(* Title: HOLCF/cfun3.ML
ID: $Id$
Author: Franz Regensburger
Copyright 1993 Technische Universitaet Muenchen
*)
open Cfun3;
(* for compatibility with old HOLCF-Version *)
qed_goal "inst_cfun_pcpo" thy "UU = fabs(%x.UU)"
(fn prems =>
[
(simp_tac (HOL_ss addsimps [UU_def,UU_cfun_def]) 1)
]);
(* ------------------------------------------------------------------------ *)
(* the contlub property for fapp its 'first' argument *)
(* ------------------------------------------------------------------------ *)
qed_goal "contlub_fapp1" thy "contlub(fapp)"
(fn prems =>
[
(rtac contlubI 1),
(strip_tac 1),
(rtac (expand_fun_eq RS iffD2) 1),
(strip_tac 1),
(stac thelub_cfun 1),
(atac 1),
(stac Cfunapp2 1),
(etac cont_lubcfun 1),
(stac thelub_fun 1),
(etac (monofun_fapp1 RS ch2ch_monofun) 1),
(rtac refl 1)
]);
(* ------------------------------------------------------------------------ *)
(* the cont property for fapp in its first argument *)
(* ------------------------------------------------------------------------ *)
qed_goal "cont_fapp1" thy "cont(fapp)"
(fn prems =>
[
(rtac monocontlub2cont 1),
(rtac monofun_fapp1 1),
(rtac contlub_fapp1 1)
]);
(* ------------------------------------------------------------------------ *)
(* contlub, cont properties of fapp in its first argument in mixfix _[_] *)
(* ------------------------------------------------------------------------ *)
qed_goal "contlub_cfun_fun" thy
"is_chain(FY) ==>\
\ lub(range FY)`x = lub(range (%i.FY(i)`x))"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac trans 1),
(etac (contlub_fapp1 RS contlubE RS spec RS mp RS fun_cong) 1),
(stac thelub_fun 1),
(etac (monofun_fapp1 RS ch2ch_monofun) 1),
(rtac refl 1)
]);
qed_goal "cont_cfun_fun" thy
"is_chain(FY) ==>\
\ range(%i.FY(i)`x) <<| lub(range FY)`x"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac thelubE 1),
(etac ch2ch_fappL 1),
(etac (contlub_cfun_fun RS sym) 1)
]);
(* ------------------------------------------------------------------------ *)
(* contlub, cont properties of fapp in both argument in mixfix _[_] *)
(* ------------------------------------------------------------------------ *)
qed_goal "contlub_cfun" thy
"[|is_chain(FY);is_chain(TY)|] ==>\
\ (lub(range FY))`(lub(range TY)) = lub(range(%i.FY(i)`(TY i)))"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac contlub_CF2 1),
(rtac cont_fapp1 1),
(rtac allI 1),
(rtac cont_fapp2 1),
(atac 1),
(atac 1)
]);
qed_goal "cont_cfun" thy
"[|is_chain(FY);is_chain(TY)|] ==>\
\ range(%i.(FY i)`(TY i)) <<| (lub (range FY))`(lub(range TY))"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac thelubE 1),
(rtac (monofun_fapp1 RS ch2ch_MF2LR) 1),
(rtac allI 1),
(rtac monofun_fapp2 1),
(atac 1),
(atac 1),
(etac (contlub_cfun RS sym) 1),
(atac 1)
]);
(* ------------------------------------------------------------------------ *)
(* cont2cont lemma for fapp *)
(* ------------------------------------------------------------------------ *)
qed_goal "cont2cont_fapp" thy
"[|cont(%x.ft x);cont(%x.tt x)|] ==> cont(%x. (ft x)`(tt x))"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac cont2cont_app2 1),
(rtac cont2cont_app2 1),
(rtac cont_const 1),
(rtac cont_fapp1 1),
(atac 1),
(rtac cont_fapp2 1),
(atac 1)
]);
