MathClasses.varieties.rings
Require
categories.varieties theory.rings.
Require Import
Ring
abstract_algebra universal_algebra ua_homomorphisms workaround_tactics.
Inductive op := plus | mult | zero | one | negate.
Definition sig: Signature := single_sorted_signature
(λ o, match o with zero | one ⇒ O | negate ⇒ 1%nat | plus | mult ⇒ 2%nat end).
Section laws.
Global Instance: Plus (Term0 sig nat tt) :=
λ x, App sig _ _ _ (App sig _ _ _ (Op sig _ plus) x).
Global Instance: Mult (Term0 sig nat tt) :=
λ x, App sig _ _ _ (App sig _ _ _ (Op sig _ mult) x).
Global Instance: Zero (Term0 sig nat tt) := Op sig _ zero.
Global Instance: One (Term0 sig nat tt) := Op sig _ one.
Global Instance: Negate (Term0 sig nat tt) := App sig _ _ _ (Op sig _ negate).
Local Notation x := (Var sig nat 0%nat tt).
Local Notation y := (Var sig nat 1%nat tt).
Local Notation z := (Var sig nat 2%nat tt).
Import notations.
Inductive Laws: EqEntailment sig → Prop :=
|e_plus_assoc: Laws (x + (y + z) === (x + y) + z)
|e_plus_comm: Laws (x + y === y + x)
|e_plus_0_l: Laws (0 + x === x)
|e_mult_assoc: Laws (x × (y × z) === (x × y) × z)
|e_mult_comm: Laws (x × y === y × x)
|e_mult_1_l: Laws (1 × x === x)
|e_mult_0_l: Laws (0 × x === 0)
|e_distr: Laws (x × (y + z) === x × y + x × z)
|e_distr_l: Laws ((x + y) × z === x × z + y × z)
|e_plus_negate_r: Laws (x + - x === 0)
|e_plus_negate_l: Laws (- x + x === 0).
End laws.
Definition theory: EquationalTheory := Build_EquationalTheory sig Laws.
Definition Object := varieties.Object theory.
Section decode_operations.
Context `{AlgebraOps theory A}.
Global Instance: Plus (A tt) := algebra_op plus.
Global Instance: Mult (A tt) := algebra_op mult.
Global Instance: Zero (A tt) := algebra_op zero.
Global Instance: One (A tt) := algebra_op one.
Global Instance: Negate (A tt) := algebra_op negate.
End decode_operations.
Section encode_with_ops.
Context A `{Ring A}.
Global Instance encode_operations: AlgebraOps sig (λ _, A) := λ o,
match o with plus ⇒ (+) | mult ⇒ (.*.) | zero ⇒ 0:A | one ⇒ 1:A | negate ⇒ (-) end.
Global Instance encode_algebra_and_ops: Algebra sig _.
Proof. constructor. intro. apply _. intro o. destruct o; simpl; try apply _; apply reflexivity. Qed.
Add Ring A: (rings.stdlib_ring_theory A).
Global Instance encode_variety_and_ops: InVariety theory (λ _, A).
Proof. constructor. apply _. intros ? [] ?; simpl; unfold algebra_op; simpl; ring. Qed.
Definition object: Object := varieties.object theory (λ _, A).
End encode_with_ops.
Lemma encode_algebra_only `{!AlgebraOps theory A} `{∀ u, Equiv (A u)} `{!Ring (A tt)}: Algebra sig A .
Proof. constructor; intros []; simpl in *; try apply _. Qed.
Instance decode_variety_and_ops `{InVariety theory A}: Ring (A tt).
