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--------------------------------------------------------------------------------
-- Agda Helium
--
-- Ring solver tactic using integers as coefficients
--------------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Helium.Algebra.Ordered.StrictTotal.Bundles using (CommutativeRing)
import Helium.Algebra.Ordered.StrictTotal.Properties.CommutativeRing as Ringₚ
open import Relation.Nullary
module Helium.Tactic.CommutativeRing.NonReflective
{ℓ₁ ℓ₂ ℓ₃}
(R : CommutativeRing ℓ₁ ℓ₂ ℓ₃)
(1≉0 : let module R = CommutativeRing R in ¬ R.1# R.≈ R.0#)
where
open import Data.Bool.Base
open import Data.Sign using (Sign) renaming (_*_ to _S*_)
open import Level using (0ℓ; _⊔_)
open import Relation.Binary.Reasoning.StrictPartialOrder (CommutativeRing.strictPartialOrder R)
module R where
open CommutativeRing R public
renaming
( trans to <-trans
; irrefl to <-irrefl
; asym to <-asym
; 0<a+0<b⇒0<ab to x>0∧y>0⇒xy>0
)
open Ringₚ R public
infixr 2 _,_
record Positive : Set (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where
constructor _,_
field
val : Carrier
proof : val ≥ 0#
open Positive
infixl 6 _*⁺_ _*′_
infix 5 _◃_
_*⁺_ : Positive → Positive → Positive
+x *⁺ +y = +x .val * +y .val , x≥0∧y≥0⇒x*y≥0 (+x .proof) (+y .proof)
_◃_ : Sign → Positive → Carrier
Sign.- ◃ +x = - (+x .val)
Sign.+ ◃ +x = +x .val
sign : Carrier → Sign
sign x with does (x <? 0#)
... | true = Sign.-
... | false = Sign.+
∣_∣ : Carrier → Positive
∣ x ∣ with x <? 0#
... | yes x<0 = - x , <⇒≤ (x<0⇒-x>0 x<0)
... | no x≮0 = x , ≮⇒≥ x≮0
◃-congˡ : ∀ s +x +y → +x .val ≈ +y .val → s ◃ +x ≈ s ◃ +y
◃-congˡ Sign.- +x +y x≈y = -‿cong x≈y
◃-congˡ Sign.+ +x +y x≈y = x≈y
_*′_ : Carrier → Carrier → Carrier
x *′ y = sign x S* sign y ◃ ∣ x ∣ *⁺ ∣ y ∣
*′≈* : ∀ x y → x *′ y ≈ x * y
*′≈* x y with x <? 0# | y <? 0#
... | yes x<0 | yes y<0 = -x*-y≈x*y x y
... | yes x<0 | no y≮0 = begin-equality
