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(* ========================================================================= *)
(* Generic iterated operations and special cases of sums over N and R. *)
(* *)
(* (c) Copyright, John Harrison 1998-2007 *)
(* (c) Copyright, Lars Schewe 2007 *)
(* ========================================================================= *)
set_jrh_lexer;;
open System;;
open Lib;;
open Fusion;;
open Basics;;
open Printer;;
open Parser;;
open Equal;;
open Bool;;
open Drule;;
open Tactics;;
open Simp;;
open Theorems;;
open Class;;
open Trivia;;
open Meson;;
open Pair;;
open Nums;;
open Arith;;
open Calc_num;;
open Realax;;
open Calc_int;;
open Realarith;;
open Reals;;
open Calc_rat;;
open Ints;;
open Sets;;
prioritize_num();;
(* ------------------------------------------------------------------------- *)
(* A natural notation for segments of the naturals. *)
(* ------------------------------------------------------------------------- *)
parse_as_infix("..",(15,"right"));;
let numseg = new_definition
`m..n = {x:num | m <= x /\ x <= n}`;;
let FINITE_NUMSEG = prove
(`!m n. FINITE(m..n)`,
REPEAT GEN_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `{x:num | x <= n}` THEN REWRITE_TAC[FINITE_NUMSEG_LE] THEN
SIMP_TAC[SUBSET; IN_ELIM_THM; numseg]);;
let NUMSEG_COMBINE_R = prove
(`!m p n. m <= p + 1 /\ p <= n ==> ((m..p) UNION ((p+1)..n) = m..n)`,
REWRITE_TAC[EXTENSION; IN_UNION; numseg; IN_ELIM_THM] THEN ARITH_TAC);;
let NUMSEG_COMBINE_L = prove
(`!m p n. m <= p /\ p <= n + 1 ==> ((m..(p-1)) UNION (p..n) = m..n)`,
REWRITE_TAC[EXTENSION; IN_UNION; numseg; IN_ELIM_THM] THEN ARITH_TAC);;
let NUMSEG_LREC = prove
(`!m n. m <= n ==> (m INSERT ((m+1)..n) = m..n)`,
REWRITE_TAC[EXTENSION; IN_INSERT; numseg; IN_ELIM_THM] THEN ARITH_TAC);;
let NUMSEG_RREC = prove
(`!m n. m <= n ==> (n INSERT (m..(n-1)) = m..n)`,
REWRITE_TAC[EXTENSION; IN_INSERT; numseg; IN_ELIM_THM] THEN ARITH_TAC);;
let NUMSEG_REC = prove
(`!m n. m <= SUC n ==> (m..SUC n = (SUC n) INSERT (m..n))`,
SIMP_TAC[GSYM NUMSEG_RREC; SUC_SUB1]);;
let IN_NUMSEG = prove
(`!m n p. p IN (m..n) <=> m <= p /\ p <= n`,
REWRITE_TAC[numseg; IN_ELIM_THM]);;
let IN_NUMSEG_0 = prove
(`!m n. m IN (0..n) <=> m <= n`,
REWRITE_TAC[IN_NUMSEG; LE_0]);;
let NUMSEG_SING = prove
(`!n. n..n = {n}`,
REWRITE_TAC[EXTENSION; IN_SING; IN_NUMSEG] THEN ARITH_TAC);;
let NUMSEG_EMPTY = prove
(`!m n. (m..n = {}) <=> n < m`,
REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; IN_NUMSEG] THEN
MESON_TAC[NOT_LE; LE_TRANS; LE_REFL]);;
let FINITE_SUBSET_NUMSEG = prove
(`!s:num->bool. FINITE s <=> ?n. s SUBSET 0..n`,
GEN_TAC THEN EQ_TAC THENL
[REWRITE_TAC[SUBSET; IN_NUMSEG; LE_0] THEN
SPEC_TAC(`s:num->bool`,`s:num->bool`) THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
MESON_TAC[LE_CASES; LE_REFL; LE_TRANS];
MESON_TAC[FINITE_SUBSET; FINITE_NUMSEG]]);;
let CARD_NUMSEG_LEMMA = prove
(`!m d. CARD(m..(m+d)) = d + 1`,
GEN_TAC THEN INDUCT_TAC THEN
ASM_SIMP_TAC[ADD_CLAUSES; NUMSEG_REC; NUMSEG_SING; FINITE_RULES;
ARITH_RULE `m <= SUC(m + d)`; CARD_CLAUSES; FINITE_NUMSEG;
NOT_IN_EMPTY; ARITH; IN_NUMSEG; ARITH_RULE `~(SUC n <= n)`]);;
let CARD_NUMSEG = prove
(`!m n. CARD(m..n) = (n + 1) - m`,
REPEAT GEN_TAC THEN
DISJ_CASES_THEN MP_TAC (ARITH_RULE `n:num < m \/ m <= n`) THENL
[ASM_MESON_TAC[NUMSEG_EMPTY; CARD_CLAUSES;
ARITH_RULE `n < m ==> ((n + 1) - m = 0)`];
SIMP_TAC[LE_EXISTS; LEFT_IMP_EXISTS_THM; CARD_NUMSEG_LEMMA] THEN
REPEAT STRIP_TAC THEN ARITH_TAC]);;
let HAS_SIZE_NUMSEG = prove
(`!m n. (m..n) HAS_SIZE ((n + 1) - m)`,
REWRITE_TAC[HAS_SIZE; FINITE_NUMSEG; CARD_NUMSEG]);;
let CARD_NUMSEG_1 = prove
(`!n. CARD(1..n) = n`,
REWRITE_TAC[CARD_NUMSEG] THEN ARITH_TAC);;
let HAS_SIZE_NUMSEG_1 = prove
(`!n. (1..n) HAS_SIZE n`,
REWRITE_TAC[CARD_NUMSEG; HAS_SIZE; FINITE_NUMSEG] THEN ARITH_TAC);;
let NUMSEG_CLAUSES = prove
(`(!m. m..0 = if m = 0 then {0} else {}) /\
(!m n. m..SUC n = if m <= SUC n then (SUC n) INSERT (m..n) else m..n)`,
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
GEN_REWRITE_TAC I [EXTENSION] THEN
REWRITE_TAC[IN_NUMSEG; NOT_IN_EMPTY; IN_INSERT] THEN
POP_ASSUM MP_TAC THEN ARITH_TAC);;
let FINITE_INDEX_NUMSEG = prove
(`!s:A->bool.
