author  huffman 
Wed, 22 Feb 2012 17:34:31 +0100  
changeset 46594  f11f332b964f 
parent 46497  89ccf66aa73d 
child 51717  9e7d1c139569 
permissions  rwrr 
33443  1 
(* Title: HOL/Library/positivstellensatz.ML 
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Author: Amine Chaieb, University of Cambridge 

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A generic arithmetic prover based on Positivstellensatz certificates 

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 also implements FourrierMotzkin elimination as a special case 

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FourrierMotzkin elimination. 

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*) 
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(* A functor for finite mappings based on Tables *) 
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signature FUNC = 
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sig 
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include TABLE 
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val apply : 'a table > key > 'a 

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val applyd :'a table > (key > 'a) > key > 'a 

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val combine : ('a > 'a > 'a) > ('a > bool) > 'a table > 'a table > 'a table 

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val dom : 'a table > key list 

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val tryapplyd : 'a table > key > 'a > 'a 

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val updatep : (key * 'a > bool) > key * 'a > 'a table > 'a table 

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val choose : 'a table > key * 'a 

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val onefunc : key * 'a > 'a table 

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end; 
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functor FuncFun(Key: KEY) : FUNC = 
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struct 
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structure Tab = Table(Key); 
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open Tab; 
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fun dom a = sort Key.ord (Tab.keys a); 
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fun applyd f d x = case Tab.lookup f x of 
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SOME y => y 
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 NONE => d x; 
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fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x; 
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fun tryapplyd f a d = applyd f (K d) a; 
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fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t 
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fun combine f z a b = 
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let 

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fun h (k,v) t = case Tab.lookup t k of 

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NONE => Tab.update (k,v) t 

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 SOME v' => let val w = f v v' 

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in if z w then Tab.delete k t else Tab.update (k,w) t end; 

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in Tab.fold h a b end; 
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fun choose f = case Tab.min_key f of 
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SOME k => (k, the (Tab.lookup f k)) 

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 NONE => error "FuncFun.choose : Completely empty function" 

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fun onefunc kv = update kv empty 
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end; 
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(* Some standard functors and utility functions for them *) 
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structure FuncUtil = 
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struct 
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structure Intfunc = FuncFun(type key = int val ord = int_ord); 
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structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord); 
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structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord); 
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structure Symfunc = FuncFun(type key = string val ord = fast_string_ord); 
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structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord); 
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val cterm_ord = Term_Ord.fast_term_ord o pairself term_of 
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structure Ctermfunc = FuncFun(type key = cterm val ord = cterm_ord); 
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type monomial = int Ctermfunc.table; 
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val monomial_ord = list_ord (prod_ord cterm_ord int_ord) o pairself Ctermfunc.dest 
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structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord) 
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type poly = Rat.rat Monomialfunc.table; 
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(* The ordering so we can create canonical HOL polynomials. *) 
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fun dest_monomial mon = sort (cterm_ord o pairself fst) (Ctermfunc.dest mon); 
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fun monomial_order (m1,m2) = 
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if Ctermfunc.is_empty m2 then LESS 
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else if Ctermfunc.is_empty m1 then GREATER 

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else 

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let 

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val mon1 = dest_monomial m1 

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val mon2 = dest_monomial m2 
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val deg1 = fold (Integer.add o snd) mon1 0 
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val deg2 = fold (Integer.add o snd) mon2 0 
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in if deg1 < deg2 then GREATER 

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else if deg1 > deg2 then LESS 

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else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2) 

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end; 

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end 
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(* positivstellensatz datatype and prover generation *) 
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signature REAL_ARITH = 
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sig 
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datatype positivstellensatz = 
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Axiom_eq of int 
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 Axiom_le of int 

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 Axiom_lt of int 

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 Rational_eq of Rat.rat 

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 Rational_le of Rat.rat 

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 Rational_lt of Rat.rat 

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 Square of FuncUtil.poly 

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 Eqmul of FuncUtil.poly * positivstellensatz 

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 Sum of positivstellensatz * positivstellensatz 

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 Product of positivstellensatz * positivstellensatz; 

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datatype pss_tree = Trivial  Cert of positivstellensatz  Branch of pss_tree * pss_tree 
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datatype tree_choice = Left  Right 
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type prover = tree_choice list > 
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(thm list * thm list * thm list > positivstellensatz > thm) > 

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thm list * thm list * thm list > thm * pss_tree 

