Structure ratTheory
signature ratTheory =
sig
type thm = Thm.thm
(* Definitions *)
val abs_rat_def : thm
val rat_0_def : thm
val rat_1_def : thm
val rat_TY_DEF : thm
val rat_add_def : thm
val rat_ainv_def : thm
val rat_bijections : thm
val rat_cons_def : thm
val rat_div_def : thm
val rat_dnm_def : thm
val rat_equiv_def : thm
val rat_geq_def : thm
val rat_gre_def : thm
val rat_leq_def : thm
val rat_les_def : thm
val rat_minv_def : thm
val rat_mul_def : thm
val rat_nmr_def : thm
val rat_of_num_primitive_def : thm
val rat_sgn_def : thm
val rat_sub_def : thm
val rep_rat_def : thm
(* Theorems *)
val RAT : thm
val RAT_0 : thm
val RAT_0LEQ_NMR : thm
val RAT_0LES_0LEQ_ADD : thm
val RAT_0LES_0LES_ADD : thm
val RAT_0LES_NMR : thm
val RAT_1 : thm
val RAT_1_NOT_0 : thm
val RAT_ABS_EQUIV : thm
val RAT_ADD_ASSOC : thm
val RAT_ADD_CALCULATE : thm
val RAT_ADD_COMM : thm
val RAT_ADD_CONG : thm
val RAT_ADD_CONG1 : thm
val RAT_ADD_CONG2 : thm
val RAT_ADD_LID : thm
val RAT_ADD_LINV : thm
val RAT_ADD_NUM_CALCULATE : thm
val RAT_ADD_ONE_ONE : thm
val RAT_ADD_RID : thm
val RAT_ADD_RINV : thm
val RAT_AINV_0 : thm
val RAT_AINV_ADD : thm
val RAT_AINV_AINV : thm
val RAT_AINV_CALCULATE : thm
val RAT_AINV_CONG : thm
val RAT_AINV_EQ : thm
val RAT_AINV_LES : thm
val RAT_AINV_LMUL : thm
val RAT_AINV_MINV : thm
val RAT_AINV_ONE_ONE : thm
val RAT_AINV_RMUL : thm
val RAT_AINV_SUB : thm
val RAT_CONS_TO_NUM : thm
val RAT_DENSE_THM : thm
val RAT_DIV_CALCULATE : thm
val RAT_DIV_CONG : thm
val RAT_DIV_CONG1 : thm
val RAT_DIV_CONG2 : thm
val RAT_DIV_MULMINV : thm
val RAT_EQ : thm
val RAT_EQ0_NMR : thm
val RAT_EQUIV : thm
val RAT_EQUIV_ALT : thm
val RAT_EQUIV_REF : thm
val RAT_EQUIV_SYM : thm
val RAT_EQUIV_TRANS : thm
val RAT_EQ_0SUB : thm
val RAT_EQ_AINV : thm
val RAT_EQ_ALT : thm
val RAT_EQ_CALCULATE : thm
val RAT_EQ_LADD : thm
val RAT_EQ_LMUL : thm
val RAT_EQ_NUM_CALCULATE : thm
val RAT_EQ_RADD : thm
val RAT_EQ_RMUL : thm
val RAT_EQ_SUB0 : thm
val RAT_LDISTRIB : thm
val RAT_LDIV_EQ : thm
val RAT_LDIV_LES_NEG : thm
val RAT_LDIV_LES_POS : thm
val RAT_LEQ0_NMR : thm
val RAT_LEQ_ANTISYM : thm
val RAT_LEQ_LES : thm
val RAT_LEQ_LES_TRANS : thm
val RAT_LEQ_REF : thm
val RAT_LEQ_TRANS : thm
val RAT_LES0_LEQ0_ADD : thm
val RAT_LES0_LES0_ADD : thm
val RAT_LES0_NMR : thm
val RAT_LES_01 : thm
val RAT_LES_0SUB : thm
val RAT_LES_AINV : thm
