(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *) (* Distributed under the terms of CeCILL-B. *) From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path. From mathcomp Require Import div fintype tuple finfun. (******************************************************************************) (* This file provides a generic definition for iterating an operator over a *) (* set of indices (bigop); this big operator is parameterized 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 *) (* 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 (semirings). 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 *) (* _ *) (* - is one of \big[op/idx], \sum, \prod, or \max (see below). *) (* - can be any expression. *) (* - binds an index variable in ; 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 | ) limits the range to the i for which *) (* holds. can be any expression that coerces to *) (* bool, and may mention the bound index i. All six kinds of *) (* ranges above can have a part. *) (* - One can use the "\big[op/idx]" notations for any operator. *) (* - BIG_F and BIG_P are pattern abbreviations for the and *) (* 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). *) (******************************************************************************) (* 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 *) (******************************************************************************) Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. Declare Scope big_scope. 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 }. Local Coercion operator : law >-> Funclass. Structure com_law := ComLaw { com_operator : law; _ : commutative com_operator }. Local Coercion com_operator : com_law >-> law. Structure mul_law := MulLaw { mul_operator : T -> T -> T; _ : left_zero idm mul_operator; _ : right_zero idm mul_operator }. Local Coercion mul_operator : mul_law >-> Funclass. Structure add_law (mul : T -> T -> T) := AddLaw { add_operator : com_law; _ : left_distributive mul add_operator; _ : right_distributive mul add_operator }. Local Coercion add_operator : add_law >-> com_law. 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. Proof. by move=> mul1x x; rewrite mulC. Qed. Lemma mulC_zero : left_zero zero mul -> right_zero zero mul. Proof. by move=> mul0x x; rewrite mulC. Qed. Lemma mulC_dist : left_distributive mul add -> right_distributive mul add. Proof. by move=> mul_addl x y z; rewrite !(mulC x). Qed. End CommutativeAxioms. Module Theory. Section Theory. Variables (T : Type) (idm : T). Section Plain. Variable mul : law idm. Lemma mul1m : left_id idm mul. Proof. by case mul. Qed. Lemma mulm1 : right_id idm mul. Proof. by case mul. Qed. Lemma mulmA : associative mul. Proof. by case mul. Qed. Lemma iteropE n x : iterop n mul x idm = iter n (mul x) idm. Proof. by case: n => // n; rewrite iterSr mulm1 iteropS. Qed. End Plain. Section Commutative. Variable mul : com_law idm. Lemma mulmC : commutative mul. Proof. by case mul. Qed. Lemma mulmCA : left_commutative mul. Proof. by move=> x y z; rewrite !mulmA (mulmC x). Qed. Lemma mulmAC : right_commutative mul. Proof. by move=> x y z; rewrite -!mulmA (mulmC y). Qed. Lemma mulmACA : interchange mul mul. Proof. by move=> x y z t; rewrite -!mulmA (mulmCA y). Qed. End Commutative. Section Mul. Variable mul : mul_law idm. Lemma mul0m : left_zero idm mul. Proof. by case mul. Qed. Lemma mulm0 : right_zero idm mul. Proof. by case mul. Qed. End Mul. Section Add. Variables (mul : T -> T -> T) (add : add_law idm mul). Lemma addmA : associative add. Proof. exact: mulmA. Qed. Lemma addmC : commutative add. Proof. exact: mulmC. Qed. Lemma addmCA : left_commutative add. Proof. exact: mulmCA. Qed. Lemma addmAC : right_commutative add. Proof. exact: mulmAC. Qed. Lemma add0m : left_id idm add. Proof. exact: mul1m. Qed. Lemma addm0 : right_id idm add. Proof. exact: mulm1. Qed. Lemma mulmDl : left_distributive mul add. Proof. by case add. Qed. Lemma mulmDr : right_distributive mul add. Proof. by case add. Qed. 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 maxnMl maxnMr. 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 or *) (* 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 , as this can give more information on the expected type of *) (* the , thus allowing for the insertion of coercions. *) Variant 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 : forall 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. Proof. by []. Qed. End BigOp. Notation bigop := BigOp.bigop (only parsing). Canonical bigop_unlock := Unlockable BigOp.bigopE. Definition index_iota m n := iota m (n - m). Lemma mem_index_iota m n i : i \in index_iota m n = (m <= i < n). Proof. rewrite mem_iota; case le_m_i: (m <= i) => //=. by rewrite -leq_subLR subSn // -subn_gt0 -subnDA subnKC // subn_gt0. Qed. (* Legacy mathcomp scripts have been relying on the fact that enum A and *) (* filter A (index_enum T) are convertible. This is likely to change in the *) (* next mathcomp release when enum, pick, subset and card are generalised to *) (* predicates with finite support in a choiceType - in fact the two will only *) (* be equal up to permutation in this new theory. *) (* It is therefore advisable to stop relying on this, and use the new *) (* facilities provided in this library: lemmas big_enumP, big_enum, big_image *) (* and such. Users wishing to test compliance should change the Defined in *) (* index_enum_key to Qed, and comment out the filter_index_enum compatibility *) (* definition below (or Import Deprecation.Reject). *) Fact index_enum_key : unit. Proof. split. Defined. (* Qed. *) Definition index_enum (T : finType) := locked_with index_enum_key (Finite.enum T). Lemma deprecated_filter_index_enum T P : filter P (index_enum T) = enum P. Proof. by rewrite [index_enum T]unlock. Qed. Lemma mem_index_enum T i : i \in index_enum T. Proof. by rewrite [index_enum T]unlock -enumT mem_enum. Qed. #[global] Hint Resolve mem_index_enum : core. Lemma index_enum_uniq T : uniq (index_enum T). Proof. by rewrite [index_enum T]unlock -enumT enum_uniq. Qed. 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. Local Notation "+%N" := addn (at level 0, only parsing). 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. Local Notation "*%N" := muln (at level 0, only parsing). 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). Proof. by case. Qed. Arguments big_load [R] K [K'] idx op [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 : forall 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). Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: K_F. Qed. Hypothesis Kop : forall 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 : forall 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). Proof. by apply: big_rec3 => i x1 x2 x3 /K_F; apply: Kop. Qed. End Elim3. Arguments big_rec3 [R1 R2 R3] K [id1 op1 id2 op2 id3 op3] _ [I r P F1 F2 F3]. Arguments big_ind3 [R1 R2 R3] K [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 : forall 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). Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: K_F. Qed. Hypothesis Kop : forall 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 : forall 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). Proof. by apply: big_rec2 => i x1 x2 /K_F; apply: Kop. Qed. 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). Proof. by rewrite unlock; elim: r => //= i r <-; rewrite -f_op -fun_if. Qed. End Elim2. Arguments big_rec2 [R1 R2] K [id1 op1 id2 op2] _ [I r P F1 F2]. Arguments big_ind2 [R1 R2] K [id1 op1 id2 op2] _ _ [I r P F1 F2]. Arguments big_morph [R1 R2] f [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 : forall i x, P i -> K x -> K (op (F i) x)) : K (\big[op/idx]_(i <- r | P i) F i). Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: Kop. Qed. Hypothesis Kop : forall x y, K x -> K y -> K (op x y). Lemma big_ind I r (P : pred I) F (K_F : forall i, P i -> K (F i)) : K (\big[op/idx]_(i <- r | P i) F i). Proof. by apply: big_rec => // i x /K_F /Kop; apply. Qed. Hypothesis Kop' : forall x y, K x -> K y -> op x y = op' x y. Lemma eq_big_op I r (P : pred I) F (K_F : forall i, P i -> K (F i)) : \big[op/idx]_(i <- r | P i) F i = \big[op'/idx]_(i <- r | P i) F i. Proof. by elim/(big_load K): _; elim/big_rec2: _ => // i _ y Pi [Ky <-]; auto. Qed. 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). Proof. exact: big_morph. Qed. End Elim1. Arguments big_rec [R] K [idx op] _ [I r P F]. Arguments big_ind [R] K [idx op] _ _ [I r P F]. Arguments eq_big_op [R] K [idx op] op' _ _ _ [I]. Arguments big_endo [R] f [idx op] _ _ [I]. Section Extensionality. Variables (R : Type) (idx : R) (op : R -> R -> R). Section SeqExtension. Variable I : Type. Lemma foldrE r : foldr op idx r = \big[op/idx]_(x <- r) x. Proof. by rewrite unlock. Qed. 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. Proof. by rewrite unlock; elim: r => //= i r <-; case (P i). Qed. 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. Proof. rewrite -big_filter -(big_filter r); congr bigop. by rewrite -filter_predI; apply: eq_filter => i; apply: andbC. Qed. 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. Proof. by move=> eqP12; rewrite -!(big_filter r) (eq_filter eqP12). Qed. (* 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. Proof. by apply: eq_bigl => i; apply: andbC. Qed. Lemma eq_bigr r (P : pred I) F1 F2 : (forall 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. Proof. by move=> eqF12; elim/big_rec2: _ => // i x _ /eqF12-> ->. Qed. Lemma eq_big r (P1 P2 : pred I) F1 F2 : P1 =1 P2 -> (forall 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. Proof. by move/eq_bigl <-; move/eq_bigr->. Qed. Lemma congr_big r1 r2 (P1 P2 : pred I) F1 F2 : r1 = r2 -> P1 =1 P2 -> (forall 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. Proof. by move=> <-{r2}; apply: eq_big. Qed. Lemma big_nil (P : pred I) F : \big[op/idx]_(i <- [::] | P i) F i = idx. Proof. by rewrite unlock. Qed. 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. Proof. by rewrite unlock. Qed. 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). Proof. by rewrite unlock; elim: r => //= j r ->. Qed. 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)). Proof. by rewrite -[r in LHS](mkseq_nth x0) big_map /index_iota subn0. Qed. Lemma big_hasC r (P : pred I) F : ~~ has P r -> \big[op/idx]_(i <- r | P i) F i = idx. Proof. by rewrite -big_filter has_count -size_filter -eqn0Ngt unlock => /nilP->. Qed. Lemma big_pred0_eq (r : seq I) F : \big[op/idx]_(i <- r | false) F i = idx. Proof. by rewrite big_hasC // has_pred0. Qed. Lemma big_pred0 r (P : pred I) F : P =1 xpred0 -> \big[op/idx]_(i <- r | P i) F i = idx. Proof. by move/eq_bigl->; apply: big_pred0_eq. Qed. 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. Proof. by rewrite unlock /reducebig foldr_cat. Qed. 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. Proof. by rewrite big_cat_nested => /big_hasC->. Qed. 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. Proof. rewrite -big_filter -(big_filter r2) filter_cat. by rewrite has_count -size_filter; case: filter. Qed. End SeqExtension. Lemma big_map_id J (h : J -> R) r (P : pred R) : \big[op/idx]_(i <- map h r | P i) i = \big[op/idx]_(j <- r | P (h j)) h j. Proof. exact: big_map. Qed. (* 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. Proof. by rewrite -!(big_filter r); congr bigop; apply: eq_in_filter => i ->. Qed. 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. Proof. by rewrite big_seq_cond big_andbC. Qed. 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. Proof. by move=> eqF; rewrite !big_seq (eq_bigr _ eqF). Qed. (* 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. Proof. by rewrite big_seq_cond; apply: eq_bigl => i; rewrite mem_index_iota. Qed. 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. Proof. by rewrite big_nat_cond big_andbC. Qed. Lemma congr_big_nat m1 n1 m2 n2 P1 P2 F1 F2 : m1 = m2 -> n1 = n2 -> (forall i, m1 <= i < n2 -> P1 i = P2 i) -> (forall 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. Proof. move=> <- <- eqP12 eqF12; rewrite big_seq_cond (big_seq_cond _ P2). apply: eq_big => i; rewrite ?inE /= !mem_index_iota. by apply: andb_id2l; apply: eqP12. by rewrite andbC; apply: eqF12. Qed. Lemma eq_big_nat m n F1 F2 : (forall i, m <= i < n -> F1 i = F2 i) -> \big[op/idx]_(m <= i < n) F1 i = \big[op/idx]_(m <= i < n) F2 i. Proof. by move=> eqF; apply: congr_big_nat. Qed. Lemma big_geq m n (P : pred nat) F : m >= n -> \big[op/idx]_(m <= i < n | P i) F i = idx. Proof. by move=> ge_m_n; rewrite /index_iota (eqnP ge_m_n) big_nil. Qed. 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. Proof. by case: n => [//|n] le_m_n; rewrite /index_iota subSn // big_cons. Qed. 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). Proof. by move=> lt_mn; apply: big_ltn_cond. Qed. 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). Proof. rewrite /index_iota -subnDA addnC iotaDl big_map. by apply: eq_big => ? *; rewrite addnC. Qed. 