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type cert_conv = cterm > thm * pss_tree 

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val gen_gen_real_arith : 
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Proof.context > (Rat.rat > cterm) * conv * conv * conv * 

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conv * conv * conv * conv * conv * conv * prover > cert_conv 

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val real_linear_prover : (thm list * thm list * thm list > positivstellensatz > thm) > 

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thm list * thm list * thm list > thm * pss_tree 

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val gen_real_arith : Proof.context > 
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(Rat.rat > cterm) * conv * conv * conv * conv * conv * conv * conv * prover > cert_conv 

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val gen_prover_real_arith : Proof.context > prover > cert_conv 
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val is_ratconst : cterm > bool 
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val dest_ratconst : cterm > Rat.rat 

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val cterm_of_rat : Rat.rat > cterm 

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end 
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structure RealArith : REAL_ARITH = 
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struct 
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open Conv 
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(*  *) 
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(* Data structure for Positivstellensatz refutations. *) 
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(*  *) 
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datatype positivstellensatz = 
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Axiom_eq of int 
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 Axiom_le of int 

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 Axiom_lt of int 

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 Rational_eq of Rat.rat 

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 Rational_le of Rat.rat 

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 Rational_lt of Rat.rat 

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 Square of FuncUtil.poly 

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 Eqmul of FuncUtil.poly * positivstellensatz 

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 Sum of positivstellensatz * positivstellensatz 

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 Product of positivstellensatz * positivstellensatz; 

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(* Theorems used in the procedure *) 
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datatype pss_tree = Trivial  Cert of positivstellensatz  Branch of pss_tree * pss_tree 
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datatype tree_choice = Left  Right 
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type prover = tree_choice list > 
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(thm list * thm list * thm list > positivstellensatz > thm) > 
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thm list * thm list * thm list > thm * pss_tree 
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type cert_conv = cterm > thm * pss_tree 
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(* Some useful derived rules *) 
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fun deduct_antisym_rule tha thb = 
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Thm.equal_intr (Thm.implies_intr (cprop_of thb) tha) 

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(Thm.implies_intr (cprop_of tha) thb); 
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fun prove_hyp tha thb = 
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if exists (curry op aconv (concl_of tha)) (Thm.hyps_of thb) (* FIXME !? *) 

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then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb; 
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val pth = @{lemma "(((x::real) < y) == (y  x > 0))" and "((x <= y) == (y  x >= 0))" and 
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"((x = y) == (x  y = 0))" and "((~(x < y)) == (x  y >= 0))" and 

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"((~(x <= y)) == (x  y > 0))" and "((~(x = y)) == (x  y > 0  (x  y) > 0))" 

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by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)}; 

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val pth_final = @{lemma "(~p ==> False) ==> p" by blast} 
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val pth_add = 
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@{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 )" and "( x = 0 ==> y >= 0 ==> x + y >= 0)" and 
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"(x = 0 ==> y > 0 ==> x + y > 0)" and "(x >= 0 ==> y = 0 ==> x + y >= 0)" and 

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"(x >= 0 ==> y >= 0 ==> x + y >= 0)" and "(x >= 0 ==> y > 0 ==> x + y > 0)" and 

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"(x > 0 ==> y = 0 ==> x + y > 0)" and "(x > 0 ==> y >= 0 ==> x + y > 0)" and 

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"(x > 0 ==> y > 0 ==> x + y > 0)" by simp_all}; 

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191 

46594  192 
val pth_mul = 
33443  193 
@{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0)" and "(x = 0 ==> y >= 0 ==> x * y = 0)" and 
194 
"(x = 0 ==> y > 0 ==> x * y = 0)" and "(x >= 0 ==> y = 0 ==> x * y = 0)" and 

195 
"(x >= 0 ==> y >= 0 ==> x * y >= 0)" and "(x >= 0 ==> y > 0 ==> x * y >= 0)" and 

196 
"(x > 0 ==> y = 0 ==> x * y = 0)" and "(x > 0 ==> y >= 0 ==> x * y >= 0)" and 

197 
"(x > 0 ==> y > 0 ==> x * y > 0)" 

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198 
by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified] 
33443  199 
mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])}; 
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200 

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201 
val pth_emul = @{lemma "y = (0::real) ==> x * y = 0" by simp}; 
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202 
val pth_square = @{lemma "x * x >= (0::real)" by simp}; 
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203 

