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(* Title: LCF/lcf.ML

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ID: $Id$

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Author: Tobias Nipkow

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Copyright 1992 University of Cambridge


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For lcf.thy. Basic lemmas about LCF


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*)


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open LCF;


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signature LCF_LEMMAS =


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sig


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val ap_term: thm


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val ap_thm: thm


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val COND_cases: thm


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val COND_cases_iff: thm


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val Contrapos: thm


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val cong: thm


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val ext: thm


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val eq_imp_less1: thm


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val eq_imp_less2: thm


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val less_anti_sym: thm


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val less_ap_term: thm


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val less_ap_thm: thm


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val less_refl: thm


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val less_UU: thm


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val not_UU_eq_TT: thm


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val not_UU_eq_FF: thm


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val not_TT_eq_UU: thm


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val not_TT_eq_FF: thm


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val not_FF_eq_UU: thm


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val not_FF_eq_TT: thm


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val rstac: thm list > int > tactic


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val stac: thm > int > tactic


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val sstac: thm list > int > tactic


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val strip_tac: int > tactic


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val tr_induct: thm


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val UU_abs: thm


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val UU_app: thm


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end;


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structure LCF_Lemmas : LCF_LEMMAS =


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struct


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(* Standard abbreviations *)


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val rstac = resolve_tac;


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fun stac th = rtac(th RS sym RS subst);


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fun sstac ths = EVERY' (map stac ths);


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fun strip_tac i = REPEAT(rstac [impI,allI] i);


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val eq_imp_less1 = prove_goal thy "x=y ==> x << y"

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(fn prems => [rtac (rewrite_rule[eq_def](hd prems) RS conjunct1) 1]);

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val eq_imp_less2 = prove_goal thy "x=y ==> y << x"

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(fn prems => [rtac (rewrite_rule[eq_def](hd prems) RS conjunct2) 1]);

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val less_refl = refl RS eq_imp_less1;


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val less_anti_sym = prove_goal thy "[ x << y; y << x ] ==> x=y"

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(fn prems => [rewtac eq_def,


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REPEAT(rstac(conjI::prems)1)]);

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val ext = prove_goal thy

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"(!!x::'a::cpo. f(x)=(g(x)::'b::cpo)) ==> (%x.f(x))=(%x.g(x))"


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(fn [prem] => [REPEAT(rstac[less_anti_sym, less_ext, allI,


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prem RS eq_imp_less1,


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prem RS eq_imp_less2]1)]);

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val cong = prove_goal thy "[ f=g; x=y ] ==> f(x)=g(y)"

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(fn prems => [cut_facts_tac prems 1, etac subst 1, etac subst 1,


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rtac refl 1]);

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val less_ap_term = less_refl RS mono;


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val less_ap_thm = less_refl RSN (2,mono);


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val ap_term = refl RS cong;


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val ap_thm = refl RSN (2,cong);


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val UU_abs = prove_goal thy "(%x::'a::cpo.UU) = UU"

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(fn _ => [rtac less_anti_sym 1, rtac minimal 2,


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rtac less_ext 1, rtac allI 1, rtac minimal 1]);

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val UU_app = UU_abs RS sym RS ap_thm;


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val less_UU = prove_goal thy "x << UU ==> x=UU"

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(fn prems=> [rtac less_anti_sym 1,rstac prems 1,rtac minimal 1]);

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val tr_induct = prove_goal thy "[ P(UU); P(TT); P(FF) ] ==> ALL b.P(b)"

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(fn prems => [rtac allI 1, rtac mp 1,


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res_inst_tac[("p","b")]tr_cases 2,


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fast_tac (FOL_cs addIs prems) 1]);

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val Contrapos = prove_goal thy "(A ==> B) ==> (~B ==> ~A)"

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(fn prems => [rtac notI 1, rtac notE 1, rstac prems 1,


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rstac prems 1, atac 1]);

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val not_less_imp_not_eq1 = eq_imp_less1 COMP Contrapos;


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val not_less_imp_not_eq2 = eq_imp_less2 COMP Contrapos;


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val not_UU_eq_TT = not_TT_less_UU RS not_less_imp_not_eq2;


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val not_UU_eq_FF = not_FF_less_UU RS not_less_imp_not_eq2;


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val not_TT_eq_UU = not_TT_less_UU RS not_less_imp_not_eq1;


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val not_TT_eq_FF = not_TT_less_FF RS not_less_imp_not_eq1;


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val not_FF_eq_UU = not_FF_less_UU RS not_less_imp_not_eq1;


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val not_FF_eq_TT = not_FF_less_TT RS not_less_imp_not_eq1;


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val COND_cases_iff = (prove_goal thy


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"ALL b. P(b=>xy) <> (b=UU>P(UU)) & (b=TT>P(x)) & (b=FF>P(y))"

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(fn _ => [cut_facts_tac [not_UU_eq_TT,not_UU_eq_FF,not_TT_eq_UU,


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not_TT_eq_FF,not_FF_eq_UU,not_FF_eq_TT]1,


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rtac tr_induct 1, stac COND_UU 1, stac COND_TT 2,


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stac COND_FF 3, REPEAT(fast_tac FOL_cs 1)])) RS spec;

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val lemma = prove_goal thy "A<>B ==> B ==> A"

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(fn prems => [cut_facts_tac prems 1, rewtac iff_def,


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fast_tac FOL_cs 1]);

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val COND_cases = conjI RSN (2,conjI RS (COND_cases_iff RS lemma));


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end;


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open LCF_Lemmas;


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