# File ‹Sum_of_Squares/positivstellensatz.ML›

```(*  Title:      HOL/Library/Sum_of_Squares/positivstellensatz.ML
Author:     Amine Chaieb, University of Cambridge

A generic arithmetic prover based on Positivstellensatz certificates
--- also implements Fourier-Motzkin elimination as a special case
Fourier-Motzkin elimination.
*)

(* A functor for finite mappings based on Tables *)

signature FUNC =
sig
include TABLE
val apply : 'a table -> key -> 'a
val applyd :'a table -> (key -> 'a) -> key -> 'a
val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
val dom : 'a table -> key list
val tryapplyd : 'a table -> key -> 'a -> 'a
val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
val choose : 'a table -> key * 'a
val onefunc : key * 'a -> 'a table
end;

functor FuncFun(Key: KEY) : FUNC =
struct

structure Tab = Table(Key);

open Tab;

fun dom a = sort Key.ord (Tab.keys a);
fun applyd f d x = case Tab.lookup f x of
SOME y => y
| NONE => d x;

fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
fun tryapplyd f a d = applyd f (K d) a;
fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
fun combine f z a b =
let
fun h (k,v) t = case Tab.lookup t k of
NONE => Tab.update (k,v) t
| SOME v' => let val w = f v v'
in if z w then Tab.delete k t else Tab.update (k,w) t end;
in Tab.fold h a b end;

fun choose f =
(case Tab.min f of
SOME entry => entry
| NONE => error "FuncFun.choose : Completely empty function")

fun onefunc kv = update kv empty

end;

(* Some standard functors and utility functions for them *)

structure FuncUtil =
struct

structure Intfunc = FuncFun(type key = int val ord = int_ord);
structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord);
structure Ctermfunc = FuncFun(type key = cterm val ord = Thm.fast_term_ord);

type monomial = int Ctermfunc.table;
val monomial_ord = list_ord (prod_ord Thm.fast_term_ord int_ord) o apply2 Ctermfunc.dest
structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)

type poly = Rat.rat Monomialfunc.table;

(* The ordering so we can create canonical HOL polynomials.                  *)

fun dest_monomial mon = sort (Thm.fast_term_ord o apply2 fst) (Ctermfunc.dest mon);

fun monomial_order (m1,m2) =
if Ctermfunc.is_empty m2 then LESS
else if Ctermfunc.is_empty m1 then GREATER
else
let
val mon1 = dest_monomial m1
val mon2 = dest_monomial m2
val deg1 = fold (Integer.add o snd) mon1 0
val deg2 = fold (Integer.add o snd) mon2 0
in if deg1 < deg2 then GREATER
else if deg1 > deg2 then LESS
else list_ord (prod_ord Thm.fast_term_ord int_ord) (mon1,mon2)
end;

end

(* positivstellensatz datatype and prover generation *)

signature REAL_ARITH =
sig

datatype positivstellensatz =
Axiom_eq of int
| Axiom_le of int
| Axiom_lt of int
| Rational_eq of Rat.rat
| Rational_le of Rat.rat
| Rational_lt of Rat.rat
| Square of FuncUtil.poly
| Eqmul of FuncUtil.poly * positivstellensatz
| Sum of positivstellensatz * positivstellensatz
| Product of positivstellensatz * positivstellensatz;

datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree

datatype tree_choice = Left | Right

type prover = tree_choice list ->
(thm list * thm list * thm list -> positivstellensatz -> thm) ->
thm list * thm list * thm list -> thm * pss_tree
