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(* Title: HOL/Decision_Procs/ferrack_tac.ML 
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Author: Amine Chaieb, TU Muenchen 
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*) 

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signature FERRACK_TAC = 
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sig 

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val trace: bool Unsynchronized.ref 
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val linr_tac: Proof.context > bool > int > tactic 
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val setup: theory > theory 

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end 

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structure Ferrack_Tac = 
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struct 
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val trace = Unsynchronized.ref false; 
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fun trace_msg s = if !trace then tracing s else (); 
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val ferrack_ss = let val ths = [@{thm real_of_int_inject}, @{thm real_of_int_less_iff}, 
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@{thm real_of_int_le_iff}] 
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in @{simpset} delsimps ths addsimps (map (fn th => th RS sym) ths) 
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end; 
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val binarith = 
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@{thms normalize_bin_simps} @ @{thms pred_bin_simps} @ @{thms succ_bin_simps} @ 
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@{thms add_bin_simps} @ @{thms minus_bin_simps} @ @{thms mult_bin_simps}; 
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val comp_arith = binarith @ simp_thms 
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val zdvd_int = @{thm zdvd_int}; 
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val zdiff_int_split = @{thm zdiff_int_split}; 
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val all_nat = @{thm all_nat}; 
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val ex_nat = @{thm ex_nat}; 
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val number_of1 = @{thm number_of1}; 
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val number_of2 = @{thm number_of2}; 
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val split_zdiv = @{thm split_zdiv}; 
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val split_zmod = @{thm split_zmod}; 
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val mod_div_equality' = @{thm mod_div_equality'}; 
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val split_div' = @{thm split_div'}; 
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val Suc_eq_plus1 = @{thm Suc_eq_plus1}; 
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val imp_le_cong = @{thm imp_le_cong}; 
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val conj_le_cong = @{thm conj_le_cong}; 
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val mod_add_left_eq = @{thm mod_add_left_eq} RS sym; 
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val mod_add_right_eq = @{thm mod_add_right_eq} RS sym; 

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val nat_div_add_eq = @{thm div_add1_eq} RS sym; 
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val int_div_add_eq = @{thm zdiv_zadd1_eq} RS sym; 
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val ZDIVISION_BY_ZERO_MOD = @{thm DIVISION_BY_ZERO} RS conjunct2; 
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val ZDIVISION_BY_ZERO_DIV = @{thm DIVISION_BY_ZERO} RS conjunct1; 
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fun prepare_for_linr sg q fm = 
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let 
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val ps = Logic.strip_params fm 
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val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm) 
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val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm) 
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fun mk_all ((s, T), (P,n)) = 
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if member (op =) (loose_bnos P) 0 then 
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(HOLogic.all_const T $ Abs (s, T, P), n) 
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else (incr_boundvars ~1 P, n1) 
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fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t; 
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val rhs = hs 
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(* val (rhs,irhs) = List.partition (relevant (rev ps)) hs *) 
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val np = length ps 
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val (fm',np) = List.foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n))) 
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(List.foldr HOLogic.mk_imp c rhs, np) ps 
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val (vs, _) = List.partition (fn t => q orelse (type_of t) = HOLogic.natT) 
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(OldTerm.term_frees fm' @ OldTerm.term_vars fm'); 
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val fm2 = List.foldr mk_all2 fm' vs 
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in (fm2, np + length vs, length rhs) end; 
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(*Object quantifier to meta *) 
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fun spec_step n th = if (n=0) then th else (spec_step (n1) th) RS spec ; 
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(* object implication to meta*) 
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fun mp_step n th = if (n=0) then th else (mp_step (n1) th) RS mp; 
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fun linr_tac ctxt q = 
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Object_Logic.atomize_prems_tac 
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THEN' (REPEAT_DETERM o split_tac [@{thm split_min}, @{thm split_max}, @{thm abs_split}]) 
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THEN' SUBGOAL (fn (g, i) => 

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let 
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val thy = ProofContext.theory_of ctxt 
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(* Transform the term*) 
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val (t,np,nh) = prepare_for_linr thy q g 
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(* Some simpsets for dealing with mod div abs and nat*) 
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val simpset0 = Simplifier.context ctxt HOL_basic_ss addsimps comp_arith 
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val ct = cterm_of thy (HOLogic.mk_Trueprop t) 
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(* Theorem for the nat > int transformation *) 
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val pre_thm = Seq.hd (EVERY 
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[simp_tac simpset0 1, 
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TRY (simp_tac (Simplifier.context ctxt ferrack_ss) 1)] 
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(Thm.trivial ct)) 
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fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i) 
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(* The result of the quantifier elimination *) 
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val (th, tac) = case prop_of pre_thm of 
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Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ => 
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let val pth = linr_oracle (ctxt, Pattern.eta_long [] t1) 
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in 
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(trace_msg ("calling procedure with term:\n" ^ 
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Syntax.string_of_term ctxt t1); 
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((pth RS iffD2) RS pre_thm, 
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assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))) 
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end 
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 _ => (pre_thm, assm_tac i) 
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in rtac ((mp_step nh o spec_step np) th) i THEN tac end); 
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val setup = 
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Method.setup @{binding rferrack} 
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(Args.mode "no_quantify" >> (fn q => fn ctxt => 

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SIMPLE_METHOD' (linr_tac ctxt (not q)))) 

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"decision procedure for linear real arithmetic"; 

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end 