Theory "rat"

Parents frac

Signature

Type	Arity
rat	0
Constant	Type
abs_rat	:frac -> rat
abs_rat_CLASS	:(frac -> bool) -> rat
rat_0	:rat
rat_1	:rat
rat_add	:rat -> rat -> rat
rat_ainv	:rat -> rat
rat_cons	:int -> int -> rat
rat_div	:rat -> rat -> rat
rat_dnm	:rat -> int
rat_equiv	:frac reln
rat_geq	:rat reln
rat_gre	:rat reln
rat_leq	:rat reln
rat_les	:rat reln
rat_minv	:rat -> rat
rat_mul	:rat -> rat -> rat
rat_nmr	:rat -> int
rat_of_num	:num -> rat
rat_sgn	:rat -> int
rat_sub	:rat -> rat -> rat
rep_rat	:rat -> frac
rep_rat_CLASS	:rat -> frac -> bool

Definitions

rat_equiv_def

|- ∀f1 f2.
     rat_equiv f1 f2 ⇔ (frac_nmr f1 * frac_dnm f2 = frac_nmr f2 * frac_dnm f1)

rat_TY_DEF

|- ∃rep. TYPE_DEFINITION (λc. ∃r. rat_equiv r r ∧ (c = rat_equiv r)) rep

rat_bijections

|- (∀a. abs_rat_CLASS (rep_rat_CLASS a) = a) ∧
   ∀r.
     (λc. ∃r. rat_equiv r r ∧ (c = rat_equiv r)) r ⇔
     (rep_rat_CLASS (abs_rat_CLASS r) = r)

rep_rat_def

|- ∀a. rep_rat a = $@ (rep_rat_CLASS a)

abs_rat_def

|- ∀r. abs_rat r = abs_rat_CLASS (rat_equiv r)

rat_nmr_def

|- ∀r. rat_nmr r = frac_nmr (rep_rat r)

rat_dnm_def

|- ∀r. rat_dnm r = frac_dnm (rep_rat r)

rat_sgn_def

|- ∀r. rat_sgn r = frac_sgn (rep_rat r)

rat_0_def

|- rat_0 = abs_rat frac_0

rat_1_def

|- rat_1 = abs_rat frac_1

rat_ainv_def

|- ∀r1. -r1 = abs_rat (frac_ainv (rep_rat r1))

rat_minv_def

|- ∀r1. rat_minv r1 = abs_rat (frac_minv (rep_rat r1))

rat_add_def

|- ∀r1 r2. r1 + r2 = abs_rat (frac_add (rep_rat r1) (rep_rat r2))

rat_sub_def

|- ∀r1 r2. r1 − r2 = abs_rat (frac_sub (rep_rat r1) (rep_rat r2))

rat_mul_def

|- ∀r1 r2. r1 * r2 = abs_rat (frac_mul (rep_rat r1) (rep_rat r2))

rat_div_def

|- ∀r1 r2. r1 / r2 = abs_rat (frac_div (rep_rat r1) (rep_rat r2))

rat_les_def

|- ∀r1 r2. r1 < r2 ⇔ (rat_sgn (r2 − r1) = 1)

rat_gre_def

|- ∀r1 r2. r1 > r2 ⇔ r2 < r1

rat_leq_def

|- ∀r1 r2. r1 ≤ r2 ⇔ r1 < r2 ∨ (r1 = r2)

rat_geq_def

|- ∀r1 r2. r1 ≥ r2 ⇔ r2 ≤ r1

rat_cons_def

|- ∀nmr dnm.
     nmr // dnm = abs_rat (abs_frac (SGN nmr * SGN dnm * ABS nmr,ABS dnm))

rat_of_num_primitive_def

|- $& =
   WFREC (@R. WF R ∧ ∀n. R (SUC n) (SUC (SUC n)))
     (λrat_of_num a.
        case a of
          0 => I rat_0
        | SUC 0 => I rat_1
        | SUC (SUC n) => I (rat_of_num (SUC n) + rat_1))

