author | clasohm |
Thu, 17 Mar 1994 11:24:31 +0100 | |
changeset 278 | 523518f44286 |
parent 8 | c3d2c6dcf3f0 |
child 280 | fb379160f4de |
permissions | -rw-r--r-- |
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(* Title: CCL/ccl |
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ID: $Id$ |
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Author: Martin Coen, Cambridge University Computer Laboratory |
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Copyright 1993 University of Cambridge |
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For ccl.thy. |
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*) |
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open CCL; |
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val ccl_data_defs = [apply_def,fix_def]; |
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val po_refl_iff_T = make_iff_T po_refl; |
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val CCL_ss = FOL_ss addcongs set_congs |
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addsimps ([po_refl_iff_T] @ mem_rews); |
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(*** Congruence Rules ***) |
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(*similar to AP_THM in Gordon's HOL*) |
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val fun_cong = prove_goal CCL.thy "(f::'a=>'b) = g ==> f(x)=g(x)" |
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(fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]); |
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(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*) |
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val arg_cong = prove_goal CCL.thy "x=y ==> f(x)=f(y)" |
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(fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]); |
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goal CCL.thy "(ALL x. f(x) = g(x)) --> (%x.f(x)) = (%x.g(x))"; |
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by (simp_tac (CCL_ss addsimps [eq_iff]) 1); |
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by (fast_tac (set_cs addIs [po_abstractn]) 1); |
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val abstractn = standard (allI RS (result() RS mp)); |
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fun type_of_terms (Const("Trueprop",_) $ |
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(Const("op =",(Type ("fun", [t,_]))) $ _ $ _)) = t; |
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fun abs_prems thm = |
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let fun do_abs n thm (Type ("fun", [_,t])) = do_abs n (abstractn RSN (n,thm)) t |
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| do_abs n thm _ = thm |
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fun do_prems n [] thm = thm |
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| do_prems n (x::xs) thm = do_prems (n+1) xs (do_abs n thm (type_of_terms x)); |
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in do_prems 1 (prems_of thm) thm |
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end; |
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val caseBs = [caseBtrue,caseBfalse,caseBpair,caseBlam,caseBbot]; |
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(*** Termination and Divergence ***) |
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goalw CCL.thy [Trm_def,Dvg_def] "Trm(t) <-> ~ t = bot"; |
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br iff_refl 1; |
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val Trm_iff = result(); |
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goalw CCL.thy [Trm_def,Dvg_def] "Dvg(t) <-> t = bot"; |
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br iff_refl 1; |
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val Dvg_iff = result(); |
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(*** Constructors are injective ***) |
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val prems = goal CCL.thy |
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"[| x=a; y=b; x=y |] ==> a=b"; |
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by (REPEAT (SOMEGOAL (ares_tac (prems@[box_equals])))); |
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val eq_lemma = result(); |
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fun mk_inj_rl thy rews s = |
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let fun mk_inj_lemmas r = ([arg_cong] RL [(r RS (r RS eq_lemma))]); |
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val inj_lemmas = flat (map mk_inj_lemmas rews); |
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val tac = REPEAT (ares_tac [iffI,allI,conjI] 1 ORELSE |
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eresolve_tac inj_lemmas 1 ORELSE |
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asm_simp_tac (CCL_ss addsimps rews) 1) |
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in prove_goal thy s (fn _ => [tac]) |