val RAT_LES_ANTISYM : thm
val RAT_LES_CALCULATE : thm
val RAT_LES_IMP_LEQ : thm
val RAT_LES_IMP_NEQ : thm
val RAT_LES_LADD : thm
val RAT_LES_LEQ : thm
val RAT_LES_LEQ2 : thm
val RAT_LES_LEQ_TRANS : thm
val RAT_LES_LMUL_NEG : thm
val RAT_LES_LMUL_POS : thm
val RAT_LES_RADD : thm
val RAT_LES_REF : thm
val RAT_LES_RMUL_NEG : thm
val RAT_LES_RMUL_POS : thm
val RAT_LES_SUB0 : thm
val RAT_LES_TOTAL : thm
val RAT_LES_TRANS : thm
val RAT_LSUB_EQ : thm
val RAT_LSUB_LES : thm
val RAT_MINV_CALCULATE : thm
val RAT_MINV_CONG : thm
val RAT_MINV_LES : thm
val RAT_MUL_ASSOC : thm
val RAT_MUL_CALCULATE : thm
val RAT_MUL_COMM : thm
val RAT_MUL_CONG : thm
val RAT_MUL_CONG1 : thm
val RAT_MUL_CONG2 : thm
val RAT_MUL_LID : thm
val RAT_MUL_LINV : thm
val RAT_MUL_LZERO : thm
val RAT_MUL_NUM_CALCULATE : thm
val RAT_MUL_ONE_ONE : thm
val RAT_MUL_RID : thm
val RAT_MUL_RINV : thm
val RAT_MUL_RZERO : thm
val RAT_MUL_SIGN_CASES : thm
val RAT_NMRDNM_EQ : thm
val RAT_NMREQ0_CONG : thm
val RAT_NMRGT0_CONG : thm
val RAT_NMRLT0_CONG : thm
val RAT_NO_IDDIV : thm
val RAT_NO_ZERODIV : thm
val RAT_NO_ZERODIV_NEG : thm
val RAT_OF_NUM : thm
val RAT_OF_NUM_CALCULATE : thm
val RAT_RDISTRIB : thm
val RAT_RDIV_EQ : thm
val RAT_RDIV_LES_NEG : thm
val RAT_RDIV_LES_POS : thm
val RAT_RSUB_EQ : thm
val RAT_RSUB_LES : thm
val RAT_SAVE : thm
val RAT_SAVE_MINV : thm
val RAT_SAVE_NUM : thm
val RAT_SAVE_TO_CONS : thm
val RAT_SGN_0 : thm
val RAT_SGN_AINV : thm
val RAT_SGN_CALCULATE : thm
val RAT_SGN_CLAUSES : thm
val RAT_SGN_COMPLEMENT : thm
val RAT_SGN_CONG : thm
val RAT_SGN_MINV : thm
val RAT_SGN_MUL : thm
val RAT_SGN_TOTAL : thm
val RAT_SUB_ADDAINV : thm
val RAT_SUB_CALCULATE : thm
val RAT_SUB_CONG : thm
val RAT_SUB_CONG1 : thm
val RAT_SUB_CONG2 : thm
val RAT_SUB_ID : thm
val RAT_SUB_LDISTRIB : thm
val RAT_SUB_LID : thm
val RAT_SUB_RDISTRIB : thm
val RAT_SUB_RID : thm
val rat_0 : thm
val rat_1 : thm
val rat_ABS_REP_CLASS : thm
val rat_QUOTIENT : thm
val rat_def : thm
val rat_of_num_def : thm
val rat_of_num_def_compute : thm
val rat_of_num_ind : thm
val rat_grammars : type_grammar.grammar * term_grammar.grammar
(*
[frac] Parent theory of "rat"
[abs_rat_def] Definition
|- ∀r. abs_rat r = abs_rat_CLASS (rat_equiv r)
[rat_0_def] Definition
|- rat_0 = abs_rat frac_0
[rat_1_def] Definition
|- rat_1 = abs_rat frac_1
[rat_TY_DEF] Definition
|- ∃rep.
