author | wenzelm |
Tue, 29 Sep 2009 16:24:36 +0200 | |
changeset 32740 | 9dd0a2f83429 |
parent 32618 | 42865636d006 |
child 33010 | 39f73a59e855 |
permissions | -rw-r--r-- |
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(* Title: HOL/SMT/Tools/smt_normalize.ML |
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Author: Sascha Boehme, TU Muenchen |
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Normalization steps on theorems required by SMT solvers: |
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* unfold trivial let expressions, |
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* replace negative numerals by negated positive numerals, |
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* embed natural numbers into integers, |
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* add extra rules specifying types and constants which occur frequently, |
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* lift lambda terms, |
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* make applications explicit for functions with varying number of arguments, |
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* fully translate into object logic, add universal closure. |
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*) |
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signature SMT_NORMALIZE = |
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sig |
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val normalize_rule: Proof.context -> thm -> thm |
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val instantiate_free: Thm.cterm * Thm.cterm -> thm -> thm |
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val discharge_definition: Thm.cterm -> thm -> thm |
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val trivial_let: Proof.context -> thm list -> thm list |
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val positive_numerals: Proof.context -> thm list -> thm list |
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val nat_as_int: Proof.context -> thm list -> thm list |
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val unfold_defs: bool Config.T |
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val add_pair_rules: Proof.context -> thm list -> thm list |
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val add_fun_upd_rules: Proof.context -> thm list -> thm list |
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val add_abs_min_max_rules: Proof.context -> thm list -> thm list |
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datatype config = |
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RewriteTrivialLets | |
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RewriteNegativeNumerals | |
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RewriteNaturalNumbers | |
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AddPairRules | |
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AddFunUpdRules | |
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AddAbsMinMaxRules |
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val normalize: config list -> Proof.context -> thm list -> |
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Thm.cterm list * thm list |
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|
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val setup: theory -> theory |
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end |
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structure SMT_Normalize: SMT_NORMALIZE = |
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struct |
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val norm_binder_conv = Conv.try_conv (More_Conv.rewrs_conv [ |
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@{lemma "All P == ALL x. P x" by (rule reflexive)}, |
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@{lemma "Ex P == EX x. P x" by (rule reflexive)}, |
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@{lemma "Let c P == let x = c in P x" by (rule reflexive)}]) |
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fun cert ctxt = Thm.cterm_of (ProofContext.theory_of ctxt) |
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fun norm_meta_def cv thm = |
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let val thm' = Thm.combination thm (Thm.reflexive cv) |
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in Thm.transitive thm' (Thm.beta_conversion false (Thm.rhs_of thm')) end |
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fun norm_def ctxt thm = |
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(case Thm.prop_of thm of |
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Const (@{const_name "=="}, _) $ _ $ Abs (_, T, _) => |
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let val v = Var ((Name.uu, #maxidx (Thm.rep_thm thm) + 1), T) |
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in norm_def ctxt (norm_meta_def (cert ctxt v) thm) end |
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| @{term Trueprop} $ (Const (@{const_name "op ="}, _) $ _ $ Abs _) => |
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norm_def ctxt (thm RS @{thm fun_cong}) |
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| _ => thm) |
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fun normalize_rule ctxt = |
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Conv.fconv_rule ( |
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Thm.beta_conversion true then_conv |
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Thm.eta_conversion then_conv |
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More_Conv.bottom_conv (K norm_binder_conv) ctxt) #> |
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norm_def ctxt #> |
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Drule.forall_intr_vars #> |
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Conv.fconv_rule ObjectLogic.atomize |
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fun instantiate_free (cv, ct) thm = |
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if Term.exists_subterm (equal (Thm.term_of cv)) (Thm.prop_of thm) |
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then Thm.forall_elim ct (Thm.forall_intr cv thm) |
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else thm |
