(* Title: HOL/Calculation.thy
ID: $Id$
Author: Markus Wenzel, TU Muenchen
Setup transitivity rules for calculational proofs. Note that in the
list below later rules have priority.
*)
theory Calculation = Int:;
theorems [trans] = HOL.ssubst; (* = x x *)
theorems [trans] = HOL.subst[COMP swap_prems_rl]; (* x = x *)
theorems [trans] = Divides.dvd_trans; (* dvd dvd dvd *)
theorem [trans]: "[| (x::'a::order) < y; y < x |] ==> P"; (* < > P *)
by (rule Ord.order_less_asym);
theorems [trans] = Ord.order_less_trans; (* < < < *)
theorems [trans] = Ord.order_le_less_trans; (* <= < < *)
theorems [trans] = Ord.order_less_le_trans; (* < <= < *)
theorems [trans] = Ord.order_trans; (* <= <= <= *)
theorems [trans] = Ord.order_antisym; (* <= >= = *)
theorem [trans]: "[| x <= y; y = z |] ==> x <= z"; (* <= = <= *)
by (rule HOL.subst);
theorem [trans]: "[| x = y; y <= z |] ==> x <= z"; (* = <= <= *)
by (rule HOL.ssubst);
theorem [trans]: "[| x < y; y = z |] ==> x < z"; (* < = < *)
by (rule HOL.subst);
theorem [trans]: "[| x = y; y < z |] ==> x < z"; (* = < < *)
by (rule HOL.ssubst);
theorems [trans] = HOL.trans (* = = = *)
end;