Datasets:

phanerozoic

/

Coq-HoTT

like 2

{R : Ring} (x : R) `{IsInvertible R x} : R := left_inverse_elem x.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Nat.Arithmetic. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Groups.Group Algebra.Groups.Subgroup. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Ring.v

inverse_elem

{R : Ring} (x : R) `{IsInvertible R x} : inverse_elem x * x = 1. Proof. apply left_inverse_eq. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Nat.Arithmetic. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Groups.Group Algebra.Groups.Subgroup. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Ring.v

rng_inv_l

{R : Ring} (x : R) `{IsInvertible R x} : x * inverse_elem x = 1. Proof. rhs_V nrapply (right_inverse_eq x). f_ap. apply path_left_inverse_elem_right_inverse_elem. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Nat.Arithmetic. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Groups.Group Algebra.Groups.Subgroup. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Ring.v

rng_inv_r

{R : Ring} (x y : R) `{IsInvertible R x} `{IsInvertible R y} (p : x = y) : inverse_elem x = inverse_elem y. Proof. destruct p. snrapply (path_left_right_inverse x). - apply rng_inv_l. - apply rng_inv_r. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Nat.Arithmetic. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Groups.Group Algebra.Groups.Subgroup. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Ring.v

isinvertible_unique

{R : Ring} (x : R) : {inv : R & prod (inv * x = 1) (x * inv = 1)} <~> IsInvertible R x. Proof. equiv_via { i : IsInvertible R x & right_inverse_elem x = left_inverse_elem x }. 1: make_equiv_contr_basedpaths. apply equiv_sigma_contr; intro i. rapply contr_inhabited_hprop. symmetry; apply path_left_inverse_elem_right_inverse_elem. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Nat.Arithmetic. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Groups.Group Algebra.Groups.Subgroup. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Ring.v

equiv_isinvertible_left_right_inverse

{R : Ring} {x y : R} `{IsInvertible R x, IsInvertible R y} : x = y <~> inverse_elem x = inverse_elem y. Proof. srapply equiv_iff_hprop. - exact (isinvertible_unique x y). - exact (isinvertible_unique (inverse_elem x) (inverse_elem y)). Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Nat.Arithmetic. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Groups.Group Algebra.Groups.Subgroup. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Ring.v

equiv_path_inverse_elem

(R : Ring) : Group. Proof. snrapply Build_Group. - exact {x : R & IsInvertible R x}. - intros [x p] [y q]. exists (x * y). exact _. - exists 1. exact _. - intros [x p]. exists (inverse_elem x). exact _. - repeat split. 1: exact _. 1-5: hnf; intros; apply path_sigma_hprop. + rapply simple_associativity. + rapply left_identity. + rapply right_identity. + apply rng_inv_l. + apply rng_inv_r. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Nat.Arithmetic. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Groups.Group Algebra.Groups.Subgroup. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Ring.v

rng_unit_group

{R : Ring} {x y z : R} `{IsInvertible R y} : y * x = z <~> x = inverse_elem y * z := equiv_moveL_equiv_V (f := (y *.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Nat.Arithmetic. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Groups.Group Algebra.Groups.Subgroup. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Ring.v

rng_inv_moveL_Vr

{R : Ring} {x y z : R} `{IsInvertible R y} : x * y = z <~> x = z * inverse_elem y := equiv_moveL_equiv_V (f := (.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Nat.Arithmetic. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Groups.Group Algebra.Groups.Subgroup. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Ring.v

rng_inv_moveL_rV

{R : Ring} {x y z : R} `{IsInvertible R y} : x = y * z <~> inverse_elem y * x = z := equiv_moveR_equiv_V (f := (y *.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Nat.Arithmetic. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Groups.Group Algebra.Groups.Subgroup. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Ring.v

rng_inv_moveR_Vr

{R : Ring} {x y z : R} `{IsInvertible R y} : x = z * y <~> x * inverse_elem y = z := equiv_moveR_equiv_V (f := (.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core Spaces.Nat.Arithmetic. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Groups.Group Algebra.Groups.Subgroup. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Ring.v

rng_inv_moveR_rV

Vector@{i|} (A : Type@{i}) (n : nat) : Type@{i} := { l : list A & length l = n }.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

Vector@

(A : Type) (n : nat) (f : forall (i : nat), (i < n)%nat -> A) : Vector A n. Proof. exists (list_map (fun '(i; Hi) => f i Hi) (seq' n)). lhs nrapply length_list_map. apply length_seq'. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

Build_Vector

{A : Type} {n : nat} (v : Vector A n) i {Hi : (i < n)%nat} : A := nth' (pr1 v) i ((pr2 v)^ # Hi).

