Structure arithmeticTheory
signature arithmeticTheory =
sig
type thm = Thm.thm
(* Definitions *)
val ABS_DIFF_def : thm
val ADD : thm
val ALT_ZERO : thm
val BIT1 : thm
val BIT2 : thm
val DIV2_def : thm
val DIVISION : thm
val DIVMOD_DEF : thm
val EVEN : thm
val EXP : thm
val FACT : thm
val FUNPOW : thm
val GREATER_DEF : thm
val GREATER_OR_EQ : thm
val LESS_OR_EQ : thm
val MAX_DEF : thm
val MIN_DEF : thm
val MODEQ_DEF : thm
val MULT : thm
val NRC : thm
val NUMERAL_DEF : thm
val ODD : thm
val SUB : thm
val findq_def : thm
val nat_elim__magic : thm
val num_case_def : thm
(* Theorems *)
val ABS_DIFF_ADD_SAME : thm
val ABS_DIFF_COMM : thm
val ABS_DIFF_EQS : thm
val ABS_DIFF_EQ_0 : thm
val ABS_DIFF_LE_SUM : thm
val ABS_DIFF_PLUS_LE : thm
val ABS_DIFF_SUC : thm
val ABS_DIFF_SUC_LE : thm
val ABS_DIFF_SUMS : thm
val ABS_DIFF_SYM : thm
val ABS_DIFF_TRIANGLE : thm
val ABS_DIFF_TRIANGLE_lem : thm
val ABS_DIFF_ZERO : thm
val ADD1 : thm
val ADD_0 : thm
val ADD_ASSOC : thm
val ADD_CLAUSES : thm
val ADD_COMM : thm
val ADD_DIV_ADD_DIV : thm
val ADD_DIV_RWT : thm
val ADD_EQ_0 : thm
val ADD_EQ_1 : thm
val ADD_EQ_SUB : thm
val ADD_INV_0 : thm
val ADD_INV_0_EQ : thm
val ADD_MOD : thm
val ADD_MODULUS : thm
val ADD_MODULUS_LEFT : thm
val ADD_MODULUS_RIGHT : thm
val ADD_MONO_LESS_EQ : thm
val ADD_SUB : thm
val ADD_SUC : thm
val ADD_SYM : thm
val BOUNDED_EXISTS_THM : thm
val BOUNDED_FORALL_THM : thm
val CANCEL_SUB : thm
val COMPLETE_INDUCTION : thm
val DA : thm
val DIV2_DOUBLE : thm
val DIVMOD_CALC : thm
val DIVMOD_CORRECT : thm
val DIVMOD_ID : thm
val DIVMOD_THM : thm
val DIV_0_IMP_LT : thm
val DIV_1 : thm
val DIV_DIV_DIV_MULT : thm
val DIV_EQ_0 : thm
val DIV_EQ_X : thm
val DIV_LESS : thm
val DIV_LESS_EQ : thm
val DIV_LE_MONOTONE : thm
val DIV_LE_X : thm
val DIV_LT_X : thm
val DIV_MOD_MOD_DIV : thm
val DIV_MULT : thm
val DIV_ONE : thm
val DIV_P : thm
val DIV_P_UNIV : thm
val DIV_SUB : thm
val DIV_UNIQUE : thm
val DOUBLE_LT : thm
val EQ_ADD_LCANCEL : thm
val EQ_ADD_RCANCEL : thm
val EQ_LESS_EQ : thm
val EQ_MONO_ADD_EQ : thm
val EQ_MULT_LCANCEL : thm
val EQ_MULT_RCANCEL : thm
val EVEN_ADD : thm
val EVEN_AND_ODD : thm
val EVEN_DOUBLE : thm
val EVEN_EXISTS : thm
val EVEN_EXP : thm
val EVEN_EXP_IFF : thm
val EVEN_MOD2 : thm
val EVEN_MULT : thm
val EVEN_ODD : thm
val EVEN_ODD_EXISTS : thm
val EVEN_OR_ODD : thm
val EVEN_SUB : thm
val EXISTS_GREATEST : thm
val EXISTS_NUM : thm
val EXP2_LT : thm
val EXP_1 : thm
val EXP_ADD : thm
val EXP_ALWAYS_BIG_ENOUGH : thm
val EXP_BASE_INJECTIVE : thm
val EXP_BASE_LEQ_MONO_IMP : thm
val EXP_BASE_LEQ_MONO_SUC_IMP : thm
val EXP_BASE_LE_IFF : thm
val EXP_BASE_LE_MONO : thm
val EXP_BASE_LT_MONO : thm
val EXP_BASE_MULT : thm
val EXP_EQ_0 : thm
val EXP_EQ_1 : thm
val EXP_EXP_INJECTIVE : thm
val EXP_EXP_LE_MONO : thm
val EXP_EXP_LT_MONO : thm
val EXP_EXP_MULT : thm
val EXP_MOD : thm
val EXP_SUB : thm
val EXP_SUB_NUMERAL : thm
val FACT_LESS : thm
val FORALL_NUM : thm
val FORALL_NUM_THM : thm
val FUNPOW_0 : thm
val FUNPOW_1 : thm
val FUNPOW_ADD : thm
val FUNPOW_SUC : thm
val GREATER_EQ : thm
val INV_PRE_EQ : thm
val INV_PRE_LESS : thm
val INV_PRE_LESS_EQ : thm
val LE : thm
val LEFT_ADD_DISTRIB : thm
val LEFT_SUB_DISTRIB : thm
val LESS_0_CASES : thm
val LESS_ADD : thm
val LESS_ADD_1 : thm
val LESS_ADD_NONZERO : thm
val LESS_ADD_SUC : thm
val LESS_ANTISYM : thm
val LESS_CASES : thm
val LESS_CASES_IMP : thm
val LESS_DIV_EQ_ZERO : thm
val LESS_EQ : thm
val LESS_EQUAL_ADD : thm
val LESS_EQUAL_ANTISYM : thm
val LESS_EQUAL_DIFF : thm
val LESS_EQ_0 : thm
val LESS_EQ_ADD : thm
val LESS_EQ_ADD_EXISTS : thm
val LESS_EQ_ADD_SUB : thm
val LESS_EQ_ANTISYM : thm
val LESS_EQ_CASES : thm
val LESS_EQ_EXISTS : thm
val LESS_EQ_IFF_LESS_SUC : thm
val LESS_EQ_IMP_LESS_SUC : thm
val LESS_EQ_LESS_EQ_MONO : thm
val LESS_EQ_LESS_TRANS : thm
val LESS_EQ_MONO : thm
val LESS_EQ_MONO_ADD_EQ : thm
val LESS_EQ_REFL : thm
val LESS_EQ_SUB_LESS : thm
val LESS_EQ_SUC_REFL : thm
val LESS_EQ_TRANS : thm
val LESS_EXP_SUC_MONO : thm
val LESS_IMP_LESS_ADD : thm
val LESS_IMP_LESS_OR_EQ : thm
val LESS_LESS_CASES : thm
val LESS_LESS_EQ_TRANS : thm
val LESS_LESS_SUC : thm
val LESS_MOD : thm
val LESS_MONO_ADD : thm
val LESS_MONO_ADD_EQ : thm
val LESS_MONO_ADD_INV : thm
val LESS_MONO_EQ : thm
val LESS_MONO_MULT : thm
val LESS_MONO_MULT2 : thm
val LESS_MONO_REV : thm
val LESS_MULT2 : thm
val LESS_MULT_MONO : thm
val LESS_NOT_SUC : thm
val LESS_OR : thm
val LESS_OR_EQ_ADD : thm
val LESS_OR_EQ_ALT : thm
val LESS_STRONG_ADD : thm
val LESS_SUB_ADD_LESS : thm
val LESS_SUC_EQ_COR : thm
val LESS_SUC_NOT : thm
val LESS_TRANS : thm
val LE_ADD_LCANCEL : thm
val LE_ADD_RCANCEL : thm
val LE_LT1 : thm
val LE_MULT_CANCEL_LBARE : thm
val LE_MULT_CANCEL_RBARE : thm
val LE_MULT_LCANCEL : thm
val LE_MULT_RCANCEL : thm
val LE_SUB_LCANCEL : thm
val LE_SUB_RCANCEL : thm
val LT_ADD_LCANCEL : thm
val LT_ADD_RCANCEL : thm
val LT_MULT_CANCEL_LBARE : thm
val LT_MULT_CANCEL_RBARE : thm
val LT_MULT_LCANCEL : thm
val LT_MULT_RCANCEL : thm
val LT_SUB_LCANCEL : thm
val LT_SUB_RCANCEL : thm
val MAX_0 : thm
val MAX_ASSOC : thm
val MAX_COMM : thm
val MAX_EQ_0 : thm
val MAX_IDEM : thm
val MAX_LE : thm
val MAX_LT : thm
val MIN_0 : thm
val MIN_ASSOC : thm
val MIN_COMM : thm
val MIN_EQ_0 : thm
val MIN_IDEM : thm
val MIN_LE : thm
val MIN_LT : thm
val MIN_MAX_EQ : thm
val MIN_MAX_LE : thm
val MIN_MAX_LT : thm
val MIN_MAX_PRED : thm
val MODEQ_0 : thm
val MODEQ_0_CONG : thm
val MODEQ_EXP_CONG : thm
val MODEQ_INTRO_CONG : thm
val MODEQ_MOD : thm
val MODEQ_MULT_CONG : thm
val MODEQ_NONZERO_MODEQUALITY : thm
val MODEQ_NUMERAL : thm
val MODEQ_PLUS_CONG : thm
val MODEQ_REFL : thm
val MODEQ_SUC_CONG : thm
val MODEQ_SYM : thm
val MODEQ_THM : thm
