author | wenzelm |
Thu, 29 May 2008 23:46:41 +0200 | |
changeset 27020 | b5b8afc9fdcd |
parent 26945 | 9cd13e810998 |
child 30406 | 15dc25f8a0e2 |
permissions | -rw-r--r-- |
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(* Title: Provers/Arith/fast_lin_arith.ML |
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ID: $Id$ |
24076 | 3 |
Author: Tobias Nipkow and Tjark Weber |
6102 | 4 |
|
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A generic linear arithmetic package. It provides two tactics |
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(cut_lin_arith_tac, lin_arith_tac) and a simplification procedure |
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(lin_arith_simproc). |
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Only take premises and conclusions into account that are already |
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(negated) (in)equations. lin_arith_simproc tries to prove or disprove |
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the term. |
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*) |
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|
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(*** Data needed for setting up the linear arithmetic package ***) |
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signature LIN_ARITH_LOGIC = |
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sig |
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val conjI : thm (* P ==> Q ==> P & Q *) |
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val ccontr : thm (* (~ P ==> False) ==> P *) |
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val notI : thm (* (P ==> False) ==> ~ P *) |
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val not_lessD : thm (* ~(m < n) ==> n <= m *) |
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val not_leD : thm (* ~(m <= n) ==> n < m *) |
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val sym : thm (* x = y ==> y = x *) |
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val mk_Eq : thm -> thm |
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val atomize : thm -> thm list |
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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 |
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val is_nat : typ list * term -> bool |
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val mk_nat_thm : theory -> term -> thm |
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end; |
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(* |
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mk_Eq(~in) = `in == False' |
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mk_Eq(in) = `in == True' |
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where `in' is an (in)equality. |
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neg_prop(t) = neg if t is wrapped up in Trueprop and neg is the |
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(logically) negated version of t (again wrapped up in Trueprop), |
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where the negation of a negative term is the term itself (no |
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double negation!); raises TERM ("neg_prop", [t]) if t is not of |
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the form 'Trueprop $ _' |
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is_nat(parameter-types,t) = t:nat |
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mk_nat_thm(t) = "0 <= t" |
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*) |
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signature LIN_ARITH_DATA = |
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sig |
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(*internal representation of linear (in-)equations:*) |
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type decomp = (term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool |
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val decomp: Proof.context -> term -> decomp option |
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val domain_is_nat: term -> bool |
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||
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(*preprocessing, performed on a representation of subgoals as list of premises:*) |
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val pre_decomp: Proof.context -> typ list * term list -> (typ list * term list) list |
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(*preprocessing, performed on the goal -- must do the same as 'pre_decomp':*) |
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val pre_tac: Proof.context -> int -> tactic |
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val number_of: int * typ -> term |
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(*the limit on the number of ~= allowed; because each ~= is split |
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into two cases, this can lead to an explosion*) |
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val fast_arith_neq_limit: int Config.T |
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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, respectively) is the decomposition of the sum term x |
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(y, respectively) into a list of summand * multiplicity |
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pairs and a constant summand and d indicates if the domain |
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is discrete. |
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domain_is_nat(`x Rel y') t should yield true iff x is of type "nat". |
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|
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The relationship between pre_decomp and pre_tac is somewhat tricky. The |
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internal representation of a subgoal and the corresponding theorem must |
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be modified by pre_decomp (pre_tac, resp.) in a corresponding way. See |
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the comment for split_items below. (This is even necessary for eta- and |
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beta-equivalent modifications, as some of the lin. arith. code is not |
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insensitive to them.) |
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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. |
aeb97860d352
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*) |
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signature FAST_LIN_ARITH = |
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sig |
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val map_data: ({add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list, |
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lessD: thm list, neqE: thm list, simpset: Simplifier.simpset} |
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-> {add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list, |
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lessD: thm list, neqE: thm list, simpset: Simplifier.simpset}) |
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-> Context.generic -> Context.generic |
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val trace: bool ref |
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val warning_count: int ref; |
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val cut_lin_arith_tac: simpset -> int -> tactic |
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val lin_arith_tac: Proof.context -> bool -> int -> tactic |
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val lin_arith_simproc: simpset -> term -> thm option |
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end; |
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functor Fast_Lin_Arith |
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(structure LA_Logic: LIN_ARITH_LOGIC and LA_Data: LIN_ARITH_DATA): FAST_LIN_ARITH = |
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struct |
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|
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(** theory data **) |
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structure Data = GenericDataFun |
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( |
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type T = |
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{add_mono_thms: thm list, |
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mult_mono_thms: thm list, |
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inj_thms: thm list, |
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lessD: thm list, |
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neqE: thm list, |
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simpset: Simplifier.simpset}; |
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val empty = {add_mono_thms = [], mult_mono_thms = [], inj_thms = [], |
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lessD = [], neqE = [], simpset = Simplifier.empty_ss}; |
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val extend = I; |
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fun merge _ |
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({add_mono_thms= add_mono_thms1, mult_mono_thms= mult_mono_thms1, inj_thms= inj_thms1, |
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lessD = lessD1, neqE=neqE1, simpset = simpset1}, |