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

entry

{A : Type} {n} (f : forall (i : nat), (i < n)%nat -> A) i {Hi : (i < n)%nat} : entry (Build_Vector A n f) i = f i Hi. Proof. snrefine (nth'_list_map _ _ _ (_^ # Hi) _ @ _). 1: nrapply length_seq'. snrapply ap011D. 1: nrapply nth'_seq'. rapply path_ishprop. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

entry_Build_Vector

path_vector@{i} (A : Type@{i}) {n : nat} (v1 v2 : Vector@{i} A n) (H : forall i (H : (i < n)%nat), entry v1 i = entry v2 i) : v1 = v2. Proof. rapply path_sigma_hprop@{i i i}. snrapply path_list_nth'. 1: exact (pr2 v1 @ (pr2 v2)^). intros i Hi. snrefine (_ @ H i (pr2 v1 # Hi) @ _). 1, 2: apply nth'_nth'. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

path_vector@

{A : Type} {n : nat} (v : Vector A n) (i j : nat) (Hi : (i < n)%nat) (Hj : (j < n)%nat) (p : i = j) : entry v i = entry v j. Proof. destruct p. apply nth'_nth'. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

path_entry_vector

{A B : Type} {n} (f : A -> B) : Vector A n -> Vector B n := fun v => Build_Vector B n (fun i _ => f (entry v i)).

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

vector_map

{A B C : Type} {n} (f : A -> B -> C) : Vector A n -> Vector B n -> Vector C n := fun v1 v2 => Build_Vector C n (fun i _ => f (entry v1 i) (entry v2 i)).

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

vector_map2

Plus (Vector A n) := vector_map2 (+).

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

vector_plus

Zero (Vector A n) := Build_Vector A n (fun _ _ => 0).

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

vector_zero

Negate (Vector A n) := vector_map (-).

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

vector_neg

Associative vector_plus. Proof. intros v1 v2 v3; apply path_vector; intros i Hi. rewrite 4 entry_Build_Vector. apply associativity. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

associative_vector_plus

Commutative vector_plus. Proof. intros v1 v2; apply path_vector; intros i Hi. rewrite 2 entry_Build_Vector. apply commutativity. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

commutative_vector_plus

LeftIdentity vector_plus vector_zero. Proof. intros v; apply path_vector; intros i Hi. rewrite 2 entry_Build_Vector. apply left_identity. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

left_identity_vector_plus

RightIdentity vector_plus vector_zero. Proof. intros v; apply path_vector; intros i Hi. rewrite 2 entry_Build_Vector. apply right_identity. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

right_identity_vector_plus

LeftInverse vector_plus vector_neg vector_zero. Proof. intros v; apply path_vector; intros i Hi. rewrite 3 entry_Build_Vector. apply left_inverse. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

left_inverse_vector_plus

RightInverse vector_plus vector_neg vector_zero. Proof. intros v; apply path_vector; intros i Hi. rewrite 3 entry_Build_Vector. apply right_inverse. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

right_inverse_vector_plus

AbGroup. Proof. snrapply Build_AbGroup. 1: snrapply Build_Group. 5: repeat split. - exact (Vector A n). - exact vector_plus. - exact vector_zero. - exact vector_neg. - exact _. - exact associative_vector_plus. - exact left_identity_vector_plus. - exact right_identity_vector_plus. - exact left_inverse_vector_plus. - exact right_inverse_vector_plus. - exact commutative_vector_plus. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

abgroup_vector

(r : R) : Vector M n -> Vector M n := vector_map (lact r).

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

vector_lact

LeftHeteroDistribute vector_lact vector_plus vector_plus. Proof. intros r v1 v2; apply path_vector; intros i Hi. rewrite 5 entry_Build_Vector. apply distribute_l. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

left_heterodistribute_vector_lact_plus

RightHeteroDistribute vector_lact (+) vector_plus. Proof. intros r1 r2 v; apply path_vector; intros i Hi. rewrite 4 entry_Build_Vector. apply distribute_r. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

right_heterodistribute_vector_lact_plus

HeteroAssociative vector_lact vector_lact vector_lact (.*.). Proof. intros r s v; apply path_vector; intros i Hi. rewrite 3 entry_Build_Vector. apply associativity. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

