author | haftmann |
Fri, 28 Oct 2005 16:35:40 +0200 | |
changeset 18011 | 685d95c793ff |
parent 17951 | ff954cc338c7 |
child 18330 | 444f16d232a2 |
permissions | -rw-r--r-- |
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(* Title: Provers/Arith/fast_lin_arith.ML |
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ID: $Id$ |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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Author: Tobias Nipkow |
aeb97860d352
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Copyright 1998 TU Munich |
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5 |
|
6062 | 6 |
A generic linear arithmetic package. |
6102 | 7 |
It provides two tactics |
8 |
||
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lin_arith_tac: int -> tactic |
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10 |
cut_lin_arith_tac: thms -> int -> tactic |
6102 | 11 |
|
12 |
and a simplification procedure |
|
13 |
||
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lin_arith_prover: theory -> simpset -> term -> thm option |
6102 | 15 |
|
16 |
Only take premises and conclusions into account that are already (negated) |
|
17 |
(in)equations. lin_arith_prover tries to prove or disprove the term. |
|
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18 |
*) |
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|
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(* Debugging: set Fast_Arith.trace *) |
7582 | 21 |
|
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(*** Data needed for setting up the linear arithmetic package ***) |
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|
6102 | 24 |
signature LIN_ARITH_LOGIC = |
25 |
sig |
|
26 |
val conjI: thm |
|
27 |
val ccontr: thm (* (~ P ==> False) ==> P *) |
|
28 |
val notI: thm (* (P ==> False) ==> ~ P *) |
|
6110 | 29 |
val not_lessD: thm (* ~(m < n) ==> n <= m *) |
6128 | 30 |
val not_leD: thm (* ~(m <= n) ==> n < m *) |
6102 | 31 |
val sym: thm (* x = y ==> y = x *) |
32 |
val mk_Eq: thm -> thm |
|
16735 | 33 |
val atomize: thm -> thm list |
6102 | 34 |
val mk_Trueprop: term -> term |
35 |
val neg_prop: term -> term |
|
36 |
val is_False: thm -> bool |
|
6128 | 37 |
val is_nat: typ list * term -> bool |
16458 | 38 |
val mk_nat_thm: theory -> term -> thm |
6102 | 39 |
end; |
40 |
(* |
|
41 |
mk_Eq(~in) = `in == False' |
|
42 |
mk_Eq(in) = `in == True' |
|
43 |
where `in' is an (in)equality. |
|
44 |
||
45 |
neg_prop(t) = neg if t is wrapped up in Trueprop and |
|
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nt is the (logically) negated version of t, where the negation |
|
47 |
of a negative term is the term itself (no double negation!); |
|
6128 | 48 |
|
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is_nat(parameter-types,t) = t:nat |
|
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mk_nat_thm(t) = "0 <= t" |
|
6102 | 51 |
*) |
52 |
||
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signature LIN_ARITH_DATA = |
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sig |
6128 | 55 |
val decomp: |
17951 | 56 |
theory -> term -> ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool) option |
16358 | 57 |
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) |
5982
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where Rel is one of "<", "~<", "<=", "~<=" and "=" and |
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p/q is the decomposition of the sum terms x/y into a list |
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of summand * multiplicity pairs and a constant summand and |
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d indicates if the domain is discrete. |
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|
9420 | 66 |
ss must reduce contradictory <= to False. |
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It should also cancel common summands to keep <= reduced; |
aeb97860d352
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otherwise <= can grow to massive proportions. |
aeb97860d352
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69 |
*) |
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|
6062 | 71 |
signature FAST_LIN_ARITH = |
72 |
sig |
|
9420 | 73 |
val setup: (theory -> theory) list |
15184 | 74 |
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} |
15184 | 76 |
-> {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}) |
10575 | 78 |
-> theory -> theory |
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val trace : bool ref |
14510 | 80 |
val fast_arith_neq_limit: int ref |
16458 | 81 |
val lin_arith_prover: theory -> simpset -> term -> thm option |
17892 | 82 |
val lin_arith_tac: bool -> int -> tactic |
17613 | 83 |
val cut_lin_arith_tac: simpset -> int -> tactic |
6062 | 84 |
end; |
85 |
||
6102 | 86 |
functor Fast_Lin_Arith(structure LA_Logic:LIN_ARITH_LOGIC |
87 |
and LA_Data:LIN_ARITH_DATA) : FAST_LIN_ARITH = |
|
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struct |
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|
9420 | 90 |
|
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(** theory data **) |
|
92 |
||
93 |
(* data kind 'Provers/fast_lin_arith' *) |
|
94 |
||
16458 | 95 |
structure Data = TheoryDataFun |
96 |
(struct |
|
9420 | 97 |
val name = "Provers/fast_lin_arith"; |
15184 | 98 |
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}; |
9420 | 100 |
|
10691 | 101 |
val empty = {add_mono_thms = [], mult_mono_thms = [], inj_thms = [], |
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102 |
lessD = [], neqE = [], simpset = Simplifier.empty_ss}; |
9420 | 103 |
val copy = I; |
16458 | 104 |
val extend = I; |
9420 | 105 |
|
16458 | 106 |
fun merge _ |
107 |
({add_mono_thms= add_mono_thms1, mult_mono_thms= mult_mono_thms1, inj_thms= inj_thms1, |
|
108 |
lessD = lessD1, neqE=neqE1, simpset = simpset1}, |
|
109 |
{add_mono_thms= add_mono_thms2, mult_mono_thms= mult_mono_thms2, inj_thms= inj_thms2, |
|
110 |
lessD = lessD2, neqE=neqE2, simpset = simpset2}) = |
|
9420 | 111 |
{add_mono_thms = Drule.merge_rules (add_mono_thms1, add_mono_thms2), |
15184 | 112 |
