Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
(* ========================================================================= *) | |
(* Define integer sums, with most theorems derived automatically. *) | |
(* ========================================================================= *) | |
let ISUM_SUPPORT = prove | |
(`!f s. isum (support (+) f s) f = isum s f`, | |
REWRITE_TAC[isum; ITERATE_SUPPORT]);; | |
let int_isum = prove | |
(`!f:A->int s. real_of_int(isum s f) = sum s (\x. real_of_int(f x))`, | |
REPEAT GEN_TAC THEN REWRITE_TAC[sum; isum] THEN | |
ONCE_REWRITE_TAC[ITERATE_EXPAND_CASES] THEN | |
SUBGOAL_THEN | |
`support(+) (\x:A. real_of_int(f x)) s = support(+) f s` | |
SUBST1_TAC THENL | |
[REWRITE_TAC[support; NEUTRAL_REAL_ADD; NEUTRAL_INT_ADD] THEN | |
REWRITE_TAC[GSYM int_of_num_th; GSYM int_eq]; | |
ALL_TAC] THEN | |
COND_CASES_TAC THEN | |
ASM_REWRITE_TAC[NEUTRAL_REAL_ADD; NEUTRAL_INT_ADD; int_of_num_th] THEN | |
POP_ASSUM MP_TAC THEN SPEC_TAC(`support(+) (f:A->int) s`,`s:A->bool`) THEN | |
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN | |
SIMP_TAC[ITERATE_CLAUSES; MONOIDAL_INT_ADD; MONOIDAL_REAL_ADD] THEN | |
SIMP_TAC[NEUTRAL_INT_ADD; NEUTRAL_REAL_ADD; int_of_num_th; int_add_th]);; | |
(* ------------------------------------------------------------------------- *) | |
(* Generalize INT_OF_REAL_THM for most common sum patterns. *) | |
(* ------------------------------------------------------------------------- *) | |
let INT_OF_REAL_THM = | |
let dest = `real_of_int` | |
and real_ty = `:real` | |
and int_ty = `:int` | |
and cond_th = prove | |
(`real_of_int(if b then x else y) = | |
if b then real_of_int x else real_of_int y`, | |
COND_CASES_TAC THEN REWRITE_TAC[]) | |
and compose_th = prove | |
(`(\x. real_of_int((f o g) x)) = (\x. real_of_int(f x)) o g`, | |
REWRITE_TAC[o_DEF]) in | |
let thlist = map GSYM | |
[int_eq; int_le; int_lt; int_ge; int_gt; | |
int_of_num_th; int_neg_th; int_add_th; int_mul_th; | |
int_sub_th; int_abs_th; int_max_th; int_min_th; int_pow_th; | |
int_isum; GSYM BETA_THM; GSYM ETA_AX; compose_th; cond_th] in | |
let REW_RULE = GEN_REWRITE_RULE REDEPTH_CONV thlist in | |
let is_fun_into_real ty = | |
try match dest_type ty with | |
"fun",[s;t] when t = real_ty -> mk_fun_ty s int_ty | |
| "real",[] -> int_ty | |
| _ -> failwith "" | |
with Failure _ -> ty in | |
let int_of_real_ty ty = | |
try match dest_type ty with | |
"real",[] -> int_ty | |
| "fun",[s;t] when t = real_ty -> mk_fun_ty s int_ty | |
| _ -> ty | |
with Failure _ -> ty in | |
let int_tm_of_real_var v = | |
let s,ty = dest_var v in | |
let tys,rty = splitlist dest_fun_ty ty in | |
if rty <> real_ty then v else | |
let ity = itlist mk_fun_ty tys int_ty in | |
let vs = map genvar tys in | |
