introduced mk/dest_numeral/number for mk/dest_binum etc.
(* Title: HOL/Integ/cooper_proof.ML
ID: $Id$
Author: Amine Chaieb and Tobias Nipkow, TU Muenchen
File containing the implementation of the proof
generation for Cooper Algorithm
*)
signature COOPER_PROOF =
sig
val qe_Not : thm
val qe_conjI : thm
val qe_disjI : thm
val qe_impI : thm
val qe_eqI : thm
val qe_exI : thm
val list_to_set : typ -> term list -> term
val qe_get_terms : thm -> term * term
val cooper_prv : theory -> term -> term -> thm
val proof_of_evalc : theory -> term -> thm
val proof_of_cnnf : theory -> term -> (term -> thm) -> thm
val proof_of_linform : theory -> string list -> term -> thm
val proof_of_adjustcoeffeq : theory -> term -> IntInf.int -> term -> thm
val prove_elementar : theory -> string -> term -> thm
val thm_of : theory -> (term -> (term list * (thm list -> thm))) -> term -> thm
end;
structure CooperProof : COOPER_PROOF =
struct
open CooperDec;
val presburger_ss = simpset ()
addsimps [diff_int_def] delsimps [thm "diff_int_def_symmetric"];
val cboolT = ctyp_of HOL.thy HOLogic.boolT;
(*Theorems that will be used later for the proofgeneration*)
val zdvd_iff_zmod_eq_0 = thm "zdvd_iff_zmod_eq_0";
val unity_coeff_ex = thm "unity_coeff_ex";
(* Theorems for proving the adjustment of the coefficients*)
val ac_lt_eq = thm "ac_lt_eq";
val ac_eq_eq = thm "ac_eq_eq";
val ac_dvd_eq = thm "ac_dvd_eq";
val ac_pi_eq = thm "ac_pi_eq";
(* The logical compination of the sythetised properties*)
val qe_Not = thm "qe_Not";
val qe_conjI = thm "qe_conjI";
val qe_disjI = thm "qe_disjI";
val qe_impI = thm "qe_impI";
val qe_eqI = thm "qe_eqI";
val qe_exI = thm "qe_exI";
val qe_ALLI = thm "qe_ALLI";
(*Modulo D property for Pminusinf an Plusinf *)
val fm_modd_minf = thm "fm_modd_minf";
val not_dvd_modd_minf = thm "not_dvd_modd_minf";
val dvd_modd_minf = thm "dvd_modd_minf";
val fm_modd_pinf = thm "fm_modd_pinf";
val not_dvd_modd_pinf = thm "not_dvd_modd_pinf";
val dvd_modd_pinf = thm "dvd_modd_pinf";
(* the minusinfinity proprty*)
val fm_eq_minf = thm "fm_eq_minf";
val neq_eq_minf = thm "neq_eq_minf";
val eq_eq_minf = thm "eq_eq_minf";
val le_eq_minf = thm "le_eq_minf";
val len_eq_minf = thm "len_eq_minf";
val not_dvd_eq_minf = thm "not_dvd_eq_minf";
val dvd_eq_minf = thm "dvd_eq_minf";
(* the Plusinfinity proprty*)
val fm_eq_pinf = thm "fm_eq_pinf";
val neq_eq_pinf = thm "neq_eq_pinf";
val eq_eq_pinf = thm "eq_eq_pinf";
val le_eq_pinf = thm "le_eq_pinf";
val len_eq_pinf = thm "len_eq_pinf";
val not_dvd_eq_pinf = thm "not_dvd_eq_pinf";
val dvd_eq_pinf = thm "dvd_eq_pinf";
(*Logical construction of the Property*)
val eq_minf_conjI = thm "eq_minf_conjI";
val eq_minf_disjI = thm "eq_minf_disjI";
val modd_minf_disjI = thm "modd_minf_disjI";
val modd_minf_conjI = thm "modd_minf_conjI";
val eq_pinf_conjI = thm "eq_pinf_conjI";
val eq_pinf_disjI = thm "eq_pinf_disjI";
val modd_pinf_disjI = thm "modd_pinf_disjI";
val modd_pinf_conjI = thm "modd_pinf_conjI";
(*Cooper Backwards...*)
(*Bset*)
val not_bst_p_fm = thm "not_bst_p_fm";
val not_bst_p_ne = thm "not_bst_p_ne";
val not_bst_p_eq = thm "not_bst_p_eq";
val not_bst_p_gt = thm "not_bst_p_gt";
val not_bst_p_lt = thm "not_bst_p_lt";
val not_bst_p_ndvd = thm "not_bst_p_ndvd";
val not_bst_p_dvd = thm "not_bst_p_dvd";
(*Aset*)
val not_ast_p_fm = thm "not_ast_p_fm";
val not_ast_p_ne = thm "not_ast_p_ne";
val not_ast_p_eq = thm "not_ast_p_eq";
val not_ast_p_gt = thm "not_ast_p_gt";
val not_ast_p_lt = thm "not_ast_p_lt";
val not_ast_p_ndvd = thm "not_ast_p_ndvd";
val not_ast_p_dvd = thm "not_ast_p_dvd";
(*Logical construction of the prop*)
(*Bset*)
val not_bst_p_conjI = thm "not_bst_p_conjI";
val not_bst_p_disjI = thm "not_bst_p_disjI";
val not_bst_p_Q_elim = thm "not_bst_p_Q_elim";
(*Aset*)
val not_ast_p_conjI = thm "not_ast_p_conjI";
val not_ast_p_disjI = thm "not_ast_p_disjI";
val not_ast_p_Q_elim = thm "not_ast_p_Q_elim";
(*Cooper*)
val cppi_eq = thm "cppi_eq";
val cpmi_eq = thm "cpmi_eq";
(*Others*)
val simp_from_to = thm "simp_from_to";
val P_eqtrue = thm "P_eqtrue";
val P_eqfalse = thm "P_eqfalse";
(*For Proving NNF*)
val nnf_nn = thm "nnf_nn";
val nnf_im = thm "nnf_im";
val nnf_eq = thm "nnf_eq";
val nnf_sdj = thm "nnf_sdj";
val nnf_ncj = thm "nnf_ncj";
val nnf_nim = thm "nnf_nim";
val nnf_neq = thm "nnf_neq";
val nnf_ndj = thm "nnf_ndj";
(*For Proving term linearizition*)
val linearize_dvd = thm "linearize_dvd";
val lf_lt = thm "lf_lt";
val lf_eq = thm "lf_eq";
val lf_dvd = thm "lf_dvd";
(* ------------------------------------------------------------------------- *)
(*This function norm_zero_one replaces the occurences of Numeral1 and Numeral0*)
(*Respectively by their abstract representation Const("HOL.one",..) and Const("HOL.zero",..)*)
(*this is necessary because the theorems use this representation.*)
(* This function should be elminated in next versions...*)
(* ------------------------------------------------------------------------- *)
fun norm_zero_one fm = case fm of
(Const ("HOL.times",_) $ c $ t) =>
if c = one then (norm_zero_one t)
else if (dest_number c = ~1)
then (Const("HOL.uminus",HOLogic.intT --> HOLogic.intT) $ (norm_zero_one t))
else (HOLogic.mk_binop "HOL.times" (norm_zero_one c,norm_zero_one t))
|(node $ rest) => ((norm_zero_one node)$(norm_zero_one rest))
|(Abs(x,T,p)) => (Abs(x,T,(norm_zero_one p)))
