author  nipkow 
Thu, 30 May 2002 10:12:11 +0200  
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parent 13105  3d1e7a199bdc 
child 13464  c98321b8d638 
permissions  rwrr 
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(* Title: Provers/Arith/fast_lin_arith.ML 
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ID: $Id$ 
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Author: Tobias Nipkow 
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Copyright 1998 TU Munich 
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6062  6 
A generic linear arithmetic package. 
6102  7 
It provides two tactics 
8 

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lin_arith_tac: int > tactic 
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cut_lin_arith_tac: thms > int > tactic 
6102  11 

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and a simplification procedure 

13 

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lin_arith_prover: Sign.sg > thm list > term > thm option 

15 

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Only take premises and conclusions into account that are already (negated) 

17 
(in)equations. lin_arith_prover tries to prove or disprove the term. 

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*) 
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9073  20 
(* Debugging: set Fast_Arith.trace *) 
7582  21 

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(*** Data needed for setting up the linear arithmetic package ***) 
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6102  24 
signature LIN_ARITH_LOGIC = 
25 
sig 

26 
val conjI: thm 

27 
val ccontr: thm (* (~ P ==> False) ==> P *) 

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val neqE: thm (* [ m ~= n; m < n ==> P; n < m ==> P ] ==> P *) 

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val notI: thm (* (P ==> False) ==> ~ P *) 

6110  30 
val not_lessD: thm (* ~(m < n) ==> n <= m *) 
6128  31 
val not_leD: thm (* ~(m <= n) ==> n < m *) 
6102  32 
val sym: thm (* x = y ==> y = x *) 
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val mk_Eq: thm > thm 

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val mk_Trueprop: term > term 

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val neg_prop: term > term 

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val is_False: thm > bool 

6128  37 
val is_nat: typ list * term > bool 
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val mk_nat_thm: Sign.sg > term > thm 

6102  39 
end; 
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(* 

41 
mk_Eq(~in) = `in == False' 

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mk_Eq(in) = `in == True' 

43 
where `in' is an (in)equality. 

44 

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neg_prop(t) = neg if t is wrapped up in Trueprop and 

46 
nt is the (logically) negated version of t, where the negation 

47 
of a negative term is the term itself (no double negation!); 

6128  48 

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is_nat(parametertypes,t) = t:nat 

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mk_nat_thm(t) = "0 <= t" 

6102  51 
*) 
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signature LIN_ARITH_DATA = 
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sig 
6128  55 
val decomp: 
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Sign.sg > term > ((term*rat)list * rat * string * (term*rat)list * rat * bool)option 
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val number_of: int * typ > term 

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end; 
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(* 
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decomp(`x Rel y') should yield (p,i,Rel,q,j,d) 
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where Rel is one of "<", "~<", "<=", "~<=" and "=" and 
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p/q is the decomposition of the sum terms x/y into a list 
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of summand * multiplicity pairs and a constant summand and 
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d indicates if the domain is discrete. 
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9420  66 
ss must reduce contradictory <= to False. 
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It should also cancel common summands to keep <= reduced; 
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otherwise <= can grow to massive proportions. 
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*) 
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6062  71 
signature FAST_LIN_ARITH = 
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sig 

9420  73 
val setup: (theory > theory) list 
10691  74 
val map_data: ({add_mono_thms: thm list, mult_mono_thms: (thm*cterm)list, inj_thms: thm list, 
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lessD: thm list, simpset: Simplifier.simpset} 
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> {add_mono_thms: thm list, mult_mono_thms: (thm*cterm)list, inj_thms: thm list, 
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lessD: thm list, simpset: Simplifier.simpset}) 
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> theory > theory 

9073  79 
val trace : bool ref 
6074  80 
val lin_arith_prover: Sign.sg > thm list > term > thm option 
6062  81 
val lin_arith_tac: int > tactic 
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val cut_lin_arith_tac: thm list > int > tactic 

83 
end; 

84 

6102  85 
functor Fast_Lin_Arith(structure LA_Logic:LIN_ARITH_LOGIC 
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and LA_Data:LIN_ARITH_DATA) : FAST_LIN_ARITH = 

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struct 
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9420  89 

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(** theory data **) 

91 

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(* data kind 'Provers/fast_lin_arith' *) 