val MODEQ_TRANS : thm
val MOD_1 : thm
val MOD_2 : thm
val MOD_COMMON_FACTOR : thm
val MOD_ELIM : thm
val MOD_EQ_0 : thm
val MOD_EQ_0_DIVISOR : thm
val MOD_LESS : thm
val MOD_LESS_EQ : thm
val MOD_LIFT_PLUS : thm
val MOD_LIFT_PLUS_IFF : thm
val MOD_MOD : thm
val MOD_MULT : thm
val MOD_MULT_MOD : thm
val MOD_ONE : thm
val MOD_P : thm
val MOD_PLUS : thm
val MOD_P_UNIV : thm
val MOD_SUB : thm
val MOD_SUC : thm
val MOD_SUC_IFF : thm
val MOD_TIMES : thm
val MOD_TIMES2 : thm
val MOD_TIMES_SUB : thm
val MOD_UNIQUE : thm
val MULT_0 : thm
val MULT_ASSOC : thm
val MULT_CLAUSES : thm
val MULT_COMM : thm
val MULT_DIV : thm
val MULT_EQ_0 : thm
val MULT_EQ_1 : thm
val MULT_EQ_DIV : thm
val MULT_EQ_ID : thm
val MULT_EXP_MONO : thm
val MULT_INCREASES : thm
val MULT_LEFT_1 : thm
val MULT_LESS_EQ_SUC : thm
val MULT_MONO_EQ : thm
val MULT_RIGHT_1 : thm
val MULT_SUC : thm
val MULT_SUC_EQ : thm
val MULT_SYM : thm
val NORM_0 : thm
val NOT_EXP_0 : thm
val NOT_GREATER : thm
val NOT_GREATER_EQ : thm
val NOT_LEQ : thm
val NOT_LESS : thm
val NOT_LESS_EQUAL : thm
val NOT_LT_ZERO_EQ_ZERO : thm
val NOT_NUM_EQ : thm
val NOT_ODD_EQ_EVEN : thm
val NOT_STRICTLY_DECREASING : thm
val NOT_SUC_ADD_LESS_EQ : thm
val NOT_SUC_LESS_EQ : thm
val NOT_SUC_LESS_EQ_0 : thm
val NOT_ZERO_LT_ZERO : thm
val NRC_0 : thm
val NRC_1 : thm
val NRC_ADD_E : thm
val NRC_ADD_EQN : thm
val NRC_ADD_I : thm
val NRC_RTC : thm
val NRC_SUC_RECURSE_LEFT : thm
val NUMERAL_MULT_EQ_DIV : thm
val ODD_ADD : thm
val ODD_DOUBLE : thm
val ODD_EVEN : thm
val ODD_EXISTS : thm
val ODD_EXP : thm
val ODD_EXP_IFF : thm
val ODD_MULT : thm
val ODD_OR_EVEN : thm
val ODD_SUB : thm
val ONE : thm
val ONE_LT_EXP : thm
val ONE_LT_MULT : thm
val ONE_LT_MULT_IMP : thm
val ONE_MOD : thm
val ONE_MOD_IFF : thm
val ONE_ONE_INV_IMAGE_BOUNDED : thm
val ONE_ONE_UNBOUNDED : thm
val OR_LESS : thm
val PRE_ELIM_THM : thm
val PRE_LESS_EQ : thm
val PRE_SUB : thm
val PRE_SUB1 : thm
val PRE_SUC_EQ : thm
val RIGHT_ADD_DISTRIB : thm
val RIGHT_SUB_DISTRIB : thm
val RTC_NRC : thm
val RTC_eq_NRC : thm
val STRICTLY_INCREASING_ONE_ONE : thm
val STRICTLY_INCREASING_TC : thm
val STRICTLY_INCREASING_UNBOUNDED : thm
val SUB_0 : thm
val SUB_ADD : thm
val SUB_CANCEL : thm
val SUB_ELIM_THM : thm
val SUB_ELIM_THM_EXISTS : thm
val SUB_EQUAL_0 : thm
val SUB_EQ_0 : thm
val SUB_EQ_EQ_0 : thm
val SUB_LEFT_ADD : thm
val SUB_LEFT_EQ : thm
val SUB_LEFT_GREATER : thm
val SUB_LEFT_GREATER_EQ : thm
val SUB_LEFT_LESS : thm
val SUB_LEFT_LESS_EQ : thm
val SUB_LEFT_SUB : thm
val SUB_LEFT_SUC : thm
val SUB_LESS : thm
val SUB_LESS_0 : thm
val SUB_LESS_EQ : thm
val SUB_LESS_EQ_ADD : thm
val SUB_LESS_OR : thm
val SUB_LESS_SUC : thm
val SUB_MOD : thm
val SUB_MONO_EQ : thm
val SUB_PLUS : thm
val SUB_RIGHT_ADD : thm
val SUB_RIGHT_EQ : thm
val SUB_RIGHT_GREATER : thm
val SUB_RIGHT_GREATER_EQ : thm
val SUB_RIGHT_LESS : thm
val SUB_RIGHT_LESS_EQ : thm
val SUB_RIGHT_SUB : thm
val SUB_SUB : thm
val SUC_ADD_SYM : thm
val SUC_ELIM_NUMERALS : thm
val SUC_ELIM_THM : thm
val SUC_MOD : thm
val SUC_NOT : thm
val SUC_ONE_ADD : thm
val SUC_PRE : thm
val SUC_SUB : thm
val SUC_SUB1 : thm
val SUM_SQUARED : thm
val TC_eq_NRC : thm
val TIMES2 : thm
val TWO : thm
val WOP : thm
val X_LE_DIV : thm
val X_LE_X_EXP : thm
val X_LE_X_SQUARED : thm
val X_LT_DIV : thm
val X_LT_EXP_X : thm
val X_LT_EXP_X_IFF : thm
val X_LT_X_SQUARED : thm
val X_MOD_Y_EQ_X : thm
val ZERO_DIV : thm
val ZERO_EXP : thm
val ZERO_LESS_ADD : thm
val ZERO_LESS_EQ : thm
val ZERO_LESS_EXP : thm
val ZERO_LESS_MULT : thm
val ZERO_LT_EXP : thm
val ZERO_MOD : thm
val binary_induct : thm
val datatype_num : thm
val findq_divisor : thm
val findq_eq_0 : thm
val findq_thm : thm
val num_CASES : thm
val num_case_compute : thm
val num_case_cong : thm
val num_case_eq : thm
val transitive_LESS : thm
val transitive_measure : thm
val transitive_monotone : thm
val arithmetic_grammars : type_grammar.grammar * term_grammar.grammar
(*
[pair] Parent theory of "arithmetic"
[prim_rec] Parent theory of "arithmetic"
[ABS_DIFF_def] Definition
⊢ ∀n m. ABS_DIFF n m = if n < m then m − n else n − m
[ADD] Definition
⊢ (∀n. 0 + n = n) ∧ ∀m n. SUC m + n = SUC (m + n)
[ALT_ZERO] Definition
⊢ ZERO = 0
[BIT1] Definition
⊢ ∀n. BIT1 n = n + (n + SUC 0)
[BIT2] Definition
⊢ ∀n. BIT2 n = n + (n + SUC (SUC 0))
[DIV2_def] Definition
⊢ ∀n. DIV2 n = n DIV 2
[DIVISION] Definition
⊢ ∀n. 0 < n ⇒ ∀k. k = k DIV n * n + k MOD n ∧ k MOD n < n
[DIVMOD_DEF] Definition
⊢ DIVMOD =
WFREC (measure (FST ∘ SND))
(λf (a,m,n).
if n = 0 then (0,0)
else if m < n then (a,m)
else (let q = findq (1,m,n) in f (a + q,m − n * q,n)))
[EVEN] Definition
⊢ (EVEN 0 ⇔ T) ∧ ∀n. EVEN (SUC n) ⇔ ¬EVEN n
[EXP] Definition
⊢ (∀m. m ** 0 = 1) ∧ ∀m n. m ** SUC n = m * m ** n
[FACT] Definition
⊢ FACT 0 = 1 ∧ ∀n. FACT (SUC n) = SUC n * FACT n
[FUNPOW] Definition
⊢ (∀f x. FUNPOW f 0 x = x) ∧
∀f n x. FUNPOW f (SUC n) x = FUNPOW f n (f x)
[GREATER_DEF] Definition
⊢ ∀m n. m > n ⇔ n < m
[GREATER_OR_EQ] Definition
⊢ ∀m n. m ≥ n ⇔ m > n ∨ m = n
[LESS_OR_EQ] Definition
⊢ ∀m n. m ≤ n ⇔ m < n ∨ m = n
[MAX_DEF] Definition
⊢ ∀m n. MAX m n = if m < n then n else m
[MIN_DEF] Definition
⊢ ∀m n. MIN m n = if m < n then m else n
[MODEQ_DEF] Definition
⊢ ∀n m1 m2. MODEQ n m1 m2 ⇔ ∃a b. a * n + m1 = b * n + m2
[MULT] Definition
⊢ (∀n. 0 * n = 0) ∧ ∀m n. SUC m * n = m * n + n
[NRC] Definition
⊢ (∀R x y. NRC R 0 x y ⇔ x = y) ∧
∀R n x y. NRC R (SUC n) x y ⇔ ∃z. R x z ∧ NRC R n z y
[NUMERAL_DEF] Definition
⊢ ∀x. NUMERAL x = x
[ODD] Definition
⊢ (ODD 0 ⇔ F) ∧ ∀n. ODD (SUC n) ⇔ ¬ODD n
[SUB] Definition
⊢ (∀m. 0 − m = 0) ∧
∀m n. SUC m − n = if m < n then 0 else SUC (m − n)
[findq_def] Definition
⊢ findq =
WFREC (measure (λ(a,m,n). m − n))
(λf (a,m,n).
