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|
%=============================================================================%
% HOL 88 Version 2.0 %
% %
% FILE NAME: list.ml %
% %
% DESCRIPTION: Defined procedures for list induction and definition %
% by primitive recursion on lists. Derived inference %
% rules for reasoning about lists. %
% %
% The induction/primitive recursion are really only for %
% compatibility with old HOL. %
% %
% AUTHOR: T. F. Melham (87.05.30) %
% W. Wong (31 Jan 94) %
% %
% USES FILES: ind.ml, prim_rec.ml, numconv.ml %
% %
% University of Cambridge %
% Hardware Verification Group %
% Computer Laboratory %
% New Museums Site %
% Pembroke Street %
% Cambridge CB2 3QG %
% England %
% %
% COPYRIGHT: T. F. Melham 1987 1990 %
% %
% REVISION HISTORY: 90.09.08 %
%=============================================================================%
if compiling then
(loadf `../ml/ind`;
loadf `../ml/prim_rec`;
loadf `../ml/numconv`;
loadf `../ml/num`);;
% --------------------------------------------------------------------- %
% LIST_INDUCT: (thm # thm) -> thm %
% %
% A1 |- t[[]] A2 |- !tl. t[tl] ==> !h. t[CONS h t] %
% ---------------------------------------------------------- %
% A1 u A2 |- !l. t[l] %
% %
% --------------------------------------------------------------------- %
let LIST_INDUCT =
let list_INDUCT = theorem `list` `list_INDUCT` in
\(base,step).
(let (tl,body) = dest_forall(concl step) in
let (asm,h,con) = (I # dest_forall) (dest_imp body) in
let P = "\^tl.^asm" and
b1 = genvar bool_ty and
b2 = genvar bool_ty in
let base' = EQ_MP (SYM(BETA_CONV "^P []")) base and
step' = DISCH asm (SPEC h (UNDISCH(SPEC tl step))) and
hypth = SYM(RIGHT_BETA(REFL "^P ^tl")) and
concth = SYM(RIGHT_BETA(REFL "^P(CONS ^h ^tl)")) and
IND = SPEC P (INST_TYPE [type_of h,":*"] list_INDUCT) in
let th1 = SUBST [hypth,b1;concth,b2]
"^(concl step') = (^b1 ==> ^b2)"
(REFL (concl step')) in
let th2 = GEN tl (DISCH "^P ^tl" (GEN h(UNDISCH (EQ_MP th1 step')))) in
let th3 = SPEC tl (MP IND (CONJ base' th2)) in
GEN tl (EQ_MP (BETA_CONV(concl th3)) th3))?failwith `LIST_INDUCT`;;
% --------------------------------------------------------------------- %
% %
% LIST_INDUCT_TAC %
% %
% [A] !l.t[l] %
% ================================ %
% [A] t[[]], [A,t[l]] !h. t[CONS h t] %
% %
% --------------------------------------------------------------------- %
let LIST_INDUCT_TAC =
let list_INDUCT = theorem `list` `list_INDUCT` in
INDUCT_THEN list_INDUCT ASSUME_TAC;;
% --------------------------------------------------------------------- %
% %
% SNOC_INDUCT_TAC %
% %
% [A] !l.t[l] %
% ================================ %
% [A] t[[]], [A,t[l]] !h. t[SNOC x t] %
% %
% --------------------------------------------------------------------- %
let SNOC_INDUCT_TAC =
let SNOC_INDUCT = theorem `list` `SNOC_INDUCT` in
INDUCT_THEN SNOC_INDUCT ASSUME_TAC;;
% ------------------------------------------------------------------------- %
% EQ_LENGTH_INDUCT_TAC : tactic %
% A ?- !l1 l2. (LENGTH l1 = LENGTH l2) ==> t[l1, l2] %
% ==================================================== EQ_LENGTH_INDUCT_TAC %
% A ?- t[ []/l1, []/l2 ] %
% A,LENGTH l1 = LENGTH l2 ?- t[(CONS h l1)/l1,(CONS h' l2)/l2] %
% ------------------------------------------------------------------------- %
let EQ_LENGTH_INDUCT_TAC =
let SUC_NOT = theorem `arithmetic` `SUC_NOT` and
NOT_SUC = theorem `num` `NOT_SUC` and
INV_SUC_EQ = theorem `prim_rec` `INV_SUC_EQ` and
LENGTH = definition `list` `LENGTH` in
LIST_INDUCT_TAC THENL[
LIST_INDUCT_TAC THENL[
REPEAT (CONV_TAC FORALL_IMP_CONV) THEN DISCH_THEN (\t.ALL_TAC);
REWRITE_TAC[LENGTH;SUC_NOT]];
GEN_TAC THEN LIST_INDUCT_TAC THEN REWRITE_TAC[LENGTH;NOT_SUC;INV_SUC_EQ]
THEN GEN_TAC THEN REPEAT (CONV_TAC FORALL_IMP_CONV) THEN DISCH_TAC];;
% ------------------------------------------------------------------------- %
% EQ_LENGTH_SNOC_INDUCT_TAC : tactic %
% A ?- !l1 l2.(LENGTH l1 = LENGTH l2) ==> t[l1,l2] %
% =============================================== EQ_LENGTH_SNOC_INDUCT_TAC %
% A ?- t[ []/l1, []/l2 ] %
% A,LENGHT l1 = LENGTH l2 ?- t[(SNOC h l1)/l1,(SNOC h' l2)/l2] %
% ------------------------------------------------------------------------- %
let EQ_LENGTH_SNOC_INDUCT_TAC =
let SUC_NOT = theorem `arithmetic` `SUC_NOT` and
NOT_SUC = theorem `num` `NOT_SUC` and
INV_SUC_EQ = theorem `prim_rec` `INV_SUC_EQ` and
LENGTH = definition `list` `LENGTH` and
LENGTH_SNOC = theorem `list` `LENGTH_SNOC` in
SNOC_INDUCT_TAC THENL[
SNOC_INDUCT_TAC THENL[
REPEAT (CONV_TAC FORALL_IMP_CONV) THEN DISCH_THEN (\t.ALL_TAC);
REWRITE_TAC[LENGTH;LENGTH_SNOC;SUC_NOT]];
GEN_TAC THEN SNOC_INDUCT_TAC
THEN REWRITE_TAC[LENGTH;LENGTH_SNOC;NOT_SUC;INV_SUC_EQ]
THEN GEN_TAC THEN REPEAT (CONV_TAC FORALL_IMP_CONV) THEN DISCH_TAC];;
% --------------------------------------------------------------------- %
% Definition by primitive recursion for lists %
% (For compatibility of new/old HOL.) %
% --------------------------------------------------------------------- %
let new_list_rec_definition =
let list_Axiom = theorem `list` `list_Axiom` in
\(name,tm). new_recursive_definition false list_Axiom name tm;;
let new_infix_list_rec_definition =