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fun discharge_definition ct thm = |
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let val (cv, cu) = Thm.dest_equals ct |
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in |
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Thm.implies_intr ct thm |
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|> instantiate_free (cv, cu) |
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|> (fn thm => Thm.implies_elim thm (Thm.reflexive cu)) |
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end |
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fun if_conv c cv1 cv2 ct = (if c (Thm.term_of ct) then cv1 else cv2) ct |
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fun if_true_conv c cv = if_conv c cv Conv.all_conv |
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(* simplification of trivial let expressions (whose bound variables occur at |
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most once) *) |
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local |
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fun count i (Bound j) = if j = i then 1 else 0 |
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| count i (t $ u) = count i t + count i u |
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| count i (Abs (_, _, t)) = count (i + 1) t |
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| count _ _ = 0 |
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fun is_trivial_let (Const (@{const_name Let}, _) $ _ $ Abs (_, _, t)) = |
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(count 0 t <= 1) |
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| is_trivial_let _ = false |
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fun let_conv _ = if_true_conv is_trivial_let (Conv.rewr_conv @{thm Let_def}) |
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fun cond_let_conv ctxt = if_true_conv (Term.exists_subterm is_trivial_let) |
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(More_Conv.top_conv let_conv ctxt) |
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in |
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fun trivial_let ctxt = map (Conv.fconv_rule (cond_let_conv ctxt)) |
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end |
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(* rewriting of negative integer numerals into positive numerals *) |
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local |
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fun neg_numeral @{term Int.Min} = true |
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| neg_numeral _ = false |
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fun is_number_sort thy T = Sign.of_sort thy (T, @{sort number_ring}) |
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fun is_neg_number ctxt (Const (@{const_name number_of}, T) $ t) = |
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Term.exists_subterm neg_numeral t andalso |
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is_number_sort (ProofContext.theory_of ctxt) (Term.body_type T) |
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| is_neg_number _ _ = false |
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fun has_neg_number ctxt = Term.exists_subterm (is_neg_number ctxt) |
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val pos_numeral_ss = HOL_ss |
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addsimps [@{thm Int.number_of_minus}, @{thm Int.number_of_Min}] |
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addsimps [@{thm Int.numeral_1_eq_1}] |
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128 |
addsimps @{thms Int.pred_bin_simps} |
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129 |
addsimps @{thms Int.normalize_bin_simps} |
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130 |
addsimps @{lemma |
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131 |
"Int.Min = - Int.Bit1 Int.Pls" |
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132 |
"Int.Bit0 (- Int.Pls) = - Int.Pls" |
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133 |
"Int.Bit0 (- k) = - Int.Bit0 k" |
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134 |
"Int.Bit1 (- k) = - Int.Bit1 (Int.pred k)" |
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135 |
by simp_all (simp add: pred_def)} |
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136 |
|
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137 |
fun pos_conv ctxt = if_conv (is_neg_number ctxt) |
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138 |
(Simplifier.rewrite (Simplifier.context ctxt pos_numeral_ss)) |
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139 |
Conv.no_conv |
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140 |
|
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141 |
fun cond_pos_conv ctxt = if_true_conv (has_neg_number ctxt) |
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142 |
(More_Conv.top_sweep_conv pos_conv ctxt) |
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143 |
in |
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144 |
fun positive_numerals ctxt = map (Conv.fconv_rule (cond_pos_conv ctxt)) |
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145 |
end |
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146 |
|
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147 |
|
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148 |
(* embedding of standard natural number operations into integer operations *) |
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149 |
|
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150 |
local |
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151 |
val nat_embedding = @{lemma |
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152 |
"nat (int n) = n" |
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153 |
"i >= 0 --> int (nat i) = i" |
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154 |
"i < 0 --> int (nat i) = 0" |
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155 |
by simp_all} |
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156 |
|
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157 |
val nat_rewriting = @{lemma |