if n = 0 then a
else (let d = 2 * n in if m < d then a else f (2 * a,m,d)))
[nat_elim__magic] Definition
⊢ ∀n. &n = n
[num_case_def] Definition
⊢ (∀v f. num_CASE 0 v f = v) ∧ ∀n v f. num_CASE (SUC n) v f = f n
[ABS_DIFF_ADD_SAME] Theorem
⊢ ∀n m p. ABS_DIFF (n + p) (m + p) = ABS_DIFF n m
[ABS_DIFF_COMM] Theorem
⊢ ∀n m. ABS_DIFF n m = ABS_DIFF m n
[ABS_DIFF_EQS] Theorem
⊢ ∀n. ABS_DIFF n n = 0
[ABS_DIFF_EQ_0] Theorem
⊢ ∀n m. ABS_DIFF n m = 0 ⇔ n = m
[ABS_DIFF_LE_SUM] Theorem
⊢ ABS_DIFF x z ≤ x + z
[ABS_DIFF_PLUS_LE] Theorem
⊢ ∀x z y. ABS_DIFF x (y + z) ≤ y + ABS_DIFF x z
[ABS_DIFF_SUC] Theorem
⊢ ∀n m. ABS_DIFF (SUC n) (SUC m) = ABS_DIFF n m
[ABS_DIFF_SUC_LE] Theorem
⊢ ∀x z. ABS_DIFF x (SUC z) ≤ SUC (ABS_DIFF x z)
[ABS_DIFF_SUMS] Theorem
⊢ ∀n1 n2 m1 m2.
ABS_DIFF (n1 + n2) (m1 + m2) ≤ ABS_DIFF n1 m1 + ABS_DIFF n2 m2
[ABS_DIFF_SYM] Theorem
⊢ ∀n m. ABS_DIFF n m = ABS_DIFF m n
[ABS_DIFF_TRIANGLE] Theorem
⊢ ∀x y z. ABS_DIFF x z ≤ ABS_DIFF x y + ABS_DIFF y z
[ABS_DIFF_TRIANGLE_lem] Theorem
⊢ ∀x y. x ≤ ABS_DIFF x y + y
[ABS_DIFF_ZERO] Theorem
⊢ ∀n. ABS_DIFF n 0 = n ∧ ABS_DIFF 0 n = n
[ADD1] Theorem
⊢ ∀m. SUC m = m + 1
[ADD_0] Theorem
⊢ ∀m. m + 0 = m
[ADD_ASSOC] Theorem
⊢ ∀m n p. m + (n + p) = m + n + p
[ADD_CLAUSES] Theorem
⊢ 0 + m = m ∧ m + 0 = m ∧ SUC m + n = SUC (m + n) ∧
m + SUC n = SUC (m + n)
[ADD_COMM] Theorem
⊢ ∀m n. m + n = n + m
[ADD_DIV_ADD_DIV] Theorem
⊢ ∀n. 0 < n ⇒ ∀x r. (x * n + r) DIV n = x + r DIV n
[ADD_DIV_RWT] Theorem
⊢ ∀n.
0 < n ⇒
∀m p.
m MOD n = 0 ∨ p MOD n = 0 ⇒
(m + p) DIV n = m DIV n + p DIV n
[ADD_EQ_0] Theorem
⊢ ∀m n. m + n = 0 ⇔ m = 0 ∧ n = 0
[ADD_EQ_1] Theorem
⊢ ∀m n. m + n = 1 ⇔ m = 1 ∧ n = 0 ∨ m = 0 ∧ n = 1
[ADD_EQ_SUB] Theorem
⊢ ∀m n p. n ≤ p ⇒ (m + n = p ⇔ m = p − n)
[ADD_INV_0] Theorem
⊢ ∀m n. m + n = m ⇒ n = 0
[ADD_INV_0_EQ] Theorem
⊢ ∀m n. m + n = m ⇔ n = 0
[ADD_MOD] Theorem
⊢ ∀n a b p.
0 < n ⇒ ((a + p) MOD n = (b + p) MOD n ⇔ a MOD n = b MOD n)
[ADD_MODULUS] Theorem
⊢ (∀n x. 0 < n ⇒ (x + n) MOD n = x MOD n) ∧
∀n x. 0 < n ⇒ (n + x) MOD n = x MOD n
[ADD_MODULUS_LEFT] Theorem
⊢ ∀n x. 0 < n ⇒ (x + n) MOD n = x MOD n
[ADD_MODULUS_RIGHT] Theorem
⊢ ∀n x. 0 < n ⇒ (n + x) MOD n = x MOD n
[ADD_MONO_LESS_EQ] Theorem
⊢ ∀m n p. m + n ≤ m + p ⇔ n ≤ p
[ADD_SUB] Theorem
⊢ ∀a c. a + c − c = a
[ADD_SUC] Theorem
⊢ ∀m n. SUC (m + n) = m + SUC n
[ADD_SYM] Theorem
⊢ ∀m n. m + n = n + m
[BOUNDED_EXISTS_THM] Theorem
⊢ ∀c. 0 < c ⇒ ((∃n. n < c ∧ P n) ⇔ P (c − 1) ∨ ∃n. n < c − 1 ∧ P n)
[BOUNDED_FORALL_THM] Theorem
⊢ ∀c. 0 < c ⇒ ((∀n. n < c ⇒ P n) ⇔ P (c − 1) ∧ ∀n. n < c − 1 ⇒ P n)
[CANCEL_SUB] Theorem
⊢ ∀p n m. p ≤ n ∧ p ≤ m ⇒ (n − p = m − p ⇔ n = m)
[COMPLETE_INDUCTION] Theorem
⊢ ∀P. (∀n. (∀m. m < n ⇒ P m) ⇒ P n) ⇒ ∀n. P n
[DA] Theorem
⊢ ∀k n. 0 < n ⇒ ∃r q. k = q * n + r ∧ r < n
[DIV2_DOUBLE] Theorem
⊢ ∀n. DIV2 (2 * n) = n
[DIVMOD_CALC] Theorem
⊢ (∀m n. 0 < n ⇒ m DIV n = FST (DIVMOD (0,m,n))) ∧
∀m n. 0 < n ⇒ m MOD n = SND (DIVMOD (0,m,n))
[DIVMOD_CORRECT] Theorem
⊢ ∀m n a. 0 < n ⇒ DIVMOD (a,m,n) = (a + m DIV n,m MOD n)
[DIVMOD_ID] Theorem
⊢ ∀n. 0 < n ⇒ n DIV n = 1 ∧ n MOD n = 0
[DIVMOD_THM] Theorem
⊢ DIVMOD (a,m,n) =
if n = 0 then (0,0)
else if m < n then (a,m)
else (let q = findq (1,m,n) in DIVMOD (a + q,m − n * q,n))
[DIV_0_IMP_LT] Theorem
⊢ ∀b n. 1 < b ∧ n DIV b = 0 ⇒ n < b
[DIV_1] Theorem
⊢ ∀q. q DIV 1 = q
[DIV_DIV_DIV_MULT] Theorem
⊢ ∀m n. 0 < m ∧ 0 < n ⇒ ∀x. x DIV m DIV n = x DIV (m * n)
[DIV_EQ_0] Theorem
⊢ 1 < b ⇒ (n DIV b = 0 ⇔ n < b)
[DIV_EQ_X] Theorem
⊢ ∀x y z. 0 < z ⇒ (y DIV z = x ⇔ x * z ≤ y ∧ y < SUC x * z)
[DIV_LESS] Theorem
⊢ ∀n d. 0 < n ∧ 1 < d ⇒ n DIV d < n
[DIV_LESS_EQ] Theorem
⊢ ∀n. 0 < n ⇒ ∀k. k DIV n ≤ k
[DIV_LE_MONOTONE] Theorem
⊢ ∀n x y. 0 < n ∧ x ≤ y ⇒ x DIV n ≤ y DIV n
[DIV_LE_X] Theorem
⊢ ∀x y z. 0 < z ⇒ (y DIV z ≤ x ⇔ y < (x + 1) * z)
[DIV_LT_X] Theorem
⊢ ∀x y z. 0 < z ⇒ (y DIV z < x ⇔ y < x * z)
[DIV_MOD_MOD_DIV] Theorem
⊢ ∀m n k. 0 < n ∧ 0 < k ⇒ (m DIV n) MOD k = m MOD (n * k) DIV n
[DIV_MULT] Theorem
⊢ ∀n r. r < n ⇒ ∀q. (q * n + r) DIV n = q
[DIV_ONE] Theorem
⊢ ∀q. q DIV SUC 0 = q
[DIV_P] Theorem
⊢ ∀P p q. 0 < q ⇒ (P (p DIV q) ⇔ ∃k r. p = k * q + r ∧ r < q ∧ P k)
[DIV_P_UNIV] Theorem
