(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
From mathcomp
From mathcomp
(******************************************************************************)
(*   The algebraic part of the Algebraic Hierarchy, as described in           *)
(*          ``Packaging mathematical structures'', TPHOLs09, by               *)
(*   Francois Garillot, Georges Gonthier, Assia Mahboubi, Laurence Rideau     *)
(*                                                                            *)
(* This file defines for each Structure (Zmodule, Ring, etc ...) its type,    *)
(* its packers and its canonical properties :                                 *)
(*                                                                            *)
(*  * Zmodule (additive abelian groups):                                      *)
(*              zmodType == interface type for Zmodule structure.             *)
(* ZmodMixin addA addC add0x addNx == builds the mixin for a Zmodule from the *)
(*                          algebraic properties of its operations.           *)
(*          ZmodType V m == packs the mixin m to build a Zmodule of type      *)
(*                          zmodType. The carrier type V must have a          *)
(*                          choiceType canonical structure.                   *)
(* [zmodType of V for S] == V-clone of the zmodType structure S: a copy of S  *)
(*                          where the sort carrier has been replaced by V,    *)
(*                          and which is therefore a zmodType structure on V. *)
(*                          The sort carrier for S must be convertible to V.  *)
(*       [zmodType of V] == clone of a canonical zmodType structure on V.     *)
(*                          Similar to the above, except S is inferred, but   *)
(*                          possibly with a syntactically different carrier.  *)
(*                     0 == the zero (additive identity) of a Zmodule.        *)
(*                 x + y == the sum of x and y (in a Zmodule).                *)
(*                   - x == the opposite (additive inverse) of x.             *)
(*                 x - y == the difference of x and y; this is only notation  *)
(*                          for x + (- y).                                    *)
(*                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.            *)
(*                x *- n == notation for - (x *+ n), the opposite of x *+ n.  *)
(*        \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].          *)
(*         oppr_closed S <-> collective predicate S is closed under opposite. *)
(*         addr_closed S <-> collective predicate S is closed under finite    *)
(*                           sums (0 and x + y in S, for x, y in S).          *)
(*         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. *)
(*         OpprPred oppS == packs oppS : oppr_closed S into an opprPred S     *)
(*                          interface structure associating this property to  *)
(*                          the canonical pred_key S, i.e. the k for which S  *)
(*                          has a Canonical keyed_pred k structure (see file  *)
(*                          ssrbool.v).                                       *)
(*         AddrPred addS == packs addS : addr_closed S into an addrPred S     *)
(*                          interface structure associating this property to  *)
(*                          the canonical pred_key S (see above).             *)
(*         ZmodPred oppS == packs oppS : oppr_closed S into an zmodPred S     *)
(*                          interface structure associating the zmod_closed   *)
(*                          property to the canonical pred_key S (see above), *)
(*                          which must already be an addrPred.                *)
(* [zmodMixin of M by <:] == zmodType mixin for a subType whose base type is  *)
(*                          a zmodType and whose predicate's canonical        *)
(*                          pred_key is a zmodPred.                           *)
(* --> Coq can be made to behave as if all predicates had canonical zmodPred  *)
(*     keys by executing Import DefaultKeying GRing.DefaultPred. The required *)
(*     oppr_closed and addr_closed assumptions will be either abstracted,     *)
(*     resolved or issued as separate proof obligations by the ssreflect      *)
(*     plugin abstraction and Prop-irrelevance functions.                     *)
(*  * Ring (non-commutative rings):                                           *)
(*              ringType == interface type for a Ring structure.              *)
(* RingMixin mulA mul1x mulx1 mulDx mulxD == builds the mixin for a Ring from *)
(*                           the algebraic properties of its multiplicative   *)
(*                           operators; the carrier type must have a zmodType *)
(*                           structure.                                       *)
(*           RingType R m == packs the ring mixin m into a ringType.          *)
(*                    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).                       *)