Theorems

RAT_EQUIV

|- ∀f1 f2. rat_equiv f1 f2 ⇔ (rat_equiv f1 = rat_equiv f2)

RAT_EQUIV_REF

|- ∀a. rat_equiv a a

RAT_EQUIV_SYM

|- ∀a b. rat_equiv a b ⇔ rat_equiv b a

RAT_EQUIV_TRANS

|- ∀a b c. rat_equiv a b ∧ rat_equiv b c ⇒ rat_equiv a c

RAT_EQUIV_ALT

|- ∀a.
     rat_equiv a =
     (λx.
        ∃b c.
          0 < b ∧ 0 < c ∧
          (frac_mul a (abs_frac (b,b)) = frac_mul x (abs_frac (c,c))))

rat_ABS_REP_CLASS

|- (∀a. abs_rat_CLASS (rep_rat_CLASS a) = a) ∧
   ∀c.
     (∃r. rat_equiv r r ∧ (c = rat_equiv r)) ⇔
     (rep_rat_CLASS (abs_rat_CLASS c) = c)

rat_QUOTIENT

|- QUOTIENT rat_equiv abs_rat rep_rat

rat_def

|- QUOTIENT rat_equiv abs_rat rep_rat

rat_of_num_ind

|- ∀P. P 0 ∧ P (SUC 0) ∧ (∀n. P (SUC n) ⇒ P (SUC (SUC n))) ⇒ ∀v. P v

rat_of_num_def

|- (0 = rat_0) ∧ (&SUC 0 = rat_1) ∧ ∀n. &SUC (SUC n) = &SUC n + rat_1

rat_of_num_def_compute

|- (0 = rat_0) ∧ (&SUC 0 = rat_1) ∧
   (∀n. &SUC (NUMERAL (BIT1 n)) = &NUMERAL (BIT1 n) + rat_1) ∧
   ∀n. &SUC (NUMERAL (BIT2 n)) = &NUMERAL (BIT2 n) + rat_1

rat_0

|- 0 = abs_rat frac_0

rat_1

|- 1 = abs_rat frac_1

RAT

|- ∀r. abs_rat (rep_rat r) = r

RAT_ABS_EQUIV

|- ∀f1 f2. (abs_rat f1 = abs_rat f2) ⇔ rat_equiv f1 f2

RAT_EQ

|- ∀f1 f2.
     (abs_rat f1 = abs_rat f2) ⇔
     (frac_nmr f1 * frac_dnm f2 = frac_nmr f2 * frac_dnm f1)

RAT_EQ_ALT

|- ∀r1 r2. (r1 = r2) ⇔ (rat_nmr r1 * rat_dnm r2 = rat_nmr r2 * rat_dnm r1)

RAT_NMREQ0_CONG

|- ∀f1. (frac_nmr (rep_rat (abs_rat f1)) = 0) ⇔ (frac_nmr f1 = 0)

RAT_NMRLT0_CONG

|- ∀f1. frac_nmr (rep_rat (abs_rat f1)) < 0 ⇔ frac_nmr f1 < 0

RAT_NMRGT0_CONG

|- ∀f1. frac_nmr (rep_rat (abs_rat f1)) > 0 ⇔ frac_nmr f1 > 0

RAT_SGN_CONG

|- ∀f1. frac_sgn (rep_rat (abs_rat f1)) = frac_sgn f1

RAT_AINV_CONG

|- ∀x. abs_rat (frac_ainv (rep_rat (abs_rat x))) = abs_rat (frac_ainv x)

RAT_MINV_CONG

|- ∀x.
     frac_nmr x ≠ 0 ⇒
     (abs_rat (frac_minv (rep_rat (abs_rat x))) = abs_rat (frac_minv x))

RAT_ADD_CONG1

|- ∀x y. abs_rat (frac_add (rep_rat (abs_rat x)) y) = abs_rat (frac_add x y)

