Library mathcomp.ssreflect.bigop
(* (c) Copyright 2006-2015 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B. *)
Require Import mathcomp.ssreflect.ssreflect.
Distributed under the terms of CeCILL-B. *)
Require Import mathcomp.ssreflect.ssreflect.
This file provides a generic definition for iterating an operator over a
set of indices (bigop); this big operator is parametrized by the return
type (R), the type of indices (I), the operator (op), the default value on
empty lists (idx), the range of indices (r), the filter applied on this
range (P) and the expression we are iterating (F). The definition is not
to be used directly, but via the wide range of notations provided and
and which support a natural use of big operators.
To improve performance of the Coq typechecker on large expressions, the
bigop constant is OPAQUE. It can however be unlocked to reveal the
transparent constant reducebig, to let Coq expand summation on an explicit
sequence with an explicit test.
The lemmas can be classified according to the operator being iterated:
1. Results independent of the operator: extensionality with respect to
the range of indices, to the filtering predicate or to the expression
being iterated; reindexing, widening or narrowing of the range of
indices; we provide lemmas for the special cases where indices are
natural numbers or bounded natural numbers ("ordinals"). We supply
several "functional" induction principles that can be used with the
ssreflect 1.3 "elim" tactic to do induction over the index range for
up to 3 bigops simultaneously.
2. Results depending on the properties of the operator:
We distinguish: monoid laws (op is associative, idx is an identity
element), abelian monoid laws (op is also commutative), and laws with
a distributive operation (semi-rings). Examples of such results are
splitting, permuting, and exchanging bigops.
A special section is dedicated to big operators on natural numbers.
Notations:
The general form for iterated operators is
<bigop>_<range> <general_term>
Tips for using lemmas in this file:
To apply a lemma for a specific operator: if no special property is
required for the operator, simply apply the lemma; if the lemma needs
certain properties for the operator, make sure the appropriate Canonical
instances are declared.
Interfaces for operator properties are packaged in the Monoid submodule:
Monoid.law idx == interface (keyed on the operator) for associative
operators with identity element idx.
Monoid.com_law idx == extension (telescope) of Monoid.law for operators
that are also commutative.
Monoid.mul_law abz == interface for operators with absorbing (zero)
element abz.
Monoid.add_law idx mop == extension of Monoid.com_law for operators over
which operation mop distributes (mop will often also
have a Monoid.mul_law idx structure).
[law of op], [com_law of op], [mul_law of op], [add_law mop of op] ==
syntax for cloning Monoid structures.
Monoid.Theory == submodule containing basic generic algebra lemmas
for operators satisfying the Monoid interfaces.
Monoid.simpm == generic monoid simplification rewrite multirule.
Monoid structures are predeclared for many basic operators: (_ && _)%B,
(_ || _)%B, (_ (+) _)%B (exclusive or) , (_ + _)%N, (_ * _)%N, maxn,
gcdn, lcmn and (_ ++ _)%SEQ (list concatenation).
Additional documentation for this file:
Y. Bertot, G. Gonthier, S. Ould Biha and I. Pasca.
Canonical Big Operators. In TPHOLs 2008, LNCS vol. 5170, Springer.
Article available at:
http://hal.inria.fr/docs/00/33/11/93/PDF/main.pdf
Examples of use in: poly.v, matrix.v
- <bigop> is one of \big[op/idx], \sum, \prod, or \max (see below).
- <general_term> can be any expression.
- <range> binds an index variable in <general_term>; <range> is one of (i <- s) i ranges over the sequence s. (m <= i < n) i ranges over the nat interval m, m+1, ..., n-1. (i < n) i ranges over the (finite) type 'I_n (i.e., ordinal n). (i : T) i ranges over the finite type T. i or (i) i ranges over its (inferred) finite type. (i in A) i ranges over the elements that satisfy the collective predicate A (the domain of A must be a finite type). (i <- s | <condition>) limits the range to the i for which <condition> holds. <condition> can be any expression that coerces to bool, and may mention the bound index i. All six kinds of ranges above can have a <condition> part.
- One can use the "\big[op/idx]" notations for any operator.
- BIG_F and BIG_P are pattern abbreviations for the <general_term> and <condition> part of a \big ... expression; for (i in A) and (i in A | C) ranges the term matched by BIG_P will include the i \in A condition.
- The (locked) head constant of a \big notation is bigop.
- The "\sum", "\prod" and "\max" notations in the %N scope are used for natural numbers with addition, multiplication and maximum (and their corresponding neutral elements), respectively.
- The "\sum" and "\prod" reserved notations are overloaded in ssralg in the %R scope; in mxalgebra, vector & falgebra in the %MS and %VS scopes; "\prod" is also overloaded in fingroup, in the %g and %G scopes.
- We reserve "\bigcup" and "\bigcap" notations for iterated union and intersection (of sets, groups, vector spaces, etc).
Set Implicit Arguments.
Reserved Notation "\big [ op / idx ]_ i F"
(at level 36, F at level 36, op, idx at level 10, i at level 0,
right associativity,
format "'[' \big [ op / idx ]_ i '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i <- r | P ) F"
(at level 36, F at level 36, op, idx at level 10, i, r at level 50,
format "'[' \big [ op / idx ]_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i <- r ) F"
(at level 36, F at level 36, op, idx at level 10, i, r at level 50,
format "'[' \big [ op / idx ]_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( m <= i < n | P ) F"
(at level 36, F at level 36, op, idx at level 10, m, i, n at level 50,
format "'[' \big [ op / idx ]_ ( m <= i < n | P ) F ']'").
Reserved Notation "\big [ op / idx ]_ ( m <= i < n ) F"
(at level 36, F at level 36, op, idx at level 10, i, m, n at level 50,
format "'[' \big [ op / idx ]_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i | P ) F"
(at level 36, F at level 36, op, idx at level 10, i at level 50,
format "'[' \big [ op / idx ]_ ( i | P ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i : t | P ) F"
(at level 36, F at level 36, op, idx at level 10, i at level 50,
format "'[' \big [ op / idx ]_ ( i : t | P ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i : t ) F"
(at level 36, F at level 36, op, idx at level 10, i at level 50,
format "'[' \big [ op / idx ]_ ( i : t ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i < n | P ) F"
(at level 36, F at level 36, op, idx at level 10, i, n at level 50,
format "'[' \big [ op / idx ]_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i < n ) F"
(at level 36, F at level 36, op, idx at level 10, i, n at level 50,
format "'[' \big [ op / idx ]_ ( i < n ) F ']'").
Reserved Notation "\big [ op / idx ]_ ( i 'in' A | P ) F"
(at level 36, F at level 36, op, idx at level 10, i, A at level 50,
format "'[' \big [ op / idx ]_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\big [ op / idx ]_ ( i 'in' A ) F"
(at level 36, F at level 36, op, idx at level 10, i, A at level 50,
format "'[' \big [ op / idx ]_ ( i 'in' A ) '/ ' F ']'").
Reserved Notation "\sum_ i F"
(at level 41, F at level 41, i at level 0,
right associativity,
format "'[' \sum_ i '/ ' F ']'").
Reserved Notation "\sum_ ( i <- r | P ) F"
(at level 41, F at level 41, i, r at level 50,
format "'[' \sum_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\sum_ ( i <- r ) F"
(at level 41, F at level 41, i, r at level 50,
format "'[' \sum_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\sum_ ( m <= i < n | P ) F"
(at level 41, F at level 41, i, m, n at level 50,
format "'[' \sum_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\sum_ ( m <= i < n ) F"
(at level 41, F at level 41, i, m, n at level 50,
format "'[' \sum_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\sum_ ( i | P ) F"
(at level 41, F at level 41, i at level 50,
format "'[' \sum_ ( i | P ) '/ ' F ']'").
Reserved Notation "\sum_ ( i : t | P ) F"
(at level 41, F at level 41, i at level 50,
only parsing).
Reserved Notation "\sum_ ( i : t ) F"
(at level 41, F at level 41, i at level 50,
only parsing).
Reserved Notation "\sum_ ( i < n | P ) F"
(at level 41, F at level 41, i, n at level 50,
format "'[' \sum_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\sum_ ( i < n ) F"
(at level 41, F at level 41, i, n at level 50,
format "'[' \sum_ ( i < n ) '/ ' F ']'").
Reserved Notation "\sum_ ( i 'in' A | P ) F"
(at level 41, F at level 41, i, A at level 50,
format "'[' \sum_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\sum_ ( i 'in' A ) F"
(at level 41, F at level 41, i, A at level 50,
format "'[' \sum_ ( i 'in' A ) '/ ' F ']'").
Reserved Notation "\max_ i F"
(at level 41, F at level 41, i at level 0,
format "'[' \max_ i '/ ' F ']'").
Reserved Notation "\max_ ( i <- r | P ) F"
(at level 41, F at level 41, i, r at level 50,
format "'[' \max_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\max_ ( i <- r ) F"
(at level 41, F at level 41, i, r at level 50,
format "'[' \max_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\max_ ( m <= i < n | P ) F"
(at level 41, F at level 41, i, m, n at level 50,
format "'[' \max_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\max_ ( m <= i < n ) F"
(at level 41, F at level 41, i, m, n at level 50,
format "'[' \max_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\max_ ( i | P ) F"
(at level 41, F at level 41, i at level 50,
format "'[' \max_ ( i | P ) '/ ' F ']'").
Reserved Notation "\max_ ( i : t | P ) F"
(at level 41, F at level 41, i at level 50,
only parsing).
Reserved Notation "\max_ ( i : t ) F"
(at level 41, F at level 41, i at level 50,
only parsing).
Reserved Notation "\max_ ( i < n | P ) F"
(at level 41, F at level 41, i, n at level 50,
format "'[' \max_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\max_ ( i < n ) F"
(at level 41, F at level 41, i, n at level 50,
format "'[' \max_ ( i < n ) F ']'").
Reserved Notation "\max_ ( i 'in' A | P ) F"
(at level 41, F at level 41, i, A at level 50,
format "'[' \max_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\max_ ( i 'in' A ) F"
(at level 41, F at level 41, i, A at level 50,
format "'[' \max_ ( i 'in' A ) '/ ' F ']'").
Reserved Notation "\prod_ i F"
(at level 36, F at level 36, i at level 0,
format "'[' \prod_ i '/ ' F ']'").
Reserved Notation "\prod_ ( i <- r | P ) F"
(at level 36, F at level 36, i, r at level 50,
format "'[' \prod_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\prod_ ( i <- r ) F"
(at level 36, F at level 36, i, r at level 50,
format "'[' \prod_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\prod_ ( m <= i < n | P ) F"
(at level 36, F at level 36, i, m, n at level 50,
format "'[' \prod_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\prod_ ( m <= i < n ) F"
(at level 36, F at level 36, i, m, n at level 50,
format "'[' \prod_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\prod_ ( i | P ) F"
(at level 36, F at level 36, i at level 50,
format "'[' \prod_ ( i | P ) '/ ' F ']'").
Reserved Notation "\prod_ ( i : t | P ) F"
(at level 36, F at level 36, i at level 50,
only parsing).
Reserved Notation "\prod_ ( i : t ) F"
(at level 36, F at level 36, i at level 50,
only parsing).
Reserved Notation "\prod_ ( i < n | P ) F"
(at level 36, F at level 36, i, n at level 50,
format "'[' \prod_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\prod_ ( i < n ) F"
(at level 36, F at level 36, i, n at level 50,
format "'[' \prod_ ( i < n ) '/ ' F ']'").
Reserved Notation "\prod_ ( i 'in' A | P ) F"
(at level 36, F at level 36, i, A at level 50,
format "'[' \prod_ ( i 'in' A | P ) F ']'").
Reserved Notation "\prod_ ( i 'in' A ) F"
(at level 36, F at level 36, i, A at level 50,
format "'[' \prod_ ( i 'in' A ) '/ ' F ']'").
Reserved Notation "\bigcup_ i F"
(at level 41, F at level 41, i at level 0,
format "'[' \bigcup_ i '/ ' F ']'").
Reserved Notation "\bigcup_ ( i <- r | P ) F"
(at level 41, F at level 41, i, r at level 50,
format "'[' \bigcup_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\bigcup_ ( i <- r ) F"
(at level 41, F at level 41, i, r at level 50,
format "'[' \bigcup_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\bigcup_ ( m <= i < n | P ) F"
(at level 41, F at level 41, m, i, n at level 50,
format "'[' \bigcup_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\bigcup_ ( m <= i < n ) F"
(at level 41, F at level 41, i, m, n at level 50,
format "'[' \bigcup_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\bigcup_ ( i | P ) F"
(at level 41, F at level 41, i at level 50,
format "'[' \bigcup_ ( i | P ) '/ ' F ']'").
Reserved Notation "\bigcup_ ( i : t | P ) F"
(at level 41, F at level 41, i at level 50,
format "'[' \bigcup_ ( i : t | P ) '/ ' F ']'").
Reserved Notation "\bigcup_ ( i : t ) F"
(at level 41, F at level 41, i at level 50,
format "'[' \bigcup_ ( i : t ) '/ ' F ']'").
Reserved Notation "\bigcup_ ( i < n | P ) F"
(at level 41, F at level 41, i, n at level 50,
format "'[' \bigcup_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\bigcup_ ( i < n ) F"
(at level 41, F at level 41, i, n at level 50,
format "'[' \bigcup_ ( i < n ) '/ ' F ']'").
Reserved Notation "\bigcup_ ( i 'in' A | P ) F"
(at level 41, F at level 41, i, A at level 50,
format "'[' \bigcup_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\bigcup_ ( i 'in' A ) F"
(at level 41, F at level 41, i, A at level 50,
format "'[' \bigcup_ ( i 'in' A ) '/ ' F ']'").
Reserved Notation "\bigcap_ i F"
(at level 41, F at level 41, i at level 0,
format "'[' \bigcap_ i '/ ' F ']'").
Reserved Notation "\bigcap_ ( i <- r | P ) F"
(at level 41, F at level 41, i, r at level 50,
format "'[' \bigcap_ ( i <- r | P ) F ']'").
