# Theory IntDiv_ZF_IML

theory IntDiv_ZF_IML
imports Int_ZF_1 IntDiv_ZF
(*
This file is a part of IsarMathLib -
a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2005, 2006  Slawomir Kolodynski

This program is free software; Redistribution and use in source and binary forms,
with or without modification, are permitted provided that the following conditions are met:

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derived from this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE AUTHOR AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES,
INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT,
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*)

theory IntDiv_ZF_IML imports Int_ZF_1 IntDiv_ZF

begin

text{*This theory translates some results form the Isabelle's
@{text "IntDiv.thy"} theory to the notation used by IsarMathLib.*}

section{*Quotient and reminder*}

text{*For any integers $m,n$ , $n>0$ there are unique integers $q,p$
such that $0\leq p < n$ and $m = n\cdot q + p$. Number $p$ in this
decompsition is usually called $m$ mod $n$. Standard Isabelle denotes numbers
$q,p$ as @{text "m zdiv n"} and @{text "m zmod n"}, resp.,
and we will use the same notation. *}

text{*The next lemma is sometimes called the "quotient-reminder theorem".*}

lemma (in int0) IntDiv_ZF_1_L1: assumes "m∈\<int>"  "n∈\<int>"
shows "m = n·(m zdiv n) \<ra> (m zmod n)"
using assms Int_ZF_1_L2 raw_zmod_zdiv_equality
by simp;

text{*If $n$ is greater than $0$ then @{text "m zmod n"} is between $0$ and $n-1$.*}

lemma (in int0) IntDiv_ZF_1_L2:
assumes A1: "m∈\<int>" and A2: "\<zero>\<lsq>n"  "n≠\<zero>"
shows
"\<zero> \<lsq> m zmod n"
"m zmod n \<lsq> n"    "m zmod n ≠ n"
"m zmod n \<lsq> n\<rs>\<one>"
proof -
from A2 have T: "n ∈ \<int>"
using Int_ZF_2_L1A by simp;
from A2 have "#0 $< n" using Int_ZF_2_L9 Int_ZF_1_L8 by auto; with T show "\<zero> \<lsq> m zmod n" "m zmod n \<lsq> n" "m zmod n ≠ n" using pos_mod Int_ZF_1_L8 Int_ZF_1_L8A zmod_type Int_ZF_2_L1 Int_ZF_2_L9AA by auto; then show "m zmod n \<lsq> n\<rs>\<one>" using Int_ZF_4_L1B by auto; qed; text{*$(m\cdot k)$div$k = m$.*} lemma (in int0) IntDiv_ZF_1_L3: assumes "m∈\<int>" "k∈\<int>" and "k≠\<zero>" shows "(m·k) zdiv k = m" "(k·m) zdiv k = m" using assms zdiv_zmult_self1 zdiv_zmult_self2 Int_ZF_1_L8 Int_ZF_1_L2 by auto; text{*The next lemma essentially translates @{text "zdiv_mono1"} from standard Isabelle to our notation.*} lemma (in int0) IntDiv_ZF_1_L4: assumes A1: "m \<lsq> k" and A2: "\<zero>\<lsq>n" "n≠\<zero>" shows "m zdiv n \<lsq> k zdiv n" proof - from A2 have "#0 \<lsq> n" "#0 ≠ n" using Int_ZF_1_L8 by auto; with A1 have "m zdiv n$≤ k zdiv n"
"m zdiv n ∈ \<int>"    "m zdiv k ∈ \<int>"
using Int_ZF_2_L1A Int_ZF_2_L9 zdiv_mono1
by auto;
then show "(m zdiv n) \<lsq> (k zdiv n)"
using Int_ZF_2_L1 by simp;
qed;

text{*A quotient-reminder theorem about integers greater than a given
product.*}

lemma (in int0) IntDiv_ZF_1_L5:
assumes A1: "n ∈ \<int>⇩+" and A2: "n \<lsq> k" and A3: "k·n \<lsq> m"
shows
"m = n·(m zdiv n) \<ra> (m zmod n)"
"m = (m zdiv n)·n \<ra> (m zmod n)"
"(m zmod n) ∈ \<zero>..(n\<rs>\<one>)"
"k \<lsq> (m zdiv n)"
"m zdiv n ∈ \<int>⇩+"
proof -
from A2 A3 have T:
"m∈\<int>"  "n∈\<int>"  "k∈\<int>"  "m zdiv n ∈ \<int>"
using Int_ZF_2_L1A by auto;
then show "m = n·(m zdiv n) \<ra> (m zmod n)"
using IntDiv_ZF_1_L1 by simp;
with T show "m = (m zdiv n)·n \<ra> (m zmod n)"
using Int_ZF_1_L4 by simp;
from A1 have I: "\<zero>\<lsq>n"  "n≠\<zero>"
using PositiveSet_def by auto;
with T show "(m zmod n) ∈ \<zero>..(n\<rs>\<one>)"
using IntDiv_ZF_1_L2 Order_ZF_2_L1
by simp;
from A3 I have "(k·n zdiv n) \<lsq> (m zdiv n)"
using IntDiv_ZF_1_L4 by simp;
with I T show "k \<lsq> (m zdiv n)"
using IntDiv_ZF_1_L3 by simp;
with A1 A2 show "m zdiv n ∈ \<int>⇩+"
using Int_ZF_1_5_L7 by blast;
qed;

end