|_ => fm;
(* ------------------------------------------------------------------------- *)
(*function list to Set, constructs a set containing all elements of a given list.*)
(* ------------------------------------------------------------------------- *)
fun list_to_set T1 l = let val T = (HOLogic.mk_setT T1) in
case l of
[] => Const ("{}",T)
|(h::t) => Const("insert", T1 --> (T --> T)) $ h $(list_to_set T1 t)
end;
(* ------------------------------------------------------------------------- *)
(* Returns both sides of an equvalence in the theorem*)
(* ------------------------------------------------------------------------- *)
fun qe_get_terms th = let val (_$(Const("op =",Type ("fun",[Type ("bool", []),_])) $ A $ B )) = prop_of th in (A,B) end;
(* ------------------------------------------------------------------------- *)
(*This function proove elementar will be used to generate proofs at
runtime*) (*It is thought to prove properties such as a dvd b
(essentially) that are only to make at runtime.*)
(* ------------------------------------------------------------------------- *)
fun prove_elementar thy s fm2 =
Goal.prove (ProofContext.init thy) [] [] (HOLogic.mk_Trueprop fm2) (fn _ => EVERY
(case s of
(*"ss" like simplification with simpset*)
"ss" =>
let val ss = presburger_ss addsimps [zdvd_iff_zmod_eq_0,unity_coeff_ex]
in [simp_tac ss 1, TRY (simple_arith_tac 1)] end
(*"bl" like blast tactic*)
(* Is only used in the harrisons like proof procedure *)
| "bl" => [blast_tac HOL_cs 1]
(*"ed" like Existence disjunctions ...*)
(* Is only used in the harrisons like proof procedure *)
| "ed" =>
let
val ex_disj_tacs =
let
val tac1 = EVERY[REPEAT(resolve_tac [disjI1,disjI2] 1), etac exI 1]
val tac2 = EVERY[etac exE 1, rtac exI 1,
REPEAT(resolve_tac [disjI1,disjI2] 1), assumption 1]
in [rtac iffI 1,
etac exE 1, REPEAT(EVERY[etac disjE 1, tac1]), tac1,
REPEAT(EVERY[etac disjE 1, tac2]), tac2]
end
in ex_disj_tacs end
| "fa" => [simple_arith_tac 1]
| "sa" =>
let val ss = presburger_ss addsimps zadd_ac
in [simp_tac ss 1, TRY (simple_arith_tac 1)] end
(* like Existance Conjunction *)
| "ec" =>
let val ss = presburger_ss addsimps zadd_ac
in [simp_tac ss 1, TRY (blast_tac HOL_cs 1)] end
| "ac" =>
let val ss = HOL_basic_ss addsimps zadd_ac
in [simp_tac ss 1] end
| "lf" =>
let val ss = presburger_ss addsimps zadd_ac
in [simp_tac ss 1, TRY (simple_arith_tac 1)] end));
(*=============================================================*)
(*-------------------------------------------------------------*)
(* The new compact model *)
(*-------------------------------------------------------------*)
(*=============================================================*)
fun thm_of sg decomp t =
let val (ts,recomb) = decomp t
in recomb (map (thm_of sg decomp) ts)
end;
(*==================================================*)
(* Compact Version for adjustcoeffeq *)
(*==================================================*)
fun decomp_adjustcoeffeq sg x l fm = case fm of
(Const("Not",_)$(Const("Orderings.less",_) $(Const("HOL.zero",_)) $(rt as (Const ("HOL.plus", _)$(Const ("HOL.times",_) $ c $ y ) $z )))) =>
let
val m = l div (dest_number c)
val n = if (x = y) then abs (m) else 1
val xtm = (HOLogic.mk_binop "HOL.times" ((mk_number ((m div n)*l) ), x))
val rs = if (x = y)
then (HOLogic.mk_binrel "Orderings.less" (zero,linear_sub [] (mk_number n) (HOLogic.mk_binop "HOL.plus" ( xtm ,( linear_cmul n z) ))))
else HOLogic.mk_binrel "Orderings.less" (zero,linear_sub [] one rt )
val ck = cterm_of sg (mk_number n)
val cc = cterm_of sg c
val ct = cterm_of sg z
val cx = cterm_of sg y
val pre = prove_elementar sg "lf"
(HOLogic.mk_binrel "Orderings.less" (Const("HOL.zero",HOLogic.intT),(mk_number n)))
val th1 = (pre RS (instantiate' [] [SOME ck,SOME cc, SOME cx, SOME ct] (ac_pi_eq)))
in ([], fn [] => [th1,(prove_elementar sg "sa" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
end
|(Const(p,_) $a $( Const ("HOL.plus", _)$(Const ("HOL.times",_) $
c $ y ) $t )) =>
if (is_arith_rel fm) andalso (x = y)
then
let val m = l div (dest_number c)
val k = (if p = "Orderings.less" then abs(m) else m)
val xtm = (HOLogic.mk_binop "HOL.times" ((mk_number ((m div k)*l) ), x))
val rs = (HOLogic.mk_binrel p ((linear_cmul k a),(HOLogic.mk_binop "HOL.plus" ( xtm ,( linear_cmul k t) ))))
val ck = cterm_of sg (mk_number k)
val cc = cterm_of sg c
val ct = cterm_of sg t
val cx = cterm_of sg x
val ca = cterm_of sg a
in
case p of
"Orderings.less" =>
let val pre = prove_elementar sg "lf"
(HOLogic.mk_binrel "Orderings.less" (Const("HOL.zero",HOLogic.intT),(mk_number k)))
val th1 = (pre RS (instantiate' [] [SOME ck,SOME ca,SOME cc, SOME cx, SOME ct] (ac_lt_eq)))
in ([], fn [] => [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
end
|"op =" =>
let val pre = prove_elementar sg "lf"
(HOLogic.Not $ (HOLogic.mk_binrel "op =" (Const("HOL.zero",HOLogic.intT),(mk_number k))))
val th1 = (pre RS(instantiate' [] [SOME ck,SOME ca,SOME cc, SOME cx, SOME ct] (ac_eq_eq)))
in ([], fn [] => [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
end
|"Divides.dvd" =>
let val pre = prove_elementar sg "lf"
(HOLogic.Not $ (HOLogic.mk_binrel "op =" (Const("HOL.zero",HOLogic.intT),(mk_number k))))
val th1 = (pre RS (instantiate' [] [SOME ck,SOME ca,SOME cc, SOME cx, SOME ct]) (ac_dvd_eq))