93 

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structure DataArgs = 

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struct 

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val name = "Provers/fast_lin_arith"; 

10691  97 
type T = {add_mono_thms: thm list, mult_mono_thms: (thm*cterm)list, inj_thms: thm list, 
10575  98 
lessD: thm list, simpset: Simplifier.simpset}; 
9420  99 

10691  100 
val empty = {add_mono_thms = [], mult_mono_thms = [], inj_thms = [], 
10575  101 
lessD = [], simpset = Simplifier.empty_ss}; 
9420  102 
val copy = I; 
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val prep_ext = I; 

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10691  105 
fun merge ({add_mono_thms= add_mono_thms1, mult_mono_thms= mult_mono_thms1, inj_thms= inj_thms1, 
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lessD = lessD1, simpset = simpset1}, 
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{add_mono_thms= add_mono_thms2, mult_mono_thms= mult_mono_thms2, inj_thms= inj_thms2, 
10575  108 
lessD = lessD2, simpset = simpset2}) = 
9420  109 
{add_mono_thms = Drule.merge_rules (add_mono_thms1, add_mono_thms2), 
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mult_mono_thms = gen_merge_lists' (Drule.eq_thm_prop o pairself fst) 
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mult_mono_thms1 mult_mono_thms2, 
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inj_thms = Drule.merge_rules (inj_thms1, inj_thms2), 
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lessD = Drule.merge_rules (lessD1, lessD2), 

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simpset = Simplifier.merge_ss (simpset1, simpset2)}; 

9420  115 

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fun print _ _ = (); 

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end; 

118 

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structure Data = TheoryDataFun(DataArgs); 

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val map_data = Data.map; 

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val setup = [Data.init]; 

122 

123 

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(*** A fast decision procedure ***) 
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(*** Code ported from HOL Light ***) 
6056  127 
(* possible optimizations: 
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use (var,coeff) rep or vector rep tp save space; 

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treat nonnegative atoms separately rather than adding 0 <= atom 

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*) 

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9073  132 
val trace = ref false; 
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datatype lineq_type = Eq  Le  Lt; 
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6056  136 
datatype injust = Asm of int 
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 Nat of int (* index of atom *) 

6128  138 
 LessD of injust 
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 NotLessD of injust 

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 NotLeD of injust 

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 NotLeDD of injust 
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 Multiplied of int * injust 
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 Multiplied2 of int * injust 
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 Added of injust * injust; 
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datatype lineq = Lineq of int * lineq_type * int list * injust; 
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(*  *) 
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(* Calculate new (in)equality type after addition. *) 
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(*  *) 
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fun find_add_type(Eq,x) = x 
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 find_add_type(x,Eq) = x 
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 find_add_type(_,Lt) = Lt 
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 find_add_type(Lt,_) = Lt 
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 find_add_type(Le,Le) = Le; 
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(*  *) 
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(* Multiply out an (in)equation. *) 
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(*  *) 
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fun multiply_ineq n (i as Lineq(k,ty,l,just)) = 
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if n = 1 then i 
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else if n = 0 andalso ty = Lt then sys_error "multiply_ineq" 
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else if n < 0 andalso (ty=Le orelse ty=Lt) then sys_error "multiply_ineq" 
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else Lineq(n * k,ty,map (apl(n,op * )) l,Multiplied(n,just)); 
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(*  *) 
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(* Add together (in)equations. *) 
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(*  *) 
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fun add_ineq (i1 as Lineq(k1,ty1,l1,just1)) (i2 as Lineq(k2,ty2,l2,just2)) = 
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let val l = map2 (op +) (l1,l2) 
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in Lineq(k1+k2,find_add_type(ty1,ty2),l,Added(just1,just2)) end; 
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(*  *) 
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(* Elimination of variable between a single pair of (in)equations. *) 
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(* If they're both inequalities, 1st coefficient must be +ve, 2nd ve. *) 
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(*  *) 
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fun el 0 (h::_) = h 
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 el n (_::t) = el (n  1) t 
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 el _ _ = sys_error "el"; 
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fun elim_var v (i1 as Lineq(k1,ty1,l1,just1)) (i2 as Lineq(k2,ty2,l2,just2)) = 
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let val c1 = el v l1 and c2 = el v l2 
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val m = lcm(abs c1,abs c2) 
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val m1 = m div (abs c1) and m2 = m div (abs c2) 
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val (n1,n2) = 
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if (c1 >= 0) = (c2 >= 0) 
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then if ty1 = Eq then (~m1,m2) 
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else if ty2 = Eq then (m1,~m2) 
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else sys_error "elim_var" 
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else (m1,m2) 
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val (p1,p2) = if ty1=Eq andalso ty2=Eq andalso (n1 = ~1 orelse n2 = ~1) 
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then (~n1,~n2) else (n1,n2) 
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in add_ineq (multiply_ineq n1 i1) (multiply_ineq n2 i2) end; 
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(*  *) 
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(* The main refutationfinding code. *) 
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(*  *) 
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fun is_trivial (Lineq(_,_,l,_)) = forall (fn i => i=0) l; 
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fun is_answer (ans as Lineq(k,ty,l,_)) = 
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case ty of Eq => k <> 0  Le => k > 0  Lt => k >= 0; 
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207 

