needs "Library/analysis.ml";;
needs "Library/transc.ml";;
needs "Library/floor.ml";;
needs "Library/calc_real.ml";;
needs "Examples/machin.ml";;

(* ----- definitions used in external theorems *)

let log2 = new_definition `
  log2 x = (ln x) / (ln (&2))
`;;

let ceil = new_definition `
  ceil x = -- floor(--x)
`;;

let o1_seq = new_definition `
  o1_seq (f:num->real) <=>
    !e:real. &0 < e ==>
      ?m:num.
        !i:num. m <= i ==> abs(f(i)) <= e
`;;

parse_as_infix("powreal",(24,"left"));;
let powreal = new_definition `
  x powreal y = exp(y * ln x)
`;;

let bkzdelta = new_definition `
  bkzdelta x = (x * ((pi*x) powreal (&1 / x))/(&2 * pi * exp(&1))) powreal (&1 / (&2 * (x - &1)))
`;;

let standardratio = new_definition `
  standardratio n q s k x = 
     ( ((n + k)*(s pow 2) + &1) powreal (&1 / &2)
     ) / ( (((n + k + &1)/x) powreal (&1 / &2))
           * ((bkzdelta x) powreal (&2 * x - (n + k + &1) - &1))
           * (q powreal (k/(n + k + &1)))
         )
`;;

(* ----- definitions used internally *)

parse_as_infix("pow_num",(24,"left"));;
let pow_num = define `
  ((p:A->A) pow_num n) =
    if n = 0 then (I:A->A) else (p o (p pow_num(n-1)))
`;;

(* ----- tactics *)

let proven_if t = ASM_CASES_TAC t THENL
[ASM_MESON_TAC [];
  ALL_TAC];;

let notetac t tac = SUBGOAL_THEN t MP_TAC THENL
[tac;
  ALL_TAC] THEN
DISCH_THEN(fun th -> ASSUME_TAC th);;

let note t why = notetac t(ASM_MESON_TAC why);;

let note_eqlambda t why = notetac t(MATCH_MP_TAC EQ_EXT THEN
  REWRITE_TAC [BETA_THM] THEN
  ASM_MESON_TAC why);;

let specialize v th = ASSUME_TAC(UNDISCH_ALL (REWRITE_RULE [IMP_CONJ] (ISPECL v th)));;

let specialize_forward v th = ASSUME_TAC(UNDISCH_ALL (fst (EQ_IMP_RULE (UNDISCH_ALL (REWRITE_RULE [IMP_CONJ] (ISPECL v th))))));;

let specialize_reverse v th = ASSUME_TAC(UNDISCH_ALL (snd (EQ_IMP_RULE (UNDISCH_ALL (REWRITE_RULE [IMP_CONJ] (ISPECL v th))))));;

let choose_specializing t v th = X_CHOOSE_TAC t(UNDISCH_ALL (REWRITE_RULE [IMP_CONJ] (ISPECL v th)));;

let specialize_forward_strip v th = STRIP_ASSUME_TAC(UNDISCH_ALL (REWRITE_RULE [IMP_CONJ] (fst (EQ_IMP_RULE (ISPECL v th)))));;

let specialize_strip v th = STRIP_ASSUME_TAC(UNDISCH_ALL (REWRITE_RULE [IMP_CONJ] (ISPECL v th)));;

let real_cancel t = specialize[](REAL_FIELD t);;

let real_linear t = specialize[](REAL_ARITH t);;

let def n d = X_CHOOSE_TAC n(MESON [] (mk_exists (n, mk_eq (n, d))));;

let choose n p why = note(mk_exists (n, p)) why THEN
X_CHOOSE_TAC n(UNDISCH (TAUT (mk_imp (mk_exists (n, p), mk_exists (n, p)))));;

(* ----- sets *)

let element_implies_nonempty = prove(`
  !S:real->bool x:real.
  S x ==> ~(S = {})
  `,
  REWRITE_TAC [GSYM EMPTY_GSPEC] THEN
  REWRITE_TAC [EXTENSION] THEN
  REWRITE_TAC [IN_ELIM_THM] THEN
  REWRITE_TAC [IN] THEN
  MESON_TAC []);;

let element_implies_nonempty2 = prove(`
  !S:real->bool x:real.
  x IN S ==> ~(S = {})
  `,
  MESON_TAC [IN;
    element_implies_nonempty]);;

let singleton_not_empty = prove(`
  !a:A. ~({a} = {})
  `,
  REWRITE_TAC [GSYM EMPTY_GSPEC] THEN
  REWRITE_TAC [GSYM SING_GSPEC] THEN
  REWRITE_TAC [INSERT] THEN
  REWRITE_TAC [IN_ELIM_THM] THEN
  REWRITE_TAC [EXTENSION] THEN
  REWRITE_TAC [IN_ELIM_THM] THEN
  MESON_TAC []);;

let singleton_union_not_empty = prove(`
  !a:A S. ~({a} UNION S = {})
  `,
  MESON_TAC [EMPTY_UNION;
    singleton_not_empty]);;

let num_chop_infinite = prove(`
  !P:num->bool m:num.
  INFINITE {i | P(i)}
  ==> INFINITE {i | m <= i /\ P(i)}
  `,
  REPEAT STRIP_TAC THEN
  notetac `{i:num | P(i)} = {i | i < m /\ P(i)} UNION {i | m <= i /\ P(i)}` (REWRITE_TAC [ARITH_RULE `i:num < m <=> ~(m <= i)`] THEN
    SET_TAC []) THEN
  note `FINITE {i:num | i < m}` [FINITE_NUMSEG_LT] THEN
  notetac `{i:num | i < m /\ P(i)} SUBSET {i:num | i < m}` (SET_TAC []) THEN
  note `FINITE {i:num | i < m /\ P(i)}` [FINITE_SUBSET] THEN
  ASM_MESON_TAC [FINITE_UNION;
    INFINITE]);;

(* ----- integer arithmetic *)

let max_le = prove(`
  !a:num b:num c:num. MAX a b <= c <=> a <= c /\ b <= c
  `,
  REWRITE_TAC [MAX] THEN
  MESON_TAC [LE_TRANS;
    NOT_LE;
    LT_IMP_LE]);;

let le_min = prove(`
  !a:num b:num c:num. c <= MIN a b <=> c <= a /\ c <= b
  `,
  REWRITE_TAC [MIN] THEN
  MESON_TAC [LE_TRANS;
    NOT_LE;
    LT_IMP_LE]);;

(* ----- real-number arithmetic *)

let pos_times_neg = prove(`
  !x y. &0 < x /\ y < &0 ==> x*y < &0
  `,
  MESON_TAC [REAL_NEG_GT0;
    REAL_NEG_RMUL;
    REAL_LT_MUL]);;

let neg_times_pos = prove(`
  !x y. &0 < x /\ y < &0 ==> y*x < &0
  `,
  MESON_TAC [REAL_NEG_GT0;
    REAL_NEG_LMUL;
    REAL_LT_MUL]);;

let neg_over_pos = prove(`
  !x y. &0 < x /\ y < &0 ==> y/x < &0
  `,
  MESON_TAC [real_div;
    neg_times_pos;
    REAL_LT_INV]);;

let recipflip = prove(`
  !x:real y:real.
  &0 < x /\ &0 < y
  /\ &1/x <= y ==> &1/y <= x
  `,
  MESON_TAC [REAL_MUL_SYM;
    REAL_LE_RDIV_EQ;
    REAL_LE_LDIV_EQ]);;

let divltflip = prove(`
  !x y z.
  &0 < x /\ &0 < y /\ z / y < x
  ==> z / x < y
  `,
  MESON_TAC [REAL_LT_RDIV_EQ;
    REAL_LT_LDIV_EQ;
    REAL_MUL_SYM]);;

let ltdivflip = prove(`
  !x y z.
  &0 < x /\ &0 < y /\ x < z / y
  ==> y < z / x
  `,
  MESON_TAC [REAL_LT_RDIV_EQ;
    REAL_LT_LDIV_EQ;
    REAL_MUL_SYM]);;

let eqdivflip = prove(`
  !x y z.
  &0 < x /\ &0 < y /\ x = z / y
  ==> y = z / x
  `,
  MESON_TAC [REAL_EQ_RDIV_EQ;
    REAL_EQ_LDIV_EQ;
    REAL_MUL_SYM]);;

let pos_1over = prove(`
  !x:real.
  &0 < x ==> &0 < &1 / x
  `,
  MESON_TAC [REAL_INV_POS;
    REAL_FIELD `&1 / x = inv x`]);;

let nz_1over = prove(`
  !x:real.
  ~(x = &0) ==> ~(&1 / x = &0)
  `,
  MESON_TAC [REAL_INV_NZ;
    REAL_FIELD `&1 / x = inv x`]);;

let lt_1over = prove(`
  !x:real y:real.
  &0 < x /\ x < y ==> &1 / y < &1 / x
  `,
  MESON_TAC [REAL_LT_INV2;
    REAL_FIELD `&1 / x = inv x`]);;

let le_1over = prove(`
  !x:real y:real.
  &0 < x /\ x <= y ==> &1 / y <= &1 / x
  `,
  MESON_TAC [REAL_LE_INV2;
    REAL_FIELD `&1 / x = inv x`]);;

let le_1over_reverse = prove(`
  !x:real y:real.
  &0 < x /\ &1 / x <= &1 / y ==> y <= x
  `,
  REPEAT STRIP_TAC THEN
  specialize [`x:real`] pos_1over THEN
  specialize [`&1/x`;
    `&1/y`] le_1over THEN
  ASM_MESON_TAC [REAL_FIELD `&1 / (&1 / z) = z`]);;

let mono_min = prove(`
  !x y f:real->real.
   (!a b. a <= b ==> f(a) <= f(b))
   ==> min (f(x)) (f(y)) = f(min x y)
  `,
  REPEAT STRIP_TAC THEN
  ASM_CASES_TAC `x <= y` THENL
  [REWRITE_TAC [real_min] THEN
    ASM_MESON_TAC [];
    ASSUME_TAC (UNDISCH (fst (EQ_IMP_RULE (ISPECL [`x:real`;
              `y:real`] REAL_NOT_LE)))) THEN
    ASSUME_TAC (UNDISCH (ISPECL [`y:real`;
          `x:real`] REAL_LT_IMP_LE)) THEN
    ONCE_REWRITE_TAC [REAL_MIN_SYM] THEN
    REWRITE_TAC [real_min] THEN
    ASM_MESON_TAC []]);;

(* ----- ceil *)

let ceil_pos = prove(`
  !x. &0 < x <=> &1 <= ceil(x)
  `,
  REWRITE_TAC [ceil] THEN
  MESON_TAC [REAL_ARITH `--(-- &1) = &1`;
    REAL_ARITH `&0 < x <=> --x - &1 < -- &1`;
    prove (`
      integer(-- &1)
      `,
      MESON_TAC [INTEGER_CASES]);
    REAL_FLOOR_LE;
    REAL_ARITH `
    floor(--x) <= -- &1 <=> &1 <= -- floor(--x)
    `]);;

let ceil_mono = prove(`
  !x y. x <= y ==> ceil x <= ceil y
  `,
  REWRITE_TAC [ceil] THEN
  MESON_TAC [FLOOR_MONO;
    REAL_LE_NEG2]);;

let ceil_isinteger = prove(`
  !x. integer(ceil x)
  `,
  REWRITE_TAC [ceil] THEN
  MESON_TAC [FLOOR;
    INTEGER_NEG]);;

let ceil_ge = prove(`
  !x. x <= ceil(x)
  `,
  REWRITE_TAC [ceil] THEN
  MESON_TAC [FLOOR;
    REAL_LE_NEG2;
    REAL_NEG_NEG]);;

let ceil_lt_plus1 = prove(`
  !x. ceil(x) < x + &1
  `,
  REWRITE_TAC [ceil] THEN
  MESON_TAC [FLOOR;
    REAL_LT_NEG2;
    REAL_ARITH `--x < y + &1 ==> --y < x + &1`]);;

(* ----- continuity *)

(* copied from CONT_MUL in Library/analysis.ml *)

(* but includes f and g explicitly for easy specialization *)

let cont_mul_fullvariables = prove(`
  !x f g. f contl x /\ g contl x ==> (\x. f(x) * g(x)) contl x
  `,
  GEN_TAC THEN
  REWRITE_TAC [CONTL_LIM] THEN
  BETA_TAC THEN
  MATCH_ACCEPT_TAC LIM_MUL);;

let cont_const_fullvariables = prove(`
  !x k. (\x. k) contl x
  `,
  REPEAT GEN_TAC THEN
  REWRITE_TAC [CONTL_LIM] THEN
  MATCH_ACCEPT_TAC LIM_CONST);;

let cont_add_fullvariables = prove(`
  !x f g. f contl x /\ g contl x ==> (\x. f(x) + g(x)) contl x
  `,
  GEN_TAC THEN
  REWRITE_TAC [CONTL_LIM] THEN
  BETA_TAC THEN
  MATCH_ACCEPT_TAC LIM_ADD);;

let cont_neg_fullvariables = prove(`
  !x f. f contl x ==> (\x. --(f(x))) contl x
  `,
  GEN_TAC THEN
  REWRITE_TAC [CONTL_LIM] THEN
  BETA_TAC THEN
  REWRITE_TAC [GSYM LIM_NEG]);;

let cont_div_fullvariables = prove(`
  !x f g. f contl x /\ g contl x /\ ~(g x = &0) ==>
        (\x. f(x) / g(x)) contl x
  `,
  GEN_TAC THEN
  REWRITE_TAC [CONTL_LIM] THEN
  BETA_TAC THEN
  MATCH_ACCEPT_TAC LIM_DIV);;

let cont_consttimes = prove(`
  !x:real c:real. (\x. c * x) contl x
  `,
  MESON_TAC [ISPECL [`x:real`;
      `\(x:real).(c:real)`;
      `\(x:real).x`] cont_mul_fullvariables;
    CONT_X;
    CONT_CONST]);;

let cont_1overcplusx = prove(`
  !x:real c:real.
  ~(c + x = &0) ==>
  (\x. &1 / (c + x)) contl x
  `,
  REPEAT GEN_TAC THEN
  MESON_TAC [ISPECL [`x:real`] CONT_X;
    ISPECL [`x:real`;
      `&1:real`] cont_const_fullvariables;
    ISPECL [`x:real`;
      `c:real`] cont_const_fullvariables;
    ISPECL [`x:real`;
      `\(x:real).(c:real)`;
      `\(x:real).x`] cont_add_fullvariables;
    ISPECL [`x:real`;
      `\(x:real).(&1:real)`;
      `\(x:real).c+x`] cont_div_fullvariables]);;

let cont_1overcplusx_at0 = prove(`
  !c:real.
  ~(c = &0) ==>
  (\x. &1 / (c + x)) contl &0
  `,
  MESON_TAC [cont_1overcplusx;
    REAL_ARITH `~(c = &0) ==> ~(c + &0 = &0)`]);;

(* ----- derivatives *)

let deriv_positive = prove(`
  !f f' a b z. a < z /\ z < b /\ (f b - f a = (b - a) * f'(z)) /\ &0 < f'(z) ==> f(a) < f(b)
  `,
  MESON_TAC [REAL_LT_TRANS;
    REAL_ARITH `a < b <=> &0 < b - a`;
    REAL_LT_MUL]);;

let deriv_negative = prove(`
  !f f' a b z. a < z /\ z < b /\ (f b - f a = (b - a) * f'(z)) /\ f'(z) < &0 ==> f(b) < f(a)
  `,
  MESON_TAC [REAL_LT_TRANS;
    REAL_ARITH `a < b <=> &0 < b - a`;
    pos_times_neg;
    REAL_ARITH `b - a < &0 ==> b < a`]);;

let deriv_positive2 = prove(`
  !f f' a b.
  (!x. a <= x /\ x <= b ==> &0 < f'(x))
  /\ (?z. a < z /\ z < b /\ (f b - f a = (b - a) * f'(z)))
  ==> f(a) < f(b)
  `,
  MESON_TAC [deriv_positive;
    REAL_LT_IMP_LE]);;

let deriv_negative2 = prove(`
  !f f' a b.
  (!x. a <= x /\ x <= b ==> f'(x) < &0)
  /\ (?z. a < z /\ z < b /\ (f b - f a = (b - a) * f'(z)))
  ==> f(b) < f(a)
  `,
  MESON_TAC [deriv_negative;
    REAL_LT_IMP_LE]);;

let deriv_positive3 = prove(`
  !f f' a b.
  a < b
  /\ (!x. a <= x /\ x <= b ==> (f diffl f'(x))(x))
  /\ (!x. a <= x /\ x <= b ==> &0 < f'(x))
  ==> f(a) < f(b)
  `,
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (ISPECL [`f:real->real`;
      `f':real->real`;
      `a:real`;
      `b:real`] MVT_ALT) THEN
  ASSUME_TAC (ISPECL [`f:real->real`;
      `f':real->real`;
      `a:real`;
      `b:real`] deriv_positive2) THEN
  ASM_MESON_TAC []);;

let deriv_negative3 = prove(`
  !f f' a b.
  a < b
  /\ (!x. a <= x /\ x <= b ==> (f diffl f'(x))(x))
  /\ (!x. a <= x /\ x <= b ==> f'(x) < &0)
  ==> f(b) < f(a)
  `,
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (ISPECL [`f:real->real`;
      `f':real->real`;
      `a:real`;
      `b:real`] MVT_ALT) THEN
  ASSUME_TAC (ISPECL [`f:real->real`;
      `f':real->real`;
      `a:real`;
      `b:real`] deriv_negative2) THEN
  ASM_MESON_TAC []);;

let deriv_negative4 = prove(`
  !f f' a b.
  a <= b
  /\ (!x. a <= x /\ x <= b ==> (f diffl f'(x))(x))
  /\ (!x. a <= x /\ x <= b ==> f'(x) < &0)
  ==> f(b) <= f(a)
  `,
  REPEAT STRIP_TAC THEN
  DISJ_CASES_TAC (ISPECL [`a:real`;
      `b:real`] REAL_LT_TOTAL) THENL
  [ASM_MESON_TAC [REAL_ARITH `f(x:real):real = f y ==> f x <= f y`];
    ASM_MESON_TAC [deriv_negative3;
      real_lt;
      REAL_LT_IMP_LE]]);;

let mvt_twosided = prove(`
  !f f' a b d.
  &0 < d
  /\ (!x. abs(x - a) < d ==> (f diffl f'(x))(x))
  /\ abs(b - a) < d
  ==> ?z. abs(z - a) <= abs(b - a) /\ (f(b) - f(a) = (b-a) * f'(z))
  `,
  REPEAT STRIP_TAC THEN
  ASM_CASES_TAC `b = a:real` THENL
  [EXISTS_TAC `a:real` THEN
    ASM_MESON_TAC [REAL_LE_REFL;
      REAL_ARITH `x - x = &0`;
      REAL_ARITH `&0 * x = &0`];
    ALL_TAC] THEN
  ASM_CASES_TAC `a < b:real` THENL
  [note `!x:real. a <= x /\ x <= b ==> (f diffl f'(x))(x)` [REAL_ARITH `abs(b - a) < d /\ a <= x /\ x <= b ==> abs(x - a) < d`] THEN
    choose `z:real` `a:real < z /\ z < b /\ (f b - f a = (b - a) * f'(z))` [MVT_ALT] THEN
    EXISTS_TAC `z:real` THEN
    ASM_MESON_TAC [REAL_ARITH `a < z /\ z < b ==> abs(z - a) <= abs(b - a)`];
    note `b < a` [REAL_ARITH `~(b = a) /\ ~(a < b) ==> b < a`] THEN
    note `!x:real. b <= x /\ x <= a ==> (f diffl f'(x))(x)` [REAL_ARITH `abs(b - a) < d /\ b <= x /\ x <= a ==> abs(x - a) < d`] THEN
    choose `z:real` `b:real < z /\ z < a /\ (f a - f b = (a - b) * f'(z))` [MVT_ALT] THEN
    EXISTS_TAC `z:real` THEN
    ASM_MESON_TAC [REAL_ARITH `b < z /\ z < a ==> abs(z - a) <= abs(b - a)`;
      REAL_ARITH `fa - fb = (b - a) * c ==> fb - fa = (a - b) * c`]]);;

(* ----- bounded sequence has convergent subsequence *)

(*
This theorem is due to 1817 Bolzano.
Weierstrass was born in 1815.

The following proof starts with the proof in 2018 Eidolon--Oman
(https://www.tandfonline.com/doi/abs/10.4169/college.math.j.48.4.288),
removes unnecessary proof-by-contradiction steps, adapts to sequences.

Define L_i = inf{s_i,s_{i+1},...}.

Case 1: For some i, L_i isn't in {s_i,s_{i+1},...}.
Then (s_i,s_{i+1},...) has a subsequence converging (down) to L_i.

Case 2: For each i, L_i is in {s_i,s_{i+1},...}.
Define f(i) = min{j in {i,i+1,...}: s_j = L_i}.
Then the infinite subsequence (s_{f(0)},s_{f(f(0)+1)},...) of s
is exactly (L_0,L_{f(0)+1},L_{f(f(0)+1)},...)
which is nondecreasing and bounded, hence converges (up) to its sup.

Q.E.D.

For comparison, in the usual "valley" proof
(which is the usual "peak" proof modulo exchanging < and >),
a "valley" is an index i for which s_i = inf {s_i,s_{i+1},...}.
That's equivalent to s_i = L_i, or in other words f(i) = i,
but the case split is defined by finitely vs. infinitely many valleys.
In the finite case, one then takes an index past all the valleys
and recursively constructs a sequence converging downwards.

There's also the "bisection" proof, which has a different flavor.

The HOL Light library has a few Bolzano variants,
but it's not clear that using them is easier
than giving a direct proof.
*)

let hasinf_union = prove(`
  !s t L.
  ~(s = {})
  /\ ~(t = {})
  /\ (s UNION t) has_inf L
  ==> L = min (inf s) (inf t)
  `,
  REPEAT STRIP_TAC THEN
  note `L:real = inf(s UNION t)` [HAS_INF_INF] THEN
  note `!x:real. x IN (s UNION t) ==> L <= x` [HAS_INF] THEN
  note `!x:real. x IN s ==> x IN (s UNION t)` [IN_UNION] THEN
  note `!x:real. x IN t ==> x IN (s UNION t)` [IN_UNION] THEN
  note `?r. !x:real. x IN s ==> r <= x` [] THEN
  note `?r. !x:real. x IN t ==> r <= x` [] THEN
  specialize [`s:real->bool`;
    `t:real->bool`] INF_UNION THEN
  ASM_MESON_TAC []);;

let hasinf_union_avoiding_finite = prove(`
  !s t L.
  ~(t = {})
  /\ (s UNION t) has_inf L
  /\ FINITE s
  /\ ~(L IN s)
  ==> t has_inf L
  `,
  REPEAT STRIP_TAC THEN
  note `s:real->bool = {} ==> s UNION t = t` [UNION_EMPTY] THEN
  proven_if `s:real->bool = {}` THEN
  note `L = min (inf s) (inf t)` [hasinf_union] THEN
  note `inf s IN s` [INF_FINITE] THEN
  note `~(L = inf s)` [] THEN
  note `L = inf t` [REAL_ARITH `L = min x y /\ ~(L = x) ==> L = y`] THEN
  note `!x:real. x IN t ==> x IN (s UNION t)` [IN_UNION] THEN
  note `!x:real. x IN (s UNION t) ==> L <= x` [HAS_INF] THEN
  ASM_MESON_TAC [HAS_INF_INF]);;

let seq_chop_range = prove(`
  !s:num->real a b.
  a <= b ==>
  {s(j) | a <= j} =
  {s(j) | a <= j /\ j < b}
  UNION {s(j) | b <= j}
  `,
  REWRITE_TAC [UNION;
    IN_ELIM_THM;
    EXTENSION] THEN
  MESON_TAC [ARITH_RULE `a <= b:num /\ b <= j ==> a <= j`;
    NOT_LT]);;

let seq_chop_hasinf = prove(`
  !s:num->real a b L.
  a <= b
  /\ {s(j) | a <= j} has_inf L
  /\ ~(L IN {s(j) | a <= j /\ j < b})
  ==> {s(j) | b <= j} has_inf L
  `,
  REPEAT STRIP_TAC THEN
  note `FINITE {j:num | j < b}` [FINITE_NUMSEG_LT] THEN
  notetac `{j:num | a <= j /\ j < b} SUBSET {j | j < b}` (SET_TAC []) THEN
  note `FINITE {j:num | a <= j /\ j < b}` [FINITE_SUBSET] THEN
  note `FINITE (IMAGE (s:num->real) {j:num | a <= j /\ j < b})` [FINITE_IMAGE] THEN
  note `{s(j:num):real | a <= j /\ j < b} = IMAGE s {j:num | a <= j /\ j < b}` [ISPECL [`s:num->real`;
      `\j:num. a <= j /\ j < b`] SIMPLE_IMAGE_GEN] THEN
  note `FINITE {s(j:num):real | a <= j /\ j < b}` [] THEN
  note `{s(j:num):real | a <= j} = {s(j) | a <= j /\ j < b} UNION {s(j) | b <= j}` [seq_chop_range] THEN
  notetac `~({s(j:num):real | b <= j} = {})` (SET_TAC [ARITH_RULE `b:num <= b`]) THEN
  note `{s(j:num):real | a <= j /\ j < b} UNION {s j | b <= j} has_inf L` [] THEN
  specialize [`{s(j:num):real | a <= j /\ j < b}`;
    `{s(j:num):real | b <= j}`;
    `L:real`] hasinf_union_avoiding_finite THEN
  ASM_MESON_TAC []);;

let seq_chop_hasinf2 = prove(`
  !s:num->real a b L.
  a <= b
  /\ {s(j) | a <= j} has_inf L
  /\ ~(L IN {s(j) | a <= j})
  ==> {s(j) | b <= j} has_inf L
  `,
  REPEAT STRIP_TAC THEN
  notetac `~(L IN {s(j:num):real | a <= j}) ==> ~(L IN {s(j) | a <= j /\ j < b})` (SET_TAC []) THEN
  ASM_MESON_TAC [seq_chop_hasinf]);;

let seq_chop_hasinf_approach = prove(`
  !s:num->real a L c.
  {s(j) | a <= j} has_inf L
  /\ ~(L IN {s(j) | a <= j})
  ==> ?k. c < k /\ s(k) < L + &1 / &(c+1)
  `,
  REPEAT STRIP_TAC THEN
  def `b:num` `MAX a (c+1)` THEN
  notetac `a <= MAX a (c+1)` ARITH_TAC THEN
  note `a:num <= b` [] THEN
  note `{s(j:num) | b <= j} has_inf L` [seq_chop_hasinf2] THEN
  notetac `c < MAX a (c+1)` ARITH_TAC THEN
  note `c:num < b` [] THEN
  notetac `0 < c + 1` ARITH_TAC THEN
  note `&0 < &(c+1)` [REAL_OF_NUM_LT] THEN
  note `&0 < &1 / &(c+1)` [pos_1over] THEN
  real_linear `&0 < &1 / &(c+1) ==> L < L + &1 / &(c+1)` THEN
  choose_specializing `x:real` [`{s(j:num):real | b <= j}`;
    `L:real`;
    `L + &1 / &(c+1)`] HAS_INF_APPROACH THEN
  notetac `x IN {s(j:num):real | b <= j} ==> ?j. b <= j /\ x = s(j)` (SET_TAC []) THEN
  ASM_MESON_TAC [LTE_TRANS]);;

let seq_increasing_jump = prove(`
  !f:num->num.
  (!n. f(n) < f(n+1))
  ==> !m n. m < n ==> f(m) < f(n)
  `,
  GEN_TAC THEN
  DISCH_TAC THEN
  GEN_TAC THEN
  INDUCT_TAC THENL
  [ARITH_TAC;
    REWRITE_TAC [ARITH_RULE `SUC n = n + 1`] THEN
    DISCH_TAC THEN
    ASM_CASES_TAC `m < n:num` THENL
    [note `f(m) < f(n:num):num` [] THEN
      ASM_MESON_TAC [LT_TRANS];
      note `m = n:num` [ARITH_RULE `m < n + 1 /\ ~(m < n) ==> m = n`] THEN
      ASM_MESON_TAC []]]);;

let seq_increasing_jump_le = prove(`
  !f:num->num.
  (!n. f(n) < f(n+1))
  ==> !m n. m <= n ==> f(m) <= f(n)
  `,
  REPEAT STRIP_TAC THEN
  note `f(m) <= f(m:num):num` [LE_REFL] THEN
  proven_if `m = n:num` THEN
  note `m < n:num` [ARITH_RULE `m <= n /\ ~(m = n) ==> m < n:num`] THEN
  ASM_MESON_TAC [seq_increasing_jump;
    LT_IMP_LE]);;

let seq_increasing_gediagonal = prove(`
  !f:num->num.
  (!n. f(n) < f(n+1))
  ==> !n. n <= f(n)
  `,
  GEN_TAC THEN
  DISCH_TAC THEN
  INDUCT_TAC THENL
  [ARITH_TAC;
    REWRITE_TAC [ARITH_RULE `SUC n = n + 1`] THEN
    ASM_MESON_TAC [ARITH_RULE `n <= x /\ x < y ==> n + 1 <= y`]]);;

let iterate_gtdiagonal = prove(`
  !f:num->num g:num->num a.
  (!n. n < f(n)) /\
  g = (\n. (f pow_num n)(a))
  ==> !n. g(n) < g(n+1)
  `,
  REPEAT STRIP_TAC THEN
  note `g(n) = (f pow_num n)(a:num)` [] THEN
  note `g(n+1) = (f pow_num (n+1))(a:num)` [] THEN
  notetac `~(n+1 = 0)` ARITH_TAC THEN
  notetac `(n+1)-1 = n` ARITH_TAC THEN
  specialize [`f:num->num`;
    `n+1`] (GENL [`p:A->A`;
      `n:num`] pow_num) THEN
  note `(f:num->num) pow_num (n+1) = f o (f pow_num n)` [] THEN
  note `(f o (f pow_num n))(a:num):num = f((f pow_num n)(a))` [o_THM] THEN
  note `g(n+1) = f(g(n:num)):num` [] THEN
  ASM_MESON_TAC []);;

let iterate_gtdiagonal_jump = prove(`
  !f:num->num g:num->num a.
  (!n. n < f(n)) /\
  g = (\n. (f pow_num n)(a))
  ==> !m n. m < n ==> g(m) < g(n)
  `,
  MESON_TAC [iterate_gtdiagonal;
    seq_increasing_jump]);;

let iterate_gtdiagonal_gediagonal = prove(`
  !f:num->num g:num->num a.
  (!n. n < f(n)) /\
  g = (\n. (f pow_num n)(a))
  ==> !n. n <= g(n)
  `,
  MESON_TAC [iterate_gtdiagonal;
    seq_increasing_gediagonal]);;

let iterate_gtdiagonal_gestart = prove(`
  !f:num->num g:num->num a.
  (!n. n < f(n)) /\
  g = (\n. (f pow_num n)(a))
  ==> !n. a <= g(n)
  `,
  REPEAT STRIP_TAC THEN
  note `!n:num. g(n):num < g(n+1)` [iterate_gtdiagonal] THEN
  notetac `0 <= n:num` ARITH_TAC THEN
  note `g(0) <= g(n):num` [seq_increasing_jump_le] THEN
  note `g(0) = I(a:num)` [pow_num] THEN
  note `g(0) = a:num` [I_THM] THEN
  ASM_MESON_TAC []);;

let seq_inf_outside_lemma = prove(`
  !s:num->real a L r:num->num p:num->num m i.
  (!c. s(r(c)) < L + &1 / &(c+1))
  /\ (!n. n < r(n))
  /\ ~(m = 0)
  /\ m <= i
  ==> s((r pow_num i)(a)) < L + &1 / &m
  `,
  REPEAT STRIP_TAC THEN
  note `~(i = 0)` [ARITH_RULE `~(m = 0) /\ m <= i ==> ~(i = 0)`] THEN
  note `(r:num->num) pow_num i = r o (r pow_num(i-1))` [pow_num] THEN
  note `(r pow_num i)(a:num) = r((r pow_num(i-1))(a))` [o_THM] THEN
  note `i-1 <= (r pow_num(i-1))(a:num)` [iterate_gtdiagonal_gediagonal] THEN
  note `m <= (r pow_num(i-1))(a:num)+1` [ARITH_RULE `~(i = 0) /\ i-1 <= x /\ m <= i ==> m <= x+1`] THEN
  note `&m <= &((r pow_num(i-1))(a:num)+1)` [REAL_OF_NUM_LE] THEN
  note `0 < m` [ARITH_RULE `~(m = 0) ==> 0 < m`] THEN
  note `&0 < &m` [REAL_OF_NUM_LT] THEN
  note `&1 / &((r pow_num(i-1))(a:num)+1) <= &1 / &m` [le_1over] THEN
  note `L + &1 / &((r pow_num(i-1))(a:num)+1) <= L + &1 / &m` [REAL_LE_LADD] THEN
  ASM_MESON_TAC [REAL_LTE_TRANS]);;

let seq_inf_outside = prove(`
  !s:num->real a L.
  {s(j) | a <= j} has_inf L
  /\ ~(L IN {s(j) | a <= j})
  ==> ?p:num->num.
      (!i j. i < j ==> p(i) < p(j))
      /\ o1_seq (\i. s(p(i)) - L)
  `,
  REPEAT STRIP_TAC THEN
  note `!x:real. x IN {s(j:num) | a <= j} ==> L <= x` [HAS_INF] THEN
  notetac `!j. a <= j ==> s(j) IN {s(j:num):real | a <= j}` (SET_TAC []) THEN
  note `!j:num. a <= j ==> L <= s(j):real` [] THEN
  def `r:num->num` `\c. minimal k. c < k /\ s(k) < L + &1 / &(c+1)` THEN
  note `!c. ?k. c < k /\ s(k) < L + &1 / &(c+1)` [seq_chop_hasinf_approach] THEN
  note `!c:num. c < r(c):num` [MINIMAL] THEN
  note `!c:num. s(r(c):num) < L + &1 / &(c+1)` [MINIMAL] THEN
  def `p:num->num` `\i:num. (r pow_num i)(a:num)` THEN
  specialize [`r:num->num`;
    `p:num->num`;
    `a:num`] iterate_gtdiagonal_jump THEN
  EXISTS_TAC `p:num->num` THEN
  CONJ_TAC THENL
  [ASM_SIMP_TAC [];
    ALL_TAC] THEN
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  choose `m:num` `~(m = 0) /\ &0 < inv(&m) /\ inv(&m) < e` [REAL_ARCH_INV] THEN
  EXISTS_TAC `m:num` THEN
  REPEAT STRIP_TAC THEN
  note `a <= p(i:num):num` [iterate_gtdiagonal_gestart] THEN
  note `s(p(i:num):num) < L + &1 / &m` [seq_inf_outside_lemma] THEN
  note `abs(s(p(i:num):num)-L) <= e` [REAL_ARITH `L <= x /\ x < L + &1 / &m /\ inv(&m) < e ==> abs(x - L) <= e`] THEN
  ASM_MESON_TAC []);;

let seq_inf_inside_lemma = prove(`
  !f:num->num f1:num->num q:num->num p:num->num.
  f1 = (\a. f(a)+1)
  /\ q = (\i:num. (f1 pow_num i)(f1(0)))
  /\ p = (\i:num. q(i)-1)
  ==> !i. p(i) = f((f1 pow_num i)(0)) /\ q(i) = p(i)+1
  `,
  REPEAT GEN_TAC THEN
  STRIP_TAC THEN
  INDUCT_TAC THENL
  [note `(f1 pow_num 0)(0) = 0` [pow_num;
      I_THM] THEN
    note `(f1 pow_num 0)(f1(0)) = f1(0)` [pow_num;
      I_THM] THEN
    note `q(0) = f(0)+1` [] THEN
    note `p(0) = f(0):num` [ARITH_RULE `(f0+1)-1 = f0`] THEN
    ASM_MESON_TAC [];
    REWRITE_TAC [ARITH_RULE `SUC n = n + 1`] THEN
    notetac `~(i + 1 = 0)` ARITH_TAC THEN
    notetac `(i + 1) - 1 = i` ARITH_TAC THEN
    note `(f1:num->num) pow_num (i+1) = f1 o (f1 pow_num i)` [pow_num] THEN
    note `((f1:num->num) pow_num (i+1))(f1(0)) = f1((f1 pow_num i)(f1(0)))` [o_THM] THEN
    note `((f1:num->num) pow_num (i+1))(0) = f1((f1 pow_num i)(0))` [o_THM] THEN
    note `q(i+1) = f(q(i))+1` [] THEN
    note `q(i) = f((f1 pow_num i)(0))+1` [] THEN
    note `q(i) = f1((f1 pow_num i)(0))` [] THEN
    note `q(i) = (f1 pow_num (i+1))(0)` [] THEN
    note `q(i+1) = f((f1 pow_num (i+1))(0))+1` [] THEN
    note `p(i+1):num = f((f1 pow_num (i+1))(0))` [ARITH_RULE `(x+1)-1 = x`] THEN
    ASM_MESON_TAC []]);;

let has_inf_subset = prove(`
  !s t l m.
   s has_inf l /\ t has_inf m /\ t SUBSET s
   ==> l <= m
  `,
  MESON_TAC [SUBSET;
    HAS_INF_LE;
    REAL_ARITH `x <= x`]);;

let seq_inf_inside_lemma2 = prove(`
  !s:num->real L:num->real.
  (!a. {s(j) | a <= j} has_inf L(a))
  ==> !a b. a <= b ==> L(a) <= L(b)
  `,
  MESON_TAC [prove (`
      a:num <= b ==> {s(j):real | b <= j} SUBSET {s(j) | a <= j}
      `,
      SET_TAC [LE_TRANS]);
    has_inf_subset]);;

let seq_inf_inside = prove(`
  !s:num->real L:num->real U:real.
  (!a. {s(j) | a <= j} has_inf L(a))
  /\ (!a. L(a) IN {s(j) | a <= j})
  /\ {L(a) | a | T} has_sup U
  ==> ?p:num->num.
      (!i j. i < j ==> p(i) < p(j))
      /\ o1_seq (\i. s(p(i)) - U)
  `,
  REPEAT STRIP_TAC THEN
  notetac `(!a:num. L(a):real IN {s(j) | a <= j}) ==> !a. ?j. a <= j /\ s(j) = L(a)` (SET_TAC []) THEN
  note `!a:num. ?j. a <= j /\ s(j) = L(a):real` [] THEN
  def `f:num->num` `\a:num. minimal j. a <= j /\ s(j) = L(a):real` THEN
  note `!a:num. a <= f(a)` [MINIMAL] THEN
  note `!a:num. s(f(a):num) = L(a):real` [MINIMAL] THEN
  def `f1:num->num` `\a:num. f(a)+1` THEN
  note `!a:num. a < f1(a)` [ARITH_RULE `a <= b ==> a < b+1`] THEN
  def `q:num->num` `\i:num. (f1 pow_num i)(f1(0))` THEN
  def `p:num->num` `\i:num. q(i)-1` THEN
  EXISTS_TAC `p:num->num` THEN
  specialize [`f1:num->num`;
    `q:num->num`;
    `f1(0):num`] iterate_gtdiagonal_jump THEN
  specialize [`f1:num->num`;
    `q:num->num`;
    `f1(0):num`] iterate_gtdiagonal_gestart THEN
  note `!m n:num. m < n ==> p(m) < p(n):num` [ARITH_RULE `f10 = f0+1 /\ qm < qn /\ f10 <= qm /\ f10 <= qn ==> qm-1 < qn-1`] THEN
  note `!i. p(i):num = f((f1 pow_num i)(0))` [seq_inf_inside_lemma] THEN
  note `!i. s(p(i):num):real = L((f1 pow_num i)(0))` [] THEN
  CONJ_TAC THENL
  [ASM_MESON_TAC [];
    ALL_TAC] THEN
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  real_linear `&0 < e ==> U - e < U` THEN
  specialize [`{L(a:num):real | a | T}`;
    `U:real`;
    `U - e:real`] HAS_SUP_APPROACH THEN
  notetac `(?x:real. x IN {L(a:num) | a:num | T} /\ U - e < x) ==> ?a:num. U - e < L(a)` (SET_TAC []) THEN
  choose `a:num` `U - e < L(a:num):real` [] THEN
  EXISTS_TAC `a:num` THEN
  REPEAT STRIP_TAC THEN
  note `i <= (f1 pow_num i)(0)` [iterate_gtdiagonal_gediagonal] THEN
  note `a <= (f1 pow_num i)(0)` [LE_TRANS] THEN
  note `L(a) <= L((f1 pow_num i)(0)):real` [seq_inf_inside_lemma2] THEN
  note `U - e < s(p(i:num):num):real` [REAL_LTE_TRANS] THEN
  note `!x:real. x IN {L(a:num):real | a:num | T} ==> x <= U:real` [HAS_SUP] THEN
  notetac `!a:num. L(a):real IN {L(a) | a | T}` (SET_TAC []) THEN
  note `L((f1 pow_num i)(0)):real <= U` [] THEN
  real_linear `L((f1 pow_num i)(0)):real <= U /\ U - e < L(a) /\ L(a) <= L((f1 pow_num i)(0)) ==> abs(L((f1 pow_num i)(0)) - U) <= e` THEN
  ASM_MESON_TAC []);;

let seq_bolzano = prove(`
  !s:num->real l u.
  (!i. l <= s(i))
  /\ (!i. s(i) <= u)
  ==> ?p:num->num t.
      (!i j. i < j ==> p(i) < p(j))
      /\ o1_seq (\i. s(p(i)) - t)
  `,
  REPEAT STRIP_TAC THEN
  def `L:num->real` `\a:num. inf{s(j) | a <= j}` THEN
  notetac `!a:num b:num. a <= b ==> ~({s(j):real | a <= j} = {})` (SET_TAC []) THEN
  notetac `!a:num. a <= a` ARITH_TAC THEN
  note `!a:num. ~({s(j):real | a <= j} = {})` [] THEN
  notetac `!a:num. !x. x IN {s(j):real | a <= j} ==> ?j. a <= j /\ x = s(j)` (SET_TAC []) THEN
  note `!a:num. !x. x IN {s(j):real | a <= j} ==> l <= x` [] THEN
  note `!a:num. {s(j) | a <= j} has_inf L(a)` [HAS_INF_INF] THEN
  ASM_CASES_TAC `!a:num. L(a):real IN {s(j) | a <= j}` THENL
  [def `U:real` `sup{L(a:num):real | a | T}` THEN
    notetac `~({L(a:num):real | a | T} = {})` (SET_TAC []) THEN
    notetac `!x. x IN {L(a:num):real | a | T} ==> ?a:num. x = L(a)` (SET_TAC []) THEN
    notetac `!a. L(a) IN {s(j:num):real | a <= j} ==> ?j:num. L(a) = s(j)` (SET_TAC []) THEN
    note `!x. x IN {L(a:num):real | a | T} ==> ?j:num. x <= u` [] THEN
    note `{L(a:num):real | a | T} has_sup U` [HAS_SUP_SUP] THEN
    ASM_MESON_TAC [seq_inf_inside];
    choose `a:num` `~(L(a:num):real IN {s(j) | a <= j})` [] THEN
    ASM_MESON_TAC [seq_inf_outside]]);;

(* ----- o1_seq basics *)

let o1_seq_lt = prove(`
  !f:num->real.
  o1_seq f <=>
    !e:real. &0 < e ==>
      ?m:num.
        !i:num. m <= i ==> abs(f(i)) < e
  `,
  REWRITE_TAC [o1_seq] THEN
  MESON_TAC [REAL_LT_IMP_LE;
    REAL_ARITH `&0 < e /\ abs x <= e / &2 ==> abs x < e`;
    REAL_ARITH `&0 < e ==> &0 < e / &2`]);;

let o1_seq_use = prove(`
  !f:num->real e:real.
  &0 < e /\ o1_seq f
  ==> ?m:num. !i:num. m <= i ==> abs(f(i)) <= e
  `,
  MESON_TAC [o1_seq]);;

let o1_seq_lt_use = prove(`
  !f:num->real e:real.
  &0 < e /\ o1_seq f
  ==> ?m:num. !i:num. m <= i ==> abs(f(i)) < e
  `,
  MESON_TAC [o1_seq_lt]);;

(* sanity check: special case of tends_num_real in library *)

let o1_seq_tends = prove(`
  !f:num->real.
  o1_seq f <=> f tends_num_real &0
  `,
  REWRITE_TAC [o1_seq_lt] THEN
  REWRITE_TAC [SEQ] THEN
  REWRITE_TAC [REAL_ARITH `y - &0 = y`] THEN
  GEN_TAC THEN
  MESON_TAC [ARITH_RULE `m <= (i:num) <=> i >= m`]);;

(* another sanity check: 1 is not o(1) *)

let o1_seq_not_1 = prove(`
  ~o1_seq (\i. &1)
  `,
  REWRITE_TAC [o1_seq] THEN
  REWRITE_TAC [NOT_FORALL_THM] THEN
  EXISTS_TAC `&1 / &2` THEN
  ASM_MESON_TAC [EQ_IMP_LE;
    REAL_ARITH `&0 < &1 / &2`;
    REAL_ARITH `~(abs(&1) <= &1 / &2)`]);;

(* i |-> 1/i is o(1) *)

let inv_is_o1_seq = prove(`
  o1_seq (\i. inv(&i))
  `,
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  X_CHOOSE_TAC `n:num` (UNDISCH (fst (EQ_IMP_RULE (ISPEC `e:real` REAL_ARCH_INV)))) THEN
  EXISTS_TAC `n:num` THEN
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (UNDISCH (ITAUT `~(n = 0) /\ &0 < inv (&n) /\ inv (&n) < e ==> ~(n = 0)`)) THEN
  ASSUME_TAC (UNDISCH (ITAUT `~(n = 0) /\ &0 < inv (&n) /\ inv (&n) < e ==> inv(&n) < e`)) THEN
  specialize [] (ARITH_RULE `~(n = 0) ==> 0 < n`) THEN
  specialize [] (ARITH_RULE `0 < n ==> n <= i ==> 0 < i`) THEN
  specialize_reverse [`0:num`;
    `i:num`] REAL_OF_NUM_LT THEN
  specialize_reverse [`0:num`;
    `n:num`] REAL_OF_NUM_LT THEN
  specialize_reverse [`n:num`;
    `i:num`] REAL_OF_NUM_LE THEN
  specialize [`&n`;
    `&i`] REAL_LE_INV2 THEN
  specialize [`&i`] REAL_LT_INV THEN
  specialize [`&0`;
    `inv(&i)`] REAL_LT_IMP_LE THEN
  specialize_reverse [`inv(&i)`] REAL_ABS_REFL THEN
  specialize [`inv(&i)`;
    `inv(&n)`;
    `e:real`] REAL_LET_TRANS THEN
  specialize [`inv(&i)`;
    `e:real`] REAL_LT_IMP_LE THEN
  ASM_SIMP_TAC []);;

let o1_seq_past = prove(`
  !f:num->real g:num->real X:num.
  (!i. X <= i ==> f(i) = g(i))
  /\ o1_seq f
  ==> o1_seq g
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  choose_specializing `Y:num` [`f:num->real`;
    `e:real`] o1_seq_use THEN
  EXISTS_TAC `MAX X Y` THEN
  REWRITE_TAC [max_le] THEN
  ASM_MESON_TAC []);;

let o1_seq_subseq = prove(`
  !f:num->real p:num->num.
  o1_seq f
  /\ (!i j. i < j ==> p(i) < p(j))
  ==> o1_seq (f o p)
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  choose_specializing `m:num` [`f:num->real`;
    `e:real`] o1_seq_use THEN
  EXISTS_TAC `m:num` THEN
  REWRITE_TAC [o_THM] THEN
  note `!i:num. i <= p(i)` [seq_increasing_gediagonal;
    ARITH_RULE `n < n + 1`] THEN
  ASM_MESON_TAC [LE_TRANS]);;

let o1_seq_0 = prove(`
  o1_seq (\i. &0)
  `,
  REWRITE_TAC [o1_seq;
    REAL_ABS_0] THEN
  MESON_TAC [REAL_LT_IMP_LE]);;

let limit_above_is_eventually_above = prove(`
  !f b c.
  b < c /\ o1_seq (\i. f(i) - c)
  ==> ?m. !i. m <= i ==> b < f(i)
  `,
  REPEAT STRIP_TAC THEN
  real_linear `b < c ==> &0 < c - b` THEN
  choose_specializing `A:num` [`\i:num. f(i)-c:real`;
    `c-b:real`] o1_seq_lt_use THEN
  EXISTS_TAC `A:num` THEN
  ASM_MESON_TAC [REAL_ARITH `b < c /\ abs(x - c) < c - b ==> b < x`]);;

let positive_limit_is_eventually_positive = prove(`
  !f c.
  &0 < c /\ o1_seq (\i. f(i) - c)
  ==> ?m. !i. m <= i ==> &0 < f(i)
  `,
  MESON_TAC [limit_above_is_eventually_above]);;

let limit_below_is_eventually_below = prove(`
  !f c d.
  c < d /\ o1_seq (\i. f(i) - c)
  ==> ?m. !i. m <= i ==> f(i) < d
  `,
  REPEAT STRIP_TAC THEN
  real_linear `c < d ==> &0 < d - c` THEN
  choose_specializing `A:num` [`\i:num. f(i)-c:real`;
    `d-c:real`] o1_seq_lt_use THEN
  EXISTS_TAC `A:num` THEN
  ASM_MESON_TAC [REAL_ARITH `c < d /\ abs(x - c) < d - c ==> x < d`]);;

let negative_limit_is_eventually_negative = prove(`
  !f c.
  c < &0 /\ o1_seq (\i. f(i) - c)
  ==> ?m. !i. m <= i ==> f(i) < &0
  `,
  MESON_TAC [limit_below_is_eventually_below]);;

(* alternative: track l <= t and t <= u through bolzano proofs *)

let seq_bolzano2 = prove(`
  !s:num->real l u.
  (!i. l <= s(i))
  /\ (!i. s(i) <= u)
  ==> ?p:num->num t.
      (!i j. i < j ==> p(i) < p(j))
      /\ o1_seq (\i. s(p(i)) - t)
      /\ l <= t
      /\ t <= u
  `,
  REPEAT STRIP_TAC THEN
  specialize [`s:num->real`;
    `l:real`;
    `u:real`] seq_bolzano THEN
  choose `p:num->num` `?t. (!i j. i < j:num ==> p(i) < p(j):num) /\ o1_seq (\i. s(p(i)) - t)` [] THEN
  choose `t:real` `(!i j. i < j:num ==> p(i) < p(j):num) /\ o1_seq (\i. s(p(i)) - t)` [] THEN
  note `o1_seq (\i:num. s(p(i):num) - t:real)` [] THEN
  EXISTS_TAC `p:num->num` THEN
  EXISTS_TAC `t:real` THEN
  ASM_CASES_TAC `t < l:real` THENL
  [specialize [`\i:num. s(p(i):num):real`;
      `t:real`;
      `l:real`] limit_below_is_eventually_below THEN
    ASM_MESON_TAC [LE_REFL;
      REAL_NOT_LE];
    ALL_TAC] THEN
  ASM_CASES_TAC `u < t:real` THENL
  [specialize [`\i:num. s(p(i):num):real`;
      `u:real`;
      `t:real`] limit_above_is_eventually_above THEN
    ASM_MESON_TAC [LE_REFL;
      REAL_NOT_LE];
    ALL_TAC] THEN
  ASM_MESON_TAC [REAL_NOT_LT]);;

let o1_seq_compose0 = prove(`
  !f:num->real g:real->real.
  g contl &0 /\ o1_seq f
  ==> o1_seq (\i. g(f(i)) - g(&0))
  `,
  REWRITE_TAC [o1_seq_lt] THEN
  REWRITE_TAC [CONTL] THEN
  REWRITE_TAC [REAL_ARITH `y - &0 = y`] THEN
  MESON_TAC []);;

let o1_seq_compose = prove(`
  !f:num->real g:real->real c:real.
  g contl c /\ o1_seq f
  ==> o1_seq (\i. g(c+f(i)) - g(c))
  `,
  REWRITE_TAC [o1_seq_lt] THEN
  REWRITE_TAC [CONTL] THEN
  REWRITE_TAC [REAL_ARITH `abs(f(i:num)) = abs((c+f(i))-c)`] THEN
  MESON_TAC []);;

let o1_seq_compose_offset = prove(`
  !f:num->real g:real->real c:real.
  g contl c /\ o1_seq (\i. f(i)-c)
  ==> o1_seq (\i. g(f(i)) - g(c))
  `,
  REWRITE_TAC [o1_seq_lt] THEN
  REWRITE_TAC [CONTL] THEN
  MESON_TAC []);;

let o1_seq_consttimes_0sub = prove(`
  !f:num->real c:real.
  o1_seq f
  ==> o1_seq (\i. c*f(i) - c * &0)
  `,
  MESON_TAC [ISPECL [`&0:real`;
      `c:real`] cont_consttimes;
    ISPECL [`f:num->real`;
      `\(x:real). c*x`] o1_seq_compose0]);;

let o1_seq_consttimes_rewrite = prove(`
  !f:num->real c:real.
  (\i. c*f(i)) = (\i. c*f(i) - c * &0)
  `,
  REWRITE_TAC [REAL_ARITH `x - c * &0 = x`]);;

let o1_seq_consttimes = prove(`
  !f:num->real c:real.
  o1_seq f
  ==> o1_seq (\i. c*f(i))
  `,
  MESON_TAC [o1_seq_consttimes_rewrite;
    o1_seq_consttimes_0sub]);;

let o1_seq_add_lemma_half = prove(`
  !o1:num->real.
  (!e:real. &0 < e ==> (?m:num. !i:num. m <= i ==> abs (o1 i) <= e))
  /\ &0 < e / &2
  ==> (?m:num. !i:num. m <= i ==> abs (o1 i) <= e / &2)
  `,
  MESON_TAC []);;

let o1_seq_add_lemma = prove(`
  !o1 e:real.
  o1_seq o1 ==> &0 < e
  ==> ?m. !i. m <= i ==> abs (o1 i) <= e / &2
  `,
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  specialize [] (REAL_ARITH `&0 < e ==> &0 < e / &2`) THEN
  specialize [`o1:num->real`] o1_seq_add_lemma_half THEN
  ASM_MESON_TAC []);;

let o1_seq_add_lemma2 = prove(`
  !Ao1 Bo1 e:real Am:num Bm:num.
  (!i. Am <= i ==> abs (Ao1 i) <= e / &2)
  /\
  (!i. Bm <= i ==> abs (Bo1 i) <= e / &2)
  ==> ?Cm. !i. Cm <= i ==> abs (Ao1 i) <= e / &2 /\ abs(Bo1 i) <= e / &2
  `,
  REPEAT STRIP_TAC THEN
  EXISTS_TAC `MAX Am Bm` THEN
  REWRITE_TAC [max_le] THEN
  ASM_MESON_TAC []);;

let o1_seq_add_lemma3 = prove(`
  !x:real y:real e:real.
  abs(x) <= e / &2 /\ abs(y) <= e / &2
  ==> abs(x + y) <= e
  `,
  MESON_TAC [REAL_ABS_TRIANGLE;
    REAL_LE_TRANS;
    REAL_LE_ADD2;
    REAL_ARITH `e / &2 + e / &2 = e`]);;

let o1_seq_add_lemma4 = prove(`
  !Ao1 Bo1 e:real Am:num Bm:num.
  (!i. Am <= i ==> abs (Ao1 i) <= e / &2)
  /\
  (!i. Bm <= i ==> abs (Bo1 i) <= e / &2)
  ==> ?Cm. !i. Cm <= i ==> abs (Ao1 i + Bo1 i) <= e
  `,
  REPEAT STRIP_TAC THEN
  specialize [`Ao1:num->real`;
    `Bo1:num->real`;
    `e:real`;
    `Am:num`;
    `Bm:num`] o1_seq_add_lemma2 THEN
  ASM_MESON_TAC [o1_seq_add_lemma3]);;

let o1_seq_add_lemma5 = prove(`
  !Ao1 Bo1 e:real.
  o1_seq Ao1 /\ o1_seq Bo1
  /\ &0 < e
  ==> ?m. !i. m <= i ==> abs (Ao1 i + Bo1 i) <= e
  `,
  REPEAT STRIP_TAC THEN
  specialize [`Ao1:num->real`;
    `e:real`] o1_seq_add_lemma THEN
  specialize [`Bo1:num->real`;
    `e:real`] o1_seq_add_lemma THEN
  ASM_MESON_TAC [o1_seq_add_lemma4]);;

let o1_seq_add_extra = prove(`
  !Ao1 Bo1.
  o1_seq Ao1 /\ o1_seq Bo1
  /\ (!i. Co1(i) = Ao1(i) + Bo1(i))
  ==> o1_seq Co1
  `,
  MESON_TAC [o1_seq_add_lemma5;
    o1_seq]);;

let o1_seq_add = prove(`
  !Ao1 Bo1.
  o1_seq Ao1 /\ o1_seq Bo1
  ==> o1_seq (\i. Ao1(i) + Bo1(i))
  `,
  MESON_TAC [o1_seq_add_extra]);;

let o1_seq_add_offset_lemma = prove(`
  !Ao1 A Bo1 B.
  (\i. (Ao1(i) - A) + (Bo1(i) - B))
  =
  (\i. (Ao1(i) + Bo1(i)) - (A + B))
  `,
  MESON_TAC [EQ_EXT;
    REAL_ARITH `(x - a) + (y - b) = (x + y) - (a + b)`]);;

let o1_seq_add_offset = prove(`
  !Ao1 A Bo1 B.
  o1_seq (\i. Ao1(i) - A)
  /\ o1_seq (\i. Bo1(i) - B)
  ==> o1_seq (\i. (Ao1(i)+Bo1(i)) - (A+B))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. Ao1(i) - A`;
    `\i:num. Bo1(i) - B`] o1_seq_add THEN
  specialize [`Ao1:num->real`;
    `A:real`;
    `Bo1:num->real`;
    `B:real`] o1_seq_add_offset_lemma THEN
  ASM_MESON_TAC []);;

let o1_seq_sub = prove(`
  !Ao1 Bo1.
  o1_seq Ao1 /\ o1_seq Bo1
  ==> o1_seq (\i. Ao1(i) - Bo1(i))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`Bo1:num->real`;
    `-- &1`] o1_seq_consttimes THEN
  specialize [`Ao1:num->real`;
    `\i:num. (-- &1) * Bo1(i)`] o1_seq_add THEN
  ONCE_REWRITE_TAC [REAL_ARITH `x - y = x + (-- &1) * y`] THEN
  ASM_MESON_TAC []);;

let o1_seq_sub_offset_lemma = prove(`
  !Ao1 A Bo1 B.
  (\i. (Ao1(i) - A) - (Bo1(i) - B))
  =
  (\i. (Ao1(i) - Bo1(i)) - (A - B))
  `,
  MESON_TAC [EQ_EXT;
    REAL_ARITH `(x - a) - (y - b) = (x - y) - (a - b)`]);;

let o1_seq_sub_offset = prove(`
  !Ao1 A Bo1 B.
  o1_seq (\i. Ao1(i) - A)
  /\ o1_seq (\i. Bo1(i) - B)
  ==> o1_seq (\i. (Ao1(i)-Bo1(i)) - (A-B))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. Ao1(i) - A`;
    `\i:num. Bo1(i) - B`] o1_seq_sub THEN
  specialize [`Ao1:num->real`;
    `A:real`;
    `Bo1:num->real`;
    `B:real`] o1_seq_sub_offset_lemma THEN
  ASM_MESON_TAC []);;

let o1_seq_1overcplusx_plus0 = prove(`
  !f:num->real c:real.
  o1_seq f
  /\ ~(c = &0)
  ==> o1_seq (\i. &1 / (c+f(i)) - &1 / (c + &0))
  `,
  MESON_TAC [ISPECL [`c:real`] cont_1overcplusx_at0;
    ISPECL [`f:num->real`;
      `\(x:real). &1 / (c+x)`] o1_seq_compose0]);;

let o1_seq_1overcplusx_rewrite = prove(`
  !f:num->real c:real.
  (\i. &1 / (c+f(i)) - &1 / (c + &0))
  = (\i. &1 / (c+f(i)) - &1 / c)
  `,
  REWRITE_TAC [REAL_ARITH `c + &0 = c`]);;

let o1_seq_1overcplusx = prove(`
  !f:num->real c:real.
  o1_seq f
  /\ ~(c = &0)
  ==> o1_seq (\i. &1 / (c+f(i)) - &1 / c)
  `,
  MESON_TAC [o1_seq_1overcplusx_plus0;
    o1_seq_1overcplusx_rewrite]);;

let o1_seq_bounded_eventually = prove(`
  !f:num->real g:num->real m:num.
  o1_seq f
  /\ (!i. m <= i ==> abs(g(i)) <= abs(f(i)))
  ==> o1_seq g
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  choose `n:num` `!i:num. n <= i ==> abs(f(i)) <= e` [o1_seq_use] THEN
  EXISTS_TAC `MAX m n` THEN
  REWRITE_TAC [max_le] THEN
  ASM_MESON_TAC [REAL_LE_TRANS]);;

let o1_seq_mul_lemma_sqrt = prove(`
  (!e:real. &0 < e ==> (?m:num. !i:num. m <= i ==> abs (o1 i) <= e)) ==> &0 < sqrt e ==> (?m:num. !i:num. m <= i ==> abs (o1 i) <= sqrt e)
  `,
  MESON_TAC []);;

let o1_seq_mul_lemma = prove(`
  !o1 e:real.
  o1_seq o1 ==> &0 < e
  ==> ?m. !i. m <= i ==> abs (o1 i) <= sqrt e
  `,
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (UNDISCH o1_seq_mul_lemma_sqrt) THEN
  ASM_MESON_TAC [SQRT_LT_0]);;

let o1_seq_mul_lemma2 = prove(`
  !Ao1 Bo1 e:real Am:num Bm:num.
  (!i. Am <= i ==> abs (Ao1 i) <= sqrt e)
  ==>
  (!i. Bm <= i ==> abs (Bo1 i) <= sqrt e)
  ==> ?Cm. !i. Cm <= i ==> abs (Ao1 i) <= sqrt e /\ abs(Bo1 i) <= sqrt e
  `,
  REPEAT STRIP_TAC THEN
  EXISTS_TAC `MAX Am Bm` THEN
  REWRITE_TAC [max_le] THEN
  ASM_MESON_TAC []);;

let o1_seq_mul_lemma3 = prove(`
  !x:real y:real e:real. abs(x) <= sqrt e /\ abs(y) <= sqrt e ==> abs(x * y) <= e
  `,
  MESON_TAC [REAL_LE_MUL2;
    REAL_ABS_MUL;
    REAL_ABS_POS;
    REAL_POW_2;
    SQRT_WORKS_GEN;
    REAL_LE_TRANS;
    SQRT_LE_0;
    REAL_ABS_REFL]);;

let o1_seq_mul_lemma4 = prove(`
  !Ao1 Bo1 e:real Am:num Bm:num.
  (!i. Am <= i ==> abs (Ao1 i) <= sqrt e)
  /\
  (!i. Bm <= i ==> abs (Bo1 i) <= sqrt e)
  ==> ?Cm. !i. Cm <= i ==> abs (Ao1 i * Bo1 i) <= e
  `,
  REPEAT STRIP_TAC THEN
  specialize [`Ao1:num->real`;
    `Bo1:num->real`;
    `e:real`;
    `Am:num`;
    `Bm:num`] o1_seq_mul_lemma2 THEN
  ASM_MESON_TAC [o1_seq_mul_lemma3]);;

let o1_seq_mul_lemma5 = prove(`
  !Ao1 Bo1 e:real.
  o1_seq Ao1 /\ o1_seq Bo1
  /\ &0 < e
  ==> ?m. !i. m <= i ==> abs (Ao1 i * Bo1 i) <= e
  `,
  REPEAT STRIP_TAC THEN
  specialize [`Ao1:num->real`;
    `e:real`] o1_seq_mul_lemma THEN
  specialize [`Bo1:num->real`;
    `e:real`] o1_seq_mul_lemma THEN
  ASM_MESON_TAC [o1_seq_mul_lemma4]);;

let o1_seq_mul_extra = prove(`
  !Ao1 Bo1 Co1.
  o1_seq Ao1 /\ o1_seq Bo1
  /\ (!i. Co1(i) = Ao1(i) * Bo1(i))
  ==> o1_seq Co1
  `,
  MESON_TAC [o1_seq_mul_lemma5;
    o1_seq]);;

let o1_seq_mul = prove(`
  !Ao1 Bo1.
  o1_seq Ao1 /\ o1_seq Bo1
  ==> o1_seq (\i. Ao1(i) * Bo1(i))
  `,
  MESON_TAC [o1_seq_mul_extra]);;

let o1_seq_offset_mul = prove(`
  !c:real co1:num->real d:real do1:num->real.
  o1_seq co1
  /\ o1_seq do1
  ==>
  ?o1.
  o1_seq o1
  /\ !i. (c + co1(i)) * (d + do1(i)) = c*d + o1(i)
  `,
  REPEAT STRIP_TAC THEN
  EXISTS_TAC `\i:num. ((c:real)*do1(i) + d*co1(i)) + co1(i)*do1(i)` THEN
  CONJ_TAC THENL
  [ASM_MESON_TAC [ISPECL [`do1:num->real`;
        `c:real`] o1_seq_consttimes;
      ISPECL [`co1:num->real`;
        `d:real`] o1_seq_consttimes;
      ISPECL [`co1:num->real`;
        `do1:num->real`] o1_seq_mul;
      ISPECL [`\i:num. (c:real)*do1(i)`;
        `\i:num. (d:real)*co1(i)`] o1_seq_add;
      ISPECL [`\i:num. (c:real)*do1(i)+(d:real)*co1(i)`;
        `\i:num. co1(i)*do1(i)`] o1_seq_add];
    REAL_ARITH_TAC]);;

let o1_seq_offset_mul2 = prove(`
  !f:num->real c:real g:num->real d:real.
  o1_seq (\i. f(i) - c)
  /\ o1_seq (\i. g(i) - d)
  ==> o1_seq (\i. f(i)*g(i) - c*d)
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [REAL_ARITH `f*g-c*d=c*(g-d)+d*(f-c)+(f-c)*(g-d)`] THEN
  specialize [`\i:num. f(i) - c:real`;
    `d:real`] o1_seq_consttimes THEN
  specialize [`\i:num. g(i) - d:real`;
    `c:real`] o1_seq_consttimes THEN
  specialize [`\i:num. f(i) - c:real`;
    `\i:num. g(i) - d:real`] o1_seq_mul THEN
  specialize [`\i:num. d*(f(i) - c):real`;
    `\i:num. (f(i) - c:real)*(g(i) - d:real)`] o1_seq_add THEN
  specialize [`\i:num. c*(g(i) - d):real`;
    `\i:num. d*(f(i) - c)+(f(i) - c:real)*(g(i) - d:real)`] o1_seq_add THEN
  ASM_MESON_TAC []);;

