Structure integerTheory
signature integerTheory =
sig
type thm = Thm.thm
(* Definitions *)
val INT_ABS : thm
val INT_DIVIDES : thm
val INT_MAX : thm
val INT_MIN : thm
val LEAST_INT_DEF : thm
val Num : thm
val int_0 : thm
val int_1 : thm
val int_ABS_def : thm
val int_REP_def : thm
val int_TY_DEF : thm
val int_add : thm
val int_bijections : thm
val int_div : thm
val int_exp : thm
val int_ge : thm
val int_gt : thm
val int_le : thm
val int_lt : thm
val int_mod : thm
val int_mul : thm
val int_neg : thm
val int_quot : thm
val int_rem : thm
val int_sub : thm
val tint_0 : thm
val tint_1 : thm
val tint_add : thm
val tint_eq : thm
val tint_lt : thm
val tint_mul : thm
val tint_neg : thm
val tint_of_num : thm
(* Theorems *)
val EQ_ADDL : thm
val EQ_LADD : thm
val INFINITE_INT_UNIV : thm
val INT : thm
val INT_0 : thm
val INT_1 : thm
val INT_10 : thm
val INT_ABS_0LT : thm
val INT_ABS_ABS : thm
val INT_ABS_EQ : thm
val INT_ABS_EQ0 : thm
val INT_ABS_EQ_ABS : thm
val INT_ABS_EQ_ID : thm
val INT_ABS_LE : thm
val INT_ABS_LE0 : thm
val INT_ABS_LT : thm
val INT_ABS_LT0 : thm
val INT_ABS_MUL : thm
val INT_ABS_NEG : thm
val INT_ABS_NUM : thm
val INT_ABS_POS : thm
val INT_ABS_QUOT : thm
val INT_ADD : thm
val INT_ADD2_SUB2 : thm
val INT_ADD_ASSOC : thm
val INT_ADD_CALCULATE : thm
val INT_ADD_COMM : thm
val INT_ADD_DIV : thm
val INT_ADD_LID : thm
val INT_ADD_LID_UNIQ : thm
val INT_ADD_LINV : thm
val INT_ADD_REDUCE : thm
val INT_ADD_RID : thm
val INT_ADD_RID_UNIQ : thm
val INT_ADD_RINV : thm
val INT_ADD_SUB : thm
val INT_ADD_SUB2 : thm
val INT_ADD_SYM : thm
val INT_DIFFSQ : thm
val INT_DISCRETE : thm
val INT_DIV : thm
val INT_DIVIDES_0 : thm
val INT_DIVIDES_1 : thm
val INT_DIVIDES_LADD : thm
val INT_DIVIDES_LMUL : thm
val INT_DIVIDES_LSUB : thm
val INT_DIVIDES_MOD0 : thm
val INT_DIVIDES_MUL : thm
val INT_DIVIDES_MUL_BOTH : thm
val INT_DIVIDES_NEG : thm
val INT_DIVIDES_RADD : thm
val INT_DIVIDES_REDUCE : thm
val INT_DIVIDES_REFL : thm
val INT_DIVIDES_RMUL : thm
val INT_DIVIDES_RSUB : thm
val INT_DIVIDES_TRANS : thm
val INT_DIVISION : thm
val INT_DIV_0 : thm
val INT_DIV_1 : thm
val INT_DIV_CALCULATE : thm
val INT_DIV_FORALL_P : thm
val INT_DIV_ID : thm
val INT_DIV_LMUL : thm
val INT_DIV_MUL_ID : thm
val INT_DIV_NEG : thm
val INT_DIV_P : thm
val INT_DIV_REDUCE : thm
val INT_DIV_RMUL : thm
val INT_DIV_UNIQUE : thm
val INT_DOUBLE : thm
val INT_ENTIRE : thm
val INT_EQ_CALCULATE : thm
val INT_EQ_IMP_LE : thm
val INT_EQ_LADD : thm
val INT_EQ_LMUL : thm
val INT_EQ_LMUL2 : thm
val INT_EQ_LMUL_IMP : thm
val INT_EQ_NEG : thm
val INT_EQ_RADD : thm
val INT_EQ_REDUCE : thm
val INT_EQ_RMUL : thm
val INT_EQ_RMUL_IMP : thm
val INT_EQ_SUB_LADD : thm
val INT_EQ_SUB_RADD : thm
val INT_EXP : thm
val INT_EXP_ADD_EXPONENTS : thm
val INT_EXP_CALCULATE : thm
val INT_EXP_EQ0 : thm
val INT_EXP_MOD : thm
val INT_EXP_MULTIPLY_EXPONENTS : thm
val INT_EXP_NEG : thm
val INT_EXP_REDUCE : thm
val INT_EXP_SUBTRACT_EXPONENTS : thm
val INT_GE_CALCULATE : thm
val INT_GE_REDUCE : thm
val INT_GT_CALCULATE : thm
val INT_GT_REDUCE : thm
val INT_INJ : thm
val INT_LDISTRIB : thm
val INT_LE : thm
val INT_LESS_MOD : thm
val INT_LET_ADD : thm
val INT_LET_ADD2 : thm
val INT_LET_ANTISYM : thm
val INT_LET_TOTAL : thm
val INT_LET_TRANS : thm
val INT_LE_01 : thm
val INT_LE_ADD : thm
val INT_LE_ADD2 : thm
val INT_LE_ADDL : thm
val INT_LE_ADDR : thm
val INT_LE_ANTISYM : thm
val INT_LE_CALCULATE : thm
val INT_LE_DOUBLE : thm
val INT_LE_LADD : thm
val INT_LE_LT : thm
val INT_LE_LT1 : thm
val INT_LE_MONO : thm
val INT_LE_MUL : thm
val INT_LE_NEG : thm
val INT_LE_NEGL : thm
val INT_LE_NEGR : thm
val INT_LE_NEGTOTAL : thm
val INT_LE_RADD : thm
val INT_LE_REDUCE : thm
val INT_LE_REFL : thm
val INT_LE_SQUARE : thm
val INT_LE_SUB_LADD : thm
val INT_LE_SUB_RADD : thm
val INT_LE_TOTAL : thm
val INT_LE_TRANS : thm
val INT_LNEG_UNIQ : thm
val INT_LT : thm
val INT_LTE_ADD : thm
val INT_LTE_ADD2 : thm
val INT_LTE_ANTSYM : thm
val INT_LTE_TOTAL : thm
val INT_LTE_TRANS : thm
val INT_LT_01 : thm
val INT_LT_ADD : thm
val INT_LT_ADD1 : thm
val INT_LT_ADD2 : thm
val INT_LT_ADDL : thm
val INT_LT_ADDNEG : thm
val INT_LT_ADDNEG2 : thm
val INT_LT_ADDR : thm
val INT_LT_ADD_SUB : thm
val INT_LT_ANTISYM : thm
val INT_LT_CALCULATE : thm
val INT_LT_GT : thm
val INT_LT_IMP_LE : thm
val INT_LT_IMP_NE : thm
val INT_LT_LADD : thm
val INT_LT_LADD_IMP : thm
val INT_LT_LE : thm
val INT_LT_LE1 : thm
val INT_LT_MONO : thm
val INT_LT_MUL : thm
val INT_LT_MUL2 : thm
val INT_LT_NEG : thm
val INT_LT_NEGTOTAL : thm
val INT_LT_NZ : thm
val INT_LT_RADD : thm
val INT_LT_REDUCE : thm
val INT_LT_REFL : thm
val INT_LT_SUB_LADD : thm
val INT_LT_SUB_RADD : thm
val INT_LT_TOTAL : thm
val INT_LT_TRANS : thm
val INT_MAX_LT : thm
val INT_MAX_NUM : thm
val INT_MIN_LT : thm
val INT_MIN_NUM : thm
val INT_MOD : thm
val INT_MOD0 : thm
val INT_MOD_1 : thm
val INT_MOD_ADD_MULTIPLES : thm
val INT_MOD_BOUNDS : thm
val INT_MOD_CALCULATE : thm
val INT_MOD_COMMON_FACTOR : thm
val INT_MOD_EQ0 : thm
val INT_MOD_FORALL_P : thm
val INT_MOD_ID : thm
val INT_MOD_MINUS1 : thm
val INT_MOD_MOD : thm
val INT_MOD_NEG : thm
val INT_MOD_NEG_NUMERATOR : thm
val INT_MOD_P : thm
val INT_MOD_PLUS : thm
val INT_MOD_REDUCE : thm
val INT_MOD_SUB : thm
val INT_MOD_UNIQUE : thm
val INT_MUL : thm
val INT_MUL_ASSOC : thm
val INT_MUL_CALCULATE : thm
val INT_MUL_COMM : thm
val INT_MUL_DIV : thm
val INT_MUL_EQ_1 : thm
val INT_MUL_LID : thm
val INT_MUL_LZERO : thm
val INT_MUL_QUOT : thm
val INT_MUL_REDUCE : thm
val INT_MUL_RID : thm
val INT_MUL_RZERO : thm
val INT_MUL_SIGN_CASES : thm
val INT_MUL_SYM : thm
val INT_NEGNEG : thm
val INT_NEG_0 : thm
val INT_NEG_ADD : thm
val INT_NEG_EQ : thm
val INT_NEG_EQ0 : thm