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{add_mono_thms= add_mono_thms2, mult_mono_thms= mult_mono_thms2, inj_thms= inj_thms2, |
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lessD = lessD2, neqE=neqE2, simpset = simpset2}) = |
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{add_mono_thms = Thm.merge_thms (add_mono_thms1, add_mono_thms2), |
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mult_mono_thms = Thm.merge_thms (mult_mono_thms1, mult_mono_thms2), |
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inj_thms = Thm.merge_thms (inj_thms1, inj_thms2), |
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lessD = Thm.merge_thms (lessD1, lessD2), |
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neqE = Thm.merge_thms (neqE1, neqE2), |
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simpset = Simplifier.merge_ss (simpset1, simpset2)}; |
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); |
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val map_data = Data.map; |
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val get_data = Data.get o Context.Proof; |
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||
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(*** A fast decision procedure ***) |
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(*** Code ported from HOL Light ***) |
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(* possible optimizations: |
142 |
use (var,coeff) rep or vector rep tp save space; |
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treat non-negative atoms separately rather than adding 0 <= atom |
|
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*) |
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val trace = ref false; |
147 |
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datatype lineq_type = Eq | Le | Lt; |
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datatype injust = Asm of int |
151 |
| Nat of int (* index of atom *) |
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| 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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(* Finding a (counter) example from the trace of a failed elimination *) |
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(* ------------------------------------------------------------------------- *) |
|
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(* Examples are represented as rational numbers, *) |
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(* Dont blame John Harrison for this code - it is entirely mine. TN *) |
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||
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exception NoEx; |
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(* Coding: (i,true,cs) means i <= cs and (i,false,cs) means i < cs. |
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In general, true means the bound is included, false means it is excluded. |
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Need to know if it is a lower or upper bound for unambiguous interpretation! |
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*) |
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|
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fun elim_eqns (Lineq (i, Le, cs, _)) = [(i, true, cs)] |
176 |
| elim_eqns (Lineq (i, Eq, cs, _)) = [(i, true, cs),(~i, true, map ~ cs)] |
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| elim_eqns (Lineq (i, Lt, cs, _)) = [(i, false, cs)]; |
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(* PRE: ex[v] must be 0! *) |
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fun eval ex v (a, le, cs) = |
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let |
182 |
val rs = map Rat.rat_of_int cs; |
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183 |
val rsum = fold2 (Rat.add oo Rat.mult) rs ex Rat.zero; |
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23063 | 184 |
in (Rat.mult (Rat.add (Rat.rat_of_int a) (Rat.neg rsum)) (Rat.inv (nth rs v)), le) end; |
185 |
(* If nth rs v < 0, le should be negated. |
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Instead this swap is taken into account in ratrelmin2. |
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187 |
*) |
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|
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fun ratrelmin2 (x as (r, ler), y as (s, les)) = |
23520 | 190 |
case Rat.ord (r, s) |
22950 | 191 |
of EQUAL => (r, (not ler) andalso (not les)) |
192 |
| LESS => x |
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193 |
| GREATER => y; |
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||
195 |
fun ratrelmax2 (x as (r, ler), y as (s, les)) = |
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case Rat.ord (r, s) |
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of EQUAL => (r, ler andalso les) |
198 |
| LESS => y |
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199 |
| GREATER => x; |
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201 |
val ratrelmin = foldr1 ratrelmin2; |
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val ratrelmax = foldr1 ratrelmax2; |
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|
22950 | 204 |
fun ratexact up (r, exact) = |
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if exact then r else |
22950 | 206 |
let |
207 |
val (p, q) = Rat.quotient_of_rat r; |
|
208 |
val nth = Rat.inv (Rat.rat_of_int q); |
|
209 |
in Rat.add r (if up then nth else Rat.neg nth) end; |
|
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|
210 |
|
22950 | 211 |
fun ratmiddle (r, s) = Rat.mult (Rat.add r s) (Rat.inv Rat.two); |
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Finally fixed the counterexample finder. Can now deal with < on real.
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|
212 |
|
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linear arithmetic splits certain operators (e.g. min, max, abs)
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|
213 |
fun choose2 d ((lb, exactl), (ub, exactu)) = |
23520 | 214 |
let val ord = Rat.sign lb in |
22950 | 215 |
if (ord = LESS orelse exactl) andalso (ord = GREATER orelse exactu) |
216 |
then Rat.zero |
|
217 |
else if not d then |
|
218 |
if ord = GREATER |
|
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|
219 |
then if exactl then lb else ratmiddle (lb, ub) |
22950 | 220 |
else if exactu then ub else ratmiddle (lb, ub) |
221 |
else (*discrete domain, both bounds must be exact*) |
|
23025 | 222 |
if ord = GREATER |
22950 | 223 |
then let val lb' = Rat.roundup lb in |
224 |
if Rat.le lb' ub then lb' else raise NoEx end |
|
225 |
else let val ub' = Rat.rounddown ub in |
|
226 |
if Rat.le lb ub' then ub' else raise NoEx end |
|
227 |
end; |
|
13498 | 228 |
|
22950 | 229 |
fun findex1 discr (v, lineqs) ex = |
230 |
let |
|
23063 | 231 |
val nz = filter (fn (Lineq (_, _, cs, _)) => nth cs v <> 0) lineqs; |
22950 | 232 |
val ineqs = maps elim_eqns nz; |
23063 | 233 |
val (ge, le) = List.partition (fn (_,_,cs) => nth cs v > 0) ineqs |
22950 | 234 |
val lb = ratrelmax (map (eval ex v) ge) |
235 |
val ub = ratrelmin (map (eval ex v) le) |
|
21109 | 236 |
in nth_map v (K (choose2 (nth discr v) (lb, ub))) ex end; |
13498 | 237 |
|
238 |
fun elim1 v x = |
|
23063 | 239 |
map (fn (a,le,bs) => (Rat.add a (Rat.neg (Rat.mult (nth bs v) x)), le, |
21109 | 240 |
nth_map v (K Rat.zero) bs)); |
13498 | 241 |
|
23520 | 242 |
fun single_var v (_, _, cs) = case filter_out (curry (op =) EQUAL o Rat.sign) cs |
23063 | 243 |
of [x] => x =/ nth cs v |
244 |
| _ => false; |
|
13498 | 245 |
|
246 |
(* The base case: |
|
247 |
all variables occur only with positive or only with negative coefficients *) |
|
248 |
fun pick_vars discr (ineqs,ex) = |
|
23520 | 249 |
let val nz = filter_out (fn (_,_,cs) => forall (curry (op =) EQUAL o Rat.sign) cs) ineqs |
14372
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Finally fixed the counterexample finder. Can now deal with < on real.
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parents:
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diff
changeset
|
250 |
in case nz of [] => ex |
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Finally fixed the counterexample finder. Can now deal with < on real.
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diff
changeset
|
251 |
| (_,_,cs) :: _ => |
23520 | 252 |
let val v = find_index (not o curry (op =) EQUAL o Rat.sign) cs |
22950 | 253 |
val d = nth discr v; |
23520 | 254 |
val pos = not (Rat.sign (nth cs v) = LESS); |
22950 | 255 |
val sv = filter (single_var v) nz; |
14372
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Finally fixed the counterexample finder. Can now deal with < on real.
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changeset
|
256 |
val minmax = |
17951 | 257 |
if pos then if d then Rat.roundup o fst o ratrelmax |
14372
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Finally fixed the counterexample finder. Can now deal with < on real.
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diff
changeset
|
258 |
else ratexact true o ratrelmax |
17951 | 259 |
else if d then Rat.rounddown o fst o ratrelmin |
14372
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parents:
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diff
changeset
|
260 |
else ratexact false o ratrelmin |
23063 | 261 |
val bnds = map (fn (a,le,bs) => (Rat.mult a (Rat.inv (nth bs v)), le)) sv |
17951 | 262 |
val x = minmax((Rat.zero,if pos then true else false)::bnds) |
14372
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Finally fixed the counterexample finder. Can now deal with < on real.