type cert_conv = cterm -> thm * pss_tree

val gen_gen_real_arith :
Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
conv * conv * conv * conv * conv * conv * prover -> cert_conv
val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
thm list * thm list * thm list -> thm * pss_tree

val gen_real_arith : Proof.context ->
(Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv

val gen_prover_real_arith : Proof.context -> prover -> cert_conv

val is_ratconst : cterm -> bool
val dest_ratconst : cterm -> Rat.rat
val cterm_of_rat : Rat.rat -> cterm

end

structure RealArith : REAL_ARITH =
struct

open Conv
(* ------------------------------------------------------------------------- *)
(* Data structure for Positivstellensatz refutations.                        *)
(* ------------------------------------------------------------------------- *)

datatype positivstellensatz =
Axiom_eq of int
| Axiom_le of int
| Axiom_lt of int
| Rational_eq of Rat.rat
| Rational_le of Rat.rat
| Rational_lt of Rat.rat
| Square of FuncUtil.poly
| Eqmul of FuncUtil.poly * positivstellensatz
| Sum of positivstellensatz * positivstellensatz
| Product of positivstellensatz * positivstellensatz;
(* Theorems used in the procedure *)

datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
datatype tree_choice = Left | Right
type prover = tree_choice list ->
(thm list * thm list * thm list -> positivstellensatz -> thm) ->
thm list * thm list * thm list -> thm * pss_tree
type cert_conv = cterm -> thm * pss_tree

(* Some useful derived rules *)
fun deduct_antisym_rule tha thb =
Thm.equal_intr (Thm.implies_intr (Thm.cprop_of thb) tha)
(Thm.implies_intr (Thm.cprop_of tha) thb);

fun prove_hyp tha thb =
if exists (curry op aconv (Thm.concl_of tha)) (Thm.hyps_of thb)  (* FIXME !? *)
then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;

val pth = @{lemma "(((x::real) < y) ≡ (y - x > 0))" and "((x ≤ y) ≡ (y - x ≥ 0))" and
"((x = y) ≡ (x - y = 0))" and "((¬(x < y)) ≡ (x - y ≥ 0))" and
"((¬(x ≤ y)) ≡ (x - y > 0))"
by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};

@{lemma "(x = (0::real) ⟹ y = 0 ⟹ x + y = 0 )" and "( x = 0 ⟹ y ≥ 0 ⟹ x + y ≥ 0)" and
"(x = 0 ⟹ y > 0 ⟹ x + y > 0)" and "(x ≥ 0 ⟹ y = 0 ⟹ x + y ≥ 0)" and
"(x ≥ 0 ⟹ y ≥ 0 ⟹ x + y ≥ 0)" and "(x ≥ 0 ⟹ y > 0 ⟹ x + y > 0)" and
"(x > 0 ⟹ y = 0 ⟹ x + y > 0)" and "(x > 0 ⟹ y ≥ 0 ⟹ x + y > 0)" and
"(x > 0 ⟹ y > 0 ⟹ x + y > 0)" by simp_all};

val pth_mul =
@{lemma "(x = (0::real) ⟹ y = 0 ⟹ x * y = 0)" and "(x = 0 ⟹ y ≥ 0 ⟹ x * y = 0)" and
"(x = 0 ⟹ y > 0 ⟹ x * y = 0)" and "(x ≥ 0 ⟹ y = 0 ⟹ x * y = 0)" and
"(x ≥ 0 ⟹ y ≥ 0 ⟹ x * y ≥ 0)" and "(x ≥ 0 ⟹ y > 0 ⟹ x * y ≥ 0)" and
"(x > 0 ⟹  y = 0 ⟹ x * y = 0)" and "(x > 0 ⟹ y ≥ 0 ⟹ x * y ≥ 0)" and
"(x > 0 ⟹  y > 0 ⟹ x * y > 0)"
by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])};

val pth_emul = @{lemma "y = (0::real) ⟹ x * y = 0"  by simp};

val weak_dnf_simps =
List.take (@{thms simp_thms}, 34) @
@{lemma "((P ∧ (Q ∨ R)) = ((P∧Q) ∨ (P∧R)))" and "((Q ∨ R) ∧ P) = ((Q∧P) ∨ (R∧P))" and
"(P ∧ Q) = (Q ∧ P)" and "((P ∨ Q) = (Q ∨ P))" by blast+};

(*
val nnfD_simps =
@{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
"((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
