(* 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 =>
[
(case_tac "a=UU|b=UU" 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_Morgan_disj RS subst) 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_Morgan_disj RS iffD2) 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_Morgan_disj RS subst) 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 =>
[
(stac strict_Ispair1 1),
(rtac strict_Isfst 1),
(rtac refl 1)
]);
qed_goal "strict_Isfst2" Sprod0.thy
"Isfst(Ispair x UU) = UU"
(fn prems =>
[
(stac strict_Ispair2 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 =>
[
(stac strict_Ispair1 1),
(rtac strict_Issnd 1),
(rtac refl 1)
]);
qed_goal "strict_Issnd2" Sprod0.thy
"Issnd(Ispair x UU) = UU"
(fn prems =>
[
(stac strict_Ispair2 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)
]);