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). Proof. by rewrite -addn1 big_addn subn1; apply: eq_big => ? *; rewrite addn1. Qed. 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). Proof. by move=> lemn; rewrite big_ltn // big_add1. Qed. 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. Proof. rewrite /index_iota subn0 -(big_map (@nat_of_ord n)). by congr bigop; rewrite /index_enum 2!unlock val_ord_enum. Qed. 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. Proof. move=> len12; symmetry; rewrite -big_filter filter_predI big_filter. have [ltn_trans eq_by_mem] := (ltn_trans, irr_sorted_eq ltn_trans ltnn). congr bigop; apply: eq_by_mem; rewrite ?sorted_filter ?iota_ltn_sorted // => i. rewrite mem_filter !mem_index_iota andbCA andbA andb_idr => // /andP[_]. by move/leq_trans->. Qed. 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. Proof. by move/big_nat_widen=> len12; rewrite -big_mkord len12 big_mkord. Qed. 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. Proof. by move=> le_n12; apply: (big_ord_widen_cond (predT)). Qed. 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). Proof. move=> len12; pose g G i := G (inord i : 'I_(n1.+1)). rewrite -(big_ord_widen_cond (g _ P) (g _ F) len12) {}/g. by apply: eq_big => i *; rewrite inord_val. Qed. Lemma big_ord0 P F : \big[op/idx]_(i < 0 | P i) F i = idx. Proof. by rewrite big_pred0 => [|[]]. Qed. Lemma big_mask_tuple I n m (t : n.-tuple I) (P : pred I) F : \big[op/idx]_(i <- mask m t | P i) F i = \big[op/idx]_(i < n | nth false m i && P (tnth t i)) F (tnth t i). Proof. rewrite [t in LHS]tuple_map_ord/= -map_mask big_map. by rewrite mask_enum_ord big_filter_cond/= enumT. Qed. Lemma big_mask I r m (P : pred I) (F : I -> R) (r_ := tnth (in_tuple r)) : \big[op/idx]_(i <- mask m r | P i) F i = \big[op/idx]_(i < size r | nth false m i && P (r_ i)) F (r_ i). Proof. exact: (big_mask_tuple _ (in_tuple r)). Qed. Lemma big_tnth I r (P : pred I) F (r_ := tnth (in_tuple r)) : \big[op/idx]_(i <- r | P i) F i = \big[op/idx]_(i < size r | P (r_ i)) (F (r_ i)). Proof. rewrite /= -[r in LHS](mask_true (leqnn (size r))) big_mask//. by apply: eq_bigl => i /=; rewrite nth_nseq ltn_ord. Qed. 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)). Proof. move=> Ur; apply/esym; rewrite big_tnth. by under [LHS]eq_bigr do rewrite index_uniq// valK. Qed. 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). Proof. by rewrite big_tnth tvalK; case: _ / (esym _). Qed. 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). Proof. case: n1 => [|n1] /= in le_n12 *. by rewrite big_ord0 big_pred0 // => i; rewrite andbF. rewrite (big_ord_widen_leq _ _ le_n12); apply: eq_big => i. by apply: andb_id2r => le_i_n1; congr P; apply: val_inj; rewrite /= inordK. by case/andP=> _ le_i_n1; congr F; apply: val_inj; rewrite /= inordK. Qed. 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). Proof. exact: (@big_ord_narrow_cond n1.+1 n2.+1). Qed. 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). Proof. exact: (big_ord_narrow_cond (predT)). Qed. 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). Proof. exact: (big_ord_narrow_cond_leq (predT)). Qed. 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)). Proof. pose G i := F (inord i); have eqFG i: F i = G i by rewrite /G inord_val. under eq_bigr do rewrite eqFG; under [in RHS]eq_bigr do rewrite eqFG. by rewrite -(big_mkord _ (fun _ => _) G) eqFG big_ltn // big_add1 /= big_mkord. Qed. 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. Proof. by rewrite unlock; elim: n => /= [|n ->]; case: (P a). Qed. Lemma big_nseq I n a (F : I -> R): \big[op/idx]_(i <- nseq n a) F i = iter n (op (F a)) idx. Proof. exact: big_nseq_cond. Qed. End Extensionality. Variant big_enum_spec (I : finType) (P : pred I) : seq I -> Type := BigEnumSpec e of forall R idx op (F : I -> R), \big[op/idx]_(i <- e) F i = \big[op/idx]_(i | P i) F i & uniq e /\ (forall i, i \in e = P i) & (let cP := [pred i | P i] in perm_eq e (enum cP) /\ size e = #|cP|) : big_enum_spec P e. (* This lemma can be used to introduce an enumeration into a non-abelian *) (* bigop, in one of three ways: *) (* have [e big_e [Ue mem_e] [e_enum size_e]] := big_enumP P. *) (* gives a permutation e of enum P alongside a equation big_e for converting *) (* between bigops iterating on (i <- e) and ones on (i | P i). Usually not *) (* all properties of e are needed, but see below the big_distr_big_dep proof *) (* where most are. *) (* rewrite -big_filter; have [e ...] := big_enumP. *) (* uses big_filter to do this conversion first, and then abstracts the *) (* resulting filter P (index_enum T) enumeration as an e with the same *) (* properties (see big_enum_cond below for an example of this usage). *) (* Finally *) (* rewrite -big_filter; case def_e: _ / big_enumP => [e ...] *) (* does the same while remembering the definition of e. *) Lemma big_enumP I P : big_enum_spec P (filter P (index_enum I)). Proof. set e := filter P _; have Ue: uniq e by apply/filter_uniq/index_enum_uniq. have mem_e i: i \in e = P i by rewrite mem_filter mem_index_enum andbT. split=> // [R idx op F | cP]; first by rewrite big_filter. suffices De: perm_eq e (enum cP) by rewrite (perm_size De) cardE. by apply/uniq_perm=> // [|i]; rewrite ?enum_uniq ?mem_enum ?mem_e. Qed. Section BigConst. Variables (R : Type) (idx : R) (op : R -> R -> R). Lemma big_const_seq I r (P : pred I) x : \big[op/idx]_(i <- r | P i) x = iter (count P r) (op x) idx. Proof. by rewrite unlock; elim: r => //= i r ->; case: (P i). Qed. Lemma big_const (I : finType) (A : {pred I}) x : \big[op/idx]_(i in A) x = iter #|A| (op x) idx. Proof. by have [e <- _ [_ <-]] := big_enumP A; rewrite big_const_seq count_predT. Qed. Lemma big_const_nat m n x : \big[op/idx]_(m <= i < n) x = iter (n - m) (op x) idx. Proof. by rewrite big_const_seq count_predT size_iota. Qed. Lemma big_const_ord n x : \big[op/idx]_(i < n) x = iter n (op x) idx. Proof. by rewrite big_const card_ord. Qed. End BigConst. Section MonoidProperties. Import Monoid.Theory. Variable R : Type. Variable idx : R. Local Notation "1" := idx. Section Plain. Variable op : Monoid.law 1. Local Notation "*%M" := op (at level 0). Local Notation "x * y" := (op x y). Lemma foldlE x r : foldl *%M x r = \big[*%M/1]_(y <- x :: r) y. Proof. by rewrite -foldrE; elim: r => [|y r IHr]/= in x *; rewrite ?mulm1 ?mulmA ?IHr. Qed. Lemma foldl_idx r : foldl *%M 1 r = \big[*%M/1]_(x <- r) x. Proof. by rewrite foldlE big_cons mul1m. Qed. 