val INT_NEG_GE0 : thm
val INT_NEG_GT0 : thm
val INT_NEG_LE0 : thm
val INT_NEG_LMUL : thm
val INT_NEG_LT0 : thm
val INT_NEG_MINUS1 : thm
val INT_NEG_MUL2 : thm
val INT_NEG_RMUL : thm
val INT_NEG_SAME_EQ : thm
val INT_NEG_SUB : thm
val INT_NOT_LE : thm
val INT_NOT_LT : thm
val INT_NUM_CASES : thm
val INT_NZ_IMP_LT : thm
val INT_OF_NUM : thm
val INT_POASQ : thm
val INT_POS : thm
val INT_POS_NZ : thm
val INT_QUOT : thm
val INT_QUOT_0 : thm
val INT_QUOT_1 : thm
val INT_QUOT_CALCULATE : thm
val INT_QUOT_ID : thm
val INT_QUOT_NEG : thm
val INT_QUOT_REDUCE : thm
val INT_QUOT_UNIQUE : thm
val INT_RDISTRIB : thm
val INT_REM : thm
val INT_REM0 : thm
val INT_REMQUOT : thm
val INT_REM_CALCULATE : thm
val INT_REM_COMMON_FACTOR : thm
val INT_REM_EQ0 : thm
val INT_REM_EQ_MOD : thm
val INT_REM_ID : thm
val INT_REM_NEG : thm
val INT_REM_REDUCE : thm
val INT_REM_UNIQUE : thm
val INT_RNEG_UNIQ : thm
val INT_SUB : thm
val INT_SUB_0 : thm
val INT_SUB_ADD : thm
val INT_SUB_ADD2 : thm
val INT_SUB_CALCULATE : thm
val INT_SUB_LDISTRIB : thm
val INT_SUB_LE : thm
val INT_SUB_LNEG : thm
val INT_SUB_LT : thm
val INT_SUB_LZERO : thm
val INT_SUB_NEG2 : thm
val INT_SUB_RDISTRIB : thm
val INT_SUB_REDUCE : thm
val INT_SUB_REFL : thm
val INT_SUB_RNEG : thm
val INT_SUB_RZERO : thm
val INT_SUB_SUB : thm
val INT_SUB_SUB2 : thm
val INT_SUB_TRIANGLE : thm
val INT_SUMSQ : thm
val LE_NUM_OF_INT : thm
val LT_ADD2 : thm
val LT_ADDL : thm
val LT_ADDR : thm
val LT_LADD : thm
val NUM_LT : thm
val NUM_NEGINT_EXISTS : thm
val NUM_OF_INT : thm
val NUM_POSINT : thm
val NUM_POSINT_EX : thm
val NUM_POSINT_EXISTS : thm
val NUM_POSTINT_EX : thm
val TINT_10 : thm
val TINT_ADD_ASSOC : thm
val TINT_ADD_LID : thm
val TINT_ADD_LINV : thm
val TINT_ADD_SYM : thm
val TINT_ADD_WELLDEF : thm
val TINT_ADD_WELLDEFR : thm
val TINT_EQ_AP : thm
val TINT_EQ_EQUIV : thm
val TINT_EQ_REFL : thm
val TINT_EQ_SYM : thm
val TINT_EQ_TRANS : thm
val TINT_INJ : thm
val TINT_LDISTRIB : thm
val TINT_LT_ADD : thm
val TINT_LT_MUL : thm
val TINT_LT_REFL : thm
val TINT_LT_TOTAL : thm
val TINT_LT_TRANS : thm
val TINT_LT_WELLDEF : thm
val TINT_LT_WELLDEFL : thm
val TINT_LT_WELLDEFR : thm
val TINT_MUL_ASSOC : thm
val TINT_MUL_LID : thm
val TINT_MUL_SYM : thm
val TINT_MUL_WELLDEF : thm
val TINT_MUL_WELLDEFR : thm
val TINT_NEG_WELLDEF : thm
val int_ABS_REP_CLASS : thm
val int_QUOTIENT : thm
val int_of_num : thm
val tint_of_num_eq : thm
val integer_grammars : type_grammar.grammar * term_grammar.grammar
(*
[divides] Parent theory of "integer"
[quotient_list] Parent theory of "integer"
[quotient_option] Parent theory of "integer"
[quotient_pair] Parent theory of "integer"
[quotient_sum] Parent theory of "integer"
[INT_ABS] Definition
⊢ ∀n. ABS n = if n < 0 then -n else n
[INT_DIVIDES] Definition
⊢ ∀p q. p int_divides q ⇔ ∃m. m * p = q
[INT_MAX] Definition
⊢ ∀x y. int_max x y = if x < y then y else x
[INT_MIN] Definition
⊢ ∀x y. int_min x y = if x < y then x else y
[LEAST_INT_DEF] Definition
⊢ ∀P. $LEAST_INT P = @i. P i ∧ ∀j. j < i ⇒ ¬P j
[Num] Definition
⊢ ∀i. Num i = @n. i = &n
[int_0] Definition
⊢ int_0 = int_ABS tint_0
[int_1] Definition
⊢ int_1 = int_ABS tint_1
[int_ABS_def] Definition
⊢ ∀r. int_ABS r = int_ABS_CLASS ($tint_eq r)
[int_REP_def] Definition
⊢ ∀a. int_REP a = $@ (int_REP_CLASS a)
[int_TY_DEF] Definition
⊢ ∃rep. TYPE_DEFINITION (λc. ∃r. r tint_eq r ∧ (c = $tint_eq r)) rep
[int_add] Definition
⊢ ∀T1 T2. T1 + T2 = int_ABS (int_REP T1 tint_add int_REP T2)
[int_bijections] Definition
⊢ (∀a. int_ABS_CLASS (int_REP_CLASS a) = a) ∧
∀r.
(λc. ∃r. r tint_eq r ∧ (c = $tint_eq r)) r ⇔
(int_REP_CLASS (int_ABS_CLASS r) = r)
[int_div] Definition
⊢ ∀i j.
j ≠ 0 ⇒
(i / j =
if 0 < j then
if 0 <= i then &(Num i DIV Num j)
else
-&(Num (-i) DIV Num j) + if Num (-i) MOD Num j = 0 then 0
else -1
else if 0 <= i then
-&(Num i DIV Num (-j)) + if Num i MOD Num (-j) = 0 then 0
else -1
else &(Num (-i) DIV Num (-j)))
[int_exp] Definition
⊢ (∀p. p ** 0 = 1) ∧ ∀p n. p ** SUC n = p * p ** n
[int_ge] Definition
⊢ ∀x y. x >= y ⇔ y <= x
[int_gt] Definition
⊢ ∀x y. x > y ⇔ y < x
[int_le] Definition
⊢ ∀x y. x <= y ⇔ ¬(y < x)
[int_lt] Definition
⊢ ∀T1 T2. T1 < T2 ⇔ int_REP T1 tint_lt int_REP T2
[int_mod] Definition
⊢ ∀i j. j ≠ 0 ⇒ (i % j = i - i / j * j)
[int_mul] Definition
⊢ ∀T1 T2. T1 * T2 = int_ABS (int_REP T1 tint_mul int_REP T2)
[int_neg] Definition
⊢ ∀T1. -T1 = int_ABS (tint_neg (int_REP T1))
[int_quot] Definition
⊢ ∀i j.
j ≠ 0 ⇒
(i quot j =
if 0 < j then
if 0 <= i then &(Num i DIV Num j)
else -&(Num (-i) DIV Num j)
else if 0 <= i then -&(Num i DIV Num (-j))
else &(Num (-i) DIV Num (-j)))
[int_rem] Definition
⊢ ∀i j. j ≠ 0 ⇒ (i rem j = i - i quot j * j)
[int_sub] Definition
⊢ ∀x y. x - y = x + -y
[tint_0] Definition
⊢ tint_0 = (1,1)
[tint_1] Definition
⊢ tint_1 = (1 + 1,1)
[tint_add] Definition
⊢ ∀x1 y1 x2 y2. (x1,y1) tint_add (x2,y2) = (x1 + x2,y1 + y2)
[tint_eq] Definition
⊢ ∀x1 y1 x2 y2. (x1,y1) tint_eq (x2,y2) ⇔ (x1 + y2 = x2 + y1)
[tint_lt] Definition
⊢ ∀x1 y1 x2 y2. (x1,y1) tint_lt (x2,y2) ⇔ x1 + y2 < x2 + y1
[tint_mul] Definition
⊢ ∀x1 y1 x2 y2.
(x1,y1) tint_mul (x2,y2) =
(x1 * x2 + y1 * y2,x1 * y2 + y1 * x2)
[tint_neg] Definition
⊢ ∀x y. tint_neg (x,y) = (y,x)
[tint_of_num] Definition
⊢ (tint_of_num 0 = tint_0) ∧
∀n. tint_of_num (SUC n) = tint_of_num n tint_add tint_1
[EQ_ADDL] Theorem
⊢ ∀x y. (x = x + y) ⇔ (y = 0)
[EQ_LADD] Theorem
⊢ ∀x y z. (x + y = x + z) ⇔ (y = z)
[INFINITE_INT_UNIV] Theorem
⊢ INFINITE 𝕌(:int)
[INT] Theorem
⊢ ∀n. &SUC n = &n + 1
[INT_0] Theorem
⊢ int_0 = 0
[INT_1] Theorem
⊢ int_1 = 1
[INT_10] Theorem
⊢ int_1 ≠ int_0
[INT_ABS_0LT] Theorem
⊢ 0 < ABS p ⇔ p ≠ 0
[INT_ABS_ABS] Theorem
⊢ ∀p. ABS (ABS p) = ABS p
[INT_ABS_EQ] Theorem
⊢ ∀p q.
