author | haftmann |
Tue, 26 Sep 2006 13:34:16 +0200 | |
changeset 20713 | 823967ef47f1 |
parent 20623 | 6ae83d153dd4 |
child 21588 | cd0dc678a205 |
permissions | -rw-r--r-- |
17516 | 1 |
(* ID: $Id$ |
2 |
Author: Amine Chaieb |
|
3 |
||
4 |
Tactic for solving equalities over commutative rings. |
|
5 |
*) |
|
6 |
||
7 |
signature COMM_RING = |
|
8 |
sig |
|
20623 | 9 |
val comm_ring_tac : Proof.context -> int -> tactic |
18708 | 10 |
val setup : theory -> theory |
17516 | 11 |
end |
12 |
||
13 |
structure CommRing: COMM_RING = |
|
14 |
struct |
|
15 |
||
16 |
(* The Cring exception for erronous uses of cring_tac *) |
|
17 |
exception CRing of string; |
|
18 |
||
19 |
(* Zero and One of the commutative ring *) |
|
20713
823967ef47f1
renamed 0 and 1 to HOL.zero and HOL.one respectivly; introduced corresponding syntactic classes
haftmann
parents:
20623
diff
changeset
|
20 |
fun cring_zero T = Const("HOL.zero",T); |
823967ef47f1
renamed 0 and 1 to HOL.zero and HOL.one respectivly; introduced corresponding syntactic classes
haftmann
parents:
20623
diff
changeset
|
21 |
fun cring_one T = Const("HOL.one",T); |
17516 | 22 |
|
23 |
(* reification functions *) |
|
24 |
(* add two polynom expressions *) |
|
25 |
fun polT t = Type ("Commutative_Ring.pol",[t]); |
|
20623 | 26 |
fun polexT t = Type("Commutative_Ring.polex",[t]); |
17516 | 27 |
val nT = HOLogic.natT; |
28 |
fun listT T = Type ("List.list",[T]); |
|
29 |
||
30 |
(* Reification of the constructors *) |
|
31 |
(* Nat*) |
|
32 |
val succ = Const("Suc",nT --> nT); |
|
20713
823967ef47f1
renamed 0 and 1 to HOL.zero and HOL.one respectivly; introduced corresponding syntactic classes
haftmann
parents:
20623
diff
changeset
|
33 |
val zero = Const("HOL.zero",nT); |
823967ef47f1
renamed 0 and 1 to HOL.zero and HOL.one respectivly; introduced corresponding syntactic classes
haftmann
parents:
20623
diff
changeset
|
34 |
val one = Const("HOL.one",nT); |
17516 | 35 |
|
36 |
(* Lists *) |
|
37 |
fun reif_list T [] = Const("List.list.Nil",listT T) |
|
38 |
| reif_list T (x::xs) = Const("List.list.Cons",[T,listT T] ---> listT T) |
|
39 |
$x$(reif_list T xs); |
|
40 |
||
20623 | 41 |
(* pol *) |
17516 | 42 |
fun pol_Pc t = Const("Commutative_Ring.pol.Pc",t --> polT t); |
43 |
fun pol_Pinj t = Const("Commutative_Ring.pol.Pinj",[nT,polT t] ---> polT t); |
|
44 |
fun pol_PX t = Const("Commutative_Ring.pol.PX",[polT t, nT, polT t] ---> polT t); |
|
45 |
||
46 |
(* polex *) |
|
47 |
fun polex_add t = Const("Commutative_Ring.polex.Add",[polexT t,polexT t] ---> polexT t); |
|
48 |
fun polex_sub t = Const("Commutative_Ring.polex.Sub",[polexT t,polexT t] ---> polexT t); |
|
49 |
fun polex_mul t = Const("Commutative_Ring.polex.Mul",[polexT t,polexT t] ---> polexT t); |
|
50 |
fun polex_neg t = Const("Commutative_Ring.polex.Neg",polexT t --> polexT t); |
|
51 |
fun polex_pol t = Const("Commutative_Ring.polex.Pol",polT t --> polexT t); |
|
52 |
fun polex_pow t = Const("Commutative_Ring.polex.Pow",[polexT t, nT] ---> polexT t); |
|
20623 | 53 |
|
17516 | 54 |
(* reification of natural numbers *) |
55 |
fun reif_nat n = |
|
56 |
if n>0 then succ$(reif_nat (n-1)) |
|
57 |
else if n=0 then zero |
|
58 |
else raise CRing "ring_tac: reif_nat negative n"; |
|
59 |
||
60 |
(* reification of polynoms : primitive cring expressions *) |
|
61 |
fun reif_pol T vs t = |
|
62 |
case t of |
|
63 |
Free(_,_) => |
|
64 |
let val i = find_index_eq t vs |
|
65 |
in if i = 0 |
|
66 |
