author | webertj |
Wed, 02 Aug 2006 03:33:28 +0200 | |
changeset 20280 | ad9fbbd01535 |
parent 20276 | d94dc40673b1 |
child 20433 | 55471f940e5c |
permissions | -rw-r--r-- |
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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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A generic linear arithmetic package. |
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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 |
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and a simplification procedure |
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lin_arith_prover: theory -> simpset -> term -> thm option |
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Only take premises and conclusions into account that are already (negated) |
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(in)equations. lin_arith_prover tries to prove or disprove the term. |
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*) |
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(* Debugging: set Fast_Arith.trace *) |
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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 |
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neg is the (logically) negated version of t, where the negation |
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of a negative term is the term itself (no double negation!); |
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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 decompT = (term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool |
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val decomp: theory -> term -> decompT option |
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val domain_is_nat : term -> bool |
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(* preprocessing, performed on a representation of subgoals as list of premises: *) |
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val pre_decomp: theory -> 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 : int -> Tactical.tactic |
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val number_of : IntInf.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, 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. |
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*) |
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signature FAST_LIN_ARITH = |
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sig |
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val setup: theory -> theory |
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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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-> theory -> theory |
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val trace: bool ref |
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val fast_arith_neq_limit: int ref |
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val lin_arith_prover: theory -> simpset -> term -> thm option |
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val lin_arith_tac: bool -> int -> tactic |
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val cut_lin_arith_tac: simpset -> int -> tactic |
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end; |
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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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(** theory data **) |
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(* data kind 'Provers/fast_lin_arith' *) |
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structure Data = TheoryDataFun |
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(struct |
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val name = "Provers/fast_lin_arith"; |
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type T = {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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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 copy = I; |
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val extend = I; |
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|
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fun merge _ |
123 |
({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 = Drule.merge_rules (add_mono_thms1, add_mono_thms2), |
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mult_mono_thms = Drule.merge_rules (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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neqE = Drule.merge_rules (neqE1, neqE2), |
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simpset = Simplifier.merge_ss (simpset1, simpset2)}; |
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fun print _ _ = (); |
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end); |
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val map_data = Data.map; |
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val setup = Data.init; |
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(*** A fast decision procedure ***) |
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(*** Code ported from HOL Light ***) |
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(* possible optimizations: |
145 |
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; |
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datatype lineq_type = Eq | Le | Lt; |
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datatype injust = Asm of int |
154 |
| 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 IntInf.int * injust |
160 |
| Multiplied2 of IntInf.int * injust |
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| Added of injust * injust; |
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datatype lineq = Lineq of IntInf.int * lineq_type * IntInf.int list * injust; |
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fun el 0 (h::_) = h |
166 |
| el n (_::t) = el (n - 1) t |
|
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| el _ _ = sys_error "el"; |
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169 |
(* ------------------------------------------------------------------------- *) |
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170 |
(* Finding a (counter) example from the trace of a failed elimination *) |
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(* ------------------------------------------------------------------------- *) |
|
172 |
(* Examples are represented as rational numbers, *) |
|
173 |
(* Dont blame John Harrison for this code - it is entirely mine. TN *) |
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175 |
exception NoEx; |
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176 |
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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(ineqs,Lineq(i,Le,cs,_)) = (i,true,cs)::ineqs |
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183 |
| elim_eqns(ineqs,Lineq(i,Eq,cs,_)) = (i,true,cs)::(~i,true,map ~ cs)::ineqs |
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| elim_eqns(ineqs,Lineq(i,Lt,cs,_)) = (i,false,cs)::ineqs; |
13498 | 185 |
|
186 |
(* PRE: ex[v] must be 0! *) |
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17951 | 187 |
fun eval (ex:Rat.rat list) v (a:IntInf.int,le,cs:IntInf.int list) = |
188 |
let val rs = map Rat.rat_of_intinf cs |
|
189 |
val rsum = Library.foldl Rat.add (Rat.zero, map Rat.mult (rs ~~ ex)) |
|
190 |
in (Rat.mult (Rat.add(Rat.rat_of_intinf a,Rat.neg rsum), Rat.inv(el v rs)), le) end; |
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(* If el v rs < 0, le should be negated. |
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192 |
Instead this swap is taken into account in ratrelmin2. |
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193 |
*) |
13498 | 194 |
|
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fun ratrelmin2(x as (r,ler),y as (s,les)) = |
17951 | 196 |
if r=s then (r, (not ler) andalso (not les)) else if Rat.le(r,s) then x else y; |
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197 |
fun ratrelmax2(x as (r,ler),y as (s,les)) = |