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parents:
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changeset
|
263 |
val ineqs' = elim1 v x nz |
21109 | 264 |
val ex' = nth_map v (K x) ex |
14372
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Finally fixed the counterexample finder. Can now deal with < on real.
nipkow
parents:
14360
diff
changeset
|
265 |
in pick_vars discr (ineqs',ex') end |
13498 | 266 |
end; |
267 |
||
268 |
fun findex0 discr n lineqs = |
|
22950 | 269 |
let val ineqs = maps elim_eqns lineqs |
270 |
val rineqs = map (fn (a,le,cs) => (Rat.rat_of_int a, le, map Rat.rat_of_int cs)) |
|
14372
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changeset
|
271 |
ineqs |
17951 | 272 |
in pick_vars discr (rineqs,replicate n Rat.zero) end; |
13498 | 273 |
|
274 |
(* ------------------------------------------------------------------------- *) |
|
23197 | 275 |
(* End of counterexample finder. The actual decision procedure starts here. *) |
13498 | 276 |
(* ------------------------------------------------------------------------- *) |
277 |
||
5982
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|
278 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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parents:
diff
changeset
|
279 |
(* Calculate new (in)equality type after addition. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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|
280 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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diff
changeset
|
281 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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parents:
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changeset
|
282 |
fun find_add_type(Eq,x) = x |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
283 |
| find_add_type(x,Eq) = x |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
284 |
| find_add_type(_,Lt) = Lt |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
285 |
| find_add_type(Lt,_) = Lt |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
286 |
| find_add_type(Le,Le) = Le; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
287 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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parents:
diff
changeset
|
288 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
289 |
(* Multiply out an (in)equation. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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parents:
diff
changeset
|
290 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
291 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
292 |
fun multiply_ineq n (i as Lineq(k,ty,l,just)) = |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
293 |
if n = 1 then i |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
294 |
else if n = 0 andalso ty = Lt then sys_error "multiply_ineq" |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
295 |
else if n < 0 andalso (ty=Le orelse ty=Lt) then sys_error "multiply_ineq" |
17524 | 296 |
else Lineq (n * k, ty, map (curry op* n) l, Multiplied (n, just)); |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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parents:
diff
changeset
|
297 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
298 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
299 |
(* Add together (in)equations. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
300 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
301 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
302 |
fun add_ineq (i1 as Lineq(k1,ty1,l1,just1)) (i2 as Lineq(k2,ty2,l2,just2)) = |
18330 | 303 |
let val l = map2 (curry (op +)) l1 l2 |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
304 |
in Lineq(k1+k2,find_add_type(ty1,ty2),l,Added(just1,just2)) end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
305 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
306 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
307 |
(* Elimination of variable between a single pair of (in)equations. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
308 |
(* If they're both inequalities, 1st coefficient must be +ve, 2nd -ve. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
309 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
310 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
311 |
fun elim_var v (i1 as Lineq(k1,ty1,l1,just1)) (i2 as Lineq(k2,ty2,l2,just2)) = |
23063 | 312 |
let val c1 = nth l1 v and c2 = nth l2 v |
23261 | 313 |
val m = Integer.lcm (abs c1) (abs c2) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
314 |
val m1 = m div (abs c1) and m2 = m div (abs c2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
315 |
val (n1,n2) = |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
316 |
if (c1 >= 0) = (c2 >= 0) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
317 |
then if ty1 = Eq then (~m1,m2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
318 |
else if ty2 = Eq then (m1,~m2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
319 |
else sys_error "elim_var" |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
320 |
else (m1,m2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
321 |
val (p1,p2) = if ty1=Eq andalso ty2=Eq andalso (n1 = ~1 orelse n2 = ~1) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
322 |
then (~n1,~n2) else (n1,n2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
323 |
in add_ineq (multiply_ineq n1 i1) (multiply_ineq n2 i2) end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
324 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
325 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
326 |
(* The main refutation-finding code. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
327 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
328 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
329 |
fun is_trivial (Lineq(_,_,l,_)) = forall (fn i => i=0) l; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
330 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
331 |
fun is_answer (ans as Lineq(k,ty,l,_)) = |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
332 |
case ty of Eq => k <> 0 | Le => k > 0 | Lt => k >= 0; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
333 |
|
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24112
diff
changeset
|
334 |
fun calc_blowup l = |
17496 | 335 |
let val (p,n) = List.partition (curry (op <) 0) (List.filter (curry (op <>) 0) l) |
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24112
diff
changeset
|
336 |
in length p * length n end; |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
337 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
338 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
339 |
(* Main elimination code: *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
340 |
(* *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
341 |
(* (1) Looks for immediate solutions (false assertions with no variables). *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
342 |
(* *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
343 |
(* (2) If there are any equations, picks a variable with the lowest absolute *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
344 |
(* coefficient in any of them, and uses it to eliminate. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
345 |
(* *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
346 |
(* (3) Otherwise, chooses a variable in the inequality to minimize the *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
347 |
(* blowup (number of consequences generated) and eliminates it. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
348 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
349 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
350 |
fun allpairs f xs ys = |
26945 | 351 |
maps (fn x => map (fn y => f x y) ys) xs; |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
352 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
353 |