heteroassociative_vector_lact_plus

LeftIdentity vector_lact 1. Proof. intros v; apply path_vector; intros i Hi. rewrite entry_Build_Vector. apply left_identity. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

left_identity_vector_lact

IsLeftModule R (abgroup_vector M n). Proof. snrapply Build_IsLeftModule. - exact vector_lact. - exact left_heterodistribute_vector_lact_plus. - exact right_heterodistribute_vector_lact_plus. - exact heteroassociative_vector_lact_plus. - exact left_identity_vector_lact. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.PathGroupoids. Require Import Types.Sigma. Require Import Algebra.AbGroups.AbelianGroup Algebra.Rings.Ring Algebra.Rings.Module. Require Import Spaces.Nat.Core. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import abstract_algebra.

Algebra\Rings\Vector.v

isleftmodule_isleftmodule_vector

CRing. Proof. snrapply Build_CRing'. - exact abgroup_Z. - exact 1%int. - exact int_mul. - exact int_mul_comm. - exact int_mul_assoc. - exact int_dist_l. - exact int_mul_1_l. Defined.

Definition

Require Import Classes.interfaces.canonical_names. Require Import Algebra.AbGroups. Require Import Algebra.Rings.CRing. Require Import Spaces.Int Spaces.Pos. Require Import WildCat.Core.

Algebra\Rings\Z.v

cring_Z

(R : Ring) := grp_pow_homo : R -> Int -> R.

Definition

Require Import Classes.interfaces.canonical_names. Require Import Algebra.AbGroups. Require Import Algebra.Rings.CRing. Require Import Spaces.Int Spaces.Pos. Require Import WildCat.Core.

Algebra\Rings\Z.v

rng_int_mult

{R : Ring} (r : R) (n : cring_Z) : rng_int_mult R r n = (rng_int_mult R 1 n) * r. Proof. cbn. rhs nrapply (grp_pow_natural (grp_homo_rng_right_mult r)); cbn. by rewrite rng_mult_one_l. Defined.

Definition

Require Import Classes.interfaces.canonical_names. Require Import Algebra.AbGroups. Require Import Algebra.Rings.CRing. Require Import Spaces.Int Spaces.Pos. Require Import WildCat.Core.

Algebra\Rings\Z.v

rng_int_mult_dist_r

{R : Ring} (r : R) (n : cring_Z) : rng_int_mult R r n = r * (rng_int_mult R 1 n). Proof. cbn. rhs nrapply (grp_pow_natural (grp_homo_rng_left_mult r)); cbn. by rewrite rng_mult_one_r. Defined.

Definition

Require Import Classes.interfaces.canonical_names. Require Import Algebra.AbGroups. Require Import Algebra.Rings.CRing. Require Import Spaces.Int Spaces.Pos. Require Import WildCat.Core.

Algebra\Rings\Z.v

rng_int_mult_dist_l

(R : Ring) : (cring_Z : Ring) $-> R. Proof. snrapply Build_RingHomomorphism. 1: exact (rng_int_mult R 1). repeat split. 1,2: exact _. apply rng_plus_zero_r. Defined.

Definition

Require Import Classes.interfaces.canonical_names. Require Import Algebra.AbGroups. Require Import Algebra.Rings.CRing. Require Import Spaces.Int Spaces.Pos. Require Import WildCat.Core.

Algebra\Rings\Z.v

rng_homo_int

{Sort : Type} := Build_SymbolTypeOf

Record

Require Import

Algebra\Universal\Algebra.v

SymbolTypeOf

Build_Signature

Record

Require Import

Algebra\Universal\Algebra.v

Signature

{σ} (A : Carriers σ) (w : SymbolType σ) : Type := DomOperation A w -> A (sort_cod w).

Definition

Require Import

Algebra\Universal\Algebra.v

Operation

{σ : Signature} : Type := Build_Algebra

Record

Require Import

Algebra\Universal\Algebra.v

Algebra

(σ : Signature) : Type := {c : Carriers σ | { _ : forall (u : Symbol σ), Operation c (σ u) | forall (s : Sort σ), IsHSet (c s) } }.

Definition

Require Import

Algebra\Universal\Algebra.v

SigAlgebra

(σ : Signature) : SigAlgebra σ <~> Algebra σ. Proof. issig. Defined.