17951 | 198 |
if r=s then (r,ler andalso les) else if Rat.le(r,s) then y else x; |
13498 | 199 |
|
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200 |
val ratrelmin = foldr1 ratrelmin2; |
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201 |
val ratrelmax = foldr1 ratrelmax2; |
13498 | 202 |
|
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203 |
fun ratexact up (r,exact) = |
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204 |
if exact then r else |
17951 | 205 |
let val (p,q) = Rat.quotient_of_rat r |
206 |
val nth = Rat.inv(Rat.rat_of_intinf q) |
|
207 |
in Rat.add(r,if up then nth else Rat.neg nth) end; |
|
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208 |
|
17951 | 209 |
fun ratmiddle(r,s) = Rat.mult(Rat.add(r,s),Rat.inv(Rat.rat_of_int 2)); |
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210 |
|
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211 |
fun choose2 d ((lb, exactl), (ub, exactu)) = |
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212 |
if Rat.le (lb, Rat.zero) andalso (lb <> Rat.zero orelse exactl) andalso |
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213 |
Rat.le (Rat.zero, ub) andalso (ub <> Rat.zero orelse exactu) |
17951 | 214 |
then Rat.zero else |
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215 |
if not d |
17951 | 216 |
then (if Rat.ge0 lb |
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then if exactl then lb else ratmiddle (lb, ub) |
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218 |
else if exactu then ub else ratmiddle (lb, ub)) |
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219 |
else (* discrete domain, both bounds must be exact *) |
17951 | 220 |
if Rat.ge0 lb then let val lb' = Rat.roundup lb |
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in if Rat.le (lb', ub) then lb' else raise NoEx end |
17951 | 222 |
else let val ub' = Rat.rounddown ub |
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223 |
in if Rat.le (lb, ub') then ub' else raise NoEx end; |
13498 | 224 |
|
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225 |
fun findex1 discr (ex, (v, lineqs)) = |
15570 | 226 |
let val nz = List.filter (fn (Lineq(_,_,cs,_)) => el v cs <> 0) lineqs; |
227 |
val ineqs = Library.foldl elim_eqns ([],nz) |
|
228 |
val (ge,le) = List.partition (fn (_,_,cs) => el v cs > 0) ineqs |
|
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229 |
val lb = ratrelmax (map (eval ex v) ge) |
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230 |
val ub = ratrelmin (map (eval ex v) le) |
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|
231 |
in nth_update (v, choose2 (nth discr v) (lb, ub)) ex end; |
13498 | 232 |
|
15570 | 233 |
fun findex discr = Library.foldl (findex1 discr); |
13498 | 234 |
|
235 |
fun elim1 v x = |
|
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236 |
map (fn (a,le,bs) => (Rat.add (a, Rat.neg (Rat.mult (el v bs, x))), le, |
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237 |
nth_update (v, Rat.zero) bs)); |
13498 | 238 |
|
17951 | 239 |
fun single_var v (_,_,cs) = (filter_out (equal Rat.zero) cs = [el v cs]); |
13498 | 240 |
|
241 |
(* The base case: |
|
242 |
all variables occur only with positive or only with negative coefficients *) |
|
243 |
fun pick_vars discr (ineqs,ex) = |
|
17951 | 244 |
let val nz = filter_out (fn (_,_,cs) => forall (equal Rat.zero) cs) ineqs |
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245 |
in case nz of [] => ex |
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246 |
| (_,_,cs) :: _ => |
17951 | 247 |
let val v = find_index (not o equal Rat.zero) cs |
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|
248 |
val d = nth discr v |
17951 | 249 |
val pos = Rat.ge0(el v cs) |
15570 | 250 |
val sv = List.filter (single_var v) nz |
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|
251 |
val minmax = |
17951 | 252 |
if pos then if d then Rat.roundup o fst o ratrelmax |
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253 |
else ratexact true o ratrelmax |
17951 | 254 |
else if d then Rat.rounddown o fst o ratrelmin |
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255 |
else ratexact false o ratrelmin |
17951 | 256 |
val bnds = map (fn (a,le,bs) => (Rat.mult(a,Rat.inv(el v bs)),le)) sv |
257 |
val x = minmax((Rat.zero,if pos then true else false)::bnds) |
|
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258 |
val ineqs' = elim1 v x nz |
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|
259 |
val ex' = nth_update (v, x) ex |
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|
260 |
in pick_vars discr (ineqs',ex') end |
13498 | 261 |
end; |
262 |
||
263 |
fun findex0 discr n lineqs = |
|
15570 | 264 |
let val ineqs = Library.foldl elim_eqns ([],lineqs) |
17951 | 265 |
val rineqs = map (fn (a,le,cs) => (Rat.rat_of_intinf a, le, map Rat.rat_of_intinf cs)) |
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266 |
ineqs |
17951 | 267 |
in pick_vars discr (rineqs,replicate n Rat.zero) end; |
13498 | 268 |
|
269 |
(* ------------------------------------------------------------------------- *) |
|
270 |
(* End of counter example finder. The actual decision procedure starts here. *) |
|
271 |
(* ------------------------------------------------------------------------- *) |
|
272 |
||
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273 |
(* ------------------------------------------------------------------------- *) |
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|
274 |
(* Calculate new (in)equality type after addition. *) |
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275 |
(* ------------------------------------------------------------------------- *) |
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276 |
|
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277 |
fun find_add_type(Eq,x) = x |
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|
278 |
| find_add_type(x,Eq) = x |
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|
279 |
| find_add_type(_,Lt) = Lt |
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|
280 |
| find_add_type(Lt,_) = Lt |
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|
281 |
| find_add_type(Le,Le) = Le; |
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|
282 |
|
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|
283 |
(* ------------------------------------------------------------------------- *) |
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|
284 |
(* Multiply out an (in)equation. *) |
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|
285 |
(* ------------------------------------------------------------------------- *) |
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|
286 |
|
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|
287 |
fun multiply_ineq n (i as Lineq(k,ty,l,just)) = |
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|
288 |
if n = 1 then i |
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|
289 |
else if n = 0 andalso ty = Lt then sys_error "multiply_ineq" |
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|
290 |
else if n < 0 andalso (ty=Le orelse ty=Lt) then sys_error "multiply_ineq" |
17524 | 291 |
else Lineq (n * k, ty, map (curry op* n) l, Multiplied (n, just)); |
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|
292 |
|
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|
293 |
(* ------------------------------------------------------------------------- *) |
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|
294 |
(* Add together (in)equations. *) |
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|
295 |
(* ------------------------------------------------------------------------- *) |
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|
296 |
|
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|
297 |
fun add_ineq (i1 as Lineq(k1,ty1,l1,just1)) (i2 as Lineq(k2,ty2,l2,just2)) = |
18330 | 298 |
let val l = map2 (curry (op +)) l1 l2 |
5982
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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|
299 |
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.