"((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
*)

val choice_iff = @{lemma "(∀x. ∃y. P x y) = (∃f. ∀x. P x (f x))" by metis};
val prenex_simps =
map (fn th => th RS sym)
([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @
@{thms "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});

val real_abs_thms1 = @{lemma
"((-1 * ¦x::real¦ ≥ r) = (-1 * x ≥ r ∧ 1 * x ≥ r))" and
"((-1 * ¦x¦ + a ≥ r) = (a + -1 * x ≥ r ∧ a + 1 * x ≥ r))" and
"((a + -1 * ¦x¦ ≥ r) = (a + -1 * x ≥ r ∧ a + 1 * x ≥ r))" and
"((a + -1 * ¦x¦ + b ≥ r) = (a + -1 * x + b ≥ r ∧ a + 1 * x + b ≥ r))" and
"((a + b + -1 * ¦x¦ ≥ r) = (a + b + -1 * x ≥ r ∧ a + b + 1 * x ≥ r))" and
"((a + b + -1 * ¦x¦ + c ≥ r) = (a + b + -1 * x + c ≥ r ∧ a + b + 1 * x + c ≥ r))" and
"((-1 * max x y ≥ r) = (-1 * x ≥ r ∧ -1 * y ≥ r))" and
"((-1 * max x y + a ≥ r) = (a + -1 * x ≥ r ∧ a + -1 * y ≥ r))" and
"((a + -1 * max x y ≥ r) = (a + -1 * x ≥ r ∧ a + -1 * y ≥ r))" and
"((a + -1 * max x y + b ≥ r) = (a + -1 * x + b ≥ r ∧ a + -1 * y  + b ≥ r))" and
"((a + b + -1 * max x y ≥ r) = (a + b + -1 * x ≥ r ∧ a + b + -1 * y ≥ r))" and
"((a + b + -1 * max x y + c ≥ r) = (a + b + -1 * x + c ≥ r ∧ a + b + -1 * y  + c ≥ r))" and
"((1 * min x y ≥ r) = (1 * x ≥ r ∧ 1 * y ≥ r))" and
"((1 * min x y + a ≥ r) = (a + 1 * x ≥ r ∧ a + 1 * y ≥ r))" and
"((a + 1 * min x y ≥ r) = (a + 1 * x ≥ r ∧ a + 1 * y ≥ r))" and
"((a + 1 * min x y + b ≥ r) = (a + 1 * x + b ≥ r ∧ a + 1 * y  + b ≥ r))" and
"((a + b + 1 * min x y ≥ r) = (a + b + 1 * x ≥ r ∧ a + b + 1 * y ≥ r))" and
"((a + b + 1 * min x y + c ≥ r) = (a + b + 1 * x + c ≥ r ∧ a + b + 1 * y  + c ≥ r))" and
"((min x y ≥ r) = (x ≥ r ∧ y ≥ r))" and
"((min x y + a ≥ r) = (a + x ≥ r ∧ a + y ≥ r))" and
"((a + min x y ≥ r) = (a + x ≥ r ∧ a + y ≥ r))" and
"((a + min x y + b ≥ r) = (a + x + b ≥ r ∧ a + y  + b ≥ r))" and
"((a + b + min x y ≥ r) = (a + b + x ≥ r ∧ a + b + y ≥ r))" and
"((a + b + min x y + c ≥ r) = (a + b + x + c ≥ r ∧ a + b + y + c ≥ r))" and
"((-1 * ¦x¦ > r) = (-1 * x > r ∧ 1 * x > r))" and
"((-1 * ¦x¦ + a > r) = (a + -1 * x > r ∧ a + 1 * x > r))" and
"((a + -1 * ¦x¦ > r) = (a + -1 * x > r ∧ a + 1 * x > r))" and
"((a + -1 * ¦x¦ + b > r) = (a + -1 * x + b > r ∧ a + 1 * x + b > r))" and
"((a + b + -1 * ¦x¦ > r) = (a + b + -1 * x > r ∧ a + b + 1 * x > r))" and
"((a + b + -1 * ¦x¦ + c > r) = (a + b + -1 * x + c > r ∧ a + b + 1 * x + c > r))" and
"((-1 * max x y > r) = ((-1 * x > r) ∧ -1 * y > r))" and
"((-1 * max x y + a > r) = (a + -1 * x > r ∧ a + -1 * y > r))" and
"((a + -1 * max x y > r) = (a + -1 * x > r ∧ a + -1 * y > r))" and
"((a + -1 * max x y + b > r) = (a + -1 * x + b > r ∧ a + -1 * y  + b > r))" and
"((a + b + -1 * max x y > r) = (a + b + -1 * x > r ∧ a + b + -1 * y > r))" and
"((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r ∧ a + b + -1 * y  + c > r))" and
"((min x y > r) = (x > r ∧ y > r))" and
"((min x y + a > r) = (a + x > r ∧ a + y > r))" and
"((a + min x y > r) = (a + x > r ∧ a + y > r))" and
"((a + min x y + b > r) = (a + x + b > r ∧ a + y  + b > r))" and
"((a + b + min x y > r) = (a + b + x > r ∧ a + b + y > r))" and
"((a + b + min x y + c > r) = (a + b + x + c > r ∧ a + b + y + c > r))"
by auto};

val pth_abs =
@{lemma "P ¦x¦ ≡ (x ≥ 0 ∧ P x ∨ x < 0 ∧ P (-x))" for x :: real