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end; |
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val ccl_injs = map (mk_inj_rl CCL.thy caseBs) |
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["<a,b> = <a',b'> <-> (a=a' & b=b')", |
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"(lam x.b(x) = lam x.b'(x)) <-> ((ALL z.b(z)=b'(z)))"]; |
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val pair_inject = ((hd ccl_injs) RS iffD1) RS conjE; |
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(*** Constructors are distinct ***) |
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local |
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fun pairs_of f x [] = [] |
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| pairs_of f x (y::ys) = (f x y) :: (f y x) :: (pairs_of f x ys); |
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fun mk_combs ff [] = [] |
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| mk_combs ff (x::xs) = (pairs_of ff x xs) @ mk_combs ff xs; |
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(* Doesn't handle binder types correctly *) |
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fun saturate thy sy name = |
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let fun arg_str 0 a s = s |
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| arg_str 1 a s = "(" ^ a ^ "a" ^ s ^ ")" |
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| arg_str n a s = arg_str (n-1) a ("," ^ a ^ (chr((ord "a")+n-1)) ^ s); |
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val sg = sign_of thy; |
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val T = case Sign.Symtab.lookup(#const_tab(Sign.rep_sg sg),sy) of |
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None => error(sy^" not declared") | Some(T) => T; |
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val arity = length (fst (strip_type T)); |
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in sy ^ (arg_str arity name "") end; |
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fun mk_thm_str thy a b = "~ " ^ (saturate thy a "a") ^ " = " ^ (saturate thy b "b"); |
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val lemma = prove_goal CCL.thy "t=t' --> case(t,b,c,d,e) = case(t',b,c,d,e)" |
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(fn _ => [simp_tac CCL_ss 1]) RS mp; |
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fun mk_lemma (ra,rb) = [lemma] RL [ra RS (rb RS eq_lemma)] RL |
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[distinctness RS notE,sym RS (distinctness RS notE)]; |
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in |
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fun mk_lemmas rls = flat (map mk_lemma (mk_combs pair rls)); |
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fun mk_dstnct_rls thy xs = mk_combs (mk_thm_str thy) xs; |
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end; |
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val caseB_lemmas = mk_lemmas caseBs; |
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val ccl_dstncts = |
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let fun mk_raw_dstnct_thm rls s = |
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prove_goal CCL.thy s (fn _=> [rtac notI 1,eresolve_tac rls 1]) |
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in map (mk_raw_dstnct_thm caseB_lemmas) |
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(mk_dstnct_rls CCL.thy ["bot","true","false","pair","lambda"]) end; |
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fun mk_dstnct_thms thy defs inj_rls xs = |
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let fun mk_dstnct_thm rls s = prove_goalw thy defs s |
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(fn _ => [simp_tac (CCL_ss addsimps (rls@inj_rls)) 1]) |
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in map (mk_dstnct_thm ccl_dstncts) (mk_dstnct_rls thy xs) end; |
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fun mkall_dstnct_thms thy defs i_rls xss = flat (map (mk_dstnct_thms thy defs i_rls) xss); |
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(*** Rewriting and Proving ***) |
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fun XH_to_I rl = rl RS iffD2; |
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fun XH_to_D rl = rl RS iffD1; |
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val XH_to_E = make_elim o XH_to_D; |
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val XH_to_Is = map XH_to_I; |
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val XH_to_Ds = map XH_to_D; |
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val XH_to_Es = map XH_to_E; |
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val ccl_rews = caseBs @ ccl_injs @ ccl_dstncts; |
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val ccl_ss = CCL_ss addsimps ccl_rews; |
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val ccl_cs = set_cs addSEs (pair_inject::(ccl_dstncts RL [notE])) |