(* ----- second-order asymptotics *)

let asymp2_asymp1_ratio = prove(`
  !X:num->real A:num->real A0:real A1:real.
  o1_seq X
  /\ (!i. ~(X(i) = &0))
  /\ o1_seq (\i. (A(i) - A0) / X(i) - A1)
  ==> o1_seq (\i. A(i) - A0)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. (A(i)-A0)/X(i)-A1`;
    `X:num->real`] o1_seq_mul THEN
  note_eqlambda `(\i:num. ((A(i)-A0)/X(i)-A1)*X(i)) = (\i:num. A(i)-A0-A1*X(i))` [REAL_FIELD `~(x = &0) ==> (b/x-c)*x = b-c*x`] THEN
  note `o1_seq (\i:num. A(i)-A0-A1*X(i))` [] THEN
  note `o1_seq (\i:num. A1*X(i))` [o1_seq_consttimes] THEN
  specialize [`\i:num. A(i)-A0-A1*X(i):real`;
    `\i:num. A1*X(i):real`] o1_seq_add THEN
  note_eqlambda `(\i:num. A(i)-A0-A1*X(i)+A1*X(i):real) = (\i:num. A(i)-A0)` [REAL_ARITH `b - c + c = b`] THEN
  ASM_MESON_TAC []);;

let asymp2_compose = prove(`
  !f:real->real f':real->real X:num->real A:num->real A0:real A1:real d:real.
  &0 < d
  /\ (!x. abs(x - A0) < d ==> (f diffl f'(x))(x))
  /\ f' contl A0
  /\ o1_seq X
  /\ (!i. ~(X(i) = &0))
  /\ o1_seq (\i. (A(i) - A0) / X(i) - A1)
  ==> o1_seq (\i. (f(A(i)) - f(A0)) / X(i) - A1 * f'(A0))
  `,
  REPEAT STRIP_TAC THEN
  note `o1_seq (\i. A(i) - A0)` [asymp2_asymp1_ratio] THEN
  choose `m:num` `!i:num. m <= i ==> abs(A(i) - A0) < d:real` [o1_seq_lt_use] THEN
  note `!i:num. m <= i ==> ?z. abs(z - A0) <= abs(A(i) - A0) /\ f(A(i)) - f(A0) = (A(i) - A0) * f'(z)` [mvt_twosided] THEN
  def `B:num->real` `\i:num. @z. abs(z - A0) <= abs(A(i) - A0) /\ f(A(i)) - f(A0) = (A(i) - A0) * f'(z)` THEN
  note `!i:num. m <= i ==> abs(B(i) - A0) <= abs(A(i) - A0)` [] THEN
  specialize [`\i:num. A(i)-A0:real`;
    `\i:num. B(i)-A0:real`;
    `m:num`] o1_seq_bounded_eventually THEN
  specialize [`\i:num. B(i):real`;
    `f':real->real`;
    `A0:real`] o1_seq_compose_offset THEN
  note `!i:num. m <= i ==> f(A(i)) - f(A0) = (A(i) - A0) * f'(B(i):real)` [] THEN
  specialize [`\i:num. (A(i)-A0)/X(i)`;
    `A1:real`;
    `\i:num. f'(B(i):real):real`;
    `f'(A0:real):real`] o1_seq_offset_mul2 THEN
  note `!i:num. m <= i ==> (A(i) - A0)/X(i) * f'(B(i):real) - A1*f'(A0) = (f(A(i)) - f(A0))/X(i) - A1*f'(A0)` [REAL_ARITH `(a*b)/x = (a/x)*b`] THEN
  specialize [`\i:num. (A(i) - A0)/X(i) * f'(B(i):real) - A1*f'(A0):real`;
    `\i:num. (f(A(i):real) - f(A0))/X(i) - A1*f'(A0):real`;
    `m:num`] o1_seq_past THEN
  ASM_MESON_TAC []);;

let asymp2_compose2 = prove(`
  !f:real->real f':real->real X:num->real A:num->real A0:real A1:real d:real.
  &0 < d
  /\ (!x. abs(x - A0) < d ==> (f diffl f'(x))(x))
  /\ f' contl A0
  /\ (!i. ~(X(i) = &0))
  /\ o1_seq (\i. &1 / X(i))
  /\ o1_seq (\i. (A(i) - A0) * X(i) - A1)
  ==> o1_seq (\i. (f(A(i)) - f(A0)) * X(i) - A1 * f'(A0))
  `,
  REPEAT STRIP_TAC THEN
  note `!i:num. ~(&1 / X(i) = &0)` [nz_1over] THEN
  note_eqlambda `(\i:num. (A(i) - A0) * X(i) - A1) = (\i:num. (A(i) - A0) / (&1 / X(i)) - A1)` [REAL_FIELD `a * b = a / (&1 / b)`] THEN
  note `o1_seq (\i. (A(i) - A0) / (&1 / X(i)) - A1)` [] THEN
  specialize [`f:real->real`;
    `f':real->real`;
    `\i:num. &1 / X(i)`;
    `A:num->real`;
    `A0:real`;
    `A1:real`;
    `d:real`] asymp2_compose THEN
  note_eqlambda `(\i:num. (f(A(i):real) - f(A0)) * X(i) - A1*f'(A0)) = (\i:num. (f(A(i)) - f(A0)) / (&1 / X(i)) - A1*f'(A0))` [REAL_FIELD `a * b = a / (&1 / b)`] THEN
  ASM_MESON_TAC []);;

let asymp2_inv_generic = prove(`
  !X:num->real A:num->real A0:real A1:real.
  (!i. ~(X(i) = &0))
  /\ o1_seq (\i. &1 / X(i))
  /\ ~(A0 = &0)
  /\ o1_seq (\i. (A(i) - A0) * X(i) - A1)
  ==> o1_seq (\i. (&1/A(i) - &1/A0) * X(i) - (--(A1)/(A0*A0)))
  `,
  REPEAT STRIP_TAC THEN
  def `d:real` `abs(A0):real` THEN
  note `&0 < d:real` [REAL_ARITH `~(x = &0) ==> &0 < abs(x)`] THEN
  note `!x. abs(x - A0) < d ==> ~(x = &0)` [REAL_ARITH `abs(x - c) < abs(c) ==> ~(x = &0)`] THEN
  note_eqlambda `(\x. inv x) = (\x. &1 / x)` [REAL_ARITH `inv x = &1 / x`] THEN
  note `!x. ~(x = &0) ==> ((\x. &1 / x) diffl --(&1 / (x*x))) x` [DIFF_XM1;
    REAL_FIELD `inv x pow 2 = &1 / (x*x)`] THEN
  note `(\x. x) contl A0` [CONT_X] THEN
  specialize [`A0:real`;
    `\x:real. x`;
    `\x:real. x`] cont_mul_fullvariables THEN
  note `(\x. x*x) contl A0` [CONT_MUL] THEN
  specialize [`A0:real`;
    `&1`] cont_const_fullvariables THEN
  note `~(A0*A0 = &0)` [REAL_LT_SQUARE;
    REAL_ARITH `&0 < x ==> ~(x = &0)`] THEN
  specialize [`A0:real`;
    `\x:real. &1`;
    `\x:real. x*x`] cont_div_fullvariables THEN
  specialize [`A0:real`;
    `\x:real. &1 / (x*x)`] cont_neg_fullvariables THEN
  note `!x. abs(x - A0) < d ==> ((\x. &1 / x) diffl --(&1 / (x*x))) x` [] THEN
  specialize [`\x. &1 / x`;
    `\x. --(&1/(x*x))`;
    `X:num->real`;
    `A:num->real`;
    `A0:real`;
    `A1:real`;
    `d:real`] asymp2_compose2 THEN
  REWRITE_TAC [REAL_FIELD `--A1/(A0*A0) = A1 * --(&1 / (A0*A0))`] THEN
  ASM_MESON_TAC []);;

let sqrt_cont = prove(`
  !x:real.
  &0 < x ==> sqrt contl x
  `,
  MESON_TAC [DIFF_SQRT;
    DIFF_CONT]);;

let asymp2_sqrt_generic = prove(`
  !X:num->real A:num->real A0:real A1:real.
  (!i. ~(X(i) = &0))
  /\ o1_seq (\i. &1 / X(i))
  /\ &0 < A0
  /\ o1_seq (\i. (A(i) - A0) * X(i) - A1)
  ==> o1_seq (\i. (sqrt(A(i)) - sqrt(A0)) * X(i) - (A1/(&2*sqrt(A0))))
  `,
  REPEAT STRIP_TAC THEN
  note `!x. abs(x - A0) < A0 ==> &0 < x` [REAL_ARITH `abs(x - c) < c ==> &0 < x`] THEN
  note `!x. abs(x - A0) < A0 ==> (sqrt diffl (&1 / (&2 * sqrt(x)))) x` [DIFF_SQRT;
    REAL_ARITH `inv x = &1 / x`] THEN
  note `sqrt contl A0` [sqrt_cont] THEN
  specialize [`A0:real`;
    `&2`] cont_const_fullvariables THEN
  specialize [`A0:real`;
    `\x:real. &2`;
    `sqrt`] cont_mul_fullvariables THEN
  specialize [`A0:real`;
    `&1`] cont_const_fullvariables THEN
  note `&0 < sqrt A0` [SQRT_LT_0] THEN
  note `~(&2 * sqrt A0 = &0)` [REAL_ARITH `&0 < x ==> ~(&2 * x = &0)`] THEN
  specialize [`A0:real`;
    `\x:real. &1`;
    `\x:real. &2 * sqrt x`] cont_div_fullvariables THEN
  specialize [`sqrt`;
    `\x. &1 / (&2 * sqrt(x))`;
    `X:num->real`;
    `A:num->real`;
    `A0:real`;
    `A1:real`;
    `A0:real`] asymp2_compose2 THEN
  ONCE_REWRITE_TAC [REAL_FIELD `A1/(&2 * sqrt(A0)) = A1 * &1 / (&2 * sqrt A0)`] THEN
  ASM_MESON_TAC []);;

(* ----- exp *)

let exp_double = prove(`
  !x. exp(&2 * x) = exp(x) * exp(x)
  `,
  MESON_TAC [REAL_EXP_ADD;
    REAL_ARITH `&2 * x = x + x`]);;

let exp_double2 = prove(`
  !x. exp(&2 * x) = exp(x) pow 2
  `,
  MESON_TAC [exp_double;
    REAL_POW_2]);;

let exp_lt_abs = prove(`
  !x y. exp(x) < y ==> abs(exp(x)) < abs(y)
  `,
  MESON_TAC [REAL_EXP_POS_LT;
    REAL_LT_TRANS;
    REAL_ABS_REFL;
    REAL_LT_IMP_LE]);;

let exp_lt_sq = prove(`
  !x y. exp(x) < y ==> exp(&2 * x) < y pow 2
  `,
  MESON_TAC [exp_double2;
    exp_lt_abs;
    REAL_LT_SQUARE_ABS]);;

let exp_min = prove(`
  !x y.
   min (exp(x)) (exp(y)) = exp(min x y)
  `,
  MESON_TAC [mono_min;
    REAL_EXP_MONO_LE]);;

(* ----- ln *)

(* better version of LN_MONO_LT *)

let ln_mono_lt = prove(`
  !x y. &0 < x /\ x < y ==> ln x < ln y
  `,
  MESON_TAC [LN_MONO_LT;
    REAL_ARITH `!x y. &0 < x /\ x < y ==> &0 < y`]);;

(* better version of LN_MONO_LE *)

let ln_mono_le = prove(`
  !x y. &0 < x /\ x <= y ==> ln x <= ln y
  `,
  MESON_TAC [LN_MONO_LE;
    REAL_ARITH `!x y. &0 < x /\ x <= y ==> &0 < y`]);;

(* like library LN_DIV but bind y *)

let ln_div = prove(`
  !x y. &0 < x /\ &0 < y ==> ln(x / y) = ln x - ln y
  `,
  MESON_TAC [LN_DIV]);;

let ln_muldiv = prove(`
  !x y z. &0 < x /\ &0 < y /\ &0 < z ==> ln(x*y/z) = ln x + ln y - ln z
  `,
  MESON_TAC [REAL_LT_DIV;
    LN_MUL;
    LN_DIV]);;

let ln_1over = prove(`
  !x:real.
  &0 < x ==> ln(&1 / x) = --ln(x)
  `,
  MESON_TAC [LN_INV;
    REAL_FIELD `&1 / x = inv x`]);;

let ln_1over_plus = prove(`
  !x.
  &0 < x ==>
  ln(&1 / x) + ln(x) = &0
  `,
  MESON_TAC [ln_1over;
    REAL_ARITH `ln(&1 / x) = --ln(x) ==> ln(&1 / x) + ln(x) = &0`]);;

let ln_gt1 = prove(`
  !y. &1 < y ==> &0 < ln y
  `,
  MESON_TAC [ln_mono_lt;
    REAL_ARITH `&0 < &1`;
    LN_1]);;

let ln_ge1 = prove(`
  !y. &1 <= y ==> &0 <= ln y
  `,
  MESON_TAC [ln_mono_le;
    REAL_ARITH `&0 < &1`;
    LN_1]);;

let ln_le1 = prove(`
  !y. &0 < y /\ y <= &1 ==> ln y <= &0
  `,
  MESON_TAC [ln_mono_le;
    LN_1]);;

let ln_continuous = prove(`
  !x. &0 < x ==> ln contl x
  `,
  MESON_TAC [DIFF_LN;
    DIFF_CONT]);;

(* ----- ln(2) *)

let ln2_positive = prove(`
  &0 < ln(&2)
  `,
  MESON_TAC [LN_POS_LT;
    REAL_ARITH `&1 < &2`]);;

let ln2_nonzero = prove(`
  ~(ln(&2) = &0)
  `,
  MESON_TAC [ln2_positive;
    REAL_LT_IMP_NZ]);;

(* ----- log2 *)

let log2ratio_lnratio = prove(`
  !x y. log2(x)/log2(y) = ln(x)/ln(y)
  `,
  REWRITE_TAC [log2] THEN
  MESON_TAC [ln2_nonzero;
    REAL_FIELD `~(d = &0) ==> (a/d)/(b/d) = a/b`]);;

let log2_2 = prove(`
  log2(&2) = &1
  `,
  MESON_TAC [log2;
    REAL_DIV_REFL;
    ln2_nonzero]);;

let log2_1 = prove(`
  log2(&1) = &0
  `,
  MESON_TAC [log2;
    LN_1;
    REAL_FIELD `&0 / y = &0`]);;

let log2_mono_le = prove(`
  !x y. &0 < x /\ x <= y ==> log2 x <= log2 y
  `,
  REWRITE_TAC [log2] THEN
  MESON_TAC [REAL_LE_DIV2_EQ;
    LN_MONO_LE;
    ln2_positive;
    REAL_ARITH `!x y. &0 < x /\ x <= y ==> &0 < y`]);;

let log2_mono_lt = prove(`
  !x y. &0 < x /\ x < y ==> log2 x < log2 y
  `,
  REWRITE_TAC [log2] THEN
  MESON_TAC [REAL_LT_DIV2_EQ;
    LN_MONO_LT;
    ln2_positive;
    REAL_ARITH `!x y. &0 < x /\ x < y ==> &0 < y`]);;

let log2_mul = prove(`
  !x y. &0 < x /\ &0 < y ==> log2(x*y) = log2(x) + log2(y)
  `,
  REWRITE_TAC [log2] THEN
  MESON_TAC [LN_MUL;
    ln2_positive;
    REAL_FIELD `&0 < L ==> (a+b)/L = a/L + b/L`]);;

let log2_ge2 = prove(`
  !x. &2 <= x ==> &1 <= log2 x
  `,
  MESON_TAC [log2_mono_le;
    REAL_ARITH `&0 < &2`;
    log2_2]);;

let log2_ge1 = prove(`
  !x. &1 <= x ==> &0 <= log2 x
  `,
  MESON_TAC [log2_mono_le;
    REAL_ARITH `&0 < &1`;
    log2_1]);;

let log2_le1 = prove(`
  !x. &0 < x /\ x <= &1 ==> log2 x <= &0
  `,
  MESON_TAC [log2_mono_le;
    REAL_ARITH `&0 < &1`;
    log2_1]);;

let log2_gt1 = prove(`
  !x. &1 < x ==> &0 < log2 x
  `,
  MESON_TAC [log2_mono_lt;
    REAL_ARITH `&0 < &1`;
    log2_1]);;

let log2_1over = prove(`
  !x:real.
  &0 < x ==> log2(&1 / x) = --log2(x)
  `,
  REWRITE_TAC [log2] THEN
  MESON_TAC [ln_1over;
    REAL_ARITH `--a/b = --(a/b)`]);;

let log2_continuous_lemma = prove(`
  log2 = \x. ln x * &1 / ln(&2)
  `,
  MESON_TAC [EQ_EXT;
    log2;
    REAL_ARITH `L / ln(&2) = L * (&1 / ln(&2))`]);;

let log2_continuous = prove(`
  !x. &0 < x ==> log2 contl x
  `,
  REPEAT STRIP_TAC THEN
  specialize [`x:real`;
    `&1 / ln(&2)`] cont_const_fullvariables THEN
  specialize [`x:real`] ln_continuous THEN
  specialize [`x:real`;
    `ln`;
    `\x:real. &1 / ln(&2)`] cont_mul_fullvariables THEN
  ASM_MESON_TAC [log2_continuous_lemma]);;

(* ----- exp(1) *)

let exp1_, exp1boundgen = REALCALC_CONV `
  exp(&1)
`;;

let exp1bound = thm_eval exp1boundgen(Int 10);;

let exp1_8732 = prove(`
  exp(&1) < &87 / &32
  `,
  MESON_TAC [exp1bound;
    REAL_ARITH `abs (&2783 - &2 pow 10 * exp (&1)) < &1 ==> exp(&1) < &87 / &32`]);;

(* ----- pi *)

let pi_30bits = PI_APPROX_BINARY_RULE 30;;

let pi_227 = prove(`
  pi < &22 / &7
  `,
  MESON_TAC [pi_30bits;
    REAL_ARITH `abs (pi - &1686629713 / &536870912) <= inv (&2 pow 30) ==> pi < &22 / &7`]);;

let pi_333106 = prove(`
  &333 / &106 < pi
  `,
  MESON_TAC [pi_30bits;
    REAL_ARITH `abs (pi - &1686629713 / &536870912) <= inv (&2 pow 30) ==> &333 / &106 < pi`]);;

(* ----- more constants *)

let exp2bound = prove(`
  exp(&2 * &1) < (&87 / &32) pow 2
  `,
  MESON_TAC [exp_lt_sq;
    exp1_8732]);;

let exp2bound2 = prove(`
  exp(&2) < &7569 / &1024
  `,
  MESON_TAC [exp2bound;
    REAL_ARITH `(&87 / &32) pow 2 = &7569 / &1024`;
    REAL_ARITH `&2 * &1 = &2`]);;

let exp2pibound = prove(`
  pi * exp(&2) < (&22 / &7) * (&7569 / &1024)
  `,
  MESON_TAC [exp2bound2;
    pi_227;
    REAL_LT_MUL2;
    REAL_EXP_POS_LE;
    PI_POS;
    REAL_LT_IMP_LE]);;

let exp2pibound2 = prove(`
  &2 * pi * exp(&2) < &60
  `,
  MESON_TAC [exp2pibound;
    REAL_ARITH `a < (&22 / &7) * (&7569 / &1024) ==> &2 * a < &60`]);;

let lnexp2pibound = prove(`
  ln(&2 * pi * exp(&2)) < ln(&60)
  `,
  MESON_TAC [ln_mono_lt;
    exp2pibound2;
    PI_POS;
    REAL_EXP_POS_LT;
    REAL_ARITH `&0 < &2`;
    REAL_LT_MUL]);;

let lnexp2piexpand = prove(`
  ln(&2 * pi * exp(&2)) = ln(&2) + ln(pi) + ln(exp(&2))
  `,
  MESON_TAC [LN_MUL;
    REAL_ARITH `&0 < &2`;
    REAL_EXP_POS_LT;
    PI_POS;
    REAL_LT_MUL]);;

let lnexp2piexpand2 = prove(`
  ln(&2 * pi * exp(&2)) = ln(&2) + ln(pi) + &2
  `,
  MESON_TAC [lnexp2piexpand;
    LN_EXP]);;

let lnexp1piexpand = prove(`
  ln(&2 * pi * exp(&1)) = ln(&2) + ln(pi) + ln(exp(&1))
  `,
  MESON_TAC [LN_MUL;
    REAL_ARITH `&0 < &2`;
    REAL_EXP_POS_LT;
    PI_POS;
    REAL_LT_MUL]);;

let lnexp1piexpand2 = prove(`
  ln(&2 * pi * exp(&1)) = ln(&2) + ln(pi) + &1
  `,
  MESON_TAC [lnexp1piexpand;
    LN_EXP]);;

let lnexp1piexp2pi = prove(`
  ln(&2 * pi * exp(&2)) = &1 + ln(&2 * pi * exp(&1))
  `,
  MESON_TAC [lnexp2piexpand2;
    lnexp1piexpand2;
    REAL_ARITH `!x y a b. x = a + b + &2 /\ y = a + b + &1 ==> x = &1 + y`]);;

let twopie_pos = prove(`
  &0 < &2 * pi * exp(&1)
  `,
  MESON_TAC [REAL_ARITH `&0 < &2`;
    REAL_EXP_POS_LT;
    PI_POS;
    REAL_LT_MUL]);;

let pi60bound2 = prove(`
  (&333 / &106) * &60 < pi * &60
  `,
  MESON_TAC [pi_333106;
    REAL_LT_RMUL;
    REAL_ARITH `&0 < &60`]);;

let exp1bound3 = prove(`
  &43 / &16 < exp(&1)
  `,
  MESON_TAC [exp1bound;
    REAL_ARITH `abs (&2783 - &2 pow 10 * exp (&1)) < &1 ==> &43 / &16 < exp(&1)`]);;

let exp30bound = prove(`
  (&43 / &16) pow 30 < exp(&1) pow 30
  `,
  MESON_TAC [exp1bound3;
    REAL_POW_LT2;
    ARITH_RULE `~(30 = 0)`;
    REAL_ARITH `&0 <= &43 / &16`]);;

let exp30bound2 = prove(`
  (&43 / &16) pow 30 < exp(&30)
  `,
  MESON_TAC [exp30bound;
    REAL_EXP_N;
    REAL_ARITH `&30 * &1 = &30`]);;

let exp30bound3 = prove(`
  &1000 < exp(&30)
  `,
  MESON_TAC [exp30bound2;
    REAL_ARITH `&1000 < (&43 / &16) pow 30`;
    REAL_LT_TRANS]);;

let pi60bound8 = prove(`
  &60 * pi < &60 * (&22 / &7)
  `,
  MESON_TAC [pi_227;
    REAL_LT_LMUL;
    REAL_ARITH `&0 < &60`]);;

let pi60bound9 = prove(`
  &60 * pi < &1000
  `,
  MESON_TAC [pi60bound8;
    REAL_ARITH `&60 * (&22 / &7) < &1000`;
    REAL_LT_TRANS]);;

let pi60exp30 = prove(`
  &60 * pi < exp(&30)
  `,
  MESON_TAC [exp30bound3;
    pi60bound9;
    REAL_LT_TRANS]);;

let lnpi60 = prove(`
  ln(pi* &60)- &60 / &2 < &0
  `,
  MESON_TAC [pi60exp30;
    LN_MONO_LT;
    LN_EXP;
    REAL_ARITH `&0 < pi ==> &0 < &60 * pi`;
    REAL_EXP_POS_LT;
    PI_POS;
    REAL_ARITH `pi * &60 = &60 * pi`;
    REAL_ARITH `a < &30 ==> a - &60 / &2 < &0`]);;

(* ----- powreal *)

let powreal_1 = prove(`
  !x. &0 < x ==> x powreal (&1) = x
  `,
  MESON_TAC [powreal;
    REAL_ARITH `&1 * y = y`;
    EXP_LN]);;

let powreal_ln = prove(`
  !x y. ln(x powreal y) = y * ln x
  `,
  MESON_TAC [powreal;
    LN_EXP]);;

let powreal_log2 = prove(`
  !x y. log2(x powreal y) = y * log2 x
  `,
  REWRITE_TAC [log2] THEN
  MESON_TAC [powreal_ln;
    ln2_positive;
    REAL_ARITH `(y*z)/L = y*(z/L)`]);;

let powreal_positive = prove(`
  !x y. &0 < x powreal y
  `,
  MESON_TAC [powreal;
    REAL_EXP_POS_LT]);;

let powreal_pow = prove(`
  !x y n. (x powreal y) pow n = x powreal (&n * y)
  `,
  REWRITE_TAC [powreal;
    GSYM REAL_EXP_N;
    REAL_MUL_ASSOC]);;

let powreal_lnratio = prove(`
  !x y. &0 < x /\ &1 < y ==> x = y powreal (ln(x)/ln(y))
  `,
  REWRITE_TAC [powreal] THEN
  MESON_TAC [LN_POS_LT;
    REAL_FIELD `&0 < a ==> b / a * a = b`;
    EXP_LN]);;

let powreal_log2ratio = prove(`
  !x y. &0 < x /\ &1 < y ==> x = y powreal (log2(x)/log2(y))
  `,
  MESON_TAC [log2ratio_lnratio;
    powreal_lnratio]);;

let powreal_cancelbase = prove(`
  !x y.
  &0 < x ==>
  ((&1 / x) powreal y) * (x powreal y) = &1
  `,
  REWRITE_TAC [powreal] THEN
  REWRITE_TAC [GSYM REAL_EXP_ADD] THEN
  REWRITE_TAC [GSYM REAL_ADD_LDISTRIB] THEN
  MESON_TAC [ln_1over_plus;
    REAL_ARITH `y * &0 = &0`;
    REAL_EXP_0]);;

(* ----- bkzdelta *)

let bkzdelta_pos = prove(`
  !x. &0 < bkzdelta x
  `,
  MESON_TAC [bkzdelta;
    powreal_positive]);;

let bkzdelta_ln = prove(`
  !x. ln(bkzdelta x) = (&1 / (&2 * (x - &1))) * ln(x * ((pi*x) powreal (&1 / x))/(&2 * pi * exp(&1)))
  `,
  REWRITE_TAC [bkzdelta;
    powreal] THEN
  MESON_TAC [LN_EXP]);;

let bkzdelta_ln2 = prove(`
  !x. ln(bkzdelta x) = ln(x * ((pi*x) powreal (&1 / x))/(&2 * pi * exp(&1))) / (&2 * (x - &1))
  `,
  MESON_TAC [bkzdelta_ln;
    REAL_FIELD (`(&1 / a) * b = b / a`)]);;

let bkzdelta_ln3 = prove(`
  !x. &60 <= x ==> ln(bkzdelta x) = (ln(x) + ln((pi*x) powreal (&1 / x)) - ln(&2 * pi * exp(&1))) / (&2 * (x - &1))
  `,
  MESON_TAC [bkzdelta_ln2;
    ln_muldiv;
    REAL_ARITH `&60 <= x ==> &0 < x`;
    twopie_pos;
    powreal_positive]);;

let bkzdelta_ln4 = prove(`
  !x. &60 <= x ==> ln(bkzdelta x) = (ln(x) + (&1 / x) * ln(pi*x) - ln(&2 * pi * exp(&1))) / (&2 * (x - &1))
  `,
  MESON_TAC [bkzdelta_ln3;
    powreal_ln]);;

let bkzdelta_ln5 = prove(`
  !x. &60 <= x ==> ln(bkzdelta x) = (ln(x) + ln(pi*x)/x - ln(&2 * pi * exp(&1))) / (&2 * (x - &1))
  `,
  MESON_TAC [bkzdelta_ln4;
    REAL_ARITH `(&1 / a) * b = b / a`]);;

let bkzdelta_log2 = prove(`
  !x. &60 <= x ==> log2(bkzdelta x) = (log2(x) + (&1 / x) * log2(pi*x) - log2(&2 * pi * exp(&1))) / (&2 * (x - &1))
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [log2] THEN
  specialize [`x:real`] bkzdelta_ln4 THEN
  ASM_MESON_TAC [ln2_positive;
    REAL_FIELD `&0 < ln(&2) ==> ( (ln(x) + (&1 / x) * ln(pi*x) - ln(&2 * pi * exp(&1))) / (&2 * (x - &1)) )/ln(&2) = (ln(x)/ln(&2) + (&1 / x) * ln(pi*x)/ln(&2) - ln(&2 * pi * exp(&1))/ln(&2)) / (&2 * (x - &1))`]);;

(* ----- standardratio *)

let standardratio_positive = prove(`
  !n q s k x.
  &0 < standardratio n q s k x
  `,
  REWRITE_TAC [standardratio] THEN
  MESON_TAC [powreal_positive;
    REAL_LT_MUL;
    REAL_LT_DIV]);;

let standardratio_ln = prove(`
  !n q s k x. ln(standardratio n q s k x) = ln(((n + k)*(s pow 2) + &1) powreal (&1 / &2)) - (ln(((n + k + &1)/x) powreal (&1 / &2)) + ln((bkzdelta x) powreal (&2 * x - (n + k + &1) - &1)) + ln(q powreal (k/(n + k + &1))))
  `,
  REWRITE_TAC [standardratio] THEN
  MESON_TAC [LN_MUL;
    LN_DIV;
    powreal_positive;
    REAL_LT_MUL]);;

let standardratio_ln2 = prove(`
  !n q s k x. ln(standardratio n q s k x) = (&1 / &2) * ln((n + k)*(s pow 2) + &1) - ((&1 / &2) * ln((n + k + &1)/x) + (&2 * x - (n + k + &1) - &1) * ln(bkzdelta x) + (k/(n + k + &1)) * ln q)
  `,
  MESON_TAC [standardratio_ln;
    powreal_ln]);;

let standardratio_ln3 = prove(`
  !n q s k x. &60 <= x ==> ln(standardratio n q s k x) = (&1 / &2) * ln((n + k)*(s pow 2) + &1) - ((&1 / &2) * ln((n + k + &1)/x) + (&2 * x - (n + k + &1) - &1) * ((ln(x) + (&1 / x) * ln(pi*x) - ln(&2 * pi * exp(&1))) / (&2 * (x - &1))) + (k/(n + k + &1)) * ln q)
  `,
  MESON_TAC [bkzdelta_ln4;
    standardratio_ln2]);;

let standardratio_ln4 = prove(`
  !n q s k x. &60 <= x ==> ln(standardratio n q s k x) = (&1 / &2) * ln((n + k)*(s pow 2) + &1) - ((&1 / &2) * ln((n + k + &1)/x) +
   (ln(x) + (&1 / x) * ln(pi*x) - ln(&2 * pi * exp(&1))) * ( (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1)))
   + (k/(n + k + &1)) * ln q)
  `,
  MESON_TAC [standardratio_ln3;
    REAL_FIELD `a * b / c = b * a / c`]);;

let standardratio_log2 = prove(`
  !n q s k x.
  &60 <= x ==>
  log2(standardratio n q s k x) =
  (&1 / &2) * log2((n + k)*(s pow 2) + &1)
  - (
      (&1 / &2) * log2((n + k + &1)/x)
      + (log2(x) + (&1 / x) * log2(pi*x) - log2(&2 * pi * exp(&1))) * ( (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1)))
      + (k/(n + k + &1)) * log2 q
    )
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [log2] THEN
  specialize [`n:real`;
    `q:real`;
    `s:real`;
    `k:real`;
    `x:real`] standardratio_ln4 THEN
  ASM_MESON_TAC [ln2_positive;
    REAL_FIELD `&0 < ln(&2) ==> ( (&1 / &2) * ln((n + k)*(s pow 2) + &1) - ((&1 / &2) * ln((n + k + &1)/x) + (ln(x) + (&1 / x) * ln(pi*x) - ln(&2 * pi * exp(&1))) * ( (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1))) + (k/(n + k + &1)) * ln q)) / ln(&2) = ( (&1 / &2) * ln((n + k)*(s pow 2) + &1)/ln(&2) - ((&1 / &2) * ln((n + k + &1)/x)/ln(&2) + (ln(x)/ln(&2) + (&1 / x) * ln(pi*x)/ln(&2) - ln(&2 * pi * exp(&1))/ln(&2)) * ( (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1))) + (k/(n + k + &1)) * ln q/ln(&2)))`]);;

(* ----- miscellany *)

let asymp1_dn_lemma = prove(`
  !n:num->real k:num->real X0:real.
  (!i. &1 < n(i))
  ==>
  (\i. &1 / n i + (k(i) / n(i) - X0))
  = (\i. (n(i) + k(i) + &1) / n(i) - (&1 + X0))
  `,
  MESON_TAC [EQ_EXT;
    REAL_FIELD `&1 < x ==> (x + y + &1) / x - (&1 + X0) = &1 / x + (y / x - X0)`]);;

let asymp1_dn = prove(`
  !n:num->real k:num->real X0:real.
  o1_seq (\i. k(i) / n(i) - X0)
  /\ (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  ==> o1_seq (\i. (n(i) + k(i) + &1) / n(i) - (&1 + X0))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. &1 / n(i)`;
    `\i:num. k(i) / n(i) - X0:real`] o1_seq_add THEN
  ASM_MESON_TAC [BETA_THM;
    asymp1_dn_lemma]);;

let asymp1_nd_lemma = prove(`
  !n:num->real k:num->real X0:real.
  (!i. &1 < n(i))
  /\ (!i. &0 < k(i))
  ==> 
  (\i. &1 / ((&1 + X0) + (n i + k i + &1) / n i - (&1 + X0)) - &1 / (&1 + X0))
  =
  (\i. n i / (n i + k i + &1) - &1 / (&1 + X0))
  `,
  REWRITE_TAC [REAL_ARITH `(&1 + x) + c - (&1 + x) = c`] THEN
  MESON_TAC [EQ_EXT;
    REAL_FIELD `&1 < n /\ &0 < k ==> &1 / ((n + k + &1) / n) = n / (n + k + &1)`]);;

let asymp1_nd_lemma2 = prove(`
  !n:num->real k:num->real X0:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &0 < k(i))
  /\ o1_seq (\i. (n(i) + k(i) + &1) / n(i) - (&1 + X0))
  /\ &0 <= X0
  ==>
  o1_seq (\i. n(i) / (n(i) + k(i) + &1) - &1 / (&1 + X0))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`X0:real`] (REAL_ARITH `!x. &0 <= x ==> ~(&1 + x = &0)`) THEN
  specialize [`\i:num. (n(i) + k(i) + &1) / n(i) - (&1 + X0)`;
    `&1 + X0`] o1_seq_1overcplusx THEN
  ASM_MESON_TAC [asymp1_nd_lemma]);;

let asymp1_nd = prove(`
  !n:num->real k:num->real X0:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &0 < k(i))
  /\ o1_seq (\i. k(i) / n(i) - X0)
  /\ &0 <= X0
  ==>
  o1_seq (\i. n(i) / (n(i) + k(i) + &1) - &1 / (&1 + X0))
  `,
  MESON_TAC [asymp1_nd_lemma2;
    asymp1_dn]);;

let asymp1_kd_lemma = prove(`
  !n:num->real k:num->real X0:real.
  (!i. &1 < n(i))
  ==> 
  (\i. (k(i) / n(i)) * (n(i) / (n(i) + k(i) + &1)) - X0 * (&1 / (&1 + X0)))
  =
  (\i. (k(i) / (n(i) + k(i) + &1)) - X0 / (&1 + X0))
  `,
  MESON_TAC [EQ_EXT;
    REAL_ARITH `a * (&1 / b) = a / b`;
    REAL_FIELD `&1 < n(i:num) ==> (k(i) / n(i)) * (n(i) / (n(i) + k(i) + &1)) = k(i)/(n(i) + k(i) + &1)`]);;

let asymp1_kd = prove(`
  !n:num->real k:num->real X0:real.
  (!i. &1 < n(i))
  /\ (!i. &0 < k(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 <= X0
  /\ o1_seq (\i. k(i) / n(i) - X0)
  ==>
  o1_seq (\i. (k(i) / (n(i) + k(i) + &1)) - X0 / (&1 + X0))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `X0:real`] asymp1_nd THEN
  specialize [`\i:num. k(i) / n(i)`;
    `X0:real`;
    `\i:num. n(i) / (n(i) + k(i) + &1)`;
    `&1 / (&1 + X0)`] o1_seq_offset_mul2 THEN
  ASM_MESON_TAC [asymp1_kd_lemma]);;

let asymp1_qn = prove(`
  !n:num->real q:num->real Q0:real qno1:num->real.
  (!i. &1 < n(i))
  /\ (!i. log2(q(i)) = (Q0 + qno1(i)) * log2(n(i)))
  ==>
  (!i. log2(q(i))/log2(n(i)) = Q0 + qno1(i))
  `,
  MESON_TAC [REAL_EQ_RDIV_EQ;
    log2_gt1]);;

let o1_seq_offset_div_lemma = prove(`
  !c:real co1:num->real d:real do1:num->real.
  o1_seq do1 /\ ~(d = &0) 
  ==> o1_seq co1
  ==>
  ?o1. o1_seq o1 /\
  (!i. (c + co1 i) * (&1 / d + (\i. &1 / (d + do1 i) - &1 / d) i) = c * &1 / d + o1 i)
  `,
  MESON_TAC [o1_seq_1overcplusx;
    ISPECL [`c:real`;
      `co1:num->real`;
      `&1 / (d:real)`;
      `\i:num. &1 / (d + do1 i) - &1 / d`] o1_seq_offset_mul]);;

let o1_seq_offset_div = prove(`
  !c:real co1:num->real d:real do1:num->real.
  o1_seq co1 ==> o1_seq do1 ==> ~(d = &0) 
  ==>
  ?o1. o1_seq o1 /\
  (!i. (c + co1 i) / (d + do1 i) = c / d + o1 i)
  `,
  REWRITE_TAC [REAL_FIELD `c / d + x = c * &1 / d + x`] THEN
  REWRITE_TAC [REAL_FIELD `(c + co1 i) / (d + do1 i) = (c + co1 i) * (&1 / d + (&1 / (d + do1 i) - &1 / d))`] THEN
  MESON_TAC [o1_seq_offset_div_lemma]);;

let o1_seq_offset_div2_lemma = prove(`
  !f:num->real c:real g:num->real d:real.
  (\i. f(i)/g(i) - c/d)
  =
  (\i. f(i) * &1 / (d + g(i) - d) - c * &1 / d)
  `,
  MESON_TAC [EQ_EXT;
    REAL_FIELD `f/g - c/d = f* &1/(d+g-d) - c * &1/d`]);;

let o1_seq_offset_div2 = prove(`
  !f:num->real c:real g:num->real d:real.
  o1_seq (\i. f(i) - c)
  /\ o1_seq (\i. g(i) - d)
  /\ ~(d = &0)
  ==> o1_seq (\i. f(i)/g(i) - c/d)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. g(i) - d`;
    `d:real`] o1_seq_1overcplusx THEN
  specialize [`\i:num. g(i) - d`;
    `d:real`] o1_seq_1overcplusx THEN
  specialize [`f:num->real`;
    `c:real`;
    `\i:num. &1 / (d + g(i) - d)`;
    `&1 / d`] o1_seq_offset_mul2 THEN
  ASM_MESON_TAC [o1_seq_offset_div2_lemma]);;

let asymp1_db_lemma = prove(`
  !n:num->real k:num->real b:num->real X0:real Y0:real.
  (!i. &1 < n(i))
  ==>
  (\i. (n(i) + k(i) + &1) / b(i) - (&1 + X0) / Y0)
  =
  (\i. ((n(i) + k(i) + &1) / n(i)) / (b(i) / n(i)) - (&1 + X0) / Y0)
  `,
  MESON_TAC [EQ_EXT;
    REAL_FIELD `&1 < x ==> (a/x)/(b/x) = a/b`]);;

let asymp1_db_lemma2 = prove(`
  !n:num->real k:num->real b:num->real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. (n(i) + k(i) + &1) / n(i) - (&1 + X0))
  /\ o1_seq (\i. b(i) / n(i) - Y0)
  /\ ~(Y0 = &0)
  ==>
  o1_seq (\i. (n(i) + k(i) + &1) / b(i) - (&1 + X0) / Y0)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. (n(i) + k(i) + &1) / n(i)`;
    `&1 + X0`;
    `\i:num. b(i) / n(i)`;
    `Y0:real`] o1_seq_offset_div2 THEN
  ASM_MESON_TAC [asymp1_db_lemma]);;

let asymp1_db = prove(`
  !n:num->real k:num->real b:num->real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. k(i) / n(i) - X0)
  /\ o1_seq (\i. b(i) / n(i) - Y0)
  /\ ~(Y0 = &0)
  ==>
  o1_seq (\i. (n(i) + k(i) + &1) / b(i) - (&1 + X0) / Y0)
  `,
  MESON_TAC [asymp1_db_lemma2;
    asymp1_dn]);;

let asymp1_sn_pow = prove(`
  !n:num->real s:num->real S0:real sno1:num->real.
  o1_seq sno1
  /\ (!i. log2(s(i))/log2(n(i)) = S0 + sno1(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &1 < n(i))
  ==>
  (!i. s(i) = n(i) powreal (S0 + sno1(i)))
  `,
  MESON_TAC [powreal_log2ratio]);;

let asymp1_sn_pow2 = prove(`
  !n:num->real s:num->real S0:real sno1:num->real.
  o1_seq sno1
  /\ (!i. log2(s(i))/log2(n(i)) = S0 + sno1(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &1 < n(i))
  ==>
  (!i. s(i) pow 2 = n(i) powreal (&2*S0 + &2*sno1(i)))
  `,
  MESON_TAC [asymp1_sn_pow;
    powreal_pow;
    REAL_ADD_LDISTRIB]);;

let lnovero1_lemma1 = prove(`
  !x:real e:real.
  &0 < e /\ &1 < x /\ &1 / x <= exp(--(&1/e))
  ==> &1 / ln(x) <= e
  `,
  MESON_TAC [REAL_ARITH `&1 < x ==> &0 < x`;
    ln_mono_le;
    LN_EXP;
    LN_POS_LT;
    ln_1over;
    pos_1over;
    REAL_ARITH `--x <= --y <=> y <= x`;
    recipflip]);;

let lnovero1_lemma2 = prove(`
  !x:real.
  &0 < e /\ &1 < x /\ abs(&1 / x) <= exp(--(&1/e))
  ==> abs(&1 / ln(x)) <= e
  `,
  MESON_TAC [lnovero1_lemma1;
    REAL_ABS_REFL;
    REAL_ARITH `&1 < x ==> &0 < x`;
    LN_POS_LT;
    pos_1over;
    REAL_ARITH `&0 < y ==> &0 <= y`]);;

let lnovero1 = prove(`
  !n:num->real.
  o1_seq (\i. &1 / n(i))
  /\ (!i. &1 < n(i))
  ==> o1_seq (\i. &1 / ln(n(i)))
  `,
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  X_CHOOSE_TAC `a:num` (UNDISCH_ALL (MESON [REAL_EXP_POS_LT] `(!e. &0 < e ==> (?m:num. !i. m <= i ==> abs (&1 / n i) <= e)) ==> (?m:num. !i. m <= i ==> abs (&1 / n i) <= exp(--(&1 / e)))`)) THEN
  EXISTS_TAC `a:num` THEN
  ASM_MESON_TAC [lnovero1_lemma2]);;

let invlog2_lnln = prove(`
  !x:real. ln(&2)/ln(x) = &1/log2(x)
  `,
  REWRITE_TAC [log2] THEN
  REWRITE_TAC [REAL_FIELD `&1/(d/e) = e/d`]);;

let invlog2_lnln2 = prove(`
  !x:real. ln(&2)*(&1/ln(x)) = &1/log2(x)
  `,
  REWRITE_TAC [REAL_FIELD `ln(&2)*(&1/x) = ln(&2)/x`] THEN
  MESON_TAC [invlog2_lnln]);;

let log2overo1 = prove(`
  !n:num->real.
  o1_seq (\i. &1 / n(i))
  /\ (!i. &1 < n(i))
  ==> o1_seq (\i. &1 / log2(n(i)))
  `,
  REWRITE_TAC [GSYM invlog2_lnln2] THEN
  MESON_TAC [ISPECL [`\i:num. &1 / ln(n(i))`;
      `ln(&2)`] o1_seq_consttimes;
    lnovero1]);;

let asymp1_nkno1 = prove(`
  !n:num->real k:num->real X0:real kno1:num->real.
  (!i. k(i) / n(i) = X0 + kno1(i))
  /\ (!i. &1 < n(i))
  ==>
  (!i. (n(i) + k(i)) / n(i) = (&1 + X0) + kno1(i))
  `,
  MESON_TAC [REAL_FIELD `&1 < b /\ a/b = c+d ==> (b+a)/b = (&1+c)+d`]);;

let o1_seq_lncplusx = prove(`
  !f:num->real c:real.
  o1_seq f /\ &0 < c
  ==> o1_seq (\i. ln(c+f(i)) - ln(c))
  `,
  MESON_TAC [ln_continuous;
    ISPECL [`f:num->real`;
      `ln`] o1_seq_compose]);;

let asymp1_ln_nkn = prove(`
  !n:num->real k:num->real X0:real kno1:num->real.
  o1_seq kno1 /\
  (!i. (n(i) + k(i)) / n(i) = (&1 + X0) + kno1(i))
  /\ &0 < &1 + X0
  ==>
  ?lnkno1.
  o1_seq lnkno1 /\
  (!i. ln((n(i) + k(i)) / n(i)) = ln(&1 + X0) + lnkno1(i))
  `,
  REPEAT STRIP_TAC THEN
  EXISTS_TAC `\i:num. ln((n(i) + k(i))/n(i)) - ln(&1 + X0:real)` THEN
  CONJ_TAC THENL
  [ASM_SIMP_TAC [ISPECL [`kno1:num->real`;
        `&1+X0:real`] o1_seq_lncplusx];
    REAL_ARITH_TAC]);;

let asymp1_ln_nkn2 = prove(`
  !n:num->real k:num->real X0:real kno1:num->real.
  o1_seq kno1 /\
  (!i. (n(i) + k(i)) / n(i) = (&1 + X0) + kno1(i))
  /\ (!i. &0 < n(i))
  /\ (!i. &0 < n(i) + k(i))
  /\ &0 < &1 + X0
  ==>
  ?lnkno1.
  o1_seq lnkno1 /\
  (!i. ln(n(i) + k(i)) - ln(n(i)) = ln(&1 + X0) + lnkno1(i))
  `,
  MESON_TAC [asymp1_ln_nkn;
    LN_DIV]);;

let asymp1_lnln_nkn_lemma2 = prove(`
  !n:num->real k:num->real X0:real lnkno1:num->real.
  (!i. ln(n(i) + k(i)) - ln(n(i)) = ln(&1 + X0) + lnkno1(i))
  /\ (!i. &1 < n(i))
  ==>
  (!i. ln(n(i) + k(i))/ln(n(i)) - &1 = ln(&1 + X0)/ln(n(i)) + lnkno1(i)/ln(n(i)))
  `,
  MESON_TAC [ln_gt1;
    REAL_FIELD `&0 < b ==> a - b = c + d ==> a/b - &1 = c/b + d/b`]);;

let asymp1_lnln_nkn_lemma3 = prove(`
  !n:num->real X0:real.
  o1_seq (\i. &1 / n(i))
  /\ (!i. &1 < n(i))
  ==> o1_seq (\i:num. ln(&1 + X0)/ln(n(i)))
  `,
  ONCE_REWRITE_TAC [REAL_FIELD `x/ln(y) = x*(&1/ln(y))`] THEN
  MESON_TAC [lnovero1;
    ISPECL [`\i:num. &1 / ln(n(i))`;
      `ln(&1 + X0):real`] o1_seq_consttimes]);;

let asymp1_lnln_nkn_lemma4 = prove(`
  !n:num->real lnkno1:num->real.
  o1_seq (\i. &1 / n(i))
  /\ (!i. &1 < n(i))
  /\ o1_seq lnkno1
  ==> o1_seq (\i:num. lnkno1(i)/ln(n(i)))
  `,
  ONCE_REWRITE_TAC [REAL_FIELD `x/ln(y) = x*(&1/ln(y))`] THEN
  MESON_TAC [lnovero1;
    ISPECL [`lnkno1:num->real`;
      `\i:num. &1 / ln(n(i))`] o1_seq_mul]);;

let asymp1_lnln_nkn_lemma5 = prove(`
  !n:num->real X0:real lnkno1:num->real.
  o1_seq (\i. &1 / n(i))
  /\ (!i. &1 < n(i))
  /\ o1_seq lnkno1
  ==> o1_seq (\i:num. ln(&1 + X0)/ln(n(i))+lnkno1(i)/ln(n(i)))
  `,
  MESON_TAC [asymp1_lnln_nkn_lemma3;
    asymp1_lnln_nkn_lemma4;
    ISPECL [`\i:num. ln (&1 + X0) / ln (n(i))`;
      `\i:num. lnkno1 i / ln (n(i))`] o1_seq_add]);;

let asymp1_lnln_nkn_lemma6 = prove(`
  !n:num->real k:num->real X0:real lnkno1:num->real.
  o1_seq (\i. &1 / n(i))
  /\ (!i. &1 < n(i))
  /\ o1_seq lnkno1
  /\ (!i. ln(n(i) + k(i)) - ln(n(i)) = ln(&1 + X0) + lnkno1(i))
  ==>
  o1_seq (\i. ln(n(i) + k(i))/ln(n(i)) - &1)
  `,
  SIMP_TAC [asymp1_lnln_nkn_lemma2;
    asymp1_lnln_nkn_lemma5]);;

let asymp1_lnln_nkn_lemma7 = prove(`
  !n:num->real k:num->real X0:real kno1:num->real.
  o1_seq kno1 /\
  (!i. (n(i) + k(i)) / n(i) = (&1 + X0) + kno1(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &1 < n(i))
  /\ (!i. &0 < n(i) + k(i))
  /\ &0 < &1 + X0
  ==>
  o1_seq (\i. ln(n(i) + k(i))/ln(n(i)) - &1)
  `,
  MESON_TAC [asymp1_lnln_nkn_lemma6;
    asymp1_ln_nkn2;
    REAL_ARITH `&1 < a ==> &0 < a`]);;

let asymp1_lnln_nkn_lemma8 = prove(`
  !n:num->real k:num->real X0:real kno1:num->real.
  o1_seq kno1 /\
  (!i. k(i) / n(i) = X0 + kno1(i))
  /\ (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &0 < n(i) + k(i))
  /\ &0 < &1 + X0
  ==>
  o1_seq (\i. ln(n(i) + k(i))/ln(n(i)) - &1)
  `,
  MESON_TAC [asymp1_lnln_nkn_lemma7;
    asymp1_nkno1]);;

let asymp1_lnln_nkn = prove(`
  !n:num->real k:num->real X0:real kno1:num->real.
  o1_seq kno1 /\
  (!i. k(i) / n(i) = X0 + kno1(i))
  /\ (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &0 < n(i) + k(i))
  /\ &0 < &1 + X0
  ==> ?lnlnkno1.
  o1_seq lnlnkno1 /\
  (!i. ln(n(i) + k(i))/ln(n(i)) = &1 + lnlnkno1(i))
  `,
  REWRITE_TAC [REAL_ARITH `ln a/ln b = &1 + c <=> ln a/ln b - &1 = c`] THEN
  MESON_TAC [asymp1_lnln_nkn_lemma8]);;

let powreal_from_lnln = prove(`
  !a b c. &0 < a /\ &1 < b ==> ln(a)/ln(b) = c ==> a = b powreal c
  `,
  REWRITE_TAC [powreal] THEN
  MESON_TAC [REAL_EQ_LDIV_EQ;
    EXP_LN;
    ln_gt1]);;

let asymp1_nk_powreal_n = prove(`
  !n:num->real k:num->real X0:real kno1:num->real.
  o1_seq kno1 /\
  (!i. k(i) / n(i) = X0 + kno1(i))
  /\ (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &0 < n(i) + k(i))
  /\ &0 < &1 + X0
  ==> ?lnlnkno1.
  o1_seq lnlnkno1 /\
  (!i. n(i)+k(i) = n(i) powreal (&1 + lnlnkno1(i)))
  `,
  MESON_TAC [asymp1_lnln_nkn;
    powreal_from_lnln]);;

let asymp1_nkss_lemma1 = prove(`
  !n:num->real s:num->real S0:real sno1:num->real k:num->real X0:real kno1:num->real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &0 < n(i) + k(i))
  /\ o1_seq kno1
  /\ (!i. k(i) / n(i) = X0 + kno1(i))
  /\ &0 < &1 + X0
  /\ (!i. &0 < s(i))
  /\ o1_seq sno1
  /\ (!i. log2(s(i))/log2(n(i)) = S0 + sno1(i))
  ==> ?lnlnkno1.
  o1_seq lnlnkno1 /\
  (!i. (n(i)+k(i))*(s(i) pow 2) = (n(i) powreal (&1 + lnlnkno1(i))) * (n(i) powreal (&2*S0 + &2*sno1(i))))
  `,
  MESON_TAC [asymp1_sn_pow2;
    asymp1_nk_powreal_n]);;

let powreal_sum = prove(`
  !x y z. x powreal (y+z) = (x powreal y)*(x powreal z)
  `,
  REWRITE_TAC [powreal] THEN
  REWRITE_TAC [REAL_ADD_RDISTRIB] THEN
  REWRITE_TAC [REAL_EXP_ADD]);;

let asymp1_nkss_lemma2 = prove(`
  !n:num->real s:num->real S0:real sno1:num->real k:num->real X0:real kno1:num->real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &0 < n(i) + k(i))
  /\ o1_seq kno1
  /\ (!i. k(i) / n(i) = X0 + kno1(i))
  /\ &0 < &1 + X0
  /\ (!i. &0 < s(i))
  /\ o1_seq sno1
  /\ (!i. log2(s(i))/log2(n(i)) = S0 + sno1(i))
  ==> ?lnlnkno1.
  o1_seq lnlnkno1 /\
  (!i. (n(i)+k(i))*(s(i) pow 2)
    = n(i) powreal ((&1 + &2*S0) + lnlnkno1(i) + &2*sno1(i)))
  `,
  ONCE_REWRITE_TAC [REAL_ARITH `(a + b) + c + d = (a + c) + (b + d)`] THEN
  ONCE_REWRITE_TAC [powreal_sum] THEN
  MESON_TAC [asymp1_nkss_lemma1]);;

let asymp1_nkss_lemma3 = prove(`
  !n:num->real s:num->real S0:real sno1:num->real k:num->real lnlnkno1:num->real.
  o1_seq sno1
  /\ (!i. log2(s(i))/log2(n(i)) = S0 + sno1(i))
  /\ o1_seq lnlnkno1
  /\ (!i. (n(i)+k(i))*(s(i) pow 2)
    = n(i) powreal ((&1 + &2*S0) + lnlnkno1(i) + &2*sno1(i)))
  ==> ?nksso1.
  o1_seq nksso1 /\
  (!i. (n(i)+k(i))*(s(i) pow 2)
    = n(i) powreal ((&1 + &2*S0) + nksso1(i)))
  `,
  REPEAT STRIP_TAC THEN
  EXISTS_TAC `\i:num. lnlnkno1(i) + &2*sno1(i)` THEN
  CONJ_TAC THENL
  [ASM_MESON_TAC [ISPECL [`sno1:num->real`;
        `&2:real`] o1_seq_consttimes;
      ISPECL [`lnlnkno1:num->real`;
        `\i:num. &2*sno1(i)`] o1_seq_add];
    ASM_MESON_TAC []]);;

let asymp1_nkss_lemma4 = prove(`
  !n:num->real s:num->real S0:real sno1:num->real k:num->real X0:real kno1:num->real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &0 < n(i) + k(i))
  /\ o1_seq kno1
  /\ (!i. k(i) / n(i) = X0 + kno1(i))
  /\ &0 < &1 + X0
  /\ (!i. &0 < s(i))
  /\ o1_seq sno1
  /\ (!i. log2(s(i))/log2(n(i)) = S0 + sno1(i))
  ==> ?nksso1.
  o1_seq nksso1 /\
  (!i. (n(i)+k(i))*(s(i) pow 2)
    = n(i) powreal ((&1 + &2*S0) + nksso1(i)))
  `,
  REPEAT GEN_TAC THEN
  DISCH_TAC THEN
  X_CHOOSE_TAC `o1:num->real` (UNDISCH_ALL (ISPECL [`n:num->real`;
        `s:num->real`;
        `S0:real`;
        `sno1:num->real`;
        `k:num->real`;
        `X0:real`;
        `kno1:num->real`] asymp1_nkss_lemma2)) THEN
  ASM_MESON_TAC [asymp1_nkss_lemma3]);;

let exp_bound1 = prove(`
  !x:real.
  &0 <= x ==> &1 + x / &2 <= exp(x / &2)
  `,
  MESON_TAC [REAL_ARITH `&0 <= (x:real) ==> &0 <= x / &2`;
    REAL_EXP_LE_X]);;

let exp_bound2 = prove(`
  !x:real.
  &0 <= x ==> x / &2 < exp(x / &2)
  `,
  MESON_TAC [exp_bound1;
    REAL_ARITH `x / &2 < &1 + x / &2`;
    REAL_LTE_TRANS]);;

let lt_square = prove(`
  !x:real y:real.
  &0 <= x /\ x < y ==> x*x < y*y
  `,
  MESON_TAC [REAL_LT_MUL2]);;

let exp_bound3 = prove(`
  !x:real.
  &0 <= x ==> x*x / &4 < exp(x / &2 + x / &2)
  `,
  REWRITE_TAC [REAL_ARITH `x*x / &4 = (x / &2) * (x / &2)`] THEN
  REWRITE_TAC [REAL_EXP_ADD] THEN
  MESON_TAC [exp_bound2;
    REAL_ARITH `&0 <= x ==> &0 <= x / &2`;
    lt_square]);;