33443  204 
val weak_dnf_simps = 
45654  205 
List.take (@{thms simp_thms}, 34) @ 
33443  206 
@{lemma "((P & (Q  R)) = ((P&Q)  (P&R)))" and "((Q  R) & P) = ((Q&P)  (R&P))" and 
207 
"(P & Q) = (Q & P)" and "((P  Q) = (Q  P))" by blast+}; 

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208 

44454  209 
(* 
33443  210 
val nnfD_simps = 
211 
@{lemma "((~(P & Q)) = (~P  ~Q))" and "((~(P  Q)) = (~P & ~Q) )" and 

212 
"((P > Q) = (~P  Q) )" and "((P = Q) = ((P & Q)  (~P & ~ Q)))" and 

213 
"((~(P = Q)) = ((P & ~ Q)  (~P & Q)) )" and "((~ ~(P)) = P)" by blast+}; 

44454  214 
*) 
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215 

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216 
val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis}; 
33443  217 
val prenex_simps = 
218 
map (fn th => th RS sym) 

219 
([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @ 

37598  220 
@{thms "HOL.all_simps"(14)} @ @{thms "ex_simps"(14)}); 
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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221 

33443  222 
val real_abs_thms1 = @{lemma 
223 
"((1 * abs(x::real) >= r) = (1 * x >= r & 1 * x >= r))" and 

224 
"((1 * abs(x) + a >= r) = (a + 1 * x >= r & a + 1 * x >= r))" and 

225 
"((a + 1 * abs(x) >= r) = (a + 1 * x >= r & a + 1 * x >= r))" and 

226 
"((a + 1 * abs(x) + b >= r) = (a + 1 * x + b >= r & a + 1 * x + b >= r))" and 

227 
"((a + b + 1 * abs(x) >= r) = (a + b + 1 * x >= r & a + b + 1 * x >= r))" and 

228 
"((a + b + 1 * abs(x) + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * x + c >= r))" and 

229 
"((1 * max x y >= r) = (1 * x >= r & 1 * y >= r))" and 

230 
"((1 * max x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and 

231 
"((a + 1 * max x y >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and 

232 
"((a + 1 * max x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y + b >= r))" and 

233 
"((a + b + 1 * max x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r))" and 

234 
"((a + b + 1 * max x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y + c >= r))" and 

235 
"((1 * min x y >= r) = (1 * x >= r & 1 * y >= r))" and 

236 
"((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and 

237 
"((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and 

238 
"((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y + b >= r))" and 

239 
"((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r))" and 

240 
"((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y + c >= r))" and 

241 
"((min x y >= r) = (x >= r & y >= r))" and 

242 
"((min x y + a >= r) = (a + x >= r & a + y >= r))" and 

243 
"((a + min x y >= r) = (a + x >= r & a + y >= r))" and 

244 
"((a + min x y + b >= r) = (a + x + b >= r & a + y + b >= r))" and 

245 
"((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r))" and 

246 
"((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r))" and 

247 
"((1 * abs(x) > r) = (1 * x > r & 1 * x > r))" and 

248 
"((1 * abs(x) + a > r) = (a + 1 * x > r & a + 1 * x > r))" and 

249 
"((a + 1 * abs(x) > r) = (a + 1 * x > r & a + 1 * x > r))" and 

250 
"((a + 1 * abs(x) + b > r) = (a + 1 * x + b > r & a + 1 * x + b > r))" and 

251 
"((a + b + 1 * abs(x) > r) = (a + b + 1 * x > r & a + b + 1 * x > r))" and 

252 
"((a + b + 1 * abs(x) + c > r) = (a + b + 1 * x + c > r & a + b + 1 * x + c > r))" and 

253 
"((1 * max x y > r) = ((1 * x > r) & 1 * y > r))" and 

254 
"((1 * max x y + a > r) = (a + 1 * x > r & a + 1 * y > r))" and 

255 
"((a + 1 * max x y > r) = (a + 1 * x > r & a + 1 * y > r))" and 

256 
"((a + 1 * max x y + b > r) = (a + 1 * x + b > r & a + 1 * y + b > r))" and 

257 
"((a + b + 1 * max x y > r) = (a + b + 1 * x > r & a + b + 1 * y > r))" and 

258 
"((a + b + 1 * max x y + c > r) = (a + b + 1 * x + c > r & a + b + 1 * y + c > r))" and 

