src/HOL/Library/Nat_Infinity.thy
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(*  Title:      HOL/Library/Nat_Infinity.thy
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    ID:         $Id$
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    Author:     David von Oheimb, TU Muenchen
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*)
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header {* Natural numbers with infinity *}
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theory Nat_Infinity
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imports Main
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begin
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subsection "Definitions"
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text {*
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  We extend the standard natural numbers by a special value indicating
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  infinity.  This includes extending the ordering relations @{term "op
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  <"} and @{term "op \<le>"}.
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*}
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datatype inat = Fin nat | Infty
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notation (xsymbols)
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  Infty  ("\<infinity>")
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notation (HTML output)
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  Infty  ("\<infinity>")
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instance inat :: "{ord, zero}" ..
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definition
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  iSuc :: "inat => inat" where
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  "iSuc i = (case i of Fin n => Fin (Suc n) | \<infinity> => \<infinity>)"
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defs (overloaded)
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  Zero_inat_def: "0 == Fin 0"
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  iless_def: "m < n ==
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    case m of Fin m1 => (case n of Fin n1 => m1 < n1 | \<infinity> => True)
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    | \<infinity>  => False"
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  ile_def: "(m::inat) \<le> n == \<not> (n < m)"
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lemmas inat_defs = Zero_inat_def iSuc_def iless_def ile_def
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lemmas inat_splits = inat.split inat.split_asm
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text {*
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  Below is a not quite complete set of theorems.  Use the method
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  @{text "(simp add: inat_defs split:inat_splits, arith?)"} to prove
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  new theorems or solve arithmetic subgoals involving @{typ inat} on
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  the fly.
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*}
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subsection "Constructors"
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lemma Fin_0: "Fin 0 = 0"
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  by (simp add: inat_defs split:inat_splits)
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lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
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  by (simp add: inat_defs split:inat_splits)
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lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
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  by (simp add: inat_defs split:inat_splits)
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lemma iSuc_Fin [simp]: "iSuc (Fin n) = Fin (Suc n)"
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  by (simp add: inat_defs split:inat_splits)
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lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
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  by (simp add: inat_defs split:inat_splits)
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lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
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  by (simp add: inat_defs split:inat_splits)
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lemma iSuc_inject [simp]: "(iSuc x = iSuc y) = (x = y)"
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  by (simp add: inat_defs split:inat_splits)
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subsection "Ordering relations"
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lemma Infty_ilessE [elim!]: "\<infinity> < Fin m ==> R"
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  by (simp add: inat_defs split:inat_splits)
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lemma iless_linear: "m < n \<or> m = n \<or> n < (m::inat)"
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  by (simp add: inat_defs split:inat_splits, arith)
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lemma iless_not_refl [simp]: "\<not> n < (n::inat)"
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  by (simp add: inat_defs split:inat_splits)
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lemma iless_trans: "i < j ==> j < k ==> i < (k::inat)"
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  by (simp add: inat_defs split:inat_splits)
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lemma iless_not_sym: "n < m ==> \<not> m < (n::inat)"
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  by (simp add: inat_defs split:inat_splits)
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lemma Fin_iless_mono [simp]: "(Fin n < Fin m) = (n < m)"
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  by (simp add: inat_defs split:inat_splits)
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lemma Fin_iless_Infty [simp]: "Fin n < \<infinity>"
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  by (simp add: inat_defs split:inat_splits)