((ABS p = q) ⇔ (p = q) ∧ 0 < q ∨ (p = -q) ∧ 0 <= q) ∧
((q = ABS p) ⇔ (p = q) ∧ 0 < q ∨ (p = -q) ∧ 0 <= q)
[INT_ABS_EQ0] Theorem
⊢ ∀p. (ABS p = 0) ⇔ (p = 0)
[INT_ABS_EQ_ABS] Theorem
⊢ (ABS x = ABS y) ⇔ (x = y) ∨ (x = -y)
[INT_ABS_EQ_ID] Theorem
⊢ ∀p. (ABS p = p) ⇔ 0 <= p
[INT_ABS_LE] Theorem
⊢ ∀p q.
(ABS p <= q ⇔ p <= q ∧ -q <= p) ∧
(q <= ABS p ⇔ q <= p ∨ p <= -q) ∧
(-ABS p <= q ⇔ -q <= p ∨ p <= q) ∧
(q <= -ABS p ⇔ p <= -q ∧ q <= p)
[INT_ABS_LE0] Theorem
⊢ ∀p. ABS p <= 0 ⇔ (p = 0)
[INT_ABS_LT] Theorem
⊢ ∀p q.
(ABS p < q ⇔ p < q ∧ -q < p) ∧ (q < ABS p ⇔ q < p ∨ p < -q) ∧
(-ABS p < q ⇔ -q < p ∨ p < q) ∧ (q < -ABS p ⇔ p < -q ∧ q < p)
[INT_ABS_LT0] Theorem
⊢ ∀p. ¬(ABS p < 0)
[INT_ABS_MUL] Theorem
⊢ ∀p q. ABS p * ABS q = ABS (p * q)
[INT_ABS_NEG] Theorem
⊢ ∀p. ABS (-p) = ABS p
[INT_ABS_NUM] Theorem
⊢ ∀n. ABS (&n) = &n
[INT_ABS_POS] Theorem
⊢ ∀p. 0 <= ABS p
[INT_ABS_QUOT] Theorem
⊢ ∀p q. q ≠ 0 ⇒ ABS (p quot q * q) <= ABS p
[INT_ADD] Theorem
⊢ ∀m n. &m + &n = &(m + n)
[INT_ADD2_SUB2] Theorem
⊢ ∀a b c d. a + b - (c + d) = a - c + (b - d)
[INT_ADD_ASSOC] Theorem
⊢ ∀z y x. x + (y + z) = x + y + z
[INT_ADD_CALCULATE] Theorem
⊢ ∀p n m.
(0 + p = p) ∧ (p + 0 = p) ∧ (&n + &m = &(n + m)) ∧
(&n + -&m = if m <= n then &(n - m) else -&(m - n)) ∧
(-&n + &m = if n <= m then &(m - n) else -&(n - m)) ∧
(-&n + -&m = -&(n + m))
[INT_ADD_COMM] Theorem
⊢ ∀y x. x + y = y + x
[INT_ADD_DIV] Theorem
⊢ ∀i j k.
k ≠ 0 ∧ ((i % k = 0) ∨ (j % k = 0)) ⇒
((i + j) / k = i / k + j / k)
[INT_ADD_LID] Theorem
⊢ ∀x. 0 + x = x
[INT_ADD_LID_UNIQ] Theorem
⊢ ∀x y. (x + y = y) ⇔ (x = 0)
[INT_ADD_LINV] Theorem
⊢ ∀x. -x + x = 0
[INT_ADD_REDUCE] Theorem
⊢ ∀p n m.
(0 + p = p) ∧ (p + 0 = p) ∧ (-0 = 0) ∧ (--p = p) ∧
(&NUMERAL n + &NUMERAL m = &NUMERAL (numeral$iZ (n + m))) ∧
(&NUMERAL n + -&NUMERAL m =
if m <= n then &NUMERAL (n - m)
else -&NUMERAL (m - n)) ∧
(-&NUMERAL n + &NUMERAL m =
if n <= m then &NUMERAL (m - n)
else -&NUMERAL (n - m)) ∧
(-&NUMERAL n + -&NUMERAL m = -&NUMERAL (numeral$iZ (n + m)))
[INT_ADD_RID] Theorem
⊢ ∀x. x + 0 = x
[INT_ADD_RID_UNIQ] Theorem
⊢ ∀x y. (x + y = x) ⇔ (y = 0)
[INT_ADD_RINV] Theorem
⊢ ∀x. x + -x = 0
[INT_ADD_SUB] Theorem
⊢ ∀x y. x + y - x = y
[INT_ADD_SUB2] Theorem
⊢ ∀x y. x - (x + y) = -y
[INT_ADD_SYM] Theorem
⊢ ∀y x. x + y = y + x
[INT_DIFFSQ] Theorem
⊢ ∀x y. (x + y) * (x - y) = x * x - y * y
[INT_DISCRETE] Theorem
⊢ ∀x y. ¬(x < y ∧ y < x + 1)
[INT_DIV] Theorem
⊢ ∀n m. m ≠ 0 ⇒ (&n / &m = &(n DIV m))
[INT_DIVIDES_0] Theorem
⊢ (∀x. x int_divides 0) ∧ ∀x. 0 int_divides x ⇔ (x = 0)
[INT_DIVIDES_1] Theorem
⊢ ∀x. 1 int_divides x ∧ (x int_divides 1 ⇔ (x = 1) ∨ (x = -1))
[INT_DIVIDES_LADD] Theorem
⊢ ∀p q r. p int_divides q ⇒ (p int_divides q + r ⇔ p int_divides r)
[INT_DIVIDES_LMUL] Theorem
⊢ ∀p q r. p int_divides q ⇒ p int_divides q * r
[INT_DIVIDES_LSUB] Theorem
⊢ ∀p q r. p int_divides q ⇒ (p int_divides q - r ⇔ p int_divides r)
[INT_DIVIDES_MOD0] Theorem
⊢ ∀p q. p int_divides q ⇔ (q % p = 0) ∧ p ≠ 0 ∨ (p = 0) ∧ (q = 0)
[INT_DIVIDES_MUL] Theorem
⊢ ∀p q. p int_divides p * q ∧ p int_divides q * p
[INT_DIVIDES_MUL_BOTH] Theorem
⊢ ∀p q r. p ≠ 0 ⇒ (p * q int_divides p * r ⇔ q int_divides r)
[INT_DIVIDES_NEG] Theorem
⊢ ∀p q.
(p int_divides -q ⇔ p int_divides q) ∧
(-p int_divides q ⇔ p int_divides q)
[INT_DIVIDES_RADD] Theorem
⊢ ∀p q r. p int_divides q ⇒ (p int_divides r + q ⇔ p int_divides r)
[INT_DIVIDES_REDUCE] Theorem
⊢ ∀n m p.
(0 int_divides 0 ⇔ T) ∧ (0 int_divides &NUMERAL (BIT1 n) ⇔ F) ∧
(0 int_divides &NUMERAL (BIT2 n) ⇔ F) ∧ (p int_divides 0 ⇔ T) ∧
(&NUMERAL (BIT1 n) int_divides &NUMERAL m ⇔
(NUMERAL m MOD NUMERAL (BIT1 n) = 0)) ∧
(&NUMERAL (BIT2 n) int_divides &NUMERAL m ⇔
(NUMERAL m MOD NUMERAL (BIT2 n) = 0)) ∧
(&NUMERAL (BIT1 n) int_divides -&NUMERAL m ⇔
(NUMERAL m MOD NUMERAL (BIT1 n) = 0)) ∧
(&NUMERAL (BIT2 n) int_divides -&NUMERAL m ⇔
(NUMERAL m MOD NUMERAL (BIT2 n) = 0)) ∧
(-&NUMERAL (BIT1 n) int_divides &NUMERAL m ⇔
(NUMERAL m MOD NUMERAL (BIT1 n) = 0)) ∧
(-&NUMERAL (BIT2 n) int_divides &NUMERAL m ⇔
(NUMERAL m MOD NUMERAL (BIT2 n) = 0)) ∧
(-&NUMERAL (BIT1 n) int_divides -&NUMERAL m ⇔
(NUMERAL m MOD NUMERAL (BIT1 n) = 0)) ∧
(-&NUMERAL (BIT2 n) int_divides -&NUMERAL m ⇔
(NUMERAL m MOD NUMERAL (BIT2 n) = 0))
[INT_DIVIDES_REFL] Theorem
⊢ ∀x. x int_divides x
[INT_DIVIDES_RMUL] Theorem
⊢ ∀p q r. p int_divides q ⇒ p int_divides r * q
[INT_DIVIDES_RSUB] Theorem
⊢ ∀p q r. p int_divides q ⇒ (p int_divides r - q ⇔ p int_divides r)
[INT_DIVIDES_TRANS] Theorem
⊢ ∀x y z. x int_divides y ∧ y int_divides z ⇒ x int_divides z
[INT_DIVISION] Theorem
⊢ ∀q.
q ≠ 0 ⇒
∀p.