Reserved Notation "\bigcap_ ( i <- r ) F"
(at level 41, F at level 41, i, r at level 50,
format "'[' \bigcap_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\bigcap_ ( m <= i < n | P ) F"
(at level 41, F at level 41, m, i, n at level 50,
format "'[' \bigcap_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\bigcap_ ( m <= i < n ) F"
(at level 41, F at level 41, i, m, n at level 50,
format "'[' \bigcap_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\bigcap_ ( i | P ) F"
(at level 41, F at level 41, i at level 50,
format "'[' \bigcap_ ( i | P ) '/ ' F ']'").
Reserved Notation "\bigcap_ ( i : t | P ) F"
(at level 41, F at level 41, i at level 50,
format "'[' \bigcap_ ( i : t | P ) '/ ' F ']'").
Reserved Notation "\bigcap_ ( i : t ) F"
(at level 41, F at level 41, i at level 50,
format "'[' \bigcap_ ( i : t ) '/ ' F ']'").
Reserved Notation "\bigcap_ ( i < n | P ) F"
(at level 41, F at level 41, i, n at level 50,
format "'[' \bigcap_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\bigcap_ ( i < n ) F"
(at level 41, F at level 41, i, n at level 50,
format "'[' \bigcap_ ( i < n ) '/ ' F ']'").
Reserved Notation "\bigcap_ ( i 'in' A | P ) F"
(at level 41, F at level 41, i, A at level 50,
format "'[' \bigcap_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\bigcap_ ( i 'in' A ) F"
(at level 41, F at level 41, i, A at level 50,
format "'[' \bigcap_ ( i 'in' A ) '/ ' F ']'").
Module Monoid.
Section Definitions.
Variables (T : Type) (idm : T).
Structure law := Law {
operator : T → T → T;
_ : associative operator;
_ : left_id idm operator;
_ : right_id idm operator
}.
Structure com_law := ComLaw {
com_operator : law;
_ : commutative com_operator
}.
Structure mul_law := MulLaw {
mul_operator : T → T → T;
_ : left_zero idm mul_operator;
_ : right_zero idm mul_operator
}.
Structure add_law (mul : T → T → T) := AddLaw {
add_operator : com_law;
_ : left_distributive mul add_operator;
_ : right_distributive mul add_operator
}.
Let op_id (op1 op2 : T → T → T) := phant_id op1 op2.
Definition clone_law op :=
fun (opL : law) & op_id opL op ⇒
fun opmA op1m opm1 (opL' := @Law op opmA op1m opm1)
& phant_id opL' opL ⇒ opL'.
Definition clone_com_law op :=
fun (opL : law) (opC : com_law) & op_id opL op & op_id opC op ⇒
fun opmC (opC' := @ComLaw opL opmC) & phant_id opC' opC ⇒ opC'.
Definition clone_mul_law op :=
fun (opM : mul_law) & op_id opM op ⇒
fun op0m opm0 (opM' := @MulLaw op op0m opm0) & phant_id opM' opM ⇒ opM'.
Definition clone_add_law mop aop :=
fun (opC : com_law) (opA : add_law mop) & op_id opC aop & op_id opA aop ⇒
fun mopDm mopmD (opA' := @AddLaw mop opC mopDm mopmD)
& phant_id opA' opA ⇒ opA'.
End Definitions.
Module Import Exports.
Coercion operator : law >-> Funclass.
Coercion com_operator : com_law >-> law.
Coercion mul_operator : mul_law >-> Funclass.
Coercion add_operator : add_law >-> com_law.
Notation "[ 'law' 'of' f ]" := (@clone_law _ _ f _ id _ _ _ id)
(at level 0, format"[ 'law' 'of' f ]") : form_scope.
Notation "[ 'com_law' 'of' f ]" := (@clone_com_law _ _ f _ _ id id _ id)
(at level 0, format "[ 'com_law' 'of' f ]") : form_scope.
Notation "[ 'mul_law' 'of' f ]" := (@clone_mul_law _ _ f _ id _ _ id)
(at level 0, format"[ 'mul_law' 'of' f ]") : form_scope.
Notation "[ 'add_law' m 'of' a ]" := (@clone_add_law _ _ m a _ _ id id _ _ id)
(at level 0, format "[ 'add_law' m 'of' a ]") : form_scope.
End Exports.
Section CommutativeAxioms.
Variable (T : Type) (zero one : T) (mul add : T → T → T) (inv : T → T).
Hypothesis mulC : commutative mul.
Lemma mulC_id : left_id one mul → right_id one mul.
Lemma mulC_zero : left_zero zero mul → right_zero zero mul.
Lemma mulC_dist : left_distributive mul add → right_distributive mul add.
End CommutativeAxioms.
Module Theory.
Section Theory.
Variables (T : Type) (idm : T).
Section Plain.
Variable mul : law idm.
Lemma mul1m : left_id idm mul.
Lemma mulm1 : right_id idm mul.
Lemma mulmA : associative mul.
Lemma iteropE n x : iterop n mul x idm = iter n (mul x) idm.
End Plain.
Section Commutative.
Variable mul : com_law idm.
Lemma mulmC : commutative mul.
Lemma mulmCA : left_commutative mul.
Lemma mulmAC : right_commutative mul.
Lemma mulmACA : interchange mul mul.
End Commutative.
Section Mul.
Variable mul : mul_law idm.
Lemma mul0m : left_zero idm mul.
Lemma mulm0 : right_zero idm mul.
End Mul.
Section Add.
Variables (mul : T → T → T) (add : add_law idm mul).
Lemma addmA : associative add.
Lemma addmC : commutative add.
Lemma addmCA : left_commutative add.
Lemma addmAC : right_commutative add.
Lemma add0m : left_id idm add.
Lemma addm0 : right_id idm add.
Lemma mulm_addl : left_distributive mul add.
Lemma mulm_addr : right_distributive mul add.
End Add.
Definition simpm := (mulm1, mulm0, mul1m, mul0m, mulmA).
End Theory.
End Theory.
Include Theory.
End Monoid.
Export Monoid.Exports.
Section PervasiveMonoids.
Import Monoid.
Canonical andb_monoid := Law andbA andTb andbT.
Canonical andb_comoid := ComLaw andbC.
Canonical andb_muloid := MulLaw andFb andbF.
Canonical orb_monoid := Law orbA orFb orbF.
Canonical orb_comoid := ComLaw orbC.
Canonical orb_muloid := MulLaw orTb orbT.
Canonical addb_monoid := Law addbA addFb addbF.
Canonical addb_comoid := ComLaw addbC.
Canonical orb_addoid := AddLaw andb_orl andb_orr.
Canonical andb_addoid := AddLaw orb_andl orb_andr.
Canonical addb_addoid := AddLaw andb_addl andb_addr.
Canonical addn_monoid := Law addnA add0n addn0.
Canonical addn_comoid := ComLaw addnC.
Canonical muln_monoid := Law mulnA mul1n muln1.
Canonical muln_comoid := ComLaw mulnC.
Canonical muln_muloid := MulLaw mul0n muln0.
Canonical addn_addoid := AddLaw mulnDl mulnDr.
Canonical maxn_monoid := Law maxnA max0n maxn0.
Canonical maxn_comoid := ComLaw maxnC.
Canonical maxn_addoid := AddLaw maxn_mull maxn_mulr.
Canonical gcdn_monoid := Law gcdnA gcd0n gcdn0.
Canonical gcdn_comoid := ComLaw gcdnC.
Canonical gcdnDoid := AddLaw muln_gcdl muln_gcdr.
Canonical lcmn_monoid := Law lcmnA lcm1n lcmn1.
Canonical lcmn_comoid := ComLaw lcmnC.
Canonical lcmn_addoid := AddLaw muln_lcml muln_lcmr.
Canonical cat_monoid T := Law (@catA T) (@cat0s T) (@cats0 T).
End PervasiveMonoids.
Unit test for the [...law of ... ] Notations
Definition myp := addn. Definition mym := muln.
Canonical myp_mon := [law of myp].
Canonical myp_cmon := [com_law of myp].
Canonical mym_mul := [mul_law of mym].
Canonical myp_add := [add_law _ of myp].
Print myp_add.
Print Canonical Projections.
Delimit Scope big_scope with BIG.
Open Scope big_scope.
The bigbody wrapper is a workaround for a quirk of the Coq pretty-printer,
which would fail to redisplay the \big notation when the <general_term> or
<condition> do not depend on the bound index. The BigBody constructor
packages both in in a term in which i occurs; it also depends on the
iterated <op>, as this can give more information on the expected type of
the <general_term>, thus allowing for the insertion of coercions.
CoInductive bigbody R I := BigBody of I & (R → R → R) & bool & R.
Definition applybig {R I} (body : bigbody R I) x :=
let: BigBody _ op b v := body in if b then op v x else x.
Definition reducebig R I idx r (body : I → bigbody R I) :=
foldr (applybig \o body) idx r.
Module Type BigOpSig.
Parameter bigop : ∀ R I, R → seq I → (I → bigbody R I) → R.
Axiom bigopE : bigop = reducebig.
End BigOpSig.
Module BigOp : BigOpSig.
Definition bigop := reducebig.
Lemma bigopE : bigop = reducebig.
End BigOp.
Notation bigop := BigOp.bigop (only parsing).
Canonical bigop_unlock := Unlockable BigOp.bigopE.
Definition index_iota m n := iota m (n - m).
Definition index_enum (T : finType) := Finite.enum T.
Lemma mem_index_iota m n i : i \in index_iota m n = (m ≤ i < n).
Lemma mem_index_enum T i : i \in index_enum T.
Hint Resolve mem_index_enum.
Lemma filter_index_enum T P : filter P (index_enum T) = enum P.
Notation "\big [ op / idx ]_ ( i <- r | P ) F" :=
(bigop idx r (fun i ⇒ BigBody i op P%B F)) : big_scope.
Notation "\big [ op / idx ]_ ( i <- r ) F" :=
(bigop idx r (fun i ⇒ BigBody i op true F)) : big_scope.
Notation "\big [ op / idx ]_ ( m <= i < n | P ) F" :=
(bigop idx (index_iota m n) (fun i : nat ⇒ BigBody i op P%B F))
: big_scope.
Notation "\big [ op / idx ]_ ( m <= i < n ) F" :=
(bigop idx (index_iota m n) (fun i : nat ⇒ BigBody i op true F))
: big_scope.
Notation "\big [ op / idx ]_ ( i | P ) F" :=
(bigop idx (index_enum _) (fun i ⇒ BigBody i op P%B F)) : big_scope.
Notation "\big [ op / idx ]_ i F" :=
(bigop idx (index_enum _) (fun i ⇒ BigBody i op true F)) : big_scope.
Notation "\big [ op / idx ]_ ( i : t | P ) F" :=
(bigop idx (index_enum _) (fun i : t ⇒ BigBody i op P%B F))
(only parsing) : big_scope.
Notation "\big [ op / idx ]_ ( i : t ) F" :=
(bigop idx (index_enum _) (fun i : t ⇒ BigBody i op true F))
(only parsing) : big_scope.
Notation "\big [ op / idx ]_ ( i < n | P ) F" :=
(\big[op/idx]_(i : ordinal n | P%B) F) : big_scope.
Notation "\big [ op / idx ]_ ( i < n ) F" :=
(\big[op/idx]_(i : ordinal n) F) : big_scope.
Notation "\big [ op / idx ]_ ( i 'in' A | P ) F" :=
(\big[op/idx]_(i | (i \in A) && P) F) : big_scope.
Notation "\big [ op / idx ]_ ( i 'in' A ) F" :=
(\big[op/idx]_(i | i \in A) F) : big_scope.
Notation BIG_F := (F in \big[_/_]_(i <- _ | _) F i)%pattern.
Notation BIG_P := (P in \big[_/_]_(i <- _ | P i) _)%pattern.
Notation "\sum_ ( i <- r | P ) F" :=
(\big[+%N/0%N]_(i <- r | P%B) F%N) : nat_scope.
Notation "\sum_ ( i <- r ) F" :=
(\big[+%N/0%N]_(i <- r) F%N) : nat_scope.
Notation "\sum_ ( m <= i < n | P ) F" :=
(\big[+%N/0%N]_(m ≤ i < n | P%B) F%N) : nat_scope.
Notation "\sum_ ( m <= i < n ) F" :=
(\big[+%N/0%N]_(m ≤ i < n) F%N) : nat_scope.
Notation "\sum_ ( i | P ) F" :=
(\big[+%N/0%N]_(i | P%B) F%N) : nat_scope.
Notation "\sum_ i F" :=
(\big[+%N/0%N]_i F%N) : nat_scope.
Notation "\sum_ ( i : t | P ) F" :=
(\big[+%N/0%N]_(i : t | P%B) F%N) (only parsing) : nat_scope.
Notation "\sum_ ( i : t ) F" :=
(\big[+%N/0%N]_(i : t) F%N) (only parsing) : nat_scope.
Notation "\sum_ ( i < n | P ) F" :=
(\big[+%N/0%N]_(i < n | P%B) F%N) : nat_scope.
Notation "\sum_ ( i < n ) F" :=
(\big[+%N/0%N]_(i < n) F%N) : nat_scope.
Notation "\sum_ ( i 'in' A | P ) F" :=
(\big[+%N/0%N]_(i in A | P%B) F%N) : nat_scope.
Notation "\sum_ ( i 'in' A ) F" :=
(\big[+%N/0%N]_(i in A) F%N) : nat_scope.
Notation "\prod_ ( i <- r | P ) F" :=
(\big[*%N/1%N]_(i <- r | P%B) F%N) : nat_scope.
Notation "\prod_ ( i <- r ) F" :=
(\big[*%N/1%N]_(i <- r) F%N) : nat_scope.
Notation "\prod_ ( m <= i < n | P ) F" :=
(\big[*%N/1%N]_(m ≤ i < n | P%B) F%N) : nat_scope.
Notation "\prod_ ( m <= i < n ) F" :=
(\big[*%N/1%N]_(m ≤ i < n) F%N) : nat_scope.
Notation "\prod_ ( i | P ) F" :=
(\big[*%N/1%N]_(i | P%B) F%N) : nat_scope.
Notation "\prod_ i F" :=
(\big[*%N/1%N]_i F%N) : nat_scope.