list_mk_abs(vs,mk_comb(dest,list_mk_comb(mk_var(s,ity),vs))) in | |
let int_of_real_var v = | |
let s,ty = dest_var v in | |
let tys,rty = splitlist dest_fun_ty ty in | |
if rty <> real_ty then v else | |
let ity = itlist mk_fun_ty tys int_ty in | |
mk_var(s,ity) in | |
let INT_OF_REAL_THM1 th = | |
let newavs = subtract (frees (concl th)) (freesl (hyp th)) in | |
let avs,bod = strip_forall(concl th) in | |
let allavs = newavs@avs in | |
let avs' = map int_tm_of_real_var allavs in | |
let avs'' = map int_of_real_var avs in | |
GENL avs'' (REW_RULE(SPECL avs' (GENL newavs th))) in | |
let rec INT_OF_REAL_THM th = | |
if is_conj(concl th) then CONJ (INT_OF_REAL_THM1 (CONJUNCT1 th)) | |
(INT_OF_REAL_THM1 (CONJUNCT2 th)) | |
else INT_OF_REAL_THM1 th in | |
INT_OF_REAL_THM;; | |
(* ------------------------------------------------------------------------- *) | |
(* Apply it in all the cases where it works. *) | |
(* ------------------------------------------------------------------------- *) | |
let CARD_EQ_ISUM = INT_OF_REAL_THM CARD_EQ_SUM;; | |
let INT_SUB_POW = INT_OF_REAL_THM REAL_SUB_POW;; | |
let ISUM_0 = INT_OF_REAL_THM SUM_0;; | |
let ISUM_ABS = INT_OF_REAL_THM SUM_ABS;; | |
let ISUM_ABS_BOUND = INT_OF_REAL_THM SUM_ABS_BOUND;; | |
let ISUM_ABS_LE = INT_OF_REAL_THM SUM_ABS_LE;; | |
let ISUM_ABS_NUMSEG = INT_OF_REAL_THM SUM_ABS_NUMSEG;; | |
let ISUM_ADD = INT_OF_REAL_THM SUM_ADD;; | |
let ISUM_ADD_NUMSEG = INT_OF_REAL_THM SUM_ADD_NUMSEG;; | |
let ISUM_ADD_SPLIT = INT_OF_REAL_THM SUM_ADD_SPLIT;; | |
let ISUM_BIJECTION = INT_OF_REAL_THM SUM_BIJECTION;; | |
let ISUM_BOUND = INT_OF_REAL_THM SUM_BOUND;; | |
let ISUM_BOUND_LT = INT_OF_REAL_THM SUM_BOUND_LT;; | |
let ISUM_BOUND_LT_ALL = INT_OF_REAL_THM SUM_BOUND_LT_ALL;; | |
let ISUM_CASES = INT_OF_REAL_THM SUM_CASES;; | |
let ISUM_CLAUSES_LEFT = INT_OF_REAL_THM SUM_CLAUSES_LEFT;; | |
let ISUM_CLAUSES_NUMSEG = INT_OF_REAL_THM SUM_CLAUSES_NUMSEG;; | |
let ISUM_CLAUSES_RIGHT = INT_OF_REAL_THM SUM_CLAUSES_RIGHT;; | |
let ISUM_COMBINE_L = INT_OF_REAL_THM SUM_COMBINE_L;; | |
let ISUM_COMBINE_R = INT_OF_REAL_THM SUM_COMBINE_R;; | |
let ISUM_CONST = INT_OF_REAL_THM SUM_CONST;; | |
let ISUM_CONST_NUMSEG = INT_OF_REAL_THM SUM_CONST_NUMSEG;; | |
let ISUM_DELETE = INT_OF_REAL_THM SUM_DELETE;; | |
let ISUM_DELETE_CASES = INT_OF_REAL_THM SUM_DELETE_CASES;; | |
let ISUM_DELTA = INT_OF_REAL_THM SUM_DELTA;; | |
let ISUM_DIFF = INT_OF_REAL_THM SUM_DIFF;; | |
let ISUM_DIFFS = INT_OF_REAL_THM SUM_DIFFS;; | |
let ISUM_EQ = INT_OF_REAL_THM SUM_EQ;; | |
let ISUM_EQ_0 = INT_OF_REAL_THM SUM_EQ_0;; | |
let ISUM_EQ_0_NUMSEG = INT_OF_REAL_THM SUM_EQ_0_NUMSEG;; | |
let ISUM_EQ_GENERAL = INT_OF_REAL_THM SUM_EQ_GENERAL;; | |
let ISUM_EQ_GENERAL_INVERSES = INT_OF_REAL_THM SUM_EQ_GENERAL_INVERSES;; | |