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|
300 |
|
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|
301 |
(* ------------------------------------------------------------------------- *) |
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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|
302 |
(* Elimination of variable between a single pair of (in)equations. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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|
303 |
(* If they're both inequalities, 1st coefficient must be +ve, 2nd -ve. *) |
aeb97860d352
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|
304 |
(* ------------------------------------------------------------------------- *) |
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|
305 |
|
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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|
306 |
fun elim_var v (i1 as Lineq(k1,ty1,l1,just1)) (i2 as Lineq(k2,ty2,l2,just2)) = |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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changeset
|
307 |
let val c1 = el v l1 and c2 = el v l2 |
16358 | 308 |
val m = lcm(abs c1, abs c2) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
309 |
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.
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changeset
|
310 |
val (n1,n2) = |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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parents:
diff
changeset
|
311 |
if (c1 >= 0) = (c2 >= 0) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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diff
changeset
|
312 |
then if ty1 = Eq then (~m1,m2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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diff
changeset
|
313 |
else if ty2 = Eq then (m1,~m2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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diff
changeset
|
314 |
else sys_error "elim_var" |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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diff
changeset
|
315 |
else (m1,m2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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parents:
diff
changeset
|
316 |
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.
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diff
changeset
|
317 |
then (~n1,~n2) else (n1,n2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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parents:
diff
changeset
|
318 |
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.
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changeset
|
319 |
|
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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|
320 |
(* ------------------------------------------------------------------------- *) |
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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changeset
|
321 |
(* The main refutation-finding code. *) |
aeb97860d352
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|
322 |
(* ------------------------------------------------------------------------- *) |
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|
323 |
|
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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|
324 |
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.
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|
325 |
|
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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|
326 |
fun is_answer (ans as Lineq(k,ty,l,_)) = |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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parents:
diff
changeset
|
327 |
case ty of Eq => k <> 0 | Le => k > 0 | Lt => k >= 0; |
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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changeset
|
328 |
|
16358 | 329 |
fun calc_blowup (l:IntInf.int list) = |
17496 | 330 |
let val (p,n) = List.partition (curry (op <) 0) (List.filter (curry (op <>) 0) l) |
5982
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|
331 |
in (length p) * (length n) end; |
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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changeset
|
332 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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changeset
|
333 |
(* ------------------------------------------------------------------------- *) |
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changeset
|
334 |
(* Main elimination code: *) |
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|
335 |
(* *) |
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Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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|
336 |
(* (1) Looks for immediate solutions (false assertions with no variables). *) |
aeb97860d352
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|
337 |
(* *) |
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changeset
|
338 |
(* (2) If there are any equations, picks a variable with the lowest absolute *) |
aeb97860d352
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|
339 |
(* coefficient in any of them, and uses it to eliminate. *) |
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|
340 |
(* *) |
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|
341 |
(* (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.
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changeset
|
342 |
(* blowup (number of consequences generated) and eliminates it. *) |
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|
343 |
(* ------------------------------------------------------------------------- *) |
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changeset
|
344 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
345 |
fun allpairs f xs ys = |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
346 |
List.concat (map (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
|
347 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
348 |
fun extract_first p = |
15531 | 349 |
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
|
350 |
else extract (y::xs) ys |
15531 | 351 |
| extract xs [] = (NONE,xs) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
352 |
in extract [] end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
353 |
|
6056 | 354 |
fun print_ineqs ineqs = |
9073 | 355 |
if !trace then |
12262 | 356 |
tracing(cat_lines(""::map (fn Lineq(c,t,l,_) => |
16358 | 357 |
IntInf.toString c ^ |
9073 | 358 |
(case t of Eq => " = " | Lt=> " < " | Le => " <= ") ^ |
16358 | 359 |
commas(map IntInf.toString l)) ineqs)) |
9073 | 360 |
else (); |
6056 | 361 |
|
13498 | 362 |