- (- x * y) ≈⟨ -‿cong (-‿*-distribˡ x y) ⟩
- - (x * y) ≈⟨ -‿involutive (x * y) ⟩
x * y ∎
... | no x≮0 | yes y<0 = begin-equality
- (x * - y) ≈⟨ -‿cong (-‿*-distribʳ x y) ⟩
- - (x * y) ≈⟨ -‿involutive (x * y) ⟩
x * y ∎
... | no x≮0 | no y≮0 = Eq.refl
open import Algebra.Bundles using (Ring)
open import Algebra.Morphism
import Data.Bool.Properties as Boolₚ
open import Data.Empty using (⊥-elim)
open import Data.Integer as ℤ hiding (suc)
open import Data.Integer.Properties as ℤₚ
open import Data.Maybe.Base
open import Data.Nat.Base as ℕ using (ℕ; suc; z≤n; s≤s)
import Data.Nat.Properties as ℕₚ
open import Data.Product
import Data.Unit
open import Data.Vec hiding (_⊛_)
open import Data.Vec.N-ary
open import Function
open import Relation.Binary.PropositionalEquality
open import Tactic.RingSolver.Core.AlmostCommutativeRing
open import Tactic.RingSolver.Core.Expression public
open import Tactic.RingSolver.Core.Polynomial.Parameters
private
T-≡ : ∀ {x} → T x ⇔ x ≡ true
T-≡ {false} = mk⇔ (λ ()) (λ ())
T-≡ {true} = mk⇔ (λ _ → refl) (λ _ → _)
T-≡′ : ∀ {x} → T (not x) ⇔ x ≡ false
T-≡′ {false} = mk⇔ (λ _ → refl) (λ _ → _)
T-≡′ {true} = mk⇔ (λ ()) λ ()
T-¬ : ∀ {x} → T (not x) ⇔ (¬ T x)
T-¬ {false} = mk⇔ (λ _ ⊥ → ⊥) (λ ¬⊥ → _)
T-¬ {true} = mk⇔ (λ ⊥ _ → ⊥) (λ ¬⊤ → ¬⊤ _)
module Ops where
ℤ-coeff : RawCoeff 0ℓ 0ℓ
ℤ-coeff = record
{ rawRing = Ring.rawRing ℤₚ.+-*-ring
; isZero = does ∘ (0ℤ ℤ.≟_)
}
ring : AlmostCommutativeRing ℓ₁ ℓ₂
ring = fromCommutativeRing R.Unordered.commutativeRing (decToMaybe ∘ (R.0# R.≟_))
ℤ⟶R : RawCoeff.rawRing ℤ-coeff -Raw-AlmostCommutative⟶ ring
ℤ⟶R = record
{ ⟦_⟧ = ⟦_⟧
; +-homo = +-homo
; *-homo = *-homo
; -‿homo = -‿homo
; 0-homo = R.Eq.refl
; 1-homo = R.Eq.refl
}
where
⟦_⟧ : ℤ → R.Carrier
⟦ + n ⟧ = n R.× R.1#
⟦ -[1+ n ] ⟧ = R.- (suc n R.× R.1#)
∸-homo : ∀ {m n} → m ℕ.≤ n → ⟦ + (n ℕ.∸ m) ⟧ R.≈ ⟦ + n ⟧ R.- ⟦ + m ⟧
∸-homo {.0} {n} z≤n = begin-equality
n R.× R.1# ≈˘⟨ R.+-identityʳ _ ⟩
n R.× R.1# R.+ R.0# ≈˘⟨ R.+-congˡ R.-0≈0 ⟩