(*  [ringType of R for S] == R-clone of the ringType structure S.             *)
(*        [ringType of R] == clone of a canonical ringType structure on 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.                               *)
(*                  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.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.       *)
(*         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 p] has a 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).       *)
(*         smulr_closed S <-> collective predicate S is closed under products *)
(*                           and opposite (-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).         *)
(*       subring_closed S <-> collective predicate S is closed under ring     *)
(*                           operations (1, x - y and x * y in S).            *)
(*          MulrPred mulS == packs mulS : mulr_closed S into a mulrPred S,    *)
(*        SmulrPred mulS     smulrPred S, semiringPred S, or subringPred S    *)
(*     SemiringPred mulS     interface structure, corresponding to the above  *)
(*      SubRingPred mulS     properties, respectively, provided S already has *)
(*                           the supplementary zmodType closure properties.   *)
(*                           The properties above coerce to subproperties so, *)
(*                           e.g., ringS : subring_closed S can be used for   *)
(*                           the proof obligations of all prerequisites.      *)
(* [ringMixin of R by <:] == ringType mixin for a subType whose base type is  *)
(*                           a ringType and whose predicate's canonical key   *)
(*                           is a SubringPred.                                *)
(*  --> As for zmodType predicates, Import DefaultKeying GRing.DefaultPred    *)
(*      turns unresolved GRing.Pred unification constraints into proof        *)
(*      obligations for basic closure assumptions.                            *)
(*                                                                            *)
(*  * ComRing (commutative Rings):                                            *)
(*            comRingType == interface type for commutative ring structure.   *)
(*     ComRingType R mulC == packs mulC into a comRingType; the carrier type  *)
(*                           R must have a ringType canonical structure.      *)
(* ComRingMixin mulA mulC mul1x mulDx == builds the mixin for a Ring (i.e., a *)
(*                           *non commutative* ring), using the commutativity *)
(*                           to reduce the number of proof obligations.       *)
(* [comRingType of R for S] == R-clone of the comRingType structure S.        *)
(*     [comRingType of R] == clone of a canonical comRingType structure on R. *)
(* [comRingMixin of R by <:] == comutativity mixin axiom for R when it is a   *)
(*                           subType of a commutative ring.                   *)
(*                                                                            *)
(*  * UnitRing (Rings whose units have computable inverses):                  *)
(*           unitRingType == interface type for the UnitRing structure.       *)
(* UnitRingMixin mulVr mulrV unitP inv0id == builds the mixin for a UnitRing  *)
(*                           from the properties of the inverse operation and *)
(*                           the boolean test for being a unit (invertible).  *)
(*                           The inverse of a non-unit x is constrained to be *)
(*                           x itself (property inv0id). The carrier type     *)
(*                           must have a ringType canonical structure.        *)
(*       UnitRingType R m == packs the unit ring mixin m into a unitRingType. *)
(*                  WARNING: while it is possible to omit R for most of the   *)
(*                           XxxType functions, R MUST be explicitly given    *)
(*                           when UnitRingType is used with a mixin produced  *)
(*                           by ComUnitRingMixin, in a Canonical definition,  *)
(*                           otherwise the resulting structure will have the  *)
(*                           WRONG sort key and will NOT BE USED during type  *)
(*                           inference.                                       *)
(* [unitRingType of R for S] == R-clone of the unitRingType structure S.      *)
(*    [unitRingType of R] == clones a canonical unitRingType structure on R.  *)
(*     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).            *)
(*         DivrPred invS == packs invS : mulr_closed S into a divrPred S,     *)
(*        SdivrPred invS    sdivrPred S or divringPred S interface structure, *)
(*      DivringPred invS    corresponding to the above properties, resp.,     *)
(*                          provided S already has the supplementary ringType *)
(*                          closure properties. The properties above coerce   *)