RAT_ADD_CONG2

|- ∀x y. abs_rat (frac_add x (rep_rat (abs_rat y))) = abs_rat (frac_add x y)

RAT_ADD_CONG

|- (∀x y.
      abs_rat (frac_add (rep_rat (abs_rat x)) y) = abs_rat (frac_add x y)) ∧
   ∀x y. abs_rat (frac_add x (rep_rat (abs_rat y))) = abs_rat (frac_add x y)

RAT_MUL_CONG1

|- ∀x y. abs_rat (frac_mul (rep_rat (abs_rat x)) y) = abs_rat (frac_mul x y)

RAT_MUL_CONG2

|- ∀x y. abs_rat (frac_mul x (rep_rat (abs_rat y))) = abs_rat (frac_mul x y)

RAT_MUL_CONG

|- (∀x y.
      abs_rat (frac_mul (rep_rat (abs_rat x)) y) = abs_rat (frac_mul x y)) ∧
   ∀x y. abs_rat (frac_mul x (rep_rat (abs_rat y))) = abs_rat (frac_mul x y)

RAT_SUB_CONG1

|- ∀x y. abs_rat (frac_sub (rep_rat (abs_rat x)) y) = abs_rat (frac_sub x y)

RAT_SUB_CONG2

|- ∀x y. abs_rat (frac_sub x (rep_rat (abs_rat y))) = abs_rat (frac_sub x y)

RAT_SUB_CONG

|- (∀x y.
      abs_rat (frac_sub (rep_rat (abs_rat x)) y) = abs_rat (frac_sub x y)) ∧
   ∀x y. abs_rat (frac_sub x (rep_rat (abs_rat y))) = abs_rat (frac_sub x y)

RAT_DIV_CONG1

|- ∀x y.
     frac_nmr y ≠ 0 ⇒
     (abs_rat (frac_div (rep_rat (abs_rat x)) y) = abs_rat (frac_div x y))

RAT_DIV_CONG2

|- ∀x y.
     frac_nmr y ≠ 0 ⇒
     (abs_rat (frac_div x (rep_rat (abs_rat y))) = abs_rat (frac_div x y))

RAT_DIV_CONG

|- (∀x y.
      frac_nmr y ≠ 0 ⇒
      (abs_rat (frac_div (rep_rat (abs_rat x)) y) = abs_rat (frac_div x y))) ∧
   ∀x y.
     frac_nmr y ≠ 0 ⇒
     (abs_rat (frac_div x (rep_rat (abs_rat y))) = abs_rat (frac_div x y))

RAT_NMRDNM_EQ

|- (abs_rat (abs_frac (frac_nmr f1,frac_dnm f1)) = 1) ⇔
   (frac_nmr f1 = frac_dnm f1)

RAT_AINV_CALCULATE

|- ∀f1. -abs_rat f1 = abs_rat (frac_ainv f1)

RAT_MINV_CALCULATE

|- ∀f1. 0 ≠ frac_nmr f1 ⇒ (rat_minv (abs_rat f1) = abs_rat (frac_minv f1))

RAT_ADD_CALCULATE

|- ∀f1 f2. abs_rat f1 + abs_rat f2 = abs_rat (frac_add f1 f2)

RAT_SUB_CALCULATE

|- ∀f1 f2. abs_rat f1 − abs_rat f2 = abs_rat (frac_sub f1 f2)

RAT_MUL_CALCULATE

|- ∀f1 f2. abs_rat f1 * abs_rat f2 = abs_rat (frac_mul f1 f2)

RAT_DIV_CALCULATE

|- ∀f1 f2.
     frac_nmr f2 ≠ 0 ⇒ (abs_rat f1 / abs_rat f2 = abs_rat (frac_div f1 f2))