FINITE s =
?f. (!i j. i IN (1..CARD(s)) /\ j IN (1..CARD(s)) /\ (f i = f j)
==> (i = j)) /\
(s = IMAGE f (1..CARD(s)))`,
GEN_TAC THEN EQ_TAC THENL
[ALL_TAC; MESON_TAC[FINITE_NUMSEG; FINITE_IMAGE]] THEN
DISCH_TAC THEN
MP_TAC(ISPECL [`s:A->bool`; `CARD(s:A->bool)`] HAS_SIZE_INDEX) THEN
ASM_REWRITE_TAC[HAS_SIZE] THEN
DISCH_THEN(X_CHOOSE_THEN `f:num->A` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\n. f(n - 1):A` THEN
ASM_REWRITE_TAC[EXTENSION; IN_IMAGE; IN_NUMSEG] THEN
CONJ_TAC THENL
[REWRITE_TAC[ARITH_RULE `1 <= i /\ i <= n <=> ~(i = 0) /\ i - 1 < n`] THEN
ASM_MESON_TAC[ARITH_RULE
`~(x = 0) /\ ~(y = 0) /\ (x - 1 = y - 1) ==> (x = y)`];
ASM_MESON_TAC
[ARITH_RULE `m < C ==> (m = (m + 1) - 1) /\ 1 <= m + 1 /\ m + 1 <= C`;
ARITH_RULE `1 <= i /\ i <= n <=> ~(i = 0) /\ i - 1 < n`]]);;
let FINITE_INDEX_NUMBERS = prove
(`!s:A->bool.
FINITE s =
?k:num->bool f. (!i j. i IN k /\ j IN k /\ (f i = f j) ==> (i = j)) /\
FINITE k /\ (s = IMAGE f k)`,
MESON_TAC[FINITE_INDEX_NUMSEG; FINITE_NUMSEG; FINITE_IMAGE]);;
let INTER_NUMSEG = prove
(`!m n p q. (m..n) INTER (p..q) = (MAX m p)..(MIN n q)`,
REWRITE_TAC[EXTENSION; IN_INTER; IN_NUMSEG] THEN ARITH_TAC);;
let DISJOINT_NUMSEG = prove
(`!m n p q. DISJOINT (m..n) (p..q) <=> n < p \/ q < m \/ n < m \/ q < p`,
REWRITE_TAC[DISJOINT; NUMSEG_EMPTY; INTER_NUMSEG] THEN ARITH_TAC);;
let NUMSEG_ADD_SPLIT = prove
(`!m n p. m <= n + 1 ==> (m..(n+p) = (m..n) UNION (n+1..n+p))`,
REWRITE_TAC[EXTENSION; IN_UNION; IN_NUMSEG] THEN ARITH_TAC);;
let NUMSEG_OFFSET_IMAGE = prove
(`!m n p. (m+p..n+p) = IMAGE (\i. i + p) (m..n)`,
REWRITE_TAC[EXTENSION; IN_IMAGE; IN_NUMSEG] THEN
REPEAT GEN_TAC THEN EQ_TAC THENL
[DISCH_THEN(fun th -> EXISTS_TAC `x - p:num` THEN MP_TAC th); ALL_TAC] THEN
ARITH_TAC);;
let SUBSET_NUMSEG = prove
(`!m n p q. (m..n) SUBSET (p..q) <=> n < m \/ p <= m /\ n <= q`,
REPEAT GEN_TAC THEN REWRITE_TAC[SUBSET; IN_NUMSEG] THEN
EQ_TAC THENL [MESON_TAC[LE_TRANS; NOT_LE; LE_REFL]; ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Equivalence with the more ad-hoc comprehension notation. *)
(* ------------------------------------------------------------------------- *)
let NUMSEG_LE = prove
(`!n. {x | x <= n} = 0..n`,
REWRITE_TAC[EXTENSION; IN_NUMSEG; IN_ELIM_THM] THEN ARITH_TAC);;
let NUMSEG_LT = prove
(`!n. {x | x < n} = if n = 0 then {} else 0..(n-1)`,
GEN_TAC THEN COND_CASES_TAC THEN
REWRITE_TAC[EXTENSION; IN_NUMSEG; IN_ELIM_THM; NOT_IN_EMPTY] THEN
ASM_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Conversion to evaluate m..n for specific numerals. *)
(* ------------------------------------------------------------------------- *)
let NUMSEG_CONV =
let pth_0 = MESON[NUMSEG_EMPTY] `n < m ==> m..n = {}`
and pth_1 = MESON[NUMSEG_SING] `m..m = {m}`
and pth_2 = MESON[NUMSEG_LREC; ADD1] `m <= n ==> m..n = m INSERT (SUC m..n)`
and ns_tm = `(..)` and m_tm = `m:num` and n_tm = `n:num` in
let rec NUMSEG_CONV tm =
let nstm,nt = dest_comb tm in
let nst,mt = dest_comb nstm in
if nst <> ns_tm then failwith "NUMSEG_CONV" else
let m = dest_numeral mt and n = dest_numeral nt in
if n </ m then MP_CONV NUM_LT_CONV (INST [mt,m_tm; nt,n_tm] pth_0)
else if n =/ m then INST [mt,m_tm] pth_1
else let th = MP_CONV NUM_LE_CONV (INST [mt,m_tm; nt,n_tm] pth_2) in
CONV_RULE(funpow 2 RAND_CONV
(LAND_CONV NUM_SUC_CONV THENC NUMSEG_CONV)) th in
NUMSEG_CONV;;
(* ------------------------------------------------------------------------- *)
(* Topological sorting of a finite set. *)
(* ------------------------------------------------------------------------- *)
let TOPOLOGICAL_SORT = prove
(`!(<<). (!x y:A. x << y /\ y << x ==> x = y) /\
(!x y z. x << y /\ y << z ==> x << z)
==> !n s. s HAS_SIZE n
==> ?f. s = IMAGE f (1..n) /\
(!j k. j IN 1..n /\ k IN 1..n /\ j < k
==> ~(f k << f j))`,
GEN_TAC THEN DISCH_TAC THEN
SUBGOAL_THEN `!n s. s HAS_SIZE n /\ ~(s = {})
==> ?a:A. a IN s /\ !b. b IN (s DELETE a) ==> ~(b << a)`
ASSUME_TAC THENL
[INDUCT_TAC THEN
REWRITE_TAC[HAS_SIZE_0; HAS_SIZE_SUC; TAUT `~(a /\ ~a)`] THEN
X_GEN_TAC `s:A->bool` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
DISCH_THEN(X_CHOOSE_TAC `a:A`) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `a:A`) THEN ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `s DELETE (a:A)`) THEN
ASM_SIMP_TAC[SET_RULE `a IN s ==> (s DELETE a = {} <=> s = {a})`] THEN
ASM_CASES_TAC `s = {a:A}` THEN ASM_REWRITE_TAC[] THENL
[EXISTS_TAC `a:A` THEN SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `b:A` STRIP_ASSUME_TAC) THEN
ASM_CASES_TAC `((a:A) << (b:A)) :bool` THENL
[EXISTS_TAC `a:A`; EXISTS_TAC `b:A`] THEN ASM SET_TAC[];
ALL_TAC] THEN
INDUCT_TAC THENL
[SIMP_TAC[HAS_SIZE_0; NUMSEG_CLAUSES; ARITH; IMAGE_CLAUSES; NOT_IN_EMPTY];
ALL_TAC] THEN
REWRITE_TAC[HAS_SIZE_SUC] THEN X_GEN_TAC `s:A->bool` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`SUC n`; `s:A->bool`]) THEN
ASM_REWRITE_TAC[HAS_SIZE_SUC] THEN
DISCH_THEN(X_CHOOSE_THEN `a:A` MP_TAC) THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `s DELETE (a:A)`) THEN ASM_SIMP_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `f:num->A` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\k. if k = 1 then a:A else f(k - 1)` THEN