type history = (int * lineq list) list; |
363 |
datatype result = Success of injust | Failure of history; |
|
364 |
||
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
365 |
fun elim (ineqs, hist) = |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
366 |
let val dummy = print_ineqs ineqs |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
367 |
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
|
368 |
if not (null triv) |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
369 |
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
|
370 |
NONE => elim (nontriv, hist) |
15531 | 371 |
| SOME(Lineq(_,_,_,j)) => Success j |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
372 |
else |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
373 |
if null nontriv then Failure hist |
13498 | 374 |
else |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
375 |
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
|
376 |
if not (null eqs) then |
15570 | 377 |
let val clist = Library.foldl (fn (cs,Lineq(_,_,l,_)) => l union cs) ([],eqs) |
16358 | 378 |
val sclist = sort (fn (x,y) => IntInf.compare(abs(x),abs(y))) |
15570 | 379 |
(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
|
380 |
val c = hd sclist |
15531 | 381 |
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
|
382 |
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
|
383 |
val v = find_index_eq c ceq |
15570 | 384 |
val (ioth,roth) = List.partition (fn (Lineq(_,_,l,_)) => el v l = 0) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
385 |
(othereqs @ noneqs) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
386 |
val others = map (elim_var v eq) roth @ ioth |
13498 | 387 |
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
|
388 |
else |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
389 |
let val lists = map (fn (Lineq(_,_,l,_)) => l) noneqs |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
390 |
val numlist = 0 upto (length(hd lists) - 1) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
391 |
val coeffs = map (fn i => map (el i) lists) numlist |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
392 |
val blows = map calc_blowup coeffs |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
393 |
val iblows = blows ~~ numlist |
15570 | 394 |
val nziblows = List.filter (fn (i,_) => i<>0) iblows |
13498 | 395 |
in if null nziblows then Failure((~1,nontriv)::hist) |
396 |
else |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
397 |
let val (c,v) = hd(sort (fn (x,y) => int_ord(fst(x),fst(y))) nziblows) |
15570 | 398 |
val (no,yes) = List.partition (fn (Lineq(_,_,l,_)) => el v l = 0) ineqs |
399 |
val (pos,neg) = List.partition(fn (Lineq(_,_,l,_)) => el v l > 0) yes |
|
13498 | 400 |
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
|
401 |
end |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
402 |
end |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
403 |
end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
404 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
405 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
406 |
(* Translate back a proof. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
407 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
408 |
|
20268 | 409 |
fun trace_thm (msg : string) (th : thm) : thm = |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
410 |
(if !trace then (tracing msg; tracing (Display.string_of_thm th)) else (); th); |
9073 | 411 |
|
20268 | 412 |
fun trace_msg (msg : string) : unit = |
12262 | 413 |
if !trace then tracing msg else (); |
9073 | 414 |
|
13498 | 415 |
(* FIXME OPTIMIZE!!!! (partly done already) |
6056 | 416 |
Addition/Multiplication need i*t representation rather than t+t+... |
10691 | 417 |
Get rid of Mulitplied(2). For Nat LA_Data.number_of should return Suc^n |
418 |
because Numerals are not known early enough. |
|
6056 | 419 |
|
420 |
Simplification may detect a contradiction 'prematurely' due to type |
|
421 |
information: n+1 <= 0 is simplified to False and does not need to be crossed |
|
422 |
with 0 <= n. |
|
423 |
*) |
|
424 |
local |
|
425 |
exception FalseE of thm |
|
426 |
in |
|
20268 | 427 |
fun mkthm (sg:theory, ss:simpset) (asms:thm list) (just:injust) : thm = |
15922
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
428 |
let val {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset, ...} = |
16458 | 429 |
Data.get sg; |
17877
67d5ab1cb0d8
Simplifier.inherit_context instead of Simplifier.inherit_bounds;
wenzelm
parents:
17613
diff
changeset
|
430 |
val simpset' = Simplifier.inherit_context ss simpset; |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
431 |
val atoms = Library.foldl (fn (ats, (lhs,_,_,rhs,_,_)) => |
6056 | 432 |
map fst lhs union (map fst rhs union ats)) |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
433 |
([], List.mapPartial (fn thm => if Thm.no_prems thm |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
434 |
then LA_Data.decomp sg (concl_of thm) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
435 |
else NONE) asms) |
6056 | 436 |
|
10575 | 437 |
fun add2 thm1 thm2 = |
6102 | 438 |
let val conj = thm1 RS (thm2 RS LA_Logic.conjI) |
15531 | 439 |
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
|
440 |
end; |
15531 | 441 |
fun try_add [] _ = NONE |
10575 | 442 |
| try_add (thm1::thm1s) thm2 = case add2 thm1 thm2 of |
15531 | 443 |
NONE => try_add thm1s thm2 | some => some; |
10575 | 444 |
|
445 |
fun addthms thm1 thm2 = |
|
446 |
case add2 thm1 thm2 of |
|
15531 | 447 |
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
|
448 |
NONE => ( the (try_add ([thm2] RL inj_thms) thm1) |
15660 | 449 |
handle Option => |
14360 | 450 |
(trace_thm "" thm1; trace_thm "" thm2; |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
451 |
sys_error "Lin.arith. failed to add thms") |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
452 |
) |
15531 | 453 |
| SOME thm => thm) |
454 |
| SOME thm => thm; |
|
10575 | 455 |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
456 |
fun multn(n,thm) = |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
457 |
let fun mul(i,th) = if i=1 then th else mul(i-1, addthms thm th) |
6102 | 458 |
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
|
459 |
|
15184 | 460 |
fun multn2(n,thm) = |
15531 | 461 |
let val SOME(mth) = |
462 |
get_first (fn th => SOME(thm RS th) handle THM _ => NONE) mult_mono_thms |
|
15184 | 463 |
fun cvar(th,_ $ (_ $ _ $ var)) = cterm_of (#sign(rep_thm th)) var; |
464 |
val cv = cvar(mth, hd(prems_of mth)); |
|
465 |
val ct = cterm_of sg (LA_Data.number_of(n,#T(rep_cterm cv))) |
|
466 |
in instantiate ([],[(cv,ct)]) mth end |
|
10691 | 467 |
|
6056 | 468 |
fun simp thm = |
17515 | 469 |
let val thm' = trace_thm "Simplified:" (full_simplify simpset' thm) |
6102 | 470 |
in if LA_Logic.is_False thm' then raise FalseE thm' else thm' end |
6056 | 471 |
|