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fun calc_blowup l = 
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let val (p,n) = partition (apl(0,op<)) (filter (apl(0,op<>)) l) 
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in (length p) * (length n) end; 
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211 

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(*  *) 
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(* Main elimination code: *) 
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(* *) 
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(* (1) Looks for immediate solutions (false assertions with no variables). *) 
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(* *) 
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(* (2) If there are any equations, picks a variable with the lowest absolute *) 
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(* coefficient in any of them, and uses it to eliminate. *) 
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(* *) 
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(* (3) Otherwise, chooses a variable in the inequality to minimize the *) 
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(* blowup (number of consequences generated) and eliminates it. *) 
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(*  *) 
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fun allpairs f xs ys = 
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flat(map (fn x => map (fn y => f x y) ys) xs); 
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226 

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fun extract_first p = 
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let fun extract xs (y::ys) = if p y then (Some y,xs@ys) 
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else extract (y::xs) ys 
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 extract xs [] = (None,xs) 
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in extract [] end; 
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6056  233 
fun print_ineqs ineqs = 
9073  234 
if !trace then 
12262  235 
tracing(cat_lines(""::map (fn Lineq(c,t,l,_) => 
9073  236 
string_of_int c ^ 
237 
(case t of Eq => " = "  Lt=> " < "  Le => " <= ") ^ 

238 
commas(map string_of_int l)) ineqs)) 

239 
else (); 

6056  240 

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fun elim ineqs = 
9073  242 
let val dummy = print_ineqs ineqs; 
6056  243 
val (triv,nontriv) = partition is_trivial ineqs in 
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if not(null triv) 
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then case Library.find_first is_answer triv of 
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None => elim nontriv  some => some 
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else 
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if null nontriv then None else 
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let val (eqs,noneqs) = partition (fn (Lineq(_,ty,_,_)) => ty=Eq) nontriv in 
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if not(null eqs) then 
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let val clist = foldl (fn (cs,Lineq(_,_,l,_)) => l union cs) ([],eqs) 
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val sclist = sort (fn (x,y) => int_ord(abs(x),abs(y))) 
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(filter (fn i => i<>0) clist) 
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val c = hd sclist 
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val (Some(eq as Lineq(_,_,ceq,_)),othereqs) = 
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extract_first (fn Lineq(_,_,l,_) => c mem l) eqs 
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val v = find_index (fn k => k=c) ceq 
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val (ioth,roth) = partition (fn (Lineq(_,_,l,_)) => el v l = 0) 
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(othereqs @ noneqs) 
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val others = map (elim_var v eq) roth @ ioth 
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in elim others end 
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else 
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let val lists = map (fn (Lineq(_,_,l,_)) => l) noneqs 
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val numlist = 0 upto (length(hd lists)  1) 
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val coeffs = map (fn i => map (el i) lists) numlist 
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val blows = map calc_blowup coeffs 
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val iblows = blows ~~ numlist 
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val nziblows = filter (fn (i,_) => i<>0) iblows 
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in if null nziblows then None else 
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let val (c,v) = hd(sort (fn (x,y) => int_ord(fst(x),fst(y))) nziblows) 
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val (no,yes) = partition (fn (Lineq(_,_,l,_)) => el v l = 0) ineqs 
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val (pos,neg) = partition(fn (Lineq(_,_,l,_)) => el v l > 0) yes 
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in elim (no @ allpairs (elim_var v) pos neg) end 
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end 
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end 
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276 
end; 
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(*  *) 
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279 
(* Translate back a proof. *) 
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(*  *) 
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281 

9073  282 
fun trace_thm msg th = 
12262  283 
if !trace then (tracing msg; tracing (Display.string_of_thm th); th) else th; 
9073  284 

285 
fun trace_msg msg = 

12262  286 
if !trace then tracing msg else (); 
9073  287 

6056  288 
(* FIXME OPTIMIZE!!!! 
289 
Addition/Multiplication need i*t representation rather than t+t+... 

