# Library ssrfun

Require Import ssreflect.

``` This file contains the basic definitions and notations for working with
functions. The definitions concern:

Pair projections
p.1  == first element of a pair
p.2  == second element of a pair

Simplifying functions, beta-reduced by simpl and /= :
[fun : T => E] == constant function from type T that returns E
[fun x => E] == unary function
[fun x : T => E] == unary function with explicit domain type
[fun x y => E] == binary function
[fun x y : T => E] == binary function with common domain type
[fun (x : T) y => E] \
[fun (x : xT) (y : yT) => E] | == binary function with (some) explicit,
[fun x (y : T) => E] / independent domain types for each argument
- partial functions using option type,
oapp f d ox == if ox is Some x returns f x,        d otherwise
odflt d ox == if ox is Some x returns x,          d otherwise
obind f ox == if ox is Some x returns f x,        None otherwise
omap f ox == if ox is Some x returns Some (f x), None otherwise
- singleton types,
all_equal_to x0 == x0 is the only value in its type, so any such value
can be rewritten to x0.
- a generic wrapper type,
wrapped T == the inductive type with values Wrap x for x : T
unwrap w == the projection of w : wrapped T on T
wrap x == the canonical injection of x : T into wrapped T; it is
equivalent to Wrap x, but is declared as a (default)
Canonical Structure, which lets the Coq HO unification
automatically expand x into unwrap (wrap x). The delta
reduction of wrap x to Wrap can be exploited to
introduce controlled nondeterminism in Canonical
Structure inference, as in the implementation of
the mxdirect predicate in matrix.v.
- identity functions
id           == NOTATION for the explicit identity function fun x => x
@id T        == notation for the explicit identity at type T
idfun        == a constant whose definition is the identity function
phant_id x y == the function type phantom _ x -> phantom _ y
*** In addition to their casual use in functional programming, identity
functions are often used to trigger static unification as part of the
construction of dependent Records and Structures. For example, if we need
a structure sT over a type T, we take as arguments T, sT, and a "dummy"
function T -> sort sT:
Definition foo T sT & T -> sort sT := ...
We can avoid specifying sT directly by calling foo (@id T), or specify
the call completely while still ensuring the consistency of T and sT, by
calling @foo T sT idfun. The phant_id type allows us to extend this trick
to non-Type canonical projections. It also allows us to sidestep
dependent type constraints when building explicit records, e.g., given
Record r := R { x; y : T(x)}.
if we need to build an r from a given y0 while inferring some x0, such
that y0 : T(x0), we pose
Definition mk_r .. y .. (x := ...) y' & phant_id y y' := R x y'.
Calling @mk_r .. y0 .. id will cause Coq to use y' := y0, while checking
the dependent type constraint y0 : T(x0).
- extensional equality for functions and relations (i.e. functions of two
arguments),
f1 =1 f2      ==  f1 x is equal to f2 x forall x
f1 =1 f2 :> A ==    ... and f2 is explicitly typed
f1 =2 f2      ==  f1 x y is equal to f2 x y forall x y
f1 =2 f2 :> A ==    ... and f2 is explicitly typed
- composition for total and partial functions,
f^~ y == function f with second argument specialised to y,
i.e., fun x => f x y
CAVEAT: conditional (non-maximal) implicit arguments
of f are NOT inserted in this context
[eta f] == the explicit eta-expansion of f, i.e., fun x => f x
CAVEAT: conditional (non-maximal) implicit arguments
of f are NOT inserted in this context
f1 \o f2  == composition of f1 and f2
note: (f1 \o f2) x simplifies to f1 (f2 x)
pcomp f1 f2 == composition of partial functions f1 and f2
- reserved notation for various arithmetic and algebraic operations
e.[a1, ..., a_n] evaluation (e.g., polynomials)
e`_i indexing (number list, integer pi-part)
x^-1 inverse (group, field)
x *+ n, x *- n integer multiplier (modules and rings)
x ^+ n, x ^- n integer exponent (groups and rings)
x *: A, A :* x external product (scaling/module product in rings,
left/right cosets in groups)
'C(A), 'C_B(A) centralisers (in groups, rings, and matrix algebras)
'Z(A) centers (in groups, rings, and matrix algebras)
m %/ d, m %% d Eclidean division and remainder (nat and polynomial)
d %| m Euclidean divisibility (nat and polynomial)
m = n %[mod d] equality mod d (also defined for <>, ==, and !-)
e^`(n) nth formal derivative (groups, polynomials)
e^`() simple formal derivative (polynomials only)
`|e| absolute value (ordered rings, nat difference)
[rec a1, ..., an] standard shorthand for hidden recursor (see prime.v)
The interpretation of these notations is not defined here, but the
declarations help maintain consistency across the library.
- properties of functions
injective f == f is injective
cancel f g == g is a left inverse of f / f is a right inverse of g
pcancel f g == g is a left inverse of f where g is partial
ocancel f g == g is a left inverse of f where f is partial
bijective f == f is bijective (has a left and right inverse)
involutive f == f is involutive
- properties for operations
left_id e op == e is a left identity for op
right_id e op == e is a right identity for op
left_inverse e i op == i is a left inverse for op with identity e
right_inverse e i op == i is a right inverse for op with identity e
self_inverse e op == each x is its own op-inverse (op x x = e)
idempotent op == op is idempotent for op
associate op == op is associative
commutative op == op is commutative
left_commutative op == op is left commutative
right_commutative op == op is right commutative
left_zero z op == z is a right zero for op
right_zero z op == z is a right zero for op
left_distributive op add == op distributes over add to the left
right_distributive op add == op distributes over add to the right
left_injective op == op is injective in its left argument
right_injective op == op is injective in its right argument
left_loop inv op == op, inv obey the inverse loop left axiom:
op (inv x) (op x y) = y for all x, y, i.e.,
op (inv x) is always a left inverse of op x
rev_left_loop inv op == op, inv obey the inverse loop reverse left
axiom: op x (op (inv x) y) = y, for all x, y
right_loop inv op == op, inv obey the inverse loop right axiom:
op (op x y) (inv y) = x for all x, y
rev_right_loop inv op == op, inv obey the inverse loop reverse right
axiom: op (op x y) (inv y) = x for all x, y
Note that familiar "cancellation" identities like x + y - y = x or
x - y + x = x are respectively instances of right_loop and rev_right_loop
The corresponding lemmas will use the K and KV suffixes, respectively.
- morphisms for functions and relations,
{morph f : x / a >-> r } == f is a morphism with respect to functions
(fun x => a) and (fun x => r)
{morph f : x / a } == f is a morphism with respect to (fun x => a)
{morph f : x y / a >-> r } == f is a morphism with respect to functions
(fun x y => a) and (fun x y => r)
{morph f : x y / a } == f is a morphism with respect to (fun x y => a)

The file also contains some basic lemmas for the above concepts.
```

