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(* ========================================================================= *)
(* Iterated application of a function, ITER n f x = f^n(x). *)
(* *)
(* (c) Marco Maggesi, Graziano Gentili and Gianni Ciolli, 2008. *)
(* ========================================================================= *)
let ITER = define
`(!f. ITER 0 f x = x) /\
(!f n. ITER (SUC n) f x = f (ITER n f x))`;;
let ITER_POINTLESS = prove
(`(!f. ITER 0 f = I) /\
(!f n. ITER (SUC n) f = f o ITER n f)`,
REWRITE_TAC [FUN_EQ_THM; I_THM; o_THM; ITER]);;
let ITER_ALT = prove
(`(!f x. ITER 0 f x = x) /\
(!f n x. ITER (SUC n) f x = ITER n f (f x))`,
REWRITE_TAC [ITER] THEN GEN_TAC THEN INDUCT_TAC THEN
ASM_REWRITE_TAC [ITER]);;
let ITER_ALT_POINTLESS = prove
(`(!f. ITER 0 f = I) /\
(!f n. ITER (SUC n) f = ITER n f o f)`,
REWRITE_TAC [FUN_EQ_THM; I_THM; o_THM; ITER_ALT]);;
let ITER_1 = prove
(`!f x. ITER 1 f x = f x`,
REWRITE_TAC[num_CONV `1`; ITER]);;
let ITER_ADD = prove
(`!f n m x. ITER n f (ITER m f x) = ITER (n + m) f x`,
GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[ITER; ADD]);;
let ITER_ADD_POINTLESS = prove
(`!m n. ITER (m + n) f = ITER m f o ITER n f`,
REWRITE_TAC[FUN_EQ_THM; o_THM; ITER_ADD]);;
let ITER_MUL = prove
(`!f n m x. ITER n (ITER m f) x = ITER (n * m) f x`,
GEN_TAC THEN INDUCT_TAC THEN
ASM_REWRITE_TAC[ITER; MULT; ITER_ADD; ADD_AC]);;
let ITER_FIXPOINT = prove
(`!f n x. f x = x ==> ITER n f x = x`,
GEN_TAC THEN INDUCT_TAC THEN ASM_SIMP_TAC [ITER_ALT]);;
(* ------------------------------------------------------------------------- *)
(* Existence of "order" or "characteristic" in a general setting. *)
(* ------------------------------------------------------------------------- *)
let ORDER_EXISTENCE_GEN = prove
(`!P f:num->A.
P(f 0) /\ (!m n. P(f m) /\ ~(m = 0) ==> (P(f(m + n)) <=> P(f n)))
==> ?d. !n. P(f n) <=> d divides n`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `!n. ~(n = 0) ==> ~P(f n:A)` THENL
[EXISTS_TAC `0` THEN REWRITE_TAC[NUMBER_RULE `0 divides n <=> n = 0`] THEN
ASM_MESON_TAC[];
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM])] THEN
GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `d:num` THEN REWRITE_TAC[NOT_IMP] THEN
REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN STRIP_TAC THEN
MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
ASM_CASES_TAC `n = 0` THENL
[ASM_MESON_TAC[NUMBER_RULE `n divides 0`]; ALL_TAC] THEN
ASM_CASES_TAC `d <= n:num` THENL
[ALL_TAC; ASM_MESON_TAC[NOT_LT; DIVIDES_LE]] THEN
SUBGOAL_THEN `n:num = (n - d) + d` SUBST1_TAC THENL
[ASM_ARITH_TAC; ABBREV_TAC `m:num = n - d`] THEN
REWRITE_TAC[NUMBER_RULE `(d:num) divides m + d <=> d divides m`] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `m:num`) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; ASM_MESON_TAC[ADD_SYM]]);;
let ORDER_EXISTENCE_ITER = prove
(`!R f z:A.