SIMP_TAC[ARITH_RULE `1 <= k ==> ~(SUC k = 1)`; SUC_SUB1] THEN
SUBGOAL_THEN `!i. i IN 1..SUC n <=> i = 1 \/ 1 < i /\ (i - 1) IN 1..n`
(fun th -> REWRITE_TAC[EXTENSION; IN_IMAGE; th])
THENL [REWRITE_TAC[IN_NUMSEG] THEN ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL
[X_GEN_TAC `b:A` THEN ASM_CASES_TAC `b:A = a` THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
FIRST_ASSUM(fun th -> ONCE_REWRITE_TAC[MATCH_MP
(SET_RULE `~(b = a) ==> (b IN s <=> b IN (s DELETE a))`) th]) THEN
ONCE_REWRITE_TAC[COND_RAND] THEN
ASM_REWRITE_TAC[IN_IMAGE; IN_NUMSEG] THEN
EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
DISCH_THEN(X_CHOOSE_TAC `i:num`) THEN EXISTS_TAC `i + 1` THEN
ASM_SIMP_TAC[ARITH_RULE `1 <= x ==> 1 < x + 1 /\ ~(x + 1 = 1)`; ADD_SUB];
MAP_EVERY X_GEN_TAC [`j:num`; `k:num`] THEN
MAP_EVERY ASM_CASES_TAC [`j = 1`; `k = 1`] THEN
ASM_REWRITE_TAC[LT_REFL] THENL
[STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM SET_TAC[];
ARITH_TAC;
STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC]]);;
(* ------------------------------------------------------------------------- *)
(* Analogous finiteness theorem for segments of integers. *)
(* ------------------------------------------------------------------------- *)
let FINITE_INTSEG = prove
(`(!l r. FINITE {x:int | l <= x /\ x <= r}) /\
(!l r. FINITE {x:int | l <= x /\ x < r}) /\
(!l r. FINITE {x:int | l < x /\ x <= r}) /\
(!l r. FINITE {x:int | l < x /\ x < r})`,
MATCH_MP_TAC(TAUT `(a ==> b) /\ a ==> a /\ b`) THEN CONJ_TAC THENL
[DISCH_TAC THEN REPEAT CONJ_TAC THEN POP_ASSUM MP_TAC THEN
REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET) THEN
REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN INT_ARITH_TAC;
REPEAT GEN_TAC THEN ASM_CASES_TAC `&0:int <= r - l` THEN
ASM_SIMP_TAC[INT_ARITH `~(&0 <= r - l:int) ==> ~(l <= x /\ x <= r)`] THEN
ASM_SIMP_TAC[EMPTY_GSPEC; FINITE_EMPTY] THEN
MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `IMAGE (\n. l + &n) (0..num_of_int(r - l))` THEN
ASM_SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG] THEN
REWRITE_TAC[SUBSET; IN_IMAGE; IN_ELIM_THM] THEN
REWRITE_TAC[GSYM INT_OF_NUM_LE; IN_NUMSEG] THEN
X_GEN_TAC `x:int` THEN STRIP_TAC THEN EXISTS_TAC `num_of_int(x - l)` THEN
ASM_SIMP_TAC[INT_OF_NUM_OF_INT; INT_SUB_LE] THEN ASM_INT_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Generic iteration of operation over set with finite support. *)
(* ------------------------------------------------------------------------- *)
let neutral = new_definition
`neutral op = @x. !y. (op x y = y) /\ (op y x = y)`;;
let monoidal = new_definition
`monoidal op <=> (!x y. op x y = op y x) /\
(!x y z. op x (op y z) = op (op x y) z) /\
(!x:A. op (neutral op) x = x)`;;
let MONOIDAL_AC = prove
(`!op. monoidal op
==> (!a. op (neutral op) a = a) /\
(!a. op a (neutral op) = a) /\
(!a b. op a b = op b a) /\
(!a b c. op (op a b) c = op a (op b c)) /\
(!a b c. op a (op b c) = op b (op a c))`,
REWRITE_TAC[monoidal] THEN MESON_TAC[]);;
let support = new_definition
`support op (f:A->B) s = {x | x IN s /\ ~(f x = neutral op)}`;;
let iterate = new_definition
`iterate op (s:A->bool) f =
if FINITE(support op f s)
then ITSET (\x a. op (f x) a) (support op f s) (neutral op)
else neutral op`;;
let IN_SUPPORT = prove
(`!op f x s. x IN (support op f s) <=> x IN s /\ ~(f x = neutral op)`,
REWRITE_TAC[support; IN_ELIM_THM]);;
let SUPPORT_SUPPORT = prove
(`!op f s. support op f (support op f s) = support op f s`,
REWRITE_TAC[support; IN_ELIM_THM; EXTENSION] THEN REWRITE_TAC[CONJ_ACI]);;
let SUPPORT_EMPTY = prove
(`!op f s. (!x. x IN s ==> (f(x) = neutral op)) <=> (support op f s = {})`,
REWRITE_TAC[IN_SUPPORT; EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN
MESON_TAC[]);;
let SUPPORT_SUBSET = prove
(`!op f s. (support op f s) SUBSET s`,
SIMP_TAC[SUBSET; IN_SUPPORT]);;
let FINITE_SUPPORT = prove
(`!op f s. FINITE s ==> FINITE(support op f s)`,
MESON_TAC[SUPPORT_SUBSET; FINITE_SUBSET]);;
let SUPPORT_CLAUSES = prove
(`(!f. support op f {} = {}) /\
(!f x s. support op f (x INSERT s) =
if f(x) = neutral op then support op f s
else x INSERT (support op f s)) /\
(!f x s. support op f (s DELETE x) = (support op f s) DELETE x) /\
(!f s t. support op f (s UNION t) =
(support op f s) UNION (support op f t)) /\
(!f s t. support op f (s INTER t) =
(support op f s) INTER (support op f t)) /\
(!f s t. support op f (s DIFF t) =
(support op f s) DIFF (support op f t)) /\
(!f g s. support op g (IMAGE f s) = IMAGE f (support op (g o f) s))`,
REWRITE_TAC[support; EXTENSION; IN_ELIM_THM; IN_INSERT; IN_DELETE; o_THM;
IN_IMAGE; NOT_IN_EMPTY; IN_UNION; IN_INTER; IN_DIFF; COND_RAND] THEN
REPEAT STRIP_TAC THEN TRY COND_CASES_TAC THEN ASM_MESON_TAC[]);;
let SUPPORT_DELTA = prove
(`!op s f a. support op (\x. if x = a then f(x) else neutral op) s =
if a IN s then support op f {a} else {}`,
REWRITE_TAC[EXTENSION; support; IN_ELIM_THM; IN_SING] THEN
REPEAT GEN_TAC THEN REPEAT COND_CASES_TAC THEN
