# Library mathcomp.algebra.ssralg

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B.                                  *)

From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq.
From mathcomp Require Import choice fintype finfun bigop prime binomial.

0 == the zero (additive identity) of a Nmodule x + y == the sum of x and y (in a Nmodule) x *+ n == n times x, with n in nat (non-negative), i.e., x + (x + .. (x + x)..) (n terms); x *+ 1 is thus convertible to x, and x *+ 2 to x + x \sum_<range> e == iterated sum for a Zmodule (cf bigop.v) e_i == nth 0 e i, when e : seq M and M has a zmodType structure support f == 0.-support f, i.e., [pred x | f x != 0] addr_closed S <-> collective predicate S is closed under finite sums (0 and x + y in S, for x, y in S) [SubChoice_isSubNmodule of U by <: ] == nmodType mixin for a subType whose base type is a nmodType and whose predicate's is a nmodClosed

• x == the opposite (additive inverse) of x
x - y == the difference of x and y; this is only notation for x + (- y) x *- n == notation for - (x *+ n), the opposite of x *+ n oppr_closed S <-> collective predicate S is closed under opposite zmod_closed S <-> collective predicate S is closed under zmodType operations (0 and x - y in S, for x, y in S) This property coerces to oppr_pred and addr_pred. [SubChoice_isSubZmodule of U by <: ] == zmodType mixin for a subType whose base type is a zmodType and whose predicate's is a zmodClosed

# SemiRing (non-commutative semirings):

R^c == the converse Ring for R: R^c is convertible to R but when R has a canonical ringType structure R^c has the converse one: if x y : R^c, then x * y = (y : R) * (x : R) 1 == the multiplicative identity element of a Ring n%:R == the ring image of an n in nat; this is just notation for 1 *+ n, so 1%:R is convertible to 1 and 2%:R to 1 + 1 <number> == <number>%:R with <number> a sequence of digits x * y == the ring product of x and y \prod_<range> e == iterated product for a ring (cf bigop.v) x ^+ n == x to the nth power with n in nat (non-negative), i.e., x * (x * .. (x * x)..) (n factors); x ^+ 1 is thus convertible to x, and x ^+ 2 to x * x GRing.comm x y <-> x and y commute, i.e., x * y = y * x GRing.lreg x <-> x if left-regular, i.e., *%R x is injective GRing.rreg x <-> x if right-regular, i.e., *%R x is injective [char R] == the characteristic of R, defined as the set of prime numbers p such that p%:R = 0 in R The set [char R] has at most one element, and is implemented as a pred_nat collective predicate (see prime.v); thus the statement p \in [char R] can be read as R has characteristic p', while [char R] =i pred0 means R has characteristic 0' when R is a field. Frobenius_aut chRp == the Frobenius automorphism mapping x in R to x ^+ p, where chRp : p \in [char R] is a proof that R has (non-zero) characteristic p mulr_closed S <-> collective predicate S is closed under finite products (1 and x * y in S for x, y in S) semiring_closed S <-> collective predicate S is closed under semiring operations (0, 1, x + y and x * y in S) [SubNmodule_isSubSemiRing of R by <: ] == [SubChoice_isSubSemiRing of R by <: ] == semiRingType mixin for a subType whose base type is a semiRingType and whose predicate's is a semiringClosed

# Ring (non-commutative rings):

GRing.sign R b := (-1) ^+ b in R : ringType, with b : bool This is a parsing-only helper notation, to be used for defining more specific instances. smulr_closed S <-> collective predicate S is closed under products and opposite (-1 and x * y in S for x, y in S) subring_closed S <-> collective predicate S is closed under ring operations (1, x - y and x * y in S) [SubZmodule_isSubRing of R by <: ] == [SubChoice_isSubRing of R by <: ] == ringType mixin for a subType whose base type is a ringType and whose predicate's is a subringClosed

# ComSemiRing (commutative SemiRings):

[SubNmodule_isSubComSemiRing of R by <: ] == [SubChoice_isSubComSemiRing of R by <: ] == comSemiRingType mixin for a subType whose base type is a comSemiRingType and whose predicate's is a semiringClosed

# ComRing (commutative Rings):

[SubZmodule_isSubComRing of R by <: ] == [SubChoice_isSubComRing of R by <: ] == comRingType mixin for a subType whose base type is a comRingType and whose predicate's is a subringClosed

# UnitRing (Rings whose units have computable inverses):

x \is a GRing.unit <=> x is a unit (i.e., has an inverse) x^-1 == the ring inverse of x, if x is a unit, else x x / y == x divided by y (notation for x * y^-1) x ^- n := notation for (x ^+ n)^-1, the inverse of x ^+ n invr_closed S <-> collective predicate S is closed under inverse divr_closed S <-> collective predicate S is closed under division (1 and x / y in S) sdivr_closed S <-> collective predicate S is closed under division and opposite (-1 and x / y in S, for x, y in S) divring_closed S <-> collective predicate S is closed under unitRing operations (1, x - y and x / y in S) [SubRing_isSubUnitRing of R by <: ] == [SubChoice_isSubUnitRing of R by <: ] == unitRingType mixin for a subType whose base type is a unitRingType and whose predicate's is a divringClosed and whose ring structure is compatible with the base type's

# ComUnitRing (commutative rings with computable inverses):

[SubChoice_isSubComUnitRing of R by <: ] == comUnitRingType mixin for a subType whose base type is a comUnitRingType and whose predicate's is a divringClosed and whose ring structure is compatible with the base type's

# IntegralDomain (integral, commutative, ring with partial inverses):

[SubComUnitRing_isSubIntegralDomain R by <: ] == [SubChoice_isSubIntegralDomain R by <: ] == mixin axiom for a idomain subType

# Field (commutative fields):

GRing.Field.axiom inv == field axiom: x != 0 -> inv x * x = 1 for all x This is equivalent to the property above, but does not require a unitRingType as inv is an explicit argument. [SubIntegralDomain_isSubField of R by <: ] == mixin axiom for a field subType

# DecidableField (fields with a decidable first order theory):

GRing.term R == the type of formal expressions in a unit ring R with formal variables 'X_k, k : nat, and manifest constants x%:T, x : R The notation of all the ring operations is redefined for terms, in scope %T. GRing.formula R == the type of first order formulas over R; the %T scope binds the logical connectives /\, \/, ~, ==>, ==, and != to formulae; GRing.True/False and GRing.Bool b denote constant formulae, and quantifiers are written 'forall/'exists 'X_k, f GRing.Unit x tests for ring units GRing.If p_f t_f e_f emulates if-then-else GRing.Pick p_f t_f e_f emulates fintype.pick foldr GRing.Exists/Forall q_f xs can be used to write iterated quantifiers GRing.eval e t == the value of term t with valuation e : seq R (e maps 'X_i to e_i) GRing.same_env e1 e2 <-> environments e1 and e2 are extensionally equal GRing.qf_form f == f is quantifier-free GRing.holds e f == the intuitionistic CiC interpretation of the formula f holds with valuation e GRing.qf_eval e f == the value (in bool) of a quantifier-free f GRing.sat e f == valuation e satisfies f (only in a decField) GRing.sol n f == a sequence e of size n such that e satisfies f, if one exists, or [:: ] if there is no such e 'exists 'X_i, u1 == 0 /\ ... /\ u_m == 0 /\ v1 != 0 ... /\ v_n != 0

# Lmodule (module with left multiplication by external scalars).

a *: v == v scaled by a, when v is in an Lmodule V and a is in the scalar Ring of V scaler_closed S <-> collective predicate S is closed under scaling linear_closed S <-> collective predicate S is closed under linear combinations (a *: u + v in S when u, v in S) submod_closed S <-> collective predicate S is closed under lmodType operations (0 and a *: u + v in S) [SubZmodule_isSubLmodule of V by <: ] == [SubChoice_isSubLmodule of V by <: ] == mixin axiom for a subType of an lmodType

# Lalgebra (left algebra, ring with scaling that associates on the left):

R^o == the regular algebra of R: R^o is convertible to R, but when R has a ringType structure then R^o extends it to an lalgType structure by letting R act on itself: if x : R and y : R^o then x *: y = x * (y : R) k%:A == the image of the scalar k in an L-algebra; this is simply notation for k *: 1 subalg_closed S <-> collective predicate S is closed under lalgType operations (1, a *: u + v and u * v in S) [lalgMixin of V by <: ] == mixin axiom for a subType of an lalgType [SubRing_SubLmodule_isSubLalgebra of V by <: ] == [SubChoice_isSubLalgebra of V by <: ] == mixin axiom for a subType of an lalgType

# Algebra (ring with scaling that associates both left and right):

[SubLalgebra_isSubAlgebra of V by <: ] == [SubChoice_isSubAlgebra of V by <: ] == mixin axiom for a subType of an algType

# UnitAlgebra (algebra with computable inverses):

divalg_closed S <-> collective predicate S is closed under all unitAlgType operations (1, a *: u + v and u / v are in S fo u, v in S) In addition to this structure hierarchy, we also develop a separate, parallel hierarchy for morphisms linking these structures:

semi_additive f <-> f of type U -> V is semi additive, i.e., f maps the Nmodule structure of U to that of V, 0 to 0 and + to + := (f 0 = 0) * {morph f : x y / x + y} additive f <-> f of type U -> V is additive, i.e., f maps the Zmodule structure of U to that of V, 0 to 0,
• to - and + to + (equivalently, binary - to -)
:= {morph f : u v / u - v} {additive U -> V} == the interface type for a Structure (keyed on a function f : U -> V) that encapsulates the semi_additive property; both U and V must have canonical nmodType instances When both U and V have zmodType instances, it is an additive function.