mult_mono_thms = Drule.merge_rules (mult_mono_thms1, mult_mono_thms2), |
10575 | 113 |
inj_thms = Drule.merge_rules (inj_thms1, inj_thms2), |
114 |
lessD = Drule.merge_rules (lessD1, lessD2), |
|
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neqE = Drule.merge_rules (neqE1, neqE2), |
10575 | 116 |
simpset = Simplifier.merge_ss (simpset1, simpset2)}; |
9420 | 117 |
|
118 |
fun print _ _ = (); |
|
16458 | 119 |
end); |
9420 | 120 |
|
121 |
val map_data = Data.map; |
|
122 |
val setup = [Data.init]; |
|
123 |
||
124 |
||
125 |
||
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126 |
(*** A fast decision procedure ***) |
aeb97860d352
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127 |
(*** Code ported from HOL Light ***) |
6056 | 128 |
(* possible optimizations: |
129 |
use (var,coeff) rep or vector rep tp save space; |
|
130 |
treat non-negative atoms separately rather than adding 0 <= atom |
|
131 |
*) |
|
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132 |
|
9073 | 133 |
val trace = ref false; |
134 |
||
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135 |
datatype lineq_type = Eq | Le | Lt; |
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136 |
|
6056 | 137 |
datatype injust = Asm of int |
138 |
| Nat of int (* index of atom *) |
|
6128 | 139 |
| LessD of injust |
140 |
| NotLessD of injust |
|
141 |
| NotLeD of injust |
|
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142 |
| NotLeDD of injust |
16358 | 143 |
| Multiplied of IntInf.int * injust |
144 |
| Multiplied2 of IntInf.int * injust |
|
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145 |
| Added of injust * injust; |
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146 |
|
16358 | 147 |
datatype lineq = Lineq of IntInf.int * lineq_type * IntInf.int list * injust; |
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148 |
|
13498 | 149 |
fun el 0 (h::_) = h |
150 |
| el n (_::t) = el (n - 1) t |
|
151 |
| el _ _ = sys_error "el"; |
|
152 |
||
153 |
(* ------------------------------------------------------------------------- *) |
|
154 |
(* Finding a (counter) example from the trace of a failed elimination *) |
|
155 |
(* ------------------------------------------------------------------------- *) |
|
156 |
(* Examples are represented as rational numbers, *) |
|
157 |
(* Dont blame John Harrison for this code - it is entirely mine. TN *) |
|
158 |
||
159 |
exception NoEx; |
|
160 |
||
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161 |
(* Coding: (i,true,cs) means i <= cs and (i,false,cs) means i < cs. |
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162 |
In general, true means the bound is included, false means it is excluded. |
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163 |
Need to know if it is a lower or upper bound for unambiguous interpretation! |
51ddf8963c95
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164 |
*) |
51ddf8963c95
Finally fixed the counterexample finder. Can now deal with < on real.
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165 |
|
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166 |
fun elim_eqns(ineqs,Lineq(i,Le,cs,_)) = (i,true,cs)::ineqs |
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167 |
| elim_eqns(ineqs,Lineq(i,Eq,cs,_)) = (i,true,cs)::(~i,true,map ~ cs)::ineqs |
51ddf8963c95
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168 |
| elim_eqns(ineqs,Lineq(i,Lt,cs,_)) = (i,false,cs)::ineqs; |
13498 | 169 |
|
170 |
(* PRE: ex[v] must be 0! *) |
|
17951 | 171 |
fun eval (ex:Rat.rat list) v (a:IntInf.int,le,cs:IntInf.int list) = |
172 |
let val rs = map Rat.rat_of_intinf cs |
|
173 |
val rsum = Library.foldl Rat.add (Rat.zero, map Rat.mult (rs ~~ ex)) |
|
174 |
in (Rat.mult (Rat.add(Rat.rat_of_intinf a,Rat.neg rsum), Rat.inv(el v rs)), le) end; |
|
14372
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175 |
(* If el v rs < 0, le should be negated. |
51ddf8963c95
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176 |
Instead this swap is taken into account in ratrelmin2. |
51ddf8963c95
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177 |
*) |
13498 | 178 |
|
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179 |
fun ratrelmin2(x as (r,ler),y as (s,les)) = |
17951 | 180 |
if r=s then (r, (not ler) andalso (not les)) else if Rat.le(r,s) then x else y; |
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181 |
fun ratrelmax2(x as (r,ler),y as (s,les)) = |
17951 | 182 |
if r=s then (r,ler andalso les) else if Rat.le(r,s) then y else x; |
13498 | 183 |
|
14372
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184 |
val ratrelmin = foldr1 ratrelmin2; |
51ddf8963c95
Finally fixed the counterexample finder. Can now deal with < on real.
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185 |
val ratrelmax = foldr1 ratrelmax2; |
13498 | 186 |
|
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187 |
fun ratexact up (r,exact) = |
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188 |
if exact then r else |
17951 | 189 |
let val (p,q) = Rat.quotient_of_rat r |
190 |
val nth = Rat.inv(Rat.rat_of_intinf q) |
|
191 |
in Rat.add(r,if up then nth else Rat.neg nth) end; |
|
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192 |
|
17951 | 193 |
fun ratmiddle(r,s) = Rat.mult(Rat.add(r,s),Rat.inv(Rat.rat_of_int 2)); |
14372
51ddf8963c95
Finally fixed the counterexample finder. Can now deal with < on real.
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194 |
|
51ddf8963c95
Finally fixed the counterexample finder. Can now deal with < on real.
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|
195 |
fun choose2 d ((lb,exactl),(ub,exactu)) = |
17951 | 196 |
if Rat.le(lb,Rat.zero) andalso (lb <> Rat.zero orelse exactl) andalso |
197 |
Rat.le(Rat.zero,ub) andalso (ub <> Rat.zero orelse exactu) |
|
198 |
then Rat.zero else |
|
14372
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|
199 |
if not d |
17951 | 200 |
then (if Rat.ge0 lb |
14372
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201 |
then if exactl then lb else ratmiddle(lb,ub) |
51ddf8963c95
Finally fixed the counterexample finder. Can now deal with < on real.
