Structure transcTheory
signature transcTheory =
sig
type thm = Thm.thm
(* Definitions *)
val Dint : thm
val acs : thm
val asn : thm
val atn : thm
val cos : thm
val division : thm
val dsize : thm
val exp : thm
val fine : thm
val gauge : thm
val ln : thm
val pi : thm
val root : thm
val rpow_def : thm
val rsum : thm
val sin : thm
val sqrt : thm
val tan : thm
val tdiv : thm
(* Theorems *)
val ACS : thm
val ACS_BOUNDS : thm
val ACS_BOUNDS_LT : thm
val ACS_COS : thm
val ASN : thm
val ASN_BOUNDS : thm
val ASN_BOUNDS_LT : thm
val ASN_SIN : thm
val ATN : thm
val ATN_BOUNDS : thm
val ATN_TAN : thm
val BASE_RPOW_LE : thm
val BASE_RPOW_LT : thm
val COS_0 : thm
val COS_2 : thm
val COS_ACS : thm
val COS_ADD : thm
val COS_ASN_NZ : thm
val COS_ATN_NZ : thm
val COS_BOUND : thm
val COS_BOUNDS : thm
val COS_CONVERGES : thm
val COS_DOUBLE : thm
val COS_FDIFF : thm
val COS_ISZERO : thm
val COS_NEG : thm
val COS_NPI : thm
val COS_PAIRED : thm
val COS_PERIODIC : thm
val COS_PERIODIC_PI : thm
val COS_PI : thm
val COS_PI2 : thm
val COS_POS_PI : thm
val COS_POS_PI2 : thm
val COS_POS_PI2_LE : thm
val COS_POS_PI_LE : thm
val COS_SIN : thm
val COS_SIN_SQ : thm
val COS_SIN_SQRT : thm
val COS_TOTAL : thm
val COS_ZERO : thm
val COS_ZERO_LEMMA : thm
val DIFF_ACS : thm
val DIFF_ACS_LEMMA : thm
val DIFF_ASN : thm
val DIFF_ASN_LEMMA : thm
val DIFF_ATN : thm
val DIFF_COMPOSITE : thm
val DIFF_COMPOSITE_EXP : thm
val DIFF_COS : thm
val DIFF_EXP : thm
val DIFF_LN : thm
val DIFF_LN_COMPOSITE : thm
val DIFF_RPOW : thm
val DIFF_SIN : thm
val DIFF_TAN : thm
val DINT_UNIQ : thm
val DIVISION_0 : thm
val DIVISION_1 : thm
val DIVISION_APPEND : thm
val DIVISION_EQ : thm
val DIVISION_EXISTS : thm
val DIVISION_GT : thm
val DIVISION_LBOUND : thm
val DIVISION_LBOUND_LT : thm
val DIVISION_LE : thm
val DIVISION_LHS : thm
val DIVISION_LT : thm
val DIVISION_LT_GEN : thm
val DIVISION_RHS : thm
val DIVISION_SINGLE : thm
val DIVISION_THM : thm
val DIVISION_UBOUND : thm
val DIVISION_UBOUND_LT : thm
val EXP_0 : thm
val EXP_ADD : thm
val EXP_ADD_MUL : thm
val EXP_CONVERGES : thm
val EXP_FDIFF : thm
val EXP_INJ : thm
val EXP_LE_X : thm
val EXP_LN : thm
val EXP_LT_1 : thm
val EXP_MONO_IMP : thm
val EXP_MONO_LE : thm
val EXP_MONO_LT : thm
val EXP_N : thm
val EXP_NEG : thm
val EXP_NEG_MUL : thm
val EXP_NEG_MUL2 : thm
val EXP_NZ : thm
val EXP_POS_LE : thm
val EXP_POS_LT : thm
val EXP_SUB : thm
val EXP_TOTAL : thm
val EXP_TOTAL_LEMMA : thm
val FINE_MIN : thm
val FTC1 : thm
val GAUGE_MIN : thm
val GEN_RPOW : thm
val INTEGRAL_NULL : thm
val LN_1 : thm
val LN_DIV : thm
val LN_EXP : thm
val LN_INJ : thm
val LN_INV : thm
val LN_LE : thm
val LN_LT_X : thm
val LN_MONO_LE : thm
val LN_MONO_LT : thm
val LN_MUL : thm
val LN_POS : thm
val LN_POW : thm
val LN_RPOW : thm
val MCLAURIN : thm
val MCLAURIN_ALL_LE : thm
val MCLAURIN_ALL_LT : thm
val MCLAURIN_EXP_LE : thm
val MCLAURIN_EXP_LT : thm
val MCLAURIN_NEG : thm
val MCLAURIN_ZERO : thm
val ONE_RPOW : thm
val PI2 : thm
val PI2_BOUNDS : thm
val PI_POS : thm
val POW_2_SQRT : thm
val POW_ROOT_POS : thm
val REAL_DIV_SQRT : thm
val ROOT_0 : thm
val ROOT_1 : thm
val ROOT_DIV : thm
val ROOT_INV : thm
val ROOT_LN : thm
val ROOT_LT_LEMMA : thm
val ROOT_MONO_LE : thm
val ROOT_MUL : thm
val ROOT_POS : thm
val ROOT_POS_LT : thm
val ROOT_POS_UNIQ : thm
val ROOT_POW_POS : thm
val RPOW_0 : thm
val RPOW_1 : thm
val RPOW_ADD : thm
val RPOW_ADD_MUL : thm
