src/HOLCF/Sprod1.ML

Works up to main theorem, then XXX...X

(*  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                                      *)
(* ------------------------------------------------------------------------ *)


qed_goalw "less_sprod1a" Sprod1.thy [less_sprod_def]
        "p1=Ispair UU UU ==> less_sprod p1 p2"
 (fn prems =>
        [
        (cut_facts_tac prems 1),
        (asm_simp_tac HOL_ss 1)
        ]);

qed_goalw "less_sprod1b" 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),
        (asm_simp_tac HOL_ss 1)
        ]);

qed_goal "less_sprod2a" 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 Sprod0_ss 1),
        (rtac (defined_Ispair RS less_sprod1b RS iffD1 RS conjunct1) 1),
        (REPEAT (fast_tac HOL_cs 1))
        ]);

qed_goal "less_sprod2b" 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)
        ]);

qed_goal "less_sprod2c" 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 (Sprod0_ss addsimps prems)1),
        (res_inst_tac [("s","Isfst(Ispair x y)"),("t","x")] subst 1),
        (simp_tac (Sprod0_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 (Sprod0_ss addsimps prems)1),
        (res_inst_tac [("s","Issnd(Ispair xa ya)"),("t","ya")] subst 1),
        (simp_tac (Sprod0_ss addsimps prems)1),
        (res_inst_tac [("s","Issnd(Ispair x y)"),("t","y")] subst 1),
        (simp_tac (Sprod0_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 (Sprod0_ss addsimps prems)1)
        ]);

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

qed_goal "refl_less_sprod" 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))
        ]);


qed_goal "antisym_less_sprod" 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)
        ]);

qed_goal "trans_less_sprod" Sprod1.thy 
 "[|less_sprod (p1::'a**'b) 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::'a)(UU::'b)")] subst 1),
        (etac less_sprod2b 1),
        (atac 1),
        (hyp_subst_tac 1),
        (res_inst_tac [("Q","p2=Ispair(UU::'a)(UU::'b)")]
                 (excluded_middle RS disjE) 1),
        (rtac (defined_Ispair RS less_sprod1b RS ssubst) 1),
        (REPEAT (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),
        (REPEAT (atac 1)),
        (rtac conjunct1 1),
        (rtac (less_sprod1b RS subst) 1),
        (REPEAT (atac 1)),
        (res_inst_tac [("y","Issnd(p2)")] trans_less 1),
        (rtac conjunct2 1),
        (rtac (less_sprod1b RS subst) 1),
        (rtac defined_Ispair 1),
        (REPEAT (atac 1)),
        (rtac conjunct2 1),
        (rtac (less_sprod1b RS subst) 1),
        (REPEAT (atac 1)),
        (hyp_subst_tac 1),
        (res_inst_tac [("s","Ispair(UU::'a)(UU::'b)"),("t","Ispair x y")] 
                subst 1),
        (etac (less_sprod2b RS sym) 1),
        (atac 1)
        ]);