(*                          to subproperties, as explained above.             *)
(* [unitRingMixin of R by <:] == unitRingType mixin for a subType whose base  *)
(*                           type is a unitRingType and whose predicate's     *)
(*                           canonical key is a divringPred and whose ring    *)
(*                           structure is compatible with the base type's.    *)
(*                                                                            *)
(*  * ComUnitRing (commutative rings with computable inverses):               *)
(*        comUnitRingType == interface type for ComUnitRing structure.        *)
(* ComUnitRingMixin mulVr unitP inv0id == builds the mixin for a UnitRing (a  *)
(*                           *non commutative* unit ring, using commutativity *)
(*                           to simplify the proof obligations; the carrier   *)
(*                           type must have a comRingType structure.          *)
(*                           WARNING: ALWAYS give an explicit type argument   *)
(*                           to UnitRingType along with a mixin produced by   *)
(*                           ComUnitRingMixin (see above).                    *)
(* [comUnitRingType of R] == a comUnitRingType structure for R created by     *)
(*                           merging canonical comRingType and unitRingType   *)
(*                           structures on R.                                 *)
(*                                                                            *)
(*  * IntegralDomain (integral, commutative, ring with partial inverses):     *)
(*            idomainType == interface type for the IntegralDomain structure. *)
(* IdomainType R mulf_eq0 == packs the integrality property into an           *)
(*                           idomainType integral domain structure; R must    *)
(*                           have a comUnitRingType canonical structure.      *)
(* [idomainType of R for S] == R-clone of the idomainType structure S.        *)
(*     [idomainType of R] == clone of a canonical idomainType structure on R. *)
(* [idomainMixin of R by <:] == mixin axiom for a idomain subType.            *)
(*                                                                            *)
(*  * Field (commutative fields):                                             *)
(*              fieldType == interface type for fields.                       *)
(*  GRing.Field.mixin_of R == the field property: x != 0 -> x \is a unit, for *)
(*                           x : R; R must be or coerce to a unitRingType.    *)
(*  GRing.Field.axiom inv == the 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.                               *)
(* FieldUnitMixin mulVf inv0 == a *non commutative unit ring* mixin, using an *)
(*                           inverse function that satisfies the field axiom  *)
(*                           and fixes 0 (arguments mulVf and inv0, resp.),   *)
(*                           and x != 0 as the Ring.unit predicate. The       *)
(*                           carrier type must be a canonical comRingType.    *)
(*    FieldIdomainMixin m == an *idomain* mixin derived from a field mixin m. *)
(* GRing.Field.IdomainType mulVf inv0 == an idomainType incorporating the two *)
(*                           mixins above, where FieldIdomainMixin is applied *)
(*                           to the trivial field mixin for FieldUnitMixin.   *)
(*  FieldMixin mulVf inv0 == the (trivial) field mixin for Field.IdomainType. *)
(*          FieldType R m == packs the field mixin M into a fieldType. The    *)
(*                           carrier type R must be an idomainType.           *)
(* --> Given proofs mulVf and inv0 as above, a non-Canonical instances        *)
(* of fieldType can be created with FieldType _ (FieldMixin mulVf inv0).      *)
(* For Canonical instances one should always specify the first (sort)         *)
(* argument of FieldType and other instance constructors, as well as pose     *)
(* Definitions for unit ring, field, and idomain mixins (in that order).      *)
(* [fieldType of F for S] == F-clone of the fieldType structure S.            *)
(*       [fieldType of F] == clone of a canonical fieldType structure on F.   *)
(*   [fieldMixin of R by <:] == mixin axiom for a field subType.              *)
(*                                                                            *)
(*  * DecidableField (fields with a decidable first order theory):            *)
(*           decFieldType == interface type for DecidableField structure.     *)
(*     DecFieldMixin satP == builds the mixin for a DecidableField from the   *)
(*                           correctness of its satisfiability predicate. The *)
(*                           carrier type must have a unitRingType structure. *)
(*       DecFieldType F m == packs the decidable field mixin m into a         *)
(*                           decFieldType; the carrier type F must have a     *)
(*                           fieldType structure.                             *)