fun extract_first p = |
15531 | 354 |
let fun extract xs (y::ys) = if p y then (SOME y,xs@ys) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
355 |
else extract (y::xs) ys |
15531 | 356 |
| extract xs [] = (NONE,xs) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
357 |
in extract [] end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
358 |
|
6056 | 359 |
fun print_ineqs ineqs = |
9073 | 360 |
if !trace then |
12262 | 361 |
tracing(cat_lines(""::map (fn Lineq(c,t,l,_) => |
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24112
diff
changeset
|
362 |
string_of_int c ^ |
9073 | 363 |
(case t of Eq => " = " | Lt=> " < " | Le => " <= ") ^ |
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24112
diff
changeset
|
364 |
commas(map string_of_int l)) ineqs)) |
9073 | 365 |
else (); |
6056 | 366 |
|
13498 | 367 |
type history = (int * lineq list) list; |
368 |
datatype result = Success of injust | Failure of history; |
|
369 |
||
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
370 |
fun elim (ineqs, hist) = |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
371 |
let val dummy = print_ineqs ineqs |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
372 |
val (triv, nontriv) = List.partition is_trivial ineqs in |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
373 |
if not (null triv) |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
374 |
then case Library.find_first is_answer triv of |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
375 |
NONE => elim (nontriv, hist) |
15531 | 376 |
| SOME(Lineq(_,_,_,j)) => Success j |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
377 |
else |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
378 |
if null nontriv then Failure hist |
13498 | 379 |
else |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
380 |
let val (eqs, noneqs) = List.partition (fn (Lineq(_,ty,_,_)) => ty=Eq) nontriv in |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
381 |
if not (null eqs) then |
15570 | 382 |
let val clist = Library.foldl (fn (cs,Lineq(_,_,l,_)) => l union cs) ([],eqs) |
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24112
diff
changeset
|
383 |
val sclist = sort (fn (x,y) => int_ord (abs x, abs y)) |
15570 | 384 |
(List.filter (fn i => i<>0) clist) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
385 |
val c = hd sclist |
15531 | 386 |
val (SOME(eq as Lineq(_,_,ceq,_)),othereqs) = |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
387 |
extract_first (fn Lineq(_,_,l,_) => c mem l) eqs |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
388 |
val v = find_index_eq c ceq |
23063 | 389 |
val (ioth,roth) = List.partition (fn (Lineq(_,_,l,_)) => nth l v = 0) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
390 |
(othereqs @ noneqs) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
391 |
val others = map (elim_var v eq) roth @ ioth |
13498 | 392 |
in elim(others,(v,nontriv)::hist) end |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
393 |
else |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
394 |
let val lists = map (fn (Lineq(_,_,l,_)) => l) noneqs |
23063 | 395 |
val numlist = 0 upto (length (hd lists) - 1) |
396 |
val coeffs = map (fn i => map (fn xs => nth xs i) lists) numlist |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
397 |
val blows = map calc_blowup coeffs |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
398 |
val iblows = blows ~~ numlist |
23063 | 399 |
val nziblows = filter_out (fn (i, _) => i = 0) iblows |
13498 | 400 |
in if null nziblows then Failure((~1,nontriv)::hist) |
401 |
else |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
402 |
let val (c,v) = hd(sort (fn (x,y) => int_ord(fst(x),fst(y))) nziblows) |
23063 | 403 |
val (no,yes) = List.partition (fn (Lineq(_,_,l,_)) => nth l v = 0) ineqs |
404 |
val (pos,neg) = List.partition(fn (Lineq(_,_,l,_)) => nth l v > 0) yes |
|
13498 | 405 |
in elim(no @ allpairs (elim_var v) pos neg, (v,nontriv)::hist) end |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
406 |
end |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
407 |
end |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
408 |
end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
409 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
410 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
411 |
(* Translate back a proof. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
412 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
413 |
|
24076 | 414 |
fun trace_thm msg th = |
415 |
(if !trace then (tracing msg; tracing (Display.string_of_thm th)) else (); th); |
|
9073 | 416 |
|
24076 | 417 |
fun trace_term ctxt msg t = |
24920 | 418 |
(if !trace then tracing (cat_lines [msg, Syntax.string_of_term ctxt t]) else (); t) |
24076 | 419 |
|
420 |
fun trace_msg msg = |
|
421 |
if !trace then tracing msg else (); |
|
9073 | 422 |
|
27020
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
423 |
val warning_count = ref 0; |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
424 |
val warning_count_max = 10; |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
425 |
|
26835
404550067389
Lookup and union operations on terms are now modulo eta conversion.
berghofe
parents:
24920
diff
changeset
|
426 |
val union_term = curry (gen_union Pattern.aeconv); |
404550067389
Lookup and union operations on terms are now modulo eta conversion.
berghofe
parents:
24920
diff
changeset
|
427 |
val union_bterm = curry (gen_union |
404550067389
Lookup and union operations on terms are now modulo eta conversion.
berghofe
parents:
24920
diff
changeset
|
428 |
(fn ((b:bool, t), (b', t')) => b = b' andalso Pattern.aeconv (t, t'))); |
404550067389
Lookup and union operations on terms are now modulo eta conversion.
berghofe
parents:
24920
diff
changeset
|
429 |
|
13498 | 430 |
(* FIXME OPTIMIZE!!!! (partly done already) |
6056 | 431 |
Addition/Multiplication need i*t representation rather than t+t+... |
10691 | 432 |
Get rid of Mulitplied(2). For Nat LA_Data.number_of should return Suc^n |
433 |
because Numerals are not known early enough. |
|
6056 | 434 |
|
435 |
Simplification may detect a contradiction 'prematurely' due to type |
|
436 |
information: n+1 <= 0 is simplified to False and does not need to be crossed |
|
437 |
with 0 <= n. |
|
438 |
*) |
|
439 |
local |
|
24076 | 440 |
exception FalseE of thm |
6056 | 441 |
in |
27020
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
442 |
|
24076 | 443 |
fun mkthm ss asms (just: injust) = |
444 |
let |
|
445 |
val ctxt = Simplifier.the_context ss; |
|
446 |
val thy = ProofContext.theory_of ctxt; |
|
447 |
val {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset, ...} = get_data ctxt; |
|
448 |
val simpset' = Simplifier.inherit_context ss simpset; |
|
449 |
val atoms = Library.foldl (fn (ats, (lhs,_,_,rhs,_,_)) => |
|
26835
404550067389
Lookup and union operations on terms are now modulo eta conversion.
berghofe
parents:
24920
diff
changeset
|
450 |
union_term (map fst lhs) (union_term (map fst rhs) ats)) |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
451 |
([], List.mapPartial (fn thm => if Thm.no_prems thm |
24076 | 452 |
then LA_Data.decomp ctxt (Thm.concl_of thm) |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
453 |
else NONE) asms) |
6056 | 454 |
|
10575 | 455 |
fun add2 thm1 thm2 = |
6102 | 456 |
let val conj = thm1 RS (thm2 RS LA_Logic.conjI) |
15531 | 457 |
in get_first (fn th => SOME(conj RS th) handle THM _ => NONE) add_mono_thms |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
458 |
end; |
15531 | 459 |
fun try_add [] _ = NONE |
10575 | 460 |
| try_add (thm1::thm1s) thm2 = case add2 thm1 thm2 of |
15531 | 461 |
NONE => try_add thm1s thm2 | some => some; |
10575 | 462 |
|
463 |
fun addthms thm1 thm2 = |
|
464 |
case add2 thm1 thm2 of |
|
15531 | 465 |
NONE => (case try_add ([thm1] RL inj_thms) thm2 of |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
466 |
NONE => ( the (try_add ([thm2] RL inj_thms) thm1) |
15660 | 467 |
handle Option => |
14360 | 468 |