let list_Axiom = theorem `list` `list_Axiom` in
\(name,tm). new_recursive_definition true list_Axiom name tm;;
% --------------------------------------------------------------------- %
% LENGTH_CONV: compute the length of a list %
% %
% A call to LENGTH_CONV "LENGTH[x1;...;xn]" returns: %
% %
% |- LENGTH [x1;...;xn] = n where n is a numeral constant %
% --------------------------------------------------------------------- %
let LENGTH_CONV =
let LEN = definition `list` `LENGTH` in
let dcons tm = snd(((\c.fst(dest_const c)=`CONS`) # I)(strip_comb tm)) in
let cend tm = (fst(dest_const tm) = `NIL` => [] | fail) in
letrec stripl tm = (let [h;t] = dcons tm in (h . stripl t)) ? cend tm in
let SUC = let suctm = "SUC" and numty = ":num" in
\(i,th). let n = mk_const(string_of_int i,numty) in
TRANS (AP_TERM suctm th) (SYM(num_CONV n)) in
let itfn cth h (i,th) =
i+1,TRANS (SPEC (rand(lhs(concl th))) (SPEC h cth)) (SUC (i,th)) in
let check = assert(curry $= `LENGTH` o fst o dest_const) in
\tm. (let _,[ty] = dest_type(type_of (snd((check # I)(dest_comb tm)))) in
let nil,cons = CONJ_PAIR (INST_TYPE [ty,":*"] LEN) in
snd(itlist (itfn cons) (stripl (rand tm)) (1,nil)))
? failwith `LENGTH_CONV`;;
% --------------------------------------------------------------------- %
% list_EQ_CONV: equality of lists. %
% %
% This conversion proves or disproves the equality of two lists, given %
% a conversion for deciding the equality of elements. %
% %
% A call to: %
% %
% list_EQ_CONV conv "[x1;...;xn] = [y1;...;ym]" %
% %
% returns: %
% %
% |- ([x1;...;xn] = [y1;...;ym]) = F %
% %
% if: %
% %
% 1: ~(n=m) or 2: conv proves |- (xi = yi) = F for any 1<=i<=n,m %
% %
% and: %
% %
% |- ([x1;...;xn] = [y1;...;ym]) = T %
% %
% if: %
% %
% 1: (n=m) and xi is syntactically identical to yi for 1<=i<=n,m, or %
% 2: (n=m) and conv proves |- (xi=yi)=T for 1<=i<=n,m %
% --------------------------------------------------------------------- %
let list_EQ_CONV =
let T = "T" and F = "F" in
let cnil = theorem `list` `NOT_CONS_NIL` in
let lne = theorem `list` `LIST_NOT_EQ` in
let nel = theorem `list` `NOT_EQ_LIST`in
let leq = theorem `list` `EQ_LIST` in
let dcons tm = snd(((\c.fst(dest_const c)=`CONS`) # I)(strip_comb tm)) in
let cend tm = (fst(dest_const tm) = `NIL` => [] | fail) in
letrec stripl tm = (let [h;t] = dcons tm in (h . stripl t)) ? cend tm in
let Cons ty = let lty = mk_type(`list`,[ty]) in
let cty = mk_type(`fun`,[ty;mk_type(`fun`,[lty;lty])]) in
\h t. mk_comb(mk_comb(mk_const(`CONS`,cty),h),t) in
let Nil ty = let lty = mk_type(`list`,[ty]) in mk_const(`NIL`,lty) in
letrec split n l =
if (n=0) then [],l else ((curry $. (hd l)) # I)(split (n-1) (tl l)) in
let itfn cnv [leq;lne;nel] (h1,h2) th =
if (is_neg (concl th)) then
let l1,l2 = dest_eq(dest_neg (concl th)) in
SPEC h2 (SPEC h1 (MP (SPEC l2 (SPEC l1 lne)) th)) else
let l1,l2 = dest_eq(concl th) in
let heq = cnv (mk_eq(h1,h2)) in
if (rand(concl heq) = T) then
let th1 = MP (SPEC h2 (SPEC h1 leq)) (EQT_ELIM heq) in
MP (SPEC l2 (SPEC l1 th1)) th else
let th1 = MP (SPEC h2 (SPEC h1 nel)) (EQF_ELIM heq) in
SPEC l2 (SPEC l1 th1) in
\cnv tm. (let l1,l2 = (stripl # stripl) (dest_eq tm) in
if (l1=l2) then EQT_INTRO(REFL (rand tm)) else
let _,[ty] = dest_type(type_of(rand tm)) in
let n = length l1 and m = length l2 in
let thms = map (INST_TYPE [ty,":*"]) [leq;lne;nel] in
let ifn = itfn cnv thms in
if (n<m) then
let exd,(x.xs) = split n l2 in
let rest = itlist (Cons ty) xs (Nil ty) in
let thm1 = SPEC rest (SPEC x (INST_TYPE [ty,":*"] cnil)) in
EQF_INTRO(itlist ifn (combine(l1,exd))(NOT_EQ_SYM thm1)) else
if (m<n) then
let exd,(x.xs) = split m l1 in
let rest = itlist (Cons ty) xs (Nil ty) in
let thm1 = SPEC rest (SPEC x (INST_TYPE [ty,":*"] cnil)) in
EQF_INTRO(itlist ifn (combine(exd,l2)) thm1) else
let thm = itlist ifn (combine(l1,l2)) (REFL (Nil ty)) in
(EQF_INTRO thm ? EQT_INTRO thm))
? failwith `list_EQ_CONV`;;
%--------------------------------------------------------------%
% Following local functions added by WW 31 Jan 94 %
begin_section `list_convs`;;
let check_const name const =
if (name = (fst (dest_const const))) then true
else failwith (`not `^name) ;;
let int_of_term = int_of_string o fst o dest_const;;
let term_of_int = let ty = ":num" in
\n. mk_const(string_of_int n, ty) ;;
%---------------------------------------------------------------------- %
% APPEND_CONV: this conversion maps terms of the form %
% %
% "APPEND [x1;...;xm] [y1;...;yn]" %
% %
% to the equation: %
% %
% |- APPEND [x1;...;xm] [y1;...;yn] = [x1;...;xm;y1;...;yn] %
% %
% ADDED: TFM 91.10.26 %
%---------------------------------------------------------------------- %
let APPEND_CONV =
let th1,th2 = CONJ_PAIR (definition `list` `APPEND`) in
let th3 = SPECL ["l1:(*)list";"l2:(*)list"] th2 in
let th4 = GENL ["l2:(*)list";"l1:(*)list"] th3 in
let itfn (cns,ath) v th =
let th1 = AP_TERM (mk_comb(cns,v)) th in
let l = rand(rator(rand(rator(concl th)))) in
TRANS (SPEC v (SPEC l ath)) th1 in
\tm. (let _,[l1;l2] = ((check_const`APPEND`) # I) (strip_comb tm) in
let els,ty = dest_list l1 in
if (null els) then ISPEC l2 th1 else
let cns = rator(rator l1) in
let step = ISPEC l2 th4 and base = ISPEC l2 th1 in
itlist (itfn (cns,step)) els base) ?