by (atomize (full)) (auto split: abs_split)}

val pth_max =
@{lemma "P (max x y) ≡ (x ≤ y ∧ P y ∨ x > y ∧ P x)" for x y :: real
by (atomize (full)) (auto simp add: max_def)}

val pth_min =
@{lemma "P (min x y) ≡ (x ≤ y ∧ P x ∨ x > y ∧ P y)" for x y :: real
by (atomize (full)) (auto simp add: min_def)}

(* Miscellaneous *)
fun literals_conv bops uops cv =
let
fun h t =
(case Thm.term_of t of
b\$_\$_ => if member (op aconv) bops b then binop_conv h t else cv t
| u\$_ => if member (op aconv) uops u then arg_conv h t else cv t
| _ => cv t)
in h end;

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 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)
| _ => Rat.of_int (HOLogic.dest_number (Thm.term_of t) |> snd)
fun is_ratconst t = can dest_ratconst t

(*
fun find_term p t = if p t then t else
case t of
a\$b => (find_term p a handle TERM _ => find_term p b)
| Abs (_,_,t') => find_term p t'
| _ => raise TERM ("find_term",[t]);
*)

fun find_cterm p t =
if p t then t else
case Thm.term_of t of
_\$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
| Abs (_,_,_) => find_cterm p (Thm.dest_abs_global t |> snd)
| _ => raise CTERM ("find_cterm",[t]);

fun is_comb t = (case Thm.term_of t of _ \$ _ => true | _ => false);

(* Map back polynomials to HOL.                         *)

fun cterm_of_varpow x k =
if k = 1 then x
else \<^instantiate>‹x and k = ‹Numeral.mk_cnumber \<^ctyp>‹nat› k› in cterm ‹x ^ k› for x :: real›

fun cterm_of_monomial m =
if FuncUtil.Ctermfunc.is_empty m then \<^cterm>‹1::real›
else
let
val m' = FuncUtil.dest_monomial m
val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
in foldr1 (fn (s, t) => \<^instantiate>‹s and t in cterm ‹s * t› for s t :: real›) vps
end

fun cterm_of_cmonomial (m,c) =
if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
else if c = @1 then cterm_of_monomial m
else \<^instantiate>‹x = ‹cterm_of_rat c› and y = ‹cterm_of_monomial m› in cterm ‹x * y› for x y :: real›;

fun cterm_of_poly p =
if FuncUtil.Monomialfunc.is_empty p then \<^cterm>‹0::real›
else
let
val cms = map cterm_of_cmonomial
(sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
in foldr1 (fn (t1, t2) => \<^instantiate>‹t1 and t2 in cterm ‹t1 + t2› for t1 t2 :: real›) cms
end;

(* A general real arithmetic prover *)

fun gen_gen_real_arith ctxt (mk_numeric,
numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
absconv1,absconv2,prover) =
let
val pre_ss = put_simpset HOL_basic_ss ctxt addsimps
@{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib
all_conj_distrib if_bool_eq_disj}
val prenex_ss = put_simpset HOL_basic_ss ctxt addsimps prenex_simps
val skolemize_ss = put_simpset HOL_basic_ss ctxt addsimps [choice_iff]
val presimp_conv = Simplifier.rewrite pre_ss
val prenex_conv = Simplifier.rewrite prenex_ss
val skolemize_conv = Simplifier.rewrite skolemize_ss
val weak_dnf_ss = put_simpset HOL_basic_ss ctxt addsimps weak_dnf_simps
val weak_dnf_conv = Simplifier.rewrite weak_dnf_ss
fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
fun oprconv cv ct =
let val g = Thm.dest_fun2 ct
in if g aconvc \<^cterm>‹(≤) :: real ⇒ _›
orelse g aconvc \<^cterm>‹(<) :: real ⇒ _›
then arg_conv cv ct else arg1_conv cv ct
end

fun real_ineq_conv th ct =
let