# RMorphism (semiring or ring morphisms):

multiplicative f <-> f of type R -> S is multiplicative, i.e., f maps 1 and * in R to 1 and * in S, respectively R ans S must have canonical semiRingType instances {rmorphism R -> S} == the interface type for semiring morphisms; both R and S must have semiRingType instances When both R and S have ringType instances, it is a ring morphism.
• > If R and S are UnitRings the f also maps units to units and inverses of units to inverses; if R is a field then f is a field isomorphism between R and its image.
• > Additive properties (raddf_suffix, see below) are duplicated and specialised for RMorphism (as rmorph_suffix). This allows more precise rewriting and cleaner chaining: although raddf lemmas will recognize RMorphism functions, the converse will not hold (we cannot add reverse inheritance rules because of incomplete backtracking in the Canonical Projection unification), so one would have to insert a /= every time one switched from additive to multiplicative rules.

# Linear (linear functions):

scalable f <-> f of type U -> V is scalable, i.e., f morphs scaling on U to scaling on V, a *: _ to a *: _ U and V must both have lmodType R structures, for the same ringType R. scalable_for s f <-> f is scalable for scaling operator s, i.e., f morphs a *: _ to s a _; the range of f only need to be a zmodType The scaling operator s should be one of *:%R (see scalable, above), *%R or a combination nu \; *%R or nu \; *:%R with nu : {rmorphism _}; otherwise some of the theory (e.g., the linearZ rule) will not apply. linear f <-> f of type U -> V is linear, i.e., f morphs linear combinations a *: u + v in U to similar linear combinations in V; U and V must both have lmodType R structures, for the same ringType R := forall a, {morph f: u v / a *: u + v} scalar f <-> f of type U -> R is a scalar function, i.e., f (a *: u + v) = a * f u + f v linear_for s f <-> f is linear for the scaling operator s, i.e., f (a *: u + v) = s a (f u) + f v The range of f only needs to be a zmodType, but s MUST be of the form described in the scalable_for paragraph above for this predicate to type check. lmorphism f <-> f is both additive and scalable This is in fact equivalent to linear f, although somewhat less convenient to prove. lmorphism_for s f <-> f is both additive and scalable for s {linear U -> V} == the interface type for linear functions, i.e., a Structure that encapsulates the linear property for functions f : U -> V; both U and V must have lmodType R structures, for the same R {scalar U} == the interface type for scalar functions, of type U -> R where U has an lmodType R structure {linear U -> V | s} == the interface type for functions linear for s (a *: u)%Rlin == transient forms that simplify to a *: u, a * u, (a * u)%Rlin nu a *: u, and nu a * u, respectively, and are (a *:^nu u)%Rlin created by rewriting with the linearZ lemma (a *^nu u)%Rlin The forms allows the RHS of linearZ to be matched reliably, using the GRing.Scale.law structure.
• > Similarly to Ring morphisms, additive properties are specialized for linear functions.
• > Although {scalar U} is convertible to {linear U -> R^o}, it does not actually use R^o, so that rewriting preserves the canonical structure of the range of scalar functions.
• > The generic linearZ lemma uses a set of bespoke interface structures to ensure that both left-to-right and right-to-left rewriting work even in the presence of scaling functions that simplify non-trivially (e.g., idfun \; *%R). Because most of the canonical instances and projections are coercions the machinery will be mostly invisible (with only the {linear ...} structure and %Rlin notations showing), but users should beware that in (a *: f u)%Rlin, a actually occurs in the f u subterm.
• > The simpler linear_LR, or more specialized linearZZ and scalarZ rules should be used instead of linearZ if there are complexity issues, as well as for explicit forward and backward application, as the main parameter of linearZ is a proper sub-interface of {linear fUV | s}.

# LRMorphism (linear ring morphisms, i.e., algebra morphisms):

lrmorphism f <-> f of type A -> B is a linear Ring (Algebra) morphism: f is both additive, multiplicative and scalable; A and B must both have lalgType R canonical structures, for the same ringType R lrmorphism_for s f <-> f a linear Ring morphism for the scaling operator s: f is additive, multiplicative and scalable for s; A must be an lalgType R, but B only needs to have a ringType structure {lrmorphism A -> B} == the interface type for linear morphisms, i.e., a Structure that encapsulates the lrmorphism property for functions f : A -> B; both A and B must have lalgType R structures, for the same R {lrmorphism A -> B | s} == the interface type for morphisms linear for s
• > Linear and rmorphism properties do not need to be specialized for as we supply inheritance join instances in both directions.
Finally we supply some helper notation for morphisms: x^f == the image of x under some morphism This notation is only reserved (not defined) here; it is bound locally in sections where some morphism is used heavily (e.g., the container morphism in the parametricity sections of poly and matrix, or the Frobenius section here) \0 == the constant null function, which has a canonical linear structure, and simplifies on application (see ssrfun.v) f \+ g == the additive composition of f and g, i.e., the function x |-> f x + g x; f \+ g is canonically linear when f and g are, and simplifies on application (see ssrfun.v) f \- g == the function x |-> f x - g x, canonically linear when f and g are, and simplifies on application \- g == the function x |-> - f x, canonically linear when f is, and simplifies on application k \*: f == the function x |-> k *: f x, which is canonically linear when f is and simplifies on application (this is a shorter alternative to *:%R k \o f) GRing.in_alg A == the ring morphism that injects R into A, where A has an lalgType R structure; GRing.in_alg A k simplifies to k%:A a \*o f == the function x |-> a * f x, canonically linear when f is and its codomain is an algType and which simplifies on application a \o* f == the function x |-> f x * a, canonically linear when f is and its codomain is an lalgType and which simplifies on application f \* g == the function x |-> f x * g x; f \* g simplifies on application The Lemmas about these structures are contained in both the GRing module and in the submodule GRing.Theory, which can be imported when unqualified access to the theory is needed (GRing.Theory also allows the unqualified use of additive, linear, Linear, etc). The main GRing module should NOT be imported. Notations are defined in scope ring_scope (delimiter %R), except term and formula notations, which are in term_scope (delimiter %T). This library also extends the conventional suffixes described in library ssrbool.v with the following: 0 -- ring 0, as in addr0 : x + 0 = x 1 -- ring 1, as in mulr1 : x * 1 = x D -- ring addition, as in linearD : f (u + v) = f u + f v B -- ring subtraction, as in opprB : - (x - y) = y - x M -- ring multiplication, as in invfM : (x * y)^-1 = x^-1 * y^-1 Mn -- ring by nat multiplication, as in raddfMn : f (x *+ n) = f x *+ n N -- ring opposite, as in mulNr : (- x) * y = - (x * y) V -- ring inverse, as in mulVr : x^-1 * x = 1 X -- ring exponentiation, as in rmorphXn : f (x ^+ n) = f x ^+ n Z -- (left) module scaling, as in linearZ : f (a *: v) = s *: f v The operator suffixes D, B, M and X are also used for the corresponding operations on nat, as in natrX : (m ^ n)%:R = m%:R ^+ n. For the binary power operator, a trailing "n" suffix is used to indicate the operator suffix applies to the left-hand ring argument, as in expr1n : 1 ^+ n = 1 vs. expr1 : x ^+ 1 = x.

Set Implicit Arguments.

Declare Scope ring_scope.
Declare Scope term_scope.
Declare Scope linear_ring_scope.

Reserved Notation "+%R" (at level 0).
Reserved Notation "-%R" (at level 0).
Reserved Notation "*%R" (at level 0, format " *%R").
Reserved Notation "*:%R" (at level 0, format " *:%R").
Reserved Notation "n %:R" (at level 2, left associativity, format "n %:R").
Reserved Notation "k %:A" (at level 2, left associativity, format "k %:A").
Reserved Notation "[ 'char' F ]" (at level 0, format "[ 'char' F ]").

Reserved Notation "x %:T" (at level 2, left associativity, format "x %:T").
Reserved Notation "''X_' i" (at level 8, i at level 2, format "''X_' i").
Patch for recurring Coq parser bug: Coq seg faults when a level 200 notation is used as a pattern.
Reserved Notation "''exists' ''X_' i , f"
(at level 199, i at level 2, right associativity,
format "'[hv' ''exists' ''X_' i , '/ ' f ']'").
Reserved Notation "''forall' ''X_' i , f"
(at level 199, i at level 2, right associativity,
format "'[hv' ''forall' ''X_' i , '/ ' f ']'").

Reserved Notation "x ^f" (at level 2, left associativity, format "x ^f").

Reserved Notation "\0" (at level 0).
Reserved Notation "f \+ g" (at level 50, left associativity).
Reserved Notation "f \- g" (at level 50, left associativity).
Reserved Notation "\- f" (at level 35, f at level 35).
Reserved Notation "a \*o f" (at level 40).
Reserved Notation "a \o* f" (at level 40).
Reserved Notation "a \*: f" (at level 40).
Reserved Notation "f \* g" (at level 40, left associativity).