let exp_bound4 = prove(`
  !x:real.
  &0 <= x ==> x*x / &4 < exp(x)
  `,
  MESON_TAC [exp_bound3;
    REAL_ARITH `x / &2 + x / &2 = x`]);;

let exp_bound5 = prove(`
  !x:real a:real.
  &0 <= x /\ &0 < a ==> a*(x*x / &4) < a*exp(x)
  `,
  MESON_TAC [exp_bound4;
    REAL_LT_LMUL]);;

let abx_lemma1 = prove(`
  !a:real b:real x:real.
  &0 < a*b*b
  /\ &4 / (a*b*b) <= x
  /\ &0 <= x
  ==> x <= a*(b*x)*(b*x) / &4
  `,
  MESON_TAC [ISPECL [`&4:real`;
      `x:real`;
      `a*b*b:real`] REAL_LE_LDIV_EQ;
    REAL_ARITH `&4 <= x*a*b*b ==> &1 <= (a*b*b)*x / &4`;
    REAL_LE_LMUL;
    REAL_ARITH `x * &1 = x`;
    REAL_ARITH `x*(a*b*b)*x / &4 = a*(b*x)*(b*x) / &4`]);;

let abx_lemma2 = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b
  /\ &4 / (a*b*b) <= x
  ==> x <= a*(b*x)*(b*x) / &4
  `,
  MESON_TAC [abx_lemma1;
    REAL_LT_MUL;
    REAL_LT_DIV;
    REAL_ARITH `&0 < &4`;
    REAL_LTE_TRANS;
    REAL_LT_IMP_LE]);;

let abx_lemma3 = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &4 / (a*b*b) <= x
  ==> &0 <= b*x
  `,
  MESON_TAC [REAL_LT_MUL;
    REAL_LT_DIV;
    REAL_ARITH `&0 < &4`;
    REAL_LTE_TRANS;
    REAL_LT_IMP_LE;
    REAL_LE_MUL]);;

let exp_beats_linear = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &4 / (a*b*b) <= x
  ==> x < a*exp(b*x)
  `,
  MESON_TAC [abx_lemma2;
    abx_lemma3;
    exp_bound5;
    REAL_LET_TRANS]);;

let exp_beats_linear_2 = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &4 * a / (b*b) <= x
  ==> a*x < exp(b*x)
  `,
  MESON_TAC [pos_1over;
    REAL_FIELD `&0 < a ==> &4 * a / (b*b) = &4 / ((&1 / a) * b*b)`;
    exp_beats_linear;
    REAL_LT_LMUL;
    REAL_FIELD `&0 < a ==> a * (&1 / a) * c = c`]);;

let linear_beats_log_lemma = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &4 * a / (b*b) <= x
  ==> &0 < x
  `,
  MESON_TAC [REAL_ARITH `&0 < &4`;
    REAL_LT_MUL;
    REAL_LT_DIV;
    REAL_LTE_TRANS]);;

let linear_beats_log_lemma2 = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &4 * a / (b*b) <= x
  ==> &0 < a*x
  `,
  MESON_TAC [REAL_LT_MUL;
    linear_beats_log_lemma]);;

let linear_beats_log = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &4 * a / (b*b) <= x
  ==> ln(a*x) < b*x
  `,
  MESON_TAC [exp_beats_linear_2;
    linear_beats_log_lemma2;
    ln_mono_lt;
    LN_EXP]);;

let linear_beats_log_2 = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &4 * a / (b*b) <= x
  ==> ln(a*x) / x < b
  `,
  MESON_TAC [linear_beats_log_lemma;
    linear_beats_log;
    ISPECL [`ln(a*x)`;
      `b*x:real`] REAL_LT_DIV2_EQ;
    REAL_FIELD `&0 < x ==> (b*x)/x = b`]);;

let abb_lemma37 = prove(`
  !a:real b:real.
  &0 < a ==> &0 < b ==> &0 < (a * b * b) / &4
  `,
  MESON_TAC [REAL_LT_MUL;
    REAL_LT_DIV;
    REAL_ARITH `&0 < &4`]);;

let abb_lemma38 = prove(`
  !a:real b:real.
  &0 < a ==> &0 < b ==> &0 < (b * b) / (&4 * a)
  `,
  MESON_TAC [REAL_LT_MUL;
    REAL_LT_DIV;
    REAL_ARITH `&0 < &4`]);;

let aby_lemma = prove(`
  !a:real b:real y:real.
  &0 < a
  /\ &0 < b
  /\ &0 < y
  /\ y <= (a*b*b) / &4
  ==> &4 / (a*b*b) <= &1 / y
  `,
  MESON_TAC [REAL_LE_INV2;
    REAL_FIELD `inv((a*b*b) / &4) = &4 / (a*b*b)`;
    REAL_FIELD `inv(y) = &1 / y`;
    ]);;

let o1_exp_beats_linear = prove(`
  !a:real b:real o1:num->real.
  o1_seq o1
  /\ (!i. &0 < o1(i))
  /\ &0 < a
  /\ &0 < b
  ==>
  ?m. !i. m <= i
          ==> &1 /o1(i) < a*exp(b/o1(i))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`a:real`;
    `b:real`] abb_lemma37 THEN
  X_CHOOSE_TAC `a:num` (UNDISCH (ISPEC `(a*b*b)/ &4:real` (UNDISCH (fst (EQ_IMP_RULE (ISPEC `o1:num->real` o1_seq)))))) THEN
  EXISTS_TAC `a:num` THEN
  ASM_MESON_TAC [REAL_LT_IMP_LE;
    REAL_ABS_REFL;
    pos_1over;
    aby_lemma;
    exp_beats_linear;
    REAL_FIELD `b / x = b * (&1 / x)`]);;

let max2_lemma = prove(`
  !a:real.
  &0 < a
  ==> &0 < ln(max (&2) (&1 / a))
  `,
  MESON_TAC [REAL_LT_MAX;
    REAL_ARITH `&1 < &2`;
    ln_gt1]);;

let o1_pospow_lemma1 = prove(`
  !x d e.
  &0 < d
  /\ &0 < x
  /\ x <= exp(ln(e) / d)
  ==> d * ln(x) <= ln(e)
  `,
  MESON_TAC [ln_mono_le;
    LN_EXP;
    REAL_LE_LMUL;
    REAL_LT_IMP_LE;
    REAL_FIELD `&0 < d ==> d * (ln(e) / d) = ln(e)`]);;

let real_ge_lmul = prove(`
  !x y z:real. x <= &0 /\ y <= z ==> x * z <= x * y
  `,
  MESON_TAC [REAL_LE_NEG2;
    REAL_ARITH `-- &0 = &0`;
    REAL_LE_LMUL;
    REAL_NEG_LMUL]);;

let real_ge_rmul = prove(`
  !x y z:real. x <= &0 /\ y <= z ==> z * x <= y * x
  `,
  MESON_TAC [REAL_LE_NEG2;
    REAL_ARITH `-- &0 = &0`;
    REAL_LE_RMUL;
    REAL_NEG_RMUL]);;

let o1_pospow_lemma2 = prove(`
  !x y d.
  d <= y
  /\ &0 < x
  /\ x <= &1
  ==> y * ln(x) <= d * ln(x)
  `,
  MESON_TAC [ln_mono_le;
    LN_1;
    real_ge_rmul]);;

let o1_pospow_lemma3 = prove(`
  !x y d e.
   &0 < d
   /\ &0 < x
   /\ x <= &1
   /\ d <= y
   /\ x <= exp(ln(e) / d)
   ==> y * ln(x) <= ln(e)
  `,
  MESON_TAC [o1_pospow_lemma1;
    o1_pospow_lemma2;
    REAL_LE_TRANS]);;

let o1_pospow_lemma4 = prove(`
  !x y d e.
   &0 < d
   /\ &0 < x
   /\ x <= min (&1) (exp(ln(e)/d))
   /\ d <= y
   ==> y * ln(x) <= ln(e)
  `,
  MESON_TAC [o1_pospow_lemma3;
    REAL_LE_MIN]);;

let o1_pospow_lemma5 = prove(`
  !x y c e.
  &0 < c
  /\ &0 < x
  /\ --c / &2 <= y
  /\ x <= min (&1) (exp(ln(e)/(c / &2)))
  ==> (c + y) * ln(x) <= ln(e)
  `,
  MESON_TAC [o1_pospow_lemma4;
    REAL_ARITH `--c / &2 <= y ==> c / &2 <= c + y`;
    REAL_ARITH `&0 < c ==> &0 < c / &2`]);;

let o1_pospow_lemma6 = prove(`
  !x y c e.
  abs(y) < c / &2
  /\ &0 < x
  /\ x <= min (&1) (exp(ln(e)/(c / &2)))
  ==> (c + y) * ln(x) <= ln(e)
  `,
  MESON_TAC [o1_pospow_lemma5;
    REAL_ARITH `abs(y) < c / &2 ==> &0 < c`;
    REAL_ARITH `abs(y) < c / &2 ==> --c / &2 <= y`]);;

let o1_pospow_lemma7 = prove(`
  !x y c e.
  &0 < e
  /\ abs(y) < c / &2
  /\ &0 < x
  /\ x <= min (&1) (exp(ln(e)/(c / &2)))
  ==> abs(x powreal (c + y)) <= e
  `,
  REWRITE_TAC [powreal] THEN
  MESON_TAC [o1_pospow_lemma6;
    REAL_EXP_MONO_LE;
    EXP_LN;
    REAL_EXP_POS_LT;
    REAL_ARITH `&0 < x /\ x <= e ==> abs(x) <= e`]);;

let o1_pospow_lemma8 = prove(`
  !x y c e.
  &0 < e
  /\ abs(y) < c / &2
  /\ &0 < x
  /\ abs(x) <= min (&1) (exp(ln(e)/(c / &2)))
  ==> abs(x powreal (c + y)) <= e
  `,
  MESON_TAC [o1_pospow_lemma7;
    REAL_ARITH `&0 < x ==> x = abs(x)`]);;

let o1_pospow_lemma9 = prove(`
  !c e.
  &0 < min (&1) (exp (ln e / (c / &2)))
  `,
  MESON_TAC [REAL_LT_MIN;
    REAL_ARITH `&0 < &1`;
    REAL_EXP_POS_LT]);;

let o1_pospow = prove(`
  !a:num->real b:num->real c:real.
  o1_seq a
  /\ (!i. &0 < a(i))
  /\ o1_seq b
  /\ &0 < c
  ==> o1_seq (\i. a(i) powreal (c+b(i)))
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (UNDISCH (REAL_ARITH `&0 < c ==> &0 < c / &2`)) THEN
  X_CHOOSE_TAC `B:num` (UNDISCH (ISPEC `c / &2:real` (UNDISCH (fst (EQ_IMP_RULE (ISPEC `b:num->real` o1_seq_lt)))))) THEN
  ASSUME_TAC (ISPECL [`c:real`;
      `e:real`] o1_pospow_lemma9) THEN
  X_CHOOSE_TAC `A:num` (UNDISCH (ISPEC `min (&1) (exp(ln(e)/(c / &2)))` (UNDISCH (fst (EQ_IMP_RULE (ISPEC `a:num->real` o1_seq)))))) THEN
  EXISTS_TAC `MAX A B` THEN
  ASM_MESON_TAC [o1_pospow_lemma8;
    max_le]);;

let o1_ln1plus_lemma1 = prove(`
  !n:num->real b:num->real c:real.
  o1_seq (\i. &1 / n(i))
  /\ (!i. &0 < &1 / n(i))
  /\ o1_seq b
  /\ &0 < c
  ==> o1_seq (\i. (&1 / n(i)) powreal (c+b(i)))
  `,
  MESON_TAC [ISPECL [`\i:num. &1 / n(i):real`;
      `b:num->real`;
      `c:real`] o1_pospow]);;

let o1_ln1plus_lemma2 = prove(`
  !n:num->real b:num->real c:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq b
  /\ &0 < c
  ==> o1_seq (\i. ((&1 / n(i)) powreal (c+b(i))) / ln(n(i)))
  `,
  ONCE_REWRITE_TAC [REAL_ARITH `x/ln(y) = x * (&1 / ln(y))`] THEN
  MESON_TAC [lnovero1;
    pos_1over;
    REAL_ARITH `&1 < x ==> &0 < x`;
    o1_ln1plus_lemma1;
    ISPECL [`\i:num. ((&1 / n(i)) powreal (c+b(i)))`;
      `\i:num. &1 / ln(n(i))`] o1_seq_mul]);;

let bounded_by_o1 = prove(`
  !f:num->real g:num->real.
  o1_seq g /\
  (!i. abs(f(i)) <= abs(g(i)))
  ==> o1_seq f
  `,
  REWRITE_TAC [o1_seq] THEN
  MESON_TAC [REAL_LE_TRANS]);;

let nonneg_le_o1 = prove(`
  !f:num->real g:num->real.
  o1_seq g /\
  (!i. &0 <= f(i)) /\
  (!i. f(i) <= g(i))
  ==> o1_seq f
  `,
  MESON_TAC [bounded_by_o1;
    REAL_ARITH `&0 <= x /\ x <= y ==> abs(x) <= abs(y)`]);;

let nonneg_lt_o1 = prove(`
  !f:num->real g:num->real.
  o1_seq g /\
  (!i. &0 <= f(i)) /\
  (!i. f(i) < g(i))
  ==> o1_seq f
  `,
  MESON_TAC [nonneg_le_o1;
    REAL_LT_IMP_LE]);;

let ln1xo1x_lemma1 = prove(`
  !x y.
  &0 < x /\ &0 < y ==> &0 <= ln(&1 + x*y)/x
  `,
  MESON_TAC [REAL_LT_MUL;
    REAL_ARITH `&0 < z ==> &1 <= &1 + z`;
    ln_ge1;
    REAL_LT_IMP_LE;
    REAL_LE_DIV]);;

let ln1xo1x_lemma2 = prove(`
  !x y.
  &0 < x /\ &0 < y ==> ln(&1 + x*y)/x <= y
  `,
  MESON_TAC [REAL_LT_MUL;
    REAL_LT_IMP_LE;
    LN_LE;
    REAL_LE_DIV2_EQ;
    prove (`
      &0 < x ==> (x*y)/x = y
      `,
      MESON_TAC [REAL_MUL_SYM;
        REAL_EQ_LDIV_EQ])]);;

let ln1xo1x = prove(`
  !o1:num->real c:num->real.
  o1_seq o1 /\
  (!i. &0 < o1(i)) /\
  (!i. &0 < c(i))
  ==> o1_seq (\i. ln(&1 + c(i)*o1(i))/c(i))
  `,
  MESON_TAC [nonneg_le_o1;
    ln1xo1x_lemma1;
    ln1xo1x_lemma2]);;

let ln1xo1x_quo_lemma1 = prove(`
  !b:num->real c:num->real.
  o1_seq (\i. b(i)/c(i))
  /\ (!i. &0 < b(i)/c(i))
  /\ (!i. &0 < c(i))
  ==> o1_seq (\i. ln(&1 + c(i)*(b(i)/c(i)))/c(i))
  `,
  SIMP_TAC [ln1xo1x]);;

let ln1xo1x_quo_lemma2 = prove(`
  !b:num->real c:num->real.
  (!i. &0 < c(i)) ==>
  (\i. ln(&1 + c(i)*(b(i)/c(i)))) = (\i. ln(&1 + b(i)))
  `,
  MESON_TAC [REAL_FIELD `&0 < c(i:num) ==> b(i) = c(i)*(b(i)/c(i))`;
    EQ_EXT]);;

let ln1xo1x_quo = prove(`
  !b:num->real c:num->real.
  (!i. &0 < b(i))
  /\ (!i. &0 < c(i))
  /\ o1_seq (\i. b(i)/c(i))
  ==> o1_seq (\i. ln(&1 + b(i))/c(i))
  `,
  MESON_TAC [REAL_LT_DIV;
    ln1xo1x_quo_lemma1;
    ln1xo1x_quo_lemma2]);;

let o1_ln1plus_lemma3 = prove(`
  !n:num->real b:num->real c:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq b
  /\ &0 < c
  /\ (!i. &0 < (&1 / n(i)) powreal (c+b(i)))
  /\ (!i. &0 < ln(n(i)))
  ==> o1_seq (\i. ln(&1 + (&1 / n(i)) powreal (c+b(i)))/(ln(n(i))))
  `,
  SIMP_TAC [o1_ln1plus_lemma2;
    ln1xo1x_quo]);;

let o1_ln1plus_lemma4 = prove(`
  !n:num->real b:num->real c:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq b
  /\ &0 < c
  ==> o1_seq (\i. ln(&1 + (&1 / n(i)) powreal (c+b(i)))/(ln(n(i))))
  `,
  MESON_TAC [o1_ln1plus_lemma3;
    ln_gt1;
    powreal_positive]);;

let pow1overbase_times_pow_2 = prove(`
  !x y.
  &0 < x ==>
  (&1 + (&1 / x) powreal y) * (x powreal y)
  = (x powreal y + &1)
  `,
  REWRITE_TAC [REAL_ADD_RDISTRIB] THEN
  MESON_TAC [powreal_cancelbase;
    REAL_ARITH `&1 * z = z`]);;

let pow1overbase_times_pow_3 = prove(`
  !x y.
  &0 < x ==>
  ln(&1 + (&1 / x) powreal y) + ln(x powreal y)
  = ln(x powreal y + &1)
  `,
  MESON_TAC [pow1overbase_times_pow_2;
    LN_MUL;
    powreal_positive;
    pow1overbase_times_pow_2;
    REAL_ARITH `&0 < z ==> &0 < &1 + z`]);;

let pow1overbase_times_pow_4 = prove(`
  !x y.
  &0 < x ==>
  ln(&1 + (&1 / x) powreal y) + y * ln(x)
  = ln(x powreal y + &1)
  `,
  MESON_TAC [pow1overbase_times_pow_3;
    powreal;
    LN_EXP]);;

let pow1overbase_times_pow_5 = prove(`
  !x y.
  &1 < x ==>
  ln(&1 + (&1 / x) powreal y) / ln(x) + y
  = ln(x powreal y + &1) / ln(x)
  `,
  MESON_TAC [pow1overbase_times_pow_4;
    REAL_ARITH `&1 < x ==> &0 < x`;
    ln_gt1;
    REAL_FIELD `&0 < c ==> (a + b*c)/c = a/c + b`]);;

let pow1overbase_times_pow_6 = prove(`
  !x y.
  &1 < x ==>
  ln(&1 + (&1 / x) powreal y) / ln(x)
  = ln(x powreal y + &1) / ln(x) - y
  `,
  MESON_TAC [pow1overbase_times_pow_5;
    REAL_ARITH `a + y = z ==> a = z - y`]);;

let o1_ln1plus_lemma5 = prove(`
  !n:num->real b:num->real c:real.
  (!i. &1 < n(i))
  ==> (\i. ln(&1 + (&1 / n(i)) powreal (c+b(i)))/(ln(n(i))))
  = (\i. ln(n(i) powreal (c+b(i)) + &1)/ln(n(i)) - (c+b(i)))
  `,
  MESON_TAC [EQ_EXT;
    pow1overbase_times_pow_6]);;

let o1_ln1plus_lemma6 = prove(`
  !n:num->real b:num->real c:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq b
  /\ &0 < c
  ==> o1_seq (\i. ln(n(i) powreal (c+b(i)) + &1)/ln(n(i)) - (c+b(i)))
  `,
  MESON_TAC [o1_ln1plus_lemma4;
    o1_ln1plus_lemma5]);;

let o1_ln1plus_lemma7 = prove(`
  !n:num->real b:num->real c:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq b
  /\ &0 < c
  ==> o1_seq (\i. (ln(n(i) powreal (c+b(i)) + &1)/ln(n(i)) - (c+b(i))) + b(i))
  `,
  MESON_TAC [o1_ln1plus_lemma6;
    o1_seq_add_extra]);;

let o1_ln1plus = prove(`
  !n:num->real b:num->real c:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq b
  /\ &0 < c
  ==> o1_seq (\i. ln(n(i) powreal (c+b(i)) + &1)/ln(n(i)) - c)
  `,
  ONCE_REWRITE_TAC [REAL_ARITH `a - c = (a - (c+b(i:num))) + b(i)`] THEN
  SIMP_TAC [o1_ln1plus_lemma7]);;

let o1_ln1plus_basen_lemma = prove(`
  !x c d.
  &1 < x ==>
  x powreal (c + d) + &1
  = x powreal (c + ln (x powreal (c + d) + &1) / ln (x) - c)
  `,
  REWRITE_TAC [powreal] THEN
  REWRITE_TAC [REAL_ARITH `c + m - c = m`] THEN
  SIMP_TAC [ln_gt1;
    REAL_FIELD `&0 < d ==> a / d * d = a`;
    ] THEN
  MESON_TAC [EXP_LN;
    REAL_EXP_POS_LT;
    REAL_ARITH `&0 < e ==> &0 < e + &1`]);;

let o1_ln1plus_basen = prove(`
  !n:num->real f:num->real c:real o1:num->real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < c
  /\ o1_seq o1
  /\ (!i. f(i) = n(i) powreal (c + o1(i)))
  ==>
  ?newo1.
  o1_seq newo1 /\
  (!i. f(i) + &1 = n(i) powreal (c + newo1(i)))
  `,
  REPEAT STRIP_TAC THEN
  EXISTS_TAC `\i:num. ln(n(i) powreal (c+o1(i)) + &1)/ln(n(i)) - c` THEN
  CONJ_TAC THEN
  ASM_SIMP_TAC [o1_ln1plus] THEN
  REWRITE_TAC [BETA_THM] THEN
  ASM_MESON_TAC [o1_ln1plus_basen_lemma]);;

let asymp1_nkss_lemma5 = prove(`
  !n:num->real k:num->real s:num->real S0:real nksso1:num->real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < &1 + &2*S0
  /\ o1_seq nksso1
  /\ (!i. (n(i)+k(i))*(s(i) pow 2)
    = n(i) powreal ((&1 + &2*S0) + nksso1(i)))
  ==>
  ?nkss1o1.
  o1_seq nkss1o1 /\
  (!i. (n(i)+k(i))*(s(i) pow 2) + &1
    = n(i) powreal ((&1 + &2*S0) + nkss1o1(i)))
  `,
  MESON_TAC [ISPECL [`n:num->real`;
      `\i:num. (n(i)+k(i))*(s(i) pow 2):real`;
      `&1 + &2*S0:real`;
      `nksso1:num->real`] o1_ln1plus_basen]);;

let asymp1_nkss_lemma6 = prove(`
  !n:num->real s:num->real S0:real sno1:num->real k:num->real X0:real kno1:num->real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &0 < n(i) + k(i))
  /\ o1_seq kno1
  /\ (!i. k(i) / n(i) = X0 + kno1(i))
  /\ &0 < &1 + X0
  /\ (!i. &0 < s(i))
  /\ o1_seq sno1
  /\ (!i. log2(s(i))/log2(n(i)) = S0 + sno1(i))
  ==> &0 < &1 + &2*S0
  ==> ?nkss1o1.
  o1_seq nkss1o1 /\
  (!i. (n(i)+k(i))*(s(i) pow 2) + &1
    = n(i) powreal ((&1 + &2*S0) + nkss1o1(i)))
  `,
  REPEAT GEN_TAC THEN
  DISCH_TAC THEN
  X_CHOOSE_TAC `o1:num->real` (UNDISCH_ALL (ISPECL [`n:num->real`;
        `s:num->real`;
        `S0:real`;
        `sno1:num->real`;
        `k:num->real`;
        `X0:real`;
        `kno1:num->real`] asymp1_nkss_lemma4)) THEN
  ASM_MESON_TAC [asymp1_nkss_lemma5]);;

let asymp1_nkss_lemma7 = prove(`
  !n x y z.
  &1 < n /\ x = n powreal (y + z)
  ==> ln(x) / ln(n) - y = z
  `,
  REWRITE_TAC [powreal] THEN
  REPEAT STRIP_TAC THEN
  specialize [`(y+z)*ln(n)`] LN_EXP THEN
  specialize [`n:real`] ln_gt1 THEN
  specialize [] (REAL_FIELD `&0 < ln(n) ==> ((y + z) * ln(n)) / ln(n) - y = z`) THEN
  ASM_MESON_TAC []);;

let asymp1_nkss_lemma8 = prove(`
  !n:num->real s:num->real k:num->real S0:real o1:num->real.
  (!i. &1 < n(i)) /\
  (!i. (n(i) + k(i)) * s(i) pow 2 + &1 = n(i) powreal ((&1 + &2 * S0) + o1(i)))
  ==>
  o1 = (\i. ln ((n(i) + k(i)) * s(i) pow 2 + &1) / ln (n(i)) - (&1 + &2 * S0))
  `,
  MESON_TAC [EQ_EXT;
    asymp1_nkss_lemma7]);;

let asymp1_nkss = prove(`
  !n:num->real s:num->real k:num->real S0:real X0:real.
  (!i. &1 < n(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < &1 + &2*S0
  /\ o1_seq (\i. log2(s(i))/log2(n(i)) - S0)
  /\ &0 <= X0
  /\ o1_seq (\i. k(i) / n(i) - X0)
  ==>
  o1_seq (\i. ln((n(i)+k(i))*(s(i) pow 2) + &1) / ln(n(i)) - (&1 + &2*S0))
  `,
  REPEAT STRIP_TAC THEN
  specialize [] (REAL_ARITH `&0 <= X0 ==> &0 < &1 + X0`) THEN
  specialize [`n:num->real`;
    `k:num->real`] (prove (`
      !n:num->real k:num->real. (!i. &1 < n(i)) /\ (!i. &0 < k(i)) ==> (!i. &0 < n(i) + k(i))
      `,
      MESON_TAC [REAL_ARITH `&1 < x /\ &0 < y ==> &0 < x + y`])) THEN
  specialize [] (REAL_ARITH `!i. k(i) / n(i) = X0 + (\i:num. k(i) / n(i) - X0)(i)`) THEN
  specialize [] (REAL_ARITH `!i. log2(s(i))/log2(n(i)) = S0 + (\i:num. log2(s(i))/log2(n(i)) - S0)(i)`) THEN
  choose_specializing `o1:num->real` [`n:num->real`;
    `s:num->real`;
    `S0:real`;
    `\i:num. log2(s(i))/log2(n(i)) - S0`;
    `k:num->real`;
    `X0:real`;
    `\i:num. k(i) / n(i) - X0`] asymp1_nkss_lemma6 THEN
  ASM_MESON_TAC [asymp1_nkss_lemma8]);;

let asymp1_nkss_2_lemma = prove(`
  !n:num->real s:num->real k:num->real S0:real X0:real.
  (\i. &1 / &2 * (ln ((n i + k i) * s i pow 2 + &1) / ln (n i) - (&1 + &2 * S0)))
  =
  (\i. &1 / &2 * ln ((n i + k i) * s i pow 2 + &1) / ln (n i) - (&1 / &2 + S0))
  `,
  MESON_TAC [EQ_EXT;
    REAL_ARITH `(&1 / &2) * (a - (&1 + &2 * b)) = &1 / &2 * a - (&1 / &2 + b)`]);;

let asymp1_nkss_2 = prove(`
  !n:num->real s:num->real k:num->real S0:real X0:real.
  (!i. &1 < n(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < &1 + &2*S0
  /\ o1_seq (\i. log2(s(i))/log2(n(i)) - S0)
  /\ &0 <= X0
  /\ o1_seq (\i. k(i) / n(i) - X0)
  ==>
  o1_seq (\i. (&1 / &2) * ln((n(i)+k(i))*(s(i) pow 2) + &1) / ln(n(i)) - (&1 / &2 + S0))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `s:num->real`;
    `k:num->real`;
    `S0:real`;
    `X0:real`] asymp1_nkss THEN
  specialize [`\i:num. ln((n(i)+k(i))*(s(i) pow 2) + &1) / ln(n(i)) - (&1 + &2*S0)`;
    `&1 / &2`] o1_seq_consttimes THEN
  ASM_MESON_TAC [asymp1_nkss_2_lemma]);;

let o1_logoverlinear_lemma = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &4 * a / (b*b) <= x
  /\ &0 <= ln(a*x) / x
  ==> abs(ln(a*x) / x) <= b
  `,
  MESON_TAC [linear_beats_log_2;
    REAL_ABS_REFL;
    REAL_LT_IMP_LE]);;

let o1_logoverlinear_lemma2 = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &4 * a / (b*b) <= x
  /\ &1 <= a*x
  ==> abs(ln(a*x) / x) <= b
  `,
  MESON_TAC [ln_ge1;
    o1_logoverlinear_lemma;
    linear_beats_log_lemma;
    REAL_LT_IMP_LE;
    REAL_LE_DIV]);;

let o1_logoverlinear_lemma3 = prove(`
  !a:real x:real.
  &0 < a /\ &1 <= a*x
  ==> &0 < &1 / x
  `,
  MESON_TAC [REAL_ARITH `&1 <= a*x ==> &0 < a*x`;
    REAL_LT_DIV;
    REAL_FIELD `&0 < a ==> (a*x)/a = x`;
    pos_1over]);;

let o1_logoverlinear_lemma4 = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &1 <= a*x
  ==> abs(&1 / x) = &1 / x
  `,
  MESON_TAC [o1_logoverlinear_lemma3;
    REAL_LT_IMP_LE;
    REAL_ABS_REFL]);;

let o1_logoverlinear_lemma5 = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &1 <= a*x
  /\ &1 / x <= (b*b) / (&4 * a)
  ==> &4 * a / (b * b) <= x
  `,
  MESON_TAC [o1_logoverlinear_lemma3;
    REAL_LT_IMP_LE;
    REAL_LE_INV2;
    REAL_FIELD `&0 < &1 / x ==> inv(&1 / x) = x`;
    REAL_FIELD `inv((b*b) / (&4 * a)) = &4 * a/(b*b)`]);;

let o1_logoverlinear_lemma6 = prove(`
  !a:real b:real x:real.
  &0 < a /\ &0 < b /\ &1 <= a*x
  /\ abs(&1 / x) <= (b*b) / (&4 * a)
  ==> abs(ln(a*x) / x) <= b
  `,
  MESON_TAC [o1_logoverlinear_lemma2;
    o1_logoverlinear_lemma4;
    o1_logoverlinear_lemma5]);;

let o1_logoverlinear = prove(`
  !a:real f:num->real.
  &0 < a
  /\ (!i. &1 <= a*f(i))
  /\ o1_seq (\i. &1 / f(i))
  ==> o1_seq (\i. ln(a*f(i)) / f(i))
  `,
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  specialize [`a:real`;
    `e:real`] abb_lemma38 THEN
  X_CHOOSE_TAC `a:num` (UNDISCH_ALL (MESON [] `&0 < (e * e) / (&4 * a) ==> (!e. &0 < e ==> (?m:num. !i. m <= i ==> abs (&1 / f i) <= e)) ==> (?m:num. !i. m <= i ==> abs (&1 / f i) <= (e*e) / (&4 * a))`)) THEN
  EXISTS_TAC `a:num` THEN
  ASM_MESON_TAC [o1_logoverlinear_lemma6]);;

let o1_logoverlinear_1 = prove(`
  !f:num->real.
  (!i. &1 <= f(i))
  /\ o1_seq (\i. &1 / f(i))
  ==> o1_seq (\i. ln(f(i)) / f(i))
  `,
  ONCE_REWRITE_TAC [REAL_ARITH `&1 <= x <=> &1 <= &1*x`] THEN
  ONCE_REWRITE_TAC [MESON [REAL_ARITH `x = &1 * x`] `ln(x) = ln(&1 * x)`] THEN
  REPEAT STRIP_TAC THEN
  real_linear `&0 < &1` THEN
  specialize [`&1`;
    `f:num->real`] o1_logoverlinear THEN
  ASM_SIMP_TAC []);;

let o1_log2overlinear_1 = prove(`
  !f:num->real.
  (!i. &1 <= f(i))
  /\ o1_seq (\i. &1 / f(i))
  ==> o1_seq (\i. log2(f(i)) / f(i))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`f:num->real`] o1_logoverlinear_1 THEN
  specialize [`\i:num. ln(f(i))/f(i)`;
    `(&1/ln(&2))`] o1_seq_consttimes THEN
  REWRITE_TAC [log2] THEN
  REWRITE_TAC [REAL_ARITH `ln(a)/b/c = (&1/b)*(ln(a)/c)`] THEN
  ASM_SIMP_TAC []);;

let asymp1_linearinfinity_lemma = prove(`
  !e Y0.
  &0 < Y0 ==> &0 < e ==> &0 < e * Y0 / &2
  `,
  MESON_TAC [REAL_ARITH `&0 < &2`;
    REAL_LT_MUL;
    REAL_LT_DIV]);;

let le_div = prove(`
  !w x y z. &0 < w /\ w <= x /\ &0 <= y /\ y <= z ==> y / x <= z / w
  `,
  MESON_TAC [le_1over;
    REAL_LE_LMUL;
    REAL_ARITH `y * (&1 / x) = y / x`;
    REAL_LE_DIV2_EQ;
    REAL_LE_TRANS]);;

let asymp1_linearinfinity_lemma2 = prove(`
  !x y c d e.
  &0 < y /\
  &0 < c / &2 /\
  x / y = c + d /\
  abs(d) <= c / &2 /\
  &1 / y <= e * c / &2
  ==> &1 / x <= (e * c / &2) / (c / &2)
  `,
  MESON_TAC [pos_1over;
    REAL_LT_IMP_LE;
    REAL_ARITH `abs(d) <= c / &2 ==> c / &2 <= c + d`;
    REAL_FIELD `&0 < y ==> (&1 / y) / (x / y) = &1 / x`;
    le_div]);;

let asymp1_linearinfinity_lemma3 = prove(`
  !x y c d e.
  &0 < y /\
  &0 < c /\
  x / y = c + d /\
  abs(d) <= c / &2 /\
  &1 / y <= e * c / &2
  ==> &1 / x <= e
  `,
  MESON_TAC [asymp1_linearinfinity_lemma2;
    REAL_FIELD `&0 < c ==> &0 < c / &2`;
    REAL_FIELD `&0 < c / &2 ==> e = (e * c / &2) / (c / &2)`]);;

let asymp1_linearinfinity_lemma4 = prove(`
  !x y c d e.
  &0 < x /\
  &0 < y /\
  &0 < c /\
  x / y = c + d /\
  abs(d) <= c / &2 /\
  abs(&1 / y) <= e * c / &2
  ==> abs(&1 / x) <= e
  `,
  MESON_TAC [asymp1_linearinfinity_lemma3;
    pos_1over;
    REAL_LT_IMP_LE;
    REAL_ABS_REFL]);;

let asymp1_linearinfinity = prove(`
  !n:num->real b:num->real Y0:real bno1:num->real.
  (!i. &0 < n(i))
  /\ (!i. &0 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < Y0
  /\ o1_seq bno1
  /\ (!i. b(i) / n(i) = Y0 + bno1(i))
  ==> o1_seq (\i. &1 / b(i))
  `,
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (UNDISCH_ALL (REAL_ARITH `&0 < Y0 ==> &0 < Y0 / &2`)) THEN
  X_CHOOSE_TAC `B:num` (UNDISCH_ALL (MESON [] `&0 < Y0 / &2 ==> (!e. &0 < e ==> (?m:num. !i. m <= i ==> abs (bno1 i) <= e)) ==> (?m:num. !i. m <= i ==> abs (bno1 i) <= Y0 / &2)`)) THEN
  specialize [`e:real`;
    `Y0:real`] asymp1_linearinfinity_lemma THEN
  X_CHOOSE_TAC `N:num` (UNDISCH_ALL (MESON [] `&0 < e * Y0 / &2 ==> (!e. &0 < e ==> (?m:num. !i. m <= i ==> abs (&1 / n i) <= e)) ==> (?m:num. !i. m <= i ==> abs (&1 / n i) <= e * Y0 / &2)`)) THEN
  EXISTS_TAC `MAX B N` THEN
  REWRITE_TAC [max_le] THEN
  ASM_MESON_TAC [asymp1_linearinfinity_lemma4]);;

let asymp1_lnbb = prove(`
  !n:num->real b:num->real Y0:real bno1:num->real.
  (!i. &1 < n(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < Y0
  /\ o1_seq bno1
  /\ (!i. b(i) / n(i) = Y0 + bno1(i))
  ==> o1_seq (\i. ln(pi*b(i)) / b(i))
  `,
  MESON_TAC [pi_333106;
    REAL_ARITH `&333 / &106 < pi ==> &0 < pi`;
    REAL_ARITH `&1 < f(i:num) ==> &0 < f(i):real`;
    ISPECL [`&333 / &106`;
      `pi`;
      `&1`;
      `b(i:num):real`] REAL_LT_MUL2;
    REAL_ARITH `&0 <= &333 / &106`;
    REAL_ARITH `&0 <= &1`;
    REAL_ARITH `(&333 / &106) * &1 < pi*b(i) ==> &1 <= pi*b(i:num)`;
    ISPECL [`pi`;
      `b:num->real`] o1_logoverlinear;
    asymp1_linearinfinity]);;

let asymp1_delta_lemma = prove(`
  !x. &60 <= x ==>
  &2 * (x - &1) * ln(bkzdelta x)
  = ln(x) + ln(pi*x)/x - ln(&2 * pi * exp(&1))
  `,
  MESON_TAC [bkzdelta_ln5;
    REAL_ARITH `&60 <= x ==> &0 < &2 * (x - &1)`;
    REAL_FIELD `&0 < &2 * (x - &1) ==> &2 * (x - &1) * (b / (&2 * (x - &1))) = b`]);;

let asymp1_delta_lemma2 = prove(`
  !x. &60 <= x ==>
  &2 * (x - &1) * ln(bkzdelta x) - ln(x) + ln(&2 * pi * exp(&1))
  = ln(pi*x)/x
  `,
  MESON_TAC [asymp1_delta_lemma;
    REAL_ARITH `(a + b - c) - a + c = b`]);;

let asymp1_delta_lemma3 = prove(`
  !x. &1 < x ==> abs(&1 / x) <= &1 / &60
  ==>
  &2 * (x - &1) * ln(bkzdelta x) - ln(x) + ln(&2 * pi * exp(&1))
  = ln(pi*x)/x
  `,
  MESON_TAC [asymp1_delta_lemma2;
    REAL_ARITH `&1 < x ==> &0 < x`;
    pos_1over;
    REAL_LT_IMP_LE;
    REAL_ABS_REFL;
    le_1over;
    REAL_ARITH `&1 / (&1 / &60) = &60`;
    REAL_FIELD `&0 < x ==> &1 / (&1 / x) = x`]);;

let asymp1_delta_lemma4 = prove(`
  !b:num->real.
  (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / b(i))
  /\ o1_seq (\i. ln(pi*b(i)) / b(i))
  ==>
  o1_seq (\i. &2 * (b(i) - &1) * ln(bkzdelta(b(i))) - ln(b(i)) + ln(&2 * pi * exp(&1)))
  `,
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (REAL_ARITH `&0 < &1 / &60`) THEN
  X_CHOOSE_TAC `B:num` (UNDISCH_ALL (MESON [] `&0 < &1 / &60 ==> (!e. &0 < e ==> (?m:num. !i. m <= i ==> abs (&1 / b i) <= e)) ==> (?m:num. !i. m <= i ==> abs (&1 / b i) <= &1 / &60)`)) THEN
  X_CHOOSE_TAC `A:num` (UNDISCH_ALL (MESON [] `&0 < e ==> (!e. &0 < e ==> (?m:num. !i. m <= i ==> abs (ln(pi*b(i)) / b(i)) <= e)) ==> (?m:num. !i. m <= i ==> abs (ln(pi*b(i)) / b(i)) <= e)`)) THEN
  EXISTS_TAC `MAX A B` THEN
  REWRITE_TAC [max_le] THEN
  ASM_MESON_TAC [asymp1_delta_lemma3]);;

let asymp1_delta_lemma5 = prove(`
  !n:num->real c:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  ==> o1_seq (\i. c/ln(n(i)))
  `,
  ONCE_REWRITE_TAC [REAL_ARITH `x / ln(y) = x * (&1 / ln(y))`] THEN
  MESON_TAC [lnovero1;
    ISPEC `\i:num. &1 / ln(n(i))` o1_seq_consttimes]);;

let asymp1_delta_lemma6 = prove(`
  !r e x.
  &0 < e /\ &1 < x /\ &1 / x <= exp(r / e)
  ==> --ln x <= r / e
  `,
  MESON_TAC [REAL_ARITH `&1 < x ==> &0 < x`;
    pos_1over;
    ln_1over;
    ln_mono_le;
    LN_EXP]);;

let asymp1_delta_lemma7 = prove(`
  !r e x.
  &0 < e /\ &1 < x /\ --ln x <= r / e
  ==> --e <= r / ln x
  `,
  MESON_TAC [REAL_ARITH `--a <= b ==> --b <= a`;
    REAL_LE_RDIV;
    ln_gt1;
    REAL_LT_DIV;
    REAL_LT_IMP_LE;
    REAL_LE_LMUL;
    REAL_FIELD `&0 < e ==> (e/ln(x))*(r/e) = r/ln(x)`;
    REAL_FIELD `&0 < ln(x) ==> (e/ln(x))*(--ln(x)) = --e`]);;

let asymp1_delta_lemma8 = prove(`
  !r e x.
  &0 < e /\ &1 < x /\ &1 / x <= exp(ln(r) / e)
  ==> --e <= ln(r) / ln x
  `,
  MESON_TAC [asymp1_delta_lemma6;
    asymp1_delta_lemma7]);;

let asymp1_delta_lemma9 = prove(`
  !r e x.
  &0 < e /\ &1 < x /\ &1 / x <= exp(-- ln(&3 * r) / e)
  ==> ln(&3 * r) / ln x <= e
  `,
  REWRITE_TAC [REAL_FIELD `-- L / y = (-- L) / y`] THEN
  MESON_TAC [asymp1_delta_lemma6;
    asymp1_delta_lemma7;
    REAL_LE_NEG2;
    REAL_FIELD `(-- L) / D = --(L / D)`]);;

let asymp1_delta_lemma10 = prove(`
  !r e x y.
  &0 < e /\ &1 < x
  /\ &1 / x <= exp(ln(r) / e)
  /\ &1 / x <= exp(-- ln(&3 * r) / e)
  /\ ln(r) / ln x <= y
  /\ y <= ln(&3 * r) / ln x
  ==> abs(y) <= e
  `,
  MESON_TAC [asymp1_delta_lemma8;
    asymp1_delta_lemma9;
    REAL_LE_TRANS;
    REAL_ABS_BOUNDS]);;

let asymp1_delta_lemma11 = prove(`
  !r e x y.
  &0 < e /\ &1 < x
  /\ &1 / x <= min (exp(ln(r) / e)) (exp(-- ln(&3 * r) / e))
  /\ ln(r) / ln x <= y
  /\ y <= ln(&3 * r) / ln x
  ==> abs(y) <= e
  `,
  MESON_TAC [REAL_LE_MIN;
    asymp1_delta_lemma10]);;

let asymp1_delta_lemma12 = prove(`
  !r e x y.
  &0 < e /\ &1 < x
  /\ &1 / x <= min (exp(ln(r) / e)) (exp(-- ln(&3 * r) / e))
  /\ ln(r) / ln x <= y
  /\ y <= ln(&3 * r) / ln x
  ==> abs(y) <= e
  `,
  MESON_TAC [exp_min;
    asymp1_delta_lemma11]);;

let asymp1_delta_lemma13 = prove(`
  !n b c d.
  &1 < n
  /\ &1 < b
  /\ &0 < c
  /\ b / n = c + d
  /\ abs(d) <= c / &2
  ==> ln(c / &2) <= ln b - ln n
  `,
  MESON_TAC [REAL_ARITH `abs(d) <= c / &2 ==> c / &2 <= c + d`;
    REAL_ARITH `&0 < c ==> &0 < c / &2`;
    REAL_ARITH `&1 < y ==> &0 < y`;
    ln_mono_le;
    LN_DIV]);;

let asymp1_delta_lemma14 = prove(`
  !n b c d.
  &1 < n
  /\ &1 < b
  /\ b / n = c + d
  /\ abs(d) <= c / &2
  ==> ln b - ln n <= ln(&3 * c / &2)
  `,
  MESON_TAC [REAL_ARITH `abs(d) <= c / &2 ==> c + d <= &3 * c / &2`;
    REAL_ARITH `&1 < y ==> &0 < y`;
    REAL_LT_DIV;
    ln_mono_le;
    LN_DIV]);;

let asymp1_delta_lemma15 = prove(`
  !n b c d.
  &1 < n
  /\ &1 < b
  /\ &0 < c
  /\ b / n = c + d
  /\ abs(d) <= c / &2
  ==> ln(c / &2) / ln n <= (ln b - ln n) / ln n
  `,
  MESON_TAC [asymp1_delta_lemma13;
    REAL_LE_DIV2_EQ;
    ln_gt1]);;

let asymp1_delta_lemma16 = prove(`
  !n b c d.
  &1 < n
  /\ &1 < b
  /\ b / n = c + d
  /\ abs(d) <= c / &2
  ==> (ln b - ln n) / ln n <= ln(&3 * c / &2) / ln n
  `,
  MESON_TAC [asymp1_delta_lemma14;
    REAL_LE_DIV2_EQ;
    ln_gt1]);;

let asymp1_delta_lemma17 = prove(`
  !n b c e.
  &1 < n
  /\ &1 < b
  /\ &0 < c
  /\ &0 < e
  /\ b / n = c + d
  /\ abs(d) <= c / &2
  /\ &1 / n <= min (exp(ln(c / &2) / e)) (exp(-- ln(&3 * (c / &2)) / e))
  ==> abs((ln b - ln n) / ln n) <= e
  `,
  MESON_TAC [REAL_ARITH `&0 < c ==> &0 < c / &2`;
    asymp1_delta_lemma12;
    asymp1_delta_lemma15;
    asymp1_delta_lemma16]);;

let asymp1_delta_lemma18 = prove(`
  !n b c e.
  &1 < n
  /\ &1 < b
  /\ &0 < c
  /\ &0 < e
  /\ b / n = c + d
  /\ abs(d) <= c / &2
  /\ abs(&1 / n) <= min (exp(ln(c / &2) / e)) (exp(-- ln(&3 * (c / &2)) / e))
  ==> abs((ln b - ln n) / ln n) <= e
  `,
  MESON_TAC [asymp1_delta_lemma17;
    REAL_ARITH `&1 < n ==> &0 < n`;
    pos_1over;
    REAL_LT_IMP_LE;
    REAL_ABS_REFL]);;

let asymp1_delta_lemma19 = prove(`
  !n b c e.
  &1 < n
  /\ &1 < b
  /\ &0 < c
  /\ &0 < e
  /\ b / n = c + d
  /\ abs(d) <= c / &2
  /\ abs(&1 / n) <= min (exp(ln(c / &2) / e)) (exp(-- ln(&3 * (c / &2)) / e))
  ==> abs((ln b)/ln(n) - &1) <= e
  `,
  MESON_TAC [asymp1_delta_lemma18;
    ln_gt1;
    REAL_FIELD `&0 < D ==> (a - D) / D = a/D - &1`]);;

let minexpexp_pos = prove(`
  !x y. &0 < min (exp(x)) (exp(y))
  `,
  MESON_TAC [exp_min;
    REAL_EXP_POS_LT]);;

let asymp1_delta_lemma20 = prove(`
  !n:num->real b:num->real Y0:real bno1:num->real.
  (!i. &1 < n(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < Y0
  /\ o1_seq bno1
  /\ (!i. b(i) / n(i) = Y0 + bno1(i))
  ==> o1_seq (\i. ln(b(i))/ln(n(i)) - &1)
  `,
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (UNDISCH_ALL (REAL_ARITH `&0 < Y0 ==> &0 < Y0 / &2`)) THEN
  X_CHOOSE_TAC `B:num` (UNDISCH_ALL (MESON [] `&0 < Y0 / &2 ==> (!e. &0 < e ==> (?m:num. !i. m <= i ==> abs (bno1 i) <= e)) ==> (?m:num. !i. m <= i ==> abs (bno1 i) <= Y0 / &2)`)) THEN
  ASSUME_TAC (ISPECL [`ln(Y0 / &2) / e`;
      `-- ln(&3 * (Y0 / &2)) / e`] minexpexp_pos) THEN
  X_CHOOSE_TAC `A:num` (UNDISCH_ALL (MESON [] `&0 < min (exp(ln(Y0 / &2) / e)) (exp(-- ln(&3 * (Y0 / &2)) / e)) ==> (!e. &0 < e ==> (?m:num. !i. m <= i ==> abs (&1 / n i) <= e)) ==> (?m:num. !i. m <= i ==> abs (&1 / n i) <= min (exp(ln(Y0 / &2) / e)) (exp(-- ln(&3 * (Y0 / &2)) / e)))`)) THEN
  EXISTS_TAC `MAX A B` THEN
  REWRITE_TAC [max_le] THEN
  ASM_MESON_TAC [asymp1_delta_lemma19]);;

let asymp1_delta_lemma21 = prove(`
  !n:num->real b:num->real.
  (!i. &0 < ln(n(i))) ==>
  (\i. &2 * (b i - &1) * ln (bkzdelta (b i)) / ln (n i) - &1)
  = (\i. (&2 * (b i - &1) * ln (bkzdelta (b i)) - ln (b i) + ln (&2 * pi * exp (&1))) * &1 / ln (n i) + ln (b i) / ln (n i) - &1 - ln (&2 * pi * exp (&1)) / ln (n i))
  `,
  MESON_TAC [EQ_EXT;
    REAL_FIELD `&0 < ln(n(i:num)) ==> &2 * (b(i) - &1) * ln(bkzdelta(b(i))) / ln(n(i)) - &1 = (&2 * (b(i) - &1) * ln(bkzdelta(b(i))) - ln(b(i)) + ln(&2 * pi * exp(&1))) * (&1 / ln(n(i))) + (ln(b(i)) / ln(n(i)) - &1) - (ln(&2 * pi * exp(&1)) / ln(n(i)))`]);;

let asymp1_delta = prove(`
  !n:num->real b:num->real Y0:real bno1:num->real.
  (!i. &1 < n(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < Y0
  /\ o1_seq bno1
  /\ (!i. b(i) / n(i) = Y0 + bno1(i))
  ==>
  o1_seq (\i. &2 * (b(i) - &1) * ln(bkzdelta(b(i))) / ln(n(i)) - &1)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`] (prove (`
      !n:num->real. (!i. &1 < n(i)) ==> (!i. &0 < n(i))
      `,
      MESON_TAC [REAL_ARITH `&1 < x ==> &0 < x`])) THEN
  specialize [`b:num->real`] (prove (`
      !n:num->real. (!i. &1 < n(i)) ==> (!i. &0 < n(i))
      `,
      MESON_TAC [REAL_ARITH `&1 < x ==> &0 < x`])) THEN
  specialize [`n:num->real`] (prove (`
      !n:num->real. (!i. &1 < n(i)) ==> (!i. &0 < ln(n(i)))
      `,
      MESON_TAC [ln_gt1])) THEN
  specialize [`n:num->real`;
    `b:num->real`;
    `Y0:real`;
    `bno1:num->real`] asymp1_linearinfinity THEN
  specialize [`n:num->real`;
    `b:num->real`;
    `Y0:real`;
    `bno1:num->real`] asymp1_lnbb THEN
  specialize [`b:num->real`] asymp1_delta_lemma4 THEN
  specialize [`n:num->real`;
    `ln(&2 * pi * exp(&1))`] asymp1_delta_lemma5 THEN
  specialize [`n:num->real`;
    `b:num->real`;
    `Y0:real`;
    `bno1:num->real`] asymp1_delta_lemma20 THEN
  specialize [`n:num->real`] lnovero1 THEN
  specialize [`\i:num. &2 * (b(i) - &1) * ln (bkzdelta (b i)) - ln (b i) + ln (&2 * pi * exp (&1))`;
    `\i:num. &1 / ln(n(i))`] o1_seq_mul THEN
  specialize [`\i:num. ln (b i) / ln (n i) - &1`;
    `\i:num. ln (&2 * pi * exp (&1)) / ln (n i)`] o1_seq_sub THEN
  specialize [`\i:num. (&2 * (b i - &1) * ln (bkzdelta (b i)) - ln (b i) + ln (&2 * pi * exp (&1))) * &1 / ln (n i)`;
    `\i:num. ln (b i) / ln (n i) - &1 - ln (&2 * pi * exp (&1)) / ln (n i)`] o1_seq_add THEN
  ASM_MESON_TAC [asymp1_delta_lemma21]);;

let asymp1_b1n_lemma = prove(`
  !n:num->real b:num->real Y0:real bno1:num->real.
  (!i. &1 < n(i))
  /\ (!i. b(i)/n(i) = Y0 + bno1(i))
  ==>
  (\i. (b(i) - &1) / n(i) - Y0)
  =
  (\i. bno1(i) - &1 / n(i))
  `,
  MESON_TAC [EQ_EXT;
    REAL_FIELD `&1 < x /\ y/x = c+d ==> (y - &1)/x - c = d - &1 / x`]);;

let asymp1_b1n = prove(`
  !n:num->real b:num->real Y0:real bno1:num->real.
  (!i. &1 < n(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < Y0
  /\ o1_seq bno1
  /\ (!i. b(i) / n(i) = Y0 + bno1(i))
  ==>
  o1_seq (\i. (b(i) - &1)/n(i) - Y0)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`bno1:num->real`;
    `\i:num. &1 / n(i)`] o1_seq_sub THEN
  ASM_MESON_TAC [asymp1_b1n_lemma]);;

let asymp1_nkb1_lemma = prove(`
  !n:num->real k:num->real b:num->real X0:real kno1:num->real Y0:real o1:num->real.
  (!i. &1 < n(i))
  /\ (!i. k(i)/n(i) = X0 + kno1(i))
  /\ ~(Y0 = &0)
  /\ (!i. ((&1 + X0) + kno1(i)) / (Y0 + (b(i) - &1) / n(i) - Y0) = (&1 + X0) / Y0 + o1(i))
  ==> !i. o1(i) = (n(i) + k(i)) / (b(i) - &1) - (&1 + X0) / Y0
  `,
  REWRITE_TAC [REAL_ARITH `Y0 + x - Y0 = x`] THEN
  MESON_TAC [REAL_FIELD `~(Y0 = &0) /\ &1 < n /\ k/n = X0 + d /\ ((&1 + X0) + d) / ((b - &1)/n) = (&1 + X0) / Y0 + e ==> e = (n + k) / (b - &1) - (&1 + X0) / Y0`]);;

let asymp1_nkb1_lemma2 = prove(`
  !n:num->real k:num->real b:num->real X0:real kno1:num->real Y0:real o1:num->real.
  (!i. &1 < n(i))
  /\ (!i. k(i)/n(i) = X0 + kno1(i))
  /\ ~(Y0 = &0)
  /\ (!i. ((&1 + X0) + kno1(i)) / (Y0 + (b(i) - &1) / n(i) - Y0) = (&1 + X0) / Y0 + o1(i))
  ==> o1 = \i. (n(i) + k(i)) / (b(i) - &1) - (&1 + X0) / Y0
  `,
  MESON_TAC [BETA_THM;
    EQ_EXT;
    asymp1_nkb1_lemma]);;

let asymp1_nkb1 = prove(`
  !n:num->real k:num->real b:num->real X0:real kno1:num->real Y0:real bno1:num->real.
  (!i. &1 < n(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < Y0
  /\ o1_seq kno1
  /\ (!i. k(i) / n(i) = X0 + kno1(i))
  /\ o1_seq bno1
  /\ (!i. b(i) / n(i) = Y0 + bno1(i))
  ==>
  o1_seq (\i. (n(i) + k(i))/(b(i) - &1) - (&1 + X0)/Y0)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `b:num->real`;
    `Y0:real`;
    `bno1:num->real`] asymp1_b1n THEN
  specialize [] (REAL_ARITH `&0 < Y0 ==> ~(Y0 = &0)`) THEN
  choose_specializing `o1:num->real` [`&1 + X0`;
    `kno1:num->real`;
    `Y0:real`;
    `\i:num. (b(i) - &1)/n(i) - Y0`] o1_seq_offset_div THEN
  ASM_MESON_TAC [asymp1_nkb1_lemma2]);;

let o1_seq_neg = prove(`
  !o1:num->real.
   o1_seq o1
   ==> o1_seq (\i. -- o1(i))
  `,
  REWRITE_TAC [o1_seq] THEN
  REWRITE_TAC [REAL_ABS_NEG]);;

let asymp1_nkb1_neg = prove(`
  !n:num->real k:num->real b:num->real X0:real kno1:num->real Y0:real bno1:num->real.
  (!i. &1 < n(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < Y0
  /\ o1_seq kno1
  /\ (!i. k(i) / n(i) = X0 + kno1(i))
  /\ o1_seq bno1
  /\ (!i. b(i) / n(i) = Y0 + bno1(i))
  ==>
  o1_seq (\i. (&1 + X0)/Y0 - (n(i) + k(i))/(b(i) - &1))
  `,
  REWRITE_TAC [REAL_ARITH `(&1 + a)/b - c/d = --(c/d - (&1 + a)/b)`] THEN
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `b:num->real`;
    `X0:real`;
    `kno1:num->real`;
    `Y0:real`;
    `bno1:num->real`] asymp1_nkb1 THEN
  specialize [`\i:num. (n(i) + k(i))/(b(i) - &1) - (&1 + X0)/Y0`] o1_seq_neg THEN
  ASM_MESON_TAC []);;

let asymp1_rho_lemma1 = prove(`
  !n:num->real k:num->real b:num->real X0:real Y0:real o1:num->real.
  (!i. &1 < b(i)) /\
  (!i.
  (
    (&2 - (&1 + X0)/Y0)
    + ((&1 + X0)/Y0 - (n(i) + k(i))/(b(i) - &1))
  )
  *
  (
    &1
    + (&2 * (b(i) - &1) * ln(bkzdelta(b(i))) / ln(n(i)) - &1)
  )
  = (&2 - (&1 + X0) / Y0) * &1 + o1(i)
  )
  ==>
  !i.
  &2 * (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i))) / ln(n(i))
  - (&2 - (&1 + X0) / Y0) = o1(i)
  `,
  REWRITE_TAC [REAL_ARITH `(&2 - c) + (c - x) = &2 - x`] THEN
  REWRITE_TAC [REAL_ARITH `&1 + (c - &1) = c`] THEN
  MESON_TAC [REAL_FIELD `&1 < b ==> (&2 - (n + k)/(b - &1)) * &2 * (b - &1) * c / d = &2 * (&2 * b - (n + k + &1) - &1) * c / d`;
    REAL_ARITH `a = b * &1 + c ==> a - b = c`]);;

let asymp1_rho_lemma2 = prove(`
  !n:num->real k:num->real b:num->real X0:real Y0:real o1:num->real.
  (!i. &1 < b(i)) /\
  (!i.
  (
    (&2 - (&1 + X0)/Y0)
    + ((&1 + X0)/Y0 - (n(i) + k(i))/(b(i) - &1))
  )
  *
  (
    &1
    + (&2 * (b(i) - &1) * ln(bkzdelta(b(i))) / ln(n(i)) - &1)
  )
  = (&2 - (&1 + X0) / Y0) * &1 + o1(i)
  )
  ==>
  o1 = \i.
  &2 * (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i))) / ln(n(i))
  - (&2 - (&1 + X0) / Y0)
  `,
  MESON_TAC [asymp1_rho_lemma1;
    BETA_THM;
    EQ_EXT]);;

let asymp1_rho_lemma3 = prove(`
  !n:num->real k:num->real b:num->real X0:real kno1:num->real Y0:real bno1:num->real.
  (!i. &1 < n(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < Y0
  /\ o1_seq kno1
  /\ (!i. k(i) / n(i) = X0 + kno1(i))
  /\ o1_seq bno1
  /\ (!i. b(i) / n(i) = Y0 + bno1(i))
  ==>
  o1_seq \i. &2 * (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i))) / ln(n(i)) - (&2 - (&1 + X0) / Y0)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `b:num->real`;
    `Y0:real`;
    `bno1:num->real`] asymp1_delta THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `b:num->real`;
    `X0:real`;
    `kno1:num->real`;
    `Y0:real`;
    `bno1:num->real`] asymp1_nkb1_neg THEN
  choose_specializing `o1:num->real` [`&2 - (&1 + X0)/Y0`;
    `\i:num. (&1 + X0) / Y0 - (n i + k i) / (b i - &1)`;
    `&1`;
    `\i:num. &2 * (b i - &1) * ln (bkzdelta (b i)) / ln (n i) - &1`] o1_seq_offset_mul THEN
  ASM_MESON_TAC [asymp1_rho_lemma2]);;

let asymp1_rho_lemma4 = prove(`
  !n:num->real k:num->real b:num->real X0:real kno1:num->real Y0:real bno1:num->real.
  (!i. &1 < n(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < Y0
  /\ o1_seq kno1
  /\ (!i. k(i) / n(i) = X0 + kno1(i))
  /\ o1_seq bno1
  /\ (!i. b(i) / n(i) = Y0 + bno1(i))
  ==>
  o1_seq \i. (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i))) / ln(n(i)) - (&1 - (&1 + X0) / (&2 * Y0))
  `,
  ONCE_REWRITE_TAC [REAL_FIELD `a * b / c - (&1 - d / (&2 * e)) = (&1 / &2) * (&2 * a * b / c - (&2 - d / e))`] THEN
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `b:num->real`;
    `X0:real`;
    `kno1:num->real`;
    `Y0:real`;
    `bno1:num->real`] asymp1_rho_lemma3 THEN
  specialize [`\i:num. &2 * (&2 * b i - (n i + k i + &1) - &1) * ln (bkzdelta (b i)) / ln (n i) - (&2 - (&1 + X0) / Y0)`;
    `&1 / &2`] o1_seq_consttimes THEN
  ASM_MESON_TAC []);;

let asymp1_rho_lemma5 = prove(`
  !n:num->real k:num->real b:num->real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < Y0
  /\ o1_seq (\i. k(i) / n(i) - X0)
  /\ o1_seq (\i. b(i) / n(i) - Y0)
  ==>
  o1_seq \i. (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i))) / ln(n(i)) - (&1 - (&1 + X0) / (&2 * Y0))
  `,
  REPEAT STRIP_TAC THEN
  specialize [] (REAL_ARITH `!i. k(i) / n(i) = X0 + (\i:num. k(i) / n(i) - X0)(i)`) THEN
  specialize [] (REAL_ARITH `!i. b(i) / n(i) = Y0 + (\i:num. b(i) / n(i) - Y0)(i)`) THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `b:num->real`;
    `X0:real`;
    `\i:num. k(i) / n(i) - X0`;
    `Y0:real`;
    `\i:num. b(i) / n(i) - Y0`] asymp1_rho_lemma4 THEN
  ASM_SIMP_TAC []);;