Lemma

Require Import

Algebra\Universal\Algebra.v

issig_algebra

`{Funext} {σ : Signature} (A B : Algebra σ) (p : carriers A = carriers B) (q : transport (fun i => forall u, Operation i (σ u)) p (operations A) = operations B) : A = B. Proof. apply (ap (issig_algebra σ)^-1)^-1; cbn. apply (path_sigma' _ p). refine (transport_sigma p _ @ _). apply path_sigma_hprop. exact q. Defined.

Lemma

Require Import

Algebra\Universal\Algebra.v

path_algebra

`{Funext} {σ} (A B : Algebra σ) (p : carriers A = carriers B) (q : transport (fun i => forall u, Operation i (σ u)) p (operations A) = operations B) : ap carriers (path_algebra A B p q) = p. Proof. destruct A as [A a ha], B as [B b hb]; cbn in p, q. destruct p, q. unfold path_algebra, path_sigma_hprop, path_sigma_uncurried. cbn -[center]. by destruct (center (ha = hb)). Defined.

Lemma

Require Import

Algebra\Universal\Algebra.v

path_ap_carriers_path_algebra

{σ : Signature} {A B : Algebra σ} (p q : A = B) (r : ap (issig_algebra σ)^-1 p = ap (issig_algebra σ)^-1 q) : p = q. Proof. set (e := (equiv_ap (issig_algebra σ)^-1 A B)). by apply (@equiv_inv _ _ (ap e) (Equivalences.isequiv_ap _ _)). Defined.

Lemma

Require Import

Algebra\Universal\Algebra.v

path_path_algebra_issig

`{Funext} {σ} {A B : Algebra σ} (p q : A = B) (r : ap carriers p = ap carriers q) : p = q. Proof. apply path_path_algebra_issig. unshelve eapply path_path_sigma. - transitivity (ap carriers p); [by destruct p |]. transitivity (ap carriers q); [exact r | by destruct q]. - apply path_ishprop. Defined.

Lemma

Require Import

Algebra\Universal\Algebra.v

path_path_algebra

{w : SymbolType σ} (f : Operation A w) : Type := forall (a b : DomOperation A w), (forall i : Arity w, Φ (sorts_dom w i) (a i) (b i)) -> Φ (sort_cod w) (f a) (f b). Class OpsCompatible : Type := ops_compatible : forall (u : Symbol σ), u.#A. Global Instance trunc_ops_compatible `{Funext} {n : trunc_index} `{!forall s x y, IsTrunc n (Φ s x y)} : IsTrunc n OpsCompatible. Proof. apply istrunc_forall. Defined.

Definition

Require Import

Algebra\Universal\Congruence.v

OpCompatible

{σ : Signature} {A B : Algebra σ} (Φ : forall s, Relation (A s)) `{!IsCongruence A Φ} (f : forall s, A s -> B s) `{!IsHomomorphism f} : Type := forall s (x y : A s), Φ s x y -> f s x = f s y.

Definition

Require Import

Algebra\Universal\Congruence.v

HomCompatible

{w : SymbolType σ} (α : Operation A w) (β : Operation B w) : Type := forall a : DomOperation A w, f (sort_cod w) (α a) = β (fun i => f (sorts_dom w i) (a i)). Global Instance hprop_oppreserving `{Funext} {w : SymbolType σ} (α : Operation A w) (β : Operation B w) : IsHProp ( α β). Proof. apply istrunc_forall. Qed.

Definition

Require Import

Algebra\Universal\Homomorphism.v

OpPreserving

{σ} {A B : Algebra σ} : Type := Build_Homomorphism

Record

Require Import

Algebra\Universal\Homomorphism.v

Homomorphism

{σ} {A B : Algebra σ} {f g : A $-> B} : f = g -> forall s, f s == g s. Proof. intro p. by destruct p. Defined.

Lemma

Require Import

Algebra\Universal\Homomorphism.v

apD10_homomorphism

{σ} (A B : Algebra σ) : Type := { def_hom : forall s, A s -> B s | IsHomomorphism def_hom }.

Definition

Require Import

Algebra\Universal\Homomorphism.v

SigHomomorphism

{σ} (A B : Algebra σ) : SigHomomorphism A B <~> (A $-> B). Proof. issig. Defined.

Lemma

Require Import

Algebra\Universal\Homomorphism.v

issig_homomorphism

`{Funext} {σ} {A B : Algebra σ} (f g : A $-> B) (p : def_homomorphism f = def_homomorphism g) : f = g. Proof. apply (ap (issig_homomorphism A B)^-1)^-1. unfold issig_homomorphism; cbn. apply path_sigma_hprop. exact p. Defined.