Notation "\prod_ ( i : t | P ) F" :=
(\big[*%N/1%N]_(i : t | P%B) F%N) (only parsing) : nat_scope.
Notation "\prod_ ( i : t ) F" :=
(\big[*%N/1%N]_(i : t) F%N) (only parsing) : nat_scope.
Notation "\prod_ ( i < n | P ) F" :=
(\big[*%N/1%N]_(i < n | P%B) F%N) : nat_scope.
Notation "\prod_ ( i < n ) F" :=
(\big[*%N/1%N]_(i < n) F%N) : nat_scope.
Notation "\prod_ ( i 'in' A | P ) F" :=
(\big[*%N/1%N]_(i in A | P%B) F%N) : nat_scope.
Notation "\prod_ ( i 'in' A ) F" :=
(\big[*%N/1%N]_(i in A) F%N) : nat_scope.
Notation "\max_ ( i <- r | P ) F" :=
(\big[maxn/0%N]_(i <- r | P%B) F%N) : nat_scope.
Notation "\max_ ( i <- r ) F" :=
(\big[maxn/0%N]_(i <- r) F%N) : nat_scope.
Notation "\max_ ( i | P ) F" :=
(\big[maxn/0%N]_(i | P%B) F%N) : nat_scope.
Notation "\max_ i F" :=
(\big[maxn/0%N]_i F%N) : nat_scope.
Notation "\max_ ( i : I | P ) F" :=
(\big[maxn/0%N]_(i : I | P%B) F%N) (only parsing) : nat_scope.
Notation "\max_ ( i : I ) F" :=
(\big[maxn/0%N]_(i : I) F%N) (only parsing) : nat_scope.
Notation "\max_ ( m <= i < n | P ) F" :=
(\big[maxn/0%N]_(m ≤ i < n | P%B) F%N) : nat_scope.
Notation "\max_ ( m <= i < n ) F" :=
(\big[maxn/0%N]_(m ≤ i < n) F%N) : nat_scope.
Notation "\max_ ( i < n | P ) F" :=
(\big[maxn/0%N]_(i < n | P%B) F%N) : nat_scope.
Notation "\max_ ( i < n ) F" :=
(\big[maxn/0%N]_(i < n) F%N) : nat_scope.
Notation "\max_ ( i 'in' A | P ) F" :=
(\big[maxn/0%N]_(i in A | P%B) F%N) : nat_scope.
Notation "\max_ ( i 'in' A ) F" :=
(\big[maxn/0%N]_(i in A) F%N) : nat_scope.
Definition applybig {R I} (body : bigbody R I) x :=
let: BigBody _ op b v := body in if b then op v x else x.
Definition reducebig R I idx r (body : I → bigbody R I) :=
foldr (applybig \o body) idx r.
Module Type BigOpSig.
Parameter bigop : ∀ R I, R → seq I → (I → bigbody R I) → R.
Axiom bigopE : bigop = reducebig.
End BigOpSig.
Module BigOp : BigOpSig.
Definition bigop := reducebig.
Lemma bigopE : bigop = reducebig.
End BigOp.
Notation bigop := BigOp.bigop (only parsing).
Canonical bigop_unlock := Unlockable BigOp.bigopE.
Definition index_iota m n := iota m (n - m).
Definition index_enum (T : finType) := Finite.enum T.
Lemma mem_index_iota m n i : i \in index_iota m n = (m ≤ i < n).
Lemma mem_index_enum T i : i \in index_enum T.
Hint Resolve mem_index_enum.
Lemma filter_index_enum T P : filter P (index_enum T) = enum P.
Notation "\big [ op / idx ]_ ( i <- r | P ) F" :=
(bigop idx r (fun i ⇒ BigBody i op P%B F)) : big_scope.
Notation "\big [ op / idx ]_ ( i <- r ) F" :=
(bigop idx r (fun i ⇒ BigBody i op true F)) : big_scope.
Notation "\big [ op / idx ]_ ( m <= i < n | P ) F" :=
(bigop idx (index_iota m n) (fun i : nat ⇒ BigBody i op P%B F))
: big_scope.
Notation "\big [ op / idx ]_ ( m <= i < n ) F" :=
(bigop idx (index_iota m n) (fun i : nat ⇒ BigBody i op true F))
: big_scope.
Notation "\big [ op / idx ]_ ( i | P ) F" :=
(bigop idx (index_enum _) (fun i ⇒ BigBody i op P%B F)) : big_scope.
Notation "\big [ op / idx ]_ i F" :=
(bigop idx (index_enum _) (fun i ⇒ BigBody i op true F)) : big_scope.
Notation "\big [ op / idx ]_ ( i : t | P ) F" :=
(bigop idx (index_enum _) (fun i : t ⇒ BigBody i op P%B F))
(only parsing) : big_scope.
Notation "\big [ op / idx ]_ ( i : t ) F" :=
(bigop idx (index_enum _) (fun i : t ⇒ BigBody i op true F))
(only parsing) : big_scope.
Notation "\big [ op / idx ]_ ( i < n | P ) F" :=
(\big[op/idx]_(i : ordinal n | P%B) F) : big_scope.
Notation "\big [ op / idx ]_ ( i < n ) F" :=
(\big[op/idx]_(i : ordinal n) F) : big_scope.
Notation "\big [ op / idx ]_ ( i 'in' A | P ) F" :=
(\big[op/idx]_(i | (i \in A) && P) F) : big_scope.
Notation "\big [ op / idx ]_ ( i 'in' A ) F" :=
(\big[op/idx]_(i | i \in A) F) : big_scope.
Notation BIG_F := (F in \big[_/_]_(i <- _ | _) F i)%pattern.
Notation BIG_P := (P in \big[_/_]_(i <- _ | P i) _)%pattern.
Notation "\sum_ ( i <- r | P ) F" :=
(\big[+%N/0%N]_(i <- r | P%B) F%N) : nat_scope.
Notation "\sum_ ( i <- r ) F" :=
(\big[+%N/0%N]_(i <- r) F%N) : nat_scope.
Notation "\sum_ ( m <= i < n | P ) F" :=
(\big[+%N/0%N]_(m ≤ i < n | P%B) F%N) : nat_scope.
Notation "\sum_ ( m <= i < n ) F" :=
(\big[+%N/0%N]_(m ≤ i < n) F%N) : nat_scope.
Notation "\sum_ ( i | P ) F" :=
(\big[+%N/0%N]_(i | P%B) F%N) : nat_scope.
Notation "\sum_ i F" :=
(\big[+%N/0%N]_i F%N) : nat_scope.
Notation "\sum_ ( i : t | P ) F" :=
(\big[+%N/0%N]_(i : t | P%B) F%N) (only parsing) : nat_scope.
Notation "\sum_ ( i : t ) F" :=
(\big[+%N/0%N]_(i : t) F%N) (only parsing) : nat_scope.
Notation "\sum_ ( i < n | P ) F" :=
(\big[+%N/0%N]_(i < n | P%B) F%N) : nat_scope.
Notation "\sum_ ( i < n ) F" :=
(\big[+%N/0%N]_(i < n) F%N) : nat_scope.
Notation "\sum_ ( i 'in' A | P ) F" :=
(\big[+%N/0%N]_(i in A | P%B) F%N) : nat_scope.
Notation "\sum_ ( i 'in' A ) F" :=
(\big[+%N/0%N]_(i in A) F%N) : nat_scope.
Notation "\prod_ ( i <- r | P ) F" :=
(\big[*%N/1%N]_(i <- r | P%B) F%N) : nat_scope.
Notation "\prod_ ( i <- r ) F" :=
(\big[*%N/1%N]_(i <- r) F%N) : nat_scope.
Notation "\prod_ ( m <= i < n | P ) F" :=
(\big[*%N/1%N]_(m ≤ i < n | P%B) F%N) : nat_scope.
Notation "\prod_ ( m <= i < n ) F" :=
(\big[*%N/1%N]_(m ≤ i < n) F%N) : nat_scope.
Notation "\prod_ ( i | P ) F" :=
(\big[*%N/1%N]_(i | P%B) F%N) : nat_scope.
Notation "\prod_ i F" :=
(\big[*%N/1%N]_i F%N) : nat_scope.
Notation "\prod_ ( i : t | P ) F" :=
(\big[*%N/1%N]_(i : t | P%B) F%N) (only parsing) : nat_scope.
Notation "\prod_ ( i : t ) F" :=
(\big[*%N/1%N]_(i : t) F%N) (only parsing) : nat_scope.
Notation "\prod_ ( i < n | P ) F" :=
(\big[*%N/1%N]_(i < n | P%B) F%N) : nat_scope.
Notation "\prod_ ( i < n ) F" :=
(\big[*%N/1%N]_(i < n) F%N) : nat_scope.
Notation "\prod_ ( i 'in' A | P ) F" :=
(\big[*%N/1%N]_(i in A | P%B) F%N) : nat_scope.
Notation "\prod_ ( i 'in' A ) F" :=
(\big[*%N/1%N]_(i in A) F%N) : nat_scope.
Notation "\max_ ( i <- r | P ) F" :=
(\big[maxn/0%N]_(i <- r | P%B) F%N) : nat_scope.
Notation "\max_ ( i <- r ) F" :=
(\big[maxn/0%N]_(i <- r) F%N) : nat_scope.
Notation "\max_ ( i | P ) F" :=
(\big[maxn/0%N]_(i | P%B) F%N) : nat_scope.
Notation "\max_ i F" :=
(\big[maxn/0%N]_i F%N) : nat_scope.
Notation "\max_ ( i : I | P ) F" :=
(\big[maxn/0%N]_(i : I | P%B) F%N) (only parsing) : nat_scope.
Notation "\max_ ( i : I ) F" :=
(\big[maxn/0%N]_(i : I) F%N) (only parsing) : nat_scope.
Notation "\max_ ( m <= i < n | P ) F" :=
(\big[maxn/0%N]_(m ≤ i < n | P%B) F%N) : nat_scope.
Notation "\max_ ( m <= i < n ) F" :=
(\big[maxn/0%N]_(m ≤ i < n) F%N) : nat_scope.
Notation "\max_ ( i < n | P ) F" :=
(\big[maxn/0%N]_(i < n | P%B) F%N) : nat_scope.
Notation "\max_ ( i < n ) F" :=
(\big[maxn/0%N]_(i < n) F%N) : nat_scope.
Notation "\max_ ( i 'in' A | P ) F" :=
(\big[maxn/0%N]_(i in A | P%B) F%N) : nat_scope.
Notation "\max_ ( i 'in' A ) F" :=
(\big[maxn/0%N]_(i in A) F%N) : nat_scope.
Induction loading
Lemma big_load R (K K' : R → Type) idx op I r (P : pred I) F :
K (\big[op/idx]_(i <- r | P i) F i) × K' (\big[op/idx]_(i <- r | P i) F i)
→ K' (\big[op/idx]_(i <- r | P i) F i).
Implicit Arguments big_load [R K' I].
Section Elim3.
Variables (R1 R2 R3 : Type) (K : R1 → R2 → R3 → Type).
Variables (id1 : R1) (op1 : R1 → R1 → R1).
Variables (id2 : R2) (op2 : R2 → R2 → R2).
Variables (id3 : R3) (op3 : R3 → R3 → R3).
Hypothesis Kid : K id1 id2 id3.
Lemma big_rec3 I r (P : pred I) F1 F2 F3
(K_F : ∀ i y1 y2 y3, P i → K y1 y2 y3 →
K (op1 (F1 i) y1) (op2 (F2 i) y2) (op3 (F3 i) y3)) :
K (\big[op1/id1]_(i <- r | P i) F1 i)
(\big[op2/id2]_(i <- r | P i) F2 i)
(\big[op3/id3]_(i <- r | P i) F3 i).
Hypothesis Kop : ∀ x1 x2 x3 y1 y2 y3,
K x1 x2 x3 → K y1 y2 y3→ K (op1 x1 y1) (op2 x2 y2) (op3 x3 y3).
Lemma big_ind3 I r (P : pred I) F1 F2 F3
(K_F : ∀ i, P i → K (F1 i) (F2 i) (F3 i)) :
K (\big[op1/id1]_(i <- r | P i) F1 i)
(\big[op2/id2]_(i <- r | P i) F2 i)
(\big[op3/id3]_(i <- r | P i) F3 i).
End Elim3.
Implicit Arguments big_rec3 [R1 R2 R3 id1 op1 id2 op2 id3 op3 I r P F1 F2 F3].
Implicit Arguments big_ind3 [R1 R2 R3 id1 op1 id2 op2 id3 op3 I r P F1 F2 F3].
Section Elim2.
Variables (R1 R2 : Type) (K : R1 → R2 → Type) (f : R2 → R1).
Variables (id1 : R1) (op1 : R1 → R1 → R1).
Variables (id2 : R2) (op2 : R2 → R2 → R2).
Hypothesis Kid : K id1 id2.
Lemma big_rec2 I r (P : pred I) F1 F2
(K_F : ∀ i y1 y2, P i → K y1 y2 →
K (op1 (F1 i) y1) (op2 (F2 i) y2)) :
K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i).
Hypothesis Kop : ∀ x1 x2 y1 y2,
K x1 x2 → K y1 y2 → K (op1 x1 y1) (op2 x2 y2).
Lemma big_ind2 I r (P : pred I) F1 F2 (K_F : ∀ i, P i → K (F1 i) (F2 i)) :
K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i).
Hypotheses (f_op : {morph f : x y / op2 x y >-> op1 x y}) (f_id : f id2 = id1).
Lemma big_morph I r (P : pred I) F :
f (\big[op2/id2]_(i <- r | P i) F i) = \big[op1/id1]_(i <- r | P i) f (F i).
End Elim2.
Implicit Arguments big_rec2 [R1 R2 id1 op1 id2 op2 I r P F1 F2].
Implicit Arguments big_ind2 [R1 R2 id1 op1 id2 op2 I r P F1 F2].
Implicit Arguments big_morph [R1 R2 id1 op1 id2 op2 I].
Section Elim1.
Variables (R : Type) (K : R → Type) (f : R → R).
Variables (idx : R) (op op' : R → R → R).
Hypothesis Kid : K idx.