TYPE_DEFINITION (λc. ∃r. rat_equiv r r ∧ (c = rat_equiv r)) rep
[rat_add_def] Definition
|- ∀r1 r2. r1 + r2 = abs_rat (frac_add (rep_rat r1) (rep_rat r2))
[rat_ainv_def] Definition
|- ∀r1. -r1 = abs_rat (frac_ainv (rep_rat r1))
[rat_bijections] Definition
|- (∀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)
[rat_cons_def] Definition
|- ∀nmr dnm.
nmr // dnm =
abs_rat (abs_frac (SGN nmr * SGN dnm * ABS nmr,ABS dnm))
[rat_div_def] Definition
|- ∀r1 r2. r1 / r2 = abs_rat (frac_div (rep_rat r1) (rep_rat r2))
[rat_dnm_def] Definition
|- ∀r. rat_dnm r = frac_dnm (rep_rat r)
[rat_equiv_def] Definition
|- ∀f1 f2.
rat_equiv f1 f2 ⇔
(frac_nmr f1 * frac_dnm f2 = frac_nmr f2 * frac_dnm f1)
[rat_geq_def] Definition
|- ∀r1 r2. r1 ≥ r2 ⇔ r2 ≤ r1
[rat_gre_def] Definition
|- ∀r1 r2. r1 > r2 ⇔ r2 < r1
[rat_leq_def] Definition
|- ∀r1 r2. r1 ≤ r2 ⇔ r1 < r2 ∨ (r1 = r2)
[rat_les_def] Definition
|- ∀r1 r2. r1 < r2 ⇔ (rat_sgn (r2 − r1) = 1)
[rat_minv_def] Definition
|- ∀r1. rat_minv r1 = abs_rat (frac_minv (rep_rat r1))
[rat_mul_def] Definition
|- ∀r1 r2. r1 * r2 = abs_rat (frac_mul (rep_rat r1) (rep_rat r2))
[rat_nmr_def] Definition
|- ∀r. rat_nmr r = frac_nmr (rep_rat r)
[rat_of_num_primitive_def] Definition
|- $& =
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))
[rat_sgn_def] Definition
|- ∀r. rat_sgn r = frac_sgn (rep_rat r)
[rat_sub_def] Definition
|- ∀r1 r2. r1 − r2 = abs_rat (frac_sub (rep_rat r1) (rep_rat r2))
[rep_rat_def] Definition
|- ∀a. rep_rat a = $@ (rep_rat_CLASS a)
[RAT] Theorem
|- ∀r. abs_rat (rep_rat r) = r
[RAT_0] Theorem
|- rat_0 = 0
[RAT_0LEQ_NMR] Theorem
|- ∀r1. 0 ≤ r1 ⇔ 0 ≤ rat_nmr r1
[RAT_0LES_0LEQ_ADD] Theorem
|- ∀r1 r2. 0 < r1 ⇒ 0 ≤ r2 ⇒ 0 < r1 + r2
[RAT_0LES_0LES_ADD] Theorem
|- ∀r1 r2. 0 < r1 ⇒ 0 < r2 ⇒ 0 < r1 + r2
[RAT_0LES_NMR] Theorem
|- ∀r1. 0 < r1 ⇔ 0 < rat_nmr r1
[RAT_1] Theorem
|- rat_1 = 1
[RAT_1_NOT_0] Theorem
|- 1 ≠ 0
[RAT_ABS_EQUIV] Theorem
|- ∀f1 f2. (abs_rat f1 = abs_rat f2) ⇔ rat_equiv f1 f2
[RAT_ADD_ASSOC] Theorem
|- ∀a b c. a + (b + c) = a + b + c
[RAT_ADD_CALCULATE] Theorem
|- ∀f1 f2. abs_rat f1 + abs_rat f2 = abs_rat (frac_add f1 f2)
[RAT_ADD_COMM] Theorem
|- ∀a b. a + b = b + a
[RAT_ADD_CONG] Theorem
|- (∀x y.
abs_rat (frac_add (rep_rat (abs_rat x)) y) =
abs_rat (frac_add x y)) ∧
∀x y.
abs_rat (frac_add x (rep_rat (abs_rat y))) =
abs_rat (frac_add x y)
[RAT_ADD_CONG1] Theorem
|- ∀x y.