failwith `APPEND_CONV`;;
% --------------------------------------------------------------------- %
% MAP_CONV conv "MAP f [e1;...;en]". [TFM 92.04.16] %
% %
% Returns |- MAP f [e1;...;en] = [r1;...;rn] %
% where conv "f ei" returns |- f ei = ri for 1 <= i <= n %
% --------------------------------------------------------------------- %
let MAP_CONV =
let mn,mc = CONJ_PAIR(definition `list` `MAP`) in
\conv tm.
(let _,[fn;l] = ((check_const`MAP`) # I) (strip_comb tm) in
let els,ty = dest_list l in
let nth = ISPEC fn mn and cth = ISPEC fn mc in
let cns = rator(rator(rand(snd(strip_forall(concl cth))))) in
let APcons t1 t2 = MK_COMB(AP_TERM cns t2,t1) in
let itfn e th =
let t = rand(rand(rator(concl th))) in
let th1 = SPEC t(SPEC e cth) in
th1 TRANS (APcons th (conv (mk_comb(fn,e)))) in
itlist itfn els nth) ? failwith `MAP_CONV`;;
%-==============================================================-%
%- CONVERSIONS added by WW 31 Jan 94 -%
%-==============================================================-%
%----------------------------------------------------------------%
%- Reductions -%
%- FOLDR_CONV conv "FOLDR f e [a0;...an]" --->
|- FOLDR f e [a0;...an] = tm
FOLDR_CONV evaluates the input expression by iteratively apply
the function f the successive element of the list starting from
the end of the list. tm is the result of the calculation.
FOLDR_CONV returns a theorem stating this fact. During each
iteration, an expression "f e' ai" is evaluated. The user
supplied conversion conv is used to derive a theorem
|- f e' ai = e'' which is then used to reduce the expression
to e''. For example,
#FOLDR_CONV conv "FOLDR ^f 0 ([x0;x1;x2;x3;x4;x5]:* list)";;
|- FOLDR(\x l'. SUC l')0[x0;x1;x2;x3;x4;x5] = 6
where f = (\x l'. SUC l') and
conv = ((RATOR_CONV BETA_CONV) THENC BETA_CONV THENC SUC_CONV))
In general, if the function f is an explicit lambda abstraction
(\x x'. t[x,x']), the conversion should be in the form
((RATOR_CONV BETA_CONV) THENC BETA_CONV THENC conv'))
where conv' applied to t[x,x'] returns the theorem |-t[x,x'] = e''.
-%
let FOLDR_CONV =
let (bthm,ithm) = CONJ_PAIR (definition `list` `FOLDR`) in
\conv tm.
let (_,[f;e;l]) = ((check_const`FOLDR`)#I)(strip_comb tm) in
let ithm' = ISPECL[f;e] ithm in
let (els,lty) = (dest_list l) in
let itfn a th =
let [f';e';l'] = snd(strip_comb(lhs(concl th))) in
let lem = SUBS [th](SPECL[a;l'] ithm') in
TRANS lem (conv (rhs (concl lem)))
in
(itlist itfn els (ISPECL [f;e] bthm)) ?\s failwith (`FOLDR_CONV: `^s);;
%----------------------------------------------------------------%
%- FOLDL_CONV conv "FOLDL f e [a0;...an]" --->
|- FOLDL f e [a0;...an] = tm
FOLDL_CONV evaluates the input expression by iteratively apply
the function f the successive element of the list starting from
the head of the list. tm is the result of the calculation.
FOLDL_CONV returns a theorem stating this fact. During each
iteration, an expression "f e' ai" is evaluated. The user
supplied conversion conv is used to derive a theorem
|- f e' ai = e'' which is then used to reduce the expression
to e''. For example,
#FOLDL_CONV conv "FOLDL ^f 0 ([x0;x1;x2;x3;x4;x5]:* list)";;
|- FOLDL(\l' x. SUC l')0[x0;x1;x2;x3;x4;x5] = 6
where f = (\l' x. SUC l') and
conv = ((RATOR_CONV BETA_CONV) THENC BETA_CONV THENC SUC_CONV))
In general, if the function f is an explicit lambda abstraction
(\x x'. t[x,x']), the conversion should be in the form
((RATOR_CONV BETA_CONV) THENC BETA_CONV THENC conv'))
where conv' applied to t[x,x'] returns the theorem |-t[x,x'] = e''.