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. Proof. move=> op_idx'; rewrite -!(big_filter _ _ r) has_count -size_filter. case/lastP: (filter P r) => {r}// r i _. by rewrite -cats1 !(big_cat_nested, big_cons, big_nil) op_idx' mulm1. Qed. 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. Proof. by move=> Pi0 op_idx'; apply: eq_big_idx_seq => //; apply/hasP; exists i0. Qed. Lemma big1_eq I r (P : pred I) : \big[*%M/1]_(i <- r | P i) 1 = 1. Proof. by rewrite big_const_seq; elim: (count _ _) => //= n ->; apply: mul1m. Qed. Lemma big1 I r (P : pred I) F : (forall i, P i -> F i = 1) -> \big[*%M/1]_(i <- r | P i) F i = 1. Proof. by move/(eq_bigr _)->; apply: big1_eq. Qed. Lemma big1_seq (I : eqType) r (P : pred I) F : (forall i, P i && (i \in r) -> F i = 1) -> \big[*%M/1]_(i <- r | P i) F i = 1. Proof. by move=> eqF1; rewrite big_seq_cond big_andbC big1. Qed. Lemma big_seq1 I (i : I) F : \big[*%M/1]_(j <- [:: i]) F j = F i. Proof. by rewrite unlock /= mulm1. Qed. 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). Proof. by rewrite unlock; elim: r => //= i r ->; case P; rewrite ?mul1m. Qed. 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). Proof. by rewrite -big_filter_cond big_mkcond big_filter. Qed. 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). Proof. by rewrite big_andbC big_mkcondr. Qed. Lemma big_rmcond I (r : seq I) (P : pred I) F : (forall i, ~~ P i -> F i = 1) -> \big[*%M/1]_(i <- r | P i) F i = \big[*%M/1]_(i <- r) F i. Proof. move=> F_eq1; rewrite big_mkcond; apply: eq_bigr => i. by case: (P i) (F_eq1 i) => // ->. Qed. Lemma big_rmcond_in (I : eqType) (r : seq I) (P : pred I) F : (forall i, i \in r -> ~~ P i -> F i = 1) -> \big[*%M/1]_(i <- r | P i) F i = \big[*%M/1]_(i <- r) F i. Proof. move=> F_eq1; rewrite big_seq_cond [RHS]big_seq_cond !big_mkcondl big_rmcond//. by move=> i /F_eq1; case: ifP => // _ ->. Qed. 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. Proof. rewrite !(big_mkcond _ P) unlock. by elim: r1 => /= [|i r1 ->]; rewrite (mul1m, mulmA). Qed. Lemma big_allpairs_dep I1 (I2 : I1 -> Type) J (h : forall i1, I2 i1 -> J) (r1 : seq I1) (r2 : forall i1, seq (I2 i1)) (F : J -> R) : \big[*%M/1]_(i <- [seq h i1 i2 | i1 <- r1, i2 <- r2 i1]) F i = \big[*%M/1]_(i1 <- r1) \big[*%M/1]_(i2 <- r2 i1) F (h i1 i2). Proof. elim: r1 => [|i1 r1 IHr1]; first by rewrite !big_nil. by rewrite big_cat IHr1 big_cons big_map. Qed. Lemma big_allpairs I1 I2 (r1 : seq I1) (r2 : seq I2) F : \big[*%M/1]_(i <- [seq (i1, i2) | i1 <- r1, i2 <- r2]) F i = \big[*%M/1]_(i1 <- r1) \big[op/idx]_(i2 <- r2) F (i1, i2). Proof. exact: big_allpairs_dep. Qed. Lemma big_pred1_eq (I : finType) (i : I) F : \big[*%M/1]_(j | j == i) F j = F i. Proof. have [e1 <- _ [e_enum _]] := big_enumP (pred1 i). by rewrite (perm_small_eq _ e_enum) enum1 ?big_seq1. Qed. Lemma big_pred1 (I : finType) i (P : pred I) F : P =1 pred1 i -> \big[*%M/1]_(j | P j) F j = F i. Proof. by move/(eq_bigl _ _)->; apply: big_pred1_eq. Qed. 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). Proof. move=> le_mn le_np; rewrite -big_cat -{2}(subnKC le_mn) -iotaD subnDA. by rewrite subnKC // leq_sub. Qed. Lemma big_nat_widenl (m1 m2 n : nat) (P : pred nat) F : m2 <= m1 -> \big[op/idx]_(m1 <= i < n | P i) F i = \big[op/idx]_(m2 <= i < n | P i && (m1 <= i)) F i. Proof. move=> le_m21; have [le_nm1|lt_m1n] := leqP n m1. rewrite big_geq// big_nat_cond big1//. by move=> i /and3P[/andP[_ /leq_trans/(_ le_nm1)/ltn_geF->]]. rewrite big_mkcond big_mkcondl (big_cat_nat _ _ le_m21) 1?ltnW//. rewrite [X in op X]big_nat_cond [X in op X]big_pred0; last first. by move=> k; case: ltnP; rewrite andbF. by rewrite Monoid.mul1m; apply: congr_big_nat => // k /andP[]. Qed. Lemma big_geq_mkord (m n : nat) (P : pred nat) F : \big[op/idx]_(m <= i < n | P i) F i = \big[op/idx]_(i < n | P i && (m <= i)) F i. Proof. by rewrite (@big_nat_widenl _ 0)// big_mkord. Qed. Lemma big_nat1 n F : \big[*%M/1]_(n <= i < n.+1) F i = F n. Proof. by rewrite big_ltn // big_geq // mulm1. Qed. 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. Proof. by move=> lemn; rewrite (@big_cat_nat n) ?leqnSn // big_nat1. Qed. Lemma big_nat_mul n k F : \big[*%M/1]_(0 <= i < n * k) F i = \big[*%M/1]_(0 <= i < n) \big[*%M/1]_(i * k <= j < i.+1 * k) F j. Proof. elim: n => [|n ih]; first by rewrite mul0n 2!big_nil. rewrite [in RHS]big_nat_recr//= -ih mulSn addnC [in LHS]/index_iota subn0 iotaD. rewrite big_cat /= [in X in _ = X * _]/index_iota subn0; congr (_ * _). by rewrite add0n /index_iota (addnC _ k) addnK. Qed. 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. Proof. transitivity (\big[*%M/1]_(0 <= i < n.+1) F (inord i)). by rewrite big_mkord; apply: eq_bigr=> i _; rewrite inord_val. rewrite big_nat_recr // big_mkord; congr (_ * F _); last first. by apply: val_inj; rewrite /= inordK. by apply: eq_bigr => [] i _; congr F; apply: ord_inj; rewrite inordK //= leqW. Qed. 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)). Proof. by rewrite ![index_enum _]unlock [@Finite.enum in LHS]unlock big_cat !big_map. Qed. 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)). Proof. rewrite -(big_map _ _ (lshift n) _ P F) -(big_map _ _ (@rshift m _) _ P F). rewrite -big_cat; congr bigop; apply: (inj_map val_inj). rewrite map_cat -!map_comp (map_comp (addn m)) /=. by rewrite ![index_enum _]unlock unlock !val_ord_enum -iotaDl addn0 iotaD. Qed. 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. Proof. by elim: rr => [|r rr IHrr]; rewrite ?big_nil //= big_cat big_cons -IHrr. Qed. Lemma big_pmap J I (h : J -> option I) (r : seq J) F : \big[op/idx]_(i <- pmap h r) F i = \big[op/idx]_(j <- r) oapp F idx (h j). Proof. elim: r => [| r0 r IHr]/=; first by rewrite !big_nil. rewrite /= big_cons; case: (h r0) => [i|] /=; last by rewrite mul1m. by rewrite big_cons IHr. Qed. Lemma telescope_big (f : nat -> nat -> R) (n m : nat) : (forall k, n < k < m -> op (f n k) (f k k.+1) = f n k.+1) -> \big[op/idx]_(n <= i < m) f i i.+1 = if n < m then f n m else idx. Proof. elim: m => [//| m IHm]; first by rewrite ltn0 big_geq. move=> tm; rewrite ltnS; case: ltnP=> // mn; first by rewrite big_geq. rewrite big_nat_recr// IHm//; last first. by move=> k /andP[nk /ltnW nm]; rewrite tm// nk. by case: ltngtP mn=> //= [nm|<-]; rewrite ?mul1m// tm// nm leqnn. Qed. End Plain. Section Abelian. Variable op : Monoid.com_law 1. Local Notation "'*%M'" := op (at level 0). Local Notation "x * y" := (op x y). Lemma perm_big (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. Proof. move/permP; rewrite !