abs_rat (frac_add (rep_rat (abs_rat x)) y) =
abs_rat (frac_add x y)
[RAT_ADD_CONG2] Theorem
|- ∀x y.
abs_rat (frac_add x (rep_rat (abs_rat y))) =
abs_rat (frac_add x y)
[RAT_ADD_LID] Theorem
|- ∀a. 0 + a = a
[RAT_ADD_LINV] Theorem
|- ∀a. -a + a = 0
[RAT_ADD_NUM_CALCULATE] Theorem
|- (∀n m. &n + &m = &(n + m)) ∧
(∀n m. -&n + &m = if n ≤ m then &(m − n) else -&(n − m)) ∧
(∀n m. &n + -&m = if m ≤ n then &(n − m) else -&(m − n)) ∧
∀n m. -&n + -&m = -&(n + m)
[RAT_ADD_ONE_ONE] Theorem
|- ∀r1. ONE_ONE ($+ r1)
[RAT_ADD_RID] Theorem
|- ∀a. a + 0 = a
[RAT_ADD_RINV] Theorem
|- ∀a. a + -a = 0
[RAT_AINV_0] Theorem
|- -0 = 0
[RAT_AINV_ADD] Theorem
|- ∀r1 r2. -(r1 + r2) = -r1 + -r2
[RAT_AINV_AINV] Theorem
|- ∀r1. --r1 = r1
[RAT_AINV_CALCULATE] Theorem
|- ∀f1. -abs_rat f1 = abs_rat (frac_ainv f1)
[RAT_AINV_CONG] Theorem
|- ∀x.
abs_rat (frac_ainv (rep_rat (abs_rat x))) =
abs_rat (frac_ainv x)
[RAT_AINV_EQ] Theorem
|- ∀r1 r2. (-r1 = r2) ⇔ (r1 = -r2)
[RAT_AINV_LES] Theorem
|- ∀r1 r2. -r1 < r2 ⇔ -r2 < r1
[RAT_AINV_LMUL] Theorem
|- ∀r1 r2. -(r1 * r2) = -r1 * r2
[RAT_AINV_MINV] Theorem
|- ∀r1. r1 ≠ 0 ⇒ (-rat_minv r1 = rat_minv (-r1))
[RAT_AINV_ONE_ONE] Theorem
|- ONE_ONE numeric_negate
[RAT_AINV_RMUL] Theorem
|- ∀r1 r2. -(r1 * r2) = r1 * -r2
[RAT_AINV_SUB] Theorem
|- ∀r1 r2. -(r1 − r2) = r2 − r1
[RAT_CONS_TO_NUM] Theorem
|- ∀n. (&n // 1 = &n) ∧ (-&n // 1 = -&n)
[RAT_DENSE_THM] Theorem
|- ∀r1 r3. r1 < r3 ⇒ ∃r2. r1 < r2 ∧ r2 < r3
[RAT_DIV_CALCULATE] Theorem
|- ∀f1 f2.
frac_nmr f2 ≠ 0 ⇒
(abs_rat f1 / abs_rat f2 = abs_rat (frac_div f1 f2))
[RAT_DIV_CONG] Theorem
|- (∀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_DIV_CONG1] Theorem
|- ∀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] Theorem
|- ∀x y.
frac_nmr y ≠ 0 ⇒
(abs_rat (frac_div x (rep_rat (abs_rat y))) =
abs_rat (frac_div x y))
[RAT_DIV_MULMINV] Theorem
|- ∀r1 r2. r1 / r2 = r1 * rat_minv r2
[RAT_EQ] Theorem
|- ∀f1 f2.