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lemma Infty_eq [simp]: "(n < \<infinity>) = (n \<noteq> \<infinity>)"
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  by (simp add: inat_defs split:inat_splits)
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lemma i0_eq [simp]: "((0::inat) < n) = (n \<noteq> 0)"
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  by (simp add: neq0_conv inat_defs split:inat_splits)
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lemma i0_iless_iSuc [simp]: "0 < iSuc n"
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  by (simp add: inat_defs split:inat_splits)
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lemma not_ilessi0 [simp]: "\<not> n < (0::inat)"
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  by (simp add: inat_defs split:inat_splits)
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lemma Fin_iless: "n < Fin m ==> \<exists>k. n = Fin k"
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  by (simp add: inat_defs split:inat_splits)
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lemma iSuc_mono [simp]: "(iSuc n < iSuc m) = (n < m)"
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  by (simp add: inat_defs split:inat_splits)
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lemma ile_def2: "(m \<le> n) = (m < n \<or> m = (n::inat))"
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  by (simp add: inat_defs split:inat_splits, arith)
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lemma ile_refl [simp]: "n \<le> (n::inat)"
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  by (simp add: inat_defs split:inat_splits)
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lemma ile_trans: "i \<le> j ==> j \<le> k ==> i \<le> (k::inat)"
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  by (simp add: inat_defs split:inat_splits)
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lemma ile_iless_trans: "i \<le> j ==> j < k ==> i < (k::inat)"
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  by (simp add: inat_defs split:inat_splits)
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lemma iless_ile_trans: "i < j ==> j \<le> k ==> i < (k::inat)"
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  by (simp add: inat_defs split:inat_splits)
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lemma Infty_ub [simp]: "n \<le> \<infinity>"
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  by (simp add: inat_defs split:inat_splits)
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lemma i0_lb [simp]: "(0::inat) \<le> n"
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  by (simp add: inat_defs split:inat_splits)
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lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m ==> R"
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  by (simp add: inat_defs split:inat_splits)
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lemma Fin_ile_mono [simp]: "(Fin n \<le> Fin m) = (n \<le> m)"
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  by (simp add: inat_defs split:inat_splits, arith)
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lemma ilessI1: "n \<le> m ==> n \<noteq> m ==> n < (m::inat)"
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  by (simp add: inat_defs split:inat_splits)
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lemma ileI1: "m < n ==> iSuc m \<le> n"
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  by (simp add: inat_defs split:inat_splits)
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lemma Suc_ile_eq: "(Fin (Suc m) \<le> n) = (Fin m < n)"
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  by (simp add: inat_defs split:inat_splits, arith)
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lemma iSuc_ile_mono [simp]: "(iSuc n \<le> iSuc m) = (n \<le> m)"
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df3a7e9ebadb tuned Proof
chaieb
parents: 21404
diff changeset
   155
  by (simp add: inat_defs split:inat_splits)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   156
11655
923e4d0d36d5 tuned parentheses in relational expressions;
wenzelm
parents: 11357
diff changeset
   157
lemma iless_Suc_eq [simp]: "(Fin m < iSuc n) = (Fin m \<le> n)"
23315
df3a7e9ebadb tuned Proof
chaieb
parents: 21404
diff changeset
   158
  by (simp add: inat_defs split:inat_splits, arith)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   159
11355
wenzelm
parents: 11351
diff changeset
   160
lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
23315
df3a7e9ebadb tuned Proof
chaieb
parents: 21404
diff changeset
   161
  by (simp add: inat_defs split:inat_splits)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   162
11355
wenzelm
parents: 11351
diff changeset
   163
lemma ile_iSuc [simp]: "n \<le> iSuc n"
23315
df3a7e9ebadb tuned Proof
chaieb
parents: 21404
diff changeset
   164
  by (simp add: inat_defs split:inat_splits)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   165
11355
wenzelm
parents: 11351
diff changeset
   166
lemma Fin_ile: "n \<le> Fin m ==> \<exists>k. n = Fin k"
23315
df3a7e9ebadb tuned Proof
chaieb
parents: 21404
diff changeset
   167
  by (simp add: inat_defs split:inat_splits)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   168
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   169
lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"
11355
wenzelm
parents: 11351
diff changeset
   170
  apply (induct_tac k)
wenzelm
parents: 11351
diff changeset
   171
   apply (simp (no_asm) only: Fin_0)
wenzelm
parents: 11351
diff changeset
   172
   apply (fast intro: ile_iless_trans i0_lb)
wenzelm
parents: 11351
diff changeset
   173
  apply (erule exE)
wenzelm
parents: 11351
diff changeset
   174
  apply (drule spec)
wenzelm
parents: 11351
diff changeset
   175
  apply (erule exE)
wenzelm
parents: 11351
diff changeset
   176
  apply (drule ileI1)
wenzelm
parents: 11351
diff changeset
   177
  apply (rule iSuc_Fin [THEN subst])
wenzelm
parents: 11351
diff changeset
   178
  apply (rule exI)
wenzelm
parents: 11351
diff changeset
   179
  apply (erule (1) ile_iless_trans)
wenzelm
parents: 11351
diff changeset
   180
  done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   181
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   182
end