ASM_REWRITE_TAC[IN_ELIM_THM; NOT_IN_EMPTY]);;
let FINITE_SUPPORT_DELTA = prove
(`!op f a. FINITE(support op (\x. if x = a then f(x) else neutral op) s)`,
REWRITE_TAC[SUPPORT_DELTA] THEN REPEAT GEN_TAC THEN
COND_CASES_TAC THEN SIMP_TAC[FINITE_RULES; FINITE_SUPPORT]);;
(* ------------------------------------------------------------------------- *)
(* Key lemmas about the generic notion. *)
(* ------------------------------------------------------------------------- *)
let ITERATE_SUPPORT = prove
(`!op f s. iterate op (support op f s) f = iterate op s f`,
SIMP_TAC[iterate; SUPPORT_SUPPORT]);;
let ITERATE_EXPAND_CASES = prove
(`!op f s. iterate op s f =
if FINITE(support op f s) then iterate op (support op f s) f
else neutral op`,
SIMP_TAC[iterate; SUPPORT_SUPPORT]);;
let ITERATE_CLAUSES_GEN = prove
(`!op. monoidal op
==> (!(f:A->B). iterate op {} f = neutral op) /\
(!f x s. monoidal op /\ FINITE(support op (f:A->B) s)
==> (iterate op (x INSERT s) f =
if x IN s then iterate op s f
else op (f x) (iterate op s f)))`,
GEN_TAC THEN STRIP_TAC THEN
ONCE_REWRITE_TAC[AND_FORALL_THM] THEN GEN_TAC THEN
MP_TAC(ISPECL [`\x a. (op:B->B->B) ((f:A->B)(x)) a`; `neutral op :B`]
FINITE_RECURSION) THEN
ANTS_TAC THENL [ASM_MESON_TAC[monoidal]; ALL_TAC] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[iterate; SUPPORT_CLAUSES; FINITE_RULES] THEN
GEN_REWRITE_TAC (LAND_CONV o RATOR_CONV o LAND_CONV) [COND_RAND] THEN
ASM_REWRITE_TAC[SUPPORT_CLAUSES; FINITE_INSERT; COND_ID] THEN
ASM_CASES_TAC `(f:A->B) x = neutral op` THEN
ASM_SIMP_TAC[IN_SUPPORT] THEN COND_CASES_TAC THEN ASM_MESON_TAC[monoidal]);;
let ITERATE_CLAUSES = prove
(`!op. monoidal op
==> (!f. iterate op {} f = neutral op) /\
(!f x s. FINITE(s)
==> (iterate op (x INSERT s) f =
if x IN s then iterate op s f
else op (f x) (iterate op s f)))`,
SIMP_TAC[ITERATE_CLAUSES_GEN; FINITE_SUPPORT]);;
let ITERATE_UNION = prove
(`!op. monoidal op
==> !f s t. FINITE s /\ FINITE t /\ DISJOINT s t
==> (iterate op (s UNION t) f =
op (iterate op s f) (iterate op t f))`,
let lemma = prove
(`(s UNION (x INSERT t) = x INSERT (s UNION t)) /\
(DISJOINT s (x INSERT t) <=> ~(x IN s) /\ DISJOINT s t)`,
SET_TAC[]) in
GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN GEN_TAC THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REPEAT DISCH_TAC THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
ASM_SIMP_TAC[ITERATE_CLAUSES; IN_UNION; UNION_EMPTY; REAL_ADD_RID; lemma;
FINITE_UNION] THEN
ASM_MESON_TAC[monoidal]);;
let ITERATE_UNION_GEN = prove
(`!op. monoidal op
==> !(f:A->B) s t. FINITE(support op f s) /\ FINITE(support op f t) /\
DISJOINT (support op f s) (support op f t)
==> (iterate op (s UNION t) f =
op (iterate op s f) (iterate op t f))`,
ONCE_REWRITE_TAC[GSYM ITERATE_SUPPORT] THEN
SIMP_TAC[SUPPORT_CLAUSES; ITERATE_UNION]);;
let ITERATE_DIFF = prove
(`!op. monoidal op
==> !f s t. FINITE s /\ t SUBSET s
==> (op (iterate op (s DIFF t) f) (iterate op t f) =
iterate op s f)`,
let lemma = prove
(`t SUBSET s ==> (s = (s DIFF t) UNION t) /\ DISJOINT (s DIFF t) t`,
SET_TAC[]) in
MESON_TAC[lemma; ITERATE_UNION; FINITE_UNION; FINITE_SUBSET; SUBSET_DIFF]);;
let ITERATE_DIFF_GEN = prove
(`!op. monoidal op
==> !f:A->B s t. FINITE (support op f s) /\
(support op f t) SUBSET (support op f s)
==> (op (iterate op (s DIFF t) f) (iterate op t f) =
iterate op s f)`,
ONCE_REWRITE_TAC[GSYM ITERATE_SUPPORT] THEN
SIMP_TAC[SUPPORT_CLAUSES; ITERATE_DIFF]);;
let ITERATE_INCL_EXCL = prove
(`!op. monoidal op
==> !s t f. FINITE s /\ FINITE t
==> op (iterate op s f) (iterate op t f) =
op (iterate op (s UNION t) f)
(iterate op (s INTER t) f)`,
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[SET_RULE
`a UNION b = ((a DIFF b) UNION (b DIFF a)) UNION (a INTER b)`] THEN
GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV)
[SET_RULE `s:A->bool = s DIFF t UNION s INTER t`] THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o ONCE_DEPTH_CONV)
[SET_RULE `t:A->bool = t DIFF s UNION s INTER t`] THEN
ASM_SIMP_TAC[ITERATE_UNION; FINITE_UNION; FINITE_DIFF; FINITE_INTER;
SET_RULE `DISJOINT (s DIFF s' UNION s' DIFF s) (s INTER s')`;
SET_RULE `DISJOINT (s DIFF s') (s' DIFF s)`;
SET_RULE `DISJOINT (s DIFF s') (s' INTER s)`;
SET_RULE `DISJOINT (s DIFF s') (s INTER s')`] THEN
FIRST_X_ASSUM(fun th -> REWRITE_TAC[MATCH_MP MONOIDAL_AC th]));;
let ITERATE_CLOSED = prove
(`!op. monoidal op
==> !P. P(neutral op) /\ (!x y. P x /\ P y ==> P (op x y))
==> !f:A->B s. (!x. x IN s /\ ~(f x = neutral op) ==> P(f x))
==> P(iterate op s f)`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[ITERATE_EXPAND_CASES] THEN
REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[GSYM IN_SUPPORT] THEN
COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN POP_ASSUM MP_TAC THEN
SPEC_TAC(`support op (f:A->B) s`,`s:A->bool`) THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
ASM_SIMP_TAC[ITERATE_CLAUSES; FINITE_INSERT; IN_INSERT]);;
let ITERATE_RELATED = prove
(`!op. monoidal op
==> !R. R (neutral op) (neutral op) /\
(!x1 y1 x2 y2. R x1 x2 /\ R y1 y2 ==> R (op x1 y1) (op x2 y2))
==> !f:A->B g s.