(abs_rat f1 = abs_rat f2) ⇔
(frac_nmr f1 * frac_dnm f2 = frac_nmr f2 * frac_dnm f1)
[RAT_EQ0_NMR] Theorem
|- ∀r1. (r1 = 0) ⇔ (rat_nmr r1 = 0)
[RAT_EQUIV] Theorem
|- ∀f1 f2. rat_equiv f1 f2 ⇔ (rat_equiv f1 = rat_equiv f2)
[RAT_EQUIV_ALT] Theorem
|- ∀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_EQUIV_REF] Theorem
|- ∀a. rat_equiv a a
[RAT_EQUIV_SYM] Theorem
|- ∀a b. rat_equiv a b ⇔ rat_equiv b a
[RAT_EQUIV_TRANS] Theorem
|- ∀a b c. rat_equiv a b ∧ rat_equiv b c ⇒ rat_equiv a c
[RAT_EQ_0SUB] Theorem
|- ∀r1 r2. (0 = r1 − r2) ⇔ (r1 = r2)
[RAT_EQ_AINV] Theorem
|- ∀r1 r2. (-r1 = -r2) ⇔ (r1 = r2)
[RAT_EQ_ALT] Theorem
|- ∀r1 r2.
(r1 = r2) ⇔ (rat_nmr r1 * rat_dnm r2 = rat_nmr r2 * rat_dnm r1)
[RAT_EQ_CALCULATE] Theorem
|- ∀f1 f2.
(abs_rat f1 = abs_rat f2) ⇔
(frac_nmr f1 * frac_dnm f2 = frac_nmr f2 * frac_dnm f1)
[RAT_EQ_LADD] Theorem
|- ∀r1 r2 r3. (r3 + r1 = r3 + r2) ⇔ (r1 = r2)
[RAT_EQ_LMUL] Theorem
|- ∀r1 r2 r3. r3 ≠ 0 ⇒ ((r3 * r1 = r3 * r2) ⇔ (r1 = r2))
[RAT_EQ_NUM_CALCULATE] Theorem
|- (∀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)
[RAT_EQ_RADD] Theorem
|- ∀r1 r2 r3. (r1 + r3 = r2 + r3) ⇔ (r1 = r2)
[RAT_EQ_RMUL] Theorem
|- ∀r1 r2 r3. r3 ≠ 0 ⇒ ((r1 * r3 = r2 * r3) ⇔ (r1 = r2))
[RAT_EQ_SUB0] Theorem
|- ∀r1 r2. (r1 − r2 = 0) ⇔ (r1 = r2)
[RAT_LDISTRIB] Theorem
|- ∀a b c. c * (a + b) = c * a + c * b
[RAT_LDIV_EQ] Theorem
|- ∀r1 r2 r3. r2 ≠ 0 ⇒ ((r1 / r2 = r3) ⇔ (r1 = r2 * r3))
[RAT_LDIV_LES_NEG] Theorem
|- ∀r1 r2 r3. r2 < 0 ⇒ (r1 / r2 < r3 ⇔ r2 * r3 < r1)
[RAT_LDIV_LES_POS] Theorem
|- ∀r1 r2 r3. 0 < r2 ⇒ (r1 / r2 < r3 ⇔ r1 < r2 * r3)
[RAT_LEQ0_NMR] Theorem
|- ∀r1. r1 ≤ 0 ⇔ rat_nmr r1 ≤ 0
[RAT_LEQ_ANTISYM] Theorem
|- ∀r1 r2. r1 ≤ r2 ∧ r2 ≤ r1 ⇒ (r1 = r2)
[RAT_LEQ_LES] Theorem
|- ∀r1 r2. ¬(r2 < r1) ⇔ r1 ≤ r2
[RAT_LEQ_LES_TRANS] Theorem
|- ∀a b c. a ≤ b ∧ b < c ⇒ a < c
[RAT_LEQ_REF] Theorem
|- ∀r1. r1 ≤ r1
[RAT_LEQ_TRANS] Theorem
|- ∀r1 r2 r3. r1 ≤ r2 ∧ r2 ≤ r3 ⇒ r1 ≤ r3
[RAT_LES0_LEQ0_ADD] Theorem
|- ∀r1 r2. r1 < 0 ⇒ r2 ≤ 0 ⇒ r1 + r2 < 0
[RAT_LES0_LES0_ADD] Theorem
|- ∀r1 r2. r1 < 0 ⇒ r2 < 0 ⇒ r1 + r2 < 0
[RAT_LES0_NMR] Theorem
|- ∀r1. r1 < 0 ⇔ rat_nmr r1 < 0
[RAT_LES_01] Theorem
|- 0 < 1
[RAT_LES_0SUB] Theorem
|- ∀r1 r2. 0 < r1 − r2 ⇔ r2 < r1
[RAT_LES_AINV] Theorem
|- ∀r1 r2. -r1 < -r2 ⇔ r2 < r1
[RAT_LES_ANTISYM] Theorem
|- ∀r1 r2. r1 < r2 ⇒ ¬(r2 < r1)
[RAT_LES_CALCULATE] Theorem
|- ∀f1 f2.