⊢ ∀P m n. 0 < n ⇒ (P (m DIV n) ⇔ ∀q r. m = q * n + r ∧ r < n ⇒ P q)
[DIV_SUB] Theorem
⊢ 0 < n ∧ n * q ≤ m ⇒ (m − n * q) DIV n = m DIV n − q
[DIV_UNIQUE] Theorem
⊢ ∀n k q. (∃r. k = q * n + r ∧ r < n) ⇒ k DIV n = q
[DOUBLE_LT] Theorem
⊢ ∀p q. 2 * p + 1 < 2 * q ⇔ 2 * p < 2 * q
[EQ_ADD_LCANCEL] Theorem
⊢ ∀m n p. m + n = m + p ⇔ n = p
[EQ_ADD_RCANCEL] Theorem
⊢ ∀m n p. m + p = n + p ⇔ m = n
[EQ_LESS_EQ] Theorem
⊢ ∀m n. m = n ⇔ m ≤ n ∧ n ≤ m
[EQ_MONO_ADD_EQ] Theorem
⊢ ∀m n p. m + p = n + p ⇔ m = n
[EQ_MULT_LCANCEL] Theorem
⊢ ∀m n p. m * n = m * p ⇔ m = 0 ∨ n = p
[EQ_MULT_RCANCEL] Theorem
⊢ ∀m n p. n * m = p * m ⇔ m = 0 ∨ n = p
[EVEN_ADD] Theorem
⊢ ∀m n. EVEN (m + n) ⇔ (EVEN m ⇔ EVEN n)
[EVEN_AND_ODD] Theorem
⊢ ∀n. ¬(EVEN n ∧ ODD n)
[EVEN_DOUBLE] Theorem
⊢ ∀n. EVEN (2 * n)
[EVEN_EXISTS] Theorem
⊢ ∀n. EVEN n ⇔ ∃m. n = 2 * m
[EVEN_EXP] Theorem
⊢ ∀m n. 0 < n ∧ EVEN m ⇒ EVEN (m ** n)
[EVEN_EXP_IFF] Theorem
⊢ ∀n m. EVEN (m ** n) ⇔ 0 < n ∧ EVEN m
[EVEN_MOD2] Theorem
⊢ ∀x. EVEN x ⇔ x MOD 2 = 0
[EVEN_MULT] Theorem
⊢ ∀m n. EVEN (m * n) ⇔ EVEN m ∨ EVEN n
[EVEN_ODD] Theorem
⊢ ∀n. EVEN n ⇔ ¬ODD n
[EVEN_ODD_EXISTS] Theorem
⊢ ∀n. (EVEN n ⇒ ∃m. n = 2 * m) ∧ (ODD n ⇒ ∃m. n = SUC (2 * m))
[EVEN_OR_ODD] Theorem
⊢ ∀n. EVEN n ∨ ODD n
[EVEN_SUB] Theorem
⊢ ∀m n. m ≤ n ⇒ (EVEN (n − m) ⇔ (EVEN n ⇔ EVEN m))
[EXISTS_GREATEST] Theorem
⊢ ∀P. (∃x. P x) ∧ (∃x. ∀y. y > x ⇒ ¬P y) ⇔ ∃x. P x ∧ ∀y. y > x ⇒ ¬P y
[EXISTS_NUM] Theorem
⊢ ∀P. (∃n. P n) ⇔ P 0 ∨ ∃m. P (SUC m)
[EXP2_LT] Theorem
⊢ ∀m n. n DIV 2 < 2 ** m ⇔ n < 2 ** SUC m
[EXP_1] Theorem
⊢ ∀n. 1 ** n = 1 ∧ n ** 1 = n
[EXP_ADD] Theorem
⊢ ∀p q n. n ** (p + q) = n ** p * n ** q
[EXP_ALWAYS_BIG_ENOUGH] Theorem
⊢ ∀b. 1 < b ⇒ ∀n. ∃m. n ≤ b ** m
[EXP_BASE_INJECTIVE] Theorem
⊢ ∀b. 1 < b ⇒ ∀n m. b ** n = b ** m ⇔ n = m
[EXP_BASE_LEQ_MONO_IMP] Theorem
⊢ ∀n m b. 0 < b ∧ m ≤ n ⇒ b ** m ≤ b ** n
[EXP_BASE_LEQ_MONO_SUC_IMP] Theorem
⊢ m ≤ n ⇒ SUC b ** m ≤ SUC b ** n
[EXP_BASE_LE_IFF] Theorem
⊢ b ** m ≤ b ** n ⇔
b = 0 ∧ n = 0 ∨ b = 0 ∧ 0 < m ∨ b = 1 ∨ 1 < b ∧ m ≤ n
[EXP_BASE_LE_MONO] Theorem
⊢ ∀b. 1 < b ⇒ ∀n m. b ** m ≤ b ** n ⇔ m ≤ n
[EXP_BASE_LT_MONO] Theorem
⊢ ∀b. 1 < b ⇒ ∀n m. b ** m < b ** n ⇔ m < n
[EXP_BASE_MULT] Theorem
⊢ ∀z x y. (x * y) ** z = x ** z * y ** z
[EXP_EQ_0] Theorem
⊢ ∀n m. n ** m = 0 ⇔ n = 0 ∧ 0 < m
[EXP_EQ_1] Theorem
⊢ ∀n m. n ** m = 1 ⇔ n = 1 ∨ m = 0
[EXP_EXP_INJECTIVE] Theorem
⊢ ∀b1 b2 x. b1 ** x = b2 ** x ⇔ x = 0 ∨ b1 = b2
[EXP_EXP_LE_MONO] Theorem
⊢ ∀a b. a ** n ≤ b ** n ⇔ a ≤ b ∨ n = 0
[EXP_EXP_LT_MONO] Theorem
⊢ ∀a b. a ** n < b ** n ⇔ a < b ∧ 0 < n
[EXP_EXP_MULT] Theorem
⊢ ∀z x y. x ** (y * z) = (x ** y) ** z
[EXP_MOD] Theorem
⊢ 0 < n ⇒ (x MOD n) ** e MOD n = x ** e MOD n
[EXP_SUB] Theorem
⊢ ∀p q n. 0 < n ∧ q ≤ p ⇒ n ** (p − q) = n ** p DIV n ** q
[EXP_SUB_NUMERAL] Theorem
⊢ 0 < n ⇒
n ** NUMERAL (BIT1 x) DIV n = n ** (NUMERAL (BIT1 x) − 1) ∧
n ** NUMERAL (BIT2 x) DIV n = n ** NUMERAL (BIT1 x)
[FACT_LESS] Theorem
⊢ ∀n. 0 < FACT n
[FORALL_NUM] Theorem
⊢ ∀P. (∀n. P n) ⇔ P 0 ∧ ∀n. P (SUC n)
[FORALL_NUM_THM] Theorem
⊢ (∀n. P n) ⇔ P 0 ∧ ∀n. P n ⇒ P (SUC n)
[FUNPOW_0] Theorem
⊢ FUNPOW f 0 x = x
[FUNPOW_1] Theorem
⊢ FUNPOW f 1 x = f x
[FUNPOW_ADD] Theorem
⊢ ∀m n. FUNPOW f (m + n) x = FUNPOW f m (FUNPOW f n x)
[FUNPOW_SUC] Theorem
⊢ ∀f n x. FUNPOW f (SUC n) x = f (FUNPOW f n x)
[GREATER_EQ] Theorem
⊢ ∀n m. n ≥ m ⇔ m ≤ n
[INV_PRE_EQ] Theorem
⊢ ∀m n. 0 < m ∧ 0 < n ⇒ (PRE m = PRE n ⇔ m = n)
[INV_PRE_LESS] Theorem
⊢ ∀m. 0 < m ⇒ ∀n. PRE m < PRE n ⇔ m < n
[INV_PRE_LESS_EQ] Theorem
⊢ ∀n. 0 < n ⇒ ∀m. PRE m ≤ PRE n ⇔ m ≤ n
[LE] Theorem
⊢ (∀n. n ≤ 0 ⇔ n = 0) ∧ ∀m n. m ≤ SUC n ⇔ m = SUC n ∨ m ≤ n
[LEFT_ADD_DISTRIB] Theorem
⊢ ∀m n p. p * (m + n) = p * m + p * n
[LEFT_SUB_DISTRIB] Theorem
⊢ ∀m n p. p * (m − n) = p * m − p * n
[LESS_0_CASES] Theorem
⊢ ∀m. 0 = m ∨ 0 < m
[LESS_ADD] Theorem
⊢ ∀m n. n < m ⇒ ∃p. p + n = m
[LESS_ADD_1] Theorem
⊢ ∀m n. n < m ⇒ ∃p. m = n + (p + 1)
[LESS_ADD_NONZERO] Theorem
⊢ ∀m n. n ≠ 0 ⇒ m < m + n
[LESS_ADD_SUC] Theorem
⊢ ∀m n. m < m + SUC n
[LESS_ANTISYM] Theorem
⊢ ∀m n. ¬(m < n ∧ n < m)
[LESS_CASES] Theorem
⊢ ∀m n. m < n ∨ n ≤ m
[LESS_CASES_IMP] Theorem
⊢ ∀m n. ¬(m < n) ∧ m ≠ n ⇒ n < m