then (pol_PX T)$((pol_Pc T)$ (cring_one T)) |
|
67 |
$one$((pol_Pc T)$(cring_zero T)) |
|
68 |
else (pol_Pinj T)$(reif_nat i)$ |
|
69 |
((pol_PX T)$((pol_Pc T)$ (cring_one T)) |
|
70 |
$one$ |
|
71 |
((pol_Pc T)$(cring_zero T))) |
|
72 |
end |
|
73 |
| _ => (pol_Pc T)$ t; |
|
74 |
||
75 |
||
76 |
(* reification of polynom expressions *) |
|
77 |
fun reif_polex T vs t = |
|
78 |
case t of |
|
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
18708
diff
changeset
|
79 |
Const("HOL.plus",_)$a$b => (polex_add T) |
17516 | 80 |
$ (reif_polex T vs a)$(reif_polex T vs b) |
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
18708
diff
changeset
|
81 |
| Const("HOL.minus",_)$a$b => (polex_sub T) |
17516 | 82 |
$ (reif_polex T vs a)$(reif_polex T vs b) |
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
18708
diff
changeset
|
83 |
| Const("HOL.times",_)$a$b => (polex_mul T) |
17516 | 84 |
$ (reif_polex T vs a)$ (reif_polex T vs b) |
19233
77ca20b0ed77
renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
18708
diff
changeset
|
85 |
| Const("HOL.uminus",_)$a => (polex_neg T) |
17516 | 86 |
$ (reif_polex T vs a) |
87 |
| (Const("Nat.power",_)$a$n) => (polex_pow T) $ (reif_polex T vs a) $ n |
|
88 |
||
89 |
| _ => (polex_pol T) $ (reif_pol T vs t); |
|
90 |
||
91 |
(* reification of the equation *) |
|
92 |
val cr_sort = Sign.read_sort (the_context ()) "{comm_ring,recpower}"; |
|
20623 | 93 |
fun reif_eq thy (eq as Const("op =",Type("fun",a::_))$lhs$rhs) = |
94 |
if Sign.of_sort thy (a,cr_sort) |
|
17516 | 95 |
then |
96 |
let val fs = term_frees eq |
|
20623 | 97 |
val cvs = cterm_of thy (reif_list a fs) |
98 |
val clhs = cterm_of thy (reif_polex a fs lhs) |
|
99 |
val crhs = cterm_of thy (reif_polex a fs rhs) |
|
100 |
val ca = ctyp_of thy a |
|
17516 | 101 |
in (ca,cvs,clhs, crhs) |
102 |
end |
|
103 |
else raise CRing "reif_eq: not an equation over comm_ring + recpower" |
|
20623 | 104 |
| reif_eq _ _ = raise CRing "reif_eq: not an equation"; |
17516 | 105 |
|
106 |
(*The cring tactic *) |
|
107 |
(* Attention: You have to make sure that no t^0 is in the goal!! *) |
|
108 |
(* Use simply rewriting t^0 = 1 *) |
|
20623 | 109 |
val cring_simps = |
110 |
map thm ["mkPX_def", "mkPinj_def","sub_def", "power_add","even_def","pow_if"] @ |
|
111 |
[sym OF [thm "power_add"]]; |
|
17516 | 112 |
|
113 |
val norm_eq = thm "norm_eq" |
|
114 |
||
20623 | 115 |
fun comm_ring_tac ctxt = SUBGOAL (fn (g, i) => |
116 |
let |
|
117 |
val thy = ProofContext.theory_of ctxt |
|
118 |
val cring_ss = Simplifier.local_simpset_of ctxt (* FIXME really the full simpset!? *) |
|
119 |
addsimps cring_simps |
|
120 |
val (ca, cvs, clhs, crhs) = reif_eq thy (HOLogic.dest_Trueprop g) |
|
121 |
val norm_eq_th = |
|
122 |
simplify cring_ss (instantiate' [SOME ca] [SOME clhs, SOME crhs, SOME cvs] norm_eq) |
|
123 |
in |
|
124 |
cut_rules_tac [norm_eq_th] i |
|
125 |
THEN (simp_tac cring_ss i) |
|
126 |
THEN (simp_tac cring_ss i) |
|
127 |
end); |
|
128 |
||
129 |
val comm_ring_meth = |
|
130 |
Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' HEADGOAL (comm_ring_tac ctxt)); |
|
17516 | 131 |
|
132 |
val setup = |
|
20623 | 133 |
Method.add_method ("comm_ring", comm_ring_meth, |
134 |
"reflective decision procedure for equalities over commutative rings") #> |
|
135 |
Method.add_method ("algebra", comm_ring_meth, |
|
136 |
"method for proving algebraic properties (same as comm_ring)"); |
|
17516 | 137 |
|
138 |
end; |