10691  290 
Get rid of Mulitplied(2). For Nat LA_Data.number_of should return Suc^n 
291 
because Numerals are not known early enough. 

6056  292 

293 
Simplification may detect a contradiction 'prematurely' due to type 

294 
information: n+1 <= 0 is simplified to False and does not need to be crossed 

295 
with 0 <= n. 

296 
*) 

297 
local 

298 
exception FalseE of thm 

299 
in 

6074  300 
fun mkthm sg asms just = 
10691  301 
let val {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} = Data.get_sg sg; 
9420  302 
val atoms = foldl (fn (ats,(lhs,_,_,rhs,_,_)) => 
6056  303 
map fst lhs union (map fst rhs union ats)) 
9420  304 
([], mapfilter (LA_Data.decomp sg o concl_of) asms) 
6056  305 

10575  306 
fun add2 thm1 thm2 = 
6102  307 
let val conj = thm1 RS (thm2 RS LA_Logic.conjI) 
10575  308 
in get_first (fn th => Some(conj RS th) handle _ => None) add_mono_thms 
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309 
end; 
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310 

10575  311 
fun try_add [] _ = None 
312 
 try_add (thm1::thm1s) thm2 = case add2 thm1 thm2 of 

313 
None => try_add thm1s thm2  some => some; 

314 

315 
fun addthms thm1 thm2 = 

316 
case add2 thm1 thm2 of 

317 
None => (case try_add ([thm1] RL inj_thms) thm2 of 

318 
None => the(try_add ([thm2] RL inj_thms) thm1) 

319 
 Some thm => thm) 

320 
 Some thm => thm; 

321 

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fun multn(n,thm) = 
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323 
let fun mul(i,th) = if i=1 then th else mul(i1, addthms thm th) 
6102  324 
in if n < 0 then mul(~n,thm) RS LA_Logic.sym else mul(n,thm) end; 
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325 

10691  326 
fun multn2(n,thm) = 
327 
let val Some(mth,cv) = 

328 
get_first (fn (th,cv) => Some(thm RS th,cv) handle _ => None) mult_mono_thms 

329 
val ct = cterm_of sg (LA_Data.number_of(n,#T(rep_cterm cv))) 

330 
in instantiate ([],[(cv,ct)]) mth end 

331 

6056  332 
fun simp thm = 
12932  333 
let val thm' = trace_thm "Simplified:" (full_simplify simpset thm) 
6102  334 
in if LA_Logic.is_False thm' then raise FalseE thm' else thm' end 
6056  335 

9073  336 
fun mk(Asm i) = trace_thm "Asm" (nth_elem(i,asms)) 
337 
 mk(Nat(i)) = (trace_msg "Nat"; 

338 
LA_Logic.mk_nat_thm sg (nth_elem(i,atoms))) 

9420  339 
 mk(LessD(j)) = trace_thm "L" (hd([mk j] RL lessD)) 
9073  340 
 mk(NotLeD(j)) = trace_thm "NLe" (mk j RS LA_Logic.not_leD) 
9420  341 
 mk(NotLeDD(j)) = trace_thm "NLeD" (hd([mk j RS LA_Logic.not_leD] RL lessD)) 
9073  342 
 mk(NotLessD(j)) = trace_thm "NL" (mk j RS LA_Logic.not_lessD) 
343 
 mk(Added(j1,j2)) = simp (trace_thm "+" (addthms (mk j1) (mk j2))) 

10717  344 
 mk(Multiplied(n,j)) = (trace_msg("*"^string_of_int n); trace_thm "*" (multn(n,mk j))) 
345 
 mk(Multiplied2(n,j)) = simp (trace_msg("**"^string_of_int n); trace_thm "**" (multn2(n,mk j))) 