[LESS_DIV_EQ_ZERO] Theorem
⊢ ∀r n. r < n ⇒ r DIV n = 0
[LESS_EQ] Theorem
⊢ ∀m n. m < n ⇔ SUC m ≤ n
[LESS_EQUAL_ADD] Theorem
⊢ ∀m n. m ≤ n ⇒ ∃p. n = m + p
[LESS_EQUAL_ANTISYM] Theorem
⊢ ∀n m. n ≤ m ∧ m ≤ n ⇒ n = m
[LESS_EQUAL_DIFF] Theorem
⊢ ∀m n. m ≤ n ⇒ ∃k. m = n − k
[LESS_EQ_0] Theorem
⊢ ∀n. n ≤ 0 ⇔ n = 0
[LESS_EQ_ADD] Theorem
⊢ ∀m n. m ≤ m + n
[LESS_EQ_ADD_EXISTS] Theorem
⊢ ∀m n. n ≤ m ⇒ ∃p. p + n = m
[LESS_EQ_ADD_SUB] Theorem
⊢ ∀c b. c ≤ b ⇒ ∀a. a + b − c = a + (b − c)
[LESS_EQ_ANTISYM] Theorem
⊢ ∀m n. ¬(m < n ∧ n ≤ m)
[LESS_EQ_CASES] Theorem
⊢ ∀m n. m ≤ n ∨ n ≤ m
[LESS_EQ_EXISTS] Theorem
⊢ ∀m n. m ≤ n ⇔ ∃p. n = m + p
[LESS_EQ_IFF_LESS_SUC] Theorem
⊢ ∀n m. n ≤ m ⇔ n < SUC m
[LESS_EQ_IMP_LESS_SUC] Theorem
⊢ ∀n m. n ≤ m ⇒ n < SUC m
[LESS_EQ_LESS_EQ_MONO] Theorem
⊢ ∀m n p q. m ≤ p ∧ n ≤ q ⇒ m + n ≤ p + q
[LESS_EQ_LESS_TRANS] Theorem
⊢ ∀m n p. m ≤ n ∧ n < p ⇒ m < p
[LESS_EQ_MONO] Theorem
⊢ ∀n m. SUC n ≤ SUC m ⇔ n ≤ m
[LESS_EQ_MONO_ADD_EQ] Theorem
⊢ ∀m n p. m + p ≤ n + p ⇔ m ≤ n
[LESS_EQ_REFL] Theorem
⊢ ∀m. m ≤ m
[LESS_EQ_SUB_LESS] Theorem
⊢ ∀a b. b ≤ a ⇒ ∀c. a − b < c ⇔ a < b + c
[LESS_EQ_SUC_REFL] Theorem
⊢ ∀m. m ≤ SUC m
[LESS_EQ_TRANS] Theorem
⊢ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p
[LESS_EXP_SUC_MONO] Theorem
⊢ ∀n m. SUC (SUC m) ** n < SUC (SUC m) ** SUC n
[LESS_IMP_LESS_ADD] Theorem
⊢ ∀n m. n < m ⇒ ∀p. n < m + p
[LESS_IMP_LESS_OR_EQ] Theorem
⊢ ∀m n. m < n ⇒ m ≤ n
[LESS_LESS_CASES] Theorem
⊢ ∀m n. m = n ∨ m < n ∨ n < m
[LESS_LESS_EQ_TRANS] Theorem
⊢ ∀m n p. m < n ∧ n ≤ p ⇒ m < p
[LESS_LESS_SUC] Theorem
⊢ ∀m n. ¬(m < n ∧ n < SUC m)
[LESS_MOD] Theorem
⊢ ∀n k. k < n ⇒ k MOD n = k
[LESS_MONO_ADD] Theorem
⊢ ∀m n p. m < n ⇒ m + p < n + p
[LESS_MONO_ADD_EQ] Theorem
⊢ ∀m n p. m + p < n + p ⇔ m < n
[LESS_MONO_ADD_INV] Theorem
⊢ ∀m n p. m + p < n + p ⇒ m < n
[LESS_MONO_EQ] Theorem
⊢ ∀m n. SUC m < SUC n ⇔ m < n
[LESS_MONO_MULT] Theorem
⊢ ∀m n p. m ≤ n ⇒ m * p ≤ n * p
[LESS_MONO_MULT2] Theorem
⊢ ∀m n i j. m ≤ i ∧ n ≤ j ⇒ m * n ≤ i * j
[LESS_MONO_REV] Theorem
⊢ ∀m n. SUC m < SUC n ⇒ m < n
[LESS_MULT2] Theorem
⊢ ∀m n. 0 < m ∧ 0 < n ⇒ 0 < m * n
[LESS_MULT_MONO] Theorem
⊢ ∀m i n. SUC n * m < SUC n * i ⇔ m < i
[LESS_NOT_SUC] Theorem
⊢ ∀m n. m < n ∧ n ≠ SUC m ⇒ SUC m < n
[LESS_OR] Theorem
⊢ ∀m n. m < n ⇒ SUC m ≤ n
[LESS_OR_EQ_ADD] Theorem
⊢ ∀n m. n < m ∨ ∃p. n = p + m
[LESS_OR_EQ_ALT] Theorem
⊢ $<= = (λx y. y = SUC x)⃰
[LESS_STRONG_ADD] Theorem
⊢ ∀m n. n < m ⇒ ∃p. SUC p + n = m
[LESS_SUB_ADD_LESS] Theorem
⊢ ∀n m i. i < n − m ⇒ i + m < n
[LESS_SUC_EQ_COR] Theorem
⊢ ∀m n. m < n ∧ SUC m ≠ n ⇒ SUC m < n
[LESS_SUC_NOT] Theorem
⊢ ∀m n. m < n ⇒ ¬(n < SUC m)
[LESS_TRANS] Theorem
⊢ ∀m n p. m < n ∧ n < p ⇒ m < p
[LE_ADD_LCANCEL] Theorem
⊢ ∀m n p. m + n ≤ m + p ⇔ n ≤ p
[LE_ADD_RCANCEL] Theorem
⊢ ∀m n p. n + m ≤ p + m ⇔ n ≤ p
[LE_LT1] Theorem
⊢ ∀x y. x ≤ y ⇔ x < y + 1
[LE_MULT_CANCEL_LBARE] Theorem
⊢ (m ≤ m * n ⇔ m = 0 ∨ 0 < n) ∧ (m ≤ n * m ⇔ m = 0 ∨ 0 < n)
[LE_MULT_CANCEL_RBARE] Theorem
⊢ (m * n ≤ m ⇔ m = 0 ∨ n ≤ 1) ∧ (m * n ≤ n ⇔ n = 0 ∨ m ≤ 1)
[LE_MULT_LCANCEL] Theorem
⊢ ∀m n p. m * n ≤ m * p ⇔ m = 0 ∨ n ≤ p
[LE_MULT_RCANCEL] Theorem
⊢ ∀m n p. m * n ≤ p * n ⇔ n = 0 ∨ m ≤ p
[LE_SUB_LCANCEL] Theorem
⊢ ∀z y x. x − y ≤ x − z ⇔ z ≤ y ∨ x ≤ y
[LE_SUB_RCANCEL] Theorem
⊢ ∀m n p. n − m ≤ p − m ⇔ n ≤ m ∨ n ≤ p
[LT_ADD_LCANCEL] Theorem
⊢ ∀m n p. p + m < p + n ⇔ m < n
[LT_ADD_RCANCEL] Theorem
⊢ ∀m n p. m + p < n + p ⇔ m < n
[LT_MULT_CANCEL_LBARE] Theorem
⊢ (m < m * n ⇔ 0 < m ∧ 1 < n) ∧ (m < n * m ⇔ 0 < m ∧ 1 < n)
[LT_MULT_CANCEL_RBARE] Theorem
⊢ (m * n < m ⇔ 0 < m ∧ n = 0) ∧ (m * n < n ⇔ 0 < n ∧ m = 0)
[LT_MULT_LCANCEL] Theorem
⊢ ∀m n p. m * n < m * p ⇔ 0 < m ∧ n < p
[LT_MULT_RCANCEL] Theorem
⊢ ∀m n p. m * n < p * n ⇔ 0 < n ∧ m < p
[LT_SUB_LCANCEL] Theorem
⊢ ∀z y x. x − y < x − z ⇔ z < y ∧ z < x
[LT_SUB_RCANCEL] Theorem
⊢ ∀m n p. n − m < p − m ⇔ n < p ∧ m < p
[MAX_0] Theorem
⊢ ∀n. MAX n 0 = n ∧ MAX 0 n = n
[MAX_ASSOC] Theorem
⊢ ∀m n p. MAX m (MAX n p) = MAX (MAX m n) p
[MAX_COMM] Theorem
⊢ ∀m n. MAX m n = MAX n m
[MAX_EQ_0] Theorem
⊢ MAX m n = 0 ⇔ m = 0 ∧ n = 0
[MAX_IDEM] Theorem
⊢ ∀n. MAX n n = n
[MAX_LE] Theorem
⊢ ∀n m p.