val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th
handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
end
val [real_lt_conv, real_le_conv, real_eq_conv,
real_not_lt_conv, real_not_le_conv] =
map real_ineq_conv pth
fun match_mp_rule ths ths' =
let
fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
| th::ths => (ths' MRS th handle THM _ => f ths ths')
in f ths ths' end
fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
(match_mp_rule pth_mul [th, th'])
fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
(Thm.instantiate' [] [SOME ct] (th RS pth_emul))
fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
\<^instantiate>‹x = t in lemma ‹x * x ≥ 0› for x :: real by simp›

fun hol_of_positivstellensatz(eqs,les,lts) proof =
let
fun translate prf =
case prf of
Axiom_eq n => nth eqs n
| Axiom_le n => nth les n
| Axiom_lt n => nth lts n
| Rational_eq x =>
eqT_elim (numeric_eq_conv
\<^instantiate>‹x = ‹mk_numeric x› in cprop ‹x = 0› for x :: real›)
| Rational_le x =>
eqT_elim (numeric_ge_conv
\<^instantiate>‹x = ‹mk_numeric x› in cprop ‹x ≥ 0› for x :: real›)
| Rational_lt x =>
eqT_elim (numeric_gt_conv
\<^instantiate>‹x = ‹mk_numeric x› in cprop ‹x > 0› for x :: real›)
| Square pt => square_rule (cterm_of_poly pt)
| Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
| Sum(p1,p2) => add_rule (translate p1) (translate p2)
| Product(p1,p2) => mul_rule (translate p1) (translate p2)
in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
(translate proof)
end

val init_conv = presimp_conv then_conv
nnf_conv ctxt then_conv skolemize_conv then_conv prenex_conv then_conv
weak_dnf_conv

val concl = Thm.dest_arg o Thm.cprop_of
fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
val is_req = is_binop \<^cterm>‹(=):: real ⇒ _›
val is_ge = is_binop \<^cterm>‹(≤):: real ⇒ _›
val is_gt = is_binop \<^cterm>‹(<):: real ⇒ _›
val is_conj = is_binop \<^cterm>‹HOL.conj›
val is_disj = is_binop \<^cterm>‹HOL.disj›
fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
fun disj_cases th th1 th2 =
let
val (p,q) = Thm.dest_binop (concl th)
val c = concl th1
val _ =
if c aconvc (concl th2) then ()
else error "disj_cases : conclusions not alpha convertible"
in Thm.implies_elim (Thm.implies_elim
(Thm.implies_elim (Thm.instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
(Thm.implies_intr \<^instantiate>‹p in cprop p› th1))
(Thm.implies_intr \<^instantiate>‹q in cprop q› th2)
end
fun overall cert_choice dun ths =
case ths of
[] =>
let
val (eq,ne) = List.partition (is_req o concl) dun
val (le,nl) = List.partition (is_ge o concl) ne
val lt = filter (is_gt o concl) nl
in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
| th::oths =>
let
val ct = concl th
in
if is_conj ct then
let
val (th1,th2) = conj_pair th
in overall cert_choice dun (th1::th2::oths) end
else if is_disj ct then
let
val (th1, cert1) =
overall (Left::cert_choice) dun
(Thm.assume (Thm.apply \<^cterm>‹Trueprop› (Thm.dest_arg1 ct))::oths)
val (th2, cert2) =
overall (Right::cert_choice) dun
(Thm.assume (Thm.apply \<^cterm>‹Trueprop› (Thm.dest_arg ct))::oths)
in (disj_cases th th1 th2, Branch (cert1, cert2)) end
else overall cert_choice (th::dun) oths
end
fun dest_binary b ct =
if is_binop b ct then Thm.dest_binop ct