val RPOW_DIV : thm
val RPOW_DIV_BASE : thm
val RPOW_INV : thm
val RPOW_LE : thm
val RPOW_LT : thm
val RPOW_MUL : thm
val RPOW_NZ : thm
val RPOW_POS_LT : thm
val RPOW_RPOW : thm
val RPOW_SUB : thm
val RPOW_SUC_N : thm
val RPOW_UNIQ_BASE : thm
val RPOW_UNIQ_EXP : thm
val SIN_0 : thm
val SIN_ACS_NZ : thm
val SIN_ADD : thm
val SIN_ASN : thm
val SIN_BOUND : thm
val SIN_BOUNDS : thm
val SIN_CIRCLE : thm
val SIN_CONVERGES : thm
val SIN_COS : thm
val SIN_COS_ADD : thm
val SIN_COS_NEG : thm
val SIN_COS_SQ : thm
val SIN_COS_SQRT : thm
val SIN_DOUBLE : thm
val SIN_FDIFF : thm
val SIN_NEG : thm
val SIN_NEGLEMMA : thm
val SIN_NPI : thm
val SIN_PAIRED : thm
val SIN_PERIODIC : thm
val SIN_PERIODIC_PI : thm
val SIN_PI : thm
val SIN_PI2 : thm
val SIN_POS : thm
val SIN_POS_PI : thm
val SIN_POS_PI2 : thm
val SIN_POS_PI2_LE : thm
val SIN_POS_PI_LE : thm
val SIN_TOTAL : thm
val SIN_ZERO : thm
val SIN_ZERO_LEMMA : thm
val SQRT_0 : thm
val SQRT_1 : thm
val SQRT_DIV : thm
val SQRT_EQ : thm
val SQRT_EVEN_POW2 : thm
val SQRT_INV : thm
val SQRT_MONO_LE : thm
val SQRT_MUL : thm
val SQRT_POS_LE : thm
val SQRT_POS_LT : thm
val SQRT_POS_UNIQ : thm
val SQRT_POW2 : thm
val SQRT_POW_2 : thm
val TAN_0 : thm
val TAN_ADD : thm
val TAN_ATN : thm
val TAN_DOUBLE : thm
val TAN_NEG : thm
val TAN_NPI : thm
val TAN_PERIODIC : thm
val TAN_PI : thm
val TAN_POS_PI2 : thm
val TAN_SEC : thm
val TAN_TOTAL : thm
val TAN_TOTAL_LEMMA : thm
val TAN_TOTAL_POS : thm
val transc_grammars : type_grammar.grammar * term_grammar.grammar
(*
[powser] Parent theory of "transc"
[Dint] Definition
⊢ ∀a b f k.
Dint (a,b) f k ⇔
∀e.
0 < e ⇒
∃g.
gauge (λx. a ≤ x ∧ x ≤ b) g ∧
∀D p.
tdiv (a,b) (D,p) ∧ fine g (D,p) ⇒
abs (rsum (D,p) f − k) < e
[acs] Definition
⊢ ∀y. acs y = @x. 0 ≤ x ∧ x ≤ pi ∧ (cos x = y)
[asn] Definition
⊢ ∀y. asn y = @x. -(pi / 2) ≤ x ∧ x ≤ pi / 2 ∧ (sin x = y)
[atn] Definition
⊢ ∀y. atn y = @x. -(pi / 2) < x ∧ x < pi / 2 ∧ (tan x = y)
[cos] Definition
⊢ ∀x.
cos x =
suminf
(λn.
(λn. if EVEN n then -1 pow (n DIV 2) / &FACT n else 0) n *
x pow n)
[division] Definition
⊢ ∀a b D.
division (a,b) D ⇔
(D 0 = a) ∧
∃N. (∀n. n < N ⇒ D n < D (SUC n)) ∧ ∀n. n ≥ N ⇒ (D n = b)
[dsize] Definition
⊢ ∀D.
dsize D =
@N. (∀n. n < N ⇒ D n < D (SUC n)) ∧ ∀n. n ≥ N ⇒ (D n = D N)
[exp] Definition
⊢ ∀x. exp x = suminf (λn. (λn. (&FACT n)⁻¹) n * x pow n)
[fine] Definition
⊢ ∀g D p. fine g (D,p) ⇔ ∀n. n < dsize D ⇒ D (SUC n) − D n < g (p n)
[gauge] Definition
⊢ ∀E g. gauge E g ⇔ ∀x. E x ⇒ 0 < g x
[ln] Definition
⊢ ∀x. ln x = @u. exp u = x
[pi] Definition
⊢ pi = 2 * @x. 0 ≤ x ∧ x ≤ 2 ∧ (cos x = 0)
[root] Definition
⊢ ∀n x. root n x = @u. (0 < x ⇒ 0 < u) ∧ (u pow n = x)
[rpow_def] Definition
⊢ ∀a b. a rpow b = exp (b * ln a)
[rsum] Definition
⊢ ∀D p f.
rsum (D,p) f =
sum (0,dsize D) (λn. f (p n) * (D (SUC n) − D n))
[sin] Definition
⊢ ∀x.
sin x =
suminf
(λn.
(λn.
if EVEN n then 0
else -1 pow ((n − 1) DIV 2) / &FACT n) n * x pow n)
[sqrt] Definition
⊢ ∀x. sqrt x = root 2 x
[tan] Definition
⊢ ∀x. tan x = sin x / cos x
[tdiv] Definition
⊢ ∀a b D p.
tdiv (a,b) (D,p) ⇔
division (a,b) D ∧ ∀n. D n ≤ p n ∧ p n ≤ D (SUC n)
[ACS] Theorem
⊢ ∀y. -1 ≤ y ∧ y ≤ 1 ⇒ 0 ≤ acs y ∧ acs y ≤ pi ∧ (cos (acs y) = y)
[ACS_BOUNDS] Theorem
⊢ ∀y. -1 ≤ y ∧ y ≤ 1 ⇒ 0 ≤ acs y ∧ acs y ≤ pi
[ACS_BOUNDS_LT] Theorem
⊢ ∀y. -1 < y ∧ y < 1 ⇒ 0 < acs y ∧ acs y < pi
[ACS_COS] Theorem
⊢ ∀y. -1 ≤ y ∧ y ≤ 1 ⇒ (cos (acs y) = y)
[ASN] Theorem
⊢ ∀y.