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158 |
"0 = nat 0" |
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159 |
"1 = nat 1" |
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160 |
"number_of i = nat (number_of i)" |
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161 |
"int (nat 0) = 0" |
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162 |
"int (nat 1) = 1" |
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163 |
"a < b = (int a < int b)" |
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164 |
"a <= b = (int a <= int b)" |
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165 |
"Suc a = nat (int a + 1)" |
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166 |
"a + b = nat (int a + int b)" |
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167 |
"a - b = nat (int a - int b)" |
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168 |
"a * b = nat (int a * int b)" |
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169 |
"a div b = nat (int a div int b)" |
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170 |
"a mod b = nat (int a mod int b)" |
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171 |
"int (nat (int a + int b)) = int a + int b" |
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172 |
"int (nat (int a * int b)) = int a * int b" |
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173 |
"int (nat (int a div int b)) = int a div int b" |
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174 |
"int (nat (int a mod int b)) = int a mod int b" |
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175 |
by (simp add: nat_mult_distrib nat_div_distrib nat_mod_distrib |
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176 |
int_mult[symmetric] zdiv_int[symmetric] zmod_int[symmetric])+} |
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177 |
|
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178 |
fun on_positive num f x = |
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179 |
(case try HOLogic.dest_number (Thm.term_of num) of |
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180 |
SOME (_, i) => if i >= 0 then SOME (f x) else NONE |
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181 |
| NONE => NONE) |
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182 |
|
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183 |
val cancel_int_nat_ss = HOL_ss |
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184 |
addsimps [@{thm Nat_Numeral.nat_number_of}] |
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185 |
addsimps [@{thm Nat_Numeral.int_nat_number_of}] |
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186 |
addsimps @{thms neg_simps} |
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187 |
|
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188 |
fun cancel_int_nat_simproc _ ss ct = |
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189 |
let |
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190 |
val num = Thm.dest_arg (Thm.dest_arg ct) |
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191 |
val goal = Thm.mk_binop @{cterm "op == :: int => _"} ct num |
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192 |
val simpset = Simplifier.inherit_context ss cancel_int_nat_ss |
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193 |
fun tac _ = Simplifier.simp_tac simpset 1 |
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194 |
in on_positive num (Goal.prove_internal [] goal) tac end |
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195 |
|
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196 |
val nat_ss = HOL_ss |
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197 |
addsimps nat_rewriting |
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198 |
addsimprocs [Simplifier.make_simproc { |
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199 |
name = "cancel_int_nat_num", lhss = [@{cpat "int (nat _)"}], |
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200 |
proc = cancel_int_nat_simproc, identifier = [] }] |
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201 |
|
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202 |
fun conv ctxt = Simplifier.rewrite (Simplifier.context ctxt nat_ss) |
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203 |
|
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204 |
val uses_nat_type = Term.exists_type (Term.exists_subtype (equal @{typ nat})) |
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205 |
in |
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206 |
fun nat_as_int ctxt thms = |
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207 |
let |
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208 |
fun norm thm uses_nat = |
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209 |
if not (uses_nat_type (Thm.prop_of thm)) then (thm, uses_nat) |
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210 |
else (Conv.fconv_rule (conv ctxt) thm, true) |
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211 |
val (thms', uses_nat) = fold_map norm thms false |
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212 |
in if uses_nat then nat_embedding @ thms' else thms' end |
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213 |
end |
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214 |
|
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215 |
|
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216 |
(* include additional rules *) |
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217 |
|
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218 |
val (unfold_defs, unfold_defs_setup) = |
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219 |
Attrib.config_bool "smt_unfold_defs" true |
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220 |
|
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221 |
local |
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222 |
val pair_rules = [@{thm fst_conv}, @{thm snd_conv}, @{thm pair_collapse}] |
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223 |
|
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224 |
val pair_type = (fn Type (@{type_name "*"}, _) => true | _ => false) |
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225 |
val exists_pair_type = Term.exists_type (Term.exists_subtype pair_type) |