FINITE s /\
(!x. x IN s ==> R (f x) (g x))
==> R (iterate op s f) (iterate op s g)`,
GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN STRIP_TAC THEN GEN_TAC THEN
GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
ASM_SIMP_TAC[ITERATE_CLAUSES; FINITE_INSERT; IN_INSERT]);;
let ITERATE_EQ_NEUTRAL = prove
(`!op. monoidal op
==> !f:A->B s. (!x. x IN s ==> (f(x) = neutral op))
==> (iterate op s f = neutral op)`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `support op (f:A->B) s = {}` ASSUME_TAC THENL
[ASM_MESON_TAC[EXTENSION; NOT_IN_EMPTY; IN_SUPPORT];
ASM_MESON_TAC[ITERATE_CLAUSES; FINITE_RULES; ITERATE_SUPPORT]]);;
let ITERATE_SING = prove
(`!op. monoidal op ==> !f:A->B x. (iterate op {x} f = f x)`,
SIMP_TAC[ITERATE_CLAUSES; FINITE_RULES; NOT_IN_EMPTY] THEN
MESON_TAC[monoidal]);;
let ITERATE_DELETE = prove
(`!op. monoidal op
==> !f:A->B s a. FINITE s /\ a IN s
==> op (f a) (iterate op (s DELETE a) f) =
iterate op s f`,
MESON_TAC[ITERATE_CLAUSES; FINITE_DELETE; IN_DELETE; INSERT_DELETE]);;
let ITERATE_DELTA = prove
(`!op. monoidal op
==> !f a s. iterate op s (\x. if x = a then f(x) else neutral op) =
if a IN s then f(a) else neutral op`,
GEN_TAC THEN DISCH_TAC THEN ONCE_REWRITE_TAC[GSYM ITERATE_SUPPORT] THEN
REWRITE_TAC[SUPPORT_DELTA] THEN REPEAT GEN_TAC THEN COND_CASES_TAC THEN
ASM_SIMP_TAC[ITERATE_CLAUSES] THEN REWRITE_TAC[SUPPORT_CLAUSES] THEN
COND_CASES_TAC THEN ASM_SIMP_TAC[ITERATE_CLAUSES; ITERATE_SING]);;
let ITERATE_IMAGE = prove
(`!op. monoidal op
==> !f:A->B g:B->C s.
(!x y. x IN s /\ y IN s /\ (f x = f y) ==> (x = y))
==> (iterate op (IMAGE f s) g = iterate op s (g o f))`,
GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN GEN_TAC THEN
SUBGOAL_THEN
`!s. FINITE s /\
(!x y:A. x IN s /\ y IN s /\ (f x = f y) ==> (x = y))
==> (iterate op (IMAGE f s) (g:B->C) = iterate op s (g o f))`
ASSUME_TAC THENL
[REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
ASM_SIMP_TAC[ITERATE_CLAUSES; IMAGE_CLAUSES; FINITE_IMAGE] THEN
REWRITE_TAC[o_THM; IN_INSERT] THEN ASM_MESON_TAC[IN_IMAGE];
GEN_TAC THEN DISCH_TAC THEN ONCE_REWRITE_TAC[ITERATE_EXPAND_CASES] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC(TAUT
`(a <=> a') /\ (a' ==> (b = b'))
==> (if a then b else c) = (if a' then b' else c)`) THEN
REWRITE_TAC[SUPPORT_CLAUSES] THEN REPEAT STRIP_TAC THENL
[MATCH_MP_TAC FINITE_IMAGE_INJ_EQ THEN ASM_MESON_TAC[IN_SUPPORT];
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[IN_SUPPORT]]]);;
let ITERATE_BIJECTION = prove
(`!op. monoidal op
==> !f:A->B p s.
(!x. x IN s ==> p(x) IN s) /\
(!y. y IN s ==> ?!x. x IN s /\ p(x) = y)
==> iterate op s f = iterate op s (f o p)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `iterate op (IMAGE (p:A->A) s) (f:A->B)` THEN CONJ_TAC THENL
[AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_IMAGE];
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
(INST_TYPE [aty,bty] ITERATE_IMAGE))] THEN
ASM_MESON_TAC[]);;
let ITERATE_ITERATE_PRODUCT = prove
(`!op. monoidal op
==> !s:A->bool t:A->B->bool x:A->B->C.
FINITE s /\ (!i. i IN s ==> FINITE(t i))
==> iterate op s (\i. iterate op (t i) (x i)) =
iterate op {i,j | i IN s /\ j IN t i} (\(i,j). x i j)`,
GEN_TAC THEN DISCH_TAC THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
ASM_SIMP_TAC[NOT_IN_EMPTY; SET_RULE `{a,b | F} = {}`; ITERATE_CLAUSES] THEN
REWRITE_TAC[SET_RULE `{i,j | i IN a INSERT s /\ j IN t i} =
IMAGE (\j. a,j) (t a) UNION {i,j | i IN s /\ j IN t i}`] THEN
ASM_SIMP_TAC[FINITE_INSERT; ITERATE_CLAUSES; IN_INSERT] THEN
REPEAT STRIP_TAC THEN
FIRST_ASSUM(fun th ->
W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP ITERATE_UNION th) o
rand o snd)) THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[FINITE_IMAGE; FINITE_PRODUCT_DEPENDENT; IN_INSERT] THEN
REWRITE_TAC[DISJOINT; EXTENSION; IN_IMAGE; IN_INTER; NOT_IN_EMPTY;
IN_ELIM_THM; EXISTS_PAIR_THM; FORALL_PAIR_THM; PAIR_EQ] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
FIRST_ASSUM(fun th ->
W(MP_TAC o PART_MATCH (lhand o rand) (MATCH_MP ITERATE_IMAGE th) o
rand o snd)) THEN
ANTS_TAC THENL
[SIMP_TAC[FORALL_PAIR_THM] THEN CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
ASM_SIMP_TAC[PAIR_EQ];
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[o_DEF] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN REWRITE_TAC[ETA_AX]]);;
let ITERATE_EQ = prove
(`!op. monoidal op
==> !f:A->B g s.