(p = p / q * q + p % q) ∧
if q < 0 then q < p % q ∧ p % q <= 0
else 0 <= p % q ∧ p % q < q
[INT_DIV_0] Theorem
⊢ ∀q. q ≠ 0 ⇒ (0 / q = 0)
[INT_DIV_1] Theorem
⊢ ∀p. p / 1 = p
[INT_DIV_CALCULATE] Theorem
⊢ (∀n m. m ≠ 0 ⇒ (&n / &m = &(n DIV m))) ∧
(∀p q. q ≠ 0 ⇒ (p / -q = -p / q)) ∧ (∀m n. (&m = &n) ⇔ (m = n)) ∧
(∀x. (-x = 0) ⇔ (x = 0)) ∧ ∀x. --x = x
[INT_DIV_FORALL_P] Theorem
⊢ ∀P x c.
c ≠ 0 ⇒
(P (x / c) ⇔
∀k r.
(x = k * c + r) ∧
(c < 0 ∧ c < r ∧ r <= 0 ∨ ¬(c < 0) ∧ 0 <= r ∧ r < c) ⇒
P k)
[INT_DIV_ID] Theorem
⊢ ∀p. p ≠ 0 ⇒ (p / p = 1)
[INT_DIV_LMUL] Theorem
⊢ ∀i j. i ≠ 0 ⇒ (i * j / i = j)
[INT_DIV_MUL_ID] Theorem
⊢ ∀p q. q ≠ 0 ∧ (p % q = 0) ⇒ (p / q * q = p)
[INT_DIV_NEG] Theorem
⊢ ∀p q. q ≠ 0 ⇒ (p / -q = -p / q)
[INT_DIV_P] Theorem
⊢ ∀P x c.
c ≠ 0 ⇒
(P (x / c) ⇔
∃k r.
(x = k * c + r) ∧
(c < 0 ∧ c < r ∧ r <= 0 ∨ ¬(c < 0) ∧ 0 <= r ∧ r < c) ∧ P k)
[INT_DIV_REDUCE] Theorem
⊢ ∀m n.
(0 / &NUMERAL (BIT1 n) = 0) ∧ (0 / &NUMERAL (BIT2 n) = 0) ∧
(&NUMERAL m / &NUMERAL (BIT1 n) =
&(NUMERAL m DIV NUMERAL (BIT1 n))) ∧
(&NUMERAL m / &NUMERAL (BIT2 n) =
&(NUMERAL m DIV NUMERAL (BIT2 n))) ∧
(-&NUMERAL m / &NUMERAL (BIT1 n) =
-&(NUMERAL m DIV NUMERAL (BIT1 n)) +
if NUMERAL m MOD NUMERAL (BIT1 n) = 0 then 0 else -1) ∧
(-&NUMERAL m / &NUMERAL (BIT2 n) =
-&(NUMERAL m DIV NUMERAL (BIT2 n)) +
if NUMERAL m MOD NUMERAL (BIT2 n) = 0 then 0 else -1) ∧
(&NUMERAL m / -&NUMERAL (BIT1 n) =
-&(NUMERAL m DIV NUMERAL (BIT1 n)) +
if NUMERAL m MOD NUMERAL (BIT1 n) = 0 then 0 else -1) ∧
(&NUMERAL m / -&NUMERAL (BIT2 n) =
-&(NUMERAL m DIV NUMERAL (BIT2 n)) +
if NUMERAL m MOD NUMERAL (BIT2 n) = 0 then 0 else -1) ∧
(-&NUMERAL m / -&NUMERAL (BIT1 n) =
&(NUMERAL m DIV NUMERAL (BIT1 n))) ∧
(-&NUMERAL m / -&NUMERAL (BIT2 n) =
&(NUMERAL m DIV NUMERAL (BIT2 n)))
[INT_DIV_RMUL] Theorem
⊢ ∀i j. i ≠ 0 ⇒ (j * i / i = j)
[INT_DIV_UNIQUE] Theorem
⊢ ∀i j q.
(∃r.
(i = q * j + r) ∧ if j < 0 then j < r ∧ r <= 0
else 0 <= r ∧ r < j) ⇒
(i / j = q)
[INT_DOUBLE] Theorem
⊢ ∀x. x + x = 2 * x
[INT_ENTIRE] Theorem
⊢ ∀x y. (x * y = 0) ⇔ (x = 0) ∨ (y = 0)
[INT_EQ_CALCULATE] Theorem
⊢ (∀m n. (&m = &n) ⇔ (m = n)) ∧ (∀x y. (-x = -y) ⇔ (x = y)) ∧
∀n m.
((&n = -&m) ⇔ (n = 0) ∧ (m = 0)) ∧
((-&n = &m) ⇔ (n = 0) ∧ (m = 0))
[INT_EQ_IMP_LE] Theorem
⊢ ∀x y. (x = y) ⇒ x <= y
[INT_EQ_LADD] Theorem
⊢ ∀x y z. (x + y = x + z) ⇔ (y = z)
[INT_EQ_LMUL] Theorem
⊢ ∀x y z. (x * y = x * z) ⇔ (x = 0) ∨ (y = z)
[INT_EQ_LMUL2] Theorem
⊢ ∀x y z. x ≠ 0 ⇒ ((y = z) ⇔ (x * y = x * z))
[INT_EQ_LMUL_IMP] Theorem
⊢ ∀x y z. x ≠ 0 ∧ (x * y = x * z) ⇒ (y = z)
[INT_EQ_NEG] Theorem
⊢ ∀x y. (-x = -y) ⇔ (x = y)
[INT_EQ_RADD] Theorem
⊢ ∀x y z. (x + z = y + z) ⇔ (x = y)
[INT_EQ_REDUCE] Theorem
⊢ ∀n m.
((0 = 0) ⇔ T) ∧ ((0 = &NUMERAL (BIT1 n)) ⇔ F) ∧
((0 = &NUMERAL (BIT2 n)) ⇔ F) ∧
((0 = -&NUMERAL (BIT1 n)) ⇔ F) ∧
((0 = -&NUMERAL (BIT2 n)) ⇔ F) ∧
((&NUMERAL (BIT1 n) = 0) ⇔ F) ∧ ((&NUMERAL (BIT2 n) = 0) ⇔ F) ∧
((-&NUMERAL (BIT1 n) = 0) ⇔ F) ∧
((-&NUMERAL (BIT2 n) = 0) ⇔ F) ∧
((&NUMERAL n = &NUMERAL m) ⇔ (n = m)) ∧
((&NUMERAL (BIT1 n) = -&NUMERAL m) ⇔ F) ∧
((&NUMERAL (BIT2 n) = -&NUMERAL m) ⇔ F) ∧
((-&NUMERAL (BIT1 n) = &NUMERAL m) ⇔ F) ∧
((-&NUMERAL (BIT2 n) = &NUMERAL m) ⇔ F) ∧
((-&NUMERAL n = -&NUMERAL m) ⇔ (n = m))
[INT_EQ_RMUL] Theorem
⊢ ∀x y z. (x * z = y * z) ⇔ (z = 0) ∨ (x = y)
[INT_EQ_RMUL_IMP] Theorem
⊢ ∀x y z. z ≠ 0 ∧ (x * z = y * z) ⇒ (x = y)
[INT_EQ_SUB_LADD] Theorem
⊢ ∀x y z. (x = y - z) ⇔ (x + z = y)
[INT_EQ_SUB_RADD] Theorem
⊢ ∀x y z. (x - y = z) ⇔ (x = z + y)
[INT_EXP] Theorem
⊢ ∀n m. &n ** m = &(n ** m)
[INT_EXP_ADD_EXPONENTS] Theorem
⊢ ∀n m p. p ** n * p ** m = p ** (n + m)
[INT_EXP_CALCULATE] Theorem
⊢ ∀p n m.
(p ** 0 = 1) ∧ (&n ** m = &(n ** m)) ∧
(-&n ** NUMERAL (BIT1 m) = -&NUMERAL (n ** NUMERAL (BIT1 m))) ∧
(-&n ** NUMERAL (BIT2 m) = &NUMERAL (n ** NUMERAL (BIT2 m)))
[INT_EXP_EQ0] Theorem
⊢ ∀p n. (p ** n = 0) ⇔ (p = 0) ∧ n ≠ 0
[INT_EXP_MOD] Theorem
⊢ ∀m n p. n <= m ∧ p ≠ 0 ⇒ (p ** m % p ** n = 0)
[INT_EXP_MULTIPLY_EXPONENTS] Theorem
⊢ ∀m n p. (p ** n) ** m = p ** (n * m)
[INT_EXP_NEG] Theorem
⊢ ∀n m.
(EVEN n ⇒ (-&m ** n = &(m ** n))) ∧
(ODD n ⇒ (-&m ** n = -&(m ** n)))
[INT_EXP_REDUCE] Theorem
⊢ ∀n m p.
(p ** 0 = 1) ∧ (&NUMERAL n ** NUMERAL m = &NUMERAL (n ** m)) ∧
(-&NUMERAL n ** NUMERAL (BIT1 m) = -&NUMERAL (n ** BIT1 m)) ∧
(-&NUMERAL n ** NUMERAL (BIT2 m) = &NUMERAL (n ** BIT2 m))
[INT_EXP_SUBTRACT_EXPONENTS] Theorem
⊢ ∀m n p. n <= m ∧ p ≠ 0 ⇒ (p ** m / p ** n = p ** (m - n))
[INT_GE_CALCULATE] Theorem
⊢ ∀x y. x >= y ⇔ y <= x
[INT_GE_REDUCE] Theorem
⊢ ∀n m.