in ([], fn [] => [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
end
end
else ([], fn [] => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] refl)
|( Const ("Not", _) $ p) => ([p], fn [th] => th RS qe_Not)
|( Const ("op &",_) $ p $ q) => ([p,q], fn [th1,th2] => [th1,th2] MRS qe_conjI)
|( Const ("op |",_) $ p $ q) =>([p,q], fn [th1,th2] => [th1,th2] MRS qe_disjI)
|_ => ([], fn [] => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] refl);
fun proof_of_adjustcoeffeq sg x l = thm_of sg (decomp_adjustcoeffeq sg x l);
(*==================================================*)
(* Finding rho for modd_minusinfinity *)
(*==================================================*)
fun rho_for_modd_minf x dlcm sg fm1 =
let
(*Some certified Terms*)
val ctrue = cterm_of sg HOLogic.true_const
val cfalse = cterm_of sg HOLogic.false_const
val fm = norm_zero_one fm1
in case fm1 of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1= zero) andalso (c2= one) then (instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_minf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z)) =>
if (is_arith_rel fm) andalso (x=y) andalso (c1= zero) andalso (c2= one)
then (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_minf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
|(Const("Orderings.less",_) $ c1 $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ pm1 $ y ) $ z )) =>
if (y=x) andalso (c1 = zero) then
if (pm1 = one) then (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_minf)) else
(instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_minf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
|Const ("Not",_) $ (Const("Divides.dvd",_)$ d $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $ c $ y ) $ z)) =>
if y=x then let val cz = cterm_of sg (norm_zero_one z)
val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.mod" (dlcm,d),norm_zero_one zero)
in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS(((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (not_dvd_modd_minf)))
end
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
|(Const("Divides.dvd",_)$ d $ (db as (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $
c $ y ) $ z))) =>
if y=x then let val cz = cterm_of sg (norm_zero_one z)
val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.mod" (dlcm,d),norm_zero_one zero)
in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (dvd_modd_minf)))
end
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
|_ => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf)
end;
(*=========================================================================*)
(*=========================================================================*)
fun rho_for_eq_minf x dlcm sg fm1 =
let
val fm = norm_zero_one fm1
in case fm1 of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1=zero) andalso (c2=one)
then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (neq_eq_minf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z)) =>
if (is_arith_rel fm) andalso (x=y) andalso ((c1=zero) orelse (c1 = norm_zero_one zero)) andalso ((c2=one) orelse (c1 = norm_zero_one one))
then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (eq_eq_minf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
|(Const("Orderings.less",_) $ c1 $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ pm1 $ y ) $ z )) =>
if (y=x) andalso (c1 =zero) then
if pm1 = one then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (le_eq_minf)) else
(instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (len_eq_minf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
|Const ("Not",_) $ (Const("Divides.dvd",_)$ d $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $ c $ y ) $ z)) =>
if y=x then let val cd = cterm_of sg (norm_zero_one d)
val cz = cterm_of sg (norm_zero_one z)
in(instantiate' [] [SOME cd, SOME cz] (not_dvd_eq_minf))
end
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
|(Const("Divides.dvd",_)$ d $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $ c $ y ) $ z)) =>
if y=x then let val cd = cterm_of sg (norm_zero_one d)
val cz = cterm_of sg (norm_zero_one z)
in(instantiate' [] [SOME cd, SOME cz ] (dvd_eq_minf))
end
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
|_ => (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
end;
(*=====================================================*)
(*=====================================================*)
(*=========== minf proofs with the compact version==========*)
fun decomp_minf_eq x dlcm sg t = case t of
Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_minf_conjI)
|Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_minf_disjI)
|_ => ([],fn [] => rho_for_eq_minf x dlcm sg t);
fun decomp_minf_modd x dlcm sg t = case t of
Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_minf_conjI)
|Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_minf_disjI)
|_ => ([],fn [] => rho_for_modd_minf x dlcm sg t);
(* -------------------------------------------------------------*)
(* Finding rho for pinf_modd *)
(* -------------------------------------------------------------*)
fun rho_for_modd_pinf x dlcm sg fm1 =
let
(*Some certified Terms*)
val ctrue = cterm_of sg HOLogic.true_const
val cfalse = cterm_of sg HOLogic.false_const
val fm = norm_zero_one fm1
in case fm1 of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z))) =>