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202 |
else if exactu then ub else ratmiddle(lb,ub)) |
51ddf8963c95
Finally fixed the counterexample finder. Can now deal with < on real.
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|
203 |
else (* discrete domain, both bounds must be exact *) |
17951 | 204 |
if Rat.ge0 lb then let val lb' = Rat.roundup lb |
205 |
in if Rat.le(lb',ub) then lb' else raise NoEx end |
|
206 |
else let val ub' = Rat.rounddown ub |
|
207 |
in if Rat.le(lb,ub') then ub' else raise NoEx end; |
|
13498 | 208 |
|
209 |
fun findex1 discr (ex,(v,lineqs)) = |
|
15570 | 210 |
let val nz = List.filter (fn (Lineq(_,_,cs,_)) => el v cs <> 0) lineqs; |
211 |
val ineqs = Library.foldl elim_eqns ([],nz) |
|
212 |
val (ge,le) = List.partition (fn (_,_,cs) => el v cs > 0) ineqs |
|
14372
51ddf8963c95
Finally fixed the counterexample finder. Can now deal with < on real.
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parents:
14360
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|
213 |
val lb = ratrelmax(map (eval ex v) ge) |
51ddf8963c95
Finally fixed the counterexample finder. Can now deal with < on real.
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14360
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|
214 |
val ub = ratrelmin(map (eval ex v) le) |
18011
685d95c793ff
cleaned up nth, nth_update, nth_map and nth_string functions
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|
215 |
in nth_update (v, choose2 (nth discr v) (lb, ub)) ex end; |
13498 | 216 |
|
15570 | 217 |
fun findex discr = Library.foldl (findex1 discr); |
13498 | 218 |
|
219 |
fun elim1 v x = |
|
17951 | 220 |
map (fn (a,le,bs) => (Rat.add(a,Rat.neg(Rat.mult(el v bs,x))), le, |
18011
685d95c793ff
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|
221 |
nth_update (v, Rat.zero) bs)); |
13498 | 222 |
|
17951 | 223 |
fun single_var v (_,_,cs) = (filter_out (equal Rat.zero) cs = [el v cs]); |
13498 | 224 |
|
225 |
(* The base case: |
|
226 |
all variables occur only with positive or only with negative coefficients *) |
|
227 |
fun pick_vars discr (ineqs,ex) = |
|
17951 | 228 |
let val nz = filter_out (fn (_,_,cs) => forall (equal Rat.zero) cs) ineqs |
14372
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|
229 |
in case nz of [] => ex |
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|
230 |
| (_,_,cs) :: _ => |
17951 | 231 |
let val v = find_index (not o equal Rat.zero) cs |
18011
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cleaned up nth, nth_update, nth_map and nth_string functions
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parents:
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|
232 |
val d = nth discr v |
17951 | 233 |
val pos = Rat.ge0(el v cs) |
15570 | 234 |
val sv = List.filter (single_var v) nz |
14372
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Finally fixed the counterexample finder. Can now deal with < on real.
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|
235 |
val minmax = |
17951 | 236 |
if pos then if d then Rat.roundup o fst o ratrelmax |
14372
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|
237 |
else ratexact true o ratrelmax |
17951 | 238 |
else if d then Rat.rounddown o fst o ratrelmin |
14372
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Finally fixed the counterexample finder. Can now deal with < on real.
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diff
changeset
|
239 |
else ratexact false o ratrelmin |
17951 | 240 |
val bnds = map (fn (a,le,bs) => (Rat.mult(a,Rat.inv(el v bs)),le)) sv |
241 |
val x = minmax((Rat.zero,if pos then true else false)::bnds) |
|
14372
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Finally fixed the counterexample finder. Can now deal with < on real.
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|
242 |
val ineqs' = elim1 v x nz |
18011
685d95c793ff
cleaned up nth, nth_update, nth_map and nth_string functions
haftmann
parents:
17951
diff
changeset
|
243 |
val ex' = nth_update (v, x) ex |
14372
51ddf8963c95
Finally fixed the counterexample finder. Can now deal with < on real.
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diff
changeset
|
244 |
in pick_vars discr (ineqs',ex') end |
13498 | 245 |
end; |
246 |
||
247 |
fun findex0 discr n lineqs = |
|
15570 | 248 |
let val ineqs = Library.foldl elim_eqns ([],lineqs) |
17951 | 249 |
val rineqs = map (fn (a,le,cs) => (Rat.rat_of_intinf a, le, map Rat.rat_of_intinf cs)) |
14372
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Finally fixed the counterexample finder. Can now deal with < on real.
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diff
changeset
|
250 |
ineqs |
17951 | 251 |
in pick_vars discr (rineqs,replicate n Rat.zero) end; |
13498 | 252 |
|
253 |
(* ------------------------------------------------------------------------- *) |
|
254 |
(* End of counter example finder. The actual decision procedure starts here. *) |
|
255 |
(* ------------------------------------------------------------------------- *) |
|
256 |
||
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|
257 |
(* ------------------------------------------------------------------------- *) |
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|
258 |
(* Calculate new (in)equality type after addition. *) |
aeb97860d352
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|
259 |
(* ------------------------------------------------------------------------- *) |
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|
260 |
|
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|
261 |
fun find_add_type(Eq,x) = x |
aeb97860d352
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diff
changeset
|
262 |
| find_add_type(x,Eq) = x |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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changeset
|
263 |
| find_add_type(_,Lt) = Lt |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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changeset
|
264 |
| find_add_type(Lt,_) = Lt |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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diff
changeset
|
265 |
| find_add_type(Le,Le) = Le; |
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|
266 |
|
aeb97860d352
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|
267 |
(* ------------------------------------------------------------------------- *) |
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|
268 |
(* Multiply out an (in)equation. *) |
aeb97860d352
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|
269 |
(* ------------------------------------------------------------------------- *) |
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|
270 |
|
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|
271 |
fun multiply_ineq n (i as Lineq(k,ty,l,just)) = |
aeb97860d352
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|
272 |
if n = 1 then i |
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changeset
|
273 |
else if n = 0 andalso ty = Lt then sys_error "multiply_ineq" |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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parents:
diff
changeset
|
274 |
else if n < 0 andalso (ty=Le orelse ty=Lt) then sys_error "multiply_ineq" |
17524 | 275 |
else Lineq (n * k, ty, map (curry op* n) l, Multiplied (n, just)); |
5982
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|
276 |
|
aeb97860d352
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|
277 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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changeset
|
278 |
(* Add together (in)equations. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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|
279 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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changeset
|
280 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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changeset
|
281 |
fun add_ineq (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
|
282 |
let val l = map2 (op +) (l1,l2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
283 |
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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changeset
|
284 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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diff
changeset
|
285 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
286 |
(* 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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parents:
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changeset
|
287 |
(* If they're both inequalities, 1st coefficient must be +ve, 2nd -ve. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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changeset
|
288 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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diff
changeset
|
289 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
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parents:
diff
changeset
|
290 |
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.