-1 ≤ y ∧ y ≤ 1 ⇒
-(pi / 2) ≤ asn y ∧ asn y ≤ pi / 2 ∧ (sin (asn y) = y)
[ASN_BOUNDS] Theorem
⊢ ∀y. -1 ≤ y ∧ y ≤ 1 ⇒ -(pi / 2) ≤ asn y ∧ asn y ≤ pi / 2
[ASN_BOUNDS_LT] Theorem
⊢ ∀y. -1 < y ∧ y < 1 ⇒ -(pi / 2) < asn y ∧ asn y < pi / 2
[ASN_SIN] Theorem
⊢ ∀y. -1 ≤ y ∧ y ≤ 1 ⇒ (sin (asn y) = y)
[ATN] Theorem
⊢ ∀y. -(pi / 2) < atn y ∧ atn y < pi / 2 ∧ (tan (atn y) = y)
[ATN_BOUNDS] Theorem
⊢ ∀y. -(pi / 2) < atn y ∧ atn y < pi / 2
[ATN_TAN] Theorem
⊢ ∀y. tan (atn y) = y
[BASE_RPOW_LE] Theorem
⊢ ∀a b c. 0 < a ∧ 0 < c ∧ 0 < b ⇒ (a rpow b ≤ c rpow b ⇔ a ≤ c)
[BASE_RPOW_LT] Theorem
⊢ ∀a b c. 0 < a ∧ 0 < c ∧ 0 < b ⇒ (a rpow b < c rpow b ⇔ a < c)
[COS_0] Theorem
⊢ cos 0 = 1
[COS_2] Theorem
⊢ cos 2 < 0
[COS_ACS] Theorem
⊢ ∀x. 0 ≤ x ∧ x ≤ pi ⇒ (acs (cos x) = x)
[COS_ADD] Theorem
⊢ ∀x y. cos (x + y) = cos x * cos y − sin x * sin y
[COS_ASN_NZ] Theorem
⊢ ∀x. -1 < x ∧ x < 1 ⇒ cos (asn x) ≠ 0
[COS_ATN_NZ] Theorem
⊢ ∀x. cos (atn x) ≠ 0
[COS_BOUND] Theorem
⊢ ∀x. abs (cos x) ≤ 1
[COS_BOUNDS] Theorem
⊢ ∀x. -1 ≤ cos x ∧ cos x ≤ 1
[COS_CONVERGES] Theorem
⊢ ∀x.
(λn.
(λn. if EVEN n then -1 pow (n DIV 2) / &FACT n else 0) n *
x pow n) sums cos x
[COS_DOUBLE] Theorem
⊢ ∀x. cos (2 * x) = cos x pow 2 − sin x pow 2
[COS_FDIFF] Theorem
⊢ diffs (λn. if EVEN n then -1 pow (n DIV 2) / &FACT n else 0) =
(λn.
-(λn. if EVEN n then 0 else -1 pow ((n − 1) DIV 2) / &FACT n)
n)
[COS_ISZERO] Theorem
⊢ ∃!x. 0 ≤ x ∧ x ≤ 2 ∧ (cos x = 0)
[COS_NEG] Theorem
⊢ ∀x. cos (-x) = cos x
[COS_NPI] Theorem
⊢ ∀n. cos (&n * pi) = -1 pow n
[COS_PAIRED] Theorem
⊢ ∀x. (λn. -1 pow n / &FACT (2 * n) * x pow (2 * n)) sums cos x
[COS_PERIODIC] Theorem
⊢ ∀x. cos (x + 2 * pi) = cos x
[COS_PERIODIC_PI] Theorem
⊢ ∀x. cos (x + pi) = -cos x
[COS_PI] Theorem
⊢ cos pi = -1
[COS_PI2] Theorem
⊢ cos (pi / 2) = 0
[COS_POS_PI] Theorem
⊢ ∀x. -(pi / 2) < x ∧ x < pi / 2 ⇒ 0 < cos x
[COS_POS_PI2] Theorem
⊢ ∀x. 0 < x ∧ x < pi / 2 ⇒ 0 < cos x
[COS_POS_PI2_LE] Theorem
⊢ ∀x. 0 ≤ x ∧ x ≤ pi / 2 ⇒ 0 ≤ cos x
[COS_POS_PI_LE] Theorem
⊢ ∀x. -(pi / 2) ≤ x ∧ x ≤ pi / 2 ⇒ 0 ≤ cos x
[COS_SIN] Theorem
⊢ ∀x. cos x = sin (pi / 2 − x)
[COS_SIN_SQ] Theorem
⊢ ∀x. -(pi / 2) ≤ x ∧ x ≤ pi / 2 ⇒ (cos x = sqrt (1 − sin x pow 2))
[COS_SIN_SQRT] Theorem
⊢ ∀x. 0 ≤ cos x ⇒ (cos x = sqrt (1 − sin x pow 2))
[COS_TOTAL] Theorem
⊢ ∀y. -1 ≤ y ∧ y ≤ 1 ⇒ ∃!x. 0 ≤ x ∧ x ≤ pi ∧ (cos x = y)
[COS_ZERO] Theorem
⊢ ∀x.