Lemma big_rec I r (P : pred I) F
(Kop : ∀ i x, P i → K x → K (op (F i) x)) :
K (\big[op/idx]_(i <- r | P i) F i).
Hypothesis Kop : ∀ x y, K x → K y → K (op x y).
Lemma big_ind I r (P : pred I) F (K_F : ∀ i, P i → K (F i)) :
K (\big[op/idx]_(i <- r | P i) F i).
Hypothesis Kop' : ∀ x y, K x → K y → op x y = op' x y.
Lemma eq_big_op I r (P : pred I) F (K_F : ∀ i, P i → K (F i)) :
\big[op/idx]_(i <- r | P i) F i = \big[op'/idx]_(i <- r | P i) F i.
Hypotheses (fM : {morph f : x y / op x y}) (f_id : f idx = idx).
Lemma big_endo I r (P : pred I) F :
f (\big[op/idx]_(i <- r | P i) F i) = \big[op/idx]_(i <- r | P i) f (F i).
End Elim1.
Implicit Arguments big_rec [R idx op I r P F].
Implicit Arguments big_ind [R idx op I r P F].
Implicit Arguments eq_big_op [R idx op I].
Implicit Arguments big_endo [R idx op I].
Section Extensionality.
Variables (R : Type) (idx : R) (op : R → R → R).
Section SeqExtension.
Variable I : Type.
Lemma big_filter r (P : pred I) F :
\big[op/idx]_(i <- filter P r) F i = \big[op/idx]_(i <- r | P i) F i.
Lemma big_filter_cond r (P1 P2 : pred I) F :
\big[op/idx]_(i <- filter P1 r | P2 i) F i
= \big[op/idx]_(i <- r | P1 i && P2 i) F i.
Lemma eq_bigl r (P1 P2 : pred I) F :
P1 =1 P2 →
\big[op/idx]_(i <- r | P1 i) F i = \big[op/idx]_(i <- r | P2 i) F i.
K (\big[op/idx]_(i <- r | P i) F i) × K' (\big[op/idx]_(i <- r | P i) F i)
→ K' (\big[op/idx]_(i <- r | P i) F i).
Implicit Arguments big_load [R K' I].
Section Elim3.
Variables (R1 R2 R3 : Type) (K : R1 → R2 → R3 → Type).
Variables (id1 : R1) (op1 : R1 → R1 → R1).
Variables (id2 : R2) (op2 : R2 → R2 → R2).
Variables (id3 : R3) (op3 : R3 → R3 → R3).
Hypothesis Kid : K id1 id2 id3.
Lemma big_rec3 I r (P : pred I) F1 F2 F3
(K_F : ∀ i y1 y2 y3, P i → K y1 y2 y3 →
K (op1 (F1 i) y1) (op2 (F2 i) y2) (op3 (F3 i) y3)) :
K (\big[op1/id1]_(i <- r | P i) F1 i)
(\big[op2/id2]_(i <- r | P i) F2 i)
(\big[op3/id3]_(i <- r | P i) F3 i).
Hypothesis Kop : ∀ x1 x2 x3 y1 y2 y3,
K x1 x2 x3 → K y1 y2 y3→ K (op1 x1 y1) (op2 x2 y2) (op3 x3 y3).
Lemma big_ind3 I r (P : pred I) F1 F2 F3
(K_F : ∀ i, P i → K (F1 i) (F2 i) (F3 i)) :
K (\big[op1/id1]_(i <- r | P i) F1 i)
(\big[op2/id2]_(i <- r | P i) F2 i)
(\big[op3/id3]_(i <- r | P i) F3 i).
End Elim3.
Implicit Arguments big_rec3 [R1 R2 R3 id1 op1 id2 op2 id3 op3 I r P F1 F2 F3].
Implicit Arguments big_ind3 [R1 R2 R3 id1 op1 id2 op2 id3 op3 I r P F1 F2 F3].
Section Elim2.
Variables (R1 R2 : Type) (K : R1 → R2 → Type) (f : R2 → R1).
Variables (id1 : R1) (op1 : R1 → R1 → R1).
Variables (id2 : R2) (op2 : R2 → R2 → R2).
Hypothesis Kid : K id1 id2.
Lemma big_rec2 I r (P : pred I) F1 F2
(K_F : ∀ i y1 y2, P i → K y1 y2 →
K (op1 (F1 i) y1) (op2 (F2 i) y2)) :
K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i).
Hypothesis Kop : ∀ x1 x2 y1 y2,
K x1 x2 → K y1 y2 → K (op1 x1 y1) (op2 x2 y2).
Lemma big_ind2 I r (P : pred I) F1 F2 (K_F : ∀ i, P i → K (F1 i) (F2 i)) :
K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i).
Hypotheses (f_op : {morph f : x y / op2 x y >-> op1 x y}) (f_id : f id2 = id1).
Lemma big_morph I r (P : pred I) F :
f (\big[op2/id2]_(i <- r | P i) F i) = \big[op1/id1]_(i <- r | P i) f (F i).
End Elim2.
Implicit Arguments big_rec2 [R1 R2 id1 op1 id2 op2 I r P F1 F2].
Implicit Arguments big_ind2 [R1 R2 id1 op1 id2 op2 I r P F1 F2].
Implicit Arguments big_morph [R1 R2 id1 op1 id2 op2 I].
Section Elim1.
Variables (R : Type) (K : R → Type) (f : R → R).
Variables (idx : R) (op op' : R → R → R).
Hypothesis Kid : K idx.
Lemma big_rec I r (P : pred I) F
(Kop : ∀ i x, P i → K x → K (op (F i) x)) :
K (\big[op/idx]_(i <- r | P i) F i).
Hypothesis Kop : ∀ x y, K x → K y → K (op x y).
Lemma big_ind I r (P : pred I) F (K_F : ∀ i, P i → K (F i)) :
K (\big[op/idx]_(i <- r | P i) F i).
Hypothesis Kop' : ∀ x y, K x → K y → op x y = op' x y.
Lemma eq_big_op I r (P : pred I) F (K_F : ∀ i, P i → K (F i)) :
\big[op/idx]_(i <- r | P i) F i = \big[op'/idx]_(i <- r | P i) F i.
Hypotheses (fM : {morph f : x y / op x y}) (f_id : f idx = idx).
Lemma big_endo I r (P : pred I) F :
f (\big[op/idx]_(i <- r | P i) F i) = \big[op/idx]_(i <- r | P i) f (F i).
End Elim1.
Implicit Arguments big_rec [R idx op I r P F].
Implicit Arguments big_ind [R idx op I r P F].
Implicit Arguments eq_big_op [R idx op I].
Implicit Arguments big_endo [R idx op I].
Section Extensionality.
Variables (R : Type) (idx : R) (op : R → R → R).
Section SeqExtension.
Variable I : Type.
Lemma big_filter r (P : pred I) F :
\big[op/idx]_(i <- filter P r) F i = \big[op/idx]_(i <- r | P i) F i.
Lemma big_filter_cond r (P1 P2 : pred I) F :
\big[op/idx]_(i <- filter P1 r | P2 i) F i
= \big[op/idx]_(i <- r | P1 i && P2 i) F i.
Lemma eq_bigl r (P1 P2 : pred I) F :
P1 =1 P2 →
\big[op/idx]_(i <- r | P1 i) F i = \big[op/idx]_(i <- r | P2 i) F i.
A lemma to permute aggregate conditions.
Lemma big_andbC r (P Q : pred I) F :
\big[op/idx]_(i <- r | P i && Q i) F i
= \big[op/idx]_(i <- r | Q i && P i) F i.
Lemma eq_bigr r (P : pred I) F1 F2 : (∀ i, P i → F1 i = F2 i) →
\big[op/idx]_(i <- r | P i) F1 i = \big[op/idx]_(i <- r | P i) F2 i.
Lemma eq_big r (P1 P2 : pred I) F1 F2 :
P1 =1 P2 → (∀ i, P1 i → F1 i = F2 i) →
\big[op/idx]_(i <- r | P1 i) F1 i = \big[op/idx]_(i <- r | P2 i) F2 i.
Lemma congr_big r1 r2 (P1 P2 : pred I) F1 F2 :
r1 = r2 → P1 =1 P2 → (∀ i, P1 i → F1 i = F2 i) →
\big[op/idx]_(i <- r1 | P1 i) F1 i = \big[op/idx]_(i <- r2 | P2 i) F2 i.
Lemma big_nil (P : pred I) F : \big[op/idx]_(i <- [::] | P i) F i = idx.
Lemma big_cons i r (P : pred I) F :
let x := \big[op/idx]_(j <- r | P j) F j in
\big[op/idx]_(j <- i :: r | P j) F j = if P i then op (F i) x else x.
Lemma big_map J (h : J → I) r (P : pred I) F :
\big[op/idx]_(i <- map h r | P i) F i
= \big[op/idx]_(j <- r | P (h j)) F (h j).
Lemma big_nth x0 r (P : pred I) F :
\big[op/idx]_(i <- r | P i) F i
= \big[op/idx]_(0 ≤ i < size r | P (nth x0 r i)) (F (nth x0 r i)).
Lemma big_hasC r (P : pred I) F :
~~ has P r → \big[op/idx]_(i <- r | P i) F i = idx.
Lemma big_pred0_eq (r : seq I) F : \big[op/idx]_(i <- r | false) F i = idx.
Lemma big_pred0 r (P : pred I) F :
P =1 xpred0 → \big[op/idx]_(i <- r | P i) F i = idx.
Lemma big_cat_nested r1 r2 (P : pred I) F :
let x := \big[op/idx]_(i <- r2 | P i) F i in
\big[op/idx]_(i <- r1 ++ r2 | P i) F i = \big[op/x]_(i <- r1 | P i) F i.
Lemma big_catl r1 r2 (P : pred I) F :
~~ has P r2 →
\big[op/idx]_(i <- r1 ++ r2 | P i) F i = \big[op/idx]_(i <- r1 | P i) F i.
Lemma big_catr r1 r2 (P : pred I) F :
~~ has P r1 →
\big[op/idx]_(i <- r1 ++ r2 | P i) F i = \big[op/idx]_(i <- r2 | P i) F i.
Lemma big_const_seq r (P : pred I) x :
\big[op/idx]_(i <- r | P i) x = iter (count P r) (op x) idx.
End SeqExtension.
\big[op/idx]_(i <- r | P i && Q i) F i
= \big[op/idx]_(i <- r | Q i && P i) F i.
Lemma eq_bigr r (P : pred I) F1 F2 : (∀ i, P i → F1 i = F2 i) →
\big[op/idx]_(i <- r | P i) F1 i = \big[op/idx]_(i <- r | P i) F2 i.
Lemma eq_big r (P1 P2 : pred I) F1 F2 :
P1 =1 P2 → (∀ i, P1 i → F1 i = F2 i) →
\big[op/idx]_(i <- r | P1 i) F1 i = \big[op/idx]_(i <- r | P2 i) F2 i.
Lemma congr_big r1 r2 (P1 P2 : pred I) F1 F2 :
r1 = r2 → P1 =1 P2 → (∀ i, P1 i → F1 i = F2 i) →
\big[op/idx]_(i <- r1 | P1 i) F1 i = \big[op/idx]_(i <- r2 | P2 i) F2 i.
Lemma big_nil (P : pred I) F : \big[op/idx]_(i <- [::] | P i) F i = idx.
Lemma big_cons i r (P : pred I) F :
let x := \big[op/idx]_(j <- r | P j) F j in
\big[op/idx]_(j <- i :: r | P j) F j = if P i then op (F i) x else x.
Lemma big_map J (h : J → I) r (P : pred I) F :
\big[op/idx]_(i <- map h r | P i) F i
= \big[op/idx]_(j <- r | P (h j)) F (h j).
Lemma big_nth x0 r (P : pred I) F :
\big[op/idx]_(i <- r | P i) F i
= \big[op/idx]_(0 ≤ i < size r | P (nth x0 r i)) (F (nth x0 r i)).
Lemma big_hasC r (P : pred I) F :
~~ has P r → \big[op/idx]_(i <- r | P i) F i = idx.
Lemma big_pred0_eq (r : seq I) F : \big[op/idx]_(i <- r | false) F i = idx.
Lemma big_pred0 r (P : pred I) F :
P =1 xpred0 → \big[op/idx]_(i <- r | P i) F i = idx.
Lemma big_cat_nested r1 r2 (P : pred I) F :
let x := \big[op/idx]_(i <- r2 | P i) F i in
\big[op/idx]_(i <- r1 ++ r2 | P i) F i = \big[op/x]_(i <- r1 | P i) F i.
Lemma big_catl r1 r2 (P : pred I) F :
~~ has P r2 →
\big[op/idx]_(i <- r1 ++ r2 | P i) F i = \big[op/idx]_(i <- r1 | P i) F i.
Lemma big_catr r1 r2 (P : pred I) F :
~~ has P r1 →
\big[op/idx]_(i <- r1 ++ r2 | P i) F i = \big[op/idx]_(i <- r2 | P i) F i.
Lemma big_const_seq r (P : pred I) x :
\big[op/idx]_(i <- r | P i) x = iter (count P r) (op x) idx.
End SeqExtension.
The following lemmas can be used to localise extensionality to a specific
index sequence. This is done by ssreflect rewriting, before applying
congruence or induction lemmas.
Lemma big_seq_cond (I : eqType) r (P : pred I) F :
\big[op/idx]_(i <- r | P i) F i
= \big[op/idx]_(i <- r | (i \in r) && P i) F i.
Lemma big_seq (I : eqType) (r : seq I) F :
\big[op/idx]_(i <- r) F i = \big[op/idx]_(i <- r | i \in r) F i.
Lemma eq_big_seq (I : eqType) (r : seq I) F1 F2 :
{in r, F1 =1 F2} → \big[op/idx]_(i <- r) F1 i = \big[op/idx]_(i <- r) F2 i.
\big[op/idx]_(i <- r | P i) F i
= \big[op/idx]_(i <- r | (i \in r) && P i) F i.
Lemma big_seq (I : eqType) (r : seq I) F :
\big[op/idx]_(i <- r) F i = \big[op/idx]_(i <- r | i \in r) F i.