let ISUM_EQ_NUMSEG = INT_OF_REAL_THM SUM_EQ_NUMSEG;; | |
let ISUM_EQ_SUPERSET = INT_OF_REAL_THM SUM_EQ_SUPERSET;; | |
let ISUM_GROUP = INT_OF_REAL_THM SUM_GROUP;; | |
let ISUM_IMAGE = INT_OF_REAL_THM SUM_IMAGE;; | |
let ISUM_IMAGE_GEN = INT_OF_REAL_THM SUM_IMAGE_GEN;; | |
let ISUM_IMAGE_LE = INT_OF_REAL_THM SUM_IMAGE_LE;; | |
let ISUM_IMAGE_NONZERO = INT_OF_REAL_THM SUM_IMAGE_NONZERO;; | |
let ISUM_INCL_EXCL = INT_OF_REAL_THM SUM_INCL_EXCL;; | |
let ISUM_INJECTION = INT_OF_REAL_THM SUM_INJECTION;; | |
let ISUM_LE = INT_OF_REAL_THM SUM_LE;; | |
let ISUM_LE_INCLUDED = INT_OF_REAL_THM SUM_LE_INCLUDED;; | |
let ISUM_LE_NUMSEG = INT_OF_REAL_THM SUM_LE_NUMSEG;; | |
let ISUM_LMUL = INT_OF_REAL_THM SUM_LMUL;; | |
let ISUM_LT = INT_OF_REAL_THM SUM_LT;; | |
let ISUM_LT_ALL = INT_OF_REAL_THM SUM_LT_ALL;; | |
let ISUM_MULTICOUNT = INT_OF_REAL_THM SUM_MULTICOUNT;; | |
let ISUM_MULTICOUNT_GEN = INT_OF_REAL_THM SUM_MULTICOUNT_GEN;; | |
let ISUM_NEG = INT_OF_REAL_THM SUM_NEG;; | |
let ISUM_OFFSET = INT_OF_REAL_THM SUM_OFFSET;; | |
let ISUM_OFFSET_0 = INT_OF_REAL_THM SUM_OFFSET_0;; | |
let ISUM_PARTIAL_PRE = INT_OF_REAL_THM SUM_PARTIAL_PRE;; | |
let ISUM_PARTIAL_SUC = INT_OF_REAL_THM SUM_PARTIAL_SUC;; | |
let ISUM_POS_BOUND = INT_OF_REAL_THM SUM_POS_BOUND;; | |
let ISUM_POS_EQ_0 = INT_OF_REAL_THM SUM_POS_EQ_0;; | |
let ISUM_POS_EQ_0_NUMSEG = INT_OF_REAL_THM SUM_POS_EQ_0_NUMSEG;; | |
let ISUM_POS_LE = INT_OF_REAL_THM SUM_POS_LE;; | |
let ISUM_POS_LE_NUMSEG = INT_OF_REAL_THM SUM_POS_LE_NUMSEG;; | |
let ISUM_RESTRICT = INT_OF_REAL_THM SUM_RESTRICT;; | |
let ISUM_RESTRICT_SET = INT_OF_REAL_THM SUM_RESTRICT_SET;; | |
let ISUM_RMUL = INT_OF_REAL_THM SUM_RMUL;; | |
let ISUM_SING = INT_OF_REAL_THM SUM_SING;; | |
let ISUM_SING_NUMSEG = INT_OF_REAL_THM SUM_SING_NUMSEG;; | |
let ISUM_SUB = INT_OF_REAL_THM SUM_SUB;; | |
let ISUM_SUBSET = INT_OF_REAL_THM SUM_SUBSET;; | |
let ISUM_SUBSET_SIMPLE = INT_OF_REAL_THM SUM_SUBSET_SIMPLE;; | |
let ISUM_SUB_NUMSEG = INT_OF_REAL_THM SUM_SUB_NUMSEG;; | |
let ISUM_ISUM_RESTRICT = INT_OF_REAL_THM SUM_SUM_RESTRICT;; | |
let ISUM_SUPERSET = INT_OF_REAL_THM SUM_SUPERSET;; | |
let ISUM_SWAP = INT_OF_REAL_THM SUM_SWAP;; | |
let ISUM_SWAP_NUMSEG = INT_OF_REAL_THM SUM_SWAP_NUMSEG;; | |
let ISUM_TRIV_NUMSEG = INT_OF_REAL_THM SUM_TRIV_NUMSEG;; | |
let ISUM_UNION = INT_OF_REAL_THM SUM_UNION;; | |
let ISUM_UNIONS_NONZERO = INT_OF_REAL_THM SUM_UNIONS_NONZERO;; | |
let ISUM_UNION_EQ = INT_OF_REAL_THM SUM_UNION_EQ;; | |
let ISUM_UNION_LZERO = INT_OF_REAL_THM SUM_UNION_LZERO;; | |
let ISUM_UNION_NONZERO = INT_OF_REAL_THM SUM_UNION_NONZERO;; | |
let ISUM_UNION_RZERO = INT_OF_REAL_THM SUM_UNION_RZERO;; | |
let ISUM_ZERO_EXISTS = INT_OF_REAL_THM SUM_ZERO_EXISTS;; | |