Reserved Notation "'{' 'additive' U '->' V '}'"
(at level 0, U at level 98, V at level 99,
format "{ 'additive' U -> V }").
Reserved Notation "'{' 'rmorphism' U '->' V '}'"
(at level 0, U at level 98, V at level 99,
format "{ 'rmorphism' U -> V }").
Reserved Notation "'{' 'lrmorphism' U '->' V '|' s '}'"
(at level 0, U at level 98, V at level 99,
format "{ 'lrmorphism' U -> V | s }").
Reserved Notation "'{' 'lrmorphism' U '->' V '}'"
(at level 0, U at level 98, V at level 99,
format "{ 'lrmorphism' U -> V }").
Reserved Notation "'{' 'linear' U '->' V '|' s '}'"
(at level 0, U at level 98, V at level 99,
format "{ 'linear' U -> V | s }").
Reserved Notation "'{' 'linear' U '->' V '}'"
(at level 0, U at level 98, V at level 99,
format "{ 'linear' U -> V }").

Declare Scope ring_scope.
Delimit Scope ring_scope with R.
Declare Scope term_scope.
Delimit Scope term_scope with T.
Local Open Scope ring_scope.

Module Import GRing.

Import Monoid.Theory.

#[short(type="nmodType")]
HB.structure Definition Nmodule := {V of isNmodule V & Choice V}.

Module NmodExports.
Bind Scope ring_scope with Nmodule.sort.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'nmodType' 'of' T 'for' cT ]" := (Nmodule.clone T cT)
(at level 0, format "[ 'nmodType' 'of' T 'for' cT ]") : form_scope.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'nmodType' 'of' T ]" := (Nmodule.clone T _)
(at level 0, format "[ 'nmodType' 'of' T ]") : form_scope.
End NmodExports.

Local Notation "0" := (@zero _) : ring_scope.
Local Notation "+%R" := (@add _) : fun_scope.
Local Notation "x + y" := (add x y) : ring_scope.

Definition natmul V x n := nosimpl iterop _ n +%R x (@zero V).

Local Notation "x *+ n" := (natmul x n) : ring_scope.

Local Notation "\sum_ ( i <- r | P ) F" := (\big[+%R/0]_(i <- r | P) F).
Local Notation "\sum_ ( m <= i < n ) F" := (\big[+%R/0]_(m i < n) F).
Local Notation "\sum_ ( i < n ) F" := (\big[+%R/0]_(i < n) F).
Local Notation "\sum_ ( i 'in' A ) F" := (\big[+%R/0]_(i in A) F).

Local Notation "s _ i" := (nth 0 s i) : ring_scope.

Section NmoduleTheory.

Variable V : nmodType.
Implicit Types x y : V.

Lemma addr0 : @right_id V V 0 +%R.

#[export]

Lemma addrCA : @left_commutative V V +%R.
Lemma addrAC : @right_commutative V V +%R.
Lemma addrACA : @interchange V +%R +%R.

Lemma mulr0n x : x *+ 0 = 0.
Lemma mulr1n x : x *+ 1 = x.
Lemma mulr2n x : x *+ 2 = x + x.

Lemma mulrS x n : x *+ n.+1 = x + x *+ n.

Lemma mulrSr x n : x *+ n.+1 = x *+ n + x.

Lemma mulrb x (b : bool) : x *+ b = (if b then x else 0).

Lemma mul0rn n : 0 *+ n = 0 :> V.

Lemma mulrnDl n : {morph (fun xx *+ n) : x y / x + y}.

Lemma mulrnDr x m n : x *+ (m + n) = x *+ m + x *+ n.

Lemma mulrnA x m n : x *+ (m × n) = x *+ m *+ n.

Lemma mulrnAC x m n : x *+ m *+ n = x *+ n *+ m.

Lemma iter_addr n x y : iter n (+%R x) y = x *+ n + y.

Lemma iter_addr_0 n x : iter n (+%R x) 0 = x *+ n.

Lemma sumrMnl I r P (F : I V) n :
\sum_(i <- r | P i) F i *+ n = (\sum_(i <- r | P i) F i) *+ n.

Lemma sumrMnr x I r P (F : I nat) :
\sum_(i <- r | P i) x *+ F i = x *+ (\sum_(i <- r | P i) F i).

Lemma sumr_const (I : finType) (A : pred I) x : \sum_(i in A) x = x *+ #|A|.

Lemma sumr_const_nat m n x : \sum_(n i < m) x = x *+ (m - n).

Section ClosedPredicates.

Variable S : {pred V}.

Definition addr_closed := 0 \in S {in S &, u v, u + v \in S}.

End ClosedPredicates.

End NmoduleTheory.

#[short(type="zmodType")]
HB.structure Definition Zmodule := {V of Nmodule_isZmodule V & Nmodule V}.

Module ZmodExports.
Bind Scope ring_scope with Zmodule.sort.
Notation ZmodMixin V := (isZmodule.Build V).
Notation "[ 'zmodType' 'of' T 'for' cT ]" := (Zmodule.clone T cT)
(at level 0, format "[ 'zmodType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'zmodType' 'of' T ]" := (Zmodule.clone T _)
(at level 0, format "[ 'zmodType' 'of' T ]") : form_scope.
End ZmodExports.

Local Notation "-%R" := (@opp _) : ring_scope.
Local Notation "- x" := (opp x) : ring_scope.
Local Notation "x - y" := (x + - y) : ring_scope.

Local Notation "x *- n" := (- (x *+ n)) : ring_scope.

Section ZmoduleTheory.

Variable V : zmodType.
Implicit Types x y : V.

Lemma addrN : @right_inverse V V V 0 -%R +%R.

Lemma addKr : @left_loop V V -%R +%R.
Lemma addNKr : @rev_left_loop V V -%R +%R.
Lemma addrK : @right_loop V V -%R +%R.
Lemma addrNK : @rev_right_loop V V -%R +%R.
Lemma subKr x : involutive (fun yx - y).
Lemma addrI : @right_injective V V V +%R.
Lemma addIr : @left_injective V V V +%R.
Lemma subrI : right_injective (fun x yx - y).
Lemma subIr : left_injective (fun x yx - y).
Lemma opprK : @involutive V -%R.
Lemma oppr_inj : @injective V V -%R.
Lemma oppr0 : -0 = 0 :> V.
Lemma oppr_eq0 x : (- x == 0) = (x == 0).

Lemma subr0 x : x - 0 = x.
Lemma sub0r x : 0 - x = - x.

Lemma opprB x y : - (x - y) = y - x.

Lemma opprD : {morph -%R: x y / x + y : V}.

Lemma addrKA z x y : (x + z) - (z + y) = x - y.

Lemma subrKA z x y : (x - z) + (z + y) = x + y.

Lemma addr0_eq x y : x + y = 0 - x = y.

Lemma subr0_eq x y : x - y = 0 x = y.

Lemma subr_eq x y z : (x - z == y) = (x == y + z).

Lemma subr_eq0 x y : (x - y == 0) = (x == y).

Lemma addr_eq0 x y : (x + y == 0) = (x == - y).

Lemma eqr_opp x y : (- x == - y) = (x == y).

Lemma eqr_oppLR x y : (- x == y) = (x == - y).

Lemma mulNrn x n : (- x) *+ n = x *- n.

Lemma mulrnBl n : {morph (fun xx *+ n) : x y / x - y}.

Lemma mulrnBr x m n : n m x *+ (m - n) = x *+ m - x *+ n.

Lemma sumrN I r P (F : I V) :
(\sum_(i <- r | P i) - F i = - (\sum_(i <- r | P i) F i)).

Lemma sumrB I r (P : pred I) (F1 F2 : I V) :
\sum_(i <- r | P i) (F1 i - F2 i)
= \sum_(i <- r | P i) F1 i - \sum_(i <- r | P i) F2 i.

Lemma telescope_sumr n m (f : nat V) : n m
\sum_(n k < m) (f k.+1 - f k) = f m - f n.

Lemma telescope_sumr_eq n m (f u : nat V) : n m
( k, (n k < m)%N u k = f k.+1 - f k)
\sum_(n k < m) u k = f m - f n.

Section ClosedPredicates.

Variable S : {pred V}.

Definition oppr_closed := {in S, u, - u \in S}.
Definition subr_2closed := {in S &, u v, u - v \in S}.
Definition zmod_closed := 0 \in S subr_2closed.

Lemma zmod_closedN : zmod_closed oppr_closed.

Lemma zmod_closedD : zmod_closed addr_closed S.

End ClosedPredicates.

End ZmoduleTheory.

Arguments addrI {V} y [x1 x2].
Arguments addIr {V} x [x1 x2].
Arguments opprK {V}.
Arguments oppr_inj {V} [x1 x2].
Arguments telescope_sumr_eq {V n m} f u.

#[short(type="semiRingType")]
HB.structure Definition SemiRing := { R of Nmodule_isSemiRing R & Nmodule R }.

Module SemiRingExports.
Bind Scope ring_scope with SemiRing.sort.
Notation "[ 'semiRingType' 'of' T 'for' cT ]" := (SemiRing.clone T cT)
(at level 0, format "[ 'semiRingType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'semiRingType' 'of' T ]" := (SemiRing.clone T _)
(at level 0, format "[ 'semiRingType' 'of' T ]") : form_scope.
End SemiRingExports.

Definition exp R x n := nosimpl iterop _ n (@mul R) x (@one R).
Definition comm R x y := @mul R x y = mul y x.
Definition lreg R x := injective (@mul R x).
Definition rreg R x := injective ((@mul R)^~ x).