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
472 |
fun mk (Asm i) = trace_thm ("Asm " ^ Int.toString i) (nth asms i) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
473 |
| mk (Nat i) = trace_thm ("Nat " ^ Int.toString i) (LA_Logic.mk_nat_thm sg (nth atoms i)) |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
474 |
| 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
|
475 |
| 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
|
476 |
| 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
|
477 |
| 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
|
478 |
| mk (Added (j1, j2)) = simp (trace_thm "+" (addthms (mk j1) (mk j2))) |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
479 |
| mk (Multiplied (n, j)) = (trace_msg ("*" ^ IntInf.toString n); trace_thm "*" (multn (n, mk j))) |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
480 |
| mk (Multiplied2 (n, j)) = simp (trace_msg ("**" ^ IntInf.toString 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
|
481 |
|
9073 | 482 |
in trace_msg "mkthm"; |
12932 | 483 |
let val thm = trace_thm "Final thm:" (mk just) |
17515 | 484 |
in let val fls = simplify simpset' thm |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
485 |
in trace_thm "After simplification:" fls; |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
486 |
if LA_Logic.is_False fls then fls |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
487 |
else |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
488 |
(tracing "Assumptions:"; List.app (tracing o Display.string_of_thm) asms; |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
489 |
tracing "Proved:"; tracing (Display.string_of_thm fls); |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
490 |
warning "Linear arithmetic should have refuted the assumptions.\n\ |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
491 |
\Please inform Tobias Nipkow (nipkow@in.tum.de)."; |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
492 |
fls) |
12932 | 493 |
end |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
494 |
end handle FalseE thm => trace_thm "False reached early:" thm |
12932 | 495 |
end |
6056 | 496 |
end; |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
497 |
|
16358 | 498 |
fun coeff poly atom : IntInf.int = |
17325 | 499 |
AList.lookup (op =) poly atom |> the_default 0; |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
500 |
|
20280
ad9fbbd01535
type annotations fixed (IntInf.int, to make SML/NJ happy)
webertj
parents:
20276
diff
changeset
|
501 |
fun lcms (is : IntInf.int list) : IntInf.int = Library.foldl lcm (1, is); |
10691 | 502 |
|
503 |
fun integ(rlhs,r,rel,rrhs,s,d) = |
|
17951 | 504 |
let val (rn,rd) = Rat.quotient_of_rat r and (sn,sd) = Rat.quotient_of_rat s |
505 |
val m = lcms(map (abs o snd o Rat.quotient_of_rat) (r :: s :: map snd rlhs @ map snd rrhs)) |
|
15965
f422f8283491
Use of IntInf.int instead of int in most numeric simprocs; avoids
paulson
parents:
15922
diff
changeset
|
506 |
fun mult(t,r) = |
17951 | 507 |
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
|
508 |
in (t,i * (m div j)) end |
12932 | 509 |
in (m,(map mult rlhs, rn*(m div rd), rel, map mult rrhs, sn*(m div sd), d)) end |
10691 | 510 |
|
13498 | 511 |
fun mklineq n atoms = |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
512 |
fn (item, k) => |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
513 |
let val (m, (lhs,i,rel,rhs,j,discrete)) = integ item |
13498 | 514 |
val lhsa = map (coeff lhs) atoms |
515 |
and rhsa = map (coeff rhs) atoms |
|
18330 | 516 |
val diff = map2 (curry (op -)) rhsa lhsa |
13498 | 517 |
val c = i-j |
518 |
val just = Asm k |
|
519 |
fun lineq(c,le,cs,j) = Lineq(c,le,cs, if m=1 then j else Multiplied2(m,j)) |
|
520 |
in case rel of |
|
521 |
"<=" => lineq(c,Le,diff,just) |
|
522 |
| "~<=" => if discrete |
|
523 |
then lineq(1-c,Le,map (op ~) diff,NotLeDD(just)) |
|
524 |
else lineq(~c,Lt,map (op ~) diff,NotLeD(just)) |
|
525 |
| "<" => if discrete |
|
526 |
then lineq(c+1,Le,diff,LessD(just)) |
|
527 |
else lineq(c,Lt,diff,just) |
|
528 |
| "~<" => lineq(~c,Le,map (op~) diff,NotLessD(just)) |
|
529 |
| "=" => lineq(c,Eq,diff,just) |
|
530 |
| _ => sys_error("mklineq" ^ rel) |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
531 |
end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
532 |
|
13498 | 533 |
(* ------------------------------------------------------------------------- *) |
534 |
(* Print (counter) example *) |
|
535 |
(* ------------------------------------------------------------------------- *) |
|
536 |
||
537 |
fun print_atom((a,d),r) = |
|
17951 | 538 |
let val (p,q) = Rat.quotient_of_rat r |
15965
f422f8283491
Use of IntInf.int instead of int in most numeric simprocs; avoids
paulson
parents:
15922
diff
changeset
|
539 |
val s = if d then IntInf.toString p else |
13498 | 540 |
if p = 0 then "0" |
15965
f422f8283491
Use of IntInf.int instead of int in most numeric simprocs; avoids
paulson
parents:
15922
diff
changeset
|
541 |
else IntInf.toString p ^ "/" ^ IntInf.toString q |
13498 | 542 |
in a ^ " = " ^ s end; |
543 |
||
19049 | 544 |
fun produce_ex sds = |
17496 | 545 |
curry (op ~~) sds |
546 |
#> map print_atom |
|
547 |
#> commas |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
548 |
#> curry (op ^) "Counter example (possibly spurious):\n"; |
13498 | 549 |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
550 |
fun trace_ex (sg, params, atoms, discr, n, hist : history) = |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
551 |
case hist of |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
552 |
[] => () |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
553 |
| (v, lineqs) :: hist' => |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
554 |
let val frees = map Free params |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
555 |
fun s_of_t t = Sign.string_of_term sg (subst_bounds (frees, t)) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
556 |
val start = if v = ~1 then (findex0 discr n lineqs, hist') |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
557 |
else (replicate n Rat.zero, hist) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
558 |
val ex = SOME (produce_ex ((map s_of_t atoms) ~~ discr) (findex discr start)) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
559 |
handle NoEx => NONE |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
560 |
in |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
561 |
case ex of |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
562 |
SOME s => (warning "arith failed - see trace for a counter example"; tracing s) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
563 |
| NONE => warning "arith failed" |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
564 |
end; |
13498 | 565 |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
566 |