Delimit Scope fun_scope with FUN.
Open Scope fun_scope.

``` Notations for argument transpose
```
Notation "f ^~ y" := (fun x => f x y)
(at level 10, y at level 8, no associativity, format "f ^~ y") : fun_scope.

Delimit Scope pair_scope with PAIR.
Open Scope pair_scope.

``` Notations for pair projections
```
Notation "p .1" := (fst p)
(at level 2, left associativity, format "p .1") : pair_scope.
Notation "p .2" := (snd p)
(at level 2, left associativity, format "p .2") : pair_scope.

``` Reserved notation for evaluation
```
Reserved Notation "e .[ x ]"
(at level 2, left associativity, format "e .[ x ]").

Reserved Notation "e .[ x1 , x2 , .. , xn ]" (at level 2, left associativity,
format "e '[ ' .[ x1 , '/' x2 , '/' .. , '/' xn ] ']'").

``` Reserved notation for subscripting and superscripting
```
Reserved Notation "s `_ i" (at level 3, i at level 2, left associativity,
format "s `_ i").
Reserved Notation "x ^-1" (at level 3, left associativity, format "x ^-1").

``` Reserved notation for integer multipliers and exponents
```
Reserved Notation "x *+ n" (at level 40, left associativity).
Reserved Notation "x *- n" (at level 40, left associativity).
Reserved Notation "x ^+ n" (at level 29, left associativity).
Reserved Notation "x ^- n" (at level 29, left associativity).

``` Reserved notation for external multiplication.
```
Reserved Notation "x *: A" (at level 40).
Reserved Notation "A :* x" (at level 40).

``` Reserved notation for centralisers and centers.
```
Reserved Notation "''C' ( A )" (at level 8, format "''C' ( A )").
Reserved Notation "''C_' B ( A )"
(at level 8, B at level 2, format "''C_' B ( A )").
Reserved Notation "''Z' ( A )" (at level 8, format "''Z' ( A )").
``` Compatibility with group action centraliser notation.
```
Reserved Notation "''C_' ( B ) ( A )" (at level 8, only parsing).

``` Reserved notation for Euclidean division
```
Reserved Notation "m %/ d" (at level 40, no associativity).
Reserved Notation "m %% d" (at level 40, no associativity).
Reserved Notation "m %| d" (at level 70, no associativity).
Reserved Notation "m = n %[mod d ]" (at level 70, n at next level,
format "'[hv ' m '/' = n '/' %[mod d ] ']'").
Reserved Notation "m == n %[mod d ]" (at level 70, n at next level,
format "'[hv ' m '/' == n '/' %[mod d ] ']'").
Reserved Notation "m <> n %[mod d ]" (at level 70, n at next level,
format "'[hv ' m '/' <> n '/' %[mod d ] ']'").
Reserved Notation "m != n %[mod d ]" (at level 70, n at next level,
format "'[hv ' m '/' != n '/' %[mod d ] ']'").

``` Reserved notation for derivatives
```
Reserved Notation "a ^` ()" (at level 8, format "a ^` ()").
Reserved Notation "a ^` ( n )" (at level 8, format "a ^` ( n )").

``` Reserved notation for absolute value
```
Reserved Notation "`| x |" (at level 0, x at level 99, format "`| x |").

``` Complements on the option type constructor, used below to
encode partial functions.
```