Local Notation "1" := (@one _) : ring_scope.
Local Notation "n %:R" := (1 *+ n) : ring_scope.
Local Notation "*%R" := (@mul _) : fun_scope.
Local Notation "x * y" := (mul x y) : ring_scope.
Local Notation "x ^+ n" := (exp x n) : ring_scope.

Local Notation "\prod_ ( i <- r | P ) F" := (\big[*%R/1]_(i <- r | P) F).
Local Notation "\prod_ ( i | P ) F" := (\big[*%R/1]_(i | P) F).
Local Notation "\prod_ ( i 'in' A ) F" := (\big[*%R/1]_(i in A) F).
Local Notation "\prod_ ( m <= i < n ) F" := (\big[*%R/1%R]_(m i < n) F%R).

The `field'' characteristic; the definition, and many of the theorems, has to apply to rings as well; indeed, we need the Frobenius automorphism results for a non commutative ring in the proof of Gorenstein 2.6.3.
Definition char (R : semiRingType) of phant R : nat_pred :=
[pred p | prime p & p%:R == 0 :> R].

Local Notation "[ 'char' R ]" := (char (Phant R)) : ring_scope.

Converse ring tag.
Definition converse R : Type := R.
Local Notation "R ^c" := (converse R) (at level 2, format "R ^c") : type_scope.

Section SemiRingTheory.

Variable R : semiRingType.
Implicit Types x y : R.

Lemma oner_eq0 : (1 == 0 :> R) = false.

#[export]
HB.instance Definition _ := Monoid.isLaw.Build R 1 *%R mulrA mul1r mulr1.
#[export]
HB.instance Definition _ := Monoid.isMulLaw.Build R 0 *%R mul0r mulr0.
#[export]
HB.instance Definition _ := Monoid.isAddLaw.Build R *%R +%R mulrDl mulrDr.

Lemma mulr_suml I r P (F : I R) x :
(\sum_(i <- r | P i) F i) × x = \sum_(i <- r | P i) F i × x.

Lemma mulr_sumr I r P (F : I R) x :
x × (\sum_(i <- r | P i) F i) = \sum_(i <- r | P i) x × F i.

Lemma mulrnAl x y n : (x *+ n) × y = (x × y) *+ n.

Lemma mulrnAr x y n : x × (y *+ n) = (x × y) *+ n.

Lemma mulr_natl x n : n%:R × x = x *+ n.

Lemma mulr_natr x n : x × n%:R = x *+ n.

Lemma natrD m n : (m + n)%:R = m%:R + n%:R :> R.

Lemma natr1 n : n%:R + 1 = n.+1%:R :> R.

Lemma nat1r n : 1 + n%:R = n.+1%:R :> R.

Definition natr_sum := big_morph (natmul 1) natrD (mulr0n 1).

Lemma natrM m n : (m × n)%:R = m%:R × n%:R :> R.

Lemma expr0 x : x ^+ 0 = 1.
Lemma expr1 x : x ^+ 1 = x.
Lemma expr2 x : x ^+ 2 = x × x.

Lemma exprS x n : x ^+ n.+1 = x × x ^+ n.

Lemma expr0n n : 0 ^+ n = (n == 0%N)%:R :> R.

Lemma expr1n n : 1 ^+ n = 1 :> R.

Lemma exprD x m n : x ^+ (m + n) = x ^+ m × x ^+ n.

Lemma exprSr x n : x ^+ n.+1 = x ^+ n × x.

Lemma expr_sum x (I : Type) (s : seq I) (P : pred I) F :
x ^+ (\sum_(i <- s | P i) F i) = \prod_(i <- s | P i) x ^+ F i :> R.

Lemma commr_sym x y : comm x y comm y x.
Lemma commr_refl x : comm x x.

Lemma commr0 x : comm x 0.

Lemma commr1 x : comm x 1.

Lemma commrD x y z : comm x y comm x z comm x (y + z).

Lemma commr_sum (I : Type) (s : seq I) (P : pred I) (F : I R) x :
( i, P i comm x (F i)) comm x (\sum_(i <- s | P i) F i).

Lemma commrMn x y n : comm x y comm x (y *+ n).

Lemma commrM x y z : comm x y comm x z comm x (y × z).

Lemma commr_prod (I : Type) (s : seq I) (P : pred I) (F : I R) x :
( i, P i comm x (F i)) comm x (\prod_(i <- s | P i) F i).

Lemma commr_nat x n : comm x n%:R.

Lemma commrX x y n : comm x y comm x (y ^+ n).

Lemma exprMn_comm x y n : comm x y (x × y) ^+ n = x ^+ n × y ^+ n.

Lemma exprMn_n x m n : (x *+ m) ^+ n = x ^+ n *+ (m ^ n) :> R.

Lemma exprM x m n : x ^+ (m × n) = x ^+ m ^+ n.

Lemma exprAC x m n : (x ^+ m) ^+ n = (x ^+ n) ^+ m.

Lemma expr_mod n x i : x ^+ n = 1 x ^+ (i %% n) = x ^+ i.

Lemma expr_dvd n x i : x ^+ n = 1 n %| i x ^+ i = 1.

Lemma natrX n k : (n ^ k)%:R = n%:R ^+ k :> R.

Lemma lastr_eq0 (s : seq R) x : x != 0 (last x s == 0) = (last 1 s == 0).

Lemma mulrI_eq0 x y : lreg x (x × y == 0) = (y == 0).

Lemma lreg_neq0 x : lreg x x != 0.

Lemma lreg1 : lreg (1 : R).

Lemma lregM x y : lreg x lreg y lreg (x × y).

Lemma lregMl (a b: R) : lreg (a × b) lreg b.

Lemma rregMr (a b: R) : rreg (a × b) rreg a.

Lemma lregX x n : lreg x lreg (x ^+ n).

Lemma iter_mulr n x y : iter n ( *%R x) y = x ^+ n × y.

Lemma iter_mulr_1 n x : iter n ( *%R x) 1 = x ^+ n.

Lemma prodr_const (I : finType) (A : pred I) x : \prod_(i in A) x = x ^+ #|A|.

Lemma prodr_const_nat n m x : \prod_(n i < m) x = x ^+ (m - n).

Lemma prodrXr x I r P (F : I nat) :
\prod_(i <- r | P i) x ^+ F i = x ^+ (\sum_(i <- r | P i) F i).

Lemma prodrMn (I : Type) (s : seq I) (P : pred I) (F : I R) (g : I nat) :
\prod_(i <- s | P i) (F i *+ g i) =
\prod_(i <- s | P i) (F i) *+ \prod_(i <- s | P i) g i.

Lemma prodrMn_const n (I : finType) (A : pred I) (F : I R) :
\prod_(i in A) (F i *+ n) = \prod_(i in A) F i *+ n ^ #|A|.

Lemma natr_prod I r P (F : I nat) :
(\prod_(i <- r | P i) F i)%:R = \prod_(i <- r | P i) (F i)%:R :> R.

Lemma exprDn_comm x y n (cxy : comm x y) :
(x + y) ^+ n = \sum_(i < n.+1) (x ^+ (n - i) × y ^+ i) *+ 'C(n, i).

Lemma exprD1n x n : (x + 1) ^+ n = \sum_(i < n.+1) x ^+ i *+ 'C(n, i).

Lemma sqrrD1 x : (x + 1) ^+ 2 = x ^+ 2 + x *+ 2 + 1.

Definition Frobenius_aut p of p \in [char R] := fun xx ^+ p.

Section FrobeniusAutomorphism.

Variable p : nat.
Hypothesis charFp : p \in [char R].

Lemma charf0 : p%:R = 0 :> R.
Lemma charf_prime : prime p.
Hint Resolve charf_prime : core.

Lemma mulrn_char x : x *+ p = 0.

Lemma natr_mod_char n : (n %% p)%:R = n%:R :> R.

Lemma dvdn_charf n : (p %| n)%N = (n%:R == 0 :> R).

Lemma charf_eq : [char R] =i (p : nat_pred).

Lemma bin_lt_charf_0 k : 0 < k < p 'C(p, k)%:R = 0 :> R.

Local Notation "x ^f" := (Frobenius_aut charFp x).

Lemma Frobenius_autE x : x^f = x ^+ p.
Local Notation fE := Frobenius_autE.

Lemma Frobenius_aut0 : 0^f = 0.

Lemma Frobenius_aut1 : 1^f = 1.

Lemma Frobenius_autD_comm x y (cxy : comm x y) : (x + y)^f = x^f + y^f.

Lemma Frobenius_autMn x n : (x *+ n)^f = x^f *+ n.

Lemma Frobenius_aut_nat n : (n%:R)^f = n%:R.

Lemma Frobenius_autM_comm x y : comm x y (x × y)^f = x^f × y^f.

Lemma Frobenius_autX x n : (x ^+ n)^f = x^f ^+ n.

End FrobeniusAutomorphism.

Section Char2.

Hypothesis charR2 : 2 \in [char R].

Lemma addrr_char2 x : x + x = 0.

End Char2.

Section ClosedPredicates.

Variable S : {pred R}.

Definition mulr_2closed := {in S &, u v, u × v \in S}.
Definition mulr_closed := 1 \in S mulr_2closed.
Definition semiring_closed := addr_closed S mulr_closed.

Lemma semiring_closedD : semiring_closed addr_closed S.

Lemma semiring_closedM : semiring_closed mulr_closed.

End ClosedPredicates.

End SemiRingTheory.

#[short(type="ringType")]
HB.structure Definition Ring := { R of SemiRing R & Zmodule R }.