(* ------------------------------------------------------------------------- *) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
567 |
|
20268 | 568 |
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
|
569 |
if LA_Logic.is_nat (pTs, atom) |
6056 | 570 |
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
|
571 |
in SOME (Lineq (0, Le, l, Nat i)) end |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset
|
572 |
else NONE; |
6056 | 573 |
|
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
574 |
(* This code is tricky. It takes a list of premises in the order they occur |
15531 | 575 |
in the subgoal. Numerical premises are coded as SOME(tuple), non-numerical |
576 |
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
|
577 |
a Lineq. The tricky bit is to convert ~= which is split into two cases < and |
13498 | 578 |
>. 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
|
579 |
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
|
580 |
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
|
581 |
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
|
582 |
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
|
583 |
|
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
584 |
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
|
585 |
*) |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
586 |
|
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
587 |
(* 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
|
588 |
(* 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
|
589 |
(* 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
|
590 |
(* 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
|
591 |
(* 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
|
592 |
|
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
593 |
fun split_items sg (Ts : typ list, terms : term list) : |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
594 |
(typ list * (LA_Data.decompT * int) list) list = |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
595 |
let |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
596 |
(* |
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
597 |
val _ = if !trace then |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
598 |
trace_msg ("split_items: Ts = " ^ string_of_list (Sign.string_of_typ sg) Ts ^ "\n" ^ |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
599 |
" terms = " ^ string_of_list (Sign.string_of_term sg) terms) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
600 |
else () |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
601 |
*) |
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
602 |
(* 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
|
603 |
(* '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
|
604 |
(* level *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
605 |
(* 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
|
606 |
(* 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
|
607 |
(* 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
|
608 |
(* 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
|
609 |
(* can be applied, and split the premise accordingly. *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
610 |
fun elim_neq (ineqs : (LA_Data.decompT option * bool) list) : |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
611 |
(LA_Data.decompT option * bool) list list = |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
612 |
let |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
613 |
fun elim_neq' nat_only ([] : (LA_Data.decompT option * bool) list) : |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
614 |
(LA_Data.decompT option * bool) list list = |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
615 |
[[]] |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
616 |
| 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
|
617 |
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
|
618 |
| 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
|
619 |
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
|
620 |
(* [| ?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
|
621 |
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
|
622 |
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
|
623 |
else |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
624 |
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
|
625 |
in |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
626 |
ineqs |> elim_neq' true |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
627 |
|> map (elim_neq' false) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
628 |
|> List.concat |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
629 |
end |
13464 | 630 |
|
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
631 |
fun number_hyps _ [] = [] |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
632 |
| 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
|
633 |
| 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
|
634 |
|
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
635 |
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
|
636 |
|> (* user-defined preprocessing of the subgoal *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
637 |
(* (typ list * term list) list *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
638 |
LA_Data.pre_decomp sg |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
639 |
|> (* compute the internal encoding of (in-)equalities *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
640 |
(* (typ list * (LA_Data.decompT option * bool) list) list *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
641 |
map (apsnd (map (fn t => (LA_Data.decomp sg t, LA_Data.domain_is_nat t)))) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
642 |
|> (* splitting of inequalities *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
643 |
(* (typ list * (LA_Data.decompT option * bool) list list) list *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
644 |
map (apsnd elim_neq) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
645 |
|> (* combine the list of lists of subgoals into a single list *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
646 |
map (fn (Ts, subgoals) => map (pair Ts o map fst) subgoals) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
647 |
|> List.concat |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
648 |
|> (* 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
|
649 |
map (apsnd (number_hyps 0)) |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
650 |
(* |
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
651 |
val _ = if !trace then |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
652 |
trace_msg ("split_items: result has " ^ Int.toString (length result) ^ |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
653 |
" subgoal(s)" ^ "\n" ^ (cat_lines o fst) (fold_map (fn (Ts, items) => fn n => |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