abs_rat f1 < abs_rat f2 ⇔
frac_nmr f1 * frac_dnm f2 < frac_nmr f2 * frac_dnm f1
[RAT_LES_IMP_LEQ] Theorem
|- ∀r1 r2. r1 < r2 ⇒ r1 ≤ r2
[RAT_LES_IMP_NEQ] Theorem
|- ∀r1 r2. r1 < r2 ⇒ r1 ≠ r2
[RAT_LES_LADD] Theorem
|- ∀r1 r2 r3. r3 + r1 < r3 + r2 ⇔ r1 < r2
[RAT_LES_LEQ] Theorem
|- ∀r1 r2. ¬(r2 ≤ r1) ⇔ r1 < r2
[RAT_LES_LEQ2] Theorem
|- ∀r1 r2. r1 < r2 ⇔ r1 ≤ r2 ∧ ¬(r2 ≤ r1)
[RAT_LES_LEQ_TRANS] Theorem
|- ∀a b c. a < b ∧ b ≤ c ⇒ a < c
[RAT_LES_LMUL_NEG] Theorem
|- ∀r1 r2 r3. r3 < 0 ⇒ (r3 * r2 < r3 * r1 ⇔ r1 < r2)
[RAT_LES_LMUL_POS] Theorem
|- ∀r1 r2 r3. 0 < r3 ⇒ (r3 * r1 < r3 * r2 ⇔ r1 < r2)
[RAT_LES_RADD] Theorem
|- ∀r1 r2 r3. r1 + r3 < r2 + r3 ⇔ r1 < r2
[RAT_LES_REF] Theorem
|- ∀r1. ¬(r1 < r1)
[RAT_LES_RMUL_NEG] Theorem
|- ∀r1 r2 r3. r3 < 0 ⇒ (r2 * r3 < r1 * r3 ⇔ r1 < r2)
[RAT_LES_RMUL_POS] Theorem
|- ∀r1 r2 r3. 0 < r3 ⇒ (r1 * r3 < r2 * r3 ⇔ r1 < r2)
[RAT_LES_SUB0] Theorem
|- ∀r1 r2. r1 − r2 < 0 ⇔ r1 < r2
[RAT_LES_TOTAL] Theorem
|- ∀r1 r2. r1 < r2 ∨ (r1 = r2) ∨ r2 < r1
[RAT_LES_TRANS] Theorem
|- ∀r1 r2 r3. r1 < r2 ∧ r2 < r3 ⇒ r1 < r3
[RAT_LSUB_EQ] Theorem
|- ∀r1 r2 r3. (r1 − r2 = r3) ⇔ (r1 = r2 + r3)
[RAT_LSUB_LES] Theorem
|- ∀r1 r2 r3. r1 − r2 < r3 ⇔ r1 < r2 + r3
[RAT_MINV_CALCULATE] Theorem
|- ∀f1.
0 ≠ frac_nmr f1 ⇒
(rat_minv (abs_rat f1) = abs_rat (frac_minv f1))
[RAT_MINV_CONG] Theorem
|- ∀x.
frac_nmr x ≠ 0 ⇒
(abs_rat (frac_minv (rep_rat (abs_rat x))) =
abs_rat (frac_minv x))
[RAT_MINV_LES] Theorem
|- ∀r1.