(trace_thm "" thm1; trace_thm "" thm2; |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
469 |
sys_error "Lin.arith. failed to add thms") |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
470 |
) |
15531 | 471 |
| SOME thm => thm) |
472 |
| SOME thm => thm; |
|
10575 | 473 |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
474 |
fun multn(n,thm) = |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
475 |
let fun mul(i,th) = if i=1 then th else mul(i-1, addthms thm th) |
6102 | 476 |
in if n < 0 then mul(~n,thm) RS LA_Logic.sym else mul(n,thm) end; |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
477 |
|
15184 | 478 |
fun multn2(n,thm) = |
15531 | 479 |
let val SOME(mth) = |
480 |
get_first (fn th => SOME(thm RS th) handle THM _ => NONE) mult_mono_thms |
|
22596 | 481 |
fun cvar(th,_ $ (_ $ _ $ var)) = cterm_of (Thm.theory_of_thm th) var; |
15184 | 482 |
val cv = cvar(mth, hd(prems_of mth)); |
24076 | 483 |
val ct = cterm_of thy (LA_Data.number_of(n,#T(rep_cterm cv))) |
15184 | 484 |
in instantiate ([],[(cv,ct)]) mth end |
10691 | 485 |
|
6056 | 486 |
fun simp thm = |
17515 | 487 |
let val thm' = trace_thm "Simplified:" (full_simplify simpset' thm) |
6102 | 488 |
in if LA_Logic.is_False thm' then raise FalseE thm' else thm' end |
6056 | 489 |
|
24076 | 490 |
fun mk (Asm i) = trace_thm ("Asm " ^ string_of_int i) (nth asms i) |
491 |
| mk (Nat i) = trace_thm ("Nat " ^ string_of_int i) (LA_Logic.mk_nat_thm thy (nth atoms i)) |
|
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
492 |
| mk (LessD j) = trace_thm "L" (hd ([mk j] RL lessD)) |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
493 |
| mk (NotLeD j) = trace_thm "NLe" (mk j RS LA_Logic.not_leD) |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
494 |
| mk (NotLeDD j) = trace_thm "NLeD" (hd ([mk j RS LA_Logic.not_leD] RL lessD)) |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
495 |
| mk (NotLessD j) = trace_thm "NL" (mk j RS LA_Logic.not_lessD) |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
496 |
| mk (Added (j1, j2)) = simp (trace_thm "+" (addthms (mk j1) (mk j2))) |
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24112
diff
changeset
|
497 |
| mk (Multiplied (n, j)) = (trace_msg ("*" ^ string_of_int n); trace_thm "*" (multn (n, mk j))) |
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24112
diff
changeset
|
498 |
| mk (Multiplied2 (n, j)) = simp (trace_msg ("**" ^ string_of_int n); trace_thm "**" (multn2 (n, mk j))) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
499 |
|
27020
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
500 |
in |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
501 |
let |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
502 |
val _ = trace_msg "mkthm"; |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
503 |
val thm = trace_thm "Final thm:" (mk just); |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
504 |
val fls = simplify simpset' thm; |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
505 |
val _ = trace_thm "After simplification:" fls; |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
506 |
val _ = |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
507 |
if LA_Logic.is_False fls then () |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
508 |
else |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
509 |
let val count = CRITICAL (fn () => inc warning_count) in |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
510 |
if count > warning_count_max then () |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
511 |
else |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
512 |
(tracing (cat_lines |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
513 |
(["Assumptions:"] @ map Display.string_of_thm asms @ [""] @ |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
514 |
["Proved:", Display.string_of_thm fls, ""] @ |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
515 |
(if count <> warning_count_max then [] |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
516 |
else ["\n(Reached maximal message count -- disabling future warnings)"]))); |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
517 |
warning "Linear arithmetic should have refuted the assumptions.\n\ |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
518 |
\Please inform Tobias Nipkow (nipkow@in.tum.de).") |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
519 |
end; |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
520 |
in fls end |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
521 |
handle FalseE thm => trace_thm "False reached early:" thm |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
522 |
end; |
b5b8afc9fdcd
added warning_count for issued reconstruction failure messages (limit 10);
wenzelm
parents:
26945
diff
changeset
|
523 |
|
6056 | 524 |
end; |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
525 |
|
23261 | 526 |
fun coeff poly atom = |
26835
404550067389
Lookup and union operations on terms are now modulo eta conversion.
berghofe
parents:
24920
diff
changeset
|
527 |
AList.lookup Pattern.aeconv poly atom |> the_default 0; |
10691 | 528 |
|
529 |
fun integ(rlhs,r,rel,rrhs,s,d) = |
|
17951 | 530 |
let val (rn,rd) = Rat.quotient_of_rat r and (sn,sd) = Rat.quotient_of_rat s |
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24112
diff
changeset
|
531 |
val m = Integer.lcms(map (abs o snd o Rat.quotient_of_rat) (r :: s :: map snd rlhs @ map snd rrhs)) |
22846 | 532 |
fun mult(t,r) = |
17951 | 533 |
let val (i,j) = Rat.quotient_of_rat r |
15965
f422f8283491
Use of IntInf.int instead of int in most numeric simprocs; avoids
paulson
parents:
15922
diff
changeset
|
534 |
in (t,i * (m div j)) end |
12932 | 535 |
in (m,(map mult rlhs, rn*(m div rd), rel, map mult rrhs, sn*(m div sd), d)) end |
10691 | 536 |
|
13498 | 537 |
fun mklineq n atoms = |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
538 |
fn (item, k) => |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
539 |
let val (m, (lhs,i,rel,rhs,j,discrete)) = integ item |
13498 | 540 |
val lhsa = map (coeff lhs) atoms |
541 |
and rhsa = map (coeff rhs) atoms |
|
18330 | 542 |
val diff = map2 (curry (op -)) rhsa lhsa |
13498 | 543 |
val c = i-j |
544 |
val just = Asm k |
|
545 |
fun lineq(c,le,cs,j) = Lineq(c,le,cs, if m=1 then j else Multiplied2(m,j)) |
|
546 |
in case rel of |
|
547 |
"<=" => lineq(c,Le,diff,just) |
|
548 |
| "~<=" => if discrete |
|
549 |
then lineq(1-c,Le,map (op ~) diff,NotLeDD(just)) |
|
550 |
else lineq(~c,Lt,map (op ~) diff,NotLeD(just)) |
|
551 |
| "<" => if discrete |
|
552 |
then lineq(c+1,Le,diff,LessD(just)) |
|
553 |
else lineq(c,Lt,diff,just) |
|
554 |
| "~<" => lineq(~c,Le,map (op~) diff,NotLessD(just)) |
|
555 |
| "=" => lineq(c,Eq,diff,just) |
|
22846 | 556 |
| _ => sys_error("mklineq" ^ rel) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
557 |
end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
558 |
|
13498 | 559 |
(* ------------------------------------------------------------------------- *) |
560 |
(* Print (counter) example *) |
|
561 |
(* ------------------------------------------------------------------------- *) |
|
562 |
||
563 |
fun print_atom((a,d),r) = |
|
17951 | 564 |
let val (p,q) = Rat.quotient_of_rat r |
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24112
diff
changeset
|
565 |
val s = if d then string_of_int p else |
13498 | 566 |
if p = 0 then "0" |
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24112
diff
changeset
|
567 |
else string_of_int p ^ "/" ^ string_of_int q |
13498 | 568 |
in a ^ " = " ^ s end; |
569 |
||
19049 | 570 |
fun produce_ex sds = |
17496 | 571 |
curry (op ~~) sds |
572 |
#> map print_atom |
|
573 |
#> commas |
|
23197 | 574 |
#> curry (op ^) "Counterexample (possibly spurious):\n"; |
13498 | 575 |
|
24076 | 576 |
fun trace_ex ctxt params atoms discr n (hist: history) = |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
577 |
case hist of |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
578 |
[] => () |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
579 |