259 
"((min x y > r) = (x > r & y > r))" and 

260 
"((min x y + a > r) = (a + x > r & a + y > r))" and 

261 
"((a + min x y > r) = (a + x > r & a + y > r))" and 

262 
"((a + min x y + b > r) = (a + x + b > r & a + y + b > r))" and 

263 
"((a + b + min x y > r) = (a + b + x > r & a + b + y > r))" and 

264 
"((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))" 

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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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265 
by auto}; 
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266 

35028
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more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
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267 
val abs_split' = @{lemma "P (abs (x::'a::linordered_idom)) == (x >= 0 & P x  x < 0 & P (x))" 
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268 
by (atomize (full)) (auto split add: abs_split)}; 
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269 

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270 
val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y  x > y & P x)" 
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271 
by (atomize (full)) (cases "x <= y", auto simp add: max_def)}; 
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272 

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273 
val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x  x > y & P y)" 
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274 
by (atomize (full)) (cases "x <= y", auto simp add: min_def)}; 
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275 

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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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276 

39920  277 
(* Miscellaneous *) 
46594  278 
fun literals_conv bops uops cv = 
279 
let 

280 
fun h t = 

281 
case (term_of t) of 

282 
b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t 

283 
 u$_ => if member (op aconv) uops u then arg_conv h t else cv t 

284 
 _ => cv t 

285 
in h end; 

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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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286 

46594  287 
fun cterm_of_rat x = 
288 
let 

289 
val (a, b) = Rat.quotient_of_rat x 

290 
in 

291 
if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a 

292 
else Thm.apply (Thm.apply @{cterm "op / :: real => _"} 

293 
(Numeral.mk_cnumber @{ctyp "real"} a)) 

294 
(Numeral.mk_cnumber @{ctyp "real"} b) 

295 
end; 

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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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296 

46594  297 
fun dest_ratconst t = 
298 
case term_of t of 

299 
Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a > snd, HOLogic.dest_number b > snd) 

300 
 _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) > snd) 

301 
fun is_ratconst t = can dest_ratconst t 

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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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302 

44454  303 
(* 
46594  304 
fun find_term p t = if p t then t else 
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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305 
case t of 
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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306 
a$b => (find_term p a handle TERM _ => find_term p b) 
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307 
 Abs (_,_,t') => find_term p t' 
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308 
 _ => raise TERM ("find_term",[t]); 
44454  309 
*) 
31120
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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310 

46594  311 
fun find_cterm p t = 
312 
if p t then t else 

313 
case term_of t of 

314 
_$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t)) 

315 
 Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t > snd) 

316 
 _ => raise CTERM ("find_cterm",[t]); 

31120
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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317 

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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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318 
(* Some conversionsrelated stuff which has been forbidden entrance into Pure/conv.ML*) 
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319 
fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms) 
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320 
fun is_comb t = case (term_of t) of _$_ => true  _ => false; 
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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321 

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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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322 
fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct')) 
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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323 
handle CTERM _ => false; 
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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324 

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sos method generates and uses proof certificates
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diff
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325 

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sos method generates and uses proof certificates
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326 
(* Map back polynomials to HOL. *) 
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327 

46594  328 
fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply @{cterm "op ^ :: real => _"} x) 
32828  329 
(Numeral.mk_cnumber @{ctyp nat} k) 
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sos method generates and uses proof certificates
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330 

46594  331 
fun cterm_of_monomial m = 
332 
if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"} 

333 
else 

334 
let 

335 
val m' = FuncUtil.dest_monomial m 

336 
val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' [] 

337 
in foldr1 (fn (s, t) => Thm.apply (Thm.apply @{cterm "op * :: real => _"} s) t) vps 

338 
end 

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parents:
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changeset

339 

46594  340 
fun cterm_of_cmonomial (m,c) = 
341 
if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c 

342 
else if c = Rat.one then cterm_of_monomial m 

343 
else Thm.apply (Thm.apply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m); 

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changeset

344 

46594  345 
fun cterm_of_poly p = 
346 
if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"} 

347 
else 

348 
let 

349 
val cms = map cterm_of_cmonomial 

350 
(sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p)) 

351 
in foldr1 (fn (t1, t2) => Thm.apply(Thm.apply @{cterm "op + :: real => _"} t1) t2) cms 

352 
end; 

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sos method generates and uses proof certificates
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353 

46594  354 
(* A general real arithmetic prover *) 
31120
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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355 

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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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356 
fun gen_gen_real_arith ctxt (mk_numeric, 
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357 
numeric_eq_conv,numeric_ge_conv,numeric_gt_conv, 
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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358 
poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv, 
46594  359 
absconv1,absconv2,prover) = 
360 
let 