RAT_EQ_CALCULATE

|- ∀f1 f2.
     (abs_rat f1 = abs_rat f2) ⇔
     (frac_nmr f1 * frac_dnm f2 = frac_nmr f2 * frac_dnm f1)

RAT_LES_CALCULATE

|- ∀f1 f2.
     abs_rat f1 < abs_rat f2 ⇔
     frac_nmr f1 * frac_dnm f2 < frac_nmr f2 * frac_dnm f1

RAT_OF_NUM_CALCULATE

|- ∀n1. &n1 = abs_rat (abs_frac (&n1,1))

RAT_EQ0_NMR

|- ∀r1. (r1 = 0) ⇔ (rat_nmr r1 = 0)

RAT_0LES_NMR

|- ∀r1. 0 < r1 ⇔ 0 < rat_nmr r1

RAT_LES0_NMR

|- ∀r1. r1 < 0 ⇔ rat_nmr r1 < 0

RAT_0LEQ_NMR

|- ∀r1. 0 ≤ r1 ⇔ 0 ≤ rat_nmr r1

RAT_LEQ0_NMR

|- ∀r1. r1 ≤ 0 ⇔ rat_nmr r1 ≤ 0

RAT_ADD_ASSOC

|- ∀a b c. a + (b + c) = a + b + c

RAT_MUL_ASSOC

|- ∀a b c. a * (b * c) = a * b * c

RAT_ADD_COMM

|- ∀a b. a + b = b + a

RAT_MUL_COMM

|- ∀a b. a * b = b * a

RAT_ADD_RID

|- ∀a. a + 0 = a

RAT_ADD_LID

|- ∀a. 0 + a = a

RAT_MUL_RID

|- ∀a. a * 1 = a

RAT_MUL_LID

|- ∀a. 1 * a = a

RAT_ADD_RINV

|- ∀a. a + -a = 0

RAT_ADD_LINV

|- ∀a. -a + a = 0

RAT_MUL_RINV

|- ∀a. a ≠ 0 ⇒ (a * rat_minv a = 1)

RAT_MUL_LINV

|- ∀a. a ≠ 0 ⇒ (rat_minv a * a = 1)

RAT_RDISTRIB

|- ∀a b c. (a + b) * c = a * c + b * c

RAT_LDISTRIB

|- ∀a b c. c * (a + b) = c * a + c * b

RAT_1_NOT_0

|- 1 ≠ 0

RAT_MUL_LZERO

|- ∀r1. 0 * r1 = 0

RAT_MUL_RZERO

|- ∀r1. r1 * 0 = 0

RAT_SUB_ADDAINV

|- ∀r1 r2. r1 − r2 = r1 + -r2

RAT_DIV_MULMINV

|- ∀r1 r2. r1 / r2 = r1 * rat_minv r2

RAT_AINV_0

|- -0 = 0

RAT_AINV_AINV

|- ∀r1. - -r1 = r1

RAT_AINV_ADD

|- ∀r1 r2. -(r1 + r2) = -r1 + -r2

RAT_AINV_SUB

|- ∀r1 r2. -(r1 − r2) = r2 − r1

RAT_AINV_RMUL

|- ∀r1 r2. -(r1 * r2) = r1 * -r2

RAT_AINV_LMUL

|- ∀r1 r2. -(r1 * r2) = -r1 * r2

RAT_AINV_MINV

|- ∀r1. r1 ≠ 0 ⇒ (-rat_minv r1 = rat_minv (-r1))

RAT_SUB_RDISTRIB

|- ∀a b c. (a − b) * c = a * c − b * c

RAT_SUB_LDISTRIB

|- ∀a b c. c * (a − b) = c * a − c * b

RAT_SUB_LID

|- ∀r1. 0 − r1 = -r1

RAT_SUB_RID

|- ∀r1. r1 − 0 = r1

RAT_SUB_ID

|- ∀r. r − r = 0

RAT_EQ_SUB0

|- ∀r1 r2. (r1 − r2 = 0) ⇔ (r1 = r2)