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226 |
|
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227 |
val fun_upd_rules = [@{thm fun_upd_same}, @{thm fun_upd_apply}] |
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228 |
val is_fun_upd = (fn Const (@{const_name fun_upd}, _) => true | _ => false) |
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229 |
val exists_fun_upd = Term.exists_subterm is_fun_upd |
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230 |
in |
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231 |
fun add_pair_rules _ thms = |
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232 |
thms |
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233 |
|> exists (exists_pair_type o Thm.prop_of) thms ? append pair_rules |
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234 |
|
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235 |
fun add_fun_upd_rules _ thms = |
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236 |
thms |
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237 |
|> exists (exists_fun_upd o Thm.prop_of) thms ? append fun_upd_rules |
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238 |
end |
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239 |
|
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240 |
|
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241 |
local |
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242 |
fun mk_entry t thm = (Term.head_of t, (thm, thm RS @{thm eq_reflection})) |
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243 |
fun prepare_def thm = |
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244 |
(case HOLogic.dest_Trueprop (Thm.prop_of thm) of |
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245 |
Const (@{const_name "op ="}, _) $ t $ _ => mk_entry t thm |
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246 |
| t => raise TERM ("prepare_def", [t])) |
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247 |
|
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248 |
val defs = map prepare_def [ |
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249 |
@{thm abs_if[where 'a = int]}, @{thm abs_if[where 'a = real]}, |
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250 |
@{thm min_def[where 'a = int]}, @{thm min_def[where 'a = real]}, |
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251 |
@{thm max_def[where 'a = int]}, @{thm max_def[where 'a = real]}] |
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252 |
|
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253 |
fun add_sym t = if AList.defined (op =) defs t then insert (op =) t else I |
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fun add_syms thms = fold (Term.fold_aterms add_sym o Thm.prop_of) thms [] |
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255 |
|
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256 |
fun unfold_conv ctxt ct = |
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257 |
(case AList.lookup (op =) defs (Term.head_of (Thm.term_of ct)) of |
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258 |
SOME (_, eq) => Conv.rewr_conv eq |
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259 |
| NONE => Conv.all_conv) ct |
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260 |
in |
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|
261 |
fun add_abs_min_max_rules ctxt thms = |
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262 |
if Config.get ctxt unfold_defs |
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|
263 |
then map (Conv.fconv_rule (More_Conv.bottom_conv unfold_conv ctxt)) thms |
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264 |
else map fst (map_filter (AList.lookup (op =) defs) (add_syms thms)) @ thms |
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265 |
end |
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266 |
|
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|
267 |
|
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|
268 |
(* lift lambda terms into additional rules *) |
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269 |
|
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270 |
local |
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271 |
val meta_eq = @{cpat "op =="} |
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272 |
val meta_eqT = hd (Thm.dest_ctyp (Thm.ctyp_of_term meta_eq)) |
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273 |
fun inst_meta cT = Thm.instantiate_cterm ([(meta_eqT, cT)], []) meta_eq |
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274 |
fun mk_meta_eq ct cu = Thm.mk_binop (inst_meta (Thm.ctyp_of_term ct)) ct cu |
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275 |
|
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276 |
fun lambda_conv conv = |
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277 |
let |
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278 |
fun sub_conv cvs ctxt ct = |
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279 |
(case Thm.term_of ct of |
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280 |
Const (@{const_name All}, _) $ Abs _ => quant_conv cvs ctxt |
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281 |
| Const (@{const_name Ex}, _) $ Abs _ => quant_conv cvs ctxt |
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282 |
| Const _ $ Abs _ => Conv.arg_conv (at_lambda_conv cvs ctxt) |
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283 |
| Const (@{const_name Let}, _) $ _ $ Abs _ => Conv.combination_conv |
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284 |
(Conv.arg_conv (sub_conv cvs ctxt)) (abs_conv cvs ctxt) |
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285 |
| Abs _ => at_lambda_conv cvs ctxt |
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286 |
| _ $ _ => Conv.comb_conv (sub_conv cvs ctxt) |
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287 |
| _ => Conv.all_conv) ct |
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288 |
and abs_conv cvs = Conv.abs_conv (fn (cv, cx) => sub_conv (cv::cvs) cx) |
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289 |
and quant_conv cvs ctxt = Conv.arg_conv (abs_conv cvs ctxt) |
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290 |