Lemma

Require Import

Algebra\Universal\Homomorphism.v

path_homomorphism

A $-> A := Build_Homomorphism (fun s (x : A s) => x). End . Arguments {σ} A%_Algebra_scope , {σ} {A}. Section homomorphism_compose. Context {σ} {A B C : Algebra σ}. Global Instance is_homomorphism_compose (g : forall s, B s -> C s) `{!IsHomomorphism g} (f : forall s, A s -> B s) `{!IsHomomorphism f} : IsHomomorphism (fun s => g s o f s). Proof. intros u a. by rewrite <- (oppreserving_hom g), (oppreserving_hom f). Qed.

Definition

Require Import

Algebra\Universal\Homomorphism.v

homomorphism_id

(g : B $-> C) (f : A $-> B) : A $-> C := Build_Homomorphism (fun s => g s o f s).

Definition

Require Import

Algebra\Universal\Homomorphism.v

homomorphism_compose

`{Funext} {σ} {A B C D : Algebra σ} (h : C $-> D) (g : B $-> C) (f : A $-> B) : (h $o g) $o f = h $o (g $o f). Proof. by apply path_homomorphism. Defined.

Lemma

Require Import

Algebra\Universal\Homomorphism.v

assoc_homomorphism_compose

`{Funext} {σ} {A B : Algebra σ} (f : A $-> B) : Id B $o f = f. Proof. by apply path_homomorphism. Defined.

Lemma

Require Import

Algebra\Universal\Homomorphism.v

left_id_homomorphism_compose

`{Funext} {σ} {A B : Algebra σ} (f : A $-> B) : f $o Id A = f. Proof. by apply path_homomorphism. Defined.

Lemma

Require Import

Algebra\Universal\Homomorphism.v

right_id_homomorphism_compose

{σ} (A : Carriers σ) {n : nat} (ss : FinSeq n.

Definition

Require Import

Algebra\Universal\Operation.v

cons_dom

{σ} (A : Carriers σ) (ss : FinSeq 0 (Sort σ)) : forall i : Fin 0, A (ss i) := Empty_ind (A o ss).

Definition

Require Import

Algebra\Universal\Operation.v

nil_dom

{σ : Signature} (A : Carriers σ) {n : nat} (ss : FinSeq n (Sort σ)) (t : Sort σ) : Type := Operation A {| Arity := Fin n; sorts_dom := ss; sort_cod := t |}.

Definition

Require Import

Algebra\Universal\Operation.v

FiniteOperation

{σ} (A : Carriers σ) {n : nat} : (FinSeq n (Sort σ)) -> Sort σ -> Type := match n with | 0 => fun ss t => A t | n'.

Fixpoint

Require Import

Algebra\Universal\Operation.v

CurriedOperation

{σ} (A : Carriers σ) {n : nat} : forall (ss : FinSeq n (Sort σ)) (t : Sort σ), CurriedOperation A ss t -> FiniteOperation A ss t := match n with | 0 => fun ss t op _ => op | n'.+1 => fun ss t op a => A (fstail ss) t (op (a fin_zero)) (a o fsucc) end. Local Example computation_example_operation_uncurry : forall (σ : Signature) (A : Carriers σ) (n : nat) (s1 s2 t : Sort σ) (ss := (fscons s1 (fscons s2 fsnil))) (op : CurriedOperation A ss t) (a : forall i, A (ss i)), A ss t op = fun a => op (a fin_zero) (a (fsucc fin_zero)). Proof. reflexivity. Qed.

Fixpoint

Require Import

Algebra\Universal\Operation.v

operation_uncurry

{σ} (A : Carriers σ) {n : nat} : forall (ss : FinSeq n (Sort σ)) (t : Sort σ), FiniteOperation A ss t -> CurriedOperation A ss t := match n with | 0 => fun ss t op => op (Empty_ind _) | n'.+1 => fun ss t op x => A (fstail ss) t (op o cons_dom A ss x) end. Local Example computation_example_operation_curry : forall (σ : Signature) (A : Carriers σ) (n : nat) (s1 s2 t : Sort σ) (ss := (fscons s1 (fscons s2 fsnil))) (op : FiniteOperation A ss t) (x1 : A s1) (x2 : A s2), A ss t op = fun x1 x2 => op (cons_dom A ss x1 (cons_dom A _ x2 (nil_dom A _))). Proof. reflexivity. Qed.

