# File ‹normarith.ML›

```(*  Title:      HOL/Analysis/normarith.ML
Author:     Amine Chaieb, University of Cambridge

Simple decision procedure for linear problems in Euclidean space.
*)

signature NORM_ARITH =
sig
val norm_arith : Proof.context -> conv
val norm_arith_tac : Proof.context -> int -> tactic
end

structure NormArith : NORM_ARITH =
struct

open Conv;
val bool_eq = op = : bool *bool -> bool
fun dest_ratconst t = case Thm.term_of t of
\<^Const_>‹divide _ for a b› => Rat.make(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
| \<^Const_>‹inverse _ for a› => Rat.make(1, HOLogic.dest_number a |> snd)
| _ => Rat.of_int (HOLogic.dest_number (Thm.term_of t) |> snd)
fun is_ratconst t = can dest_ratconst t
fun augment_norm b t acc = case Thm.term_of t of
\<^Const_>‹norm _ for _› => insert (eq_pair bool_eq (op aconvc)) (b,Thm.dest_arg t) acc
| _ => acc
fun find_normedterms t acc = case Thm.term_of t of
\<^Const_>‹plus \<^typ>‹real› for _ _› =>
find_normedterms (Thm.dest_arg1 t) (find_normedterms (Thm.dest_arg t) acc)
| \<^Const_>‹times \<^typ>‹real› for _ _› =>
if not (is_ratconst (Thm.dest_arg1 t)) then acc else
augment_norm (dest_ratconst (Thm.dest_arg1 t) >= @0)
(Thm.dest_arg t) acc
| _ => augment_norm true t acc

val cterm_lincomb_neg = FuncUtil.Ctermfunc.map (K ~)
fun cterm_lincomb_cmul c t =
if c = @0 then FuncUtil.Ctermfunc.empty else FuncUtil.Ctermfunc.map (fn _ => fn x => x * c) t
fun cterm_lincomb_add l r = FuncUtil.Ctermfunc.combine (curry op +) (fn x => x = @0) l r
fun cterm_lincomb_sub l r = cterm_lincomb_add l (cterm_lincomb_neg r)
fun cterm_lincomb_eq l r = FuncUtil.Ctermfunc.is_empty (cterm_lincomb_sub l r)

(*
val int_lincomb_neg = FuncUtil.Intfunc.map (K ~)
*)
fun int_lincomb_cmul c t =
if c = @0 then FuncUtil.Intfunc.empty else FuncUtil.Intfunc.map (fn _ => fn x => x * c) t
fun int_lincomb_add l r = FuncUtil.Intfunc.combine (curry op +) (fn x => x = @0) l r
(*
fun int_lincomb_sub l r = int_lincomb_add l (int_lincomb_neg r)
fun int_lincomb_eq l r = FuncUtil.Intfunc.is_empty (int_lincomb_sub l r)
*)

fun vector_lincomb t = case Thm.term_of t of
\<^Const_>‹plus _ for _ _› =>
cterm_lincomb_add (vector_lincomb (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
| \<^Const_>‹minus _ for _ _› =>
cterm_lincomb_sub (vector_lincomb (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
| \<^Const_>‹scaleR _ for _ _› =>
cterm_lincomb_cmul (dest_ratconst (Thm.dest_arg1 t)) (vector_lincomb (Thm.dest_arg t))
| \<^Const_>‹uminus _ for _› =>
cterm_lincomb_neg (vector_lincomb (Thm.dest_arg t))
(* FIXME: how should we handle numerals?