(big_mkcond _ _ P). elim: r1 r2 => [|i r1 IHr1] r2 eq_r12. by case: r2 eq_r12 => // i r2 /(_ (pred1 i)); rewrite /= eqxx. have r2i: i \in r2 by rewrite -has_pred1 has_count -eq_r12 /= eqxx. case/splitPr: r2 / r2i => [r3 r4] in eq_r12 *; rewrite big_cat /= !big_cons. rewrite mulmCA; congr (_ * _); rewrite -big_cat; apply: IHr1 => a. by move/(_ a): eq_r12; rewrite !count_cat /= addnCA; apply: addnI. Qed. Lemma big_enum_cond (I : finType) (A : {pred I}) (P : pred I) F : \big[*%M/1]_(i <- enum A | P i) F i = \big[*%M/1]_(i in A | P i) F i. Proof. by rewrite -big_filter_cond; have [e _ _ [/perm_big->]] := big_enumP. Qed. Lemma big_enum (I : finType) (A : {pred I}) F : \big[*%M/1]_(i <- enum A) F i = \big[*%M/1]_(i in A) F i. Proof. by rewrite big_enum_cond big_andbC. Qed. 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. Proof. move=> uniq_r; rewrite -big_enum; apply: perm_big. by rewrite uniq_perm ?enum_uniq // => i; rewrite mem_enum. Qed. 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. Proof. by move/perm_to_rem/(perm_big _)->; rewrite !(big_mkcond _ _ P) big_cons. Qed. 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. Proof. move=> idM; rewrite -!(big_filter _ _ _ P) filter_undup. elim: {P r}(filter P r) => //= i r IHr. case: ifP => [r_i | _]; rewrite !big_cons {}IHr //. by rewrite (big_rem _ _ r_i) mulmA idM. Qed. 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. Proof. move=> idM eq_r; rewrite -big_undup // -(big_undup r2) //; apply/perm_big. by rewrite uniq_perm ?undup_uniq // => i; rewrite !mem_undup eq_r. Qed. 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. Proof. rewrite -[RHS](perm_big _ F (perm_count_undup _)) big_flatten big_map. by rewrite big_mkcond; apply: eq_bigr => i _; rewrite big_nseq_cond iteropE. Qed. 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. Proof. by elim/big_rec3: _ => [|i x y _ _ ->]; rewrite ?mulm1 // mulmCA -!mulmA mulmCA. Qed. 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. Proof. rewrite !(big_mkcond _ _ _ F) -big_split. by apply: eq_bigr => i; case: (a i); rewrite !simpm. Qed. 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). Proof. move=> dAB; rewrite (bigID (mem A)). congr (_ * _); apply: eq_bigl => i; first by rewrite orbK. by have:= pred0P dAB i; rewrite andbC /= !inE; case: (i \in A). Qed. 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. Proof. move=> Pj; rewrite (bigID (pred1 j)); congr (_ * _). by apply: big_pred1 => i; rewrite /= andbC; case: eqP => // ->. Qed. Arguments bigD1 [I] j [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. Proof. by move=> /big_rem-> /rem_filter->; rewrite big_filter. Qed. Lemma cardD1x (I : finType) (A : pred I) j : A j -> #|SimplPred A| = 1 + #|[pred i | A i & i != j]|. Proof. move=> Aj; rewrite (cardD1 j) [j \in A]Aj; congr (_ + _). by apply: eq_card => i; rewrite inE /= andbC. Qed. Arguments cardD1x [I A]. Lemma partition_big I (s : seq I) (J : finType) (P : pred I) (p : I -> J) (Q : pred J) F : (forall i, P i -> Q (p i)) -> \big[*%M/1]_(i <- s | P i) F i = \big[*%M/1]_(j : J | Q j) \big[*%M/1]_(i <- s | (P i) && (p i == j)) F i. Proof. move=> Qp; transitivity (\big[*%M/1]_(i <- s | P i && Q (p i)) F i). by apply: eq_bigl => i; case Pi: (P i); rewrite // Qp. have [n leQn] := ubnP #|Q|; elim: n => // n IHn in Q {Qp} leQn *. case: (pickP Q) => [j Qj | Q0]; last first. by rewrite !big_pred0 // => i; rewrite Q0 andbF. rewrite (bigD1 j) // -IHn; last by rewrite ltnS (cardD1x j Qj) in leQn. rewrite (bigID (fun i => p i == j)); congr (_ * _); apply: eq_bigl => i. by case: eqP => [-> | _]; rewrite !(Qj, simpm). by rewrite andbA. Qed. Arguments partition_big [I s J P] p Q [F]. Lemma big_image_cond I (J : finType) (h : J -> I) (A : pred J) (P : pred I) F : \big[*%M/1]_(i <- [seq h j | j in A] | P i) F i = \big[*%M/1]_(j in A | P (h j)) F (h j). Proof. by rewrite big_map big_enum_cond. Qed. Lemma big_image I (J : finType) (h : J -> I) (A : pred J) F : \big[*%M/1]_(i <- [seq h j | j in A]) F i = \big[*%M/1]_(j in A) F (h j). Proof. by rewrite big_map big_enum. Qed. Lemma big_image_cond_id (J : finType) (h : J -> R) (A : pred J) (P : pred R) : \big[*%M/1]_(i <- [seq h j | j in A] | P i) i = \big[*%M/1]_(j in A | P (h j)) h j. Proof. exact: big_image_cond. Qed. Lemma big_image_id (J : finType) (h : J -> R) (A : pred J) : \big[*%M/1]_(i <- [seq h j | j in A]) i = \big[*%M/1]_(j in A) h j. Proof. exact: big_image. Qed. Lemma reindex_omap (I J : finType) (h : J -> I) h' (P : pred I) F : (forall i, P i -> omap h (h' i) = some i) -> \big[*%M/1]_(i | P i) F i = \big[*%M/1]_(j | P (h j) && (h' (h j) == some j)) F (h j). Proof. move=> h'K; have [n lePn] := ubnP #|P|; elim: n => // n IHn in P h'K lePn *. case: (pickP P) => [i Pi | P0]; last first. by rewrite !big_pred0 // => j; rewrite P0. have := h'K i Pi; case h'i_eq : (h' i) => [/= j|//] [hj_eq]. rewrite (bigD1 i Pi) (bigD1 j) hj_eq ?Pi ?h'i_eq ?eqxx //=; congr (_ * _). rewrite {}IHn => [|k /andP[]|]; [|by auto | by rewrite (cardD1x i) in lePn]. apply: eq_bigl => k; rewrite andbC -andbA (andbCA (P _)); case: eqP => //= hK. congr (_ && ~~ _); apply/eqP/eqP => [|->//]. by move=> /(congr1 h'); rewrite h'i_eq hK => -[]. Qed. Arguments reindex_omap [I J] h h' [P F]. Lemma reindex_onto (I J : finType) (h : J -> I) h' (P : pred I) F : (forall 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). Proof. by move=> h'K; rewrite (reindex_omap h (some \o h'))//= => i Pi; rewrite h'K. Qed. Arguments reindex_onto [I J] h h' [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). Proof. case=> h' hK h'K; rewrite (reindex_onto h h' h'K). by apply: eq_bigl => j /[!inE]; case Pi: (P _); rewrite //= hK ?eqxx. Qed. Arguments reindex [I J] h [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). Proof. by move=> injh; apply: reindex (onW_bij _ (injF_bij injh)). Qed. Arguments reindex_inj [I h P F]. Lemma bigD1_ord n j (P : pred 'I_n) F : P j -> \big[*%M/1]_(i < n | P i) F i = F j * \big[*%M/1]_(i < n.