(cos x = 0) ⇔
(∃n. ¬EVEN n ∧ (x = &n * (pi / 2))) ∨
∃n. ¬EVEN n ∧ (x = -(&n * (pi / 2)))
[COS_ZERO_LEMMA] Theorem
⊢ ∀x. 0 ≤ x ∧ (cos x = 0) ⇒ ∃n. ¬EVEN n ∧ (x = &n * (pi / 2))
[DIFF_ACS] Theorem
⊢ ∀x. -1 < x ∧ x < 1 ⇒ (acs diffl -(sqrt (1 − x pow 2))⁻¹) x
[DIFF_ACS_LEMMA] Theorem
⊢ ∀x. -1 < x ∧ x < 1 ⇒ (acs diffl (-sin (acs x))⁻¹) x
[DIFF_ASN] Theorem
⊢ ∀x. -1 < x ∧ x < 1 ⇒ (asn diffl (sqrt (1 − x pow 2))⁻¹) x
[DIFF_ASN_LEMMA] Theorem
⊢ ∀x. -1 < x ∧ x < 1 ⇒ (asn diffl (cos (asn x))⁻¹) x
[DIFF_ATN] Theorem
⊢ ∀x. (atn diffl (1 + x pow 2)⁻¹) x
[DIFF_COMPOSITE] Theorem
⊢ ((f diffl l) x ∧ f x ≠ 0 ⇒ ((λx. (f x)⁻¹) diffl -(l / f x pow 2)) x) ∧
((f diffl l) x ∧ (g diffl m) x ∧ g x ≠ 0 ⇒
((λx. f x / g x) diffl ((l * g x − m * f x) / g x pow 2)) x) ∧
((f diffl l) x ∧ (g diffl m) x ⇒ ((λx. f x + g x) diffl (l + m)) x) ∧
((f diffl l) x ∧ (g diffl m) x ⇒
((λx. f x * g x) diffl (l * g x + m * f x)) x) ∧
((f diffl l) x ∧ (g diffl m) x ⇒ ((λx. f x − g x) diffl (l − m)) x) ∧
((f diffl l) x ⇒ ((λx. -f x) diffl -l) x) ∧
((g diffl m) x ⇒
((λx. g x pow n) diffl (&n * g x pow (n − 1) * m)) x) ∧
((g diffl m) x ⇒ ((λx. exp (g x)) diffl (exp (g x) * m)) x) ∧
((g diffl m) x ⇒ ((λx. sin (g x)) diffl (cos (g x) * m)) x) ∧
((g diffl m) x ⇒ ((λx. cos (g x)) diffl (-sin (g x) * m)) x)
[DIFF_COMPOSITE_EXP] Theorem
⊢ ∀g m x. (g diffl m) x ⇒ ((λx. exp (g x)) diffl (exp (g x) * m)) x
[DIFF_COS] Theorem
⊢ ∀x. (cos diffl -sin x) x
[DIFF_EXP] Theorem
⊢ ∀x. (exp diffl exp x) x
[DIFF_LN] Theorem
⊢ ∀x. 0 < x ⇒ (ln diffl x⁻¹) x
[DIFF_LN_COMPOSITE] Theorem
⊢ ∀g m x.
(g diffl m) x ∧ 0 < g x ⇒
((λx. ln (g x)) diffl ((g x)⁻¹ * m)) x
[DIFF_RPOW] Theorem
⊢ ∀x y. 0 < x ⇒ ((λx. x rpow y) diffl (y * x rpow (y − 1))) x
[DIFF_SIN] Theorem
⊢ ∀x. (sin diffl cos x) x
[DIFF_TAN] Theorem
⊢ ∀x. cos x ≠ 0 ⇒ (tan diffl (cos x pow 2)⁻¹) x
[DINT_UNIQ] Theorem
⊢ ∀a b f k1 k2. a ≤ b ∧ Dint (a,b) f k1 ∧ Dint (a,b) f k2 ⇒ (k1 = k2)
[DIVISION_0] Theorem
⊢ ∀a b. (a = b) ⇒ (dsize (λn. if n = 0 then a else b) = 0)
[DIVISION_1] Theorem
⊢ ∀a b. a < b ⇒ (dsize (λn. if n = 0 then a else b) = 1)
[DIVISION_APPEND] Theorem
⊢ ∀a b c.
(∃D1 p1. tdiv (a,b) (D1,p1) ∧ fine g (D1,p1)) ∧
(∃D2 p2. tdiv (b,c) (D2,p2) ∧ fine g (D2,p2)) ⇒
∃D p. tdiv (a,c) (D,p) ∧ fine g (D,p)
[DIVISION_EQ] Theorem
⊢ ∀D a b. division (a,b) D ⇒ ((a = b) ⇔ (dsize D = 0))
[DIVISION_EXISTS] Theorem
⊢ ∀a b g.