nipkow
parents:
diff
changeset
|
291 |
let val c1 = el v l1 and c2 = el v l2 |
16358 | 292 |
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
|
293 |
val m1 = m div (abs c1) and m2 = m div (abs c2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
294 |
val (n1,n2) = |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
295 |
if (c1 >= 0) = (c2 >= 0) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
296 |
then if ty1 = Eq then (~m1,m2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
297 |
else if ty2 = Eq then (m1,~m2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
298 |
else sys_error "elim_var" |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
299 |
else (m1,m2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
300 |
val (p1,p2) = if ty1=Eq andalso ty2=Eq andalso (n1 = ~1 orelse n2 = ~1) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
301 |
then (~n1,~n2) else (n1,n2) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
302 |
in add_ineq (multiply_ineq n1 i1) (multiply_ineq n2 i2) end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
303 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
304 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
305 |
(* The main refutation-finding code. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
306 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
307 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
308 |
fun is_trivial (Lineq(_,_,l,_)) = forall (fn i => i=0) l; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
309 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
310 |
fun is_answer (ans as Lineq(k,ty,l,_)) = |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
311 |
case ty of Eq => k <> 0 | Le => k > 0 | Lt => k >= 0; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
312 |
|
16358 | 313 |
fun calc_blowup (l:IntInf.int list) = |
17496 | 314 |
let val (p,n) = List.partition (curry (op <) 0) (List.filter (curry (op <>) 0) l) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
315 |
in (length p) * (length n) end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
316 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
317 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
318 |
(* Main elimination code: *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
319 |
(* *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
320 |
(* (1) Looks for immediate solutions (false assertions with no variables). *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
321 |
(* *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
322 |
(* (2) If there are any equations, picks a variable with the lowest absolute *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
323 |
(* coefficient in any of them, and uses it to eliminate. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
324 |
(* *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
325 |
(* (3) Otherwise, chooses a variable in the inequality to minimize the *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
326 |
(* blowup (number of consequences generated) and eliminates it. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
327 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
328 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
329 |
fun allpairs f xs ys = |
15570 | 330 |
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
|
331 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
332 |
fun extract_first p = |
15531 | 333 |
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
|
334 |
else extract (y::xs) ys |
15531 | 335 |
| extract xs [] = (NONE,xs) |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
336 |
in extract [] end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
337 |
|
6056 | 338 |
fun print_ineqs ineqs = |
9073 | 339 |
if !trace then |
12262 | 340 |
tracing(cat_lines(""::map (fn Lineq(c,t,l,_) => |
16358 | 341 |
IntInf.toString c ^ |
9073 | 342 |
(case t of Eq => " = " | Lt=> " < " | Le => " <= ") ^ |
16358 | 343 |
commas(map IntInf.toString l)) ineqs)) |
9073 | 344 |
else (); |
6056 | 345 |
|
13498 | 346 |
type history = (int * lineq list) list; |
347 |
datatype result = Success of injust | Failure of history; |
|
348 |
||
349 |
fun elim(ineqs,hist) = |
|
9073 | 350 |
let val dummy = print_ineqs ineqs; |
15570 | 351 |
val (triv,nontriv) = List.partition is_trivial ineqs in |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
352 |
if not(null triv) |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
353 |
then case Library.find_first is_answer triv of |
15531 | 354 |
NONE => elim(nontriv,hist) |
355 |
| SOME(Lineq(_,_,_,j)) => Success j |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
356 |
else |
13498 | 357 |
if null nontriv then Failure(hist) |
358 |
else |
|
15570 | 359 |
let val (eqs,noneqs) = List.partition (fn (Lineq(_,ty,_,_)) => ty=Eq) nontriv in |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
360 |
if not(null eqs) then |
15570 | 361 |
let val clist = Library.foldl (fn (cs,Lineq(_,_,l,_)) => l union cs) ([],eqs) |
16358 | 362 |
val sclist = sort (fn (x,y) => IntInf.compare(abs(x),abs(y))) |
15570 | 363 |
(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
|
364 |
val c = hd sclist |
15531 | 365 |
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
|
366 |
extract_first (fn Lineq(_,_,l,_) => c mem l) eqs |
13498 | 367 |
val v = find_index_eq c ceq |
15570 | 368 |
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
|
369 |
(othereqs @ noneqs) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
370 |
val others = map (elim_var v eq) roth @ ioth |