else raise CTERM ("dest_binary",[b,ct])
val dest_eq = dest_binary \<^cterm>‹(=) :: real ⇒ _›
fun real_not_eq_conv ct =
let
val (l,r) = dest_eq (Thm.dest_arg ct)
val th =
\<^instantiate>‹x = l and y = r in lemma ‹x ≠ y ≡ x - y > 0 ∨ - (x - y) > 0› for x y :: real
by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)›;
val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
val th_x = Drule.arg_cong_rule \<^cterm>‹uminus :: real ⇒ _› th_p
val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
val th' = Drule.binop_cong_rule \<^cterm>‹HOL.disj›
(Drule.arg_cong_rule \<^cterm>‹(<) (0::real)› th_p)
(Drule.arg_cong_rule \<^cterm>‹(<) (0::real)› th_n)
in Thm.transitive th th'
end
fun equal_implies_1_rule PQ =
let
val P = Thm.lhs_of PQ
in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
end
(*FIXME!!! Copied from groebner.ml*)
val strip_exists =
let
fun h (acc, t) =
case Thm.term_of t of
\<^Const_>‹Ex _ for ‹Abs _›› =>
h (Thm.dest_abs_global (Thm.dest_arg t) |>> (fn v => v::acc))
| _ => (acc,t)
in fn t => h ([],t)
end
fun name_of x =
case Thm.term_of x of
Free(s,_) => s
| Var ((s,_),_) => s
| _ => "x"

fun mk_forall x th =
let
val T = Thm.ctyp_of_cterm x
val all = \<^instantiate>‹'a = T in cterm All›
in Drule.arg_cong_rule all (Thm.abstract_rule (name_of x) x th) end

val specl = fold_rev (fn x => fn th => Thm.instantiate' [] [SOME x] (th RS spec));

fun mk_ex x t =
\<^instantiate>‹'a = ‹Thm.ctyp_of_cterm x› and P = ‹Thm.lambda x t›
in cprop ‹Ex P› for P :: ‹'a ⇒ bool››

fun choose x th th' =
case Thm.concl_of th of
\<^Const_>‹Trueprop for \<^Const_>‹Ex _ for _›› =>
let
val P = Thm.dest_arg (Thm.dest_arg (Thm.cprop_of th))
val T = Thm.dest_ctyp0 (Thm.ctyp_of_cterm P)
val Q = Thm.dest_arg (Thm.cprop_of th')
val th0 =
\<^instantiate>‹'a = T and P and Q in
lemma "∃x::'a. P x ⟹ (⋀x. P x ⟹ Q) ⟹ Q" by (fact exE)›
val Px =
\<^instantiate>‹'a = T and P and x in cprop ‹P x› for x :: 'a›
val th1 = Thm.forall_intr x (Thm.implies_intr Px th')
in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
| _ => raise THM ("choose",0,[th, th'])

fun simple_choose x th =
choose x (Thm.assume (mk_ex x (Thm.dest_arg (hd (Thm.chyps_of th))))) th

val strip_forall =
let
fun h (acc, t) =
case Thm.term_of t of
\<^Const_>‹All _ for ‹Abs _›› =>
h (Thm.dest_abs_global (Thm.dest_arg t) |>> (fn v => v::acc))
| _ => (acc,t)
in fn t => h ([],t)
end
in
fn A =>
let
val nnf_norm_conv' =
nnf_conv ctxt then_conv
literals_conv [\<^Const>‹conj›, \<^Const>‹disj›] []
(Conv.cache_conv
(first_conv [real_lt_conv, real_le_conv,
real_eq_conv, real_not_lt_conv,
real_not_le_conv, real_not_eq_conv, all_conv]))
fun absremover ct = (literals_conv [\<^Const>‹conj›, \<^Const>‹disj›] []
(try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
val not_A = \<^instantiate>‹A in cprop ‹¬ A››
val th0 = (init_conv then_conv arg_conv nnf_norm_conv') not_A
val tm0 = Thm.dest_arg (Thm.rhs_of th0)
val (th, certificates) =
if tm0 aconvc \<^cterm>‹False› then (equal_implies_1_rule th0, Trivial) else
let
val (evs,bod) = strip_exists tm0
val (avs,ibod) = strip_forall bod
val th1 = Drule.arg_cong_rule \<^cterm>‹Trueprop› (fold mk_forall avs (absremover ibod))
val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
val th3 =
fold simple_choose evs
(prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply \<^cterm>‹Trueprop› bod))) th2)
in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume not_A)) th3), certs)
end
in
(Thm.implies_elim \<^instantiate>‹A in lemma ‹(¬ A ⟹ False) ⟹ A› by blast› th,
certificates)
end
end;

(* A linear arithmetic prover *)
local
val linear_add = FuncUtil.Ctermfunc.combine (curry op +) (fn z => z = @0)
fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c * x)
val one_tm = \<^cterm>‹1::real›
fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p @0)) orelse
((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
not(p(FuncUtil.Ctermfunc.apply e one_tm)))

fun linear_ineqs vars (les,lts) =
case find_first (contradictory (fn x => x > @0)) lts of
SOME r => r
| NONE =>
(case find_first (contradictory (fn x => x > @0)) les of
SOME r => r
| NONE =>
if null vars then error "linear_ineqs: no contradiction" else
let
val ineqs = les @ lts
fun blowup v =
length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) ineqs) +
length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) ineqs) *
length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 < @0) ineqs)
val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
(map (fn v => (v,blowup v)) vars)))
fun addup (e1,p1) (e2,p2) acc =
let
val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v @0
val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v @0
in
if c1 * c2 >= @0 then acc else
let
val e1' = linear_cmul (abs c2) e1
val e2' = linear_cmul (abs c1) e2
val p1' = Product(Rational_lt (abs c2), p1)
val p2' = Product(Rational_lt (abs c1), p2)
end
end
val (les0,les1) =
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) les
val (lts0,lts1) =
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) lts
val (lesp,lesn) =
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) les1
val (ltsp,ltsn) =
List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) lts1
val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
(fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
in linear_ineqs (remove (op aconvc) v vars) (les',lts')
end)

fun linear_eqs(eqs,les,lts) =
case find_first (contradictory (fn x => x = @0)) eqs of
SOME r => r
| NONE =>
(case eqs of
[] =>
let val vars = remove (op aconvc) one_tm
(fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) [])
in linear_ineqs vars (les,lts) end
| (e,p)::es =>
if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
let
val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
fun xform (inp as (t,q)) =
let val d = FuncUtil.Ctermfunc.tryapplyd t x @0 in
if d = @0 then inp else
let
val k = ~ d * abs c / c
val e' = linear_cmul k e
val t' = linear_cmul (abs c) t
val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
val q' = Product(Rational_lt (abs c), q)
end
end
in linear_eqs(map xform es,map xform les,map xform lts)
end)

fun linear_prover (eq,le,lt) =
let
val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
val les = map_index (fn (n, p) => (p,Axiom_le n)) le
val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
in linear_eqs(eqs,les,lts)
end

fun lin_of_hol ct =
if ct aconvc \<^cterm>‹0::real› then FuncUtil.Ctermfunc.empty
else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, @1)