Fixpoint

Require Import

Algebra\Universal\Operation.v

operation_curry

expand_cons_dom' {σ} (A : Carriers σ) (n : nat) : forall (i : Fin n) (ss : FinSeq n (Sort σ)) (N : n > 0) (a : forall i, A (ss i)), cons_dom' A i ss N (head_dom' A n N ss a) (tail_dom' A n ss a) = a i. Proof. intro i. induction i using fin_ind; intros ss N a. - unfold cons_dom'. rewrite fin_ind_beta_zero. reflexivity. - unfold cons_dom'. by rewrite fin_ind_beta_fsucc. Qed.

Lemma

Require Import

Algebra\Universal\Operation.v

expand_cons_dom'

`{Funext} {σ} (A : Carriers σ) {n : nat} (ss : FinSeq n.+1 (Sort σ)) (a : forall i, A (ss i)) : cons_dom A ss (head_dom A ss a) (tail_dom A ss a) = a. Proof. funext i. apply '. Defined.

Lemma

Require Import

Algebra\Universal\Operation.v

expand_cons_dom

`{Funext} {σ} (A : Carriers σ) {n : nat} (ss : FinSeq n (Sort σ)) (t : Sort σ) : operation_uncurry A ss t o operation_curry A ss t == idmap. Proof. intro a. induction n as [| n IHn]. - funext d. refine (ap a _). apply path_contr. - funext a'. refine (ap (fun x => x _) (IHn _ _) @ _). refine (ap a _). apply expand_cons_dom. Qed.

Lemma

Require Import

Algebra\Universal\Operation.v

path_operation_curry_to_cunurry

`{Funext} {σ} (A : Carriers σ) {n : nat} (ss : FinSeq n (Sort σ)) (t : Sort σ) : operation_curry A ss t o operation_uncurry A ss t == idmap. Proof. intro a. induction n; [reflexivity|]. funext x. refine (_ @ IHn (fstail ss) (a x)). refine (ap (operation_curry A (fstail ss) t) _). funext a'. simpl. unfold cons_dom, cons_dom'. rewrite fin_ind_beta_zero. refine (ap (operation_uncurry A (fstail ss) t (a x)) _). funext i'. by rewrite fin_ind_beta_fsucc. Qed.

Lemma

Require Import

Algebra\Universal\Operation.v

path_operation_uncurry_to_curry

`{Funext} {σ} (A : Carriers σ) {n : nat} (ss : FinSeq n (Sort σ)) (t : Sort σ) : FiniteOperation A ss t <~> CurriedOperation A ss t := Build_Equiv _ _ (operation_curry A ss t) _.

Definition

Require Import

Algebra\Universal\Operation.v

equiv_operation_curry

{σ} (C : Carriers σ) : Carriers σ := | var_term_algebra : forall s, C s -> C s | ops_term_algebra : forall (u : Symbol σ), DomOperation ( C) (σ u) -> C (sort_cod (σ u)).

Inductive

Require Import

Algebra\Universal\TermAlgebra.v

CarriersTermAlgebra

{σ} := @CarriersTermAlgebra_ind σ.

Definition

Require Import

Algebra\Universal\TermAlgebra.v

CarriersTermAlgebra_rect

{σ : Signature} (C : Carriers σ) (P : Sort σ -> Type) (vs : forall (s : Sort σ), C s -> P s) (os : forall (u : Symbol σ) (c : DomOperation (CarriersTermAlgebra C) (σ u)), (forall i : Arity (σ u), P (sorts_dom (σ u) i)) -> P (sort_cod (σ u))) (s : Sort σ) (T : CarriersTermAlgebra C s) : P s := CarriersTermAlgebra_ind C (fun s _ => P s) vs os s T.

Definition

Require Import

Algebra\Universal\TermAlgebra.v

CarriersTermAlgebra_rec

{σ : Signature} {C : Carriers σ} (R : forall s, Relation (C s)) {s1 s2 : Sort σ} (S : CarriersTermAlgebra C s1) (T : CarriersTermAlgebra C s2) : Type := match S, T with | var_term_algebra s1 x, var_term_algebra s2 y => {p : s1 = s2 | R s2 (p # x) y} | ops_term_algebra u1 a, ops_term_algebra u2 b => { p : u1 = u2 | forall i : Arity (σ u1), R (a i) (b (transport (fun v => Arity (σ v)) p i))} | _, _ => Empty end.