-%
let FOLDL_CONV =
let (bthm,ithm) = CONJ_PAIR (definition `list` `FOLDL`) in
\conv tm.
let (_,[f;e;l]) = ((check_const `FOLDL`)#I)(strip_comb tm) in
let ithm' = ISPEC f ithm in
letrec itfn (term) =
let (_,[f;e;l]) = strip_comb term in
if (is_const l)
then let (nil,_) = dest_const l in
if not(nil = `NIL`)
then failwith `expecting null list`
else (ISPECL[f;e]bthm)
else
let [h;t] = snd(strip_comb l) in
let th = ISPECL[e;h;t] ithm' in
let lem = CONV_RULE
((RAND_CONV o RATOR_CONV o RAND_CONV) conv) th in
(TRANS lem (itfn (rhs(concl lem)))) in
(itfn tm) ?\s failwith (`FOLDL_CONV: `^s);;
% --------------------------------------------------------------------- %
% list_FOLD_CONV : thm -> conv -> conv %
% list_FOLD_CONV foldthm conv tm %
% where canme is the name of constant and foldthm is a theorem of the %
% the following form: %
% |- !x0 ... xn. CONST x0 ... xn = FOLD[LR] f e l %
% and conv is a conversion which will be passed to FOLDR_CONV or %
% FOLDL_CONV to reduce the right-hand side of the above theorem %
% --------------------------------------------------------------------- %
let list_FOLD_CONV =
\foldthm conv tm.
(let (cname,args) = (strip_comb tm) in
let fthm = ISPECL args foldthm in
let left,right = (dest_eq(concl fthm)) in
let const = fst(strip_comb left) in
let f = fst(dest_const(fst(strip_comb right))) in
if not(cname = const) then failwith `theorem and term are different`
else if (f = `FOLDL`) then
TRANS fthm (FOLDL_CONV conv right)
else if (f = `FOLDR`) then
TRANS fthm (FOLDR_CONV conv right)
else failwith `not FOLD theorem`)
?\s failwith (`list_FOLD_CONV: `^s);;
let SUM_CONV =
list_FOLD_CONV (theorem `list` `SUM_FOLDR`) ADD_CONV;;
%----------------------------------------------------------------%
%- Filter -%
%- FILTER_CONV conv "FILTER P [a0;...an]" --->
|- FILTER P [a0,...;an] = [...;ai;...]
where conv "P ai" returns a theorem |- P ai = T for all ai
in the resulting list.
-%
let FILTER_CONV =
let (bth,ith) = CONJ_PAIR (definition `list` `FILTER`) in
\conv tm.
(let (_,[P;l]) =
((check_const `FILTER`) # I) (strip_comb tm) in
let bth' = ISPEC P bth and ith' = ISPEC P ith in
let lis = fst(dest_list l) in
let ffn x th =
let (left,right) = dest_eq(concl th) in
let (_,[p;ls]) = strip_comb left in
let fthm = SPECL [x;ls] ith' and cthm = conv "^P ^x" in
(CONV_RULE (RAND_CONV COND_CONV) (SUBS[cthm;th]fthm))
in
(itlist ffn lis bth')) ?\s failwith (`FILTER_CONV: `^s);;
%----------------------------------------------------------------%
%- SNOC_CONV : conv
SNOC_CONV "SNOC x [x0;...xn]" --->
|- SNOC x [x0;...xn] = [x0;...;xn;x] -%
%----------------------------------------------------------------%
let SNOC_CONV =
let bthm,sthm = CONJ_PAIR (definition `list` `SNOC`) in
\tm.
(let _,[d;lst] =
((check_const `SNOC`) # I) (strip_comb tm) in
let ty = type_of lst in
let lst',ety = (dest_list lst) in
let EMP = "[]:^ty" and CONS = "CONS:^ety -> ^ty ->^ty" in
let itfn x (lst,ithm) =
mk_comb(mk_comb(CONS,x),lst), (SUBS[ithm](ISPECL[d;x;lst]sthm)) in
snd(itlist itfn lst' (EMP,(ISPEC d bthm))))
?\s failwith(`SNOC_CONV: `^s);;
%----------------------------------------------------------------%
%- REVERSE_CONV : conv
REVERSE_CONV "REVERSE [x0;...;xn]" --->
|- REVERSE [x0;...;xn] = [xn;...;x0] -%
%----------------------------------------------------------------%
let REVERSE_CONV =
let fthm = theorem `list` `REVERSE_FOLDL` in
let conv = ((RATOR_CONV BETA_CONV) THENC BETA_CONV) in
\tm.
(let _,lst =
((check_const `REVERSE`) # I) (dest_comb tm) in
let fthm' = ISPEC lst fthm in
TRANS fthm' (FOLDL_CONV conv (rhs(concl fthm'))))
?\s failwith (`REVERSE_CONV: `^s);;
%----------------------------------------------------------------%
%- FLAT_CONV : conv
FLAT_CONV "FLAT [[x00;...;x0n];...;[xm0;...xmn]]" --->
|- "FLAT [[x00;...;x0n];...;[xm0;...xmn]]" =
[x00;...;x0n;...;xm0;...xmn] -%
%----------------------------------------------------------------%
let FLAT_CONV =
let lem = PROVE("APPEND = (\x1 x2:* list. APPEND x1 x2)",
CONV_TAC FUN_EQ_CONV THEN GEN_TAC THEN BETA_TAC
THEN CONV_TAC FUN_EQ_CONV THEN GEN_TAC
THEN BETA_TAC THEN REFL_TAC) in
let ffthm = theorem `list` `FLAT_FOLDR` in
let afthm = theorem `list` `APPEND_FOLDR` in
let fthm = REWRITE_RULE[afthm](SUBS[lem] ffthm) in
let conv = (RAND_CONV (FOLDR_CONV ((RATOR_CONV BETA_CONV)
THENC BETA_CONV THENC (FOLDR_CONV ALL_CONV)))) in
\tm.
