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@ -117,8 +117,7 @@ module rec T : sig
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val comparator : (t, comparator_witness) Comparator.t
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end = struct
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include T0
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include Comparator.Make (T0)
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include T0 include Comparator.Make (T0)
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end
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(* auxiliary definition for safe recursive module initialization *)
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@ -725,8 +724,7 @@ let rec sum_to_exp typ sum =
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| _ -> Add {typ; args= sum} )
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| _ -> Add {typ; args= sum}
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and rational Q.({num; den}) typ =
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simp_div (integer num typ) (integer den typ)
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and rational Q.{num; den} typ = simp_div (integer num typ) (integer den typ)
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and simp_div x y =
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match (x, y) with
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@ -771,11 +769,10 @@ let simp_urem x y =
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| _, Integer {data; typ} when Z.equal Z.one data -> integer Z.zero typ
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| _ -> App {op= App {op= Urem; arg= x}; arg= y}
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(* Sums of polynomial terms represented by multisets. A sum ∑ᵢ cᵢ ×
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Xᵢ of monomials Xᵢ with coefficients cᵢ is represented by a
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multiset where the elements are Xᵢ with multiplicities cᵢ. A constant
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is treated as the coefficient of the empty monomial, which is the unit of
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multiplication 1. *)
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(* Sums of polynomial terms represented by multisets. A sum ∑ᵢ cᵢ × Xᵢ of
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monomials Xᵢ with coefficients cᵢ is represented by a multiset where the
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elements are Xᵢ with multiplicities cᵢ. A constant is treated as the
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coefficient of the empty monomial, which is the unit of multiplication 1. *)
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module Sum = struct
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let empty = empty_qset
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@ -809,8 +806,7 @@ let rec simp_add_ typ es poly =
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rational Q.((coeff * of_z i) + of_z j) typ
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(* (c × ∑ᵢ cᵢ × Xᵢ) + s ==> (∑ᵢ (c × cᵢ) × Xᵢ) + s *)
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| Add {args}, _ -> simp_add_ typ (Sum.mul_const coeff args) poly
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(* (c₀ × X₀) + (∑ᵢ₌₁ⁿ cᵢ × Xᵢ) ==> ∑ᵢ₌₀ⁿ
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cᵢ × Xᵢ *)
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(* (c₀ × X₀) + (∑ᵢ₌₁ⁿ cᵢ × Xᵢ) ==> ∑ᵢ₌₀ⁿ cᵢ × Xᵢ *)
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| _, Add {args} -> Sum.to_exp typ (Sum.add coeff exp args)
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(* (c₁ × X₁) + X₂ ==> ∑ᵢ₌₁² cᵢ × Xᵢ for c₂ = 1 *)
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| _ -> Sum.to_exp typ (Sum.add coeff exp (Sum.singleton poly))
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@ -820,9 +816,9 @@ let rec simp_add_ typ es poly =
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let simp_add typ es = simp_add_ typ es (integer Z.zero typ)
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let simp_add2 typ e f = simp_add_ typ (Sum.singleton e) f
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(* Products of indeterminants represented by multisets. A product ∏ᵢ
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xᵢ^nᵢ of indeterminates xᵢ is represented by a multiset where the
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elements are xᵢ and the multiplicities are the exponents nᵢ. *)
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(* Products of indeterminants represented by multisets. A product ∏ᵢ xᵢ^nᵢ
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of indeterminates xᵢ is represented by a multiset where the elements are
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xᵢ and the multiplicities are the exponents nᵢ. *)
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module Prod = struct
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let empty = empty_qset
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let add exp prod = Qset.add prod exp Q.one
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@ -849,26 +845,22 @@ let rec simp_mul2 typ e f =
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| Integer {data}, _ when Z.equal Z.zero data -> e
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(* e × 0 ==> 0 *)
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| _, Integer {data} when Z.equal Z.zero data -> f
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(* c × (∑ᵤ cᵤ × ∏ⱼ yᵤⱼ) ==> ∑ᵤ c × cᵤ × ∏ⱼ
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yᵤⱼ *)
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(* c × (∑ᵤ cᵤ × ∏ⱼ yᵤⱼ) ==> ∑ᵤ c × cᵤ × ∏ⱼ yᵤⱼ *)
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| Integer {data}, Add {args} | Add {args}, Integer {data} ->
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Sum.to_exp typ (Sum.mul_const (Q.of_z data) args)
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(* c₁ × x₁ ==> ∑ᵢ₌₁ cᵢ × xᵢ *)
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| Integer {data= c}, x | x, Integer {data= c} ->
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Sum.to_exp typ (Sum.singleton ~coeff:(Q.of_z c) x)
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(* (∏ᵤ₌₀ⁱ xᵤ) × (∏ᵥ₌ᵢ₊₁ⁿ xᵥ) ==>
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∏ⱼ₌₀ⁿ xⱼ *)
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(* (∏ᵤ₌₀ⁱ xᵤ) × (∏ᵥ₌ᵢ₊₁ⁿ xᵥ) ==> ∏ⱼ₌₀ⁿ xⱼ *)
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| Mul {typ; args= xs1}, Mul {args= xs2} ->
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Mul {typ; args= Prod.union xs1 xs2}
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(* (∏ᵢ xᵢ) × (∑ᵤ cᵤ × ∏ⱼ yᵤⱼ) ==> ∑ᵤ cᵤ ×
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∏ᵢ xᵢ × ∏ⱼ yᵤⱼ *)
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(* (∏ᵢ xᵢ) × (∑ᵤ cᵤ × ∏ⱼ yᵤⱼ) ==> ∑ᵤ cᵤ × ∏ᵢ xᵢ × ∏ⱼ yᵤⱼ *)
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| Mul {args= prod}, (Add _ as poly) | (Add _ as poly), Mul {args= prod} ->
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poly_map_monos ~f:(Prod.union prod) poly
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(* x₀ × (∏ᵢ₌₁ⁿ xᵢ) ==> ∏ᵢ₌₀ⁿ xᵢ *)
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| Mul {typ; args= xs1}, x | x, Mul {typ; args= xs1} ->
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Mul {typ; args= Prod.add x xs1}
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(* e × (∑ᵤ cᵤ × ∏ⱼ yᵤⱼ) ==> ∑ᵤ e × cᵤ × ∏ⱼ
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yᵤⱼ *)
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(* e × (∑ᵤ cᵤ × ∏ⱼ yᵤⱼ) ==> ∑ᵤ e × cᵤ × ∏ⱼ yᵤⱼ *)
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| Add {args}, e | e, Add {args} ->
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simp_add typ (Sum.map ~f:(fun m -> simp_mul2 typ e m) args)
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(* x₁ × x₂ ==> ∏ᵢ₌₁² xᵢ *)
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Josh Berdine
6 years ago
committed by
Facebook Github Bot