let asymp1_kqdn_lemma = prove(`
  !n:num->real q:num->real k:num->real Q0:real X0:real.
  (!i. &1 < n(i))
  /\ (!i. &0 < k(i))
  ==>
  (\i. (k(i) / (n(i) + k(i) + &1)) * ln(q(i))/ln(n(i)) - (X0 / (&1 + X0)) * Q0)
  = (\i. k(i) / (n(i) + k(i) + &1) * log2(q(i))/log2(n(i)) - X0 / (&1 + X0) * Q0)
  `,
  REWRITE_TAC [log2ratio_lnratio] THEN
  MESON_TAC []);;

let asymp1_kqdn = prove(`
  !n:num->real q:num->real k:num->real Q0:real X0:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &1 < q(i))
  /\ o1_seq (\i. log2(q(i))/log2(n(i)) - Q0)
  /\ (!i. &0 < k(i))
  /\ &0 <= X0
  /\ o1_seq (\i. k(i)/n(i) - X0)
  ==> 
  o1_seq (\i. (k(i) / (n(i) + k(i) + &1)) * ln(q(i))/ln(n(i)) - (X0 / (&1 + X0)) * Q0)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `X0:real`] asymp1_kd THEN
  specialize [`\i:num. k(i) / (n(i) + k(i) + &1)`;
    `X0 / (&1 + X0)`;
    `\i:num. log2(q(i))/log2(n(i))`;
    `Q0:real`] o1_seq_offset_mul2 THEN
  ASM_MESON_TAC [asymp1_kqdn_lemma]);;

let asymp1_rho_lemma6 = prove(`
  !n:num->real q:num->real k:num->real b:num->real Q0:real X0:real Y0:real.
  (\i. (&2 * b i - (n i + k i + &1) - &1) * ln (bkzdelta (b i)) / ln (n i) - (&1 - (&1 + X0) / (&2 * Y0)) + k i / (n i + k i + &1) * ln (q i) / ln (n i) - X0 / (&1 + X0) * Q0)
  =
  (\i. ((&2 * b i - (n i + k i + &1) - &1) * ln (bkzdelta (b i)) / ln (n i) + k i / (n i + k i + &1) * ln (q i) / ln (n i)) - (&1 - (&1 + X0) / (&2 * Y0) + X0 / (&1 + X0) * Q0))
  `,
  MESON_TAC [EQ_EXT;
    REAL_ARITH `(a-b)+c-d = (a+c)-(b+d)`]);;

let asymp1_rho_lemma7 = prove(`
  !n:num->real q:num->real k:num->real b:num->real Q0:real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. log2(q(i))/log2(n(i)) - Q0)
  /\ &0 <= X0
  /\ o1_seq (\i. k(i)/n(i) - X0)
  /\ &0 < Y0
  /\ o1_seq (\i. b(i)/n(i) - Y0)
  ==>
  o1_seq \i.
    ( (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i))) / ln(n(i))
      + (k(i) / (n(i) + k(i) + &1)) * ln(q(i)) / ln(n(i))
    ) - (
      (&1 - (&1 + X0) / (&2 * Y0))
      + (X0 / (&1 + X0)) * Q0
    )
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `q:num->real`;
    `k:num->real`;
    `Q0:real`;
    `X0:real`] asymp1_kqdn THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `b:num->real`;
    `X0:real`;
    `Y0:real`] asymp1_rho_lemma5 THEN
  specialize [`\i:num. (&2 * b i - (n i + k i + &1) - &1) * ln (bkzdelta (b i)) / ln (n i) - (&1 - (&1 + X0) / (&2 * Y0))`;
    `\i:num. k i / (n i + k i + &1) * ln (q i) / ln (n i) - X0 / (&1 + X0) * Q0`] o1_seq_add THEN
  ASM_MESON_TAC [asymp1_rho_lemma6]);;

let o1_seq_timesconvergent_lemma = prove(`
  !x:real y:real e:real c:real.
  abs(x - c) <= &1
  /\ abs(y) <= e / (&1 + abs(c))
  ==> abs(x * y) <= e
  `,
  REWRITE_TAC [REAL_ABS_MUL] THEN
  MESON_TAC [REAL_ABS_TRIANGLE;
    REAL_ARITH `abs((x-c)+c) = abs(x)`;
    REAL_ARITH `abs(x-c) <= &1 ==> abs(x-c)+abs(c) <= &1+abs(c)`;
    REAL_LE_TRANS;
    REAL_ABS_POS;
    REAL_FIELD `(&1 + abs(c)) * (e / (&1 + abs(c))) = e`;
    REAL_LE_MUL2]);;

let o1_seq_timesconvergent = prove(`
  !f:num->real c:real g:num->real.
  o1_seq (\i. f(i) - c)
  /\ o1_seq g
  ==> o1_seq (\i. f(i)*g(i))
  `,
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  specialize [] (REAL_ARITH `&0 < &1`) THEN
  specialize [] (REAL_ARITH `&0 < &1 + abs(c)`) THEN
  specialize [`e:real`;
    `&1 + abs(c)`] REAL_LT_DIV THEN
  X_CHOOSE_TAC `A:num` (UNDISCH_ALL (MESON [] `&0 < &1 ==> (!e. &0 < e ==> (?m:num. !i. m <= i ==> abs (f(i) - c) <= e)) ==> (?m:num. !i. m <= i ==> abs (f(i) - c) <= &1)`)) THEN
  X_CHOOSE_TAC `B:num` (UNDISCH_ALL (MESON [] `&0 < e / (&1 + abs(c)) ==> (!e. &0 < e ==> (?m:num. !i. m <= i ==> abs (g(i)) <= e)) ==> (?m:num. !i. m <= i ==> abs (g(i)) <= e / (&1 + abs(c)))`)) THEN
  EXISTS_TAC `MAX A B` THEN
  REWRITE_TAC [max_le] THEN
  ASM_MESON_TAC [o1_seq_timesconvergent_lemma]);;

let o1_seq_compose_ln = prove(`
  !f:num->real c.
  &0 < c
  /\ o1_seq (\i. f(i) - c)
  ==> o1_seq (\i. ln(f(i)) - ln(c))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`c:real`] ln_continuous THEN
  specialize [`f:num->real`;
    `ln`;
    `c:real`] o1_seq_compose_offset THEN
  ASM_SIMP_TAC []);;

let o1_seq_compose_log2 = prove(`
  !f:num->real c.
  &0 < c
  /\ o1_seq (\i. f(i) - c)
  ==> o1_seq (\i. log2(f(i)) - log2(c))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`c:real`] log2_continuous THEN
  specialize [`f:num->real`;
    `log2`;
    `c:real`] o1_seq_compose_offset THEN
  ASM_SIMP_TAC []);;

let o1_seq_timesconvergent_ln = prove(`
  !f:num->real c:real g:num->real.
  o1_seq (\i. f(i) - c)
  /\ &0 < c
  /\ o1_seq g
  ==> o1_seq (\i. ln(f(i))*g(i))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`f:num->real`;
    `c:real`] o1_seq_compose_ln THEN
  specialize [`\i:num. ln(f(i))`;
    `ln(c)`;
    `g:num->real`] o1_seq_timesconvergent THEN
  ASM_SIMP_TAC []);;

let asymp0_db = prove(`
  !n:num->real k:num->real b:num->real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. k(i) / n(i) - X0)
  /\ o1_seq (\i. b(i) / n(i) - Y0)
  /\ &0 <= X0
  /\ &0 < Y0
  ==>
  o1_seq (\i. ln((n(i) + k(i) + &1) / b(i))/ln(n(i)))
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [REAL_FIELD `ln(x)/ln(y) = ln(x) * (&1 / ln(y))`] THEN
  specialize [] (REAL_ARITH `&0 < Y0 ==> ~(Y0 = &0)`) THEN
  specialize [] (REAL_ARITH `&0 <= X0 ==> &0 < &1 + X0`) THEN
  specialize [`&1 + X0`;
    `Y0:real`] REAL_LT_DIV THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `b:num->real`;
    `X0:real`;
    `Y0:real`] asymp1_db THEN
  specialize [`n:num->real`] lnovero1 THEN
  specialize [`\i:num. (n(i)+k(i) + &1)/b(i)`;
    `(&1 + X0) / Y0`;
    `\i:num. &1 / ln(n(i))`] o1_seq_timesconvergent_ln THEN
  ASM_SIMP_TAC []);;

let asymp0_db_half = prove(`
  !n:num->real k:num->real b:num->real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. k(i) / n(i) - X0)
  /\ o1_seq (\i. b(i) / n(i) - Y0)
  /\ &0 <= X0
  /\ &0 < Y0
  ==>
  o1_seq (\i. (&1 / &2) * ln((n(i) + k(i) + &1) / b(i))/ln(n(i)))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `b:num->real`;
    `X0:real`;
    `Y0:real`] asymp0_db THEN
  specialize [`\i:num. ln((n(i) + k(i) + &1) / b(i))/ln(n(i))`;
    `&1 / &2`] o1_seq_consttimes THEN
  ASM_SIMP_TAC []);;

let asymp1_rho_lemma8 = prove(`
  !n:num->real q:num->real k:num->real b:num->real Q0:real X0:real Y0:real.
  (\i. (&1 / &2) * ln ((n i + k i + &1) / b i) / ln (n i) + ((&2 * b i - (n i + k i + &1) - &1) * ln (bkzdelta (b i)) / ln (n i) + k i / (n i + k i + &1) * ln (q i) / ln (n i)) - (&1 - (&1 + X0) / (&2 * Y0) + X0 / (&1 + X0) * Q0))
  = 
  (\i. ((&1 / &2) * ln ((n i + k i + &1) / b i) / ln (n i) + (&2 * b i - (n i + k i + &1) - &1) * ln (bkzdelta (b i)) / ln (n i) + k i / (n i + k i + &1) * ln (q i) / ln (n i)) - (&1 - (&1 + X0) / (&2 * Y0) + X0 / (&1 + X0) * Q0))
  `,
  MESON_TAC [EQ_EXT;
    REAL_ARITH `a + (b - c) = (a + b) - c`]);;

let asymp1_rho_lemma9 = prove(`
  !n:num->real q:num->real k:num->real b:num->real Q0:real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. log2(q(i))/log2(n(i)) - Q0)
  /\ &0 <= X0
  /\ o1_seq (\i. k(i)/n(i) - X0)
  /\ &0 < Y0
  /\ o1_seq (\i. b(i)/n(i) - Y0)
  ==>
  o1_seq \i.
    ( (&1 / &2) * ln((n(i) + k(i) + &1) / b(i)) / ln(n(i))
      + (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i))) / ln(n(i))
      + (k(i) / (n(i) + k(i) + &1)) * ln(q(i)) / ln(n(i))
    ) - (
      (&1 - (&1 + X0) / (&2 * Y0))
      + (X0 / (&1 + X0)) * Q0
    )
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `q:num->real`;
    `k:num->real`;
    `b:num->real`;
    `Q0:real`;
    `X0:real`;
    `Y0:real`] asymp1_rho_lemma7 THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `b:num->real`;
    `X0:real`;
    `Y0:real`] asymp0_db_half THEN
  specialize [`\i:num. (&1 / &2) * ln((n(i) + k(i) + &1) / b(i))/ln(n(i))`;
    `\i:num. ( (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i))) / ln(n(i)) + (k(i) / (n(i) + k(i) + &1)) * ln(q(i)) / ln(n(i))) - ( (&1 - (&1 + X0) / (&2 * Y0)) + (X0 / (&1 + X0)) * Q0)`] o1_seq_add THEN
  ASM_MESON_TAC [asymp1_rho_lemma8]);;

let asymp1_rho_lemma10 = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real Q0:real S0:real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. log2(q(i))/log2(n(i)) - Q0)
  /\ &0 < &1 + &2*S0
  /\ o1_seq (\i. log2(s(i))/log2(n(i)) - S0)
  /\ &0 <= X0
  /\ o1_seq (\i. k(i)/n(i) - X0)
  /\ &0 < Y0
  /\ o1_seq (\i. b(i)/n(i) - Y0)
  ==>
  o1_seq \i.
    ( (&1 / &2) * ln((n(i)+k(i))*(s(i) pow 2) + &1) / ln(n(i))
      - (
          (&1 / &2) * ln((n(i) + k(i) + &1) / b(i)) / ln(n(i))
          + (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i))) / ln(n(i))
          + (k(i) / (n(i) + k(i) + &1)) * ln(q(i)) / ln(n(i))
        )
    ) - (
      (&1 / &2 + S0)
      - (
          (&1 - (&1 + X0) / (&2 * Y0))
          + (X0 / (&1 + X0)) * Q0
        )
    )
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `q:num->real`;
    `k:num->real`;
    `b:num->real`;
    `Q0:real`;
    `X0:real`;
    `Y0:real`] asymp1_rho_lemma9 THEN
  specialize [`n:num->real`;
    `s:num->real`;
    `k:num->real`;
    `S0:real`;
    `X0:real`] asymp1_nkss_2 THEN
  specialize [`\i:num. (&1 / &2) * ln((n(i)+k(i))*(s(i) pow 2) + &1) / ln(n(i))`;
    `&1 / &2 + S0`;
    `\i:num. (&1 / &2) * ln((n(i) + k(i) + &1) / b(i)) / ln(n(i)) + (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i))) / ln(n(i)) + (k(i) / (n(i) + k(i) + &1)) * ln(q(i)) / ln(n(i))`;
    `(&1 - (&1 + X0) / (&2 * Y0)) + (X0 / (&1 + X0)) * Q0`] o1_seq_sub_offset THEN
  ASM_SIMP_TAC []);;

let asymp1_rho_lemma11 = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real Q0:real S0:real X0:real Y0:real.
  (!i. &1 < n(i))
  ==>
  (\i.
    ( (&1 / &2) * ln((n(i)+k(i))*(s(i) pow 2) + &1)
      - (
          (&1 / &2) * ln((n(i) + k(i) + &1) / b(i))
          + (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i)))
          + (k(i) / (n(i) + k(i) + &1)) * ln(q(i))
        )
    ) / ln(n(i)) - (
      S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0
    )
  )
  =
  (\i.
    ( (&1 / &2) * ln((n(i)+k(i))*(s(i) pow 2) + &1) / ln(n(i))
      - (
          (&1 / &2) * ln((n(i) + k(i) + &1) / b(i)) / ln(n(i))
          + (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i))) / ln(n(i))
          + (k(i) / (n(i) + k(i) + &1)) * ln(q(i)) / ln(n(i))
        )
    ) - (
      (&1 / &2 + S0)
      - (
          (&1 - (&1 + X0) / (&2 * Y0))
          + (X0 / (&1 + X0)) * Q0
        )
    )
  )
  `,
  MESON_TAC [EQ_EXT;
    ln_gt1;
    REAL_FIELD `&0 < a ==> C*c/a - (D*d/a + E*e/a + G*g/a) = (C*c - (D*d + E*e + G*g))/a`;
    REAL_ARITH `(&1 / &2 + S) - ((&1 - a) + b) = S - &1 / &2 + a - b`]);;

let asymp1_rho = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real Q0:real S0:real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. log2(q(i))/log2(n(i)) - Q0)
  /\ &0 < &1 + &2*S0
  /\ o1_seq (\i. log2(s(i))/log2(n(i)) - S0)
  /\ &0 <= X0
  /\ o1_seq (\i. k(i)/n(i) - X0)
  /\ &0 < Y0
  /\ o1_seq (\i. b(i)/n(i) - Y0)
  ==>
  o1_seq \i.
    ( (&1 / &2) * ln((n(i)+k(i))*(s(i) pow 2) + &1)
      - (
          (&1 / &2) * ln((n(i) + k(i) + &1) / b(i))
          + (&2 * b(i) - (n(i) + k(i) + &1) - &1) * ln(bkzdelta(b(i)))
          + (k(i) / (n(i) + k(i) + &1)) * ln(q(i))
        )
    ) / ln(n(i)) - (
      S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0
    )
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `q:num->real`;
    `s:num->real`;
    `k:num->real`;
    `b:num->real`;
    `Q0:real`;
    `S0:real`;
    `X0:real`;
    `Y0:real`] asymp1_rho_lemma10 THEN
  ASM_MESON_TAC [asymp1_rho_lemma11]);;

let asymp1_standardratio = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real Q0:real S0:real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. log2(q(i))/log2(n(i)) - Q0)
  /\ &0 < &1 + &2*S0
  /\ o1_seq (\i. log2(s(i))/log2(n(i)) - S0)
  /\ &0 <= X0
  /\ o1_seq (\i. k(i)/n(i) - X0)
  /\ &0 < Y0
  /\ o1_seq (\i. b(i)/n(i) - Y0)
  ==>
  o1_seq \i.
    ln(standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i)))
    / ln(n(i))
    - (
      S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0
    )
  `,
  REWRITE_TAC [standardratio_ln2] THEN
  SIMP_TAC [asymp1_rho]);;

let standardratio_below_lemma1 = prove(`
  !X0 Y0 D.
  &0 <= X0 /\ &0 < D /\ (&1 + X0) / D < Y0
  ==> (&1 + X0) / Y0 < D
  `,
  MESON_TAC [REAL_ARITH `&0 <= X0 ==> &0 < &1 + X0`;
    REAL_LT_TRANS;
    REAL_LT_DIV;
    divltflip]);;

let standardratio_below_lemma2 = prove(`
  !Q0 S0 X0 Y0.
  &0 <= X0
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0)) < Y0
  ==> 
  S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0 < &0
  `,
  MESON_TAC [standardratio_below_lemma1;
    REAL_FIELD `(&1 + X0) / Y0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0) ==> S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0 < &0`]);;

let standardratio_below_lemma1_alt = prove(`
  !X0 Y0 D.
  &0 < &1 + X0 /\ &0 < D /\ (&1 + X0) / D < Y0
  ==> (&1 + X0) / Y0 < D
  `,
  MESON_TAC [REAL_LT_TRANS;
    REAL_LT_DIV;
    divltflip]);;

let standardratio_below_lemma2_alt = prove(`
  !Q0 S0 X0 Y0.
  &0 < &1 + X0
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0)) < Y0
  ==> 
  S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0 < &0
  `,
  MESON_TAC [standardratio_below_lemma1_alt;
    REAL_FIELD `(&1 + X0) / Y0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0) ==> S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0 < &0`]);;

let standardratio_below_lemma3 = prove(`
  !x y.
  &1 < y /\ ln(x) / ln(y) < &0 ==> x < &1
  `,
  MESON_TAC [ln_gt1;
    ln_ge1;
    REAL_LT_IMP_LE;
    REAL_LE_DIV;
    REAL_NOT_LT]);;

let standardratio_below = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real Q0:real S0:real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. log2(q(i))/log2(n(i)) - Q0)
  /\ &0 < &1 + &2*S0
  /\ o1_seq (\i. log2(s(i))/log2(n(i)) - S0)
  /\ &0 <= X0
  /\ o1_seq (\i. k(i)/n(i) - X0)
  /\ &0 < Y0
  /\ o1_seq (\i. b(i)/n(i) - Y0)
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0)) < Y0
  ==>
  ?m. !i. m <= i ==> standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i)) < &1
  `,
  REPEAT STRIP_TAC THEN
  specialize [`Q0:real`;
    `S0:real`;
    `X0:real`;
    `Y0:real`] standardratio_below_lemma2 THEN
  specialize [`n:num->real`;
    `q:num->real`;
    `s:num->real`;
    `k:num->real`;
    `b:num->real`;
    `Q0:real`;
    `S0:real`;
    `X0:real`;
    `Y0:real`] asymp1_standardratio THEN
  specialize [`\i:num. ln (standardratio (n i) (q i) (s i) (k i) (b i)) / ln (n i)`;
    `S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - X0 / (&1 + X0) * Q0`] negative_limit_is_eventually_negative THEN
  ASM_MESON_TAC [standardratio_below_lemma3]);;

let standardratio_above_lemma2 = prove(`
  !Q0 S0 X0 Y0.
  &0 < Y0
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ Y0 < (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0))
  ==> 
  &0 < S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0
  `,
  MESON_TAC [ltdivflip;
    REAL_FIELD `&1 - &2*S0 + &2*Q0*X0/(&1 + X0) < (&1 + X0) / Y0 ==> &0 < S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0`]);;

let standardratio_above_lemma3 = prove(`
  !x y.
  &0 < x /\ &1 < y /\ &0 < ln(x) / ln(y)
  ==> &1 < x
  `,
  REPEAT STRIP_TAC THEN
  specialize [`y:real`] ln_gt1 THEN
  specialize [`(ln x)/ln y`;
    `ln y`] REAL_LT_MUL THEN
  specialize [] (REAL_ARITH `&0 < ln y ==> ~(ln y = &0)`) THEN
  specialize [`ln x`;
    `ln y`] REAL_DIV_RMUL THEN
  ASM_MESON_TAC [REAL_NOT_LT;
    ln_le1]);;

let standardratio_above = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real Q0:real S0:real X0:real Y0:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. log2(q(i))/log2(n(i)) - Q0)
  /\ &0 < &1 + &2*S0
  /\ o1_seq (\i. log2(s(i))/log2(n(i)) - S0)
  /\ &0 <= X0
  /\ o1_seq (\i. k(i)/n(i) - X0)
  /\ &0 < Y0
  /\ o1_seq (\i. b(i)/n(i) - Y0)
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ Y0 < (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0))
  ==>
  ?m. !i. m <= i ==> &1 < standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`Q0:real`;
    `S0:real`;
    `X0:real`;
    `Y0:real`] standardratio_above_lemma2 THEN
  specialize [`n:num->real`;
    `q:num->real`;
    `s:num->real`;
    `k:num->real`;
    `b:num->real`;
    `Q0:real`;
    `S0:real`;
    `X0:real`;
    `Y0:real`] asymp1_standardratio THEN
  specialize [`\i:num. ln (standardratio (n i) (q i) (s i) (k i) (b i)) / ln (n i)`;
    `S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - X0 / (&1 + X0) * Q0`] positive_limit_is_eventually_positive THEN
  ASM_MESON_TAC [standardratio_above_lemma3;
    standardratio_positive]);;

let asymp1_const = prove(`
  !n:num->real c:real.
  o1_seq (\i. (c) - (c))
  `,
  REWRITE_TAC [REAL_ARITH `c-c = &0`;
    REAL_ARITH `&0 * x = &0`] THEN
  MESON_TAC [o1_seq_0]);;

let asymp1_add = prove(`
  !n:num->real A:num->real A0:real B:num->real B0:real.
  o1_seq (\i. (A(i)) - (A0))
  /\ o1_seq (\i. (B(i)) - (B0))
  ==> o1_seq (\i. ((A(i))+(B(i))) - ((A0)+(B0)))
  `,
  MESON_TAC [o1_seq_add_offset]);;

let asymp1_sub = prove(`
  !n:num->real A:num->real A0:real B:num->real B0:real.
  o1_seq (\i. (A(i)) - (A0))
  /\ o1_seq (\i. (B(i)) - (B0))
  ==> o1_seq (\i. ((A(i))-(B(i))) - ((A0)-(B0)))
  `,
  MESON_TAC [o1_seq_sub_offset]);;

let asymp1_mul = prove(`
  !n:num->real A:num->real A0:real B:num->real B0:real.
  o1_seq (\i. (A(i)) - (A0))
  /\ o1_seq (\i. (B(i)) - (B0))
  ==> o1_seq (\i. ((A(i))*(B(i))) - ((A0)*(B0)))
  `,
  MESON_TAC [o1_seq_offset_mul2]);;

let asymp1_div = prove(`
  !n:num->real A:num->real A0:real B:num->real B0:real.
  ~(B0 = &0)
  /\ o1_seq (\i. (A(i)) - (A0))
  /\ o1_seq (\i. (B(i)) - (B0))
  ==> o1_seq (\i. ((A(i))/(B(i))) - ((A0)*((&1)/(B0))))
  `,
  REWRITE_TAC [REAL_FIELD `a * (&1 / b) = a / b`] THEN
  MESON_TAC [o1_seq_offset_div2]);;

let asymp1_logn = prove(`
  !n:num->real A:num->real A0:real.
  &0 < A0
  /\ (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. (A(i)) - (A0))
  ==> o1_seq (\i. (log2(A(i))/log2(n(i))) - (&0))
  `,
  REWRITE_TAC [log2ratio_lnratio] THEN
  REPEAT STRIP_TAC THEN
  ONCE_REWRITE_TAC [REAL_FIELD `x / y - &0 = x * (&1 / y)`] THEN
  specialize [`n:num->real`] lnovero1 THEN
  specialize [`\i:num. A(i):real`;
    `A0:real`;
    `\i:num. &1 / ln(n(i))`] o1_seq_timesconvergent_ln THEN
  ASM_MESON_TAC []);;

let a_times_n_powreal_b = prove(`
  !a:real n:real b:real.
   &0 < a /\ &1 < n ==>
   a*(n powreal b) = n powreal(ln(a)/ln(n)+b)
  `,
  REWRITE_TAC [powreal] THEN
  REPEAT STRIP_TAC THEN
  specialize [`n:real`] ln_gt1 THEN
  REWRITE_TAC [REAL_ADD_RDISTRIB] THEN
  REWRITE_TAC [REAL_EXP_ADD] THEN
  specialize [] (REAL_FIELD `&0 < ln n ==> ln a / ln n * ln n = ln a`) THEN
  ASM_MESON_TAC [EXP_LN]);;

let asymp1_logn1plusAtimesntotheB_lemma = prove(`
  !n:num->real A:num->real A0:real B:num->real B0:real.
  (!i. &0 < A(i))
  /\ (!i. &1 < n(i))
  ==>
  (\i. (ln(&1+(A(i))*(n(i))powreal(B(i)))/ln(n(i))) - (B0))
  =
  (\i. ln(n(i) powreal(B0 + ((ln(A(i))/ln(n(i))+B(i)) - B0)) + &1)/ln(n(i)) - B0)
  `,
  REWRITE_TAC [REAL_ARITH `y + (a - y) = a`] THEN
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC EQ_EXT THEN
  ASM_MESON_TAC [a_times_n_powreal_b;
    REAL_ARITH `&1 + x = x + &1`]);;

let asymp1_logn1plusAtimesntotheB_lemma2 = prove(`
  !n:num->real A:num->real A0:real B:num->real B0:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < A0
  /\ o1_seq (\i. (A(i)) - (A0))
  /\ o1_seq (\i. (B(i)) - (B0))
  ==> o1_seq (\i. (ln(A(i))/ln(n(i)) + B(i)) - B0)
  `,
  REWRITE_TAC [GSYM log2ratio_lnratio] THEN
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `A:num->real`;
    `A0:real`] asymp1_logn THEN
  specialize [`n:num->real`;
    `\i:num. log2(A(i))/log2(n(i))`;
    `&0`;
    `B:num->real`;
    `B0:real`] asymp1_add THEN
  ASM_MESON_TAC [REAL_ARITH `&0 + B0 = B0`]);;

let asymp1_logn1plusAtimesntotheB = prove(`
  !n:num->real A:num->real A0:real B:num->real B0:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < A0
  /\ &0 < B0
  /\ (!i. &0 < A(i))
  /\ o1_seq (\i. (A(i)) - (A0))
  /\ o1_seq (\i. (B(i)) - (B0))
  ==> o1_seq (\i. (log2(&1+(A(i))*(n(i))powreal(B(i)))/log2(n(i))) - (B0))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `A:num->real`;
    `A0:real`;
    `B:num->real`;
    `B0:real`] asymp1_logn1plusAtimesntotheB_lemma2 THEN
  specialize [`n:num->real`;
    `\i:num. (ln(A(i))/ln(n(i))+B(i)) - B0`;
    `B0:real`] o1_ln1plus THEN
  REWRITE_TAC [log2ratio_lnratio] THEN
  ASM_MESON_TAC [asymp1_logn1plusAtimesntotheB_lemma]);;

let asymp2_const = prove(`
  !n:num->real c:real.
  o1_seq (\i. (c - c) * log2(n(i)) - &0)
  `,
  REWRITE_TAC [REAL_ARITH `c-c = &0`;
    REAL_ARITH `&0 * x = &0`;
    REAL_ARITH `&0 - &0 = &0`] THEN
  MESON_TAC [o1_seq_0]);;

let asymp2_add = prove(`
  !n:num->real A:num->real A0:real A1:real B:num->real B0:real B1:real.
  o1_seq (\i. (((A(i)) - (A0)) * log2(n(i))) - (A1))
  /\ o1_seq (\i. (((B(i)) - (B0)) * log2(n(i))) - (B1))
  ==> o1_seq (\i. ((((A(i))+(B(i))) - ((A0)+(B0))) * log2(n(i))) - ((A1)+(B1)))
  `,
  REWRITE_TAC [REAL_FIELD `((a+b)-(a0+b0))*L-(a1+b1) = ((a-a0)*L-a1)+((b-b0)*L-b1)`] THEN
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. (((A(i)) - (A0)) * log2(n(i))) - (A1)`;
    `\i:num. (((B(i)) - (B0)) * log2(n(i))) - (B1)`] o1_seq_add THEN
  ASM_MESON_TAC []);;

let asymp2_sub = prove(`
  !n:num->real A:num->real A0:real A1:real B:num->real B0:real B1:real.
  o1_seq (\i. (((A(i)) - (A0)) * log2(n(i))) - (A1))
  /\ o1_seq (\i. (((B(i)) - (B0)) * log2(n(i))) - (B1))
  ==> o1_seq (\i. ((((A(i))-(B(i))) - ((A0)-(B0))) * log2(n(i))) - ((A1)-(B1)))
  `,
  REWRITE_TAC [REAL_FIELD `((a-b)-(a0-b0))*L-(a1-b1) = ((a-a0)*L-a1)-((b-b0)*L-b1)`] THEN
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. (((A(i)) - (A0)) * log2(n(i))) - (A1)`;
    `\i:num. (((B(i)) - (B0)) * log2(n(i))) - (B1)`] o1_seq_sub THEN
  ASM_MESON_TAC []);;

let asymp2_asymp1_lemma = prove(`
  !n:real A:real A0:real A1:real.
  &1 < n
  ==> A - A0
  = ((A - A0) * log2 (n) - A1) * &1 / log2 (n) + A1 * &1 / log2 (n)
  `,
  MESON_TAC [log2_gt1;
    REAL_FIELD `&0 < L ==> A - A0 = ((A - A0) * L - A1) * &1 / L + A1 * &1 / L`]);;

let asymp2_asymp1_lemma2 = prove(`
  !n:num->real A:num->real A0:real A1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  ==>
  (\i. A(i) - A0)
  =
  (\i. ((A i - A0) * log2 (n i) - A1) * &1 / log2 (n i) + A1 * &1 / log2 (n i))
  `,
  MESON_TAC [EQ_EXT;
    asymp2_asymp1_lemma]);;

let asymp2_asymp1 = prove(`
  !n:num->real A:num->real A0:real A1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. (((A(i)) - (A0)) * log2(n(i))) - (A1))
  ==> o1_seq (\i. A(i) - A0)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`] log2overo1 THEN
  specialize [`\i:num. (A i - A0) * log2 (n i) - A1`;
    `\i:num. &1 / log2(n i)`] o1_seq_mul THEN
  specialize [`\i:num. &1 / log2(n i)`;
    `A1:real`] o1_seq_consttimes THEN
  specialize [`\i:num. ((A i - A0) * log2 (n i) - A1) * &1 / log2 (n i)`;
    `\i:num. A1 * &1 / log2 (n i)`] o1_seq_add THEN
  ASM_MESON_TAC [asymp2_asymp1_lemma2]);;

let asymp2_mul = prove(`
  !n:num->real A:num->real A0:real A1:real B:num->real B0:real B1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. (((A(i)) - (A0)) * log2(n(i))) - (A1))
  /\ o1_seq (\i. (((B(i)) - (B0)) * log2(n(i))) - (B1))
  ==> o1_seq (\i. ((((A(i))*(B(i))) - ((A0)*(B0))) * log2(n(i))) - (((A0)*(B1))+((A1)*(B0))))
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [REAL_ARITH `(A(i:num) * B i - A0 * B0) * log2 (n i) - (A0 * B1 + A1 * B0) = (((A i - A0) * log2 (n i) - A1) * (B i - B0) + B0 * ((A i - A0) * log2 (n i) - A1)) + A0 * ((B i - B0) * log2 (n i) - B1) + A1 * (B i - B0)`] THEN
  specialize [`n:num->real`;
    `B:num->real`;
    `B0:real`;
    `B1:real`] asymp2_asymp1 THEN
  specialize [`\i:num. (A(i)-A0)*log2(n(i))-A1`;
    `\i:num. B(i)-B0`] o1_seq_mul THEN
  specialize [`\i:num. (A(i)-A0)*log2(n(i))-A1`;
    `B0:real`] o1_seq_consttimes THEN
  specialize [`\i:num. (B(i)-B0)*log2(n(i))-B1`;
    `A0:real`] o1_seq_consttimes THEN
  specialize [`\i:num. B(i)-B0`;
    `A1:real`] o1_seq_consttimes THEN
  specialize [`\i:num. ((A i - A0) * log2 (n i) - A1) * (B i - B0)`;
    `\i:num. B0 * ((A i - A0) * log2 (n i) - A1)`] o1_seq_add THEN
  specialize [`\i:num. A0 * ((B i - B0) * log2 (n i) - B1)`;
    `\i:num. A1 * (B i - B0)`] o1_seq_add THEN
  specialize [`\i:num. ((A i - A0) * log2 (n i) - A1) * (B i - B0) + B0 * ((A i - A0) * log2 (n i) - A1)`;
    `\i:num. A0 * ((B i - B0) * log2 (n i) - B1) + A1 * (B i - B0)`] o1_seq_add THEN
  ASM_SIMP_TAC []);;

let asymp2_mulconst = prove(`
  !n:num->real A:num->real A0:real A1:real c:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. (A(i) - A0) * log2(n(i)) - A1)
  ==> o1_seq (\i. (c*A(i) - c*A0) * log2(n(i)) - c*A1)
  `,
  REWRITE_TAC [REAL_ARITH `(c*A(i:num) - c*A0) * log2(n(i)) - c*A1 = c*((A(i) - A0) * log2(n(i)) - A1)`] THEN
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. (A(i) - A0) * log2(n(i)) - A1`;
    `c:real`] o1_seq_consttimes THEN
  ASM_MESON_TAC []);;

let asymp2_scaleto1_lemma = prove(`
  !n:num->real A:num->real A0:real A1:real c:real.
  (!i. &1 < n(i))
  /\ ~(A0 = &0)
  ==>
  (\i. (&1 / A0 * A i - &1 / A0 * A0) * log2 (n i) - &1 / A0 * A1)
  =
  (\i. (A i / A0 - &1) * log2 (n i) - A1 / A0)
  `,
  REWRITE_TAC [REAL_ARITH `&1/x*y = y/x`] THEN
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC EQ_EXT THEN
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [BETA_THM] THEN
  ASM_MESON_TAC [REAL_FIELD `~(x = &0) ==> x/x = &1`]);;

let asymp2_scaleto1 = prove(`
  !n:num->real A:num->real A0:real A1:real c:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ ~(A0 = &0)
  /\ o1_seq (\i. (A(i) - A0) * log2(n(i)) - A1)
  ==> o1_seq (\i. (A(i)/A0 - &1) * log2(n(i)) - A1/A0)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `A:num->real`;
    `A0:real`;
    `A1:real`;
    `&1/A0`] asymp2_mulconst THEN
  ASM_MESON_TAC [asymp2_scaleto1_lemma]);;

let asymp2_inv = prove(`
  !n:num->real B:num->real B0:real B1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ ~(B0 = &0)
  /\ o1_seq (\i. (B(i) - B0) * log2(n(i)) - B1)
  ==> o1_seq (\i. (&1/B(i) - &1/B0) * log2(n(i)) - (--(B1)/(B0*B0)))
  `,
  REPEAT STRIP_TAC THEN
  note `!i:num. ~(log2(n(i)) = &0)` [log2_gt1;
    REAL_LT_IMP_NZ] THEN
  specialize [`n:num->real`] log2overo1 THEN
  specialize [`\i:num. log2(n(i))`;
    `B:num->real`;
    `B0:real`;
    `B1:real`] asymp2_inv_generic THEN
  ASM_MESON_TAC []);;

let asymp2_sqrt = prove(`
  !n:num->real A:num->real A0:real A1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < A0
  /\ o1_seq (\i. (A(i) - A0) * log2(n(i)) - A1)
  ==> o1_seq (\i. (sqrt(A(i)) - sqrt(A0)) * log2(n(i)) - (A1/(&2*sqrt(A0))))
  `,
  REPEAT STRIP_TAC THEN
  note `!i:num. ~(log2(n(i)) = &0)` [log2_gt1;
    REAL_LT_IMP_NZ] THEN
  specialize [`n:num->real`] log2overo1 THEN
  specialize [`\i:num. log2(n(i))`;
    `A:num->real`;
    `A0:real`;
    `A1:real`] asymp2_sqrt_generic THEN
  ASM_MESON_TAC []);;

let asymp2_div_lemma = prove(`
  !n:num->real A:num->real A0:real A1:real B:num->real B0:real B1:real.
  (\i. (((A(i) * (&1 / B(i)) - ((A0)*((&1)/(B0)))) * log2(n(i))) - (((A0)*((--(B1))/((B0)*(B0))))+((A1)*((&1)/(B0))))))
  = 
  (\i. (((A(i) / B(i) - ((A0)*((&1)/(B0)))) * log2(n(i))) - (((A0)*((--(B1))/((B0)*(B0))))+((A1)*((&1)/(B0))))))
  `,
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC EQ_EXT THEN
  STRIP_TAC THEN
  REWRITE_TAC [BETA_THM] THEN
  ASM_MESON_TAC [REAL_FIELD `A(i:num) / B(i) = A(i) * (&1 / B(i))`]);;

let asymp2_div = prove(`
  !n:num->real A:num->real A0:real A1:real B:num->real B0:real B1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ ~(B0 = &0)
  /\ (!i. &0 < B(i))
  /\ o1_seq (\i. (((A(i)) - (A0)) * log2(n(i))) - (A1))
  /\ o1_seq (\i. (((B(i)) - (B0)) * log2(n(i))) - (B1))
  ==> o1_seq (\i. (((A(i) / B(i) - ((A0)*((&1)/(B0)))) * log2(n(i))) - (((A0)*((--(B1))/((B0)*(B0))))+((A1)*((&1)/(B0))))))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `B:num->real`;
    `B0:real`;
    `B1:real`] asymp2_inv THEN
  specialize [`n:num->real`;
    `A:num->real`;
    `A0:real`;
    `A1:real`;
    `\i:num. &1 / B i`;
    `&1 / B0`;
    `--B1/(B0*B0)`] asymp2_mul THEN
  ASM_MESON_TAC [asymp2_div_lemma]);;

let asymp2_logn_lemma = prove(`
  !n:num->real A:num->real A0:real A1:real.
  (!i. &1 < n(i))
  ==>
  (\i. (((log2(A(i))/log2(n(i))) - (&0)) * log2(n(i))) - (log2(A0)))
  =
  (\i. log2(A(i)) - log2(A0))
  `,
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC EQ_EXT THEN
  STRIP_TAC THEN
  REWRITE_TAC [BETA_THM] THEN
  REWRITE_TAC [REAL_ARITH `x - &0 = x`] THEN
  specialize [] (MESON [] `(!i:num. &1 < n(i)) ==> &1 < n(x)`) THEN
  specialize [`n(x:num):real`] log2_gt1 THEN
  ASM_MESON_TAC [REAL_FIELD `&0 < L ==> B / L * L = B`]);;

let asymp2_logn_mini = prove(`
  !n:num->real A:num->real A0:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < A0
  /\ o1_seq (\i. A(i) - A0)
  ==> o1_seq (\i. (((log2(A(i))/log2(n(i))) - (&0)) * log2(n(i))) - (log2(A0)))
  `,
  SIMP_TAC [asymp2_logn_lemma] THEN
  REPEAT STRIP_TAC THEN
  specialize [`A:num->real`;
    `A0:real`] o1_seq_compose_log2 THEN
  ASM_SIMP_TAC []);;

let asymp2_logn = prove(`
  !n:num->real A:num->real A0:real A1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < A0
  /\ o1_seq (\i. (((A(i)) - (A0)) * log2(n(i))) - (A1))
  ==> o1_seq (\i. (((log2(A(i))/log2(n(i))) - (&0)) * log2(n(i))) - (log2(A0)))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `A:num->real`;
    `A0:real`;
    `A1:real`] asymp2_asymp1 THEN
  specialize [`n:num->real`;
    `A:num->real`;
    `A0:real`] asymp2_logn_mini THEN
  ASM_SIMP_TAC []);;

let ln1o1_lemma = prove(`
  (!i:num. &0 < (\i. &1) i)
  `,
  MESON_TAC [REAL_ARITH `&0 < &1`]);;

let ln1o1_lemma2 = prove(`
  !o1:num->real.
  (\i. ln (&1 + &1 * o1(i)) / &1)
  =
  (\i. ln(&1 + o1(i)))
  `,
  REWRITE_TAC [REAL_ARITH `&1 * x = x`] THEN
  REWRITE_TAC [REAL_ARITH `x / &1 = x`]);;

let ln1o1 = prove(`
  !o1:num->real.
  o1_seq o1 /\
  (!i. &0 < o1(i))
  ==> o1_seq (\i. ln(&1 + o1(i)))
  `,
  REPEAT STRIP_TAC THEN
  specialize [] ln1o1_lemma THEN
  specialize [`o1:num->real`;
    `\i:num. &1`] ln1xo1x THEN
  ASM_MESON_TAC [ln1o1_lemma2]);;

let log2_1o1_lemma = prove(`
  !o1:num->real.
  (\i. (&1 / ln(&2)) * ln(&1 + o1(i)))
  =
  (\i. log2(&1 + o1(i)))
  `,
  MESON_TAC [EQ_EXT;
    REAL_ARITH `(&1 / ln(&2))*x = x/ln(&2)`;
    log2]);;

let log2_1o1 = prove(`
  !o1:num->real.
  o1_seq o1 /\
  (!i. &0 < o1(i))
  ==> o1_seq (\i. log2(&1 + o1(i)))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`o1:num->real`] ln1o1 THEN
  specialize [`\i:num. ln(&1 + o1(i))`;
    `&1 / ln(&2)`] o1_seq_consttimes THEN
  ASM_MESON_TAC [log2_1o1_lemma]);;

let ngt1_imp_overngt0 = prove(`
  !n:num->real.
  (!i. &1 < n(i))
  ==> (!i. &0 < &1 / n(i))
  `,
  MESON_TAC [REAL_ARITH `&1 < x ==> &0 < x`;
    pos_1over]);;

let asymp2_logn1plusAtimesntotheB_lemma1 = prove(`
  !n:real A:real B:real B0:real.
  &1 < n
  /\ &0 < A
  ==> 
  log2 (n powreal (ln A / ln n + B)) + log2 (&1 + (&1 / n) powreal (ln A / ln n + B))
  =
  log2 (&1 + n powreal (ln A / ln n + B))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:real`;
    `ln A / ln n + B`] powreal_positive THEN
  specialize [`&1 / n`;
    `ln A / ln n + B`] powreal_positive THEN
  real_linear `&0 < n powreal (ln A / ln n + B) ==> &0 < &1 + n powreal (ln A / ln n + B)` THEN
  real_linear `&0 < (&1 / n) powreal (ln A / ln n + B) ==> &0 < &1 + (&1 / n) powreal (ln A / ln n + B)` THEN
  specialize [`n powreal (ln A / ln n + B)`;
    `&1 + (&1 / n) powreal (ln A / ln n + B)`] (GSYM log2_mul) THEN
  real_linear `n powreal (ln A / ln n + B) * (&1 + (&1 / n) powreal (ln A / ln n + B)) = (&1 / n) powreal (ln A / ln n + B) * n powreal (ln A / ln n + B) + n powreal (ln A / ln n + B)` THEN
  specialize [] (MESON [] `n powreal (ln A / ln n + B) * (&1 + (&1 / n) powreal (ln A / ln n + B)) = (&1 / n) powreal (ln A / ln n + B) * n powreal (ln A / ln n + B) + n powreal (ln A / ln n + B) ==> log2(n powreal (ln A / ln n + B) * (&1 + (&1 / n) powreal (ln A / ln n + B))) = log2((&1 / n) powreal (ln A / ln n + B) * n powreal (ln A / ln n + B) + n powreal (ln A / ln n + B))`) THEN
  real_linear `&1 < n ==> &0 < n` THEN
  specialize [`n:real`;
    `ln A / ln n + B`] powreal_cancelbase THEN
  specialize [] (MESON [] `log2 (n powreal (ln A / ln n + B)) + log2 (&1 + (&1 / n) powreal (ln A / ln n + B)) = log2 (n powreal (ln A / ln n + B) * (&1 + (&1 / n) powreal (ln A / ln n + B)))
    /\ log2(n powreal (ln A / ln n + B) * (&1 + (&1 / n) powreal (ln A / ln n + B))) = log2((&1 / n) powreal (ln A / ln n + B) * n powreal (ln A / ln n + B) + n powreal (ln A / ln n + B))
    ==> 
    log2 (n powreal (ln A / ln n + B)) + log2 (&1 + (&1 / n) powreal (ln A / ln n + B)) = log2((&1 / n) powreal (ln A / ln n + B) * n powreal (ln A / ln n + B) + n powreal (ln A / ln n + B))`) THEN
  specialize [] (MESON [] `log2 (n powreal (ln A / ln n + B)) + log2 (&1 + (&1 / n) powreal (ln A / ln n + B)) = log2((&1 / n) powreal (ln A / ln n + B) * n powreal (ln A / ln n + B) + n powreal (ln A / ln n + B))
    /\ (&1 / n) powreal (ln A / ln n + B) * n powreal (ln A / ln n + B) = &1
    ==> 
    log2 (n powreal (ln A / ln n + B)) + log2 (&1 + (&1 / n) powreal (ln A / ln n + B)) = log2(&1 + n powreal (ln A / ln n + B))`) THEN
  ASM_MESON_TAC []);;

let asymp2_logn1plusAtimesntotheB_lemma2 = prove(`
  !n:real A:real A0:real A1:real B:real B0:real B1:real.
  &1 < n /\ &0 < A /\ &0 < A0
  ==>
  (log2 (A) - log2 A0 + (B - B0) * log2 (n) - B1) + log2 (&1 + (&1 / n) powreal ((&0 + B0) + (log2 (A) / log2 (n) + B) - (&0 + B0)))
  =
  (log2 (&1 + A * n powreal B) / log2 (n) - B0) * log2 (n) - (log2 A0 + B1)
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [REAL_ARITH `((&0 + B0) + (log2 A / log2 n + B) - (&0 + B0)) = log2(A) / log2(n) + B`] THEN
  specialize [`A:real`;
    `n:real`;
    `B:real`] a_times_n_powreal_b THEN
  specialize [`n:real`] log2_gt1 THEN
  real_cancel `&0 < log2(n) ==> (log2 (&1 + n powreal (ln A / ln n + B)) / log2 n - B0) * log2 n - (log2(A0)+B1) = log2(&1 + n powreal (ln A / ln n + B)) - B0 * log2 n - (log2(A0)+B1)` THEN
  real_linear `(log2 A - log2 A0 + (B - B0) * log2 n - B1) + log2 (&1 + (&1 / n) powreal (log2 A / log2 n + B)) = ((log2 A + B * log2 n) + log2 (&1 + (&1 / n) powreal (log2 A / log2 n + B))) - B0 * log2 n - (log2 A0 + B1)` THEN
  specialize [`n:real`;
    `B:real`] (GSYM powreal_log2) THEN
  specialize [`n:real`;
    `B:real`] powreal_positive THEN
  specialize [`n:real`;
    `ln A / ln n + B`] powreal_positive THEN
  specialize [`A:real`;
    `n powreal B`] (GSYM log2_mul) THEN
  specialize [`n:real`;
    `A:real`;
    `B:real`;
    `B0:real`] asymp2_logn1plusAtimesntotheB_lemma1 THEN
  specialize [`A:real`;
    `n:real`] log2ratio_lnratio THEN
  specialize [] (MESON [] `B * log2 n = log2(n powreal B) /\ log2 A + log2 (n powreal B) = log2 (A * n powreal B)
    ==>
    ((log2 A + B * log2 n) + log2 (&1 + (&1 / n) powreal (log2 A / log2 n + B))) - B0 * log2 n - (log2 A0 + B1)
    = (log2(A * n powreal B) + log2 (&1 + (&1 / n) powreal (log2 A / log2 n + B))) - B0 * log2 n - (log2 A0 + B1)`) THEN
  specialize [] (MESON [] `A * n powreal B = n powreal (ln A / ln n + B)
    /\ log2 A / log2 n = ln A / ln n
    ==>
    (log2(A * n powreal B) + log2 (&1 + (&1 / n) powreal (log2 A / log2 n + B))) - B0 * log2 n - (log2 A0 + B1)
    = (log2 (n powreal (ln A / ln n + B)) + log2 (&1 + (&1 / n) powreal (ln A / ln n + B))) - B0 * log2 n - (log2 A0 + B1)`) THEN
  specialize [] (MESON [] `log2 (n powreal (ln A / ln n + B)) + log2 (&1 + (&1 / n) powreal (ln A / ln n + B)) = log2 (&1 + n powreal (ln A / ln n + B))
    /\
    (log2(A * n powreal B) + log2 (&1 + (&1 / n) powreal (log2 A / log2 n + B))) - B0 * log2 n - (log2 A0 + B1)
    = (log2 (n powreal (ln A / ln n + B)) + log2 (&1 + (&1 / n) powreal (ln A / ln n + B))) - B0 * log2 n - (log2 A0 + B1)
    ==>
    (log2(A * n powreal B) + log2 (&1 + (&1 / n) powreal (log2 A / log2 n + B))) - B0 * log2 n - (log2 A0 + B1)
    = (log2 (&1 + n powreal (ln A / ln n + B))) - B0 * log2 n - (log2 A0 + B1)`) THEN
  specialize [] (MESON [] `A * n powreal B = n powreal (ln A / ln n + B)
    /\ (log2 (&1 + n powreal (ln A / ln n + B)) / log2 n - B0) * log2 n - (log2(A0)+B1) = log2(&1 + n powreal (ln A / ln n + B)) - B0 * log2 n - (log2(A0)+B1)
    ==>
    (log2 (&1 + A * n powreal B) / log2 n - B0) * log2 n - (log2 A0 + B1)
    = log2(&1 + n powreal (ln A / ln n + B)) - B0 * log2 n - (log2(A0)+B1)`) THEN
  ASM_MESON_TAC []);;

let asymp2_logn1plusAtimesntotheB_lemma3 = prove(`
  !n:num->real A:num->real B:num->real B0:real.
   !i. &0 < (&1 / n i) powreal ((&0 + B0) + (log2 (A i) / log2 (n i) + B i) - (&0 + B0))
  `,
  MESON_TAC [powreal_positive]);;

let asymp2_logn1plusAtimesntotheB_lemma4 = prove(`
  !n:num->real A:num->real A0:real A1:real B:num->real B0:real B1:real.
  (!i. &1 < n(i))
  /\ (!i. &0 < A(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < A0
  ==>
  (\i. (log2 (A i) - log2 A0 + (B i - B0) * log2 (n i) - B1) + log2 (&1 + (&1 / n i) powreal ((&0 + B0) + (log2 (A i) / log2 (n i) + B i) - (&0 + B0))))
  = (\i. (log2 (&1 + A i * n i powreal B i) / log2 (n i) - B0) * log2 (n i) - (log2 A0 + B1))
  `,
  MESON_TAC [EQ_EXT;
    asymp2_logn1plusAtimesntotheB_lemma2]);;

let asymp2_logn1plusAtimesntotheB_mini = prove(`
  !n:num->real A:num->real A0:real B:num->real B0:real B1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < A0
  /\ &0 < B0
  /\ (!i. &0 < A(i))
  /\ o1_seq (\i. A(i) - A0)
  /\ o1_seq (\i. (((B(i)) - (B0)) * log2(n(i))) - (B1))
  ==> o1_seq (\i. (((log2(&1+(A(i))*(n(i))powreal(B(i)))/log2(n(i))) - (B0)) * log2(n(i))) - ((log2(A0))+(B1)))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `B:num->real`;
    `B0:real`;
    `B1:real`] asymp2_asymp1 THEN
  specialize [`n:num->real`;
    `A:num->real`;
    `A0:real`] asymp1_logn THEN
  specialize [`A:num->real`;
    `A0:real`] o1_seq_compose_log2 THEN
  specialize [`n:num->real`;
    `\i:num. log2(A(i))/log2(n(i))`;
    `&0`;
    `B:num->real`;
    `B0:real`] asymp1_add THEN
  real_linear `&0 < B0 ==> &0 < &0 + B0` THEN
  specialize [`n:num->real`] ngt1_imp_overngt0 THEN
  specialize [`n:num->real`;
    `\i:num. (log2 (A i) / log2 (n i) + B i) - (&0 + B0)`;
    `&0 + B0`] o1_ln1plus_lemma1 THEN
  specialize [`n:num->real`;
    `A:num->real`;
    `B:num->real`;
    `B0:real`] asymp2_logn1plusAtimesntotheB_lemma3 THEN
  specialize [`\i:num. (&1 / n i) powreal ((&0 + B0) + (log2 (A i) / log2 (n i) + B i) - (&0 + B0))`] log2_1o1 THEN
  specialize [`\i:num. log2 (A i) - log2 A0`;
    `\i:num. (B i - B0) * log2 (n i) - B1`] o1_seq_add THEN
  specialize [`\i:num. log2 (A i) - log2 A0 + (B i - B0) * log2 (n i) - B1`;
    `\i:num. log2 (&1 + (&1 / n i) powreal ((&0 + B0) + (log2 (A i) / log2 (n i) + B i) - (&0 + B0)))`] o1_seq_add THEN
  ASM_MESON_TAC [asymp2_logn1plusAtimesntotheB_lemma4]);;

let asymp2_logn1plusAtimesntotheB = prove(`
  !n:num->real A:num->real A0:real A1:real B:num->real B0:real B1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < A0
  /\ &0 < B0
  /\ (!i. &0 < A(i))
  /\ o1_seq (\i. (((A(i)) - (A0)) * log2(n(i))) - (A1))
  /\ o1_seq (\i. (((B(i)) - (B0)) * log2(n(i))) - (B1))
  ==> o1_seq (\i. (((log2(&1+(A(i))*(n(i))powreal(B(i)))/log2(n(i))) - (B0)) * log2(n(i))) - ((log2(A0))+(B1)))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `A:num->real`;
    `A0:real`;
    `A1:real`] asymp2_asymp1 THEN
  specialize [`n:num->real`;
    `A:num->real`;
    `A0:real`;
    `B:num->real`;
    `B0:real`;
    `B1:real`] asymp2_logn1plusAtimesntotheB_mini THEN
  ASM_MESON_TAC []);;