Lemma eq_big_seq (I : eqType) (r : seq I) F1 F2 :
{in r, F1 =1 F2} → \big[op/idx]_(i <- r) F1 i = \big[op/idx]_(i <- r) F2 i.
Similar lemmas for exposing integer indexing in the predicate.
Lemma big_nat_cond m n (P : pred nat) F :
\big[op/idx]_(m ≤ i < n | P i) F i
= \big[op/idx]_(m ≤ i < n | (m ≤ i < n) && P i) F i.
Lemma big_nat m n F :
\big[op/idx]_(m ≤ i < n) F i = \big[op/idx]_(m ≤ i < n | m ≤ i < n) F i.
Lemma congr_big_nat m1 n1 m2 n2 P1 P2 F1 F2 :
m1 = m2 → n1 = n2 →
(∀ i, m1 ≤ i < n2 → P1 i = P2 i) →
(∀ i, P1 i && (m1 ≤ i < n2) → F1 i = F2 i) →
\big[op/idx]_(m1 ≤ i < n1 | P1 i) F1 i
= \big[op/idx]_(m2 ≤ i < n2 | P2 i) F2 i.
Lemma eq_big_nat m n F1 F2 :
(∀ i, m ≤ i < n → F1 i = F2 i) →
\big[op/idx]_(m ≤ i < n) F1 i = \big[op/idx]_(m ≤ i < n) F2 i.
Lemma big_geq m n (P : pred nat) F :
m ≥ n → \big[op/idx]_(m ≤ i < n | P i) F i = idx.
Lemma big_ltn_cond m n (P : pred nat) F :
m < n → let x := \big[op/idx]_(m.+1 ≤ i < n | P i) F i in
\big[op/idx]_(m ≤ i < n | P i) F i = if P m then op (F m) x else x.
Lemma big_ltn m n F :
m < n →
\big[op/idx]_(m ≤ i < n) F i = op (F m) (\big[op/idx]_(m.+1 ≤ i < n) F i).
Lemma big_addn m n a (P : pred nat) F :
\big[op/idx]_(m + a ≤ i < n | P i) F i =
\big[op/idx]_(m ≤ i < n - a | P (i + a)) F (i + a).
Lemma big_add1 m n (P : pred nat) F :
\big[op/idx]_(m.+1 ≤ i < n | P i) F i =
\big[op/idx]_(m ≤ i < n.-1 | P (i.+1)) F (i.+1).
Lemma big_nat_recl n m F : m ≤ n →
\big[op/idx]_(m ≤ i < n.+1) F i =
op (F m) (\big[op/idx]_(m ≤ i < n) F i.+1).
Lemma big_mkord n (P : pred nat) F :
\big[op/idx]_(0 ≤ i < n | P i) F i = \big[op/idx]_(i < n | P i) F i.
Lemma big_nat_widen m n1 n2 (P : pred nat) F :
n1 ≤ n2 →
\big[op/idx]_(m ≤ i < n1 | P i) F i
= \big[op/idx]_(m ≤ i < n2 | P i && (i < n1)) F i.
Lemma big_ord_widen_cond n1 n2 (P : pred nat) (F : nat → R) :
n1 ≤ n2 →
\big[op/idx]_(i < n1 | P i) F i
= \big[op/idx]_(i < n2 | P i && (i < n1)) F i.
Lemma big_ord_widen n1 n2 (F : nat → R) :
n1 ≤ n2 →
\big[op/idx]_(i < n1) F i = \big[op/idx]_(i < n2 | i < n1) F i.
Lemma big_ord_widen_leq n1 n2 (P : pred 'I_(n1.+1)) F :
n1 < n2 →
\big[op/idx]_(i < n1.+1 | P i) F i
= \big[op/idx]_(i < n2 | P (inord i) && (i ≤ n1)) F (inord i).
Lemma big_ord0 P F : \big[op/idx]_(i < 0 | P i) F i = idx.
Lemma big_tnth I r (P : pred I) F :
let r_ := tnth (in_tuple r) in
\big[op/idx]_(i <- r | P i) F i
= \big[op/idx]_(i < size r | P (r_ i)) (F (r_ i)).
Lemma big_index_uniq (I : eqType) (r : seq I) (E : 'I_(size r) → R) :
uniq r →
\big[op/idx]_i E i = \big[op/idx]_(x <- r) oapp E idx (insub (index x r)).
Lemma big_tuple I n (t : n.-tuple I) (P : pred I) F :
\big[op/idx]_(i <- t | P i) F i
= \big[op/idx]_(i < n | P (tnth t i)) F (tnth t i).
Lemma big_ord_narrow_cond n1 n2 (P : pred 'I_n2) F (le_n12 : n1 ≤ n2) :
let w := widen_ord le_n12 in
\big[op/idx]_(i < n2 | P i && (i < n1)) F i
= \big[op/idx]_(i < n1 | P (w i)) F (w i).
Lemma big_ord_narrow_cond_leq n1 n2 (P : pred _) F (le_n12 : n1 ≤ n2) :
let w := @widen_ord n1.+1 n2.+1 le_n12 in
\big[op/idx]_(i < n2.+1 | P i && (i ≤ n1)) F i
= \big[op/idx]_(i < n1.+1 | P (w i)) F (w i).
Lemma big_ord_narrow n1 n2 F (le_n12 : n1 ≤ n2) :
let w := widen_ord le_n12 in
\big[op/idx]_(i < n2 | i < n1) F i = \big[op/idx]_(i < n1) F (w i).
Lemma big_ord_narrow_leq n1 n2 F (le_n12 : n1 ≤ n2) :
let w := @widen_ord n1.+1 n2.+1 le_n12 in
\big[op/idx]_(i < n2.+1 | i ≤ n1) F i = \big[op/idx]_(i < n1.+1) F (w i).
Lemma big_ord_recl n F :
\big[op/idx]_(i < n.+1) F i =
op (F ord0) (\big[op/idx]_(i < n) F (@lift n.+1 ord0 i)).
Lemma big_const (I : finType) (A : pred I) x :
\big[op/idx]_(i in A) x = iter #|A| (op x) idx.
Lemma big_const_nat m n x :
\big[op/idx]_(m ≤ i < n) x = iter (n - m) (op x) idx.
Lemma big_const_ord n x :
\big[op/idx]_(i < n) x = iter n (op x) idx.
Lemma big_nseq_cond I n a (P : pred I) F :
\big[op/idx]_(i <- nseq n a | P i) F i = if P a then iter n (op (F a)) idx else idx.
Lemma big_nseq I n a (F : I → R):
\big[op/idx]_(i <- nseq n a) F i = iter n (op (F a)) idx.
End Extensionality.
Section MonoidProperties.
Import Monoid.Theory.
Variable R : Type.
Variable idx : R.
Notation Local "1" := idx.
Section Plain.
Variable op : Monoid.law 1.
Notation Local "*%M" := op (at level 0).
Notation Local "x * y" := (op x y).
Lemma eq_big_idx_seq idx' I r (P : pred I) F :
right_id idx' *%M → has P r →
\big[*%M/idx']_(i <- r | P i) F i =\big[*%M/1]_(i <- r | P i) F i.
Lemma eq_big_idx idx' (I : finType) i0 (P : pred I) F :
P i0 → right_id idx' *%M →
\big[*%M/idx']_(i | P i) F i =\big[*%M/1]_(i | P i) F i.
Lemma big1_eq I r (P : pred I) : \big[*%M/1]_(i <- r | P i) 1 = 1.
Lemma big1 I r (P : pred I) F :
(∀ i, P i → F i = 1) → \big[*%M/1]_(i <- r | P i) F i = 1.
Lemma big1_seq (I : eqType) r (P : pred I) F :
(∀ i, P i && (i \in r) → F i = 1) →
\big[*%M/1]_(i <- r | P i) F i = 1.
Lemma big_seq1 I (i : I) F : \big[*%M/1]_(j <- [:: i]) F j = F i.
Lemma big_mkcond I r (P : pred I) F :
\big[*%M/1]_(i <- r | P i) F i =
\big[*%M/1]_(i <- r) (if P i then F i else 1).
Lemma big_mkcondr I r (P Q : pred I) F :
\big[*%M/1]_(i <- r | P i && Q i) F i =
\big[*%M/1]_(i <- r | P i) (if Q i then F i else 1).
Lemma big_mkcondl I r (P Q : pred I) F :
\big[*%M/1]_(i <- r | P i && Q i) F i =
\big[*%M/1]_(i <- r | Q i) (if P i then F i else 1).
Lemma big_cat I r1 r2 (P : pred I) F :
\big[*%M/1]_(i <- r1 ++ r2 | P i) F i =
\big[*%M/1]_(i <- r1 | P i) F i × \big[*%M/1]_(i <- r2 | P i) F i.
Lemma big_pred1_eq (I : finType) (i : I) F :
\big[*%M/1]_(j | j == i) F j = F i.
Lemma big_pred1 (I : finType) i (P : pred I) F :
P =1 pred1 i → \big[*%M/1]_(j | P j) F j = F i.
Lemma big_cat_nat n m p (P : pred nat) F : m ≤ n → n ≤ p →
\big[*%M/1]_(m ≤ i < p | P i) F i =
(\big[*%M/1]_(m ≤ i < n | P i) F i) × (\big[*%M/1]_(n ≤ i < p | P i) F i).
Lemma big_nat1 n F : \big[*%M/1]_(n ≤ i < n.+1) F i = F n.
Lemma big_nat_recr n m F : m ≤ n →
\big[*%M/1]_(m ≤ i < n.+1) F i = (\big[*%M/1]_(m ≤ i < n) F i) × F n.
Lemma big_ord_recr n F :
\big[*%M/1]_(i < n.+1) F i =
(\big[*%M/1]_(i < n) F (widen_ord (leqnSn n) i)) × F ord_max.
Lemma big_sumType (I1 I2 : finType) (P : pred (I1 + I2)) F :
\big[*%M/1]_(i | P i) F i =
(\big[*%M/1]_(i | P (inl _ i)) F (inl _ i))
× (\big[*%M/1]_(i | P (inr _ i)) F (inr _ i)).
Lemma big_split_ord m n (P : pred 'I_(m + n)) F :
\big[*%M/1]_(i | P i) F i =
(\big[*%M/1]_(i | P (lshift n i)) F (lshift n i))
× (\big[*%M/1]_(i | P (rshift m i)) F (rshift m i)).
Lemma big_flatten I rr (P : pred I) F :
\big[*%M/1]_(i <- flatten rr | P i) F i
= \big[*%M/1]_(r <- rr) \big[*%M/1]_(i <- r | P i) F i.
End Plain.
Section Abelian.
Variable op : Monoid.com_law 1.
Notation Local "'*%M'" := op (at level 0).
Notation Local "x * y" := (op x y).
Lemma eq_big_perm (I : eqType) r1 r2 (P : pred I) F :
perm_eq r1 r2 →
\big[*%M/1]_(i <- r1 | P i) F i = \big[*%M/1]_(i <- r2 | P i) F i.
Lemma big_uniq (I : finType) (r : seq I) F :
uniq r → \big[*%M/1]_(i <- r) F i = \big[*%M/1]_(i in r) F i.
Lemma big_rem (I : eqType) r x (P : pred I) F :
x \in r →
\big[*%M/1]_(y <- r | P y) F y
= (if P x then F x else 1) × \big[*%M/1]_(y <- rem x r | P y) F y.
Lemma big_undup (I : eqType) (r : seq I) (P : pred I) F :
idempotent *%M →
\big[*%M/1]_(i <- undup r | P i) F i = \big[*%M/1]_(i <- r | P i) F i.
Lemma eq_big_idem (I : eqType) (r1 r2 : seq I) (P : pred I) F :
idempotent *%M → r1 =i r2 →
\big[*%M/1]_(i <- r1 | P i) F i = \big[*%M/1]_(i <- r2 | P i) F i.
Lemma big_undup_iterop_count (I : eqType) (r : seq I) (P : pred I) F :
\big[*%M/1]_(i <- undup r | P i) iterop (count_mem i r) *%M (F i) 1
= \big[*%M/1]_(i <- r | P i) F i.
Lemma big_split I r (P : pred I) F1 F2 :
\big[*%M/1]_(i <- r | P i) (F1 i × F2 i) =
\big[*%M/1]_(i <- r | P i) F1 i × \big[*%M/1]_(i <- r | P i) F2 i.
Lemma bigID I r (a P : pred I) F :
\big[*%M/1]_(i <- r | P i) F i =
\big[*%M/1]_(i <- r | P i && a i) F i ×
\big[*%M/1]_(i <- r | P i && ~~ a i) F i.
Implicit Arguments bigID [I r].
Lemma bigU (I : finType) (A B : pred I) F :
[disjoint A & B] →
\big[*%M/1]_(i in [predU A & B]) F i =
(\big[*%M/1]_(i in A) F i) × (\big[*%M/1]_(i in B) F i).
Lemma bigD1 (I : finType) j (P : pred I) F :
P j → \big[*%M/1]_(i | P i) F i
= F j × \big[*%M/1]_(i | P i && (i != j)) F i.
Implicit Arguments bigD1 [I P F].
Lemma bigD1_seq (I : eqType) (r : seq I) j F :
j \in r → uniq r →
\big[*%M/1]_(i <- r) F i = F j × \big[*%M/1]_(i <- r | i != j) F i.
Lemma cardD1x (I : finType) (A : pred I) j :
A j → #|SimplPred A| = 1 + #|[pred i | A i & i != j]|.
Implicit Arguments cardD1x [I A].
Lemma partition_big (I J : finType) (P : pred I) p (Q : pred J) F :
(∀ i, P i → Q (p i)) →
\big[*%M/1]_(i | P i) F i =
\big[*%M/1]_(j | Q j) \big[*%M/1]_(i | P i && (p i == j)) F i.
Implicit Arguments partition_big [I J P F].
Lemma reindex_onto (I J : finType) (h : J → I) h' (P : pred I) F :
(∀ i, P i → h (h' i) = i) →
\big[*%M/1]_(i | P i) F i =
\big[*%M/1]_(j | P (h j) && (h' (h j) == j)) F (h j).
Implicit Arguments reindex_onto [I J P F].