(0 >= 0 ⇔ T) ∧ (&NUMERAL n >= 0 ⇔ T) ∧
(-&NUMERAL (BIT1 n) >= 0 ⇔ F) ∧ (-&NUMERAL (BIT2 n) >= 0 ⇔ F) ∧
(0 >= &NUMERAL (BIT1 n) ⇔ F) ∧ (0 >= &NUMERAL (BIT2 n) ⇔ F) ∧
(0 >= -&NUMERAL (BIT1 n) ⇔ T) ∧ (0 >= -&NUMERAL (BIT2 n) ⇔ T) ∧
(&NUMERAL m >= &NUMERAL n ⇔ n <= m) ∧
(-&NUMERAL (BIT1 m) >= &NUMERAL n ⇔ F) ∧
(-&NUMERAL (BIT2 m) >= &NUMERAL n ⇔ F) ∧
(&NUMERAL m >= -&NUMERAL n ⇔ T) ∧
(-&NUMERAL m >= -&NUMERAL n ⇔ m <= n)
[INT_GT_CALCULATE] Theorem
⊢ ∀x y. x > y ⇔ y < x
[INT_GT_REDUCE] Theorem
⊢ ∀n m.
(&NUMERAL (BIT1 n) > 0 ⇔ T) ∧ (&NUMERAL (BIT2 n) > 0 ⇔ T) ∧
(0 > 0 ⇔ F) ∧ (-&NUMERAL n > 0 ⇔ F) ∧ (0 > &NUMERAL n ⇔ F) ∧
(0 > -&NUMERAL (BIT1 n) ⇔ T) ∧ (0 > -&NUMERAL (BIT2 n) ⇔ T) ∧
(&NUMERAL m > &NUMERAL n ⇔ n < m) ∧
(&NUMERAL m > -&NUMERAL (BIT1 n) ⇔ T) ∧
(&NUMERAL m > -&NUMERAL (BIT2 n) ⇔ T) ∧
(-&NUMERAL m > &NUMERAL n ⇔ F) ∧
(-&NUMERAL m > -&NUMERAL n ⇔ m < n)
[INT_INJ] Theorem
⊢ ∀m n. (&m = &n) ⇔ (m = n)
[INT_LDISTRIB] Theorem
⊢ ∀z y x. x * (y + z) = x * y + x * z
[INT_LE] Theorem
⊢ ∀m n. &m <= &n ⇔ m <= n
[INT_LESS_MOD] Theorem
⊢ ∀i j. 0 <= i ∧ i < j ⇒ (i % j = i)
[INT_LET_ADD] Theorem
⊢ ∀x y. 0 <= x ∧ 0 < y ⇒ 0 < x + y
[INT_LET_ADD2] Theorem
⊢ ∀w x y z. w <= x ∧ y < z ⇒ w + y < x + z
[INT_LET_ANTISYM] Theorem
⊢ ∀x y. ¬(x < y ∧ y <= x)
[INT_LET_TOTAL] Theorem
⊢ ∀x y. x <= y ∨ y < x
[INT_LET_TRANS] Theorem
⊢ ∀x y z. x <= y ∧ y < z ⇒ x < z
[INT_LE_01] Theorem
⊢ 0 <= 1
[INT_LE_ADD] Theorem
⊢ ∀x y. 0 <= x ∧ 0 <= y ⇒ 0 <= x + y
[INT_LE_ADD2] Theorem
⊢ ∀w x y z. w <= x ∧ y <= z ⇒ w + y <= x + z
[INT_LE_ADDL] Theorem
⊢ ∀x y. y <= x + y ⇔ 0 <= x
[INT_LE_ADDR] Theorem
⊢ ∀x y. x <= x + y ⇔ 0 <= y
[INT_LE_ANTISYM] Theorem
⊢ ∀x y. x <= y ∧ y <= x ⇔ (x = y)
[INT_LE_CALCULATE] Theorem
⊢ ∀x y. x <= y ⇔ x < y ∨ (x = y)
[INT_LE_DOUBLE] Theorem
⊢ ∀x. 0 <= x + x ⇔ 0 <= x
[INT_LE_LADD] Theorem
⊢ ∀x y z. x + y <= x + z ⇔ y <= z
[INT_LE_LT] Theorem
⊢ ∀x y. x <= y ⇔ x < y ∨ (x = y)
[INT_LE_LT1] Theorem
⊢ x <= y ⇔ x < y + 1
[INT_LE_MONO] Theorem
⊢ ∀x y z. 0 < x ⇒ (x * y <= x * z ⇔ y <= z)
[INT_LE_MUL] Theorem
⊢ ∀x y. 0 <= x ∧ 0 <= y ⇒ 0 <= x * y
[INT_LE_NEG] Theorem
⊢ ∀x y. -x <= -y ⇔ y <= x
[INT_LE_NEGL] Theorem
⊢ ∀x. -x <= x ⇔ 0 <= x
[INT_LE_NEGR] Theorem
⊢ ∀x. x <= -x ⇔ x <= 0
[INT_LE_NEGTOTAL] Theorem
⊢ ∀x. 0 <= x ∨ 0 <= -x
[INT_LE_RADD] Theorem
⊢ ∀x y z. x + z <= y + z ⇔ x <= y
[INT_LE_REDUCE] Theorem
⊢ ∀n m.
(0 <= 0 ⇔ T) ∧ (0 <= &NUMERAL n ⇔ T) ∧
(0 <= -&NUMERAL (BIT1 n) ⇔ F) ∧ (0 <= -&NUMERAL (BIT2 n) ⇔ F) ∧
(&NUMERAL (BIT1 n) <= 0 ⇔ F) ∧ (&NUMERAL (BIT2 n) <= 0 ⇔ F) ∧
(-&NUMERAL (BIT1 n) <= 0 ⇔ T) ∧ (-&NUMERAL (BIT2 n) <= 0 ⇔ T) ∧
(&NUMERAL n <= &NUMERAL m ⇔ n <= m) ∧
(&NUMERAL n <= -&NUMERAL (BIT1 m) ⇔ F) ∧
(&NUMERAL n <= -&NUMERAL (BIT2 m) ⇔ F) ∧
(-&NUMERAL n <= &NUMERAL m ⇔ T) ∧
(-&NUMERAL n <= -&NUMERAL m ⇔ m <= n)
[INT_LE_REFL] Theorem
⊢ ∀x. x <= x
[INT_LE_SQUARE] Theorem
⊢ ∀x. 0 <= x * x
[INT_LE_SUB_LADD] Theorem
⊢ ∀x y z. x <= y - z ⇔ x + z <= y
[INT_LE_SUB_RADD] Theorem
⊢ ∀x y z. x - y <= z ⇔ x <= z + y
[INT_LE_TOTAL] Theorem
⊢ ∀x y. x <= y ∨ y <= x
[INT_LE_TRANS] Theorem
⊢ ∀x y z. x <= y ∧ y <= z ⇒ x <= z
[INT_LNEG_UNIQ] Theorem
⊢ ∀x y. (x + y = 0) ⇔ (x = -y)
[INT_LT] Theorem
⊢ ∀m n. &m < &n ⇔ m < n
[INT_LTE_ADD] Theorem
⊢ ∀x y. 0 < x ∧ 0 <= y ⇒ 0 < x + y
[INT_LTE_ADD2] Theorem
⊢ ∀w x y z. w < x ∧ y <= z ⇒ w + y < x + z
[INT_LTE_ANTSYM] Theorem
⊢ ∀x y. ¬(x <= y ∧ y < x)
[INT_LTE_TOTAL] Theorem
⊢ ∀x y. x < y ∨ y <= x
[INT_LTE_TRANS] Theorem
⊢ ∀x y z. x < y ∧ y <= z ⇒ x < z
[INT_LT_01] Theorem
⊢ 0 < 1
[INT_LT_ADD] Theorem
⊢ ∀x y. 0 < x ∧ 0 < y ⇒ 0 < x + y
[INT_LT_ADD1] Theorem
⊢ ∀x y. x <= y ⇒ x < y + 1
[INT_LT_ADD2] Theorem
⊢ ∀w x y z. w < x ∧ y < z ⇒ w + y < x + z
[INT_LT_ADDL] Theorem
⊢ ∀x y. y < x + y ⇔ 0 < x
[INT_LT_ADDNEG] Theorem
⊢ ∀x y z. y < x + -z ⇔ y + z < x
[INT_LT_ADDNEG2] Theorem
⊢ ∀x y z. x + -y < z ⇔ x < z + y
[INT_LT_ADDR] Theorem
⊢ ∀x y. x < x + y ⇔ 0 < y
[INT_LT_ADD_SUB] Theorem
⊢ ∀x y z. x + y < z ⇔ x < z - y
[INT_LT_ANTISYM] Theorem
⊢ ∀x y. ¬(x < y ∧ y < x)
[INT_LT_CALCULATE] Theorem
⊢ ∀n m.