(let _,lst = ((check_const `FLAT`) # I) (dest_comb tm) in
let fthm' = ISPEC lst fthm in
CONV_RULE conv fthm') ?\s failwith (`FLAT_CONV: `^s);;
%-----------------------------------------------------------------------%
% EL_CONV : conv %
% The argument to this conversion should be in the form of %
% "EL k [x0; x1; ...; xk; ...; xn]" %
% It returns a theorem %
% |- EL k [x0; x1; ...; xk; ...; xn] = xk %
% iff 0 <= k <= n, otherwise failure occurs. %
%-----------------------------------------------------------------------%
let EL_CONV =
let bthm,ithm = CONJ_PAIR (definition `list` `EL`) in
let HD = definition `list` `HD` and TL = definition `list``TL` in
let dec n = let nn = int_of_term n in
mk_const(string_of_int(nn - 1), ":num") in
let tail lst = hd(tl(snd(strip_comb lst))) in
let iter ct N bits =
letref n',m',lst' = ct-1, (dec N), (tail bits) in
letref sthm = PURE_ONCE_REWRITE_RULE[TL](ISPECL [bits; m'] ithm) in
if (n' = 0) then
(TRANS sthm (SUBS[ISPECL(snd(strip_comb lst'))HD](ISPEC lst' bthm)))
loop
(n' := n' -1;
sthm := TRANS (RIGHT_CONV_RULE(RATOR_CONV(RAND_CONV num_CONV)) sthm)
(SUBS[ISPECL(snd(strip_comb lst'))TL](ISPECL[lst';(dec m')] ithm));
lst' := tail lst';
m' := dec m') in
\tm.
(let _,[N;bits] = ((check_const `EL`) # I) (strip_comb tm) in
let n = int_of_term N in
let lst = bits and m = N in
if (n = 0) then
(PURE_ONCE_REWRITE_RULE[HD](ISPEC bits bthm))
else if (n < length(fst(dest_list bits))) then
(SUBS [SYM (num_CONV N)](iter n N bits))
else failwith `index too large` )?\s failwith(`EL_CONV: `^s);;
%-----------------------------------------------------------------------%
% ELL_CONV : conv %
% It takes a term of the form "ELL k [x(n-1); ... x0]" and returns %
% |- ELL k [x(n-1); ...; x0] = x(k) %
%-----------------------------------------------------------------------%
let ELL_CONV =
let bthm = theorem `list` `ELL_0_SNOC` and
ithm = theorem `list` `ELL_SUC_SNOC` in
let iter count (d,lst) elty =
letref n = count and x = d and l = lst in
letref th = (ISPECL[(term_of_int n); x; mk_list(l,elty)]ithm) in
if (n = 0) then
(x := last l; l := butlast l;
(th := TRANS th (CONV_RULE ((RATOR_CONV o RAND_CONV o RAND_CONV)
SNOC_CONV) (ISPECL [mk_list(l,elty);x] bthm))))
loop
(n := n - 1;
x := (last l);
l := butlast l;
th := TRANS (RIGHT_CONV_RULE ((RATOR_CONV o RAND_CONV) num_CONV) th)
(CONV_RULE ((RATOR_CONV o RAND_CONV o RAND_CONV)SNOC_CONV)
(ISPECL[(term_of_int n); x; mk_list(l,elty)]ithm))) in
\tm.
(let _,[N;lst] = ((check_const`ELL`) # I)(strip_comb tm) in
let ty = type_of lst in
let lst',ety = (dest_list lst) in
let n = int_of_term N in
if not(n < (length lst')) then failwith `index too large`
else if (n = 0) then
(CONV_RULE ((RATOR_CONV o RAND_CONV o RAND_CONV)SNOC_CONV)
(ISPECL[mk_list(butlast lst', ety);(last lst')]bthm))
else
SUBS_OCCS[[1],(SYM (num_CONV N))]
(CONV_RULE ((RATOR_CONV o RAND_CONV o RAND_CONV)SNOC_CONV)
(iter (n - 1) ((last lst'), (butlast lst')) ety)))
?\s failwith(`ELL_CONV: `^s);;
% --------------------------------------------------------------------- %
% MAP2_CONV conv "MAP2 f [x1;...;xn] [y1;...;yn]" %
% %
% Returns |- MAP2 f [x1;...;xn] [y1;...;yn] = [r1;...;rn] %
% where conv "f xi yi" returns |- f xi yi = ri for 1 <= i <= n %
% --------------------------------------------------------------------- %
let MAP2_CONV =
let mn,mc = CONJ_PAIR(definition `list` `MAP2`) in
\conv tm.
(let _,[fn;l1;l2] = ((check_const`MAP2`) # I) (strip_comb tm) in
let el1s,ty1 = dest_list l1 and el2s,ty2 = dest_list l2 in
let els = combine (el1s,el2s) in
let nth = ISPEC fn mn and cth = ISPEC fn mc in
let cns = rator(rator(rand(snd(strip_forall(concl cth))))) in
let itfn (e1,e2) th =
let _,[f;t1;t2] = strip_comb(lhs(concl th)) in
let th1 = SPECL [e1; t1; e2; t2] cth in
let r = conv (mk_comb(mk_comb(fn,e1),e2)) in
(SUBS[r;th]th1) in
itlist itfn els nth) ?\s failwith (`MAP2_CONV: `^s);;
% --------------------------------------------------------------------- %
% ALL_EL_CONV : conv -> conv %
% ALL_EL_CONV conv "ALL_EL P [x0;...;xn]" ---> %
% |- ALL_EL P [x0;...;xn] = T iff conv "P xi"---> |- P xi = T for all i %
% |- ALL_EL P [x0;...;xn] = F otherwise %
% --------------------------------------------------------------------- %
let ALL_EL_CONV =
let bth,ith = CONJ_PAIR (definition `list` `ALL_EL`) in
let AND_THM = setify(flat(map (CONJ_LIST 5)
[(SPEC "T" AND_CLAUSES);(SPEC "F" AND_CLAUSES)])) in
\conv tm.