Lemma reindex (I J : finType) (h : J → I) (P : pred I) F :
{on [pred i | P i], bijective h} →
\big[*%M/1]_(i | P i) F i = \big[*%M/1]_(j | P (h j)) F (h j).
Implicit Arguments reindex [I J P F].
Lemma reindex_inj (I : finType) (h : I → I) (P : pred I) F :
injective h → \big[*%M/1]_(i | P i) F i = \big[*%M/1]_(j | P (h j)) F (h j).
Implicit Arguments reindex_inj [I h P F].
Lemma big_nat_rev m n P F :
\big[*%M/1]_(m ≤ i < n | P i) F i
= \big[*%M/1]_(m ≤ i < n | P (m + n - i.+1)) F (m + n - i.+1).
Lemma pair_big_dep (I J : finType) (P : pred I) (Q : I → pred J) F :
\big[*%M/1]_(i | P i) \big[*%M/1]_(j | Q i j) F i j =
\big[*%M/1]_(p | P p.1 && Q p.1 p.2) F p.1 p.2.
Lemma pair_big (I J : finType) (P : pred I) (Q : pred J) F :
\big[*%M/1]_(i | P i) \big[*%M/1]_(j | Q j) F i j =
\big[*%M/1]_(p | P p.1 && Q p.2) F p.1 p.2.
Lemma pair_bigA (I J : finType) (F : I → J → R) :
\big[*%M/1]_i \big[*%M/1]_j F i j = \big[*%M/1]_p F p.1 p.2.
Lemma exchange_big_dep I J rI rJ (P : pred I) (Q : I → pred J)
(xQ : pred J) F :
(∀ i j, P i → Q i j → xQ j) →
\big[*%M/1]_(i <- rI | P i) \big[*%M/1]_(j <- rJ | Q i j) F i j =
\big[*%M/1]_(j <- rJ | xQ j) \big[*%M/1]_(i <- rI | P i && Q i j) F i j.
Implicit Arguments exchange_big_dep [I J rI rJ P Q F].
Lemma exchange_big I J rI rJ (P : pred I) (Q : pred J) F :
\big[*%M/1]_(i <- rI | P i) \big[*%M/1]_(j <- rJ | Q j) F i j =
\big[*%M/1]_(j <- rJ | Q j) \big[*%M/1]_(i <- rI | P i) F i j.
Lemma exchange_big_dep_nat m1 n1 m2 n2 (P : pred nat) (Q : rel nat)
(xQ : pred nat) F :
(∀ i j, m1 ≤ i < n1 → m2 ≤ j < n2 → P i → Q i j → xQ j) →
\big[*%M/1]_(m1 ≤ i < n1 | P i) \big[*%M/1]_(m2 ≤ j < n2 | Q i j) F i j =
\big[*%M/1]_(m2 ≤ j < n2 | xQ j)
\big[*%M/1]_(m1 ≤ i < n1 | P i && Q i j) F i j.
Implicit Arguments exchange_big_dep_nat [m1 n1 m2 n2 P Q F].
Lemma exchange_big_nat m1 n1 m2 n2 (P Q : pred nat) F :
\big[*%M/1]_(m1 ≤ i < n1 | P i) \big[*%M/1]_(m2 ≤ j < n2 | Q j) F i j =
\big[*%M/1]_(m2 ≤ j < n2 | Q j) \big[*%M/1]_(m1 ≤ i < n1 | P i) F i j.
End Abelian.
End MonoidProperties.
Implicit Arguments big_filter [R op idx I].
Implicit Arguments big_filter_cond [R op idx I].
Implicit Arguments congr_big [R op idx I r1 P1 F1].
Implicit Arguments eq_big [R op idx I r P1 F1].
Implicit Arguments eq_bigl [R op idx I r P1].
Implicit Arguments eq_bigr [R op idx I r P F1].
Implicit Arguments eq_big_idx [R op idx idx' I P F].
Implicit Arguments big_seq_cond [R op idx I r].
Implicit Arguments eq_big_seq [R op idx I r F1].
Implicit Arguments congr_big_nat [R op idx m1 n1 P1 F1].
Implicit Arguments big_map [R op idx I J r].
Implicit Arguments big_nth [R op idx I r].
Implicit Arguments big_catl [R op idx I r1 r2 P F].
Implicit Arguments big_catr [R op idx I r1 r2 P F].
Implicit Arguments big_geq [R op idx m n P F].
Implicit Arguments big_ltn_cond [R op idx m n P F].
Implicit Arguments big_ltn [R op idx m n F].
Implicit Arguments big_addn [R op idx].
Implicit Arguments big_mkord [R op idx n].
Implicit Arguments big_nat_widen [R op idx] .
Implicit Arguments big_ord_widen_cond [R op idx n1].
Implicit Arguments big_ord_widen [R op idx n1].
Implicit Arguments big_ord_widen_leq [R op idx n1].
Implicit Arguments big_ord_narrow_cond [R op idx n1 n2 P F].
Implicit Arguments big_ord_narrow_cond_leq [R op idx n1 n2 P F].
Implicit Arguments big_ord_narrow [R op idx n1 n2 F].
Implicit Arguments big_ord_narrow_leq [R op idx n1 n2 F].
Implicit Arguments big_mkcond [R op idx I r].
Implicit Arguments big1_eq [R op idx I].
Implicit Arguments big1_seq [R op idx I].
Implicit Arguments big1 [R op idx I].
Implicit Arguments big_pred1 [R op idx I P F].
Implicit Arguments eq_big_perm [R op idx I r1 P F].
Implicit Arguments big_uniq [R op idx I F].
Implicit Arguments big_rem [R op idx I r P F].
Implicit Arguments bigID [R op idx I r].
Implicit Arguments bigU [R op idx I].
Implicit Arguments bigD1 [R op idx I P F].
Implicit Arguments bigD1_seq [R op idx I r F].
Implicit Arguments partition_big [R op idx I J P F].
Implicit Arguments reindex_onto [R op idx I J P F].
Implicit Arguments reindex [R op idx I J P F].
Implicit Arguments reindex_inj [R op idx I h P F].
Implicit Arguments pair_big_dep [R op idx I J].
Implicit Arguments pair_big [R op idx I J].
Implicit Arguments exchange_big_dep [R op idx I J rI rJ P Q F].
Implicit Arguments exchange_big_dep_nat [R op idx m1 n1 m2 n2 P Q F].
Implicit Arguments big_ord_recl [R op idx].
Implicit Arguments big_ord_recr [R op idx].
Implicit Arguments big_nat_recl [R op idx].
Implicit Arguments big_nat_recr [R op idx].
Section Distributivity.
Import Monoid.Theory.
Variable R : Type.
Variables zero one : R.
Notation Local "0" := zero.
Notation Local "1" := one.
Variable times : Monoid.mul_law 0.
Notation Local "*%M" := times (at level 0).
Notation Local "x * y" := (times x y).
Variable plus : Monoid.add_law 0 *%M.
Notation Local "+%M" := plus (at level 0).
Notation Local "x + y" := (plus x y).
Lemma big_distrl I r a (P : pred I) F :
\big[+%M/0]_(i <- r | P i) F i × a = \big[+%M/0]_(i <- r | P i) (F i × a).
Lemma big_distrr I r a (P : pred I) F :
a × \big[+%M/0]_(i <- r | P i) F i = \big[+%M/0]_(i <- r | P i) (a × F i).
Lemma big_distrlr I J rI rJ (pI : pred I) (pJ : pred J) F G :
(\big[+%M/0]_(i <- rI | pI i) F i) × (\big[+%M/0]_(j <- rJ | pJ j) G j)
= \big[+%M/0]_(i <- rI | pI i) \big[+%M/0]_(j <- rJ | pJ j) (F i × G j).
Lemma big_distr_big_dep (I J : finType) j0 (P : pred I) (Q : I → pred J) F :
\big[*%M/1]_(i | P i) \big[+%M/0]_(j | Q i j) F i j =
\big[+%M/0]_(f in pfamily j0 P Q) \big[*%M/1]_(i | P i) F i (f i).
Lemma big_distr_big (I J : finType) j0 (P : pred I) (Q : pred J) F :
\big[*%M/1]_(i | P i) \big[+%M/0]_(j | Q j) F i j =
\big[+%M/0]_(f in pffun_on j0 P Q) \big[*%M/1]_(i | P i) F i (f i).
Lemma bigA_distr_big_dep (I J : finType) (Q : I → pred J) F :
\big[*%M/1]_i \big[+%M/0]_(j | Q i j) F i j
= \big[+%M/0]_(f in family Q) \big[*%M/1]_i F i (f i).
Lemma bigA_distr_big (I J : finType) (Q : pred J) (F : I → J → R) :
\big[*%M/1]_i \big[+%M/0]_(j | Q j) F i j
= \big[+%M/0]_(f in ffun_on Q) \big[*%M/1]_i F i (f i).
Lemma bigA_distr_bigA (I J : finType) F :
\big[*%M/1]_(i : I) \big[+%M/0]_(j : J) F i j
= \big[+%M/0]_(f : {ffun I → J}) \big[*%M/1]_i F i (f i).
End Distributivity.
Implicit Arguments big_distrl [R zero times plus I r].
Implicit Arguments big_distrr [R zero times plus I r].
Implicit Arguments big_distr_big_dep [R zero one times plus I J].
Implicit Arguments big_distr_big [R zero one times plus I J].
Implicit Arguments bigA_distr_big_dep [R zero one times plus I J].
Implicit Arguments bigA_distr_big [R zero one times plus I J].
Implicit Arguments bigA_distr_bigA [R zero one times plus I J].
Section BigBool.
Section Seq.
Variables (I : Type) (r : seq I) (P B : pred I).
Lemma big_has : \big[orb/false]_(i <- r) B i = has B r.
Lemma big_all : \big[andb/true]_(i <- r) B i = all B r.
Lemma big_has_cond : \big[orb/false]_(i <- r | P i) B i = has (predI P B) r.
Lemma big_all_cond :
\big[andb/true]_(i <- r | P i) B i = all [pred i | P i ==> B i] r.
End Seq.
Section FinType.
Variables (I : finType) (P B : pred I).
Lemma big_orE : \big[orb/false]_(i | P i) B i = [∃ (i | P i), B i].
Lemma big_andE : \big[andb/true]_(i | P i) B i = [∀ (i | P i), B i].
End FinType.
End BigBool.
Section NatConst.
Variables (I : finType) (A : pred I).
Lemma sum_nat_const n : \sum_(i in A) n = #|A| × n.
Lemma sum1_card : \sum_(i in A) 1 = #|A|.
Lemma sum1_count J (r : seq J) (a : pred J) : \sum_(j <- r | a j) 1 = count a r.
Lemma sum1_size J (r : seq J) : \sum_(j <- r) 1 = size r.
Lemma prod_nat_const n : \prod_(i in A) n = n ^ #|A|.
Lemma sum_nat_const_nat n1 n2 n : \sum_(n1 ≤ i < n2) n = (n2 - n1) × n.
Lemma prod_nat_const_nat n1 n2 n : \prod_(n1 ≤ i < n2) n = n ^ (n2 - n1).
End NatConst.
Lemma leqif_sum (I : finType) (P C : pred I) (E1 E2 : I → nat) :
(∀ i, P i → E1 i ≤ E2 i ?= iff C i) →
\sum_(i | P i) E1 i ≤ \sum_(i | P i) E2 i ?= iff [∀ (i | P i), C i].
Lemma leq_sum I r (P : pred I) (E1 E2 : I → nat) :
(∀ i, P i → E1 i ≤ E2 i) →
\sum_(i <- r | P i) E1 i ≤ \sum_(i <- r | P i) E2 i.
Lemma sum_nat_eq0 (I : finType) (P : pred I) (E : I → nat) :
(\sum_(i | P i) E i == 0)%N = [∀ (i | P i), E i == 0%N].
Lemma prodn_cond_gt0 I r (P : pred I) F :
(∀ i, P i → 0 < F i) → 0 < \prod_(i <- r | P i) F i.
Lemma prodn_gt0 I r (P : pred I) F :
(∀ i, 0 < F i) → 0 < \prod_(i <- r | P i) F i.
Lemma leq_bigmax_cond (I : finType) (P : pred I) F i0 :
P i0 → F i0 ≤ \max_(i | P i) F i.
Implicit Arguments leq_bigmax_cond [I P F].
Lemma leq_bigmax (I : finType) F (i0 : I) : F i0 ≤ \max_i F i.
Implicit Arguments leq_bigmax [I F].
Lemma bigmax_leqP (I : finType) (P : pred I) m F :
reflect (∀ i, P i → F i ≤ m) (\max_(i | P i) F i ≤ m).
Lemma bigmax_sup (I : finType) i0 (P : pred I) m F :
P i0 → m ≤ F i0 → m ≤ \max_(i | P i) F i.
Implicit Arguments bigmax_sup [I P m F].
Lemma bigmax_eq_arg (I : finType) i0 (P : pred I) F :
P i0 → \max_(i | P i) F i = F [arg max_(i > i0 | P i) F i].
Implicit Arguments bigmax_eq_arg [I P F].
Lemma eq_bigmax_cond (I : finType) (A : pred I) F :
#|A| > 0 → {i0 | i0 \in A & \max_(i in A) F i = F i0}.
Lemma eq_bigmax (I : finType) F : #|I| > 0 → {i0 : I | \max_i F i = F i0}.
Lemma expn_sum m I r (P : pred I) F :
(m ^ (\sum_(i <- r | P i) F i) = \prod_(i <- r | P i) m ^ F i)%N.
Lemma dvdn_biglcmP (I : finType) (P : pred I) F m :
reflect (∀ i, P i → F i %| m) (\big[lcmn/1%N]_(i | P i) F i %| m).
Lemma biglcmn_sup (I : finType) i0 (P : pred I) F m :
P i0 → m %| F i0 → m %| \big[lcmn/1%N]_(i | P i) F i.
Implicit Arguments biglcmn_sup [I P F m].
Lemma dvdn_biggcdP (I : finType) (P : pred I) F m :
reflect (∀ i, P i → m %| F i) (m %| \big[gcdn/0]_(i | P i) F i).
Lemma biggcdn_inf (I : finType) i0 (P : pred I) F m :
P i0 → F i0 %| m → \big[gcdn/0]_(i | P i) F i %| m.