Local Notation "1" := one.
Local Notation "x * y" := (mul x y).
Lemma mul0r : @left_zero R R 0 mul.
Lemma mulr0 : @right_zero R R 0 mul.

Module RingExports.
Bind Scope ring_scope with Ring.sort.
Notation "[ 'ringType' 'of' T 'for' cT ]" := (Ring.clone T cT)
(at level 0, format "[ 'ringType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'ringType' 'of' T ]" := (Ring.clone T _)
(at level 0, format "[ 'ringType' 'of' T ]") : form_scope.
End RingExports.

Notation sign R b := (exp (- @one R) (nat_of_bool b)) (only parsing).

Local Notation "- 1" := (- (1)) : ring_scope.

Section RingTheory.

Variable R : ringType.
Implicit Types x y : R.

Lemma mulrN x y : x × (- y) = - (x × y).
Lemma mulNr x y : (- x) × y = - (x × y).
Lemma mulrNN x y : (- x) × (- y) = x × y.
Lemma mulN1r x : -1 × x = - x.
Lemma mulrN1 x : x × -1 = - x.

Lemma mulrBl x y z : (y - z) × x = y × x - z × x.

Lemma mulrBr x y z : x × (y - z) = x × y - x × z.

Lemma natrB m n : n m (m - n)%:R = m%:R - n%:R :> R.

Lemma commrN x y : comm x y comm x (- y).

Lemma commrN1 x : comm x (-1).

Lemma commrB x y z : comm x y comm x z comm x (y - z).

Lemma commr_sign x n : comm x ((-1) ^+ n).

Lemma signr_odd n : (-1) ^+ (odd n) = (-1) ^+ n :> R.

Lemma signr_eq0 n : ((-1) ^+ n == 0 :> R) = false.

Lemma mulr_sign (b : bool) x : (-1) ^+ b × x = (if b then - x else x).

Lemma signr_addb b1 b2 : (-1) ^+ (b1 (+) b2) = (-1) ^+ b1 × (-1) ^+ b2 :> R.

Lemma signrE (b : bool) : (-1) ^+ b = 1 - b.*2%:R :> R.

Lemma signrN b : (-1) ^+ (~~ b) = - (-1) ^+ b :> R.

Lemma mulr_signM (b1 b2 : bool) x1 x2 :
((-1) ^+ b1 × x1) × ((-1) ^+ b2 × x2) = (-1) ^+ (b1 (+) b2) × (x1 × x2).

Lemma exprNn x n : (- x) ^+ n = (-1) ^+ n × x ^+ n :> R.

Lemma sqrrN x : (- x) ^+ 2 = x ^+ 2.

Lemma sqrr_sign n : ((-1) ^+ n) ^+ 2 = 1 :> R.

Lemma signrMK n : @involutive R ( *%R ((-1) ^+ n)).

Lemma mulrI0_lreg x : ( y, x × y = 0 y = 0) lreg x.

Lemma lregN x : lreg x lreg (- x).

Lemma lreg_sign n : lreg ((-1) ^+ n : R).

Lemma prodrN (I : finType) (A : pred I) (F : I R) :
\prod_(i in A) - F i = (- 1) ^+ #|A| × \prod_(i in A) F i.

Lemma exprBn_comm x y n (cxy : comm x y) :
(x - y) ^+ n =
\sum_(i < n.+1) ((-1) ^+ i × x ^+ (n - i) × y ^+ i) *+ 'C(n, i).

Lemma subrXX_comm x y n (cxy : comm x y) :
x ^+ n - y ^+ n = (x - y) × (\sum_(i < n) x ^+ (n.-1 - i) × y ^+ i).

Lemma subrX1 x n : x ^+ n - 1 = (x - 1) × (\sum_(i < n) x ^+ i).

Lemma sqrrB1 x : (x - 1) ^+ 2 = x ^+ 2 - x *+ 2 + 1.

Lemma subr_sqr_1 x : x ^+ 2 - 1 = (x - 1) × (x + 1).

Section FrobeniusAutomorphism.

Variable p : nat.
Hypothesis charFp : p \in [char R].

Hint Resolve charf_prime : core.

Local Notation "x ^f" := (Frobenius_aut charFp x).

Lemma Frobenius_autN x : (- x)^f = - x^f.

Lemma Frobenius_autB_comm x y : comm x y (x - y)^f = x^f - y^f.

End FrobeniusAutomorphism.

Lemma exprNn_char x n : [char R].-nat n (- x) ^+ n = - (x ^+ n).

Section Char2.

Hypothesis charR2 : 2 \in [char R].

Lemma oppr_char2 x : - x = x.

Lemma subr_char2 x y : x - y = x + y.

Lemma addrK_char2 x : involutive (+%R^~ x).

Lemma addKr_char2 x : involutive (+%R x).

End Char2.

Section ClosedPredicates.

Variable S : {pred R}.

Definition smulr_closed := -1 \in S mulr_2closed S.
Definition subring_closed := [/\ 1 \in S, subr_2closed S & mulr_2closed S].

Lemma smulr_closedM : smulr_closed mulr_closed S.

Lemma smulr_closedN : smulr_closed oppr_closed S.

Lemma subring_closedB : subring_closed zmod_closed S.

Lemma subring_closedM : subring_closed smulr_closed.

Lemma subring_closed_semi : subring_closed semiring_closed S.

End ClosedPredicates.

End RingTheory.

Module ConverseRingExports.

End ConverseRingExports.

Section SemiRightRegular.

Variable R : semiRingType.
Implicit Types x y : R.

Lemma mulIr_eq0 x y : rreg x (y × x == 0) = (y == 0).

Lemma rreg_neq0 x : rreg x x != 0.

Lemma rreg1 : rreg (1 : R).

Lemma rregM x y : rreg x rreg y rreg (x × y).

Lemma revrX x n : (x : R^c) ^+ n = (x : R) ^+ n.

Lemma rregX x n : rreg x rreg (x ^+ n).

End SemiRightRegular.

Section RightRegular.

Variable R : ringType.
Implicit Types x y : R.

Lemma mulIr0_rreg x : ( y, y × x = 0 y = 0) rreg x.

Lemma rregN x : rreg x rreg (- x).

End RightRegular.

#[short(type="lmodType")]
HB.structure Definition Lmodule (R : ringType) :=
{M of Zmodule M & Zmodule_isLmodule R M}.

Module LmodExports.
Bind Scope ring_scope with Lmodule.sort.
Notation "[ 'lmodType' R 'of' T 'for' cT ]" := (Lmodule.clone R T%type cT)
(at level 0, format "[ 'lmodType' R 'of' T 'for' cT ]") : form_scope.
Notation "[ 'lmodType' R 'of' T ]" := (Lmodule.clone R T%type _)
(at level 0, format "[ 'lmodType' R 'of' T ]") : form_scope.
End LmodExports.

Local Notation "*:%R" := (@scale _ _) : fun_scope.
Local Notation "a *: v" := (scale a v) : ring_scope.

Section LmoduleTheory.

Variables (R : ringType) (V : lmodType R).
Implicit Types (a b c : R) (u v : V).

TODO: fix the type for the exported operation
Lemma scalerA a b v : scale a (scale b v) = scale (a × b) v.

Lemma scale0r v : 0 *: v = 0.

Lemma scaler0 a : a *: 0 = 0 :> V.

Lemma scaleNr a v : - a *: v = - (a *: v).

Lemma scaleN1r v : (- 1) *: v = - v.

Lemma scalerN a v : a *: (- v) = - (a *: v).

Lemma scalerBl a b v : (a - b) *: v = a *: v - b *: v.

Lemma scalerBr a u v : a *: (u - v) = a *: u - a *: v.

Lemma scaler_nat n v : n%:R *: v = v *+ n.

Lemma scaler_sign (b : bool) v: (-1) ^+ b *: v = (if b then - v else v).

Lemma signrZK n : @involutive V ( *:%R ((-1) ^+ n)).

Lemma scalerMnl a v n : a *: v *+ n = (a *+ n) *: v.

Lemma scalerMnr a v n : a *: v *+ n = a *: (v *+ n).

Lemma scaler_suml v I r (P : pred I) F :
(\sum_(i <- r | P i) F i) *: v = \sum_(i <- r | P i) F i *: v.

Lemma scaler_sumr a I r (P : pred I) (F : I V) :
a *: (\sum_(i <- r | P i) F i) = \sum_(i <- r | P i) a *: F i.

Section ClosedPredicates.

Variable S : {pred V}.

Definition scaler_closed := a, {in S, v, a *: v \in S}.
Definition linear_closed := a, {in S &, u v, a *: u + v \in S}.
Definition submod_closed := 0 \in S linear_closed.

Lemma linear_closedB : linear_closed subr_2closed S.

Lemma submod_closedB : submod_closed zmod_closed S.

Lemma submod_closedZ : submod_closed scaler_closed.

End ClosedPredicates.

End LmoduleTheory.

#[short(type="lalgType")]
HB.structure Definition Lalgebra R :=
{A of Lmodule_isLalgebra R A & Ring A & Lmodule R A}.

Module LalgExports.
Bind Scope ring_scope with Lalgebra.sort.
Notation "[ 'lalgType' R 'of' T 'for' cT ]" := (Lalgebra.clone R T%type cT)
(at level 0, format "[ 'lalgType' R 'of' T 'for' cT ]") : form_scope.
Notation "[ 'lalgType' R 'of' T ]" := (Lalgebra.clone R T%type _)
(at level 0, format "[ 'lalgType' R 'of' T ]") : form_scope.
End LalgExports.