if ((x=y) andalso (c1= zero) andalso (c2= one))
then (instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_pinf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z)) =>
if ((is_arith_rel fm) andalso (x = y) andalso (c1 = zero) andalso (c2 = one))
then (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_pinf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
|(Const("Orderings.less",_) $ c1 $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ pm1 $ y ) $ z )) =>
if ((y=x) andalso (c1 = zero)) then
if (pm1 = one)
then (instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_pinf))
else (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_pinf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
|Const ("Not",_) $ (Const("Divides.dvd",_)$ d $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $ c $ y ) $ z)) =>
if y=x then let val cz = cterm_of sg (norm_zero_one z)
val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.mod" (dlcm,d),norm_zero_one zero)
in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS(((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (not_dvd_modd_pinf)))
end
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
|(Const("Divides.dvd",_)$ d $ (db as (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $
c $ y ) $ z))) =>
if y=x then let val cz = cterm_of sg (norm_zero_one z)
val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.mod" (dlcm,d),norm_zero_one zero)
in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (dvd_modd_pinf)))
end
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
|_ => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf)
end;
(* -------------------------------------------------------------*)
(* Finding rho for pinf_eq *)
(* -------------------------------------------------------------*)
fun rho_for_eq_pinf x dlcm sg fm1 =
let
val fm = norm_zero_one fm1
in case fm1 of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1=zero) andalso (c2=one)
then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (neq_eq_pinf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z)) =>
if (is_arith_rel fm) andalso (x=y) andalso ((c1=zero) orelse (c1 = norm_zero_one zero)) andalso ((c2=one) orelse (c1 = norm_zero_one one))
then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (eq_eq_pinf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
|(Const("Orderings.less",_) $ c1 $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ pm1 $ y ) $ z )) =>
if (y=x) andalso (c1 =zero) then
if pm1 = one then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (le_eq_pinf)) else
(instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (len_eq_pinf))
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
|Const ("Not",_) $ (Const("Divides.dvd",_)$ d $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $ c $ y ) $ z)) =>
if y=x then let val cd = cterm_of sg (norm_zero_one d)
val cz = cterm_of sg (norm_zero_one z)
in(instantiate' [] [SOME cd, SOME cz] (not_dvd_eq_pinf))
end
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
|(Const("Divides.dvd",_)$ d $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $ c $ y ) $ z)) =>
if y=x then let val cd = cterm_of sg (norm_zero_one d)
val cz = cterm_of sg (norm_zero_one z)
in(instantiate' [] [SOME cd, SOME cz ] (dvd_eq_pinf))
end
else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
|_ => (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
end;
fun minf_proof_of_c sg x dlcm t =
let val minf_eqth = thm_of sg (decomp_minf_eq x dlcm sg) t
val minf_moddth = thm_of sg (decomp_minf_modd x dlcm sg) t
in (minf_eqth, minf_moddth)
end;
(*=========== pinf proofs with the compact version==========*)
fun decomp_pinf_eq x dlcm sg t = case t of
Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_pinf_conjI)
|Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_pinf_disjI)
|_ =>([],fn [] => rho_for_eq_pinf x dlcm sg t) ;
fun decomp_pinf_modd x dlcm sg t = case t of
Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_pinf_conjI)
|Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_pinf_disjI)
|_ => ([],fn [] => rho_for_modd_pinf x dlcm sg t);
fun pinf_proof_of_c sg x dlcm t =
let val pinf_eqth = thm_of sg (decomp_pinf_eq x dlcm sg) t
val pinf_moddth = thm_of sg (decomp_pinf_modd x dlcm sg) t
in (pinf_eqth,pinf_moddth)
end;
(* ------------------------------------------------------------------------- *)
(* Here we generate the theorem for the Bset Property in the simple direction*)
(* It is just an instantiation*)
(* ------------------------------------------------------------------------- *)
(*
fun bsetproof_of sg (x as Free(xn,xT)) fm bs dlcm =
let
val cp = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
val cdlcm = cterm_of sg dlcm
val cB = cterm_of sg (list_to_set HOLogic.intT (map norm_zero_one bs))
in instantiate' [] [SOME cdlcm,SOME cB, SOME cp] (bst_thm)
end;
fun asetproof_of sg (x as Free(xn,xT)) fm ast dlcm =
let
val cp = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
val cdlcm = cterm_of sg dlcm
val cA = cterm_of sg (list_to_set HOLogic.intT (map norm_zero_one ast))
in instantiate' [] [SOME cdlcm,SOME cA, SOME cp] (ast_thm)
end;
*)
(* For the generation of atomic Theorems*)
(* Prove the premisses on runtime and then make RS*)
(* ------------------------------------------------------------------------- *)