let asymp2_main = prove(`
  !n:num->real
   q:num->real
   s:num->real
   k:num->real
   b:num->real
   Q0:real
   Q1:real
   S0:real
   S1:real
   X0:real
   X1:real
   Y0:real
   Y1:real
   twopie:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i:num. ((((&1)/(n(i))) - (&0)) * log2(n(i))) - (&0))
  /\ o1_seq (\i:num. (((log2(q(i))/log2(n(i))) - (Q0)) * log2(n(i))) - (Q1))
  /\ o1_seq (\i:num. (((log2(s(i))/log2(n(i))) - (S0)) * log2(n(i))) - (S1))
  /\ o1_seq (\i:num. ((((k(i))/(n(i))) - (X0)) * log2(n(i))) - (X1))
  /\ o1_seq (\i:num. ((((b(i))/(n(i))) - (Y0)) * log2(n(i))) - (Y1))
  /\ &0 < &2
  /\ (!i:num. &0 < &2)
  /\ ~(&2 = &0)
  /\ (!i:num. &0 < (&1)+((k(i))/(n(i))))
  /\ &0 < (&1)+(X0)
  /\ &0 < (&1)+((&2)*(S0))
  /\ &0 < Y0
  /\ (!i:num. &0 < (b(i))/(n(i)))
  /\ ~(Y0 = &0)
  /\ &0 < ((&1)+((X0)+(&0)))*((&1)/(Y0))
  /\ &0 < (pi)*(Y0)
  /\ &0 < twopie
  /\ &0 < (&2)*((Y0)-(&0))
  /\ (!i:num. &0 < (&2)*(((b(i))/(n(i)))-((&1)/(n(i)))))
  /\ ~((&2)*((Y0)-(&0)) = &0)
  /\ &0 < (&1)+((X0)+(&0))
  /\ (!i:num. &0 < (&1)+(((k(i))/(n(i)))+((&1)/(n(i)))))
  /\ ~((&1)+((X0)+(&0)) = &0)
  ==> o1_seq (\i:num. ((((((&1)/(&2))*(log2(&1+((&1)+((k(i))/(n(i))))*(n(i))powreal((&1)+((&2)*(log2(s(i))/log2(n(i))))))/log2(n(i))))-((((&1)/(&2))*(log2(((&1)+(((k(i))/(n(i)))+((&1)/(n(i)))))/((b(i))/(n(i))))/log2(n(i))))+((((&1)+((log2((b(i))/(n(i)))/log2(n(i)))+((((&1)/(n(i)))*(((&1)/((b(i))/(n(i))))*((log2((pi)*((b(i))/(n(i))))/log2(n(i)))+(&1))))-(log2(twopie)/log2(n(i))))))*(((((&2)*((b(i))/(n(i))))-((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))-((&1)/(n(i))))/((&2)*(((b(i))/(n(i)))-((&1)/(n(i)))))))+((((k(i))/(n(i)))/((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))*(log2(q(i))/log2(n(i))))))) - ((((&1)*((&1)/(&2)))*((&1)+((&2)*(S0))))-((((&1)*((&1)/(&2)))*(&0))+((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0))))))+(((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q0)))))) * log2(n(i))) - (((((&1)*((&1)/(&2)))*((log2((&1)+(X0)))+((&0)+(((&2)*(S1))+((&0)*(S0))))))+((((&1)*((--(&0))/((&2)*(&2))))+((&0)*((&1)/(&2))))*((&1)+((&2)*(S0)))))-(((((&1)*((&1)/(&2)))*(log2(((&1)+((X0)+(&0)))*((&1)/(Y0)))))+((((&1)*((--(&0))/((&2)*(&2))))+((&0)*((&1)/(&2))))*(&0)))+(((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*((((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((--(((&2)*((Y1)-(&0)))+((&0)*((Y0)-(&0)))))/(((&2)*((Y0)-(&0)))*((&2)*((Y0)-(&0))))))+((((((&2)*(Y1))+((&0)*(Y0)))-((&0)+((X1)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))))+((((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q1))+((((X0)*((--((&0)+((X1)+(&0))))/(((&1)+((X0)+(&0)))*((&1)+((X0)+(&0))))))+((X1)*((&1)/((&1)+((X0)+(&0))))))*(Q0)))))))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. n(i):real`;
    `&1:real`] asymp2_const THEN
  specialize [`\i:num. n(i):real`;
    `&2:real`] asymp2_const THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. &2:real`;
    `&2:real`;
    `&0:real`] asymp2_div THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. (k(i))/(n(i)):real`;
    `X0:real`;
    `X1:real`] asymp2_add THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. &2:real`;
    `&2:real`;
    `&0:real`;
    `\i:num. log2(s(i))/log2(n(i)):real`;
    `S0:real`;
    `S1:real`] asymp2_mul THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. (&2)*(log2(s(i))/log2(n(i))):real`;
    `(&2)*(S0):real`;
    `((&2)*(S1))+((&0)*(S0)):real`] asymp2_add THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (&1)+((k(i))/(n(i))):real`;
    `(&1)+(X0):real`;
    `(&0)+(X1):real`;
    `\i:num. (&1)+((&2)*(log2(s(i))/log2(n(i)))):real`;
    `(&1)+((&2)*(S0)):real`;
    `(&0)+(((&2)*(S1))+((&0)*(S0))):real`] asymp2_logn1plusAtimesntotheB THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (&1)/(&2):real`;
    `(&1)*((&1)/(&2)):real`;
    `((&1)*((--(&0))/((&2)*(&2))))+((&0)*((&1)/(&2))):real`;
    `\i:num. log2(&1+((&1)+((k(i))/(n(i))))*(n(i))powreal((&1)+((&2)*(log2(s(i))/log2(n(i))))))/log2(n(i)):real`;
    `(&1)+((&2)*(S0)):real`;
    `(log2((&1)+(X0)))+((&0)+(((&2)*(S1))+((&0)*(S0)))):real`] asymp2_mul THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (k(i))/(n(i)):real`;
    `X0:real`;
    `X1:real`;
    `\i:num. (&1)/(n(i)):real`;
    `&0:real`;
    `&0:real`] asymp2_add THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. ((k(i))/(n(i)))+((&1)/(n(i))):real`;
    `(X0)+(&0):real`;
    `(X1)+(&0):real`] asymp2_add THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (&1)+(((k(i))/(n(i)))+((&1)/(n(i)))):real`;
    `(&1)+((X0)+(&0)):real`;
    `(&0)+((X1)+(&0)):real`;
    `\i:num. (b(i))/(n(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_div THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. ((&1)+(((k(i))/(n(i)))+((&1)/(n(i)))))/((b(i))/(n(i))):real`;
    `((&1)+((X0)+(&0)))*((&1)/(Y0)):real`;
    `(((&1)+((X0)+(&0)))*((--(Y1))/((Y0)*(Y0))))+(((&0)+((X1)+(&0)))*((&1)/(Y0))):real`] asymp2_logn THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (&1)/(&2):real`;
    `(&1)*((&1)/(&2)):real`;
    `((&1)*((--(&0))/((&2)*(&2))))+((&0)*((&1)/(&2))):real`;
    `\i:num. log2(((&1)+(((k(i))/(n(i)))+((&1)/(n(i)))))/((b(i))/(n(i))))/log2(n(i)):real`;
    `&0:real`;
    `log2(((&1)+((X0)+(&0)))*((&1)/(Y0))):real`] asymp2_mul THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (b(i))/(n(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_logn THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. (b(i))/(n(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_div THEN
  specialize [`\i:num. n(i):real`;
    `pi:real`] asymp2_const THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. pi:real`;
    `pi:real`;
    `&0:real`;
    `\i:num. (b(i))/(n(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_mul THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (pi)*((b(i))/(n(i))):real`;
    `(pi)*(Y0):real`;
    `((pi)*(Y1))+((&0)*(Y0)):real`] asymp2_logn THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. log2((pi)*((b(i))/(n(i))))/log2(n(i)):real`;
    `&0:real`;
    `log2((pi)*(Y0)):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`] asymp2_add THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (&1)/((b(i))/(n(i))):real`;
    `(&1)*((&1)/(Y0)):real`;
    `((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))):real`;
    `\i:num. (log2((pi)*((b(i))/(n(i))))/log2(n(i)))+(&1):real`;
    `(&0)+(&1):real`;
    `(log2((pi)*(Y0)))+(&0):real`] asymp2_mul THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (&1)/(n(i)):real`;
    `&0:real`;
    `&0:real`;
    `\i:num. ((&1)/((b(i))/(n(i))))*((log2((pi)*((b(i))/(n(i))))/log2(n(i)))+(&1)):real`;
    `((&1)*((&1)/(Y0)))*((&0)+(&1)):real`;
    `(((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1))):real`] asymp2_mul THEN
  specialize [`\i:num. n(i):real`;
    `twopie:real`] asymp2_const THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. twopie:real`;
    `twopie:real`;
    `&0:real`] asymp2_logn THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. ((&1)/(n(i)))*(((&1)/((b(i))/(n(i))))*((log2((pi)*((b(i))/(n(i))))/log2(n(i)))+(&1))):real`;
    `(&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))):real`;
    `((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))):real`;
    `\i:num. log2(twopie)/log2(n(i)):real`;
    `&0:real`;
    `log2(twopie):real`] asymp2_sub THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. log2((b(i))/(n(i)))/log2(n(i)):real`;
    `&0:real`;
    `log2(Y0):real`;
    `\i:num. (((&1)/(n(i)))*(((&1)/((b(i))/(n(i))))*((log2((pi)*((b(i))/(n(i))))/log2(n(i)))+(&1))))-(log2(twopie)/log2(n(i))):real`;
    `((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0):real`;
    `(((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)):real`] asymp2_add THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. (log2((b(i))/(n(i)))/log2(n(i)))+((((&1)/(n(i)))*(((&1)/((b(i))/(n(i))))*((log2((pi)*((b(i))/(n(i))))/log2(n(i)))+(&1))))-(log2(twopie)/log2(n(i)))):real`;
    `(&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0)):real`;
    `(log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie))):real`] asymp2_add THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. &2:real`;
    `&2:real`;
    `&0:real`;
    `\i:num. (b(i))/(n(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_mul THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (&2)*((b(i))/(n(i))):real`;
    `(&2)*(Y0):real`;
    `((&2)*(Y1))+((&0)*(Y0)):real`;
    `\i:num. (&1)+(((k(i))/(n(i)))+((&1)/(n(i)))):real`;
    `(&1)+((X0)+(&0)):real`;
    `(&0)+((X1)+(&0)):real`] asymp2_sub THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. ((&2)*((b(i))/(n(i))))-((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))):real`;
    `((&2)*(Y0))-((&1)+((X0)+(&0))):real`;
    `(((&2)*(Y1))+((&0)*(Y0)))-((&0)+((X1)+(&0))):real`;
    `\i:num. (&1)/(n(i)):real`;
    `&0:real`;
    `&0:real`] asymp2_sub THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (b(i))/(n(i)):real`;
    `Y0:real`;
    `Y1:real`;
    `\i:num. (&1)/(n(i)):real`;
    `&0:real`;
    `&0:real`] asymp2_sub THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. &2:real`;
    `&2:real`;
    `&0:real`;
    `\i:num. ((b(i))/(n(i)))-((&1)/(n(i))):real`;
    `(Y0)-(&0):real`;
    `(Y1)-(&0):real`] asymp2_mul THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (((&2)*((b(i))/(n(i))))-((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))-((&1)/(n(i))):real`;
    `(((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0):real`;
    `((((&2)*(Y1))+((&0)*(Y0)))-((&0)+((X1)+(&0))))-(&0):real`;
    `\i:num. (&2)*(((b(i))/(n(i)))-((&1)/(n(i)))):real`;
    `(&2)*((Y0)-(&0)):real`;
    `((&2)*((Y1)-(&0)))+((&0)*((Y0)-(&0))):real`] asymp2_div THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (&1)+((log2((b(i))/(n(i)))/log2(n(i)))+((((&1)/(n(i)))*(((&1)/((b(i))/(n(i))))*((log2((pi)*((b(i))/(n(i))))/log2(n(i)))+(&1))))-(log2(twopie)/log2(n(i))))):real`;
    `(&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))):real`;
    `(&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)))):real`;
    `\i:num. ((((&2)*((b(i))/(n(i))))-((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))-((&1)/(n(i))))/((&2)*(((b(i))/(n(i)))-((&1)/(n(i))))):real`;
    `((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))):real`;
    `(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((--(((&2)*((Y1)-(&0)))+((&0)*((Y0)-(&0)))))/(((&2)*((Y0)-(&0)))*((&2)*((Y0)-(&0))))))+((((((&2)*(Y1))+((&0)*(Y0)))-((&0)+((X1)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0))))):real`] asymp2_mul THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. (k(i))/(n(i)):real`;
    `X0:real`;
    `X1:real`;
    `\i:num. (&1)+(((k(i))/(n(i)))+((&1)/(n(i)))):real`;
    `(&1)+((X0)+(&0)):real`;
    `(&0)+((X1)+(&0)):real`] asymp2_div THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. ((k(i))/(n(i)))/((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))):real`;
    `(X0)*((&1)/((&1)+((X0)+(&0)))):real`;
    `((X0)*((--((&0)+((X1)+(&0))))/(((&1)+((X0)+(&0)))*((&1)+((X0)+(&0))))))+((X1)*((&1)/((&1)+((X0)+(&0))))):real`;
    `\i:num. log2(q(i))/log2(n(i)):real`;
    `Q0:real`;
    `Q1:real`] asymp2_mul THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. ((&1)+((log2((b(i))/(n(i)))/log2(n(i)))+((((&1)/(n(i)))*(((&1)/((b(i))/(n(i))))*((log2((pi)*((b(i))/(n(i))))/log2(n(i)))+(&1))))-(log2(twopie)/log2(n(i))))))*(((((&2)*((b(i))/(n(i))))-((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))-((&1)/(n(i))))/((&2)*(((b(i))/(n(i)))-((&1)/(n(i)))))):real`;
    `((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0))))):real`;
    `(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*((((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((--(((&2)*((Y1)-(&0)))+((&0)*((Y0)-(&0)))))/(((&2)*((Y0)-(&0)))*((&2)*((Y0)-(&0))))))+((((((&2)*(Y1))+((&0)*(Y0)))-((&0)+((X1)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))):real`;
    `\i:num. (((k(i))/(n(i)))/((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))*(log2(q(i))/log2(n(i))):real`;
    `((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q0):real`;
    `(((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q1))+((((X0)*((--((&0)+((X1)+(&0))))/(((&1)+((X0)+(&0)))*((&1)+((X0)+(&0))))))+((X1)*((&1)/((&1)+((X0)+(&0))))))*(Q0)):real`] asymp2_add THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. ((&1)/(&2))*(log2(((&1)+(((k(i))/(n(i)))+((&1)/(n(i)))))/((b(i))/(n(i))))/log2(n(i))):real`;
    `((&1)*((&1)/(&2)))*(&0):real`;
    `(((&1)*((&1)/(&2)))*(log2(((&1)+((X0)+(&0)))*((&1)/(Y0)))))+((((&1)*((--(&0))/((&2)*(&2))))+((&0)*((&1)/(&2))))*(&0)):real`;
    `\i:num. (((&1)+((log2((b(i))/(n(i)))/log2(n(i)))+((((&1)/(n(i)))*(((&1)/((b(i))/(n(i))))*((log2((pi)*((b(i))/(n(i))))/log2(n(i)))+(&1))))-(log2(twopie)/log2(n(i))))))*(((((&2)*((b(i))/(n(i))))-((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))-((&1)/(n(i))))/((&2)*(((b(i))/(n(i)))-((&1)/(n(i)))))))+((((k(i))/(n(i)))/((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))*(log2(q(i))/log2(n(i)))):real`;
    `(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0))))))+(((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q0)):real`;
    `((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*((((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((--(((&2)*((Y1)-(&0)))+((&0)*((Y0)-(&0)))))/(((&2)*((Y0)-(&0)))*((&2)*((Y0)-(&0))))))+((((((&2)*(Y1))+((&0)*(Y0)))-((&0)+((X1)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))))+((((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q1))+((((X0)*((--((&0)+((X1)+(&0))))/(((&1)+((X0)+(&0)))*((&1)+((X0)+(&0))))))+((X1)*((&1)/((&1)+((X0)+(&0))))))*(Q0))):real`] asymp2_add THEN
  specialize [`\i:num. n(i):real`;
    `\i:num. ((&1)/(&2))*(log2(&1+((&1)+((k(i))/(n(i))))*(n(i))powreal((&1)+((&2)*(log2(s(i))/log2(n(i))))))/log2(n(i))):real`;
    `((&1)*((&1)/(&2)))*((&1)+((&2)*(S0))):real`;
    `(((&1)*((&1)/(&2)))*((log2((&1)+(X0)))+((&0)+(((&2)*(S1))+((&0)*(S0))))))+((((&1)*((--(&0))/((&2)*(&2))))+((&0)*((&1)/(&2))))*((&1)+((&2)*(S0)))):real`;
    `\i:num. (((&1)/(&2))*(log2(((&1)+(((k(i))/(n(i)))+((&1)/(n(i)))))/((b(i))/(n(i))))/log2(n(i))))+((((&1)+((log2((b(i))/(n(i)))/log2(n(i)))+((((&1)/(n(i)))*(((&1)/((b(i))/(n(i))))*((log2((pi)*((b(i))/(n(i))))/log2(n(i)))+(&1))))-(log2(twopie)/log2(n(i))))))*(((((&2)*((b(i))/(n(i))))-((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))-((&1)/(n(i))))/((&2)*(((b(i))/(n(i)))-((&1)/(n(i)))))))+((((k(i))/(n(i)))/((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))*(log2(q(i))/log2(n(i))))):real`;
    `(((&1)*((&1)/(&2)))*(&0))+((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0))))))+(((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q0))):real`;
    `((((&1)*((&1)/(&2)))*(log2(((&1)+((X0)+(&0)))*((&1)/(Y0)))))+((((&1)*((--(&0))/((&2)*(&2))))+((&0)*((&1)/(&2))))*(&0)))+(((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*((((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((--(((&2)*((Y1)-(&0)))+((&0)*((Y0)-(&0)))))/(((&2)*((Y0)-(&0)))*((&2)*((Y0)-(&0))))))+((((((&2)*(Y1))+((&0)*(Y0)))-((&0)+((X1)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))))+((((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q1))+((((X0)*((--((&0)+((X1)+(&0))))/(((&1)+((X0)+(&0)))*((&1)+((X0)+(&0))))))+((X1)*((&1)/((&1)+((X0)+(&0))))))*(Q0)))):real`] asymp2_sub THEN
  ASM_MESON_TAC []);;

let asymp2_simple_lemma = prove(`
  !n:num->real b:num->real.
  (!i. &1 < n(i))
  /\ (!i. &1 < b(i))
  ==> 
  (!i. &0 < (&2)*(((b(i))/(n(i)))-((&1)/(n(i)))))
  `,
  MESON_TAC [REAL_ARITH `&1 < x ==> &0 < x - &1`;
    REAL_ARITH `&1 < y ==> &0 < y`;
    REAL_LT_DIV;
    REAL_FIELD `&1 < y ==> (x - &1) / y = x/y - &1 / y`;
    REAL_ARITH `&0 < z ==> &0 < &2 * z`]);;

let asymp2_simple_lemma2 = prove(`
  !n:num->real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  ==>
  o1_seq (\i:num. ((((&1)/(n(i))) - (&0)) * log2(n(i))) - (&0))
  `,
  REWRITE_TAC [REAL_ARITH `(&1/x - &0)*y - &0 = y/x`] THEN
  REPEAT STRIP_TAC THEN
  real_linear `&0 < &1` THEN
  specialize [] (MESON [REAL_ARITH `&1 < x ==> &1 <= x`] `(!i:num. &1 < n(i)) ==> (!i. &1 <= n(i))`) THEN
  specialize [`n:num->real`] o1_log2overlinear_1 THEN
  ASM_SIMP_TAC []);;

let asymp2_simple = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real
   Q0:real Q1:real S0:real S1:real X0:real X1:real Y0:real Y1:real twopie:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i:num. (((log2(q(i))/log2(n(i))) - (Q0)) * log2(n(i))) - (Q1))
  /\ o1_seq (\i:num. (((log2(s(i))/log2(n(i))) - (S0)) * log2(n(i))) - (S1))
  /\ o1_seq (\i:num. ((((k(i))/(n(i))) - (X0)) * log2(n(i))) - (X1))
  /\ o1_seq (\i:num. ((((b(i))/(n(i))) - (Y0)) * log2(n(i))) - (Y1))
  /\ &0 < (&1)+(X0)
  /\ &0 < (&1)+((&2)*(S0))
  /\ &0 < Y0
  /\ &0 < twopie
  ==> o1_seq (\i:num. ((((((&1)/(&2))*(log2(&1+((&1)+((k(i))/(n(i))))*(n(i))powreal((&1)+((&2)*(log2(s(i))/log2(n(i))))))/log2(n(i))))-((((&1)/(&2))*(log2(((&1)+(((k(i))/(n(i)))+((&1)/(n(i)))))/((b(i))/(n(i))))/log2(n(i))))+((((&1)+((log2((b(i))/(n(i)))/log2(n(i)))+((((&1)/(n(i)))*(((&1)/((b(i))/(n(i))))*((log2((pi)*((b(i))/(n(i))))/log2(n(i)))+(&1))))-(log2(twopie)/log2(n(i))))))*(((((&2)*((b(i))/(n(i))))-((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))-((&1)/(n(i))))/((&2)*(((b(i))/(n(i)))-((&1)/(n(i)))))))+((((k(i))/(n(i)))/((&1)+(((k(i))/(n(i)))+((&1)/(n(i))))))*(log2(q(i))/log2(n(i))))))) - ((((&1)*((&1)/(&2)))*((&1)+((&2)*(S0))))-((((&1)*((&1)/(&2)))*(&0))+((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0))))))+(((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q0)))))) * log2(n(i))) - (((((&1)*((&1)/(&2)))*((log2((&1)+(X0)))+((&0)+(((&2)*(S1))+((&0)*(S0))))))+((((&1)*((--(&0))/((&2)*(&2))))+((&0)*((&1)/(&2))))*((&1)+((&2)*(S0)))))-(((((&1)*((&1)/(&2)))*(log2(((&1)+((X0)+(&0)))*((&1)/(Y0)))))+((((&1)*((--(&0))/((&2)*(&2))))+((&0)*((&1)/(&2))))*(&0)))+(((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*((((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((--(((&2)*((Y1)-(&0)))+((&0)*((Y0)-(&0)))))/(((&2)*((Y0)-(&0)))*((&2)*((Y0)-(&0))))))+((((((&2)*(Y1))+((&0)*(Y0)))-((&0)+((X1)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))))+((((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q1))+((((X0)*((--((&0)+((X1)+(&0))))/(((&1)+((X0)+(&0)))*((&1)+((X0)+(&0))))))+((X1)*((&1)/((&1)+((X0)+(&0))))))*(Q0)))))))
  `,
  REPEAT STRIP_TAC THEN
  specialize [] PI_POS THEN
  specialize [`pi`;
    `Y0:real`] REAL_LT_MUL THEN
  real_linear `&0 < &1` THEN
  specialize [`&1`;
    `Y0:real`] REAL_LT_DIV THEN
  real_linear `&0 < &1+X0 ==> &0 < &1+(X0 + &0)` THEN
  real_linear `&0 < &1+X0 ==> ~(&1+(X0 + &0) = &0)` THEN
  specialize [`&1+(X0 + &0):real`;
    `&1/Y0`] REAL_LT_MUL THEN
  real_linear `&0 < &2` THEN
  ASSUME_TAC (REAL_ARITH `!i:num. &0 < &2`) THEN
  real_linear `~(&2 = &0)` THEN
  real_linear `&0 < Y0 ==> ~(Y0 = &0)` THEN
  real_linear `&0 < Y0 ==> &0 < (&2)*(Y0 - &0)` THEN
  real_linear `&0 < Y0 ==> ~((&2)*((Y0)-(&0)) = &0)` THEN
  specialize [] (MESON [REAL_ARITH `&1 < x ==> &0 < x`;
      REAL_LT_DIV] `(!i. &1 < n(i)) /\ (!i. &1 < b(i)) ==> (!i:num. &0 < (b(i))/(n(i)))`) THEN
  specialize [] (MESON [REAL_ARITH `&1 < x ==> &0 < x`;
      REAL_LT_DIV;
      REAL_ARITH `&0 < x ==> &0 < &1 + x`] `(!i. &1 < n(i)) /\ (!i. &0 < k(i)) ==> (!i:num. &0 < &1 + k(i)/n(i))`) THEN
  specialize [] (MESON [REAL_ARITH `&1 < x ==> &0 < x`;
      REAL_LT_DIV;
      REAL_LT_ADD;
      REAL_ARITH `&0 < &1`;
      REAL_ARITH `&0 < x ==> &0 < &1 + x`] `(!i. &1 < n(i)) /\ (!i. &0 < k(i)) ==> (!i:num. &0 < &1 + (k(i)/n(i) + &1/n(i)))`) THEN
  specialize [`n:num->real`;
    `q:num->real`;
    `s:num->real`;
    `k:num->real`;
    `b:num->real`;
    `Q0:real`;
    `Q1:real`;
    `S0:real`;
    `S1:real`;
    `X0:real`;
    `X1:real`;
    `Y0:real`;
    `Y1:real`;
    `twopie:real`] asymp2_main THEN
  specialize [`n:num->real`;
    `b:num->real`] asymp2_simple_lemma THEN
  specialize [`n:num->real`] asymp2_simple_lemma2 THEN
  ASM_SIMP_TAC []);;

let asymp2_simple2_lemma = prove(`
  !n b.
  &1 < n
  /\ &1 < b
  ==>
  ln (b * exp (&1 / b * ln (pi * b)) / (&2 * pi * exp (&1))) / ln n
  = (&1 +
   ln (b / n) / ln n +
   &1 / n * (n * &1) / b * (ln (pi * b / n) / ln n + &1) -
   ln (&2 * pi * exp (&1)) / ln n)
  `,
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> &1 / n * (n * &1) / x * y = y / x`] THEN
  REPEAT STRIP_TAC THEN
  real_linear `&1 < b ==> &0 < b` THEN
  specialize [`&1 / b * ln(pi*b)`] REAL_EXP_POS_LT THEN
  real_linear `&0 < &2` THEN
  specialize [] PI_POS THEN
  specialize [`&1`] REAL_EXP_POS_LT THEN
  specialize [`pi`;
    `exp(&1)`] REAL_LT_MUL THEN
  specialize [`&2`;
    `pi*exp(&1)`] REAL_LT_MUL THEN
  specialize [`exp(&1 / b * ln(pi*b))`;
    `&2*pi*exp(&1)`] REAL_LT_DIV THEN
  specialize [`b:real`;
    `exp(&1 / b * ln(pi*b)) / (&2*pi*exp(&1))`] LN_MUL THEN
  specialize [`exp(&1 / b * ln(pi*b))`;
    `(&2*pi*exp(&1))`] ln_div THEN
  specialize [`&1 / b * ln (pi * b)`] LN_EXP THEN
  specialize [`n:real`] ln_gt1 THEN
  ASM_SIMP_TAC [REAL_FIELD `&0 < ln(n) ==> (ln (b) + (&1 / b * ln (pi * b)) - ln (&2 * pi * exp (&1))) / ln n = ln (b)/ ln(n) + (ln (pi * b) / ln n) / b - ln (&2 * pi * exp (&1)) / ln n`] THEN
  real_linear `&1 < n ==> &0 < n` THEN
  REWRITE_TAC [REAL_ARITH `pi*b/n = (pi*b)/n`] THEN
  specialize [`pi`;
    `b:real`] REAL_LT_MUL THEN
  ASM_SIMP_TAC [ln_div] THEN
  ASM_SIMP_TAC [REAL_FIELD `&0 < ln(n) ==> (x - ln(n))/ln(n) + &1 = x / ln(n)`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&0 < ln(n) ==> &1 + (x - ln(n))/ln(n) + y = x/ln(n) + y`]);;

let asymp2_simple2_standardratio = prove(`
  !n q s k b.
  &1 < n /\
  &1 < b /\
  &0 < s ==>
  ln(standardratio n q s k b)/ln(n) = 
  &1 / &2 * log2 (&1 + (&1 + k / n) * n powreal (&1 + &2 * log2 s / log2 n)) / log2 n - (&1 / &2 * log2 ((&1 + k / n + &1 / n) / (b / n)) / log2 n + (&1 + log2 (b / n) / log2 n + &1 / n * &1 / (b / n) * (log2 (pi * b / n) / log2 n + &1) - log2 (&2 * pi * exp(&1)) / log2 n) * (&2 * b / n - (&1 + k / n + &1 / n) - &1 / n) / (&2 * (b / n - &1 / n)) + k / n / (&1 + k / n + &1 / n) * log2 q / log2 n)
  `,
  REWRITE_TAC [log2ratio_lnratio] THEN
  REWRITE_TAC [standardratio_ln2] THEN
  REWRITE_TAC [bkzdelta_ln] THEN
  REWRITE_TAC [powreal] THEN
  REPEAT STRIP_TAC THEN
  specialize [`n:real`] ln_gt1 THEN
  ASM_SIMP_TAC [REAL_FIELD `&0 < ln(n) ==> (a + b)/ln(n) = a/ln(n) + b/ln(n)`;
    REAL_FIELD `&0 < ln(n) ==> (a - b)/ln(n) = a/ln(n) - b/ln(n)`] THEN
  REWRITE_TAC [REAL_FIELD `a / b / c = a / (b * c)`] THEN
  REWRITE_TAC [REAL_FIELD `a/(b/n) = (n*a)/b`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> n * (&1 + k / n + &1 / n) = n + k + &1`] THEN
  REWRITE_TAC [REAL_FIELD `(a*b)/ln n = a*(b/ln n)`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&0 < ln(n) ==> (&1 + &2 * L / ln(n)) * ln(n) = ln(n) + &2 * L`] THEN
  REWRITE_TAC [REAL_EXP_ADD] THEN
  ASM_SIMP_TAC [EXP_LN;
    REAL_ARITH `&1 < n ==> &0 < n`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> (&1 + k / n) * n * X = (n + k) * X`] THEN
  REWRITE_TAC [REAL_EXP_N] THEN
  ASM_SIMP_TAC [EXP_LN] THEN
  REWRITE_TAC [REAL_ARITH `&1 + x * y pow 2 = x * y pow 2 + &1`] THEN
  ASM_SIMP_TAC [REAL_ARITH `&1 < n ==> &2 * (b / n - &1 / n) = (&2 * (b - &1)) / n`] THEN
  REWRITE_TAC [REAL_FIELD `a/(b/n) = (n*a)/b`] THEN
  REWRITE_TAC [REAL_ARITH `n*(&2*b/n-(&1+k/n+ &1/n)- &1/n) = &2*(n*(b/n))-n*(&1+k/n+ &1/n) - n*(&1/n)`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> n * (&1 + k / n + &1 / n) = n + k + &1`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> n*(b/n) = b`] THEN
  REWRITE_TAC [REAL_ARITH `(&2 * b - (n + k + &1) - &1) * &1 / (&2 * (b - &1)) * X = X * (&2 * b - (n + k + &1) - &1) / (&2 * (b - &1))`] THEN
  specialize [`n:real`;
    `b:real`] asymp2_simple2_lemma THEN
  ASM_SIMP_TAC []);;

let asymp2_simple2_standardratio2 = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real.
  (!i. &1 < n(i)) /\
  (!i. &1 < b(i)) /\
  (!i. &0 < s(i)) ==>
  !i.
  ln(standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i))) / ln(n(i)) =
  &1 / &2 * log2 (&1 + (&1 + k(i) / n(i)) * n(i) powreal (&1 + &2 * log2(s(i)) / log2(n(i)))) / log2(n(i)) - (&1 / &2 * log2 ((&1 + k(i) / n(i) + &1 / n(i)) / (b(i) / n(i))) / log2(n(i)) + (&1 + log2 (b(i) / n(i)) / log2(n(i)) + &1 / n(i) * &1 / (b(i) / n(i)) * (log2 (pi * b(i) / n(i)) / log2(n(i)) + &1) - log2 (&2 * pi * exp(&1)) / log2(n(i))) * (&2 * b(i) / n(i) - (&1 + k(i) / n(i) + &1 / n(i)) - &1 / n(i)) / (&2 * (b(i) / n(i) - &1 / n(i))) + k(i) / n(i) / (&1 + k(i) / n(i) + &1 / n(i)) * log2(q(i))/ log2(n(i)))
  `,
  MESON_TAC [asymp2_simple2_standardratio]);;

let asymp2_simple2 = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real
   Q0:real Q1:real S0:real S1:real X0:real X1:real Y0:real Y1:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i:num. (((log2(q(i))/log2(n(i))) - (Q0)) * log2(n(i))) - (Q1))
  /\ o1_seq (\i:num. (((log2(s(i))/log2(n(i))) - (S0)) * log2(n(i))) - (S1))
  /\ o1_seq (\i:num. ((((k(i))/(n(i))) - (X0)) * log2(n(i))) - (X1))
  /\ o1_seq (\i:num. ((((b(i))/(n(i))) - (Y0)) * log2(n(i))) - (Y1))
  /\ &0 < (&1)+(X0)
  /\ &0 < (&1)+((&2)*(S0))
  /\ &0 < Y0
  ==> o1_seq (\i:num. ((
    log2(standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i))) / log2(n(i))
    - ((((&1)*((&1)/(&2)))*((&1)+((&2)*(S0))))-((((&1)*((&1)/(&2)))*(&0))+((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0))))))+(((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q0)))))) * log2(n(i))) - (((((&1)*((&1)/(&2)))*((log2((&1)+(X0)))+((&0)+(((&2)*(S1))+((&0)*(S0))))))+((((&1)*((--(&0))/((&2)*(&2))))+((&0)*((&1)/(&2))))*((&1)+((&2)*(S0)))))-(((((&1)*((&1)/(&2)))*(log2(((&1)+((X0)+(&0)))*((&1)/(Y0)))))+((((&1)*((--(&0))/((&2)*(&2))))+((&0)*((&1)/(&2))))*(&0)))+(((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*((((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((--(((&2)*((Y1)-(&0)))+((&0)*((Y0)-(&0)))))/(((&2)*((Y0)-(&0)))*((&2)*((Y0)-(&0))))))+((((((&2)*(Y1))+((&0)*(Y0)))-((&0)+((X1)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(&2 * pi * exp(&1))))))*(((((&2)*(Y0))-((&1)+((X0)+(&0))))-(&0))*((&1)/((&2)*((Y0)-(&0)))))))+((((X0)*((&1)/((&1)+((X0)+(&0)))))*(Q1))+((((X0)*((--((&0)+((X1)+(&0))))/(((&1)+((X0)+(&0)))*((&1)+((X0)+(&0))))))+((X1)*((&1)/((&1)+((X0)+(&0))))))*(Q0)))))))
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [log2ratio_lnratio] THEN
  real_linear `&0 < &2` THEN
  specialize [] PI_POS THEN
  specialize [`&1`] REAL_EXP_POS_LT THEN
  specialize [`pi`;
    `exp(&1)`] REAL_LT_MUL THEN
  specialize [`&2`;
    `pi*exp(&1)`] REAL_LT_MUL THEN
  specialize [`n:num->real`;
    `q:num->real`;
    `s:num->real`;
    `k:num->real`;
    `b:num->real`;
    `Q0:real`;
    `Q1:real`;
    `S0:real`;
    `S1:real`;
    `X0:real`;
    `X1:real`;
    `Y0:real`;
    `Y1:real`;
    `&2 * pi * exp(&1)`] asymp2_simple THEN
  specialize [`n:num->real`;
    `q:num->real`;
    `s:num->real`;
    `k:num->real`;
    `b:num->real`] asymp2_simple2_standardratio2 THEN
  ASM_SIMP_TAC []);;

let asymp2_simple3_lemma = REAL_FIELD `
  !Q0 S0 X0 Y0.
  &0 < Y0 ==> 
  (&1 * &1 / &2) * (&1 + &2 * S0) - ((&1 * &1 / &2) * &0 + (&1 + &0 + &0 * (&1 * &1 / Y0) * (&0 + &1) - &0) * (&2 * Y0 - (&1 + X0 + &0) - &0) * &1 / (&2 * (Y0 - &0)) + (X0 * &1 / (&1 + X0 + &0)) * Q0)
  = S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0
`;;

let asymp2_simple3_lemma2 = prove(`
  !Q0 S0 X0 Y0 Q1 S1 X1 Y1.
  &0 < Y0 /\ &0 < &1 + X0 ==>
  (((&1 * &1 / &2) * (log2 (&1 + X0) + &0 + &2 * S1 + &0 * S0) + (&1 * -- &0 / (&2 * &2) + &0 * &1 / &2) * (&1 + &2 * S0)) - (((&1 * &1 / &2) * log2 ((&1 + X0 + &0) * (&1 / Y0)) + (&1 * -- &0 / (&2 * &2) + &0 * &1 / &2) * &0) + ((&1 + &0 + &0 * (&1 * &1 / Y0) * (&0 + &1) - &0) * ((&2 * Y0 - (&1 + X0 + &0) - &0) * --(&2 * (Y1 - &0) + &0 * (Y0 - &0)) / ((&2 * (Y0 - &0)) * &2 * (Y0 - &0)) + ((&2 * Y1 + &0 * Y0) - (&0 + X1 + &0) - &0) * &1 / (&2 * (Y0 - &0))) + (&0 + log2 Y0 + (&0 * ((&1 * &1 / Y0) * (log2 (pi * Y0) + &0) + (&1 * --Y1 / (Y0 * Y0) + &0 * &1 / Y0) * (&0 + &1)) + &0 * (&1 * &1 / Y0) * (&0 + &1)) - log2 (&2 * pi * exp (&1))) * (&2 * Y0 - (&1 + X0 + &0) - &0) * &1 / (&2 * (Y0 - &0))) + (X0 * &1 / (&1 + X0 + &0)) * Q1 + (X0 * --(&0 + X1 + &0) / ((&1 + X0 + &0) * (&1 + X0 + &0)) + X1 * &1 / (&1 + X0 + &0)) * Q0))
  = S1 + (&1 / &2) * (log2(Y0) + X1/Y0 - (&1 + X0)*Y1/(Y0*Y0) - (&2 - (&1 + X0)/Y0)*(log2(Y0) - log2(&2 * pi * exp(&1)))) - (Q1 * X0/(&1 + X0) + Q0 * X1/((&1 + X0)*(&1 + X0)))
  `,
  SIMP_TAC [MESON [REAL_ARITH `&1 + X0 = &1 + X0 + &0`;
      log2_mul;
      pos_1over] `&0 < Y0 /\ &0 < &1 + X0 ==> log2 ((&1 + X0 + &0) * (&1 / Y0)) = log2(&1 + X0 + &0) + log2(&1 / Y0)`] THEN
  REWRITE_TAC [REAL_ARITH `x - &0 = x`;
    REAL_ARITH `x + &0 = x`;
    REAL_ARITH `&0 + x = x`;
    REAL_ARITH `x * &0 = &0`;
    REAL_ARITH `&0 * x = &0`;
    REAL_ARITH `&0 / x = &0`;
    REAL_ARITH `-- &0 / x = &0`;
    REAL_ARITH `&1 * x = x`;
    REAL_ARITH `x * &1 = x`;
    REAL_ARITH `x + &0 - y = x - y`;
    REAL_ARITH `x * &1 / y = x / y`] THEN
  SIMP_TAC [REAL_FIELD `&0 < &1 + X0 ==> (X0 * --X1 / ((&1 + X0) * (&1 + X0)) + X1 / (&1 + X0)) = X1 / ((&1 + X0) * (&1 + X0))`] THEN
  SIMP_TAC [log2_1over] THEN
  SIMP_TAC [REAL_FIELD `!Q0 X0 Y0 Q1 S1 X1 Y1. &0 < Y0 /\ &0 < &1 + X0 ==> &1 / &2 * (log2 (&1 + X0) + &2 * S1) - (&1 / &2 * (log2 (&1 + X0) + --log2 Y0) + (((&2 * Y0 - (&1 + X0)) * --(&2 * Y1) / ((&2 * Y0) * &2 * Y0) + (&2 * Y1 - X1) / (&2 * Y0)) + (log2 Y0 - log2 (&2 * pi * exp (&1))) * (&2 * Y0 - (&1 + X0)) / (&2 * Y0)) + X0 / (&1 + X0) * Q1 + X1 / ((&1 + X0) * (&1 + X0)) * Q0) = S1 + &1 / &2 * (log2 Y0 + X1 / Y0 - (&1 + X0) * Y1 / (Y0 * Y0) - (&2 - (&1 + X0) / Y0) * (log2 Y0 - log2 (&2 * pi * exp (&1)))) - (Q1 * X0 / (&1 + X0) + Q0 * X1 / ((&1 + X0) * (&1 + X0)))`]);;

let asymp2_simple3 = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real
   Q0:real Q1:real S0:real S1:real X0:real X1:real Y0:real Y1:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i:num. (((log2(q(i))/log2(n(i))) - (Q0)) * log2(n(i))) - (Q1))
  /\ o1_seq (\i:num. (((log2(s(i))/log2(n(i))) - (S0)) * log2(n(i))) - (S1))
  /\ o1_seq (\i:num. ((((k(i))/(n(i))) - (X0)) * log2(n(i))) - (X1))
  /\ o1_seq (\i:num. ((((b(i))/(n(i))) - (Y0)) * log2(n(i))) - (Y1))
  /\ &0 < &1 + X0
  /\ &0 < &1 + &2*S0
  /\ &0 < Y0
  ==> o1_seq (\i:num.
    (
      log2(standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i))) / log2(n(i))
      - (S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0)
    ) * log2(n(i))
    - (S1 + (&1 / &2) * (log2(Y0) + X1/Y0 - (&1 + X0)*Y1/(Y0*Y0) - (&2 - (&1 + X0)/Y0)*(log2(Y0) - log2(&2 * pi * exp(&1)))) - (Q1 * X0/(&1 + X0) + Q0 * X1/((&1 + X0)*(&1 + X0))))
    )
  `,
  SIMP_TAC [GSYM asymp2_simple3_lemma] THEN
  SIMP_TAC [GSYM asymp2_simple3_lemma2] THEN
  MESON_TAC [asymp2_simple2]);;

let asymp2_eventually_negative = prove(`
  !n f c0 c1.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. (f(i) - c0)*log2(n(i)) - c1)
  /\ c0 < &0 
  ==> ?m. !i. m <= i ==> f(i) < &0
  `,
  MESON_TAC [asymp2_asymp1;
    negative_limit_is_eventually_negative]);;

let asymp2_eventually_positive = prove(`
  !n f c0 c1.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. (f(i) - c0)*log2(n(i)) - c1)
  /\ &0 < c0
  ==> ?m. !i. m <= i ==> &0 < f(i)
  `,
  MESON_TAC [asymp2_asymp1;
    positive_limit_is_eventually_positive]);;

let asymp2_eventually_negative2_lemma = prove(`
  !n x.
  &1 < n
  /\ x*log2(n) < &0
  ==> x < &0
  `,
  MESON_TAC [log2_gt1;
    REAL_LT_DIV2_EQ;
    REAL_ARITH `&0 / L = &0`;
    REAL_FIELD `&0 < L ==> (x*L)/L = x`]);;

let asymp2_eventually_negative2 = prove(`
  !n f c1.
  (!i. &1 < n(i))
  /\ o1_seq (\i. (f(i) - &0)*log2(n(i)) - c1)
  /\ c1 < &0
  ==> ?m. !i. m <= i ==> f(i) < &0
  `,
  REWRITE_TAC [REAL_ARITH `x - &0 = x`] THEN
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. f(i)*log2(n(i))`;
    `c1:real`] negative_limit_is_eventually_negative THEN
  ASM_MESON_TAC [asymp2_eventually_negative2_lemma]);;

let asymp2_eventually_positive2_lemma = prove(`
  !n x.
  &1 < n
  /\ &0 < x*log2(n)
  ==> &0 < x
  `,
  MESON_TAC [log2_gt1;
    REAL_LT_DIV2_EQ;
    REAL_ARITH `&0 / L = &0`;
    REAL_FIELD `&0 < L ==> (x*L)/L = x`]);;

let asymp2_eventually_positive2 = prove(`
  !n f c1.
  (!i. &1 < n(i))
  /\ o1_seq (\i. (f(i) - &0)*log2(n(i)) - c1)
  /\ &0 < c1
  ==> ?m. !i. m <= i ==> &0 < f(i)
  `,
  REWRITE_TAC [REAL_ARITH `x - &0 = x`] THEN
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. f(i)*log2(n(i))`;
    `c1:real`] positive_limit_is_eventually_positive THEN
  ASM_MESON_TAC [asymp2_eventually_positive2_lemma]);;

let asymp2_abovebelow_lemma1 = prove(`
  !Q0 S0 X0 Y0 Z0.
  &0 < Y0
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ Y0 = Z0
  /\ Z0 = (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0))
  ==> 
  &0 = S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0
  `,
  MESON_TAC [eqdivflip;
    REAL_FIELD `&1 - &2*S0 + &2*Q0*X0/(&1 + X0) = (&1 + X0) / Y0 ==> &0 = S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0`]);;

let asymp2_abovebelow_lemma2 = prove(`
  !Q0 Q1 S0 S1 X0 X1 Y0 Y1 Z0 Z1.
  &0 < &1 + X0
  /\ &0 < Y0
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ Z0 = (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0))
  /\ Z1 = ((S1 + (&1 / &2) * (log2(Z0) + X1/Z0 - (&2 - (&1 + X0)/Z0)*(log2(Z0) - log2(&2 * pi * exp(&1)))) - (Q1 * X0/(&1 + X0) + Q0 * X1/((&1 + X0)*(&1 + X0)))) * (&2 * Z0 * Z0)) / (&1 + X0)
  /\ Y0 = Z0
  ==> 
  (S1 + (&1 / &2) * (log2(Y0) + X1/Y0 - (&1 + X0)*Y1/(Y0*Y0) - (&2 - (&1 + X0)/Y0)*(log2(Y0) - log2(&2 * pi * exp(&1)))) - (Q1 * X0/(&1 + X0) + Q0 * X1/((&1 + X0)*(&1 + X0))))
  = (&1 + X0) * (Z1 - Y1) / (&2 * Z0 * Z0)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`&1 + X0`;
    `&1 - &2*S0 + &2*Q0*X0/(&1 + X0)`] REAL_LT_DIV THEN
  real_linear `&0 < (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0)) /\ Z0 = (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0)) ==> &0 < Z0` THEN
  real_linear `&0 < &2` THEN
  specialize [`Z0:real`;
    `Z0:real`] REAL_LT_MUL THEN
  specialize [`&2`;
    `Z0*Z0:real`] REAL_LT_MUL THEN
  specialize_forward [`Z1:real`;
    `((S1 + (&1 / &2) * (log2(Z0) + X1/Z0 - (&2 - (&1 + X0)/Z0)*(log2(Z0) - log2(&2 * pi * exp(&1)))) - (Q1 * X0/(&1 + X0) + Q0 * X1/((&1 + X0)*(&1 + X0)))) * (&2 * Z0 * Z0))`;
    `&1+X0`] REAL_EQ_RDIV_EQ THEN
  specialize_reverse [`Z1*(&1+X0)`;
    `S1 + &1 / &2 * (log2 Z0 + X1 / Z0 - (&2 - (&1 + X0) / Z0) * (log2 Z0 - log2 (&2 * pi * exp (&1)))) - (Q1 * X0 / (&1 + X0) + Q0 * X1 / ((&1 + X0) * (&1 + X0)))`;
    `&2 * Z0 * Z0`] REAL_EQ_LDIV_EQ THEN
  real_cancel `&0 < Z0 ==> (&1 + X0) * (Z1 - Y1) / (&2 * Z0 * Z0) = (Z1 * (&1 + X0)) / (&2 * Z0 * Z0) - (&1 + X0)*Y1 / (&2 *Z0 * Z0)` THEN
  real_cancel `S1 + &1 / &2 * (log2 Z0 + X1 / Z0 - (&1 + X0) * Y1 / (Z0 * Z0) - (&2 - (&1 + X0) / Z0) * (log2 Z0 - log2 (&2 * pi * exp (&1)))) - (Q1 * X0 / (&1 + X0) + Q0 * X1 / ((&1 + X0) * (&1 + X0))) = (S1 + &1 / &2 * (log2 Z0 + X1 / Z0 - (&2 - (&1 + X0) / Z0) * (log2 Z0 - log2 (&2 * pi * exp (&1)))) - (Q1 * X0 / (&1 + X0) + Q0 * X1 / ((&1 + X0) * (&1 + X0)))) - (&1 + X0)*Y1 / (&2 *Z0 * Z0)` THEN
  ASM_MESON_TAC []);;

let asymp2_abovebelow_lemma3 = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real
   Q0:real Q1:real S0:real S1:real X0:real X1:real Y0:real Y1:real Z0:real Z1:real.
  &0 < &1 + X0
  /\ &0 < Y0
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ Z0 = (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0))
  /\ Z1 = ((S1 + (&1 / &2) * (log2(Z0) + X1/Z0 - (&2 - (&1 + X0)/Z0)*(log2(Z0) - log2(&2 * pi * exp(&1)))) - (Q1 * X0/(&1 + X0) + Q0 * X1/((&1 + X0)*(&1 + X0)))) * (&2 * Z0 * Z0)) / (&1 + X0)
  /\ Y0 = Z0
  ==>
  (\i:num.
    (
      log2(standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i))) / log2(n(i))
      - (S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0)
    ) * log2(n(i))
    - (S1 + (&1 / &2) * (log2(Y0) + X1/Y0 - (&1 + X0)*Y1/(Y0*Y0) - (&2 - (&1 + X0)/Y0)*(log2(Y0) - log2(&2 * pi * exp(&1)))) - (Q1 * X0/(&1 + X0) + Q0 * X1/((&1 + X0)*(&1 + X0))))
  ) =
  (\i:num.
    (
      log2(standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i))) / log2(n(i))
      - &0
    ) * log2(n(i))
    - ((&1 + X0) * (Z1 - Y1) / (&2 * Z0 * Z0))
  )
  `,
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC EQ_EXT THEN
  specialize [`Q0:real`;
    `S0:real`;
    `X0:real`;
    `Y0:real`] asymp2_abovebelow_lemma1 THEN
  specialize [`Q0:real`;
    `Q1:real`;
    `S0:real`;
    `S1:real`;
    `X0:real`;
    `X1:real`;
    `Y0:real`;
    `Y1:real`;
    `Z0:real`;
    `Z1:real`] asymp2_abovebelow_lemma2 THEN
  ASM_MESON_TAC []);;

let asymp2_abovebelow_lemma4 = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real
   Q0:real Q1:real S0:real S1:real X0:real X1:real Y0:real Y1:real Z0:real Z1:real.
  &0 < &1 + X0
  /\ &0 < Y0
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ Z0 = (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0))
  /\ Z1 = ((S1 + (&1 / &2) * (log2(Z0) + X1/Z0 - (&2 - (&1 + X0)/Z0)*(log2(Z0) - log2(&2 * pi * exp(&1)))) - (Q1 * X0/(&1 + X0) + Q0 * X1/((&1 + X0)*(&1 + X0)))) * (&2 * Z0 * Z0)) / (&1 + X0)
  /\ Y0 = Z0
  /\
  o1_seq (\i:num.
    (
      log2(standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i))) / log2(n(i))
      - (S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - (X0 / (&1 + X0)) * Q0)
    ) * log2(n(i))
    - (S1 + (&1 / &2) * (log2(Y0) + X1/Y0 - (&1 + X0)*Y1/(Y0*Y0) - (&2 - (&1 + X0)/Y0)*(log2(Y0) - log2(&2 * pi * exp(&1)))) - (Q1 * X0/(&1 + X0) + Q0 * X1/((&1 + X0)*(&1 + X0))))
  )
  ==>
  o1_seq (\i:num.
    (
      log2(standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i))) / log2(n(i))
      - &0
    ) * log2(n(i))
    - ((&1 + X0) * (Z1 - Y1) / (&2 * Z0 * Z0))
  )
  `,
  MESON_TAC [asymp2_abovebelow_lemma3]);;

let asymp2_abovebelow_lemma5 = prove(`
  !Q0:real Q1:real S0:real S1:real X0:real X1:real Y0:real Y1:real Z0:real Z1:real.
  &0 < &1 + X0
  /\ &0 < Y0
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ Z0 = (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0))
  /\ Y0 = Z0
  /\ Y1 < Z1
  ==> &0 < (&1 + X0) * (Z1 - Y1) / (&2 * Z0 * Z0)
  `,
  REPEAT STRIP_TAC THEN
  real_linear `Y1 < Z1 ==> &0 < Z1 - Y1` THEN
  specialize [`&1 + X0`;
    `&1 - &2*S0 + &2*Q0*X0/(&1 + X0)`] REAL_LT_DIV THEN
  real_linear `&0 < (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0)) /\ Z0 = (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0)) ==> &0 < Z0` THEN
  real_linear `&0 < &2` THEN
  specialize [`Z0:real`;
    `Z0:real`] REAL_LT_MUL THEN
  specialize [`&2`;
    `Z0*Z0:real`] REAL_LT_MUL THEN
  specialize [`Z1-Y1:real`;
    `&2*Z0*Z0`] REAL_LT_DIV THEN
  specialize [`&1 + X0`;
    `(Z1-Y1) / (&2*Z0*Z0)`] REAL_LT_MUL THEN
  ASM_SIMP_TAC []);;

let asymp2_abovebelow_lemma6 = prove(`
  !Q0:real Q1:real S0:real S1:real X0:real X1:real Y0:real Y1:real Z0:real Z1:real.
  &0 < &1 + X0
  /\ &0 < Y0
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ Z0 = (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0))
  /\ Y0 = Z0
  /\ Z1 < Y1
  ==> (&1 + X0) * (Z1 - Y1) / (&2 * Z0 * Z0) < &0
  `,
  REPEAT STRIP_TAC THEN
  real_linear `Z1 < Y1 ==> &0 < Y1 - Z1` THEN
  specialize [`&1 + X0`;
    `&1 - &2*S0 + &2*Q0*X0/(&1 + X0)`] REAL_LT_DIV THEN
  real_linear `&0 < (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0)) /\ Z0 = (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0)) ==> &0 < Z0` THEN
  real_linear `&0 < &2` THEN
  specialize [`Z0:real`;
    `Z0:real`] REAL_LT_MUL THEN
  specialize [`&2`;
    `Z0*Z0:real`] REAL_LT_MUL THEN
  specialize [`Y1-Z1:real`;
    `&2*Z0*Z0`] REAL_LT_DIV THEN
  specialize [`&1 + X0`;
    `(Y1-Z1) / (&2*Z0*Z0)`] REAL_LT_MUL THEN
  specialize_reverse [`(&1 + X0) * (Y1 - Z1) / (&2 * Z0 * Z0)`;
    `&0`] REAL_LT_NEG2 THEN
  real_linear `-- &0 = &0` THEN
  real_linear `--((&1 + X0) * (Y1 - Z1) / (&2 * Z0 * Z0)) = (&1 + X0) * (Z1 - Y1) / (&2 * Z0 * Z0)` THEN
  ASM_MESON_TAC []);;

let asymp2_abovebelow = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->real
   Q0:real Q1:real S0:real S1:real X0:real X1:real Y0:real Y1:real Z0:real Z1:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ (!i. &1 < b(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i:num. (((log2(q(i))/log2(n(i))) - (Q0)) * log2(n(i))) - (Q1))
  /\ o1_seq (\i:num. (((log2(s(i))/log2(n(i))) - (S0)) * log2(n(i))) - (S1))
  /\ o1_seq (\i:num. ((((k(i))/(n(i))) - (X0)) * log2(n(i))) - (X1))
  /\ o1_seq (\i:num. ((((b(i))/(n(i))) - (Y0)) * log2(n(i))) - (Y1))
  /\ &0 < &1 + X0
  /\ &0 < &1 + &2*S0
  /\ &0 < Y0
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ Z0 = (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0))
  /\ Z1 = ((S1 + (&1 / &2) * (log2(Z0) + X1/Z0 - (&2 - (&1 + X0)/Z0)*(log2(Z0) - log2(&2 * pi * exp(&1)))) - (Q1 * X0/(&1 + X0) + Q0 * X1/((&1 + X0)*(&1 + X0)))) * (&2 * Z0 * Z0)) / (&1 + X0)
  ==>
  ( Y0 < Z0            ==> ?m. !i. m <= i ==> &1 < standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i))      ) /\
  ( Y0 = Z0 /\ Y1 < Z1 ==> ?m. !i. m <= i ==> &1 < standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i))      ) /\
  ( Y0 = Z0 /\ Z1 < Y1 ==> ?m. !i. m <= i ==>      standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i)) < &1 ) /\
  ( Z0 < Y0            ==> ?m. !i. m <= i ==>      standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i)) < &1 )
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `q:num->real`;
    `s:num->real`;
    `k:num->real`;
    `b:num->real`;
    `Q0:real`;
    `Q1:real`;
    `S0:real`;
    `S1:real`;
    `X0:real`;
    `X1:real`;
    `Y0:real`;
    `Y1:real`] asymp2_simple3 THENL
  [real_linear `Y0 < Z0 /\ Z0 = (&1 + X0) / (&1 - &2 * S0 + &2 * Q0 * X0 / (&1 + X0)) ==> Y0 < (&1 + X0) / (&1 - &2 * S0 + &2 * Q0 * X0 / (&1 + X0))` THEN
    specialize [`Q0:real`;
      `S0:real`;
      `X0:real`;
      `Y0:real`] standardratio_above_lemma2 THEN
    specialize [`n:num->real`;
      `\i:num. log2 (standardratio (n(i)) (q i) (s i) (k i) (b i)) / log2 (n i)`;
      `S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - X0 / (&1 + X0) * Q0`;
      `S1 + &1 / &2 * (log2 Y0 + X1 / Y0 - (&1 + X0) * Y1 / (Y0 * Y0) - (&2 - (&1 + X0) / Y0) * (log2 Y0 - log2 (&2 * pi * exp (&1)))) - (Q1 * X0 / (&1 + X0) + Q0 * X1 / ((&1 + X0) * (&1 + X0)))`] asymp2_eventually_positive;
    specialize [`n:num->real`;
      `q:num->real`;
      `s:num->real`;
      `k:num->real`;
      `b:num->real`;
      `Q0:real`;
      `Q1:real`;
      `S0:real`;
      `S1:real`;
      `X0:real`;
      `X1:real`;
      `Y0:real`;
      `Y1:real`] asymp2_simple3 THEN
    specialize [`n:num->real`;
      `q:num->real`;
      `s:num->real`;
      `k:num->real`;
      `b:num->real`;
      `Q0:real`;
      `Q1:real`;
      `S0:real`;
      `S1:real`;
      `X0:real`;
      `X1:real`;
      `Y0:real`;
      `Y1:real`;
      `Z0:real`;
      `Z1:real`] asymp2_abovebelow_lemma4 THEN
    specialize [`Q0:real`;
      `Q1:real`;
      `S0:real`;
      `S1:real`;
      `X0:real`;
      `X1:real`;
      `Y0:real`;
      `Y1:real`;
      `Z0:real`;
      `Z1:real`] asymp2_abovebelow_lemma5 THEN
    specialize [`n:num->real`;
      `\i:num. log2 (standardratio (n(i)) (q i) (s i) (k i) (b i)) / log2 (n i)`;
      `(&1 + X0) * (Z1 - Y1) / (&2 * Z0 * Z0)`] asymp2_eventually_positive2;
    specialize [`n:num->real`;
      `q:num->real`;
      `s:num->real`;
      `k:num->real`;
      `b:num->real`;
      `Q0:real`;
      `Q1:real`;
      `S0:real`;
      `S1:real`;
      `X0:real`;
      `X1:real`;
      `Y0:real`;
      `Y1:real`] asymp2_simple3 THEN
    specialize [`n:num->real`;
      `q:num->real`;
      `s:num->real`;
      `k:num->real`;
      `b:num->real`;
      `Q0:real`;
      `Q1:real`;
      `S0:real`;
      `S1:real`;
      `X0:real`;
      `X1:real`;
      `Y0:real`;
      `Y1:real`;
      `Z0:real`;
      `Z1:real`] asymp2_abovebelow_lemma4 THEN
    specialize [`Q0:real`;
      `Q1:real`;
      `S0:real`;
      `S1:real`;
      `X0:real`;
      `X1:real`;
      `Y0:real`;
      `Y1:real`;
      `Z0:real`;
      `Z1:real`] asymp2_abovebelow_lemma6 THEN
    specialize [`n:num->real`;
      `\i:num. log2 (standardratio (n(i)) (q i) (s i) (k i) (b i)) / log2 (n i)`;
      `(&1 + X0) * (Z1 - Y1) / (&2 * Z0 * Z0)`] asymp2_eventually_negative2;
    specialize [`n:num->real`;
      `q:num->real`;
      `s:num->real`;
      `k:num->real`;
      `b:num->real`;
      `Q0:real`;
      `Q1:real`;
      `S0:real`;
      `S1:real`;
      `X0:real`;
      `X1:real`;
      `Y0:real`;
      `Y1:real`] asymp2_simple3 THEN
    real_linear `Z0 < Y0 /\ Z0 = (&1 + X0) / (&1 - &2 * S0 + &2 * Q0 * X0 / (&1 + X0)) ==> (&1 + X0) / (&1 - &2 * S0 + &2 * Q0 * X0 / (&1 + X0)) < Y0` THEN
    specialize [`Q0:real`;
      `S0:real`;
      `X0:real`;
      `Y0:real`] standardratio_below_lemma2_alt THEN
    specialize [`n:num->real`;
      `\i:num. log2 (standardratio (n(i)) (q i) (s i) (k i) (b i)) / log2 (n i)`;
      `S0 - &1 / &2 + (&1 + X0) / (&2 * Y0) - X0 / (&1 + X0) * Q0`;
      `S1 + &1 / &2 * (log2 Y0 + X1 / Y0 - (&1 + X0) * Y1 / (Y0 * Y0) - (&2 - (&1 + X0) / Y0) * (log2 Y0 - log2 (&2 * pi * exp (&1)))) - (Q1 * X0 / (&1 + X0) + Q0 * X1 / ((&1 + X0) * (&1 + X0)))`] asymp2_eventually_negative] THEN
  ASM_MESON_TAC [log2ratio_lnratio;
    standardratio_above_lemma3;
    standardratio_below_lemma3;
    standardratio_positive]);;

let asymp2_below_j = prove(`
  !n:num->real q:num->real s:num->real k:num->real b:num->num->real
   Q0:real Q1:real S0:real S1:real X0:real X1:real Y1:num->real Z0:real Z1:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ (!i. &0 < k(i))
  /\ (!j i. &1 < b j i)
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i:num. (((log2(q(i))/log2(n(i))) - (Q0)) * log2(n(i))) - (Q1))
  /\ o1_seq (\i:num. (((log2(s(i))/log2(n(i))) - (S0)) * log2(n(i))) - (S1))
  /\ o1_seq (\i:num. ((((k(i))/(n(i))) - (X0)) * log2(n(i))) - (X1))
  /\ (!j. o1_seq (\i:num. ((((b j i)/(n(i))) - (Z0)) * log2(n(i))) - (Y1 j)))
  /\ &0 < &1 + X0
  /\ &0 < &1 + &2*S0
  /\ &0 < Z0
  /\ &0 < &1 - &2*S0 + &2*Q0*X0/(&1 + X0)
  /\ Z0 = (&1 + X0) / (&1 - &2*S0 + &2*Q0*X0/(&1 + X0))
  /\ Z1 = ((S1 + (&1 / &2) * (log2(Z0) + X1/Z0 - (&2 - (&1 + X0)/Z0)*(log2(Z0) - log2(&2 * pi * exp(&1)))) - (Q1 * X0/(&1 + X0) + Q0 * X1/((&1 + X0)*(&1 + X0)))) * (&2 * Z0 * Z0)) / (&1 + X0)
  /\ (!j. Z1 < Y1(j))
  ==>
  !j. ?m. !i. m <= i ==>      standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b j i) < &1
  `,
  MESON_TAC [asymp2_abovebelow]);;

let main_x0_positive = prove(`
  !Q0:real S0:real x0:real.
  -- &1 / &2 < Q0 - S0
  /\ &1 / &2 < Q0 + S0
  /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2)
  ==> &0 < x0
  `,
  MESON_TAC [REAL_ARITH `-- &1 / &2 < Q0 - S0 ==> &0 < Q0 - S0 + &1 / &2`;
    REAL_ARITH `&1 / &2 < Q0 + S0 ==> &0 < Q0 + S0 - &1 / &2`;
    REAL_LT_DIV]);;

let main_x0_le1 = prove(`
  !Q0:real S0:real x0:real.
  -- &1 / &2 < Q0 - S0
  /\ S0 <= &1 / &2
  /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2)
  ==> x0 <= &1
  `,
  MESON_TAC [REAL_ARITH `-- &1 / &2 < Q0 - S0 ==> &0 < Q0 - S0 + &1 / &2`;
    REAL_ARITH `S0 <= &1 / &2 ==> (Q0 + S0 - &1 / &2) <= &1 * (Q0 - S0 + &1 / &2)`;
    REAL_LE_LDIV]);;

let main_k_integer = prove(`
  !n:num->real K:num->real x0:real.
  K = (\i. ceil(x0 * n(i)))
  ==> (!i. integer(K(i)))
  `,
  MESON_TAC [ceil_isinteger]);;

let main_k_positive_lemma = prove(`
  !n:real K:real x0:real.
  &1 < n
  /\ &0 < x0
  /\ K = ceil(x0 * n)
  ==> &0 < K
  `,
  REPEAT STRIP_TAC THEN
  real_linear `&1 < n ==> &0 < n` THEN
  specialize [`x0:real`;
    `n:real`] REAL_LT_MUL THEN
  specialize_forward [`x0*n:real`] ceil_pos THEN
  real_linear `&1 <= ceil(x0*n) ==> &0 < ceil(x0*n)` THEN
  ASM_SIMP_TAC []);;

let main_k_positive = prove(`
  !n:num->real K:num->real x0:real.
  (!i. &1 < n(i))
  /\ &0 < x0
  /\ K = (\i. ceil(x0 * n(i)))
  ==> (!i. &0 < K(i))
  `,
  MESON_TAC [main_k_positive_lemma]);;

let main_k_leceiln_lemma = prove(`
  !n:real K:real x0:real.
  &1 < n
  /\ x0 <= &1
  /\ K = ceil(x0 * n)
  ==> K <= ceil(n)
  `,
  REPEAT STRIP_TAC THEN
  real_linear `&1 < n ==> &0 <= n` THEN
  specialize [`x0:real`;
    `&1`;
    `n:real`] REAL_LE_RMUL THEN
  real_linear `&1 * n = n` THEN
  ASM_MESON_TAC [ceil_mono]);;

let main_k_leceiln = prove(`
  !n:num->real K:num->real x0:real.
  (!i. &1 < n(i))
  /\ x0 <= &1
  /\ K = (\i. ceil(x0 * n(i)))
  ==> (!i. K(i) <= ceil(n(i)))
  `,
  MESON_TAC [main_k_leceiln_lemma]);;

let main_kn_o1_lemma = prove(`
  !n:real K:real x0:real.
  &1 < n
  /\ K = ceil(x0 * n)
  ==> &0 <= (K/n - x0)*log2(n)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`x0*n:real`] ceil_ge THEN
  real_linear `&1 < n ==> &0 < n` THEN
  specialize [`n:real`] log2_gt1 THEN
  real_linear `&0 < log2(n) ==> &0 <= log2(n)` THEN
  ASM_MESON_TAC [REAL_LE_RDIV_EQ;
    REAL_ARITH `x0 <= K/n ==> &0 <= K/n - x0`;
    REAL_LE_MUL]);;

let main_kn_o1_lemma2 = prove(`
  !n:real K:real x0:real.
  &1 < n
  /\ K = ceil(x0 * n)
  ==> (K/n - x0)*log2(n) < log2(n)/n
  `,
  REPEAT STRIP_TAC THEN
  specialize [`x0*n:real`] ceil_lt_plus1 THEN
  real_linear `&1 < n ==> &0 < n` THEN
  real_linear `ceil(x0*n) < x0*n + &1 ==> ceil(x0*n)-x0*n < &1` THEN
  specialize [`n:real`] log2_gt1 THEN
  specialize [`log2 n`;
    `n:real`] REAL_LT_DIV THEN
  specialize [`ceil(x0*n)-x0*n`;
    `&1`;
    `log2(n)/n`] REAL_LT_RMUL THEN
  real_cancel `&1 < n ==> (ceil(x0*n) - x0*n)*log2(n)/n = (ceil(x0*n)/n - x0)*log2(n)` THEN
  real_linear `&1 * log2(n) / n = log2(n) / n` THEN
  ASM_MESON_TAC []);;

let main_kn_o1_lemma3 = prove(`
  !n:num->real K:num->real x0:real.
  (!i. &1 < n(i))
  /\ K = (\i. ceil(x0 * n(i)))
  ==> !i. &0 <= (K(i)/n(i) - x0)*log2(n(i))
  `,
  MESON_TAC [main_kn_o1_lemma]);;

let main_kn_o1_lemma4 = prove(`
  !n:num->real K:num->real x0:real.
  (!i. &1 < n(i))
  /\ K = (\i. ceil(x0 * n(i)))
  ==> !i. (K(i)/n(i) - x0)*log2(n(i)) < log2(n(i))/n(i)
  `,
  MESON_TAC [main_kn_o1_lemma2]);;

let main_kn_o1_lemma5 = prove(`
  !n:num->real.
  (!i. &1 < n(i))
  ==> (!i. &1 <= n(i))
  `,
  MESON_TAC [REAL_LT_IMP_LE]);;