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addSDs (XH_to_Ds ccl_injs); |
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(****** Facts from gfp Definition of [= and = ******) |
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val major::prems = goal Set.thy "[| A=B; a:B <-> P |] ==> a:A <-> P"; |
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brs (prems RL [major RS ssubst]) 1; |
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val XHlemma1 = result(); |
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goal CCL.thy "(P(t,t') <-> Q) --> (<t,t'> : {p.EX t t'.p=<t,t'> & P(t,t')} <-> Q)"; |
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by (fast_tac ccl_cs 1); |
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val XHlemma2 = result() RS mp; |
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(*** Pre-Order ***) |
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goalw CCL.thy [POgen_def,SIM_def] "mono(%X.POgen(X))"; |
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br monoI 1; |
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by (safe_tac ccl_cs); |
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by (REPEAT_SOME (resolve_tac [exI,conjI,refl])); |
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by (ALLGOALS (simp_tac ccl_ss)); |
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by (ALLGOALS (fast_tac set_cs)); |
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val POgen_mono = result(); |
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goalw CCL.thy [POgen_def,SIM_def] |
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"<t,t'> : POgen(R) <-> t= bot | (t=true & t'=true) | (t=false & t'=false) | \ |
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\ (EX a a' b b'.t=<a,b> & t'=<a',b'> & <a,a'> : R & <b,b'> : R) | \ |
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\ (EX f f'.t=lam x.f(x) & t'=lam x.f'(x) & (ALL x.<f(x),f'(x)> : R))"; |
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br (iff_refl RS XHlemma2) 1; |
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val POgenXH = result(); |
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goal CCL.thy |
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"t [= t' <-> t=bot | (t=true & t'=true) | (t=false & t'=false) | \ |
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\ (EX a a' b b'.t=<a,b> & t'=<a',b'> & a [= a' & b [= b') | \ |
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\ (EX f f'.t=lam x.f(x) & t'=lam x.f'(x) & (ALL x.f(x) [= f'(x)))"; |
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by (simp_tac (ccl_ss addsimps [PO_iff]) 1); |
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br (rewrite_rule [POgen_def,SIM_def] |
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(POgen_mono RS (PO_def RS def_gfp_Tarski) RS XHlemma1)) 1; |
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br (iff_refl RS XHlemma2) 1; |
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val poXH = result(); |
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goal CCL.thy "bot [= b"; |
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br (poXH RS iffD2) 1; |
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by (simp_tac ccl_ss 1); |
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val po_bot = result(); |
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goal CCL.thy "a [= bot --> a=bot"; |
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br impI 1; |
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bd (poXH RS iffD1) 1; |
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be rev_mp 1; |
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by (simp_tac ccl_ss 1); |
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val bot_poleast = result() RS mp; |
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goal CCL.thy "<a,b> [= <a',b'> <-> a [= a' & b [= b'"; |
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br (poXH RS iff_trans) 1; |
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by (simp_tac ccl_ss 1); |
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by (fast_tac ccl_cs 1); |
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val po_pair = result(); |
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goal CCL.thy "lam x.f(x) [= lam x.f'(x) <-> (ALL x. f(x) [= f'(x))"; |
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br (poXH RS iff_trans) 1; |
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by (simp_tac ccl_ss 1); |
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by (REPEAT (ares_tac [iffI,allI] 1 ORELSE eresolve_tac [exE,conjE] 1)); |
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by (asm_simp_tac ccl_ss 1); |
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by (fast_tac ccl_cs 1); |
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val po_lam = result(); |
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val ccl_porews = [po_bot,po_pair,po_lam]; |