| (v, lineqs) :: hist' => |
24076 | 580 |
let |
581 |
val frees = map Free params |
|
24920 | 582 |
fun show_term t = Syntax.string_of_term ctxt (subst_bounds (frees, t)) |
24076 | 583 |
val start = |
584 |
if v = ~1 then (hist', findex0 discr n lineqs) |
|
22950 | 585 |
else (hist, replicate n Rat.zero) |
24076 | 586 |
val ex = SOME (produce_ex (map show_term atoms ~~ discr) |
587 |
(uncurry (fold (findex1 discr)) start)) |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
588 |
handle NoEx => NONE |
24076 | 589 |
in |
590 |
case ex of |
|
591 |
SOME s => (warning "arith failed - see trace for a counterexample"; tracing s) |
|
592 |
| NONE => warning "arith failed" |
|
593 |
end; |
|
13498 | 594 |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
595 |
(* ------------------------------------------------------------------------- *) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
596 |
|
20268 | 597 |
fun mknat (pTs : typ list) (ixs : int list) (atom : term, i : int) : lineq option = |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
598 |
if LA_Logic.is_nat (pTs, atom) |
6056 | 599 |
then let val l = map (fn j => if j=i then 1 else 0) ixs |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
600 |
in SOME (Lineq (0, Le, l, Nat i)) end |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
601 |
else NONE; |
6056 | 602 |
|
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
603 |
(* This code is tricky. It takes a list of premises in the order they occur |
15531 | 604 |
in the subgoal. Numerical premises are coded as SOME(tuple), non-numerical |
605 |
ones as NONE. Going through the premises, each numeric one is converted into |
|
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
606 |
a Lineq. The tricky bit is to convert ~= which is split into two cases < and |
13498 | 607 |
>. Thus split_items returns a list of equation systems. This may blow up if |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
608 |
there are many ~=, but in practice it does not seem to happen. The really |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
609 |
tricky bit is to arrange the order of the cases such that they coincide with |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
610 |
the order in which the cases are in the end generated by the tactic that |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
611 |
applies the generated refutation thms (see function 'refute_tac'). |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
612 |
|
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
613 |
For variables n of type nat, a constraint 0 <= n is added. |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
614 |
*) |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
615 |
|
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
616 |
(* FIXME: To optimize, the splitting of cases and the search for refutations *) |
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
617 |
(* could be intertwined: separate the first (fully split) case, *) |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
618 |
(* refute it, continue with splitting and refuting. Terminate with *) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
619 |
(* failure as soon as a case could not be refuted; i.e. delay further *) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
620 |
(* splitting until after a refutation for other cases has been found. *) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
621 |
|
26945 | 622 |
fun split_items ctxt do_pre (Ts, terms) : (typ list * (LA_Data.decomp * int) list) list = |
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
623 |
let |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
624 |
(* splits inequalities '~=' into '<' and '>'; this corresponds to *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
625 |
(* 'REPEAT_DETERM (eresolve_tac neqE i)' at the theorem/tactic *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
626 |
(* level *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
627 |
(* FIXME: this is currently sensitive to the order of theorems in *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
628 |
(* neqE: The theorem for type "nat" must come first. A *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
629 |
(* better (i.e. less likely to break when neqE changes) *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
630 |
(* implementation should *test* which theorem from neqE *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
631 |
(* can be applied, and split the premise accordingly. *) |
26945 | 632 |
fun elim_neq (ineqs : (LA_Data.decomp option * bool) list) : |
633 |
(LA_Data.decomp option * bool) list list = |
|
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
634 |
let |
26945 | 635 |
fun elim_neq' nat_only ([] : (LA_Data.decomp option * bool) list) : |
636 |
(LA_Data.decomp option * bool) list list = |
|
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
637 |
[[]] |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
638 |
| elim_neq' nat_only ((NONE, is_nat) :: ineqs) = |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
639 |
map (cons (NONE, is_nat)) (elim_neq' nat_only ineqs) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
640 |
| elim_neq' nat_only ((ineq as (SOME (l, i, rel, r, j, d), is_nat)) :: ineqs) = |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
641 |
if rel = "~=" andalso (not nat_only orelse is_nat) then |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
642 |
(* [| ?l ~= ?r; ?l < ?r ==> ?R; ?r < ?l ==> ?R |] ==> ?R *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
643 |
elim_neq' nat_only (ineqs @ [(SOME (l, i, "<", r, j, d), is_nat)]) @ |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
644 |
elim_neq' nat_only (ineqs @ [(SOME (r, j, "<", l, i, d), is_nat)]) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
645 |
else |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
646 |
map (cons ineq) (elim_neq' nat_only ineqs) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
647 |
in |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
648 |
ineqs |> elim_neq' true |
26945 | 649 |
|> maps (elim_neq' false) |
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
650 |
end |
13464 | 651 |
|
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
652 |
fun number_hyps _ [] = [] |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
653 |
| number_hyps n (NONE::xs) = number_hyps (n+1) xs |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
654 |
| number_hyps n ((SOME x)::xs) = (x, n) :: number_hyps (n+1) xs |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
655 |
|
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
656 |
val result = (Ts, terms) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
657 |
|> (* user-defined preprocessing of the subgoal *) |
24076 | 658 |
(if do_pre then LA_Data.pre_decomp ctxt else Library.single) |
23195 | 659 |
|> tap (fn subgoals => trace_msg ("Preprocessing yields " ^ |
660 |
string_of_int (length subgoals) ^ " subgoal(s) total.")) |
|
22846 | 661 |
|> (* produce the internal encoding of (in-)equalities *) |
24076 | 662 |
map (apsnd (map (fn t => (LA_Data.decomp ctxt t, LA_Data.domain_is_nat t)))) |
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
663 |
|> (* splitting of inequalities *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
664 |
map (apsnd elim_neq) |
22846 | 665 |
|> maps (fn (Ts, subgoals) => map (pair Ts o map fst) subgoals) |
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
666 |
|> (* numbering of hypotheses, ignoring irrelevant ones *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
667 |
map (apsnd (number_hyps 0)) |
23195 | 668 |
in |
669 |
trace_msg ("Splitting of inequalities yields " ^ |
|
670 |
string_of_int (length result) ^ " subgoal(s) total."); |
|
671 |
result |
|
672 |
end; |
|
13464 | 673 |
|
26945 | 674 |
fun add_atoms (ats : term list, ((lhs,_,_,rhs,_,_) : LA_Data.decomp, _)) : term list = |
26835
404550067389
Lookup and union operations on terms are now modulo eta conversion.