(* [decFieldType of F for S] == F-clone of the decFieldType structure S.      *)
(*    [decFieldType of F] == clone of a canonical decFieldType structure on F *)
(*           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.    *)
(* QEdecFieldMixin wfP okP == a decidable field Mixin built from a quantifier *)
(*                           eliminator p and proofs wfP : GRing.wf_QE_proj p *)
(*                           and okP : GRing.valid_QE_proj p that p returns   *)
(*                           well-formed and valid formulae, i.e., p i (u, v) *)
(*                           is a quantifier-free formula equivalent to       *)
(*        'exists 'X_i, u1 == 0 /\ ... /\ u_m == 0 /\ v1 != 0 ... /\ v_n != 0 *)
(*                                                                            *)
(*  * ClosedField (algebraically closed fields):                              *)
(*        closedFieldType == interface type for the ClosedField structure.    *)
(*    ClosedFieldType F m == packs the closed field mixin m into a            *)
(*                           closedFieldType. The carrier F must have a       *)
(*                           decFieldType structure.                          *)
(* [closedFieldType of F on S] == F-clone of a closedFieldType structure S.   *)
(* [closedFieldType of F] == clone of a canonicalclosedFieldType structure    *)
(*                           on F.                                            *)
(*                                                                            *)
(*  * Lmodule (module with left multiplication by external scalars).          *)
(*             lmodType R == interface type for an Lmodule structure with     *)
(*                           scalars of type R; R must have a ringType        *)
(*                           structure.                                       *)
(* LmodMixin scalA scal1v scalxD scalDv == builds an Lmodule mixin from the   *)
(*                           algebraic properties of the scaling operation;   *)
(*                           the module carrier type must have a zmodType     *)
(*                           structure, and the scalar carrier must have a    *)
(*                           ringType structure.                              *)
(*         LmodType R V m == packs the mixin v to build an Lmodule of type    *)
(*                           lmodType R. The carrier type V must have a       *)
(*                           zmodType structure.                              *)
(* [lmodType R of V for S] == V-clone of an lmodType R structure S.           *)
(*      [lmodType R of V] == clone of a canonical lmodType R structure on V.  *)
(*                 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).              *)
(*      SubmodPred scaleS == packs scaleS : scaler_closed S in a submodPred S *)
(*                           interface structure corresponding to the above   *)
(*                           property, provided S's key is a zmodPred;        *)
(*                           submod_closed coerces to all the prerequisites.  *)
(* [lmodMixin of V by <:] == mixin for a subType of an lmodType, whose        *)
(*                           predicate's key is a submodPred.                 *)
(*                                                                            *)
(*  * Lalgebra (left algebra, ring with scaling that associates on the left): *)
(*             lalgType R == interface type for Lalgebra structures with      *)
(*                           scalars in R; R must have ringType structure.    *)
(*    LalgType R V scalAl == packs scalAl : k (x y) = (k x) y into an         *)
(*                           Lalgebra of type lalgType R. The carrier type V  *)
(*                           must have both lmodType R and ringType canonical *)
(*                           structures.                                      *)
(*                    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.                   *)
(* [lalgType R of V for S] == V-clone the lalgType R structure S.             *)
(*      [lalgType R of V] == clone of a canonical lalgType R structure on V.  *)
(*        subalg_closed S <-> collective predicate S is closed under lalgType *)
(*                           operations (1, a *: u + v and u * v in S).       *)
(*      SubalgPred scaleS == packs scaleS : scaler_closed S in a subalgPred S *)
(*                           interface structure corresponding to the above   *)
(*                           property, provided S's key is a subringPred;     *)
(*                           subalg_closed coerces to all the prerequisites.  *)
(* [lalgMixin of V by <:] == mixin axiom for a subType of an lalgType.        *)
(*                                                                            *)
(*  * Algebra (ring with scaling that associates both left and right):        *)
(*              algType R == type for Algebra structure with scalars in R.    *)
(*                           R should be a commutative ring.                  *)
(*     AlgType R A scalAr == packs scalAr : k (x y) = x (k y) into an Algebra *)