(*========= this is rho ============*)
fun generate_atomic_not_bst_p sg (x as Free(xn,xT)) fm dlcm B at =
let
val cdlcm = cterm_of sg dlcm
val cB = cterm_of sg B
val cfma = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
val cat = cterm_of sg (norm_zero_one at)
in
case at of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1=zero) andalso (c2=one)
then let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ (norm_zero_one (linear_cmul ~1 z)) $ B)
val th2 = prove_elementar sg "ss" (HOLogic.mk_eq ((norm_zero_one (linear_cmul ~1 z)),Const("HOL.uminus",HOLogic.intT --> HOLogic.intT) $(norm_zero_one z)))
val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "Orderings.less" (Const("HOL.zero",HOLogic.intT),dlcm))
in (instantiate' [] [SOME cfma]([th3,th1,th2] MRS (not_bst_p_ne)))
end
else (instantiate' [] [SOME cfma, SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("HOL.plus", T) $(Const ("HOL.times",_) $ c2 $ y) $z)) =>
if (is_arith_rel at) andalso (x=y)
then let val bst_z = norm_zero_one (linear_neg (linear_add [] z (mk_number 1)))
in let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ bst_z $ B)
val th2 = prove_elementar sg "ss" (HOLogic.mk_eq (bst_z,Const("HOL.minus",T) $ (Const("HOL.uminus",HOLogic.intT --> HOLogic.intT) $(norm_zero_one z)) $ (Const("HOL.one",HOLogic.intT))))
val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "Orderings.less" (Const("HOL.zero",HOLogic.intT),dlcm))
in (instantiate' [] [SOME cfma] ([th3,th1,th2] MRS (not_bst_p_eq)))
end
end
else (instantiate' [] [SOME cfma, SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
|(Const("Orderings.less",_) $ c1 $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ pm1 $ y ) $ z )) =>
if (y=x) andalso (c1 =zero) then
if pm1 = one then
let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ (norm_zero_one (linear_cmul ~1 z)) $ B)
val th2 = prove_elementar sg "ss" (HOLogic.mk_eq ((norm_zero_one (linear_cmul ~1 z)),Const("HOL.uminus",HOLogic.intT --> HOLogic.intT) $(norm_zero_one z)))
in (instantiate' [] [SOME cfma, SOME cdlcm]([th1,th2] MRS (not_bst_p_gt)))
end
else let val th1 = prove_elementar sg "ss" (HOLogic.mk_binrel "Orderings.less" (Const("HOL.zero",HOLogic.intT),dlcm))
in (instantiate' [] [SOME cfma, SOME cB,SOME (cterm_of sg (norm_zero_one z))] (th1 RS (not_bst_p_lt)))
end
else (instantiate' [] [SOME cfma, SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
|Const ("Not",_) $ (Const("Divides.dvd",_)$ d $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $ c $ y ) $ z)) =>
if y=x then
let val cz = cterm_of sg (norm_zero_one z)
val th1 = (prove_elementar sg "ss" (HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.mod" (dlcm,d),norm_zero_one zero))) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1)
in (instantiate' [] [SOME cfma, SOME cB,SOME cz] (th1 RS (not_bst_p_ndvd)))
end
else (instantiate' [] [SOME cfma, SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
|(Const("Divides.dvd",_)$ d $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $ c $ y ) $ z)) =>
if y=x then
let val cz = cterm_of sg (norm_zero_one z)
val th1 = (prove_elementar sg "ss" (HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.mod" (dlcm,d),norm_zero_one zero))) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1)
in (instantiate' [] [SOME cfma,SOME cB,SOME cz] (th1 RS (not_bst_p_dvd)))
end
else (instantiate' [] [SOME cfma, SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
|_ => (instantiate' [] [SOME cfma, SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
end;
(* ------------------------------------------------------------------------- *)
(* Main interpretation function for this backwards dirction*)
(* if atomic do generate atomis formulae else Construct theorems and then make RS with the construction theorems*)
(*Help Function*)
(* ------------------------------------------------------------------------- *)
(*==================== Proof with the compact version *)
fun decomp_nbstp sg x dlcm B fm t = case t of
Const("op &",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_bst_p_conjI )
|Const("op |",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_bst_p_disjI)
|_ => ([], fn [] => generate_atomic_not_bst_p sg x fm dlcm B t);
fun not_bst_p_proof_of_c sg (x as Free(xn,xT)) fm dlcm B t =
let
val th = thm_of sg (decomp_nbstp sg x dlcm (list_to_set xT (map norm_zero_one B)) fm) t
val fma = absfree (xn,xT, norm_zero_one fm)
in let val th1 = prove_elementar sg "ss" (HOLogic.mk_eq (fma,fma))
in [th,th1] MRS (not_bst_p_Q_elim)
end
end;
(* ------------------------------------------------------------------------- *)
(* Protokol interpretation function for the backwards direction for cooper's Theorem*)
(* For the generation of atomic Theorems*)
(* Prove the premisses on runtime and then make RS*)
(* ------------------------------------------------------------------------- *)
(*========= this is rho ============*)
fun generate_atomic_not_ast_p sg (x as Free(xn,xT)) fm dlcm A at =
let
val cdlcm = cterm_of sg dlcm
val cA = cterm_of sg A
val cfma = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
val cat = cterm_of sg (norm_zero_one at)
in
case at of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("HOL.plus", _) $(Const ("HOL.times",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1=zero) andalso (c2=one)