Proof with simpl; auto.
pose proof (λ law lawgood x y z, variety_laws law lawgood (λ s n,
match s with tt ⇒ match n with 0 ⇒ x | 1 ⇒ y | _ ⇒ z end end)) as laws.
repeat (constructor; try apply _); repeat intro.
apply_simplified (laws _ e_plus_assoc).
apply (algebra_propers plus)...
apply_simplified (laws _ e_plus_0_l)...
transitivity (algebra_op plus (algebra_op zero) x).
apply_simplified (laws _ e_plus_comm)...
apply_simplified (laws _ e_plus_0_l)...
apply (algebra_propers negate)...
apply_simplified (laws _ e_plus_negate_l)...
apply_simplified (laws _ e_plus_negate_r)...
apply_simplified (laws _ e_plus_comm)...
apply_simplified (laws _ e_mult_assoc)...
apply (algebra_propers mult)...
apply_simplified (laws _ e_mult_1_l)...
transitivity (algebra_op mult (algebra_op one) x).
apply_simplified (laws _ e_mult_comm)...
apply_simplified (laws _ e_mult_1_l)...
apply_simplified (laws _ e_mult_comm)...
apply_simplified (laws _ e_distr)...
Qed.
Lemma encode_morphism_only
`{AlgebraOps theory A} `{∀ u, Equiv (A u)}
`{AlgebraOps theory B} `{∀ u, Equiv (B u)}
(f: ∀ u, A u → B u) `{!Ring (A tt)} `{!Ring (B tt)} `{!SemiRing_Morphism (f tt)}: HomoMorphism sig A B f.
Proof.
constructor.
intros []. apply _.
intros []; simpl.
apply rings.preserves_plus.
apply rings.preserves_mult.
apply rings.preserves_0.
apply rings.preserves_1.
apply rings.preserves_negate.
apply encode_algebra_only.
apply encode_algebra_only.
Qed.
Lemma encode_morphism_and_ops `{Ring A} `{Ring B} {f : A → B} `{!SemiRing_Morphism f}:
@HomoMorphism sig (λ _, A) (λ _, B) _ _ _ _ (λ _, f).
Proof. intros. apply (encode_morphism_only _). Qed.
Lemma decode_morphism_and_ops
`{InVariety theory x} `{InVariety theory y} `{!HomoMorphism theory x y f}:
SemiRing_Morphism (f tt).
Proof.
pose proof (preserves theory x y f) as P.
repeat (constructor; try apply _)
; [ apply (P plus) | apply (P zero) | apply (P mult) | apply (P one) ].
Qed.
categories.varieties theory.rings.
Require Import
Ring
abstract_algebra universal_algebra ua_homomorphisms workaround_tactics.
Inductive op := plus | mult | zero | one | negate.
Definition sig: Signature := single_sorted_signature
(λ o, match o with zero | one ⇒ O | negate ⇒ 1%nat | plus | mult ⇒ 2%nat end).
Section laws.
Global Instance: Plus (Term0 sig nat tt) :=
λ x, App sig _ _ _ (App sig _ _ _ (Op sig _ plus) x).
Global Instance: Mult (Term0 sig nat tt) :=
λ x, App sig _ _ _ (App sig _ _ _ (Op sig _ mult) x).
Global Instance: Zero (Term0 sig nat tt) := Op sig _ zero.
Global Instance: One (Term0 sig nat tt) := Op sig _ one.
Global Instance: Negate (Term0 sig nat tt) := App sig _ _ _ (Op sig _ negate).
Local Notation x := (Var sig nat 0%nat tt).
Local Notation y := (Var sig nat 1%nat tt).
Local Notation z := (Var sig nat 2%nat tt).
Import notations.
Inductive Laws: EqEntailment sig → Prop :=
|e_plus_assoc: Laws (x + (y + z) === (x + y) + z)
|e_plus_comm: Laws (x + y === y + x)
|e_plus_0_l: Laws (0 + x === x)
|e_mult_assoc: Laws (x × (y × z) === (x × y) × z)
|e_mult_comm: Laws (x × y === y × x)
|e_mult_1_l: Laws (1 × x === x)
|e_mult_0_l: Laws (0 × x === 0)
|e_distr: Laws (x × (y + z) === x × y + x × z)
|e_distr_l: Laws ((x + y) × z === x × z + y × z)
|e_plus_negate_r: Laws (x + - x === 0)
|e_plus_negate_l: Laws (- x + x === 0).
End laws.
Definition theory: EquationalTheory := Build_EquationalTheory sig Laws.
Definition Object := varieties.Object theory.
