src/HOLCF/Sprod0.ML

corrected some errors that occurred after introduction of local simpsets

(*  Title: 	HOLCF/sprod0.thy
    ID:         $Id$
    Author: 	Franz Regensburger
    Copyright   1993  Technische Universitaet Muenchen

Lemmas for theory sprod0.thy
*)

open Sprod0;

(* ------------------------------------------------------------------------ *)
(* A non-emptyness result for Sprod                                         *)
(* ------------------------------------------------------------------------ *)

qed_goalw "SprodI" Sprod0.thy [Sprod_def]
	"(Spair_Rep a b):Sprod"
(fn prems =>
	[
	(EVERY1 [rtac CollectI, rtac exI,rtac exI, rtac refl])
	]);


qed_goal "inj_onto_Abs_Sprod" Sprod0.thy 
	"inj_onto Abs_Sprod Sprod"
(fn prems =>
	[
	(rtac inj_onto_inverseI 1),
	(etac Abs_Sprod_inverse 1)
	]);


(* ------------------------------------------------------------------------ *)
(* Strictness and definedness of Spair_Rep                                  *)
(* ------------------------------------------------------------------------ *)


qed_goalw "strict_Spair_Rep" Sprod0.thy [Spair_Rep_def]
 "(a=UU | b=UU) ==> (Spair_Rep a b) = (Spair_Rep UU UU)"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac ext 1),
	(rtac ext 1),
	(rtac iffI 1),
	(fast_tac HOL_cs 1),
	(fast_tac HOL_cs 1)
	]);

qed_goalw "defined_Spair_Rep_rev" Sprod0.thy [Spair_Rep_def]
 "(Spair_Rep a b) = (Spair_Rep UU UU) ==> (a=UU | b=UU)"
 (fn prems =>
	[
	(res_inst_tac [("Q","a=UU|b=UU")] classical2 1),
	(atac 1),
	(rtac disjI1 1),
	(rtac ((hd prems) RS fun_cong RS fun_cong RS iffD2 RS mp RS 
	conjunct1 RS sym) 1),
	(fast_tac HOL_cs 1),
	(fast_tac HOL_cs 1)
	]);


(* ------------------------------------------------------------------------ *)
(* injectivity of Spair_Rep and Ispair                                      *)
(* ------------------------------------------------------------------------ *)

qed_goalw "inject_Spair_Rep" Sprod0.thy [Spair_Rep_def]
"[|~aa=UU ; ~ba=UU ; Spair_Rep a b = Spair_Rep aa ba |] ==> a=aa & b=ba"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac ((nth_elem (2,prems)) RS fun_cong  RS fun_cong 
		RS iffD1 RS mp) 1),
	(fast_tac HOL_cs 1),
	(fast_tac HOL_cs 1)
	]);


qed_goalw "inject_Ispair" Sprod0.thy [Ispair_def]
	"[|~aa=UU ; ~ba=UU ; Ispair a b = Ispair aa ba |] ==> a=aa & b=ba"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(etac inject_Spair_Rep 1),
	(atac 1),
	(etac (inj_onto_Abs_Sprod  RS inj_ontoD) 1),
	(rtac SprodI 1),
	(rtac SprodI 1)
	]);


(* ------------------------------------------------------------------------ *)
(* strictness and definedness of Ispair                                     *)
(* ------------------------------------------------------------------------ *)

qed_goalw "strict_Ispair" Sprod0.thy [Ispair_def] 
 "(a=UU | b=UU) ==> Ispair a b = Ispair UU UU"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(etac (strict_Spair_Rep RS arg_cong) 1)
	]);

qed_goalw "strict_Ispair1" Sprod0.thy [Ispair_def]
	"Ispair UU b  = Ispair UU UU"
(fn prems =>
	[
	(rtac (strict_Spair_Rep RS arg_cong) 1),
	(rtac disjI1 1),
	(rtac refl 1)
	]);

qed_goalw "strict_Ispair2" Sprod0.thy [Ispair_def]
	"Ispair a UU = Ispair UU UU"
(fn prems =>
	[
	(rtac (strict_Spair_Rep RS arg_cong) 1),
	(rtac disjI2 1),
	(rtac refl 1)
	]);