| Const(@ {const_name vec},_)\$_ =>
let
val b = ((snd o HOLogic.dest_number o term_of o Thm.dest_arg) t = 0
handle TERM _=> false)
in if b then FuncUtil.Ctermfunc.onefunc (t,@1)
else FuncUtil.Ctermfunc.empty
end
*)
| _ => FuncUtil.Ctermfunc.onefunc (t,@1)

fun vector_lincombs ts =
fold_rev
(fn t => fn fns => case AList.lookup (op aconvc) fns t of
NONE =>
let val f = vector_lincomb t
in case find_first (fn (_,f') => cterm_lincomb_eq f f') fns of
SOME (_,f') => (t,f') :: fns
| NONE => (t,f) :: fns
end
| SOME _ => fns) ts []

fun replacenegnorms cv t = case Thm.term_of t of
\<^Const_>‹plus \<^typ>‹real› for _ _› => binop_conv (replacenegnorms cv) t
| \<^Const_>‹times \<^typ>‹real› for _ _› =>
if dest_ratconst (Thm.dest_arg1 t) < @0 then arg_conv cv t else Thm.reflexive t
| _ => Thm.reflexive t
(*
fun flip v eq =
if FuncUtil.Ctermfunc.defined eq v
then FuncUtil.Ctermfunc.update (v, ~ (FuncUtil.Ctermfunc.apply eq v)) eq else eq
*)
fun allsubsets s = case s of
[] => [[]]
|(a::t) => let val res = allsubsets t in
map (cons a) res @ res end
fun evaluate env lin =
FuncUtil.Intfunc.fold (fn (x,c) => fn s => s + c * (FuncUtil.Intfunc.apply env x))
lin @0

fun solve (vs,eqs) = case (vs,eqs) of
([],[]) => SOME (FuncUtil.Intfunc.onefunc (0,@1))
|(_,eq::oeqs) =>
(case filter (member (op =) vs) (FuncUtil.Intfunc.dom eq) of (*FIXME use find_first here*)
[] => NONE
| v::_ =>
if FuncUtil.Intfunc.defined eq v
then
let
val c = FuncUtil.Intfunc.apply eq v
val vdef = int_lincomb_cmul (~ (Rat.inv c)) eq
fun eliminate eqn = if not (FuncUtil.Intfunc.defined eqn v) then eqn
else int_lincomb_add (int_lincomb_cmul (FuncUtil.Intfunc.apply eqn v) vdef) eqn
in (case solve (remove (op =) v vs, map eliminate oeqs) of
NONE => NONE
| SOME soln => SOME (FuncUtil.Intfunc.update (v, evaluate soln (FuncUtil.Intfunc.delete_safe v vdef)) soln))
end
else NONE)

fun combinations k l = if k = 0 then [[]] else
case l of
[] => []
| h::t => map (cons h) (combinations (k - 1) t) @ combinations k t

fun vertices vs eqs =
let
fun vertex cmb = case solve(vs,cmb) of
NONE => NONE
| SOME soln => SOME (map (fn v => FuncUtil.Intfunc.tryapplyd soln v @0) vs)
val rawvs = map_filter vertex (combinations (length vs) eqs)
val unset = filter (forall (fn c => c >= @0)) rawvs
in fold_rev (insert (eq_list op =)) unset []
end

val subsumes = eq_list (fn (x, y) => Rat.abs x <= Rat.abs y)

fun subsume todo dun = case todo of
[] => dun
|v::ovs =>
let val dun' = if exists (fn w => subsumes (w, v)) dun then dun
else v:: filter (fn w => not (subsumes (v, w))) dun
in subsume ovs dun'
end;

fun match_mp PQ P = P RS PQ;

fun cterm_of_rat x =
let val (a, b) = Rat.dest x
in
if b = 1 then Numeral.mk_cnumber \<^ctyp>‹real› a
else
\<^instantiate>‹
a = ‹Numeral.mk_cnumber \<^ctyp>‹real› a› and
b = ‹Numeral.mk_cnumber \<^ctyp>‹real› b›
in cterm ‹a / b› for a b :: real›
end;

fun norm_cmul_rule c th = Thm.instantiate' [] [SOME (cterm_of_rat c)] (th RS @{thm norm_cmul_rule_thm});

(* I think here the static context should be sufficient!! *)
fun inequality_canon_rule ctxt =
let
(* FIXME : Should be computed statically!! *)
val real_poly_conv =
Semiring_Normalizer.semiring_normalize_wrapper ctxt
(the (Semiring_Normalizer.match ctxt \<^cterm>‹(0::real) + 1›))
in
fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv
arg_conv (Numeral_Simprocs.field_comp_conv ctxt then_conv real_poly_conv)))
end;

val apply_pth1 = rewr_conv @{thm pth_1};
val apply_pth2 = rewr_conv @{thm pth_2};
val apply_pth3 = rewr_conv @{thm pth_3};
val apply_pth4 = rewrs_conv @{thms pth_4};
val apply_pth5 = rewr_conv @{thm pth_5};
val apply_pth6 = rewr_conv @{thm pth_6};
val apply_pth7 = rewrs_conv @{thms pth_7};
fun apply_pth8 ctxt =
rewr_conv @{thm pth_8} then_conv
arg1_conv (Numeral_Simprocs.field_comp_conv ctxt) then_conv
(try_conv (rewr_conv (mk_meta_eq @{thm scaleR_zero_left})));
fun apply_pth9 ctxt =
rewrs_conv @{thms pth_9} then_conv
arg1_conv (arg1_conv (Numeral_Simprocs.field_comp_conv ctxt));
val apply_ptha = rewr_conv @{thm pth_a};
val apply_pthb = rewrs_conv @{thms pth_b};
val apply_pthc = rewrs_conv @{thms pth_c};
val apply_pthd = try_conv (rewr_conv @{thm pth_d});

fun headvector t = case t of
\<^Const_>‹plus _ for \<^Const_>‹scaleR _ for _ v› _› => v
| \<^Const_>‹scaleR _ for _ v› => v
| _ => error "headvector: non-canonical term"

fun vector_cmul_conv ctxt ct =
((apply_pth5 then_conv arg1_conv (Numeral_Simprocs.field_comp_conv ctxt)) else_conv
(apply_pth6 then_conv binop_conv (vector_cmul_conv ctxt))) ct

fun vector_add_conv ctxt ct = apply_pth7 ct
handle CTERM _ =>
(apply_pth8 ctxt ct
handle CTERM _ =>
(case Thm.term_of ct of
\<^Const_>‹plus _ for lt rt› =>
let
in (case Term_Ord.fast_term_ord (l,r) of
LESS => (apply_pthb then_conv arg_conv (vector_add_conv ctxt)
then_conv apply_pthd) ct
| GREATER => (apply_pthc then_conv arg_conv (vector_add_conv ctxt)
then_conv apply_pthd) ct
| EQUAL => (apply_pth9 ctxt then_conv
arg_conv (vector_add_conv ctxt) then_conv apply_pthd)) ct)
end
| _ => Thm.reflexive ct))

fun vector_canon_conv ctxt ct = case Thm.term_of ct of
\<^Const_>‹plus _ for _ _› =>
let
val ((p,l),r) = Thm.dest_comb ct |>> Thm.dest_comb
val lth = vector_canon_conv ctxt l
val rth = vector_canon_conv ctxt r
val th = Drule.binop_cong_rule p lth rth
in fconv_rule (arg_conv (vector_add_conv ctxt)) th end

| \<^Const_>‹scaleR _ for _ _› =>
let
val (p,r) = Thm.dest_comb ct
val rth = Drule.arg_cong_rule p (vector_canon_conv ctxt r)
in fconv_rule (arg_conv (apply_pth4 else_conv (vector_cmul_conv ctxt))) rth
end

| \<^Const_>‹minus _ for _ _› => (apply_pth2 then_conv (vector_canon_conv ctxt)) ct

| \<^Const_>‹uminus _ for _› => (apply_pth3 then_conv (vector_canon_conv ctxt)) ct

(* FIXME
| Const(@{const_name vec},_)\$n =>
let val n = Thm.dest_arg ct
in if is_ratconst n andalso not (dest_ratconst n =/ @0)
then Thm.reflexive ct else apply_pth1 ct
end
*)
| _ => apply_pth1 ct

fun norm_canon_conv ctxt ct = case Thm.term_of ct of
\<^Const_>‹norm _ for _› => arg_conv (vector_canon_conv ctxt) ct
| _ => raise CTERM ("norm_canon_conv", [ct])

fun int_flip v eq =
if FuncUtil.Intfunc.defined eq v
then FuncUtil.Intfunc.update (v, ~ (FuncUtil.Intfunc.apply eq v)) eq else eq;