13498 | 371 |
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
|
372 |
else |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
373 |
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
|
374 |
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
|
375 |
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
|
376 |
val blows = map calc_blowup coeffs |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
377 |
val iblows = blows ~~ numlist |
15570 | 378 |
val nziblows = List.filter (fn (i,_) => i<>0) iblows |
13498 | 379 |
in if null nziblows then Failure((~1,nontriv)::hist) |
380 |
else |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
381 |
let val (c,v) = hd(sort (fn (x,y) => int_ord(fst(x),fst(y))) nziblows) |
15570 | 382 |
val (no,yes) = List.partition (fn (Lineq(_,_,l,_)) => el v l = 0) ineqs |
383 |
val (pos,neg) = List.partition(fn (Lineq(_,_,l,_)) => el v l > 0) yes |
|
13498 | 384 |
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
|
385 |
end |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
386 |
end |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
387 |
end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
388 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
389 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
390 |
(* Translate back a proof. *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
391 |
(* ------------------------------------------------------------------------- *) |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
392 |
|
9073 | 393 |
fun trace_thm msg th = |
12262 | 394 |
if !trace then (tracing msg; tracing (Display.string_of_thm th); th) else th; |
9073 | 395 |
|
396 |
fun trace_msg msg = |
|
12262 | 397 |
if !trace then tracing msg else (); |
9073 | 398 |
|
13498 | 399 |
(* FIXME OPTIMIZE!!!! (partly done already) |
6056 | 400 |
Addition/Multiplication need i*t representation rather than t+t+... |
10691 | 401 |
Get rid of Mulitplied(2). For Nat LA_Data.number_of should return Suc^n |
402 |
because Numerals are not known early enough. |
|
6056 | 403 |
|
404 |
Simplification may detect a contradiction 'prematurely' due to type |
|
405 |
information: n+1 <= 0 is simplified to False and does not need to be crossed |
|
406 |
with 0 <= n. |
|
407 |
*) |
|
408 |
local |
|
409 |
exception FalseE of thm |
|
410 |
in |
|
17515 | 411 |
fun mkthm (sg, ss) asms just = |
15922
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
412 |
let val {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset, ...} = |
16458 | 413 |
Data.get sg; |
17877
67d5ab1cb0d8
Simplifier.inherit_context instead of Simplifier.inherit_bounds;
wenzelm
parents:
17613
diff
changeset
|
414 |
val simpset' = Simplifier.inherit_context ss simpset; |
15570 | 415 |
val atoms = Library.foldl (fn (ats,(lhs,_,_,rhs,_,_)) => |
6056 | 416 |
map fst lhs union (map fst rhs union ats)) |
15570 | 417 |
([], List.mapPartial (fn thm => if Thm.no_prems thm |
13464 | 418 |
then LA_Data.decomp sg (concl_of thm) |
15531 | 419 |
else NONE) asms) |
6056 | 420 |
|
10575 | 421 |
fun add2 thm1 thm2 = |
6102 | 422 |
let val conj = thm1 RS (thm2 RS LA_Logic.conjI) |
15531 | 423 |
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
|
424 |
end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
425 |
|
15531 | 426 |
fun try_add [] _ = NONE |
10575 | 427 |
| try_add (thm1::thm1s) thm2 = case add2 thm1 thm2 of |
15531 | 428 |
NONE => try_add thm1s thm2 | some => some; |
10575 | 429 |
|
430 |
fun addthms thm1 thm2 = |
|
431 |
case add2 thm1 thm2 of |
|
15531 | 432 |
NONE => (case try_add ([thm1] RL inj_thms) thm2 of |
15570 | 433 |
NONE => ( valOf(try_add ([thm2] RL inj_thms) thm1) |
15660 | 434 |
handle Option => |
14360 | 435 |
(trace_thm "" thm1; trace_thm "" thm2; |
436 |
sys_error "Lin.arith. failed to add thms") |
|
437 |
) |
|
15531 | 438 |
| SOME thm => thm) |
439 |
| SOME thm => thm; |
|
10575 | 440 |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
441 |
fun multn(n,thm) = |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
442 |
let fun mul(i,th) = if i=1 then th else mul(i-1, addthms thm th) |
6102 | 443 |
in if n < 0 then mul(~n,thm) RS LA_Logic.sym else mul(n,thm) end; |
15184 | 444 |
(* |
10691 | 445 |
fun multn2(n,thm) = |
15531 | 446 |
let val SOME(mth,cv) = |
447 |
get_first (fn (th,cv) => SOME(thm RS th,cv) handle THM _ => NONE) mult_mono_thms |
|
10691 | 448 |
val ct = cterm_of sg (LA_Data.number_of(n,#T(rep_cterm cv))) |
449 |
in instantiate ([],[(cv,ct)]) mth end |
|
15184 | 450 |
*) |
451 |
fun multn2(n,thm) = |
|
15531 | 452 |
let val SOME(mth) = |
453 |
get_first (fn th => SOME(thm RS th) handle THM _ => NONE) mult_mono_thms |
|
15184 | 454 |
fun cvar(th,_ $ (_ $ _ $ var)) = cterm_of (#sign(rep_thm th)) var; |
455 |
val cv = cvar(mth, hd(prems_of mth)); |
|
456 |
val ct = cterm_of sg (LA_Data.number_of(n,#T(rep_cterm cv))) |
|
457 |
in instantiate ([],[(cv,ct)]) mth end |
|
10691 | 458 |
|
6056 | 459 |
fun simp thm = |
17515 | 460 |
let val thm' = trace_thm "Simplified:" (full_simplify simpset' thm) |
6102 | 461 |
in if LA_Logic.is_False thm' then raise FalseE thm' else thm' end |
6056 | 462 |
|
18011
685d95c793ff
cleaned up nth, nth_update, nth_map and nth_string functions
haftmann
parents:
17951
diff
changeset
|
463 |
fun mk(Asm i) = trace_thm "Asm" (nth asms i) |
685d95c793ff
cleaned up nth, nth_update, nth_map and nth_string functions
haftmann
parents:
17951
diff
changeset
|
464 |
| mk(Nat i) = (trace_msg "Nat"; LA_Logic.mk_nat_thm sg (nth atoms i)) |
9420 | 465 |
| mk(LessD(j)) = trace_thm "L" (hd([mk j] RL lessD)) |