Fixpoint

Require Import

Algebra\Universal\TermAlgebra.v

ExtendDRelTermAlgebra

{σ : Signature} {C : Carriers σ} (R : forall s, Relation (C s)) {s : Sort σ} : CarriersTermAlgebra C s -> CarriersTermAlgebra C s -> Type := ExtendDRelTermAlgebra R. Section extend_rel_term_algebra. Context `{Funext} {σ : Signature} {C : Carriers σ} (R : forall s, Relation (C s)) `{!forall s, is_mere_relation (C s) (R s)}. Global Instance hprop_extend_drel_term_algebra {s1 s2 : Sort σ} (S : CarriersTermAlgebra C s1) (T : CarriersTermAlgebra C s2) : IsHProp (ExtendDRelTermAlgebra R S T). Proof. generalize dependent s2. induction S; intros s2 T; destruct T; exact _. Qed.

Definition

Require Import

Algebra\Universal\TermAlgebra.v

ExtendRelTermAlgebra

`{!forall s, Symmetric (R s)} {s1 s2 : Sort σ} (S : CarriersTermAlgebra C s1) (T : CarriersTermAlgebra C s2) (h : ExtendDRelTermAlgebra R S T) : ExtendDRelTermAlgebra R T S. Proof. generalize dependent s2. induction S as [| u c h]; intros s2 [] p. - destruct p as [p1 p2]. induction p1. exists idpath. by symmetry. - elim p. - elim p. - destruct p as [p f]. induction p. exists idpath. intro i. apply h. apply f. Qed.

Lemma

Require Import

Algebra\Universal\TermAlgebra.v

symmetric_extend_drel_term_algebra

`{!forall s, Transitive (R s)} {s1 s2 s3 : Sort σ} (S : CarriersTermAlgebra C s1) (T : CarriersTermAlgebra C s2) (U : CarriersTermAlgebra C s3) (h1 : ExtendDRelTermAlgebra R S T) (h2 : ExtendDRelTermAlgebra R T U) : ExtendDRelTermAlgebra R S U. Proof. generalize dependent s3. generalize dependent s2. induction S as [| u c h]; intros s2 [? d | ? d] h2 s3 [] h3; destruct h2 as [p2 P2], h3 as [p3 P3] || by (elim h2 || elim h3). - exists (p2 @ p3). rewrite transport_pp. induction p2, p3. by transitivity d. - exists (p2 @ p3). intro i. induction p2. apply (h i _ (d i)). + apply P2. + rewrite concat_1p. apply P3. Qed.

Lemma

Require Import

Algebra\Universal\TermAlgebra.v

transitive_extend_drel_term_algebra

{s : Sort σ} (S : CarriersTermAlgebra C s) (T : CarriersTermAlgebra C s) : Type := ExtendRelTermAlgebra (fun s => paths) S T. Global Instance reflexive_extend_path_term_algebra : forall s : Sort σ, Reflexive (@ s). Proof. by apply reflexive_extend_rel_term_algebra. Defined.

Definition

Require Import

Algebra\Universal\TermAlgebra.v

ExtendPathTermAlgebra

{s : Sort σ} {S T : CarriersTermAlgebra C s} (p : S = T) : ExtendPathTermAlgebra S T. Proof. induction p. apply reflexive_extend_path_term_algebra. Defined.

Lemma

Require Import

Algebra\Universal\TermAlgebra.v

reflexive_extend_path_term_algebra_path

{s1 s2 : Sort σ} (S : CarriersTermAlgebra C s1) (T : CarriersTermAlgebra C s2) (e : ExtendDRelTermAlgebra (fun s => paths) S T) : {p : s1 = s2 | p # S = T}. Proof. generalize dependent s2. induction S as [| u c h]; intros s2 [? d | ? d] e; solve [elim e] || destruct e as [p e]. - exists p. by induction p, e. - induction p. exists idpath. cbn. f_ap. funext a. destruct (h a _ (d a) (e a)) as [p q]. by induction (hset_path2 idpath p). Defined.

Lemma

Require Import

Algebra\Universal\TermAlgebra.v

dependent_path_extend_path_term_algebra

{s : Sort σ} (S T : CarriersTermAlgebra C s) (e : ExtendPathTermAlgebra S T) : S = T. Proof. destruct (dependent_path_extend_path_term_algebra S T e) as [p q]. by induction (hset_path2 idpath p). Defined.

Lemma

Require Import

Algebra\Universal\TermAlgebra.v

path_extend_path_term_algebra

{s : Sort σ} (S T : CarriersTermAlgebra C s) : ExtendPathTermAlgebra S T <~> (S = T) := equiv_iff_hprop (path_extend_path_term_algebra S T) reflexive_extend_path_term_algebra_path.