Section decode_operations.
Context `{AlgebraOps theory A}.
Global Instance: Plus (A tt) := algebra_op plus.
Global Instance: Mult (A tt) := algebra_op mult.
Global Instance: Zero (A tt) := algebra_op zero.
Global Instance: One (A tt) := algebra_op one.
Global Instance: Negate (A tt) := algebra_op negate.
End decode_operations.
Section encode_with_ops.
Context A `{Ring A}.
Global Instance encode_operations: AlgebraOps sig (λ _, A) := λ o,
match o with plus ⇒ (+) | mult ⇒ (.*.) | zero ⇒ 0:A | one ⇒ 1:A | negate ⇒ (-) end.
Global Instance encode_algebra_and_ops: Algebra sig _.
Proof. constructor. intro. apply _. intro o. destruct o; simpl; try apply _; apply reflexivity. Qed.
Add Ring A: (rings.stdlib_ring_theory A).
Global Instance encode_variety_and_ops: InVariety theory (λ _, A).
Proof. constructor. apply _. intros ? [] ?; simpl; unfold algebra_op; simpl; ring. Qed.
Definition object: Object := varieties.object theory (λ _, A).
End encode_with_ops.
Lemma encode_algebra_only `{!AlgebraOps theory A} `{∀ u, Equiv (A u)} `{!Ring (A tt)}: Algebra sig A .
Proof. constructor; intros []; simpl in *; try apply _. Qed.
Instance decode_variety_and_ops `{InVariety theory A}: Ring (A tt).
Proof with simpl; auto.
pose proof (λ law lawgood x y z, variety_laws law lawgood (λ s n,
match s with tt ⇒ match n with 0 ⇒ x | 1 ⇒ y | _ ⇒ z end end)) as laws.
repeat (constructor; try apply _); repeat intro.
apply_simplified (laws _ e_plus_assoc).
apply (algebra_propers plus)...
apply_simplified (laws _ e_plus_0_l)...
transitivity (algebra_op plus (algebra_op zero) x).
apply_simplified (laws _ e_plus_comm)...
apply_simplified (laws _ e_plus_0_l)...
apply (algebra_propers negate)...
apply_simplified (laws _ e_plus_negate_l)...
apply_simplified (laws _ e_plus_negate_r)...
apply_simplified (laws _ e_plus_comm)...
apply_simplified (laws _ e_mult_assoc)...
apply (algebra_propers mult)...
apply_simplified (laws _ e_mult_1_l)...
transitivity (algebra_op mult (algebra_op one) x).
apply_simplified (laws _ e_mult_comm)...
apply_simplified (laws _ e_mult_1_l)...
apply_simplified (laws _ e_mult_comm)...
apply_simplified (laws _ e_distr)...
Qed.
Lemma encode_morphism_only
`{AlgebraOps theory A} `{∀ u, Equiv (A u)}
`{AlgebraOps theory B} `{∀ u, Equiv (B u)}
(f: ∀ u, A u → B u) `{!Ring (A tt)} `{!Ring (B tt)} `{!SemiRing_Morphism (f tt)}: HomoMorphism sig A B f.
Proof.
constructor.
intros []. apply _.
intros []; simpl.
apply rings.preserves_plus.
apply rings.preserves_mult.
apply rings.preserves_0.
apply rings.preserves_1.
apply rings.preserves_negate.
apply encode_algebra_only.
apply encode_algebra_only.
Qed.
Lemma encode_morphism_and_ops `{Ring A} `{Ring B} {f : A → B} `{!SemiRing_Morphism f}:
@HomoMorphism sig (λ _, A) (λ _, B) _ _ _ _ (λ _, f).
Proof. intros. apply (encode_morphism_only _). Qed.
Lemma decode_morphism_and_ops
`{InVariety theory x} `{InVariety theory y} `{!HomoMorphism theory x y f}:
SemiRing_Morphism (f tt).
Proof.
pose proof (preserves theory x y f) as P.
repeat (constructor; try apply _)
; [ apply (P plus) | apply (P zero) | apply (P mult) | apply (P one) ].
Qed.