0 < r1 ⇒ (rat_minv r1 < 0 ⇔ r1 < 0) ∧ (0 < rat_minv r1 ⇔ 0 < r1)
[RAT_MUL_ASSOC] Theorem
|- ∀a b c. a * (b * c) = a * b * c
[RAT_MUL_CALCULATE] Theorem
|- ∀f1 f2. abs_rat f1 * abs_rat f2 = abs_rat (frac_mul f1 f2)
[RAT_MUL_COMM] Theorem
|- ∀a b. a * b = b * a
[RAT_MUL_CONG] Theorem
|- (∀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_MUL_CONG1] Theorem
|- ∀x y.
abs_rat (frac_mul (rep_rat (abs_rat x)) y) =
abs_rat (frac_mul x y)
[RAT_MUL_CONG2] Theorem
|- ∀x y.
abs_rat (frac_mul x (rep_rat (abs_rat y))) =
abs_rat (frac_mul x y)
[RAT_MUL_LID] Theorem
|- ∀a. 1 * a = a
[RAT_MUL_LINV] Theorem
|- ∀a. a ≠ 0 ⇒ (rat_minv a * a = 1)
[RAT_MUL_LZERO] Theorem
|- ∀r1. 0 * r1 = 0
[RAT_MUL_NUM_CALCULATE] Theorem
|- (∀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_MUL_ONE_ONE] Theorem
|- ∀r1. r1 ≠ 0 ⇔ ONE_ONE ($* r1)
[RAT_MUL_RID] Theorem
|- ∀a. a * 1 = a
[RAT_MUL_RINV] Theorem
|- ∀a. a ≠ 0 ⇒ (a * rat_minv a = 1)
[RAT_MUL_RZERO] Theorem
|- ∀r1. r1 * 0 = 0
[RAT_MUL_SIGN_CASES] Theorem
|- ∀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_NMRDNM_EQ] Theorem
|- (abs_rat (abs_frac (frac_nmr f1,frac_dnm f1)) = 1) ⇔
(frac_nmr f1 = frac_dnm f1)
[RAT_NMREQ0_CONG] Theorem
|- ∀f1. (frac_nmr (rep_rat (abs_rat f1)) = 0) ⇔ (frac_nmr f1 = 0)
[RAT_NMRGT0_CONG] Theorem
|- ∀f1. frac_nmr (rep_rat (abs_rat f1)) > 0 ⇔ frac_nmr f1 > 0
[RAT_NMRLT0_CONG] Theorem
|- ∀f1. frac_nmr (rep_rat (abs_rat f1)) < 0 ⇔ frac_nmr f1 < 0
[RAT_NO_IDDIV] Theorem
|- ∀r1 r2. (r1 * r2 = r2) ⇔ (r1 = 1) ∨ (r2 = 0)
[RAT_NO_ZERODIV] Theorem
|- ∀r1 r2. (r1 = 0) ∨ (r2 = 0) ⇔ (r1 * r2 = 0)
[RAT_NO_ZERODIV_NEG] Theorem
|- ∀r1 r2. r1 * r2 ≠ 0 ⇔ r1 ≠ 0 ∧ r2 ≠ 0
[RAT_OF_NUM] Theorem
|- ∀n. (0 = rat_0) ∧ ∀n. &SUC n = &n + rat_1
[RAT_OF_NUM_CALCULATE] Theorem
|- ∀n1. &n1 = abs_rat (abs_frac (&n1,1))
[RAT_RDISTRIB] Theorem
|- ∀a b c. (a + b) * c = a * c + b * c
[RAT_RDIV_EQ] Theorem
|- ∀r1 r2 r3. r3 ≠ 0 ⇒ ((r1 = r2 / r3) ⇔ (r1 * r3 = r2))
[RAT_RDIV_LES_NEG] Theorem
|- ∀r1 r2 r3. r3 < 0 ⇒ (r1 < r2 / r3 ⇔ r2 < r1 * r3)
[RAT_RDIV_LES_POS] Theorem
|- ∀r1 r2 r3. 0 < r3 ⇒ (r1 < r2 / r3 ⇔ r1 * r3 < r2)
[RAT_RSUB_EQ] Theorem
|- ∀r1 r2 r3. (r1 = r2 − r3) ⇔ (r1 + r3 = r2)
[RAT_RSUB_LES] Theorem
|- ∀r1 r2 r3. r1 < r2 − r3 ⇔ r1 + r3 < r2
[RAT_SAVE] Theorem
|- ∀r1. ∃a1 b1. r1 = abs_rat (frac_save a1 b1)
[RAT_SAVE_MINV] Theorem
|- ∀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_NUM] Theorem
|- ∀n. &n = abs_rat (frac_save (&n) 0)
[RAT_SAVE_TO_CONS] Theorem
|- ∀a1 b1. abs_rat (frac_save a1 b1) = a1 // (&b1 + 1)
[RAT_SGN_0] Theorem
|- rat_sgn 0 = 0
[RAT_SGN_AINV] Theorem
|- ∀r1. -rat_sgn (-r1) = rat_sgn r1
[RAT_SGN_CALCULATE] Theorem
|- rat_sgn (abs_rat f1) = frac_sgn f1
[RAT_SGN_CLAUSES] Theorem
|- ∀r1.