Definition

Require Import

Algebra\Universal\TermAlgebra.v

equiv_path_extend_path_term_algebra

`{Funext} {σ : Signature} (C : Carriers σ) `{!forall s, IsHSet (C s)} : Algebra σ := Build_Algebra (CarriersTermAlgebra C) (@ops_term_algebra _ C).

Definition

Require Import

Algebra\Universal\TermAlgebra.v

TermAlgebra

{σ} (C : Carriers σ) (s : Sort σ) (x y : C s) : var_term_algebra C s x = var_term_algebra C s y -> x = y. Proof. intro p. apply reflexive_extend_path_term_algebra_path in p. destruct p as [p1 p2]. by destruct (hset_path2 p1 idpath)^. Qed.

Lemma

Require Import

Algebra\Universal\TermAlgebra.v

isinj_var_term_algebra

`{Funext} {σ} (C : Carriers σ) (u : Symbol σ) (a b : DomOperation (CarriersTermAlgebra C) (σ u)) : ops_term_algebra C u a = ops_term_algebra C u b -> a = b. Proof. intro p. apply reflexive_extend_path_term_algebra_path in p. destruct p as [p1 p2]. destruct (hset_path2 p1 idpath)^. funext i. apply path_extend_path_term_algebra. apply p2. Qed.

Lemma

Require Import

Algebra\Universal\TermAlgebra.v

isinj_ops_term_algebra

{σ} {C : Carriers σ} (A : Algebra σ) (f : forall s, C s -> A s) (s : Sort σ) (T : CarriersTermAlgebra C s) : A s := CarriersTermAlgebra_rec C A f (fun u _ r => u.#A r) s T. Global Instance is_homomorphism_map_term_algebra : @IsHomomorphism σ (TermAlgebra C) A ( A f). Proof. intros u a. by refine (ap u.#A _). Qed.

Definition

Require Import

Algebra\Universal\TermAlgebra.v

map_term_algebra

TermAlgebra C $-> A := @Build_Homomorphism σ (TermAlgebra C) A (map_term_algebra A f) _.

Definition

Require Import

Algebra\Universal\TermAlgebra.v

hom_term_algebra

(f : TermAlgebra C $-> A) : forall s, C s -> A s := fun s x => f s (var_term_algebra C s x).

Definition

Require Import

Algebra\Universal\TermAlgebra.v

precomp_var_term_algebra

forall (f : TermAlgebra C $-> A), hom_term_algebra A (precomp_var_term_algebra f) = f. Proof. intro f. apply path_homomorphism. funext s T. induction T as [|u c h]. - reflexivity. - refine (_ @ (is_homomorphism f u c)^). refine (ap u.#A _). funext i. apply h. Defined.

Lemma

Require Import

Algebra\Universal\TermAlgebra.v

path_precomp_var_term_algebra_to_hom_term_algebra

forall (f : forall s, C s -> A s), precomp_var_term_algebra (hom_term_algebra A f) = f. Proof. intro f. by funext s a. Defined.

Lemma

Require Import

Algebra\Universal\TermAlgebra.v

path_hom_term_algebra_to_precomp_var_term_algebra

(TermAlgebra C $-> A) <~> (forall s, C s -> A s). Proof. exact (Build_Equiv _ _ precomp_var_term_algebra _). Defined.

Theorem

Require Import

Algebra\Universal\TermAlgebra.v

ump_term_algebra

Cast Q F := rationals_to_field Q F.

Instance

Require Import

Analysis\Locator.v

qinc

(x : F) := forall q r : Q, q < r -> (' q < x) + (x < ' r).

Definition

Require Import

Analysis\Locator.v

locator

locator' (x : F) :=

Record

Require Import

Analysis\Locator.v

locator'

{x : F} (l : locator' x) {q r : Q} : q < r -> DHProp := fun nu => Build_DHProp (Build_HProp (~ (locates_right l nu))) _.

Definition

Require Import

Analysis\Locator.v

locates_left

(x : F) : locator x. Proof. intros q r ltqr. case (LEM (' q < x)). - apply _. - exact inl. - intros notlt. apply inr. assert (ltqr' : ' q < ' r) by auto. exact (nlt_lt_trans notlt ltqr'). Qed.

Lemma

Require Import

Analysis\Locator.v

all_reals_locators