(!x. x IN s ==> f x = g x) ==> iterate op s f = iterate op s g`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[ITERATE_EXPAND_CASES] THEN
SUBGOAL_THEN `support op g s = support op (f:A->B) s` SUBST1_TAC THENL
[REWRITE_TAC[EXTENSION; IN_SUPPORT] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN
`FINITE(support op (f:A->B) s) /\
(!x. x IN (support op f s) ==> f x = g x)`
MP_TAC THENL [ASM_MESON_TAC[IN_SUPPORT]; REWRITE_TAC[IMP_CONJ]] THEN
SPEC_TAC(`support op (f:A->B) s`,`t:A->bool`) THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN ASM_SIMP_TAC[ITERATE_CLAUSES] THEN
MESON_TAC[IN_INSERT]);;
let ITERATE_EQ_GENERAL = prove
(`!op. monoidal op
==> !s:A->bool t:B->bool f:A->C g h.
(!y. y IN t ==> ?!x. x IN s /\ h(x) = y) /\
(!x. x IN s ==> h(x) IN t /\ g(h x) = f x)
==> iterate op s f = iterate op t g`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `t = IMAGE (h:A->B) s` SUBST1_TAC THENL
[REWRITE_TAC[EXTENSION; IN_IMAGE] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `iterate op s ((g:B->C) o (h:A->B))` THEN CONJ_TAC THENL
[ASM_MESON_TAC[ITERATE_EQ; o_THM];
CONV_TAC SYM_CONV THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP ITERATE_IMAGE) THEN
ASM_MESON_TAC[]]);;
let ITERATE_EQ_GENERAL_INVERSES = prove
(`!op. monoidal op
==> !s:A->bool t:B->bool f:A->C g h k.
(!y. y IN t ==> k(y) IN s /\ h(k y) = y) /\
(!x. x IN s ==> h(x) IN t /\ k(h x) = x /\ g(h x) = f x)
==> iterate op s f = iterate op t g`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP ITERATE_EQ_GENERAL) THEN
EXISTS_TAC `h:A->B` THEN ASM_MESON_TAC[]);;
let ITERATE_INJECTION = prove
(`!op. monoidal op
==> !f:A->B p:A->A s.
FINITE s /\
(!x. x IN s ==> p x IN s) /\
(!x y. x IN s /\ y IN s /\ p x = p y ==> x = y)
==> iterate op s (f o p) = iterate op s f`,
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP ITERATE_BIJECTION) THEN
MP_TAC(ISPECL [`s:A->bool`; `p:A->A`] SURJECTIVE_IFF_INJECTIVE) THEN
ASM_REWRITE_TAC[SUBSET; IN_IMAGE] THEN ASM_MESON_TAC[]);;
let ITERATE_UNION_NONZERO = prove
(`!op. monoidal op
==> !f:A->B s t.
FINITE(s) /\ FINITE(t) /\
(!x. x IN (s INTER t) ==> f x = neutral(op))
==> iterate op (s UNION t) f =
op (iterate op s f) (iterate op t f)`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM ITERATE_SUPPORT] THEN
REWRITE_TAC[SUPPORT_CLAUSES] THEN
FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP ITERATE_UNION) THEN
ASM_SIMP_TAC[FINITE_SUPPORT; DISJOINT; IN_INTER; IN_SUPPORT; EXTENSION] THEN
ASM_MESON_TAC[IN_INTER; NOT_IN_EMPTY]);;
let ITERATE_OP = prove
(`!op. monoidal op
==> !f g s. FINITE s
==> iterate op s (\x. op (f x) (g x)) =
op (iterate op s f) (iterate op s g)`,
GEN_TAC THEN DISCH_TAC THEN
GEN_TAC THEN GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
ASM_SIMP_TAC[ITERATE_CLAUSES; MONOIDAL_AC]);;
let ITERATE_SUPERSET = prove
(`!op. monoidal op
==> !f:A->B u v.
u SUBSET v /\
(!x. x IN v /\ ~(x IN u) ==> f(x) = neutral op)
==> iterate op v f = iterate op u f`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM ITERATE_SUPPORT] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[support; EXTENSION; IN_ELIM_THM] THEN ASM_MESON_TAC[SUBSET]);;
let ITERATE_UNIV = prove
(`!op. monoidal op
==> !f:A->B s. support op f UNIV SUBSET s
==> iterate op s f = iterate op UNIV f`,
REWRITE_TAC[support; SUBSET; IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP ITERATE_SUPERSET) THEN
ASM SET_TAC[]);;
let ITERATE_IMAGE_NONZERO = prove
(`!op. monoidal op
==> !g:B->C f:A->B s.
FINITE s /\
(!x y. x IN s /\ y IN s /\ ~(x = y) /\ f x = f y
==> g(f x) = neutral op)
==> iterate op (IMAGE f s) g = iterate op s (g o f)`,
GEN_TAC THEN DISCH_TAC THEN
GEN_TAC THEN GEN_TAC THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
ASM_SIMP_TAC[IMAGE_CLAUSES; ITERATE_CLAUSES; FINITE_IMAGE] THEN
MAP_EVERY X_GEN_TAC [`a:A`; `s:A->bool`] THEN
REWRITE_TAC[IN_INSERT] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `iterate op s ((g:B->C) o (f:A->B)) = iterate op (IMAGE f s) g`
SUBST1_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[IN_IMAGE] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[o_THM] THEN
SUBGOAL_THEN `(g:B->C) ((f:A->B) a) = neutral op` SUBST1_TAC THEN
ASM_MESON_TAC[MONOIDAL_AC]);;
let ITERATE_CASES = prove
(`!op. monoidal op
==> !s P f g:A->B.
FINITE s
==> iterate op s (\x. if P x then f x else g x) =
op (iterate op {x | x IN s /\ P x} f)
(iterate op {x | x IN s /\ ~P x} g)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC
`op (iterate op {x | x IN s /\ P x} (\x. if P x then f x else (g:A->B) x))
(iterate op {x | x IN s /\ ~P x} (\x. if P x then f x else g x))` THEN
CONJ_TAC THENL
[FIRST_ASSUM(fun th -> ASM_SIMP_TAC[GSYM(MATCH_MP ITERATE_UNION th);
FINITE_RESTRICT;
SET_RULE `DISJOINT {x | x IN s /\ P x} {x | x IN s /\ ~P x}`]) THEN
AP_THM_TAC THEN AP_TERM_TAC THEN SET_TAC[];
BINOP_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP ITERATE_EQ) THEN
SIMP_TAC[IN_ELIM_THM]]);;
let ITERATE_OP_GEN = prove
(`!op. monoidal op
==> !f g:A->B s.