a ≤ b ∧ gauge (λx. a ≤ x ∧ x ≤ b) g ⇒
∃D p. tdiv (a,b) (D,p) ∧ fine g (D,p)
[DIVISION_GT] Theorem
⊢ ∀D a b. division (a,b) D ⇒ ∀n. n < dsize D ⇒ D n < D (dsize D)
[DIVISION_LBOUND] Theorem
⊢ ∀D a b. division (a,b) D ⇒ ∀r. a ≤ D r
[DIVISION_LBOUND_LT] Theorem
⊢ ∀D a b. division (a,b) D ∧ dsize D ≠ 0 ⇒ ∀n. a < D (SUC n)
[DIVISION_LE] Theorem
⊢ ∀D a b. division (a,b) D ⇒ a ≤ b
[DIVISION_LHS] Theorem
⊢ ∀D a b. division (a,b) D ⇒ (D 0 = a)
[DIVISION_LT] Theorem
⊢ ∀D a b. division (a,b) D ⇒ ∀n. n < dsize D ⇒ D 0 < D (SUC n)
[DIVISION_LT_GEN] Theorem
⊢ ∀D a b m n. division (a,b) D ∧ m < n ∧ n ≤ dsize D ⇒ D m < D n
[DIVISION_RHS] Theorem
⊢ ∀D a b. division (a,b) D ⇒ (D (dsize D) = b)
[DIVISION_SINGLE] Theorem
⊢ ∀a b. a ≤ b ⇒ division (a,b) (λn. if n = 0 then a else b)
[DIVISION_THM] Theorem
⊢ ∀D a b.
division (a,b) D ⇔
(D 0 = a) ∧ (∀n. n < dsize D ⇒ D n < D (SUC n)) ∧
∀n. n ≥ dsize D ⇒ (D n = b)
[DIVISION_UBOUND] Theorem
⊢ ∀D a b. division (a,b) D ⇒ ∀r. D r ≤ b
[DIVISION_UBOUND_LT] Theorem
⊢ ∀D a b n. division (a,b) D ∧ n < dsize D ⇒ D n < b
[EXP_0] Theorem
⊢ exp 0 = 1
[EXP_ADD] Theorem
⊢ ∀x y. exp (x + y) = exp x * exp y
[EXP_ADD_MUL] Theorem
⊢ ∀x y. exp (x + y) * exp (-x) = exp y
[EXP_CONVERGES] Theorem
⊢ ∀x. (λn. (λn. (&FACT n)⁻¹) n * x pow n) sums exp x
[EXP_FDIFF] Theorem
⊢ diffs (λn. (&FACT n)⁻¹) = (λn. (&FACT n)⁻¹)
[EXP_INJ] Theorem
⊢ ∀x y. (exp x = exp y) ⇔ (x = y)
[EXP_LE_X] Theorem
⊢ ∀x. 0 ≤ x ⇒ 1 + x ≤ exp x
[EXP_LN] Theorem
⊢ ∀x. (exp (ln x) = x) ⇔ 0 < x
[EXP_LT_1] Theorem
⊢ ∀x. 0 < x ⇒ 1 < exp x
[EXP_MONO_IMP] Theorem
⊢ ∀x y. x < y ⇒ exp x < exp y
[EXP_MONO_LE] Theorem
⊢ ∀x y. exp x ≤ exp y ⇔ x ≤ y
[EXP_MONO_LT] Theorem
⊢ ∀x y. exp x < exp y ⇔ x < y
[EXP_N] Theorem
⊢ ∀n x. exp (&n * x) = exp x pow n
[EXP_NEG] Theorem
⊢ ∀x. exp (-x) = (exp x)⁻¹
[EXP_NEG_MUL] Theorem
⊢ ∀x. exp x * exp (-x) = 1
[EXP_NEG_MUL2] Theorem
⊢ ∀x. exp (-x) * exp x = 1
[EXP_NZ] Theorem
⊢ ∀x. exp x ≠ 0
[EXP_POS_LE] Theorem
⊢ ∀x. 0 ≤ exp x
[EXP_POS_LT] Theorem
⊢ ∀x. 0 < exp x
[EXP_SUB] Theorem
⊢ ∀x y. exp (x − y) = exp x / exp y
[EXP_TOTAL] Theorem
⊢ ∀y. 0 < y ⇒ ∃x. exp x = y
[EXP_TOTAL_LEMMA] Theorem
⊢ ∀y. 1 ≤ y ⇒ ∃x. 0 ≤ x ∧ x ≤ y − 1 ∧ (exp x = y)
[FINE_MIN] Theorem
⊢ ∀g1 g2 D p.
fine (λx. if g1 x < g2 x then g1 x else g2 x) (D,p) ⇒
fine g1 (D,p) ∧ fine g2 (D,p)
[FTC1] Theorem
⊢ ∀f f' a b.
a ≤ b ∧ (∀x. a ≤ x ∧ x ≤ b ⇒ (f diffl f' x) x) ⇒
Dint (a,b) f' (f b − f a)
[GAUGE_MIN] Theorem
⊢ ∀E g1 g2.
gauge E g1 ∧ gauge E g2 ⇒
gauge E (λx. if g1 x < g2 x then g1 x else g2 x)
[GEN_RPOW] Theorem
⊢ ∀a n. 0 < a ⇒ (a pow n = a rpow &n)
[INTEGRAL_NULL] Theorem
⊢ ∀f a. Dint (a,a) f 0
[LN_1] Theorem
⊢ ln 1 = 0
[LN_DIV] Theorem
⊢ ∀x y. 0 < x ∧ 0 < y ⇒ (ln (x / y) = ln x − ln y)
[LN_EXP] Theorem
⊢ ∀x. ln (exp x) = x
[LN_INJ] Theorem
⊢ ∀x y. 0 < x ∧ 0 < y ⇒ ((ln x = ln y) ⇔ (x = y))
[LN_INV] Theorem
⊢ ∀x. 0 < x ⇒ (ln x⁻¹ = -ln x)
[LN_LE] Theorem
⊢ ∀x. 0 ≤ x ⇒ ln (1 + x) ≤ x
[LN_LT_X] Theorem
⊢ ∀x. 0 < x ⇒ ln x < x
[LN_MONO_LE] Theorem
⊢ ∀x y. 0 < x ∧ 0 < y ⇒ (ln x ≤ ln y ⇔ x ≤ y)
[LN_MONO_LT] Theorem
⊢ ∀x y. 0 < x ∧ 0 < y ⇒ (ln x < ln y ⇔ x < y)
[LN_MUL] Theorem
⊢ ∀x y. 0 < x ∧ 0 < y ⇒ (ln (x * y) = ln x + ln y)
[LN_POS] Theorem
⊢ ∀x. 1 ≤ x ⇒ 0 ≤ ln x
[LN_POW] Theorem
⊢ ∀n x. 0 < x ⇒ (ln (x pow n) = &n * ln x)
[LN_RPOW] Theorem
⊢ ∀a b. 0 < a ⇒ (ln (a rpow b) = b * ln a)
[MCLAURIN] Theorem
⊢ ∀f diff h n.