Module Option.

Definition apply aT rT (f : aT -> rT) x u := if u is Some y then f y else x.

Definition default T := apply (fun x : T => x).

Definition bind aT rT (f : aT -> option rT) := apply f None.

Definition map aT rT (f : aT -> rT) := bind (fun x => Some (f x)).

End Option.

Notation oapp := Option.apply.
Notation odflt := Option.default.
Notation obind := Option.bind.
Notation omap := Option.map.
Notation some := (@Some _) (only parsing).

``` Syntax for defining auxiliary recursive function.
Usage:
Section FooDefinition.
Variables (g1 : T1) (g2 : T2).  (globals)
Fixoint foo_auxiliary (a3 : T3) ... :=
body, using [rec e3, ...] for recursive calls
where "[ 'rec' a3 , a4 , ... ]" := foo_auxiliary.
Definition foo x y .. := [rec e1, ...].
+ proofs about foo
End FooDefinition.
```

Reserved Notation "[ 'rec' a0 ]"
(at level 0).
Reserved Notation "[ 'rec' a0 , a1 ]"
(at level 0).
Reserved Notation "[ 'rec' a0 , a1 , a2 ]"
(at level 0).
Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 ]"
(at level 0).
Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 ]"
(at level 0).
Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 ]"
(at level 0).
Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 ]"
(at level 0).
Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ]"
(at level 0).
Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 ]"
(at level 0).

``` Definitions and notation for explicit functions with simplification,
i.e., which simpl and /= beta expand (this is complementary to nosimpl).
```

Section SimplFun.

Variables aT rT : Type.

CoInductive simpl_fun : Type := SimplFun of aT -> rT.

Definition fun_of_simpl := fun f x => let: SimplFun lam := f in lam x.

Coercion fun_of_simpl : simpl_fun >-> Funclass.

End SimplFun.

Notation "[ 'fun' : T => E ]" := (SimplFun (fun _ : T => E))
(at level 0,
format "'[hv' [ 'fun' : T => '/ ' E ] ']'") : fun_scope.

Notation "[ 'fun' x => E ]" := (SimplFun (fun x => E))
(at level 0, x ident,
format "'[hv' [ 'fun' x => '/ ' E ] ']'") : fun_scope.

Notation "[ 'fun' x : T => E ]" := (SimplFun (fun x : T => E))
(at level 0, x ident, only parsing) : fun_scope.

Notation "[ 'fun' x y => E ]" := (fun x => [fun y => E])
(at level 0, x ident, y ident,
format "'[hv' [ 'fun' x y => '/ ' E ] ']'") : fun_scope.

Notation "[ 'fun' x y : T => E ]" := (fun x : T => [fun y : T => E])
(at level 0, x ident, y ident, only parsing) : fun_scope.

Notation "[ 'fun' ( x : T ) y => E ]" := (fun x : T => [fun y => E])
(at level 0, x ident, y ident, only parsing) : fun_scope.

Notation "[ 'fun' x ( y : T ) => E ]" := (fun x => [fun y : T => E])
(at level 0, x ident, y ident, only parsing) : fun_scope.

Notation "[ 'fun' ( x : xT ) ( y : yT ) => E ]" :=
(fun x : xT => [fun y : yT => E])
(at level 0, x ident, y ident, only parsing) : fun_scope.

``` For delta functions in eqtype.v.
```
Definition SimplFunDelta aT rT (f : aT -> aT -> rT) := [fun z => f z z].

``` Shorthand for some basic equality lemmas.
```

Definition erefl := refl_equal.
Definition esym := sym_eq.
Definition nesym := sym_not_eq.
Definition etrans := trans_eq.
Definition congr1 := f_equal.
Definition congr2 := f_equal2.
``` Force at least one implicit when used as a view.
```

``` A predicate for singleton types.
```
Definition all_equal_to T (x0 : T) := forall x, x = x0.