berghofe
parents:
24920
diff
changeset
|
675 |
union_term (map fst lhs) (union_term (map fst rhs) ats); |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
676 |
|
26945 | 677 |
fun add_datoms (dats : (bool * term) list, ((lhs,_,_,rhs,_,d) : LA_Data.decomp, _)) : |
20268 | 678 |
(bool * term) list = |
26835
404550067389
Lookup and union operations on terms are now modulo eta conversion.
berghofe
parents:
24920
diff
changeset
|
679 |
union_bterm (map (pair d o fst) lhs) (union_bterm (map (pair d o fst) rhs) dats); |
13498 | 680 |
|
26945 | 681 |
fun discr (initems : (LA_Data.decomp * int) list) : bool list = |
20268 | 682 |
map fst (Library.foldl add_datoms ([],initems)); |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
683 |
|
24076 | 684 |
fun refutes ctxt params show_ex : |
26945 | 685 |
(typ list * (LA_Data.decomp * int) list) list -> injust list -> injust list option = |
686 |
let |
|
687 |
fun refute ((Ts, initems : (LA_Data.decomp * int) list) :: initemss) (js: injust list) = |
|
688 |
let |
|
689 |
val atoms = Library.foldl add_atoms ([], initems) |
|
690 |
val n = length atoms |
|
691 |
val mkleq = mklineq n atoms |
|
692 |
val ixs = 0 upto (n - 1) |
|
693 |
val iatoms = atoms ~~ ixs |
|
694 |
val natlineqs = List.mapPartial (mknat Ts ixs) iatoms |
|
695 |
val ineqs = map mkleq initems @ natlineqs |
|
696 |
in case elim (ineqs, []) of |
|
697 |
Success j => |
|
698 |
(trace_msg ("Contradiction! (" ^ string_of_int (length js + 1) ^ ")"); |
|
699 |
refute initemss (js @ [j])) |
|
700 |
| Failure hist => |
|
701 |
(if not show_ex then () |
|
702 |
else |
|
703 |
let |
|
704 |
val (param_names, ctxt') = ctxt |> Variable.variant_fixes (map fst params) |
|
705 |
val (more_names, ctxt'') = ctxt' |> Variable.variant_fixes |
|
706 |
(Name.invents (Variable.names_of ctxt') Name.uu (length Ts - length params)) |
|
707 |
val params' = (more_names @ param_names) ~~ Ts |
|
708 |
in |
|
709 |
trace_ex ctxt'' params' atoms (discr initems) n hist |
|
710 |
end; NONE) |
|
711 |
end |
|
712 |
| refute [] js = SOME js |
|
713 |
in refute end; |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
714 |
|
24076 | 715 |
fun refute ctxt params show_ex do_pre terms : injust list option = |
716 |
refutes ctxt params show_ex (split_items ctxt do_pre (map snd params, terms)) []; |
|
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
717 |
|
22950 | 718 |
fun count P xs = length (filter P xs); |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
719 |
|
26945 | 720 |
fun prove ctxt params show_ex do_pre Hs concl : injust list option = |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
721 |
let |
23190 | 722 |
val _ = trace_msg "prove:" |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
723 |
(* append the negated conclusion to 'Hs' -- this corresponds to *) |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
724 |
(* 'DETERM (resolve_tac [LA_Logic.notI, LA_Logic.ccontr] i)' at the *) |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
725 |
(* theorem/tactic level *) |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
726 |
val Hs' = Hs @ [LA_Logic.neg_prop concl] |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
727 |
fun is_neq NONE = false |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
728 |
| is_neq (SOME (_,_,r,_,_,_)) = (r = "~=") |
24112 | 729 |
val neq_limit = Config.get ctxt LA_Data.fast_arith_neq_limit; |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
730 |
in |
24076 | 731 |
if count is_neq (map (LA_Data.decomp ctxt) Hs') > neq_limit then |
732 |
(trace_msg ("fast_arith_neq_limit exceeded (current value is " ^ |
|
733 |
string_of_int neq_limit ^ ")"); NONE) |
|
734 |
else |
|
735 |
refute ctxt params show_ex do_pre Hs' |
|
23190 | 736 |
end handle TERM ("neg_prop", _) => |
737 |
(* since no meta-logic negation is available, we can only fail if *) |
|
738 |
(* the conclusion is not of the form 'Trueprop $ _' (simply *) |
|
739 |
(* dropping the conclusion doesn't work either, because even *) |
|
740 |
(* 'False' does not imply arbitrary 'concl::prop') *) |
|
741 |
(trace_msg "prove failed (cannot negate conclusion)."; NONE); |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
742 |
|
22846 | 743 |
fun refute_tac ss (i, justs) = |
6074 | 744 |
fn state => |
24076 | 745 |
let |
746 |
val ctxt = Simplifier.the_context ss; |
|
747 |
val _ = trace_thm ("refute_tac (on subgoal " ^ string_of_int i ^ ", with " ^ |
|
748 |
string_of_int (length justs) ^ " justification(s)):") state |
|
749 |
val {neqE, ...} = get_data ctxt; |
|
750 |
fun just1 j = |
|
751 |
(* eliminate inequalities *) |
|
752 |
REPEAT_DETERM (eresolve_tac neqE i) THEN |
|
753 |
PRIMITIVE (trace_thm "State after neqE:") THEN |
|
754 |
(* use theorems generated from the actual justifications *) |
|
755 |
METAHYPS (fn asms => rtac (mkthm ss asms j) 1) i |
|
756 |
in |
|
757 |
(* rewrite "[| A1; ...; An |] ==> B" to "[| A1; ...; An; ~B |] ==> False" *) |
|
758 |
DETERM (resolve_tac [LA_Logic.notI, LA_Logic.ccontr] i) THEN |
|
759 |
(* user-defined preprocessing of the subgoal *) |
|
760 |
DETERM (LA_Data.pre_tac ctxt i) THEN |
|
761 |
PRIMITIVE (trace_thm "State after pre_tac:") THEN |
|
762 |
(* prove every resulting subgoal, using its justification *) |
|
763 |
EVERY (map just1 justs) |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
764 |
end state; |
6074 | 765 |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
766 |
(* |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
767 |
Fast but very incomplete decider. Only premises and conclusions |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
768 |
that are already (negated) (in)equations are taken into account. |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
769 |
*) |
24076 | 770 |
fun simpset_lin_arith_tac ss show_ex = SUBGOAL (fn (A, i) => |
771 |
let |
|
772 |
val ctxt = Simplifier.the_context ss |
|
773 |
val params = rev (Logic.strip_params A) |
|
774 |
val Hs = Logic.strip_assums_hyp A |
|
775 |
val concl = Logic.strip_assums_concl A |
|
776 |
val _ = trace_term ctxt ("Trying to refute subgoal " ^ string_of_int i) A |
|
777 |
in |
|
778 |
case prove ctxt params show_ex true Hs concl of |
|
779 |
NONE => (trace_msg "Refutation failed."; no_tac) |
|
780 |
| SOME js => (trace_msg "Refutation succeeded."; refute_tac ss (i, js)) |
|
781 |
end); |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
782 |
|
24076 | 783 |
fun cut_lin_arith_tac ss = |
784 |