654 |
(" " ^ Int.toString n ^ ". Ts = " ^ |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
655 |
string_of_list (Sign.string_of_typ sg) Ts ^ "\n" ^ |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
656 |
" items = " ^ string_of_list (string_of_pair |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
657 |
(fn (l, i, rel, r, j, d) => enclose "(" ")" (commas |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
658 |
[string_of_list |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
659 |
(string_of_pair (Sign.string_of_term sg) Rat.string_of_rat) l, |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
660 |
Rat.string_of_rat i, |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
661 |
rel, |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
662 |
string_of_list |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
663 |
(string_of_pair (Sign.string_of_term sg) Rat.string_of_rat) r, |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
664 |
Rat.string_of_rat j, |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
665 |
Bool.toString d])) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
666 |
Int.toString) items, n+1)) result 1)) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
667 |
else () |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
668 |
*) |
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
669 |
in result end; |
13464 | 670 |
|
20268 | 671 |
fun add_atoms (ats : term list, ((lhs,_,_,rhs,_,_) : LA_Data.decompT, _)) : term list = |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
672 |
(map fst lhs) union ((map fst rhs) union ats); |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
673 |
|
20268 | 674 |
fun add_datoms (dats : (bool * term) list, ((lhs,_,_,rhs,_,d) : LA_Data.decompT, _)) : |
675 |
(bool * term) list = |
|
676 |
(map (pair d o fst) lhs) union ((map (pair d o fst) rhs) union dats); |
|
13498 | 677 |
|
20268 | 678 |
fun discr (initems : (LA_Data.decompT * int) list) : bool list = |
679 |
map fst (Library.foldl add_datoms ([],initems)); |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
680 |
|
20268 | 681 |
fun refutes (sg : theory) (params : (string * typ) list) (show_ex : bool) : |
682 |
(typ list * (LA_Data.decompT * int) list) list -> injust list -> injust list option = |
|
13498 | 683 |
let |
20268 | 684 |
fun refute ((Ts : typ list, initems : (LA_Data.decompT * int) list)::initemss) |
685 |
(js : injust list) : injust list option = |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
686 |
let val atoms = Library.foldl add_atoms ([], initems) |
13498 | 687 |
val n = length atoms |
688 |
val mkleq = mklineq n atoms |
|
689 |
val ixs = 0 upto (n-1) |
|
690 |
val iatoms = atoms ~~ ixs |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
691 |
val natlineqs = List.mapPartial (mknat Ts ixs) iatoms |
13498 | 692 |
val ineqs = map mkleq initems @ natlineqs |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
693 |
in case elim (ineqs, []) of |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
694 |
Success j => |
20268 | 695 |
(trace_msg ("Contradiction! (" ^ Int.toString (length js + 1) ^ ")"); |
696 |
refute initemss (js@[j])) |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
697 |
| Failure hist => |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
698 |
(if not show_ex then |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
699 |
() |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
700 |
else let |
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
701 |
(* invent names for bound variables that are new, i.e. in Ts, *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
702 |
(* but not in params; we assume that Ts still contains (map *) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
703 |
(* snd params) as a suffix *) |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
704 |
val new_count = length Ts - length params - 1 |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
705 |
val new_names = map Name.bound (0 upto new_count) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
706 |
val params' = (new_names @ map fst params) ~~ Ts |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
707 |
in |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
708 |
trace_ex (sg, params', atoms, discr initems, n, hist) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
709 |
end; NONE) |
13498 | 710 |
end |
15531 | 711 |
| refute [] js = SOME js |
13498 | 712 |
in refute end; |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
713 |
|
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
714 |
fun refute (sg : theory) (params : (string * Term.typ) list) (show_ex : bool) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
715 |
(terms : term list) : injust list option = |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
716 |
refutes sg params show_ex (split_items sg (map snd params, terms)) []; |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
717 |
|
20268 | 718 |
fun count (P : 'a -> bool) (xs : 'a list) : int = length (List.filter P xs); |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
719 |
|
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
720 |
(* The limit on the number of ~= allowed. |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
721 |
Because each ~= is split into two cases, this can lead to an explosion. |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
722 |
*) |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
723 |
val fast_arith_neq_limit = ref 9; |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
724 |
|
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
725 |
fun prove (sg : theory) (params : (string * Term.typ) list) (show_ex : bool) |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
726 |
(Hs : term list) (concl : term) : injust list option = |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
727 |
let |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
728 |
(* 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
|
729 |
(* '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
|
730 |
(* theorem/tactic level *) |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
731 |
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
|
732 |
fun is_neq NONE = false |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
733 |
| is_neq (SOME (_,_,r,_,_,_)) = (r = "~=") |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
734 |
in |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
735 |
trace_msg "prove"; |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
736 |
if count is_neq (map (LA_Data.decomp sg) Hs') |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
737 |
> !fast_arith_neq_limit then ( |
20268 | 738 |
trace_msg ("fast_arith_neq_limit exceeded (current value is " ^ |
739 |
string_of_int (!fast_arith_neq_limit) ^ ")"); |
