(*

This file is a part of IsarMathLib -

a library of formalized mathematics written for Isabelle/Isar.

Copyright (C) 2011 Victor Porton

This program is free software; Redistribution and use in source and binary forms,

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header {*\isaheader{NatGenIntEx\_ZF.thy} *}

theory NatGenIntEx_ZF imports Int_ZF Generalization_ZF

begin

text{*This theory shows an example application of of the setup for

generalization presented in @{text "Generalization_ZF."}*}

text {* In this example I show that integers can be considered

as a generalization of natural numbers. The next @{text "interpretion"}

shows that we can use theorems proven in the @{text "generalization"}

locale to sets @{text "nat"}, @{text "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 "∀n∈nat. ?f`(n)= int_of(n)"

using ZF_fun_from_tot_val by simp;

moreover have "∀n∈nat.∀m∈nat. 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 @{text "generalization1"} locale

from within @{text "generalization"} locale.*}

sublocale

generalization ⊆ generalization1 small big embed zf_newbig zf_move

proof

show "zf_move∈bij(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