/usr/share/hol88-2.02.19940316/Library/reduce/arithconv.ml is in hol88-library-source 2.02.19940316-31.
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** LIBRARY: reduce (part II) **
** **
** DESCRIPTION: Conversions to reduce arithmetic constant expressions **
** **
** AUTHOR: John Harrison **
** University of Cambridge Computer Laboratory **
** New Museums Site **
** Pembroke Street **
** Cambridge CB2 3QG **
** England. **
** **
** jrh@cl.cam.ac.uk **
** **
** DATE: 18th May 1991 **
******************************************************************************%
%-----------------------------------------------------------------------%
% dest_op - Split application down spine, checking head operator %
%-----------------------------------------------------------------------%
let dest_op op tm = snd ((assert (curry $= op) # I) (strip_comb tm));;
%-----------------------------------------------------------------------%
% term_of_int - Convert ML integer to object level numeric constant %
%-----------------------------------------------------------------------%
let term_of_int =
let ty = ":num" in
\n. mk_const(string_of_int n, ty);;
%-----------------------------------------------------------------------%
% int_of_term - Convert object level numeric constant to ML integer %
%-----------------------------------------------------------------------%
let int_of_term =
int_of_string o fst o dest_const;;
%-----------------------------------------------------------------------%
% provelt x y = |- [x] < [y], if true, else undefined. %
%-----------------------------------------------------------------------%
let provelt =
let ltstep = PROVE("!x. (z = SUC y) ==> (x < y) ==> (x < z)",
GEN_TAC THEN DISCH_THEN SUBST1_TAC THEN
MATCH_ACCEPT_TAC (theorem `prim_rec` `LESS_SUC`))
and ltbase = PROVE("(y = SUC x) ==> (x < y)",
DISCH_THEN SUBST1_TAC THEN
MATCH_ACCEPT_TAC (theorem `prim_rec` `LESS_SUC_REFL`))
and bistep = PROVE("(SUC x < SUC y) = (x < y)",
MATCH_ACCEPT_TAC (theorem `arithmetic` `LESS_MONO_EQ`))
and bibase = PROVE("!x. 0 < (SUC x)",
MATCH_ACCEPT_TAC (theorem `prim_rec` `LESS_0`))
and ltop = "$<" and eqop = "$=:bool->bool->bool" and rhs = "x < y"
and xv = "x:num" and yv = "y:num" and zv = "z:num" and Lo = "$< 0" in
\x y. let xn = term_of_int x and yn = term_of_int y in
if 4*(y - x) < 5*x then
let x' = x + 1 in let xn' = term_of_int x' in
let step = SPEC xn ltstep in
letref z,zn,zn',th = x',xn',xn', MP (INST [(xn,xv);(xn',yv)] ltbase)
(num_CONV xn') in
while z < y do
(zn':=term_of_int(z:=z+1);
th := MP (MP (INST [(zn,yv); (zn',zv)] step) (num_CONV zn')) th;
zn:=zn');
th
else
let lhs = mk_comb(mk_comb(ltop,xn),yn) in
let pat = mk_comb(mk_comb(eqop,lhs),rhs) in
letref w, z, wn, zn, th = x, y, xn, yn, REFL lhs in
while w > 0 do
(th :=
let tran = TRANS (SUBST [(num_CONV wn,xv); (num_CONV zn,yv)] pat th)
in tran (INST[((wn:=term_of_int(w:=w-1)),xv);
((zn:=term_of_int(z:=z-1)),yv)] bistep));
EQ_MP (SYM (TRANS th (AP_TERM Lo (num_CONV zn))))
(SPEC (term_of_int(z-1)) bibase);;
%-----------------------------------------------------------------------%
% NEQ_CONV "[x] = [y]" = |- ([x] = [y]) = [x=y -> T | F] %
%-----------------------------------------------------------------------%
let NEQ_CONV =
let eq1 = PROVE
("(x < y) ==> ((x = y) = F)",