RAT_EQ_0SUB

|- ∀r1 r2. (0 = r1 − r2) ⇔ (r1 = r2)

RAT_SGN_CALCULATE

|- rat_sgn (abs_rat f1) = frac_sgn f1

RAT_SGN_CLAUSES

|- ∀r1.
     ((rat_sgn r1 = -1) ⇔ r1 < 0) ∧ ((rat_sgn r1 = 0) ⇔ (r1 = 0)) ∧
     ((rat_sgn r1 = 1) ⇔ r1 > 0)

RAT_SGN_0

|- rat_sgn 0 = 0

RAT_SGN_AINV

|- ∀r1. -rat_sgn (-r1) = rat_sgn r1

RAT_SGN_MUL

|- ∀r1 r2. rat_sgn (r1 * r2) = rat_sgn r1 * rat_sgn r2

RAT_SGN_MINV

|- ∀r1. r1 ≠ 0 ⇒ (rat_sgn (rat_minv r1) = rat_sgn r1)

RAT_SGN_TOTAL

|- ∀r1. (rat_sgn r1 = -1) ∨ (rat_sgn r1 = 0) ∨ (rat_sgn r1 = 1)

RAT_SGN_COMPLEMENT

|- ∀r1.
     (rat_sgn r1 ≠ -1 ⇔ (rat_sgn r1 = 0) ∨ (rat_sgn r1 = 1)) ∧
     (rat_sgn r1 ≠ 0 ⇔ (rat_sgn r1 = -1) ∨ (rat_sgn r1 = 1)) ∧
     (rat_sgn r1 ≠ 1 ⇔ (rat_sgn r1 = -1) ∨ (rat_sgn r1 = 0))

RAT_LES_REF

|- ∀r1. ¬(r1 < r1)

RAT_LES_ANTISYM

|- ∀r1 r2. r1 < r2 ⇒ ¬(r2 < r1)

RAT_LES_TRANS

|- ∀r1 r2 r3. r1 < r2 ∧ r2 < r3 ⇒ r1 < r3

RAT_LES_TOTAL

|- ∀r1 r2. r1 < r2 ∨ (r1 = r2) ∨ r2 < r1

RAT_LEQ_REF

|- ∀r1. r1 ≤ r1

RAT_LEQ_ANTISYM

|- ∀r1 r2. r1 ≤ r2 ∧ r2 ≤ r1 ⇒ (r1 = r2)

RAT_LEQ_TRANS

|- ∀r1 r2 r3. r1 ≤ r2 ∧ r2 ≤ r3 ⇒ r1 ≤ r3

RAT_LES_01

|- 0 < 1

RAT_LES_IMP_LEQ

|- ∀r1 r2. r1 < r2 ⇒ r1 ≤ r2

RAT_LES_IMP_NEQ

|- ∀r1 r2. r1 < r2 ⇒ r1 ≠ r2

RAT_LEQ_LES

|- ∀r1 r2. ¬(r2 < r1) ⇔ r1 ≤ r2

RAT_LES_LEQ

|- ∀r1 r2. ¬(r2 ≤ r1) ⇔ r1 < r2

RAT_LES_LEQ2

|- ∀r1 r2. r1 < r2 ⇔ r1 ≤ r2 ∧ ¬(r2 ≤ r1)

RAT_LES_LEQ_TRANS

|- ∀a b c. a < b ∧ b ≤ c ⇒ a < c

RAT_LEQ_LES_TRANS

|- ∀a b c. a ≤ b ∧ b < c ⇒ a < c

RAT_0LES_0LES_ADD

|- ∀r1 r2. 0 < r1 ⇒ 0 < r2 ⇒ 0 < r1 + r2

RAT_LES0_LES0_ADD

|- ∀r1 r2. r1 < 0 ⇒ r2 < 0 ⇒ r1 + r2 < 0

RAT_0LES_0LEQ_ADD

|- ∀r1 r2. 0 < r1 ⇒ 0 ≤ r2 ⇒ 0 < r1 + r2

RAT_LES0_LEQ0_ADD

|- ∀r1 r2. r1 < 0 ⇒ r2 ≤ 0 ⇒ r1 + r2 < 0

RAT_AINV_ONE_ONE

|- ONE_ONE numeric_negate

RAT_ADD_ONE_ONE

|- ∀r1. ONE_ONE ($+ r1)