361 
val pre_ss = HOL_basic_ss addsimps 

362 
@{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib all_conj_distrib if_bool_eq_disj} 

363 
val prenex_ss = HOL_basic_ss addsimps prenex_simps 

364 
val skolemize_ss = HOL_basic_ss addsimps [choice_iff] 

365 
val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss) 

366 
val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss) 

367 
val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss) 

368 
val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps 

369 
val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss) 

370 
fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI} 

371 
fun oprconv cv ct = 

372 
let val g = Thm.dest_fun2 ct 

373 
in if g aconvc @{cterm "op <= :: real => _"} 

374 
orelse g aconvc @{cterm "op < :: real => _"} 

375 
then arg_conv cv ct else arg1_conv cv ct 

376 
end 

31120
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
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changeset

377 

46594  378 
fun real_ineq_conv th ct = 
379 
let 

380 
val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th 

381 
handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct])) 

382 
in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th')) 

383 
end 

384 
val [real_lt_conv, real_le_conv, real_eq_conv, 

385 
real_not_lt_conv, real_not_le_conv, _] = 

386 
map real_ineq_conv pth 

387 
fun match_mp_rule ths ths' = 

388 
let 

389 
fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths) 

390 
 th::ths => (ths' MRS th handle THM _ => f ths ths') 

391 
in f ths ths' end 

392 
fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv)) 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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changeset

393 
(match_mp_rule pth_mul [th, th']) 
46594  394 
fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv)) 
31120
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A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
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395 
(match_mp_rule pth_add [th, th']) 
46594  396 
fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) 
397 
(instantiate' [] [SOME ct] (th RS pth_emul)) 

398 
fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv)) 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

399 
(instantiate' [] [SOME t] pth_square) 
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

400 

46594  401 
fun hol_of_positivstellensatz(eqs,les,lts) proof = 
402 
let 

403 
fun translate prf = 

404 
case prf of 

405 
Axiom_eq n => nth eqs n 

406 
 Axiom_le n => nth les n 

407 
 Axiom_lt n => nth lts n 

408 
 Rational_eq x => eqT_elim(numeric_eq_conv(Thm.apply @{cterm Trueprop} 

409 
(Thm.apply (Thm.apply @{cterm "op =::real => _"} (mk_numeric x)) 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

410 
@{cterm "0::real"}))) 
46594  411 
 Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply @{cterm Trueprop} 
412 
(Thm.apply (Thm.apply @{cterm "op <=::real => _"} 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

413 
@{cterm "0::real"}) (mk_numeric x)))) 
46594  414 
 Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply @{cterm Trueprop} 
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45654
diff
changeset

415 
(Thm.apply (Thm.apply @{cterm "op <::real => _"} @{cterm "0::real"}) 
31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

416 
(mk_numeric x)))) 
46594  417 
 Square pt => square_rule (cterm_of_poly pt) 
418 
 Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p) 

419 
 Sum(p1,p2) => add_rule (translate p1) (translate p2) 

420 
 Product(p1,p2) => mul_rule (translate p1) (translate p2) 

421 
in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

422 
(translate proof) 
46594  423 
end 
424 

425 
val init_conv = presimp_conv then_conv 

426 
nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv 

427 
weak_dnf_conv 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

428 

46594  429 
val concl = Thm.dest_arg o cprop_of 
430 
fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false) 

431 
val is_req = is_binop @{cterm "op =:: real => _"} 

432 
val is_ge = is_binop @{cterm "op <=:: real => _"} 

433 
val is_gt = is_binop @{cterm "op <:: real => _"} 

434 
val is_conj = is_binop @{cterm HOL.conj} 

435 
val is_disj = is_binop @{cterm HOL.disj} 

436 
fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2}) 

437 
fun disj_cases th th1 th2 = 

438 
let 

439 
val (p,q) = Thm.dest_binop (concl th) 

440 
val c = concl th1 

441 
val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible" 

442 
in Thm.implies_elim (Thm.implies_elim 

36945  443 
(Thm.implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th) 
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45654
diff
changeset

444 
(Thm.implies_intr (Thm.apply @{cterm Trueprop} p) th1)) 
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45654
diff
changeset

445 
(Thm.implies_intr (Thm.apply @{cterm Trueprop} q) th2) 
46594  446 
end 
447 
fun overall cert_choice dun ths = 

