
diff -r 16d4213d4cbc -r 856492b0f755 src/Doc/Logics-ZF/ZF_examples.thy
--- a/src/Doc/Logics-ZF/ZF_examples.thy	Mon Apr 07 16:37:57 2014 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,202 +0,0 @@
-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 [elim_format]
-
-
-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