FINITE(support op f s) /\ FINITE(support op g s)
==> iterate op s (\x. op (f x) (g x)) =
op (iterate op s f) (iterate op s g)`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM ITERATE_SUPPORT] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `iterate op (support op f s UNION support op g s)
(\x. op ((f:A->B) x) (g x))` THEN
CONJ_TAC THENL
[CONV_TAC SYM_CONV;
ASM_SIMP_TAC[ITERATE_OP; FINITE_UNION] THEN BINOP_TAC] THEN
FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP ITERATE_SUPERSET) THEN
REWRITE_TAC[support; IN_ELIM_THM; SUBSET; IN_UNION] THEN
ASM_MESON_TAC[monoidal]);;
let ITERATE_CLAUSES_NUMSEG = prove
(`!op. monoidal op
==> (!m. iterate op (m..0) f = if m = 0 then f(0) else neutral op) /\
(!m n. iterate op (m..SUC n) f =
if m <= SUC n then op (iterate op (m..n) f) (f(SUC n))
else iterate op (m..n) f)`,
REWRITE_TAC[NUMSEG_CLAUSES] THEN REPEAT STRIP_TAC THEN
COND_CASES_TAC THEN
ASM_SIMP_TAC[ITERATE_CLAUSES; FINITE_NUMSEG; IN_NUMSEG; FINITE_EMPTY] THEN
REWRITE_TAC[ARITH_RULE `~(SUC n <= n)`; NOT_IN_EMPTY] THEN
ASM_MESON_TAC[monoidal]);;
let ITERATE_PAIR = prove
(`!op. monoidal op
==> !f m n. iterate op (2*m..2*n+1) f =
iterate op (m..n) (\i. op (f(2*i)) (f(2*i+1)))`,
GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN GEN_TAC THEN
INDUCT_TAC THEN CONV_TAC NUM_REDUCE_CONV THENL
[ASM_SIMP_TAC[num_CONV `1`; ITERATE_CLAUSES_NUMSEG] THEN
REWRITE_TAC[ARITH_RULE `2 * m <= SUC 0 <=> m = 0`] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[MULT_EQ_0; ARITH];
REWRITE_TAC[ARITH_RULE `2 * SUC n + 1 = SUC(SUC(2 * n + 1))`] THEN
ASM_SIMP_TAC[ITERATE_CLAUSES_NUMSEG] THEN
REWRITE_TAC[ARITH_RULE `2 * m <= SUC(SUC(2 * n + 1)) <=> m <= SUC n`;
ARITH_RULE `2 * m <= SUC(2 * n + 1) <=> m <= SUC n`] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[ARITH_RULE `2 * SUC n = SUC(2 * n + 1)`;
ARITH_RULE `2 * SUC n + 1 = SUC(SUC(2 * n + 1))`] THEN
ASM_MESON_TAC[monoidal]]);;
let ITERATE_REFLECT = prove
(`!op:A->A->A.
monoidal op
==> !x m n. iterate op (m..n) x =
if n < m then neutral op
else iterate op (0..n-m) (\i. x(n - i))`,
REWRITE_TAC[GSYM NUMSEG_EMPTY] THEN REPEAT STRIP_TAC THEN
COND_CASES_TAC THENL
[ASM_MESON_TAC[ITERATE_CLAUSES];
RULE_ASSUM_TAC(REWRITE_RULE[NUMSEG_EMPTY; NOT_LT])] THEN
FIRST_ASSUM(MP_TAC o
ISPECL [`\i:num. n - i`; `x:num->A`; `0..n-m`] o
MATCH_MP (INST_TYPE [`:X`,`:A`] ITERATE_IMAGE)) THEN
REWRITE_TAC[o_DEF; IN_NUMSEG] THEN
ANTS_TAC THENL [ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[EXTENSION; IN_IMAGE; IN_NUMSEG] THEN
REWRITE_TAC[UNWIND_THM2; ARITH_RULE
`x = n - y /\ 0 <= y /\ y <= n - m <=>
y = n - x /\ x <= n /\ y <= n - m`] THEN
ASM_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Sums of natural numbers. *)
(* ------------------------------------------------------------------------- *)
prioritize_num();;
let nsum = new_definition
`nsum = iterate (+)`;;
let NEUTRAL_ADD = prove
(`neutral((+):num->num->num) = 0`,
REWRITE_TAC[neutral] THEN MATCH_MP_TAC SELECT_UNIQUE THEN
MESON_TAC[ADD_CLAUSES]);;
let NEUTRAL_MUL = prove
(`neutral(( * ):num->num->num) = 1`,
REWRITE_TAC[neutral] THEN MATCH_MP_TAC SELECT_UNIQUE THEN
MESON_TAC[MULT_CLAUSES; MULT_EQ_1]);;
let MONOIDAL_ADD = prove
(`monoidal((+):num->num->num)`,
REWRITE_TAC[monoidal; NEUTRAL_ADD] THEN ARITH_TAC);;
let MONOIDAL_MUL = prove
(`monoidal(( * ):num->num->num)`,
REWRITE_TAC[monoidal; NEUTRAL_MUL] THEN ARITH_TAC);;
let NSUM_DEGENERATE = prove
(`!f s. ~(FINITE {x | x IN s /\ ~(f x = 0)}) ==> nsum s f = 0`,
REPEAT GEN_TAC THEN REWRITE_TAC[nsum] THEN
SIMP_TAC[iterate; support; NEUTRAL_ADD]);;
let NSUM_CLAUSES = prove
(`(!f. nsum {} f = 0) /\
(!x f s. FINITE(s)
==> (nsum (x INSERT s) f =
if x IN s then nsum s f else f(x) + nsum s f))`,
REWRITE_TAC[nsum; GSYM NEUTRAL_ADD] THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
MATCH_MP_TAC ITERATE_CLAUSES THEN REWRITE_TAC[MONOIDAL_ADD]);;
let NSUM_UNION = prove
(`!f s t. FINITE s /\ FINITE t /\ DISJOINT s t
==> (nsum (s UNION t) f = nsum s f + nsum t f)`,
SIMP_TAC[nsum; ITERATE_UNION; MONOIDAL_ADD]);;
let NSUM_DIFF = prove
(`!f s t. FINITE s /\ t SUBSET s
==> (nsum (s DIFF t) f = nsum s f - nsum t f)`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC(ARITH_RULE `(x + z = y:num) ==> (x = y - z)`) THEN
ASM_SIMP_TAC[nsum; ITERATE_DIFF; MONOIDAL_ADD]);;
let NSUM_INCL_EXCL = prove
(`!s t (f:A->num).
FINITE s /\ FINITE t
==> nsum s f + nsum t f = nsum (s UNION t) f + nsum (s INTER t) f`,
REWRITE_TAC[nsum; GSYM NEUTRAL_ADD] THEN
MATCH_MP_TAC ITERATE_INCL_EXCL THEN REWRITE_TAC[MONOIDAL_ADD]);;
let NSUM_SUPPORT = prove
(`!f s. nsum (support (+) f s) f = nsum s f`,
SIMP_TAC[nsum; iterate; SUPPORT_SUPPORT]);;
let NSUM_ADD = prove
(`!f g s. FINITE s ==> (nsum s (\x. f(x) + g(x)) = nsum s f + nsum s g)`,
SIMP_TAC[nsum; ITERATE_OP; MONOIDAL_ADD]);;
let NSUM_ADD_GEN = prove
(`!f g s.