else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
else
let val (lop,r) = Thm.dest_comb ct
in
if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, @1)
else
let val (opr,l) = Thm.dest_comb lop
in
if opr aconvc \<^cterm>‹(+) :: real ⇒ _›
then linear_add (lin_of_hol l) (lin_of_hol r)
else if opr aconvc \<^cterm>‹(*) :: real ⇒ _›
andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
else FuncUtil.Ctermfunc.onefunc (ct, @1)
end
end

fun is_alien ct =
case Thm.term_of ct of
\<^Const_>‹of_nat _ for n› => not (can HOLogic.dest_number n)
| \<^Const_>‹of_int _ for n› => not (can HOLogic.dest_number n)
| _ => false
in
fun real_linear_prover translator (eq,le,lt) =
let
val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o Thm.cprop_of
val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o Thm.cprop_of
val eq_pols = map lhs eq
val le_pols = map rhs le
val lt_pols = map rhs lt
val aliens = filter is_alien
(fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom)
(eq_pols @ le_pols @ lt_pols) [])
val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,@1)) aliens
val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
val le' = le @ map (fn a => Thm.instantiate' [] [SOME (Thm.dest_arg a)] @{thm of_nat_0_le_iff}) aliens
in ((translator (eq,le',lt) proof), Trivial)
end
end;

(* A less general generic arithmetic prover dealing with abs,max and min*)

local
val absmaxmin_elim_ss1 =
simpset_of (put_simpset HOL_basic_ss \<^context> addsimps real_abs_thms1)
fun absmaxmin_elim_conv1 ctxt =
Simplifier.rewrite (put_simpset absmaxmin_elim_ss1 ctxt)

val absmaxmin_elim_conv2 =
let
fun elim_construct pred conv tm =
let
val a = find_cterm (pred o Thm.term_of) tm
val P = Thm.lambda a tm
in conv P a end

val elim_abs = elim_construct (fn \<^Const_>‹abs \<^Type>‹real› for _› => true | _ => false)
(fn P => fn a =>
let val x = Thm.dest_arg a in
\<^instantiate>‹P and x in
lemma ‹P ¦x¦ ≡ (x ≥ 0 ∧ P x ∨ x < 0 ∧ P (- x))› for x :: real
by (atomize (full)) (auto split: abs_split)›
end)
val elim_max = elim_construct (fn \<^Const_>‹max \<^Type>‹real› for _ _› => true | _ => false)
(fn P => fn a =>
let val (x, y) = Thm.dest_binop a in
\<^instantiate>‹P and x and y in
lemma ‹P (max x y) ≡ (x ≤ y ∧ P y ∨ x > y ∧ P x)› for x y :: real
by (atomize (full)) (auto simp add: max_def)›
end)
val elim_min = elim_construct (fn \<^Const_>‹min \<^Type>‹real› for _ _› => true | _ => false)
(fn P => fn a =>
let val (x, y) = Thm.dest_binop a in
\<^instantiate>‹P and x and y in
lemma ‹P (min x y) ≡ (x ≤ y ∧ P x ∨ x > y ∧ P y)› for x y :: real
by (atomize (full)) (auto simp add: min_def)›
end)
in first_conv [elim_abs, elim_max, elim_min, all_conv] end;
in
gen_gen_real_arith ctxt
absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
end;

(* An instance for reals*)

fun gen_prover_real_arith ctxt prover =
let
val {add, mul, neg, pow = _, sub = _, main} =
Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
(the (Semiring_Normalizer.match ctxt \<^cterm>‹(0::real) + 1›))
Thm.term_ord
in gen_real_arith ctxt
(cterm_of_rat,
Numeral_Simprocs.field_comp_conv ctxt,
Numeral_Simprocs.field_comp_conv ctxt,
Numeral_Simprocs.field_comp_conv ctxt,
main ctxt, neg ctxt, add ctxt, mul ctxt, prover)
end;

end
```