let main_kn_o1 = prove(`
  !n:num->real K:num->real x0:real.
  (!i. &1 < n(i))
  /\ K = (\i. ceil(x0 * n(i)))
  /\ o1_seq (\i. &1 / n(i))
  ==> o1_seq (\i. (K(i)/n(i) - x0)*log2(n(i)) - &0)
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [REAL_ARITH `x - &0 = x`] THEN
  specialize [`n:num->real`;
    `K:num->real`;
    `x0:real`] main_kn_o1_lemma3 THEN
  specialize [`n:num->real`;
    `K:num->real`;
    `x0:real`] main_kn_o1_lemma4 THEN
  specialize [`n:num->real`] main_kn_o1_lemma5 THEN
  specialize [`n:num->real`] o1_log2overlinear_1 THEN
  specialize [`\i:num. (K(i)/n(i) - x0)*log2(n(i))`;
    `\i:num. log2(n(i))/n(i)`] nonneg_lt_o1 THEN
  ASM_SIMP_TAC []);;

let main_b_integer = prove(`
  !n:num->real k:num->real z0:real z1:real b:num->num->real.
  b = (\j:num i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))))
  ==> !j i. integer(b j i)
  `,
  MESON_TAC [ceil_isinteger]);;

let main_b_gt1_lemma = prove(`
  !n:real k:real z0:real z1:real b:real.
  &1 < n
  /\ &0 < k
  /\ b = ceil(min (n + k + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n))*n)))
  ==> &1 < b
  `,
  REPEAT STRIP_TAC THEN
  real_linear `&1 < n /\ &0 < k ==> &2 <= n + k + &1` THEN
  real_linear `&2 <= n + k + &1 ==> &2 <= min (n + k + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n))*n))` THEN
  ASM_MESON_TAC [ceil_ge;
    REAL_LE_TRANS;
    REAL_ARITH `&2 <= b ==> &1 < b`]);;

let main_b_gt1 = prove(`
  !n:num->real k:num->real z0:real z1:real b:num->num->real.
  (!i. &1 < n(i))
  /\ (!i. &0 < k(i))
  /\ b = (\j:num i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))))
  ==> !j i. &1 < b j i
  `,
  MESON_TAC [main_b_gt1_lemma]);;

let main_b_lenk1_lemma = prove(`
  !n:real k:real z0:real z1:real b:real.
  &1 < n
  /\ &0 < k
  /\ b = ceil(min (n + k + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n))*n)))
  ==> b <= ceil(n + k + &1)
  `,
  MESON_TAC [REAL_ARITH `min (n + k + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n))*n)) <= n + k + &1`;
    ceil_mono]);;

let main_b_lenk1 = prove(`
  !n:num->real k:num->real z0:real z1:real b:num->num->real.
  (!i. &1 < n(i))
  /\ (!i. &0 < k(i))
  /\ b = (\j:num i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))))
  ==> !j i. b j i <= ceil(n(i) + k(i) + &1)
  `,
  MESON_TAC [main_b_lenk1_lemma]);;

let main_b_growth_lemma1 = prove(`
  !L z0 z1.
  &0 < z1 /\
  abs(&1 / L) < z0 / (&2 * z1)
  ==> --z0 / &2 < z1/L
  `,
  REPEAT STRIP_TAC THEN
  real_cancel `abs(&1 / L) < z0 / (&2 * z1) ==> (-- z0 / &2) / z1 < &1 / L` THEN
  specialize [`(--z0 / &2)/z1`;
    `&1/L`;
    `z1:real`] REAL_LT_RMUL THEN
  real_cancel `&0 < z1 ==> --z0 / &2 / z1 * z1 = --z0 / &2` THEN
  real_linear `&1 / L * z1 = z1 / L` THEN
  ASM_MESON_TAC []);;

let main_b_growth_lemma2 = prove(`
  !L z0 z1.
  &0 < --z1 /\
  abs(&1 / L) < z0 / (-- &2 * z1)
  ==> --z0 / &2 < z1/L
  `,
  REPEAT STRIP_TAC THEN
  real_cancel `abs(&1 / L) < z0 / (-- &2 * z1) ==> --z0 / (-- &2 * z1) < -- &1 / L` THEN
  specialize [`--z0 / (-- &2 * z1)`;
    `-- &1 / L`;
    `--z1:real`] REAL_LT_RMUL THEN
  real_cancel `&0 < --z1 ==> --z0 / (-- &2 * z1) * --z1 = --z0 / &2` THEN
  real_linear `-- &1 / L * --z1 = z1 / L` THEN
  ASM_MESON_TAC []);;

let main_b_growth_lemma3 = prove(`
  !n:num->real z0:real z1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < z0
  ==> ?Z. !i. Z <= i ==> -- z0 / &2 < z1/log2(n(i))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`] log2overo1 THEN
  STRUCT_CASES_TAC (SPEC `z1:real` REAL_LT_NEGTOTAL) THENL
  [EXISTS_TAC `0` THEN
    REPEAT STRIP_TAC THEN
    real_linear `&0 < z0 ==> --z0 / &2 < &0 / log2(n(i:num))` THEN
    ASM_MESON_TAC [];
    real_linear `&0 < z1 ==> &0 < &2 * z1` THEN
    specialize [`z0:real`;
      `&2 * z1`] REAL_LT_DIV THEN
    choose_specializing `Z:num` [`\i:num. &1 / log2(n i)`;
      `z0 / (&2 * z1)`] o1_seq_lt_use THEN
    EXISTS_TAC `Z:num` THEN
    ASM_MESON_TAC [main_b_growth_lemma1];
    real_linear `&0 < --z1 ==> &0 < -- &2 * z1` THEN
    specialize [`z0:real`;
      `-- &2 * z1`] REAL_LT_DIV THEN
    choose_specializing `Z:num` [`\i:num. &1 / log2(n i)`;
      `z0 / (-- &2 * z1)`] o1_seq_lt_use THEN
    EXISTS_TAC `Z:num` THEN
    ASM_MESON_TAC [main_b_growth_lemma2]]);;

let main_b_growth_lemma4 = prove(`
  !n:num->real z0:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < z0
  ==> ?Y. !i. Y <= i ==> &1 / n(i) <= z0 / &4
  `,
  REPEAT STRIP_TAC THEN
  real_linear `&0 < &4` THEN
  specialize [`z0:real`;
    `&4`] REAL_LT_DIV THEN
  choose_specializing `Y:num` [`\i:num. &1 / n(i)`;
    `z0 / &4`] o1_seq_use THEN
  EXISTS_TAC `Y:num` THEN
  ASM_MESON_TAC [REAL_ARITH `abs(a) <= b ==> a <= b`]);;

let main_b_growth_lemma5 = prove(`
  !n:real z0:real z1:real j:num.
  &1 < n
  /\ --z0 / &2 < z1/log2(n)
  /\ &1 / n <= z0 / &4
  ==>
  max (&2) ((z0 + (z1 + exp(-- &j))/log2(n))*n)
  = (z0 + (z1 + exp(-- &j))/log2(n))*n
  `,
  REPEAT STRIP_TAC THEN
  specialize [`-- &j`] REAL_EXP_POS_LT THEN
  specialize [`n:real`] log2_gt1 THEN
  specialize [`exp(-- &j)`;
    `log2(n)`] REAL_LT_DIV THEN
  specialize [`--z0 / &2`;
    `z1/log2(n)`;
    `&0`;
    `exp(-- &j) / log2(n)`] REAL_LT_ADD2 THEN
  real_linear `--z0 / &2 + &0 < z1 / log2 n + exp (-- &j) / log2 n ==> z0 / &2 < z0 + (z1 + exp (-- &j)) / log2 n` THEN
  real_linear `&1 < n ==> &0 < n` THEN
  specialize [`z0 / &2`;
    `z0 + (z1 + exp (-- &j)) / log2 n`;
    `n:real`] REAL_LT_RMUL THEN
  real_linear `&1 < n ==> &0 <= &2 * n` THEN
  specialize [`&1 / n`;
    `z0 / &4`;
    `&2 * n`] REAL_LE_RMUL THEN
  real_cancel `&1 < n /\ &1 / n * &2 * n <= z0 / &4 * &2 * n ==> &2 <= z0 / &2 * n` THEN
  specialize [`&2`;
    `z0 / &2 * n`;
    `(z0 + (z1 + exp (-- &j)) / log2 n) * n`] REAL_LET_TRANS THEN
  real_linear `&2 < (z0 + (z1 + exp (-- &j)) / log2 n) * n ==> max (&2) ((z0 + (z1 + exp (-- &j)) / log2 n) * n) = ((z0 + (z1 + exp (-- &j)) / log2 n) * n)` THEN
  ASM_SIMP_TAC []);;

let main_b_growth_lemma6 = prove(`
  !n:num->real k:num->real z0:real z1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ (!i. &0 < k(i))
  /\ &0 < z0
  ==> 
  ?X. !j. !i. X <= i ==> 
  ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i))))
  = ceil(min (n(i) + k(i) + &1) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))
  `,
  REPEAT STRIP_TAC THEN
  choose_specializing `Z:num` [`n:num->real`;
    `z0:real`;
    `z1:real`] main_b_growth_lemma3 THEN
  choose_specializing `Y:num` [`n:num->real`;
    `z0:real`] main_b_growth_lemma4 THEN
  EXISTS_TAC `MAX Z Y` THEN
  REWRITE_TAC [max_le] THEN
  ASM_MESON_TAC [main_b_growth_lemma5]);;

let main_b_growth_lemma7 = prove(`
  !L x0 z0 z1.
  &0 < z1 + &1 /\
  abs(&1 / L) <= (&1 + x0 - z0)/(z1 + &1)
  ==> (z1 + &1) / L <= &1 + x0 - z0
  `,
  MESON_TAC [REAL_ABS_LE;
    REAL_LE_TRANS;
    REAL_LE_LMUL;
    REAL_LT_IMP_LE;
    REAL_ARITH `(z1 + &1) * &1 / L = (z1 + &1) / L`;
    REAL_FIELD `&0 < z1 + &1 ==> (z1 + &1) * (&1 + x0 - z0)/(z1 + &1) = &1 + x0 - z0`]);;

let main_b_growth_lemma8 = prove(`
  !L x0 z0 z1.
  &0 < --(z1 + &1) /\
  abs(&1 / L) <= (&1 + x0 - z0)/(--(z1 + &1))
  ==> (z1 + &1) / L <= &1 + x0 - z0
  `,
  REPEAT STRIP_TAC THEN
  real_linear `abs(&1 / L) <= (&1 + x0 - z0)/(--(z1 + &1)) ==> --(&1 / L) <= (&1 + x0 - z0)/(--(z1 + &1))` THEN
  real_linear `&0 < --(z1 + &1) ==> &0 <= --(z1 + &1)` THEN
  specialize [`--(z1 + &1)`;
    `--(&1 / L)`;
    `(&1 + x0 - z0) / --(z1 + &1)`] REAL_LE_LMUL THEN
  real_linear `--(z1 + &1) * --(&1 / L) = (z1 + &1) / L` THEN
  real_cancel `&0 < --(z1 + &1) ==> --(z1 + &1) * (&1 + x0 - z0) / --(z1 + &1) = &1 + x0 - z0` THEN
  ASM_MESON_TAC []);;

let main_b_growth_lemma9 = prove(`
  !n:num->real x0:real z0:real z1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ &0 < &1 + x0 - z0
  ==> ?Z. !i. Z <= i ==> (z1 + &1)/log2(n(i)) <= &1 + x0 - z0
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`] log2overo1 THEN
  STRUCT_CASES_TAC (SPEC `z1 + &1` REAL_LT_NEGTOTAL) THENL
  [EXISTS_TAC `0` THEN
    REPEAT STRIP_TAC THEN
    real_linear `&0 < &1 + x0 - z0 ==> &0 / log2(n(i:num)) <= &1 + x0 - z0` THEN
    ASM_MESON_TAC [];
    specialize [`&1 + x0 - z0`;
      `z1 + &1`] REAL_LT_DIV THEN
    choose_specializing `Z:num` [`\i:num. &1 / log2(n i)`;
      `(&1 + x0 - z0)/(z1 + &1)`] o1_seq_use THEN
    EXISTS_TAC `Z:num` THEN
    ASM_MESON_TAC [main_b_growth_lemma7];
    specialize [`&1 + x0 - z0`;
      `--(z1 + &1)`] REAL_LT_DIV THEN
    choose_specializing `Z:num` [`\i:num. &1 / log2(n i)`;
      `(&1 + x0 - z0)/(-- (z1 + &1))`] o1_seq_use THEN
    EXISTS_TAC `Z:num` THEN
    ASM_MESON_TAC [main_b_growth_lemma8]]);;

let main_b_growth_lemma10 = prove(`
  !j. exp(-- &j) <= &1
  `,
  MESON_TAC [REAL_EXP_MONO_LE;
    REAL_EXP_0;
    REAL_ARITH `-- &j <= &0`]);;

let main_b_growth_lemma11 = prove(`
  !n k x0 z0 z1 j.
  &1 < n /\
  k = ceil(x0*n) /\
  (z1 + &1) / log2(n) <= &1 + x0 - z0
  ==>
  min (n + k + &1) ((z0 + (z1 + exp(-- &j))/log2(n))*n)
  = (z0 + (z1 + exp(-- &j))/log2(n))*n
  `,
  REPEAT STRIP_TAC THEN
  specialize [`j:num`] main_b_growth_lemma10 THEN
  specialize [`n:real`] log2_gt1 THEN
  specialize_reverse [`exp(-- &j)`;
    `&1`;
    `log2(n)`] REAL_LE_DIV2_EQ THEN
  real_linear `(z1 + &1) / log2 n <= &1 + x0 - z0 /\ exp (-- &j) / log2 n <= &1 / log2 n ==> z0 + (z1 + exp(-- &j)) / log2 n <= &1 + x0` THEN
  real_linear `&1 < n ==> &0 <= n` THEN
  specialize [`z0 + (z1 + exp (-- &j)) / log2 n`;
    `&1 + x0`;
    `n:real`] REAL_LE_RMUL THEN
  specialize [`x0*n:real`] ceil_ge THEN
  real_linear `k = ceil(x0 * n) /\ x0 * n <= ceil(x0 * n) ==> (&1 + x0) * n <= n + k + &1` THEN
  ASM_MESON_TAC [REAL_LE_TRANS;
    REAL_ARITH `a <= b ==> min b a = a`]);;

let main_b_growth_lemma12 = prove(`
  !n:num->real k:num->real x0:real z0:real z1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ k = (\i. ceil(x0*n(i)))
  /\ &0 < &1 + x0 - z0
  ==> 
  ?Y. !j. !i. Y <= i ==> 
  ceil(min (n(i) + k(i) + &1) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))
  = ceil((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i))
  `,
  REPEAT STRIP_TAC THEN
  choose_specializing `Z:num` [`n:num->real`;
    `x0:real`;
    `z0:real`;
    `z1:real`] main_b_growth_lemma9 THEN
  EXISTS_TAC `Z:num` THEN
  ASM_MESON_TAC [main_b_growth_lemma11]);;

let main_b_growth_lemma13 = prove(`
  !n:num->real k:num->real x0:real z0:real z1:real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ k = (\i. ceil(x0*n(i)))
  /\ (!i. &0 < k(i))
  /\ &0 < z0
  /\ &0 < &1 + x0 - z0
  ==>
  ?Z. !j. !i. Z <= i ==> 
  ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i))))
  = ceil((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i))
  `,
  REPEAT STRIP_TAC THEN
  choose_specializing `X:num` [`n:num->real`;
    `k:num->real`;
    `z0:real`;
    `z1:real`] main_b_growth_lemma6 THEN
  choose_specializing `Y:num` [`n:num->real`;
    `k:num->real`;
    `x0:real`;
    `z0:real`;
    `z1:real`] main_b_growth_lemma12 THEN
  EXISTS_TAC `MAX X Y` THEN
  REWRITE_TAC [max_le] THEN
  ASM_MESON_TAC []);;

let main_b_growth_lemma14 = prove(`
  !n:real z0:real z1:real j:num.
  &1 < n
  ==>
  &0 <= (ceil((z0 + (z1 + exp(-- &j))/log2(n))*n)/n - z0) * log2(n) - (z1 + exp(-- &j))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`(z0 + (z1 + exp(-- &j))/log2(n))*n`] ceil_ge THEN
  specialize [`n:real`] log2_gt1 THEN
  real_linear `&1 < n ==> &0 < n` THEN
  real_linear `&0 < log2 n ==> &0 <= log2 n` THEN
  specialize_reverse [`(z0 + (z1 + exp (-- &j)) / log2 n) * n`;
    `ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n)`;
    `n:real`] REAL_LE_DIV2_EQ THEN
  real_cancel `&0 < n ==> ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n = z0 + (z1 + exp (-- &j)) / log2 n` THEN
  real_linear `((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n = z0 + (z1 + exp (-- &j)) / log2 n /\ ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n <= ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n ==> (z1 + exp (-- &j)) / log2 n <= ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n - z0` THEN
  specialize [`(z1 + exp (-- &j)) / log2 n`;
    `ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n - z0`;
    `log2 n`] REAL_LE_RMUL THEN
  real_cancel `&0 < log2 n ==> (z1 + exp (-- &j)) / log2 n * log2 n = z1 + exp(-- &j)` THEN
  real_linear `(z1 + exp (-- &j)) / log2 n * log2 n = z1 + exp(-- &j) /\ (z1 + exp (-- &j)) / log2 n * log2 n <= (ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n - z0) * log2 n ==> &0 <= (ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n - z0) * log2 n - (z1 + exp(-- &j))` THEN
  ASM_SIMP_TAC []);;

let main_b_growth_lemma15 = prove(`
  !n:real z0:real z1:real j:num.
  &1 < n
  ==>
  (ceil((z0 + (z1 + exp(-- &j))/log2(n))*n)/n - z0) * log2(n) - (z1 + exp(-- &j))
  < log2(n)/n
  `,
  REPEAT STRIP_TAC THEN
  specialize [`(z0 + (z1 + exp(-- &j))/log2(n))*n`] ceil_lt_plus1 THEN
  specialize [`n:real`] log2_gt1 THEN
  real_linear `&1 < n ==> &0 < n` THEN
  specialize_reverse [`ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n)`;
    `(z0 + (z1 + exp (-- &j)) / log2 n) * n + &1`;
    `n:real`] REAL_LT_DIV2_EQ THEN
  real_cancel `&0 < n ==> ((z0 + (z1 + exp (-- &j)) / log2 n) * n + &1) / n = (z0 + (z1 + exp (-- &j)) / log2 n) + &1 / n` THEN
  real_linear `((z0 + (z1 + exp (-- &j)) / log2 n) * n + &1) / n = (z0 + (z1 + exp (-- &j)) / log2 n) + &1 / n /\ ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n < ((z0 + (z1 + exp (-- &j)) / log2 n) * n + &1) / n ==> ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n - z0 < (z1 + exp (-- &j)) / log2 n + &1 / n` THEN
  specialize [`ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n - z0`;
    `(z1 + exp (-- &j)) / log2 n + &1 / n`;
    `log2 n`] REAL_LT_RMUL THEN
  real_cancel `&0 < log2 n ==> ((z1 + exp (-- &j)) / log2 n + &1 / n) * log2 n = (z1 + exp (-- &j)) + log2 n / n` THEN
  real_linear `((z1 + exp (-- &j)) / log2 n + &1 / n) * log2 n = (z1 + exp (-- &j)) + log2 n / n /\ (ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n - z0) * log2 n < ((z1 + exp (-- &j)) / log2 n + &1 / n) * log2 n ==> (ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n - z0) * log2 n - (z1 + exp (-- &j)) < log2 n / n` THEN
  ASM_SIMP_TAC []);;

let main_b_growth_lemma16 = prove(`
  !n:num->real z0:real z1:real j:num.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  ==>
  o1_seq (\i. (ceil((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i))/n(i) - z0) * log2(n(i)) - (z1 + exp(-- &j)))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`] main_kn_o1_lemma5 THEN
  specialize [`n:num->real`] o1_log2overlinear_1 THEN
  ASM_MESON_TAC [main_b_growth_lemma14;
    main_b_growth_lemma15;
    nonneg_lt_o1]);;

let main_b_growth_lemma17 = prove(`
  !n:num->real k:num->real z0:real z1:real b:num->real j:num.
  b = (\i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))))
  ==>
  (\i. (b i / n i - z0) * log2 (n i) - (z1 + exp (-- &j)))
  =
  (\i. (ceil (min (n i + k i + &1) (max (&2) ((z0 + (z1 + exp (-- &j)) / log2 (n i)) * n i))) / n i - z0) * log2 (n i) - (z1 + exp (-- &j)))
  `,
  SIMP_TAC []);;

let main_b_growth_lemma18 = prove(`
  !n:num->real k:num->real x0:real z0:real z1:real b:num->real j:num.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ k = (\i. ceil(x0 * n(i)))
  /\ (!i. &0 < k(i))
  /\ &0 < z0 /\
  &0 < &1 + x0 - z0
  /\ b = (\i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))))
  ==> o1_seq (\i. (b(i)/n(i) - z0) * log2(n(i)) - (z1 + exp(-- &j)))
  `,
  REPEAT STRIP_TAC THEN
  choose_specializing `Z:num` [`n:num->real`;
    `k:num->real`;
    `x0:real`;
    `z0:real`;
    `z1:real`] main_b_growth_lemma13 THEN
  specialize [`n:num->real`;
    `z0:real`;
    `z1:real`;
    `j:num`] main_b_growth_lemma16 THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `z0:real`;
    `z1:real`;
    `b:num->real`;
    `j:num`] main_b_growth_lemma17 THEN
  specialize [] (MESON [] `(!j i:num.  Z <= i ==> ceil (min (n i + k i + &1) (max (&2) ((z0 + (z1 + exp (-- &j)) / log2 (n i)) * n i))) = ceil ((z0 + (z1 + exp (-- &j)) / log2 (n i)) * n i)) ==> !i. Z <= i ==> (ceil ((z0 + (z1 + exp (-- &j)) / log2 (n i)) * n i) / n i - z0) * log2 (n i) - (z1 + exp (-- &j)) = (ceil (min (n i + k i + &1) (max (&2) ((z0 + (z1 + exp (-- &j)) / log2 (n i)) * n i))) / n i - z0) * log2 (n i) - (z1 + exp (-- &j))`) THEN
  specialize [`\i:num. (ceil ((z0 + (z1 + exp (-- &j)) / log2 (n i)) * n i) / n i - z0) * log2 (n i) - (z1 + exp (-- &j))`;
    `\i:num. (ceil (min (n i + k i + &1) (max (&2) ((z0 + (z1 + exp (-- &j)) / log2 (n i)) * n i))) / n i - z0) * log2 (n i) - (z1 + exp (-- &j))`;
    `Z:num`] o1_seq_past THEN
  ASM_MESON_TAC []);;

let main_b_growth = prove(`
  !n:num->real k:num->real x0:real z0:real z1:real b:num->num->real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ k = (\i. ceil(x0 * n(i)))
  /\ (!i. &0 < k(i))
  /\ &0 < z0
  /\ &0 < &1 + x0 - z0
  /\ b = (\j:num i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))))
  ==> !j. o1_seq (\i. ((b j i)/n(i) - z0) * log2(n(i)) - (z1 + exp(-- &j)))
  `,
  REPEAT STRIP_TAC THEN
  specialize [] (MESON [] `b = (\j:num i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i))))) ==> b(j) = (\i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))))`) THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `x0:real`;
    `z0:real`;
    `z1:real`;
    `(b:num->num->real) j`;
    `j:num`] main_b_growth_lemma18 THEN
  ASM_SIMP_TAC []);;

let main_z0_positive = prove(`
  !Q0:real S0:real z0:real.
  -- &1 / &2 < Q0 - S0
  /\ &1 / &2 < Q0 + S0
  /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2)
  ==> &0 < z0
  `,
  REPEAT STRIP_TAC THEN
  real_linear `-- &1 / &2 < Q0 - S0 /\ &1 / &2 < Q0 + S0 ==> &0 < &2 * Q0` THEN
  real_linear `-- &1 / &2 < Q0 - S0 ==> &0 < Q0 - S0 + &1 / &2` THEN
  specialize [`Q0 - S0 + &1 / &2`;
    `Q0 - S0 + &1 / &2`] REAL_LT_MUL THEN
  ASM_MESON_TAC [REAL_POW_2;
    REAL_LT_DIV;
    REAL_ARITH `&2 * a / b = (&2 * a) / b`]);;

let main_1x0z0_positive = prove(`
  !Q0:real S0:real x0:real z0:real.
  &1 / &2 < Q0 - S0
  /\ &1 / &2 < Q0 + S0
  /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2)
  /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2)
  ==> &0 < &1 + x0 - z0
  `,
  REPEAT STRIP_TAC THEN
  real_linear `&1 / &2 < Q0 - S0 /\ &1 / &2 < Q0 + S0 ==> &0 < &2 * Q0` THEN
  real_linear `&1 / &2 < Q0 - S0 ==> &0 < Q0 - S0 + &1 / &2` THEN
  real_linear `&1 / &2 < Q0 - S0 ==> &0 < Q0 - S0 - &1 / &2` THEN
  real_cancel `&0 < Q0 - S0 + &1 / &2 /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2) /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2) ==> &1 + x0 - z0 = (&2 * Q0) * (Q0 - S0 - &1 / &2) / ((Q0 - S0 + &1 / &2) * (Q0 - S0 + &1 / &2))` THEN
  ASM_MESON_TAC [REAL_LT_MUL;
    REAL_LT_DIV]);;

let main_SQx_positive_variant = prove(`
  !Q0:real S0:real x0:real.
  S0 < &1 / &2
  /\ -- &1 / &2 < Q0 - S0
  /\ &1 / &2 < Q0 + S0
  /\ &0 <= x0
  ==> &0 < &1 - &2*S0 + &2*Q0*x0/(&1 + x0)
  `,
  REPEAT STRIP_TAC THEN
  real_linear `-- &1 / &2 < Q0 - S0 /\ &1 / &2 < Q0 + S0 ==> &0 <= Q0` THEN
  real_linear `&0 <= &2` THEN
  real_linear `&0 <= x0 ==> &0 <= &1 + x0` THEN
  note `&0 <= x0/(&1 + x0)` [REAL_LE_DIV] THEN
  note `&0 <= &2*Q0*x0/(&1 + x0)` [REAL_LE_MUL] THEN
  real_linear `S0 < &1 / &2 /\ &0 <= &2*Q0*x0/(&1 + x0) ==> &0 < &1 - &2*S0 + &2*Q0*x0/(&1 + x0)` THEN
  ASM_MESON_TAC []);;

let main_SQx_positive = prove(`
  !Q0:real S0:real x0:real.
  S0 <= &1 / &2
  /\ &1 / &2 < Q0 - S0
  /\ &1 / &2 < Q0 + S0
  /\ &0 < x0
  ==> &0 < &1 - &2*S0 + &2*Q0*x0/(&1 + x0)
  `,
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC (REAL_ARITH `S0 <= &1 / &2 /\ &0 < &2 * Q0 * x0/(&1 + x0) ==> &0 < &1 - &2*S0 + &2*Q0*x0/(&1 + x0)`) THEN
  real_linear `&0 < x0 ==> &0 < &1 + x0` THEN
  real_linear `&0 < &2` THEN
  ASM_MESON_TAC [REAL_ARITH `&1 / &2 < Q0 - S0 /\ &1 / &2 < Q0 + S0 ==> &0 < Q0`;
    REAL_LT_MUL;
    REAL_LT_DIV]);;

let main_zxSQ = REAL_FIELD `
  !Q0:real S0:real x0:real z0:real.
  &1 / &2 < Q0 - S0
  /\ &1 / &2 < Q0 + S0
  /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2)
  /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2)
  ==>
  z0 = (&1 + x0) / (&1 - &2*S0 + &2*Q0*x0/(&1 + x0))
`;;

let main_z1formula = prove(`
  !Q0:real Q1:real S0:real S1:real x0:real z0:real z1:real.
  &1 / &2 < Q0 - S0
  /\ &1 / &2 < Q0 + S0
  /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2)
  /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2)
  /\ z1 = (&2 * S1 + log2(z0) - (S0 - Q0 + &3 / &2)*(log2(z0) - log2(&2 * pi * exp(&1))) - Q1*(Q0 + S0 - &1 / &2)/Q0)*(&2 * Q0)/(Q0 - S0 + &1 / &2) pow 3
  ==> z1 = ((S1 + (&1 / &2) * (log2(z0) + &0/z0 - (&2 - (&1 + x0)/z0)*(log2(z0) - log2(&2 * pi * exp(&1)))) - (Q1 * x0/(&1 + x0) + Q0 * &0/((&1 + x0)*(&1 + x0)))) * (&2 * z0 * z0)) / (&1 + x0)
  `,
  REPEAT STRIP_TAC THEN
  real_linear `&1 / &2 < Q0 - S0 /\ &1 / &2 < Q0 + S0 ==> &0 < Q0` THEN
  real_cancel `&0 < Q0 /\ &1 / &2 < Q0 - S0 /\ &1 / &2 < Q0 + S0 /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2) ==> x0/(&1 + x0) = (Q0 + S0 - &1 / &2) / (&2 * Q0)` THEN
  real_cancel `&0 < Q0 /\ &1 / &2 < Q0 - S0 /\ &1 / &2 < Q0 + S0 /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2) /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2) ==> (&1 + x0)/z0 = Q0 - S0 + &1 / &2` THEN
  real_cancel `&1 / &2 < Q0 - S0 /\ &1 / &2 < Q0 + S0 /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2) /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2) ==> &2*z0*z0/(&1+x0) = &4*Q0/(Q0-S0 + &1 / &2) pow 3` THEN
  real_cancel `x0 / (&1 + x0) = (Q0 + S0 - &1 / &2) / (&2 * Q0) /\ (&1 + x0)/z0 = Q0 - S0 + &1 / &2 /\ &2*z0*z0/(&1+x0) = &4*Q0/(Q0-S0 + &1 / &2) pow 3 /\ z1 = (&2 * S1 + log2(z0) - (S0 - Q0 + &3 / &2)*(log2(z0) - log2(&2 * pi * exp(&1))) - Q1*(Q0 + S0 - &1 / &2)/Q0)*(&2 * Q0)/(Q0 - S0 + &1 / &2) pow 3 ==> z1 = ((S1 + (&1 / &2) * (log2(z0) + &0/z0 - (&2 - (&1 + x0)/z0)*(log2(z0) - log2(&2 * pi * exp(&1)))) - (Q1 * x0/(&1 + x0) + Q0 * &0/((&1 + x0)*(&1 + x0)))) * (&2 * z0 * z0)) / (&1 + x0)` THEN
  ASM_MESON_TAC []);;

let main_z1lt = prove(`
  !z1:real j.
  z1 < z1 + exp(-- &j)
  `,
  MESON_TAC [REAL_EXP_POS_LT;
    REAL_ARITH `&0 < e ==> x < x + e`]);;

let main_m_le_M = prove(`
  !m:num->num M:num->num j:num.
  M = (\j. j + nsum (0..j) m)
  ==> m(j) <= M(j)
  `,
  REPEAT STRIP_TAC THEN
  DISJ_CASES_TAC (SPEC `j:num` num_CASES) THENL
  [ASM_MESON_TAC [NSUM_CLAUSES_NUMSEG;
      ARITH_RULE `x <= 0 + x`];
    ASM_MESON_TAC [NSUM_CLAUSES_NUMSEG;
      ARITH_RULE `0 <= SUC n`;
      ARITH_RULE `(x:num) <= z + y + x`]]);;

let main_M_mono = prove(`
  !m:num->num M:num->num.
  M = (\j. j + nsum (0..j) m)
  ==> !j k. j < k ==> M(j) < M(k)
  `,
  REPEAT STRIP_TAC THEN
  ASM_SIMP_TAC [] THEN
  specialize [] (ARITH_RULE `0 <= j + 1`) THEN
  specialize [`m:num->num`;
    `0`;
    `j:num`;
    `k-j:num`] NSUM_ADD_SPLIT THEN
  specialize [] (ARITH_RULE `j < k ==> j + k - j = k:num`) THEN
  specialize [] (ARITH_RULE `j < k ==> j + nsum(0..j)m < k + nsum(0..j)m + nsum(j+1..j+k-j) m`) THEN
  ASM_MESON_TAC []);;

let num_mono_ge = prove(`
  !f:num->num.
  (!j k. j < k ==> f(j) < f(k))
  ==> !t. t <= f(t)
  `,
  STRIP_TAC THEN
  DISCH_TAC THEN
  INDUCT_TAC THENL
  [ARITH_TAC;
    ASM_MESON_TAC [ARITH_RULE `t < SUC t`;
      ARITH_RULE `x <= y /\ y < z ==> SUC x <= z`]]);;

let num_mono_gt = prove(`
  !f:num->num t.
  (!j k. j < k ==> f(j) < f(k))
  ==> t < f(t+1)
  `,
  MESON_TAC [num_mono_ge;
    ARITH_RULE `t < t + 1`;
    LET_TRANS]);;

let main_M_le = prove(`
  !M:num->num i:num.
  M(0) <= i
  /\ x = (minimal j. i < M(j + 1))
  ==> M(x) <= i
  `,
  REPEAT STRIP_TAC THEN
  ASM_CASES_TAC `x = 0` THENL
  [ASM_MESON_TAC [];
    choose_specializing `y:num` [] (MESON [num_CASES;
        ARITH_RULE `SUC n = n + 1`] `~(x = 0) ==> ?y. x = y + 1`) THEN
    specialize [] (ARITH_RULE `x = y + 1 ==> ~(x <= y)`) THEN
    specialize [] (MESON [MINIMAL_UBOUND] `x = (minimal j. i:num < M(j + 1)) /\ ~(x <= y) ==> ~(i < M(y + 1))`) THEN
    ASM_MESON_TAC [NOT_LT]]);;

let gluing_pattern = prove(`
  !m:num->num M:num->num b:num->num->real B:num->real P:num->real->bool.
  (!j. ?A. !i. A <= i ==> P i (b j i))
  /\ m = (\j:num. minimal A. !i. A <= i ==> P i (b j i))
  /\ M = (\j. j + nsum (0..j) m)
  /\ B = (\i:num. b(minimal j. i < M(j+1)) (i))
  ==> !i. M(0) <= i ==> P i (B i)
  `,
  REPEAT STRIP_TAC THEN
  def `x:num` `minimal j. i:num < M(j+1)` THEN
  specialize [`M:num->num`;
    `i:num`] main_M_le THEN
  specialize [`m:num->num`;
    `M:num->num`;
    `x:num`] main_m_le_M THEN
  specialize [`m(x:num):num`;
    `M(x:num):num`;
    `i:num`] LE_TRANS THEN
  specialize [] (MESON [] `(!j:num. ?A:num. !i:num. A <= i ==> P i (b j i:real)) ==> ?A. !i. A <= i ==> P i (b x i)`) THEN
  specialize [`\A:num. !i. A <= i ==> P i (b (x:num) i:real)`] (MESON [MINIMAL] `!P. (?n. P n) ==> P((minimal) P)`) THEN
  ASM_MESON_TAC []);;

let main_B_gt1 = prove(`
  !b:num->num->real m:num->num B:num->real.
  (!j i. &1 < b j i)
  /\ B = (\i. b (minimal j. i < M (j + 1)) i)
  ==> !i. &1 < B i
  `,
  MESON_TAC []);;

let main_B_lenk1 = prove(`
  !n:num->real k:num->real b:num->num->real m:num->num B:num->real.
  (!j i. b j i <= ceil(n i + k i + &1))
  /\ B = (\i. b (minimal j. i < M (j + 1)) i)
  ==> !i. B i <= ceil(n i + k i + &1)
  `,
  MESON_TAC []);;

let main_B_growth_lemma = prove(`
  !n:real z0:real z1:real j:num.
  &1 < n ==>
  &0 <= (ceil((z0 + (z1 + exp(-- &j))/log2(n))*n)/n - z0)*log2(n) - z1
  `,
  REPEAT STRIP_TAC THEN
  specialize [`-- &j`] REAL_EXP_POS_LE THEN
  specialize [`n:real`] log2_gt1 THEN
  specialize [`(z0 + (z1 + exp(-- &j))/log2(n))*n`] ceil_ge THEN
  real_linear `&1 < n ==> &0 < n` THEN
  specialize [`log2 n`;
    `n:real`] REAL_LT_DIV THEN
  real_linear `&0 < log2 n / n ==> &0 <= log2 n / n` THEN
  specialize [`(z0 + (z1 + exp (-- &j)) / log2 n) * n`;
    `ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n)`;
    `(log2 n)/n`] REAL_LE_RMUL THEN
  real_cancel `&0 < n /\ &0 < log2 n ==> ((z0 + (z1 + exp (-- &j)) / log2 n) * n) * log2 n / n = z0 * log2 n + z1 + exp(-- &j)` THEN
  specialize [] (MESON [] `((z0 + (z1 + exp (-- &j)) / log2 n) * n) * log2 n / n <= ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) * log2 n / n /\ ((z0 + (z1 + exp (-- &j)) / log2 n) * n) * log2 n / n = z0 * log2 n + z1 + exp (-- &j) ==> z0 * log2 n + z1 + exp (-- &j) <= ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) * log2 n / n`) THEN
  real_linear `&0 <= exp(-- &j) /\ z0 * log2 n + z1 + exp (-- &j) <= ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) * log2 n / n ==> &0 <= (ceil ((z0 + (z1 + exp (-- &j)) / log2 n) * n) / n - z0) * log2 n - z1` THEN
  ASM_SIMP_TAC []);;

let main_B_growth_lemma2 = prove(`
  !n:num->real k:num->real x0:real z0:real z1:real b:num->num->real M:num->num B:num->real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ k = (\i. ceil(x0*n(i)))
  /\ (!i. &0 < k(i))
  /\ &0 < z0
  /\ &0 < &1 + x0 - z0
  /\ b = (\j:num i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))))
  /\ B = (\i:num. b (minimal j. i < M(j+1)) (i))
  ==> ?Z. !i. Z <= i ==> &0 <= (B(i)/n(i) - z0)*log2(n(i)) - z1
  `,
  REPEAT STRIP_TAC THEN
  choose_specializing `Z:num` [`n:num->real`;
    `k:num->real`;
    `x0:real`;
    `z0:real`;
    `z1:real`] main_b_growth_lemma13 THEN
  EXISTS_TAC `Z:num` THEN
  REPEAT STRIP_TAC THEN
  specialize [] (MESON [] `(!i:num. &1 < n i) ==> &1 < n i`) THEN
  specialize [`n(i:num):real`;
    `z0:real`;
    `z1:real`;
    `minimal j. i:num < M(j+1)`] main_B_growth_lemma THEN
  specialize [] (MESON [] `(!j:num i:num. Z <= i ==> ceil (min (n i + k i + &1) (max (&2) ((z0 + (z1 + exp (-- &j)) / log2 (n i)) * n i))) = ceil ((z0 + (z1 + exp (-- &j)) / log2 (n i)) * n i)) /\ Z <= i ==> (ceil (min (n i + k i + &1) (max (&2) ((z0 + (z1 + exp (-- &(minimal j. i < M(j + 1)))) / log2 (n i)) * n i))) - z0)*log2(n i) - z1 = (ceil ((z0 + (z1 + exp (-- &(minimal j. i < M(j + 1)))) / log2 (n i)) * n i) - z0)*log2(n i) - z1`) THEN
  specialize [] (MESON [] `B = (\i:num. b (minimal j. i < M (j + 1)) i) ==> (B i/n i - z0)*log2(n i) - z1 = ((b (minimal j. i < M (j + 1)) i)/n i - z0)*log2(n i) - z1`) THEN
  specialize [] (MESON [] `b = (\j:num i:num. ceil (min (n i + k i + &1) (max (&2) ((z0 + (z1 + exp (-- &j)) / log2 (n i)) * n i)))) ==> (b (minimal j. i < M (j + 1)) i/n i - z0) * log2 (n i) - z1 = (ceil (min (n i + k i + &1) (max (&2) ((z0 + (z1 + exp (-- &(minimal j. i < M (j + 1)))) / log2 (n i)) * n i)))/n i - z0) * log2(n i) - z1`) THEN
  ASM_SIMP_TAC []);;

let main_b_mono = prove(`
  !n:num->real k:num->real z0:real z1:real h:num j:num.
  (!i. &1 < n(i))
  /\ b = (\j:num i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))))
  /\ h <= j
  ==> (!i. b j i <= b h i)
  `,
  REPEAT STRIP_TAC THEN
  note `&1 < n(i:num)` [] THEN
  real_linear `&1 < n(i:num) ==> &0 <= n(i)` THEN
  note `&0 < log2(n(i:num))` [log2_gt1] THEN
  note `&h <= &j` [REAL_OF_NUM_LE] THEN
  note `-- &j <= -- &h` [REAL_LE_NEG2] THEN
  note `exp(-- &j) <= exp(-- &h)` [REAL_EXP_MONO_LE] THEN
  note `z1 + exp(-- &j) <= z1 + exp(-- &h)` [REAL_LE_REFL;
    REAL_LE_ADD2] THEN
  note `(z1 + exp(-- &j))/log2(n(i:num)) <= (z1 + exp(-- &h))/log2(n(i))` [REAL_LE_DIV2_EQ] THEN
  note `z0 + (z1 + exp(-- &j))/log2(n(i:num)) <= z0 + (z1 + exp(-- &h))/log2(n(i))` [REAL_LE_REFL;
    REAL_LE_ADD2] THEN
  note `(z0 + (z1 + exp(-- &j))/log2(n(i:num)))*n(i) <= (z0 + (z1 + exp(-- &h))/log2(n(i)))*n(i)` [REAL_LE_RMUL] THEN
  note `max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i:num)))*n(i)) <= max (&2) ((z0 + (z1 + exp(-- &h))/log2(n(i)))*n(i))` [REAL_ARITH `!x y z. y <= z ==> max x y <= max x z`] THEN
  note `min (n(i)+k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i:num)))*n(i))) <= min (n(i)+k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &h))/log2(n(i)))*n(i)))` [REAL_ARITH `!x y z. y <= z ==> min x y <= min x z`] THEN
  note `ceil(min (n(i)+k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i:num)))*n(i)))) <= ceil(min (n(i)+k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &h))/log2(n(i)))*n(i))))` [ceil_mono] THEN
  ASM_MESON_TAC []);;

let mono_lt_imp_le = prove(`
  !f:num->num.
  (!j k. j < k ==> f(j) < f(k))
  ==> (!j k. j <= k ==> f(j) <= f(k))
  `,
  REPEAT STRIP_TAC THEN
  ASM_CASES_TAC `j:num < k` THENL
  [ASM_MESON_TAC [LT_IMP_LE];
    note `j = k:num` [ARITH_RULE `j <= k /\ ~(j < k) ==> j = k:num`] THEN
    ASM_MESON_TAC [LE_REFL]]);;

let gluing_mono = prove(`
  !b:num->num->real M:num->num B:num->real i:num t:num.
  (!h j. h <= j ==> !c. b j c <= b h c)
  /\ (!j k. j < k ==> M j < M k)
  /\ B = (\c:num. b (minimal j. c < M(j+1)) (c))
  /\ M(t) <= i ==>
  B(i) <= b t i
  `,
  REPEAT STRIP_TAC THEN
  note `i < M(i + 1)` [num_mono_gt] THEN
  specialize [`M:num->num`] mono_lt_imp_le THEN
  note `!m:num. m < t ==> ~(i:num < M(m+1))` [ARITH_RULE `!m:num. m < t ==> m+1 <= t`;
    LE_TRANS;
    NOT_LT] THEN
  note `?r. i:num < M(r+1)` [] THEN
  specialize [`\j. i:num < M(j+1)`;
    `t:num`] MINIMAL_LBOUND THEN
  ASM_MESON_TAC [main_b_mono]);;

let o1_expminus_lemma = prove(`
  !e:real.
  &0 < e
  ==> ?j. exp(-- &j) <= e
  `,
  REPEAT STRIP_TAC THEN
  choose_specializing `j:num` [`--ln(e)`] REAL_ARCH_SIMPLE THEN
  EXISTS_TAC `j:num` THEN
  real_linear `--ln(e) <= &j ==> -- &j <= ln(e)` THEN
  note `exp(-- &j) <= exp(ln(e))` [REAL_EXP_MONO_LE] THEN
  ASM_MESON_TAC [EXP_LN]);;

let main_B_growth_lemma3 = prove(`
  !n:real z0:real z1:real e:real X:num b:real B:real.
  &1 < n
  /\ &0 < e
  /\ &0 <= (B / n - z0) * log2 (n) - z1
  /\ B <= b
  /\ abs ((b / n - z0) * log2 (n) - (z1 + exp (-- &X))) <= e / &2
  /\ exp(-- &X) <= e / &2
  ==> abs ((B / n - z0) * log2 (n) - z1) <= e
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [REAL_ABS_BOUNDS] THEN
  note `&0 < log2 n` [log2_gt1] THEN
  real_linear `&0 < e /\ &0 <= (B / n - z0) * log2 (n) - z1 ==> --e <= (B / n - z0) * log2 (n) - z1` THEN
  real_linear `&1 < n ==> &0 < n` THEN
  note `B / n <= b / n` [REAL_LE_DIV2_EQ] THEN
  real_linear `B/n <= b/n ==> B / n - z0 <= b / n - z0` THEN
  real_linear `&0 < log2 n ==> &0 <= log2 n` THEN
  note `(B / n - z0) * log2 n <= (b / n - z0) * log2 n` [REAL_LE_RMUL] THEN
  real_linear `(B / n - z0) * log2 n <= (b / n - z0) * log2 n ==> (B / n - z0) * log2 n - z1 <= (b / n - z0) * log2 n - z1` THEN
  real_linear `abs ((b / n - z0) * log2 (n) - (z1 + exp (-- &X))) <= e / &2 ==> (b / n - z0) * log2 (n) - z1 <= e / &2 + exp (-- &X)` THEN
  note `(B / n - z0) * log2 n - z1 <= e / &2 + exp(-- &X)` [REAL_LE_TRANS] THEN
  real_linear `exp(-- &X) <= e / &2 ==> e / &2 + exp(-- &X) <= e` THEN
  ASM_MESON_TAC [REAL_LE_TRANS]);;

let main_B_growth = prove(`
  !n:num->real k:num->real x0:real z0:real z1:real b:num->num->real M:num->num B:num->real.
  (!i. &1 < n(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ k = (\i. ceil(x0 * n(i)))
  /\ (!i. &0 < k(i))
  /\ &0 < z0
  /\ &0 < &1 + x0 - z0
  /\ b = (\j:num i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i)))))
  /\ (!j k. j < k ==> M j < M k)
  /\ B = (\i:num. b (minimal j. i < M(j+1)) (i))
  ==>
  o1_seq (\i. (B(i)/n(i) - z0) * log2(n(i)) - z1)
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [o1_seq] THEN
  REPEAT STRIP_TAC THEN
  real_linear `&0 < e ==> &0 < e / &2` THEN
  choose_specializing `X:num` [`e / &2`] o1_expminus_lemma THEN
  note `o1_seq (\i:num. ((b X i)/n(i) - z0) * log2(n(i)) - (z1 + exp(-- &X)))` [main_b_growth] THEN
  choose_specializing `Y:num` [`\i:num. ((b X i)/n(i) - z0) * log2(n(i)) - (z1 + exp(-- &X))`;
    `e / &2`] o1_seq_use THEN
  choose_specializing `Z:num` [`n:num->real`;
    `k:num->real`;
    `x0:real`;
    `z0:real`;
    `z1:real`;
    `b:num->num->real`;
    `M:num->num`;
    `B:num->real`] main_B_growth_lemma2 THEN
  EXISTS_TAC `MAX (MAX Y Z) (M(X:num))` THEN
  REWRITE_TAC [max_le] THEN
  REPEAT STRIP_TAC THEN
  note `(!h j:num. h <= j ==> !i:num. b j i <= b h i:real)` [main_b_mono] THEN
  specialize [`b:num->num->real`;
    `M:num->num`;
    `B:num->real`;
    `i:num`;
    `X:num`] gluing_mono THEN
  ASM_MESON_TAC [main_B_growth_lemma3]);;

let forward_main = prove(`
  !n:num->real q:num->real s:num->real
   Q0:real Q1:real S0:real S1:real
   x0:real z0:real z1:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. (log2(q(i))/log2(n(i)) - Q0) * log2(n(i)) - Q1)
  /\ o1_seq (\i. (log2(s(i))/log2(n(i)) - S0) * log2(n(i)) - S1)
  /\ -- &1 / &2 < S0
  /\ S0 <= &1 / &2
  /\ &1 / &2 < Q0 - S0
  /\ &1 / &2 < Q0 + S0
  /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2)
  /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2)
  /\ z1 =   (&2 * S1 + log2(z0)
             - (S0 - Q0 + &3 / &2) * (log2(z0) - log2(&2 * pi * exp(&1)))
             - Q1 * (Q0 + S0 - &1 / &2)/Q0
            )
          * (&2 * Q0) / ((Q0 - S0 + &1 / &2) pow 3)
  ==> ?k:num->real b:num->real.
    (!i. integer(k(i)))
    /\ (!i. &0 < k(i))
    /\ (!i. k(i) <= ceil(n(i)))
    /\ o1_seq (\i. (k(i)/n(i) - x0) * log2(n(i)) - &0)
    /\ (!i. integer(b(i)))
    /\ (!i. &1 < b(i))
    /\ (!i. b(i) <= ceil(n(i) + k(i) + &1))
    /\ o1_seq (\i. (b(i)/n(i) - z0) * log2(n(i)) - z1)
    /\ ?m. !i. m <= i ==> standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i)) < &1
  `,
  REPEAT STRIP_TAC THEN
  def `k:num->real` `\i:num. ceil(x0*n(i))` THEN
  EXISTS_TAC `k:num->real` THEN
  real_linear `-- &1 / &2 < S0 ==> &0 < &1 + &2 * S0` THEN
  real_linear `&1 / &2 < Q0 - S0 ==> -- &1 / &2 < Q0 - S0` THEN
  specialize [`Q0:real`;
    `S0:real`;
    `x0:real`] main_x0_positive THEN
  specialize [`Q0:real`;
    `S0:real`;
    `x0:real`] main_x0_le1 THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `x0:real`] main_k_integer THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `x0:real`] main_k_positive THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `x0:real`] main_k_leceiln THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `x0:real`] main_kn_o1 THEN
  def `b:num->num->real` `\j:num i:num. ceil(min (n(i) + k(i) + &1) (max (&2) ((z0 + (z1 + exp(-- &j))/log2(n(i)))*n(i))))` THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `z0:real`;
    `z1:real`;
    `b:num->num->real`] main_b_integer THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `z0:real`;
    `z1:real`;
    `b:num->num->real`] main_b_gt1 THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `z0:real`;
    `z1:real`;
    `b:num->num->real`] main_b_lenk1 THEN
  specialize [`Q0:real`;
    `S0:real`;
    `z0:real`] main_z0_positive THEN
  specialize [`Q0:real`;
    `S0:real`;
    `x0:real`;
    `z0:real`] main_1x0z0_positive THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `x0:real`;
    `z0:real`;
    `z1:real`;
    `b:num->num->real`] main_b_growth THEN
  real_linear `&0 < x0 ==> &0 < &1 + x0` THEN
  specialize [`Q0:real`;
    `S0:real`;
    `x0:real`] main_SQx_positive THEN
  specialize [`Q0:real`;
    `S0:real`;
    `x0:real`;
    `z0:real`] main_zxSQ THEN
  specialize [`Q0:real`;
    `Q1:real`;
    `S0:real`;
    `S1:real`;
    `x0:real`;
    `z0:real`;
    `z1:real`] main_z1formula THEN
  specialize [`z1:real`] main_z1lt THEN
  specialize [`n:num->real`;
    `q:num->real`;
    `s:num->real`;
    `k:num->real`;
    `b:num->num->real`;
    `Q0:real`;
    `Q1:real`;
    `S0:real`;
    `S1:real`;
    `x0:real`;
    `&0`;
    `\j. z1+exp(-- &j)`;
    `z0:real`;
    `z1:real`] asymp2_below_j THEN
  def `m:num->num` `\j:num. minimal A. !i. A <= i ==> standardratio (n i) (q i) (s i) (k i) (b j i) < &1` THEN
  def `M:num->num` `\j:num. j + nsum(0..j) m` THEN
  def `B:num->real` `\i:num. b (minimal j. i < M(j+1)) (i):real` THEN
  EXISTS_TAC `B:num->real` THEN
  specialize [`m:num->num`;
    `M:num->num`;
    `b:num->num->real`;
    `B:num->real`;
    `\i:num beta:real. standardratio (n(i)) (q(i)) (s(i)) (k(i)) (beta) < &1`] gluing_pattern THEN
  specialize [`b:num->num->real`;
    `m:num->num`;
    `B:num->real`] main_B_gt1 THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `b:num->num->real`;
    `m:num->num`;
    `B:num->real`] main_B_lenk1 THEN
  specialize [`m:num->num`;
    `M:num->num`] main_M_mono THEN
  specialize [`n:num->real`;
    `k:num->real`;
    `x0:real`;
    `z0:real`;
    `z1:real`;
    `b:num->num->real`;
    `M:num->num`;
    `B:num->real`] main_B_growth THEN
  ASM_MESON_TAC []);;

