|
1 structure LinrTac = |
|
2 struct |
|
3 |
|
4 val trace = ref false; |
|
5 fun trace_msg s = if !trace then tracing s else (); |
|
6 |
|
7 val ferrack_ss = let val ths = [@{thm real_of_int_inject}, @{thm real_of_int_less_iff}, |
|
8 @{thm real_of_int_le_iff}] |
|
9 in @{simpset} delsimps ths addsimps (map (fn th => th RS sym) ths) |
|
10 end; |
|
11 |
|
12 val binarith = |
|
13 @{thms normalize_bin_simps} @ @{thms pred_bin_simps} @ @{thms succ_bin_simps} @ |
|
14 @{thms add_bin_simps} @ @{thms minus_bin_simps} @ @{thms mult_bin_simps}; |
|
15 val comp_arith = binarith @ simp_thms |
|
16 |
|
17 val zdvd_int = @{thm zdvd_int}; |
|
18 val zdiff_int_split = @{thm zdiff_int_split}; |
|
19 val all_nat = @{thm all_nat}; |
|
20 val ex_nat = @{thm ex_nat}; |
|
21 val number_of1 = @{thm number_of1}; |
|
22 val number_of2 = @{thm number_of2}; |
|
23 val split_zdiv = @{thm split_zdiv}; |
|
24 val split_zmod = @{thm split_zmod}; |
|
25 val mod_div_equality' = @{thm mod_div_equality'}; |
|
26 val split_div' = @{thm split_div'}; |
|
27 val Suc_plus1 = @{thm Suc_plus1}; |
|
28 val imp_le_cong = @{thm imp_le_cong}; |
|
29 val conj_le_cong = @{thm conj_le_cong}; |
|
30 val nat_mod_add_eq = @{thm mod_add1_eq} RS sym; |
|
31 val nat_mod_add_left_eq = @{thm mod_add_left_eq} RS sym; |
|
32 val nat_mod_add_right_eq = @{thm mod_add_right_eq} RS sym; |
|
33 val int_mod_add_eq = @{thm zmod_zadd1_eq} RS sym; |
|
34 val int_mod_add_left_eq = @{thm zmod_zadd_left_eq} RS sym; |
|
35 val int_mod_add_right_eq = @{thm zmod_zadd_right_eq} RS sym; |
|
36 val nat_div_add_eq = @{thm div_add1_eq} RS sym; |
|
37 val int_div_add_eq = @{thm zdiv_zadd1_eq} RS sym; |
|
38 val ZDIVISION_BY_ZERO_MOD = @{thm DIVISION_BY_ZERO} RS conjunct2; |
|
39 val ZDIVISION_BY_ZERO_DIV = @{thm DIVISION_BY_ZERO} RS conjunct1; |
|
40 |
|
41 fun prepare_for_linr sg q fm = |
|
42 let |
|
43 val ps = Logic.strip_params fm |
|
44 val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm) |
|
45 val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm) |
|
46 fun mk_all ((s, T), (P,n)) = |
|
47 if 0 mem loose_bnos P then |
|
48 (HOLogic.all_const T $ Abs (s, T, P), n) |
|
49 else (incr_boundvars ~1 P, n-1) |
|
50 fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t; |
|
51 val rhs = hs |
|
52 (* val (rhs,irhs) = List.partition (relevant (rev ps)) hs *) |
|
53 val np = length ps |
|
54 val (fm',np) = foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n))) |
|
55 (foldr HOLogic.mk_imp c rhs, np) ps |
|
56 val (vs, _) = List.partition (fn t => q orelse (type_of t) = HOLogic.natT) |
|
57 (term_frees fm' @ term_vars fm'); |
|
58 val fm2 = foldr mk_all2 fm' vs |
|
59 in (fm2, np + length vs, length rhs) end; |
|
60 |
|
61 (*Object quantifier to meta --*) |
|
62 fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ; |
|
63 |
|
64 (* object implication to meta---*) |
|
65 fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp; |
|
66 |
|
67 |
|
68 fun linr_tac ctxt q i = |
|
69 (ObjectLogic.atomize_prems_tac i) |
|
70 THEN (REPEAT_DETERM (split_tac [@{thm split_min}, @{thm split_max}, @{thm abs_split}] i)) |
|
71 THEN (fn st => |
|
72 let |
|
73 val g = List.nth (prems_of st, i - 1) |
|
74 val thy = ProofContext.theory_of ctxt |
|
75 (* Transform the term*) |
|
76 val (t,np,nh) = prepare_for_linr thy q g |
|
77 (* Some simpsets for dealing with mod div abs and nat*) |
|
78 val simpset0 = Simplifier.theory_context thy HOL_basic_ss addsimps comp_arith |
|
79 val ct = cterm_of thy (HOLogic.mk_Trueprop t) |
|
80 (* Theorem for the nat --> int transformation *) |
|
81 val pre_thm = Seq.hd (EVERY |
|
82 [simp_tac simpset0 1, |
|
83 TRY (simp_tac (Simplifier.theory_context thy ferrack_ss) 1)] |
|
84 (trivial ct)) |
|
85 fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i) |
|
86 (* The result of the quantifier elimination *) |
|
87 val (th, tac) = case (prop_of pre_thm) of |
|
88 Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ => |
|
89 let val pth = linr_oracle (cterm_of thy (Pattern.eta_long [] t1)) |
|
90 in |
|
91 (trace_msg ("calling procedure with term:\n" ^ |
|
92 Syntax.string_of_term ctxt t1); |
|
93 ((pth RS iffD2) RS pre_thm, |
|
94 assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))) |
|
95 end |
|
96 | _ => (pre_thm, assm_tac i) |
|
97 in (rtac (((mp_step nh) o (spec_step np)) th) i |
|
98 THEN tac) st |
|
99 end handle Subscript => no_tac st); |
|
100 |
|
101 fun linr_meth src = |
|
102 Method.syntax (Args.mode "no_quantify") src |
|
103 #> (fn (q, ctxt) => Method.SIMPLE_METHOD' (linr_tac ctxt (not q))); |
|
104 |
|
105 val setup = |
|
106 Method.add_method ("rferrack", linr_meth, |
|
107 "decision procedure for linear real arithmetic"); |
|
108 |
|
109 end |