(p ≤ MAX m n ⇔ p ≤ m ∨ p ≤ n) ∧ (MAX m n ≤ p ⇔ m ≤ p ∧ n ≤ p)
[MAX_LT] Theorem
⊢ ∀n m p.
(p < MAX m n ⇔ p < m ∨ p < n) ∧ (MAX m n < p ⇔ m < p ∧ n < p)
[MIN_0] Theorem
⊢ ∀n. MIN n 0 = 0 ∧ MIN 0 n = 0
[MIN_ASSOC] Theorem
⊢ ∀m n p. MIN m (MIN n p) = MIN (MIN m n) p
[MIN_COMM] Theorem
⊢ ∀m n. MIN m n = MIN n m
[MIN_EQ_0] Theorem
⊢ MIN m n = 0 ⇔ m = 0 ∨ n = 0
[MIN_IDEM] Theorem
⊢ ∀n. MIN n n = n
[MIN_LE] Theorem
⊢ ∀n m p.
(MIN m n ≤ p ⇔ m ≤ p ∨ n ≤ p) ∧ (p ≤ MIN m n ⇔ p ≤ m ∧ p ≤ n)
[MIN_LT] Theorem
⊢ ∀n m p.
(MIN m n < p ⇔ m < p ∨ n < p) ∧ (p < MIN m n ⇔ p < m ∧ p < n)
[MIN_MAX_EQ] Theorem
⊢ ∀m n. MIN m n = MAX m n ⇔ m = n
[MIN_MAX_LE] Theorem
⊢ ∀m n. MIN m n ≤ MAX m n
[MIN_MAX_LT] Theorem
⊢ ∀m n. MIN m n < MAX m n ⇔ m ≠ n
[MIN_MAX_PRED] Theorem
⊢ ∀P m n. P m ∧ P n ⇒ P (MIN m n) ∧ P (MAX m n)
[MODEQ_0] Theorem
⊢ 0 < n ⇒ MODEQ n n 0
[MODEQ_0_CONG] Theorem
⊢ MODEQ 0 m1 m2 ⇔ m1 = m2
[MODEQ_EXP_CONG] Theorem
⊢ MODEQ n x y ⇒ MODEQ n (x ** e) (y ** e)
[MODEQ_INTRO_CONG] Theorem
⊢ 0 < n ⇒ MODEQ n e0 e1 ⇒ e0 MOD n = e1 MOD n
[MODEQ_MOD] Theorem
⊢ 0 < n ⇒ MODEQ n (x MOD n) x
[MODEQ_MULT_CONG] Theorem
⊢ MODEQ n x0 x1 ⇒ MODEQ n y0 y1 ⇒ MODEQ n (x0 * y0) (x1 * y1)
[MODEQ_NONZERO_MODEQUALITY] Theorem
⊢ 0 < n ⇒ (MODEQ n m1 m2 ⇔ m1 MOD n = m2 MOD n)
[MODEQ_NUMERAL] Theorem
⊢ (NUMERAL n ≤ NUMERAL m ⇒
MODEQ (NUMERAL (BIT1 n)) (NUMERAL (BIT1 m))
(NUMERAL (BIT1 m) MOD NUMERAL (BIT1 n))) ∧
(NUMERAL n ≤ NUMERAL m ⇒
MODEQ (NUMERAL (BIT1 n)) (NUMERAL (BIT2 m))
(NUMERAL (BIT2 m) MOD NUMERAL (BIT1 n))) ∧
(NUMERAL n ≤ NUMERAL m ⇒
MODEQ (NUMERAL (BIT2 n)) (NUMERAL (BIT2 m))
(NUMERAL (BIT2 m) MOD NUMERAL (BIT2 n))) ∧
(NUMERAL n < NUMERAL m ⇒
MODEQ (NUMERAL (BIT2 n)) (NUMERAL (BIT1 m))
(NUMERAL (BIT1 m) MOD NUMERAL (BIT2 n)))
[MODEQ_PLUS_CONG] Theorem
⊢ MODEQ n x0 x1 ⇒ MODEQ n y0 y1 ⇒ MODEQ n (x0 + y0) (x1 + y1)
[MODEQ_REFL] Theorem
⊢ ∀x. MODEQ n x x
[MODEQ_SUC_CONG] Theorem
⊢ MODEQ n x y ⇒ MODEQ n (SUC x) (SUC y)
[MODEQ_SYM] Theorem
⊢ MODEQ n x y ⇔ MODEQ n y x
[MODEQ_THM] Theorem
⊢ MODEQ n m1 m2 ⇔ n = 0 ∧ m1 = m2 ∨ 0 < n ∧ m1 MOD n = m2 MOD n
[MODEQ_TRANS] Theorem
⊢ ∀x y z. MODEQ n x y ∧ MODEQ n y z ⇒ MODEQ n x z
[MOD_1] Theorem
⊢ ∀k. k MOD 1 = 0
[MOD_2] Theorem
⊢ ∀n. n MOD 2 = if EVEN n then 0 else 1
[MOD_COMMON_FACTOR] Theorem
⊢ ∀n p q. 0 < n ∧ 0 < q ⇒ n * p MOD q = (n * p) MOD (n * q)
[MOD_ELIM] Theorem
⊢ ∀P x n. 0 < n ∧ P x ∧ (∀y. P (y + n) ⇒ P y) ⇒ P (x MOD n)
[MOD_EQ_0] Theorem
⊢ ∀n. 0 < n ⇒ ∀k. (k * n) MOD n = 0
[MOD_EQ_0_DIVISOR] Theorem
⊢ 0 < n ⇒ (k MOD n = 0 ⇔ ∃d. k = d * n)
[MOD_LESS] Theorem
⊢ ∀m n. 0 < n ⇒ m MOD n < n
[MOD_LESS_EQ] Theorem
⊢ 0 < y ⇒ x MOD y ≤ x
[MOD_LIFT_PLUS] Theorem
⊢ 0 < n ∧ k < n − x MOD n ⇒ (x + k) MOD n = x MOD n + k
[MOD_LIFT_PLUS_IFF] Theorem
⊢ 0 < n ⇒ ((x + k) MOD n = x MOD n + k ⇔ k < n − x MOD n)
[MOD_MOD] Theorem
⊢ ∀n. 0 < n ⇒ ∀k. k MOD n MOD n = k MOD n
[MOD_MULT] Theorem
⊢ ∀n r. r < n ⇒ ∀q. (q * n + r) MOD n = r
[MOD_MULT_MOD] Theorem
⊢ ∀m n. 0 < n ∧ 0 < m ⇒ ∀x. x MOD (n * m) MOD n = x MOD n
[MOD_ONE] Theorem
⊢ ∀k. k MOD SUC 0 = 0
[MOD_P] Theorem
⊢ ∀P p q. 0 < q ⇒ (P (p MOD q) ⇔ ∃k r. p = k * q + r ∧ r < q ∧ P r)
[MOD_PLUS] Theorem
⊢ ∀n. 0 < n ⇒ ∀j k. (j MOD n + k MOD n) MOD n = (j + k) MOD n
[MOD_P_UNIV] Theorem
⊢ ∀P m n. 0 < n ⇒ (P (m MOD n) ⇔ ∀q r. m = q * n + r ∧ r < n ⇒ P r)
[MOD_SUB] Theorem
⊢ 0 < n ∧ n * q ≤ m ⇒ (m − n * q) MOD n = m MOD n
[MOD_SUC] Theorem
⊢ 0 < y ∧ SUC x ≠ SUC (x DIV y) * y ⇒ SUC x MOD y = SUC (x MOD y)
[MOD_SUC_IFF] Theorem
⊢ 0 < y ⇒ (SUC x MOD y = SUC (x MOD y) ⇔ SUC x ≠ SUC (x DIV y) * y)
[MOD_TIMES] Theorem
⊢ ∀n. 0 < n ⇒ ∀q r. (q * n + r) MOD n = r MOD n
[MOD_TIMES2] Theorem
⊢ ∀n. 0 < n ⇒ ∀j k. (j MOD n * k MOD n) MOD n = (j * k) MOD n
[MOD_TIMES_SUB] Theorem
⊢ ∀n q r. 0 < n ∧ 0 < q ∧ r ≤ n ⇒ (q * n − r) MOD n = (n − r) MOD n
[MOD_UNIQUE] Theorem
⊢ ∀n k r. (∃q. k = q * n + r ∧ r < n) ⇒ k MOD n = r
[MULT_0] Theorem
⊢ ∀m. m * 0 = 0
[MULT_ASSOC] Theorem
⊢ ∀m n p. m * (n * p) = m * n * p
[MULT_CLAUSES] Theorem
⊢ ∀m n.