(* ----- and working towards converse *)

let o1_lower_1 = prove(`
  !f:num->real. (!e:real. &0 < e ==> FINITE {n | f n < --e}) ==> FINITE {n | f n < -- &1}
  `,
  MESON_TAC [REAL_ARITH `&0 < &1`]);;

let o1_lower_1_nonempty = prove(`
  !f:num->real. ~(!n. -- &1 <= f n) ==> ~({n | f n < -- &1} = {})
  `,
  REWRITE_TAC [GSYM EMPTY_GSPEC] THEN
  REWRITE_TAC [EXTENSION] THEN
  REWRITE_TAC [IN_ELIM_THM] THEN
  MESON_TAC [REAL_NOT_LT]);;

let o1_lower_1_nonempty_image = prove(`
  !f:num->real. ~(!n. -- &1 <= f n) ==> ~(IMAGE f {n | f n < -- &1} = {})
  `,
  MESON_TAC [o1_lower_1_nonempty;
    IMAGE_EQ_EMPTY;
    SIMPLE_IMAGE_GEN]);;

let in_image_rewrite = prove(`
  !f:num->real L:real. (!x. x IN IMAGE f { n | f n < -- &1} ==> L <= x) ==> !n. f n < -- &1 ==> L <= f n
  `,
  REWRITE_TAC [IMAGE] THEN
  REWRITE_TAC [IN_ELIM_THM] THEN
  MESON_TAC []);;

let o1_lower_1_bound_lemma = prove(`
  !f:num->real. FINITE {n | f n < -- &1} ==> ?L. !n. f n < -- &1 ==> L <= f n
  `,
  REPEAT STRIP_TAC THEN
  ASM_CASES_TAC `!n:num. -- &1 <= f n` THENL
  [EXISTS_TAC `-- &1` THEN
    ASM_MESON_TAC [];
    EXISTS_TAC `inf(IMAGE f {n:num | f n < -- &1})` THEN
    ASSUME_TAC (ISPECL [`f:num->real`] o1_lower_1_nonempty_image) THEN
    ASSUME_TAC (ISPECL [`IMAGE f {n:num | f n < -- &1}`] INF_FINITE) THEN
    ASSUME_TAC (ISPECL [`f:num->real`;
        `{n:num | f n < -- &1}`] FINITE_IMAGE) THEN
    ASSUME_TAC (ISPECL [`f:num->real`;
        `inf(IMAGE f {n:num | f n < -- &1})`] in_image_rewrite) THEN
    ASM_MESON_TAC []]);;

let o1_lower_1_bound_lemma2 = prove(`
  !f:num->real L:real. (!n. f n < -- &1 ==> L <= f n) ==> ?M:real. M <= -- &1 /\ (!n. M <= f n)
  `,
  REPEAT STRIP_TAC THEN
  ASM_CASES_TAC `L <= -- &1` THENL
  [EXISTS_TAC `L:real` THEN
    ASM_MESON_TAC [REAL_NOT_LT;
      REAL_LE_TRANS];
    EXISTS_TAC `-- &1:real` THEN
    ASM_MESON_TAC [REAL_NOT_LT;
      REAL_NOT_LE;
      REAL_LTE_TRANS;
      REAL_LT_IMP_LE;
      REAL_LE_REFL]]);;

let o1_lower_bound = prove(`
  !f:num->real.
  FINITE {n | f n < -- &1}
  ==> ?L. L <= -- &1 /\ !n. L <= f n
  `,
  MESON_TAC [o1_lower_1_bound_lemma;
    o1_lower_1_bound_lemma2]);;

let o1_lower_fn_in_Tn = prove(`
  !f:num->real n.
  f n IN ({&0} UNION (IMAGE f {x | n <= x}))
  `,
  REWRITE_TAC [UNION] THEN
  REWRITE_TAC [IMAGE] THEN
  REWRITE_TAC [IN_ELIM_THM] THEN
  ASM_MESON_TAC [LE_REFL]);;

let o1_lower_Tn_bound = prove(`
  !f:num->real L:real n:num x:real.
  L <= -- &1
  /\ (!n. L <= f n)
  /\ (x IN ({&0} UNION (IMAGE f {x | n <= x})))
  ==> L <= x
  `,
  REWRITE_TAC [UNION] THEN
  REWRITE_TAC [IMAGE] THEN
  REWRITE_TAC [GSYM SING_GSPEC] THEN
  REWRITE_TAC [IN_ELIM_THM] THEN
  MESON_TAC [REAL_ARITH `-- &1 <= &0`;
    REAL_LE_TRANS]);;

let o1_lower_gn_le_fn = prove(`
  !f:num->real L:real.
  L <= -- &1
  /\ (!n. L <= f n)
  ==> 
  !n. inf ({&0} UNION (IMAGE f {x | n <= x})) <= f(n)
  `,
  MESON_TAC [INF;
    singleton_union_not_empty;
    o1_lower_fn_in_Tn;
    o1_lower_Tn_bound]);;

let o1_lower_gn_le_fn_2 = prove(`
  !f:num->real.
  (!e:real. &0 < e ==> FINITE {n | f n < --e})
  ==> 
  !n. inf ({&0} UNION (IMAGE f {x | n <= x})) <= f(n)
  `,
  MESON_TAC [o1_lower_gn_le_fn;
    o1_lower_bound;
    REAL_ARITH `&0 < &1`]);;

let o1_lower_0_in_Tn = prove(`
  !f:num->real n:num.
  &0 IN ({&0} UNION (IMAGE f {x | n <= x}))
  `,
  MESON_TAC [IN_SING;
    IN_UNION]);;

let o1_lower_Tn_nonempty = prove(`
  !f:num->real n:num.
  ~({&0} UNION (IMAGE f {x | n <= x}) = {})
  `,
  MESON_TAC [o1_lower_0_in_Tn;
    element_implies_nonempty2]);;

let o1_lower_gn_le_0 = prove(`
  !f:num->real L:real.
  L <= -- &1
  /\ (!n. L <= f n)
  ==> 
  !n. inf ({&0} UNION (IMAGE f {x | n <= x})) <= &0
  `,
  MESON_TAC [INF;
    singleton_union_not_empty;
    o1_lower_fn_in_Tn;
    o1_lower_Tn_bound;
    o1_lower_0_in_Tn]);;

let o1_lower_gn_le_0_2 = prove(`
  !f:num->real.
  (!e:real. &0 < e ==> FINITE {n | f n < --e})
  ==> 
  !n. inf ({&0} UNION (IMAGE f {x | n <= x})) <= &0
  `,
  MESON_TAC [o1_lower_gn_le_0;
    o1_lower_bound;
    REAL_ARITH `&0 < &1`]);;

let o1_lower_above_minuse = prove(`
  !f:num->real e:real a:num n:num y:real.
  &0 < e /\
  SUC a <= n /\
  (!x. f x < --e ==> x <= a) /\
  y IN ({&0} UNION (IMAGE f {x | n <= x}))
  ==> --e <= y
  `,
  REWRITE_TAC [GSYM SING_GSPEC] THEN
  REWRITE_TAC [IMAGE] THEN
  REWRITE_TAC [UNION] THEN
  REWRITE_TAC [IN_ELIM_THM] THEN
  MESON_TAC [REAL_ARITH `&0 < e ==> --e <= &0`;
    REAL_NOT_LT;
    ARITH_RULE `x <= a /\ SUC a <= n ==> ~(n <= x)`]);;

let o1_lower_above_something = prove(`
  !f:num->real e:real a:num n:num.
  &0 < e /\
  SUC a <= n /\
  (!x. f x < --e ==> x <= a) ==>
  ?b:real. !y:real. y IN ({&0} UNION (IMAGE f {x | n <= x})) ==> b <= y
  `,
  REPEAT STRIP_TAC THEN
  EXISTS_TAC `--e:real` THEN
  ASM_MESON_TAC [o1_lower_above_minuse]);;

let o1_lower_gn_above_minuse = prove(`
  !f:num->real e:real a:num n:num.
  &0 < e /\
  SUC a <= n /\
  (!x. f x < --e ==> x <= a)
  ==>
  --e <= inf ({&0} UNION (IMAGE f {x | n <= x}))
  `,
  STRIP_TAC THEN
  STRIP_TAC THEN
  STRIP_TAC THEN
  STRIP_TAC THEN
  STRIP_TAC THEN
  ASSUME_TAC (ISPECL [`f:num->real`;
      `e:real`;
      `a:num`;
      `n:num`] o1_lower_above_something) THEN
  ASSUME_TAC (ISPECL [`f:num->real`;
      `n:num`] o1_lower_Tn_nonempty) THEN
  ASSUME_TAC (ISPECL [`({&0:real} UNION (IMAGE (f:num->real) {x | n <= x}))`] INF) THEN
  ASSUME_TAC (ISPECL [`f:num->real`;
      `e:real`;
      `a:num`;
      `n:num`] o1_lower_above_minuse) THEN
  ASM_MESON_TAC []);;

let o1_lower_gn_above_minuse_2 = prove(`
  !f:num->real e:real L:real.
  &0 < e
  /\ FINITE {n | f n < --e}
  ==>
  ?m. !n. m <= n ==>
  --e <= inf ({&0} UNION (IMAGE f {x | n <= x}))
  `,
  REWRITE_TAC [num_FINITE] THEN
  REWRITE_TAC [IN_ELIM_THM] THEN
  REPEAT STRIP_TAC THEN
  EXISTS_TAC `SUC a` THEN
  ASM_MESON_TAC [o1_lower_gn_above_minuse]);;

let o1_lower_gn_above_minuse_3 = prove(`
  !f:num->real e:real L:real.
  &0 < e
  /\ FINITE {n | f n < --e}
  /\ L <= -- &1
  /\ (!n. L <= f n)
  ==>
  ?m. !n. m <= n ==>
  abs(inf ({&0} UNION (IMAGE f {x | n <= x}))) <= e
  `,
  MESON_TAC [REAL_ARITH `--e <= x /\ x <= &0 ==> abs(x) <= e`;
    o1_lower_gn_le_0;
    o1_lower_gn_above_minuse_2]);;

let o1_lower_gn_o1 = prove(`
  !f:num->real.
  (!e:real. &0 < e ==> FINITE {n | f n < --e})
  ==> 
  o1_seq (\n. inf ({&0} UNION (IMAGE f {x | n <= x })))
  `,
  REWRITE_TAC [o1_seq] THEN
  MESON_TAC [o1_lower_gn_above_minuse_3;
    o1_lower_bound;
    REAL_ARITH `&0 < &1`]);;

let o1_lower_main = prove(`
  !f:num->real.
  (!e:real. &0 < e ==> FINITE {n | f n < --e})
  ==>
  ?g:num->real.
    (!n. g n <= f n) /\
    (!n. g n <= &0) /\
    o1_seq g
  `,
  REPEAT STRIP_TAC THEN
  EXISTS_TAC `\n:num. inf ({&0:real} UNION (IMAGE (f:num->real) {x | n <= x}))` THEN
  REWRITE_TAC [BETA_THM] THEN
  ASM_MESON_TAC [o1_lower_gn_le_0_2;
    o1_lower_gn_le_fn_2;
    o1_lower_gn_o1] THEN
  ASSUME_TAC (ISPECL [`f:num->real`] o1_lower_gn_le_fn_2) THEN
  ASSUME_TAC (ISPECL [`f:num->real`] o1_lower_gn_le_0_2) THEN
  ASSUME_TAC (ISPECL [`f:num->real`] o1_lower_gn_o1) THEN
  ASM_MESON_TAC []);;

let lnexp2pibound2 = prove(`
  !x. &60 <= x ==> ln(&2 * pi * exp(&2)) < ln x
  `,
  MESON_TAC [ln_mono_le;
    REAL_ARITH `&0 < &60`;
    lnexp2pibound;
    REAL_LTE_TRANS]);;

let ln60bound = prove(`
  !x. &60 <= x ==> &1 < ln x - ln(&2 * pi * exp(&1))
  `,
  MESON_TAC [lnexp2pibound2;
    lnexp1piexp2pi;
    REAL_ARITH `&1 + a < b ==> &1 < b - a`]);;

let pi60bound = prove(`
  !x. &60 <= x ==> pi * &60 <= pi * x
  `,
  MESON_TAC [REAL_LE_LMUL;
    PI_POS;
    REAL_LT_IMP_LE]);;

let pi60bound3 = prove(`
  !x. &60 <= x ==> (&333 / &106) * &60 < pi * x
  `,
  MESON_TAC [pi60bound;
    pi60bound2;
    REAL_LTE_TRANS]);;

let pi60bound4 = prove(`
  !x. &60 <= x ==> &1 < pi * x
  `,
  MESON_TAC [pi60bound3;
    REAL_ARITH `&1 < (&333 / &106) * &60`;
    REAL_LT_TRANS]);;

let pi60bound5 = prove(`
  !x. &60 <= x ==> ln(&1) < ln(pi * x)
  `,
  MESON_TAC [pi60bound4;
    ln_mono_lt;
    REAL_ARITH `&0 < &1`]);;

let pi60bound6 = prove(`
  !x. &60 <= x ==> &0 < ln(pi * x)
  `,
  MESON_TAC [pi60bound5;
    LN_1]);;

let pi60bound7 = prove(`
  !x. &60 <= x ==> &0 < ln(pi * x) / x
  `,
  MESON_TAC [pi60bound6;
    REAL_LT_DIV;
    REAL_ARITH `&60 <= x ==> &0 < x`]);;

let deltabound = prove(`
  !x. &60 <= x ==> &1 < ln x + ln(pi * x) / x - ln(&2 * pi * exp(&1))
  `,
  MESON_TAC [ln60bound;
    pi60bound7;
    REAL_ARITH `&0 < a /\ &1 < b - c ==> &1 < b + a - c`]);;

let deltabound2 = prove(`
  !x. &60 <= x ==> &0 < ln x + ln(pi * x) / x - ln(&2 * pi * exp(&1))
  `,
  MESON_TAC [deltabound;
    REAL_ARITH `&1 < a ==> &0 < a`]);;

let deltabound3 = prove(`
  !x. &60 <= x ==> &0 < (ln x + ln(pi * x) / x - ln(&2 * pi * exp(&1))) / (&2 * (x - &1))
  `,
  MESON_TAC [deltabound2;
    REAL_ARITH `&60 <= x ==> &0 < &2 * (x - &1)`;
    REAL_LT_DIV]);;

let deltabound4 = prove(`
  !x. &60 <= x ==> &0 < ln(bkzdelta x)
  `,
  MESON_TAC [bkzdelta_ln5;
    deltabound3]);;

let deltabound5 = prove(`
  !x. &60 <= x ==> &1 < bkzdelta x
  `,
  MESON_TAC [deltabound4;
    LN_1;
    LN_MONO_LT;
    REAL_ARITH `&0 < &1`;
    bkzdelta_pos]);;

let deltabound6 = prove(`
  !x. &60 <= x ==> &1 / (&2 * (x - &1)) < (ln(x) + ln(pi*x) / x - ln(&2 * pi * exp(&1))) / (&2 * (x - &1))
  `,
  MESON_TAC [deltabound;
    REAL_ARITH `&60 <= x ==> &0 < &2 * (x - &1)`;
    REAL_LT_DIV2_EQ]);;

let bkzdelta_ln6 = prove(`
  !x. &60 <= x ==> &1 / (&2 * (x - &1)) < ln(bkzdelta x)
  `,
  MESON_TAC [deltabound6;
    bkzdelta_ln5]);;

let bkzdelta_ln7 = prove(`
  !x. &60 <= x ==> &1 / (x - &1) < &2 * ln(bkzdelta x)
  `,
  MESON_TAC [bkzdelta_ln6;
    REAL_FIELD `&1 / (&2 * x) < y ==> &1 / x < &2 * y`]);;

let bkzdelta_ln8 = prove(`
  !x. &60 <= x ==> ln(bkzdelta x) * (&2 * (x - &1)) = ln(x) + ln(pi*x)/x - ln(&2 * pi * exp(&1))
  `,
  MESON_TAC [bkzdelta_ln5;
    REAL_DIV_RMUL;
    REAL_ARITH `&60 <= x ==> ~(&2 * (x - &1) = &0)`]);;

let xx1_step1 = REAL_ARITH `
  (x - &1)*(x + &1) < x pow 2
`;;

let x2_positive = prove(`
  !x. &60 <= x ==> &0 < x pow 2
  `,
  MESON_TAC [REAL_LT_MUL;
    REAL_ARITH `&60 <= x ==> &0 < x`;
    REAL_ARITH `x * x = x pow 2`]);;

let xx1_step2 = prove(`
  !x. &60 <= x ==> ((x - &1)*(x + &1))/(x pow 2) < &1
  `,
  MESON_TAC [REAL_LT_LDIV_EQ;
    x2_positive;
    xx1_step1;
    REAL_ARITH `&1 * (x pow 2) = x pow 2`]);;

let xx1_step3 = prove(`
  !x. &60 <= x ==> ((x + &1)/(x pow 2))*(x - &1) < &1
  `,
  MESON_TAC [xx1_step2;
    REAL_ARITH `(a*b)/c = (b/c)*a`]);;

let xx1_step4 = prove(`
  !x. &60 <= x ==> (x + &1)/(x pow 2) < &1/(x - &1)
  `,
  MESON_TAC [REAL_LT_RDIV_EQ;
    REAL_ARITH `&60 <= x ==> &0 < x - &1`;
    xx1_step3]);;

let xx1_step5 = prove(`
  !x. &60 <= x ==> (x + &1)/(x pow 2) < &2 * ln(bkzdelta x)
  `,
  MESON_TAC [bkzdelta_ln7;
    xx1_step4;
    REAL_LT_TRANS]);;

let delta_deriv = DIFF_CONV `
  \x. (ln(x) + ln(pi*x)/x - ln(&2 * pi * exp(&1))) / (&2 * (x - &1))
`;;

let delta_nonzero_2x1 = REAL_ARITH `
  !x. &60 <= x ==> ~(&2 * (x - &1) = &0)
`;;

let delta_nonzero_x = REAL_ARITH `
  !x. &60 <= x ==> ~(x = &0)
`;;

let delta_sign_x = REAL_ARITH `
  !x. &60 <= x ==> &0 < x
`;;

let delta_sign_pix = prove(`
  !x. &60 <= x ==> &0 < pi * x
  `,
  MESON_TAC [delta_sign_x;
    PI_POS;
    REAL_LT_MUL]);;

let delta_deriv2 = prove(`
  !x. &60 <= x ==>
    ((\x. (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1))) /
                   (&2 * (x - &1))) diffl
              ((inv x * &1 +
                ((inv (pi * x) * (&0 * x + &1 * pi)) * x - &1 * ln (pi * x)) /
                x pow 2 -
                &0) *
               &2 *
               (x - &1) -
               (&0 * (x - &1) + (&1 - &0) * &2) *
               (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1)))) /
              (&2 * (x - &1)) pow 2)
             x
  `,
  MESON_TAC [delta_deriv;
    delta_nonzero_2x1;
    delta_nonzero_x;
    delta_sign_x;
    delta_sign_pix]);;

let sign_lnpixx2 = prove(`
  !x. &60 <= x ==> &0 < ln(pi*x)/(x pow 2)
  `,
  MESON_TAC [pi60bound6;
    REAL_LT_DIV;
    x2_positive]);;

let xx1_step6 = prove(`
  !x. &60 <= x ==> (x + &1)/(x pow 2) - ln(pi*x)/(x pow 2) < &2 * ln(bkzdelta x)
  `,
  MESON_TAC [sign_lnpixx2;
    xx1_step5;
    REAL_ARITH `a < b /\ &0 < c ==> a - c < b`]);;

let xx1_step7 = prove(`
  !x. &60 <= x ==> ((x + &1)/(x pow 2) - ln(pi*x)/(x pow 2)) * &2 * (x - &1) < (&2 * ln(bkzdelta x)) * &2 * (x - &1)
  `,
  MESON_TAC [xx1_step6;
    REAL_LT_RMUL;
    REAL_ARITH `&60 <= x ==> &0 < &2 * (x - &1)`]);;

let xx1_step8 = prove(`
  !x. &60 <= x ==> (&2 * ln(bkzdelta x)) * &2 * (x - &1) = (&0 * (x - &1) + (&1 - &0) * &2) * (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1)))
  `,
  SIMP_TAC [bkzdelta_ln5] THEN
  CONV_TAC REAL_FIELD);;

let xx1_step9 = prove(`
  !x. &60 <= x ==> ((x + &1)/(x pow 2) - ln(pi*x)/(x pow 2)) * &2 * (x - &1) < (&0 * (x - &1) + (&1 - &0) * &2) * (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1)))
  `,
  MESON_TAC [xx1_step7;
    xx1_step8]);;

let xx1_step10 = REAL_FIELD `
  !x. &60 <= x /\ &0 < pi ==> ((x + &1)/(x pow 2) - ln(pi*x)/(x pow 2)) * &2 * (x - &1) =
  (inv x * &1 + ((inv (pi * x) * (&0 * x + &1 * pi)) * x - &1 * ln (pi * x)) / x pow 2 - &0)
  * &2 * (x - &1)
`;;

let xx1_step11 = prove(`
  !x. &60 <= x ==> 
  (inv x * &1 + ((inv (pi * x) * (&0 * x + &1 * pi)) * x - &1 * ln (pi * x)) / x pow 2 - &0)
  * &2 * (x - &1) < (&0 * (x - &1) + (&1 - &0) * &2) * (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1)))
  `,
  MESON_TAC [xx1_step9;
    xx1_step10;
    PI_POS]);;

let xx1_step12 = prove(`
  !x. &60 <= x ==> 
  (inv x * &1 + ((inv (pi * x) * (&0 * x + &1 * pi)) * x - &1 * ln (pi * x)) / x pow 2 - &0)
  * &2 * (x - &1) - (&0 * (x - &1) + (&1 - &0) * &2) * (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1))) < &0
  `,
  MESON_TAC [xx1_step11;
    REAL_ARITH `x < y ==> x - y < &0`]);;

let delta_sign_denom = prove(`
  !x. &60 <= x ==> &0 < (&2 * (x - &1)) pow 2
  `,
  MESON_TAC [REAL_ARITH `&60 <= x ==> &0 < &2 * (x - &1)`;
    REAL_LT_MUL;
    REAL_POW_2]);;

let delta_deriv_negative = prove(`
  !x. &60 <= x ==> (
  (inv x * &1 + ((inv (pi * x) * (&0 * x + &1 * pi)) * x - &1 * ln (pi * x)) / x pow 2 - &0)
  * &2 * (x - &1) - (&0 * (x - &1) + (&1 - &0) * &2) * (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1)))) / ((&2 * (x - &1)) pow 2) < &0
  `,
  MESON_TAC [xx1_step12;
    delta_sign_denom;
    neg_over_pos]);;

let delta_decreasing = prove(`
  !a b. &60 <= a /\ a <= b ==> (\x. (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1))) / (&2 * (x - &1)))(b) <= (\x. (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1))) / (&2 * (x - &1)))(a)
  `,
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC deriv_negative4 THEN
  EXISTS_TAC `\x. (
  (inv x * &1 + ((inv (pi * x) * (&0 * x + &1 * pi)) * x - &1 * ln (pi * x)) / x pow 2 - &0)
  * &2 * (x - &1) - (&0 * (x - &1) + (&1 - &0) * &2) * (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1)))) / ((&2 * (x - &1)) pow 2)` THEN
  REWRITE_TAC [BETA_THM] THEN
  CONJ_TAC THEN
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (REAL_ARITH `&60 <= a /\ a <= x ==> &60 <= x`) THENL
  [ASM_REWRITE_TAC [];
    ASSUME_TAC (ISPECL [`x:real`] delta_deriv2) THEN
    ASM_MESON_TAC [];
    ASSUME_TAC (ISPECL [`x:real`] delta_deriv_negative) THEN
    ASM_MESON_TAC []]);;

let rho_deriv = DIFF_CONV `
  \x. (&1 / &2) * ln((n + k)*(s pow 2) + &1) - ((&1 / &2) * ln((n + k + &1)/x) + (ln(x) + (&1 / x) * ln(pi*x) - ln(&2 * pi * exp(&1))) * ( (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1))) + (k/(n + k + &1)) * ln q)
`;;

let rho_deriv2_lemma = REAL_ARITH `
  !n k. &1 < n /\ &0 < k ==> &0 < n + k + &1
`;;

let rho_deriv2_lemma2 = prove(`
  !n k x. &60 <= x /\ &1 < n /\ &0 < k ==> &0 < (n + k + &1) / x
  `,
  MESON_TAC [rho_deriv2_lemma;
    delta_sign_x;
    REAL_LT_DIV]);;

let rho_deriv2 = prove(`
  !n q s k x. &60 <= x /\ &1 < n /\ &0 < k ==>
  ((\x. &1 / &2 * ln ((n + k) * s pow 2 + &1) -
                   (&1 / &2 * ln ((n + k + &1) / x) +
                    (ln x + &1 / x * ln (pi * x) - ln (&2 * pi * exp (&1))) *
                    (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1)) +
                    k / (n + k + &1) * ln q)) diffl
              &0 -
              ((&0 * ln ((n + k + &1) / x) +
                (inv ((n + k + &1) / x) *
                 (&0 * x - &1 * (n + k + &1)) / x pow 2) *
                &1 / &2) +
               ((inv x * &1 +
                 ((&0 * x - &1 * &1) / x pow 2 * ln (pi * x) +
                  (inv (pi * x) * (&0 * x + &1 * pi)) * &1 / x) -
                 &0) *
                (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1)) +
                (((&0 * x + &1 * &2) - &0 - &0) * &2 * (x - &1) -
                 (&0 * (x - &1) + (&1 - &0) * &2) *
                 (&2 * x - (n + k + &1) - &1)) /
                (&2 * (x - &1)) pow 2 *
                (ln x + &1 / x * ln (pi * x) - ln (&2 * pi * exp (&1)))) +
               &0))
             x
  `,
  MESON_TAC [rho_deriv;
    delta_sign_x;
    delta_sign_pix;
    delta_nonzero_x;
    delta_nonzero_2x1;
    rho_deriv2_lemma2]);;

let lnpix_deriv = DIFF_CONV `
  \x. ln(pi*x)-x / &2
`;;

let lnpix_deriv2 = prove(`
  !x. &60 <= x ==> &1 / x - &1 / &2 < &0
  `,
  MESON_TAC [REAL_LE_INV2;
    REAL_ARITH `&0 < &60`;
    REAL_ARITH `inv x <= inv(&60) ==> &1 / x - &1 / &2 < &0`]);;

let lnpix_deriv3 = REAL_FIELD `
  ~(x = &0) /\ &0 < pi ==> &1 / x - &1 / &2 = inv (pi * x) * (&0 * x + &1 * pi) - (&1 * &2 - &0 * x) / &2 pow 2
`;;

let lnpix_deriv4 = prove(`
  !x. &60 <= x ==> inv (pi * x) * (&0 * x + &1 * pi) - (&1 * &2 - &0 * x) / &2 pow 2 < &0
  `,
  MESON_TAC [lnpix_deriv3;
    lnpix_deriv2;
    PI_POS;
    REAL_ARITH `&60 <= x ==> ~(x = &0)`]);;

let lnpix_decreasing = prove(`
  !a b. &60 <= a /\ a <= b ==> (\x. ln(pi*x)-x / &2)(b) <= (\x. ln(pi*x)-x / &2)(a)
  `,
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC deriv_negative4 THEN
  EXISTS_TAC `\x. inv (pi * x) * (&0 * x + &1 * pi) - (&1 * &2 - &0 * x) / &2 pow 2` THEN
  REWRITE_TAC [BETA_THM] THEN
  CONJ_TAC THEN
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (REAL_ARITH `~(&2 = &0)`) THEN
  ASSUME_TAC (pi60bound4) THEN
  ASSUME_TAC (REAL_ARITH `&1 < pi * x ==> &0 < pi * x`) THEN
  ASSUME_TAC (REAL_ARITH `&60 <= a /\ a <= x ==> &60 <= x`) THENL
  [ASM_REWRITE_TAC [];
    ASSUME_TAC (ISPECL [`x:real`] lnpix_deriv) THEN
    ASM_MESON_TAC [];
    ASSUME_TAC (ISPECL [`x:real`] lnpix_deriv4) THEN
    ASM_MESON_TAC []]);;

let lnpix_decreasing2 = prove(`
  !a b. &60 <= a /\ a <= b ==> ln(pi*b)-b / &2 <= ln(pi*a)-a / &2
  `,
  MESON_TAC [lnpix_decreasing;
    BETA_THM]);;

let lnpix_decreasing3 = prove(`
  !b. &60 <= b ==> ln(pi*b)-b / &2 <= ln(pi* &60)- &60 / &2
  `,
  MESON_TAC [lnpix_decreasing2;
    REAL_ARITH `&60 <= &60`]);;

let lnpix_negative = prove(`
  !b. &60 <= b ==> ln(pi*b)-b / &2 < &0
  `,
  MESON_TAC [lnpix_decreasing3;
    lnpi60;
    REAL_LET_TRANS]);;

let lnpix_negative2 = prove(`
  !x. &60 <= x ==> (ln(pi*x) - x / &2) / x pow 2 < &0
  `,
  MESON_TAC [neg_over_pos;
    x2_positive;
    lnpix_negative]);;

let lnpix_negative3 = prove(`
  !x. &60 <= x ==> ln(pi*x) / x pow 2 - &1 / (&2 * x) < &0
  `,
  MESON_TAC [lnpix_negative2;
    REAL_FIELD `&60 <= x ==> (ln(pi*x) - x / &2) / x pow 2 = ln(pi*x) / x pow 2 - &1 / (&2 * x)`]);;

let overx2_positive = prove(`
  !x. &60 <= x ==> &0 < inv(x pow 2)
  `,
  MESON_TAC [x2_positive;
    REAL_LT_INV;
    REAL_ARITH `&60 <= x ==> &0 < x`]);;

let nkdelta_deriv_negative = prove(`
  !n k x. &60 <= x /\ &1 < n /\ &0 < k ==> 
  (n + k) * (
  (inv x * &1 + ((inv (pi * x) * (&0 * x + &1 * pi)) * x - &1 * ln (pi * x)) / x pow 2 - &0)
  * &2 * (x - &1) - (&0 * (x - &1) + (&1 - &0) * &2) * (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1)))) / ((&2 * (x - &1)) pow 2)
  < &0
  `,
  MESON_TAC [delta_deriv_negative;
    REAL_ARITH `&1 < n /\ &0 < k ==> &0 < n+k`;
    pos_times_neg]);;

let rho_deriv_negative = prove(`
  !n k x. &60 <= x /\ &1 < n /\ &0 < k ==> (
  (ln(pi*x) / x pow 2 - &1 / (&2 * x))
  - inv(x pow 2)
  + (n + k) * (
  (inv x * &1 + ((inv (pi * x) * (&0 * x + &1 * pi)) * x - &1 * ln (pi * x)) / x pow 2 - &0)
  * &2 * (x - &1) - (&0 * (x - &1) + (&1 - &0) * &2) * (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1)))) / ((&2 * (x - &1)) pow 2)
  )
  < &0
  `,
  MESON_TAC [lnpix_negative3;
    overx2_positive;
    nkdelta_deriv_negative;
    REAL_ARITH `a < &0 /\ &0 < b /\ c < &0 ==> a - b + c < &0`]);;

let rho_deriv_mess = REAL_FIELD `
  !n k x. ~(x = &0) /\ ~(n + k + &1 = &0) /\ &0 < pi /\ ~(x - &1 = &0) ==>
  (ln(pi*x) / x pow 2 - &1 / (&2 * x))
  - inv(x pow 2)
  + (n + k) * (
  (inv x * &1 + ((inv (pi * x) * (&0 * x + &1 * pi)) * x - &1 * ln (pi * x)) / x pow 2 - &0)
  * &2 * (x - &1) - (&0 * (x - &1) + (&1 - &0) * &2) * (ln x + ln (pi * x) / x - ln (&2 * pi * exp (&1)))) / ((&2 * (x - &1)) pow 2)
  = &0 -
              ((&0 * ln ((n + k + &1) / x) +
                (inv ((n + k + &1) / x) *
                 (&0 * x - &1 * (n + k + &1)) / x pow 2) *
                &1 / &2) +
               ((inv x * &1 +
                 ((&0 * x - &1 * &1) / x pow 2 * ln (pi * x) +
                  (inv (pi * x) * (&0 * x + &1 * pi)) * &1 / x) -
                 &0) *
                (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1)) +
                (((&0 * x + &1 * &2) - &0 - &0) * &2 * (x - &1) -
                 (&0 * (x - &1) + (&1 - &0) * &2) *
                 (&2 * x - (n + k + &1) - &1)) /
                (&2 * (x - &1)) pow 2 *
                (ln x + &1 / x * ln (pi * x) - ln (&2 * pi * exp (&1)))) +
               &0)
`;;

let rho_deriv_negative2 = prove(`
  !n k x. &60 <= x /\ &1 < n /\ &0 < k ==>
  &0 -
              ((&0 * ln ((n + k + &1) / x) +
                (inv ((n + k + &1) / x) *
                 (&0 * x - &1 * (n + k + &1)) / x pow 2) *
                &1 / &2) +
               ((inv x * &1 +
                 ((&0 * x - &1 * &1) / x pow 2 * ln (pi * x) +
                  (inv (pi * x) * (&0 * x + &1 * pi)) * &1 / x) -
                 &0) *
                (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1)) +
                (((&0 * x + &1 * &2) - &0 - &0) * &2 * (x - &1) -
                 (&0 * (x - &1) + (&1 - &0) * &2) *
                 (&2 * x - (n + k + &1) - &1)) /
                (&2 * (x - &1)) pow 2 *
                (ln x + &1 / x * ln (pi * x) - ln (&2 * pi * exp (&1)))) +
               &0) < &0
  `,
  MESON_TAC [rho_deriv_mess;
    rho_deriv_negative;
    REAL_ARITH `&60 <= x ==> ~(x = &0)`;
    REAL_ARITH `&1 < n /\ &0 < k ==> ~(n + k + &1 = &0)`;
    PI_POS;
    REAL_ARITH `&60 <= x ==> ~(x - &1 = &0)`]);;

let rho_decreasing = prove(`
  !n k a b. &60 <= a /\ a <= b /\ &1 < n /\ &0 < k ==>
  (\x. &1 / &2 * ln ((n + k) * s pow 2 + &1) -
                   (&1 / &2 * ln ((n + k + &1) / x) +
                    (ln x + &1 / x * ln (pi * x) - ln (&2 * pi * exp (&1))) *
                    (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1)) +
                    k / (n + k + &1) * ln q))
  (b) <=
  (\x. &1 / &2 * ln ((n + k) * s pow 2 + &1) -
                   (&1 / &2 * ln ((n + k + &1) / x) +
                    (ln x + &1 / x * ln (pi * x) - ln (&2 * pi * exp (&1))) *
                    (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1)) +
                    k / (n + k + &1) * ln q))
  (a)
  `,
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC deriv_negative4 THEN
  EXISTS_TAC `\x.
     &0 -
              ((&0 * ln ((n + k + &1) / x) +
                (inv ((n + k + &1) / x) *
                 (&0 * x - &1 * (n + k + &1)) / x pow 2) *
                &1 / &2) +
               ((inv x * &1 +
                 ((&0 * x - &1 * &1) / x pow 2 * ln (pi * x) +
                  (inv (pi * x) * (&0 * x + &1 * pi)) * &1 / x) -
                 &0) *
                (&2 * x - (n + k + &1) - &1) / (&2 * (x - &1)) +
                (((&0 * x + &1 * &2) - &0 - &0) * &2 * (x - &1) -
                 (&0 * (x - &1) + (&1 - &0) * &2) *
                 (&2 * x - (n + k + &1) - &1)) /
                (&2 * (x - &1)) pow 2 *
                (ln x + &1 / x * ln (pi * x) - ln (&2 * pi * exp (&1)))) +
               &0)` THEN
  REWRITE_TAC [BETA_THM] THEN
  CONJ_TAC THEN
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (REAL_ARITH `&60 <= a /\ a <= x ==> &60 <= x`) THENL
  [ASM_REWRITE_TAC [];
    ASSUME_TAC (ISPECL [`n:real`;
        `q:real`;
        `s:real`;
        `k:real`;
        `x:real`] rho_deriv2) THEN
    ASM_MESON_TAC [];
    ASSUME_TAC (ISPECL [`n:real`;
        `k:real`;
        `x:real`] rho_deriv_negative2) THEN
    ASM_MESON_TAC []]);;

let rho_decreasing2 = prove(`
  !n k a b. &60 <= a /\ a <= b /\ &1 < n /\ &0 < k ==>
  (&1 / &2 * ln ((n + k) * s pow 2 + &1) -
                   (&1 / &2 * ln ((n + k + &1) / b) +
                    (ln b + &1 / b * ln (pi * b) - ln (&2 * pi * exp (&1))) *
                    (&2 * b - (n + k + &1) - &1) / (&2 * (b - &1)) +
                    k / (n + k + &1) * ln q))
  <=
  (&1 / &2 * ln ((n + k) * s pow 2 + &1) -
                   (&1 / &2 * ln ((n + k + &1) / a) +
                    (ln a + &1 / a * ln (pi * a) - ln (&2 * pi * exp (&1))) *
                    (&2 * a - (n + k + &1) - &1) / (&2 * (a - &1)) +
                    k / (n + k + &1) * ln q))
  `,
  MESON_TAC [rho_decreasing]);;

let rho_decreasing3 = prove(`
  !n k a b. &60 <= a /\ a <= b /\ &1 < n /\ &0 < k ==>
  ln(standardratio n q s k b) <= ln(standardratio n q s k a)
  `,
  MESON_TAC [rho_decreasing2;
    standardratio_ln4;
    REAL_ARITH `&60 <= a /\ a <= b ==> &60 <= b`]);;

let rho_decreasing4 = prove(`
  !n k a b. &60 <= a /\ a <= b /\ &1 < n /\ &0 < k ==>
  standardratio n q s k b <= standardratio n q s k a
  `,
  MESON_TAC [rho_decreasing3;
    LN_MONO_LE;
    standardratio_positive]);;

let sqrtprod_nonneg = prove(`
  !x y. &0 <= (sqrt x - sqrt y) pow 2
  `,
  MESON_TAC [REAL_LE_POW_2]);;

let sqrtprod_pow2 = REAL_ARITH `
  !x y. (sqrt x - sqrt y) pow 2 = (sqrt x) pow 2 + (sqrt y) pow 2 - &2 * sqrt x * sqrt y
`;;

let sqrtprod_pow2_simplify = prove(`
  !x y. &0 <= x /\ &0 <= y ==> (sqrt x - sqrt y) pow 2 = x + y - &2 * sqrt(x*y)
  `,
  MESON_TAC [sqrtprod_pow2;
    SQRT_POW_2;
    SQRT_MUL]);;

let sqrtprod_nonneg2 = prove(`
  !x y. &0 <= x /\ &0 <= y ==> &0 <= x + y - &2 * sqrt(x*y)
  `,
  MESON_TAC [sqrtprod_nonneg;
    sqrtprod_pow2_simplify]);;

let sqrtprod_le = prove(`
  !x y. &0 <= x /\ &0 <= y ==> &2 * sqrt(x*y) <= x + y
  `,
  MESON_TAC [sqrtprod_nonneg2;
    REAL_ARITH `!a b c. &0 <= a + b - c ==> c <= a + b`]);;

let kappa_main_sqrt_le = prove(`
  !n k D Q. ~(n + k + &1 = &0) /\ &0 <= D*(n + k + &1) /\ &0 <= Q*(n + &1)/(n + k + &1) ==> &2 * sqrt((n + &1)*D*Q) <= D*(n + k + &1) + Q*(n + &1)/(n + k + &1)
  `,
  MESON_TAC [sqrtprod_le;
    REAL_FIELD `!n k D Q. ~(n + k + &1 = &0) ==> (D*(n + k + &1))*(Q*(n + &1)/(n + k + &1)) = (n + &1)*D*Q`]);;

let kappa_main_reshuffle = REAL_FIELD `
  !n k b D Q. ~(n + k + &1 = &0) /\ &2 * sqrt((n + &1)*D*Q) <= D*(n + k + &1) + Q*(n + &1)/(n + k + &1) ==> (&2 * b - n - k - &2)*D + (k / (n + k + &1))*Q <= (&2 * b - &1)*D + (Q*(n + &1)/(n + k + &1) + (k / (n + k + &1))*Q) - &2 * sqrt((n + &1)*D*Q)
`;;

let kappa_main_reshuffle2 = prove(`
  !n k b D Q. ~(n + k + &1 = &0) /\ &2 * sqrt((n + &1)*D*Q) <= D*(n + k + &1) + Q*(n + &1)/(n + k + &1) ==> (&2 * b - n - k - &2)*D + (k / (n + k + &1))*Q <= (&2 * b - &1)*D + Q - &2 * sqrt((n + &1)*D*Q)
  `,
  REPEAT STRIP_TAC THEN
  ASSUME_TAC (REAL_FIELD `~(n + k + &1 = &0) ==> Q*(n + &1)/(n + k + &1) + (k / (n + k + &1))*Q = Q`) THEN
  ASM_MESON_TAC [kappa_main_reshuffle]);;

let kappa_main_reshuffle3 = prove(`
  !n k b D Q. ~(n + k + &1 = &0) /\ &0 <= D*(n + k + &1) /\ &0 <= Q*(n + &1)/(n + k + &1) ==> (&2 * b - n - k - &2)*D + (k / (n + k + &1))*Q <= (&2 * b - &1)*D + Q - &2 * sqrt((n + &1)*D*Q)
  `,
  MESON_TAC [kappa_main_sqrt_le;
    kappa_main_reshuffle2]);;

let kappa_n1pos = REAL_ARITH `
  !n. &1 < n ==> &0 < n + &1
`;;

let kappa_nk1nonzero = REAL_ARITH `
  !n k. &1 < n /\ &0 < k ==> ~(n + k + &1 = &0)
`;;

let kappa_nk1pos = REAL_ARITH `
  !n k. &1 < n /\ &0 < k ==> &0 < n + k + &1
`;;

let kappa_Dnk1pos = prove(`
  !n k D. &1 < n /\ &0 < k /\ &0 <= D ==> &0 <= D*(n + k + &1)
  `,
  MESON_TAC [kappa_nk1pos;
    REAL_LE_MUL_EQ]);;

let kappa_nk1ratiopos = prove(`
  !n k. &1 < n /\ &0 < k ==> &0 < (n + &1)/(n + k + &1)
  `,
  MESON_TAC [kappa_n1pos;
    kappa_nk1pos;
    REAL_LT_DIV]);;

let kappa_Qnk1ratiopos = prove(`
  !n k Q. &1 < n /\ &0 < k /\ &0 < Q ==> &0 <= Q*(n + &1)/(n + k + &1)
  `,
  MESON_TAC [kappa_nk1ratiopos;
    REAL_LT_MUL;
    REAL_LT_IMP_LE]);;

let kappa_main_reshuffle4 = prove(`
  !n k b D Q. &1 < n /\ &0 < Q /\ &0 <= D /\ &0 < k ==> (&2 * b - n - k - &2)*D + (k / (n + k + &1))*Q <= (&2 * b - &1)*D + Q - &2 * sqrt((n + &1)*D*Q)
  `,
  MESON_TAC [kappa_nk1nonzero;
    kappa_Dnk1pos;
    kappa_Qnk1ratiopos;
    kappa_main_reshuffle3]);;

let kappa_main = prove(`
  !n k b d q. &1 < n /\ &1 < q /\ &1 <= d /\ &0 < k ==> (&2 * b - n - k - &2)*log2 d + (k / (n + k + &1))*log2 q <= (&2 * b - &1)*log2 d + log2 q - &2 * sqrt((n + &1)*log2 d*log2 q)
  `,
  MESON_TAC [kappa_main_reshuffle4;
    log2_gt1;
    log2_ge1]);;

let asymp2_2B1D_lemma = prove(`
  !N:num->real
   B:num->real
   Y0:real
   Y1:real
   twopie:real.
  (!i. &1 < N(i))
  /\ (!i. &1 < B(i))
  /\ o1_seq (\i. &1 / N(i))
  /\ o1_seq (\i:num. ((((&1)/(N(i))) - (&0)) * log2(N(i))) - (&0))
  /\ o1_seq (\i:num. ((((B(i))/(N(i))) - (Y0)) * log2(N(i))) - (Y1))
  /\ &0 < ((&2)*(Y0))-((&2)*(&0))
  /\ (!i:num. &0 < ((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i)))))
  /\ ~(((&2)*(Y0))-((&2)*(&0)) = &0)
  /\ &0 < Y0
  /\ (!i:num. &0 < (B(i))/(N(i)))
  /\ ~(Y0 = &0)
  /\ &0 < (pi)*(Y0)
  /\ &0 < twopie
  ==> o1_seq (\i:num. (((((((&2)*((B(i))/(N(i))))-((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))))))) - (((((&2)*(Y0))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0)))))) * log2(N(i))) - ((((((&2)*(Y0))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie))))))+((((((&2)*(Y0))-(&0))*((--((((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0)))))/((((&2)*(Y0))-((&2)*(&0)))*(((&2)*(Y0))-((&2)*(&0))))))+(((((&2)*(Y1))+((&0)*(Y0)))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0))))))*((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0)))))))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. N(i):real`;
    `&2:real`] asymp2_const THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &2:real`;
    `&2:real`;
    `&0:real`;
    `\i:num. (B(i))/(N(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (&2)*((B(i))/(N(i))):real`;
    `(&2)*(Y0):real`;
    `((&2)*(Y1))+((&0)*(Y0)):real`;
    `\i:num. (&1)/(N(i)):real`;
    `&0:real`;
    `&0:real`] asymp2_sub THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &2:real`;
    `&2:real`;
    `&0:real`;
    `\i:num. (&1)/(N(i)):real`;
    `&0:real`;
    `&0:real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (&2)*((B(i))/(N(i))):real`;
    `(&2)*(Y0):real`;
    `((&2)*(Y1))+((&0)*(Y0)):real`;
    `\i:num. (&2)*((&1)/(N(i))):real`;
    `(&2)*(&0):real`;
    `((&2)*(&0))+((&0)*(&0)):real`] asymp2_sub THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. ((&2)*((B(i))/(N(i))))-((&1)/(N(i))):real`;
    `((&2)*(Y0))-(&0):real`;
    `(((&2)*(Y1))+((&0)*(Y0)))-(&0):real`;
    `\i:num. ((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i)))):real`;
    `((&2)*(Y0))-((&2)*(&0)):real`;
    `(((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0))):real`] asymp2_div THEN
  specialize [`\i:num. N(i):real`;
    `&1:real`] asymp2_const THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (B(i))/(N(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_logn THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. (B(i))/(N(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_div THEN
  specialize [`\i:num. N(i):real`;
    `pi:real`] asymp2_const THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. pi:real`;
    `pi:real`;
    `&0:real`;
    `\i:num. (B(i))/(N(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (pi)*((B(i))/(N(i))):real`;
    `(pi)*(Y0):real`;
    `((pi)*(Y1))+((&0)*(Y0)):real`] asymp2_logn THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. log2((pi)*((B(i))/(N(i))))/log2(N(i)):real`;
    `&0:real`;
    `log2((pi)*(Y0)):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`] asymp2_add THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (&1)/((B(i))/(N(i))):real`;
    `(&1)*((&1)/(Y0)):real`;
    `((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))):real`;
    `\i:num. (log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1):real`;
    `(&0)+(&1):real`;
    `(log2((pi)*(Y0)))+(&0):real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (&1)/(N(i)):real`;
    `&0:real`;
    `&0:real`;
    `\i:num. ((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1)):real`;
    `((&1)*((&1)/(Y0)))*((&0)+(&1)):real`;
    `(((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1))):real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `twopie:real`] asymp2_const THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. twopie:real`;
    `twopie:real`;
    `&0:real`] asymp2_logn THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. ((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))):real`;
    `(&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))):real`;
    `((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))):real`;
    `\i:num. log2(twopie)/log2(N(i)):real`;
    `&0:real`;
    `log2(twopie):real`] asymp2_sub THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. log2((B(i))/(N(i)))/log2(N(i)):real`;
    `&0:real`;
    `log2(Y0):real`;
    `\i:num. (((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))):real`;
    `((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0):real`;
    `(((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)):real`] asymp2_add THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. (log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i)))):real`;
    `(&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0)):real`;
    `(log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie))):real`] asymp2_add THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (((&2)*((B(i))/(N(i))))-((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))):real`;
    `(((&2)*(Y0))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))):real`;
    `((((&2)*(Y0))-(&0))*((--((((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0)))))/((((&2)*(Y0))-((&2)*(&0)))*(((&2)*(Y0))-((&2)*(&0))))))+(((((&2)*(Y1))+((&0)*(Y0)))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0))))):real`;
    `\i:num. (&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))))):real`;
    `(&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))):real`;
    `(&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)))):real`] asymp2_mul THEN
  ASM_MESON_TAC []);;

let asymp2_2B1D_simpler = prove(`
  !N:num->real
   B:num->real
   Y0:real
   Y1:real
   twopie:real.
  (!i. &1 < N(i))
  /\ o1_seq (\i. &1 / N(i))
  /\ (!i. &1 < B(i))
  /\ o1_seq (\i:num. (B(i)/N(i) - Y0) * log2(N(i)) - Y1)
  /\ &0 < Y0
  /\ &0 < twopie
  ==> o1_seq (\i:num. (((((((&2)*((B(i))/(N(i))))-((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))))))) - &1) * log2(N(i))) - (log2(Y0) - log2(twopie)))
  `,
  REPEAT STRIP_TAC THEN
  note `&0 < pi*Y0` [PI_POS;
    REAL_LT_MUL] THEN
  real_linear `&0 < Y0 ==> ~(Y0 = &0)` THEN
  real_linear `&0 < Y0 ==> ~(((&2)*(Y0))-((&2)*(&0)) = &0)` THEN
  real_linear `&0 < Y0 ==> &0 < ((&2)*(Y0))-((&2)*(&0))` THEN
  note `!i:num. &1 <= N(i)` [REAL_LT_IMP_LE] THEN
  note `!i:num. &0 < N(i)` [REAL_ARITH `&1 < n ==> &0 < n`] THEN
  note `!i:num. &0 < B(i)` [REAL_ARITH `&1 < n ==> &0 < n`] THEN
  note `!i:num. &0 < B(i)/N(i)` [REAL_LT_DIV] THEN
  note `!i:num. &0 < &2*(B(i) - &1)` [REAL_ARITH `&1 < b ==> &0 < &2 * (b - &1)`] THEN
  note `!i:num. &0 < (&2*(B(i) - &1))/N(i)` [REAL_LT_DIV] THEN
  note `!i:num. &0 < (&2)*(B(i)/N(i))-(&2)*((&1)/N(i))` [REAL_FIELD `&1 < n ==> (&2*(b - &1))/n = (&2)*(b/n) - (&2)*(&1/n)`] THEN
  note `o1_seq (\i:num. log2(N(i))/N(i))` [o1_log2overlinear_1] THEN
  notetac `(\i:num. log2(N(i))/N(i)) = (\i:num. (&1/N(i) - &0) * log2(N(i)) - &0)` (REWRITE_TAC [EQ_EXT;
      REAL_ARITH `x - &0 = x`;
      REAL_ARITH `(&1/x)*y = y/x`]) THEN
  note `o1_seq (\i:num. (&1/N(i) - &0) * log2(N(i)) - &0)` [] THEN
  specialize [`N:num->real`;
    `B:num->real`;
    `Y0:real`;
    `Y1:real`;
    `twopie:real`] asymp2_2B1D_lemma THEN
  real_cancel `&0 < Y0 ==> log2(Y0) - log2(twopie) = ((((((&2)*(Y0))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie))))))+((((((&2)*(Y0))-(&0))*((--((((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0)))))/((((&2)*(Y0))-((&2)*(&0)))*(((&2)*(Y0))-((&2)*(&0))))))+(((((&2)*(Y1))+((&0)*(Y0)))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0))))))*((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))))` THEN
  notetac `(\i:num. (((((((&2)*((B(i))/(N(i))))-((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))))))) - (((((&2)*(Y0))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0)))))) * log2(N(i))) - (log2(Y0) - log2(twopie))) = (\i:num. (((((((&2)*((B(i))/(N(i))))-((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))))))) - (((((&2)*(Y0))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0)))))) * log2(N(i))) - ((((((&2)*(Y0))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie))))))+((((((&2)*(Y0))-(&0))*((--((((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0)))))/((((&2)*(Y0))-((&2)*(&0)))*(((&2)*(Y0))-((&2)*(&0))))))+(((((&2)*(Y1))+((&0)*(Y0)))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0))))))*((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0)))))))` (ASM_MESON_TAC [EQ_EXT]) THEN
  note `o1_seq (\i:num. (((((((&2)*((B(i))/(N(i))))-((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))))))) - (((((&2)*(Y0))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0)))))) * log2(N(i))) - (log2(Y0) - log2(twopie)))` [] THEN
  real_cancel `&0 < Y0 ==> &1 = (((((&2)*(Y0))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0)))))` THEN
  notetac `(\i:num. (((((((&2)*((B(i))/(N(i))))-((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))))))) - &1) * log2(N(i))) - (log2(Y0) - log2(twopie))) = (\i:num. (((((((&2)*((B(i))/(N(i))))-((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))))))) - (((((&2)*(Y0))-(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0)))))) * log2(N(i))) - (log2(Y0) - log2(twopie)))` (ASM_MESON_TAC [EQ_EXT]) THEN
  ASM_MESON_TAC []);;

let asymp2_2B1D_lemma3 = prove(`
  !n:real x:real.
  &1 < n /\ &60 <= x
  ==>
  (&2 * x - &1) * log2(bkzdelta(x)) / log2(n) = (((((&2)*((x)/(n)))-((&1)/(n)))/(((&2)*((x)/(n)))-((&2)*((&1)/(n)))))*((&1)+((log2((x)/(n))/log2(n))+((((&1)/(n))*(((&1)/((x)/(n)))*((log2((pi)*((x)/(n)))/log2(n))+(&1))))-(log2(&2 * pi * exp(&1))/log2(n))))))
  `,
  REPEAT STRIP_TAC THEN
  ASM_SIMP_TAC [log2ratio_lnratio] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> ((&2)*((x)/(n)))-((&1)/(n)) = (&2*x - &1)/n`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> &2 * x / n - &2 * &1 / n = (&2 * (x - &1)) / n`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> (a / n) / (b / n) = a / b`] THEN
  real_linear `&1 < n ==> &0 < n` THEN
  real_linear `&60 <= x ==> &0 < x` THEN
  ASM_SIMP_TAC [ln_div] THEN
  ASM_SIMP_TAC [ln_muldiv;
    PI_POS] THEN
  note `&0 < ln n` [ln_gt1] THEN
  ASM_SIMP_TAC [REAL_FIELD `&0 < ln n ==> (ln pi + ln x - ln n) / ln n + &1 = (ln pi + ln x) / ln n`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&0 < n ==> &1 / n * &1 / (x / n) * y = &1 / x * y`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&0 < ln n ==> &1 + (ln x - ln n) / ln n + b * c / ln n - d / ln n = (ln x + b * c - d) / ln n`] THEN
  ASM_SIMP_TAC [REAL_ARITH `b / c * e / f = b * (e / c) / f`] THEN
  ASM_MESON_TAC [bkzdelta_ln4;
    LN_MUL;
    PI_POS]);;

let asymp2_2B1D_lemma4 = prove(`
  !N:num->real
   B:num->real
   Y0:real.
  (!i. &1 < N(i))
  /\ (!i. &60 <= B(i))
  ==>
  (\i:num. (((&2 * B(i) - &1) * log2(bkzdelta(B(i))) / log2(N(i)) - &1) * log2(N(i))) - (log2(Y0) - log2(&2 * pi * exp(&1))))
  = (\i:num. (((((((&2)*((B(i))/(N(i))))-((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(&2 * pi * exp(&1))/log2(N(i))))))) - &1) * log2(N(i))) - (log2(Y0) - log2(&2 * pi * exp(&1))))
  `,
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC EQ_EXT THEN
  REWRITE_TAC [BETA_THM] THEN
  ASM_MESON_TAC [asymp2_2B1D_lemma3]);;

let asymp2_2B1D = prove(`
  !N:num->real
   B:num->real
   Y0:real
   Y1:real.
  (!i. &1 < N(i))
  /\ o1_seq (\i. &1 / N(i))
  /\ (!i. &60 <= B(i))
  /\ o1_seq (\i:num. (B(i)/N(i) - Y0) * log2(N(i)) - Y1)
  /\ &0 < Y0
  ==> o1_seq (\i:num. (((&2 * B(i) - &1) * log2(bkzdelta(B(i))) / log2(N(i)) - &1) * log2(N(i))) - (log2(Y0) - log2(&2 * pi * exp(&1))))
  `,
  REPEAT STRIP_TAC THEN
  note `&0 < &2 * pi * exp(&1)` [twopie_pos] THEN
  note `!i:num. &1 < B(i)` [REAL_ARITH `&60 <= b ==> &1 < b`] THEN
  note `(\i:num. (((&2 * B(i) - &1) * log2(bkzdelta(B(i))) / log2(N(i)) - &1) * log2(N(i))) - (log2(Y0) - log2(&2 * pi * exp(&1)))) = (\i:num. (((((((&2)*((B(i))/(N(i))))-((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(&2 * pi * exp(&1))/log2(N(i))))))) - &1) * log2(N(i))) - (log2(Y0) - log2(&2 * pi * exp(&1))))` [asymp2_2B1D_lemma4] THEN
  specialize [`N:num->real`;
    `B:num->real`;
    `Y0:real`;
    `Y1:real`;
    `&2 * pi * exp(&1)`] asymp2_2B1D_simpler THEN
  ASM_MESON_TAC []);;

let asymp2_ndq_lemma = prove(`
  !N:num->real
   Q:num->real
   B:num->real
   Q0:real
   Q1:real
   Y0:real
   Y1:real
   twopie:real.
  (!i. &1 < N(i))
  /\ (!i. &1 < Q(i))
  /\ (!i. &1 < B(i))
  /\ o1_seq (\i. &1 / N(i))
  /\ o1_seq (\i:num. ((((&1)/(N(i))) - (&0)) * log2(N(i))) - (&0))
  /\ o1_seq (\i:num. (((log2(Q(i))/log2(N(i))) - (Q0)) * log2(N(i))) - (Q1))
  /\ o1_seq (\i:num. ((((B(i))/(N(i))) - (Y0)) * log2(N(i))) - (Y1))
  /\ &0 < ((&2)*(Y0))-((&2)*(&0))
  /\ (!i:num. &0 < ((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i)))))
  /\ ~(((&2)*(Y0))-((&2)*(&0)) = &0)
  /\ &0 < Y0
  /\ (!i:num. &0 < (B(i))/(N(i)))
  /\ ~(Y0 = &0)
  /\ &0 < (pi)*(Y0)
  /\ &0 < twopie
  ==> o1_seq (\i:num. ((((((&1)+((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*(((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))))))*(log2(Q(i))/log2(N(i))))) - ((((&1)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q0)))) * log2(N(i))) - (((((&1)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q1))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)))))*(Q0))))+(((((&1)+(&0))*((--((((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0)))))/((((&2)*(Y0))-((&2)*(&0)))*(((&2)*(Y0))-((&2)*(&0))))))+(((&0)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0))))))*(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q0)))))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. N(i):real`;
    `&1:real`] asymp2_const THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. (&1)/(N(i)):real`;
    `&0:real`;
    `&0:real`] asymp2_add THEN
  specialize [`\i:num. N(i):real`;
    `&2:real`] asymp2_const THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &2:real`;
    `&2:real`;
    `&0:real`;
    `\i:num. (B(i))/(N(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &2:real`;
    `&2:real`;
    `&0:real`;
    `\i:num. (&1)/(N(i)):real`;
    `&0:real`;
    `&0:real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (&2)*((B(i))/(N(i))):real`;
    `(&2)*(Y0):real`;
    `((&2)*(Y1))+((&0)*(Y0)):real`;
    `\i:num. (&2)*((&1)/(N(i))):real`;
    `(&2)*(&0):real`;
    `((&2)*(&0))+((&0)*(&0)):real`] asymp2_sub THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (&1)+((&1)/(N(i))):real`;
    `(&1)+(&0):real`;
    `(&0)+(&0):real`;
    `\i:num. ((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i)))):real`;
    `((&2)*(Y0))-((&2)*(&0)):real`;
    `(((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0))):real`] asymp2_div THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (B(i))/(N(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_logn THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. (B(i))/(N(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_div THEN
  specialize [`\i:num. N(i):real`;
    `pi:real`] asymp2_const THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. pi:real`;
    `pi:real`;
    `&0:real`;
    `\i:num. (B(i))/(N(i)):real`;
    `Y0:real`;
    `Y1:real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (pi)*((B(i))/(N(i))):real`;
    `(pi)*(Y0):real`;
    `((pi)*(Y1))+((&0)*(Y0)):real`] asymp2_logn THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. log2((pi)*((B(i))/(N(i))))/log2(N(i)):real`;
    `&0:real`;
    `log2((pi)*(Y0)):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`] asymp2_add THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (&1)/((B(i))/(N(i))):real`;
    `(&1)*((&1)/(Y0)):real`;
    `((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))):real`;
    `\i:num. (log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1):real`;
    `(&0)+(&1):real`;
    `(log2((pi)*(Y0)))+(&0):real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (&1)/(N(i)):real`;
    `&0:real`;
    `&0:real`;
    `\i:num. ((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1)):real`;
    `((&1)*((&1)/(Y0)))*((&0)+(&1)):real`;
    `(((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1))):real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `twopie:real`] asymp2_const THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. twopie:real`;
    `twopie:real`;
    `&0:real`] asymp2_logn THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. ((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))):real`;
    `(&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))):real`;
    `((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))):real`;
    `\i:num. log2(twopie)/log2(N(i)):real`;
    `&0:real`;
    `log2(twopie):real`] asymp2_sub THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. log2((B(i))/(N(i)))/log2(N(i)):real`;
    `&0:real`;
    `log2(Y0):real`;
    `\i:num. (((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))):real`;
    `((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0):real`;
    `(((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)):real`] asymp2_add THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. (log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i)))):real`;
    `(&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0)):real`;
    `(log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie))):real`] asymp2_add THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))))):real`;
    `(&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))):real`;
    `(&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)))):real`;
    `\i:num. log2(Q(i))/log2(N(i)):real`;
    `Q0:real`;
    `Q1:real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. ((&1)+((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))):real`;
    `((&1)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))):real`;
    `(((&1)+(&0))*((--((((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0)))))/((((&2)*(Y0))-((&2)*(&0)))*(((&2)*(Y0))-((&2)*(&0))))))+(((&0)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0))))):real`;
    `\i:num. ((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(twopie)/log2(N(i))))))*(log2(Q(i))/log2(N(i))):real`;
    `((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q0):real`;
    `(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q1))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(twopie)))))*(Q0)):real`] asymp2_mul THEN
  ASM_MESON_TAC []);;

let asymp2_ndq_lemma2 = prove(`
  !N:num->real Q:num->real B:num->real
   Q0:real Q1:real Y0:real Y1:real.
  (!i. &1 < N(i))
  /\ (!i. &1 < Q(i))
  /\ (!i. &1 < B(i))
  /\ o1_seq (\i. &1 / N(i))
  /\ o1_seq (\i:num. (log2(Q(i))/log2(N(i)) - Q0) * log2(N(i)) - Q1)
  /\ o1_seq (\i:num. (B(i)/N(i) - Y0) * log2(N(i)) - Y1)
  /\ &0 < Y0
  ==> o1_seq (\i:num. ((((((&1)+((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*(((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(&2 * pi * exp(&1))/log2(N(i))))))*(log2(Q(i))/log2(N(i))))) - ((((&1)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q0)))) * log2(N(i))) - (((((&1)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q1))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(&2 * pi * exp(&1))))))*(Q0))))+(((((&1)+(&0))*((--((((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0)))))/((((&2)*(Y0))-((&2)*(&0)))*(((&2)*(Y0))-((&2)*(&0))))))+(((&0)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0))))))*(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q0)))))
  `,
  REPEAT STRIP_TAC THEN
  note `&0 < pi*Y0` [PI_POS;
    REAL_LT_MUL] THEN
  note `&0 < &2 * pi * exp(&1)` [twopie_pos] THEN
  real_linear `&0 < Y0 ==> ~(Y0 = &0)` THEN
  real_linear `&0 < Y0 ==> ~(((&2)*(Y0))-((&2)*(&0)) = &0)` THEN
  real_linear `&0 < Y0 ==> &0 < ((&2)*(Y0))-((&2)*(&0))` THEN
  note `!i:num. &1 <= N(i)` [REAL_LT_IMP_LE] THEN
  note `o1_seq (\i:num. log2(N(i))/N(i))` [o1_log2overlinear_1] THEN
  notetac `(\i:num. log2(N(i))/N(i)) = (\i:num. (&1/N(i) - &0) * log2(N(i)) - &0)` (REWRITE_TAC [EQ_EXT;
      REAL_ARITH `x - &0 = x`;
      REAL_ARITH `(&1/x)*y = y/x`]) THEN
  note `o1_seq (\i:num. (&1/N(i) - &0) * log2(N(i)) - &0)` [] THEN
  note `!i:num. &0 < N(i)` [REAL_ARITH `&1 < n ==> &0 < n`] THEN
  note `!i:num. &0 < B(i)` [REAL_ARITH `&1 < n ==> &0 < n`] THEN
  note `!i:num. &0 < B(i)/N(i)` [REAL_LT_DIV] THEN
  note `!i:num. &0 < &2*(B(i) - &1)` [REAL_ARITH `&1 < b ==> &0 < &2 * (b - &1)`] THEN
  note `!i:num. &0 < (&2*(B(i) - &1))/N(i)` [REAL_LT_DIV] THEN
  note `!i:num. &0 < (&2)*(B(i)/N(i))-(&2)*((&1)/N(i))` [REAL_FIELD `&1 < n ==> (&2*(b - &1))/n = (&2)*(b/n) - (&2)*(&1/n)`] THEN
  specialize [`N:num->real`;
    `Q:num->real`;
    `B:num->real`;
    `Q0:real`;
    `Q1:real`;
    `Y0:real`;
    `Y1:real`;
    `&2 * pi * exp(&1)`] asymp2_ndq_lemma THEN
  ASM_MESON_TAC []);;

let asymp2_ndq_lemma3 = prove(`
  !N:num->real Q:num->real B:num->real
   Q0:real Q1:real Y0:real Y1:real.
  (!i. &1 < N(i))
  /\ (!i. &1 < Q(i))
  /\ (!i. &1 < B(i))
  /\ o1_seq (\i. &1 / N(i))
  /\ o1_seq (\i:num. (log2(Q(i))/log2(N(i)) - Q0) * log2(N(i)) - Q1)
  /\ o1_seq (\i:num. (B(i)/N(i) - Y0) * log2(N(i)) - Y1)
  /\ &0 < Y0
  ==> o1_seq (\i:num. ((((((&1)+((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*(((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(&2 * pi * exp(&1))/log2(N(i))))))*(log2(Q(i))/log2(N(i))))) - ((((&1)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q0)))) * log2(N(i))) - (((((&1)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q1))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(&2 * pi * exp(&1))))))*(Q0))))+(((((&1)+(&0))*((--((((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0)))))/((((&2)*(Y0))-((&2)*(&0)))*(((&2)*(Y0))-((&2)*(&0))))))+(((&0)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0))))))*(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q0)))))
  `,
  REPEAT STRIP_TAC THEN
  note `&0 < pi*Y0` [PI_POS;
    REAL_LT_MUL] THEN
  note `&0 < &2 * pi * exp(&1)` [twopie_pos] THEN
  real_linear `&0 < Y0 ==> ~(Y0 = &0)` THEN
  real_linear `&0 < Y0 ==> ~(((&2)*(Y0))-((&2)*(&0)) = &0)` THEN
  real_linear `&0 < Y0 ==> &0 < ((&2)*(Y0))-((&2)*(&0))` THEN
  note `!i:num. &1 <= N(i)` [REAL_LT_IMP_LE] THEN
  note `o1_seq (\i:num. log2(N(i))/N(i))` [o1_log2overlinear_1] THEN
  notetac `(\i:num. log2(N(i))/N(i)) = (\i:num. (&1/N(i) - &0) * log2(N(i)) - &0)` (REWRITE_TAC [EQ_EXT;
      REAL_ARITH `x - &0 = x`;
      REAL_ARITH `(&1/x)*y = y/x`]) THEN
  note `o1_seq (\i:num. (&1/N(i) - &0) * log2(N(i)) - &0)` [] THEN
  note `!i:num. &0 < N(i)` [REAL_ARITH `&1 < n ==> &0 < n`] THEN
  note `!i:num. &0 < B(i)` [REAL_ARITH `&1 < n ==> &0 < n`] THEN
  note `!i:num. &0 < B(i)/N(i)` [REAL_LT_DIV] THEN
  note `!i:num. &0 < &2*(B(i) - &1)` [REAL_ARITH `&1 < b ==> &0 < &2 * (b - &1)`] THEN
  note `!i:num. &0 < (&2*(B(i) - &1))/N(i)` [REAL_LT_DIV] THEN
  note `!i:num. &0 < (&2)*(B(i)/N(i))-(&2)*((&1)/N(i))` [REAL_FIELD `&1 < n ==> (&2*(b - &1))/n = (&2)*(b/n) - (&2)*(&1/n)`] THEN
  specialize [`N:num->real`;
    `Q:num->real`;
    `B:num->real`;
    `Q0:real`;
    `Q1:real`;
    `Y0:real`;
    `Y1:real`;
    `&2 * pi * exp(&1)`] asymp2_ndq_lemma THEN
  ASM_MESON_TAC []);;

let asymp2_ndq_lemma4 = REAL_FIELD `
  !Q0:real Y0:real.
  &0 < Y0
  ==> 
  ((((&1)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q0)))
  = Q0 / (&2 * Y0)
`;;

let asymp2_ndq_lemma5 = REAL_FIELD `
  !Q0 Q1 Y0 Y1.
  &0 < Y0
  ==>
  (((((&1)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q1))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(&2 * pi * exp(&1))))))*(Q0))))+(((((&1)+(&0))*((--((((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0)))))/((((&2)*(Y0))-((&2)*(&0)))*(((&2)*(Y0))-((&2)*(&0))))))+(((&0)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0))))))*(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q0))))
  =
  ((Q1 + (Q0*log2(Y0) - Q0*log2(&2 * pi * exp(&1))))*Y0 - Q0*Y1)/(&2 * Y0 pow 2)
`;;

let asymp2_ndq_lemma6 = prove(`
  !N:num->real Q:num->real B:num->real
   Q0:real Q1:real Y0:real Y1:real.
  &0 < Y0
  ==>
  (\i:num. ((((((&1)+((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*(((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(&2 * pi * exp(&1))/log2(N(i))))))*(log2(Q(i))/log2(N(i))))) - ((((&1)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q0)))) * log2(N(i))) - (((((&1)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0)))))*((((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q1))+(((&0)+((log2(Y0))+((((&0)*((((&1)*((&1)/(Y0)))*((log2((pi)*(Y0)))+(&0)))+((((&1)*((--(Y1))/((Y0)*(Y0))))+((&0)*((&1)/(Y0))))*((&0)+(&1)))))+((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1)))))-(log2(&2 * pi * exp(&1))))))*(Q0))))+(((((&1)+(&0))*((--((((&2)*(Y1))+((&0)*(Y0)))-(((&2)*(&0))+((&0)*(&0)))))/((((&2)*(Y0))-((&2)*(&0)))*(((&2)*(Y0))-((&2)*(&0))))))+(((&0)+(&0))*((&1)/(((&2)*(Y0))-((&2)*(&0))))))*(((&1)+((&0)+(((&0)*(((&1)*((&1)/(Y0)))*((&0)+(&1))))-(&0))))*(Q0)))))
  =
  (\i:num. ((((((&1)+((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*(((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(&2 * pi * exp(&1))/log2(N(i))))))*(log2(Q(i))/log2(N(i))))) - Q0 / (&2 * Y0)) * log2(N(i))) - ((Q1 + (Q0*log2(Y0) - Q0*log2(&2 * pi * exp(&1))))*Y0 - Q0*Y1)/(&2 * Y0 pow 2))
  `,
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC EQ_EXT THEN
  REWRITE_TAC [BETA_THM] THEN
  ASM_MESON_TAC [asymp2_ndq_lemma4;
    asymp2_ndq_lemma5]);;