FINITE {x | x IN s /\ ~(f x = 0)} /\ FINITE {x | x IN s /\ ~(g x = 0)}
==> nsum s (\x. f x + g x) = nsum s f + nsum s g`,
REWRITE_TAC[GSYM NEUTRAL_ADD; GSYM support; nsum] THEN
MATCH_MP_TAC ITERATE_OP_GEN THEN ACCEPT_TAC MONOIDAL_ADD);;
let NSUM_EQ_0 = prove
(`!f s. (!x:A. x IN s ==> (f(x) = 0)) ==> (nsum s f = 0)`,
REWRITE_TAC[nsum; GSYM NEUTRAL_ADD] THEN
SIMP_TAC[ITERATE_EQ_NEUTRAL; MONOIDAL_ADD]);;
let NSUM_0 = prove
(`!s:A->bool. nsum s (\n. 0) = 0`,
SIMP_TAC[NSUM_EQ_0]);;
let NSUM_LMUL = prove
(`!f c s:A->bool. nsum s (\x. c * f(x)) = c * nsum s f`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `c = 0` THEN
ASM_REWRITE_TAC[MULT_CLAUSES; NSUM_0] THEN REWRITE_TAC[nsum] THEN
ONCE_REWRITE_TAC[ITERATE_EXPAND_CASES] THEN
SUBGOAL_THEN `support (+) (\x:A. c * f(x)) s = support (+) f s` SUBST1_TAC
THENL [ASM_SIMP_TAC[support; MULT_EQ_0; NEUTRAL_ADD]; ALL_TAC] THEN
COND_CASES_TAC THEN REWRITE_TAC[NEUTRAL_ADD; MULT_CLAUSES] THEN
UNDISCH_TAC `FINITE (support (+) f (s:A->bool))` THEN
SPEC_TAC(`support (+) f (s:A->bool)`,`t:A->bool`) THEN
REWRITE_TAC[GSYM nsum] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
SIMP_TAC[NSUM_CLAUSES; MULT_CLAUSES; LEFT_ADD_DISTRIB]);;
let NSUM_RMUL = prove
(`!f c s:A->bool. nsum s (\x. f(x) * c) = nsum s f * c`,
ONCE_REWRITE_TAC[MULT_SYM] THEN REWRITE_TAC[NSUM_LMUL]);;
let NSUM_LE = prove
(`!f g s. FINITE(s) /\ (!x. x IN s ==> f(x) <= g(x))
==> nsum s f <= nsum s g`,
ONCE_REWRITE_TAC[IMP_CONJ] THEN
GEN_TAC THEN GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
SIMP_TAC[NSUM_CLAUSES; LE_REFL; LE_ADD2; IN_INSERT]);;
let NSUM_LT = prove
(`!f g s:A->bool.
FINITE(s) /\ (!x. x IN s ==> f(x) <= g(x)) /\
(?x. x IN s /\ f(x) < g(x))
==> nsum s f < nsum s g`,
REPEAT GEN_TAC THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `a:A` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `s = (a:A) INSERT (s DELETE a)` SUBST1_TAC THENL
[UNDISCH_TAC `a:A IN s` THEN SET_TAC[]; ALL_TAC] THEN
ASM_SIMP_TAC[NSUM_CLAUSES; FINITE_DELETE; IN_DELETE] THEN
ASM_SIMP_TAC[LTE_ADD2; NSUM_LE; IN_DELETE; FINITE_DELETE]);;
let NSUM_LT_ALL = prove
(`!f g s. FINITE s /\ ~(s = {}) /\ (!x. x IN s ==> f(x) < g(x))
==> nsum s f < nsum s g`,
MESON_TAC[MEMBER_NOT_EMPTY; LT_IMP_LE; NSUM_LT]);;
let NSUM_EQ = prove
(`!f g s. (!x. x IN s ==> (f x = g x)) ==> (nsum s f = nsum s g)`,
REWRITE_TAC[nsum] THEN
MATCH_MP_TAC ITERATE_EQ THEN REWRITE_TAC[MONOIDAL_ADD]);;
let NSUM_CONST = prove
(`!c s. FINITE s ==> (nsum s (\n. c) = (CARD s) * c)`,
GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
SIMP_TAC[NSUM_CLAUSES; CARD_CLAUSES] THEN
REPEAT STRIP_TAC THEN ARITH_TAC);;
let NSUM_POS_BOUND = prove
(`!f b s. FINITE s /\ nsum s f <= b ==> !x:A. x IN s ==> f x <= b`,
GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
SIMP_TAC[NSUM_CLAUSES; NOT_IN_EMPTY; IN_INSERT] THEN
MESON_TAC[LE_0; ARITH_RULE
`0 <= x /\ 0 <= y /\ x + y <= b ==> x <= b /\ y <= b`]);;
let NSUM_EQ_0_IFF = prove
(`!s. FINITE s ==> (nsum s f = 0 <=> !x. x IN s ==> f x = 0)`,
REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[NSUM_EQ_0] THEN
ASM_MESON_TAC[ARITH_RULE `n = 0 <=> n <= 0`; NSUM_POS_BOUND]);;
let NSUM_POS_LT = prove
(`!f s:A->bool.
FINITE s /\ (?x. x IN s /\ 0 < f x)
==> 0 < nsum s f`,
SIMP_TAC[ARITH_RULE `0 < n <=> ~(n = 0)`; NSUM_EQ_0_IFF] THEN MESON_TAC[]);;
let NSUM_POS_LT_ALL = prove
(`!s f:A->num.
FINITE s /\ ~(s = {}) /\ (!i. i IN s ==> 0 < f i) ==> 0 < nsum s f`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC NSUM_POS_LT THEN
ASM_MESON_TAC[MEMBER_NOT_EMPTY; REAL_LT_IMP_LE]);;
let NSUM_DELETE = prove
(`!f s a. FINITE s /\ a IN s ==> f(a) + nsum(s DELETE a) f = nsum s f`,
SIMP_TAC[nsum; ITERATE_DELETE; MONOIDAL_ADD]);;
let NSUM_SING = prove
(`!f x. nsum {x} f = f(x)`,
SIMP_TAC[NSUM_CLAUSES; FINITE_RULES; NOT_IN_EMPTY; ADD_CLAUSES]);;
let NSUM_DELTA = prove
(`!s a. nsum s (\x. if x = a:A then b else 0) = if a IN s then b else 0`,
REWRITE_TAC[nsum; GSYM NEUTRAL_ADD] THEN
SIMP_TAC[ITERATE_DELTA; MONOIDAL_ADD]);;
let NSUM_SWAP = prove
(`!f:A->B->num s t.
FINITE(s) /\ FINITE(t)
==> (nsum s (\i. nsum t (f i)) = nsum t (\j. nsum s (\i. f i j)))`,
GEN_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN