src/HOLCF/sprod1.ML

add_defs(_i): overloaded option;

(*  Title: 	HOLCF/sprod1.ML
    ID:         $Id$
    Author: 	Franz Regensburger
    Copyright   1993  Technische Universitaet Muenchen

Lemmas for theory sprod1.thy
*)

open Sprod1;

(* ------------------------------------------------------------------------ *)
(* reduction properties for less_sprod                                      *)
(* ------------------------------------------------------------------------ *)


val less_sprod1a = prove_goalw Sprod1.thy [less_sprod_def]
	"p1=Ispair(UU,UU) ==> less_sprod(p1,p2)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac eqTrueE 1),
	(rtac select_equality 1),
	(rtac conjI 1),
	(fast_tac HOL_cs 1),
	(strip_tac 1),
	(contr_tac 1),
	(dtac conjunct1 1),
	(etac rev_mp 1),
	(atac 1)
	]);

val less_sprod1b = prove_goalw Sprod1.thy [less_sprod_def]
 "~p1=Ispair(UU,UU) ==> \
\ less_sprod(p1,p2) = ( Isfst(p1) << Isfst(p2) & Issnd(p1) << Issnd(p2))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac select_equality 1),
	(rtac conjI 1),
	(strip_tac 1),
	(contr_tac 1),
	(fast_tac HOL_cs 1),
	(dtac conjunct2 1),
	(etac rev_mp 1),
	(atac 1)
	]);

val less_sprod2a = prove_goal Sprod1.thy
	"less_sprod(Ispair(x,y),Ispair(UU,UU)) ==> x = UU | y = UU"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (excluded_middle RS disjE) 1),
	(atac 2),
	(rtac disjI1 1),
	(rtac antisym_less 1),
	(rtac minimal 2),
	(res_inst_tac [("s","Isfst(Ispair(x,y))"),("t","x")] subst 1),
	(rtac Isfst 1),
	(fast_tac HOL_cs 1),
	(fast_tac HOL_cs 1),
	(res_inst_tac [("s","Isfst(Ispair(UU,UU))"),("t","UU")] subst 1),
	(simp_tac Sprod_ss 1),
	(rtac (defined_Ispair RS less_sprod1b RS iffD1 RS conjunct1) 1),
	(REPEAT (fast_tac HOL_cs 1))
	]);

val less_sprod2b = prove_goal Sprod1.thy
 "less_sprod(p,Ispair(UU,UU)) ==> p = Ispair(UU,UU)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(res_inst_tac [("p","p")] IsprodE 1),
	(atac 1),
	(hyp_subst_tac 1),
	(rtac strict_Ispair 1),
	(etac less_sprod2a 1)
	]);

val less_sprod2c = prove_goal Sprod1.thy 
 "[|less_sprod(Ispair(xa,ya),Ispair(x,y));\
\~ xa = UU ; ~ ya = UU;~ x = UU ; ~ y = UU |] ==> xa << x & ya << y"
(fn prems =>
	[
	(rtac conjI 1),
	(res_inst_tac [("s","Isfst(Ispair(xa,ya))"),("t","xa")] subst 1),
	(simp_tac (Sprod_ss addsimps prems)1),
	(res_inst_tac [("s","Isfst(Ispair(x,y))"),("t","x")] subst 1),
	(simp_tac (Sprod_ss addsimps prems)1),
	(rtac (defined_Ispair RS less_sprod1b RS iffD1 RS conjunct1) 1),
	(resolve_tac prems 1),
	(resolve_tac prems 1),
	(simp_tac (Sprod_ss addsimps prems)1),
	(res_inst_tac [("s","Issnd(Ispair(xa,ya))"),("t","ya")] subst 1),
	(simp_tac (Sprod_ss addsimps prems)1),
	(res_inst_tac [("s","Issnd(Ispair(x,y))"),("t","y")] subst 1),
	(simp_tac (Sprod_ss addsimps prems)1),
	(rtac (defined_Ispair RS less_sprod1b RS iffD1 RS conjunct2) 1),
	(resolve_tac prems 1),
	(resolve_tac prems 1),
	(simp_tac (Sprod_ss addsimps prems)1)
	]);