448 
case ths of 

449 
[] => 

450 
let 

451 
val (eq,ne) = List.partition (is_req o concl) dun 

452 
val (le,nl) = List.partition (is_ge o concl) ne 

453 
val lt = filter (is_gt o concl) nl 

454 
in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end 

455 
 th::oths => 

456 
let 

457 
val ct = concl th 

458 
in 

459 
if is_conj ct then 

460 
let 

461 
val (th1,th2) = conj_pair th 

462 
in overall cert_choice dun (th1::th2::oths) end 

463 
else if is_disj ct then 

464 
let 

465 
val (th1, cert1) = overall (Left::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths) 

466 
val (th2, cert2) = overall (Right::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg ct))::oths) 

467 
in (disj_cases th th1 th2, Branch (cert1, cert2)) end 

468 
else overall cert_choice (th::dun) oths 

469 
end 

470 
fun dest_binary b ct = 

471 
if is_binop b ct then Thm.dest_binop ct 

472 
else raise CTERM ("dest_binary",[b,ct]) 

473 
val dest_eq = dest_binary @{cterm "op = :: real => _"} 

474 
val neq_th = nth pth 5 

475 
fun real_not_eq_conv ct = 

476 
let 

477 
val (l,r) = dest_eq (Thm.dest_arg ct) 

478 
val th = Thm.instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th 

479 
val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th))) 

480 
val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p 

481 
val th_n = fconv_rule (arg_conv poly_neg_conv) th_x 

482 
val th' = Drule.binop_cong_rule @{cterm HOL.disj} 

483 
(Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p) 

484 
(Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n) 

485 
in Thm.transitive th th' 

486 
end 

487 
fun equal_implies_1_rule PQ = 

488 
let 

489 
val P = Thm.lhs_of PQ 

490 
in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P)) 

491 
end 

492 
(* FIXME!!! Copied from groebner.ml *) 

493 
val strip_exists = 

494 
let 

495 
fun h (acc, t) = 

496 
case (term_of t) of 

497 
Const(@{const_name Ex},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) >> (fn v => v::acc)) 

498 
 _ => (acc,t) 

499 
in fn t => h ([],t) 

500 
end 

501 
fun name_of x = 

502 
case term_of x of 

503 
Free(s,_) => s 

504 
 Var ((s,_),_) => s 

505 
 _ => "x" 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

506 

46594  507 
fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (Thm.abstract_rule (name_of x) x th) 
31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

508 

46594  509 
val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec)); 
31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

510 

46594  511 
fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex} 
512 
fun mk_ex v t = Thm.apply (ext (ctyp_of_term v)) (Thm.lambda v t) 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

513 

46594  514 
fun choose v th th' = 
515 
case concl_of th of 

516 
@{term Trueprop} $ (Const(@{const_name Ex},_)$_) => 

517 
let 

518 
val p = (funpow 2 Thm.dest_arg o cprop_of) th 

519 
val T = (hd o Thm.dest_ctyp o ctyp_of_term) p 

520 
val th0 = fconv_rule (Thm.beta_conversion true) 

521 
(instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE) 

522 
val pv = (Thm.rhs_of o Thm.beta_conversion true) 

523 
(Thm.apply @{cterm Trueprop} (Thm.apply p v)) 

524 
val th1 = Thm.forall_intr v (Thm.implies_intr pv th') 

525 
in Thm.implies_elim (Thm.implies_elim th0 th) th1 end 

526 
 _ => raise THM ("choose",0,[th, th']) 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

527 

46594  528 
fun simple_choose v th = 
529 
choose v (Thm.assume ((Thm.apply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

530 

46594  531 
val strip_forall = 
532 
let 

533 
fun h (acc, t) = 

534 
case (term_of t) of 

535 
Const(@{const_name All},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) >> (fn v => v::acc)) 

536 
 _ => (acc,t) 

537 
in fn t => h ([],t) 

538 
end 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

539 

46594  540 
fun f ct = 
541 
let 

542 
val nnf_norm_conv' = 

543 
nnf_conv then_conv 

544 
literals_conv [@{term HOL.conj}, @{term HOL.disj}] [] 

545 
(Conv.cache_conv 

546 
(first_conv [real_lt_conv, real_le_conv, 

547 
real_eq_conv, real_not_lt_conv, 

548 
real_not_le_conv, real_not_eq_conv, all_conv])) 