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346 

9073  347 
in trace_msg "mkthm"; 
12932  348 
let val thm = trace_thm "Final thm:" (mk just) 
349 
in let val fls = simplify simpset thm 

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350 
in trace_thm "After simplification:" fls; 
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351 
if LA_Logic.is_False fls then fls 
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352 
else 
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353 
(tracing "Assumptions:"; seq print_thm asms; 
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354 
tracing "Proved:"; print_thm fls; 
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355 
warning "Linear arithmetic should have refuted the assumptions.\n\ 
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356 
\Please inform Tobias Nipkow (nipkow@in.tum.de)."; 
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357 
fls) 
12932  358 
end 
359 
end handle FalseE thm => (trace_thm "False reached early:" thm; thm) 

360 
end 

6056  361 
end; 
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362 

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363 
fun coeff poly atom = case assoc(poly,atom) of None => 0  Some i => i; 
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364 

10691  365 
fun lcms is = foldl lcm (1,is); 
366 

367 
fun integ(rlhs,r,rel,rrhs,s,d) = 

368 
let val (rn,rd) = rep_rat r and (sn,sd) = rep_rat s 

369 
val m = lcms(map (abs o snd o rep_rat) (r :: s :: map snd rlhs @ map snd rrhs)) 

370 
fun mult(t,r) = let val (i,j) = rep_rat r in (t,i * (m div j)) end 

12932  371 
in (m,(map mult rlhs, rn*(m div rd), rel, map mult rrhs, sn*(m div sd), d)) end 
10691  372 

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373 
fun mklineq atoms = 
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374 
let val n = length atoms in 
10691  375 
fn (item,k) => 
376 
let val (m,(lhs,i,rel,rhs,j,discrete)) = integ item 

377 
val lhsa = map (coeff lhs) atoms 

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378 
and rhsa = map (coeff rhs) atoms 
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379 
val diff = map2 (op ) (rhsa,lhsa) 
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380 
val c = ij 
6056  381 
val just = Asm k 
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382 
fun lineq(c,le,cs,j) = Lineq(c,le,cs, if m=1 then j else Multiplied2(m,j)) 
5982
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383 
in case rel of 
10691  384 
"<=" => lineq(c,Le,diff,just) 
7551
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385 
 "~<=" => if discrete 
10691  386 
then lineq(1c,Le,map (op ~) diff,NotLeDD(just)) 
387 
else lineq(~c,Lt,map (op ~) diff,NotLeD(just)) 

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388 
 "<" => if discrete 
10691  389 
then lineq(c+1,Le,diff,LessD(just)) 
390 
else lineq(c,Lt,diff,just) 

391 
 "~<" => lineq(~c,Le,map (op~) diff,NotLessD(just)) 

392 
 "=" => lineq(c,Eq,diff,just) 

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393 
 _ => sys_error("mklineq" ^ rel) 
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394 
end 
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395 
end; 
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396 

6056  397 
fun mknat pTs ixs (atom,i) = 
6128  398 
if LA_Logic.is_nat(pTs,atom) 
6056  399 
then let val l = map (fn j => if j=i then 1 else 0) ixs 
400 
in Some(Lineq(0,Le,l,Nat(i))) end 

401 
else None 

402 

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403 
(* This code is tricky. It takes a list of premises in the order they occur 
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404 
in the subgoal. Numerical premises are coded as Some(tuple), nonnumerical 
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405 
ones as None. Going through the premises, each numeric one is converted into 
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406 
a Lineq. The tricky bit is to convert ~= which is split into two cases < and 
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407 
>. Thus mklineqss returns a list of equation systems. This may blow up if 
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408 
there are many ~=, but in practice it does not seem to happen. The really 
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409 
tricky bit is to arrange the order of the cases such that they coincide with 
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410 
the order in which the cases are in the end generated by the tactic that 
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411 
applies the generated refutation thms (see function 'refute_tac'). 
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412 

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413 
For variables n of type nat, a constraint 0 <= n is added. 
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414 
*) 
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415 
fun mklineqss(pTs,items) = 
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416 
let fun add(ats,None) = ats 
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417 
 add(ats,Some(lhs,_,_,rhs,_,_)) = 
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418 
(map fst lhs) union ((map fst rhs) union ats) 
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419 
val atoms = foldl add ([],items) 
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420 
val mkleq = mklineq atoms 
6056  421 
val ixs = 0 upto (length(atoms)1) 
422 
val iatoms = atoms ~~ ixs 