|
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
740 |
NONE |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
741 |
) else |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
742 |
refute sg params show_ex Hs' |
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
743 |
end; |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
744 |
|
20268 | 745 |
fun refute_tac (ss : simpset) (i : int, justs : injust list) : tactic = |
6074 | 746 |
fn state => |
20268 | 747 |
let val _ = trace_thm ("refute_tac (on subgoal " ^ Int.toString i ^ ", with " ^ |
748 |
Int.toString (length justs) ^ " justification(s)):") state |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
749 |
val sg = theory_of_thm state |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
750 |
val {neqE, ...} = Data.get sg |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
751 |
fun just1 j = |
20268 | 752 |
(* eliminate inequalities *) |
753 |
REPEAT_DETERM (eresolve_tac neqE i) THEN |
|
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
754 |
PRIMITIVE (trace_thm "State after neqE:") THEN |
20268 | 755 |
(* use theorems generated from the actual justifications *) |
756 |
METAHYPS (fn asms => rtac (mkthm (sg, ss) asms j) 1) i |
|
757 |
in (* 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 i) THEN |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
761 |
PRIMITIVE (trace_thm "State after pre_tac:") THEN |
20268 | 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 |
*) |
20268 | 770 |
fun simpset_lin_arith_tac (ss : simpset) (show_ex : bool) (i : int) (st : thm) = |
771 |
SUBGOAL (fn (A,_) => |
|
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
772 |
let val params = rev (Logic.strip_params A) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
773 |
val Hs = Logic.strip_assums_hyp A |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
774 |
val concl = Logic.strip_assums_concl A |
12932 | 775 |
in trace_thm ("Trying to refute subgoal " ^ string_of_int i) st; |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
776 |
case prove (Thm.sign_of_thm st) params show_ex Hs concl of |
15531 | 777 |
NONE => (trace_msg "Refutation failed."; no_tac) |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
778 |
| SOME js => (trace_msg "Refutation succeeded."; refute_tac ss (i, js)) |
9420 | 779 |
end) i st; |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
780 |
|
20268 | 781 |
fun lin_arith_tac (show_ex : bool) (i : int) (st : thm) = |
20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
782 |
simpset_lin_arith_tac |
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20268
diff
changeset
|
783 |
(Simplifier.theory_context (Thm.theory_of_thm st) Simplifier.empty_ss) |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
784 |
show_ex i st; |
17613 | 785 |
|
20268 | 786 |
fun cut_lin_arith_tac (ss : simpset) (i : int) = |
17613 | 787 |
cut_facts_tac (Simplifier.prems_of_ss ss) i THEN |
788 |
simpset_lin_arith_tac ss false i; |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
789 |
|
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
790 |
(** Forward proof from theorems **) |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
791 |
|
20268 | 792 |
fun prover (ctxt as (sg, ss)) thms (Tconcl : term) (js : injust list) (pos : bool) : thm option = |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
793 |
let |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
794 |
(* There is no "forward version" of 'pre_tac'. Therefore we combine the *) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
795 |
(* available theorems into a single proof state and perform "backward proof" *) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
796 |
(* using 'refute_tac'. *) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
797 |
val Hs = map prop_of thms |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
798 |
val Prop = fold (curry Logic.mk_implies) (rev Hs) Tconcl |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
799 |
val cProp = cterm_of sg Prop |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
800 |
val concl = Goal.init cProp |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
801 |
|> refute_tac ss (1, js) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
802 |
|> Seq.hd |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
803 |
|> Goal.finish |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
804 |
|> fold (fn thA => fn thAB => implies_elim thAB thA) thms |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
805 |
in SOME (trace_thm "Proved by lin. arith. prover:" |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
806 |
(LA_Logic.mk_Eq concl)) end |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
807 |
(* in case concl contains ?-var, which makes assume fail: *) |
15531 | 808 |
handle THM _ => NONE; |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
809 |
|
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
810 |
(* PRE: concl is not negated! |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
811 |
This assumption is OK because |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
812 |
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:
13105
diff
changeset
|
813 |
2. lin_arith_prover is applied by the simplifier which |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
814 |
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
|
815 |
*) |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
816 |
|
20268 | 817 |
fun lin_arith_prover sg ss (concl : term) : thm option = |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
818 |
let val thms = List.concat (map LA_Logic.atomize (prems_of_ss ss)); |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
819 |
val Hs = map prop_of thms |
6102 | 820 |
val Tconcl = LA_Logic.mk_Trueprop concl |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
821 |
(* |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
822 |
val _ = trace_msg "lin_arith_prover" |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
823 |
val _ = map (trace_thm "thms:") thms |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
824 |
val _ = trace_msg ("concl:" ^ Sign.string_of_term sg concl) |
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19618
diff
changeset
|
825 |
*) |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
826 |
in case prove sg [] false Hs Tconcl of (* concl provable? *) |
17515 | 827 |
SOME js => prover (sg, ss) thms Tconcl js true |
15531 | 828 |
| NONE => let val nTconcl = LA_Logic.neg_prop Tconcl |
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset
|
829 |
in case prove sg [] false Hs nTconcl of (* ~concl provable? *) |
17515 | 830 |
SOME js => prover (sg, ss) thms nTconcl js false |
15531 | 831 |
| NONE => NONE |
6079 | 832 |
end |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
833 |
end; |
6074 | 834 |
|
835 |
end; |