0 * m = 0 ∧ m * 0 = 0 ∧ 1 * m = m ∧ m * 1 = m ∧
SUC m * n = m * n + n ∧ m * SUC n = m + m * n
[MULT_COMM] Theorem
⊢ ∀m n. m * n = n * m
[MULT_DIV] Theorem
⊢ ∀n q. 0 < n ⇒ q * n DIV n = q
[MULT_EQ_0] Theorem
⊢ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0
[MULT_EQ_1] Theorem
⊢ ∀x y. x * y = 1 ⇔ x = 1 ∧ y = 1
[MULT_EQ_DIV] Theorem
⊢ 0 < x ⇒ (x * y = z ⇔ y = z DIV x ∧ z MOD x = 0)
[MULT_EQ_ID] Theorem
⊢ ∀m n. m * n = n ⇔ m = 1 ∨ n = 0
[MULT_EXP_MONO] Theorem
⊢ ∀p q n m. n * SUC q ** p = m * SUC q ** p ⇔ n = m
[MULT_INCREASES] Theorem
⊢ ∀m n. 1 < m ∧ 0 < n ⇒ SUC n ≤ m * n
[MULT_LEFT_1] Theorem
⊢ ∀m. 1 * m = m
[MULT_LESS_EQ_SUC] Theorem
⊢ ∀m n p. m ≤ n ⇔ SUC p * m ≤ SUC p * n
[MULT_MONO_EQ] Theorem
⊢ ∀m i n. SUC n * m = SUC n * i ⇔ m = i
[MULT_RIGHT_1] Theorem
⊢ ∀m. m * 1 = m
[MULT_SUC] Theorem
⊢ ∀m n. m * SUC n = m + m * n
[MULT_SUC_EQ] Theorem
⊢ ∀p m n. n * SUC p = m * SUC p ⇔ n = m
[MULT_SYM] Theorem
⊢ ∀m n. m * n = n * m
[NORM_0] Theorem
⊢ 0 = 0
[NOT_EXP_0] Theorem
⊢ ∀m n. SUC n ** m ≠ 0
[NOT_GREATER] Theorem
⊢ ∀m n. ¬(m > n) ⇔ m ≤ n
[NOT_GREATER_EQ] Theorem
⊢ ∀m n. ¬(m ≥ n) ⇔ SUC m ≤ n
[NOT_LEQ] Theorem
⊢ ∀m n. ¬(m ≤ n) ⇔ SUC n ≤ m
[NOT_LESS] Theorem
⊢ ∀m n. ¬(m < n) ⇔ n ≤ m
[NOT_LESS_EQUAL] Theorem
⊢ ∀m n. ¬(m ≤ n) ⇔ n < m
[NOT_LT_ZERO_EQ_ZERO] Theorem
⊢ ∀n. ¬(0 < n) ⇔ n = 0
[NOT_NUM_EQ] Theorem
⊢ ∀m n. m ≠ n ⇔ SUC m ≤ n ∨ SUC n ≤ m
[NOT_ODD_EQ_EVEN] Theorem
⊢ ∀n m. SUC (n + n) ≠ m + m
[NOT_STRICTLY_DECREASING] Theorem
⊢ ∀f. ¬∀n. f (SUC n) < f n
[NOT_SUC_ADD_LESS_EQ] Theorem
⊢ ∀m n. ¬(SUC (m + n) ≤ m)
[NOT_SUC_LESS_EQ] Theorem
⊢ ∀n m. ¬(SUC n ≤ m) ⇔ m ≤ n
[NOT_SUC_LESS_EQ_0] Theorem
⊢ ∀n. ¬(SUC n ≤ 0)
[NOT_ZERO_LT_ZERO] Theorem
⊢ ∀n. n ≠ 0 ⇔ 0 < n
[NRC_0] Theorem
⊢ ∀R x y. NRC R 0 x y ⇔ x = y
[NRC_1] Theorem
⊢ NRC R 1 x y ⇔ R x y
[NRC_ADD_E] Theorem
⊢ ∀m n x z. NRC R (m + n) x z ⇒ ∃y. NRC R m x y ∧ NRC R n y z
[NRC_ADD_EQN] Theorem
⊢ NRC R (m + n) x z ⇔ ∃y. NRC R m x y ∧ NRC R n y z
[NRC_ADD_I] Theorem
⊢ ∀m n x y z. NRC R m x y ∧ NRC R n y z ⇒ NRC R (m + n) x z
[NRC_RTC] Theorem
⊢ ∀n x y. NRC R n x y ⇒ R⃰ x y
[NRC_SUC_RECURSE_LEFT] Theorem
⊢ NRC R (SUC n) x y ⇔ ∃z. NRC R n x z ∧ R z y
[NUMERAL_MULT_EQ_DIV] Theorem
⊢ (NUMERAL (BIT1 x) * y = NUMERAL z ⇔
y = NUMERAL z DIV NUMERAL (BIT1 x) ∧
NUMERAL z MOD NUMERAL (BIT1 x) = 0) ∧
(NUMERAL (BIT2 x) * y = NUMERAL z ⇔
y = NUMERAL z DIV NUMERAL (BIT2 x) ∧
NUMERAL z MOD NUMERAL (BIT2 x) = 0)
[ODD_ADD] Theorem
⊢ ∀m n. ODD (m + n) ⇔ (ODD m ⇎ ODD n)
[ODD_DOUBLE] Theorem
⊢ ∀n. ODD (SUC (2 * n))
[ODD_EVEN] Theorem
⊢ ∀n. ODD n ⇔ ¬EVEN n
[ODD_EXISTS] Theorem
⊢ ∀n. ODD n ⇔ ∃m. n = SUC (2 * m)
[ODD_EXP] Theorem
⊢ ∀m n. 0 < n ∧ ODD m ⇒ ODD (m ** n)
[ODD_EXP_IFF] Theorem
⊢ ∀n m. ODD (m ** n) ⇔ n = 0 ∨ ODD m
[ODD_MULT] Theorem
⊢ ∀m n. ODD (m * n) ⇔ ODD m ∧ ODD n
[ODD_OR_EVEN] Theorem
⊢ ∀n. ∃m. n = SUC (SUC 0) * m ∨ n = SUC (SUC 0) * m + 1
[ODD_SUB] Theorem
⊢ ∀m n. m ≤ n ⇒ (ODD (n − m) ⇔ (ODD n ⇎ ODD m))
[ONE] Theorem
⊢ 1 = SUC 0
[ONE_LT_EXP] Theorem
⊢ ∀x y. 1 < x ** y ⇔ 1 < x ∧ 0 < y
[ONE_LT_MULT] Theorem
⊢ ∀x y. 1 < x * y ⇔ 0 < x ∧ 1 < y ∨ 0 < y ∧ 1 < x
[ONE_LT_MULT_IMP] Theorem
⊢ ∀p q. 1 < p ∧ 0 < q ⇒ 1 < p * q
[ONE_MOD] Theorem
⊢ 1 < n ⇒ 1 MOD n = 1
[ONE_MOD_IFF] Theorem
⊢ 1 < n ⇔ 0 < n ∧ 1 MOD n = 1
[ONE_ONE_INV_IMAGE_BOUNDED] Theorem
⊢ ONE_ONE f ⇒ ∀b. ∃a. ∀x. f x ≤ b ⇒ x ≤ a
[ONE_ONE_UNBOUNDED] Theorem
⊢ ∀f. ONE_ONE f ⇒ ∀b. ∃n. b < f n
[OR_LESS] Theorem
⊢ ∀m n. SUC m ≤ n ⇒ m < n
[PRE_ELIM_THM] Theorem
⊢ P (PRE n) ⇔ ∀m. (n = 0 ⇒ P 0) ∧ (n = SUC m ⇒ P m)
[PRE_LESS_EQ] Theorem
⊢ ∀n. m ≤ n ⇒ PRE m ≤ PRE n
[PRE_SUB] Theorem
⊢ ∀m n. PRE (m − n) = PRE m − n
[PRE_SUB1] Theorem
⊢ ∀m. PRE m = m − 1
[PRE_SUC_EQ] Theorem
⊢ ∀m n. 0 < n ⇒ (m = PRE n ⇔ SUC m = n)
[RIGHT_ADD_DISTRIB] Theorem
⊢ ∀m n p. (m + n) * p = m * p + n * p
[RIGHT_SUB_DISTRIB] Theorem
⊢ ∀m n p. (m − n) * p = m * p − n * p
[RTC_NRC] Theorem
⊢ ∀x y. R⃰ x y ⇒ ∃n. NRC R n x y
[RTC_eq_NRC] Theorem
⊢ ∀R x y. R⃰ x y ⇔ ∃n. NRC R n x y
[STRICTLY_INCREASING_ONE_ONE] Theorem
⊢ ∀f. (∀n. f n < f (SUC n)) ⇒ ONE_ONE f
[STRICTLY_INCREASING_TC] Theorem
⊢ ∀f. (∀n. f n < f (SUC n)) ⇒ ∀m n. m < n ⇒ f m < f n
[STRICTLY_INCREASING_UNBOUNDED] Theorem
⊢ ∀f. (∀n. f n < f (SUC n)) ⇒ ∀b. ∃n. b < f n
[SUB_0] Theorem
⊢ ∀m. 0 − m = 0 ∧ m − 0 = m
[SUB_ADD] Theorem
⊢ ∀m n. n ≤ m ⇒ m − n + n = m
[SUB_CANCEL] Theorem
⊢ ∀p n m. n ≤ p ∧ m ≤ p ⇒ (p − n = p − m ⇔ n = m)
[SUB_ELIM_THM] Theorem
⊢ P (a − b) ⇔ ∀d. (b = a + d ⇒ P 0) ∧ (a = b + d ⇒ P d)