Scalar injection (see the definition of in_alg A below).
Local Notation "k %:A" := (k *: 1) : ring_scope.

Regular ring algebra tag.
Definition regular R : Type := R.
Local Notation "R ^o" := (regular R) (at level 2, format "R ^o") : type_scope.

Module RegularLalgExports.
Section LalgebraTheory.

Variables (R : ringType) (A : lalgType R).
Implicit Types x y : A.

End LalgebraTheory.
End RegularLalgExports.

Section LalgebraTheory.

Variables (R : ringType) (A : lalgType R).
Implicit Types x y : A.

Lemma mulr_algl a x : (a *: 1) × x = a *: x.

Section ClosedPredicates.

Variable S : {pred A}.

Definition subalg_closed := [/\ 1 \in S, linear_closed S & mulr_2closed S].

Lemma subalg_closedZ : subalg_closed submod_closed S.

Lemma subalg_closedBM : subalg_closed subring_closed S.

End ClosedPredicates.

End LalgebraTheory.

Morphism hierarchy.

Definition semi_additive (U V : nmodType) (f : U V) : Prop :=
(f 0 = 0) × {morph f : x y / x + y}.

HB.structure Definition Additive (U V : nmodType) :=
{f of isSemiAdditive U V f}.

Definition additive (U V : zmodType) (f : U V) := {morph f : x y / x - y}.

Local Lemma raddf0 : apply 0 = 0.

Local Lemma raddfD : {morph apply : x y / x + y}.

Definition apply_deprecated (U V : nmodType) (phUV : phant (U V)) :=
Notation apply := apply_deprecated.
Notation "{ 'additive' U -> V }" := (Additive.type U%type V%type) : type_scope.
Notation "[ 'additive' 'of' f 'as' g ]" := (Additive.clone _ _ f%function g)
(at level 0, format "[ 'additive' 'of' f 'as' g ]") : form_scope.
Notation "[ 'additive' 'of' f ]" := (Additive.clone _ _ f%function _)
(at level 0, format "[ 'additive' 'of' f ]") : form_scope.

Section LiftedNmod.
Variables (U : Type) (V : nmodType).
Definition null_fun_head (phV : phant V) of U : V := let: Phant := phV in 0.
Definition add_fun (f g : U V) x := f x + g x.
End LiftedNmod.
Section LiftedZmod.
Variables (U : Type) (V : zmodType).
Definition sub_fun (f g : U V) x := f x - g x.
Definition opp_fun (f : U V) x := - f x.
End LiftedZmod.

Lifted multiplication.
Section LiftedSemiRing.
Variables (R : semiRingType) (T : Type).
Implicit Type f : T R.
Definition mull_fun a f x := a × f x.
Definition mulr_fun a f x := f x × a.
Definition mul_fun f g x := f x × g x.
End LiftedSemiRing.

Lifted linear operations.
Section LiftedScale.
Variables (R : ringType) (U : Type) (V : lmodType R) (A : lalgType R).
Definition scale_fun a (f : U V) x := a *: f x.
Definition in_alg_head (phA : phant A) k : A := let: Phant := phA in k%:A.
End LiftedScale.

Notation null_fun V := (null_fun_head (Phant V)) (only parsing).
The real in_alg notation is declared after GRing.Theory so that at least in Coq 8.2 it gets precedence when GRing.Theory is not imported.
Local Notation in_alg_loc A := (in_alg_head (Phant A)) (only parsing).

Local Notation "\0" := (null_fun _) : ring_scope.
Local Notation "f \+ g" := (add_fun f g) : ring_scope.
Local Notation "f \- g" := (sub_fun f g) : ring_scope.
Local Notation "\- f" := (opp_fun f) : ring_scope.
Local Notation "a \*: f" := (scale_fun a f) : ring_scope.
Local Notation "x \*o f" := (mull_fun x f) : ring_scope.
Local Notation "x \o* f" := (mulr_fun x f) : ring_scope.
Local Notation "f \* g" := (mul_fun f g) : ring_scope.

Arguments add_fun {_ _} f g _ /.
Arguments sub_fun {_ _} f g _ /.
Arguments opp_fun {_ _} f _ /.
Arguments mull_fun {_ _} a f _ /.
Arguments mulr_fun {_ _} a f _ /.
Arguments scale_fun {_ _ _} a f _ /.
Arguments mul_fun {_ _} f g _ /.

Section Properties.

Variables (U V : nmodType) (k : unit) (f : {additive U V}).

Lemma raddf0 : f 0 = 0.

Lemma raddf_eq0 x : injective f (f x == 0) = (x == 0).

Lemma raddfD : {morph f : x y / x + y}.

Lemma raddfMn n : {morph f : x / x *+ n}.

Lemma raddf_sum I r (P : pred I) E :
f (\sum_(i <- r | P i) E i) = \sum_(i <- r | P i) f (E i).

Lemma can2_semi_additive f' : cancel f f' cancel f' f semi_additive f'.

End Properties.

Section SemiRingProperties.

Variables (R S : semiRingType) (f : {additive R S}).

Lemma raddfMnat n x : f (n%:R × x) = n%:R × f x.

End SemiRingProperties.

Variables (U V W : nmodType).
Variables (f g : {additive V W}) (h : {additive U V}).

#[export]
HB.instance Definition _ := isSemiAdditive.Build U U idfun

#[export]
HB.instance Definition _ := isSemiAdditive.Build U W (f \o h)

#[export]
HB.instance Definition _ := isSemiAdditive.Build U V \0

#[export]
HB.instance Definition _ := isSemiAdditive.Build V W (f \+ g)

Section MulFun.

Variables (R : semiRingType) (U : nmodType) (a : R) (f : {additive U R}).

#[export]
HB.instance Definition _ := isSemiAdditive.Build U R (a \*o f)

#[export]
HB.instance Definition _ := isSemiAdditive.Build U R (a \o× f)

End MulFun.

Section Properties.

Variables (U V : zmodType) (k : unit) (f : {additive U V}).

Lemma raddfN : {morph f : x / - x}.

Lemma raddfB : {morph f : x y / x - y}.

Lemma raddf_inj : ( x, f x = 0 x = 0) injective f.

Lemma raddfMNn n : {morph f : x / x *- n}.

Lemma can2_additive f' : cancel f f' cancel f' f additive f'.

End Properties.

Section RingProperties.

Variables (R S : ringType) (f : {additive R S}).

Lemma raddfMsign n x : f ((-1) ^+ n × x) = (-1) ^+ n × f x.

Variables (U : lmodType R) (V : lmodType S) (h : {additive U V}).

Lemma raddfZnat n u : h (n%:R *: u) = n%:R *: h u.

Lemma raddfZsign n u : h ((-1) ^+ n *: u) = (-1) ^+ n *: h u.

End RingProperties.

Variables (U V W : zmodType) (f g : {additive V W}) (h : {additive U V}).

#[export]

#[export]

#[export]

Section ScaleFun.

Variables (R : ringType) (U : zmodType) (V : lmodType R).
Variables (a : R) (f : {additive U V}).

#[export]
HB.instance Definition _ := isAdditive.Build V V ( *:%R a) (@scalerBr R V a).
#[export]
HB.instance Definition _ := Additive.copy (a \*: f) (f \; *:%R a).

End ScaleFun.

Definition multiplicative (R S : semiRingType) (f : R S) : Prop :=
{morph f : x y / x × y}%R × (f 1 = 1).

FIXME: remove the @ once https://github.com/math-comp/hierarchy-builder/issues/319 is fixed

Module RMorphismExports.
Notation "{ 'rmorphism' U -> V }" := (RMorphism.type U%type V%type)
: type_scope.
Notation "[ 'rmorphism' 'of' f 'as' g ]" := (RMorphism.clone _ _ f%function g)
(at level 0, format "[ 'rmorphism' 'of' f 'as' g ]") : form_scope.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'rmorphism' 'of' f ]" := (RMorphism.clone _ _ f%function _)
(at level 0, format "[ 'rmorphism' 'of' f ]") : form_scope.
End RMorphismExports.

Section RmorphismTheory.

Section Properties.

Variables (R S : semiRingType) (k : unit) (f : {rmorphism R S}).

Lemma rmorph0 : f 0 = 0.
Lemma rmorphD : {morph f : x y / x + y}.
Lemma rmorphMn n : {morph f : x / x *+ n}.
Lemma rmorph_sum I r (P : pred I) E :
f (\sum_(i <- r | P i) E i) = \sum_(i <- r | P i) f (E i).

Lemma rmorphismMP : multiplicative f.
Lemma rmorph1 : f 1 = 1.
Lemma rmorphM : {morph f: x y / x × y}.

Lemma rmorph_prod I r (P : pred I) E :
f (\prod_(i <- r | P i) E i) = \prod_(i <- r | P i) f (E i).

Lemma rmorphXn n : {morph f : x / x ^+ n}.

Lemma rmorph_nat n : f n%:R = n%:R.

Lemma rmorph_char p : p \in [char R] p \in [char S].

Lemma rmorph_eq_nat x n : injective f (f x == n%:R) = (x == n%:R).

Lemma rmorph_eq1 x : injective f (f x == 1) = (x == 1).

Lemma can2_rmorphism f' : cancel f f' cancel f' f multiplicative f'.

End Properties.

