# HG changeset patch # User wenzelm # Date 991341298 -7200 # Node ID 908b761cdfb039920726091cdabb7089ab7a0de8 # Parent 8fbb19b84f9474508f32002a4b0b3eca85c6cf07 tuned diff -r 8fbb19b84f94 -r 908b761cdfb0 src/HOL/Library/Nat_Infinity.thy --- 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]: "\ \ 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 \ \" - 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 \ = \" - 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 \ 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!]: "\ < 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 \ m = n \ 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]: "\ 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 ==> \ 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 < \" - by (simp add:inat_defs split:inat_splits, arith?) + by (simp add: inat_defs split:inat_splits, arith?) lemma Infty_eq [simp]: "n < \ = (n \ \)" - 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 \ 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]: "\ 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 ==> \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 \ n = (m < n \ 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 \ (n::inat)" - by (simp add:inat_defs split:inat_splits, arith?) + by (simp add: inat_defs split:inat_splits, arith?) lemma ile_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 ile_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_ile_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 Infty_ub [simp]: "n \ \" - by (simp add:inat_defs split:inat_splits, arith?) + by (simp add: inat_defs split:inat_splits, arith?) lemma i0_lb [simp]: "(0::inat) \ n" - by (simp add:inat_defs split:inat_splits, arith?) + by (simp add: inat_defs split:inat_splits, arith?) lemma Infty_ileE [elim!]: "\ \ 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 \ Fin m) = (n \ m)" - by (simp add:inat_defs split:inat_splits, arith?) + by (simp add: inat_defs split:inat_splits, arith?) lemma ilessI1: "n \ m ==> n \ 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 \ 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) \ 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 \ iSuc m = (n \ 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 \ n)" - by (simp add:inat_defs split:inat_splits, arith?) + by (simp add: inat_defs split:inat_splits, arith?) lemma not_iSuc_ilei0 [simp]: "\ iSuc n \ 0" - by (simp add:inat_defs split:inat_splits, arith?) + by (simp add: inat_defs split:inat_splits, arith?) lemma ile_iSuc [simp]: "n \ iSuc n" - by (simp add:inat_defs split:inat_splits, arith?) + by (simp add: inat_defs split:inat_splits, arith?) lemma Fin_ile: "n \ Fin m ==> \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: "\i. \j. Y i < Y j ==> \j. Fin k < Y j" apply (induct_tac k)