(&n < &m ⇔ n < m) ∧ (-&n < -&m ⇔ m < n) ∧
(-&n < &m ⇔ n ≠ 0 ∨ m ≠ 0) ∧ (&n < -&m ⇔ F)
[INT_LT_GT] Theorem
⊢ ∀x y. x < y ⇒ ¬(y < x)
[INT_LT_IMP_LE] Theorem
⊢ ∀x y. x < y ⇒ x <= y
[INT_LT_IMP_NE] Theorem
⊢ ∀x y. x < y ⇒ x ≠ y
[INT_LT_LADD] Theorem
⊢ ∀x y z. x + y < x + z ⇔ y < z
[INT_LT_LADD_IMP] Theorem
⊢ ∀x y z. y < z ⇒ x + y < x + z
[INT_LT_LE] Theorem
⊢ ∀x y. x < y ⇔ x <= y ∧ x ≠ y
[INT_LT_LE1] Theorem
⊢ x < y ⇔ x + 1 <= y
[INT_LT_MONO] Theorem
⊢ ∀x y z. 0 < x ⇒ (x * y < x * z ⇔ y < z)
[INT_LT_MUL] Theorem
⊢ ∀x y. int_0 < x ∧ int_0 < y ⇒ int_0 < x * y
[INT_LT_MUL2] Theorem
⊢ ∀x1 x2 y1 y2.
0 <= x1 ∧ 0 <= y1 ∧ x1 < x2 ∧ y1 < y2 ⇒ x1 * y1 < x2 * y2
[INT_LT_NEG] Theorem
⊢ ∀x y. -x < -y ⇔ y < x
[INT_LT_NEGTOTAL] Theorem
⊢ ∀x. (x = 0) ∨ 0 < x ∨ 0 < -x
[INT_LT_NZ] Theorem
⊢ ∀n. &n ≠ 0 ⇔ 0 < &n
[INT_LT_RADD] Theorem
⊢ ∀x y z. x + z < y + z ⇔ x < y
[INT_LT_REDUCE] Theorem
⊢ ∀n m.
(0 < &NUMERAL (BIT1 n) ⇔ T) ∧ (0 < &NUMERAL (BIT2 n) ⇔ T) ∧
(0 < 0 ⇔ F) ∧ (0 < -&NUMERAL n ⇔ F) ∧ (&NUMERAL n < 0 ⇔ F) ∧
(-&NUMERAL (BIT1 n) < 0 ⇔ T) ∧ (-&NUMERAL (BIT2 n) < 0 ⇔ T) ∧
(&NUMERAL n < &NUMERAL m ⇔ n < m) ∧
(-&NUMERAL (BIT1 n) < &NUMERAL m ⇔ T) ∧
(-&NUMERAL (BIT2 n) < &NUMERAL m ⇔ T) ∧
(&NUMERAL n < -&NUMERAL m ⇔ F) ∧
(-&NUMERAL n < -&NUMERAL m ⇔ m < n)
[INT_LT_REFL] Theorem
⊢ ∀x. ¬(x < x)
[INT_LT_SUB_LADD] Theorem
⊢ ∀x y z. x < y - z ⇔ x + z < y
[INT_LT_SUB_RADD] Theorem
⊢ ∀x y z. x - y < z ⇔ x < z + y
[INT_LT_TOTAL] Theorem
⊢ ∀x y. (x = y) ∨ x < y ∨ y < x
[INT_LT_TRANS] Theorem
⊢ ∀x y z. x < y ∧ y < z ⇒ x < z
[INT_MAX_LT] Theorem
⊢ ∀x y z. int_max x y < z ⇒ x < z ∧ y < z
[INT_MAX_NUM] Theorem
⊢ ∀m n. int_max (&m) (&n) = &MAX m n
[INT_MIN_LT] Theorem
⊢ ∀x y z. x < int_min y z ⇒ x < y ∧ x < z
[INT_MIN_NUM] Theorem
⊢ ∀m n. int_min (&m) (&n) = &MIN m n
[INT_MOD] Theorem
⊢ ∀n m. m ≠ 0 ⇒ (&n % &m = &(n MOD m))
[INT_MOD0] Theorem
⊢ ∀p. p ≠ 0 ⇒ (0 % p = 0)
[INT_MOD_1] Theorem
⊢ ∀i. i % 1 = 0
[INT_MOD_ADD_MULTIPLES] Theorem
⊢ k ≠ 0 ⇒ ((q * k + r) % k = r % k)
[INT_MOD_BOUNDS] Theorem
⊢ ∀p q.
q ≠ 0 ⇒
if q < 0 then q < p % q ∧ p % q <= 0
else 0 <= p % q ∧ p % q < q
[INT_MOD_CALCULATE] Theorem
⊢ (∀n m. m ≠ 0 ⇒ (&n % &m = &(n MOD m))) ∧
(∀p q. q ≠ 0 ⇒ (p % -q = -(-p % q))) ∧ (∀x. --x = x) ∧
(∀m n. (&m = &n) ⇔ (m = n)) ∧ ∀x. (-x = 0) ⇔ (x = 0)
[INT_MOD_COMMON_FACTOR] Theorem
⊢ ∀p. p ≠ 0 ⇒ ∀q. (q * p) % p = 0
[INT_MOD_EQ0] Theorem
⊢ ∀q. q ≠ 0 ⇒ ∀p. (p % q = 0) ⇔ ∃k. p = k * q
[INT_MOD_FORALL_P] Theorem
⊢ ∀P x c.
c ≠ 0 ⇒
(P (x % c) ⇔
∀q r.
(x = q * c + r) ∧
(c < 0 ∧ c < r ∧ r <= 0 ∨ ¬(c < 0) ∧ 0 <= r ∧ r < c) ⇒
P r)
[INT_MOD_ID] Theorem
⊢ ∀i. i ≠ 0 ⇒ (i % i = 0)
[INT_MOD_MINUS1] Theorem
⊢ ∀n. 0 < n ⇒ (-1 % n = n - 1)
[INT_MOD_MOD] Theorem
⊢ k ≠ 0 ⇒ (j % k % k = j % k)
[INT_MOD_NEG] Theorem
⊢ ∀p q. q ≠ 0 ⇒ (p % -q = -(-p % q))
[INT_MOD_NEG_NUMERATOR] Theorem
⊢ k ≠ 0 ⇒ (-x % k = (k - x) % k)
[INT_MOD_P] Theorem
⊢ ∀P x c.
c ≠ 0 ⇒
(P (x % c) ⇔
∃k r.
(x = k * c + r) ∧
(c < 0 ∧ c < r ∧ r <= 0 ∨ ¬(c < 0) ∧ 0 <= r ∧ r < c) ∧ P r)
[INT_MOD_PLUS] Theorem
⊢ k ≠ 0 ⇒ ((i % k + j % k) % k = (i + j) % k)
[INT_MOD_REDUCE] Theorem
⊢ ∀m n.
(0 % &NUMERAL (BIT1 n) = 0) ∧ (0 % &NUMERAL (BIT2 n) = 0) ∧
(0 % -&NUMERAL (BIT1 n) = 0) ∧ (0 % -&NUMERAL (BIT2 n) = 0) ∧
(&NUMERAL m % &NUMERAL (BIT1 n) =
&(NUMERAL m MOD NUMERAL (BIT1 n))) ∧
(&NUMERAL m % &NUMERAL (BIT2 n) =
&(NUMERAL m MOD NUMERAL (BIT2 n))) ∧
(&NUMERAL m % -&NUMERAL (BIT1 n) =
-(-&NUMERAL m % &NUMERAL (BIT1 n))) ∧
(&NUMERAL m % -&NUMERAL (BIT2 n) =
-(-&NUMERAL m % &NUMERAL (BIT2 n))) ∧
(x % &NUMERAL (BIT1 n) =
x - x / &NUMERAL (BIT1 n) * &NUMERAL (BIT1 n)) ∧
(x % &NUMERAL (BIT2 n) =
x - x / &NUMERAL (BIT2 n) * &NUMERAL (BIT2 n)) ∧
(x % -&NUMERAL (BIT1 n) =
-x / &NUMERAL (BIT1 n) * &NUMERAL (BIT1 n) + x) ∧
(x % -&NUMERAL (BIT2 n) =
-x / &NUMERAL (BIT2 n) * &NUMERAL (BIT2 n) + x)
[INT_MOD_SUB] Theorem
⊢ k ≠ 0 ⇒ ((i % k - j % k) % k = (i - j) % k)
[INT_MOD_UNIQUE] Theorem
⊢ ∀i j m.
(∃q.
(i = q * j + m) ∧ if j < 0 then j < m ∧ m <= 0
else 0 <= m ∧ m < j) ⇒
(i % j = m)
[INT_MUL] Theorem
⊢ ∀m n. &m * &n = &(m * n)
[INT_MUL_ASSOC] Theorem
⊢ ∀z y x. x * (y * z) = x * y * z
[INT_MUL_CALCULATE] Theorem
⊢ (∀m n. &m * &n = &(m * n)) ∧ (∀x y. -x * y = -(x * y)) ∧
(∀x y. x * -y = -(x * y)) ∧ ∀x. --x = x
[INT_MUL_COMM] Theorem
⊢ ∀y x. x * y = y * x
[INT_MUL_DIV] Theorem
⊢ ∀p q k. q ≠ 0 ∧ (p % q = 0) ⇒ (k * p / q = k * (p / q))
[INT_MUL_EQ_1] Theorem
⊢ ∀x y. (x * y = 1) ⇔ (x = 1) ∧ (y = 1) ∨ (x = -1) ∧ (y = -1)
[INT_MUL_LID] Theorem
⊢ ∀x. 1 * x = x
[INT_MUL_LZERO] Theorem
⊢ ∀x. 0 * x = 0
[INT_MUL_QUOT] Theorem
⊢ ∀p q k. q ≠ 0 ∧ (p rem q = 0) ⇒ (k * p quot q = k * (p quot q))
[INT_MUL_REDUCE] Theorem
⊢ ∀m n p.