then let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ (norm_zero_one (linear_cmul ~1 z)) $ A)
val th2 = prove_elementar sg "ss" (HOLogic.mk_eq ((norm_zero_one (linear_cmul ~1 z)),Const("HOL.uminus",HOLogic.intT --> HOLogic.intT) $(norm_zero_one z)))
val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "Orderings.less" (Const("HOL.zero",HOLogic.intT),dlcm))
in (instantiate' [] [SOME cfma]([th3,th1,th2] MRS (not_ast_p_ne)))
end
else (instantiate' [] [SOME cfma, SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("HOL.plus", T) $(Const ("HOL.times",_) $ c2 $ y) $z)) =>
if (is_arith_rel at) andalso (x=y)
then let val ast_z = norm_zero_one (linear_sub [] one z )
val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ ast_z $ A)
val th2 = prove_elementar sg "ss" (HOLogic.mk_eq (ast_z,Const("HOL.plus",T) $ (Const("HOL.uminus",HOLogic.intT --> HOLogic.intT) $(norm_zero_one z)) $ (Const("HOL.one",HOLogic.intT))))
val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "Orderings.less" (Const("HOL.zero",HOLogic.intT),dlcm))
in (instantiate' [] [SOME cfma] ([th3,th1,th2] MRS (not_ast_p_eq)))
end
else (instantiate' [] [SOME cfma, SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
|(Const("Orderings.less",_) $ c1 $(Const ("HOL.plus", _) $(Const ("HOL.times",_) $ pm1 $ y ) $ z )) =>
if (y=x) andalso (c1 =zero) then
if pm1 = (mk_number ~1) then
let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ (norm_zero_one z) $ A)
val th2 = prove_elementar sg "ss" (HOLogic.mk_binrel "Orderings.less" (zero,dlcm))
in (instantiate' [] [SOME cfma]([th2,th1] MRS (not_ast_p_lt)))
end
else let val th1 = prove_elementar sg "ss" (HOLogic.mk_binrel "Orderings.less" (Const("HOL.zero",HOLogic.intT),dlcm))
in (instantiate' [] [SOME cfma, SOME cA,SOME (cterm_of sg (norm_zero_one z))] (th1 RS (not_ast_p_gt)))
end
else (instantiate' [] [SOME cfma, SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
|Const ("Not",_) $ (Const("Divides.dvd",_)$ d $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $ c $ y ) $ z)) =>
if y=x then
let val cz = cterm_of sg (norm_zero_one z)
val th1 = (prove_elementar sg "ss" (HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.mod" (dlcm,d),norm_zero_one zero))) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1)
in (instantiate' [] [SOME cfma, SOME cA,SOME cz] (th1 RS (not_ast_p_ndvd)))
end
else (instantiate' [] [SOME cfma, SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
|(Const("Divides.dvd",_)$ d $ (Const ("HOL.plus",_) $ (Const ("HOL.times",_) $ c $ y ) $ z)) =>
if y=x then
let val cz = cterm_of sg (norm_zero_one z)
val th1 = (prove_elementar sg "ss" (HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.mod" (dlcm,d),norm_zero_one zero))) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1)
in (instantiate' [] [SOME cfma,SOME cA,SOME cz] (th1 RS (not_ast_p_dvd)))
end
else (instantiate' [] [SOME cfma, SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
|_ => (instantiate' [] [SOME cfma, SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
end;
(* ------------------------------------------------------------------------ *)
(* Main interpretation function for this backwards dirction*)
(* if atomic do generate atomis formulae else Construct theorems and then make RS with the construction theorems*)
(*Help Function*)
(* ------------------------------------------------------------------------- *)
fun decomp_nastp sg x dlcm A fm t = case t of
Const("op &",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_ast_p_conjI )
|Const("op |",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_ast_p_disjI)
|_ => ([], fn [] => generate_atomic_not_ast_p sg x fm dlcm A t);
fun not_ast_p_proof_of_c sg (x as Free(xn,xT)) fm dlcm A t =
let
val th = thm_of sg (decomp_nastp sg x dlcm (list_to_set xT (map norm_zero_one A)) fm) t
val fma = absfree (xn,xT, norm_zero_one fm)
in let val th1 = prove_elementar sg "ss" (HOLogic.mk_eq (fma,fma))
in [th,th1] MRS (not_ast_p_Q_elim)
end
end;
(* -------------------------------*)
(* Finding rho and beta for evalc *)
(* -------------------------------*)
fun rho_for_evalc sg at = case at of
(Const (p,_) $ s $ t) =>(
case AList.lookup (op =) operations p of
SOME f =>
((if (f ((dest_number s),(dest_number t)))
then prove_elementar sg "ss" (HOLogic.mk_eq(at,HOLogic.true_const))
else prove_elementar sg "ss" (HOLogic.mk_eq(at, HOLogic.false_const)))
handle _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl)
| _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl )
|Const("Not",_)$(Const (p,_) $ s $ t) =>(
case AList.lookup (op =) operations p of
SOME f =>
((if (f ((dest_number s),(dest_number t)))
then prove_elementar sg "ss" (HOLogic.mk_eq(at, HOLogic.false_const))
else prove_elementar sg "ss" (HOLogic.mk_eq(at,HOLogic.true_const)))
handle _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl)
| _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl )
| _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl;
(*=========================================================*)
fun decomp_evalc sg t = case t of
(Const("op &",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_conjI)
|(Const("op |",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_disjI)
|(Const("op -->",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_impI)