ONCE_REWRITE_TAC[] THEN
MATCH_ACCEPT_TAC (theorem `prim_rec` `LESS_NOT_EQ`))
and eq2 = PROVE
("(y < x) ==> ((x = y) = F)",
ONCE_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN
MATCH_ACCEPT_TAC (theorem `prim_rec` `LESS_NOT_EQ`))
and neqop = "=:num->num->bool" and xv = "x:num" and yv = "y:num" in
\tm. (let [xn;yn] = dest_op neqop tm in
let x = int_of_term xn and y = int_of_term yn in
if x = y then EQT_INTRO (REFL xn) else
if x < y then MP (INST [(xn,xv);(yn,yv)] eq1)
(provelt x y)
else MP (INST [(xn,xv);(yn,yv)] eq2)
(provelt y x))
? failwith `NEQ_CONV`;;
%-----------------------------------------------------------------------%
% LT_CONV "[x] < [y]" = |- ([x] < [y]) = [x<y -> T | F] %
%-----------------------------------------------------------------------%
let LT_CONV =
let lt1 = PROVE("!x. (x < x) = F",
REWRITE_TAC[theorem `prim_rec` `LESS_REFL`])
and lt2 = PROVE("(y < x) ==> ((x < y) = F)",
PURE_ONCE_REWRITE_TAC[EQ_CLAUSES] THEN REPEAT DISCH_TAC THEN
IMP_RES_TAC (theorem `arithmetic` `LESS_ANTISYM`))
and ltop = "$<" and xv = "x:num" and yv = "y:num" in
\tm. (let [xn;yn] = dest_op ltop tm in
let x = int_of_term xn and y = int_of_term yn in
if x < y then EQT_INTRO (provelt x y) else
if x = y then SPEC xn lt1
else MP (INST [(xn,xv);(yn,yv)] lt2)
(provelt y x))
? failwith `LT_CONV`;;
%-----------------------------------------------------------------------%
% GT_CONV "[x] > [y]" = |- ([x] > [y]) = [x>y -> T | F] %
%-----------------------------------------------------------------------%
let GT_CONV =
let gt1 = PROVE("!x. (x > x) = F",
REWRITE_TAC[theorem `prim_rec` `LESS_REFL`;
definition `arithmetic` `GREATER`])
and gt2 = PROVE("(x < y) ==> ((x > y) = F)",
PURE_REWRITE_TAC
[EQ_CLAUSES; definition `arithmetic` `GREATER`]
THEN REPEAT DISCH_TAC THEN
IMP_RES_TAC (theorem `arithmetic` `LESS_ANTISYM`))
and gt3 = PROVE("(y < x) ==> ((x > y) = T)",
DISCH_THEN (SUBST1_TAC o SYM o EQT_INTRO) THEN
MATCH_ACCEPT_TAC (definition `arithmetic` `GREATER`))
and gtop = "$>" and xv = "x:num" and yv = "y:num" in
\tm. (let [xn;yn] = dest_op gtop tm in
let x = int_of_term xn and y = int_of_term yn in
if x = y then SPEC xn gt1 else
if x < y then MP (INST [(xn,xv);(yn,yv)] gt2)
(provelt x y)
else MP (INST [(xn,xv); (yn,yv)] gt3)
(provelt y x))
? failwith `GT_CONV`;;
%-----------------------------------------------------------------------%
% LE_CONV "[x] <= [y]" = |- ([x]<=> [y]) = [x<=y -> T | F] %
%-----------------------------------------------------------------------%
let LE_CONV =
let le1 = PROVE("!x. (x <= x) = T",
REWRITE_TAC[theorem `arithmetic` `LESS_EQ_REFL`])
and le2 = PROVE("(x < y) ==> ((x <= y) = T)",
DISCH_THEN (ACCEPT_TAC o EQT_INTRO o MATCH_MP
(theorem `arithmetic` `LESS_IMP_LESS_OR_EQ`)))
and le3 = PROVE("(y < x) ==> ((x <= y) = F)",
PURE_ONCE_REWRITE_TAC[EQ_CLAUSES] THEN
REPEAT DISCH_TAC THEN
IMP_RES_TAC (theorem `arithmetic` `LESS_EQ_ANTISYM`))
and leop = "$<=" and xv = "x:num" and yv = "y:num" in
\tm. (let [xn;yn] = dest_op leop tm in
let x = int_of_term xn and y = int_of_term yn in
if x = y then SPEC xn le1 else
if x < y then MP (INST [(xn,xv);(yn,yv)] le2)
(provelt x y)
else MP (INST [(xn,xv);(yn,yv)] le3)
(provelt y x))
? failwith `LE_CONV`;;
%-----------------------------------------------------------------------%