(* ------------------------------------------------------------------------ *)
(* cont2mono Lemma for %x. LAM y. c1(x)(y) *)
(* ------------------------------------------------------------------------ *)
qed_goal "cont2mono_LAM" thy
"[|!x.cont(c1 x); !y.monofun(%x.c1 x y)|] ==>\
\ monofun(%x. LAM y. c1 x y)"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac monofunI 1),
(strip_tac 1),
(stac less_cfun 1),
(stac less_fun 1),
(rtac allI 1),
(stac beta_cfun 1),
(etac spec 1),
(stac beta_cfun 1),
(etac spec 1),
(etac ((hd (tl prems)) RS spec RS monofunE RS spec RS spec RS mp) 1)
]);
(* ------------------------------------------------------------------------ *)
(* cont2cont Lemma for %x. LAM y. c1 x y) *)
(* ------------------------------------------------------------------------ *)
qed_goal "cont2cont_LAM" thy
"[| !x.cont(c1 x); !y.cont(%x.c1 x y) |] ==> cont(%x. LAM y. c1 x y)"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac monocontlub2cont 1),
(etac cont2mono_LAM 1),
(rtac (cont2mono RS allI) 1),
(etac spec 1),
(rtac contlubI 1),
(strip_tac 1),
(stac thelub_cfun 1),
(rtac (cont2mono_LAM RS ch2ch_monofun) 1),
(atac 1),
(rtac (cont2mono RS allI) 1),
(etac spec 1),
(atac 1),
(res_inst_tac [("f","fabs")] arg_cong 1),
(rtac ext 1),
(stac (beta_cfun RS ext) 1),
(etac spec 1),
(rtac (cont2contlub RS contlubE
RS spec RS mp ) 1),
(etac spec 1),
(atac 1)
]);
(* ------------------------------------------------------------------------ *)
(* elimination of quantifier in premisses of cont2cont_LAM yields good *)
(* lemma for the cont tactic *)
(* ------------------------------------------------------------------------ *)
bind_thm ("cont2cont_LAM2", allI RSN (2,(allI RS cont2cont_LAM)));
(*
[| !!x. cont (?c1.0 x);
!!y. cont (%x. ?c1.0 x y) |] ==> cont (%x. LAM y. ?c1.0 x y)
*)
(* ------------------------------------------------------------------------ *)
(* cont2cont tactic *)
(* ------------------------------------------------------------------------ *)
val cont_lemmas1 = [cont_const, cont_id, cont_fapp2,
cont2cont_fapp,cont2cont_LAM2];
Addsimps cont_lemmas1;
(*val cont_tac = (fn i => (resolve_tac cont_lemmas i));*)
(*val cont_tacR = (fn i => (REPEAT (cont_tac i)));*)
(* ------------------------------------------------------------------------ *)
(* function application _[_] is strict in its first arguments *)
(* ------------------------------------------------------------------------ *)
qed_goal "strict_fapp1" thy "(UU::'a->'b)`x = (UU::'b)"
(fn prems =>
[
(stac inst_cfun_pcpo 1),
(stac beta_cfun 1),
(Simp_tac 1),
(rtac refl 1)
]);
(* ------------------------------------------------------------------------ *)
(* results about strictify *)
(* ------------------------------------------------------------------------ *)
qed_goalw "Istrictify1" thy [Istrictify_def]
"Istrictify(f)(UU)= (UU)"
(fn prems =>
[
(Simp_tac 1)
]);
qed_goalw "Istrictify2" thy [Istrictify_def]
"~x=UU ==> Istrictify(f)(x)=f`x"
(fn prems =>
[
(cut_facts_tac prems 1),
(Asm_simp_tac 1)
]);
qed_goal "monofun_Istrictify1" thy "monofun(Istrictify)"
(fn prems =>
[
(rtac monofunI 1),
(strip_tac 1),
(rtac (less_fun RS iffD2) 1),
(strip_tac 1),
(res_inst_tac [("Q","xa=UU")] (excluded_middle RS disjE) 1),
(stac Istrictify2 1),
(atac 1),