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val [p1,p2,p3,p4,p5] = goal CCL.thy |
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"[| t [= t'; a [= a'; b [= b'; !!x y.c(x,y) [= c'(x,y); \ |
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\ !!u.d(u) [= d'(u) |] ==> case(t,a,b,c,d) [= case(t',a',b',c',d')"; |
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br (p1 RS po_cong RS po_trans) 1; |
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br (p2 RS po_cong RS po_trans) 1; |
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br (p3 RS po_cong RS po_trans) 1; |
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br (p4 RS po_abstractn RS po_abstractn RS po_cong RS po_trans) 1; |
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by (res_inst_tac [("f1","%d.case(t',a',b',c',d)")] |
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(p5 RS po_abstractn RS po_cong RS po_trans) 1); |
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br po_refl 1; |
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val case_pocong = result(); |
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val [p1,p2] = goalw CCL.thy ccl_data_defs |
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"[| f [= f'; a [= a' |] ==> f ` a [= f' ` a'"; |
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by (REPEAT (ares_tac [po_refl,case_pocong,p1,p2 RS po_cong] 1)); |
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val apply_pocong = result(); |
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val prems = goal CCL.thy "~ lam x.b(x) [= bot"; |
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br notI 1; |
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bd bot_poleast 1; |
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be (distinctness RS notE) 1; |
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val npo_lam_bot = result(); |
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val eq1::eq2::prems = goal CCL.thy |
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"[| x=a; y=b; x[=y |] ==> a[=b"; |
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br (eq1 RS subst) 1; |
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br (eq2 RS subst) 1; |
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brs prems 1; |
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val po_lemma = result(); |
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goal CCL.thy "~ <a,b> [= lam x.f(x)"; |
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br notI 1; |
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br (npo_lam_bot RS notE) 1; |
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be (case_pocong RS (caseBlam RS (caseBpair RS po_lemma))) 1; |
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by (REPEAT (resolve_tac [po_refl,npo_lam_bot] 1)); |
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val npo_pair_lam = result(); |
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goal CCL.thy "~ lam x.f(x) [= <a,b>"; |
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br notI 1; |
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br (npo_lam_bot RS notE) 1; |
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be (case_pocong RS (caseBpair RS (caseBlam RS po_lemma))) 1; |
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by (REPEAT (resolve_tac [po_refl,npo_lam_bot] 1)); |
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val npo_lam_pair = result(); |
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249 |
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fun mk_thm s = prove_goal CCL.thy s (fn _ => |
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[rtac notI 1,dtac case_pocong 1,etac rev_mp 5, |
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ALLGOALS (simp_tac ccl_ss), |
0 | 253 |
REPEAT (resolve_tac [po_refl,npo_lam_bot] 1)]); |
254 |
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val npo_rls = [npo_pair_lam,npo_lam_pair] @ map mk_thm |
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["~ true [= false", "~ false [= true", |
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257 |
"~ true [= <a,b>", "~ <a,b> [= true", |
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"~ true [= lam x.f(x)","~ lam x.f(x) [= true", |
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"~ false [= <a,b>", "~ <a,b> [= false", |
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260 |
"~ false [= lam x.f(x)","~ lam x.f(x) [= false"]; |
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261 |
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262 |
(* Coinduction for [= *) |
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263 |
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264 |
val prems = goal CCL.thy "[| <t,u> : R; R <= POgen(R) |] ==> t [= u"; |
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265 |
br (PO_def RS def_coinduct RS (PO_iff RS iffD2)) 1; |
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266 |