n R.× R.1# R.- R.0# ∎
∸-homo {.suc m} {.suc n} (s≤s m≤n) = begin-equality
(n ℕ.∸ m) R.× R.1# ≈⟨ ∸-homo m≤n ⟩
n R.× R.1# R.- m R.× R.1# ≈˘⟨ R.+-congʳ (R.+-identityʳ _) ⟩
n R.× R.1# R.+ R.0# R.- m R.× R.1# ≈˘⟨ R.+-congʳ (R.+-congˡ (R.-‿inverseʳ R.1#)) ⟩
n R.× R.1# R.+ (R.1# R.- R.1#) R.- m R.× R.1# ≈˘⟨ R.+-congʳ (R.+-assoc _ _ _) ⟩
n R.× R.1# R.+ R.1# R.- R.1# R.- m R.× R.1# ≈⟨ R.+-assoc _ _ _ ⟩
n R.× R.1# R.+ R.1# R.+ (R.- R.1# R.- m R.× R.1#) ≈⟨ R.+-cong (R.+-comm _ R.1#) (R.-‿+-comm R.1# _) ⟩
R.1# R.+ n R.× R.1# R.- (R.1# R.+ m R.× R.1#) ≈˘⟨ R.+-cong (R.1+× n R.1#) (R.-‿cong (R.1+× m R.1#)) ⟩
suc n R.× R.1# R.- suc m R.× R.1# ∎
-‿homo : ∀ n → ⟦ - n ⟧ R.≈ R.- ⟦ n ⟧
-‿homo -[1+ n ] = R.Eq.sym (R.-‿involutive _)
-‿homo +0 = R.Eq.sym R.-0≈0
-‿homo +[1+ n ] = R.Eq.refl
⊖-homo : ∀ m n → ⟦ m ⊖ n ⟧ R.≈ ⟦ + m ⟧ R.- ⟦ + n ⟧
⊖-homo m n with m ℕ.<ᵇ n in eq
... | true = begin-equality
⟦ - (+ (n ℕ.∸ m)) ⟧ ≈⟨ -‿homo (+ (n ℕ.∸ m)) ⟩
R.- ⟦ + (n ℕ.∸ m) ⟧ ≈⟨ R.-‿cong (∸-homo (ℕₚ.<⇒≤ (ℕₚ.<ᵇ⇒< m n (Equivalence.g T-≡ eq)))) ⟩
R.- (⟦ + n ⟧ R.- ⟦ + m ⟧) ≈˘⟨ R.-‿+-comm ⟦ + n ⟧ (R.- ⟦ + m ⟧) ⟩
R.- ⟦ + n ⟧ R.- R.- ⟦ + m ⟧ ≈⟨ R.+-congˡ (R.-‿involutive ⟦ + m ⟧) ⟩
R.- ⟦ + n ⟧ R.+ ⟦ + m ⟧ ≈⟨ R.+-comm (R.- ⟦ + n ⟧) ⟦ + m ⟧ ⟩
⟦ + m ⟧ R.- ⟦ + n ⟧ ∎
... | false = ∸-homo (ℕₚ.≮⇒≥ {m} {n} (Equivalence.f T-¬ (Equivalence.g T-≡′ eq) ∘ ℕₚ.<⇒<ᵇ))
+-homo : ∀ m n → ⟦ m + n ⟧ R.≈ ⟦ m ⟧ R.+ ⟦ n ⟧
+-homo -[1+ m ] -[1+ n ] = begin-equality
R.- (suc (suc (m ℕ.+ n)) R.× R.1#) ≡˘⟨ cong (λ x → R.- (x R.× R.1#)) (ℕₚ.+-suc (suc m) n) ⟩
R.- ((suc m ℕ.+ suc n) R.× R.1#) ≈⟨ R.-‿cong (R.×-homo-+ R.1# (suc m) (suc n)) ⟩
R.- (suc m R.× R.1# R.+ suc n R.× R.1#) ≈˘⟨ R.-‿+-comm _ _ ⟩
R.- (suc m R.× R.1#) R.- (suc n R.× R.1#) ∎
+-homo -[1+ m ] (+ n) = begin-equality
⟦ n ⊖ suc m ⟧ ≈⟨ ⊖-homo n (suc m) ⟩
⟦ + n ⟧ R.+ ⟦ -[1+ m ] ⟧ ≈⟨ R.+-comm ⟦ + n ⟧ ⟦ -[1+ m ] ⟧ ⟩
⟦ -[1+ m ] ⟧ R.+ ⟦ + n ⟧ ∎
+-homo (+ m) -[1+ n ] = ⊖-homo m (suc n)
+-homo (+ m) (+ n) = R.×-homo-+ R.1# m n