RAT_MUL_ONE_ONE

|- ∀r1. r1 ≠ 0 ⇔ ONE_ONE ($* r1)

RAT_EQ_AINV

|- ∀r1 r2. (-r1 = -r2) ⇔ (r1 = r2)

RAT_EQ_LADD

|- ∀r1 r2 r3. (r3 + r1 = r3 + r2) ⇔ (r1 = r2)

RAT_EQ_RADD

|- ∀r1 r2 r3. (r1 + r3 = r2 + r3) ⇔ (r1 = r2)

RAT_EQ_RMUL

|- ∀r1 r2 r3. r3 ≠ 0 ⇒ ((r1 * r3 = r2 * r3) ⇔ (r1 = r2))

RAT_EQ_LMUL

|- ∀r1 r2 r3. r3 ≠ 0 ⇒ ((r3 * r1 = r3 * r2) ⇔ (r1 = r2))

RAT_AINV_EQ

|- ∀r1 r2. (-r1 = r2) ⇔ (r1 = -r2)

RAT_LSUB_EQ

|- ∀r1 r2 r3. (r1 − r2 = r3) ⇔ (r1 = r2 + r3)

RAT_RSUB_EQ

|- ∀r1 r2 r3. (r1 = r2 − r3) ⇔ (r1 + r3 = r2)

RAT_LDIV_EQ

|- ∀r1 r2 r3. r2 ≠ 0 ⇒ ((r1 / r2 = r3) ⇔ (r1 = r2 * r3))

RAT_RDIV_EQ

|- ∀r1 r2 r3. r3 ≠ 0 ⇒ ((r1 = r2 / r3) ⇔ (r1 * r3 = r2))

RAT_LES_RADD

|- ∀r1 r2 r3. r1 + r3 < r2 + r3 ⇔ r1 < r2

RAT_LES_LADD

|- ∀r1 r2 r3. r3 + r1 < r3 + r2 ⇔ r1 < r2

RAT_LES_AINV

|- ∀r1 r2. -r1 < -r2 ⇔ r2 < r1

RAT_LSUB_LES

|- ∀r1 r2 r3. r1 − r2 < r3 ⇔ r1 < r2 + r3

RAT_RSUB_LES

|- ∀r1 r2 r3. r1 < r2 − r3 ⇔ r1 + r3 < r2

RAT_LES_RMUL_POS

|- ∀r1 r2 r3. 0 < r3 ⇒ (r1 * r3 < r2 * r3 ⇔ r1 < r2)

RAT_LES_LMUL_POS

|- ∀r1 r2 r3. 0 < r3 ⇒ (r3 * r1 < r3 * r2 ⇔ r1 < r2)

RAT_LES_RMUL_NEG

|- ∀r1 r2 r3. r3 < 0 ⇒ (r2 * r3 < r1 * r3 ⇔ r1 < r2)

RAT_LES_LMUL_NEG

|- ∀r1 r2 r3. r3 < 0 ⇒ (r3 * r2 < r3 * r1 ⇔ r1 < r2)

RAT_AINV_LES

|- ∀r1 r2. -r1 < r2 ⇔ -r2 < r1

RAT_LDIV_LES_POS

|- ∀r1 r2 r3. 0 < r2 ⇒ (r1 / r2 < r3 ⇔ r1 < r2 * r3)

RAT_LDIV_LES_NEG

|- ∀r1 r2 r3. r2 < 0 ⇒ (r1 / r2 < r3 ⇔ r2 * r3 < r1)