(*                           Structure of type algType R. The carrier type A  *)
(*                           must have an lalgType R structure.               *)
(*        CommAlgType R A == creates an Algebra structure for an A that has   *)
(*                           both lalgType R and comRingType structures.      *)
(* [algType R of V for S] == V-clone of an algType R structure on S.          *)
(*       [algType R of V] == clone of a canonical algType R structure on V.   *)
(*  [algMixin of V by <:] == mixin axiom for a subType of an algType.         *)
(*                                                                            *)
(*  * UnitAlgebra (algebra with computable inverses):                         *)
(*          unitAlgType R == interface type for UnitAlgebra structure with    *)
(*                           scalars in R; R should have a unitRingType       *)
(*                           structure.                                       *)
(*   [unitAlgType R of V] == a unitAlgType R structure for V created by       *)
(*                           merging canonical algType and unitRingType on V. *)
(*        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).                          *)
(*      DivalgPred scaleS == packs scaleS : scaler_closed S in a divalgPred S *)
(*                           interface structure corresponding to the above   *)
(*                           property, provided S's key is a divringPred;     *)
(*                           divalg_closed coerces to all the prerequisites.  *)
(*                                                                            *)
(*   In addition to this structure hierarchy, we also develop a separate,     *)
(* parallel hierarchy for morphisms linking these structures:                 *)
(*                                                                            *)
(* * Additive (additive functions):                                           *)
(*             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     *)
(*                           additive property; both U and V must have        *)
(*                           zmodType canonical structures.                   *)
(*         Additive add_f == packs add_f : additive f into an additive        *)
(*                           function structure of type {additive U -> V}.    *)
(*   [additive of f as g] == an f-clone of the additive structure on the      *)
(*                           function g -- f and g must be convertible.       *)
(*        [additive of f] == a clone of an existing additive structure on f.  *)
(*                                                                            *)
(* * RMorphism (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 ringType structures. *)
(*            rmorphism f <-> f is a ring morphism, i.e., f is both additive  *)
(*                           and multiplicative.                              *)
(*     {rmorphism R -> S} == the interface type for ring morphisms, i.e.,     *)
(*                           a Structure that encapsulates the rmorphism      *)
(*                           property for functions f : R -> S; both R and S  *)
(*                           must have ringType structures.                   *)
(*      RMorphism morph_f == packs morph_f : rmorphism f into a Ring morphism *)
(*                           structure of type {rmorphism R -> S}.            *)
(*     AddRMorphism mul_f == packs mul_f : multiplicative f into an rmorphism *)
(*                           structure of type {rmorphism R -> S}; f must     *)
(*                           already have an {additive R -> S} structure.     *)
(*  [rmorphism of f as g] == an f-clone of the rmorphism structure of g.      *)
(*       [rmorphism of f] == a clone of an existing additive structure on f.  *)
(*  -> 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 if a field isomorphism    *)
(*     between R and its image.                                               *)
(*  -> As rmorphism coerces to both additive and multiplicative, all          *)
(*     structures for f can be built from a single proof of rmorphism f.      *)
(*  -> 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.      *)
(*  -> The property duplication also means that it is not strictly necessary  *)
(*     to declare all Additive instances.                                     *)
(*                                                                            *)
(* * 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 in scalable_for paragraph  *)
(*                           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.   *)
(*           Linear lin_f == packs lin_f : lmorphism_for s f into a linear    *)
(*                           function structure of type {linear U -> V | s}.  *)
(*                           As linear_for s f coerces to lmorphism_for s f,  *)
(*                           Linear can be used with lin_f : linear_for s f   *)
(*                           (indeed, that is the recommended usage). Note    *)
