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