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423 
val natlineqs = mapfilter (mknat pTs ixs) iatoms 
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424 

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425 
fun elim_neq front _ [] = [front] 
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426 
 elim_neq front n (None::ineqs) = elim_neq front (n+1) ineqs 
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427 
 elim_neq front n (Some(ineq as (l,i,rel,r,j,d))::ineqs) = 
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428 
if rel = "~=" then elim_neq front n (ineqs @ [Some(l,i,"<",r,j,d)]) @ 
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429 
elim_neq front n (ineqs @ [Some(r,j,"<",l,i,d)]) 
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430 
else elim_neq (mkleq(ineq,n) :: front) (n+1) ineqs 
5982
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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431 

13186
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432 
in elim_neq natlineqs 0 items end; 
6074  433 

13186
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434 
fun elim_all (ineqs::ineqss) js = 
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435 
(case elim ineqs of None => None 
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436 
 Some(Lineq(_,_,_,j)) => elim_all ineqss (js@[j])) 
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437 
 elim_all [] js = Some js 
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438 

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439 
fun refute(pTsitems) = elim_all (mklineqss pTsitems) []; 
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440 

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441 
fun refute_tac(i,justs) = 
6074  442 
fn state => 
443 
let val sg = #sign(rep_thm state) 

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444 
fun just1 j = REPEAT_DETERM(etac LA_Logic.neqE i) THEN 
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445 
METAHYPS (fn asms => rtac (mkthm sg asms j) 1) i 
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446 
in DETERM(resolve_tac [LA_Logic.notI,LA_Logic.ccontr] i) THEN 
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447 
EVERY(map just1 justs) 
6074  448 
end 
449 
state; 

450 

9420  451 
fun prove sg (pTs,Hs,concl) = 
13186
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452 
let val Hitems = map (LA_Data.decomp sg) Hs 
9420  453 
in case LA_Data.decomp sg concl of 
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454 
None => refute(pTs,Hitems@[None]) 
7551
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Now distinguishes discrete from nondistrete types.
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455 
 Some(citem as (r,i,rel,l,j,d)) => 
13186
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456 
let val neg::rel0 = explode rel 
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457 
val nrel = if neg = "~" then implode rel0 else "~"^rel 
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458 
in refute(pTs, Hitems @ [Some(r,i,nrel,l,j,d)]) end 
6074  459 
end; 
5982
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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460 

aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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461 
(* 
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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462 
Fast but very incomplete decider. Only premises and conclusions 
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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463 
that are already (negated) (in)equations are taken into account. 
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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diff
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464 
*) 
13186
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465 
fun lin_arith_tac i st = SUBGOAL (fn (A,_) => 
6056  466 
let val pTs = rev(map snd (Logic.strip_params A)) 
467 
val Hs = Logic.strip_assums_hyp A 

6074  468 
val concl = Logic.strip_assums_concl A 
12932  469 
in trace_thm ("Trying to refute subgoal " ^ string_of_int i) st; 
470 
case prove (Thm.sign_of_thm st) (pTs,Hs,concl) of 

13186
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471 
None => no_tac 
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472 
 Some js => refute_tac(i,js) 
9420  473 
end) i st; 
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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474 

aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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475 
fun cut_lin_arith_tac thms i = cut_facts_tac thms i THEN lin_arith_tac i; 
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476 

13186
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changeset

477 
(** Forward proof from theorems **) 
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478 

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479 
(* More tricky code. Needs to arrange the proofs of the multiple cases (due 
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Big update. Allows case splitting on ~= now (trying both < and >).
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480 
to splits of ~= premises) such that it coincides with the order of the cases 
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Big update. Allows case splitting on ~= now (trying both < and >).
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481 
generated by function mklineqss. *) 
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Big update. Allows case splitting on ~= now (trying both < and >).
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482 

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483 
datatype splittree = Tip of thm list 
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484 
 Spl of thm * cterm * splittree * cterm * splittree 
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Big update. Allows case splitting on ~= now (trying both < and >).
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485 