9073 | 466 |
| mk(NotLeD(j)) = trace_thm "NLe" (mk j RS LA_Logic.not_leD) |
9420 | 467 |
| mk(NotLeDD(j)) = trace_thm "NLeD" (hd([mk j RS LA_Logic.not_leD] RL lessD)) |
9073 | 468 |
| mk(NotLessD(j)) = trace_thm "NL" (mk j RS LA_Logic.not_lessD) |
469 |
| mk(Added(j1,j2)) = simp (trace_thm "+" (addthms (mk j1) (mk j2))) |
|
16358 | 470 |
| mk(Multiplied(n,j)) = (trace_msg("*"^IntInf.toString n); trace_thm "*" (multn(n,mk j))) |
471 |
| 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
|
472 |
|
9073 | 473 |
in trace_msg "mkthm"; |
12932 | 474 |
let val thm = trace_thm "Final thm:" (mk just) |
17515 | 475 |
in let val fls = simplify simpset' thm |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
476 |
in trace_thm "After simplification:" fls; |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
477 |
if LA_Logic.is_False fls then fls |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
478 |
else |
15570 | 479 |
(tracing "Assumptions:"; List.app print_thm asms; |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
480 |
tracing "Proved:"; print_thm fls; |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
481 |
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
|
482 |
\Please inform Tobias Nipkow (nipkow@in.tum.de)."; |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
483 |
fls) |
12932 | 484 |
end |
485 |
end handle FalseE thm => (trace_thm "False reached early:" thm; thm) |
|
486 |
end |
|
6056 | 487 |
end; |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
488 |
|
16358 | 489 |
fun coeff poly atom : IntInf.int = |
17325 | 490 |
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
|
491 |
|
16358 | 492 |
fun lcms is = Library.foldl lcm (1, is); |
10691 | 493 |
|
494 |
fun integ(rlhs,r,rel,rrhs,s,d) = |
|
17951 | 495 |
let val (rn,rd) = Rat.quotient_of_rat r and (sn,sd) = Rat.quotient_of_rat s |
496 |
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
|
497 |
fun mult(t,r) = |
17951 | 498 |
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
|
499 |
in (t,i * (m div j)) end |
12932 | 500 |
in (m,(map mult rlhs, rn*(m div rd), rel, map mult rrhs, sn*(m div sd), d)) end |
10691 | 501 |
|
13498 | 502 |
fun mklineq n atoms = |
503 |
fn (item,k) => |
|
504 |
let val (m,(lhs,i,rel,rhs,j,discrete)) = integ item |
|
505 |
val lhsa = map (coeff lhs) atoms |
|
506 |
and rhsa = map (coeff rhs) atoms |
|
507 |
val diff = map2 (op -) (rhsa,lhsa) |
|
508 |
val c = i-j |
|
509 |
val just = Asm k |
|
510 |
fun lineq(c,le,cs,j) = Lineq(c,le,cs, if m=1 then j else Multiplied2(m,j)) |
|
511 |
in case rel of |
|
512 |
"<=" => lineq(c,Le,diff,just) |
|
513 |
| "~<=" => if discrete |
|
514 |
then lineq(1-c,Le,map (op ~) diff,NotLeDD(just)) |
|
515 |
else lineq(~c,Lt,map (op ~) diff,NotLeD(just)) |
|
516 |
| "<" => if discrete |
|
517 |
then lineq(c+1,Le,diff,LessD(just)) |
|
518 |
else lineq(c,Lt,diff,just) |
|
519 |
| "~<" => lineq(~c,Le,map (op~) diff,NotLessD(just)) |
|
520 |
| "=" => lineq(c,Eq,diff,just) |
|
521 |
| _ => sys_error("mklineq" ^ rel) |
|
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
522 |
end; |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
523 |
|
13498 | 524 |
(* ------------------------------------------------------------------------- *) |
525 |
(* Print (counter) example *) |
|
526 |
(* ------------------------------------------------------------------------- *) |
|
527 |
||
528 |
fun print_atom((a,d),r) = |
|
17951 | 529 |
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
|
530 |
val s = if d then IntInf.toString p else |
13498 | 531 |
if p = 0 then "0" |
15965
f422f8283491
Use of IntInf.int instead of int in most numeric simprocs; avoids
paulson
parents:
15922
diff
changeset
|
532 |
else IntInf.toString p ^ "/" ^ IntInf.toString q |
13498 | 533 |
in a ^ " = " ^ s end; |
534 |
||
535 |
fun print_ex sds = |
|
17496 | 536 |
curry (op ~~) sds |
537 |
#> map print_atom |
|
538 |
#> commas |
|
539 |
#> curry (op ^) "Counter example:\n" |
|
540 |
#> tracing; |
|
13498 | 541 |
|
542 |
fun trace_ex(sg,params,atoms,discr,n,hist:history) = |
|
543 |
if null hist then () |
|
544 |
else let val frees = map Free params; |
|
545 |
fun s_of_t t = Sign.string_of_term sg (subst_bounds(frees,t)); |
|
546 |
val (v,lineqs) :: hist' = hist |
|
547 |
val start = if v = ~1 then (findex0 discr n lineqs,hist') |
|
17951 | 548 |
else (replicate n Rat.zero,hist) |
13516 | 549 |
in warning "arith failed - see trace for a counter example"; |
550 |
print_ex ((map s_of_t atoms)~~discr) (findex discr start) |
|
13498 | 551 |
handle NoEx => |
552 |
(tracing "The decision procedure failed to prove your proposition\n\ |
|
553 |
\but could not construct a counter example either.\n\ |
|
554 |
\Probably the proposition is true but cannot be proved\n\ |
|
555 |
\by the incomplete decision procedure.") |
|
14386 | 556 |
end; |
13498 | 557 |
|
6056 | 558 |
fun mknat pTs ixs (atom,i) = |
6128 | 559 |
if LA_Logic.is_nat(pTs,atom) |
6056 | 560 |
then let val l = map (fn j => if j=i then 1 else 0) ixs |
15531 | 561 |
in SOME(Lineq(0,Le,l,Nat(i))) end |
562 |
else NONE |
|
6056 | 563 |
|
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
564 |
(* This code is tricky. It takes a list of premises in the order they occur |
15531 | 565 |
in the subgoal. Numerical premises are coded as SOME(tuple), non-numerical |
566 |
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
|
567 |
a Lineq. The tricky bit is to convert ~= which is split into two cases < and |
13498 | 568 |
>. 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
|
569 |
