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signature primitivesArithTheory =
sig
type thm = Thm.thm
(* Theorems *)
val add_sub_same_eq : thm
val ge_eq_le : thm
val gt_eq_lt : thm
val int_add : thm
val int_add_minus_same_eq : thm
val int_induction : thm
val int_induction_ideal : thm
val int_of_num_id : thm
val int_of_num_inj : thm
val le_eq_ge : thm
val lt_eq_gt : thm
val mul_pos_left_le : thm
val mul_pos_left_lt : thm
val mul_pos_right_le : thm
val mul_pos_right_lt : thm
val not_ge_eq_lt : thm
val not_gt_eq_le : thm
val not_le_eq_gt : thm
val not_lt_eq_ge : thm
val num_sub_1_eq : thm
val num_sub_eq : thm
val pos_div_pos_ge : thm
val pos_div_pos_is_pos : thm
val pos_div_pos_le : thm
val pos_div_pos_le_init : thm
val pos_div_pos_mul_le : thm
val pos_mod_pos_ineqs : thm
val pos_mod_pos_is_pos : thm
val pos_mod_pos_le_init : thm
val pos_mod_pos_lt : thm
val pos_mul_2_div_pos_decompose : thm
val pos_mul_left_pos_le : thm
val pos_mul_left_pos_lt : thm
val pos_mul_pos_div_pos_decompose : thm
val pos_mul_pos_is_pos : thm
val pos_mul_right_pos_le : thm
val pos_mul_right_pos_lt : thm
val primitivesArith_grammars : type_grammar.grammar * term_grammar.grammar
(*
[Omega] Parent theory of "primitivesArith"
[int_arith] Parent theory of "primitivesArith"
[add_sub_same_eq] Theorem
⊢ ∀i j. i + j − j = i
[ge_eq_le] Theorem
⊢ ∀x y. x ≥ y ⇔ y ≤ x
[gt_eq_lt] Theorem
⊢ ∀x y. x > y ⇔ y < x
[int_add] Theorem
⊢ ∀m n. &(m + n) = &m + &n
[int_add_minus_same_eq] Theorem
⊢ ∀i j. i + j − j = i
[int_induction] Theorem
⊢ ∀P. (∀i. i < 0 ⇒ P i) ∧ P 0 ∧ (∀i. P i ⇒ P (i + 1)) ⇒ ∀i. P i
[int_induction_ideal] Theorem
⊢ ∀P. P 0 ∧ (∀i. 0 ≤ i ∧ P i ⇒ P (i + 1)) ⇒ ∀i. 0 ≤ i ⇒ P i
[int_of_num_id] Theorem
⊢ ∀i. 0 ≤ i ⇒ &Num i = i
[int_of_num_inj] Theorem
⊢ ∀n m. &n = &m ⇒ n = m
[le_eq_ge] Theorem
⊢ ∀x y. x ≤ y ⇔ y ≥ x
[lt_eq_gt] Theorem
⊢ ∀x y. x < y ⇔ y > x
[mul_pos_left_le] Theorem
⊢ ∀a x y. 0 ≤ a ⇒ x ≤ y ⇒ a * x ≤ a * y
[mul_pos_left_lt] Theorem
⊢ ∀a x y. 0 < a ⇒ x < y ⇒ a * x < a * y
[mul_pos_right_le] Theorem
⊢ ∀a x y. 0 ≤ a ⇒ x ≤ y ⇒ x * a ≤ y * a
[mul_pos_right_lt] Theorem
⊢ ∀a x y. 0 < a ⇒ x < y ⇒ x * a < y * a
[not_ge_eq_lt] Theorem
⊢ ∀x y. ¬(x ≥ y) ⇔ x < y
[not_gt_eq_le] Theorem
⊢ ∀x y. ¬(x > y) ⇔ x ≤ y
[not_le_eq_gt] Theorem
⊢ ∀x y. ¬(x ≤ y) ⇔ x > y
[not_lt_eq_ge] Theorem
⊢ ∀x y. ¬(x < y) ⇔ x ≥ y
[num_sub_1_eq] Theorem
⊢ ∀x y. x = y − 1 ⇒ 0 ≤ x ⇒ Num y = SUC (Num x)
[num_sub_eq] Theorem
⊢ ∀x y z. x = y − z ⇒ 0 ≤ x ⇒ 0 ≤ z ⇒ Num y = Num z + Num x
[pos_div_pos_ge] Theorem
⊢ ∀x y d. 0 ≤ x ⇒ 0 ≤ y ⇒ 0 < d ⇒ x ≥ y ⇒ x / d ≥ y / d
[pos_div_pos_is_pos] Theorem
⊢ ∀x y. 0 ≤ x ⇒ 0 < y ⇒ 0 ≤ x / y
[pos_div_pos_le] Theorem
⊢ ∀x y d. 0 ≤ x ⇒ 0 ≤ y ⇒ 0 < d ⇒ x ≤ y ⇒ x / d ≤ y / d
[pos_div_pos_le_init] Theorem
⊢ ∀x y. 0 ≤ x ⇒ 0 < y ⇒ x / y ≤ x
[pos_div_pos_mul_le] Theorem
⊢ ∀x y. 0 ≤ x ⇒ 0 < y ⇒ x / y * y ≤ x
[pos_mod_pos_ineqs] Theorem
⊢ ∀x y. 0 ≤ x ⇒ 0 < y ⇒ 0 ≤ x % y ∧ x % y < y
[pos_mod_pos_is_pos] Theorem
⊢ ∀x y. 0 ≤ x ⇒ 0 < y ⇒ 0 ≤ x % y
[pos_mod_pos_le_init] Theorem
⊢ ∀x y. 0 ≤ x ⇒ 0 < y ⇒ x % y ≤ x
[pos_mod_pos_lt] Theorem
⊢ ∀x y. 0 ≤ x ⇒ 0 < y ⇒ x % y < y
[pos_mul_2_div_pos_decompose] Theorem
⊢ ∀x y. 0 ≤ x ⇒ 0 < y ⇒ x / y + x / y ≤ x * 2 / y
[pos_mul_left_pos_le] Theorem
⊢ ∀a x. 0 < a ⇒ 0 ≤ x ⇒ x ≤ a * x
[pos_mul_left_pos_lt] Theorem
⊢ ∀a x. 1 < a ⇒ 0 < x ⇒ x < a * x
[pos_mul_pos_div_pos_decompose] Theorem
⊢ ∀x y z. 0 ≤ x ⇒ 0 ≤ y ⇒ 0 < z ⇒ x / z + y / z ≤ (x + y) / z
[pos_mul_pos_is_pos] Theorem
⊢ ∀x y. 0 ≤ x ⇒ 0 ≤ y ⇒ 0 ≤ x * y
[pos_mul_right_pos_le] Theorem
⊢ ∀a x. 0 < a ⇒ 0 ≤ x ⇒ x ≤ x * a
[pos_mul_right_pos_lt] Theorem
⊢ ∀a x. 1 < a ⇒ 0 < x ⇒ x < x * a
*)
end
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