0 < h ∧ 0 < n ∧ (diff 0 = f) ∧
(∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h ⇒ (diff m diffl diff (SUC m) t) t) ⇒
∃t.
0 < t ∧ t < h ∧
(f h =
sum (0,n) (λm. diff m 0 / &FACT m * h pow m) +
diff n t / &FACT n * h pow n)
[MCLAURIN_ALL_LE] Theorem
⊢ ∀f diff.
(diff 0 = f) ∧ (∀m x. (diff m diffl diff (SUC m) x) x) ⇒
∀x n.
∃t.
abs t ≤ abs x ∧
(f x =
sum (0,n) (λm. diff m 0 / &FACT m * x pow m) +
diff n t / &FACT n * x pow n)
[MCLAURIN_ALL_LT] Theorem
⊢ ∀f diff.
(diff 0 = f) ∧ (∀m x. (diff m diffl diff (SUC m) x) x) ⇒
∀x n.
x ≠ 0 ∧ 0 < n ⇒
∃t.
0 < abs t ∧ abs t < abs x ∧
(f x =
sum (0,n) (λm. diff m 0 / &FACT m * x pow m) +
diff n t / &FACT n * x pow n)
[MCLAURIN_EXP_LE] Theorem
⊢ ∀x n.
∃t.
abs t ≤ abs x ∧
(exp x =
sum (0,n) (λm. x pow m / &FACT m) +
exp t / &FACT n * x pow n)
[MCLAURIN_EXP_LT] Theorem
⊢ ∀x n.
x ≠ 0 ∧ 0 < n ⇒
∃t.
0 < abs t ∧ abs t < abs x ∧
(exp x =
sum (0,n) (λm. x pow m / &FACT m) +
exp t / &FACT n * x pow n)
[MCLAURIN_NEG] Theorem
⊢ ∀f diff h n.
h < 0 ∧ 0 < n ∧ (diff 0 = f) ∧
(∀m t. m < n ∧ h ≤ t ∧ t ≤ 0 ⇒ (diff m diffl diff (SUC m) t) t) ⇒
∃t.
h < t ∧ t < 0 ∧
(f h =
sum (0,n) (λm. diff m 0 / &FACT m * h pow m) +
diff n t / &FACT n * h pow n)
[MCLAURIN_ZERO] Theorem
⊢ ∀diff n x.
(x = 0) ∧ 0 < n ⇒
(sum (0,n) (λm. diff m 0 / &FACT m * x pow m) = diff 0 0)
[ONE_RPOW] Theorem
⊢ ∀a. 0 < a ⇒ (1 rpow a = 1)
[PI2] Theorem
⊢ pi / 2 = @x. 0 ≤ x ∧ x ≤ 2 ∧ (cos x = 0)
[PI2_BOUNDS] Theorem
⊢ 0 < pi / 2 ∧ pi / 2 < 2
[PI_POS] Theorem
⊢ 0 < pi
[POW_2_SQRT] Theorem
⊢ 0 ≤ x ⇒ (sqrt (x pow 2) = x)
[POW_ROOT_POS] Theorem
⊢ ∀n x. 0 ≤ x ⇒ (root (SUC n) (x pow SUC n) = x)
[REAL_DIV_SQRT] Theorem
⊢ ∀x. 0 ≤ x ⇒ (x / sqrt x = sqrt x)
[ROOT_0] Theorem
⊢ ∀n. root (SUC n) 0 = 0
[ROOT_1] Theorem
⊢ ∀n. root (SUC n) 1 = 1
[ROOT_DIV] Theorem
⊢ ∀n x y.
0 ≤ x ∧ 0 ≤ y ⇒
(root (SUC n) (x / y) = root (SUC n) x / root (SUC n) y)
[ROOT_INV] Theorem
⊢ ∀n x. 0 ≤ x ⇒ (root (SUC n) x⁻¹ = (root (SUC n) x)⁻¹)
[ROOT_LN] Theorem
⊢ ∀n x. 0 < x ⇒ (root (SUC n) x = exp (ln x / &SUC n))
[ROOT_LT_LEMMA] Theorem
⊢ ∀n x. 0 < x ⇒ (exp (ln x / &SUC n) pow SUC n = x)
[ROOT_MONO_LE] Theorem
⊢ ∀x y. 0 ≤ x ∧ x ≤ y ⇒ root (SUC n) x ≤ root (SUC n) y
[ROOT_MUL] Theorem
⊢ ∀n x y.
