Theory NatGenIntEx_ZF

(* 
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section Generalizations - an example application

theory NatGenIntEx_ZF imports ZF.Int Generalization_ZF

begin

textThis theory shows an example application of of the setup for 
  generalization presented in Generalization_ZF.›

text In this example I show that integers can be considered 
  as a generalization of natural numbers. The next interpretion›
  shows that we can use theorems proven in the generalization›
  locale to sets nat›, int› and the natural embedding
  of natural numbers into integers.

interpretation int_interpr: 
  generalization "nat" "int" "{n,int_of(n). n  nat}"
proof -
  let ?f = "{n,int_of(n). n  nat}"
  have "?f  inj(nat,int)"
  proof -
    have I: "?f: nat  int" using ZF_fun_from_total by simp
    moreover from I have "nnat. ?f`(n)= int_of(n)" 
      using ZF_fun_from_tot_val by simp
    moreover have "nnat.mnat. int_of(n)=int_of(m)  n=m"
      using int_of_inject by simp
    ultimately show ?thesis using inj_def by simp
  qed
  then show "generalization(nat,int,?f)" using generalization_def by simp
qed


text Next we prove that ZF generalization is an arbitrary generalization.
  This allows to access notions defined in generalization1› locale 
  from within generalization› locale.

sublocale 
  generalization  generalization1 small big embed zf_newbig zf_move
proof
  show "zf_movebij(big, zf_newbig)" using zf_move_bij by auto
  show "zf_move O embed = id(small)" using zf_embed_move by auto
qed

abbreviation "int_obj  int_interpr.zf_newbig"

text Naturals are a subset of integers.

lemma "nat  int_obj" using int_interpr.small_less_zf_newbig by auto

text An example of defining an operation on the generalization set.

definition add where
  "add(x,y)  int_interpr.zf_move`(int_interpr.ret`x $+ int_interpr.ret`y)"

end