qed_goal "strict_Ispair_rev" Sprod0.thy 
	"~Ispair x y = Ispair UU UU ==> ~x=UU & ~y=UU"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (de_morgan1 RS ssubst) 1),
	(etac contrapos 1),
	(etac strict_Ispair 1)
	]);

qed_goalw "defined_Ispair_rev" Sprod0.thy [Ispair_def]
	"Ispair a b  = Ispair UU UU ==> (a = UU | b = UU)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac defined_Spair_Rep_rev 1),
	(rtac (inj_onto_Abs_Sprod  RS inj_ontoD) 1),
	(atac 1),
	(rtac SprodI 1),
	(rtac SprodI 1)
	]);

qed_goal "defined_Ispair" Sprod0.thy  
"[|a~=UU; b~=UU|] ==> (Ispair a b) ~= (Ispair UU UU)" 
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac contrapos 1),
	(etac defined_Ispair_rev 2),
	(rtac (de_morgan1 RS iffD1) 1),
	(etac conjI 1),
	(atac 1)
	]);


(* ------------------------------------------------------------------------ *)
(* Exhaustion of the strict product **                                      *)
(* ------------------------------------------------------------------------ *)

qed_goalw "Exh_Sprod" Sprod0.thy [Ispair_def]
	"z=Ispair UU UU | (? a b. z=Ispair a b & a~=UU & b~=UU)"
(fn prems =>
	[
	(rtac (rewrite_rule [Sprod_def] Rep_Sprod RS CollectE) 1),
	(etac exE 1),
	(etac exE 1),
	(rtac (excluded_middle RS disjE) 1),
	(rtac disjI2 1),
	(rtac exI 1),
	(rtac exI 1),
	(rtac conjI 1),
	(rtac (Rep_Sprod_inverse RS sym RS trans) 1),
	(etac arg_cong 1),
	(rtac (de_morgan1 RS ssubst) 1),
	(atac 1),
	(rtac disjI1 1),
	(rtac (Rep_Sprod_inverse RS sym RS trans) 1),
	(res_inst_tac [("f","Abs_Sprod")] arg_cong 1),
	(etac trans 1),
	(etac strict_Spair_Rep 1)
	]);

(* ------------------------------------------------------------------------ *)
(* general elimination rule for strict product                              *)
(* ------------------------------------------------------------------------ *)

qed_goal "IsprodE" Sprod0.thy
"[|p=Ispair UU UU ==> Q ;!!x y. [|p=Ispair x y; x~=UU ; y~=UU|] ==> Q|] ==> Q"
(fn prems =>
	[
	(rtac (Exh_Sprod RS disjE) 1),
	(etac (hd prems) 1),
	(etac exE 1),
	(etac exE 1),
	(etac conjE 1),
	(etac conjE 1),
	(etac (hd (tl prems)) 1),
	(atac 1),
	(atac 1)
	]);


(* ------------------------------------------------------------------------ *)
(* some results about the selectors Isfst, Issnd                            *)
(* ------------------------------------------------------------------------ *)

qed_goalw "strict_Isfst" Sprod0.thy [Isfst_def] 
	"p=Ispair UU UU ==> Isfst p = UU"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac  select_equality 1),
	(rtac conjI 1),
	(fast_tac HOL_cs  1),
	(strip_tac 1),
	(res_inst_tac [("P","Ispair UU UU = Ispair a b")] notE 1),
	(rtac not_sym 1),
	(rtac defined_Ispair 1),
	(REPEAT (fast_tac HOL_cs  1))
	]);


qed_goal "strict_Isfst1" Sprod0.thy
	"Isfst(Ispair UU y) = UU"
(fn prems =>
	[
	(rtac (strict_Ispair1 RS ssubst) 1),
	(rtac strict_Isfst 1),
	(rtac refl 1)
	]);

qed_goal "strict_Isfst2" Sprod0.thy
	"Isfst(Ispair x UU) = UU"
(fn prems =>
	[
	(rtac (strict_Ispair2 RS ssubst) 1),
	(rtac strict_Isfst 1),
	(rtac refl 1)
	]);