0 ≤ x ∧ 0 ≤ y ⇒
(root (SUC n) (x * y) = root (SUC n) x * root (SUC n) y)
[ROOT_POS] Theorem
⊢ ∀n x. 0 ≤ x ⇒ 0 ≤ root (SUC n) x
[ROOT_POS_LT] Theorem
⊢ ∀n x. 0 < x ⇒ 0 < root (SUC n) x
[ROOT_POS_UNIQ] Theorem
⊢ ∀n x y. 0 ≤ x ∧ 0 ≤ y ∧ (y pow SUC n = x) ⇒ (root (SUC n) x = y)
[ROOT_POW_POS] Theorem
⊢ ∀n x. 0 ≤ x ⇒ (root (SUC n) x pow SUC n = x)
[RPOW_0] Theorem
⊢ ∀a. 0 < a ⇒ (a rpow 0 = 1)
[RPOW_1] Theorem
⊢ ∀a. 0 < a ⇒ (a rpow 1 = a)
[RPOW_ADD] Theorem
⊢ ∀a b c. a rpow (b + c) = a rpow b * a rpow c
[RPOW_ADD_MUL] Theorem
⊢ ∀a b c. a rpow (b + c) * a rpow -b = a rpow c
[RPOW_DIV] Theorem
⊢ ∀a b c. 0 < a ∧ 0 < b ⇒ ((a / b) rpow c = a rpow c / b rpow c)
[RPOW_DIV_BASE] Theorem
⊢ ∀x t. 0 < x ⇒ (x rpow t / x = x rpow (t − 1))
[RPOW_INV] Theorem
⊢ ∀a b. 0 < a ⇒ (a⁻¹ rpow b = (a rpow b)⁻¹)
[RPOW_LE] Theorem
⊢ ∀a b c. 1 < a ⇒ (a rpow b ≤ a rpow c ⇔ b ≤ c)
[RPOW_LT] Theorem
⊢ ∀a b c. 1 < a ⇒ (a rpow b < a rpow c ⇔ b < c)
[RPOW_MUL] Theorem
⊢ ∀a b c. 0 < a ∧ 0 < b ⇒ ((a * b) rpow c = a rpow c * b rpow c)
[RPOW_NZ] Theorem
⊢ ∀a b. 0 ≠ a ⇒ a rpow b ≠ 0
[RPOW_POS_LT] Theorem
⊢ ∀a b. 0 < a ⇒ 0 < a rpow b
[RPOW_RPOW] Theorem
⊢ ∀a b c. 0 < a ⇒ ((a rpow b) rpow c = a rpow (b * c))
[RPOW_SUB] Theorem
⊢ ∀a b c. a rpow (b − c) = a rpow b / a rpow c
[RPOW_SUC_N] Theorem
⊢ ∀a n. 0 < a ⇒ (a rpow (&n + 1) = a pow SUC n)
[RPOW_UNIQ_BASE] Theorem
⊢ ∀a b c. 0 < a ∧ 0 < c ∧ 0 ≠ b ∧ (a rpow b = c rpow b) ⇒ (a = c)
[RPOW_UNIQ_EXP] Theorem
⊢ ∀a b c. 1 < a ∧ 0 < c ∧ 0 < b ∧ (a rpow b = a rpow c) ⇒ (b = c)
[SIN_0] Theorem
⊢ sin 0 = 0
[SIN_ACS_NZ] Theorem
⊢ ∀x. -1 < x ∧ x < 1 ⇒ sin (acs x) ≠ 0
[SIN_ADD] Theorem
⊢ ∀x y. sin (x + y) = sin x * cos y + cos x * sin y
[SIN_ASN] Theorem
⊢ ∀x. -(pi / 2) ≤ x ∧ x ≤ pi / 2 ⇒ (asn (sin x) = x)
[SIN_BOUND] Theorem
⊢ ∀x. abs (sin x) ≤ 1
[SIN_BOUNDS] Theorem
⊢ ∀x. -1 ≤ sin x ∧ sin x ≤ 1
[SIN_CIRCLE] Theorem
⊢ ∀x. sin x pow 2 + cos x pow 2 = 1
[SIN_CONVERGES] Theorem
⊢ ∀x.
(λn.
(λn.
if EVEN n then 0
else -1 pow ((n − 1) DIV 2) / &FACT n) n * x pow n) sums
sin x
[SIN_COS] Theorem
⊢ ∀x. sin x = cos (pi / 2 − x)
[SIN_COS_ADD] Theorem
⊢ ∀x y.
(sin (x + y) − (sin x * cos y + cos x * sin y)) pow 2 +
(cos (x + y) − (cos x * cos y − sin x * sin y)) pow 2 = 0
[SIN_COS_NEG] Theorem
⊢ ∀x. (sin (-x) + sin x) pow 2 + (cos (-x) − cos x) pow 2 = 0
[SIN_COS_SQ] Theorem
⊢ ∀x. 0 ≤ x ∧ x ≤ pi ⇒ (sin x = sqrt (1 − cos x pow 2))
[SIN_COS_SQRT] Theorem
⊢ ∀x. 0 ≤ sin x ⇒ (sin x = sqrt (1 − cos x pow 2))
[SIN_DOUBLE] Theorem
⊢ ∀x. sin (2 * x) = 2 * (sin x * cos x)
[SIN_FDIFF] Theorem
⊢ diffs (λn. if EVEN n then 0 else -1 pow ((n − 1) DIV 2) / &FACT n) =
(λn. if EVEN n then -1 pow (n DIV 2) / &FACT n else 0)
[SIN_NEG] Theorem
⊢ ∀x. sin (-x) = -sin x
[SIN_NEGLEMMA] Theorem
⊢ ∀x.
-sin x =
suminf
(λn.
-((λn.
if EVEN n then 0
else -1 pow ((n − 1) DIV 2) / &FACT n) n * x pow n))
[SIN_NPI] Theorem
⊢ ∀n. sin (&n * pi) = 0
[SIN_PAIRED] Theorem
⊢ ∀x.