and at_lambda_conv cvs ctxt = abs_conv cvs ctxt then_conv conv cvs ctxt |
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291 |
in sub_conv [] end |
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292 |
|
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293 |
fun used_vars cvs ct = |
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294 |
let |
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|
295 |
val lookup = AList.lookup (op aconv) (map (` Thm.term_of) cvs) |
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296 |
val add = (fn (SOME ct) => insert (op aconvc) ct | _ => I) |
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297 |
in Term.fold_aterms (add o lookup) (Thm.term_of ct) [] end |
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|
298 |
|
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299 |
val rev_int_fst_ord = rev_order o int_ord o pairself fst |
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300 |
fun ordered_values tab = |
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|
301 |
Termtab.fold (fn (_, x) => OrdList.insert rev_int_fst_ord x) tab [] |
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|
302 |
|> map snd |
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|
303 |
in |
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|
304 |
fun lift_lambdas ctxt thms = |
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|
305 |
let |
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|
306 |
val declare_frees = fold (Thm.fold_terms Term.declare_term_frees) |
32740 | 307 |
val names = Unsynchronized.ref (declare_frees thms (Name.make_context [])) |
308 |
val fresh_name = Unsynchronized.change_result names o yield_singleton Name.variants |
|
32618
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|
309 |
|
32740 | 310 |
val defs = Unsynchronized.ref (Termtab.empty : (int * thm) Termtab.table) |
311 |
fun add_def t thm = Unsynchronized.change defs (Termtab.update (t, (serial (), thm))) |
|
32618
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312 |
fun make_def cvs eq = Thm.symmetric (fold norm_meta_def cvs eq) |
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|
313 |
fun def_conv cvs ctxt ct = |
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|
314 |
let |
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|
315 |
val cvs' = used_vars cvs ct |
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|
316 |
val ct' = fold Thm.cabs cvs' ct |
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|
317 |
in |
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|
318 |
(case Termtab.lookup (!defs) (Thm.term_of ct') of |
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|
319 |
SOME (_, eq) => make_def cvs' eq |
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|
320 |
| NONE => |
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|
321 |
let |
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|
322 |
val {t, T, ...} = Thm.rep_cterm ct' |
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|
323 |
val eq = mk_meta_eq (cert ctxt (Free (fresh_name "", T))) ct' |
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|
324 |
val thm = Thm.assume eq |
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|
325 |
in (add_def t thm; make_def cvs' thm) end) |
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|
326 |
end |
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|
327 |
val thms' = map (Conv.fconv_rule (lambda_conv def_conv ctxt)) thms |
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|
328 |
val eqs = ordered_values (!defs) |
42865636d006
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|
329 |
in |
42865636d006
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|
330 |
(maps (#hyps o Thm.crep_thm) eqs, map (normalize_rule ctxt) eqs @ thms') |
42865636d006
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|
331 |
end |
42865636d006
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|
332 |
end |
42865636d006
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|
333 |
|
42865636d006
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|
334 |
|
42865636d006
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changeset
|
335 |
(* make application explicit for functions with varying number of arguments *) |
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|
336 |
|
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|
337 |
local |
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|
338 |
val const = prefix "c" and free = prefix "f" |
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|
339 |
fun min i (e as (_, j)) = if i <> j then (true, Int.min (i, j)) else e |
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|
340 |
fun add t i = Symtab.map_default (t, (false, i)) (min i) |
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|
341 |
fun traverse t = |
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|
342 |
(case Term.strip_comb t of |
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|
343 |
(Const (n, _), ts) => add (const n) (length ts) #> fold traverse ts |
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|
344 |
| (Free (n, _), ts) => add (free n) (length ts) #> fold traverse ts |
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|
345 |
| (Abs (_, _, u), ts) => fold traverse (u :: ts) |
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|
346 |
| (_, ts) => fold traverse ts) |
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|
347 |
val prune = (fn (n, (true, i)) => Symtab.update (n, i) | _ => I) |
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|
348 |
fun prune_tab tab = Symtab.fold prune tab Symtab.empty |
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|
349 |
|
42865636d006
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|
350 |
fun binop_conv cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2 |
42865636d006
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changeset
|
351 |
fun nary_conv conv1 conv2 ct = |
42865636d006
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|
352 |
(Conv.combination_conv (nary_conv conv1 conv2) conv2 else_conv conv1) ct |
42865636d006
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|
353 |
fun abs_conv conv tb = Conv.abs_conv (fn (cv, cx) => |