local
val concl = Thm.dest_arg o Thm.cprop_of
fun real_vector_combo_prover ctxt translator (nubs,ges,gts) =
let
(* FIXME: Should be computed statically!!*)
val real_poly_conv =
Semiring_Normalizer.semiring_normalize_wrapper ctxt
(the (Semiring_Normalizer.match ctxt \<^cterm>‹(0::real) + 1›))
val sources = map (Thm.dest_arg o Thm.dest_arg1 o concl) nubs
val rawdests = fold_rev (find_normedterms o Thm.dest_arg o concl) (ges @ gts) []
val _ = if not (forall fst rawdests) then error "real_vector_combo_prover: Sanity check"
else ()
val dests = distinct (op aconvc) (map snd rawdests)
val srcfuns = map vector_lincomb sources
val destfuns = map vector_lincomb dests
val vvs = fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom) (srcfuns @ destfuns) []
val n = length srcfuns
val nvs = 1 upto n
val srccombs = srcfuns ~~ nvs
fun consider d =
let
fun coefficients x =
let
val inp =
if FuncUtil.Ctermfunc.defined d x
then FuncUtil.Intfunc.onefunc (0, ~ (FuncUtil.Ctermfunc.apply d x))
else FuncUtil.Intfunc.empty
in fold_rev (fn (f,v) => fn g => if FuncUtil.Ctermfunc.defined f x then FuncUtil.Intfunc.update (v, FuncUtil.Ctermfunc.apply f x) g else g) srccombs inp
end
val equations = map coefficients vvs
val inequalities = map (fn n => FuncUtil.Intfunc.onefunc (n,@1)) nvs
fun plausiblevertices f =
let
val flippedequations = map (fold_rev int_flip f) equations
val constraints = flippedequations @ inequalities
val rawverts = vertices nvs constraints
fun check_solution v =
let
val f = fold_rev FuncUtil.Intfunc.update (nvs ~~ v) (FuncUtil.Intfunc.onefunc (0, @1))
in forall (fn e => evaluate f e = @0) flippedequations
end
val goodverts = filter check_solution rawverts
val signfixups = map (fn n => if member (op =) f n then ~1 else 1) nvs
in map (map2 (fn s => fn c => Rat.of_int s * c) signfixups) goodverts
end
val allverts = fold_rev append (map plausiblevertices (allsubsets nvs)) []
in subsume allverts []
end
fun compute_ineq v =
let
val ths = map_filter (fn (v,t) => if v = @0 then NONE
else SOME(norm_cmul_rule v t))
(v ~~ nubs)
fun end_itlist f xs = split_last xs |> uncurry (fold_rev f)
in inequality_canon_rule ctxt (end_itlist norm_add_rule ths)
end
val ges' = map_filter (try compute_ineq) (fold_rev (append o consider) destfuns []) @
map (inequality_canon_rule ctxt) nubs @ ges
val zerodests = filter
(fn t => null (FuncUtil.Ctermfunc.dom (vector_lincomb t))) (map snd rawdests)

in fst (RealArith.real_linear_prover translator
(zerodests |> map (fn t =>
\<^instantiate>‹'a = ‹Thm.ctyp_of_cterm t› in
lemma ‹norm (0::'a::real_normed_vector) = 0› by simp›),
map (fconv_rule (try_conv (Conv.top_sweep_conv norm_canon_conv ctxt) then_conv
arg_conv (arg_conv real_poly_conv))) ges',
map (fconv_rule (try_conv (Conv.top_sweep_conv norm_canon_conv ctxt) then_conv
arg_conv (arg_conv real_poly_conv))) gts))
end
in val real_vector_combo_prover = real_vector_combo_prover
end;

local
val pth = @{thm norm_imp_pos_and_ge}
val norm_mp = match_mp pth
val concl = Thm.dest_arg o Thm.cprop_of
fun conjunct1 th = th RS @{thm conjunct1}
