 inductive

   domains "acc(r)" \<subseteq> "field(r)"

   intros

     vimage:  "[| r-``{a}: Pow(acc(r)); a \<in> field(r) |] ==> a \<in> acc(r)"

   monos      Pow_mono

 consts

   llist  :: "i=>i";

 codatatype

   "llist(A)" = LNil | LCons ("a \<in> A", "l \<in> llist(A)")

 (*Coinductive definition of equality*)

 consts

   lleq :: "i=>i"

 (*Previously used <*> in the domain and variant pairs as elements.  But

   standard pairs work just as well.  To use variant pairs, must change prefix

   a q/Q to the Sigma, Pair and converse rules.*)

 coinductive

   domains "lleq(A)" \<subseteq> "llist(A) * llist(A)"

   intros

     LNil:  "<LNil, LNil> \<in> lleq(A)"

     LCons: "[| a \<in> A; <l,l'> \<in> lleq(A) |] 

             ==> <LCons(a,l), LCons(a,l')> \<in> lleq(A)"

   type_intros  llist.intros

 subsection{*Powerset example*}

 lemma Pow_mono: "A\<subseteq>B  ==>  Pow(A) \<subseteq> Pow(B)"

 apply (rule subsetI)

 apply (rule PowI)

 apply (drule PowD)

 apply (erule subset_trans, assumption)

 done

 lemma "Pow(A Int B) = Pow(A) Int Pow(B)"

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule equalityI)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule Int_greatest)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule Int_lower1 [THEN Pow_mono])

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule Int_lower2 [THEN Pow_mono])

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule subsetI)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (erule IntE)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule PowI)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (drule PowD)+

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule Int_greatest)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (assumption+)

 done

 text{*Trying again from the beginning in order to use @{text blast}*}

 lemma "Pow(A Int B) = Pow(A) Int Pow(B)"

 by blast

 lemma "C\<subseteq>D ==> Union(C) \<subseteq> Union(D)"

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule subsetI)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (erule UnionE)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule UnionI)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (erule subsetD)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply assumption 

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply assumption 

 done

 text{*A more abstract version of the same proof*}

 lemma "C\<subseteq>D ==> Union(C) \<subseteq> Union(D)"

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule Union_least)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule Union_upper)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (erule subsetD, assumption)

 done

 lemma "[| a \<in> A;  f \<in> A->B;  g \<in> C->D;  A \<inter> C = 0 |] ==> (f \<union> g)`a = f`a"

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule apply_equality)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule UnI1)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule apply_Pair)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply assumption 

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply assumption 

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply (rule fun_disjoint_Un)

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply assumption 

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply assumption 

   --{* @{subgoals[display,indent=0,margin=65]} *}

 apply assumption 

 done

 end