-1 | P (lift j i)) F (lift j i). Proof. move=> Pj; rewrite (bigD1 j Pj) (reindex_omap (lift j) (unlift j))/=. by under eq_bigl do rewrite liftK eq_sym eqxx neq_lift ?andbT. by move=> i; case: unliftP => [k ->|->]; rewrite ?eqxx ?andbF. Qed. Lemma big_enum_val_cond (I : finType) (A : pred I) (P : pred I) F : \big[op/idx]_(x in A | P x) F x = \big[op/idx]_(i < #|A| | P (enum_val i)) F (enum_val i). Proof. have [A_eq0|/card_gt0P[x0 x0A]] := posnP #|A|. rewrite !big_pred0 // => i; last by rewrite card0_eq. by have: false by move: i => []; rewrite A_eq0. rewrite (reindex (enum_val : 'I_#|A| -> I)). by apply: eq_big => [x|x Px]; rewrite ?enum_valP. by apply: subon_bij (enum_val_bij_in x0A) => y /andP[]. Qed. Arguments big_enum_val_cond [I A] P F. Lemma big_enum_rank_cond (I : finType) (A : pred I) x (xA : x \in A) P F (h := enum_rank_in xA) : \big[op/idx]_(i < #|A| | P i) F i = \big[op/idx]_(s in A | P (h s)) F (h s). Proof. rewrite big_enum_val_cond {}/h. by apply: eq_big => [i|i Pi]; rewrite ?enum_valK_in. Qed. Arguments big_enum_rank_cond [I A x] xA P F. Lemma big_enum_val (I : finType) (A : pred I) F : \big[op/idx]_(x in A) F x = \big[op/idx]_(i < #|A|) F (enum_val i). Proof. by rewrite -(big_enum_val_cond predT) big_mkcondr. Qed. Arguments big_enum_val [I A] F. Lemma big_enum_rank (I : finType) (A : pred I) x (xA : x \in A) F (h := enum_rank_in xA) : \big[op/idx]_(i < #|A|) F i = \big[op/idx]_(s in A) F (h s). Proof. by rewrite (big_enum_rank_cond xA) big_mkcondr. Qed. Arguments big_enum_rank [I A x] xA 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). Proof. case: (ltnP m n) => ltmn; last by rewrite !big_geq. rewrite -{3 4}(subnK (ltnW ltmn)) addnA. do 2!rewrite (big_addn _ _ 0) big_mkord; rewrite (reindex_inj rev_ord_inj) /=. by apply: eq_big => [i | i _]; rewrite /= -addSn subnDr addnC addnBA. Qed. Lemma big_rev_mkord m n P F : \big[op/idx]_(m <= k < n | P k) F k = \big[op/idx]_(k < n - m | P (n - k.+1)) F (n - k.+1). Proof. rewrite big_nat_rev (big_addn _ _ 0) big_mkord. by apply: eq_big => [i|i _]; rewrite -addSn addnC subnDr. Qed. Lemma sig_big_dep (I : finType) (J : I -> finType) (P : pred I) (Q : forall {i}, pred (J i)) (F : forall {i}, J i -> R) : \big[op/idx]_(i | P i) \big[op/idx]_(j : J i | Q j) F j = \big[op/idx]_(p : {i : I & J i} | P (tag p) && Q (tagged p)) F (tagged p). Proof. pose s := [seq Tagged J j | i <- index_enum I, j <- index_enum (J i)]. rewrite [LHS]big_mkcond big_mkcondl [RHS]big_mkcond -[RHS](@perm_big _ s). rewrite big_allpairs_dep/=; apply: eq_bigr => i _; rewrite -big_mkcond/=. by case: P; rewrite // big1. rewrite uniq_perm ?index_enum_uniq//. by rewrite allpairs_uniq_dep// => [|i|[i j] []]; rewrite ?index_enum_uniq. by move=> [i j]; rewrite ?mem_index_enum; apply/allpairsPdep; exists i, j. Qed. 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. Proof. rewrite sig_big_dep; apply: (reindex (fun x => Tagged (fun=> J) x.2)). by exists (fun x => (projT1 x, projT2 x)) => -[]. Qed. 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. Proof. exact: pair_big_dep. Qed. 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. Proof. exact: pair_big_dep. Qed. Lemma exchange_big_dep I J rI rJ (P : pred I) (Q : I -> pred J) (xQ : pred J) F : (forall 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. Proof. move=> PQxQ; pose p u := (u.2, u.1). under [LHS]eq_bigr do rewrite big_tnth; rewrite [LHS]big_tnth. under [RHS]eq_bigr do rewrite big_tnth; rewrite [RHS]big_tnth. rewrite !pair_big_dep (reindex_onto (p _ _) (p _ _)) => [|[]] //=. apply: eq_big => [] [j i] //=; symmetry; rewrite eqxx andbT andb_idl //. by case/andP; apply: PQxQ. Qed. Arguments exchange_big_dep [I J rI rJ P Q] xQ [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. Proof. rewrite (exchange_big_dep Q) //. by under eq_bigr => i Qi do under eq_bigl do rewrite Qi andbT. Qed. Lemma exchange_big_dep_nat m1 n1 m2 n2 (P : pred nat) (Q : rel nat) (xQ : pred nat) F : (forall 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. Proof. move=> PQxQ; under eq_bigr do rewrite big_seq_cond. rewrite big_seq_cond /= (exchange_big_dep xQ) => [|i j]; last first. by rewrite !mem_index_iota => /andP[mn_i Pi] /andP[mn_j /PQxQ->]. rewrite 2!(big_seq_cond _ _ _ xQ); apply: eq_bigr => j /andP[-> _] /=. by rewrite [rhs in _ = rhs]big_seq_cond; apply: eq_bigl => i; rewrite -andbA. Qed. Arguments exchange_big_dep_nat [m1 n1 m2 n2 P Q] xQ [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. Proof. rewrite (exchange_big_dep_nat Q) //. by under eq_bigr => i Qi do under eq_bigl do rewrite Qi andbT. Qed. End Abelian. End MonoidProperties. Arguments big_filter [R idx op I]. Arguments big_filter_cond [R idx op I]. Arguments congr_big [R idx op I r1] r2 [P1] P2 [F1] F2. Arguments eq_big [R idx op I r P1] P2 [F1] F2. Arguments eq_bigl [R idx op I r P1] P2. Arguments eq_bigr [R idx op I r P F1] F2. Arguments eq_big_idx [R idx op idx' I] i0 [P F]. Arguments big_seq_cond [R idx op I r]. Arguments eq_big_seq [R idx op I r F1] F2. Arguments congr_big_nat [R idx op m1 n1] m2 n2 [P1] P2 [F1] F2. Arguments big_map [R idx op I J] h [r]. Arguments big_nth [R idx op I] x0 [r]. Arguments big_catl [R idx op I r1 r2 P F]. Arguments big_catr [R idx op I r1 r2 P F]. Arguments big_geq [R idx op m n P F]. Arguments big_ltn_cond [R idx op m n P F]. Arguments big_ltn [R idx op m n F]. Arguments big_addn [R idx op]. Arguments big_mkord [R idx op n]. Arguments big_nat_widen [R idx op]. Arguments big_nat_widenl [R idx op]. Arguments big_geq_mkord [R idx op]. Arguments big_ord_widen_cond [R idx op n1]. Arguments big_ord_widen [R idx op n1]. Arguments big_ord_widen_leq [R idx op n1]. Arguments big_ord_narrow_cond [R idx op n1 n2 P F]. Arguments big_ord_narrow_cond_leq [R idx op n1 n2 P F]. Arguments big_ord_narrow [R idx op n1 n2 F]. Arguments big_ord_narrow_leq [R idx op n1 n2 F]. Arguments big_mkcond [R idx op I r]. Arguments big1_eq [R idx op I]. Arguments big1_seq [R idx op I]. Arguments big1 [R idx op I]. Arguments big_pred1 [R idx op I] i [P F]. Arguments perm_big [R idx op I r1] r2 [P F]. Arguments big_uniq [R idx op I] r [F]. Arguments big_rem [R idx op I r] x [P F]. Arguments bigID [R idx op I r]. Arguments bigU [R idx op I]. Arguments bigD1 [R idx op I] j [P F]. Arguments bigD1_seq [R idx op I r] j [F]. Arguments bigD1_ord [R idx op n] j [P F]. Arguments partition_big [R idx op I s J P] p Q [F]. Arguments reindex_omap [R idx op I J] h h' [P F]. Arguments reindex_onto [R idx op I J] h h' [P F]. Arguments reindex [R idx op I J] h [P F]. Arguments reindex_inj [R idx op I h P F]. Arguments big_enum_val_cond [R idx op I A] P F. Arguments big_enum_rank_cond [R idx op I A x] xA P F. Arguments big_enum_val [R idx op I A] F. Arguments big_enum_rank [R idx op I A x] xA F. Arguments sig_big_dep [R idx op I J]. Arguments pair_big_dep [R idx op I J]. Arguments pair_big [R idx op I J]. Arguments big_allpairs_dep {R idx op I1 I2 J h r1 r2 F}. Arguments big_allpairs {R idx op I1 I2 r1 r2 F}. Arguments exchange_big_dep [R idx op I J rI rJ P Q] xQ [F]. Arguments exchange_big_dep_nat [R idx op m1 n1 m2 n2 P Q] xQ [F]. Arguments big_ord_recl [R idx op]. Arguments big_ord_recr [R idx op]. Arguments big_nat_recl [R idx op]. Arguments big_nat_recr [R idx op]. Arguments big_pmap [R idx op J I] h [r]. Arguments telescope_big [R idx op] f [n m]. Section EqSupport. Variables (R : eqType) (idx : R). Section MonoidSupport. Variables (op : Monoid.law idx) (I : Type). Lemma eq_bigl_supp (r : seq I) (P1 : pred I) (P2 : pred I) (F : I -> R) : {in [pred x | F x != idx], P1 =1 P2} -> \big[op/idx]_(i <- r | P1 i) F i = \big[op/idx]_(i <- r | P2 i) F i. Proof. move=> P12; rewrite big_mkcond [RHS]big_mkcond; apply: eq_bigr => i _. by case: (eqVneq (F i) idx) => [->|/P12->]; rewrite ?if_same. Qed. End MonoidSupport. Section ComoidSupport. Variables (op : Monoid.com_law idx) (I : eqType). Lemma perm_big_supp_cond [r s : seq I] [P : pred I] (F : I -> R) : perm_eq [seq i <- r | P i && (F i != idx)] [seq i <- s | P i && (F i != idx)] -> \big[op/idx]_(i <- r | P i) F i = \big[op/idx]_(i <- s | P i) F i. Proof. move=> prs; rewrite !(bigID [pred i | F i == idx] P F)/=. rewrite big1 ?Monoid.mul1m; last by move=> i /andP[_ /eqP->]. rewrite [in RHS]big1 ?Monoid.mul1m; last by move=> i /andP[_ /eqP->]. by rewrite -[in LHS]big_filter -[in RHS]big_filter; apply perm_big. Qed. Lemma perm_big_supp [r s : seq I] [P : pred I] (F : I -> R) : perm_eq [seq i <- r | F i != idx] [seq i <- s | F i != idx] -> \big[op/idx]_(i <- r | P i) F i = \big[op/idx]_(i <- s | P i) F i. Proof. by move=> ?; apply: perm_big_supp_cond; rewrite !filter_predI perm_filter. Qed. End ComoidSupport. End EqSupport. Arguments eq_bigl_supp [R idx op I r P1]. Arguments perm_big_supp_cond [R idx op I r s P]. Arguments perm_big_supp [R idx op I r s P]. Section Distributivity. Import Monoid.Theory. Variable R : Type. Variables zero one : R. Local Notation "0" := zero. Local Notation "1" := one. Variable times : Monoid.mul_law 0. Local Notation "*%M" := times (at level 0). Local Notation "x * y" := (times x y). Variable plus : Monoid.add_law 0 *%M. Local Notation "+%M" := plus (at level 0). Local Notation "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). Proof. by rewrite (big_endo ( *%M^~ a)) ?mul0m // => x y; apply: mulmDl. Qed. 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). Proof. by rewrite big_endo ?mulm0 // => x y; apply: mulmDr. Qed. 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). Proof. by rewrite big_distrl; under eq_bigr do rewrite big_distrr. Qed. 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). Proof. pose fIJ := {ffun I -> J}; pose Pf := pfamily j0 (_ : seq I) Q. have [r big_r [Ur mem_r] _] := big_enumP P. symmetry; transitivity (\big[+%M/0]_(f in Pf r) \big[*%M/1]_(i <- r) F i (f i)). by apply: eq_big => // f; apply: eq_forallb => i; rewrite /= mem_r. rewrite -{P mem_r}big_r; elim: r Ur => /= [_ | i r IHr]. rewrite (big_pred1 [ffun=> j0]) ?big_nil //= => f. apply/familyP/eqP=> /= [Df |->{f} i]; last by rewrite ffunE !inE. by apply/ffunP=> i; rewrite ffunE; apply/eqP/Df. case/andP=> /negbTE nri; rewrite big_cons big_distrl => {}/IHr<-. rewrite (partition_big (fun f : fIJ => f i) (Q i)) => [|f]; last first. by move/familyP/(_ i); rewrite /= inE /= eqxx. pose seti j (f : fIJ) := [ffun k => if k == i then j else f k]. apply: eq_bigr => j Qij. rewrite (reindex_onto (seti j) (seti j0)) => [|f /andP[_ /eqP fi]]; last first. by apply/ffunP=> k; rewrite !ffunE; case: eqP => // ->. rewrite big_distrr; apply: eq_big => [f | f eq_f]; last first. rewrite big_cons ffunE eqxx !big_seq; congr (_ * _). by apply: eq_bigr => k; rewrite ffunE; case: eqP nri => // -> ->. rewrite !ffunE !eqxx andbT; apply/andP/familyP=> /= [[Pjf fij0] k | Pff]. have /[!(ffunE, inE)] := familyP Pjf k; case: eqP => // -> _. by rewrite nri -(eqP fij0) !ffunE !inE !eqxx. split; [apply/familyP | apply/eqP/ffunP] => k; have /[!(ffunE, inE)]:= Pff k. by case: eqP => // ->. by case: eqP => // ->; rewrite nri /= => /eqP. Qed. 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). Proof. rewrite (big_distr_big_dep j0); apply: eq_bigl => f. by apply/familyP/familyP=> Pf i; case: ifP (Pf i). Qed. 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). Proof. have [j _ | J0] := pickP J; first by rewrite (big_distr_big_dep j). have Q0 i: Q i =i pred0 by move=> /J0/esym/notF[]. transitivity (iter #|I| ( *%M 0) 1). by rewrite -big_const; apply/eq_bigr=> i; have /(big_pred0 _)-> := Q0 i. have [i _ | I0] := pickP I. rewrite (cardD1 i) //= mul0m big_pred0 // => f. by apply/familyP=> /(_ i); rewrite Q0. have f: I -> J by move=> /I0/esym/notF[]. rewrite eq_card0 // (big_pred1 (finfun f)) ?big_pred0 // => g. by apply/familyP/eqP=> _; first apply/ffunP; move=> /I0/esym/notF[]. Qed. 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). Proof. exact: bigA_distr_big_dep. Qed. 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). Proof. by rewrite bigA_distr_big; apply: eq_bigl => ?; apply/familyP. Qed. End Distributivity. Arguments big_distrl [R zero times plus I r]. Arguments big_distrr [R zero times plus I r]. Arguments big_distr_big_dep [R zero one times plus I J]. Arguments big_distr_big [R zero one times plus I J]. Arguments bigA_distr_big_dep [R zero one times plus I J]. Arguments bigA_distr_big [R zero one times plus I J]. 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. Proof. by rewrite unlock. Qed. Lemma big_all : \big[andb/true]_(i <- r) B i = all B r. Proof. by rewrite unlock. Qed. Lemma big_has_cond : \big[orb/false]_(i <- r | P i) B i = has (predI P B) r. Proof. by rewrite big_mkcond unlock. Qed. Lemma big_all_cond : \big[andb/true]_(i <- r | P i) B i = all [pred i | P i ==> B i] r. Proof. by rewrite big_mkcond unlock. Qed. Lemma big_bool R (idx : R) (op : Monoid.com_law idx) (F : bool -> R): \big[op/idx]_(i : bool) F i = op (F true) (F false). Proof. by rewrite /index_enum !unlock /= Monoid.mulm1. Qed. End Seq. Section FinType. Variables (I : finType) (P B : pred I). Lemma big_orE : \big[orb/false]_(i | P i) B i = [exists (i | P i), B i]. Proof. by rewrite big_has_cond; apply/hasP/existsP=> [] [i]; exists i. Qed. Lemma big_andE : \big[andb/true]_(i | P i) B i = [forall (i | P i), B i]. Proof. rewrite big_all_cond; apply/allP/forallP=> /= allB i; rewrite allB //. exact: mem_index_enum. Qed. End FinType. End BigBool. Section NatConst. Variables (I : finType) (A : pred I). Lemma sum_nat_const n : \sum_(i in A) n = #|A| * n. Proof. by rewrite big_const iter_addn_0 mulnC. Qed. Lemma sum1_card : \sum_(i in A) 1 = #|A|. Proof. by rewrite sum_nat_const muln1. Qed. Lemma sum1_count J (r : seq J) (a : pred J) : \sum_(j <- r | a j) 1 = count a r. Proof. by rewrite big_const_seq iter_addn_0 mul1n. Qed. Lemma sum1_size J (r : seq J) : \sum_(j <- r) 1 = size r. Proof. by rewrite sum1_count count_predT. Qed. Lemma prod_nat_const n : \prod_(i in A) n = n ^ #|A|. Proof. by rewrite big_const -Monoid.iteropE. Qed. Lemma sum_nat_const_nat n1 n2 n : \sum_(n1 <= i < n2) n = (n2 - n1) * n. Proof. by rewrite big_const_nat iter_addn_0 mulnC. Qed. Lemma prod_nat_const_nat n1 n2 n : \prod_(n1 <= i < n2) n = n ^ (n2 - n1). Proof. by rewrite big_const_nat -Monoid.iteropE. Qed. End NatConst. Lemma telescope_sumn_in n m f : n <= m -> {in [pred i | n <= i <= m], {homo f : x y / x <= y}} -> \sum_(n <= k < m) (f k.