(p * 0 = 0) ∧ (0 * p = 0) ∧
(&NUMERAL m * &NUMERAL n = &NUMERAL (m * n)) ∧
(-&NUMERAL m * &NUMERAL n = -&NUMERAL (m * n)) ∧
(&NUMERAL m * -&NUMERAL n = -&NUMERAL (m * n)) ∧
(-&NUMERAL m * -&NUMERAL n = &NUMERAL (m * n))
[INT_MUL_RID] Theorem
⊢ ∀x. x * 1 = x
[INT_MUL_RZERO] Theorem
⊢ ∀x. x * 0 = 0
[INT_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)
[INT_MUL_SYM] Theorem
⊢ ∀y x. x * y = y * x
[INT_NEGNEG] Theorem
⊢ ∀x. --x = x
[INT_NEG_0] Theorem
⊢ -0 = 0
[INT_NEG_ADD] Theorem
⊢ ∀x y. -(x + y) = -x + -y
[INT_NEG_EQ] Theorem
⊢ ∀x y. (-x = y) ⇔ (x = -y)
[INT_NEG_EQ0] Theorem
⊢ ∀x. (-x = 0) ⇔ (x = 0)
[INT_NEG_GE0] Theorem
⊢ ∀x. 0 <= -x ⇔ x <= 0
[INT_NEG_GT0] Theorem
⊢ ∀x. 0 < -x ⇔ x < 0
[INT_NEG_LE0] Theorem
⊢ ∀x. -x <= 0 ⇔ 0 <= x
[INT_NEG_LMUL] Theorem
⊢ ∀x y. -(x * y) = -x * y
[INT_NEG_LT0] Theorem
⊢ ∀x. -x < 0 ⇔ 0 < x
[INT_NEG_MINUS1] Theorem
⊢ ∀x. -x = -1 * x
[INT_NEG_MUL2] Theorem
⊢ ∀x y. -x * -y = x * y
[INT_NEG_RMUL] Theorem
⊢ ∀x y. -(x * y) = x * -y
[INT_NEG_SAME_EQ] Theorem
⊢ ∀p. (p = -p) ⇔ (p = 0)
[INT_NEG_SUB] Theorem
⊢ ∀x y. -(x - y) = y - x
[INT_NOT_LE] Theorem
⊢ ∀x y. ¬(x <= y) ⇔ y < x
[INT_NOT_LT] Theorem
⊢ ∀x y. ¬(x < y) ⇔ y <= x
[INT_NUM_CASES] Theorem
⊢ ∀p. (∃n. (p = &n) ∧ n ≠ 0) ∨ (∃n. (p = -&n) ∧ n ≠ 0) ∨ (p = 0)
[INT_NZ_IMP_LT] Theorem
⊢ ∀n. n ≠ 0 ⇒ 0 < &n
[INT_OF_NUM] Theorem
⊢ ∀i. (&Num i = i) ⇔ 0 <= i
[INT_POASQ] Theorem
⊢ ∀x. 0 < x * x ⇔ x ≠ 0
[INT_POS] Theorem
⊢ ∀n. 0 <= &n
[INT_POS_NZ] Theorem
⊢ ∀x. 0 < x ⇒ x ≠ 0
[INT_QUOT] Theorem
⊢ ∀p q. q ≠ 0 ⇒ (&p quot &q = &(p DIV q))
[INT_QUOT_0] Theorem
⊢ ∀q. q ≠ 0 ⇒ (0 quot q = 0)
[INT_QUOT_1] Theorem
⊢ ∀p. p quot 1 = p
[INT_QUOT_CALCULATE] Theorem
⊢ (∀p q. q ≠ 0 ⇒ (&p quot &q = &(p DIV q))) ∧
(∀p q.
q ≠ 0 ⇒ (-p quot q = -(p quot q)) ∧ (p quot -q = -(p quot q))) ∧
(∀m n. (&m = &n) ⇔ (m = n)) ∧ (∀x. (-x = 0) ⇔ (x = 0)) ∧
∀x. --x = x
[INT_QUOT_ID] Theorem
⊢ ∀p. p ≠ 0 ⇒ (p quot p = 1)
[INT_QUOT_NEG] Theorem
⊢ ∀p q. q ≠ 0 ⇒ (-p quot q = -(p quot q)) ∧ (p quot -q = -(p quot q))
[INT_QUOT_REDUCE] Theorem
⊢ ∀m n.
(0 quot &NUMERAL (BIT1 n) = 0) ∧
(0 quot &NUMERAL (BIT2 n) = 0) ∧
(&NUMERAL m quot &NUMERAL (BIT1 n) =
&(NUMERAL m DIV NUMERAL (BIT1 n))) ∧
(&NUMERAL m quot &NUMERAL (BIT2 n) =
&(NUMERAL m DIV NUMERAL (BIT2 n))) ∧
(-&NUMERAL m quot &NUMERAL (BIT1 n) =
-&(NUMERAL m DIV NUMERAL (BIT1 n))) ∧
(-&NUMERAL m quot &NUMERAL (BIT2 n) =
-&(NUMERAL m DIV NUMERAL (BIT2 n))) ∧
(&NUMERAL m quot -&NUMERAL (BIT1 n) =
-&(NUMERAL m DIV NUMERAL (BIT1 n))) ∧
(&NUMERAL m quot -&NUMERAL (BIT2 n) =
-&(NUMERAL m DIV NUMERAL (BIT2 n))) ∧
(-&NUMERAL m quot -&NUMERAL (BIT1 n) =
&(NUMERAL m DIV NUMERAL (BIT1 n))) ∧
(-&NUMERAL m quot -&NUMERAL (BIT2 n) =
&(NUMERAL m DIV NUMERAL (BIT2 n)))
[INT_QUOT_UNIQUE] Theorem
⊢ ∀p q k.
(∃r.
(p = k * q + r) ∧ (if 0 < p then 0 <= r else r <= 0) ∧
ABS r < ABS q) ⇒
(p quot q = k)
[INT_RDISTRIB] Theorem
⊢ ∀x y z. (x + y) * z = x * z + y * z
[INT_REM] Theorem
⊢ ∀p q. q ≠ 0 ⇒ (&p rem &q = &(p MOD q))
[INT_REM0] Theorem
⊢ ∀q. q ≠ 0 ⇒ (0 rem q = 0)
[INT_REMQUOT] Theorem
⊢ ∀q.
q ≠ 0 ⇒
∀p.
(p = p quot q * q + p rem q) ∧
(if 0 < p then 0 <= p rem q else p rem q <= 0) ∧
ABS (p rem q) < ABS q
[INT_REM_CALCULATE] Theorem
⊢ (∀p q. q ≠ 0 ⇒ (&p rem &q = &(p MOD q))) ∧
(∀p q. q ≠ 0 ⇒ (-p rem q = -(p rem q)) ∧ (p rem -q = p rem q)) ∧
(∀x. --x = x) ∧ (∀m n. (&m = &n) ⇔ (m = n)) ∧
∀x. (-x = 0) ⇔ (x = 0)
[INT_REM_COMMON_FACTOR] Theorem
⊢ ∀p. p ≠ 0 ⇒ ∀q. (q * p) rem p = 0
[INT_REM_EQ0] Theorem
⊢ ∀q. q ≠ 0 ⇒ ∀p. (p rem q = 0) ⇔ ∃k. p = k * q
[INT_REM_EQ_MOD] Theorem
⊢ ∀i n.
0 < n ⇒
(i rem n = if i < 0 then (i - 1) % n - n + 1 else i % n)
[INT_REM_ID] Theorem
⊢ ∀p. p ≠ 0 ⇒ (p rem p = 0)
[INT_REM_NEG] Theorem
⊢ ∀p q. q ≠ 0 ⇒ (-p rem q = -(p rem q)) ∧ (p rem -q = p rem q)
[INT_REM_REDUCE] Theorem
⊢ ∀m n.
(0 rem &NUMERAL (BIT1 n) = 0) ∧ (0 rem &NUMERAL (BIT2 n) = 0) ∧
(&NUMERAL m rem &NUMERAL (BIT1 n) =
&(NUMERAL m MOD NUMERAL (BIT1 n))) ∧
(&NUMERAL m rem &NUMERAL (BIT2 n) =
&(NUMERAL m MOD NUMERAL (BIT2 n))) ∧
(-&NUMERAL m rem &NUMERAL (BIT1 n) =
-&(NUMERAL m MOD NUMERAL (BIT1 n))) ∧
(-&NUMERAL m rem &NUMERAL (BIT2 n) =
-&(NUMERAL m MOD NUMERAL (BIT2 n))) ∧
(&NUMERAL m rem -&NUMERAL (BIT1 n) =
&(NUMERAL m MOD NUMERAL (BIT1 n))) ∧
(&NUMERAL m rem -&NUMERAL (BIT2 n) =
&(NUMERAL m MOD NUMERAL (BIT2 n))) ∧
(-&NUMERAL m rem -&NUMERAL (BIT1 n) =
-&(NUMERAL m MOD NUMERAL (BIT1 n))) ∧
(-&NUMERAL m rem -&NUMERAL (BIT2 n) =
-&(NUMERAL m MOD NUMERAL (BIT2 n)))
[INT_REM_UNIQUE] Theorem
⊢ ∀p q r.