Section Projections.

Variables (R S T : semiRingType).
Variables (f : {rmorphism S T}) (g : {rmorphism R S}).

Fact idfun_is_multiplicative : multiplicative (@idfun R).
#[export]
HB.instance Definition _ := isMultiplicative.Build R R idfun
idfun_is_multiplicative.

Fact comp_is_multiplicative : multiplicative (f \o g).
#[export]
HB.instance Definition _ := isMultiplicative.Build R T (f \o g)
comp_is_multiplicative.

End Projections.

Section Properties.

Variables (R S : ringType) (k : unit) (f : {rmorphism R S}).

Lemma rmorphN : {morph f : x / - x}.
Lemma rmorphB : {morph f: x y / x - y}.
Lemma rmorphMNn n : {morph f : x / x *- n}.
Lemma rmorphMsign n : {morph f : x / (- 1) ^+ n × x}.

Lemma rmorphN1 : f (- 1) = (- 1).

Lemma rmorph_sign n : f ((- 1) ^+ n) = (- 1) ^+ n.

End Properties.

Section InAlgebra.

Variables (R : ringType) (A : lalgType R).

#[export]
HB.instance Definition _ := isAdditive.Build R A (in_alg_loc A)

Fact in_alg_is_rmorphism : multiplicative (in_alg_loc A).
#[export]
HB.instance Definition _ := isMultiplicative.Build R A (in_alg_loc A)
in_alg_is_rmorphism.

Lemma in_algE a : in_alg_loc A a = a%:A.

End InAlgebra.

End RmorphismTheory.

Module Scale.

#[export]
HB.structure Definition Law R V := {op of isLaw R V op}.
Definition law := Law.type.

Section ScaleLaw.

Variables (R : ringType) (V : zmodType) (s_law : law R V).
Local Notation s_op := (Law.sort s_law).

Lemma N1op : s_op (-1) =1 -%R.
Fact opB a : additive (s_op a).

Variables (aR : ringType) (nu : {rmorphism aR R}).
Fact compN1op : (nu \; s_op) (-1) =1 -%R.

End ScaleLaw.

Module Exports. End Exports.

End Scale.
Export Scale.Exports.

#[export]
HB.instance Definition _ (R : ringType) := Scale.isLaw.Build R R *%R
(@mulN1r R) (@mulrBr R).

#[export]
HB.instance Definition _ (R : ringType) (U : lmodType R) :=
Scale.isLaw.Build R U *:%R (@scaleN1r R U) (@scalerBr R U).

#[export]
HB.instance Definition _ (R : ringType) (V : zmodType) (s : Scale.law R V)
(aR : ringType) (nu : {rmorphism aR R}) :=
Scale.isLaw.Build aR V (nu \; s)
(@Scale.compN1op _ _ s _ nu) (fun a Scale.opB _ _).

#[export, non_forgetful_inheritance]
HB.instance Definition _ (R : ringType) (V : zmodType) (s : Scale.law R V) a :=
isAdditive.Build V V (s a) (Scale.opB s a).

Definition scalable_for (R : ringType) (U : lmodType R) (V : zmodType)
(s : R V V) (f : U V) :=
a, {morph f : u / a *: u >-> s a u}.

Definition linear_for (R : ringType) (U : lmodType R) (V : zmodType)
(s : R V V) (f : U V) :=
a, {morph f : u v / a *: u + v >-> s a u + v}.

Lemma additive_linear (R : ringType) (U : lmodType R) V
(s : Scale.law R V) (f : U V) : linear_for s f additive f.

Lemma scalable_linear (R : ringType) (U : lmodType R) V
(s : Scale.law R V) (f : U V) : linear_for s f scalable_for s f.

Module LinearExports.
Notation scalable f := (scalable_for *:%R f).
Notation linear f := (linear_for *:%R f).
Notation scalar f := (linear_for *%R f).
Module Linear.
Section Linear.
Variables (R : ringType) (U : lmodType R) (V : zmodType) (s : R V V).
Definition apply_deprecated (phUV : phant (U V)) := @Linear.sort R U V s.
Notation apply := apply_deprecated.
Support for right-to-left rewriting with the generic linearZ rule.
Local Notation mapUV := (@Linear.type R U V s).
Definition map_class := mapUV.
Definition map_at (a : R) := mapUV.
Structure map_for a s_a := MapFor {map_for_map : mapUV; _ : s a = s_a}.
Definition unify_map_at a (f : map_at a) := MapFor f (erefl (s a)).
Structure wrapped := Wrap {unwrap : mapUV}.
Definition wrap (f : map_class) := Wrap f.
End Linear.
End Linear.
Notation "{ 'linear' U -> V | s }" := (@Linear.type _ U%type V%type s)
: type_scope.
Notation "{ 'linear' U -> V }" := {linear U%type V%type | *:%R}
: type_scope.
Notation "{ 'scalar' U }" := {linear U _ | *%R}
(at level 0, format "{ 'scalar' U }") : type_scope.
Notation "[ 'linear' 'of' f 'as' g ]" := (Linear.clone _ _ _ _ f%function g)
(at level 0, format "[ 'linear' 'of' f 'as' g ]") : form_scope.
Notation "[ 'linear' 'of' f ]" := (Linear.clone _ _ _ _ f%function _)
(at level 0, format "[ 'linear' 'of' f ]") : form_scope.
Support for right-to-left rewriting with the generic linearZ rule.
Coercion Linear.map_for_map : Linear.map_for >-> Linear.type.
Coercion Linear.unify_map_at : Linear.map_at >-> Linear.map_for.
Canonical Linear.unify_map_at.
Coercion Linear.unwrap : Linear.wrapped >-> Linear.type.
Coercion Linear.wrap : Linear.map_class >-> Linear.wrapped.
Canonical Linear.wrap.
End LinearExports.

Section LinearTheory.

Variable R : ringType.

Section GenericProperties.

Variables (U : lmodType R) (V : zmodType) (s : R V V) (k : unit).
Variable f : {linear U V | s}.

Lemma linear0 : f 0 = 0.
Lemma linearN : {morph f : x / - x}.
Lemma linearD : {morph f : x y / x + y}.
Lemma linearB : {morph f : x y / x - y}.
Lemma linearMn n : {morph f : x / x *+ n}.
Lemma linearMNn n : {morph f : x / x *- n}.
Lemma linear_sum I r (P : pred I) E :
f (\sum_(i <- r | P i) E i) = \sum_(i <- r | P i) f (E i).

Lemma linearZ_LR : scalable_for s f.
Lemma linearP a : {morph f : u v / a *: u + v >-> s a u + v}.

End GenericProperties.

Section BidirectionalLinearZ.

Variables (U : lmodType R) (V : zmodType) (s : R V V).

The general form of the linearZ lemma uses some bespoke interfaces to allow right-to-left rewriting when a composite scaling operation such as conjC \; *%R has been expanded, say in a^* * f u. This redex is matched by using the Scale.law interface to recognize a "head" scaling operation h (here *%R), stow away its "scalar" c, then reconcile h c and s a, once s is known, that is, once the Linear.map structure for f has been found. In general, s and a need not be equal to h and c; indeed they need not have the same type! The unification is performed by the unify_map_at default instance for the Linear.map_for U s a h_c sub-interface of Linear.map; the h_c pattern uses the Scale.law structure to insure it is inferred when rewriting right-to-left. The wrap on the rhs allows rewriting f (a *: b *: u) into a *: b *: f u with rewrite !linearZ /= instead of rewrite linearZ /= linearZ /=. Without it, the first rewrite linearZ would produce (a *: apply (map_for_map (@check_map_at .. a f)) (b *: u)%R)%Rlin and matching the second rewrite LHS would bypass the unify_map_at default instance for b, reuse the one for a, and subsequently fail to match the b *: u argument. The extra wrap / unwrap ensures that this can't happen. In the RL direction, the wrap / unwrap will be inserted on the redex side as needed, without causing unnecessary delta-expansion: using an explicit identity function would have Coq normalize the redex to head normal, then reduce the identity to expose the map_for_map projection, and the expanded Linear.map structure would then be exposed in the result. Most of this machinery will be invisible to a casual user, because all the projections and default instances involved are declared as coercions.

Variables (S : ringType) (h : Scale.law S V).

Lemma linearZ c a (h_c := h c) (f : Linear.map_for U s a h_c) u :
f (a *: u) = h_c (Linear.wrap f u).

End BidirectionalLinearZ.

Section LmodProperties.

Variables (U V : lmodType R) (f : {linear U V}).

Lemma linearZZ : scalable f.
Lemma linearPZ : linear f.

Lemma can2_scalable f' : cancel f f' cancel f' f scalable f'.

Lemma can2_linear f' : cancel f f' cancel f' f linear f'.

End LmodProperties.

Section ScalarProperties.

Variable (U : lmodType R) (f : {scalar U}).

Lemma scalarZ : scalable_for *%R f.
Lemma scalarP : scalar f.

End ScalarProperties.

Section LinearLmod.

Variables (W U : lmodType R) (V : zmodType).

Section Plain.

Variable (s : R V V).
Variables (f : {linear U V | s}) (h : {linear W U}).

Lemma idfun_is_scalable : scalable (@idfun U).
#[export]
HB.instance Definition _ := isScalable.Build R U U *:%R idfun idfun_is_scalable.

Lemma opp_is_scalable : scalable (-%R : U U).
#[export]
HB.instance Definition _ := isScalable.Build R U U *:%R -%R opp_is_scalable.

