Theory Nat_ZF

(* 
    This file is a part of IsarMathLib - 
    a library of formalized mathematics for Isabelle/Isar.

    Copyright (C) 2005, 2006  Slawomir Kolodynski

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

theory Nat_ZF imports Nat 

begin

text{*This theory contains lemmas that are missing from the standard 
  Isabelle's Nat.thy file.*}

section{*Induction*}

text{*The induction lemmas in the standard Isabelle's Nat.thy file like 
  for example @{text "nat_induct"} require the induction step to 
  be a higher order 
  statement (the one that uses the $\Longrightarrow$ sign). I found it 
  difficult to apply from Isar, which is perhaps more of an indication of 
  my Isar skills than anything else. Anyway, here we provide a first order
  version that is easier to reference in Isar declarative style proofs.*}

text{*The induction step for the first order induction.*}

lemma Nat_ZF_1_L1: assumes "x∈nat" "P(x)" 
  and "∀k∈nat. P(k)-->P(succ(k))"
  shows "P(succ(x))" using prems by simp;

text{*The actual first order induction on natural numbers.*}

lemma Nat_ZF_1_L2: 
  assumes A1: "n∈nat" and A2: "P(0)" and A3: "∀k∈nat. P(k)-->P(succ(k))"
  shows "P(n)"
proof -;
  from A1 A2 have "n∈nat" "P(0)" by auto
  then show "P(n)" using Nat_ZF_1_L1 by (rule nat_induct);
qed;

text{*A nonzero natural number has a predecessor.*}

lemma Nat_ZF_1_L3: assumes A1: "n∈nat" and A2: "n≠0"
  shows "∃k∈nat. n = succ(k)"
proof -
  from A1 have "n ∈ {0} ∪ {succ(k). k∈nat}"
    using nat_unfold by simp;
  with A2 show ?thesis by simp
qed;

end;