fun conjunct2 th = th RS @{thm conjunct2}
fun real_vector_ineq_prover ctxt translator (ges,gts) =
let
(*   val _ = error "real_vector_ineq_prover: pause" *)
val ntms = fold_rev find_normedterms (map (Thm.dest_arg o concl) (ges @ gts)) []
val lctab = vector_lincombs (map snd (filter (not o fst) ntms))
val (fxns, ctxt') = Variable.variant_fixes (replicate (length lctab) "x") ctxt
fun mk_norm t =
let val T = Thm.ctyp_of_cterm t
in \<^instantiate>‹'a = T and t in cterm ‹norm t›› end
fun mk_equals l r =
let val T = Thm.ctyp_of_cterm l
in \<^instantiate>‹'a = T and l and r in cterm ‹l ≡ r›› end
val asl = map2 (fn (t,_) => fn n => Thm.assume (mk_equals (mk_norm t) (Thm.cterm_of ctxt' (Free(n,\<^typ>‹real›))))) lctab fxns
val replace_conv = try_conv (rewrs_conv asl)
val replace_rule = fconv_rule (funpow 2 arg_conv (replacenegnorms replace_conv))
val ges' =
fold_rev (fn th => fn ths => conjunct1(norm_mp th)::ths)
asl (map replace_rule ges)
val gts' = map replace_rule gts
val nubs = map (conjunct2 o norm_mp) asl
val th1 = real_vector_combo_prover ctxt' translator (nubs,ges',gts')
val shs = filter (member (fn (t,th) => t aconvc Thm.cprop_of th) asl) (Thm.chyps_of th1)
val th11 = hd (Variable.export ctxt' ctxt [fold Thm.implies_intr shs th1])
val cps = map (swap o Thm.dest_equals) (cprems_of th11)
val th12 = Drule.instantiate_normalize (TVars.empty, Vars.make (map (apfst (dest_Var o Thm.term_of)) cps)) th11
val th13 = fold Thm.elim_implies (map (Thm.reflexive o snd) cps) th12;
in hd (Variable.export ctxt' ctxt [th13])
end
in val real_vector_ineq_prover = real_vector_ineq_prover
end;

local
val rawrule = fconv_rule (arg_conv (rewr_conv @{thm real_eq_0_iff_le_ge_0}))
fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
(* FIXME: Lookup in the context every time!!! Fix this !!!*)
fun splitequation ctxt th acc =
let
val real_poly_neg_conv = #neg
(Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
(the (Semiring_Normalizer.match ctxt \<^cterm>‹(0::real) + 1›)) Thm.term_ord)
val (th1,th2) = conj_pair(rawrule th)
in th1::fconv_rule (arg_conv (arg_conv (real_poly_neg_conv ctxt))) th2::acc
end
in fun real_vector_prover ctxt _ translator (eqs,ges,gts) =
(real_vector_ineq_prover ctxt translator
(fold_rev (splitequation ctxt) eqs ges,gts), RealArith.Trivial)
end;

fun init_conv ctxt =
Simplifier.rewrite (put_simpset HOL_basic_ss ctxt
addsimps ([(*@{thm vec_0}, @{thm vec_1},*) @{thm dist_norm}, @{thm right_minus},
@{thm diff_self}, @{thm norm_zero}] @ @{thms arithmetic_simps} @ @{thms norm_pths}))
then_conv Numeral_Simprocs.field_comp_conv ctxt
then_conv nnf_conv ctxt

fun pure ctxt = fst o RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt);
fun norm_arith ctxt ct =
let
val ctxt' = Variable.declare_term (Thm.term_of ct) ctxt
val th = init_conv ctxt' ct
in Thm.equal_elim (Drule.arg_cong_rule \<^cterm>‹Trueprop› (Thm.symmetric th))
(pure ctxt' (Thm.rhs_of th))
end

fun norm_arith_tac ctxt =
clarify_tac (put_claset HOL_cs ctxt) THEN'
Object_Logic.full_atomize_tac ctxt THEN'
CSUBGOAL ( fn (p,i) => resolve_tac ctxt [norm_arith ctxt (Thm.dest_arg p )] i);

end;
```