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Big update. Allows case splitting on ~= now (trying both < and >).
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486 
fun extract imp = 
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487 
let val (Il,r) = Thm.dest_comb imp 
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Big update. Allows case splitting on ~= now (trying both < and >).
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488 
val (_,imp1) = Thm.dest_comb Il 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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489 
val (Ict1,_) = Thm.dest_comb imp1 
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Big update. Allows case splitting on ~= now (trying both < and >).
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490 
val (_,ct1) = Thm.dest_comb Ict1 
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parents:
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491 
val (Ir,_) = Thm.dest_comb r 
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changeset

492 
val (_,Ict2r) = Thm.dest_comb Ir 
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493 
val (Ict2,_) = Thm.dest_comb Ict2r 
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Big update. Allows case splitting on ~= now (trying both < and >).
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494 
val (_,ct2) = Thm.dest_comb Ict2 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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495 
in (ct1,ct2) end; 
6074  496 

13186
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Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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changeset

497 
fun splitasms asms = 
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498 
let fun split(asms',[]) = Tip(rev asms') 
ef8ed6adcb38
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parents:
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diff
changeset

499 
 split(asms',asm::asms) = 
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Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

500 
let val spl = asm COMP LA_Logic.neqE 
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Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

501 
val (ct1,ct2) = extract(cprop_of spl) 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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502 
val thm1 = assume ct1 and thm2 = assume ct2 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

503 
in Spl(spl,ct1,split(asms',asms@[thm1]),ct2,split(asms',asms@[thm2])) end 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

504 
handle THM _ => split(asm::asms', asms) 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

505 
in split([],asms) end; 
6074  506 

13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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507 
fun fwdproof sg (Tip asms) (j::js) = (mkthm sg asms j, js) 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

508 
 fwdproof sg (Spl(thm,ct1,tree1,ct2,tree2)) js = 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
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diff
changeset

509 
let val (thm1,js1) = fwdproof sg tree1 js 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
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510 
val (thm2,js2) = fwdproof sg tree2 js1 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

511 
val thm1' = implies_intr ct1 thm1 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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diff
changeset

512 
val thm2' = implies_intr ct2 thm2 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

513 
in (thm2' COMP (thm1' COMP thm), js2) end; 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

514 
(* needs handle _ => None ? *) 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
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diff
changeset

515 

ef8ed6adcb38
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parents:
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changeset

516 
fun prover sg thms Tconcl js pos = 
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Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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changeset

517 
let val nTconcl = LA_Logic.neg_prop Tconcl 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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518 
val cnTconcl = cterm_of sg nTconcl 
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Big update. Allows case splitting on ~= now (trying both < and >).
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519 
val nTconclthm = assume cnTconcl 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

520 
val tree = splitasms (thms @ [nTconclthm]) 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

521 
val (thm,_) = fwdproof sg tree js 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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changeset

522 
val contr = if pos then LA_Logic.ccontr else LA_Logic.notI 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

523 
in Some(LA_Logic.mk_Eq((implies_intr cnTconcl thm) COMP contr)) end 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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524 
(* in case concl contains ?var, which makes assume fail: *) 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

525 
handle THM _ => None; 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

526 

ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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527 
(* PRE: concl is not negated! 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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changeset

528 
This assumption is OK because 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

529 
1. lin_arith_prover tries both to prove and disprove concl and 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
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diff
changeset

530 
2. lin_arith_prover is applied by the simplifier which 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
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diff
changeset

531 
dives into terms and will thus try the nonnegated concl anyway. 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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532 
*) 
6074  533 
fun lin_arith_prover sg thms concl = 
534 
let val Hs = map (#prop o rep_thm) thms 

6102  535 
val Tconcl = LA_Logic.mk_Trueprop concl 
13186
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Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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changeset

536 
in case prove sg ([],Hs,Tconcl) of (* concl provable? *) 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

537 
Some js => prover sg thms Tconcl js true 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
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parents:
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diff
changeset

538 
 None => let val nTconcl = LA_Logic.neg_prop Tconcl 
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Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
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diff
changeset

539 
in case prove sg ([],Hs,nTconcl) of (* ~concl provable? *) 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset

540 
Some js => prover sg thms nTconcl js false 
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset

541 
 None => None 
6079  542 
end 
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset

543 
end; 
6074  544 

545 
end; 