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
|
570 |
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
|
571 |
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
|
572 |
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
|
573 |
|
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
574 |
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
|
575 |
*) |
13464 | 576 |
fun split_items(items) = |
577 |
let fun elim_neq front _ [] = [rev front] |
|
15531 | 578 |
| elim_neq front n (NONE::ineqs) = elim_neq front (n+1) ineqs |
579 |
| elim_neq front n (SOME(ineq as (l,i,rel,r,j,d))::ineqs) = |
|
580 |
if rel = "~=" then elim_neq front n (ineqs @ [SOME(l,i,"<",r,j,d)]) @ |
|
581 |
elim_neq front n (ineqs @ [SOME(r,j,"<",l,i,d)]) |
|
13464 | 582 |
else elim_neq ((ineq,n) :: front) (n+1) ineqs |
583 |
in elim_neq [] 0 items end; |
|
584 |
||
13498 | 585 |
fun add_atoms(ats,((lhs,_,_,rhs,_,_),_)) = |
586 |
(map fst lhs) union ((map fst rhs) union ats) |
|
13464 | 587 |
|
13498 | 588 |
fun add_datoms(dats,((lhs,_,_,rhs,_,d),_)) = |
589 |
(map (pair d o fst) lhs) union ((map (pair d o fst) rhs) union dats) |
|
590 |
||
15570 | 591 |
fun discr initems = map fst (Library.foldl add_datoms ([],initems)); |
13464 | 592 |
|
13498 | 593 |
fun refutes sg (pTs,params) ex = |
594 |
let |
|
595 |
fun refute (initems::initemss) js = |
|
15570 | 596 |
let val atoms = Library.foldl add_atoms ([],initems) |
13498 | 597 |
val n = length atoms |
598 |
val mkleq = mklineq n atoms |
|
599 |
val ixs = 0 upto (n-1) |
|
600 |
val iatoms = atoms ~~ ixs |
|
15570 | 601 |
val natlineqs = List.mapPartial (mknat pTs ixs) iatoms |
13498 | 602 |
val ineqs = map mkleq initems @ natlineqs |
603 |
in case elim(ineqs,[]) of |
|
604 |
Success(j) => |
|
605 |
(trace_msg "Contradiction!"; refute initemss (js@[j])) |
|
606 |
| Failure(hist) => |
|
607 |
(if not ex then () |
|
608 |
else trace_ex(sg,params,atoms,discr initems,n,hist); |
|
15531 | 609 |
NONE) |
13498 | 610 |
end |
15531 | 611 |
| refute [] js = SOME js |
13498 | 612 |
in refute end; |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
613 |
|
13498 | 614 |
fun refute sg ps ex items = refutes sg ps ex (split_items items) []; |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
615 |
|
17613 | 616 |
fun refute_tac ss (i,justs) = |
6074 | 617 |
fn state => |
618 |
let val sg = #sign(rep_thm state) |
|
16458 | 619 |
val {neqE, ...} = Data.get sg; |
15922
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
620 |
fun just1 j = REPEAT_DETERM(eresolve_tac neqE i) THEN |
17613 | 621 |
METAHYPS (fn asms => rtac (mkthm (sg, ss) asms j) 1) i |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
622 |
in DETERM(resolve_tac [LA_Logic.notI,LA_Logic.ccontr] i) THEN |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
623 |
EVERY(map just1 justs) |
6074 | 624 |
end |
625 |
state; |
|
626 |
||
15570 | 627 |
fun count P xs = length(List.filter P xs); |
14510 | 628 |
|
629 |
(* The limit on the number of ~= allowed. |
|
630 |
Because each ~= is split into two cases, this can lead to an explosion. |
|
631 |
*) |
|
632 |
val fast_arith_neq_limit = ref 9; |
|
633 |
||
13498 | 634 |
fun prove sg ps ex Hs concl = |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
635 |
let val Hitems = map (LA_Data.decomp sg) Hs |
15531 | 636 |
in if count (fn NONE => false | SOME(_,_,r,_,_,_) => r = "~=") Hitems |
637 |
> !fast_arith_neq_limit then NONE |
|
14510 | 638 |
else |
639 |
case LA_Data.decomp sg concl of |
|
15531 | 640 |
NONE => refute sg ps ex (Hitems@[NONE]) |
641 |
| SOME(citem as (r,i,rel,l,j,d)) => |
|
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
642 |
let val neg::rel0 = explode rel |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
643 |
val nrel = if neg = "~" then implode rel0 else "~"^rel |
15531 | 644 |
in refute sg ps ex (Hitems @ [SOME(r,i,nrel,l,j,d)]) end |
6074 | 645 |
end; |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
646 |
|
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
647 |
(* |
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
648 |
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
|
649 |
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
|
650 |
*) |
17613 | 651 |
fun simpset_lin_arith_tac ss ex i st = SUBGOAL (fn (A,_) => |
13498 | 652 |
let val params = rev(Logic.strip_params A) |
653 |
val pTs = map snd params |
|
6056 | 654 |
val Hs = Logic.strip_assums_hyp A |
6074 | 655 |
val concl = Logic.strip_assums_concl A |
12932 | 656 |
in trace_thm ("Trying to refute subgoal " ^ string_of_int i) st; |
13498 | 657 |
case prove (Thm.sign_of_thm st) (pTs,params) ex Hs concl of |
15531 | 658 |
NONE => (trace_msg "Refutation failed."; no_tac) |
17613 | 659 |
| SOME js => (trace_msg "Refutation succeeded."; refute_tac ss (i,js)) |
9420 | 660 |
end) i st; |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
661 |
|
17892 | 662 |
fun lin_arith_tac ex i st = |
663 |
simpset_lin_arith_tac (Simplifier.theory_context (Thm.theory_of_thm st) Simplifier.empty_ss) |
|
664 |
ex i st; |
|
17613 | 665 |
|
666 |
fun cut_lin_arith_tac ss i = |
|
667 |
cut_facts_tac (Simplifier.prems_of_ss ss) i THEN |
|
668 |
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
|
669 |
|
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
670 |
(** Forward proof from theorems **) |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
671 |
|
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
672 |
(* More tricky code. Needs to arrange the proofs of the multiple cases (due |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