|(Const("op =", Type ("fun",[Type ("bool", []),_]))$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_eqI)
|_ => ([], fn [] => rho_for_evalc sg t);
fun proof_of_evalc sg fm = thm_of sg (decomp_evalc sg) fm;
(*==================================================*)
(* Proof of linform with the compact model *)
(*==================================================*)
fun decomp_linform sg vars t = case t of
(Const("op &",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_conjI)
|(Const("op |",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_disjI)
|(Const("op -->",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_impI)
|(Const("op =", Type ("fun",[Type ("bool", []),_]))$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_eqI)
|(Const("Not",_)$p) => ([p],fn [th] => th RS qe_Not)
|(Const("Divides.dvd",_)$d$r) =>
if is_numeral d then ([], fn [] => (prove_elementar sg "lf" (HOLogic.mk_eq (r, lint vars r))) RS (instantiate' [] [NONE , NONE, SOME (cterm_of sg d)](linearize_dvd)))
else (warning "Nonlinear Term --- Non numeral leftside at dvd";
raise COOPER)
|_ => ([], fn [] => prove_elementar sg "lf" (HOLogic.mk_eq (t, linform vars t)));
fun proof_of_linform sg vars f = thm_of sg (decomp_linform sg vars) f;
(* ------------------------------------------------------------------------- *)
(* Interpretaion of Protocols of the cooper procedure : minusinfinity version*)
(* ------------------------------------------------------------------------- *)
fun coopermi_proof_of sg (x as Free(xn,xT)) fm B dlcm =
(* Get the Bset thm*)
let val (minf_eqth, minf_moddth) = minf_proof_of_c sg x dlcm fm
val dpos = prove_elementar sg "ss" (HOLogic.mk_binrel "Orderings.less" (zero,dlcm));
val nbstpthm = not_bst_p_proof_of_c sg x fm dlcm B fm
in (dpos,minf_eqth,nbstpthm,minf_moddth)
end;
(* ------------------------------------------------------------------------- *)
(* Interpretaion of Protocols of the cooper procedure : plusinfinity version *)
(* ------------------------------------------------------------------------- *)
fun cooperpi_proof_of sg (x as Free(xn,xT)) fm A dlcm =
let val (pinf_eqth,pinf_moddth) = pinf_proof_of_c sg x dlcm fm
val dpos = prove_elementar sg "ss" (HOLogic.mk_binrel "Orderings.less" (zero,dlcm));
val nastpthm = not_ast_p_proof_of_c sg x fm dlcm A fm
in (dpos,pinf_eqth,nastpthm,pinf_moddth)
end;
(* ------------------------------------------------------------------------- *)
(* Interpretaion of Protocols of the cooper procedure : full version*)
(* ------------------------------------------------------------------------- *)
fun cooper_thm sg s (x as Free(xn,xT)) cfm dlcm ast bst= case s of
"pi" => let val (dpsthm,pinf_eqth,nbpth,pinf_moddth) = cooperpi_proof_of sg x cfm ast dlcm
in [dpsthm,pinf_eqth,nbpth,pinf_moddth] MRS (cppi_eq)
end
|"mi" => let val (dpsthm,minf_eqth,nbpth,minf_moddth) = coopermi_proof_of sg x cfm bst dlcm
in [dpsthm,minf_eqth,nbpth,minf_moddth] MRS (cpmi_eq)
end
|_ => error "parameter error";
(* ------------------------------------------------------------------------- *)
(* This function should evoluate to the end prove Procedure for one quantifier elimination for Presburger arithmetic*)
(* It shoud be plugged in the qfnp argument of the quantifier elimination proof function*)
(* ------------------------------------------------------------------------- *)
(* val (timef:(unit->thm) -> thm,prtime,time_reset) = gen_timer();*)
(* val (timef2:(unit->thm) -> thm,prtime2,time_reset2) = gen_timer(); *)
fun cooper_prv sg (x as Free(xn,xT)) efm = let
(* lfm_thm : efm = linearized form of efm*)
val lfm_thm = proof_of_linform sg [xn] efm
(*efm2 is the linearized form of efm *)
val efm2 = snd(qe_get_terms lfm_thm)
(* l is the lcm of all coefficients of x *)
val l = formlcm x efm2
(*ac_thm: efm = efm2 with adjusted coefficients of x *)
val ac_thm = [lfm_thm , (proof_of_adjustcoeffeq sg x l efm2)] MRS trans
(* fm is efm2 with adjusted coefficients of x *)
val fm = snd (qe_get_terms ac_thm)
(* cfm is l dvd x & fm' where fm' is fm where l*x is replaced by x*)
val cfm = unitycoeff x fm
(*afm is fm where c*x is replaced by 1*x or -1*x *)
val afm = adjustcoeff x l fm
(* P = %x.afm*)
val P = absfree(xn,xT,afm)
(* This simpset allows the elimination of the sets in bex {1..d} *)
val ss = presburger_ss addsimps
[simp_from_to] delsimps [P_eqtrue, P_eqfalse, bex_triv, insert_iff]
(* uth : EX x.P(l*x) = EX x. l dvd x & P x*)
val uth = instantiate' [] [SOME (cterm_of sg P) , SOME (cterm_of sg (mk_number l))] (unity_coeff_ex)
(* e_ac_thm : Ex x. efm = EX x. fm*)
val e_ac_thm = (forall_intr (cterm_of sg x) ac_thm) COMP (qe_exI)
(* A and B set of the formula*)
val A = aset x cfm
val B = bset x cfm
(* the divlcm (delta) of the formula*)
val dlcm = mk_number (divlcm x cfm)
(* Which set is smaller to generate the (hoepfully) shorter proof*)
val cms = if ((length A) < (length B )) then "pi" else "mi"
(* val _ = if cms = "pi" then writeln "Plusinfinity" else writeln "Minusinfinity"*)
(* synthesize the proof of cooper's theorem*)
(* cp_thm: EX x. cfm = Q*)
val cp_thm = cooper_thm sg cms x cfm dlcm A B
(* Exxpand the right hand side to get rid of EX j : {1..d} to get a huge disjunction*)
(* exp_cp_thm: EX x.cfm = Q' , where Q' is a simplified version of Q*)
(*
val _ = prth cp_thm
val _ = writeln "Expanding the bounded EX..."