RAT_RDIV_LES_POS

|- ∀r1 r2 r3. 0 < r3 ⇒ (r1 < r2 / r3 ⇔ r1 * r3 < r2)

RAT_RDIV_LES_NEG

|- ∀r1 r2 r3. r3 < 0 ⇒ (r1 < r2 / r3 ⇔ r2 < r1 * r3)

RAT_LES_SUB0

|- ∀r1 r2. r1 − r2 < 0 ⇔ r1 < r2

RAT_LES_0SUB

|- ∀r1 r2. 0 < r1 − r2 ⇔ r2 < r1

RAT_MINV_LES

|- ∀r1. 0 < r1 ⇒ (rat_minv r1 < 0 ⇔ r1 < 0) ∧ (0 < rat_minv r1 ⇔ 0 < r1)

RAT_MUL_SIGN_CASES

|- ∀p q.
     (0 < p * q ⇔ 0 < p ∧ 0 < q ∨ p < 0 ∧ q < 0) ∧
     (p * q < 0 ⇔ 0 < p ∧ q < 0 ∨ p < 0 ∧ 0 < q)

RAT_NO_ZERODIV

|- ∀r1 r2. (r1 = 0) ∨ (r2 = 0) ⇔ (r1 * r2 = 0)

RAT_NO_ZERODIV_NEG

|- ∀r1 r2. r1 * r2 ≠ 0 ⇔ r1 ≠ 0 ∧ r2 ≠ 0

RAT_NO_IDDIV

|- ∀r1 r2. (r1 * r2 = r2) ⇔ (r1 = 1) ∨ (r2 = 0)

RAT_DENSE_THM

|- ∀r1 r3. r1 < r3 ⇒ ∃r2. r1 < r2 ∧ r2 < r3

RAT_SAVE

|- ∀r1. ∃a1 b1. r1 = abs_rat (frac_save a1 b1)

RAT_SAVE_MINV

|- ∀a1 b1.
     abs_rat (frac_save a1 b1) ≠ 0 ⇒
     (rat_minv (abs_rat (frac_save a1 b1)) =
      abs_rat (frac_save (SGN a1 * (&b1 + 1)) (Num (ABS a1 − 1))))

RAT_SAVE_TO_CONS

|- ∀a1 b1. abs_rat (frac_save a1 b1) = a1 // (&b1 + 1)

RAT_OF_NUM

|- ∀n. (0 = rat_0) ∧ ∀n. &SUC n = &n + rat_1

RAT_SAVE_NUM

|- ∀n. &n = abs_rat (frac_save (&n) 0)

RAT_CONS_TO_NUM

|- ∀n. (&n // 1 = &n) ∧ (-&n // 1 = -&n)

RAT_0

|- rat_0 = 0

RAT_1

|- rat_1 = 1

RAT_ADD_NUM_CALCULATE

|- (∀n m. &n + &m = &(n + m)) ∧
   (∀n m. -&n + &m = if n ≤ m then &(m − n) else -&(n − m)) ∧
   (∀n m. &n + -&m = if m ≤ n then &(n − m) else -&(m − n)) ∧
   ∀n m. -&n + -&m = -&(n + m)

RAT_MUL_NUM_CALCULATE

|- (∀n m. &n * &m = &(n * m)) ∧ (∀n m. -&n * &m = -&(n * m)) ∧
   (∀n m. &n * -&m = -&(n * m)) ∧ ∀n m. -&n * -&m = &(n * m)

RAT_EQ_NUM_CALCULATE

|- (∀n m. (&n = &m) ⇔ (n = m)) ∧ (∀n m. (&n = -&m) ⇔ (n = 0) ∧ (m = 0)) ∧
   (∀n m. (-&n = &m) ⇔ (n = 0) ∧ (m = 0)) ∧ ∀n m. (-&n = -&m) ⇔ (n = m)