673 |
to splits of ~= premises) such that it coincides with the order of the cases |
13498 | 674 |
generated by function split_items. *) |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
675 |
|
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
676 |
datatype splittree = Tip of thm list |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
677 |
| Spl of thm * cterm * splittree * cterm * splittree |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
678 |
|
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
679 |
fun extract imp = |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
680 |
let val (Il,r) = Thm.dest_comb imp |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
681 |
val (_,imp1) = Thm.dest_comb Il |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
682 |
val (Ict1,_) = Thm.dest_comb imp1 |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
683 |
val (_,ct1) = Thm.dest_comb Ict1 |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
684 |
val (Ir,_) = Thm.dest_comb r |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
685 |
val (_,Ict2r) = Thm.dest_comb Ir |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
686 |
val (Ict2,_) = Thm.dest_comb Ict2r |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
687 |
val (_,ct2) = Thm.dest_comb Ict2 |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
688 |
in (ct1,ct2) end; |
6074 | 689 |
|
15922
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
690 |
fun splitasms sg asms = |
16458 | 691 |
let val {neqE, ...} = Data.get sg; |
15922
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
692 |
fun split(asms',[]) = Tip(rev asms') |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
693 |
| split(asms',asm::asms) = |
15922
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
694 |
(case get_first (fn th => SOME(asm COMP th) handle THM _ => NONE) neqE |
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
695 |
of SOME spl => |
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
696 |
let val (ct1,ct2) = extract(cprop_of spl) |
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
697 |
val thm1 = assume ct1 and thm2 = assume ct2 |
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
698 |
in Spl(spl,ct1,split(asms',asms@[thm1]),ct2,split(asms',asms@[thm2])) |
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
699 |
end |
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
700 |
| NONE => split(asm::asms', asms)) |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
701 |
in split([],asms) end; |
6074 | 702 |
|
17515 | 703 |
fun fwdproof ctxt (Tip asms) (j::js) = (mkthm ctxt asms j, js) |
704 |
| fwdproof ctxt (Spl(thm,ct1,tree1,ct2,tree2)) js = |
|
705 |
let val (thm1,js1) = fwdproof ctxt tree1 js |
|
706 |
val (thm2,js2) = fwdproof ctxt tree2 js1 |
|
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
707 |
val thm1' = implies_intr ct1 thm1 |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
708 |
val thm2' = implies_intr ct2 thm2 |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
709 |
in (thm2' COMP (thm1' COMP thm), js2) end; |
15531 | 710 |
(* needs handle THM _ => NONE ? *) |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
711 |
|
17515 | 712 |
fun prover (ctxt as (sg, _)) thms Tconcl js pos = |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
713 |
let val nTconcl = LA_Logic.neg_prop Tconcl |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
714 |
val cnTconcl = cterm_of sg nTconcl |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
715 |
val nTconclthm = assume cnTconcl |
15922
7ef183f3fc98
neqE applies even if the type is not one which partakes in linear arithmetic.
nipkow
parents:
15660
diff
changeset
|
716 |
val tree = splitasms sg (thms @ [nTconclthm]) |
17515 | 717 |
val (thm,_) = fwdproof ctxt tree js |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
718 |
val contr = if pos then LA_Logic.ccontr else LA_Logic.notI |
15531 | 719 |
in SOME(LA_Logic.mk_Eq((implies_intr cnTconcl thm) COMP contr)) end |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
720 |
(* in case concl contains ?-var, which makes assume fail: *) |
15531 | 721 |
handle THM _ => NONE; |
13186
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
722 |
|
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
723 |
(* PRE: concl is not negated! |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
724 |
This assumption is OK because |
ef8ed6adcb38
Big update. Allows case splitting on ~= now (trying both < and >).
nipkow
parents:
13105
diff
changeset
|
725 |
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
|
726 |
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
|
727 |
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
|
728 |
*) |
15027 | 729 |
fun lin_arith_prover sg ss concl = |
730 |
let |
|
16735 | 731 |
val thms = List.concat(map LA_Logic.atomize (prems_of_ss ss)); |
15027 | 732 |
val Hs = map (#prop o rep_thm) thms |
6102 | 733 |
val Tconcl = LA_Logic.mk_Trueprop concl |
13498 | 734 |
in case prove sg ([],[]) false Hs Tconcl of (* concl provable? *) |
17515 | 735 |
SOME js => prover (sg, ss) thms Tconcl js true |
15531 | 736 |
| NONE => let val nTconcl = LA_Logic.neg_prop Tconcl |
13498 | 737 |
in case prove sg ([],[]) false Hs nTconcl of (* ~concl provable? *) |
17515 | 738 |
SOME js => prover (sg, ss) thms nTconcl js false |
15531 | 739 |
| NONE => NONE |
6079 | 740 |
end |
5982
aeb97860d352
Replaced the puny nat_transitive.ML by the general fast_lin_arith.ML.
nipkow
parents:
diff
changeset
|
741 |
end; |
6074 | 742 |
|
743 |
end; |