((rat_sgn r1 = -1) ⇔ r1 < 0) ∧ ((rat_sgn r1 = 0) ⇔ (r1 = 0)) ∧
((rat_sgn r1 = 1) ⇔ r1 > 0)
[RAT_SGN_COMPLEMENT] Theorem
|- ∀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_SGN_CONG] Theorem
|- ∀f1. frac_sgn (rep_rat (abs_rat f1)) = frac_sgn f1
[RAT_SGN_MINV] Theorem
|- ∀r1. r1 ≠ 0 ⇒ (rat_sgn (rat_minv r1) = rat_sgn r1)
[RAT_SGN_MUL] Theorem
|- ∀r1 r2. rat_sgn (r1 * r2) = rat_sgn r1 * rat_sgn r2
[RAT_SGN_TOTAL] Theorem
|- ∀r1. (rat_sgn r1 = -1) ∨ (rat_sgn r1 = 0) ∨ (rat_sgn r1 = 1)
[RAT_SUB_ADDAINV] Theorem
|- ∀r1 r2. r1 − r2 = r1 + -r2
[RAT_SUB_CALCULATE] Theorem
|- ∀f1 f2. abs_rat f1 − abs_rat f2 = abs_rat (frac_sub f1 f2)
[RAT_SUB_CONG] Theorem
|- (∀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_SUB_CONG1] Theorem
|- ∀x y.
abs_rat (frac_sub (rep_rat (abs_rat x)) y) =
abs_rat (frac_sub x y)
[RAT_SUB_CONG2] Theorem
|- ∀x y.
abs_rat (frac_sub x (rep_rat (abs_rat y))) =
abs_rat (frac_sub x y)
[RAT_SUB_ID] Theorem
|- ∀r. r − r = 0
[RAT_SUB_LDISTRIB] Theorem
|- ∀a b c. c * (a − b) = c * a − c * b
[RAT_SUB_LID] Theorem
|- ∀r1. 0 − r1 = -r1
[RAT_SUB_RDISTRIB] Theorem
|- ∀a b c. (a − b) * c = a * c − b * c
[RAT_SUB_RID] Theorem
|- ∀r1. r1 − 0 = r1
[rat_0] Theorem
|- 0 = abs_rat frac_0
[rat_1] Theorem
|- 1 = abs_rat frac_1
[rat_ABS_REP_CLASS] Theorem
|- (∀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] Theorem
|- QUOTIENT rat_equiv abs_rat rep_rat
[rat_def] Theorem
|- QUOTIENT rat_equiv abs_rat rep_rat
[rat_of_num_def] Theorem
|- (0 = rat_0) ∧ (&SUC 0 = rat_1) ∧ ∀n. &SUC (SUC n) = &SUC n + rat_1
[rat_of_num_def_compute] Theorem
|- (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_of_num_ind] Theorem
|- ∀P. P 0 ∧ P (SUC 0) ∧ (∀n. P (SUC n) ⇒ P (SUC (SUC n))) ⇒ ∀v. P v
*)
end
HOL 4, Kananaskis-10