42865636d006
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|
354 |
let val n = fst (Term.dest_Free (Thm.term_of cv)) |
42865636d006
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|
355 |
in conv (Symtab.update (free n, 0) tb) cx end) |
42865636d006
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changeset
|
356 |
val apply_rule = @{lemma "f x == apply f x" by (simp add: apply_def)} |
42865636d006
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changeset
|
357 |
in |
42865636d006
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boehmes
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changeset
|
358 |
fun explicit_application ctxt thms = |
42865636d006
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boehmes
parents:
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changeset
|
359 |
let |
42865636d006
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boehmes
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diff
changeset
|
360 |
fun sub_conv tb ctxt ct = |
42865636d006
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boehmes
parents:
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changeset
|
361 |
(case Term.strip_comb (Thm.term_of ct) of |
42865636d006
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changeset
|
362 |
(Const (n, _), ts) => app_conv tb (const n) (length ts) ctxt |
42865636d006
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changeset
|
363 |
| (Free (n, _), ts) => app_conv tb (free n) (length ts) ctxt |
42865636d006
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changeset
|
364 |
| (Abs _, ts) => nary_conv (abs_conv sub_conv tb ctxt) (sub_conv tb ctxt) |
42865636d006
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boehmes
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changeset
|
365 |
| (_, ts) => nary_conv Conv.all_conv (sub_conv tb ctxt)) ct |
42865636d006
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boehmes
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changeset
|
366 |
and app_conv tb n i ctxt = |
42865636d006
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changeset
|
367 |
(case Symtab.lookup tb n of |
42865636d006
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changeset
|
368 |
NONE => nary_conv Conv.all_conv (sub_conv tb ctxt) |
42865636d006
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changeset
|
369 |
| SOME j => apply_conv tb ctxt (i - j)) |
42865636d006
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boehmes
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changeset
|
370 |
and apply_conv tb ctxt i ct = ( |
42865636d006
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boehmes
parents:
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changeset
|
371 |
if i = 0 then nary_conv Conv.all_conv (sub_conv tb ctxt) |
42865636d006
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boehmes
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changeset
|
372 |
else |
42865636d006
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boehmes
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changeset
|
373 |
Conv.rewr_conv apply_rule then_conv |
42865636d006
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changeset
|
374 |
binop_conv (apply_conv tb ctxt (i-1)) (sub_conv tb ctxt)) ct |
42865636d006
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changeset
|
375 |
|
42865636d006
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boehmes
parents:
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changeset
|
376 |
val tab = prune_tab (fold (traverse o Thm.prop_of) thms Symtab.empty) |
42865636d006
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changeset
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377 |
in map (Conv.fconv_rule (sub_conv tab ctxt)) thms end |
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end |
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|
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380 |
|
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(* combined normalization *) |
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|
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datatype config = |
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RewriteTrivialLets | |
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385 |
RewriteNegativeNumerals | |
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386 |
RewriteNaturalNumbers | |
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387 |
AddPairRules | |
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388 |
AddFunUpdRules | |
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389 |
AddAbsMinMaxRules |
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390 |
|
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391 |
fun normalize config ctxt thms = |
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let fun if_enabled c f = member (op =) config c ? f ctxt |
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393 |
in |
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394 |
thms |
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|> if_enabled RewriteTrivialLets trivial_let |
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|> if_enabled RewriteNegativeNumerals positive_numerals |
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|> if_enabled RewriteNaturalNumbers nat_as_int |
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|> if_enabled AddPairRules add_pair_rules |
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|> if_enabled AddFunUpdRules add_fun_upd_rules |
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|> if_enabled AddAbsMinMaxRules add_abs_min_max_rules |
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|> map (normalize_rule ctxt) |
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|> lift_lambdas ctxt |
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|> apsnd (explicit_application ctxt) |
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404 |
end |
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|
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val setup = unfold_defs_setup |
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407 |
|
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408 |
end |