by (REPEAT (ares_tac prems 1)); |
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267 |
val po_coinduct = result(); |
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268 |
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269 |
fun po_coinduct_tac s i = res_inst_tac [("R",s)] po_coinduct i; |
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270 |
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271 |
(*************** EQUALITY *******************) |
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272 |
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273 |
goalw CCL.thy [EQgen_def,SIM_def] "mono(%X.EQgen(X))"; |
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274 |
br monoI 1; |
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275 |
by (safe_tac set_cs); |
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by (REPEAT_SOME (resolve_tac [exI,conjI,refl])); |
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277 |
by (ALLGOALS (simp_tac ccl_ss)); |
0 | 278 |
by (ALLGOALS (fast_tac set_cs)); |
279 |
val EQgen_mono = result(); |
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280 |
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281 |
goalw CCL.thy [EQgen_def,SIM_def] |
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282 |
"<t,t'> : EQgen(R) <-> (t=bot & t'=bot) | (t=true & t'=true) | \ |
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283 |
\ (t=false & t'=false) | \ |
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284 |
\ (EX a a' b b'.t=<a,b> & t'=<a',b'> & <a,a'> : R & <b,b'> : R) | \ |
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285 |
\ (EX f f'.t=lam x.f(x) & t'=lam x.f'(x) & (ALL x.<f(x),f'(x)> : R))"; |
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286 |
br (iff_refl RS XHlemma2) 1; |
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287 |
val EQgenXH = result(); |
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288 |
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289 |
goal CCL.thy |
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290 |
"t=t' <-> (t=bot & t'=bot) | (t=true & t'=true) | (t=false & t'=false) | \ |
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291 |
\ (EX a a' b b'.t=<a,b> & t'=<a',b'> & a=a' & b=b') | \ |
|
292 |
\ (EX f f'.t=lam x.f(x) & t'=lam x.f'(x) & (ALL x.f(x)=f'(x)))"; |
|
293 |
by (subgoal_tac |
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294 |
"<t,t'> : EQ <-> (t=bot & t'=bot) | (t=true & t'=true) | (t=false & t'=false) | \ |
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295 |
\ (EX a a' b b'.t=<a,b> & t'=<a',b'> & <a,a'> : EQ & <b,b'> : EQ) | \ |
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296 |
\ (EX f f'.t=lam x.f(x) & t'=lam x.f'(x) & (ALL x.<f(x),f'(x)> : EQ))" 1); |
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297 |
be rev_mp 1; |
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298 |
by (simp_tac (CCL_ss addsimps [EQ_iff RS iff_sym]) 1); |
0 | 299 |
br (rewrite_rule [EQgen_def,SIM_def] |
300 |
(EQgen_mono RS (EQ_def RS def_gfp_Tarski) RS XHlemma1)) 1; |
|
301 |
br (iff_refl RS XHlemma2) 1; |
|
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val eqXH = result(); |
|
303 |
||
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val prems = goal CCL.thy "[| <t,u> : R; R <= EQgen(R) |] ==> t = u"; |
|
305 |
br (EQ_def RS def_coinduct RS (EQ_iff RS iffD2)) 1; |
|
306 |
by (REPEAT (ares_tac prems 1)); |
|
307 |
val eq_coinduct = result(); |
|
308 |
||
309 |
val prems = goal CCL.thy |
|
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"[| <t,u> : R; R <= EQgen(lfp(%x.EQgen(x) Un R Un EQ)) |] ==> t = u"; |
|
311 |
br (EQ_def RS def_coinduct3 RS (EQ_iff RS iffD2)) 1; |
|
312 |
by (REPEAT (ares_tac (EQgen_mono::prems) 1)); |
|
313 |
val eq_coinduct3 = result(); |
|
314 |
||
315 |
fun eq_coinduct_tac s i = res_inst_tac [("R",s)] eq_coinduct i; |
|
316 |
fun eq_coinduct3_tac s i = res_inst_tac [("R",s)] eq_coinduct3 i; |
|
317 |
||
318 |
(*** Untyped Case Analysis and Other Facts ***) |
|
319 |
||
320 |
goalw CCL.thy [apply_def] "(EX f.t=lam x.f(x)) --> t = lam x.(t ` x)"; |
|
321 |
by (safe_tac ccl_cs); |
|
8
c3d2c6dcf3f0
Installation of new simplfier. Previously appeared to set up the old
lcp
parents:
0
diff
changeset
|
322 |
by (simp_tac ccl_ss 1); |
0 | 323 |
val cond_eta = result() RS mp; |
324 |
||
325 |
goal CCL.thy "(t=bot) | (t=true) | (t=false) | (EX a b.t=<a,b>) | (EX f.t=lam x.f(x))"; |
|
326 |
by (cut_facts_tac [refl RS (eqXH RS iffD1)] 1); |
|
327 |
by (fast_tac set_cs 1); |
|
328 |
val exhaustion = result(); |
|
329 |
||
330 |
val prems = goal CCL.thy |
|
331 |
"[| P(bot); P(true); P(false); !!x y.P(<x,y>); !!b.P(lam x.b(x)) |] ==> P(t)"; |
|
332 |
by (cut_facts_tac [exhaustion] 1); |
|
333 |
by (REPEAT_SOME (ares_tac prems ORELSE' eresolve_tac [disjE,exE,ssubst])); |
|
334 |
val term_case = result(); |
|
335 |
||
336 |
fun term_case_tac a i = res_inst_tac [("t",a)] term_case i; |