(λn. -1 pow n / &FACT (2 * n + 1) * x pow (2 * n + 1)) sums
sin x
[SIN_PERIODIC] Theorem
⊢ ∀x. sin (x + 2 * pi) = sin x
[SIN_PERIODIC_PI] Theorem
⊢ ∀x. sin (x + pi) = -sin x
[SIN_PI] Theorem
⊢ sin pi = 0
[SIN_PI2] Theorem
⊢ sin (pi / 2) = 1
[SIN_POS] Theorem
⊢ ∀x. 0 < x ∧ x < 2 ⇒ 0 < sin x
[SIN_POS_PI] Theorem
⊢ ∀x. 0 < x ∧ x < pi ⇒ 0 < sin x
[SIN_POS_PI2] Theorem
⊢ ∀x. 0 < x ∧ x < pi / 2 ⇒ 0 < sin x
[SIN_POS_PI2_LE] Theorem
⊢ ∀x. 0 ≤ x ∧ x ≤ pi / 2 ⇒ 0 ≤ sin x
[SIN_POS_PI_LE] Theorem
⊢ ∀x. 0 ≤ x ∧ x ≤ pi ⇒ 0 ≤ sin x
[SIN_TOTAL] Theorem
⊢ ∀y. -1 ≤ y ∧ y ≤ 1 ⇒ ∃!x. -(pi / 2) ≤ x ∧ x ≤ pi / 2 ∧ (sin x = y)
[SIN_ZERO] Theorem
⊢ ∀x.
(sin x = 0) ⇔
(∃n. EVEN n ∧ (x = &n * (pi / 2))) ∨
∃n. EVEN n ∧ (x = -(&n * (pi / 2)))
[SIN_ZERO_LEMMA] Theorem
⊢ ∀x. 0 ≤ x ∧ (sin x = 0) ⇒ ∃n. EVEN n ∧ (x = &n * (pi / 2))
[SQRT_0] Theorem
⊢ sqrt 0 = 0
[SQRT_1] Theorem
⊢ sqrt 1 = 1
[SQRT_DIV] Theorem
⊢ ∀x y. 0 ≤ x ∧ 0 ≤ y ⇒ (sqrt (x / y) = sqrt x / sqrt y)
[SQRT_EQ] Theorem
⊢ ∀x y. (x pow 2 = y) ∧ 0 ≤ x ⇒ (x = sqrt y)
[SQRT_EVEN_POW2] Theorem
⊢ ∀n. EVEN n ⇒ (sqrt (2 pow n) = 2 pow (n DIV 2))
[SQRT_INV] Theorem
⊢ ∀x. 0 ≤ x ⇒ (sqrt x⁻¹ = (sqrt x)⁻¹)
[SQRT_MONO_LE] Theorem
⊢ ∀x y. 0 ≤ x ∧ x ≤ y ⇒ sqrt x ≤ sqrt y
[SQRT_MUL] Theorem
⊢ ∀x y. 0 ≤ x ∧ 0 ≤ y ⇒ (sqrt (x * y) = sqrt x * sqrt y)
[SQRT_POS_LE] Theorem
⊢ ∀x. 0 ≤ x ⇒ 0 ≤ sqrt x
[SQRT_POS_LT] Theorem
⊢ ∀x. 0 < x ⇒ 0 < sqrt x
[SQRT_POS_UNIQ] Theorem
⊢ ∀x y. 0 ≤ x ∧ 0 ≤ y ∧ (y pow 2 = x) ⇒ (sqrt x = y)
[SQRT_POW2] Theorem
⊢ ∀x. (sqrt x pow 2 = x) ⇔ 0 ≤ x
[SQRT_POW_2] Theorem
⊢ ∀x. 0 ≤ x ⇒ (sqrt x pow 2 = x)
[TAN_0] Theorem
⊢ tan 0 = 0
[TAN_ADD] Theorem
⊢ ∀x y.
cos x ≠ 0 ∧ cos y ≠ 0 ∧ cos (x + y) ≠ 0 ⇒
(tan (x + y) = (tan x + tan y) / (1 − tan x * tan y))
[TAN_ATN] Theorem
⊢ ∀x. -(pi / 2) < x ∧ x < pi / 2 ⇒ (atn (tan x) = x)
[TAN_DOUBLE] Theorem
⊢ ∀x.
cos x ≠ 0 ∧ cos (2 * x) ≠ 0 ⇒
(tan (2 * x) = 2 * tan x / (1 − tan x pow 2))
[TAN_NEG] Theorem
⊢ ∀x. tan (-x) = -tan x
[TAN_NPI] Theorem
⊢ ∀n. tan (&n * pi) = 0
[TAN_PERIODIC] Theorem
⊢ ∀x. tan (x + 2 * pi) = tan x
[TAN_PI] Theorem
⊢ tan pi = 0
[TAN_POS_PI2] Theorem
⊢ ∀x. 0 < x ∧ x < pi / 2 ⇒ 0 < tan x
[TAN_SEC] Theorem
⊢ ∀x. cos x ≠ 0 ⇒ (1 + tan x pow 2 = (cos x)⁻¹ pow 2)
[TAN_TOTAL] Theorem
⊢ ∀y. ∃!x. -(pi / 2) < x ∧ x < pi / 2 ∧ (tan x = y)
[TAN_TOTAL_LEMMA] Theorem
⊢ ∀y. 0 < y ⇒ ∃x. 0 < x ∧ x < pi / 2 ∧ y < tan x
[TAN_TOTAL_POS] Theorem
⊢ ∀y. 0 ≤ y ⇒ ∃x. 0 ≤ x ∧ x < pi / 2 ∧ (tan x = y)
*)
end
HOL 4, Kananaskis-13