549 
fun absremover ct = (literals_conv [@{term HOL.conj}, @{term HOL.disj}] [] 

550 
(try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv 

551 
try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct 

552 
val nct = Thm.apply @{cterm Trueprop} (Thm.apply @{cterm "Not"} ct) 

553 
val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct 

554 
val tm0 = Thm.dest_arg (Thm.rhs_of th0) 

555 
val (th, certificates) = 

556 
if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else 

557 
let 

558 
val (evs,bod) = strip_exists tm0 

559 
val (avs,ibod) = strip_forall bod 

560 
val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod)) 

561 
val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))] 

562 
val th3 = fold simple_choose evs (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply @{cterm Trueprop} bod))) th2) 

563 
in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs) 

564 
end 

565 
in (Thm.implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates) 

566 
end 

567 
in f 

568 
end; 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

569 

fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

570 
(* A linear arithmetic prover *) 
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

571 
local 
32828  572 
val linear_add = FuncUtil.Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero) 
39027  573 
fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c */ x) 
31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

574 
val one_tm = @{cterm "1::real"} 
32829
671eb46eb0a3
tuned FuncFun and FuncUtil structure in positivstellensatz.ML
Philipp Meyer
parents:
32828
diff
changeset

575 
fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p Rat.zero)) orelse 
33038  576 
((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso 
32829
671eb46eb0a3
tuned FuncFun and FuncUtil structure in positivstellensatz.ML
Philipp Meyer
parents:
32828
diff
changeset

577 
not(p(FuncUtil.Ctermfunc.apply e one_tm))) 
31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

578 

46594  579 
fun linear_ineqs vars (les,lts) = 
580 
case find_first (contradictory (fn x => x >/ Rat.zero)) lts of 

581 
SOME r => r 

582 
 NONE => 

583 
(case find_first (contradictory (fn x => x >/ Rat.zero)) les of 

584 
SOME r => r 

585 
 NONE => 

586 
if null vars then error "linear_ineqs: no contradiction" else 

587 
let 

588 
val ineqs = les @ lts 

589 
fun blowup v = 

590 
length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) + 

591 
length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) * 

592 
length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs) 

593 
val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j)) 

594 
(map (fn v => (v,blowup v)) vars))) 

595 
fun addup (e1,p1) (e2,p2) acc = 

596 
let 

597 
val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero 

598 
val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero 

599 
in 

600 
if c1 */ c2 >=/ Rat.zero then acc else 

601 
let 

602 
val e1' = linear_cmul (Rat.abs c2) e1 

603 
val e2' = linear_cmul (Rat.abs c1) e2 

604 
val p1' = Product(Rational_lt(Rat.abs c2),p1) 

605 
val p2' = Product(Rational_lt(Rat.abs c1),p2) 

606 
in (linear_add e1' e2',Sum(p1',p2'))::acc 

607 
end 

608 
end 

609 
val (les0,les1) = 

610 
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les 

611 
val (lts0,lts1) = 

612 
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts 

613 
val (lesp,lesn) = 

614 
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1 

615 
val (ltsp,ltsn) = 

616 
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1 

617 
val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0 

618 
val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

619 
(fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0) 
46594  620 
in linear_ineqs (remove (op aconvc) v vars) (les',lts') 
621 
end) 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

622 

46594  623 
fun linear_eqs(eqs,les,lts) = 
624 
case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of 

625 
SOME r => r 

626 
 NONE => 

627 
(case eqs of 

628 
[] => 

629 
let val vars = remove (op aconvc) one_tm 

630 
(fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) []) 

631 
in linear_ineqs vars (les,lts) end 

632 
 (e,p)::es => 

633 
if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else 

634 
let 

635 
val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e) 

636 
fun xform (inp as (t,q)) = 

637 
let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in 

638 
if d =/ Rat.zero then inp else 

639 
let 

640 
val k = (Rat.neg d) */ Rat.abs c // c 

641 
val e' = linear_cmul k e 

642 
val t' = linear_cmul (Rat.abs c) t 

643 
val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p) 

644 
val q' = Product(Rational_lt(Rat.abs c),q) 

645 
in (linear_add e' t',Sum(p',q')) 

646 
end 

647 
end 

648 
in linear_eqs(map xform es,map xform les,map xform lts) 

649 
end) 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

650 

46594  651 
fun linear_prover (eq,le,lt) = 
652 
let 

653 
val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq 

654 
val les = map_index (fn (n, p) => (p,Axiom_le n)) le 

655 
val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt 

656 
in linear_eqs(eqs,les,lts) 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