[SUB_ELIM_THM_EXISTS] Theorem
⊢ P (a − b) ⇔ (∃d. b = a + d ∧ P 0) ∨ ∃d. a = b + d ∧ P d
[SUB_EQUAL_0] Theorem
⊢ ∀c. c − c = 0
[SUB_EQ_0] Theorem
⊢ ∀m n. m − n = 0 ⇔ m ≤ n
[SUB_EQ_EQ_0] Theorem
⊢ ∀m n. m − n = m ⇔ m = 0 ∨ n = 0
[SUB_LEFT_ADD] Theorem
⊢ ∀m n p. m + (n − p) = if n ≤ p then m else m + n − p
[SUB_LEFT_EQ] Theorem
⊢ ∀m n p. m = n − p ⇔ m + p = n ∨ m ≤ 0 ∧ n ≤ p
[SUB_LEFT_GREATER] Theorem
⊢ ∀m n p. m > n − p ⇔ m + p > n ∧ m > 0
[SUB_LEFT_GREATER_EQ] Theorem
⊢ ∀m n p. m ≥ n − p ⇔ m + p ≥ n
[SUB_LEFT_LESS] Theorem
⊢ ∀m n p. m < n − p ⇔ m + p < n
[SUB_LEFT_LESS_EQ] Theorem
⊢ ∀m n p. m ≤ n − p ⇔ m + p ≤ n ∨ m ≤ 0
[SUB_LEFT_SUB] Theorem
⊢ ∀m n p. m − (n − p) = if n ≤ p then m else m + p − n
[SUB_LEFT_SUC] Theorem
⊢ ∀m n. SUC (m − n) = if m ≤ n then SUC 0 else SUC m − n
[SUB_LESS] Theorem
⊢ ∀m n. 0 < n ∧ n ≤ m ⇒ m − n < m
[SUB_LESS_0] Theorem
⊢ ∀n m. m < n ⇔ 0 < n − m
[SUB_LESS_EQ] Theorem
⊢ ∀n m. n − m ≤ n
[SUB_LESS_EQ_ADD] Theorem
⊢ ∀m p. m ≤ p ⇒ ∀n. p − m ≤ n ⇔ p ≤ m + n
[SUB_LESS_OR] Theorem
⊢ ∀m n. n < m ⇒ n ≤ m − 1
[SUB_LESS_SUC] Theorem
⊢ ∀p m. p − m < SUC p
[SUB_MOD] Theorem
⊢ ∀m n. 0 < n ∧ n ≤ m ⇒ (m − n) MOD n = m MOD n
[SUB_MONO_EQ] Theorem
⊢ ∀n m. SUC n − SUC m = n − m
[SUB_PLUS] Theorem
⊢ ∀a b c. a − (b + c) = a − b − c
[SUB_RIGHT_ADD] Theorem
⊢ ∀m n p. m − n + p = if m ≤ n then p else m + p − n
[SUB_RIGHT_EQ] Theorem
⊢ ∀m n p. m − n = p ⇔ m = n + p ∨ m ≤ n ∧ p ≤ 0
[SUB_RIGHT_GREATER] Theorem
⊢ ∀m n p. m − n > p ⇔ m > n + p
[SUB_RIGHT_GREATER_EQ] Theorem
⊢ ∀m n p. m − n ≥ p ⇔ m ≥ n + p ∨ 0 ≥ p
[SUB_RIGHT_LESS] Theorem
⊢ ∀m n p. m − n < p ⇔ m < n + p ∧ 0 < p
[SUB_RIGHT_LESS_EQ] Theorem
⊢ ∀m n p. m − n ≤ p ⇔ m ≤ n + p
[SUB_RIGHT_SUB] Theorem
⊢ ∀m n p. m − n − p = m − (n + p)
[SUB_SUB] Theorem
⊢ ∀b c. c ≤ b ⇒ ∀a. a − (b − c) = a + c − b
[SUC_ADD_SYM] Theorem
⊢ ∀m n. SUC (m + n) = SUC n + m
[SUC_ELIM_NUMERALS] Theorem
⊢ ∀f g.
(∀n. g (SUC n) = f n (SUC n)) ⇔
(∀n.
g (NUMERAL (BIT1 n)) =
f (NUMERAL (BIT1 n) − 1) (NUMERAL (BIT1 n))) ∧
∀n.
g (NUMERAL (BIT2 n)) =
f (NUMERAL (BIT1 n)) (NUMERAL (BIT2 n))
[SUC_ELIM_THM] Theorem
⊢ ∀P. (∀n. P (SUC n) n) ⇔ ∀n. 0 < n ⇒ P n (n − 1)
[SUC_MOD] Theorem
⊢ ∀n a b. 0 < n ⇒ (SUC a MOD n = SUC b MOD n ⇔ a MOD n = b MOD n)
[SUC_NOT] Theorem
⊢ ∀n. 0 ≠ SUC n
[SUC_ONE_ADD] Theorem
⊢ ∀n. SUC n = 1 + n
[SUC_PRE] Theorem
⊢ 0 < m ⇔ SUC (PRE m) = m
[SUC_SUB] Theorem
⊢ ∀a. SUC a − a = 1
[SUC_SUB1] Theorem
⊢ ∀m. SUC m − 1 = m
[SUM_SQUARED] Theorem
⊢ (x + y)² = x² + 2 * x * y + y²
[TC_eq_NRC] Theorem
⊢ ∀R x y. R⁺ x y ⇔ ∃n. NRC R (SUC n) x y
[TIMES2] Theorem
⊢ ∀n. 2 * n = n + n
[TWO] Theorem
⊢ 2 = SUC 1
[WOP] Theorem
⊢ ∀P. (∃n. P n) ⇒ ∃n. P n ∧ ∀m. m < n ⇒ ¬P m
[X_LE_DIV] Theorem
⊢ ∀x y z. 0 < z ⇒ (x ≤ y DIV z ⇔ x * z ≤ y)
[X_LE_X_EXP] Theorem
⊢ 0 < n ⇒ x ≤ x ** n
[X_LE_X_SQUARED] Theorem
⊢ x ≤ x²
[X_LT_DIV] Theorem
⊢ ∀x y z. 0 < z ⇒ (x < y DIV z ⇔ (x + 1) * z ≤ y)
[X_LT_EXP_X] Theorem
⊢ 1 < b ⇒ x < b ** x
[X_LT_EXP_X_IFF] Theorem
⊢ x < b ** x ⇔ 1 < b ∨ x = 0
[X_LT_X_SQUARED] Theorem
⊢ x < x² ⇔ 1 < x
[X_MOD_Y_EQ_X] Theorem
⊢ ∀x y. 0 < y ⇒ (x MOD y = x ⇔ x < y)
[ZERO_DIV] Theorem
⊢ ∀n. 0 < n ⇒ 0 DIV n = 0
[ZERO_EXP] Theorem
⊢ 0 ** x = if x = 0 then 1 else 0
[ZERO_LESS_ADD] Theorem
⊢ ∀m n. 0 < m + n ⇔ 0 < m ∨ 0 < n
[ZERO_LESS_EQ] Theorem
⊢ ∀n. 0 ≤ n
[ZERO_LESS_EXP] Theorem
⊢ ∀m n. 0 < SUC n ** m
[ZERO_LESS_MULT] Theorem
⊢ ∀m n. 0 < m * n ⇔ 0 < m ∧ 0 < n
[ZERO_LT_EXP] Theorem
⊢ 0 < x ** y ⇔ 0 < x ∨ y = 0
[ZERO_MOD] Theorem
⊢ ∀n. 0 < n ⇒ 0 MOD n = 0
[binary_induct] Theorem
⊢ ∀P. P 0 ∧ (∀n. P n ⇒ P (2 * n) ∧ P (2 * n + 1)) ⇒ ∀n. P n
[datatype_num] Theorem
⊢ DATATYPE (num 0 SUC)
[findq_divisor] Theorem
⊢ n ≤ m ⇒ findq (a,m,n) * n ≤ a * m
[findq_eq_0] Theorem
⊢ ∀a m n. findq (a,m,n) = 0 ⇔ a = 0
[findq_thm] Theorem
⊢ findq (a,m,n) =
if n = 0 then a
else (let d = 2 * n in if m < d then a else findq (2 * a,m,d))
[num_CASES] Theorem
⊢ ∀m. m = 0 ∨ ∃n. m = SUC n
[num_case_compute] Theorem
⊢ ∀n. num_CASE n f g = if n = 0 then f else g (PRE n)
[num_case_cong] Theorem
⊢ ∀M M' v f.
M = M' ∧ (M' = 0 ⇒ v = v') ∧ (∀n. M' = SUC n ⇒ f n = f' n) ⇒
num_CASE M v f = num_CASE M' v' f'
[num_case_eq] Theorem
⊢ num_CASE n zc sc = v ⇔ n = 0 ∧ zc = v ∨ ∃x. n = SUC x ∧ sc x = v
[transitive_LESS] Theorem
⊢ transitive $<
[transitive_measure] Theorem
⊢ ∀f. transitive (measure f)
[transitive_monotone] Theorem
⊢ ∀R f.
transitive R ∧ (∀n. R (f n) (f (SUC n))) ⇒
∀m n. m < n ⇒ R (f m) (f n)
*)
end
HOL 4, Kananaskis-13