% GE_CONV "[x] >= [y]" = |- ([x] >= [y]) = [x>=y -> T | F] %
%-----------------------------------------------------------------------%
let GE_CONV =
let ge1 = PROVE("!x. (x >= x) = T",
REWRITE_TAC[definition `arithmetic` `GREATER_OR_EQ`])
and ge2 = PROVE("(y < x) ==> ((x >= y) = T)",
DISCH_TAC THEN
ASM_REWRITE_TAC[definition `arithmetic` `GREATER_OR_EQ`;
definition `arithmetic` `GREATER`])
and ge3 = PROVE("(x < y) ==> ((x >= y) = F)",
PURE_REWRITE_TAC (EQ_CLAUSES. (map (definition `arithmetic`)
[`GREATER_OR_EQ`; `GREATER`])) THEN
PURE_ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN
REPEAT STRIP_TAC THEN IMP_RES_TAC (PURE_REWRITE_RULE
[definition `arithmetic` `LESS_OR_EQ`]
(theorem `arithmetic` `LESS_EQ_ANTISYM`)))
and geop = "$>=" and xv = "x:num" and yv = "y:num" in
\tm. (let [xn;yn] = dest_op geop tm in
let x = int_of_term xn and y = int_of_term yn in
if x = y then SPEC xn ge1 else
if x < y then MP (INST [(xn,xv);(yn,yv)] ge3)
(provelt x y)
else MP (INST [(xn,xv);(yn,yv)] ge2)
(provelt y x))
? failwith `GE_CONV`;;
%-----------------------------------------------------------------------%
% SUC_CONV "SUC [x]" = |- SUC [x] = [x+1] %
%-----------------------------------------------------------------------%
let SUC_CONV =
let sucop = "SUC" in
\tm. (let [xn] = dest_op sucop tm in
SYM (num_CONV (term_of_int (1 + (int_of_term xn)))))
? failwith `SUC_CONV`;;
%-----------------------------------------------------------------------%
% PRE_CONV "PRE [n]" = |- PRE [n] = [n-1] %
%-----------------------------------------------------------------------%
let PRE_CONV =
let preop = "PRE" and zero = "0" and xv = "x:num" and yv = "y:num"
and spree = PROVE("(x = SUC y) ==> (PRE x = y)",
DISCH_TAC THEN ASM_REWRITE_TAC[theorem `prim_rec` `PRE`])
and szero = PROVE("PRE 0 = 0",REWRITE_TAC[theorem `prim_rec` `PRE`]) in
\tm. (let [xn] = dest_op preop tm in
if xn = zero then szero
else MP (INST[(xn,xv);(term_of_int((int_of_term xn) - 1),yv)] spree)
(num_CONV xn))
? failwith `PRE_CONV`;;
%-----------------------------------------------------------------------%
% SBC_CONV "[x] - [y]" = |- ([x] - [y]) = [x - y] %
%-----------------------------------------------------------------------%
let SBC_CONV =
let subm = PROVE("(x < y) ==> (x - y = 0)",
PURE_ONCE_REWRITE_TAC[theorem `arithmetic` `SUB_EQ_0`]
THEN MATCH_ACCEPT_TAC
(theorem `arithmetic` `LESS_IMP_LESS_OR_EQ`))
and step = PROVE("(SUC x) - (SUC y) = x - y",
MATCH_ACCEPT_TAC (theorem `arithmetic` `SUB_MONO_EQ`))
and base1 = PROVE("!x. x - 0 = x",
REWRITE_TAC[theorem `arithmetic` `SUB_0`])
and base2 = PROVE("!x. x - x = 0",
MATCH_ACCEPT_TAC(theorem `arithmetic` `SUB_EQUAL_0`))
and less0 = PROVE("!x. 0 < SUC x",
REWRITE_TAC[theorem `prim_rec` `LESS_0`])
and swap = PROVE("(x - z = y) ==> (0 < y) ==> (x - y = z)",
let [sub_less_0; sub_sub; less_imp_less_or_eq; add_sym; add_sub] =
map (theorem `arithmetic`)
[`SUB_LESS_0`; `SUB_SUB`; `LESS_IMP_LESS_OR_EQ`; `ADD_SYM`; `ADD_SUB`] in
DISCH_THEN (SUBST1_TAC o SYM) THEN PURE_ONCE_REWRITE_TAC
[SYM (SPEC_ALL sub_less_0)] THEN DISCH_THEN (SUBST1_TAC o SPEC "x:num" o
MATCH_MP sub_sub o MATCH_MP less_imp_less_or_eq) THEN PURE_ONCE_REWRITE_TAC
[add_sym] THEN PURE_ONCE_REWRITE_TAC[add_sub] THEN REFL_TAC)
and lop = "$< 0" and minusop = "$-" and eqop = "$=:num->num->bool"