let asymp2_ndq_lemma7 = prove(`
  !N:num->real Q:num->real B:num->real
   Q0:real Q1:real Y0:real Y1:real.
  (!i. &1 < N(i))
  /\ (!i. &1 < Q(i))
  /\ (!i. &1 < B(i))
  /\ o1_seq (\i. &1 / N(i))
  /\ o1_seq (\i:num. (log2(Q(i))/log2(N(i)) - Q0) * log2(N(i)) - Q1)
  /\ o1_seq (\i:num. (B(i)/N(i) - Y0) * log2(N(i)) - Y1)
  /\ &0 < Y0
  ==> o1_seq (\i:num. ((((((&1)+((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*(((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(&2 * pi * exp(&1))/log2(N(i))))))*(log2(Q(i))/log2(N(i))))) - Q0 / (&2 * Y0)) * log2(N(i))) - ((Q1 + (Q0*log2(Y0) - Q0*log2(&2 * pi * exp(&1))))*Y0 - Q0*Y1)/(&2 * Y0 pow 2))
  `,
  MESON_TAC [asymp2_ndq_lemma3;
    asymp2_ndq_lemma6]);;

let asymp2_ndq_lemma8 = prove(`
  !n:real q:real x:real.
  &1 < n /\ &1 < q /\ &60 <= x
  ==>
  (n + &1) * log2(bkzdelta(x)) / log2(n) * log2(q) / log2(n) =
  ((((&1)+((&1)/(n)))/(((&2)*((x)/(n)))-((&2)*((&1)/(n)))))*(((&1)+((log2((x)/(n))/log2(n))+((((&1)/(n))*(((&1)/((x)/(n)))*((log2((pi)*((x)/(n)))/log2(n))+(&1))))-(log2(&2 * pi * exp(&1))/log2(n)))))*(log2(q)/log2(n))))
  `,
  REPEAT STRIP_TAC THEN
  ASM_SIMP_TAC [log2ratio_lnratio] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> &1 + &1 / n = (n + &1) / n`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> &2 * x / n - &2 * &1 / n = (&2 * (x - &1)) / n`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> (a / n) / (b / n) = a / b`] THEN
  real_linear `&1 < n ==> &0 < n` THEN
  real_linear `&60 <= x ==> &0 < x` THEN
  ASM_SIMP_TAC [ln_div] THEN
  ASM_SIMP_TAC [ln_muldiv;
    PI_POS] THEN
  note `&0 < ln n` [ln_gt1] THEN
  ASM_SIMP_TAC [REAL_FIELD `&0 < ln n ==> (ln pi + ln x - ln n) / ln n + &1 = (ln pi + ln x) / ln n`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&0 < n ==> &1 / n * &1 / (x / n) * y = &1 / x * y`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&0 < ln n ==> &1 + (ln x - ln n) / ln n + b * c / ln n - d / ln n = (ln x + b * c - d) / ln n`] THEN
  ASM_SIMP_TAC [REAL_ARITH `b / c * e / f * g / h = b * (e / c) / f * g / h`] THEN
  ASM_MESON_TAC [bkzdelta_ln4;
    LN_MUL;
    PI_POS]);;

let asymp2_ndq_lemma9 = prove(`
  !N:num->real Q:num->real B:num->real
   Q0:real Q1:real Y0:real Y1:real.
  (!i. &1 < N(i))
  /\ (!i. &1 < Q(i))
  /\ (!i. &60 <= B(i))
  ==>
  (\i:num. ((((((&1)+((&1)/(N(i))))/(((&2)*((B(i))/(N(i))))-((&2)*((&1)/(N(i))))))*(((&1)+((log2((B(i))/(N(i)))/log2(N(i)))+((((&1)/(N(i)))*(((&1)/((B(i))/(N(i))))*((log2((pi)*((B(i))/(N(i))))/log2(N(i)))+(&1))))-(log2(&2 * pi * exp(&1))/log2(N(i))))))*(log2(Q(i))/log2(N(i))))) - Q0 / (&2 * Y0)) * log2(N(i))) - ((Q1 + (Q0*log2(Y0) - Q0*log2(&2 * pi * exp(&1))))*Y0 - Q0*Y1)/(&2 * Y0 pow 2))
  =
  (\i:num. (((N(i) + &1) * log2(bkzdelta(B(i))) / log2(N(i)) * log2(Q(i)) / log2(N(i)) - Q0 / (&2 * Y0)) * log2(N(i))) - ((Q1 + (Q0*log2(Y0) - Q0*log2(&2 * pi * exp(&1))))*Y0 - Q0*Y1)/(&2 * Y0 pow 2))
  `,
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC EQ_EXT THEN
  REWRITE_TAC [BETA_THM] THEN
  ASM_MESON_TAC [asymp2_ndq_lemma8]);;

let asymp2_ndq = prove(`
  !N:num->real Q:num->real B:num->real
   Q0:real Q1:real Y0:real Y1:real.
  (!i. &1 < N(i))
  /\ (!i. &1 < Q(i))
  /\ (!i. &60 <= B(i))
  /\ o1_seq (\i. &1 / N(i))
  /\ o1_seq (\i:num. (log2(Q(i))/log2(N(i)) - Q0) * log2(N(i)) - Q1)
  /\ o1_seq (\i:num. (B(i)/N(i) - Y0) * log2(N(i)) - Y1)
  /\ &0 < Y0
  ==>
  o1_seq (\i:num. (((N(i) + &1) * log2(bkzdelta(B(i))) / log2(N(i)) * log2(Q(i)) / log2(N(i)) - Q0 / (&2 * Y0)) * log2(N(i))) - ((Q1 + (Q0*log2(Y0) - Q0*log2(&2 * pi * exp(&1))))*Y0 - Q0*Y1)/(&2 * Y0 pow 2))
  `,
  REPEAT STRIP_TAC THEN
  note `!i:num. &1 < B(i)` [REAL_ARITH `&60 <= b ==> &1 < b`] THEN
  ASM_MESON_TAC [asymp2_ndq_lemma7;
    asymp2_ndq_lemma9]);;

let converse_X_gt = prove(`
  !c:real x:num->real X:num->real r:num->num r2:num->num.
  (!i. c < x(i))
  /\ X = x o r o r2
  ==> (!i. c < X(i))
  `,
  MESON_TAC [o_THM]);;

let converse_X_ge = prove(`
  !c:real x:num->real X:num->real r:num->num r2:num->num.
  (!i. c <= x(i))
  /\ X = x o r o r2
  ==> (!i. c <= X(i))
  `,
  MESON_TAC [o_THM]);;

let converse_rr2_subst = prove(`
  !x:num->real X:num->real r:num->num r2:num->num f:real->real.
  X = x o r o r2 ==>
  ((\i. f(x(i))) o r) o r2
  =
  (\i. f(X(i)))
  `,
  MESON_TAC [o_THM;
    EQ_EXT]);;

let converse_rr2_subst2 = prove(`
  !x:num->real X:num->real y:num->real Y:num->real r:num->num r2:num->num f:real->real->real.
  X = x o r o r2 /\
  Y = y o r o r2 ==>
  ((\i. f (x(i)) (y(i))) o r) o r2
  =
  (\i. f (X(i)) (Y(i)))
  `,
  MESON_TAC [o_THM;
    EQ_EXT]);;

let converse_overN_o1_lemma = prove(`
  !n:num->real N:num->real r:num->num r2:num->num.
  N = n o r o r2 ==>
  ((\i. &1 / n(i)) o r) o r2
  =
  (\i. &1 / N(i))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `N:num->real`;
    `r:num->num`;
    `r2:num->num`;
    `\x. &1 / x`] converse_rr2_subst THEN
  ASM_MESON_TAC []);;

let converse_overN_o1 = prove(`
  !n:num->real N:num->real r:num->num r2:num->num.
  o1_seq (\i. &1 / n(i))
  /\ (!i j. i < j ==> r(i) < r(j))
  /\ (!i j. i < j ==> r2(i) < r2(j))
  /\ N = n o r o r2
  ==> o1_seq (\i. &1 / N(i))
  `,
  REPEAT STRIP_TAC THEN
  specialize [`\i:num. &1 / n(i)`;
    `r:num->num`] o1_seq_subseq THEN
  specialize [`(\i:num. &1 / n(i)) o (r:num->num)`;
    `r2:num->num`] o1_seq_subseq THEN
  ASM_MESON_TAC [converse_overN_o1_lemma]);;

let converse_X_o1_lemma = prove(`
  !n:num->real N:num->real x:num->real X:num->real X0:real X1:real r:num->num r2:num->num.
  N = n o r o r2 /\
  X = x o r o r2 ==>
  ((\i:num. (log2(x(i))/log2(n(i)) - X0) * log2(n(i)) - X1) o r) o r2
  =
  (\i:num. (log2(X(i))/log2(N(i)) - X0) * log2(N(i)) - X1)
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:num->real`;
    `N:num->real`;
    `x:num->real`;
    `X:num->real`;
    `r:num->num`;
    `r2:num->num`;
    `\n x. (log2(x)/log2(n) - X0)*log2(n) - X1`] converse_rr2_subst2 THEN
  ASM_MESON_TAC []);;

let converse_X_o1 = prove(`
  !n:num->real N:num->real x:num->real X:num->real X0:real X1:real r:num->num r2:num->num.
  o1_seq (\i:num. (log2(x(i))/log2(n(i)) - X0) * log2(n(i)) - X1)
  /\ (!i j. i < j ==> r(i) < r(j))
  /\ (!i j. i < j ==> r2(i) < r2(j))
  /\ X = x o r o r2
  /\ N = n o r o r2
  ==> o1_seq (\i:num. (log2(X(i))/log2(N(i)) - X0) * log2(N(i)) - X1)
  `,
  REPEAT STRIP_TAC THEN
  ASM_MESON_TAC [o1_seq_subseq;
    converse_X_o1_lemma]);;

let converse_K_X0 = prove(`
  !n:num->real N:num->real k:num->real K:num->real X0:real r:num->num r2:num->num.
  o1_seq (\i. k(r(r2(i)))/n(r(r2(i))) - X0)
  /\ N = n o r o r2
  /\ K = k o r o r2
  ==>
  o1_seq (\i. K(i)/N(i) - X0)
  `,
  REPEAT STRIP_TAC THEN
  notetac `(\i. k(r(r2(i:num):num):num)/n(r(r2(i))) - X0) = (\i. K(i)/N(i) - X0)` (ASM_REWRITE_TAC [EQ_EXT;
      o_THM] THEN
    ASM_MESON_TAC []) THEN
  ASM_MESON_TAC []);;

let converse_beta_le_B = prove(`
  !n:num->real N:num->real b:num->real beta:num->real B:num->real z0:real z1:real e:real r:num->num r2:num->num.
  (!i. &1 < n(i))
  /\ &0 < e
  /\ (!i. (b(r(i))/n(r(i)) - z0)*log2(n(r(i))) - z1 < --e)
  /\ beta = b o r o r2
  /\ N = n o r o r2
  /\ B = (\i. (z0 + (z1 - e)/log2(N(i)))*N(i))
  ==> !i. beta(i) <= B(i)
  `,
  REPEAT STRIP_TAC THEN
  ASM_REWRITE_TAC [o_THM] THEN
  def `j:num` `r(r2(i:num):num):num` THEN
  note `&1 < n(j:num)` [] THEN
  note `&0 < log2(n(j:num))` [log2_gt1] THEN
  note `(b(j:num)/n(j) - z0)*log2(n(j)) - z1 < --e` [] THEN
  real_linear `(b(j:num)/n(j) - z0)*log2(n(j)) - z1 < --e ==> (b(j:num)/n(j) - z0)*log2(n(j)) < z1 - e` THEN
  real_cancel `&0 < log2(n(j)) ==> ((b(j:num)/n(j) - z0)*log2(n(j)))/log2(n(j)) = b(j)/n(j) - z0` THEN
  note `b(j:num)/n(j) - z0 < (z1 - e)/log2(n(j))` [REAL_LT_DIV2_EQ] THEN
  real_linear `b(j:num)/n(j) - z0 < (z1 - e)/log2(n(j)) ==> b(j:num)/n(j) < z0 + (z1 - e)/log2(n(j))` THEN
  real_cancel `&1 < n(j:num) ==> b(j:num)/n(j)*n(j) = b(j:num)` THEN
  real_linear `&1 < n(j:num) ==> &0 < n(j)` THEN
  note `b(j:num) < (z0 + (z1 - e)/log2(n(j)))*n(j)` [REAL_LT_RMUL] THEN
  ASM_MESON_TAC [REAL_LT_IMP_LE]);;

let converse_standardratio_rr2 = prove(`
  !n:num->real q:num->real s:num->real
   k:num->real b:num->real
   N:num->real Q:num->real S:num->real
   K:num->real beta:num->real
   r:num->num r2:num->num
   z0:real z1:real e:real m:num.
  (!i. m <= i ==> standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i)) <= &1)
  /\ IMAGE r (:num) = {i | m <= i /\ (b(i)/n(i) - z0)*log2(n(i)) - z1 < --e}
  /\ N = n o r o r2
  /\ Q = q o r o r2
  /\ S = s o r o r2
  /\ K = k o r o r2
  /\ beta = b o r o r2
  ==> 
  !i. standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (beta(i)) <= &1
  `,
  REPEAT STRIP_TAC THEN
  ASM_REWRITE_TAC [o_THM] THEN
  notetac `IMAGE r (:num) = {i:num | m <= i /\ (b(i)/n(i) - z0)*log2(n(i)) - z1 < --e} ==> !i:num. m <= r(i)` (SET_TAC []) THEN
  note `!i:num. m <= r(i):num` [] THEN
  ASM_MESON_TAC []);;

let converse_B_o1 = prove(`
  !N:num->real B:num->real z0:real z1:real e:real.
  (!i. &1 < N(i))
  /\ B = (\i. (z0 + (z1 - e)/log2(N(i)))*N(i))
  ==> o1_seq (\i. (B(i)/N(i) - z0)*log2(N(i)) - (z1 - e))
  `,
  REPEAT STRIP_TAC THEN
  note `!i:num. &0 < log2(N(i))` [log2_gt1] THEN
  note `!i:num. &0 < N(i)` [REAL_ARITH `&1 < x ==> &0 < x`] THEN
  real_cancel `!i:num. &1 < N(i) ==> ((z0 + (z1 - e)/log2(N(i)))*N(i))/N(i) - z0 = (z1 - e)/log2(N(i))` THEN
  note `!i:num. B(i)/N(i) - z0 = (z1 - e)/log2(N(i))` [] THEN
  real_cancel `!i:num. &0 < log2(N(i)) ==> ((z1 - e)/log2(N(i)))*log2(N(i)) - (z1 - e) = &0` THEN
  note `!i:num. (B(i)/N(i) - z0)*log2(N(i)) - (z1 - e) = &0` [] THEN
  ASM_REWRITE_TAC [o1_seq_0]);;

let converse_asymp1_lemma = prove(`
  !N:num->real Q:num->real S:num->real
   K:num->real B:num->real c:real.
  (!i. &1 < N(i))
  /\ (!i. standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (B(i)) <= &1)
  /\ o1_seq (\i:num. ln (standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (B(i))) / ln (N(i)) - c)
  ==> c <= &0
  `,
  REPEAT STRIP_TAC THEN
  note `!i:num. &0 < ln(N(i))` [ln_gt1] THEN
  note `!i:num. ln(standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (B(i))) <= &0` [ln_le1;
    standardratio_positive] THEN
  note `!i:num. ln(standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (B(i)))/ln(N(i)) <= &0` [REAL_LE_DIV2_EQ;
    REAL_ARITH `&0/x = &0`] THEN
  ASM_CASES_TAC `&0 < c` THENL
  [choose_specializing `m:num` [`\i:num. ln(standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (B(i)))/ln(N(i))`;
      `c:real`] positive_limit_is_eventually_positive THEN
    ASM_MESON_TAC [LE_REFL;
      REAL_NOT_LT];
    ASM_MESON_TAC [REAL_NOT_LT]]);;

let converse_asymp1_lemma2 = prove(`
  !Q0 S0 z0 X0.
  &0 <= X0
  /\ &0 < Q0
  /\ -- &1 / &2 < Q0 - S0
  /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2)
  /\ S0 - &1 / &2 + (&1 + X0) / (&2 * z0) - X0 / (&1 + X0) * Q0 <= &0
  ==> 
  X0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2)
  `,
  REPEAT STRIP_TAC THEN
  note `&0 <= Q0` [REAL_LT_IMP_LE] THEN
  note `(S0 - &1 / &2 + (&1 + X0) / (&2 * z0) - X0 / (&1 + X0) * Q0)*Q0 <= &0` [REAL_LE_RMUL;
    REAL_ARITH `&0 * x = &0`] THEN
  real_linear `(S0 - &1 / &2 + (&1 + X0) / (&2 * z0) - X0 / (&1 + X0) * Q0)*Q0 <= &0 ==> (S0 - &1 / &2)*Q0 + (&1 + X0) * Q0 / (&2 * z0) - X0 / (&1 + X0) * Q0*Q0 <= &0` THEN
  real_cancel `&0 < Q0 /\ -- &1 / &2 < Q0 - S0 /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2) ==> Q0 / (&2 * z0) = ((Q0 - S0 + &1 / &2) pow 2) / &4` THEN
  note `(S0 - &1 / &2)*Q0 + (&1 + X0) * (((Q0 - S0 + &1 / &2) pow 2) / &4) - X0 / (&1 + X0) * Q0*Q0 <= &0` [] THEN
  real_linear `&0 <= X0 ==> &0 <= &1 + X0` THEN
  note `((S0 - &1 / &2)*Q0 + (&1 + X0) * (((Q0 - S0 + &1 / &2) pow 2) / &4) - X0 / (&1 + X0) * Q0*Q0)*(&1 + X0) <= &0` [REAL_LE_RMUL;
    REAL_ARITH `&0 * x = &0`] THEN
  real_cancel `&0 <= X0 ==> ((S0 - &1 / &2)*Q0 + (&1 + X0) * (((Q0 - S0 + &1 / &2) pow 2) / &4) - X0 / (&1 + X0) * Q0*Q0)*(&1 + X0) = (S0 - &1 / &2)*Q0*(&1 + X0) + (&1 + X0) pow 2 * (((Q0 - S0 + &1 / &2) pow 2) / &4) - X0 * Q0*Q0` THEN
  note `(S0 - &1 / &2)*Q0*(&1 + X0) + (&1 + X0) pow 2 * (((Q0 - S0 + &1 / &2) pow 2) / &4) - X0 * Q0*Q0 <= &0` [] THEN
  real_linear `(S0 - &1 / &2)*Q0*(&1 + X0) + (&1 + X0) pow 2 * (((Q0 - S0 + &1 / &2) pow 2) / &4) - X0 * Q0*Q0 <= &0 ==> ((Q0 - S0 + &1 / &2) * X0 - (Q0 + S0 - &1 / &2)) pow 2 <= &0` THEN
  note `(Q0 - S0 + &1 / &2) * X0 - (Q0 + S0 - &1 / &2) = &0` [REAL_LT_POW_2;
    REAL_NOT_LT] THEN
  real_linear `(Q0 - S0 + &1 / &2) * X0 - (Q0 + S0 - &1 / &2) = &0 ==> (Q0 - S0 + &1 / &2) * X0 = Q0 + S0 - &1 / &2` THEN
  real_linear `-- &1 / &2 < Q0 - S0 ==> &0 < Q0 - S0 + &1 / &2` THEN
  ASM_MESON_TAC [REAL_EQ_LDIV_EQ;
    REAL_MUL_SYM]);;

let converse_asymp1 = prove(`
  !N:num->real Q:num->real S:num->real
   K:num->real B:num->real
   Q0:real Q1:real S0:real S1:real
   x0:real z0:real z1:real X0:real e:real.
  (!i. &1 < N(i))
  /\ o1_seq (\i. &1 / N(i))
  /\ (!i. &1 < Q(i))
  /\ o1_seq (\i. (log2(Q(i))/log2(N(i)) - Q0) * log2(N(i)) - Q1)
  /\ (!i. &0 < S(i))
  /\ o1_seq (\i. (log2(S(i))/log2(N(i)) - S0) * log2(N(i)) - S1)
  /\ (!i. &0 < K(i))
  /\ &0 <= X0
  /\ o1_seq (\i. K(i)/N(i) - X0)
  /\ (!i. &60 <= B(i))
  /\ o1_seq (\i. (B(i)/N(i) - z0) * log2(N(i)) - (z1 - e))
  /\ &0 < &1 + &2 * S0
  /\ S0 <= &1 / &2
  /\ -- &1 / &2 < Q0 - S0
  /\ &1 / &2 < Q0 + S0
  /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2)
  /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2)
  /\ z1 =   (&2 * S1 + log2(z0)
             - (S0 - Q0 + &3 / &2) * (log2(z0) - log2(&2 * pi * exp(&1)))
             - Q1 * (Q0 + S0 - &1 / &2)/Q0
            )
          * (&2 * Q0) / ((Q0 - S0 + &1 / &2) pow 3)
  /\ &0 < e
  /\ (!i. standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (B(i)) <= &1)
  ==>
  x0 = X0
  `,
  REPEAT STRIP_TAC THEN
  specialize [`N:num->real`;
    `\i:num. log2(Q(i))/log2(N(i))`;
    `Q0:real`;
    `Q1:real`] asymp2_asymp1 THEN
  specialize [`N:num->real`;
    `\i:num. log2(S(i))/log2(N(i))`;
    `S0:real`;
    `S1:real`] asymp2_asymp1 THEN
  specialize [`N:num->real`;
    `\i:num. B(i)/N(i):real`;
    `z0:real`;
    `z1-e:real`] asymp2_asymp1 THEN
  note `!i:num. &1 < B(i)` [REAL_ARITH `&60 <= x ==> &1 < x`] THEN
  real_linear `-- &1 / &2 < Q0 - S0 /\ &1 / &2 < Q0 + S0 ==> &0 < Q0` THEN
  note `&0 < z0` [main_z0_positive] THEN
  note `o1_seq (\i:num. ln (standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (B(i))) / ln (N(i)) - (S0 - &1 / &2 + (&1 + X0) / (&2 * z0) - X0 / (&1 + X0) * Q0))` [asymp1_standardratio] THEN
  note `S0 - &1 / &2 + (&1 + X0) / (&2 * z0) - X0 / (&1 + X0) * Q0 <= &0` [converse_asymp1_lemma] THEN
  ASM_MESON_TAC [converse_asymp1_lemma2]);;

let z1_lemma1 = prove(`
  !Q0 Q1 S0 S1 e z0 log2z0 log21x0 log2twopie.
  &0 < z0
  /\ &0 < Q0 - S0 + &1 / &2
  /\ &0 < Q0
  /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2)
  /\ z1 = (&2 * S1 + log2z0 - (S0 - Q0 + &3 / &2) * (log2z0 - log2twopie) - Q1 * (Q0 + S0 - &1 / &2)/Q0) * (&2 * Q0) / ((Q0 - S0 + &1 / &2) pow 3)
  ==>
  (Q0 - S0 + &1 / &2) * e / (&2 * z0)
  =
  &1 / &2 * (log21x0 + &0 + &2 * S1 + &0 * S0)
  - &1 / &2 * (log21x0 - log2z0)
  - ((log2z0 - log2twopie + Q1) - &2 * ((Q1 + Q0 * log2z0 - Q0 * log2twopie) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2) / (Q0 - S0 + &1 / &2))
  `,
  REPEAT STRIP_TAC THEN
  REWRITE_TAC [REAL_ARITH `&1 / &2 * (a + b) - &1 / &2 * (a - c) - d = &1 / &2 * b + &1 / &2 *c - d`] THEN
  REWRITE_TAC [REAL_ARITH `&0 * x = &0`;
    REAL_ARITH `&0 + x = x`;
    REAL_ARITH `x + &0 = x`;
    REAL_ARITH `&1 / &2 * &2 * s = s`] THEN
  ASM_REWRITE_TAC [] THEN
  real_cancel `&0 < Q0 /\ &0 < (Q0 - S0 + &1 / &2) ==> (Q0 - S0 + &1 / &2) * e / (&2 * &2 * Q0 / (Q0 - S0 + &1 / &2) pow 2) = S1 + &1 / &2 * log2z0 - ((log2z0 - log2twopie + Q1) - &2 * ((Q1 + Q0 * log2z0 - Q0 * log2twopie) * &2 * Q0 / (Q0 - S0 + &1 / &2) pow 2 - Q0 * ((&2 * S1 + log2z0 - (S0 - Q0 + &3 / &2) * (log2z0 - log2twopie) - Q1 * (Q0 + S0 - &1 / &2) / Q0) * (&2 * Q0) / (Q0 - S0 + &1 / &2) pow 3 - e)) / (&2 * (&2 * Q0 / (Q0 - S0 + &1 / &2) pow 2) pow 2) / (Q0 - S0 + &1 / &2))` THEN
  ASM_REWRITE_TAC []);;

let z1_lemma2 = prove(`
  !Q0 Q1 S0 S1 e x0 z0.
  &0 < z0
  /\ &0 < &1 + x0
  /\ &0 < Q0 - S0 + &1 / &2
  /\ &0 < Q0
  /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2)
  /\ z1 =   (&2 * S1 + log2(z0)
             - (S0 - Q0 + &3 / &2) * (log2(z0) - log2(&2 * pi * exp(&1)))
             - Q1 * (Q0 + S0 - &1 / &2)/Q0
            )
          * (&2 * Q0) / ((Q0 - S0 + &1 / &2) pow 3)
  ==>
  (Q0 - S0 + &1 / &2) * e / (&2 * z0)
  =
  &1 / &2 * (log2 (&1 + x0) + &0 + &2 * S1 + &0 * S0)
  - &1 / &2 * log2 (((&1 + x0) + &0) * &1 / z0)
  - ((log2 z0 - log2 (&2 * pi * exp (&1)) + Q1) - &2 * ((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2) / (Q0 - S0 + &1 / &2))
  `,
  REPEAT STRIP_TAC THEN
  note `&0 < &1 / z0` [pos_1over] THEN
  note `log2 (((&1 + x0) + &0) * &1 / z0) = log2(&1 + x0) - log2(z0)` [REAL_ARITH `(&1 + x0) + &0 = &1 + x0`;
    log2_mul;
    log2_1over;
    REAL_ARITH `a + --b = a - b`] THEN
  specialize [`Q0:real`;
    `Q1:real`;
    `S0:real`;
    `S1:real`;
    `e:real`;
    `z0:real`;
    `log2(z0)`;
    `log2(&1 + x0)`;
    `log2(&2 * pi * exp(&1))`] z1_lemma1 THEN
  ASM_MESON_TAC []);;

let sqrt_misc_lemma = prove(`
  &0 < L ==> sqrt(a * b / L * c / L) = sqrt(a * b * c) / L
  `,
  REWRITE_TAC [REAL_FIELD `a * b / L * c / L = (a*b*c)/(L*L)`] THEN
  MESON_TAC [REAL_LT_IMP_LE;
    SQRT_DIV;
    REAL_POW_2;
    POW_2_SQRT;
    REAL_LE_SQUARE]);;

let converse_asymp2_lemma1 = REAL_FIELD `
  !Q0:real S0:real z0:real.
  &0 < Q0
  /\ -- &1 / &2 < Q0 - S0
  /\ z0 = &2 * Q0 / (Q0 - S0 + &1 / &2) pow 2
  ==> Q0 / (&2 * z0) =
      ((Q0 - S0 + &1 / &2) / &2) pow 2
`;;

let converse_asymp2_lemma2 = prove(`
  !N:num->real Q:num->real S:num->real K:num->real B:num->real Q0:real S0:real z0:real e:real.
  (!i. &0 < log2(N(i)))
  ==>
  (\i:num. (&1 / &2 * log2 (&1 + (&1 + K i / N i) * N i powreal (&1 + &2 * log2 (S i) / log2 (N i))) / log2 (N i) - &1 / &2 * log2 (((&1 + K i / N i) + &1 / N i) / (B i / N i)) / log2 (N i) - (((&2 * B i - &1) * log2 (bkzdelta (B i)) / log2 (N i) + log2 (Q i) / log2 (N i)) - &2 * sqrt ((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i))) - &0) * log2 (N i) - (Q0 - S0 + &1 / &2) * e / (&2 * z0))
  =
  (\i:num. (&1 / &2 * log2 (&1 + (&1 + K i / N i) * N i powreal (&1 + &2 * log2 (S i) / log2 (N i))) - &1 / &2 * log2 (((&1 + K i / N i) + &1 / N i) / (B i / N i)) - (((&2 * B i - &1) * log2 (bkzdelta (B i)) + log2 (Q i)) - &2 * sqrt ((N i + &1) * log2 (bkzdelta (B i)) * log2 (Q i)))) - (Q0 - S0 + &1 / &2) * e / (&2 * z0))
  `,
  REPEAT STRIP_TAC THEN
  MATCH_MP_TAC EQ_EXT THEN
  REWRITE_TAC [BETA_THM] THEN
  ASM_MESON_TAC [sqrt_misc_lemma;
    REAL_FIELD `&0 < L ==> ((&1 / &2 * log2x / L - &1 / &2 * log2y / L - (((&2 * bi - &1) * log2d / L + log2q / L) - &2 * s / L)) - &0) * L = &1 / &2 * log2x - &1 / &2 * log2y - (((&2 * bi - &1) * log2d + log2q ) - &2 * s)`]);;

let converse_asymp2_lemma3 = prove(`
  !n q s k b.
  &1 < n
  /\ &0 < k
  /\ &1 < q
  /\ &0 < s
  /\ &60 <= b
  ==>
  &1 / &2 * log2 (&1 + (&1 + k / n) * n powreal (&1 + &2 * log2 (s) / log2 (n))) - &1 / &2 * log2 (((&1 + k / n) + &1 / n) / (b / n)) - (((&2 * b - &1) * log2 (bkzdelta (b)) + log2 (q)) - &2 * sqrt ((n + &1) * log2 (bkzdelta (b)) * log2 (q)))
  <= log2(standardratio n q s k b)
  `,
  REPEAT STRIP_TAC THEN
  ASM_SIMP_TAC [standardratio_log2] THEN
  REWRITE_TAC [powreal_sum] THEN
  real_linear `&1 < n ==> &0 < n` THEN
  ASM_SIMP_TAC [powreal_1] THEN
  ASM_REWRITE_TAC [GSYM powreal_pow] THEN
  ASM_SIMP_TAC [GSYM powreal_log2ratio] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> &1 + (&1 + k / n) * (n * x) = (n + k)*x + &1`] THEN
  ASM_SIMP_TAC [REAL_FIELD `&1 < n ==> ((&1 + k / n) + &1 / n) / (b / n) = (n + k + &1)/b`] THEN
  def `d:real` `bkzdelta(b)` THEN
  note `&1 < d` [deltabound5] THEN
  real_linear `&1 < d ==> &1 <= d` THEN
  note `(&2 * b - n - k - &2) * log2 d + k / (n + k + &1) * log2 q <= (&2 * b - &1) * log2 d + log2 q - &2 * sqrt ((n + &1) * log2 d * log2 q)` [kappa_main] THEN
  real_linear `(&2 * b - n - k - &2) * log2 d + k / (n + k + &1) * log2 q <= (&2 * b - &1) * log2 d + log2 q - &2 * sqrt ((n + &1) * log2 d * log2 q) ==> &1 / &2 * log2 ((n + k) * s pow 2 + &1) - &1 / &2 * log2 ((n + k + &1) / b) - (((&2 * b - &1) * log2 d + log2 q) - &2 * sqrt ((n + &1) * log2 d * log2 q)) <= &1 / &2 * log2 ((n + k) * s pow 2 + &1) - (&1 / &2 * log2 ((n + k + &1) / b) + ( (&2 * b - n - k - &2) * log2 d + k / (n + k + &1) * log2 q))` THEN
  note `log2 (bkzdelta b) = (log2 b + &1 / b * log2 (pi * b) - log2 (&2 * pi * exp (&1))) / (&2 * (b - &1))` [bkzdelta_log2] THEN
  note `(&2 * b - n - k - &2) * log2 d = (log2 b + &1 / b * log2 (pi * b) - log2 (&2 * pi * exp (&1))) * (&2 * b - (n + k + &1) - &1) / (&2 * (b - &1))` [REAL_ARITH `c * (L / x) = L * (c / x)`;
    REAL_ARITH `&2 * b - n - k - &2 = &2 * b - (n + k + &1) - &1`] THEN
  ASM_MESON_TAC []);;

let converse_asymp2_lemma4 = prove(`
  !n q s k b.
  &1 < n
  /\ &0 < k
  /\ &1 < q
  /\ &0 < s
  /\ &60 <= b
  /\
  &0 < &1 / &2 * log2 (&1 + (&1 + k / n) * n powreal (&1 + &2 * log2 (s) / log2 (n))) - &1 / &2 * log2 (((&1 + k / n) + &1 / n) / (b / n)) - (((&2 * b - &1) * log2 (bkzdelta (b)) + log2 (q)) - &2 * sqrt ((n + &1) * log2 (bkzdelta (b)) * log2 (q)))
  ==> &1 < standardratio n q s k b
  `,
  REPEAT STRIP_TAC THEN
  specialize [`n:real`;
    `q:real`;
    `s:real`;
    `k:real`;
    `b:real`] converse_asymp2_lemma3 THEN
  note `&0 < log2(standardratio n q s k b)` [REAL_LTE_TRANS] THEN
  ASM_MESON_TAC [REAL_NOT_LT;
    standardratio_positive;
    log2_le1]);;

let converse_asymp2 = prove(`
  !N:num->real Q:num->real S:num->real
   K:num->real B:num->real
   Q0:real Q1:real S0:real S1:real
   x0:real z0:real z1:real e:real.
  (!i. &1 < N(i))
  /\ o1_seq (\i. &1 / N(i))
  /\ (!i. &1 < Q(i))
  /\ o1_seq (\i. (log2(Q(i))/log2(N(i)) - Q0) * log2(N(i)) - Q1)
  /\ (!i. &0 < S(i))
  /\ o1_seq (\i. (log2(S(i))/log2(N(i)) - S0) * log2(N(i)) - S1)
  /\ (!i. &0 < K(i))
  /\ o1_seq (\i. K(i)/N(i) - x0)
  /\ (!i. &60 <= B(i))
  /\ o1_seq (\i. (B(i)/N(i) - z0) * log2(N(i)) - (z1 - e))
  /\ -- &1 / &2 < S0
  /\ S0 <= &1 / &2
  /\ -- &1 / &2 < Q0 - S0
  /\ &1 / &2 < Q0 + S0
  /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2)
  /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2)
  /\ z1 =   (&2 * S1 + log2(z0)
             - (S0 - Q0 + &3 / &2) * (log2(z0) - log2(&2 * pi * exp(&1)))
             - Q1 * (Q0 + S0 - &1 / &2)/Q0
            )
          * (&2 * Q0) / ((Q0 - S0 + &1 / &2) pow 3)
  /\ &0 < e
  ==>
  ?m. !i. m <= i ==> &1 < standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (B(i))
  `,
  REPEAT STRIP_TAC THEN
  real_linear `-- &1 / &2 < S0 ==> &0 < &1 + &2 * S0` THEN
  note `&0 < x0` [main_x0_positive] THEN
  note `&0 < z0` [main_z0_positive] THEN
  specialize [`N:num->real`;
    `B:num->real`;
    `z0:real`;
    `z1 - e:real`] asymp2_2B1D THEN
  specialize [`N:num->real`;
    `Q:num->real`;
    `B:num->real`;
    `Q0:real`;
    `Q1:real`;
    `z0:real`;
    `z1 - e:real`] asymp2_ndq THEN
  real_linear `-- &1 / &2 < Q0 - S0 /\ &1 / &2 < Q0 + S0 ==> &0 < Q0` THEN
  note `Q0 / (&2 * z0) = ((Q0 - S0 + &1 / &2) / &2) pow 2` [converse_asymp2_lemma1] THEN
  note_eqlambda `(\i:num. ((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i) - Q0 / (&2 * z0)) * log2 (N i) - ((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2)) = (\i:num. ((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i) - ((Q0 - S0 + &1 / &2) / &2) pow 2) * log2 (N i) - ((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2))` [] THEN
  note `o1_seq (\i:num. ((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i) - ((Q0 - S0 + &1 / &2) / &2) pow 2) * log2 (N i) - ((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2))` [] THEN
  real_linear `-- &1 / &2 < Q0 - S0 ==> &0 < Q0 - S0 + &1 / &2` THEN
  real_linear `-- &1 / &2 < Q0 - S0 ==> &0 < (Q0 - S0 + &1 / &2) / &2` THEN
  note `&0 < ((Q0 - S0 + &1 / &2) / &2) pow 2` [REAL_LT_IMP_NZ;
    REAL_LT_POW_2] THEN
  specialize [`N:num->real`;
    `\i:num. (N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i)`;
    `((Q0 - S0 + &1 / &2) / &2) pow 2`;
    `((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2)`] asymp2_sqrt THEN
  note `sqrt (((Q0 - S0 + &1 / &2) / &2) pow 2) = (Q0 - S0 + &1 / &2) / &2` [REAL_LT_IMP_LE;
    POW_2_SQRT] THEN
  note `&2 * sqrt (((Q0 - S0 + &1 / &2) / &2) pow 2) = Q0 - S0 + &1 / &2` [REAL_ARITH `&2 * (a / &2) = a`] THEN
  note_eqlambda `(\i:num. (sqrt ((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i)) - sqrt (((Q0 - S0 + &1 / &2) / &2) pow 2)) * log2 (N i) - ((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2) / (&2 * sqrt (((Q0 - S0 + &1 / &2) / &2) pow 2))) = (\i. (sqrt ((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i)) - (Q0 - S0 + &1 / &2) / &2) * log2 (N i) - ((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2) / (Q0 - S0 + &1 / &2))` [] THEN
  note `o1_seq (\i:num. (sqrt ((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i)) - (Q0 - S0 + &1 / &2) / &2) * log2 (N i) - ((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2) / (Q0 - S0 + &1 / &2))` [] THEN
  specialize [`N:num->real`;
    `\i:num. sqrt((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i))`;
    `(Q0 - S0 + &1 / &2) / &2`;
    `((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2) / (Q0 - S0 + &1 / &2)`;
    `&2`] asymp2_mulconst THEN
  specialize [`N:num->real`;
    `\i:num. (&2 * B i - &1) * log2 (bkzdelta (B i)) / log2 (N i)`;
    `&1`;
    `log2 z0 - log2 (&2 * pi * exp (&1))`;
    `\i:num. log2(Q(i))/log2(N(i))`;
    `Q0:real`;
    `Q1:real`] asymp2_add THEN
  specialize [`N:num->real`;
    `\i:num. (&2 * B i - &1) * log2 (bkzdelta (B i)) / log2 (N i) + log2 (Q i) / log2 (N i)`;
    `&1 + Q0`;
    `log2 z0 - log2 (&2 * pi * exp (&1)) + Q1`;
    `\i:num. &2 * sqrt((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i))`;
    `&2 * ((Q0 - S0 + &1 / &2) / &2)`;
    `&2 * ((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2) / (Q0 - S0 + &1 / &2)`] asymp2_sub THEN
  specialize [`N:num->real`] asymp2_simple_lemma2 THEN
  specialize [`\i:num. N(i):real`;
    `&1:real`] asymp1_const THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `\i:num. (K(i))/(N(i)):real`;
    `x0:real`] asymp1_add THEN
  specialize [`\i:num. N(i):real`;
    `&1:real`] asymp2_const THEN
  specialize [`\i:num. N(i):real`;
    `&2:real`] asymp2_const THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &2:real`;
    `&2:real`;
    `&0:real`;
    `\i:num. log2(S(i))/log2(N(i)):real`;
    `S0:real`;
    `S1:real`] asymp2_mul THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. &1:real`;
    `&1:real`;
    `&0:real`;
    `\i:num. (&2)*(log2(S(i))/log2(N(i))):real`;
    `(&2)*(S0):real`;
    `((&2)*(S1))+((&0)*(S0)):real`] asymp2_add THEN
  note `!i:num. &0 < N(i)` [REAL_ARITH `&1 < x ==> &0 < x`] THEN
  note `!i:num. &0 < K(i)/N(i)` [REAL_LT_DIV] THEN
  note `!i:num. &0 < &1 + K(i)/N(i)` [REAL_ARITH `&0 < x ==> &0 < &1 + x`] THEN
  real_linear `&0 < x0 ==> &0 < &1 + x0` THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (&1)+((K(i))/(N(i))):real`;
    `(&1)+(x0):real`;
    `\i:num. (&1)+((&2)*(log2(S(i))/log2(N(i)))):real`;
    `(&1)+((&2)*(S0)):real`;
    `(&0)+(((&2)*(S1))+((&0)*(S0))):real`] asymp2_logn1plusAtimesntotheB_mini THEN
  specialize [`N:num->real`;
    `\i:num. log2 (&1 + (&1 + K i / N i) * N i powreal (&1 + &2 * log2 (S i) / log2 (N i))) / log2 (N i)`;
    `&1 + &2 * S0`;
    `(log2 (&1 + x0) + &0 + &2 * S1 + &0 * S0)`;
    `&1 / &2`] asymp2_mulconst THEN
  note_eqlambda `(\i:num. &1 / N(i)) = (\i:num. &1 / N(i) - &0)` [REAL_ARITH `x - &0 = x`] THEN
  note `o1_seq (\i:num. &1 / N(i) - &0)` [] THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. (&1)+((K(i))/(N(i))):real`;
    `(&1)+(x0):real`;
    `\i:num. (&1)/(N(i)):real`;
    `&0:real`] asymp1_add THEN
  specialize [`N:num->real`;
    `\i:num. B(i)/N(i)`;
    `z0:real`;
    `z1-e:real`] asymp2_asymp1 THEN
  real_linear `&0 < z0 ==> ~(z0 = &0)` THEN
  specialize [`\i:num. N(i):real`;
    `\i:num. ((&1)+((K(i))/(N(i))))+((&1)/(N(i))):real`;
    `((&1)+(x0))+(&0):real`;
    `\i:num. (B(i))/(N(i)):real`;
    `z0:real`] asymp1_div THEN
  note `&0 < ((&1 + x0) + &0) * &1 / z0` [pos_1over;
    REAL_ARITH `x + &0 = x`;
    REAL_LT_MUL] THEN
  specialize [`N:num->real`;
    `\i:num. ((&1 + K i / N i) + &1 / N i) / (B i / N i)`;
    `((&1 + x0) + &0) * &1 / z0`] asymp2_logn_mini THEN
  specialize [`N:num->real`;
    `\i:num. log2 (((&1 + K i / N i) + &1 / N i) / (B i / N i)) / log2 (N i)`;
    `&0`;
    `log2 (((&1 + x0) + &0) * &1 / z0)`;
    `&1 / &2`] asymp2_mulconst THEN
  specialize [`N:num->real`;
    `\i:num. &1 / &2 * log2 (&1 + (&1 + K i / N i) * N i powreal (&1 + &2 * log2 (S i) / log2 (N i))) / log2 (N i)`;
    `&1 / &2 * (&1 + &2 * S0)`;
    `&1 / &2 * (log2 (&1 + x0) + &0 + &2 * S1 + &0 * S0)`;
    `\i:num. &1 / &2 * log2 (((&1 + K i / N i) + &1 / N i) / (B i / N i)) / log2 (N i)`;
    `&1 / &2 * &0`;
    `&1 / &2 * log2 (((&1 + x0) + &0) * &1 / z0)`] asymp2_sub THEN
  specialize [`N:num->real`;
    `\i:num. &1 / &2 * log2 (&1 + (&1 + K i / N i) * N i powreal (&1 + &2 * log2 (S i) / log2 (N i))) / log2 (N i) - &1 / &2 * log2 (((&1 + K i / N i) + &1 / N i) / (B i / N i)) / log2 (N i)`;
    `&1 / &2 * (&1 + &2 * S0) - &1 / &2 * &0`;
    `&1 / &2 * (log2 (&1 + x0) + &0 + &2 * S1 + &0 * S0) - &1 / &2 * log2 (((&1 + x0) + &0) * &1 / z0)`;
    `\i:num. ((&2 * B i - &1) * log2 (bkzdelta (B i)) / log2 (N i) + log2 (Q i) / log2 (N i)) - &2 * sqrt ((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i))`;
    `(&1 + Q0) - &2 * (Q0 - S0 + &1 / &2) / &2`;
    `(log2 z0 - log2 (&2 * pi * exp (&1)) + Q1) - &2 * ((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2) / (Q0 - S0 + &1 / &2)`] asymp2_sub THEN
  real_linear `&0 = (&1 / &2 * (&1 + &2 * S0) - &1 / &2 * &0 - ((&1 + Q0) - &2 * (Q0 - S0 + &1 / &2) / &2))` THEN
  note `(Q0 - S0 + &1 / &2) * e / (&2 * z0) = &1 / &2 * (log2 (&1 + x0) + &0 + &2 * S1 + &0 * S0) - &1 / &2 * log2 (((&1 + x0) + &0) * &1 / z0) - ((log2 z0 - log2 (&2 * pi * exp (&1)) + Q1) - &2 * ((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2) / (Q0 - S0 + &1 / &2))` [z1_lemma2] THEN
  note_eqlambda `(\i:num. (&1 / &2 * log2 (&1 + (&1 + K i / N i) * N i powreal (&1 + &2 * log2 (S i) / log2 (N i))) / log2 (N i) - &1 / &2 * log2 (((&1 + K i / N i) + &1 / N i) / (B i / N i)) / log2 (N i) - (((&2 * B i - &1) * log2 (bkzdelta (B i)) / log2 (N i) + log2 (Q i) / log2 (N i)) - &2 * sqrt ((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i))) - (&1 / &2 * (&1 + &2 * S0) - &1 / &2 * &0 - ((&1 + Q0) - &2 * (Q0 - S0 + &1 / &2) / &2))) * log2 (N i) - (&1 / &2 * (log2 (&1 + x0) + &0 + &2 * S1 + &0 * S0) - &1 / &2 * log2 (((&1 + x0) + &0) * &1 / z0) - ((log2 z0 - log2 (&2 * pi * exp (&1)) + Q1) - &2 * ((Q1 + Q0 * log2 z0 - Q0 * log2 (&2 * pi * exp (&1))) * z0 - Q0 * (z1 - e)) / (&2 * z0 pow 2) / (Q0 - S0 + &1 / &2)))) = (\i:num. (&1 / &2 * log2 (&1 + (&1 + K i / N i) * N i powreal (&1 + &2 * log2 (S i) / log2 (N i))) / log2 (N i) - &1 / &2 * log2 (((&1 + K i / N i) + &1 / N i) / (B i / N i)) / log2 (N i) - (((&2 * B i - &1) * log2 (bkzdelta (B i)) / log2 (N i) + log2 (Q i) / log2 (N i)) - &2 * sqrt ((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i))) - &0) * log2 (N i) - (Q0 - S0 + &1 / &2) * e / (&2 * z0))` [] THEN
  note `o1_seq (\i:num. (&1 / &2 * log2 (&1 + (&1 + K i / N i) * N i powreal (&1 + &2 * log2 (S i) / log2 (N i))) / log2 (N i) - &1 / &2 * log2 (((&1 + K i / N i) + &1 / N i) / (B i / N i)) / log2 (N i) - (((&2 * B i - &1) * log2 (bkzdelta (B i)) / log2 (N i) + log2 (Q i) / log2 (N i)) - &2 * sqrt ((N i + &1) * log2 (bkzdelta (B i)) / log2 (N i) * log2 (Q i) / log2 (N i))) - &0) * log2 (N i) - (Q0 - S0 + &1 / &2) * e / (&2 * z0))` [] THEN
  note `!i:num. &0 < log2(N(i))` [log2_gt1] THEN
  specialize [`N:num->real`;
    `Q:num->real`;
    `S:num->real`;
    `K:num->real`;
    `B:num->real`;
    `Q0:real`;
    `S0:real`;
    `z0:real`;
    `e:real`] converse_asymp2_lemma2 THEN
  note `o1_seq (\i:num. ( &1 / &2 * log2 (&1 + (&1 + K i / N i) * N i powreal (&1 + &2 * log2 (S i) / log2 (N i))) - &1 / &2 * log2 (((&1 + K i / N i) + &1 / N i) / (B i / N i)) - (((&2 * B i - &1) * log2 (bkzdelta (B i)) + log2 (Q i)) - &2 * sqrt ((N i + &1) * log2 (bkzdelta (B i)) * log2 (Q i)))) - (Q0 - S0 + &1 / &2) * e / (&2 * z0))` [] THEN
  note `&0 < (Q0 - S0 + &1 / &2) * e / (&2 * z0)` [REAL_LT_MUL;
    REAL_LT_DIV;
    REAL_ARITH `&0 < &2`] THEN
  choose_specializing `M:num` [`\i:num. &1 / &2 * log2 (&1 + (&1 + K i / N i) * N i powreal (&1 + &2 * log2 (S i) / log2 (N i))) - &1 / &2 * log2 (((&1 + K i / N i) + &1 / N i) / (B i / N i)) - (((&2 * B i - &1) * log2 (bkzdelta (B i)) + log2 (Q i)) - &2 * sqrt ((N i + &1) * log2 (bkzdelta (B i)) * log2 (Q i)))`;
    `(Q0 - S0 + &1 / &2) * e / (&2 * z0)`] positive_limit_is_eventually_positive THEN
  EXISTS_TAC `M:num` THEN
  ASM_MESON_TAC [converse_asymp2_lemma4]);;

let converse_denouement = prove(`
  !n:num->real k:num->real b:num->real
  z0:real z1:real g:num->real L:num->real.
  (!i. &1 < n(i))
  /\ (!i. g(i) <= (b(i)/n(i) - z0)*log2(n(i)) - z1)
  /\ o1_seq g
  /\ L = (\i. (z0 + (z1 + g(i))/log2(n(i)))*n(i))
  ==> (!i. L(i) <= b(i))
  /\ o1_seq (\i. (L(i)/n(i) - z0)*log2(n(i)) - z1)
  `,
  REPEAT STRIP_TAC THENL
  [note `g(i:num) <= (b(i)/n(i) - z0)*log2(n(i)) - z1` [] THEN
    real_linear `g(i:num) <= (b(i)/n(i) - z0)*log2(n(i)) - z1 ==> z1 + g(i) <= (b(i)/n(i) - z0)*log2(n(i))` THEN
    note `&0 < log2(n(i:num))` [log2_gt1] THEN
    note `(z1 + g(i))/log2(n(i:num)) <= ((b(i)/n(i) - z0)*log2(n(i)))/log2(n(i))` [REAL_LE_DIV2_EQ] THEN
    real_cancel `&0 < log2(n(i:num)) ==> ((b i / n i - z0) * log2 (n i)) / log2 (n i) = b(i)/n(i) - z0` THEN
    note `(z1 + g(i))/log2(n(i:num)) <= b(i)/n(i) - z0` [] THEN
    real_linear `(z1 + g(i:num))/log2(n(i:num)) <= b(i)/n(i) - z0 ==> z0 + (z1 + g(i))/log2(n(i:num)) <= b(i)/n(i)` THEN
    note `&1 < n(i:num)` [] THEN
    real_linear `&1 < n(i:num) ==> &0 <= n(i)` THEN
    note `(z0 + (z1 + g(i))/log2(n(i:num)))*n(i) <= (b(i)/n(i))*n(i)` [REAL_LE_RMUL] THEN
    real_cancel `&1 < n(i:num) ==> (b(i)/n(i))*n(i) = b(i)` THEN
    ASM_MESON_TAC [];
    SUBGOAL_THEN `!i:num. g(i) = (L(i) / n(i) - z0) * log2(n(i)) - z1` MP_TAC THENL
    [STRIP_TAC THEN
      note `&1 < n(i:num)` [] THEN
      note `L(i:num) = (z0 + (z1 + g(i))/log2(n(i)))*n(i)` [] THEN
      real_cancel `&1 < n(i) /\ L(i:num) = (z0 + (z1 + g(i))/log2(n(i)))*n(i) ==> L(i)/n(i) - z0 = (z1 + g(i))/log2(n(i))` THEN
      note `&0 < log2(n(i:num))` [log2_gt1] THEN
      real_cancel `&0 < log2(n(i:num)) /\ L(i)/n(i) - z0 = (z1 + g(i))/log2(n(i)) ==> (L(i)/n(i) - z0) * log2(n(i)) - z1 = g(i)` THEN
      ASM_MESON_TAC [];
      ASM_MESON_TAC [EQ_EXT]]]);;

let converse_main = prove(`
  !n:num->real q:num->real s:num->real
   k:num->real b:num->real
   Q0:real Q1:real S0:real S1:real
   x0:real z0:real z1:real.
  (!i. &1 < n(i))
  /\ (!i. &1 < q(i))
  /\ (!i. &0 < s(i))
  /\ o1_seq (\i. &1 / n(i))
  /\ o1_seq (\i. (log2(q(i))/log2(n(i)) - Q0) * log2(n(i)) - Q1)
  /\ o1_seq (\i. (log2(s(i))/log2(n(i)) - S0) * log2(n(i)) - S1)
  /\ -- &1 / &2 < S0
  /\ S0 <= &1 / &2
  /\ -- &1 / &2 < Q0 - S0
  /\ &1 / &2 < Q0 + S0
  /\ x0 = (Q0 + S0 - &1 / &2)/(Q0 - S0 + &1 / &2)
  /\ z0 = &2 * Q0/((Q0 - S0 + &1 / &2) pow 2)
  /\ z1 =   (&2 * S1 + log2(z0)
             - (S0 - Q0 + &3 / &2) * (log2(z0) - log2(&2 * pi * exp(&1)))
             - Q1 * (Q0 + S0 - &1 / &2)/Q0
            )
          * (&2 * Q0) / ((Q0 - S0 + &1 / &2) pow 3)
  /\ (!i. &0 < k(i))
  /\ (!i. k(i) <= &100 * n(i))
  /\ (!i. &60 <= b(i))
  /\ (!i. b(i) <= ceil(n(i) + k(i) + &1))
  /\ (?m. !i. m <= i ==> standardratio (n(i)) (q(i)) (s(i)) (k(i)) (b(i)) <= &1)
  ==> 
  ?L. (!i. L(i) <= b(i))
  /\ o1_seq (\i. (L(i)/n(i) - z0) * log2(n(i)) - z1)
  `,
  REPEAT STRIP_TAC THEN
  ASM_CASES_TAC `!e:real. &0 < e ==> FINITE {i:num | (b(i)/n(i) - z0)*log2(n(i)) - z1 < --e}` THENL
  [choose_specializing `g:num->real` [`\i:num. (b(i)/n(i) - z0)*log2(n(i)) - z1`] o1_lower_main THEN
    def `L:num->real` `\i:num. (z0 + (z1 + g(i))/log2(n(i)))*n(i)` THEN
    ASM_MESON_TAC [converse_denouement];
    ALL_TAC] THEN
  choose `e:real` `&0 < e /\ INFINITE {i:num | (b(i)/n(i) - z0)*log2(n(i)) - z1 < --e}` [INFINITE] THEN
  note `INFINITE {i:num | (b(i)/n(i) - z0)*log2(n(i)) - z1 < --e}` [] THEN
  specialize [`\i:num. (b(i)/n(i) - z0)*log2(n(i)) - z1 < --e`;
    `m:num`] num_chop_infinite THEN
  choose `r:num->num` `(!m n:num. m < n ==> r(m) < r(n)) /\ IMAGE r (:num) = {i:num | m <= i /\ (b(i)/n(i) - z0)*log2(n(i)) - z1 < --e}` [INFINITE_ENUMERATE] THEN
  notetac `IMAGE r (:num) = {i:num | m <= i /\ (b(i)/n(i) - z0)*log2(n(i)) - z1 < --e} ==> !i. m <= r(i) /\ (b(r(i))/n(r(i)) - z0)*log2(n(r(i))) - z1 < --e` (SET_TAC []) THEN
  note `!i:num. m <= r(i) /\ (b(r(i):num)/n(r(i)) - z0)*log2(n(r(i))) - z1 < --e` [] THEN
  note `!i:num. &0 <= k(r(i):num)/n(r(i))` [REAL_ARITH `&1 < x ==> &0 < x`;
    REAL_LT_DIV;
    REAL_LT_IMP_LE] THEN
  note `!i:num. k(r(i):num)/n(r(i)) <= &100` [REAL_LE_DIV2_EQ;
    REAL_ARITH `&1 < x ==> &0 < x`;
    REAL_FIELD `&0 < x ==> (&100 * x) / x = &100`] THEN
  specialize [`\i:num. k(r(i):num):real / n(r(i))`;
    `&0`;
    `&100`] seq_bolzano2 THEN
  choose `r2:num->num` `?X0:real. (!i j. i < j ==> r2(i) < r2(j)) /\ o1_seq (\i. k(r(r2(i:num):num):num)/n(r(r2(i))) - X0) /\ &0 <= X0` [] THEN
  choose `X0:real` `(!i j. i < j ==> r2(i) < r2(j)) /\ o1_seq (\i. k(r(r2(i:num):num):num)/n(r(r2(i))) - X0) /\ &0 <= X0` [] THEN
  def `N:num->real` `(n:num->real) o (r:num->num) o (r2:num->num)` THEN
  def `K:num->real` `(k:num->real) o (r:num->num) o (r2:num->num)` THEN
  def `Q:num->real` `(q:num->real) o (r:num->num) o (r2:num->num)` THEN
  def `S:num->real` `(s:num->real) o (r:num->num) o (r2:num->num)` THEN
  def `beta:num->real` `(b:num->real) o (r:num->num) o (r2:num->num)` THEN
  def `B:num->real` `\i:num. (z0 + (z1 - e)/log2(N(i)))*N(i)` THEN
  note `!i j. i < j:num ==> r(i) < r(j):num` [] THEN
  note `!i j. i < j:num ==> r2(i) < r2(j):num` [] THEN
  specialize [`&1`;
    `n:num->real`;
    `N:num->real`;
    `r:num->num`;
    `r2:num->num`] converse_X_gt THEN
  specialize [`&1`;
    `q:num->real`;
    `Q:num->real`;
    `r:num->num`;
    `r2:num->num`] converse_X_gt THEN
  specialize [`&0`;
    `s:num->real`;
    `S:num->real`;
    `r:num->num`;
    `r2:num->num`] converse_X_gt THEN
  specialize [`&0`;
    `k:num->real`;
    `K:num->real`;
    `r:num->num`;
    `r2:num->num`] converse_X_gt THEN
  specialize [`&60`;
    `b:num->real`;
    `beta:num->real`;
    `r:num->num`;
    `r2:num->num`] converse_X_ge THEN
  specialize [`n:num->real`;
    `N:num->real`;
    `r:num->num`;
    `r2:num->num`] converse_overN_o1 THEN
  note `!i:num. beta(i) <= B(i)` [converse_beta_le_B] THEN
  note `!i:num. &60 <= B(i)` [REAL_LE_TRANS] THEN
  note `!i:num. standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (B(i)) <= standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (beta(i))` [rho_decreasing4] THEN
  note `!i:num. standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (beta(i)) <= &1` [converse_standardratio_rr2] THEN
  note `!i:num. standardratio (N(i)) (Q(i)) (S(i)) (K(i)) (B(i)) <= &1` [REAL_LE_TRANS] THEN
  specialize [`n:num->real`;
    `N:num->real`;
    `q:num->real`;
    `Q:num->real`;
    `Q0:real`;
    `Q1:real`;
    `r:num->num`;
    `r2:num->num`] converse_X_o1 THEN
  specialize [`n:num->real`;
    `N:num->real`;
    `s:num->real`;
    `S:num->real`;
    `S0:real`;
    `S1:real`;
    `r:num->num`;
    `r2:num->num`] converse_X_o1 THEN
  note `o1_seq (\i:num. K(i)/N(i) - X0:real)` [converse_K_X0] THEN
  note `o1_seq (\i:num. (B(i)/N(i) - z0)*log2(N(i)) - (z1 - e))` [converse_B_o1] THEN
  real_linear `-- &1 / &2 < S0 ==> &0 < &1 + &2 * S0` THEN
  note `&0 <= X0` [] THEN
  note `&0 < e` [] THEN
  specialize [`N:num->real`;
    `Q:num->real`;
    `S:num->real`;
    `K:num->real`;
    `B:num->real`;
    `Q0:real`;
    `Q1:real`;
    `S0:real`;
    `S1:real`;
    `x0:real`;
    `z0:real`;
    `z1:real`;
    `X0:real`;
    `e:real`] converse_asymp1 THEN
  note `o1_seq (\i. K(i)/N(i) - x0)` [] THEN
  choose_specializing `M:num` [`N:num->real`;
    `Q:num->real`;
    `S:num->real`;
    `K:num->real`;
    `B:num->real`;
    `Q0:real`;
    `Q1:real`;
    `S0:real`;
    `S1:real`;
    `x0:real`;
    `z0:real`;
    `z1:real`;
    `e:real`] converse_asymp2 THEN
  def `U:num` `MAX m M` THEN
  note `m <= U:num` [max_le;
    LE_REFL] THEN
  note `M <= U:num` [max_le;
    LE_REFL] THEN
  note `!i:num. m <= i ==> standardratio (N i) (Q i) (S i) (K i) (B i) <= &1` [o_THM] THEN
  note `&1 < standardratio (N (U:num)) (Q U) (S U) (K U) (B U)` [] THEN
  note `standardratio (N (U:num)) (Q U) (S U) (K U) (B U) <= &1` [] THEN
  ASM_MESON_TAC [REAL_NOT_LT]);;

(* ----- review external theorems *)

let forward_main = forward_main;;

let converse_main = converse_main;;