657 
end 
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

658 

46594  659 
fun lin_of_hol ct = 
660 
if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty 

661 
else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one) 

662 
else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct) 

663 
else 

664 
let val (lop,r) = Thm.dest_comb ct 

665 
in 

666 
if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one) 

667 
else 

668 
let val (opr,l) = Thm.dest_comb lop 

669 
in 

670 
if opr aconvc @{cterm "op + :: real =>_"} 

671 
then linear_add (lin_of_hol l) (lin_of_hol r) 

672 
else if opr aconvc @{cterm "op * :: real =>_"} 

673 
andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l) 

674 
else FuncUtil.Ctermfunc.onefunc (ct, Rat.one) 

675 
end 

676 
end 

677 

678 
fun is_alien ct = 

679 
case term_of ct of 

680 
Const(@{const_name "real"}, _)$ n => 

681 
if can HOLogic.dest_number n then false else true 

682 
 _ => false 

683 
in 

684 
fun real_linear_prover translator (eq,le,lt) = 

685 
let 

686 
val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o cprop_of 

687 
val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o cprop_of 

688 
val eq_pols = map lhs eq 

689 
val le_pols = map rhs le 

690 
val lt_pols = map rhs lt 

691 
val aliens = filter is_alien 

692 
(fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom) 

693 
(eq_pols @ le_pols @ lt_pols) []) 

694 
val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens 

695 
val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols) 

696 
val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens 

697 
in ((translator (eq,le',lt) proof), Trivial) 

698 
end 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

699 
end; 
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

700 

fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

701 
(* A less general generic arithmetic prover dealing with abs,max and min*) 
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

702 

fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

703 
local 
46594  704 
val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1 
705 
fun absmaxmin_elim_conv1 ctxt = 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

706 
Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1) 
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

707 

46594  708 
val absmaxmin_elim_conv2 = 
709 
let 

710 
val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split' 

711 
val pth_max = instantiate' [SOME @{ctyp real}] [] max_split 

712 
val pth_min = instantiate' [SOME @{ctyp real}] [] min_split 

713 
val abs_tm = @{cterm "abs :: real => _"} 

714 
val p_tm = @{cpat "?P :: real => bool"} 

715 
val x_tm = @{cpat "?x :: real"} 

716 
val y_tm = @{cpat "?y::real"} 

717 
val is_max = is_binop @{cterm "max :: real => _"} 

718 
val is_min = is_binop @{cterm "min :: real => _"} 

719 
fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm 

720 
fun eliminate_construct p c tm = 

721 
let 

722 
val t = find_cterm p tm 

723 
val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.apply (Thm.lambda t tm) t) 

724 
val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0 

725 
in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false)))) 

726 
(Thm.transitive th0 (c p ax)) 

727 
end 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

728 

46594  729 
val elim_abs = eliminate_construct is_abs 
730 
(fn p => fn ax => 

731 
Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax)]) pth_abs) 

732 
val elim_max = eliminate_construct is_max 

733 
(fn p => fn ax => 

734 
let val (ax,y) = Thm.dest_comb ax 

735 
in Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)]) 

736 
pth_max end) 

737 
val elim_min = eliminate_construct is_min 

738 
(fn p => fn ax => 

739 
let val (ax,y) = Thm.dest_comb ax 

740 
in Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)]) 

741 
pth_min end) 

742 
in first_conv [elim_abs, elim_max, elim_min, all_conv] 

743 
end; 

744 
in 

745 
fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) = 

746 
gen_gen_real_arith ctxt 

747 
(mkconst,eq,ge,gt,norm,neg,add,mul, 

748 
absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover) 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

749 
end; 
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

750 

46594  751 
(* An instance for reals*) 
31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

752 

46594  753 
fun gen_prover_real_arith ctxt prover = 
754 
let 

755 
fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS 

756 
val {add, mul, neg, pow = _, sub = _, main} = 

757 
Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt 

758 
(the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"})) 

759 
simple_cterm_ord 

760 
in gen_real_arith ctxt 

761 
(cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, 

762 
main,neg,add,mul, prover) 

763 
end; 

31120
fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

764 

fc654c95c29e
A generic arithmetic prover based on Positivstellensatz certificates  also implements FourrierMotzkin elimination as a special case FourrierMotzkin elimination
chaieb
parents:
diff
changeset

765 
end 