(* ------------------------------------------------------------------------ *)
(* less_sprod is a partial order on Sprod                                   *)
(* ------------------------------------------------------------------------ *)

val refl_less_sprod = prove_goal Sprod1.thy "less_sprod(p,p)"
(fn prems =>
	[
	(res_inst_tac [("p","p")] IsprodE 1),
	(etac less_sprod1a 1),
	(hyp_subst_tac 1),
	(rtac (less_sprod1b RS ssubst) 1),
	(rtac defined_Ispair 1),
	(REPEAT (fast_tac (HOL_cs addIs [refl_less]) 1))
	]);


val antisym_less_sprod = prove_goal Sprod1.thy 
 "[|less_sprod(p1,p2);less_sprod(p2,p1)|] ==> p1=p2"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(res_inst_tac [("p","p1")] IsprodE 1),
	(hyp_subst_tac 1),
	(res_inst_tac [("p","p2")] IsprodE 1),
	(hyp_subst_tac 1),
	(rtac refl 1),
	(hyp_subst_tac 1),
	(rtac (strict_Ispair RS sym) 1),
	(etac less_sprod2a 1),
	(hyp_subst_tac 1),
	(res_inst_tac [("p","p2")] IsprodE 1),
	(hyp_subst_tac 1),
	(rtac (strict_Ispair) 1),
	(etac less_sprod2a 1),
	(hyp_subst_tac 1),
	(res_inst_tac [("x1","x"),("y1","xa"),("x","y"),("y","ya")] (arg_cong RS cong) 1),
	(rtac antisym_less 1),
	(asm_simp_tac (HOL_ss addsimps [less_sprod2c RS conjunct1]) 1),
	(asm_simp_tac (HOL_ss addsimps [less_sprod2c RS conjunct1]) 1),
	(rtac antisym_less 1),
	(asm_simp_tac (HOL_ss addsimps [less_sprod2c RS conjunct2]) 1),
	(asm_simp_tac (HOL_ss addsimps [less_sprod2c RS conjunct2]) 1)
	]);

val trans_less_sprod = prove_goal Sprod1.thy 
 "[|less_sprod(p1,p2);less_sprod(p2,p3)|] ==> less_sprod(p1,p3)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(res_inst_tac [("p","p1")] IsprodE 1),
	(etac less_sprod1a 1),
	(hyp_subst_tac 1),
	(res_inst_tac [("p","p3")] IsprodE 1),
	(hyp_subst_tac 1),
	(res_inst_tac [("s","p2"),("t","Ispair(UU,UU)")] subst 1),
	(etac less_sprod2b 1),
	(atac 1),
	(hyp_subst_tac 1),
	(res_inst_tac [("Q","p2=Ispair(UU,UU)")]  
		(excluded_middle RS disjE) 1),
	(rtac (defined_Ispair RS less_sprod1b RS ssubst) 1),
	(atac 1),
	(atac 1),
	(rtac conjI 1),
	(res_inst_tac [("y","Isfst(p2)")] trans_less 1),
	(rtac conjunct1 1),
	(rtac (less_sprod1b RS subst) 1),
	(rtac defined_Ispair 1),
	(atac 1),
	(atac 1),
	(atac 1),
	(rtac conjunct1 1),
	(rtac (less_sprod1b RS subst) 1),
	(atac 1),
	(atac 1),
	(res_inst_tac [("y","Issnd(p2)")] trans_less 1),
	(rtac conjunct2 1),
	(rtac (less_sprod1b RS subst) 1),
	(rtac defined_Ispair 1),
	(atac 1),
	(atac 1),
	(atac 1),
	(rtac conjunct2 1),
	(rtac (less_sprod1b RS subst) 1),
	(atac 1),
	(atac 1),
	(hyp_subst_tac 1),
	(res_inst_tac [("s","Ispair(UU,UU)"),("t","Ispair(x,y)")] subst 1),
	(etac (less_sprod2b RS sym) 1),
	(atac 1)
	]);