ABS r < ABS q ∧ (if 0 < p then 0 <= r else r <= 0) ∧
(∃k. p = k * q + r) ⇒
(p rem q = r)
[INT_RNEG_UNIQ] Theorem
⊢ ∀x y. (x + y = 0) ⇔ (y = -x)
[INT_SUB] Theorem
⊢ ∀n m. m <= n ⇒ (&n - &m = &(n - m))
[INT_SUB_0] Theorem
⊢ ∀x y. (x - y = 0) ⇔ (x = y)
[INT_SUB_ADD] Theorem
⊢ ∀x y. x - y + y = x
[INT_SUB_ADD2] Theorem
⊢ ∀x y. y + (x - y) = x
[INT_SUB_CALCULATE] Theorem
⊢ ∀x y. x - y = x + -y
[INT_SUB_LDISTRIB] Theorem
⊢ ∀x y z. x * (y - z) = x * y - x * z
[INT_SUB_LE] Theorem
⊢ ∀x y. 0 <= x - y ⇔ y <= x
[INT_SUB_LNEG] Theorem
⊢ ∀x y. -x - y = -(x + y)
[INT_SUB_LT] Theorem
⊢ ∀x y. 0 < x - y ⇔ y < x
[INT_SUB_LZERO] Theorem
⊢ ∀x. 0 - x = -x
[INT_SUB_NEG2] Theorem
⊢ ∀x y. -x - -y = y - x
[INT_SUB_RDISTRIB] Theorem
⊢ ∀x y z. (x - y) * z = x * z - y * z
[INT_SUB_REDUCE] Theorem
⊢ ∀m n p.
(p - 0 = p) ∧ (0 - p = -p) ∧
(&NUMERAL m - &NUMERAL n = &NUMERAL m + -&NUMERAL n) ∧
(-&NUMERAL m - &NUMERAL n = -&NUMERAL m + -&NUMERAL n) ∧
(&NUMERAL m - -&NUMERAL n = &NUMERAL m + &NUMERAL n) ∧
(-&NUMERAL m - -&NUMERAL n = -&NUMERAL m + &NUMERAL n)
[INT_SUB_REFL] Theorem
⊢ ∀x. x - x = 0
[INT_SUB_RNEG] Theorem
⊢ ∀x y. x - -y = x + y
[INT_SUB_RZERO] Theorem
⊢ ∀x. x - 0 = x
[INT_SUB_SUB] Theorem
⊢ ∀x y. x - y - x = -y
[INT_SUB_SUB2] Theorem
⊢ ∀x y. x - (x - y) = y
[INT_SUB_TRIANGLE] Theorem
⊢ ∀a b c. a - b + (b - c) = a - c
[INT_SUMSQ] Theorem
⊢ ∀x y. (x * x + y * y = 0) ⇔ (x = 0) ∧ (y = 0)
[LE_NUM_OF_INT] Theorem
⊢ ∀n i. &n <= i ⇒ n <= Num i
[LT_ADD2] Theorem
⊢ ∀x1 x2 y1 y2. x1 < y1 ∧ x2 < y2 ⇒ x1 + x2 < y1 + y2
[LT_ADDL] Theorem
⊢ ∀x y. x < x + y ⇔ 0 < y
[LT_ADDR] Theorem
⊢ ∀x y. ¬(x + y < x)
[LT_LADD] Theorem
⊢ ∀x y z. x + y < x + z ⇔ y < z
[NUM_LT] Theorem
⊢ 0 <= x ∧ 0 <= y ⇒ (Num x < Num y ⇔ x < y)
[NUM_NEGINT_EXISTS] Theorem
⊢ ∀i. i <= 0 ⇒ ∃n. i = -&n
[NUM_OF_INT] Theorem
⊢ ∀n. Num (&n) = n
[NUM_POSINT] Theorem
⊢ ∀i. 0 <= i ⇒ ∃!n. i = &n
[NUM_POSINT_EX] Theorem
⊢ ∀t. ¬(t < int_0) ⇒ ∃n. t = &n
[NUM_POSINT_EXISTS] Theorem
⊢ ∀i. 0 <= i ⇒ ∃n. i = &n
[NUM_POSTINT_EX] Theorem
⊢ ∀t. ¬(t tint_lt tint_0) ⇒ ∃n. t tint_eq tint_of_num n
[TINT_10] Theorem
⊢ ¬(tint_1 tint_eq tint_0)
[TINT_ADD_ASSOC] Theorem
⊢ ∀x y z. x tint_add (y tint_add z) = x tint_add y tint_add z
[TINT_ADD_LID] Theorem
⊢ ∀x. tint_0 tint_add x tint_eq x
[TINT_ADD_LINV] Theorem
⊢ ∀x. tint_neg x tint_add x tint_eq tint_0
[TINT_ADD_SYM] Theorem
⊢ ∀x y. x tint_add y = y tint_add x
[TINT_ADD_WELLDEF] Theorem
⊢ ∀x1 x2 y1 y2.
x1 tint_eq x2 ∧ y1 tint_eq y2 ⇒
x1 tint_add y1 tint_eq x2 tint_add y2
[TINT_ADD_WELLDEFR] Theorem
⊢ ∀x1 x2 y. x1 tint_eq x2 ⇒ x1 tint_add y tint_eq x2 tint_add y
[TINT_EQ_AP] Theorem
⊢ ∀p q. (p = q) ⇒ p tint_eq q
[TINT_EQ_EQUIV] Theorem
⊢ ∀p q. p tint_eq q ⇔ ($tint_eq p = $tint_eq q)
[TINT_EQ_REFL] Theorem
⊢ ∀x. x tint_eq x
[TINT_EQ_SYM] Theorem
⊢ ∀x y. x tint_eq y ⇔ y tint_eq x
[TINT_EQ_TRANS] Theorem
⊢ ∀x y z. x tint_eq y ∧ y tint_eq z ⇒ x tint_eq z
[TINT_INJ] Theorem
⊢ ∀m n. tint_of_num m tint_eq tint_of_num n ⇔ (m = n)
[TINT_LDISTRIB] Theorem
⊢ ∀x y z.
x tint_mul (y tint_add z) = x tint_mul y tint_add x tint_mul z
[TINT_LT_ADD] Theorem
⊢ ∀x y z. y tint_lt z ⇒ x tint_add y tint_lt x tint_add z
[TINT_LT_MUL] Theorem
⊢ ∀x y.
tint_0 tint_lt x ∧ tint_0 tint_lt y ⇒
tint_0 tint_lt x tint_mul y
[TINT_LT_REFL] Theorem
⊢ ∀x. ¬(x tint_lt x)
[TINT_LT_TOTAL] Theorem
⊢ ∀x y. x tint_eq y ∨ x tint_lt y ∨ y tint_lt x
[TINT_LT_TRANS] Theorem
⊢ ∀x y z. x tint_lt y ∧ y tint_lt z ⇒ x tint_lt z
[TINT_LT_WELLDEF] Theorem
⊢ ∀x1 x2 y1 y2.
x1 tint_eq x2 ∧ y1 tint_eq y2 ⇒ (x1 tint_lt y1 ⇔ x2 tint_lt y2)
[TINT_LT_WELLDEFL] Theorem
⊢ ∀x y1 y2. y1 tint_eq y2 ⇒ (x tint_lt y1 ⇔ x tint_lt y2)
[TINT_LT_WELLDEFR] Theorem
⊢ ∀x1 x2 y. x1 tint_eq x2 ⇒ (x1 tint_lt y ⇔ x2 tint_lt y)
[TINT_MUL_ASSOC] Theorem
⊢ ∀x y z. x tint_mul (y tint_mul z) = x tint_mul y tint_mul z
[TINT_MUL_LID] Theorem
⊢ ∀x. tint_1 tint_mul x tint_eq x
[TINT_MUL_SYM] Theorem
⊢ ∀x y. x tint_mul y = y tint_mul x
[TINT_MUL_WELLDEF] Theorem
⊢ ∀x1 x2 y1 y2.
x1 tint_eq x2 ∧ y1 tint_eq y2 ⇒
x1 tint_mul y1 tint_eq x2 tint_mul y2
[TINT_MUL_WELLDEFR] Theorem
⊢ ∀x1 x2 y. x1 tint_eq x2 ⇒ x1 tint_mul y tint_eq x2 tint_mul y
[TINT_NEG_WELLDEF] Theorem
⊢ ∀x1 x2. x1 tint_eq x2 ⇒ tint_neg x1 tint_eq tint_neg x2
[int_ABS_REP_CLASS] Theorem
⊢ (∀a. int_ABS_CLASS (int_REP_CLASS a) = a) ∧
∀c.
(∃r. r tint_eq r ∧ (c = $tint_eq r)) ⇔
(int_REP_CLASS (int_ABS_CLASS c) = c)
[int_QUOTIENT] Theorem
⊢ quotient$QUOTIENT $tint_eq int_ABS int_REP
[int_of_num] Theorem
⊢ (0 = int_0) ∧ ∀n. &SUC n = &n + int_1
[tint_of_num_eq] Theorem
⊢ ∀n. FST (tint_of_num n) = SND (tint_of_num n) + n
*)
end
HOL 4, Kananaskis-11