Lemma comp_is_scalable : scalable_for s (f \o h).
#[export]
HB.instance Definition _ := isScalable.Build R W V s (f \o h) comp_is_scalable.

End Plain.

Section Scale.

Variable (s : Scale.law R V).
Variables (f : {linear U V | s}) (g : {linear U V | s}).

Lemma null_fun_is_scalable : scalable_for s (\0 : U V).
#[export]
HB.instance Definition _ := isScalable.Build R U V s \0 null_fun_is_scalable.

Lemma add_fun_is_scalable : scalable_for s (f \+ g).
#[export]
HB.instance Definition _ := isScalable.Build R U V s (f \+ g) add_fun_is_scalable.

Lemma sub_fun_is_scalable : scalable_for s (f \- g).
#[export]
HB.instance Definition _ := isScalable.Build R U V s (f \- g) sub_fun_is_scalable.

Lemma opp_fun_is_scalable : scalable_for s (\- g).
#[export]
HB.instance Definition _ := isScalable.Build R U V s (\- g) opp_fun_is_scalable.

End Scale.

End LinearLmod.

Section LinearLalg.

Variables (A : lalgType R) (U : lmodType R).

Variables (a : A) (f : {linear U A}).

Fact mulr_fun_is_scalable : scalable (a \o× f).
#[export]
HB.instance Definition _ := isScalable.Build R U A *:%R (a \o× f)
mulr_fun_is_scalable.

End LinearLalg.

End LinearTheory.

FIXME: remove the @ once https://github.com/math-comp/hierarchy-builder/issues/319 is fixed

Module LRMorphismExports.
Module LRMorphism.
Definition apply_deprecated (R : ringType) (A : lalgType R) (B : ringType)
(s : R B B) (phAB : phant (A B)) := @LRMorphism.sort R A B s.
Notation apply := apply_deprecated.
End LRMorphism.
Notation "{ 'lrmorphism' A -> B | s }" := (@LRMorphism.type _ A%type B%type s)
: type_scope.
Notation "{ 'lrmorphism' A -> B }" := {lrmorphism A%type B%type | *:%R}
: type_scope.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'lrmorphism' 'of' f ]" := (LRMorphism.clone _ _ _ _ f%function _)
(at level 0, format "[ 'lrmorphism' 'of' f ]") : form_scope.
End LRMorphismExports.

Section LRMorphismTheory.

Variables (R : ringType) (A B : lalgType R) (C : ringType) (s : R C C).
Variables (k : unit) (f : {lrmorphism A B}) (g : {lrmorphism B C | s}).

#[export] HB.instance Definition _ := RMorphism.on (@idfun A).
#[export] HB.instance Definition _ := RMorphism.on (g \o f).

Lemma rmorph_alg a : f a%:A = a%:A.

End LRMorphismTheory.

#[short(type="comSemiRingType")]
HB.structure Definition ComSemiRing :=
{R of SemiRing R & SemiRing_hasCommutativeMul R}.

Module ComSemiRingExports.
Bind Scope ring_scope with ComSemiRing.sort.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'comSemiRingType' 'of' T 'for' cT ]" := (ComSemiRing.clone T cT)
(at level 0, format "[ 'comSemiRingType' 'of' T 'for' cT ]") : form_scope.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'comSemiRingType' 'of' T ]" := (ComSemiRing.clone T _)
(at level 0, format "[ 'comSemiRingType' 'of' T ]") : form_scope.
End ComSemiRingExports.

Definition mulr1 := Monoid.mulC_id mulrC mul1r.
Definition mulrDr := Monoid.mulC_dist mulrC mulrDl.
Lemma mulr0 : right_zero zero mul.

Section ComSemiRingTheory.

Variable R : comSemiRingType.
Implicit Types x y : R.

#[export]
HB.instance Definition _ := SemiGroup.isCommutativeLaw.Build R *%R mulrC.
Lemma mulrCA : @left_commutative R R *%R.
Lemma mulrAC : @right_commutative R R *%R.
Lemma mulrACA : @interchange R *%R *%R.

Lemma exprMn n : {morph (fun xx ^+ n) : x y / x × y}.

Lemma prodrXl n I r (P : pred I) (F : I R) :
\prod_(i <- r | P i) F i ^+ n = (\prod_(i <- r | P i) F i) ^+ n.

Lemma prodr_undup_exp_count (I : eqType) r (P : pred I) (F : I R) :
\prod_(i <- undup r | P i) F i ^+ count_mem i r = \prod_(i <- r | P i) F i.

Lemma exprDn x y n :
(x + y) ^+ n = \sum_(i < n.+1) (x ^+ (n - i) × y ^+ i) *+ 'C(n, i).

Lemma sqrrD x y : (x + y) ^+ 2 = x ^+ 2 + x × y *+ 2 + y ^+ 2.

End ComSemiRingTheory.

#[short(type="comRingType")]
HB.structure Definition ComRing := {R of Ring R & ComSemiRing R}.

Definition mulr1 := Monoid.mulC_id mulrC mul1r.
Definition mulrDr := Monoid.mulC_dist mulrC mulrDl.

Module ComRingExports.
Bind Scope ring_scope with ComRing.sort.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'comRingType' 'of' T 'for' cT ]" := (ComRing.clone T%type cT)
(at level 0, format "[ 'comRingType' 'of' T 'for' cT ]") : form_scope.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'comRingType' 'of' T ]" := (ComRing.clone T%type _)
(at level 0, format "[ 'comRingType' 'of' T ]") : form_scope.
End ComRingExports.

Section ComRingTheory.

Variable R : comRingType.
Implicit Types x y : R.

Lemma exprBn x y n :
(x - y) ^+ n =
\sum_(i < n.+1) ((-1) ^+ i × x ^+ (n - i) × y ^+ i) *+ 'C(n, i).

Lemma subrXX x y n :
x ^+ n - y ^+ n = (x - y) × (\sum_(i < n) x ^+ (n.-1 - i) × y ^+ i).

Lemma sqrrB x y : (x - y) ^+ 2 = x ^+ 2 - x × y *+ 2 + y ^+ 2.

Lemma subr_sqr x y : x ^+ 2 - y ^+ 2 = (x - y) × (x + y).

Lemma subr_sqrDB x y : (x + y) ^+ 2 - (x - y) ^+ 2 = x × y *+ 4.

Section FrobeniusAutomorphism.

Variables (p : nat) (charRp : p \in [char R]).

Lemma Frobenius_aut_is_multiplicative : multiplicative (Frobenius_aut charRp).

#[export]
HB.instance Definition _ := isAdditive.Build R R (Frobenius_aut charRp)
#[export]
HB.instance Definition _ := isMultiplicative.Build R R (Frobenius_aut charRp)
Frobenius_aut_is_multiplicative.

End FrobeniusAutomorphism.

Lemma exprDn_char x y n : [char R].-nat n (x + y) ^+ n = x ^+ n + y ^+ n.

Lemma rmorph_comm (S : ringType) (f : {rmorphism R S}) x y :
comm (f x) (f y).

Section ScaleLinear.

Variables (U V : lmodType R) (b : R) (f : {linear U V}).

Lemma scale_is_scalable : scalable ( *:%R b : V V).
#[export]
HB.instance Definition _ := isScalable.Build R V V *:%R ( *:%R b)
scale_is_scalable.

Lemma scale_fun_is_scalable : scalable (b \*: f).
#[export]
HB.instance Definition _ := isScalable.Build R U V *:%R (b \*: f)
scale_fun_is_scalable.

End ScaleLinear.

End ComRingTheory.

#[short(type="algType")]
HB.structure Definition Algebra (R : ringType) :=
{A of Lalgebra_isAlgebra R A & Lalgebra R A}.

Module AlgExports.
Bind Scope ring_scope with Algebra.sort.
Notation "[ 'algType' R 'of' T 'for' cT ]" := (Algebra.clone R T%type cT)
(at level 0, format "[ 'algType' R 'of' T 'for' cT ]")
: form_scope.
Notation "[ 'algType' R 'of' T ]" := (Algebra.clone R T%type _)
(at level 0, format "[ 'algType' R 'of' T ]") : form_scope.
End AlgExports.

Lemma scalarAr k (x y : V) : k *: (x × y) = x × (k *: y).

#[short(type="comAlgType")]
HB.structure Definition ComAlgebra R := {V of ComRing V & Algebra R V}.

Module ComAlgExports.
Bind Scope ring_scope with ComAlgebra.sort.
#[deprecated(since="mathcomp 2.0.0",
Notation "[ 'comAlgType' R 'of' T ]" := (ComAlgebra.clone R T%type _)
(at level 0, format "[ 'comAlgType' R 'of' T ]") : form_scope.
End ComAlgExports.

Section AlgebraTheory.
Variables (R : comRingType) (A : algType R).
#[export]
HB.instance Definition converse_ : SemiRing_hasCommutativeMul R^c :=
SemiRing_hasCommutativeMul.Build R^c (fun _ _ mulrC _ _).
#[export]
HB.instance Definition regular_comSemiRingType :
SemiRing_hasCommutativeMul R^o := ComSemiRing.on R^o.
#[export]
HB.instance Definition regular_comAlgType : Lalgebra_isComAlgebra R R^o :=
Lalgebra_isComAlgebra.Build R R^o.
End AlgebraTheory.

Section AlgebraTheory.

Variables (R : comRingType) (A : algType R).
Implicit