+1 - f k) = f m - f n. Proof. move=> nm fle; rewrite (telescope_big (fun i j => f j - f i)). by case: ltngtP nm => // ->; rewrite subnn. move=> k /andP[nk km] /=; rewrite addnBAC ?fle 1?ltnW// ?subnKC// ?fle// inE. - by rewrite (ltnW nk) ltnW. - by rewrite leqnn ltnW// (ltn_trans nk). Qed. Lemma telescope_sumn n m f : {homo f : x y / x <= y} -> \sum_(n <= k < m) (f k.+1 - f k) = f m - f n. Proof. move=> fle; case: (ltnP n m) => nm. apply: (telescope_sumn_in (ltnW nm)) => ? ?; exact: fle. by apply/esym/eqP; rewrite big_geq// subn_eq0 fle. Qed. Lemma sumnE r : sumn r = \sum_(i <- r) i. Proof. exact: foldrE. Qed. Lemma card_bseq n (T : finType) : #|{bseq n of T}| = \sum_(i < n.+1) #|T| ^ i. Proof. rewrite (bij_eq_card bseq_tagged_tuple_bij) card_tagged sumnE big_map big_enum. by under eq_bigr do rewrite card_tuple. Qed. Lemma leqif_sum (I : finType) (P C : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i ?= iff C i) -> \sum_(i | P i) E1 i <= \sum_(i | P i) E2 i ?= iff [forall (i | P i), C i]. Proof. move=> leE12; rewrite -big_andE. by elim/big_rec3: _ => // i Ci m1 m2 /leE12; apply: leqif_add. Qed. Lemma leq_sum I r (P : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i) -> \sum_(i <- r | P i) E1 i <= \sum_(i <- r | P i) E2 i. Proof. by move=> leE12; elim/big_ind2: _ => // m1 m2 n1 n2; apply: leq_add. Qed. Lemma sumnB I r (P : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i) -> \sum_(i <- r | P i) (E2 i - E1 i) = \sum_(i <- r | P i) E2 i - \sum_(i <- r | P i) E1 i. Proof. by move=> /(_ _ _)/subnK-/(eq_bigr _)<-; rewrite big_split addnK. Qed. Lemma sum_nat_eq0 (I : finType) (P : pred I) (E : I -> nat) : (\sum_(i | P i) E i == 0)%N = [forall (i | P i), E i == 0%N]. Proof. by rewrite eq_sym -(@leqif_sum I P _ (fun _ => 0%N) E) ?big1_eq. Qed. Lemma leq_prod I r (P : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i) -> \prod_(i <- r | P i) E1 i <= \prod_(i <- r | P i) E2 i. Proof. by move=> leE12; elim/big_ind2: _ => // m1 m2 n1 n2; apply: leq_mul. Qed. Lemma prodn_cond_gt0 I r (P : pred I) F : (forall i, P i -> 0 < F i) -> 0 < \prod_(i <- r | P i) F i. Proof. by move=> Fpos; elim/big_ind: _ => // n1 n2; rewrite muln_gt0 => ->. Qed. Lemma prodn_gt0 I r (P : pred I) F : (forall i, 0 < F i) -> 0 < \prod_(i <- r | P i) F i. Proof. by move=> Fpos; apply: prodn_cond_gt0. Qed. Lemma leq_bigmax_seq (I : eqType) r (P : pred I) F i0 : i0 \in r -> P i0 -> F i0 <= \max_(i <- r | P i) F i. Proof. move=> + Pi0; elim: r => // h t ih; rewrite inE big_cons. move=> /predU1P[<-|i0t]; first by rewrite Pi0 leq_maxl. by case: ifPn => Ph; [rewrite leq_max ih// orbT|rewrite ih]. Qed. Arguments leq_bigmax_seq [I r P F]. Lemma leq_bigmax_cond (I : finType) (P : pred I) F i0 : P i0 -> F i0 <= \max_(i | P i) F i. Proof. exact: leq_bigmax_seq. Qed. Arguments leq_bigmax_cond [I P F]. Lemma leq_bigmax (I : finType) F (i0 : I) : F i0 <= \max_i F i. Proof. exact: leq_bigmax_cond. Qed. Arguments leq_bigmax [I F]. Lemma bigmax_leqP (I : finType) (P : pred I) m F : reflect (forall i, P i -> F i <= m) (\max_(i | P i) F i <= m). Proof. apply: (iffP idP) => leFm => [i Pi|]. by apply: leq_trans leFm; apply: leq_bigmax_cond. by elim/big_ind: _ => // m1 m2; rewrite geq_max => ->. Qed. Lemma bigmax_leqP_seq (I : eqType) r (P : pred I) m F : reflect (forall i, i \in r -> P i -> F i <= m) (\max_(i <- r | P i) F i <= m). Proof. apply: (iffP idP) => leFm => [i ri Pi|]. exact/(leq_trans _ leFm)/leq_bigmax_seq. rewrite big_seq_cond; elim/big_ind: _ => // [m1 m2|i /andP[ri]]. by rewrite geq_max => ->. exact: leFm. Qed. Lemma bigmax_sup (I : finType) i0 (P : pred I) m F : P i0 -> m <= F i0 -> m <= \max_(i | P i) F i. Proof. by move=> Pi0 le_m_Fi0; apply: leq_trans (leq_bigmax_cond i0 Pi0). Qed. Arguments bigmax_sup [I] i0 [P m F]. Lemma bigmax_sup_seq (I : eqType) r i0 (P : pred I) m F : i0 \in r -> P i0 -> m <= F i0 -> m <= \max_(i <- r | P i) F i. Proof. by move=> i0r Pi0 ?; apply: leq_trans (leq_bigmax_seq i0 _ _). Qed. Arguments bigmax_sup_seq [I r] i0 [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]. Proof. move=> Pi0; case: arg_maxnP => //= i Pi maxFi. by apply/eqP; rewrite eqn_leq leq_bigmax_cond // andbT; apply/bigmax_leqP. Qed. Arguments bigmax_eq_arg [I] i0 [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}. Proof. case: (pickP A) => [i0 Ai0 _ | ]; last by move/eq_card0->. by exists [arg max_(i > i0 in A) F i]; [case: arg_maxnP | apply: bigmax_eq_arg]. Qed. Lemma eq_bigmax (I : finType) F : #|I| > 0 -> {i0 : I | \max_i F i = F i0}. Proof. by case/(eq_bigmax_cond F) => x _ ->; exists x. Qed. 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. Proof. exact: (big_morph _ (expnD m)). Qed. Lemma dvdn_biglcmP (I : finType) (P : pred I) F m : reflect (forall i, P i -> F i %| m) (\big[lcmn/1%N]_(i | P i) F i %| m). Proof. apply: (iffP idP) => [dvFm i Pi | dvFm]. by rewrite (bigD1 i) // dvdn_lcm in dvFm; case/andP: dvFm. by elim/big_ind: _ => // p q p_m; rewrite dvdn_lcm p_m. Qed. 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. Proof. by move=> Pi0 m_Fi0; rewrite (dvdn_trans m_Fi0) // (bigD1 i0) ?dvdn_lcml. Qed. Arguments biglcmn_sup [I] i0 [P F m]. Lemma dvdn_biggcdP (I : finType) (P : pred I) F m : reflect (forall i, P i -> m %| F i) (m %| \big[gcdn/0]_(i | P i) F i). Proof. apply: (iffP idP) => [dvmF i Pi | dvmF]. by rewrite (bigD1 i) // dvdn_gcd in dvmF; case/andP: dvmF. by elim/big_ind: _ => // p q m_p; rewrite dvdn_gcd m_p. Qed. 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. Proof. by move=> Pi0; apply: dvdn_trans; rewrite (bigD1 i0) ?dvdn_gcdl. Qed. Arguments biggcdn_inf [I] i0 [P F m]. #[deprecated(since="mathcomp 1.13.0", note="Use big_rmcond instead.")] Notation big_uncond := big_rmcond (only parsing).