qed_goalw "strict_Issnd" Sprod0.thy [Issnd_def] 
	"p=Ispair UU UU ==>Issnd p=UU"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac  select_equality 1),
	(rtac conjI 1),
	(fast_tac HOL_cs  1),
	(strip_tac 1),
	(res_inst_tac [("P","Ispair UU UU = Ispair a b")] notE 1),
	(rtac not_sym 1),
	(rtac defined_Ispair 1),
	(REPEAT (fast_tac HOL_cs  1))
	]);

qed_goal "strict_Issnd1" Sprod0.thy
	"Issnd(Ispair UU y) = UU"
(fn prems =>
	[
	(rtac (strict_Ispair1 RS ssubst) 1),
	(rtac strict_Issnd 1),
	(rtac refl 1)
	]);

qed_goal "strict_Issnd2" Sprod0.thy
	"Issnd(Ispair x UU) = UU"
(fn prems =>
	[
	(rtac (strict_Ispair2 RS ssubst) 1),
	(rtac strict_Issnd 1),
	(rtac refl 1)
	]);

qed_goalw "Isfst" Sprod0.thy [Isfst_def]
	"[|x~=UU ;y~=UU |] ==> Isfst(Ispair x y) = x"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac  select_equality 1),
	(rtac conjI 1),
	(strip_tac 1),
	(res_inst_tac [("P","Ispair x y = Ispair UU UU")] notE 1),
	(etac defined_Ispair 1),
	(atac 1),
	(atac 1),
	(strip_tac 1),
	(rtac (inject_Ispair RS conjunct1) 1),
	(fast_tac HOL_cs  3),
	(fast_tac HOL_cs  1),
	(fast_tac HOL_cs  1),
	(fast_tac HOL_cs  1)
	]);

qed_goalw "Issnd" Sprod0.thy [Issnd_def]
	"[|x~=UU ;y~=UU |] ==> Issnd(Ispair x y) = y"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac  select_equality 1),
	(rtac conjI 1),
	(strip_tac 1),
	(res_inst_tac [("P","Ispair x y = Ispair UU UU")] notE 1),
	(etac defined_Ispair 1),
	(atac 1),
	(atac 1),
	(strip_tac 1),
	(rtac (inject_Ispair RS conjunct2) 1),
	(fast_tac HOL_cs  3),
	(fast_tac HOL_cs  1),
	(fast_tac HOL_cs  1),
	(fast_tac HOL_cs  1)
	]);

qed_goal "Isfst2" Sprod0.thy "y~=UU ==>Isfst(Ispair x y)=x"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1),
	(etac Isfst 1),
	(atac 1),
	(hyp_subst_tac 1),
	(rtac strict_Isfst1 1)
	]);

qed_goal "Issnd2" Sprod0.thy "~x=UU ==>Issnd(Ispair x y)=y"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(res_inst_tac [("Q","y=UU")] (excluded_middle RS disjE) 1),
	(etac Issnd 1),
	(atac 1),
	(hyp_subst_tac 1),
	(rtac strict_Issnd2 1)
	]);


(* ------------------------------------------------------------------------ *)
(* instantiate the simplifier                                               *)
(* ------------------------------------------------------------------------ *)

val Sprod0_ss = 
        HOL_ss 
        addsimps [strict_Isfst1,strict_Isfst2,strict_Issnd1,strict_Issnd2,
                 Isfst2,Issnd2];

qed_goal "defined_IsfstIssnd" Sprod0.thy 
	"p~=Ispair UU UU ==> Isfst p ~= UU & Issnd p ~= UU"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(res_inst_tac [("p","p")] IsprodE 1),
	(contr_tac 1),
	(hyp_subst_tac 1),
	(rtac conjI 1),
        (asm_simp_tac Sprod0_ss 1),
        (asm_simp_tac Sprod0_ss 1)
	]);


(* ------------------------------------------------------------------------ *)
(* Surjective pairing: equivalent to Exh_Sprod                              *)
(* ------------------------------------------------------------------------ *)

qed_goal "surjective_pairing_Sprod" Sprod0.thy 
	"z = Ispair(Isfst z)(Issnd z)"
(fn prems =>
	[
	(res_inst_tac [("z1","z")] (Exh_Sprod RS disjE) 1),
        (asm_simp_tac Sprod0_ss 1),
	(etac exE 1),
	(etac exE 1),
        (asm_simp_tac Sprod0_ss 1)
	]);