let REAL_OF_NUM_ISUM = INT_OF_REAL_THM REAL_OF_NUM_SUM;; | |
let REAL_OF_NUM_ISUM_NUMSEG = INT_OF_REAL_THM REAL_OF_NUM_SUM_NUMSEG;; | |
(* ------------------------------------------------------------------------- *) | |
(* Manually derive the few cases where it doesn't. *) | |
(* *) | |
(* Note that SUM_BOUND_GEN and SUM_BOUND_LT_GEN don't seem to have immediate *) | |
(* analogs over the integers since they involve division. *) | |
(* *) | |
(* Should really roll ADMISSIBLE_ISUM into "define" as well. *) | |
(* ------------------------------------------------------------------------- *) | |
let ISUM_ISUM_PRODUCT = prove | |
(`!s:A->bool t:A->B->bool x. | |
FINITE s /\ (!i. i IN s ==> FINITE(t i)) | |
==> isum s (\i. isum (t i) (x i)) = | |
isum {i,j | i IN s /\ j IN t i} (\(i,j). x i j)`, | |
REWRITE_TAC[isum] THEN MATCH_MP_TAC ITERATE_ITERATE_PRODUCT THEN | |
REWRITE_TAC[MONOIDAL_INT_ADD]);; | |
let ADMISSIBLE_ISUM = prove | |
(`!(<<) p:(B->C)->P->bool s:P->A h a b. | |
admissible(<<) (\f (k,x). a(x) <= k /\ k <= b(x) /\ p f x) | |
(\(k,x). s x) (\f (k,x). h f x k) | |
==> admissible(<<) p s (\f x. isum(a(x)..b(x)) (h f x))`, | |
REWRITE_TAC[admissible; FORALL_PAIR_THM] THEN REPEAT STRIP_TAC THEN | |
MATCH_MP_TAC ISUM_EQ_NUMSEG THEN ASM_MESON_TAC[]);; | |
let INT_SUB_POW_L1 = prove | |
(`!x n. 1 <= n ==> &1 - x pow n = (&1 - x) * isum (0..n - 1) (\i. x pow i)`, | |
SIMP_TAC[INT_OF_REAL_THM REAL_SUB_POW_L1; ETA_AX]);; | |
let INT_SUB_POW_R1 = prove | |
(`!x n. 1 <= n ==> x pow n - &1 = (x - &1) * isum (0..n - 1) (\i. x pow i)`, | |
SIMP_TAC[INT_OF_REAL_THM REAL_SUB_POW_R1; ETA_AX]);; | |
let ISUM_UNIV = prove | |
(`!f:A->int s. support (+) f (:A) SUBSET s ==> isum s f = isum (:A) f`, | |
REWRITE_TAC[isum] THEN MATCH_MP_TAC ITERATE_UNIV THEN | |
REWRITE_TAC[MONOIDAL_INT_ADD]);; | |
let ISUM_CLOSED = prove | |
(`!P f:A->int s. | |
P(&0) /\ (!x y. P x /\ P y ==> P(x + y)) /\ (!a. a IN s ==> P(f a)) | |
==> P(isum s f)`, | |
REPEAT STRIP_TAC THEN MP_TAC(MATCH_MP ITERATE_CLOSED MONOIDAL_INT_ADD) THEN | |
DISCH_THEN(MP_TAC o SPEC `P:int->bool`) THEN | |
ASM_SIMP_TAC[NEUTRAL_INT_ADD; GSYM isum]);; | |
let ISUM_RELATED = prove | |
(`!R (f:A->int) g s. | |
R (&0) (&0) /\ | |
(!m n m' n'. R m n /\ R m' n' ==> R (m + m') (n + n')) /\ | |
FINITE s /\ (!x. x IN s ==> R (f x) (g x)) | |
==> R (isum s f) (isum s g)`, | |
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN | |
GEN_TAC THEN REPEAT DISCH_TAC THEN | |
MP_TAC(ISPEC `R:int->int->bool` | |
(MATCH_MP ITERATE_RELATED MONOIDAL_INT_ADD)) THEN | |
ASM_REWRITE_TAC[GSYM isum; NEUTRAL_INT_ADD] THEN ASM_MESON_TAC[]);; | |
let ISUM_CLOSED_NONEMPTY = prove | |
(`!P f:A->int s. | |
FINITE s /\ ~(s = {}) /\ | |
(!x y. P x /\ P y ==> P(x + y)) /\ (!a. a IN s ==> P(f a)) | |