(let (_,[P;l]) = ((check_const`ALL_EL`) # I)(strip_comb tm) in
let bth' = ISPEC P bth and ith' = ISPEC P ith in
let lis = fst(dest_list l) in
let ffn x th =
let (left,right) = dest_eq(concl th) in
let (_,[p;ls]) = strip_comb left in
let fthm = SPECL [x;ls] ith' and cthm = conv "^P ^x" in
SUBS AND_THM (SUBS[cthm;th]fthm)
in
(itlist ffn lis bth')) ?\s failwith (`ALL_EL_CONV: `^s);;
% --------------------------------------------------------------------- %
% SOME_EL_CONV : conv -> conv %
% SOME_EL_CONV conv "SOME_EL P [x0;...;xn]" ---> %
% |- SOME_EL P [x0;...;xn] = F iff conv "P xi"---> |- P xi = F for all i%
% |- SOME_EL P [x0;...;xn] = F otherwise %
% --------------------------------------------------------------------- %
let SOME_EL_CONV =
let bth,ith = CONJ_PAIR (definition `list` `SOME_EL`) in
let OR_THM = setify(flat(map (CONJ_LIST 5)
[(SPEC "T" OR_CLAUSES);(SPEC "F" OR_CLAUSES)])) in
\conv tm.
(let (_,[P;l]) = ((check_const`SOME_EL`) # I)(strip_comb tm) in
let bth' = ISPEC P bth and ith' = ISPEC P ith in
let lis = fst(dest_list l) in
let ffn x th =
let (left,right) = dest_eq(concl th) in
let (_,[p;ls]) = strip_comb left in
let fthm = SPECL [x;ls] ith' and cthm = conv "^P ^x" in
SUBS OR_THM (SUBS[cthm;th]fthm)
in
(itlist ffn lis bth')) ?\s failwith (`SOME_EL_CONV: `^s);;
% --------------------------------------------------------------------- %
% IS_EL_CONV : conv -> conv %
% IS_EL_CONV conv "IS_EL P [x0;...;xn]" ---> %
% |- IS_EL x [x0;...;xn] = T iff conv "x = xi" ---> %
% |- (x = xi) = F for an i %
% |- IS_EL x [x0;...;xn] = F otherwise %
% --------------------------------------------------------------------- %
let IS_EL_CONV =
let bth = (definition `list` `IS_EL_DEF`) in
\conv tm.
(let (_,[x;l]) = ((check_const`IS_EL`) # I)(strip_comb tm) in
let bth' = ISPECL[x;l] bth in
let right = rhs (concl bth') in
TRANS bth' (SOME_EL_CONV conv right))
?\s failwith (`IS_EL_CONV: `^s);;
% --------------------------------------------------------------------- %
% LAST_CONV : conv %
% LAST_CONV "LAST [x0;...;xn]" ---> |- LAST [x0;...;xn] = xn %
% --------------------------------------------------------------------- %
let LAST_CONV =
let bth = theorem `list` `LAST` in
\tm.
(let _,l = ((check_const`LAST`) # I) (dest_comb tm) in
let l',lty = dest_list l in
if ((length l') = 0) then failwith `empty list`
else
(let x = last l' and lis = mk_list((butlast l'),lty) in
let bth' = ISPECL[x;lis] bth in
CONV_RULE ((RATOR_CONV o RAND_CONV o RAND_CONV)SNOC_CONV) bth'))
?\s failwith (`LAST_CONV: `^s);;
% --------------------------------------------------------------------- %
% BUTLAST_CONV : conv %
% BUTLAST_CONV "BUTLAST [x0;...;xn-1;xn]" ---> %
% |- BUTLAST [x0;...;xn-1;xn] = [x0;...;xn-1] %
% --------------------------------------------------------------------- %
let BUTLAST_CONV =
let bth = theorem `list` `BUTLAST` in
\tm.
(let _,l = ((check_const`BUTLAST`) # I) (dest_comb tm) in
let l',lty = dest_list l in
if ((length l') = 0) then failwith `empty list`
else
(let x = last l' and lis = mk_list((butlast l'),lty) in
let bth' = ISPECL[x;lis] bth in
CONV_RULE ((RATOR_CONV o RAND_CONV o RAND_CONV)SNOC_CONV) bth'))
?\s failwith (`BUTLAST_CONV: `^s);;
%----------------------------------------------------------------%
let SUC_CONV =
let numty = mk_type(`num`,[]) in
\tm.
let (SUC,(n,_)) = (I # dest_const)(dest_comb tm) in
let n' = string_of_int(1 + (int_of_string n)) in
SYM (num_CONV (mk_const(n', numty)));;
%---------------------------------------------------------------%
% SEG_CONV : conv %
% SEG_CONV "SEG m k [x0;...;xk;...;xm+k;...xn]" ---> %
% |- SEG m k [x0;...;xk;...;xm+k;...xn] = [xk;...xm+k-1] %
%---------------------------------------------------------------%
let SEG_CONV =
let [bthm;mthm;kthm] = CONJ_LIST 3 (definition `list` `SEG`) in
let SUC = "SUC" in
let mifn mthm' x th =
let [M';_;L] = snd(strip_comb(lhs(concl th))) in
SUBS[(SUC_CONV(mk_comb(SUC,M')));th](SPECL[M';x;L]mthm') in
let kifn kthm' x th =
let [_;K';L] = snd(strip_comb(lhs(concl th))) in
SUBS[(SUC_CONV(mk_comb(SUC,K')));th](SPECL[K';x;L]kthm') in
\tm.
(let _,[M;K;L] = ((check_const`SEG`)# I)(strip_comb tm) in
let lis,lty = dest_list L in
let m = int_of_term M and k = int_of_term K in
if ((m + k) > (length lis)) then failwith `indexes too large`
else if (m = 0) then (ISPECL[K;L]bthm)
else let mthm' = INST_TYPE [(lty,":*")] mthm in
if (k = 0) then
let (ls,lt) = chop_list m lis in
let bthm' = ISPECL["0";(mk_list(lt,lty))] bthm in
(itlist (mifn mthm') ls bthm')
else
let lk,(ls,lt) = (I #(chop_list m))(chop_list k lis) in
let bthm' = ISPECL["0";(mk_list(lt,lty))] bthm in
let kthm' = SUBS[SYM(num_CONV M)]
(INST_TYPE[(lty,":*")](SPEC (term_of_int(m-1)) kthm)) in
let bbthm = itlist (mifn mthm') ls bthm' in
(itlist (kifn kthm') lk bbthm))
?\s failwith (`SEG_CONV: `^s);;
%-----------------------------------------------------------------------%
% LASTN_CONV : conv %
% It takes a term of the form "LASTN k [x0; ...; x(n-k); ...; x(n-1)]" %
% and returns the following theorem: %
% |- LASTN k [x0; ...; x(n-k); ...; x(n-1)] = [x(n-k); ...; x(n-1)] %
%-----------------------------------------------------------------------%
let LASTN_CONV =
let LASTN_LENGTH_APPEND = theorem `list` `LASTN_LENGTH_APPEND` and
bthm = CONJUNCT1 (definition `list` `LASTN`) and
ithm = (theorem `list` `LASTN_LENGTH_ID`) in
let len_conv ty lst =
LENGTH_CONV(mk_comb("LENGTH:(^ty)list -> num",lst)) in
\tm.