(*                           that as linear f, scalar f, {linear U -> V} and  *)
(*                           {scalar U} are simply notation for corresponding *)
(*                           generic "_for" forms, Linear can be used for any *)
(*                           of these special cases, transparently.           *)
(*       AddLinear scal_f == packs scal_f : scalable_for s f into a           *)
(*                           {linear U -> V | s} structure; f must already    *)
(*                           have an additive structure; as with Linear,      *)
(*                           AddLinear can be used with lin_f : linear f, etc *)
(*     [linear of f as g] == an f-clone of the linear structure of g.         *)
(*          [linear of f] == a clone of an existing linear structure on f.    *)
(*          (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. The *)
(*        (a *^nu u)%Rlin    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.  *)
(*   LRmorphism lrmorph_f == packs lrmorph_f : lrmorphism_for s f into a      *)
(*                           linear morphism structure of type                *)
(*                           {lrmorphism A -> B | s}. Like Linear, LRmorphism *)
(*                           can be used transparently for lrmorphism f.      *)
(*   AddLRmorphism scal_f == packs scal_f : scalable_for s f into a linear    *)
(*                           morphism structure of type                       *)
(*                           {lrmorphism A -> B | s}; f must already have an  *)
(*                           {rmorphism A -> B} structure, and AddLRmorphism  *)
(*                           can be applied to a linear_for s f, linear f,    *)
(*                           scalar f, etc argument, like AddLinear.          *)
(*      [lrmorphism of f] == creates an lrmorphism structure from existing    *)
(*                           rmorphism and linear structures on f; this is    *)
(*                           the preferred way of creating lrmorphism         *)
(*                           structures.                                      *)
(*  -> 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.                                     *)
(*                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   *)
(*                           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   *)
(*                           linear when f is and its codomain is an lalgType *)
(*                           and which 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 rmorphX : 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.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
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 "a \*o f" (at level 40).
Reserved Notation "a \o* f" (at level 40).
Reserved Notation "a \*: f" (at level 40).
Delimit Scope ring_scope with R.
Delimit Scope term_scope with T.
Local Open Scope ring_scope.
Module Import GRing.
Module Zmodule.
Record mixin_of (V : Type) : Type := Mixin {
zero : V;
opp : V -> V;
add : V -> V -> V;
_ : associative add;
_ : commutative add;
_ : left_id zero add;
_ : left_inverse zero opp add
}.
Section ClassDef.
Record class_of T := Class { base : Choice.class_of T; mixin : mixin_of T }.
Local Coercion base : class_of >-> Choice.class_of.
Structure type := Pack {sort; _ : class_of sort}.
Local Coercion sort : type >-> Sortclass.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack T c.
Let xT := let: Pack T _ := cT in T.
Notation xclass := (class : class_of xT).
Definition pack m :=
fun bT b & phant_id (Choice.class bT) b => Pack (@Class T b m).
Definition eqType := @Equality.Pack cT xclass.
Definition choiceType := @Choice.Pack cT xclass.
End ClassDef.
Module Exports.
Coercion base : class_of >-> Choice.class_of.
Coercion mixin : class_of >-> mixin_of.
Coercion sort : type >-> Sortclass.
Bind Scope ring_scope with sort.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Coercion choiceType : type >-> Choice.type.
Canonical choiceType.
Notation zmodType := type.
Notation ZmodType T m := (@pack T m _ _ id).
Notation ZmodMixin := Mixin.
Notation "[ 'zmodType' 'of' T 'for' cT ]" := (@clone T cT _ idfun)
(at level 0, format "[ 'zmodType'  'of'  T  'for'  cT ]") : form_scope.
Notation "[ 'zmodType' 'of' T ]" := (@clone T _ _ id)
(at level 0, format "[ 'zmodType'  'of'  T ]") : form_scope.
End Exports.
End Zmodule.
Definition zero V := Zmodule.zero (Zmodule.class V).
Definition opp V := Zmodule.opp (Zmodule.class V).
Definition add V := Zmodule.add (Zmodule.class V).
Local Notation "0" := (zero _) : ring_scope.
Local Notation "-%R" := (@opp _) : ring_scope.
Local Notation "- x" := (opp x) : ring_scope.
Local Notation "+%R" := (@add _) : ring_scope.
Local Notation "x + y" := (add x y) : ring_scope.
Local Notation "x - y" := (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 "x *- n" := (- (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 ZmoduleTheory.
Variable V : zmodType.
Implicit Types x y : V.

Lemma addrC : @commutative V V +%R. Proof. by case V => T [? []]. Qed.