∣∙∣-homo : ∀ n → ⟦ + ∣ n ∣ ⟧ R.≈ R.Positive.val R.∣ ⟦ n ⟧ ∣
∣∙∣-homo n with ⟦ n ⟧ R.<? R.0#
∣∙∣-homo (+ n) | yes n<0 = ⊥-elim (R.<⇒≱ n<0 (R.x≥0⇒n×x≥0 n R.0≤1))
∣∙∣-homo (+ n) | no n≮0 = R.Eq.refl
∣∙∣-homo -[1+ n ] | yes n<0 = R.Eq.sym (R.-‿involutive _)
∣∙∣-homo -[1+ n ] | no n≮0 = ⊥-elim (n≮0 (R.x>0⇒-x<0 (R.n≢0∧x>0⇒n×x>0 (suc n) (R.x≉y⇒0<1 1≉0))))
sign-homo : ∀ n → sign n ≡ R.sign ⟦ n ⟧
sign-homo n with ⟦ n ⟧ R.<? R.0#
sign-homo (+ n) | yes n<0 = ⊥-elim (R.<⇒≱ n<0 (R.x≥0⇒n×x≥0 n R.0≤1))
sign-homo (+ n) | no n≮0 = refl
sign-homo -[1+ n ] | yes n<0 = refl
sign-homo -[1+ n ] | no n≮0 = ⊥-elim (n≮0 (R.x>0⇒-x<0 (R.n≢0∧x>0⇒n×x>0 (suc n) (R.x≉y⇒0<1 1≉0))))
◃-homo : ∀ s n → ⟦ s ◃ n ⟧ R.≈ s R.◃ (⟦ + n ⟧ R., R.x≥0⇒n×x≥0 n R.0≤1)
◃-homo Sign.- 0 = R.Eq.sym R.-0≈0
◃-homo Sign.- (suc n) = R.Eq.refl
◃-homo Sign.+ 0 = R.Eq.refl
◃-homo Sign.+ (suc n) = R.Eq.refl
*ⁿ-homo : ∀ m n → ⟦ + (m ℕ.* n) ⟧ R.≈ ⟦ + m ⟧ R.* ⟦ + n ⟧
*ⁿ-homo 0 n = R.Eq.sym (R.zeroˡ ⟦ + n ⟧)
*ⁿ-homo (suc m) n = begin-equality
(n ℕ.+ m ℕ.* n) R.× R.1# ≈⟨ R.×-homo-+ R.1# n (m ℕ.* n) ⟩
n R.× R.1# R.+ (m ℕ.* n) R.× R.1# ≈⟨ R.+-congˡ (*ⁿ-homo m n) ⟩
n R.× R.1# R.+ m R.× R.1# R.* n R.× R.1# ≈˘⟨ R.+-congʳ (R.*-identityˡ _) ⟩
R.1# R.* n R.× R.1# R.+ m R.× R.1# R.* n R.× R.1# ≈˘⟨ R.distribʳ _ R.1# _ ⟩
(R.1# R.+ m R.× R.1#) R.* n R.× R.1# ≈˘⟨ R.*-congʳ (R.1+× m R.1#) ⟩
suc m R.× R.1# R.* n R.× R.1# ∎
*-homo : ∀ m n → ⟦ m * n ⟧ R.≈ ⟦ m ⟧ R.* ⟦ n ⟧
*-homo m n = begin-equality
⟦ m * n ⟧ ≡⟨⟩
⟦ sm*sn ◃ ∣ m ∣ ℕ.* ∣ n ∣ ⟧ ≈⟨ ◃-homo sm*sn (∣ m ∣ ℕ.* ∣ n ∣) ⟩
sm*sn R.◃ (⟦ + (∣ m ∣ ℕ.* ∣ n ∣) ⟧ R., ∣m∣∣n∣≥0) ≈⟨ R.◃-congˡ sm*sn _ _ (*ⁿ-homo ∣ m ∣ ∣ n ∣) ⟩
sm*sn R.◃ (⟦ + ∣ m ∣ ⟧ R.* ⟦ + ∣ n ∣ ⟧ R., ∣mn∣≥0) ≈⟨ R.◃-congˡ sm*sn _ _ (R.*-cong (∣∙∣-homo m) (∣∙∣-homo n)) ⟩
sm*sn R.◃ R.∣ ⟦ m ⟧ ∣ R.*⁺ R.∣ ⟦ n ⟧ ∣ ≡⟨ cong₂ (λ x y → x S* y R.◃ R.∣ ⟦ m ⟧ ∣ R.*⁺ R.∣ ⟦ n ⟧ ∣) (sign-homo m) (sign-homo n) ⟩
Rsm*Rsn R.◃ R.∣ ⟦ m ⟧ ∣ R.*⁺ R.∣ ⟦ n ⟧ ∣ ≡⟨⟩