Lemma unitE : all_equal_to tt.

``` A generic wrapper type
```

Structure wrapped T := Wrap {unwrap : T}.
Canonical Structure wrap T x := @Wrap T x.

``` Extensional equality, for unary and binary functions, including syntactic
sugar.
```

Section ExtensionalEquality.

Variables A B C : Type.

Definition eqfun (f g : B -> A) : Prop := forall x, f x = g x.

Definition eqrel (r s : C -> B -> A) : Prop := forall x y, r x y = s x y.

Lemma frefl : forall f, eqfun f f.

Lemma fsym : forall f g, eqfun f g -> eqfun g f.

Lemma ftrans : forall f g h, eqfun f g -> eqfun g h -> eqfun f h.

Lemma rrefl : forall r, eqrel r r.

End ExtensionalEquality.

Hint Resolve frefl rrefl.

Notation "f1 =1 f2" := (eqfun f1 f2)
(at level 70, no associativity) : fun_scope.
Notation "f1 =1 f2 :> A" := (f1 =1 (f2 : A))
(at level 70, f2 at next level, A at level 90) : fun_scope.
Notation "f1 =2 f2" := (eqrel f1 f2)
(at level 70, no associativity) : fun_scope.
Notation "f1 =2 f2 :> A" := (f1 =2 (f2 : A))
(at level 70, f2 at next level, A, B at level 90) : fun_scope.

Section Composition.

Variables A B C : Type.

Definition funcomp u (f : B -> A) (g : C -> B) x := let: tt := u in f (g x).
Local Notation comp := (funcomp tt).

Definition pcomp (f : B -> option A) (g : C -> option B) x := obind f (g x).

Lemma eq_comp : forall f f' g g', f =1 f' -> g =1 g' -> comp f g =1 comp f' g'.

End Composition.

Notation "[ 'eta' f ]" := (fun x => f x)
(at level 0, format "[ 'eta' f ]") : fun_scope.

Notation id := (fun x => x).
Notation "@ 'id' T " := (fun x : T => x)
(at level 10, T at level 8, only parsing) : fun_scope.

Notation comp := (funcomp tt).
Notation "@ 'comp'" := (fun A B C => @funcomp A B C tt).
Notation "f1 \o f2" := (comp f1 f2) (at level 50) : fun_scope.

Definition idfun T := @id T.

Definition phant_id T1 T2 v1 v2 := phantom T1 v1 -> phantom T2 v2.

Section Morphism.

Variables (aT rT sT : Type) (f : aT -> rT).

Definition morphism_1 aF rF := forall x, f (aF x) = rF (f x).
Definition morphism_2 aOp rOp := forall x y, f (aOp x y) = rOp (f x) (f y).

End Morphism.

Notation "{ 'morph' f : x / a >-> r }" :=
(morphism_1 f (fun x => a) (fun x => r))
(at level 0, f at level 99, x ident,
format "{ 'morph' f : x / a >-> r }") : type_scope.

Notation "{ 'morph' f : x / a }" :=
(morphism_1 f (fun x => a) (fun x => a))
(at level 0, f at level 99, x ident,
format "{ 'morph' f : x / a }") : type_scope.

Notation "{ 'morph' f : x y / a >-> r }" :=
(morphism_2 f (fun x y => a) (fun x y => r))
(at level 0, f at level 99, x ident, y ident,
format "{ 'morph' f : x y / a >-> r }") : type_scope.

Notation "{ 'morph' f : x y / a }" :=
(morphism_2 f (fun x y => a) (fun x y => a))
(at level 0, f at level 99, x ident, y ident,
format "{ 'morph' f : x y / a }") : type_scope.

``` In an intuitionistic setting, we have two degrees of injectivity. The
weaker one gives only simplification, and the strong one provides a left
inverse (we show in `fintype' that they coincide for finite types).
We also define an intermediate version where the left inverse is only a
partial function.
```

Section Injections.

``` rT must come first so we can use @ to mitigate the Coq 1st order
unification bug (e..g., Coq can't infer rT from a "cancel" lemma).
```
Variables (rT aT : Type) (f : aT -> rT).

Definition injective := forall x1 x2, f x1 = f x2 -> x1 = x2.

Definition cancel g := forall x, g (f x) = x.

Definition pcancel g := forall x, g (f x) = Some x.