Implicit Arguments biggcdn_inf [I P F m].
Unset Implicit Arguments.
\big[op/idx]_(m ≤ i < n | P i) F i
= \big[op/idx]_(m ≤ i < n | (m ≤ i < n) && P i) F i.
Lemma big_nat m n F :
\big[op/idx]_(m ≤ i < n) F i = \big[op/idx]_(m ≤ i < n | m ≤ i < n) F i.
Lemma congr_big_nat m1 n1 m2 n2 P1 P2 F1 F2 :
m1 = m2 → n1 = n2 →
(∀ i, m1 ≤ i < n2 → P1 i = P2 i) →
(∀ i, P1 i && (m1 ≤ i < n2) → F1 i = F2 i) →
\big[op/idx]_(m1 ≤ i < n1 | P1 i) F1 i
= \big[op/idx]_(m2 ≤ i < n2 | P2 i) F2 i.
Lemma eq_big_nat m n F1 F2 :
(∀ i, m ≤ i < n → F1 i = F2 i) →
\big[op/idx]_(m ≤ i < n) F1 i = \big[op/idx]_(m ≤ i < n) F2 i.
Lemma big_geq m n (P : pred nat) F :
m ≥ n → \big[op/idx]_(m ≤ i < n | P i) F i = idx.
Lemma big_ltn_cond m n (P : pred nat) F :
m < n → let x := \big[op/idx]_(m.+1 ≤ i < n | P i) F i in
\big[op/idx]_(m ≤ i < n | P i) F i = if P m then op (F m) x else x.
Lemma big_ltn m n F :
m < n →
\big[op/idx]_(m ≤ i < n) F i = op (F m) (\big[op/idx]_(m.+1 ≤ i < n) F i).
Lemma big_addn m n a (P : pred nat) F :
\big[op/idx]_(m + a ≤ i < n | P i) F i =
\big[op/idx]_(m ≤ i < n - a | P (i + a)) F (i + a).
Lemma big_add1 m n (P : pred nat) F :
\big[op/idx]_(m.+1 ≤ i < n | P i) F i =
\big[op/idx]_(m ≤ i < n.-1 | P (i.+1)) F (i.+1).
Lemma big_nat_recl n m F : m ≤ n →
\big[op/idx]_(m ≤ i < n.+1) F i =
op (F m) (\big[op/idx]_(m ≤ i < n) F i.+1).
Lemma big_mkord n (P : pred nat) F :
\big[op/idx]_(0 ≤ i < n | P i) F i = \big[op/idx]_(i < n | P i) F i.
Lemma big_nat_widen m n1 n2 (P : pred nat) F :
n1 ≤ n2 →
\big[op/idx]_(m ≤ i < n1 | P i) F i
= \big[op/idx]_(m ≤ i < n2 | P i && (i < n1)) F i.
Lemma big_ord_widen_cond n1 n2 (P : pred nat) (F : nat → R) :
n1 ≤ n2 →
\big[op/idx]_(i < n1 | P i) F i
= \big[op/idx]_(i < n2 | P i && (i < n1)) F i.
Lemma big_ord_widen n1 n2 (F : nat → R) :
n1 ≤ n2 →
\big[op/idx]_(i < n1) F i = \big[op/idx]_(i < n2 | i < n1) F i.
Lemma big_ord_widen_leq n1 n2 (P : pred 'I_(n1.+1)) F :
n1 < n2 →
\big[op/idx]_(i < n1.+1 | P i) F i
= \big[op/idx]_(i < n2 | P (inord i) && (i ≤ n1)) F (inord i).
Lemma big_ord0 P F : \big[op/idx]_(i < 0 | P i) F i = idx.
Lemma big_tnth I r (P : pred I) F :
let r_ := tnth (in_tuple r) in
\big[op/idx]_(i <- r | P i) F i
= \big[op/idx]_(i < size r | P (r_ i)) (F (r_ i)).
Lemma big_index_uniq (I : eqType) (r : seq I) (E : 'I_(size r) → R) :
uniq r →
\big[op/idx]_i E i = \big[op/idx]_(x <- r) oapp E idx (insub (index x r)).
Lemma big_tuple I n (t : n.-tuple I) (P : pred I) F :
\big[op/idx]_(i <- t | P i) F i
= \big[op/idx]_(i < n | P (tnth t i)) F (tnth t i).
Lemma big_ord_narrow_cond n1 n2 (P : pred 'I_n2) F (le_n12 : n1 ≤ n2) :
let w := widen_ord le_n12 in
\big[op/idx]_(i < n2 | P i && (i < n1)) F i
= \big[op/idx]_(i < n1 | P (w i)) F (w i).
Lemma big_ord_narrow_cond_leq n1 n2 (P : pred _) F (le_n12 : n1 ≤ n2) :
let w := @widen_ord n1.+1 n2.+1 le_n12 in
\big[op/idx]_(i < n2.+1 | P i && (i ≤ n1)) F i
= \big[op/idx]_(i < n1.+1 | P (w i)) F (w i).
Lemma big_ord_narrow n1 n2 F (le_n12 : n1 ≤ n2) :
let w := widen_ord le_n12 in
\big[op/idx]_(i < n2 | i < n1) F i = \big[op/idx]_(i < n1) F (w i).
Lemma big_ord_narrow_leq n1 n2 F (le_n12 : n1 ≤ n2) :
let w := @widen_ord n1.+1 n2.+1 le_n12 in
\big[op/idx]_(i < n2.+1 | i ≤ n1) F i = \big[op/idx]_(i < n1.+1) F (w i).
Lemma big_ord_recl n F :
\big[op/idx]_(i < n.+1) F i =
op (F ord0) (\big[op/idx]_(i < n) F (@lift n.+1 ord0 i)).
Lemma big_const (I : finType) (A : pred I) x :
\big[op/idx]_(i in A) x = iter #|A| (op x) idx.
Lemma big_const_nat m n x :
\big[op/idx]_(m ≤ i < n) x = iter (n - m) (op x) idx.
Lemma big_const_ord n x :
\big[op/idx]_(i < n) x = iter n (op x) idx.
Lemma big_nseq_cond I n a (P : pred I) F :
\big[op/idx]_(i <- nseq n a | P i) F i = if P a then iter n (op (F a)) idx else idx.
Lemma big_nseq I n a (F : I → R):
\big[op/idx]_(i <- nseq n a) F i = iter n (op (F a)) idx.
End Extensionality.
Section MonoidProperties.
Import Monoid.Theory.
Variable R : Type.
Variable idx : R.
Notation Local "1" := idx.
Section Plain.
Variable op : Monoid.law 1.
Notation Local "*%M" := op (at level 0).
Notation Local "x * y" := (op x y).
Lemma eq_big_idx_seq idx' I r (P : pred I) F :
right_id idx' *%M → has P r →
\big[*%M/idx']_(i <- r | P i) F i =\big[*%M/1]_(i <- r | P i) F i.
Lemma eq_big_idx idx' (I : finType) i0 (P : pred I) F :
P i0 → right_id idx' *%M →
\big[*%M/idx']_(i | P i) F i =\big[*%M/1]_(i | P i) F i.
Lemma big1_eq I r (P : pred I) : \big[*%M/1]_(i <- r | P i) 1 = 1.
Lemma big1 I r (P : pred I) F :
(∀ i, P i → F i = 1) → \big[*%M/1]_(i <- r | P i) F i = 1.
Lemma big1_seq (I : eqType) r (P : pred I) F :
(∀ i, P i && (i \in r) → F i = 1) →
\big[*%M/1]_(i <- r | P i) F i = 1.
Lemma big_seq1 I (i : I) F : \big[*%M/1]_(j <- [:: i]) F j = F i.
Lemma big_mkcond I r (P : pred I) F :
\big[*%M/1]_(i <- r | P i) F i =
\big[*%M/1]_(i <- r) (if P i then F i else 1).
Lemma big_mkcondr I r (P Q : pred I) F :
\big[*%M/1]_(i <- r | P i && Q i) F i =
\big[*%M/1]_(i <- r | P i) (if Q i then F i else 1).
Lemma big_mkcondl I r (P Q : pred I) F :
\big[*%M/1]_(i <- r | P i && Q i) F i =
\big[*%M/1]_(i <- r | Q i) (if P i then F i else 1).
Lemma big_cat I r1 r2 (P : pred I) F :
\big[*%M/1]_(i <- r1 ++ r2 | P i) F i =
\big[*%M/1]_(i <- r1 | P i) F i × \big[*%M/1]_(i <- r2 | P i) F i.
Lemma big_pred1_eq (I : finType) (i : I) F :
\big[*%M/1]_(j | j == i) F j = F i.
Lemma big_pred1 (I : finType) i (P : pred I) F :
P =1 pred1 i → \big[*%M/1]_(j | P j) F j = F i.
Lemma big_cat_nat n m p (P : pred nat) F : m ≤ n → n ≤ p →
\big[*%M/1]_(m ≤ i < p | P i) F i =
(\big[*%M/1]_(m ≤ i < n | P i) F i) × (\big[*%M/1]_(n ≤ i < p | P i) F i).
Lemma big_nat1 n F : \big[*%M/1]_(n ≤ i < n.+1) F i = F n.
Lemma big_nat_recr n m F : m ≤ n →
\big[*%M/1]_(m ≤ i < n.+1) F i = (\big[*%M/1]_(m ≤ i < n) F i) × F n.
Lemma big_ord_recr n F :
\big[*%M/1]_(i < n.+1) F i =
(\big[*%M/1]_(i < n) F (widen_ord (leqnSn n) i)) × F ord_max.
Lemma big_sumType (I1 I2 : finType) (P : pred (I1 + I2)) F :
\big[*%M/1]_(i | P i) F i =
(\big[*%M/1]_(i | P (inl _ i)) F (inl _ i))
× (\big[*%M/1]_(i | P (inr _ i)) F (inr _ i)).
Lemma big_split_ord m n (P : pred 'I_(m + n)) F :
\big[*%M/1]_(i | P i) F i =
(\big[*%M/1]_(i | P (lshift n i)) F (lshift n i))
× (\big[*%M/1]_(i | P (rshift m i)) F (rshift m i)).
Lemma big_flatten I rr (P : pred I) F :
\big[*%M/1]_(i <- flatten rr | P i) F i
= \big[*%M/1]_(r <- rr) \big[*%M/1]_(i <- r | P i) F i.
End Plain.
Section Abelian.
Variable op : Monoid.com_law 1.
Notation Local "'*%M'" := op (at level 0).
Notation Local "x * y" := (op x y).
Lemma eq_big_perm (I : eqType) r1 r2 (P : pred I) F :
perm_eq r1 r2 →
\big[*%M/1]_(i <- r1 | P i) F i = \big[*%M/1]_(i <- r2 | P i) F i.
Lemma big_uniq (I : finType) (r : seq I) F :
uniq r → \big[*%M/1]_(i <- r) F i = \big[*%M/1]_(i in r) F i.
Lemma big_rem (I : eqType) r x (P : pred I) F :
x \in r →
\big[*%M/1]_(y <- r | P y) F y
= (if P x then F x else 1) × \big[*%M/1]_(y <- rem x r | P y) F y.
Lemma big_undup (I : eqType) (r : seq I) (P : pred I) F :
idempotent *%M →
\big[*%M/1]_(i <- undup r | P i) F i = \big[*%M/1]_(i <- r | P i) F i.
Lemma eq_big_idem (I : eqType) (r1 r2 : seq I) (P : pred I) F :
idempotent *%M → r1 =i r2 →
\big[*%M/1]_(i <- r1 | P i) F i = \big[*%M/1]_(i <- r2 | P i) F i.
Lemma big_undup_iterop_count (I : eqType) (r : seq I) (P : pred I) F :
\big[*%M/1]_(i <- undup r | P i) iterop (count_mem i r) *%M (F i) 1
= \big[*%M/1]_(i <- r | P i) F i.
Lemma big_split I r (P : pred I) F1 F2 :
\big[*%M/1]_(i <- r | P i) (F1 i × F2 i) =
\big[*%M/1]_(i <- r | P i) F1 i × \big[*%M/1]_(i <- r | P i) F2 i.
Lemma bigID I r (a P : pred I) F :
\big[*%M/1]_(i <- r | P i) F i =
\big[*%M/1]_(i <- r | P i && a i) F i ×
\big[*%M/1]_(i <- r | P i && ~~ a i) F i.
Implicit Arguments bigID [I r].
Lemma bigU (I : finType) (A B : pred I) F :
[disjoint A & B] →
\big[*%M/1]_(i in [predU A & B]) F i =
(\big[*%M/1]_(i in A) F i) × (\big[*%M/1]_(i in B) F i).
Lemma bigD1 (I : finType) j (P : pred I) F :
P j → \big[*%M/1]_(i | P i) F i
= F j × \big[*%M/1]_(i | P i && (i != j)) F i.
Implicit Arguments bigD1 [I P F].
Lemma bigD1_seq (I : eqType) (r : seq I) j F :
j \in r → uniq r →
\big[*%M/1]_(i <- r) F i = F j × \big[*%M/1]_(i <- r | i != j) F i.
Lemma cardD1x (I : finType) (A : pred I) j :
A j → #|SimplPred A| = 1 + #|[pred i | A i & i != j]|.
Implicit Arguments cardD1x [I A].
Lemma partition_big (I J : finType) (P : pred I) p (Q : pred J) F :
(∀ i, P i → Q (p i)) →
\big[*%M/1]_(i | P i) F i =
\big[*%M/1]_(j | Q j) \big[*%M/1]_(i | P i && (p i == j)) F i.
Implicit Arguments partition_big [I J P F].
Lemma reindex_onto (I J : finType) (h : J → I) h' (P : pred I) F :
(∀ i, P i → h (h' i) = i) →
\big[*%M/1]_(i | P i) F i =
\big[*%M/1]_(j | P (h j) && (h' (h j) == j)) F (h j).
Implicit Arguments reindex_onto [I J P F].
Lemma reindex (I J : finType) (h : J → I) (P : pred I) F :
{on [pred i | P i], bijective h} →
\big[*%M/1]_(i | P i) F i = \big[*%M/1]_(j | P (h j)) F (h j).