and rhs = "x - y" and xv = "x:num" and yv = "y:num" and zv = "z:num" in
let sprove x y =
let xn = term_of_int x and yn = term_of_int y in
let lhs = mk_comb(mk_comb(minusop,xn),yn) in
let pat = mk_comb(mk_comb(eqop,lhs),rhs) in
letref w, z, wn, zn, th = x, y, xn, yn, REFL lhs in
while (z > 0) do
(th :=
let tran = TRANS (SUBST [(num_CONV wn,xv); (num_CONV zn,yv)] pat th) in
tran (INST [((wn := term_of_int(w:=w-1)),xv);
((zn := term_of_int(z:=z-1)),yv)] step));
TRANS th (SPEC wn base1) in
\tm. (let [xn;yn] = dest_op minusop tm in
let x = int_of_term xn and y = int_of_term yn in
if x < y then MP (INST[(xn,xv);(yn,yv)] subm)
(provelt x y) else
if y = 0 then SPEC xn base1 else
if x = y then SPEC xn base2 else
if y < (x - y) then sprove x y
else
let z = x - y in let zn = term_of_int z in
MP (MP
(INST[(xn,xv);(yn,yv);(zn,zv)] swap)
(sprove x z))
(EQ_MP (AP_TERM lop (SYM (num_CONV yn)))
(SPEC (term_of_int (y-1)) less0)))
? failwith `SBC_CONV`;;
%-----------------------------------------------------------------------%
% ADD_CONV "[x] + [y]" = |- [x] + [y] = [x+y] %
%-----------------------------------------------------------------------%
let ADD_CONV =
let subadd = PROVE
("(z - y = x) ==> 0 < x ==> (x + y = z)",
DISCH_THEN (SUBST1_TAC o SYM) THEN
PURE_ONCE_REWRITE_TAC[SYM (SPEC_ALL (theorem `arithmetic` `SUB_LESS_0`))]
THEN DISCH_THEN (SUBST1_TAC o MATCH_MP (theorem `arithmetic` `SUB_ADD`) o
MATCH_MP (theorem `arithmetic` `LESS_IMP_LESS_OR_EQ`)) THEN REFL_TAC)
and [raz; laz] = CONJUNCTS(PROVE("(!x. x + 0 = x) /\ (!y. 0 + y = y)",
REWRITE_TAC[definition `arithmetic` `ADD`; theorem `arithmetic` `ADD_0`]))
and lo = PROVE("!n. 0 < SUC n",MATCH_ACCEPT_TAC(theorem `prim_rec` `LESS_0`))
and plusop = "$+" and minusop = "$-" and lop = "$< 0"
and xv = "x:num" and yv = "y:num" and zv = "z:num" in
\tm. (let [xn;yn] = dest_op plusop tm in
let x = int_of_term xn and y = int_of_term yn in
if x = 0 then SPEC yn laz else
if y = 0 then SPEC xn raz else
let zn = term_of_int(x + y) in
let p1 = SBC_CONV (mk_comb(mk_comb(minusop,zn),yn))
and p2 = EQ_MP (AP_TERM lop (SYM (num_CONV xn)))
(SPEC (term_of_int (int_of_term xn - 1)) lo)
and chain = INST[(xn,xv); (yn,yv); (zn,zv)] subadd in
MP (MP chain p1) p2)
? failwith `ADD_CONV`;;
%-----------------------------------------------------------------------%
% MUL_CONV "[x] * [y]" = |- [x] * [y] = [x*y] %
%-----------------------------------------------------------------------%
let MUL_CONV =
let [mbase; mstep; mzero] = CONJUNCTS (PROVE
("(!y. 0 * y = 0) /\ (!y x. (SUC x) * y = (x * y) + y) /\ (!n. n * 0 = 0)",
REWRITE_TAC[definition `arithmetic` `MULT`;
theorem `arithmetic` `MULT_0`]))
and msym = PROVE("!m n. m * n = n * m",
MATCH_ACCEPT_TAC (theorem `arithmetic` `MULT_SYM`))
and multop = "$*" and xv = "x:num" and pv = "p:num" and zero = "0"
and plusop = "$+" and eqop = "=:num->num->bool" in
let mulpr x y =
let xn = term_of_int x and yn = term_of_int y in
let step = SPEC yn mstep in
let pat = mk_comb(mk_comb(eqop,(mk_comb(mk_comb(multop,xv),yn))),
mk_comb(mk_comb(plusop,pv),yn)) in
letref w, wn, p, th = 0, zero, 0, SPEC yn mbase in
while w < x do
(th := TRANS
(let st = SPEC wn step in
SUBST [(SYM (num_CONV(wn:=term_of_int(w:=w+1))),xv);
(th,pv)] pat st)
(ADD_CONV (mk_comb(mk_comb(plusop,(term_of_int p)),yn)));
p := p + y);
th in
\tm. (let [xn;yn] = dest_op multop tm in
let x = int_of_term xn and y = int_of_term yn in
if x = 0 then SPEC yn mbase else
if y = 0 then SPEC xn mzero else
if x < y then mulpr x y
else TRANS (SPECL [xn;yn] msym) (mulpr y x))
? failwith `MUL_CONV`;;
%-----------------------------------------------------------------------%
% EXP_CONV "[x] EXP [y]" = |- [x] EXP [y] = [x**y] %
%-----------------------------------------------------------------------%
let EXP_CONV =
let [ebase; estep] = CONJUNCTS (PROVE
("(!m. m EXP 0 = 1) /\ (!m n. m EXP (SUC n) = m * (m EXP n))",
REWRITE_TAC[definition `arithmetic` `EXP`]))
and expop = "EXP" and multop = "$*" and zero = "0" and ev = "e:num"
and eqop = "$=:num->num->bool" and yv = "y:num" in
\tm. (let [xn;yn] = dest_op expop tm in
let x = int_of_term xn and y = int_of_term yn
and step = SPEC xn estep in
let pat = mk_comb(mk_comb(eqop,mk_comb(mk_comb(expop,xn),yv)),
mk_comb(mk_comb(multop,xn),ev)) in
letref z, zn, e, th = 0, zero, 1, SPEC xn ebase in
while z < y do
(th := TRANS
(let st = SPEC zn step in
SUBST [(SYM (num_CONV(zn:=term_of_int(z:=z+1))),yv);
(th,ev)] pat st)
(MUL_CONV (mk_comb(mk_comb(multop,xn),term_of_int e)));
e := x * e);
th)
? failwith `EXP_CONV`;;
%-----------------------------------------------------------------------%
% DIV_CONV "[x] DIV [y]" = |- [x] DIV [y] = [x div y] %
%-----------------------------------------------------------------------%
let DIV_CONV =
let divt = PROVE("(q * y = p) ==> (p + r = x) ==> (r < y) ==> (x DIV y = q)",
REPEAT DISCH_TAC THEN MATCH_MP_TAC (theorem `arithmetic` `DIV_UNIQUE`) THEN
EXISTS_TAC "r:num" THEN ASM_REWRITE_TAC[])
and divop = "$DIV" and multop = "$*" and plusop = "$+"
and xv,yv,pv,qv,rv = "x:num","y:num","p:num","q:num","r:num" in
\tm. (let [xn;yn] = dest_op divop tm in
let x = int_of_term xn and y = int_of_term yn in
let q = x / y in
let p = q * y in
let r = x - p in
let pn = term_of_int p and qn = term_of_int q and rn = term_of_int r in
let p1 = MUL_CONV (mk_comb(mk_comb(multop,qn),yn))
and p2 = ADD_CONV (mk_comb(mk_comb(plusop,pn),rn))
and p3 = provelt r y
and chain = INST[(xn,xv); (yn,yv); (pn,pv); (qn,qv); (rn,rv)] divt
in MP (MP (MP chain p1) p2) p3)
? failwith `DIV_CONV`;;
%-----------------------------------------------------------------------%
% MOD_CONV "[x] MOD [y]" = |- [x] MOD [y] = [x mod y] %
%-----------------------------------------------------------------------%
let MOD_CONV =
let modt = PROVE("(q * y = p) ==> (p + r = x) ==> (r < y) ==> (x MOD y = r)",
REPEAT DISCH_TAC THEN MATCH_MP_TAC (theorem `arithmetic` `MOD_UNIQUE`) THEN
EXISTS_TAC "q:num" THEN ASM_REWRITE_TAC[])
and modop = "$MOD" and multop = "$*" and plusop = "$+"
and xv,yv,pv,qv,rv = "x:num","y:num","p:num","q:num","r:num" in
\tm. (let [xn;yn] = dest_op modop tm in
let x = int_of_term xn and y = int_of_term yn in
let q = x / y in
let p = q * y in
let r = x - p in
let pn = term_of_int p and qn = term_of_int q and rn = term_of_int r in
let p1 = MUL_CONV (mk_comb(mk_comb(multop,qn),yn))
and p2 = ADD_CONV (mk_comb(mk_comb(plusop,pn),rn))
and p3 = provelt r y
and chain = INST[(xn,xv); (yn,yv); (pn,pv); (qn,qv); (rn,rv)] modt
in MP (MP (MP chain p1) p2) p3)
? failwith `MOD_CONV`;;
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