⟦ m ⟧ R.*′ ⟦ n ⟧ ≈⟨ R.*′≈* ⟦ m ⟧ ⟦ n ⟧ ⟩
⟦ m ⟧ R.* ⟦ n ⟧ ∎
where
∣m∣∣n∣≥0 = R.x≥0⇒n×x≥0 (∣ m ∣ ℕ.* ∣ n ∣) R.0≤1
∣mn∣≥0 = R.x≥0∧y≥0⇒x*y≥0 (R.x≥0⇒n×x≥0 ∣ m ∣ R.0≤1) (R.x≥0⇒n×x≥0 ∣ n ∣ R.0≤1)
sm*sn = sign m S* sign n
Rsm*Rsn = R.sign ⟦ m ⟧ S* R.sign ⟦ n ⟧
homo : Homomorphism 0ℓ 0ℓ ℓ₁ ℓ₂
homo = record
{ from = ℤ-coeff
; to = ring
; morphism = ℤ⟶R
; Zero-C⟶Zero-R = λ { +0 _ → R.Eq.refl }
}
open Eval R.Unordered.rawRing (_-Raw-AlmostCommutative⟶_.⟦_⟧ ℤ⟶R) public
open import Tactic.RingSolver.Core.Polynomial.Base (Homomorphism.from homo)
norm : ∀ {n} → Expr ℤ n → Poly n
norm (Κ x) = κ x
norm (Ι x) = ι x
norm (x ⊕ y) = norm x ⊞ norm y
norm (x ⊗ y) = norm x ⊠ norm y
norm (x ⊛ i) = norm x ⊡ i
norm (⊝ x) = ⊟ norm x
⟦_⇓⟧ : ∀ {n} → Expr ℤ n → Vec R.Carrier n → R.Carrier
⟦ expr ⇓⟧ = ⟦ norm expr ⟧ₚ
where open import Tactic.RingSolver.Core.Polynomial.Semantics homo renaming (⟦_⟧ to ⟦_⟧ₚ)
correct : ∀ {n} (expr : Expr ℤ n) ρ → ⟦ expr ⇓⟧ ρ R.≈ ⟦ expr ⟧ ρ
correct {n = n} = go
where
open import Tactic.RingSolver.Core.Polynomial.Homomorphism homo
go : ∀ (expr : Expr ℤ n) ρ → ⟦ expr ⇓⟧ ρ R.≈ ⟦ expr ⟧ ρ
go (Κ x) ρ = κ-hom x ρ
go (Ι x) ρ = ι-hom x ρ
go (x ⊕ y) ρ = ⊞-hom (norm x) (norm y) ρ ⟨ R.Eq.trans ⟩ (go x ρ ⟨ R.+-cong ⟩ go y ρ)
go (x ⊗ y) ρ = ⊠-hom (norm x) (norm y) ρ ⟨ R.Eq.trans ⟩ (go x ρ ⟨ R.*-cong ⟩ go y ρ)
go (x ⊛ i) ρ = ⊡-hom (norm x) i ρ ⟨ R.Eq.trans ⟩ R.^-congˡ i (go x ρ)
go (⊝ x) ρ = ⊟-hom (norm x) ρ ⟨ R.Eq.trans ⟩ R.-‿cong (go x ρ)
open import Relation.Binary.Reflection R.strictTotalOrder.Eq.setoid Ι ⟦_⟧ ⟦_⇓⟧ correct public
solve : ∀ (n : ℕ) →
(f : N-ary n (Expr ℤ n) (Expr ℤ n × Expr ℤ n)) →
Eqʰ n R._≈_ (curryⁿ (Ops.⟦_⇓⟧ (proj₁ (Ops.close n f)))) (curryⁿ (Ops.⟦_⇓⟧ (proj₂ (Ops.close n f)))) →
Eq n R._≈_ (curryⁿ (Ops.⟦_⟧ (proj₁ (Ops.close n f)))) (curryⁿ (Ops.⟦_⟧ (proj₂ (Ops.close n f))))
solve = Ops.solve
{-# INLINE solve #-}
infixl 6 _⊜_
_⊜_ : ∀ {n} → Expr ℤ n → Expr ℤ n → Expr ℤ n × Expr ℤ n
_⊜_ = _,_
{-# INLINE _⊜_ #-}
|