Implicit Arguments reindex [I J P F].
Lemma reindex_inj (I : finType) (h : I → I) (P : pred I) F :
injective h → \big[*%M/1]_(i | P i) F i = \big[*%M/1]_(j | P (h j)) F (h j).
Implicit Arguments reindex_inj [I h P F].
Lemma big_nat_rev m n P F :
\big[*%M/1]_(m ≤ i < n | P i) F i
= \big[*%M/1]_(m ≤ i < n | P (m + n - i.+1)) F (m + n - i.+1).
Lemma pair_big_dep (I J : finType) (P : pred I) (Q : I → pred J) F :
\big[*%M/1]_(i | P i) \big[*%M/1]_(j | Q i j) F i j =
\big[*%M/1]_(p | P p.1 && Q p.1 p.2) F p.1 p.2.
Lemma pair_big (I J : finType) (P : pred I) (Q : pred J) F :
\big[*%M/1]_(i | P i) \big[*%M/1]_(j | Q j) F i j =
\big[*%M/1]_(p | P p.1 && Q p.2) F p.1 p.2.
Lemma pair_bigA (I J : finType) (F : I → J → R) :
\big[*%M/1]_i \big[*%M/1]_j F i j = \big[*%M/1]_p F p.1 p.2.
Lemma exchange_big_dep I J rI rJ (P : pred I) (Q : I → pred J)
(xQ : pred J) F :
(∀ i j, P i → Q i j → xQ j) →
\big[*%M/1]_(i <- rI | P i) \big[*%M/1]_(j <- rJ | Q i j) F i j =
\big[*%M/1]_(j <- rJ | xQ j) \big[*%M/1]_(i <- rI | P i && Q i j) F i j.
Implicit Arguments exchange_big_dep [I J rI rJ P Q F].
Lemma exchange_big I J rI rJ (P : pred I) (Q : pred J) F :
\big[*%M/1]_(i <- rI | P i) \big[*%M/1]_(j <- rJ | Q j) F i j =
\big[*%M/1]_(j <- rJ | Q j) \big[*%M/1]_(i <- rI | P i) F i j.
Lemma exchange_big_dep_nat m1 n1 m2 n2 (P : pred nat) (Q : rel nat)
(xQ : pred nat) F :
(∀ i j, m1 ≤ i < n1 → m2 ≤ j < n2 → P i → Q i j → xQ j) →
\big[*%M/1]_(m1 ≤ i < n1 | P i) \big[*%M/1]_(m2 ≤ j < n2 | Q i j) F i j =
\big[*%M/1]_(m2 ≤ j < n2 | xQ j)
\big[*%M/1]_(m1 ≤ i < n1 | P i && Q i j) F i j.
Implicit Arguments exchange_big_dep_nat [m1 n1 m2 n2 P Q F].
Lemma exchange_big_nat m1 n1 m2 n2 (P Q : pred nat) F :
\big[*%M/1]_(m1 ≤ i < n1 | P i) \big[*%M/1]_(m2 ≤ j < n2 | Q j) F i j =
\big[*%M/1]_(m2 ≤ j < n2 | Q j) \big[*%M/1]_(m1 ≤ i < n1 | P i) F i j.
End Abelian.
End MonoidProperties.
Implicit Arguments big_filter [R op idx I].
Implicit Arguments big_filter_cond [R op idx I].
Implicit Arguments congr_big [R op idx I r1 P1 F1].
Implicit Arguments eq_big [R op idx I r P1 F1].
Implicit Arguments eq_bigl [R op idx I r P1].
Implicit Arguments eq_bigr [R op idx I r P F1].
Implicit Arguments eq_big_idx [R op idx idx' I P F].
Implicit Arguments big_seq_cond [R op idx I r].
Implicit Arguments eq_big_seq [R op idx I r F1].
Implicit Arguments congr_big_nat [R op idx m1 n1 P1 F1].
Implicit Arguments big_map [R op idx I J r].
Implicit Arguments big_nth [R op idx I r].
Implicit Arguments big_catl [R op idx I r1 r2 P F].
Implicit Arguments big_catr [R op idx I r1 r2 P F].
Implicit Arguments big_geq [R op idx m n P F].
Implicit Arguments big_ltn_cond [R op idx m n P F].
Implicit Arguments big_ltn [R op idx m n F].
Implicit Arguments big_addn [R op idx].
Implicit Arguments big_mkord [R op idx n].
Implicit Arguments big_nat_widen [R op idx] .
Implicit Arguments big_ord_widen_cond [R op idx n1].
Implicit Arguments big_ord_widen [R op idx n1].
Implicit Arguments big_ord_widen_leq [R op idx n1].
Implicit Arguments big_ord_narrow_cond [R op idx n1 n2 P F].
Implicit Arguments big_ord_narrow_cond_leq [R op idx n1 n2 P F].
Implicit Arguments big_ord_narrow [R op idx n1 n2 F].
Implicit Arguments big_ord_narrow_leq [R op idx n1 n2 F].
Implicit Arguments big_mkcond [R op idx I r].
Implicit Arguments big1_eq [R op idx I].
Implicit Arguments big1_seq [R op idx I].
Implicit Arguments big1 [R op idx I].
Implicit Arguments big_pred1 [R op idx I P F].
Implicit Arguments eq_big_perm [R op idx I r1 P F].
Implicit Arguments big_uniq [R op idx I F].
Implicit Arguments big_rem [R op idx I r P F].
Implicit Arguments bigID [R op idx I r].
Implicit Arguments bigU [R op idx I].
Implicit Arguments bigD1 [R op idx I P F].
Implicit Arguments bigD1_seq [R op idx I r F].
Implicit Arguments partition_big [R op idx I J P F].
Implicit Arguments reindex_onto [R op idx I J P F].
Implicit Arguments reindex [R op idx I J P F].
Implicit Arguments reindex_inj [R op idx I h P F].
Implicit Arguments pair_big_dep [R op idx I J].
Implicit Arguments pair_big [R op idx I J].
Implicit Arguments exchange_big_dep [R op idx I J rI rJ P Q F].
Implicit Arguments exchange_big_dep_nat [R op idx m1 n1 m2 n2 P Q F].
Implicit Arguments big_ord_recl [R op idx].
Implicit Arguments big_ord_recr [R op idx].
Implicit Arguments big_nat_recl [R op idx].
Implicit Arguments big_nat_recr [R op idx].
Section Distributivity.
Import Monoid.Theory.
Variable R : Type.
Variables zero one : R.
Notation Local "0" := zero.
Notation Local "1" := one.
Variable times : Monoid.mul_law 0.
Notation Local "*%M" := times (at level 0).
Notation Local "x * y" := (times x y).
Variable plus : Monoid.add_law 0 *%M.
Notation Local "+%M" := plus (at level 0).
Notation Local "x + y" := (plus x y).
Lemma big_distrl I r a (P : pred I) F :
\big[+%M/0]_(i <- r | P i) F i × a = \big[+%M/0]_(i <- r | P i) (F i × a).
Lemma big_distrr I r a (P : pred I) F :
a × \big[+%M/0]_(i <- r | P i) F i = \big[+%M/0]_(i <- r | P i) (a × F i).
Lemma big_distrlr I J rI rJ (pI : pred I) (pJ : pred J) F G :
(\big[+%M/0]_(i <- rI | pI i) F i) × (\big[+%M/0]_(j <- rJ | pJ j) G j)
= \big[+%M/0]_(i <- rI | pI i) \big[+%M/0]_(j <- rJ | pJ j) (F i × G j).
Lemma big_distr_big_dep (I J : finType) j0 (P : pred I) (Q : I → pred J) F :
\big[*%M/1]_(i | P i) \big[+%M/0]_(j | Q i j) F i j =
\big[+%M/0]_(f in pfamily j0 P Q) \big[*%M/1]_(i | P i) F i (f i).
Lemma big_distr_big (I J : finType) j0 (P : pred I) (Q : pred J) F :
\big[*%M/1]_(i | P i) \big[+%M/0]_(j | Q j) F i j =
\big[+%M/0]_(f in pffun_on j0 P Q) \big[*%M/1]_(i | P i) F i (f i).
Lemma bigA_distr_big_dep (I J : finType) (Q : I → pred J) F :
\big[*%M/1]_i \big[+%M/0]_(j | Q i j) F i j
= \big[+%M/0]_(f in family Q) \big[*%M/1]_i F i (f i).
Lemma bigA_distr_big (I J : finType) (Q : pred J) (F : I → J → R) :
\big[*%M/1]_i \big[+%M/0]_(j | Q j) F i j
= \big[+%M/0]_(f in ffun_on Q) \big[*%M/1]_i F i (f i).
Lemma bigA_distr_bigA (I J : finType) F :
\big[*%M/1]_(i : I) \big[+%M/0]_(j : J) F i j
= \big[+%M/0]_(f : {ffun I → J}) \big[*%M/1]_i F i (f i).
End Distributivity.
Implicit Arguments big_distrl [R zero times plus I r].
Implicit Arguments big_distrr [R zero times plus I r].
Implicit Arguments big_distr_big_dep [R zero one times plus I J].
Implicit Arguments big_distr_big [R zero one times plus I J].
Implicit Arguments bigA_distr_big_dep [R zero one times plus I J].
Implicit Arguments bigA_distr_big [R zero one times plus I J].
Implicit Arguments bigA_distr_bigA [R zero one times plus I J].
Section BigBool.
Section Seq.
Variables (I : Type) (r : seq I) (P B : pred I).
Lemma big_has : \big[orb/false]_(i <- r) B i = has B r.
Lemma big_all : \big[andb/true]_(i <- r) B i = all B r.
Lemma big_has_cond : \big[orb/false]_(i <- r | P i) B i = has (predI P B) r.
Lemma big_all_cond :
\big[andb/true]_(i <- r | P i) B i = all [pred i | P i ==> B i] r.
End Seq.
Section FinType.
Variables (I : finType) (P B : pred I).
Lemma big_orE : \big[orb/false]_(i | P i) B i = [∃ (i | P i), B i].
Lemma big_andE : \big[andb/true]_(i | P i) B i = [∀ (i | P i), B i].
End FinType.
End BigBool.
Section NatConst.
Variables (I : finType) (A : pred I).
Lemma sum_nat_const n : \sum_(i in A) n = #|A| × n.
Lemma sum1_card : \sum_(i in A) 1 = #|A|.
Lemma sum1_count J (r : seq J) (a : pred J) : \sum_(j <- r | a j) 1 = count a r.
Lemma sum1_size J (r : seq J) : \sum_(j <- r) 1 = size r.
Lemma prod_nat_const n : \prod_(i in A) n = n ^ #|A|.
Lemma sum_nat_const_nat n1 n2 n : \sum_(n1 ≤ i < n2) n = (n2 - n1) × n.
Lemma prod_nat_const_nat n1 n2 n : \prod_(n1 ≤ i < n2) n = n ^ (n2 - n1).
End NatConst.
Lemma leqif_sum (I : finType) (P C : pred I) (E1 E2 : I → nat) :
(∀ i, P i → E1 i ≤ E2 i ?= iff C i) →
\sum_(i | P i) E1 i ≤ \sum_(i | P i) E2 i ?= iff [∀ (i | P i), C i].
Lemma leq_sum I r (P : pred I) (E1 E2 : I → nat) :
(∀ i, P i → E1 i ≤ E2 i) →
\sum_(i <- r | P i) E1 i ≤ \sum_(i <- r | P i) E2 i.
Lemma sum_nat_eq0 (I : finType) (P : pred I) (E : I → nat) :
(\sum_(i | P i) E i == 0)%N = [∀ (i | P i), E i == 0%N].
Lemma prodn_cond_gt0 I r (P : pred I) F :
(∀ i, P i → 0 < F i) → 0 < \prod_(i <- r | P i) F i.
Lemma prodn_gt0 I r (P : pred I) F :
(∀ i, 0 < F i) → 0 < \prod_(i <- r | P i) F i.
Lemma leq_bigmax_cond (I : finType) (P : pred I) F i0 :
P i0 → F i0 ≤ \max_(i | P i) F i.
Implicit Arguments leq_bigmax_cond [I P F].
Lemma leq_bigmax (I : finType) F (i0 : I) : F i0 ≤ \max_i F i.
Implicit Arguments leq_bigmax [I F].
Lemma bigmax_leqP (I : finType) (P : pred I) m F :
reflect (∀ i, P i → F i ≤ m) (\max_(i | P i) F i ≤ m).
Lemma bigmax_sup (I : finType) i0 (P : pred I) m F :
P i0 → m ≤ F i0 → m ≤ \max_(i | P i) F i.
Implicit Arguments bigmax_sup [I P m F].
Lemma bigmax_eq_arg (I : finType) i0 (P : pred I) F :
P i0 → \max_(i | P i) F i = F [arg max_(i > i0 | P i) F i].
Implicit Arguments bigmax_eq_arg [I P F].
Lemma eq_bigmax_cond (I : finType) (A : pred I) F :
#|A| > 0 → {i0 | i0 \in A & \max_(i in A) F i = F i0}.
Lemma eq_bigmax (I : finType) F : #|I| > 0 → {i0 : I | \max_i F i = F i0}.
Lemma expn_sum m I r (P : pred I) F :
(m ^ (\sum_(i <- r | P i) F i) = \prod_(i <- r | P i) m ^ F i)%N.
Lemma dvdn_biglcmP (I : finType) (P : pred I) F m :
reflect (∀ i, P i → F i %| m) (\big[lcmn/1%N]_(i | P i) F i %| m).
Lemma biglcmn_sup (I : finType) i0 (P : pred I) F m :
P i0 → m %| F i0 → m %| \big[lcmn/1%N]_(i | P i) F i.
Implicit Arguments biglcmn_sup [I P F m].
Lemma dvdn_biggcdP (I : finType) (P : pred I) F m :
reflect (∀ i, P i → m %| F i) (m %| \big[gcdn/0]_(i | P i) F i).
Lemma biggcdn_inf (I : finType) i0 (P : pred I) F m :
P i0 → F i0 %| m → \big[gcdn/0]_(i | P i) F i %| m.
Implicit Arguments biggcdn_inf [I P F m].
Unset Implicit Arguments.