*)
val exp_cp_thm = refl RS (simplify ss (cp_thm RSN (2,trans)))
(*
val _ = writeln "Expanded" *)
(* lsuth = EX.P(l*x) ; rsuth = EX x. l dvd x & P x*)
val (lsuth,rsuth) = qe_get_terms (uth)
(* lseacth = EX x. efm; rseacth = EX x. fm*)
val (lseacth,rseacth) = qe_get_terms(e_ac_thm)
(* lscth = EX x. cfm; rscth = Q' *)
val (lscth,rscth) = qe_get_terms (exp_cp_thm)
(* u_c_thm: EX x. P(l*x) = Q'*)
val u_c_thm = [([uth,prove_elementar sg "ss" (HOLogic.mk_eq (rsuth,lscth))] MRS trans),exp_cp_thm] MRS trans
(* result: EX x. efm = Q'*)
in ([e_ac_thm,[(prove_elementar sg "ss" (HOLogic.mk_eq (rseacth,lsuth))),u_c_thm] MRS trans] MRS trans)
end
|cooper_prv _ _ _ = error "Parameters format";
(* **************************************** *)
(* An Other Version of cooper proving *)
(* by giving a withness for EX *)
(* **************************************** *)
fun cooper_prv_w sg (x as Free(xn,xT)) efm = let
(* lfm_thm : efm = linearized form of efm*)
val lfm_thm = proof_of_linform sg [xn] efm
(*efm2 is the linearized form of efm *)
val efm2 = snd(qe_get_terms lfm_thm)
(* l is the lcm of all coefficients of x *)
val l = formlcm x efm2
(*ac_thm: efm = efm2 with adjusted coefficients of x *)
val ac_thm = [lfm_thm , (proof_of_adjustcoeffeq sg x l efm2)] MRS trans
(* fm is efm2 with adjusted coefficients of x *)
val fm = snd (qe_get_terms ac_thm)
(* cfm is l dvd x & fm' where fm' is fm where l*x is replaced by x*)
val cfm = unitycoeff x fm
(*afm is fm where c*x is replaced by 1*x or -1*x *)
val afm = adjustcoeff x l fm
(* P = %x.afm*)
val P = absfree(xn,xT,afm)
(* This simpset allows the elimination of the sets in bex {1..d} *)
val ss = presburger_ss addsimps
[simp_from_to] delsimps [P_eqtrue, P_eqfalse, bex_triv, insert_iff]
(* uth : EX x.P(l*x) = EX x. l dvd x & P x*)
val uth = instantiate' [] [SOME (cterm_of sg P) , SOME (cterm_of sg (mk_number l))] (unity_coeff_ex)
(* e_ac_thm : Ex x. efm = EX x. fm*)
val e_ac_thm = (forall_intr (cterm_of sg x) ac_thm) COMP (qe_exI)
(* lsuth = EX.P(l*x) ; rsuth = EX x. l dvd x & P x*)
val (lsuth,rsuth) = qe_get_terms (uth)
(* lseacth = EX x. efm; rseacth = EX x. fm*)
val (lseacth,rseacth) = qe_get_terms(e_ac_thm)
val (w,rs) = cooper_w [] cfm
val exp_cp_thm = case w of
(* FIXME - e_ac_thm just tipped to test syntactical correctness of the program!!!!*)
SOME n => e_ac_thm (* Prove cfm (n) and use exI and then Eq_TrueI*)
|_ => let
(* A and B set of the formula*)
val A = aset x cfm
val B = bset x cfm
(* the divlcm (delta) of the formula*)
val dlcm = mk_number (divlcm x cfm)
(* Which set is smaller to generate the (hoepfully) shorter proof*)
val cms = if ((length A) < (length B )) then "pi" else "mi"
(* synthesize the proof of cooper's theorem*)
(* cp_thm: EX x. cfm = Q*)
val cp_thm = cooper_thm sg cms x cfm dlcm A B
(* Exxpand the right hand side to get rid of EX j : {1..d} to get a huge disjunction*)
(* exp_cp_thm: EX x.cfm = Q' , where Q' is a simplified version of Q*)
in refl RS (simplify ss (cp_thm RSN (2,trans)))
end
(* lscth = EX x. cfm; rscth = Q' *)
val (lscth,rscth) = qe_get_terms (exp_cp_thm)
(* u_c_thm: EX x. P(l*x) = Q'*)
val u_c_thm = [([uth,prove_elementar sg "ss" (HOLogic.mk_eq (rsuth,lscth))] MRS trans),exp_cp_thm] MRS trans
(* result: EX x. efm = Q'*)
in ([e_ac_thm,[(prove_elementar sg "ss" (HOLogic.mk_eq (rseacth,lsuth))),u_c_thm] MRS trans] MRS trans)
end
|cooper_prv_w _ _ _ = error "Parameters format";
fun decomp_cnnf sg lfnp P = case P of
Const ("op &",_) $ p $q => ([p,q] , fn [th1,th2] => [th1,th2] MRS qe_conjI )
|Const ("op |",_) $ p $q => ([p,q] , fn [th1,th2] => [th1,th2] MRS qe_disjI)
|Const ("Not",_) $ (Const("Not",_) $ p) => ([p], fn [th] => th RS nnf_nn)
|Const("Not",_) $ (Const(opn,T) $ p $ q) =>
if opn = "op |"
then case (p,q) of
(A as (Const ("op &",_) $ r $ s),B as (Const ("op &",_) $ r1 $ t)) =>
if r1 = negate r
then ([r,HOLogic.Not$s,r1,HOLogic.Not$t],fn [th1_1,th1_2,th2_1,th2_2] => [[th1_1,th1_1] MRS qe_conjI,[th2_1,th2_2] MRS qe_conjI] MRS nnf_sdj)
else ([HOLogic.Not $ p,HOLogic.Not $ q ], fn [th1,th2] => [th1,th2] MRS nnf_ndj)
|(_,_) => ([HOLogic.Not $ p,HOLogic.Not $ q ], fn [th1,th2] => [th1,th2] MRS nnf_ndj)
else (
case (opn,T) of
("op &",_) => ([HOLogic.Not $ p,HOLogic.Not $ q ], fn [th1,th2] =>[th1,th2] MRS nnf_ncj )
|("op -->",_) => ([p,HOLogic.Not $ q ], fn [th1,th2] =>[th1,th2] MRS nnf_nim )
|("op =",Type ("fun",[Type ("bool", []),_])) =>
([HOLogic.conj $ p $ (HOLogic.Not $ q),HOLogic.conj $ (HOLogic.Not $ p) $ q], fn [th1,th2] => [th1,th2] MRS nnf_neq)
|(_,_) => ([], fn [] => lfnp P)
)
|(Const ("op -->",_) $ p $ q) => ([HOLogic.Not$p,q], fn [th1,th2] => [th1,th2] MRS nnf_im)
|(Const ("op =", Type ("fun",[Type ("bool", []),_])) $ p $ q) =>
([HOLogic.conj $ p $ q,HOLogic.conj $ (HOLogic.Not $ p) $ (HOLogic.Not $ q) ], fn [th1,th2] =>[th1,th2] MRS nnf_eq )
|_ => ([], fn [] => lfnp P);
fun proof_of_cnnf sg p lfnp =
let val th1 = thm_of sg (decomp_cnnf sg lfnp) p
val rs = snd(qe_get_terms th1)
val th2 = prove_elementar sg "ss" (HOLogic.mk_eq(rs,simpl rs))
in [th1,th2] MRS trans
end;
end;