R z z /\
(!x y. R x y ==> R y x) /\
(!x y z. R x y /\ R y z ==> R x z) /\
(!x y. R x y ==> R (f x) (f y))
==> ?d. !n. R (ITER n f z) z <=> d divides n`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`\x. (R:A->A->bool) x z`;
`\n. ITER n f (z:A)`] ORDER_EXISTENCE_GEN) THEN
ASM_REWRITE_TAC[ITER] THEN DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN
ONCE_REWRITE_TAC[ADD_SYM] THEN REWRITE_TAC[GSYM ITER_ADD] THEN
MP_TAC(MESON[]
`!a b:num->A. (!x y. R x y ==> R y x) /\
(!x y z. R x y /\ R y z ==> R x z) /\
(!n. R (a n) (b n))
==> (!n. R (a n) z <=> R (b n) z)`) THEN
DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
INDUCT_TAC THEN ASM_REWRITE_TAC[ITER] THEN
ASM_MESON_TAC[]);;
let ORDER_EXISTENCE_CARD = prove
(`!R f z:A k.
FINITE { R(ITER n f z) | n IN (:num)} /\
CARD { R(ITER n f z) | n IN (:num)} <= k /\
R z z /\
(!x y. R x y ==> R y x) /\
(!x y z. R x y /\ R y z ==> R x z) /\
(!x y. R (f x) (f y) <=> R x y)
==> ?d. 0 < d /\ d <= k /\ !n. R (ITER n f z) z <=> d divides n`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`?m. 0 < m /\ m <= k /\ (R:A->A->bool) (ITER m f z) z`
STRIP_ASSUME_TAC THENL
[MP_TAC(ISPECL [`\n. (R:A->A->bool) (ITER n f z)`; `0..k`]
CARD_IMAGE_EQ_INJ) THEN
REWRITE_TAC[FINITE_NUMSEG; CARD_NUMSEG; SUB_0] THEN
MATCH_MP_TAC(TAUT `~p /\ (~q ==> r) ==> (p <=> q) ==> r`) THEN
CONJ_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ARITH_RULE
`c <= k ==> s <= c ==> ~(s = k + 1)`)) THEN
MATCH_MP_TAC CARD_SUBSET THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[];
REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN
MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN
CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`p:num`; `q:num`] THEN
REWRITE_TAC[IN_NUMSEG] THEN REPEAT STRIP_TAC THEN
EXISTS_TAC `q - p:num` THEN
REPEAT(CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN
SUBGOAL_THEN
`!d. d <= p
==> (R:A->A->bool) (ITER (p - d) f z) (ITER (q - d) f z)`
MP_TAC THENL
[INDUCT_TAC THEN ASM_REWRITE_TAC[SUB_0] THENL
[SPEC_TAC(`q:num`,`q:num`) THEN INDUCT_TAC THEN
ASM_REWRITE_TAC[ITER];
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
SUBGOAL_THEN `q - d = SUC(q - SUC d) /\ p - d = SUC(p - SUC d)`
(fun th -> REWRITE_TAC[th]) THENL [ASM_ARITH_TAC; ALL_TAC] THEN
ASM_REWRITE_TAC[ITER]];
DISCH_THEN(MP_TAC o SPEC `p:num`) THEN
REWRITE_TAC[LE_REFL; SUB_REFL; ITER] THEN ASM_MESON_TAC[]]];
MP_TAC(ISPECL [`R:A->A->bool`; `f:A->A`; `z:A`] ORDER_EXISTENCE_ITER) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `d:num` THEN ASM_CASES_TAC `d = 0` THEN ASM_SIMP_TAC[] THEN
DISCH_THEN(MP_TAC o SPEC `m:num`) THEN
ASM_SIMP_TAC[LE_1; NUMBER_RULE `!n. 0 divides n <=> n = 0`] THEN
DISCH_THEN(MP_TAC o MATCH_MP DIVIDES_LE) THEN ASM_ARITH_TAC]);;
let ORDER_EXISTENCE_FINITE = prove
(`!R f z:A.
FINITE { R(ITER n f z) | n IN (:num)} /\
R z z /\
(!x y. R x y ==> R y x) /\
(!x y z. R x y /\ R y z ==> R x z) /\
(!x y. R (f x) (f y) <=> R x y)
==> ?d. 0 < d /\ !n. R (ITER n f z) z <=> d divides n`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`R:A->A->bool`; `f:A->A`; `z:A`;
`CARD {(R:A->A->bool)(ITER n f z) | n IN (:num)}`]
ORDER_EXISTENCE_CARD) THEN ASM_REWRITE_TAC[LE_REFL] THEN MESON_TAC[]);;
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