(let _,[N;lst] = ((check_const`LASTN`) # I) (strip_comb tm) in
let n = int_of_term N in
if (n = 0) then (ISPEC lst bthm)
else
(let bits,lty = (dest_list lst) in
let len = (length bits) in
if (n > len) then failwith `index too large`
else if (n = len) then
(SUBS[(len_conv lty lst)](ISPEC lst ithm))
else
(let l1,l2 = (chop_list (len - n) bits) in
let l1' = mk_list(l1, lty) and l2' = mk_list(l2, lty) in
let APP = "APPEND:(^lty)list -> (^lty)list -> (^lty)list" in
let thm2 = len_conv lty l2' in
let thm3 = APPEND_CONV (mk_comb(mk_comb(APP, l1'),l2')) in
(SUBS[thm2;thm3](ISPECL [l1';l2'] LASTN_LENGTH_APPEND)) )))
?\s failwith (`LASTN_CONV: `^s);;
%-----------------------------------------------------------------------%
% BUTLASTN_CONV : conv %
% It takes a term of the form "BUTLASTN k [x0;x1;...;x(n-k);...;x(n-1)]"%
% and returns the following theorem: %
% |- BUTLASTN k [x0; x1; ...; x(n-k);...;x(n-1)] = [x0; ...; x(n-k-1)] %
%-----------------------------------------------------------------------%
let BUTLASTN_CONV =
let bthm = CONJUNCT1 (definition `list` `BUTLASTN`) in
let lthm = (theorem `list` `BUTLASTN_LENGTH_NIL`) in
let athm = (theorem `list` `BUTLASTN_LENGTH_APPEND`) in
let len_conv ty lst =
LENGTH_CONV(mk_comb("LENGTH:(^ty)list -> num",lst)) in
\tm.
(let _,[N;lst] = ((check_const`BUTLASTN`) # I) (strip_comb tm) in
let n = int_of_term N in
if (n = 0) then (ISPEC lst bthm)
else
(let bits,lty = (dest_list lst) in
let len = (length bits) in
if (n > len) then failwith `index too large`
else if (n = len) then
let thm1 = len_conv lty lst in (SUBS[thm1](ISPEC lst lthm))
else
(let l1,l2 = (chop_list (len - n) bits) in
let l1' = mk_list(l1, lty) and l2' = mk_list(l2, lty) in
let APP = "APPEND:(^lty)list -> (^lty)list -> (^lty)list" in
let thm2 = len_conv lty l2' in
let thm3 = APPEND_CONV (mk_comb(mk_comb(APP, l1'),l2')) in
(SUBS[thm2;thm3](ISPECL [l2';l1'] athm)) )))
?\s failwith (`BUTLASTN_CONV: `^s);;
%-----------------------------------------------------------------------%
% BUTFIRSTN_CONV : conv %
% BUTFIRSTN_CONV "BUTFIRSTN k [x0;...;xk;...;xn]" ---> %
% |- BUTFIRSTN k [x0;...;xk;...;xn] = [xk;...;xn] %
%-----------------------------------------------------------------------%
let BUTFIRSTN_CONV =
let bthm,ithm = CONJ_PAIR (definition `list` `BUTFIRSTN`) in
let SUC = "SUC" in
let itfn ithm' x th =
let _,[N';L'] = strip_comb(lhs(concl th)) in
SUBS[(SUC_CONV(mk_comb(SUC,N')));th](SPECL[N';x;L']ithm') in
\tm.
(let _,[K;L] = ((check_const`BUTFIRSTN`)# I)(strip_comb tm) in
let k = int_of_term K and lis,lty = dest_list L in
if (k > (length lis)) then failwith `index too large`
else if (k = 0) then (ISPEC L bthm)
else
let ll,lr = chop_list k lis in
let bthm' = ISPEC (mk_list(lr,lty)) bthm in
let ithm' = INST_TYPE[(lty,":*")]ithm in
itlist (itfn ithm') ll bthm')
?\s failwith (`BUTFIRSTN_CONV: `^s);;
%-----------------------------------------------------------------------%
% FIRSTN_CONV : conv %
% FIRSTN_CONV "FIRSTN k [x0;...;xk;...;xn]" ---> %
% |- FIRSTN k [x0;...;xk;...;xn] = [x0;...;xk] %
%-----------------------------------------------------------------------%
let FIRSTN_CONV =
let bthm,ithm = CONJ_PAIR (definition `list` `FIRSTN`) in
let SUC = "SUC" in
let itfn ithm' x th =
let _,[N';L'] = strip_comb(lhs(concl th)) in
SUBS[(SUC_CONV(mk_comb(SUC,N')));th](SPECL[N';x;L']ithm') in
\tm.
(let _,[K;L] = ((check_const`FIRSTN`)# I)(strip_comb tm) in
let k = int_of_term K and lis,lty = dest_list L in
if (k > (length lis)) then failwith `index too large`
else if (k = 0) then (ISPEC L bthm)
else
let ll,lr = chop_list k lis in
let bthm' = ISPEC (mk_list(lr,lty)) bthm in
let ithm' = INST_TYPE[(lty,":*")]ithm in
itlist (itfn ithm') ll bthm')
?\s failwith (`FIRSTN_CONV: `^s);;
%-----------------------------------------------------------------------%
% SCANL_CONV : conv -> conv %
% SCANL_CONV conv "SCANL f e [x0;...;xn]" ---> %
% |- SCANL f e [x0;...;xn] = [e; t0; ...; tn] %
% where conv "f ei xi" ---> |- f ei xi = ti %
%-----------------------------------------------------------------------%
let SCANL_CONV =
let bthm,ithm = CONJ_PAIR (definition `list` `SCANL`) in
\conv tm.
(let _,[f;e;l] = ((check_const`SCANL`)#I)(strip_comb tm) in
let bthm' = ISPEC f bthm and ithm' = ISPEC f ithm in
letrec scan_conv tm' =
let [_;E;L] = snd(strip_comb tm') in
if (is_const L) then (SPEC E bthm')
else
let [x;l] = snd(strip_comb L) in
let th1 = conv (mk_comb(mk_comb(f,E),x)) in
let th2 = SUBS[th1](SPECL[E;x;l] ithm') in
let th3 = scan_conv (last(snd(strip_comb(rhs(concl th2))))) in
SUBS[th3]th2
in
(scan_conv tm)) ?\s failwith (`SCANL_CONV: `^s);;
%-----------------------------------------------------------------------%
% SCANR_CONV : conv -> conv %
% SCANR_CONV conv "SCANR f e [x0;...;xn]" ---> %
% |- SCANR f e [x0;...;xn] = [t0; ...; tn; e] %
% where conv "f xi ei" ---> |- f xi ei = ti %
%-----------------------------------------------------------------------%
let SCANR_CONV =
let bthm,ithm = CONJ_PAIR (definition `list` `SCANR`) in
let HD = definition `list` `HD` in
\conv tm.
(let _,[f;e;l] = ((check_const`SCANR`)#I)(strip_comb tm) in
let bthm' = ISPEC f bthm and ithm' = ISPEC f ithm in
letrec scan_conv tm' =
let [_;E;L] = snd(strip_comb tm') in
if (is_const L) then (SPEC E bthm')
else
let [x;l] = snd(strip_comb L) in
let th2 = (SPECL[E;x;l] ithm') in
let th3 = scan_conv (last(snd(strip_comb(rhs(concl th2))))) in
let th4 = PURE_ONCE_REWRITE_RULE[HD](SUBS[th3]th2) in
let th5 = conv (hd(snd(strip_comb(rhs(concl th4))))) in
SUBS[th5]th4
in
(scan_conv tm)) ?\s failwith (`SCANR_CONV: `^s);;
%-----------------------------------------------------------------------%
% REPLICATE_CONV : conv %
% REPLICATE conv "REPLICATE f n" ---> %
% |- REPLICATE n x = [x; ...; x] %
%-----------------------------------------------------------------------%
let REPLICATE_CONV =
let (bthm,ithm) = CONJ_PAIR (definition `list` `REPLICATE`) in
let dec n = term_of_int((int_of_term n) - 1) in
letrec repconv (bthm, ithm) tm =
let [n;x] = snd(strip_comb tm) in
if ((int_of_term n) = 0) then bthm
else (let th1 = SUBS[SYM (num_CONV n)](SPEC (dec n) ithm) in
CONV_RULE ((RAND_CONV o RAND_CONV) (repconv(bthm,ithm))) th1) in
\tm.
(let _,[n;x] = ((check_const`REPLICATE`)#I)(strip_comb tm) in
let xty = type_of x in
let bthm' = ISPEC x bthm and
ithm' = GEN_ALL(ISPECL[mk_var(`n`,xty);x] ithm) in
(repconv (bthm',ithm') tm)) ?\s failwith (`REPLICATE_CONV: `^s);;
%-----------------------------------------------------------------------%
% GENLIST_CONV : conv -> conv %
% GENLIST conv "GENLIST f n" ---> |- GENLIST f n = [f 0;f 1; ...;f(n-1)]%
%-----------------------------------------------------------------------%
let GENLIST_CONV =
let (bthm,ithm) = CONJ_PAIR (definition `list` `GENLIST`) in
let dec n = term_of_int((int_of_term n) - 1) in
letrec genconv (bthm,ithm) conv tm =
let n = last(snd(strip_comb tm)) in
if ((int_of_term n) = 0) then CONV_RULE(ONCE_DEPTH_CONV conv) bthm
else (let th1 = SUBS[SYM (num_CONV n)](SPEC (dec n) ithm) in
let th2 = RIGHT_CONV_RULE ((RATOR_CONV o RAND_CONV) conv) th1 in
RIGHT_CONV_RULE (RAND_CONV (genconv (bthm,ithm) conv)) th2) in
\conv tm.
(let _,[f;n] = ((check_const`GENLIST`)# I)(strip_comb tm) in
let bthm' = ISPEC f bthm and ithm' = ISPEC f ithm in
RIGHT_CONV_RULE (TOP_DEPTH_CONV SNOC_CONV)(genconv (bthm',ithm') conv tm))
?\s failwith (`GENLIST_CONV: `^s);;
(APPEND_CONV, MAP_CONV, FOLDR_CONV, FOLDL_CONV, list_FOLD_CONV,
SUM_CONV, FILTER_CONV, SNOC_CONV, REVERSE_CONV, FLAT_CONV,
EL_CONV, ELL_CONV, MAP2_CONV, ALL_EL_CONV, SOME_EL_CONV,
IS_EL_CONV, LAST_CONV, BUTLAST_CONV, SEG_CONV, LASTN_CONV,
BUTLASTN_CONV, BUTFIRSTN_CONV, FIRSTN_CONV, SCANL_CONV, SCANR_CONV,
REPLICATE_CONV,GENLIST_CONV);;
end_section `list_convs`;;
let (APPEND_CONV, MAP_CONV, FOLDR_CONV, FOLDL_CONV, list_FOLD_CONV,
SUM_CONV, FILTER_CONV, SNOC_CONV, REVERSE_CONV, FLAT_CONV,
EL_CONV, ELL_CONV, MAP2_CONV, ALL_EL_CONV, SOME_EL_CONV,
IS_EL_CONV, LAST_CONV, BUTLAST_CONV, SEG_CONV, LASTN_CONV,
BUTLASTN_CONV, BUTFIRSTN_CONV, FIRSTN_CONV, SCANL_CONV, SCANR_CONV,
REPLICATE_CONV,GENLIST_CONV)
= it;;
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