(stac Istrictify2 1),
(atac 1),
(rtac monofun_cfun_fun 1),
(atac 1),
(hyp_subst_tac 1),
(stac Istrictify1 1),
(stac Istrictify1 1),
(rtac refl_less 1)
]);
qed_goal "monofun_Istrictify2" thy "monofun(Istrictify(f))"
(fn prems =>
[
(rtac monofunI 1),
(strip_tac 1),
(res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1),
(stac Istrictify2 1),
(etac notUU_I 1),
(atac 1),
(stac Istrictify2 1),
(atac 1),
(rtac monofun_cfun_arg 1),
(atac 1),
(hyp_subst_tac 1),
(stac Istrictify1 1),
(rtac minimal 1)
]);
qed_goal "contlub_Istrictify1" thy "contlub(Istrictify)"
(fn prems =>
[
(rtac contlubI 1),
(strip_tac 1),
(rtac (expand_fun_eq RS iffD2) 1),
(strip_tac 1),
(stac thelub_fun 1),
(etac (monofun_Istrictify1 RS ch2ch_monofun) 1),
(res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1),
(stac Istrictify2 1),
(atac 1),
(stac (Istrictify2 RS ext) 1),
(atac 1),
(stac thelub_cfun 1),
(atac 1),
(stac beta_cfun 1),
(rtac cont_lubcfun 1),
(atac 1),
(rtac refl 1),
(hyp_subst_tac 1),
(stac Istrictify1 1),
(stac (Istrictify1 RS ext) 1),
(rtac (chain_UU_I_inverse RS sym) 1),
(rtac (refl RS allI) 1)
]);
qed_goal "contlub_Istrictify2" thy "contlub(Istrictify(f::'a -> 'b))"
(fn prems =>
[
(rtac contlubI 1),
(strip_tac 1),
(case_tac "lub(range(Y))=(UU::'a)" 1),
(res_inst_tac [("t","lub(range(Y))")] subst 1),
(rtac sym 1),
(atac 1),
(stac Istrictify1 1),
(rtac sym 1),
(rtac chain_UU_I_inverse 1),
(strip_tac 1),
(res_inst_tac [("t","Y(i)"),("s","UU::'a")] subst 1),
(rtac sym 1),
(rtac (chain_UU_I RS spec) 1),
(atac 1),
(atac 1),
(rtac Istrictify1 1),
(stac Istrictify2 1),
(atac 1),
(res_inst_tac [("s","lub(range(%i. f`(Y i)))")] trans 1),
(rtac contlub_cfun_arg 1),
(atac 1),
(rtac lub_equal2 1),
(rtac (chain_mono2 RS exE) 1),
(atac 2),
(rtac chain_UU_I_inverse2 1),
(atac 1),
(rtac exI 1),
(strip_tac 1),
(rtac (Istrictify2 RS sym) 1),
(fast_tac HOL_cs 1),
(rtac ch2ch_monofun 1),
(rtac monofun_fapp2 1),
(atac 1),
(rtac ch2ch_monofun 1),
(rtac monofun_Istrictify2 1),
(atac 1)
]);
bind_thm ("cont_Istrictify1", contlub_Istrictify1 RS
(monofun_Istrictify1 RS monocontlub2cont));
bind_thm ("cont_Istrictify2", contlub_Istrictify2 RS
(monofun_Istrictify2 RS monocontlub2cont));
qed_goalw "strictify1" thy [strictify_def] "strictify`f`UU=UU" (fn _ => [
(stac beta_cfun 1),
(simp_tac (!simpset addsimps [cont_Istrictify2,cont_Istrictify1,
cont2cont_CF1L]) 1),
(stac beta_cfun 1),
(rtac cont_Istrictify2 1),
(rtac Istrictify1 1)
]);
qed_goalw "strictify2" thy [strictify_def]
"~x=UU ==> strictify`f`x=f`x" (fn prems => [
(stac beta_cfun 1),
(simp_tac (!simpset addsimps [cont_Istrictify2,cont_Istrictify1,
cont2cont_CF1L]) 1),
(stac beta_cfun 1),
(rtac cont_Istrictify2 1),
(rtac Istrictify2 1),
(resolve_tac prems 1)
]);
(* ------------------------------------------------------------------------ *)
(* Instantiate the simplifier *)
(* ------------------------------------------------------------------------ *)
Addsimps [minimal,refl_less,beta_cfun,strict_fapp1,strictify1, strictify2];
(* ------------------------------------------------------------------------ *)
(* use cont_tac as autotac. *)
(* ------------------------------------------------------------------------ *)
(*simpset := !simpset addsolver (K (DEPTH_SOLVE_1 o cont_tac));*)