doc-src/ZF/ZF_examples.thy
 author nipkow Sun, 22 Feb 2009 17:25:28 +0100 changeset 30056 0a35bee25c20 parent 16417 9bc16273c2d4 child 35416 d8d7d1b785af permissions -rw-r--r--
```
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)

constdefs  n_nodes_tail :: "i => i"
"n_nodes_tail(t) == n_nodes_aux(t) ` 0"

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

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
```