Definition ocancel (g : aT -> option rT) h := forall x, oapp h x (g x) = x.

Lemma can_pcan : forall g, cancel g -> pcancel (fun y => Some (g y)).

Lemma pcan_inj : forall g, pcancel g -> injective.

Lemma can_inj : forall g, cancel g -> injective.

Lemma canLR : forall g x y, cancel g -> x = f y -> g x = y.

Lemma canRL : forall g x y, cancel g -> f x = y -> x = g y.

End Injections.

``` cancellation lemmas for dependent type casts.
```
Lemma esymK : forall T x y, cancel (@esym T x y) (@esym T y x).

Lemma etrans_id : forall T x y (eqxy : x = y :> T),
etrans (erefl x) eqxy = eqxy.

Section InjectionsTheory.

Variables (A B C : Type) (f g : B -> A) (h : C -> B).

Lemma inj_id : injective (@id A).

Lemma inj_can_sym : forall f', cancel f f' -> injective f' -> cancel f' f.

Lemma inj_comp : injective f -> injective h -> injective (f \o h).

Lemma can_comp : forall f' h',
cancel f f' -> cancel h h' -> cancel (f \o h) (h' \o f').

Lemma pcan_pcomp : forall f' h',
pcancel f f' -> pcancel h h' -> pcancel (f \o h) (pcomp h' f').

Lemma eq_inj : injective f -> f =1 g -> injective g.

Lemma eq_can : forall f' g', cancel f f' -> f =1 g -> f' =1 g' -> cancel g g'.

Lemma inj_can_eq : forall f',
cancel f f' -> injective f' -> cancel g f' -> f =1 g.

End InjectionsTheory.

Section Bijections.

Variables (A B : Type) (f : B -> A).

CoInductive bijective : Prop := Bijective g of cancel f g & cancel g f.

Hypothesis bijf : bijective.

Lemma bij_inj : injective f.

Lemma bij_can_sym : forall f', cancel f' f <-> cancel f f'.

Lemma bij_can_eq : forall f' f'', cancel f f' -> cancel f f'' -> f' =1 f''.

End Bijections.

Section BijectionsTheory.

Variables (A B C : Type) (f : B -> A) (h : C -> B).

Lemma eq_bij : bijective f -> forall g, f =1 g -> bijective g.

Lemma bij_comp : bijective f -> bijective h -> bijective (f \o h).

Lemma bij_can_bij : bijective f -> forall f', cancel f f' -> bijective f'.

End BijectionsTheory.

Section Involutions.

Variables (A : Type) (f : A -> A).

Definition involutive := cancel f f.

Hypothesis Hf : involutive.

Lemma inv_inj : injective f.

Lemma inv_bij : bijective f.

End Involutions.

Section OperationProperties.

Variables S T R : Type.

Section SopTisR.
Implicit Type op : S -> T -> R.
Definition left_inverse e inv op := forall x, op (inv x) x = e.
Definition right_inverse e inv op := forall x, op x (inv x) = e.
Definition left_injective op := forall x, injective (op^~ x).
Definition right_injective op := forall y, injective (op y).
End SopTisR.

Section SopTisS.
Implicit Type op : S -> T -> S.
Definition right_id e op := forall x, op x e = x.
Definition left_zero z op := forall x, op z x = z.
Definition right_commutative op := forall x y z, op (op x y) z = op (op x z) y.
Definition left_distributive op add :=
forall x y z, op (add x y) z = add (op x z) (op y z).
Definition right_loop inv op := forall y, cancel (op^~ y) (op^~ (inv y)).
Definition rev_right_loop inv op := forall y, cancel (op^~ (inv y)) (op^~ y).
End SopTisS.

Section SopTisT.
Implicit Type op : S -> T -> T.
Definition left_id e op := forall x, op e x = x.
Definition right_zero z op := forall x, op x z = z.
Definition left_commutative op := forall x y z, op x (op y z) = op y (op x z).
Definition right_distributive op add :=
forall x y z, op x (add y z) = add (op x y) (op x z).
Definition left_loop inv op := forall x, cancel (op x) (op (inv x)).
Definition rev_left_loop inv op := forall x, cancel (op (inv x)) (op x).
End SopTisT.

Section SopSisT.
Implicit Type op : S -> S -> T.
Definition self_inverse e op := forall x, op x x = e.
Definition commutative op := forall x y, op x y = op y x.
End SopSisT.

Section SopSisS.
Implicit Type op : S -> S -> S.
Definition idempotent op := forall x, op x x = x.
Definition associative op := forall x y z, op x (op y z) = op (op x y) z.
End SopSisS.

End OperationProperties.