cut_facts_tac (Simplifier.prems_of_ss ss) THEN' |
|
785 |
simpset_lin_arith_tac ss false; |
|
17613 | 786 |
|
24076 | 787 |
fun lin_arith_tac ctxt = |
788 |
simpset_lin_arith_tac (Simplifier.context ctxt Simplifier.empty_ss); |
|
789 |
||
790 |
||
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
791 |
|
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
792 |
(** Forward proof from theorems **) |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
793 |
|
20433
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
794 |
(* More tricky code. Needs to arrange the proofs of the multiple cases (due |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
795 |
to splits of ~= premises) such that it coincides with the order of the cases |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
796 |
generated by function split_items. *) |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
797 |
|
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
798 |
datatype splittree = Tip of thm list |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
799 |
| Spl of thm * cterm * splittree * cterm * splittree; |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
800 |
|
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
801 |
(* "(ct1 ==> ?R) ==> (ct2 ==> ?R) ==> ?R" is taken to (ct1, ct2) *) |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
802 |
|
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
803 |
fun extract (imp : cterm) : cterm * cterm = |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
804 |
let val (Il, r) = Thm.dest_comb imp |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
805 |
val (_, imp1) = Thm.dest_comb Il |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
806 |
val (Ict1, _) = Thm.dest_comb imp1 |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
807 |
val (_, ct1) = Thm.dest_comb Ict1 |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
808 |
val (Ir, _) = Thm.dest_comb r |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
809 |
val (_, Ict2r) = Thm.dest_comb Ir |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
810 |
val (Ict2, _) = Thm.dest_comb Ict2r |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
811 |
val (_, ct2) = Thm.dest_comb Ict2 |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
812 |
in (ct1, ct2) end; |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
813 |
|
24076 | 814 |
fun splitasms ctxt (asms : thm list) : splittree = |
815 |
let val {neqE, ...} = get_data ctxt |
|
20433
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
816 |
fun elim_neq (asms', []) = Tip (rev asms') |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
817 |
| elim_neq (asms', asm::asms) = |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
818 |
(case get_first (fn th => SOME (asm COMP th) handle THM _ => NONE) neqE of |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
819 |
SOME spl => |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
820 |
let val (ct1, ct2) = extract (cprop_of spl) |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
821 |
val thm1 = assume ct1 |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
822 |
val thm2 = assume ct2 |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
823 |
in Spl (spl, ct1, elim_neq (asms', asms@[thm1]), ct2, elim_neq (asms', asms@[thm2])) |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
824 |
end |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
825 |
| NONE => elim_neq (asm::asms', asms)) |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
826 |
in elim_neq ([], asms) end; |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
827 |
|
24076 | 828 |
fun fwdproof ss (Tip asms : splittree) (j::js : injust list) = (mkthm ss asms j, js) |
829 |
| fwdproof ss (Spl (thm, ct1, tree1, ct2, tree2)) js = |
|
830 |
let |
|
831 |
val (thm1, js1) = fwdproof ss tree1 js |
|
832 |
val (thm2, js2) = fwdproof ss tree2 js1 |
|
20433
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
833 |
val thm1' = implies_intr ct1 thm1 |
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
834 |
val thm2' = implies_intr ct2 thm2 |
24076 | 835 |
in (thm2' COMP (thm1' COMP thm), js2) end; |
836 |
(* FIXME needs handle THM _ => NONE ? *) |
|
20433
55471f940e5c
lin_arith_prover: splitting reverted because of performance loss
webertj
parents:
20280
diff
changeset
|
837 |
|
24076 | 838 |
fun prover ss thms Tconcl (js : injust list) pos : thm option = |
839 |
let |
|
840 |
val ctxt = Simplifier.the_context ss |
|
841 |
val thy = ProofContext.theory_of ctxt |
|
842 |
val nTconcl = LA_Logic.neg_prop Tconcl |
|
843 |
val cnTconcl = cterm_of thy nTconcl |
|
844 |
val nTconclthm = assume cnTconcl |
|
845 |
val tree = splitasms ctxt (thms @ [nTconclthm]) |
|
846 |
val (Falsethm, _) = fwdproof ss tree js |
|
847 |
val contr = if pos then LA_Logic.ccontr else LA_Logic.notI |
|
848 |
val concl = implies_intr cnTconcl Falsethm COMP contr |
|
849 |
in SOME (trace_thm "Proved by lin. arith. prover:" (LA_Logic.mk_Eq concl)) end |
|
850 |
(*in case concl contains ?-var, which makes assume fail:*) (* FIXME Variable.import_terms *) |
|
851 |
handle THM _ => NONE; |
|
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
852 |
|
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
853 |
(* PRE: concl is not negated! |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
854 |
This assumption is OK because |
24076 | 855 |
1. lin_arith_simproc tries both to prove and disprove concl and |
856 |
2. lin_arith_simproc is applied by the Simplifier which |
|
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
857 |
dives into terms and will thus try the non-negated concl anyway. |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
858 |
*) |
24076 | 859 |
fun lin_arith_simproc ss concl = |
860 |
let |
|
861 |
val ctxt = Simplifier.the_context ss |
|
26945 | 862 |
val thms = maps LA_Logic.atomize (Simplifier.prems_of_ss ss) |
24076 | 863 |
val Hs = map Thm.prop_of thms |
6102 | 864 |
val Tconcl = LA_Logic.mk_Trueprop concl |
24076 | 865 |
in |
866 |
case prove ctxt [] false false Hs Tconcl of (* concl provable? *) |
|
867 |
SOME js => prover ss thms Tconcl js true |
|
868 |
| NONE => |
|
869 |
let val nTconcl = LA_Logic.neg_prop Tconcl in |
|
870 |
case prove ctxt [] false false Hs nTconcl of (* ~concl provable? *) |
|
871 |
SOME js => prover ss thms nTconcl js false |
|
872 |
| NONE => NONE |
|
873 |
end |
|
874 |
end; |
|
6074 | 875 |
|
876 |
end; |