--- a/src/HOL/Library/Nat_Infinity.thy Thu May 31 20:53:49 2001 +0200
+++ b/src/HOL/Library/Nat_Infinity.thy Thu May 31 22:34:58 2001 +0200
@@ -42,130 +42,130 @@
lemmas inat_splits = inat.split inat.split_asm
text {*
- Below is a not quite complete set of theorems. Use method @{text
- "(simp add: inat_defs split:inat_splits, arith?)"} to prove new
- theorems or solve arithmetic subgoals involving @{typ inat} on the
- fly.
+ Below is a not quite complete set of theorems. Use the method
+ @{text "(simp add: inat_defs split:inat_splits, arith?)"} to prove
+ new theorems or solve arithmetic subgoals involving @{typ inat} on
+ the fly.
*}
subsection "Constructors"
lemma Fin_0: "Fin 0 = 0"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iSuc_Fin [simp]: "iSuc (Fin n) = Fin (Suc n)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iSuc_inject [simp]: "(iSuc x = iSuc y) = (x = y)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
subsection "Ordering relations"
lemma Infty_ilessE [elim!]: "\<infinity> < Fin m ==> R"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iless_linear: "m < n \<or> m = n \<or> n < (m::inat)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iless_not_refl [simp]: "\<not> n < (n::inat)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iless_trans: "i < j ==> j < k ==> i < (k::inat)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iless_not_sym: "n < m ==> \<not> m < (n::inat)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma Fin_iless_mono [simp]: "(Fin n < Fin m) = (n < m)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma Fin_iless_Infty [simp]: "Fin n < \<infinity>"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma Infty_eq [simp]: "n < \<infinity> = (n \<noteq> \<infinity>)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma i0_eq [simp]: "((0::inat) < n) = (n \<noteq> 0)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma i0_iless_iSuc [simp]: "0 < iSuc n"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma not_ilessi0 [simp]: "\<not> n < (0::inat)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma Fin_iless: "n < Fin m ==> \<exists>k. n = Fin k"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iSuc_mono [simp]: "iSuc n < iSuc m = (n < m)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
(* ----------------------------------------------------------------------- *)
lemma ile_def2: "m \<le> n = (m < n \<or> m = (n::inat))"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma ile_refl [simp]: "n \<le> (n::inat)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma ile_trans: "i \<le> j ==> j \<le> k ==> i \<le> (k::inat)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma ile_iless_trans: "i \<le> j ==> j < k ==> i < (k::inat)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iless_ile_trans: "i < j ==> j \<le> k ==> i < (k::inat)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma Infty_ub [simp]: "n \<le> \<infinity>"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma i0_lb [simp]: "(0::inat) \<le> n"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m ==> R"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma Fin_ile_mono [simp]: "(Fin n \<le> Fin m) = (n \<le> m)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma ilessI1: "n \<le> m ==> n \<noteq> m ==> n < (m::inat)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma ileI1: "m < n ==> iSuc m \<le> n"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma Suc_ile_eq: "Fin (Suc m) \<le> n = (Fin m < n)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m = (n \<le> m)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma iless_Suc_eq [simp]: "Fin m < iSuc n = (Fin m \<le> n)"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma ile_iSuc [simp]: "n \<le> iSuc n"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma Fin_ile: "n \<le> Fin m ==> \<exists>k. n = Fin k"
- by (simp add:inat_defs split:inat_splits, arith?)
+ by (simp add: inat_defs split:inat_splits, arith?)
lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"
apply (induct_tac k)