==> P(isum s f)`, | |
REPEAT STRIP_TAC THEN | |
MP_TAC(MATCH_MP ITERATE_CLOSED_NONEMPTY MONOIDAL_INT_ADD) THEN | |
DISCH_THEN(MP_TAC o SPEC `P:int->bool`) THEN | |
ASM_SIMP_TAC[NEUTRAL_INT_ADD; GSYM isum]);; | |
let ISUM_RELATED_NONEMPTY = prove | |
(`!R (f:A->int) g s. | |
(!m n m' n'. R m n /\ R m' n' ==> R (m + m') (n + n')) /\ | |
FINITE s /\ ~(s = {}) /\ (!x. x IN s ==> R (f x) (g x)) | |
==> R (isum s f) (isum s g)`, | |
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN | |
GEN_TAC THEN REPEAT DISCH_TAC THEN | |
MP_TAC(ISPEC `R:int->int->bool` | |
(MATCH_MP ITERATE_RELATED_NONEMPTY MONOIDAL_INT_ADD)) THEN | |
ASM_REWRITE_TAC[GSYM isum; NEUTRAL_INT_ADD] THEN ASM_MESON_TAC[]);; | |
(* ------------------------------------------------------------------------- *) | |
(* Expand "isum (m..n) f" where m and n are numerals. *) | |
(* ------------------------------------------------------------------------- *) | |
let EXPAND_ISUM_CONV = | |
let [pth_0; pth_1; pth_2] = (CONJUNCTS o prove) | |
(`(n < m ==> isum(m..n) f = &0) /\ | |
isum(m..m) f = f m /\ | |
(m <= n ==> isum (m..n) f = f m + isum (m + 1..n) f)`, | |
REWRITE_TAC[ISUM_CLAUSES_LEFT; ISUM_SING_NUMSEG; ISUM_TRIV_NUMSEG]) | |
and ns_tm = `..` and f_tm = `f:num->int` | |
and m_tm = `m:num` and n_tm = `n:num` in | |
let rec conv tm = | |
let smn,ftm = dest_comb tm in | |
let s,mn = dest_comb smn in | |
if not(is_const s && fst(dest_const s) = "isum") | |
then failwith "EXPAND_ISUM_CONV" else | |
let mtm,ntm = dest_binop ns_tm mn in | |
let m = dest_numeral mtm and n = dest_numeral ntm in | |
if n < m then | |
let th1 = INST [ftm,f_tm; mtm,m_tm; ntm,n_tm] pth_0 in | |
MP th1 (EQT_ELIM(NUM_LT_CONV(lhand(concl th1)))) | |
else if n = m then CONV_RULE (RAND_CONV(TRY_CONV BETA_CONV)) | |
(INST [ftm,f_tm; mtm,m_tm] pth_1) | |
else | |
let th1 = INST [ftm,f_tm; mtm,m_tm; ntm,n_tm] pth_2 in | |
let th2 = MP th1 (EQT_ELIM(NUM_LE_CONV(lhand(concl th1)))) in | |
CONV_RULE (RAND_CONV(COMB2_CONV (RAND_CONV(TRY_CONV BETA_CONV)) | |
(LAND_CONV(LAND_CONV NUM_ADD_CONV) THENC conv))) th2 in | |
conv;; | |
(* ------------------------------------------------------------------------- *) | |
(* Extend the congruences. *) | |
(* ------------------------------------------------------------------------- *) | |
let th = prove | |
(`(!f g s. (!x. x IN s ==> f(x) = g(x)) | |
==> isum s (\i. f(i)) = isum s g) /\ | |
(!f g a b. (!i. a <= i /\ i <= b ==> f(i) = g(i)) | |
==> isum(a..b) (\i. f(i)) = isum(a..b) g) /\ | |
(!f g p. (!x. p x ==> f x = g x) | |
==> isum {y | p y} (\i. f(i)) = isum {y | p y} g)`, | |
REPEAT STRIP_TAC THEN MATCH_MP_TAC ISUM_EQ THEN | |
ASM_SIMP_TAC[IN_ELIM_THM; IN_NUMSEG]) in | |
extend_basic_congs (map SPEC_ALL (CONJUNCTS th));; | |